Fuzzy Logic in Geologyfa.mie.sut.ac.ir/Downloads/AcademicStaff/5/Courses...book is to make researchers in fuzzy logic aware of the emerging opportunities for the application of their

Fuzzy Logic in Geology

Edited by

Robert V. Demiccoand

George J. KlirCENTER FOR INTELLIGENT SYSTEMSBINGHAMTON UNIVERSITY (SUNY)

BINGHAMTON, NEW YORK, USA

Copyright © 2004, Elsevier Science (USA)

Academic PressAn imprint of Elsevier Science

525 B Street, Suite 1900, San Diego, California 92101-4495, USAhttp://www.academicpress.com

Academic Press84 Theobald’s Road, London WC1X 8RR, UK

http://www.academicpress.com

Library of Congress Cataloging-in-Publication Data

ISBN 0-12-415146-9

PRINTED IN THE UNITED STATES OF AMERICA

Contents

Contributors vii

Foreword by Lotfi A. Zadeh ix

Preface xiii

Glossary of Symbols xv

Chapter 1 Introduction 1

Chapter 2 Fuzzy Logic: A Specialized Tutorial 11

Chapter 3 Fuzzy Logic and Earth Science: An Overview 63

Chapter 4 Fuzzy Logic in Geological Sciences: A Literature Review 103

Chapter 5 Applications of Fuzzy Logic to Stratigraphic Modeling 121

Chapter 6 Fuzzy Logic in Hydrology and Water Resources 153

Chapter 7 Formal Concept Analysis in Geology 191

Chapter 8 Fuzzy Logic and Earthquake Research 239

Chapter 9 Fuzzy Transform: Application to the Reef Growth Problem 275

Chapter 10 Ancient Sea Level Estimation 301

Acknowledgments 337

Index 339

v

Foreword

In October 1999, at the invitation of my eminent friend, Professor George Klir, Ivisited the Binghamton campus of the State University of New York. In the courseof my visit, I became aware of the fact that Professor Klir, a leading contributorto fuzzy logic and theories of uncertainty, was collaborating with Professor RobertDemicco, a leading contributor to geology and an expert on sedimentology, on anNSF-supported research project involving an exploration of possible applications offuzzy logic to geology. What could be more obvious than suggesting to ProfessorsKlir and Demicco to edit a book entitled “Fuzzy Logic in Geology.” No such bookwas in existence at the time.

I was delighted when Professors Klir and Demicco accepted my suggestion. And,needless to say, I am gratified that the book has become a reality. But, what is reallyimportant is that Professors Klir and Demicco, the contributors and the publisher,Academic Press, have produced a book that is superlative in all respects.

As the editors state in the preface, Fuzzy Logic in Geology is intended to serve threeprincipal purposes: (1) to examine what has been done in this field; (2) to explorenew directions; and (3) to expand the use of fuzzy logic in geology and related fieldsthrough exposition of new tools.

To say that Fuzzy Logic in Geology achieves its aims with distinction is an under-statement. The excellence of organization, the wealth of new material, the profusionof applications, and the high expository skill of contributors, including Professors Klirand Demicco, combine to make the book an invaluable reference and an importantsource of new ideas. There is no doubt that Fuzzy Logic in Geology will be viewedas a landmark in its field.

In the preface, Professors Klir and Demicco note that applications of fuzzy logicin science are far less visible than in engineering and, especially, in the realm ofconsumer products. Is there an explanation?

In science, there is a deep-seated tradition of striving for the ultimate in rigor andprecision. Although fuzzy logic is a mathematically based theory, as is seen in Chapter2, there is a misperception, reflecting the connotation of its label, that fuzzy logic isimprecise and not well-founded. In fact, fuzzy logic may be viewed as an attemptto deal precisely with imprecision, just as probability theory may be viewed as anattempt to deal precisely with uncertainty.

ix

x Foreword

A related point is that in many of its applications, a concept which plays a keyrole is that of a linguistic variable, that is, a variable where values are words ratherthan numbers. Words are less precise than numbers. That is why the use of linguisticvariables in fuzzy logic drew critical comments from some of the leading membersof the scientific establishment. As an illustration, when I gave my first lecture onlinguistic variables in 1972, Professor Rudolf Kalman, a brilliant scientist/engineer,had this to say:

I would like to comment briefly on Professor Zadeh’s presentation. His proposals could beseverely, ferociously, even brutally criticized from a technical point of view. This would beout of place here. But a blunt question remains: Is Professor Zadeh presenting importantideas or is he indulging in wishful thinking? No doubt Professor Zadeh’s enthusiasm forfuzziness has been reinforced by the prevailing climate in the US—one of unprecedentedpermissiveness. ‘Fuzzification’ is a kind of scientific permissiveness; it tends to result insocially appealing slogans unaccompanied by the discipline of hard scientific work and patientobservation.

In a similar vein, a colleague of mine at UCB and a friend, Professor WilliamKahan, wrote:

Fuzzy theory is wrong, wrong, and pernicious. I cannot think of any problem that could notbe solved better by ordinary logic. . . . What Zadeh is saying is the same sort of things as,‘Technology got us into this mess and now it can’t get us out’. Well, technology did not get usinto this mess. Greed and weakness and ambivalence got us into this mess. What we need ismore logical thinking, not less. The danger of fuzzy theory is that it will encourage the sort ofimprecise thinking that has brought us so much trouble.

What Professors Kalman, Kahan, and other prominent members of the scientificestablishment did not realize is that mathematically based use of words enhances theability of scientific theories to deal with real-world problems. In particular, in bothscience and engineering, the use of words makes it possible to exploit the tolerancefor imprecision to achieve tractability, robustness, simplicity and low cost of solution.The use of linguistic variable is the basis for the calculus of fuzzy if-then rules—acalculus which plays a key role in many of the applications of fuzzy logic—includingits applications in geology.

During the past few years, the use of words in fuzzy logic has evolved into method-ology labeled computing with words and perceptions (CWP)—a methodology whichcasts a new light on fuzzy logic and may lead to a radical enlargement of the role ofnatural languages in science and engineering.

Computing with words and perceptions is inspired by the remarkable human capa-bility to perform a wide variety of physical and mental tasks, e.g., driving a car incity traffic or playing tennis, without any measurements and any computations. Inperforming such tasks, humans employ perceptions—perceptions of distance, speed,direction, intent, likelihood, and other attributes of physical and mental objects.

Foreword xi

There is an enormous literature on perceptions, spanning psychology, philosophy,linguistics, and other fields. But what has not been in existence is a theory in whichperceptions can be operated on as objects of computation. Fuzzy logic provides abasis for such a theory—a theory which is referred to as the computational theory of

perceptions (CTP).In the computational theory of perceptions, perceptions are dealt with not as patterns

of brain activity, but through their descriptions in a natural language. In this sense,a natural language may be viewed as a system for describing perceptions. Thus, ifclassical, bivalent logic is viewed as the logic of measurements, then fuzzy logic maybe viewed as the logic of perceptions.

Although the methodology of computing with words and perceptions is not treatedexplicitly in the book, the basic ideas which underlie it are in evidence throughout.Furthermore, Fuzzy Logic in Geology ventures beyond well-established techniquesand presents authoritative expositions of methods which lie on the frontiers offuzzy logic. In this respect, particularly worthy of note are the chapters on for-mal concept analysis (R. Belohlávek), F-transformation (I. Perfilieva), and linguistictheory (V. Novák).

In sum, Fuzzy Logic in Geology is a true role model. It is a high quality workwhich opens the door to application of new methods and new viewpoints to a varietyof basic problems in geology, geophysics, and related fields. It is well-organized andreader-friendly. The editors, the contributors, and the publisher deserve our thanksand accolades.

Lotfi A. ZadehMay 13, 2003Berkeley, CA

This book has three purposes. Its first purpose is to demonstrate that fuzzy logic opensa radically new way to represent geological knowledge and to deal with geologicalproblems, and that this new approach has been surprisingly successful in many areasof geology. This book’s second purpose is to help geologists understand the mainfacets of fuzzy logic and the role of these facets in geology. The final purpose of thisbook is to make researchers in fuzzy logic aware of the emerging opportunities forthe application of their expertise in geology.

This book is a chimera in that it is oriented not only at theoreticians, practitioners,and teachers of geology, but also at members of the fuzzy-set community. For geol-ogists, the book contains a specialized tutorial on fuzzy logic (Chapter 2), a basic intro-duction to the application of fuzzy logic to model geological situations (Chapter 3), anoverview of currently known applications of fuzzy logic in geology (Chapter 4), andsix additional chapters with more extensive examples of applications of fuzzy logicto problems in a broad range of geological disciplines. For fuzzy logicians, the bookis an overview of areas of geology in which fuzzy logic is already well established oris promising. Thus, our overall aim in preparing this book is to provide a useful linkbetween the two communities and further stimulate interdisciplinary research.

The book is a product of a close cooperation between the editors and the severalcontributing authors. The authors were commissioned to write chapters on specifictopics. Great care has been taken to assure that the mathematical terminology andnotation are uniform throughout the book. Moreover, care was also taken to assurethat the structure of individual chapters and the style of referencing were consistentthroughout. Furthermore, authors were requested to focus on clarity of presentation,adding summaries of technical content where appropriate. All these features makethe book attractive and appropriate as a text for graduate courses and seminars.

The book is written, by and large, in a narrative style, with the exception of afew sections in Chapters 7 and 9. These chapters are dependent on fairly complexmathematical preliminaries. It is far more efficient to introduce these preliminaries ina more formal style, typical of mathematical literature, using numbered definitions,lemmas, theorems, and examples. Although this formal presentation in Chapters 7and 9 is essential for understanding operational details of the described methods, itis not necessary for a conceptual understanding of the methods and their geologicalapplications. In fact, these chapters are structured conceptually. With this structure,

xiii

Preface

xiv Preface

the reader may still get the gist of the chapter without studying the details of theformal presentation.

The idea of preparing a book on fuzzy logic in geology was suggested to theeditors by Lotfi Zadeh, the founder of fuzzy logic, during his visit to BinghamtonUniversity in October 1999. Our opinion then, and now, is that it was a good idea.While fuzzy logic is now well established as an important tool in engineering, itsapplications in science are far less developed. Nevertheless, the utility of fuzzy logicin various areas of science has been increasingly recognized since at least the mid1990s. A good example is in chemistry, where the role of fuzzy logic is examinedin the excellent book Fuzzy Logic in Chemistry, edited by Dennis H. Rouvray andpublished by Academic Press in 1997. It thus seemed natural to propose this book,which examines the role of fuzzy logic in geology, to Academic Press, with an eyetoward obtaining a synergistic effect. We hope that this book will not only serve itspurpose well, but that it will stimulate publication of other books exploring the roleof fuzzy logic in other areas of natural sciences such as biology and physics as wellas in the social sciences such as geography and economics.

Robert V. Demicco and George J. KlirBinghamton, New York

December 2002

General Symbols

{x, y, . . .} Set of elements x, y, . . .

{x | p{x}} Set determined by property p〈x1, x2, . . . , xn〉 n-tuple[xij ] Matrix[x1, x2, . . . , xn] Vector[a, b] Closed interval of real numbers between a and b

[a, b), (b, a] Interval of real numbers closed in a and open in b

(a, b) Open interval of real numbersA, B, C . . . . Arbitrary sets (crisp or fuzzy)x ∈A Element x belongs to crisp set A

A(x) or μA(X) Membership grade of x in fuzzy set AαA α-cut of fuzzy set Aα+A Strong α-cut of fuzzy set A

A = B Set equalityA �= B Set inequalityA− B Set differenceA ⊆ B Set inclusionA ⊂ B Proper set inclusion (A ⊆ B and A �= B)SUB(A, B) Degree of subsethood of A in B

P(X) Set of all crisp subsets of X (power set)F(X) Set of all standard fuzzy subsets of X (fuzzy power set)|A| Cardinality of crisp or fuzzy set A (sigma count)hA Height of fuzzy set A

A Complement of set A

A ∩ B Set intersectionA ∪ B Set unionA× B Cartesian product of sets A and B

A2 Cartesian product A× A

f : X → Y Function from X to Y

f−1 Inverse of function f

R ◦Q Standard composition of fuzzy relations R and Q

xv

Symbols

xvi Glossary of Symbols

R ∗Q Join of fuzzy relations R and Q

R−1 Inverse of a binary fuzzy relation< Less than≤ Less than or equal to (also used for a partial ordering)x | y x given y

x ⇒ y x implies y

x ⇔ y x if and only if y∑Summation

� Productmax(a1, a2, . . . , an) Maximum of (a1, a2, . . . , an)min(a1, a2, . . . , an) Minimum of (a1, a2, . . . , an)N Set of positive integers (natural numbers)Nn Set {1, 2, . . . , n}

R Set of all real numbers

Special Symbols

B(X, Y, I ) The set of all fuzzy concepts in a given context 〈X, Y, I 〉

c Fuzzy complementd(A) Defuzzified value of fuzzy set A

E Similarity relation (fuzzy equivalence)h Averaging operationhp Generalized meansi Fuzzy intersection or t-normimin Drastic fuzzy intersectioniw Fuzzy intersection of Yager classJ Fuzzy implication operatorL Set of truth degreesL Complete residuated latticeLX The set of all fuzzy sets in X with truth values in L

m Fuzzy modifierNecE Necessity measure corresponding to PosE

pA Fuzzy propositional form and truth assignmentp Fuzzy probability qualifierPosE Possibility measure associated with a proposition “ν is E”S(Q, R) Solution set of fuzzy relation equation R ◦Q = R

T Fuzzy truth qualifierX , Y Variables〈X, Z, I 〉 Fuzzy contextu Fuzzy union or t-conormumax Drastic fuzzy union

Glossary of Symbols xvii

uw Fuzzy union of Yager classW Set of possible worldsX Universal set (universe of discourse)Ø Empty set⊗ Operation on L corresponding to conjunction (t-norm)→ Operation on L corresponding to implication∧ Classical operation of conjunction or minimum operation∨ Classical operation of disjunction or maximum operation

Chapter 1 Introduction

Robert V. Demicco and George J. Klir

Traditionally, science, engineering, and mathematics showed virtually no interest instudying uncertainty. It was considered undesirable and the ideal was to eliminateit. In fact, eliminating uncertainty from science was viewed as one manifestationof progress. This attitude towards uncertainty, prevalent prior to the 20th century,was seriously challenged by some developments in the first half of that century.Among them were the emergence of statistical mechanics, Heisenberg’s uncertaintyprinciple in quantum mechanics, and Gödel’s theorems that established an inher-ent uncertainty in formal mathematical systems. In spite of these developments, thetraditional attitude towards uncertainty changed too little and too slowly during thefirst half of the century. While uncertainty became recognized as useful, or evenessential, in statistical mechanics and in some other areas (such as the actuarial pro-fession or the design of large-scale telephone exchanges), it was for a long timetacitly assumed that probability theory was capable of capturing the full scope ofuncertainty.

The presumed equality between uncertainty and probability was challenged onlyin the second half of the 20th century. The challenge came from two important gen-eralizations in mathematics. The first one was the generalization of classical measuretheory [Halmos, 1950] to the theory of monotone measures, which was first suggestedby Choquet [1953] in his theory of capacities. The second one was the generalizationof classical set theory to fuzzy set theory, which was introduced by Zadeh [1965]. Inthe theory of monotone measures, the additivity requirement of classical measures isreplaced with a weaker requirement of monotonicity with respect to set inclusion. Infuzzy set theory, the requirement of sharp boundaries of classical sets is abandoned.That is, the membership of an object in a fuzzy set is not a matter of either affirma-tion or denial, as it is in the case of any classical set, but it is in general a matter ofdegree.

For historical reasons of little significance, monotone measures are often referredto in the literature as fuzzy measures [Wang & Klir, 1992]. This name is somewhatconfusing since no fuzzy sets are involved in the definition of monotone measures.However, monotone measures can be fuzzified (i.e., defined on fuzzy sets), whichresults in a more general class of monotone measures—fuzzy monotone measures

[Wang & Klir, 1992, Appendix E].

1

2 1 Introduction

As is well known, probability theory is based on classical measure theory which,in turn, is based on classical set theory [Halmos, 1950]. When classical measures arereplaced with monotone measures of some type and classical sets are replaced withfuzzy sets of some type, a framework is obtained for formalizing some new typesof uncertainty, distinct from probability. This indicates that the two generalizationshave opened a vast territory for formalizing uncertainty. At this time, only a rathersmall part of this territory has been adequately explored [Klir & Wierman, 1999;Klir, 2002].

Liberating uncertainty from its narrow confines of probability theory opens new,more expressive ways of representing scientific knowledge. As is increasingly rec-ognized, scientific knowledge is organized, by and large, in terms of systems ofvarious types (or categories in the sense of mathematical theory of categories)[Klir & Rozehnal, 1996; Klir & Elias, 2003]. In general, systems are viewed asrelations between states of some variables. They are constructed for various purposes(prediction, retrodiction, prescription, diagnosis, control, etc.). In each system, itsrelations are utilized, in a given purposeful way, for determining unknown statesof some variables on the basis of known states of some other variables. Systems inwhich the unknown states are determined uniquely are called deterministic; all othersystems are called nondeterministic.

By definition, each nondeterministic system involves uncertainty of some type.This uncertainty pertains to the purpose for which the system was constructed. It isthus natural to distinguish between predictive uncertainty, retrodictive uncertainty,diagnostic uncertainty, etc. In each nondeterministic system, the relevant uncertaintymust be properly incorporated into the description of the system in some formalizedlanguage. To understand the full scope of uncertainty is thus essential for dealing withnondeterministic systems.

When constructing a system for some given purpose, our ultimate goal is to obtain asystem that is as useful as possible for this purpose. This means, in turn, to construct asystem with a proper blend of the three most fundamental characteristics of systems:credibility, complexity, and uncertainty. Ideally, we would like to obtain a systemwith high credibility, low complexity, and low uncertainty. Unfortunately, these threecriteria conflict with one another. To achieve high usefulness of the system, we needto find the right trade-off among them.

The relationship between credibility, complexity and uncertainty is quite intri-cate and is not fully understood yet. However, it is already well established thatuncertainty has a pivotal role in any efforts to maximize the usefulness of constructedsystems. Although usually undesirable in systems when considered alone, uncertaintybecomes very valuable when considered in connection with credibility and complex-ity of systems. A slight increase in relevant uncertainty may often significantly reducecomplexity and, at the same time, increase credibility of the system. Uncertainty isthus an important commodity in the knowledge business, a commodity that can betraded for gains in the other essential characteristics of systems by which we represent

1 Introduction 3

knowledge. Because of this important role, uncertainty is no longer viewed in scienceand engineering as an unavoidable plague, but rather as an important resource thatallow us to deal effectively with problems involving very complex systems.

It is our contention that monotone measures and fuzzy sets (as well as the variousuncertainty theories opened by these two profound generalizations in mathematics)are highly relevant to geology, and that their utility in geology should be seriouslystudied in the years ahead. The aim of this book is to demonstrate this point by focusingon the role of fuzzy set theory, and especially the associated fuzzy logic, in geology.

The term “fuzzy logic” has in fact two distinct meanings. In a narrow sense, it isviewed as a generalization of classical multivalued logics. It is concerned with thedevelopment of syntactic aspects (based on the notion of proof ) and semantic aspects

(based on the notion of truth) of a relevant logic calculus. In order to be acceptable,the calculus must be sound (provability implies truth) and complete (truth impliesprovability). These issues have successfully been addressed for fuzzy logic in thenarrow sense by Hájek [1998].

In a broad sense, fuzzy logic is viewed as a system of concepts, principles, andmethods for dealing with modes of reasoning that are approximate rather than exact.The two meanings are connected since the very purpose of research on fuzzy logic inthe narrow sense is to provide fuzzy logic in the broad sense with sound foundations.In this book, we are concerned only with fuzzy logic in the broad sense, which issurveyed in Chapter 2, and its role in geology, which is the subject of Chapters 3–10.

From the standpoint of science, as it is still predominantly understood, the ideasof a fuzzy set and a fuzzy proposition are extremely radical. When accepted, onehas to give up classical bivalent logic, generally presumed to be the principal pillarof science. Instead, we obtain a logic in which propositions are not required to beeither true or false, but may be true or false to different degrees. As a consequence,some laws of bivalent logic no longer hold, such as the law of excluded middle or thelaw of contradiction. At first sight, this seems to be at odds with the very purpose ofscience. However, this is not the case. There are at least the following four reasonswhy allowing membership degrees in sets and degrees of truth in propositions in factenhances scientific methodology quite considerably:

1. Fuzzy sets and fuzzy propositions possess far greater capabilities than their classi-cal counterparts to capture irreducible measurement uncertainties in their variousmanifestations. As a consequence, their use improves the bridge between mathe-

matical models and the associated physical reality considerably. It is paradoxicalthat, in the face of the inevitable measurement errors, fuzzy data are always moreaccurate than their crisp (i.e., nonfuzzy) counterparts. Crisp data of each vari-able are based on a partition of the state set of the variable. The coarseness ofthis partition is determined by the resolution power of the measuring instrumentemployed. Measurements falling into the same block of the partition are not dis-tinguished in crisp data, regardless of their position within the block. Thus, for

4 1 Introduction

example, a measurement that is at the mid-point of the block is not distinguishedfrom those at the borders with adjacent blocks. While the former is uncertaintyfree, provided that the block is sufficiently large relative to the resolution power ofthe measuring instrument employed, the latter involves considerable uncertaintydue to the inevitability of measurement errors. This fundamental distinction isnot captured at all in crisp data. On the contrary, fuzzy data can capture thisand other measurement distinctions of this kind in terms of distinct member-ship degrees. Fuzzy data are thus more accurate than crisp data in this sense.Membership degrees that accompany fuzzy data express indirectly pertinent mea-surement uncertainties. When fuzzy data are processed, the membership degreesare processed as well. This implies that any results obtained by this processingare again more accurate (in the empirical sense) than their counterparts obtainedby processing the less accurate crisp data.

2. An important feature of fuzzy logic in the broad sense is its capability to capturethe vagueness of linguistic terms in statements that are expressed in natural lan-guages. Vagueness of a symbol (a linguistic term) in a given language results fromthe existence of objects for which it is intrinsically impossible to decide whetherthe symbol does or does not apply to them according to linguistic habits of somespeech community using the language. That is, vagueness is a kind of uncertaintythat does not result from information deficiency, but rather from imprecise mean-ings of linguistic terms, which are particularly abundant in natural languages.Classical set theory and classical bivalent logic are not capable of expressing theimprecision in meanings of vague terms. Hence, propositions in natural languagethat contain vague terms were traditionally viewed as unscientific. However, thisview is extremely restrictive. As has increasingly been recognized in many areasof science, including especially geology, natural language is often the only wayin which meaningful knowledge can be expressed.

3. Fuzzy sets and fuzzy propositions are powerful tools for managing complexity andcontrolling computational cost. This is primarily due to granulation of systemsvariables, which is a fuzzy counterpart of the classical quantization of variables.In quantization, states of a given variable are grouped into subsets (quanta) thatare pairwise disjoint. In granulation, they are grouped into suitable fuzzy subsets(granules). The aim of both quantization and granulation is to make precisioncompatible with a given task. The advantage of granulation is that, contraryto quantization, it allows us to express gradual transitions from each granuleto its neighbors. In quantization, the transition from one quantum to another isalways abrupt and, hence, rather superficial. Granulation is thus a better way thanquantization to adjust precision of systems as needed.

4. The apparatus of fuzzy set theory and fuzzy logic enhances our capabilities ofmodeling human common-sense reasoning, decision-making, and other aspectsof human cognition. These capabilities are essential for acquiring knowledgefrom human experts, for representating and manipulating knowledge in expert

1 Introduction 5

systems in a human-like manner, and, generally, for designing and buildinghuman-friendly machines with high intelligence. Fuzzy sets and fuzzy propo-sitions are also essential for studying human reasoning, decision making, andacting that are based on perceptions rather than measurements.

It is the synergy of all these capabilities that has made fuzzy set theory and fuzzylogic highly successful in many engineering applications over the last two decadesor so. The most visible of these applications have been in the area of control, rangingfrom simple control systems in consumer products (intelligent washing machines,vacuum cleaners, camcorders, etc.) to highly challenging control systems, such asthe one for controlling a pilotless helicopter via wireless communication of commandsexpressed in natural language. Less visible but equally successful applications havebeen demonstrated in the areas of database and information retrieval systems, expertsystems, decision making, pattern recognition and clustering, image processing andcomputer vision, manufacturing, robotics, transportation, risk and reliability analyses,and many other engineering areas. In fact, every field of engineering has already beenpositively affected, in one way or another, by fuzzy set theory and fuzzy logic [Ruspiniet al., 1998].

In science, applications of fuzzy set theory and fuzzy logic have developed ata considerably slower pace than in engineering and only in some areas of sciencethus far. This is understandable if we realize how extremely radical the ideas offuzzy sets and fuzzy propositions actually are. Nevertheless, successful applicationshave already been demonstrated in many areas of science. Examples are applicationsin quantum physics [Pykacz, 1993; Cattaneo, 1993], chemistry [Rouvray, 1997],biology [Von Sternberg & Klir, 1998], geography [Gale, 1972], ecology [Libelli& Cianchi, 1996], linguistics [Rieger, 2001], economics [Billot, 1992], psychology[Zétényi, 1988], and social sciences [Smithson, 1987]. In geology, the utility of fuzzyset theory and fuzzy logic was recognized, by and large, only in the late 1990s, but thenumber of publications dealing with applications of fuzzy logic in geology is alreadysubstantial and is growing fast (Chapter 4). This is a clear indicator that the use offuzzy logic in geology has a great potential. Our motivation for publishing this bookis to help to develop this potential.

It is important to realize that fuzzy set theory and fuzzy logic are not only toolsthat help us to deal with some difficult problems in science, engineering, and otherprofessional areas, but they also provide us with a conceptual framework for a rad-ically new way of thinking. Sharp boundaries of classical sets and absolute truthsor falsities of classical propositions are still possible under the new thinking, whenjustifiable, but they are viewed as limiting cases rather than the only possibilities.Thinking in absolute terms is replaced with thinking in relative terms. Everythingbecomes a matter of degree. This change in our thinking will undoubtedly open new,more refined ways of looking at old issues of epistemology, ethics, law, social policy,and other areas that affect our lives.

6 1 Introduction

The emergence of fuzzy set theory and fuzzy logic and their impact on mathematicsand logic as well as on science and science-dependent areas of human affairs possessall distinctive features that are characteristic of a paradigm shift, as introduced inthe highly influential book by Thomas Kuhn [1962]. Since logic is fundamental tovirtually all branches of mathematics as well as science, this paradigm shift has muchbroader implications than those generally recognized in the history of science andmathematics, each of which affects only a particular area of science or mathemat-ics. It is thus appropriate to refer to it as a “grand paradigm shift.” Various specialcharacteristics of this paradigm shift, which is still ongoing, are discussed by Klir[1995, 1997, 2000]. It is generally agreed that this paradigm shift was initiated by thepublication of the seminal paper by Zadeh [1965]. However, many ideas pertainingto fuzzy logic had appeared in the literature prior to the publication of that paper.Unfortunately, these ideas were by and large ignored at that time [Klir, 2001].

The purpose of this book is threefold: (i) to examine how fuzzy logic has alreadybeen applied in some areas of geology; (ii) to stimulate the development of applica-tions of fuzzy logic in other areas of geology; and (iii) to stimulate the use of additionaltools of fuzzy logic in geology. Material covered in Chapters 2–10 was carefullyselected to accomplish this purpose. The following is a brief preview of this material.

Chapter 2 is an overview of fuzzy logic in the broad sense. It is written as a tutorialfor those readers who are not familiar with fuzzy logic. This chapter covers not onlythose components of fuzzy logic that are employed in subsequent chapters, but alsosome additional ones which offer new application possibilities for geology. More-over, this chapter introduces terminology and notation that are followed consistentlythroughout the whole book.

The aim of Chapter 3 is twofold: (i) to discuss reasons for using fuzzy logic ingeology; and (ii) to illustrate the use of fuzzy logic in geology by simple examples.For geologists, some of the notions of fuzzy logic introduced in Chapter 2 are furtherdiscussed in terms of simple geological interpretations. For researchers in fuzzy logic,the chapter is a sort of tutorial which introduces them to some issues that are of concernto geology.

Chapter 4 is a comprehensive overview of currently known applications of fuzzylogic in geology. It is primarily an annotated bibliography that is grouped into thefollowing nine categories: (1) surface hydrology; (2) subsurface hydrology; (3)groundwater risk assessment; (4) geotechnical engineering; (5) hydrocarbon explo-ration; (6) seismology; (7) soil science and landscape development; (8) depositionof sediments; and (9) miscellaneous applications. In addition, the role of fuzzy logicwithin the broader area of soft computing is briefly characterized. The aim of thischapter is to provide readers with a useful resource for further study of establishedapplications of fuzzy logic in geology, sometimes in the broader context of softcomputing.

Each of the remaining six chapters of this book covers in greater depth applica-tions of fuzzy logic in some specific area of geology. The utility of fuzzy logic to

1 Introduction 7

stratigraphic modeling is demonstrated in Chapter 5 via several case studies. Thechapter describes two-dimensional and three-dimensional stratigraphic simulationsthat use fuzzy logic to model sediment production, sediment erosion, sediment trans-port, and sediment deposition. It is shown that fuzzy logic offers a robust, easilyadaptable, and computationally efficient alternative to the traditional numerical solu-tion of complex, coupled differential equations commonly used to model sedimentdispersal in stratigraphic models.

Chapter 6 examines the utility of fuzzy logic in hydrology and water resources.These are areas of geology where applications of fuzzy logic are well established.After the various applications of fuzzy logic in these areas are surveyed, one majorarea of hydrology is chosen to describe the use of fuzzy logic in detail: the areaof hydro-climatic modeling of hydrological extremes (i.e., droughts and intensiveprecipitation). Results over four regions (Arizona, Nebraska, Germany, and Hungary)and under three different climates (semiarid, dry, and wet continental) suggest thatthe use of fuzzy logic is successful in predicting statistical properties of monthlyprecipitation and drought index from the joint forcing of macrocirculation patternsand ENSO information.

The purpose of Chapter 7 is to present formal concept analysis of fuzzy data andto explore its prospective applications in geology and paleontology. Formal conceptanalysis is concerned with analyzing data in terms of objects and their attributes. Itis capable of answering questions such as: (i) What are the natural concepts that arehidden in the object-attribute data (e.g., important classes of organisms, minerals, orfossils)?; or (ii) What are the dependencies that are implicit in the object-attribute data?Fuzzified formal concept analysis, which is a relatively new methodological tool, isdescribed in detail in the chapter and is illustrated by an example from paleontology.

Chapter 8 is a comprehensive overview of the role of fuzzy logic in seismologyand some closely related areas. Basic terminology of seismology is introduced to helpreaders who are not familiar with this area of geology. The focus in the chapter ison applications of fuzzy logic and other areas of fuzzy mathematics to earthquakeprediction, assessment of earthquake intensity, assessment of earthquake damage,and study of the relationship between isoseismal area and earthquake magnitude.

The last two chapters of the book explore some new ideas emerging from fuzzy logicthat can be applied to a broad range of geological problems. These chapters requiresome mathematical sophistication, but they are self-contained in the sense that thereader is provided with the relevant preliminaries and specific examples of applica-tions. Chapter 9 describes a new numerical technique—fuzzy transformation—thatallows complex functions to be approximated to a high order. Moreover, useful manip-ulations (such as numerical integration) are, in a number of cases, easier for thetransformed expressions than for the originals. This technique is then applied to asolution of an ordinary differential equation used to model long-term reef growthunder a variable sea level regime. Chapter 10 provides an example of how fuzzylogic can mathematically formalize what heretofore were primarily only linguistic

8 1 Introduction

descriptions and interpretations of geologic phenomena. In this case, a computer pro-gram using specialized fuzzy-set based “evaluating expressions” is taught to mimicthe linguistic geologic “rules” for both the division of Paleozoic measured sections oflimestone into a hierarchy of different cycles, and the interpretation of those cyclesin terms of ancient sea level.

References

Billot, A. [1992], Economic Theory of Fuzzy Equilibria. Springer-Verlag, New York.Cattaneo, G. [1993], “Fuzzy quantum logic II: The logics of unsharp quantum mechanics.”

International Journal of Theoretical Physics, 32(10), 1709–1734.Choquet, G. [1953–54], “ Theory of capacities.” Annales de L’Institut Fourier, 5, 131–295.Gale, S. [1972], “Inexactness, fuzzy sets and the foundations of behavioral geography.”

Geographical Analysis, 4, 337–349.Hájek, P. [1998], Metamathematics of Fuzzy Logic. Kluwer, Boston, MA.Halmos, P. R. [1950], Measure Theory. Van Nostrand, Princeton, NJ.Klir, G. J. [1995], “From classical sets to fuzzy sets: a grand paradigm shift.” In: Wang,

P. P. (ed.), Advances in Fuzzy Theory and Technology, Vol. III, pp. 3–30. Duke University,Durham, NC.

Klir, G. J. [1997], “From classical mathematics to fuzzy mathematics: emergence of a newparadigm for theoretical science.” In: Rouvray, D. H. (ed.), Fuzzy Logic in Chemistry,pp. 31–63. Academic Press, San Diego, CA.

Klir, G. J. [2000], Fuzzy Sets: An Overview of Fundamentals, Applications, and Personal

Views. Beijing Normal University Press, Beijing, China.Klir, G. J. [2001], “Foundations of fuzzy set theory and fuzzy logic: A historical overview.”

International Journal of General Systems, 30(2), 91–132.Klir, G. J. [2002], “Uncertainty-based information.” In: Melo-Pinto and H.-N. Teodorescu

(eds.), Systemic Organisation of Information in Fuzzy Systems, pp. 21–52. IOS Press,Amsterdam.

Klir, G. J., & Elias, D. [2003], Architecture of Systems Problem Solving (2nd edition).Kluwer/Plenum, New York.

Klir, G. J., & Rozehnal, I. [1996], “Epistemological categories of systems: an overview.”International Journal of General Systems, 24(1–2), 207–224.

Klir, G. J., & Wierman, M. J. [1999], Uncertainty-Based Information: Elements of Gener-

alized Information Theory (2nd edition). Physica-Verlag/Springer-Verlag, Heidelberg andNew York.

Kuhn, T. S. [1962], The Structure of Scientific Revolutions. University of Chicago Press,Chicago, IL.

Libelli, S. M., & Cianchi, P. [1996], “Fuzzy ecological models.” In: Pedrycz, W. (ed.), Fuzzy

Modelling Paradigms and Practice, pp.141–164. Kluwer, Boston, MA.Pykacz, J. [1993], “Fuzzy quantum logic I.” International Journal of Theoretical Physics,

32(10), 1691–1707.Rieger, B. B. [2001], “Computing granular word meanings: A fuzzy linguistic approach in

computational semiotics.” In: Wang, P. P. (ed.), Computing with Words, pp. 147–208.John Wiley, New York.

References 9

Rouvray, D. H. (ed.) [1997], Fuzzy Logic in Chemistry. Academic Press, San Diego, CA.Ruspini, E. H., Bonissone, P. P., & Pedrycz, W. (eds.) [1988], Handbook of Fuzzy Computation.

Institute of Physics Publishing, Bristol (UK) and Philadelphia, PA.Smithson, M. [1987], Fuzzy Set Analysis for Behavioral and Social Sciences. Springer-Verlag,

New York.Von Sternberg, R., & Klir, G. J. [1998], “Generative archetypes and taxa: A fuzzy set

formalization.” Biology Forum, 91, 403–424.Wang, Z., & Klir, G. J. [1992], Fuzzy Measure Theory. Plenum Press, New York.Zadeh, L. A. [1965], “Fuzzy sets.” Information and Control, 8(3), 338–353.Zétényi, T. (ed.) [1988], Fuzzy Sets in Psychology. North-Holland, Amsterdam and New York.

Chapter 2 Fuzzy Logic: A Specialized Tutorial

George J. Klir

2.1 Introduction 112.2 Basic Concepts of Fuzzy Sets 142.3 Operations on Fuzzy Sets 19

2.3.1 Modifiers 19

2.3.2 Complements 21

2.3.3 Intersections and unions 22

2.3.4 Averaging operations 25

2.3.5 Arithmetic operations 28

2.4 Fuzzy Relations 312.4.1 Projections, cylindric extensions, and cylindric closures 32

2.4.2 Inverses, compositions, and joins 33

2.4.3 Fuzzy relation equations 34

2.4.4 Fuzzy relations on a single set 36

2.5 Fuzzy Logic 382.5.1 Basic types of propositional forms 41

2.5.2 Approximate reasoning 44

2.6 Possibility Theory 462.7 Fuzzy Systems 492.8 Constructing Fuzzy Sets and Operations 532.9 Nonstandard Fuzzy Sets 552.10 Principal Sources for Further Study 57References 59

2.1 Introduction

The term “fuzzy logic,” as currently used in the literature, has two distinct meanings.In the narrow sense, it is viewed as a generalization of the various many-valued log-ics that have been investigated in the area of mathematical logic since the beginningof the 20th century. An excellent historical overview of the emergence and devel-opment of many-valued logics was prepared by Rescher [1969]; the various issues

11

12 2 Fuzzy Logic: A Specialized Tutorial

involved in generalizing many-valued logics into fuzzy logic are thoroughly coveredin monographs by Hájek [1998] and Novák et al. [1999].

In the alternative, broad sense, fuzzy logic is viewed as a system of concepts,principles, and methods for dealing with modes of reasoning that are approximaterather than exact [Novák & Perfilieva, 2000]. In this book, we are interested in fuzzylogic only in this broad sense. In this sense, fuzzy logic is based upon fuzzy set theory.It utilizes the apparatus of fuzzy set theory for formulating various forms of soundapproximate reasoning in natural language. It is thus essential to begin our tutorialwith an overview of basic concepts of fuzzy set theory.

Fuzzy set theory, introduced by Zadeh [1965], is an outgrowth of classical settheory. Contrary to the classical concept of a set, or crisp set, the boundary of afuzzy set is not precise. That is, the change from nonmembership to membershipin a fuzzy set may be gradual rather than abrupt. This gradual change is expressedby a membership function, which completely and uniquely characterizes a particularfuzzy set.

Every geologist is familiar with the terms clay, silt, and gravel, terms used todescribe the “size” of sedimentary particles (Figure 2.1a). These terms stand for crispsets as they are most commonly used, insofar as a grain can only belong to one sizegrade at a time. Thus, in the traditional “pigeon hole” view of grain sizes, a sphericalgrain with a diameter of 1.999 mm would be sand whereas a grain 2.001 mm in diam-eter would be gravel. An alternative representation of the crisp set “sand” would be toassign a value of 1 to grain diameters that are members of the set “sand” (the domaininterval (0.0625–2] mm) and a 0 to grain diameters that are not sand. In contrast,

Figure 2.1 Comparison of crisp-set (a) versus fuzzy-set (b) representation of the geologicvariable “grain size.”

2.1 Introduction 13

one possible representation of the sedimentary size terms clay, silt, sand, and gravelwith fuzzy sets is shown in Figure 2.1b. In a fuzzy set representation the range ofmembership in a given set (e.g., “sand”) is not limited to 0 or 1 but can take on anyvalue between and including [0, 1]. Our hypothetical 1.999 and 2.001 mm diame-ter grains are simultaneously members of both sets, sand and gravel, to a degree ofabout 0.5. The simple trapezoids represent the membership functions.

Two distinct notations are most commonly employed in the literature to denotemembership functions. In one of them, the membership function of a fuzzy set A isdenoted by μA(x) and usually has the form

μA: X → [0, 1], (2.1)

where X denotes the universal set under consideration and A is a label of the fuzzyset defined by this function. The universal set is always assumed to be a crisp set.For each x ∈ X, the value μA(x) expresses the degree (or grade) of membership ofelement x of X in fuzzy set A.

In the second notation, the symbol A of a fuzzy set is also used to denote themembership function of A. However, no ambiguity results from this double use ofthe same symbol since each fuzzy set is completely and uniquely defined by oneparticular membership function. That is, A(x) in the second notation has the samemeaning as μA in the first notation; (2.1) is thus written in the second notation as

A: X → [0, 1]. (2.2)

In this book, the second notation is adopted. It is simpler and, by and large,more popular in current literature on fuzzy set theory. Classical (crisp) sets maybe viewed from the standpoint of fuzzy set theory as special fuzzy sets, in whichA(x) is either 0 or 1 for each x ∈ X. Hence, we use the same notation for fuzzy setsand crisp sets.

Fuzzy sets whose membership functions have the form (2.2), which are calledstandard fuzzy sets, do not capture the full variety of fuzzy sets. Since standard fuzzysets are currently predominant in the literature, this tutorial is largely devoted tothem. However, basic properties of several nonstandard types of fuzzy sets, whoseimportance in some applications has lately been recognized, are introduced inSection 2.9.

Additional examples of membership functions are shown in Figure 2.2. These func-tions may be considered as candidates for representing the meaning of the linguisticexpression “around 3” in the context of a given application. The width of each of thesefunctions is, of course, strongly dependent on the application context. In general, amembership function that is supposed to capture the intended meaning of a linguisticexpression in the context of a particular application must be somehow constructed.This issue is discussed in Section 2.8.


Figure 2.2 Possible shapes of membership functions whose purpose is to capture the meaningof the linguistic expression “around 3” in the context of a given application.

2.2 Basic Concepts of Fuzzy Sets

Given two fuzzy sets A, B defined on the same universal set X, A is said to be asubset of B if and only if

A(x) ≤ B(x)

for all x ∈ X. The usual notation, A ⊆ B, is used to signify the subsethood relation.The set of all fuzzy subsets of X is called the fuzzy power set of X and is denotedby F(X). Observe that this set is crisp, even though its members are fuzzy sets.Moreover, this set is always infinite, even if X is finite. It is also useful to define adegree of subsethood, SUB(A, B), of A in B. When the sets are defined on a finite

2.2 Basic Concepts of Fuzzy Sets 15

universal set X, we have

SUB(A, B) =

∑x∈X

A(x)−∑x∈X

max[0, A(x)− B(x)]

∑x∈X

A(x). (2.3)

The negative term in the numerator describes the sum of the degrees to which thesubset inequality A(x)≤B(x) is violated, the positive term describes the largestpossible violation of the inequality, the difference in the numerator describes the sumof the degrees to which the inequality is not violated, and the term in the denominatoris a normalizing factor to obtain the range

0 ≤ SUB(A, B) ≤ 1.

When sets A and B are defined on a bounded subset of real numbers (i.e., X is aclosed interval of real numbers), the three � terms in (2.3) are replaced with integralsover X.

For any fuzzy set A defined on a finite universal set X, its scalar cardinality, |A|,is defined by the formula

|A| =∑

x∈X

A(x).

Scalar cardinality is sometimes referred to in the literature as a sigma count.Among the most important concepts of standard fuzzy sets are the concepts of an

α-cut and a strong α-cut. Given a fuzzy set A defined on X and a particular numberα in the unit interval [0, 1], the α-cut of A, denoted by αA, is a crisp set that consistsof all elements of X whose membership degrees in A are greater than or equal to α.This can formally be written as

αA = {x|A(x) ≥ α}.

The strong α-cut, α+A, has a similar meaning, but the condition “greater than or equalto” is replaced with the stronger condition “greater than.” Formally,

α+A = {x|A(x) > α}.

The set 0+A is called the support of A and the set 1A is called the core of A. Whenthe core A is not empty, A is called normal; otherwise, it is called subnormal. Thelargest value of A is called the height of A and it is denoted by hA. The set of distinctvalues A(x) for all x ∈ X is called the level set of A and is denoted by ΛA.


Figure 2.3 Illustration of some basic characteristics of fuzzy sets.

All the introduced concepts are illustrated in Figure 2.3. We can see that

α1A ⊆ α2A and α1+A ⊆ α2+A2

when α1 ≥ α2. This implies that the set of all distinct α-cuts (as well as strong α-cuts)is always a nested family of crisp sets. When α is increased, the new α-cut (strongα-cut) is always a subset of the previous one. Clearly, 0A = X and 1+A = ∅.

It is well established [Klir & Yuan, 1995] that each fuzzy set is uniquely representedby the associated family of its α-cuts via the formula

A(x) = sup {α · αA(x)|α ∈ [0, 1]}, (2.4)

or by the associated family of its strong α-cuts via the formula

A(x) = sup {α · α+A(x)|α ∈ [0, 1]}, (2.5)

where sup denotes the supremum of the respective set and αA (or α+A) denotes foreach α ∈ [0, 1] the special membership function (characteristic function) of the α-cut(or strong α-cut, respectively).

The significance of the α-cut (or strong α-cut) representation of fuzzy sets is thatit connects fuzzy sets with crisp sets. While each crisp set is a collection of objectsthat are conceived as a whole, each fuzzy set is a collection of nested crisp sets thatare also conceived as a whole. Fuzzy sets are thus wholes of a higher category.

The α-cut representation of fuzzy sets allows us to extend the various propertiesof crisp sets, established in classical set theory, into their fuzzy counterparts. Thisis accomplished by requiring that the classical property be satisfied by all α-cuts ofthe fuzzy set concerned. Any property that is extended in this way from classical

2.2 Basic Concepts of Fuzzy Sets 17

set theory into the domain of fuzzy set theory is called a cutworthy property. Forexample, when convexity of fuzzy sets is defined by the requirement that all α-cutsof a fuzzy convex set be convex in the classical sense, this conception of fuzzyconvexity is cutworthy. Other important examples are the concepts of a fuzzy partition,fuzzy equivalence, fuzzy compatibility, and various kinds of fuzzy orderings that arecutworthy (Section 2.4).

It is important to realize that many (perhaps most) properties of fuzzy sets, perfectlymeaningful and useful, are not cutworthy. These properties cannot be derived fromclassical set theory.

Another way of connecting classical set theory and fuzzy set theory is to fuzzifyfunctions. Given a function

f : X → Y,

where X and Y are crisp sets, we say that the function is fuzzified when it is extendedto act on fuzzy sets defined on X and Y . That is, the fuzzified function maps, ingeneral, fuzzy sets defined on X to fuzzy sets defined on Y . Formally, the fuzzifiedfunction, F , has the form

F : F(X)→ F(Y ),

where F(X) and F(Y ) denote the fuzzy power set (the set of all fuzzy subsets) of X

and Y , respectively. To qualify as a fuzzified version of f , function F must conformto f within the extended domain F(X) and F(Y ). This is guaranteed when a principleis employed that is called an extension principle. According to this principle,

B = F(A)

is determined for any given fuzzy set A ∈ F(X) via the formula

B(y) = maxx|y=f (x)

A(x) (2.6)

for all y ∈ Y . Clearly, when the maximum in (2.6) does not exist, it is replaced withthe supremum.

The inverse function

F−1: F(Y )→ F(X),

of F is defined, according to the extension principle, for any given B ∈ F(Y ), by theformula

[F−1(B)](x) = B(y), (2.7)

for all x ∈ X, where y = f (x). Clearly,

F−1[F(A)] ⊇ A


Figure 2.4 Illustration of the extension principle.

for all A ∈ F(X), where the equality is obtained when f is a one-to-one function.The use of the extension principle is illustrated in Figure 2.4, where it is shown

how fuzzy set A is mapped to fuzzy set B via function F that is consistent with thegiven function f . That is, B = F(A). For example, since

b = f (a1) = f (a2) = f (a3),

we have

B(b) = max[A(a1), A(a2), A(a3)]

by Equation (2.6). Conversely,

F−1(B)(a1) = F−1(B)(a2) = F−1(B)(a3) = B(b)

by (2.7).The introduced extension principle, by which functions are fuzzified, is basically

described by Equations (2.6) and (2.7). These equations are direct generalizationsof similar equations describing the extension principle of classical set theory. In thelatter, symbols A and B denote characteristic functions of crisp sets.

2.3 Operations on Fuzzy Sets 19

2.3 Operations on Fuzzy Sets

Operations on fuzzy sets possess a considerably greater variety than those on classicalsets. In fact, most operations on fuzzy sets do not have any counterparts in classical settheory. The following five types of operations on fuzzy sets are currently recognized:

(a) modifiers;(b) complements;(c) intersections;(d) unions;(e) averaging operations.

Modifiers and complements operate on one fuzzy set. Intersections and unions oper-ate on two fuzzy sets, but their application can be extended to any number of fuzzysets via their property of associativity. The averaging operations, which are not asso-ciative, operate, in general, on n fuzzy sets (n ≥ 2). In addition to these five typesof operations, special fuzzy sets referred to as fuzzy intervals are also subject toarithmetic operations.

As can be seen from this overall characterization of operations on fuzzy sets, thissubject is very extensive. It is also a subject that has been investigated by manyresearchers, and that is now quite well developed. Due to the enormous scope ofthe subject, we are able to present in this section only a very brief characterizationof each of the introduced types of operations, but we provide the reader with amplereferences for further study.

2.3.1 Modifiers

Modifiers are unary operations whose primary purpose is to modify fuzzy sets toaccount for linguistic hedges, such as very, fairly, extremely, moderately, etc., inrepresenting expressions of natural language. Each modifier, m, is an increasing(and usually continuous) one-to-one function of the form

m: [0, 1] → [0, 1],

which assigns to each membership grade A(x) of a given fuzzy set A a modifiedgrade m(A(x)). The modified grades for all x ∈ X define a new, modified fuzzy set.Denoting conveniently this modified set by MA, we have

m(A(x)) = MA(x)

Observe that function m is totally independent of elements x to which values A(x)

are assigned; it depends only on the values themselves. In describing its formalproperties, we may thus ignore x and assume that the argument of m is an arbitrarynumber a in the unit interval [0, 1].


In general, a modifier increases or decreases values of the membership functions towhich it is applied, but preserves the order. That is, if a ≤ b then m(a) ≤ m(b) for alla, b∈ [0, 1]or, recognizing the meaning of a and b, if A(x) ≤ A(y) for some x, y ∈X,then MA(x) ≤ MA(y). Sometimes, it is also required that m(0) = 0 and m(1) = 1.

Modifiers are basically of three types, depending on which values of themembership functions they increase or decrease:

(i) modifiers that increase all values;(ii) modifiers that decrease all values;

(iii) modifiers that increase some values and decrease other values.

To illustrate these types of modifiers, let us consider the fuzzy set A in Figure 2.2.For each x ∈ R, A is clearly defined by the formula

A(x) =

⎧⎪⎪⎨⎪⎪⎩

(x − 1)/2 when x ∈ [1,3]

(5− x)/2 when x ∈ [3,5]

0 otherwise

Assume that this fuzzy set represents, in a given application context, the linguisticconcept “close to 3.” To modify A for representing the concept “very close to 3,”we need to reduce in some way the values of A. This can be done by choosing anappropriate modifier from the class of functions

mλ(a) = aλ, (2.8)

where a is the value of A to which mλ is applied and λ is a parameter whose valuedetermines how strongly mλ modifies A. For each value of λ, which must be in thiscase greater than 1, we obtain a particular modifier. When applying the modifier toA, we obtain a new membership function, mλ[A(x)], a composite of functions A andm, which for each x ∈ R is defined by the formula

mλ[A(x)] =

⎧⎪⎪⎨⎪⎪⎩

[(x − 1)/2]λ when x ∈ [1,3]

[(5− x)/2]λ when x ∈ [3,5]

0 otherwise

This modified membership function has a shape exemplified by the function labeledas C in Figure 2.2. Its width is determined by the value λ of the chosen modifier: thelarger the value, the narrower the function. The proper value of λ must be determinedin the context of each particular application.

Assume now that we want to modify the same set A for representing the concept“fairly close to 3.” In this case, we need to increase the values of A. This can be donewith modifiers of the form (2.8), provided that λ ∈ (0, 1). Applying these modifiers to


A results in a new membership function whose shape is exemplified by the functionlabeled as F in Figure 2.2. The smaller the value of λ, the wider is the modifiedmembership function.

It should be mentioned at this point that (2.8) is given here solely as an example ofa possible class of modifiers of fuzzy sets. As is well known, these modifiers do notalways properly capture the meaning of linguistic hedges in natural language. A morecomprehensive treatment of linguistic hedges is presented in Chapter 10; see alsoNovák [1989].

2.3.2 Complements

Similarly to modifiers, complements of fuzzy sets may be defined via appropri-ate unary operations on [0, 1]. While modifiers preserve the order of membershipdegrees, complements reverse the order. In particular, each fuzzy complement, c,must satisfy at least the following two requirements:

(c1) c(0) = 1 and c(1) = 0;(c2) for all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b).

Requirement (c1) guarantees that all fuzzy complements collapse to the unique clas-sical complement for crisp sets. Requirement (c2) guarantees that increases in thedegree of membership in A do not result in increases in the degree of membershipin the complement of A. This is essential since any increase in the degree of mem-bership of an object in a fuzzy set cannot simultaneously increase the degree ofnonmembership of the same object in the same fuzzy set.

When used as a fuzzy complement, function c is always applied to membershipdegrees A(x) of some fuzzy set A. It depends only on the values A(x) and not on theobjects x to which the values are assigned. For the purpose of characterizing fuzzycomplements, we may thus ignore these objects and observe only how function c

depends on numbers in [0, 1]. This is the reason why no reference is made to specificdegrees A(x) in the requirements (c1) and (c2). However, when function c defines acomplement of a particular fuzzy set A, we must keep track of the relevant objects x

to make the connection between A(x) and c[A(x)].Although requirements (c1) and (c2) are sufficient to characterize the largest class

of acceptable fuzzy complements, two additional requirements are imposed on fuzzycomplements by most applications of fuzzy set theory:

(c3) c is a continuous function;(c4) c(c(a)) = a for all a ∈ [0, 1].

Requirement (c3) guarantees that infinitesimal changes in the argument do not resultin discontinuous changes in the function. Requirement (c4) guarantees that fuzzy sets


are not changed by double complementation. Fuzzy complements that satisfy (c4)are called involutive.

A practical class of fuzzy complements that satisfy requirements (c1)–(c4) isdefined for each a ∈ [0, 1] by the formula

cλ(a) = (1− aλ)1/λ, (2.9)

where λ ∈ (0,∞); it is called the Yager class of fuzzy complements. One particularfuzzy complement is obtained for each value of the parameter λ. The complementobtained for λ = 1, which is called a standard fuzzy complement, is the most com-mon complement in applications of fuzzy set theory. Clearly, the standard fuzzycomplement of a fuzzy set A, usually denoted by �A, is defined for each x ∈X by theequation

�A(x) = 1− A(x).

Other parameter-based formulas for describing classes of fuzzy complements havebeen proposed in the literature. In fact, some procedures have been developed bywhich new classes of fuzzy complements can be generated [Klir & Yuan, 1995].However, this theoretical topic is beyond the scope of this tutorial.

To determine the most fitting complement in the context of each particular appli-cation is a problem of knowledge acquisition, somewhat similar to the problem ofconstructing membership functions. Given a class of fuzzy complements, such as theYager class, the constructing problem reduces to the problem of determining the rightvalue of the relevant parameter.

2.3.3 Intersections and unions

Intersections and unions of fuzzy sets, denoted by i and u respectively, are general-izations of the classical operations of intersections and unions of crisp sets. They maybe defined via appropriate functions that map each pair of real numbers from [0, 1](representing degrees A(x) and B(x) of given fuzzy sets A and B for some x ∈ X)into a single number in [0, 1] (representing membership degree (A ∩ B)(x) of theintersection of A and B or membership degree of the union of A and B for the givenx). Hence,

(A ∩ B)(x) = i[A(x), B(x)]

and

(A ∪ B)(x) = u[A(x), B(x)]

for all x ∈ X. To discuss properties of functions i and u, which do not depend on x,we may view i and u as functions from [0, 1] × [0, 1] to [0, 1].


Contrary to their classical counterparts, fuzzy intersections and unions are notunique. This is a natural consequence of the well-established fact that the linguisticexpressions “x is a member of A and B” and “x is a member of A or B” have differentmeanings when applied by human beings to different vague concepts in differentcontexts. To be able to capture the different meanings, we need to characterize theclasses of fuzzy intersections and fuzzy unions as broadly as possible.

It has been established that operations known in the literature as triangular

norms or t-norms and triangular conorms or t-conorms, which have been exten-sively studied in mathematics, possess exactly those properties that are requisite, onintuitive grounds, for fuzzy intersections and fuzzy unions, respectively. The classof t-norms/fuzzy intersections is characterized by four requirements; the class oft-conorms/fuzzy unions is also characterized by four requirements, three of which areidentical with the requirements for t-norms. In the following list, the requirements fort-norms/fuzzy intersections i are paired with their counterparts for t-conorms/fuzzyunions u, and must be satisfied for all a, b, d ∈ [0, 1]:

(i1) i(a, 1) = a (boundary requirement for i);(u1) u(a, 0) = a (boundary requirement for u);(i2) b ≤ d implies i(a, b) ≤ i(a, d)

}(monotonicity);

(u2) b ≤ d implies u(a, b) ≤ u(a, d)

(i3) i(a, b) = i(b, a)}

(commutativity);(u3) u(a, b) = u(b, a)

(i4) i(a, i(b, d)) = i(i(a, b), d)}

(associativity).(u4) u(a, u(b, d)) = u(u(a, b), d)

It is easy to see that the first three requirements for i ensure that fuzzy intersectionscollapse to the classical set intersection when applied to crisp sets: i(0, 1) = 0 andi(1, 1) = 1 follow directly from the boundary requirement; i(1, 0) = 0 and i(0, 0) =

0 follow then from commutativity and monotonicity, respectively. Similarly, thefirst three requirements for u ensure that fuzzy unions collapse to the classical setunion when applied to crisp sets. Commutativity requirements ensure that fuzzyintersections and unions are symmetric operations, indifferent to the order in whichsets to be combined are considered; together with monotonicity requirements, theyguarantee that fuzzy intersections and unions do not decrease when any of theirarguments are increased, and do not increase when any arguments are decreased.Associativity requirements allow us to extend fuzzy intersections and unions to morethan two sets, in perfect analogy with their classical counterparts.

The following are examples of some common fuzzy intersections and fuzzy unionswith their usual names (each defined for all a, b ∈ [0, 1]).

Standard fuzzy intersection: i(a, b) = min(a, b)

Algebraic product: i(a, b) = ab


Bounded difference: i(a, b) = max(0, a + b − 1)

Drastic intersection: imin(a, b) =

⎧⎨⎩

a when b = 1b when a = 10 otherwise

Standard fuzzy union: u(a, b) = max(a, b)

Algebraic sum: u(a, b) = a + b − ab

Bounded sum: u(a, b) = min(1, a + b)

Drastic union: umax(a, b) =

⎧⎨⎩

a when b = 0b when a = 01 otherwise

It is easy to verify that the inequalities

imin(a, b) ≤ i(a, b) ≤ min(a, b)

max(a, b) ≤ u(a, b) ≤ umax(a, b)

are satisfied for all a, b ∈ [0, 1] by any fuzzy intersection i and any fuzzy union u,respectively. These inequalities specify, in effect, the full ranges of fuzzy intersectionsand fuzzy unions.

Examples of classes of fuzzy intersections, iw, and fuzzy union, uw, that cover thefull ranges of these operations are defined for all a, b ∈ [0, 1] by the formulas

iw(a, b) = 1−min{1, [(1− a)w + (1− b)w]1/w}

uw(a, b) = min[1, (aw + bw)1/w]

}, (2.10)

where w is a parameter whose range is (0, ∞). One particular fuzzy intersectionand one particular fuzzy union are obtained for each value of the parameter. Theseoperations are often referred to in the literature as the Yager classes of intersections

and unions. Although it is not obvious from the formulas, it is relatively easy to provethat the standard fuzzy operations are obtained in the limit for w →∞.

Since Yager intersections increase as the value of w increases, they become lessrestrictive or weaker with increasing w. The drastic intersection is the strongest andthe standard intersection is the weakest. For Yager unions, this pattern is inverted;they become more restrictive or stronger with increasing w. The standard union isthe strongest, the drastic union the weakest.

It should be mentioned that various other classes of fuzzy intersections andunions have been examined in the literature. Moreover, special procedures are nowavailable by which new classes of fuzzy intersection and unions can be generated[Klir & Yuan, 1995].

Among the great variety of fuzzy intersections and unions, the standard operationspossess certain properties that give them special significance. First, we recognizethat they are located at opposite ends of the respected ranges of these operations.


While the standard intersection is the weakest one among all fuzzy intersections,the standard union is the strongest one among all fuzzy unions. Second, the standardoperations are the only cutworthy operations among all fuzzy intersections and unions.Third, they are also the only operations among fuzzy intersections and unions thatare idempotent. This means that is(a, a) = us(a, a) = a for all a ∈ [0, 1]. Non-standard fuzzy intersections are only subidempotent, while nonstandard fuzzy unionsare superidempotent; this means that

i(a, a) < a and u(a, a) > a

for all a ∈ (0, 1). In addition, when using the standard fuzzy operations, errors of theoperands do not compound. This is a desirable property from the computational pointof view, which other fuzzy operations do not possess.

Whatever combination of fuzzy counterparts of the three classical set-theoreticoperations (complement, intersection, union) we choose, some properties of the clas-sical operations (properties of the underlying Boolean algebra) are inevitably violated.This is a consequence of imprecise boundaries of fuzzy sets. The standard fuzzy oper-ations violate only the law of excluded middle and the law of contradiction. Someother combinations preserve these laws, but violate distributivity and idempotence[Klir & Yuan, 1995].

2.3.4 Averaging operations

Fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are special types of oper-ations for aggregating fuzzy sets: given two or more fuzzy sets, they produce a singlefuzzy set, an aggregate of the given sets. While they do not cover all aggregating oper-ations, they cover all aggregating operations that are associative. Because of the lackof associativity, the remaining aggregating operations must be defined as functionsof n arguments for each n ≥ 2. These remaining aggregation operations are calledaveraging operations. As the name suggests, they average in various ways member-ship functions of two or more fuzzy sets defined on the same universal set. Theydo not have any counterparts in classical set theory. Indeed, an average of severalcharacteristic functions of classical sets is not, in general, a characteristic function!However, classical sets can be averaged if they are viewed as special fuzzy sets.

For each n ≥ 2, an averaging operation, h, aggregates n fuzzy sets defined on thesame universal set X, say sets A1, A2, . . . , An. Denoting conveniently the aggregatefuzzy set by H(A1, A2, . . . , An), we have

H(A1, A2, . . . , An)(x) = h[A1(x), A2(x), . . . , An(x)]

for all x ∈X. Since properties of various averaging operations h do not depend onx, but only on the membership degrees A1(x), A2(x), . . . , An(x) ∈ [0, 1], we mayview these operations as functions from [0, 1]n to [0, 1].


The following two requirements are requisite for any averaging operation h withn arguments (n ≥ 2):

(h1) for all a ∈ [0, 1], h(a, a, . . . , a) = a (idempotency);(h2) for any pair of n-tuples of real numbers in [0,1], 〈a1, a2, . . . , an〉 and

〈b1, b2, . . . , bn〉, if ai ≤ bi for all i ∈ Nn, then

h(a1, a2, . . . , an) ≤ h(b1, b2, . . . , bn) (monotonicity).

Requirement (h1) expresses our intuition that an average of equal numbers must resultin the same number. Requirement (h2) guarantees that the average does not decreasewhen any of the arguments increase.

In addition to these essential and easily understood requirements, averagingoperations on fuzzy sets are usually expected to satisfy two additional requirements:

(h3) h is a continuous function;(h4) h is a symmetric function in all its arguments, which means that

h(a1, a2, . . . , an) = h(ap(1), ap(2), . . . , ap(n))

for any permutation p on Nn. Requirement (h3) guarantees that small changes in anyof the arguments do not result in discontinuous changes in the average. Requirement(h4) captures the usual assumption that the aggregated fuzzy sets are equally impor-tant. If this assumption is not warranted in some application contexts, the symmetryrequirement must be dropped.

It is significant that any averaging operation h that satisfies the two basic require-ments (h1) and (h2) produces numbers that for each n-tuple 〈a1, a2, . . . , an〉 ∈ [0, 1]n

lie within the interval defined by the inequalities

min(a1, a2, . . . , an) ≤ h(a1, a2, . . . , an) ≤ max(a1, a2, . . . , an).

To see this, let

a∗ = min(a1, a2, . . . , an) and a∗ = max(a1, a2, . . . , an).

If h satisfies requirements (h1) and (h2), then a∗=h(a∗, a∗, . . . , a∗)≤h(a1,a2, . . . , an) ≤ h(a∗1 , a∗2 , . . . , a∗n) = a∗. Conversely, if h produces numbers within theinterval bounded by the min and max operations, then it must also satisfy requirement(h1) of idempotency; indeed,

a = min(a, a, . . . , a) ≤ h(a, a, . . . , a) ≤ max(a, a, . . . , a) = a

for all a ∈ [0, 1]. That is, averaging operations cover the whole range between thestandard fuzzy intersection and the standard fuzzy union. The standard operations


play a pivotal role in the three types of aggregating operations of fuzzy sets. Owing totheir idempotency, they qualify not only as extensions of the classical set intersectionand union, but also as extreme averaging operations.

One class of averaging operations that covers the entire interval between min andmax operations consists of generalized means, which are defined by the formula

hp(a1, a2, . . . , an) =

(a

p

1 + ap

2 + · · · + apn

n

)1/p

, (2.11)

where p is a parameter whose range is the set of all real numbers except 0. Oneparticular averaging operation is obtained for each value of the parameter. For p = 0,function hp is not defined by the formula but by the limit,

limp→0

hp(a1a2 · · · an) = (a1a2 · · · an)1/n,

which is the well-known geometric mean. Moreover,

limp→−∞

hp(a1, a2, . . . , an) = min(a1, a2, . . . , an)

limp→∞

hp(a1, a2, . . . , an) = max(a1, a2, . . . , an).

For p = 1, hp yields the arithmetic mean

h1(a1, a2, . . . , an) =a1 + a2 + · · · + an

n;

and for p = −1, it yields the harmonic mean

h−1(a1, a2, . . . , an) =n

1

a1+

1

a2+ · · · +

1

an

.

It seems reasonable to consider the arithmetic mean as the standard averaging

operation.Generalized means are symmetric averaging operations. When symmetry is not

desirable, they may be replaced with weighted generalized means, whp, defined bythe formula

whp (a1, a2, . . . , an, w1, w2, . . . , wn) =

(n∑

i=1

wi api

)1/p

, (2.12)


Figure 2.5 An overview of the three basic classes of aggregation operations for fuzzy sets.

where wi(i ∈ Nn) are nonnegative real numbers, called weights, for which

n∑

i=1

wi = 1

The role of the weights is to express the relative importance of the sets to be aggregated.It is worth mentioning that other classes of averaging operations have been proposed

and studied in the literature. Also, some more sophisticated classes of functions havebeen proposed, which cover more than one of the three basic types of aggregationoperations [Klir & Yuan, 1995].

The full scope of aggregation operations is summarized in Figure 2.5 in termsof the Yager intersections iw, Yager union uw, and the generalized means hp. Thethree types of aggregation operations of fuzzy sets are illustrated by the two fuzzysets in Figure 2.6, which may represent, for example, silt and sand, as conceived inFigure 2.1. In each case, the bold lines represent the result of the standard operationand the shaded area indicates the range of all operations of that type.

2.3.5 Arithmetic operations

Arithmetic operations are applicable only to special fuzzy sets that are called fuzzyintervals. These are standard and normal fuzzy sets defined on the set of real numbers,R, whose α-cuts for all α ∈ (0, 1] are closed intervals of real numbers and whosesupports are bounded. Any fuzzy interval A for which A(x)= 1 for exactly one x ∈R


Figure 2.6 An illustration of the three basic classes of aggregation operations.

is called a fuzzy number. Clearly, all fuzzy sets specified in Figure 2.2 are fuzzynumbers, and those in Figure 2.1b are fuzzy intervals.

Every fuzzy interval A may conveniently be expressed for all x ∈ R in the canonicalform

A(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

fA(x) when x ∈ [a, b)

1 when x ∈ [b, c]

gA(x) when x ∈ (c, d]

0 otherwise,

(2.13)


where a, b, c, d are specific real numbers such that a ≤ b ≤ c ≤ d, fA is a real-valuedfunction that is increasing and right-continuous, and gA is a real-valued function thatis decreasing and left continuous.

For any fuzzy interval A expressed in the canonical form (2.13), the α-cuts of A

are expressed for all α ∈ (0, 1] by the formula

αA =

{[f−1

A (α), g−1A (α)] when α ∈ (0, 1)

[b, c] when α = 1,(2.14)

where f−1A and g−1

A are the inverse functions of fA and gA, respectively.Employing the α-cut representation, arithmetic operations on fuzzy intervals are

defined in terms of the well-established arithmetic operations on closed intervals ofreal numbers [Moore, 1966; Neumaier, 1990]. Given any pair of fuzzy intervals, A

and B, the four basic arithmetic operations on the α-cuts of A and B are defined forall α ∈ (0, 1] by the general formula

α(A ∗ B) = {a ∗ b|〈a, b〉 ∈ αA× αB}, (2.15)

where ∗ denotes any of the four basic arithmetic operations; when the operation isdivision of A by B, it is required that 0 �= αB for any α ∈ (0, 1]. Let

αA = [a(α), a(α)]

αB = [b(α), b(α)].

Then, the individual arithmetic operations on the α-cuts of A and B can be definedmore specifically in terms of these endpoints by the following formulas [Kaufmann& Gupta, 1985]:

αA+ αB = [a(α)+ b(α), a(α)+ b(α)]

αA− αB = [a(α)− b(α), a(α)− b(α)]

αA · αB = [a, b],

where

a = min{a(α) · b(α), a(α) · b(α), a(α) · b(α), a(α) · b(α)}

b = max{a(α) · b(α), a(α) · b(α), a(α) · b(α), a(α) · b(α)}

αA/αB = [a(α), a(α)] · [1/b(α), 1/b(α)],

provided that 0 �∈ [b(α), b(α)] for all α ∈ (0, 1].Fuzzy arithmetic described by these formulas is usually referred to as standard

fuzzy arithmetic. It turns out that this arithmetic does not take into account constraints

2.4 Fuzzy Relations 31

among fuzzy numbers that exist in various applications. As a consequence, it leads toresults that are, in general, deficient of information, even though they are principallycorrect. To avoid this information deficiency, we need to revise standard fuzzy arith-metic to take all existing constraints among fuzzy numbers in each application intoaccount. This leads to constrained fuzzy arithmetic [Klir, 1997; Klir & Pan, 1998].

Fuzzy arithmetic is essential for evaluating algebraic expressions in which valuesof variables are fuzzy intervals or fuzzy numbers. It is also essential for dealing withfuzzy algebraic equations. These are equations in which coefficients and unknownsare fuzzy numbers and algebraic expressions are formed by operations of fuzzy arith-metic. Furthermore, fuzzy arithmetic is a basis for developing fuzzy calculus and forfuzzifying any area of mathematics that involves numbers. Although a lot of workhas already been done along these lines, enormous research effort is still needed tofully develop the mathematical areas mentioned.

2.4 Fuzzy Relations

When fuzzy sets are defined on universal sets that are Cartesian products of two ormore sets, we refer to them as fuzzy relations. A fuzzy relation R is thus defined by amembership function of the general form

R: X1 ×X2 × · · · ×Xn → [0, 1].

The membership degree R(x1, x2, . . . , xn) of a particular n-tuple 〈x1, x2, . . . , xn〉,where xi ∈Xi for all i ∈Nn = {1, 2, . . . , n}, indicates the strength of relation amongelements of the n-tuple. The individual sets in the Cartesian product are called dimen-

sions of the relation. With n sets in the Cartesian product, the relation is calledn-dimensional. Relations that are 2-dimensional have special significance; they areusually called binary relations.

When all dimensions of a fuzzy relation are finite sets, which is the usual case, anyn-dimensional fuzzy relation may conveniently be represented by an n-dimensionalarray whose entries are real numbers in the unit interval [0, 1]. This representationis particularly important for dealing with fuzzy relations on the computer. For binaryrelations, clearly, the arrays become matrices.

From the standpoint of fuzzy relations, ordinary fuzzy sets may be viewed asdegenerate, 1-dimensional fuzzy relations. This implies that all concepts introducedfor ordinary fuzzy sets are also applicable to fuzzy relations. The various types ofaggregating operations introduced in Section 2.3 are applicable to fuzzy relationsas well. However, fuzzy relations involve additional operations that emerge fromtheir multidimensionality. These additional operations include projections, exten-sions, compositions, joins, and inverses of fuzzy relations. Projections and extensionsare applicable to any fuzzy relations, whereas compositions, joins, and inverses areapplicable only to binary relations.


2.4.1 Projections, cylindric extensions, and cylindric closures

To define projections and extensions of fuzzy relations, we assume that the relationsare n-dimensional, where n ≥ 2. Let

x = 〈xi |i ∈ Ni〉

denote elements (n-tuples) of the Cartesian product

X =×i∈Nn

Xi

and let

y = 〈yj |j ∈ J 〉,

where J ⊂Nn and |J | = r < n, denote the elements (r-tuples) of the Cartesianproduct

Y =×j∈J

Xj .

Furthermore, let y < x denote that yj = xj for all j ∈ J . Then, given a fuzzy relationR on X, a projection, RY , of R on Y is defined for all y ∈ Y by the formula

RY (y) = maxx>y

R(x). (2.16)

As can be seen from this formula, the operation of projection converts a givenn-dimensional relation R into an r-dimensional relation RY (r < n) that

(i) is consistent with R in all dimensions included in Y ; and(ii) in which all dimensions that are not included in Y are suppressed (not recognized).

The maximum operator in (2.16) represents the standard fuzzy union of all singletonsets {x} for which x > y. Since the standard fuzzy union is the only union (t-conorm)that is cutworthy, the operation of projection defined by (2.16) is cutworthy as well.This means that, for each α ∈ (0, 1], αRE is the projection of R in the sense ofclassical set theory.

An operation that is inverse to a projection is called a cylindric extension. Bythis operation, a given r-dimensional projection RY of an n-dimensional relationR(r < n) is converted to an n-dimensional relation REY , a cylindric extension of RY ,by the formula

REY (x) = RY (y) (2.17)


for each x such that x > y. This definition, which again is a cutworthy generalizationof the classical concept of cylindric extension, guarantees that R ⊆ REY . Clearly,cylindric extension REY of RY is the largest fuzzy relation that is consistent with RY .

Given a set {RYk|k ∈ K} of projections of R, a cylindric closure, Cyl{RYk

| k ∈ K},of these projections is the standard fuzzy intersection of their cylindric extensions.That is,

Cyl{RYk| k ∈ K}(x) = min

k∈K{REYk

(x)} (2.18)

for all x∈X. Again, by using the standard fuzzy intersection, this definition is acutworthy generalization of the classical concept of cylindric closure. It produces thelargest fuzzy relation that is consistent with all the projections involved. Consequently,it guarantees that this relation, which is reconstructed from the given projections ofR, always contains R. That is,

R ⊆ Cyl{RYk| k ∈ K}

for any given set{RYk| k ∈ K

}.

Projections, cylindric extension, and cylindric closures are the main operations fordealing with n-dimensional relations. Some additional operations are important fordealing with binary relations. The rest of this section is devoted to these operationsas well as to properties of some important types of binary fuzzy relations.

2.4.2 Inverses, compositions, and joins

The inverse of a binary fuzzy relation R on X × Y , denoted by R−1, is a relation onY ×X such that

R−1(y, x) = R(x, y)

for all pairs 〈y, x〉 ∈ Y ×X. When R is represented by a matrix, R−1 is representedby the transpose of this matrix. This means that rows are replaced with columns andvice versa. Clearly,

(R−1)−1 = R

holds for any binary relation.Consider now two binary fuzzy relations P and S that are defined on set X × Y

and Y ×Z, respectively. Any such relations, which are connected via the commonset Y , can be composed to yield a relation on Y × Z. The standard composition of


these relations, which is denoted by P ◦ S, produces a relation R on Y × Z definedby the formula

R(x, z) = (P ◦ S)(x, z) = maxy∈Y

min[P(x, y), S(y, z)] (2.19)

for all pairs 〈x, z〉 ∈ X × Z.Other definitions of a composition of fuzzy relations, in which the min and max

operations are replaced with other t-norms and t-conorms, respectively, are possibleand useful in some applications. All compositions possess the following importantproperties:

(P ◦ S) ◦Q = P ◦ (S ◦Q)

(P ◦ S)−1 = S−1 ◦ P−1.

However, the standard fuzzy composition is the only one that is cutworthy.A similar operation on two connected binary relations, which differs from the

composition in that it yields a 3-dimensional relation instead of a binary one, isknown as the relational join. For the same fuzzy relations P and S, the standardrelational join, P ∗ S, is a 3-dimensional relation X× Y ×Z defined by the formula

R(x, y, z) = (P ∗ S)(x, y, z) = min[P(x, y), S(y, z)] (2.20)

for all triples 〈x, y, z〉 ∈ X× Y ×Z. Again, the min operation in this definition maybe replaced with another t-norm. However, the relational join defined by (2.20) is theonly one that is cutworthy.

2.4.3 Fuzzy relation equations

Consider binary relations P, Q, R, defined on sets X×Y, Y ×Z, and X×Z,respectively, for which

P ◦Q = R,

where ◦ denotes the standard composition. This means that a set of equations of theform

maxy∈Y

min[P(x, y), Q(y, z)] = R(x, z)

is satisfied for all x ∈X and z∈Z. These equations are called fuzzy relation equations.The problem of solving fuzzy relation equations is any problem in which two of the

relations are given and the third is to be determined via the equations. When P and Q


are given, the problem of determining R is trivial. It is solved by performing thecomposition P ◦Q, usually in terms of the matrix representations of P and Q. WhenR and Q (or R and P ) are given, the problem of determining P (or Q) is considerablymore difficult, but it is very important for many applications. It is reasonable to viewthis problem as a decomposition of R with respect to Q or P .

Several efficient methods have been developed for solving the decompositionproblem of fuzzy relation equations. While these methods are rather tedious forhuman beings, they can easily be implemented on the computer and, moreover,they are highly suitable for parallel processing. Although details of these methodsare beyond the scope of this tutorial, it seems useful to describe basic characteristicsof the solutions obtained by them.

Let S(Q, R) denote the solution set obtained by solving the problem of decompos-ing R with respect to Q. That is, members of the solution set are all versions of therelation P for which the fuzzy relation equations are satisfied, given relations Q andR. Any member �P of the solution set S(Q, R) is called a maximal solution if, for allP ∈ S(Q, R), �P ⊆ P implies P = �P . Similarly, any member P of S(Q, R) is calleda minimal solution if, for all P ∈ S(Q, R), P ⊆ P implies P = P .

It is well established that, whenever the equations are solvable, the solution setalways contains a unique maximum solution, �P , and it may contain several minimalsolutions, 1P , 2P , . . . , nP . Moreover, the solution set can be fully characterized byits maximum and minimal solutions. To describe this characterization, let

iP = {P | iP ⊆ P ⊆ �P }

denote the family of relations that are between the maximum solution �P and theminimal iP for each i ∈ Nn. The solution set is then described by taking the union ofthese families iP for all the minimal solutions iP . That is,

S(Q, R) =

n⋃

i=1

iP.

This convenient way of characterizing the solution set is illustrated visually inFigure 2.7.

In all the proposed methods for solving fuzzy relation equations, it is computation-ally very simple to determine the maximum solution. This is of great advantage to anyapplication of fuzzy relation equations in which the maximum solution is sufficient.When the solution to fuzzy relation equations is unique, which is a rather rare case,this unique solution is identical with the maximum solution.

It may also happen that the given fuzzy relation equations are not solvable. If asolution is essential in some application, then it is important to be able to find a rea-sonable approximate solution. Important results regarding approximate solutions offuzzy relation equations have already been obtained, but this problem is still a subject


Figure 2.7 Structure of the solution set S(Q, R).

of active research. Another area of current research is concerned with fuzzy relationequations that are based on compositions distinct from the standard composition.These more general fuzzy relation equations play a useful role in some applications,particularly in the area of approximate reasoning.

It is well recognized that many problems emanating from diverse applications offuzzy set theory can be formulated in terms of fuzzy relation equations. Methodsfor solving these equations have thus a broad utility. Constructing rules of infer-ence in fuzzy knowledge-based systems, knowledge acquisition, the problem ofidentifying fuzzy systems from input–output observations, and the problem of decom-posing fuzzy systems are just a few examples illustrating this utility. The principalsource for fuzzy relation equations is the book by Di Nola et al. [1989]; other bookswith a thorough coverage of this subject include Pedrycz [1989] and Klir & Yuan[1995].

2.4.4 Fuzzy relations on a single set

Binary relations in which elements of a set are related to themselves have specialsignificance and utility. For example, they allow us to rigorously define equivalence,compatibility, and various kinds of orderings among elements of the set of concern.


Figure 2.8 Diagram and matrix representation of a fuzzy relation on N6.

Although membership functions of relations of this kind have the form

R : X ×X → [0, 1],

they are usually referred to as relations on X rather than relations on X × X. Theirmatrix representations are square matrices in which rows and columns are assigned tothe same elements. Other useful representations of these relations are simple diagramswith the following properties: (i) each element of the set X is represented by a singlenode in the diagram; (ii) directed connections between nodes are included in thediagram only for pairs of elements of X that are contained in the support of therelation; (iii) each connection in the diagram is labeled by the membership degree ofthe corresponding pair in the relation. An example of the diagram representation of arelation R defined on N6 is shown in Figure 2.8, where it is compared with the matrixrepresentation.

Three of the most important classes of classical binary relations on a set—equivalence, compatibility, and ordering relations—are characterized in terms offour distinctive properties: reflexivity, symmetry, antisymmetry, and transitivity. Toreformulate reflexivity, symmetry, and antisymmetry for fuzzy relations R on X israther trivial. We say that a fuzzy relation is:

● reflexive if and only if R(x,x) = 1 for all x ∈ X;● symmetric if and only if R(x,y) = R(y,x) for all x,y ∈ X;


● antisymmetric if and only if R(x,y) > 0 and R(y,x) > 0 implies that x = y forall x,y ∈ X.

The property of fuzzy transitivity can be defined in numerous ways, all of whichcollapse to the classical definition of transitivity for crisp relations. According to themost common definition, R is transitive if

R(x,z) ≥ maxy ∈X

min[R(x,y), R(y,z)] (2.21)

for all x,z ∈ X. This definition, which is based on the standard fuzzy intersection andunion, is the only one that is cutworthy. This particular definition of fuzzy transitivityis often referred to as max-min transitivity. Alternative definitions of transitivity,based upon other fuzzy intersections and unions, are possible and useful in someapplications. However, they do not result in fuzzy relations that are cutworthy.

Employing these definitions, fuzzy equivalence relations are reflexive, symmetric,and transitive (in the fuzzified sense of these properties), fuzzy compatibility relations

are reflexive and symmetric, and fuzzy partial orderings are reflexive, antisymmetric,and transitive. Each of these types of fuzzy relations is cutworthy. That is, eachα-cut of a fuzzy relation of a particular type is a crisp relation of the same type.Hence, fuzzy equivalence, compatibility, and partial ordering are properties that arepreserved in each α-cut in the classical sense. Moreover, by increasing α, equivalenceand compatibility classes in α-cuts become more refined, while α-cuts of fuzzy partialorderings increase the number of noncomparable pairs.

Examples of simple fuzzy equivalence and compatibility relations are shown inFigure 2.9. Since both relations are reflexive and symmetric, the diagrams are sim-plified: the connections of each node to itself (required by reflexivity) are omitted,and the bidirectional connections of nodes (required by symmetry) are replaced withundirected connections. Equivalence classes and maximal compatibility classes in allα-cuts of these relations are also shown in the figure. The increasing refinements ofthese classes with increasing values of α are clearly visible.

An example of a fuzzy partial ordering is shown in Figure 2.10. Also shown in thefigure are all its α-cuts, which are crisp partial orderings. The α-cuts are representedby simplified diagrams, in which connections are made only to immediate successorsand immediate predecessors. Diagrams of this sort, which are called Hasse diagrams,are common for crisp partial orderings.

2.5 Fuzzy Logic

In order to use the apparatus of fuzzy set theory in the domain of fuzzy logic, it isnecessary to establish a connection between degrees of membership in fuzzy sets anddegrees of truth of fuzzy propositions. This is fairly straightforward, provided that

2.5 Fuzzy Logic 39

Figure 2.9 Examples of fuzzy equivalence and fuzzy compatibility.

the degrees of membership and the degrees of truth to be connected refer to the sameobjects. Let X denote the universal set of these common objects.

To establish a meaningful connection between the two kinds of degrees, let usconsider first the simplest propositional form

pA : X is A,

where X is a variable whose range is X and A is a fuzzy set representing an inherentlyvague linguistic expression (such as low, high, small, large, shallow, very deep,subtidal, etc.) in a given context. A proposition is obtained when a particular objectx from X is substituted for the variable X in the propositional form.

Let pA(x) denote the degree of truth of the respective proposition. This meansthat the symbol pA, which denotes the propositional form, is also employed fordenoting a function by which degrees of truth are assigned to propositions based onthe propositional form. Let this function be called a truth assignment function.

The double use of the symbol pA, employed here for the sake of simplicity, doesnot create any confusion since there is only one truth assignment function for eachpropositional form. This is analogous to the use of the same symbol for a fuzzy setand its membership function.


Figure 2.10 An example of fuzzy partial ordering.

2.5 Fuzzy Logic 41

Using this simplified notation, we can formulate the connection between fuzzy setsand fuzzy propositions as follows. Given a fuzzy set A, its membership degree A(x)

for each x ∈ X may be interpreted as the degree of truth of the proposition obtainedfrom the propositional form

pA : X is a member of A

for the same x ∈ X. That is,

pA(x) = A(x) (2.22)

for all x ∈ X. Conversely, given an arbitrary propositional form

pA : X is A,

the degree of truth pA(x) for each x ∈X may be interpreted as the degree of com-patibility, A(x), of x with the concept represented by A. That is, we obtain againEquation (2.22).

In summary, a fuzzy set and a fuzzy propositional form are connected wheneverboth are defined in terms of the same set of objects and both represent the samemeaning of a linguistic expression. Given a fuzzy set and a fuzzy propositional formthat are connected in this sense, the degrees of membership in the fuzzy set and thedegrees of truth of fuzzy propositions defined for the same objects are numericallyequal, as expressed by (2.22). As a consequence, logic operations of negation, con-junction, and disjunction are defined in exactly the same way as the operations ofcomplementation, intersection, and union, respectively.

We should emphasize at this point that Equation (2.22) applies only to the simplestpropositional form: X is A. For more complex forms (quantified, truth-qualified,conditional, etc.), the equation must be properly modified. To explain how to modify it,we need to introduce basic types of propositional forms from which fuzzy propositionscan be obtained.

2.5.1 Basic types of propositional forms

The principal aim of fuzzy logic is to formalize reasoning with propositions in naturallanguage. The linguistic expressions involved may contain fuzzy linguistic terms ofany of the following types:

● fuzzy predicates—tall, young, expensive, low, high , normal, etc.;● fuzzy truth values—true, fairly true, very true, false, etc.;● fuzzy probabilities—likely, very likely, highly unlikely, etc.;● fuzzy quantifiers—most, few, almost all, usually, often, etc.


All of these linguistic terms are represented in each context by appropriate fuzzy sets.Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements towhich the predicates apply. Fuzzy truth values and fuzzy probabilities are representedby fuzzy sets defined on the unit interval [0, 1]. Fuzzy quantifiers are either absoluteor relative; they are represented by appropriate fuzzy numbers defined either on theset of natural numbers or on the interval [0, 1].

Observe that simple linguistic terms of any of the mentioned types are sometimesmodified by special linguistic terms such as very, fairly, extremely, more or less,

and the like. These linguistic terms are called linguistic hedges. Contrary to theother linguistic terms, they are not represented by fuzzy sets, but rather by specialoperations on fuzzy sets. These operations are called modifiers and are discussed inSection 2.3.1, on p. 19.

In a crude way, it is useful to distinguish the following four types of fuzzy pro-positional forms and fuzzy propositions based on them. Each of these forms may, inaddition, be quantified by an appropriate fuzzy quantifier.

1. Unconditional and unqualified propositions are expressed by the canonical form

pA : X is A.

As already explained in this section, the truth values of propositions based on thisform are given by Equation (2.22).

2. Unconditional and qualified propositions are characterized by either thecanonical form

pT (A): X is A is T

or the canonical form

pP(A): Pro{X is A} is P,

where X and A have the same meaning as before, Pro {X is A} denotes the prob-

ability of a fuzzy event defined by the expression “X is A,” T is a fuzzy truth

qualifier, and P is a fuzzy probability qualifier. Both T and P are represented byfuzzy sets defined on [0, 1]. For any given probability distribution function f onX, Pro {X is A} is determined by the formula

Pro {X is F } =∑

x∈X

f (x)A(x).

The first canonical form is called a truth-qualified form, the second one is calleda probability-qualified form. To obtain the degree of truth of a fuzzy propositionbased on the truth-qualified form, we need to compose membership functionA (representing a fuzzy predicate) with the membership function of the truth

2.5 Fuzzy Logic 43

qualifier T . That is,

pT (A)(x) = T (A(x)) (2.23)

for all x ∈X. Similarly, for fuzzy propositions based on probability-qualifiedforms, we need to compose membership function A with the membership functionof the probability qualifier P . That is,

pP(A)(x) = P(A(x)) (2.24)

for all x ∈X. Propositional forms in which both kinds of qualification are involved,

pTP(A)(x) : X is A is P is T

are also meaningful. To obtain the degree of truth of a fuzzy proposition based onthis form, we need to compose A with P first and, then, to compose the resultingfunction with T . That is,

pTP(A)(x) = T (P (A(x))) (2.25)

for all x ∈ X.3. Conditional and unqualified fuzzy propositions have the canonical form

pB|A: If X is A, then Y is B,

where X , Y are variables whose ranges consist of objects in some universal setsX, Y , respectively, and A, B are relevant fuzzy predicates represented by appro-priate fuzzy sets. These propositions may also be expressed in an alternative butequivalent form

pB|A: (X , Y) is R,

where R is a fuzzy relation on X× Y that is determined for each x ∈ X and eachy ∈ Y by the formula

R(x, y) = J [A(x), B(y)].

The symbol J stands for a binary operation on [0, 1] that represents a suitablefuzzy implication in the given context. Clearly,

pB|A(x, y) = R(x, y). (2.26)

It should be mentioned at this point that it is essential in fuzzy logic (as in clas-sical logic) to distinguish implications (as well as other logic connectives) on twolevels, the syntactic level and the semantic level. On the syntactic level, implication(or another logic connective) is represented by a symbol (usually denoted by⇒).


On the semantic level, it is represented by a suitable operation (usually denotedby→) on the set of truth values [0, 1]. This distinction is followed in Chapter 10.

Operations that qualify as fuzzy implications form a class of binary operationson [0, 1], similarly to fuzzy intersections and unions [Klir & Yuan, 1995]. In somesense, this class can be characterized in terms of fuzzy intersections, unions, andcomplements. The most common fuzzy implication, referred to as the Łukasiewiczimplication, is defined by the formula

J (a, b) = min[1, 1− a + b].

Conditional fuzzy propositions are essential components of fuzzy rules ofinference. Hence, they play a fundamental role in approximate reasoning.

4. Conditional and qualified fuzzy propositions have either the canonical form

pT (B|A): If X is A, then Y is B is T ,

if they are truth qualified, or the canonical form

pP(B|A): Pro {Y is B|X is A} is P,

if they are probability qualified. These forms are basically combinations of theprevious forms.

Fuzzy propositions of any of the introduced types may also be quantified. Ingeneral, fuzzy quantifiers are fuzzy numbers. Fuzzy quantifiers of one type, whichare called absolute quantifiers, are expressed by fuzzy numbers defined on theset of real numbers or on the set of integers. They characterize linguistic termssuch as about 10, at least about 500, much more than a dozen, etc. Quantifiers ofanother type, which are called relative quantifiers, are expressed by fuzzy num-bers defined on [0,1]. They characterize linguistic terms such as almost all, about

half, no more than about 20%, most, etc.Various procedures for determining degrees of truth of quantified fuzzy proposi-

tions are described in the literature, but they are not covered here due to the limitedspace. This fairly complex subject is perhaps most extensively covered in variouspapers by Zadeh [Yager et al., 1987; Klir &Yuan, 1996] andYager [1983, 1985–86,1991], but its basic ideas are also summarized in the text by Klir & Yuan [1995].

2.5.2 Approximate reasoning

Reasoning based on fuzzy propositions of the four types, possibly quantified by var-ious fuzzy quantifiers, is usually referred to as approximate reasoning. Althoughapproximate reasoning is currently a subject of intensive research, its basic prin-ciples are already well established. In general, approximate reasoning draws upon

2.5 Fuzzy Logic 45

methodological apparatus of fuzzy set theory, such as operations on fuzzy sets,manipulations of fuzzy relations, and fuzzy arithmetic.

The most fundamental components of approximate reasoning are conditional fuzzypropositions, which may also be truth qualified, probability qualified, quantified, orany combination of these. Special procedures are needed for each of these types offuzzy propositions. This great variety of fuzzy propositions makes approximate rea-soning methodologically rather intricate. This reflects the richness of natural languageand the many intricacies of common-sense reasoning, which approximate reasoningbased upon fuzzy set theory attempts to model.

The essence of approximate reasoning is illustrated in this section by explaininghow the most common inference rules of classical logic—modus ponens and modus

tollens—can be generalized within the framework of fuzzy logic. For the sake ofsimplicity, the explanation is restricted to unqualified fuzzy propositions withoutquantifiers.

Consider variables X and Y , the ranges of which consist of objects in some givensets X and Y , respectively. Assume that the variables are constrained by a fuzzyrelation R on X × Y . Then, knowing that X is A, where A is a fuzzy set on X, wecan infer that Y is B, where B is a fuzzy set on Y , by the formula

B(y) = supx∈X

min[A(x), R(x, y)]) (2.27)

for all y ∈ Y . This formula, which is called a compositional rule of inference, is abasis for the generalized modus ponens as well as the generalized modus tollens; itcan also be written, more concisely, as

B = A ◦ R.

Assume now that the relation R is not given explicitly, but it is embedded in theconditional propositional form

pG|F : if X is F, then Y is G.

In this case, the relation is determined via the formula

R(x, y) = J [F(x), G(x)], (2.28)

which, in turn, is determined by the choice of a suitable operation of fuzzy implicationJ . Using this relation, obtained from the given fuzzy propositional form pG|F , andgiven another propositional form

pA : X is A,

regarding variable X , we may conclude that Y is B by the compositional rule ofinference (2.27). This procedure is called a generalized modus ponens.


Viewing the conditional propositional form pG|F as a fuzzy rule and the simpleconditional form pA as a fuzzy fact, the generalized modus ponens is expressed bythe schema:

Fuzzy rule: If X is F, then Y is G

Fuzzy fact: X is A

Fuzzy conclusion: Y is B

In a similar way, the generalized modus tollens is expressed by the schema:

Fuzzy rule: If X is F, then Y is G

Fuzzy fact: Y is B

Fuzzy conclusion: X is A

In this case, the compositional rule of inference has the form

A(x) = supy∈Y

min[B(y), R−1(y, x)]. (2.29)

To use the compositional rule of inference, we need to choose a fitting fuzzy impli-cation in each application context and express it in terms of a fuzzy relation R byEquation (2.28). There are several ways in which this can be done. One way is toderive from the application context (by observations or experts’ judgments) pairsA, B of fuzzy sets that are supposed to be inferentially connected. Relation R, whichrepresents a fuzzy implication, is thus determined by solving sets of fuzzy relationequations of either the form (2.27) or the form (2.29). This and other issues regardingfuzzy implications in approximate reasoning are discussed fairly thoroughly in thetext by Klir & Yuan [1995].

2.6 Possibility Theory

It was first recognized by Zadeh [1978] that possibility theory is a natural tool forrepresenting and manipulating information expressed in terms of fuzzy propositions.In this interpretation of possibility theory, the classical (crisp) possibility and necessitymeasures based upon modal logic [Hughes & Cresswell, 1996] are extended to theirfuzzy counterparts via the α-cut representation.

Consider a set of alternatives, X. One of the alternatives is true, but we are notcertain which one it is, due to limited evidence. Assume that we only know, accordingto all evidence available, that it is not possible that the true alternative could beoutside a given set E, where ∅ �=E⊂X. This simple evidence can be expressed by

2.6 Possibility Theory 47

a possibility measure, PosE , defined on X by the formulas

PosE({x}) =

{1 when x ∈ E

0 when x ∈ �E

for all x ∈ X and

PosE(A) = supx∈A

PosE({x})

for all A∈P(X ). In this equation, as well in other equations in this section, thesupremum may be replaced with the maximum if the latter exists. Associated withthe possibility measure PosE is a necessity measure, NecE , defined via the equation

NecE(A) = 1− PosE(�A)

for all A ∈ P(X).Assume now that the set E, in terms of which the evidence is expressed, is a

standard fuzzy set. Then, the previous formulas are still applicable to the α-cuts ofE, provided that ∅ �= αE ⊆ X for all α ∈ [0, 1]. For each α ∈ [0, 1], we can define apossibility measure αPosE in the same way as before:

αPosE({x}) =

{1 when x ∈ αE

0 when x ∈ α�E(2.30)

for all x ∈ X and

αPosE(A) = supx∈A

αPosE({x})

for all A ∈ P(X). Now, using the α-cut representation of E, we have

E(x) = supα∈[0,1]

α · αE(x)

for all x ∈X, where αE denotes here the characteristic function of the α-cut of E.Since Equation (2.30) can be rewritten as

αPosE({x}) = αE(x)

for all α ∈ [0, 1] and x ∈X, it is natural to define a possibility measure, PosE , interms of the possibility measures αPosE (α ∈ [0, 1]) via the α-cut representation ofE. Hence,

PosE({x}) = supα∈[0,1]

α · αPosE({x})


for all x ∈ X, which can be rewritten as

PosE({x}) = E(x). (2.31)

This definition of possibility measures, which is due to Zadeh [1978], is usuallyreferred to in the literature as the standard fuzzy-set interpretation of possibility theory.Given PosE({x}) for all x ∈ X, PosE(A) is then calculated by the equation

PosE(A) = supx∈A

PosE({x}) (2.32)

for all A ∈ P(X). When A is a fuzzy set, Equation (2.32) must be replaced with themore general equation

PosE(A) = supx∈X

min[A(x), PosE({x})]. (2.33)

The associated necessity measure, NecE , is again defined by the equation

NecE(A) = 1− PosE(�A) (2.34)

for each set A, which may be crisp or fuzzy.Possibility measures and the associated necessity measures that represent evidence

expressed in terms of standard fuzzy set via Equations (2.31) to (2.33) are thus cut-worthy. They form a coherent theory of evidence that is referred to as possibility

theory [Dubois & Prade, 1988; De Cooman, 1997], provided that the requirementthat

∅ �= αE ⊆ X

is satisfied for all α ∈ [0, 1]. This means that E must be a normal fuzzy set.When evidence is expressed in terms of a subnormal fuzzy set, the coherence of the

standard fuzzy-set interpretation of possibility theory is lost. Observe, for example,that

PosE(0+E) = hE and NecE(0+E) = 1.

When hE < 1, then NecE(0+E) ≥ PosE(0+E). This violates the fundamental inequal-ity of possibility theory, NecE(A)≤PosE(A), which is required to hold for allA ∈ P(X).

The fact that the standard fuzzy-set interpretation of possibility theory, defined byEquation (2.31), is not applicable to subnormal fuzzy sets has been recognized inthe literature since the mid-1980s. In a recent paper [Klir, 1999], it is shown thatthe only way to make a fuzzy-set interpretation applicable to all standard fuzzy sets

2.7 Fuzzy Systems 49

without distorting given evidence is to replace Equation (2.31) with the more generalequation

PosE({x}) = E(x)+ 1− hE

for all α ∈ [0, 1] and x ∈ X, where hE denotes the height of E.

2.7 Fuzzy Systems

The term “fuzzy system” refers to any system whose variables (or at least some ofthem) range over states that are fuzzy sets. For each variable, the fuzzy sets are definedon some relevant universal set. In most typical systems, the universal sets are specificintervals of real numbers. In this special but important case, states of the variablesare fuzzy intervals.

Representing states of variables by appropriate fuzzy sets is a fuzzy quantizationor, using a more common term, granulation. Each fuzzy set representing a state of avariable is called a granul. If each granul represents a linguistic term (such as verysmall, small, medium, etc.), the variable is called a linguistic variable. Fuzzy systemsare thus usually systems of linguistic variables.

Each linguistic variable is defined in terms of a base variable, whose values areassumed to be real numbers within a specific interval of real numbers. A base variableis a variable in the classical sense. Examples of geological variables are: distancefrom source, tidal range, depth, grain size, and percentage of coral cover. Linguisticterms involved in a linguistic variable are used for approximating the actual valuesof the associated base variable. Their meanings are captured, in the context of eachparticular application, by appropriate fuzzy intervals. That is, each linguistic variableconsists of:

● a name, which should reflect the meaning of the base variable involved;● a base variable with its range of values (a closed interval of real numbers);● a set of linguistic terms that refer to values of the base variable;● a set of semantic rules, which assign to each linguistic term its meaning in terms

of an appropriate fuzzy interval defined on the range of the base variable.

An example of a linguistic variable is shown in Figure 2.11. Its name is “tidal range,”which captures the meaning of the associated base variable—a variable that expressesthe tidal range (in meters) at the place under study. The range of the base variable is[−10, 10]. Five states (values) are distinguished by the linguistic terms subtidal, low-

intertidal, medium-intertidal, high-intertidal, and supratidal. Each of these linguisticterms is assigned one of the trapezoidal-shape fuzzy intervals shown in Figure 2.11.These fuzzy intervals are supposed to approximate the meaning of the linguistic termsin a given application context.


Figure 2.11 An example of a linguistic variable.

In principle, fuzzy systems can be knowledge-based, model-based, or hybrid. Inknowledge-based fuzzy systems, relationships between variables are described bycollections of if-then rules (conditional fuzzy propositions). These rules attempt tocapture the knowledge of a human expert, expressed often in natural language. Model-

based fuzzy systems are based on traditional systems modeling, but they employappropriate areas of fuzzy mathematics (fuzzy analysis, fuzzy geometry, etc.). Hybrid

fuzzy systems are combinations of knowledge-based and model-based fuzzy systems.At this time, knowledge-based fuzzy systems are more developed than model-basedor hybrid fuzzy systems.

In knowledge-based systems, the relation between input and output linguistic vari-ables is expressed in terms of a set of fuzzy if–then rules (conditional propositionalforms). From these rules and any fact describing actual states of input variables, theactual states of output variables are derived by an appropriate compositional rule ofinference. Assuming that the input variables are X1, X2, . . . , and the output variables

2.7 Fuzzy Systems 51

are Y1, Y2, . . . , we have the following general scheme of inference to represent theinput–output relation of the system:

Rule 1: If X1 is A11 and X2 is A21 and · · ·, then Y1 is B11 andY2 is B21 and · · ·

Rule 2: If X1 is A12 and X2 is A22 and · · ·, then Y1 is B12 andY2 is B22 and · · ·

…………………………………………………………………………

Rule n: If X1 is A1n and X2 is A2n and · · ·, then Y1 is B1n andY2 is B2n and · · ·

Fact: X1 is C1 and X2 is C2 and · · ·

Conclusion: Y1 is D1 and Y2 is D2 and · · ·

This overall scheme can be broken down into several schemes, one for each outputvariable. For output variable Yk , for example, the ith rule (i ∈ Nn) becomes

Rule i: If X1 is A1i and X2 is A2i and · · ·, then Yk is Bki

This rule can be rewritten as

Rule i: If 〈X1, X2, . . .〉 is Qi, then Yk is Bki,

where Qi is the cylindric closure of A1i, A2i, . . . . Similarly, the fact in the overallscheme can be rewritten as

Fact: 〈X1, X2, . . .〉 is P ,

where P is the cylindric closure of C1, C2, . . . . Rule i can be further rewritten as

Rule i: 〈〈X1, X2, . . .〉, Yk〉 is Ri,

where Ri is a binary relation that expresses the chosen fuzzy implication, as explainedin Section 2.5. Hence, the inference scheme for variable Yk can be rewritten in thefollowing form:

Rule 1: 〈〈X1, X2, . . .〉, Yk〉 is R1

Rule 2: 〈〈X1, X2, . . .〉, Yk〉 is R2

…………………………………………………

Rule n: 〈〈X1, X2, . . .〉, Yk〉 is Rn

Fact: 〈X1, X2, . . .〉 is P

Conclusion: Yk is Dk


Since the rules are interpreted as disjunctive, Dk is determined by the formula

Dk =⋃

i∈Nn

(P ◦ Ri), (2.35)

where⋃

and ◦ stand usually for the standard fuzzy union and the standard (max–min)composition. However, other fuzzy unions and compositions may be employed whendesirable.

The result of each fuzzy inference is clearly a fuzzy set. This set can be convertedto a single real number, if this is needed, by a defuzzification method. The outcomeof any defuzzification of a given fuzzy set should be the best representation, in thecontext of each application, of the elastic constraint imposed on possible values ofthe output variable by the fuzzy set.

Among the various defuzzification methods described in the literature, each ofwhich is based on some rationale, the most frequently used method is called a centroid

method. To describe it, let us assume that we want to defuzzify a given fuzzy set A onX=R. The defuzzified value of A, d(A), obtained by the centroid method is definedby the formula

d(A) =

∫R

xA(x)dx∫R

A(x)dx. (2.36)

It is clear that d(A) is in this case the value for which the area under the graph of mem-bership function A is divided into two equal subareas. Following this interpretation,the centroid method is sometimes called the center of area method.

Assume now that A is defined on a finite universal set X = {x1, x2, . . . , xn}. Thenthe formula

d(A) =

n∑i=1

xiA(xi)

n∑i=1

A(xi)

(2.37)

is a discrete counterpart of (2.36).It is now increasingly recognized that the centroid defuzzification method and other

methods proposed in the literature may be viewed as special members of parametrizedfamilies of defuzzification methods. For the discrete case, an interesting family isdefined by the formula

dδ(A) =

n∑i=1

xiAδ(xi)

n∑i=1

Aδ(xi)

, (2.38)

2.8 Constructing Fuzzy Sets and Operations 53

where δ ∈ (0,∞) is a parameter by which different defuzzification methods aredistinguished.

A good overview of the various defuzzification methods with references to theoriginal publications was prepared by Van Leekwijck & Kerre [1999].

Although fuzzy systems based on numerical variables have special significance, dueto their extensive applicability, these are not the only fuzzy systems. Any classical(crisp) systems whose variables are not numerical (e.g., ordinal-scale or nominal-scale variables) can be fuzzified as well. One type of fuzzy systems, based in generalon nominal-scale variables (whose states are nonnumerical or even unordered),comprises finite-state fuzzy automata [Klir & Yuan, 1995].

Literature on fuzzy systems is extensive. An important early book on fuzzy systemswas written by Negoita & Ralescu [1975]; more recent books with broad coverageof fuzzy systems are by Yager & Filev [1994] and Piegat [2001].

2.8 Constructing Fuzzy Sets and Operations

Fuzzy set theory provides us with a broad spectrum of tools for representing propo-sitions expressed in natural language and for reasoning based on this representation.Most linguistic terms in natural language are not only predominantly vague, but theirmeanings are almost invariably dependent on context as well. A prerequisite for usingthe tools of fuzzy set theory in each application is to determine the intended meaningsof relevant linguistic terms in the context of that particular application. Some linguis-tic terms are represented by fuzzy sets (predicates, truth or probability qualifiers,quantifiers), others are represented by operations on fuzzy sets (logical connectives,linguistic hedges). To capture the intended meanings of linguistic terms involved inan application, we need to construct appropriate membership functions or operationson membership functions.

The problem of constructing fuzzy sets and operations on fuzzy sets in the contextof various applications is not a problem of fuzzy set theory per se. It is a problemof knowledge acquisition, which is a subject of a relatively new field referred to asknowledge engineering. The process of knowledge acquisition involves one or moreexperts in a specific domain of interest, and a knowledge engineer. The role of theknowledge engineer is to elicit the knowledge of interest from the experts, and expressit in some operational form of a required type.

In applications of fuzzy set theory, knowledge acquisition involves basically twostages. In the first stage, the knowledge engineer attempts to elicit relevant knowledgein terms of propositions expressed in natural language. In the second stage, he orshe attempts to determine the meaning of each linguistic term employed in thesepropositions. It is during this second stage of knowledge acquisition that membershipfunctions of fuzzy sets as well as appropriate operations on these fuzzy sets areconstructed.


Many methods for constructing membership functions are described in the litera-ture. It is useful to classify them into direct methods and indirect methods. In directmethods, the expert is expected either to define a membership function completelyor to exemplify it for some selected individuals in the universal set. To request acomplete definition from the expert, usually in terms of a justifiable mathematicalformula, is feasible only for a concept that is perfectly represented by some objectsof the universal set, called ideal prototypes of the concept, and the compatibilityof other objects in the universal set with these ideal prototypes can be expressedmathematically by a meaningful similarity relation.

If it is not feasible to define the membership function in question completely, theexpert should at least be able to exemplify it for some representative objects of theuniversal set. The exemplification may be facilitated by asking the expert questionsregarding the compatibility of individual objects x with the linguistic term that is tobe represented by fuzzy set A. These questions, regardless of their form, result ina set of pairs 〈x, A(x)〉 that exemplify the membership function under construction.This set is then used for constructing the full membership function. One way to dothat is to select an appropriate class of functions (triangular, trapezoidal, S-shaped,bell-shaped, etc.) and employ some relevant curve-fitting method to determine thefunction that fits best the given samples. Another way is to use an appropriate neuralnetwork to construct the membership function by learning from the given samples.This approach has been so successful that neural networks are now viewed as astandard tool for constructing membership functions.

When a direct method is extended from one expert to multiple experts, the opinionsof individual experts must be properly combined. Any averaging operation, includingthose introduced in Section 2.3.4, can be used for this purpose. The most commonoperation is the simple weighted average

A(x) =

n∑

i=1

ciAi(x),

where Ai(x) denotes the valuation of the proposition “x belongs to A” by expert i, n

denotes the number of experts involved, and ci denote weights by which the relativesignificance of individual experts can be expressed; it is assumed that

n∑

i=1

c1 = 1.

Experts are instructed either to value each proposition by a number in [0, 1] or tovalue it as true or false.

Direct methods based on exemplification have one fundamental disadvantage.They require the expert (or experts) to give answers that are overly precise and, hence,unrealistic as expressions of their qualitative subjective judgments. As a consequence,

2.9 Nonstandard Fuzzy Sets 55

the answers are always somewhat arbitrary. Indirect methods attempt to reduce thisarbitrariness by replacing the requested direct estimates of degrees of membershipwith simpler tasks.

In indirect methods, experts are usually asked to compare elements of the universalset in pairs according to their relative standing with respect to their membershipin the fuzzy set to be constructed. The pairwise comparisons are often easier toestimate than the direct values, but they have to be somehow connected to the directvalues. Numerous methods have been developed for dealing with this problem. Theyhave to take into account possible inconsistencies in the pairwise estimates. Most ofthese methods deal with pairwise comparisons obtained from one expert, but a fewmethods are described in the literature that aggregate pairwise estimates from multipleexperts. The latter methods are particularly powerful since they allow the knowledgeengineer to determine the degrees of competence of the participating experts, whichare then utilized, together with the expert’s judgments, for calculating the degrees ofmembership in question. The coverage of these various methods is beyond the scopeof this tutorial.

The problem of constructing membership functions and operations on them hasbeen addressed by many authors. A good review of the various methods, with ref-erences to the original publications, was prepared by Sancho-Royo & Verdegay[1999].

2.9 Nonstandard Fuzzy Sets

Since the introduction of standard fuzzy sets by Zadeh [1965], several other types offuzzy sets have been introduced in the literature. Each of them leads to a particularformalized language, which may be viewed as a branch of the overall fuzzy set theory.The following are definitions of the most visible types of nonstandard fuzzy sets. Ineach of them, symbols X and A denote, respectively, the universal set of concern andthe fuzzy set defined.

1. Interval-valued fuzzy sets: A : X → CI([0, 1])CI ([0,1]) denotes here the set of all closed intervals contained in [0, 1]. That is,A(x) is a closed interval of real numbers in [0, 1] for each x ∈X. An alternativeformulation is

A = 〈A, �A〉,

where A and �A are standard fuzzy sets such that A(x)≤ �A(x) for all x ∈X. Fuzzysets defined in this way are usually called gray fuzzy sets. For each x ∈ X, functionsA and �A clearly form an interval [A(x), �A(x)] ∈ CI[0, 1]. Interval-valued fuzzysets have been investigated since the early 1970s and are used in many applications[Gorzalczany, 1987].


2. Fuzzy sets of type 2: A : X → FI([0, 1])FI([0, 1]) denotes the set of all fuzzy intervals defined on [0, 1]. Fuzzy setsof type 2, which are generalizations of interval-valued fuzzy sets, have beeninvestigated since the mid-1970s. Their theory is now well developed and utilizedin many applications. A recent appraisal of the theory was prepared by John[1998]; advanced developments, including those regarding computer software forcomputing with type-2 fuzzy sets, are reported in a paper by Karnik et al. [1999]and in a book by Mendel [2001].

3. Fuzzy sets of type k(k>2): A : X → FIk−1([0, 1])FIk−1([0, 1]) denotes the set of all fuzzy sets of type k− 1. For each x ∈X, A(x)

is a fuzzy set of type k − 1. These sets were introduced as theoretically possiblegeneralizations of type 2 fuzzy sets in the mid-1970s. Their theory is not fullydeveloped as yet, and their practical utility remains to be seen.

4. Fuzzy sets of level 2: A : F(X)→ [0, 1]F(X) is a family of fuzzy sets defined on X. That is, a fuzzy set of level 2 isdefined on a family of fuzzy sets, each of which is defined, in turn, on a givenuniversal set X. This mathematical structure allows us to represent a higher levelconcept by lower level concepts, all expressed in imprecise linguistic terms ofnatural language. Thus far, fuzzy sets of level 2 have been rather neglected in theliterature, even though they were already recognized in the early 1970s.

5. Fuzzy sets of level k(k>2): A : F k−1(X)→ [0, 1]Fk−1(X) denotes a family of fuzzy sets of level k− 1. Sets of this type arenatural generalizations of fuzzy sets of level 2. They are sufficiently expressiveto facilitate representation of high level concepts embedded in natural language.Notwithstanding their importance, no adequate theory has yet been developed forfuzzy sets of this type.

6. L-fuzzy sets: A : X → L

L denotes a recognized set of membership grades which is required to be at leastpartially ordered. Usually, L is assumed to be a complete lattice. This importanttype of fuzzy set was introduced very early in the history of fuzzy set theory byGoguen [1967].

7. Intuitionistic fuzzy sets: A = 〈AM, AN〉

Symbols AM and AN denote standard fuzzy sets on X such that

0 ≤ AM(x)+ AN(x) ≤ 1

for all x ∈X. The values AM(x) and AN(x) are interpreted for each x ∈X as,respectively, the degree of membership and the degree of nonmembership of x inA. Intuitionist fuzzy sets have been investigated since the early 1980s. Althoughtheir theory is now fairly well developed, primarily due to work by Atanassov[2000], their utility remains to be established.

8. Rough fuzzy sets: AR = 〈AR, �AR〉

2.10 Principal Sources for Further Study 57

These are fuzzy sets whose α-cuts are approximated by rough sets [Pawlak, 1991].That is, AR is a rough approximation of a fuzzy set A based on an equivalencerelation R on X. Symbols AR and �AR denote, respectively, the lower and upperapproximations of A in which the set of equivalence classes X/R is employedinstead of the universal set X; for each α ∈ [0, 1], the α-cuts of AR and �AR aredefined by the formulas

αAR =⋃{[x]R|[x]R ⊆

αA, x ∈ X}

αAR =⋃{[x]R|[x]R ∩

αA �= ∅, x ∈ X},

where [x]R denotes the equivalence class in X/R that contains x. This combinationof fuzzy sets with rough sets must be distinguished from another combination, inwhich a fuzzy equivalence relation is employed in the definition of a rough set. Itis appropriate to refer to the sets that are based on the latter combination as fuzzy

rough sets. These combinations, which have been discussed in the literature sincethe early 1990s, seem to be of great utility in some application areas.

Observe that the introduced types of fuzzy sets are interrelated in numerous ways.For example, a fuzzy set of any type that employs the unit interval [0, 1] can begeneralized by replacing [0, 1] with a complete lattice L; some of the types (e.g.,standard, interval-valued, or type 2 fuzzy sets) can be viewed as special cases ofL-fuzzy sets; or rough fuzzy sets can be viewed as special interval-valued sets.The overall fuzzy set theory is thus a broad formalized language based upon anappreciable inventory of interrelated types of fuzzy sets, each associated with itsown variety of concepts, operations, methods of computation, interpretations, andapplications.

2.10 Principal Sources for Further Study

For further study of fuzzy set theory and fuzzy logic, a graduate text by Klir and Yuan[1995] is recommended since it is a natural extension of this tutorial. It employs thesame terminology and notation, and covers virtually all aspects of fuzzy set theory andrelated areas in a thorough and mathematically rigorous fashion. It also contains a bib-liography of over 1700 entries and a bibliographical index. In addition, the followinggeneral textbooks are also recommended: Lin & Lee [1996]—excellent coverage ofthe role of neural networks in fuzzy systems, with a focus on integrated fuzzy-neuralintelligent systems; Nguyen & Walker [1997]—a well-written, rigorous presentationwith many examples and exercises; Pedrycz & Gomide [1998]—a comprehensivecoverage with a good balance of theory and applications. Furthermore, the two vol-umes of collected papers by Lotfi Zadeh [Yager et al., 1987; Klir & Yuan, 1996]


are indispensable for proper comprehension of the four facets of fuzziness—theset-theoretic, relational, logical, and epistemological facets.

In the category of reference books, two important handbooks—Ruspini et al.[1998] and Dubois & Prade [1999]—are recommended as convenient sources ofinformation on virtually any aspect of fuzzy set theory and related areas; the latter isthe first volume of a multi-volume handbook on fuzzy sets.

In addition to references made in previous sections of this tutorial, several importantbooks that seem to be relevant to geological modeling should be mentioned. Amongthe many books on knowledge-based systems, most of which are oriented to control,two excellent books with a broader coverage of fuzzy modeling are recommended, onewritten by Babuška [1998], and one edited by Hellendoorn and Driankov [1997]. Animportant resource also is a book of selected papers by Sugeno, edited by Nguyen andPrasad [1999]. Fuzzy modeling in which neural networks or genetic algorithms playimportant roles is well covered in the books by Nauck et al. [1997] and Rutkowska[2002] and the one edited by Sanchez et al. [1997], respectively. Finally, the relatedareas of fuzzy classification, pattern recognition, and clustering, which are also ofinterest to geologists, are covered by several books. The books by Bezdek [1981],Kandel [1982], and Pal & Majumder [1986] are classics in these areas; the ones bySato et al. [1997] and Pal & Mitra [1999] are more up to date.

Two valuable books were written in a popular genre by McNeill & Freiberger[1993] and Kosko [1993]. Although both books characterize the relatively short butdramatic history of fuzzy set theory and discuss the significance of the theory, theyhave different foci. While the former book focuses on the impact of fuzzy set theory onhigh technology, the latter is concerned more with philosophical and cultural aspects;these issues are further explored in a more recent book by Kosko [1999]. Anotherbook of popular genre, which is worth reading, was written by DeBono [1991]. Heargues that fuzzy logic (called in the book water logic) is important in virtually allaspects of human affairs.

Fuzzy logic is a field that is currently developing extremely rapidly. The followingjournals are the principal sources (in English) of these developments:

1. Fuzzy Sets and Systems

2. IEEE Transactions on Fuzzy Systems

3. Journal of Intelligent and Fuzzy Systems

4. International Journal of Approximate Reasoning

5. Journal of Fuzzy Mathematics

6. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

7. Japanese Journal of Fuzzy Theory and Systems (English translation by AllertonPress)

8. International Journal of Intelligent Systems

9. Journal of Intelligent Information Systems

10. Fuzzy Systems and A.I. Reports and Letters

References 59

11. Soft Computing

12. Fuzzy Economic Review

13. International Journal of Fuzzy Systems

14. Fuzzy Optimization and Decision Making

Finally, research and education in fuzzy logic are now supported by numerousprofessional organizations. Many of them cooperate in a federation-like manner viathe International Fuzzy Systems Association (IFSA), which publishes the prime jour-nal in the field, Fuzzy Sets and Systems, and has organized the biennial World IFSACongress since 1985. The oldest professional organization supporting fuzzy logicis the North American Fuzzy Information Processing Society (NAFIPS); founded in1981, NAFIPS publishes the Journal of Approximate Reasoning and organizes annualmeetings.

References

Atanassov, K. T. [2000], Intuitionistic Fuzzy Sets. Springer-Verlag, New York.Babuška, R. [1998], Fuzzy Modeling for Control. Kluwer, Boston.Bezdek, J. C. [1981], Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum

Press, New York.De Bono, E. [1991], I Am Right, You Are Wrong: From Rock Logic to Water Logic. Viking

Penguin, New York.De Cooman, G. [1997], “Possibility theory—I, II, III.” International Journal of General

Systems, 25(4), 291–371.Di Nola, A., Sessa, S., Pedrycz, W., & Sanchez, E. [1989], Fuzzy Relation Equations and Their

Applications to Knowledge Engineering. Kluwer, Boston.Dubois, D., & Prade, H. [1988], Possibility Theory. Plenum Press, New York.Dubois, D., & Prade, H. [1999], Fundamentals of Fuzzy Sets (Vol. 1 of Handbook of Fuzzy

Sets.) Kluwer, Boston.Goguen, J. A. [1967], “L-fuzzy sets.” Journal of Mathematical Analysis and Applications,

18(1), 145–174.Gorzalczany, M. B. [1987], “A method for inference in approximate reasoning based on

interval-valued fuzzy sets.” Fuzzy Sets and Systems, 21(1), 1–17.Hájek, P. [1998], Metamathematics of Fuzzy Logic. Kluwer, Boston.Hellendoorn, H., & Driankov, D. (eds.) [1997], Fuzzy Model Identification: Selected

Approaches. Springer-Verlag, New York.Hughes, G. E., & Cresswell, M. J. [1996], A New Introduction to Modal Logic. Routledge,

London and New York.John, R. [1998], “Type 2 fuzzy sets: An appraisal of theory and applications.” International

Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 6(6), 563–576.Kandel, A. [1982], Fuzzy Techniques in Pattern Recognition. John Wiley, New York.Karnik, N. N., Mendel, J. M., & Liang, Q. [1999], “Type-2 fuzzy logic systems.” IEEE

Transactions on Fuzzy Systems, 7(6), 643–658.


Kaufmann, A., & Gupta, M. M. [1985], Introduction to Fuzzy Arithmetic: Theory and

Applications. Van Nostrand, New York.Klir, G. J. [1997], “Fuzzy arithmetic with requisite constraints.” Fuzzy Sets and Systems, 91(2),

165–175.Klir, G. J. [1999], “On fuzzy-set interpretation of possibility theory.” Fuzzy Sets and Systems,

108(3), 263–273.Klir, G. J., & Pan, Y. [1998], “Constrained fuzzy arithmetic: Basic questions and some

answers.” Soft Computing, 2(2), 100–108.Klir, G. J., & Yuan, B. [1995], Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-

Hall, Upper Saddle River, NJ.Klir, G. J., & Yuan, B. (eds.) [1996], Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected

Papers by Lotfi A. Zadeh. World Scientific, Singapore.Kosko, B. [1993], Fuzzy Thinking: The New Science of Fuzzy Logic. Hyperion, New York.Kosko, B. [1999], The Fuzzy Future. Harmony Books, New York.Lin, C. T., & Lee, C. S. G. [1996], Neural Fuzzy Systems: A Neuro Fuzzy Synergism to

Intelligent Systems. Prentice-Hall, Upper Saddle River, NJ.McNeill, D., & Freiberger, P. [1993], Fuzzy Logic: The Discovery of a Revolutionary Computer

Technology—and How It Is Changing Our World. Simon & Schuster, New York.Mendel, J. M. [2001], Uncertain Rule-Based Fuzzy Logic Systems. Prentice-Hall, Upper Saddle

River, NJ.Moore, R. E. [1966], Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ.Nauck, D., Klawonn, F., & Kruse, R. [1997], Foundations of Neuro-Fuzzy Systems. John Wiley,

New York.Negoita, C. V., & Ralescu, D. A. [1975], Applications of Fuzzy Sets to Systems Analysis.

Birkhäuser, Basel–Stuttgart, and Halsted Press, New York.Neumaier, A. [1990], Interval Methods for Systems of Equations. Cambridge University

Press, Cambridge (UK) and New York.Nguyen, H. T., & Prasad, N. R. (eds.) [1999], Fuzzy Modeling and Control: Selected Works of

M. Sugeno. CRC Press, Boca Raton, FL.Nguyen, H. T., & Walker, E. A. [1997], A First Course in Fuzzy Logic. CRC Press, Boca

Raton, FL.Novák, V. [1989], Fuzzy Sets and Their Applications. Adam Hilger, Bristol and Philadelphia.Novák, V., & Perfilieva, I. (eds.) [2000], Discovering the World with Fuzzy Logic. Physica-

Verlag/Springer-Verlag, Heidelberg and New York.Novák, V., Perfilieva, I., & Mockor, J. [1999], Mathematical Principles of Fuzzy Logic. Kluwer,

Boston.Pal, S. K., & Dutta Majumder, D. K. [1986], Fuzzy Mathematical Approach to Pattern

Recognition. Wiley Eastern Limited, New Delhi.Pal, S. K., & Mitra, S. [1999], Neuro-Fuzzy Pattern Recognition: Methods in Soft Computing.

John Wiley, New York.Pawlak, Z. [1991], Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer, Boston.Pedrycz, W. [1989], Fuzzy Control and Fuzzy Systems. Research Studies Press, Rayton, UK.Pedrycz, W., & Gomide, F. [1998], An Introduction to Fuzzy Sets: Analysis and Design. MIT

Press, Cambridge, MA.Piegat, A. [2001], Fuzzy Modelling and Control. Physica-Verlag/Springer-Verlag, Heidelberg

and New York.

References 61

Rescher, N. [1969], Many-Valued Logic. McGraw-Hill, New York.Ruspini, E. H., Bonissone, P. P., & Pedrycz, W. (eds.) [1998], Handbook of Fuzzy Computation.

Institute of Physics Publications, Bristol (UK) and Philadelphia.Rutkowska, D. [2002], Neuro-Fuzzy Architectures and Hybrid Learning. Physica-Verlag/

Springer-Verlag, Heidelberg and New York.Sanchez, E., Shibata, T., & Zadeh, L. A. (eds.) [1997], Genetic Algorithms and Fuzzy Logic

Systems: Soft Computing Perspectives. World Scientific, Singapore.Sancho-Royo, A., & Verdegay, J. L. [1999], “Methods for the construction of membership

functions.” International Journal of Intelligent Systems, 14(12), 1213–1230.Sato, M., Sato, Y., & Jain, L. C. [1997], Fuzzy Clustering Models and Applications. Physica-

Verlag/Springer-Verlag, Heidelberg and New York.Van Leekwijck, W., & Kerre, E. E. [1999], “Defuzzification: criteria and classification.”

Fuzzy Sets and Systems, 108(2), 159–178.Yager, R. R. [1983], “Quantified Propositions in a Linguistic Logic.” International Journal of

Man–Machine Studies, 19(2), 195–227.Yager, R. R. [1985–86], “Reasoning with fuzzy quantified statements.” Kybernetes, 14,15

(Part I, Part II), 233–240, 111–120.Yager, R. R. [1991], “Connectives and quantifiers in fuzzy sets.” Fuzzy Sets and Systems, 40(1),

39–75.Yager, R. R., & Filev, D. P. [1994], Essentials of Fuzzy Modeling and Control. John Wiley,

New York.Yager, R. R., Orchinnikov, S., Tong, R. M., & Nguyen, H. T. (eds.) [1987], Fuzzy Sets and

Applications—Selected Papers by L. A. Zadeh. John Wiley, New York.Zadeh, L. A. [1965], “Fuzzy Sets.” Information and Control, 8(3), 338–353.Zadeh, L. A. [1978], “Fuzzy sets as a basis for a theory of possibility.” Fuzzy Sets and Systems,

1(1), 3–28.

Chapter 3 Fuzzy Logic and Earth Science:An Overview

Robert V. Demicco

3.1 Introduction 633.2 Crisp Sets and Geology 663.3 Fuzzy Sets in Geology 683.4 Fuzzy Logic Systems 73

3.4.1 Application of standard (“Mamdani”) inference rules to compaction

curves 74

3.4.2 Application of standard (“Mamdani”) inference rules to coral reef

growth 78

3.4.3 Application of self-adjusting inference rules to calculation of

exposure index 82

3.4.4 Carbonate production as a function of depth and distance to

platform edge 88

3.4.5 Permeability as a function of grain size and sorting using

fuzzy clustering 93

3.4.6 Adding more antecedent variables: permeability revisited 99

3.5 Summary and Conclusions 100References 101

3.1 Introduction

We are sure that most geologists are aware of the trend toward explicit use of the termsEarth, System, and Science together in the titles of an increasing number of geologytextbooks. We need to look no further than introductory texts in “geology” including:The Blue Planet: an Introduction to Earth Systems Science [Skinner & Porter, 1999];Earth’s Dynamic Systems [Hamblin & Christiansen, 2001]; The Earth System [Kumpet al., 1999]; Earth System History [Stanley, 1999]; and Earth Systems [Ernst, 2000]to name a few. Indeed, the first ten chapters of the influential textbook Understanding

Earth by Press & Siever [2001] are organized into a unit entitled “Understanding the

Earth System.” This trend (although many of our more traditional colleagues wouldsay “fad”), acknowledges the fact that there is an independent field of inquiry known

63

64 3 Fuzzy Logic and Earth Science: An Overview

as “Systems Science” that has rapidly evolved over the second half of the 20th andbeginning of the 21st centuries.

Although the roots of system science are based in antiquity, it did not becomea recognized discipline until the latter half of the 20th century. It arose from threeprincipal roots: (i) successful efforts in mathematics to introduce and develop moreexpressive formalized languages, such as fuzzy set theory, fuzzy measure theory,fractal geometry, cellular automata, etc.; (ii) the emergence of computer technology,which opened new methodological possibilities as well as laboratory tools for theprospective systems science; and (iii) a host of ideas, often captured by the generalterm systems thinking, which emerged in the 20th century. Systems thinking includedideas emanating from the renewed interest in holism in science, the emergence ofinterdisciplinary areas in science, and some developments in engineering (controltheory, information theory, similarity theory, etc.). It is not our intent here to review theorigin, scope, and methodology of system science or to comment on whether the use ofEarth System Science in the books mentioned above is really appropriate (it is in somecases, not in others). The interested reader is referred to Klir [2001]. If geologists aregoing to use the term “system” in the same context as system scientists, then we need tounderstand this term in the sense that system scientists use it. System science seeks tocategorize, understand, and exploit the interactions and linkages among componentsof some arbitrary division of either the artificial world (such as telephone networks)or the naturally occurring world. In the system science perspective it is not so muchthe components of the arbitrary division but the interactions between and amongcomponents. The most widely known concepts of system science that have becomegenerally used in the more traditional sciences (including geology) are positive andnegative feedback loops as process controls. Use of the term system in this specificway allows geologists to tap into the paradigms, methodologies, insights, etc. ofsystem science.

Mathematics is clearly at the core of system science. It has been widely appreci-ated that the most important tool produced by mathematics prior to the 20th centurywas the calculus. Indeed, Newtonian mechanics, which is a major outcome of thecalculus, still comprises the bedrock of many Earth sciences (physical oceanography,seismology, climatology, whole-Earth geophysics, etc.). For example, the “diffusionequation” has application in heat flow and nearly all current groundwater flow model-ing involves piecewise finite difference or finite element approximate solutions overa grid of solution points. In spite of the widespread and successful use of Newtonianmechanics in the Earth sciences, Newtonian mechanics is best at dealing only withrather simple problems involving deterministic and predominantly linear systems thathave only a limited number of variables. Along with this crown jewel of pre-twentiethcentury mathematics there arose a traditional view that uncertainty (imprecision,nonspecificity, vagueness, inconsistency, etc.) was unscientific and had to be avoided.

In the late nineteenth century, science turned to the study of physical processesat the molecular level. It rapidly became clear that, although the precise laws of

3.1 Introduction 65

Newtonian mechanics were relevant to physical processes at this level, they were notapplicable in practice due to the sheer number of entities involved and the numberof calculations that would have to be made. Statistical mechanics arose from thesepractical difficulties. It developed to deal with systems wherein there were manyindividuals (whether the individuals were molecules or discrete telephones), manyvariables, and where the variables interacted with a very high degree of randomness.The role played in Newtonian mechanics by calculus (where there is no uncertainty)was replaced by probability theory in statistical mechanics. Along with the rise ofstatistical mechanics came the realization that uncertainty is not only welcome inscience, but also essential to disciplines such as statistical mechanics and quantumtheory.

Newtonian mechanics and statistical mechanics are highly complementary. Thedifferential equations at the heart of Newtonian mechanics excel in modeling sys-tems involving relatively small numbers of variables that are related to each otherin predictable ways. On the other hand, statistical mechanics has the exact oppositecharacteristics: an ability to model large numbers of variables with a high degree ofrandomness in their interactions. It is now generally agreed that these mathematicaltools only cover problems at the opposite ends of the complexity and randomnessscales. In a well-known paper, Warren Weaver [1948] referred to these as problemsof organized simplicity and disorganized complexity. He argued that most problemsin the sciences as well as in modern technology lie somewhere between these twoextremes. They involve nondeterministic and highly nonlinear systems with largenumbers of components and rich interactions among the components. Furthermore,the non-deterministic nature of these systems does not arise out of randomness that canyield meaningful statistical averages and be tackled by statistical methods. Weavercalled them systems of organized complexity. We would maintain that most currentresearch in geology and all research in the area of Earth systems science focus onnatural systems of organized complexity. This research has, at its heart, the desire toconstruct rigorous mathematical models of the behavior of Earth systems.

One of the mathematical tools initially developed by systems scientists to deal withproblems of organized complexity is fuzzy logic. Fuzzy logic arose out of a funda-mentally different way of dealing with uncertainty. Zadeh [1965] introduced a theoryof mathematical objects he called fuzzy sets—sets wherein the boundaries are notprecise. Fuzzy logic (based on the mathematical manipulation of fuzzy sets) providesanother approach toward modeling complex systems, an approach based on commonsense, intuition, and natural language, where precise mathematical formulations ofchemical and physical components of a system are replaced by natural, linguisticrules based on expert human understanding of the natural system.

This chapter has two purposes. First, we would like to point out why fuzzy logicconcepts naturally lend themselves to applications in the Earth sciences. Second, wewould like to show how the basic concepts of fuzzy logic could be applied to theEarth sciences by way of a few simple examples. The chapters that follow this one


have more complicated, explicit case histories of application of fuzzy logic in Earthsciences.

3.2 Crisp Sets and Geology

It is obvious that almost all of the variables at the core of the Earth sciences arecontinuous. Obvious examples include temperature, pressure, depth in the ocean,etc. Less obvious examples are solid solution composition of feldspars, amount ofquartz in a plutonic igneous rock, etc. Moreover, it is also true that many of thesecontinua vary over many orders of magnitude. The size of sedimentary particles, forexample, ranges at least from 10−4 mm through 104 mm. Likewise, permeability,a parameter from Darcy’s Law1 which depends only on the material properties of aporous medium, varies from approximately 10−16 to 10−3 cm2 for Earth materials.

It is also true that most of these variables are, more often than not, broken upinto arbitrary “pigeon holes” by geologists seeking to “classify.” Such “pigeon hole”classification schemes can be represented mathematically as conventional crisp sets.In a crisp set, an individual is either included in a given set or not included in it. Thisdistinction is often described by a characteristic function. The value of either 1 or0 is assigned by this function to each individual of concern, thereby discriminatingbetween individuals that either are members of the set (the assigned value is 1) orare not members of the set (the assigned value is 0). Figure 3.1a is an example:the crisp set concept of “water depth” applied to a typical, shallow-marine setting.The domain of this variable ranges from 2 m below mean sea level to 2 m abovemean sea level. This continuum is generally partitioned into a number of crisp sets:subtidal, intertidal, and supratidal, with the intertidal being further subdivided intohigh-intertidal, mid-intertidal, and low-intertidal areas [Reading & Collinson, 1996,p. 213]. In the example shown in Figure 3.1a these crisp sets are the following closedor left-open intervals of real numbers (expressing measurements in meters):

Subtidal = [−2,−0.75]

Low-intertidal = (−0.75,−0.25]

Mid-intertidal = (−0.25, 0.25]

1Darcy’s Law can be expressed by the formula

v =−kρg

μ

dh

dl

where v is the specific discharge, ρ is the fluid density, g is the gravitational acceleration, μ is the viscosityof the fluid, h is the hydraulic head (a proxy for a fluid potential field made up of potential energy andkinetic energy terms), l is length over which the potential change is measured, and k is the permeability.

3.2 Crisp Sets and Geology 67

Figure 3.1 Comparison of a crisp set description of the variable “tidal range” (a), with a fuzzyset description (b). In (a): “mean low water” = −1.25 m, “mean sea level” = 0 m, and “meanhigh water” = 0.75 m. The fuzzy set representation better captures natural variations (impliedby the adjective “mean”) due to periodic tidal curve changes resulting from the ebb–neap–ebbcycle, and non-periodic, random variations such as storm flooding, etc.

High-intertidal = (0.25, 0.75]

Supratidal = (0.75, 2].

Each of these sets (intervals) may also be expressed by a characteristic function.Denoting, for example, the characteristic function of the set (interval) representingmid-intertidal water depth by A, we have

A(x) =

{1 when x ∈(−0.25, 0.25]

0 otherwise.(3.1)

However, on modern tidal flats, these boundaries are constantly changing due to peri-odic variations in over a dozen principal tidal harmonic components [see Table 11.1in Knauss, 1978]. More importantly, it is commonly flooding due to anomalous“wind tides” and “barometric tides” [Knauss, 1978] that is important for erosion anddeposition in beaches, tidal flats, etc.


3.3 Fuzzy Sets in Geology

Zadeh [1965] introduced a concept that has come to be called a standard fuzzy set inorder to convey the inherent imprecision of arbitrary “pigeon hole” boundaries. Theimprecision of these boundaries results from both the precision of the measurementand, as in the case of tidal flats, the accuracy of trying to pin down an ever-changinglocation. In a standard fuzzy set the characteristic function is generalized by allowingus to assign not only 0 or 1 to each individual of concern, but also any value between0 and 1. This generalized characteristic function is called a membership function

(Figure 3.1b). The value assigned to an individual by the membership function of afuzzy set is interpreted as the degree of membership of the individual in the standardfuzzy set. The membership function B(x) of the standard fuzzy set “mid-intertidal”represented in Figure 3.1b is

B(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 when x ≤ −0.5 mx + 0.5

0.5when − 0.5 ≤ x ≤ 0 m

0.5− x

0.5when 0 ≤ x ≤ 0.5 m

0 when 0.5 m ≤ x

(3.2)

The fuzzy set description of tidal range given in Figure 3.1b better captures theessence of the gradations between locations on beaches, tidal flats, etc. Similarly, 1 to2 meters below sea level is certainly shallow, but where does a carbonate platform orsiliciclastic shelf become “deep” or “open” (see Nordlund [1996])? Using fuzzy sets,there can be a complete gradation between all these depth ranges. Each membershipfunction is represented by a curve that indicates the assignment of a membershipdegree in a fuzzy set to each value of a variable within the domain of the variableinvolved (e.g. the variable “water depth”). The membership degree may also beinterpreted as the degree of compatibility of each value of the variable with theconcept represented by the fuzzy set (e.g. subtidal, low-intertidal, etc.). Curves ofthe membership functions can be simple triangles, trapezoids, bell-shaped curves, orhave more complicated shapes.

Contrary to the symbolic role of numbers 1 and 0 in characteristic functions of crispsets, numbers assigned to individuals by membership functions of standard fuzzy setshave a clear numerical significance. This significance is preserved when crisp setsare viewed (from the standpoint of fuzzy set theory) as special fuzzy sets.

Another example of the difference between crisp and fuzzy sets is provided by theconcept of “grain size” (also mentioned in Section 2.1). The domain of this variableranges over at least eight orders of magnitude from particles that are sub-micron sizeto particles that are meter size. Because of this spread in the domain of the variable,grain size is usually represented over a base 2 logarithmic domain. This continuum

3.3 Fuzzy Sets in Geology 69

Figure 3.2 Comparison of a crisp set description of the variable “grain size” (a) with a fuzzyset description (b) of part of the range of the variable. (The “phi” scale (σ ) = the negative logof the size of the particle with base 2.)

is generally divided into six crisp sets2; clay, silt, sand, gravel, cobbles and boulders(Figure 3.2a). The characteristic function A(x) of sand is, for example,

A(x) =

⎧⎨⎩

1 when1

16mm ≤ x ≤ 2 mm

0 otherwise(3.3)

In this crisp set representation of grain size a grain with diameter of 1.9999 mm wouldbe classified as sand, whereas a grain with diameter of 2.0001 mm would be classifiedas gravel. If fuzzy sets are used instead of crisp sets (Figure 3.2b), than the artificialclassification boundaries are replaced by gradational boundaries and the two grainsdescribed would share membership in both sets, described by the linguistic terms“coarse sand” and “gravel.” With increasing diameter of grains, the membership in“gravel” will increase and the membership in “sand” will decrease in some way thatdepends on the application context. The basic idea is that the membership in a fuzzy setis not a matter of affirmation or denial, as it is in a classical set, but a matter of degree.

2However, for some usages “sand” is further subdivided into up to 20 “pigeon holes.”


The membership functions in this case have complicated formulas because the domainis represented on a logarithmic scale whereas the range is on an arithmetic scale. Otherexamples of fuzzy sets relevant to geologic systems might include the thickness ofsediments eroded and deposited, the anorthite content in plagioclase feldspar, andthe velocity of flow in an aquifer. In these contexts, terms such as “produce some,”“erode a little,” or “about 30% anorthite” or “very slow fluid flow” have meaning.

We can apply the concept of fuzzy sets to combinations of variables. For exam-ple, Figure 3.3a is the classification of intrusive igneous rocks recommended by theInternational Union of Geological Sciences [Streckeisen, 1974] and features a classi-fication space comprising of two back-to-back triangles of the kind every student ofgeology will remember from petrography classes. In this scheme, rock name is basedon the ratios of three minerals: (1) feldspar containing variable amounts of Na or Ca(symbolized by an A at the left apex of the diamond); (2) feldspar containing K (sym-bolized by a P at the right apex of the diamond); and (3) either quartz (symbolized bya Q at the upper apex of the diamond) or minerals referred to as “feldspathoids” (sym-bolized by an F at the lower apex of the diamond). The numbers along the straight-linejoins between the four apices indicate the relative percentages of either A–P–Q orA–P–F where the percentages of these minerals have been normalized to 100%. Forexample, “granite” contains between 20% to 60% quartz and therefore contains 80%to 40% feldspar. The feldspars in turn must lie between a 10%–90% A–F mixture anda 90%–10% A–F mixture.

In a crisp set representation of the concept “granite” (Figure 3.3b) we would assigna 1 to those combinations of variables that fit the definition of granite, and a 0 else-where. A fuzzy set representation of granite (Figure 3.3c) would take into accountthe obviously transitional nature of the boundaries where, depending on the context,values between 0 and 1 would be assigned. In other words, a particular igneous rockcan be simultaneously a member of the fuzzy set quartz syenite and granite. Adoptingsuch a scheme would go a long way toward settling the debates of where, exactly, theboundaries of the pigeon holes should be. Needless to say, fuzzy set concepts couldbe applied to the classification of sedimentary and metamorphic rocks as well.

Other, nonstandard types of fuzzy sets have been introduced in the literature (seeoverview in Section 2.9). In this chapter, however, we consider only standard fuzzysets in which degrees of membership are characterized by numbers between 0 and 1.Therefore the adjective “standard” is omitted.

Fuzzy sets are a powerful tool for relating independent to dependent variables, asis demonstrated below. However, there are some instances in which the use of crispsets is quite adequate. For example, the bed forms (and by extension the sedimentarystructures) that form under steady, uniform flows in flumes or in nature are wellrepresented by crisp sets. There is no significant transitional form between ripplesand dunes where steady states obtain. Thus, it would be inappropriate to use a fuzzyset description of bed forms. However, we should recognize that crisp sets are specialfuzzy sets.

3.3 Fuzzy Sets in Geology 71

Figure 3.3 (a) The International Union of Geological Sciences recommended classificationof plutonic igneous rocks [Streckeisen, 1974]. See text for details. (b) Perspective view of theupper triangular portion of (a) showing a crisp set representation of the term granite. A 1 isassigned to all values within the trapezoid boundaries of granite. (c) Perspective view of theupper triangular portion of A showing a possible fuzzy set representation of the rock namegranite.


Figure 3.3 Continued

3.4 Fuzzy Logic Systems 73

3.4 Fuzzy Logic Systems

There is a wealth of observational and experimental data in the geological sciences.Moreover, a complicated mix of quantitative and qualitative data types usually charac-terizes any given geologic system. For example, in a metamorphic terrain we may beable to measure isotopes and elemental compositions of a few hundred samples withtremendous accuracy and repeatability. However, these samples are usually scatteredover many hundreds or even thousands of kilometers in arbitrarily located, three-dimensional outcrops. Nordlund [1996] refers to qualitative information as “soft”information. Other examples of qualitative information would be “beach sands tendto be well sorted and are coarser than offshore sands,” or “carbonate sediment isproduced in an offshore carbonate ‘factory’ and is transported and deposited in tidalflats,” or “basaltic magmas have a lower viscosity than more siliceous magmas.” Suchstatements carry information, but are not easily quantified. Indeed, these types of qual-itative statement are commonly very important sources of information that is obtainedby field studies. Other examples of soft information would include descriptions ofrock types, interpretations of depositional settings and their entombed fossils. Thesequalitative, “soft data” are usually admixed with what we might refer to as “hard data.”Hard data might include seismic (or outcrop-scale) geometric patterns of reflectorsor bedding geometries, isotopic ratios along a closely spaced sampling line, etc.

Fuzzy logic allows us to formalize and treat such “soft” information in a rigorous,mathematical way and it also allows quantitative information to be treated in a morenatural, continuous fashion. We would like to suggest that fuzzy logic might be apowerful and computationally efficient alternative technique to numerical modelingof geological systems. It has the distinct advantage in that models based on fuzzylogic are robust, easily adaptable, more attuned to common sense, computationallyefficient, and in a sensitivity analysis can be easily altered, allowing many differentcombinations of input parameters to be run in a quick and efficient way.

The primary purpose of fuzzy logic is to formalize reasoning in natural language.This requires that propositions expressed in natural language be properly formalized.In fuzzy logic, the various components of natural-language propositions (predicates,logical connectives, truth qualifiers, quantifiers, linguistic hedges, etc.) are repre-sented by appropriate fuzzy sets and operations on fuzzy sets [Zadeh, 1975–76]. Eachof these fuzzy sets and operations is strongly context dependent and, consequently,must be determined in the context of each application [Klir & Yuan, 1995]. Themost common fuzzy logic systems are sets of fuzzy inference rules, or “if–then” rules(see also Section 2.5). These are conditional and usually unqualified fuzzy proposi-tions that describe dependence of one or more output-variable fuzzy sets to one ormore input-variable fuzzy sets. A simple fuzzy if–then rule assumes the canonicalform

If x is A then y is B


where A and B are linguistic values defined by fuzzy sets on the universal sets X andY , respectively. The “if” part of the rule “x is A” is referred to as the antecedent orpremise whereas the “then” part of the rule “y is B” is referred to as the consequent

or conclusion.

3.4.1 Application of standard (“Mamdani”) inference rules to

compaction curves

Where fine-textured sediments are progressively buried in subsiding sedimentarybasins, their porosity (the volume fraction of connected voids that allow fluidmovement) is sharply reduced. Figure 3.4a [Goldhammer, 1997] summarizes thedata available for fine-grained or muddy carbonate sediments from a variety ofsources. Goldhammer [1997] fits two empirically derived exponential curves to thesedata:

φ = 70e−z/263(z < 150 m) (3.4)

φ = 40e−z/6500(z > 150 m), (3.5)

where φ = porosity and z = depth in meters. The dot-dash lines in Figure 3.4 showthe solutions of these equations. This is a fairly typical example of “curve fitting”to geological data and, as written, these equations tell the reader that the change inporosity of muddy carbonate sediments with depth can be modeled by an exponentialfunction.

A fuzzy logic approach to this same data set could start with the straightforwardstatement: “Very near the surface, the porosity is high; around 100 meters or so theporosity decreases to intermediate values of around 40 percent; and then the porositysteadily decreases to low values at 4500 meters or so.” Both burial depth and porosityin this context are fuzzy sets. The plot in Figure 3.5a shows three possible membershipfunctions for the fuzzy sets “surface,” “shallow,” and “deep” for the input variabledepth of burial over the domain 0 to 4500 m. The plot in Figure 3.5b shows threepossible membership functions for the fuzzy sets “low,” “medium,” and “high” forthe dependent variable porosity over the domain 10% to 90%. With these fuzzy setscharacterizing the input and output variables, we can formally break our statementabove into three “if–then” rules:

1. If the burial depth is near-surface, then the porosity is high;2. If the burial depth is shallow, then the porosity is medium;3. If the burial depth is deep, then the porosity is low.

The standard (so-called “Mamdani”) interpretation [Mamdani & Assilian, 1975]of these if–then rules is shown in Figure 3.6 for a burial depth of 100 m. The left-handcolumn represents the input variable burial depth whereas the right-hand column


Figure 3.4 Application of fuzzy logic to compaction of lime mud with increasing burial depth.Data points shown by asterisks in (a) and (b) are from Goldhammer (1997). Dot-dash curvesin (a) and (b) are empirical fits described by Equations (3.4) and (3.5). The solid lines in (a)and (b) are fuzzy logic system approximations to data. (a) Output of fuzzy logic system shownin Figures 3.5a and 3.5b. (b) Output of fuzzy logic system shown in Figures 3.5c and 3.5d.


Figure 3.5 (a)Antecedent membership functions for the variable burial depth: close to surface;shallow; and deep. (b) Consequent membership functions of the output variable porosity: low;medium; and high. The triangular shapes of fuzzy logic systems (a) and (b) produces the outputcurve in Figure 3.4a. The “bell-shaped” membership functions of the same variables (burialdepth and porosity) in (c) and (d) produce the output curve in Figure 3.4b.

represents the output variable porosity. The upper row is the rule: “if depth is near-surface, then porosity is high.” The second row is the rule: “if depth is shallow,then porosity is medium.” Finally, the third row represents the rule: “if depth isdeep, then porosity is low.” The input variable (100 m in this example) is evaluatedsimultaneously for each water depth and a truth value= degree of membership of theinput variable in each of the potential input sets (“surface,” “shallow,” and “deep”)is calculated. In this case, the burial depth of 100 m only “fires” the first two rules.This is because a depth of 100 m is not part of the fuzzy set “deep,” so the “truthvalue” of the proposition “100 m is a member of the fuzzy set ‘deep”’ is 0. Thesetruth values truncate the membership functions of the appropriate output variable.For each burial depth, the maximum of the three truncated membership functions of


Figure 3.6 Standard (“Mamdani”) interpretation of the “if–then” rules (“if the burial depth isnear surface, then porosity is high”; “if the burial depth is shallow, then the porosity is medium”;and “if the burial depth, is deep, then the porosity is low”) is shown for a burial depth of 100 m.The input variable is evaluated for each depth and a truth value = degree of membershipof the input variable in each of the potential input sets (“surface,” “shallow,” and “deep”) iscalculated. These truth values truncate the membership functions of the appropriate outputvariable. For each burial depth, the truncated membership functions of the output variable aresummed, and the centroid of the appropriate curve is taken as the “defuzzified” output value.

the output variable is taken, and the centroid of the appropriate curve is taken as the“defuzzified” output value. The solid curves in Figure 3.4a shows this fuzzy inferencesystem evaluated over the depth 0 to 4500 m.

There are two distinct advantages in the approach of using fuzzy logic to char-acterize geologic systems rather than empirical equations. First, fuzzy sets describesystems in natural language. More importantly, the shapes of the membership func-tions can easily be changed by small increments, thereby allowing rapid “sensitivityanalysis” of the effects of changing the boundaries of the fuzzy sets. An example ofthis is shown in Figures 3.5c and 3.5d. When the shapes and boundaries of the mem-bership functions are slightly changed, the output function is also slightly changed.In this manner, by “trial and error,” the output values of a fuzzy inference systemare changed in order to more nearly match ground truth. Figure 3.5c shows differ-ently shaped membership functions with slightly different boundaries to those of the


triangular-shaped membership functions of Figure 3.5a. The result of using the samerules as before, evaluated with these membership functions, is shown in Figure 3.4bby the solid curve (the dot-dash curve is the empirical fit suggested by Goldhammer[1997]). In robot control algorithms, where fuzzy logic was first developed, systemscould self-adjust the shapes of the membership functions and set boundaries untilthe required task was flawlessly performed. This aspect of fuzzy systems, commonlyfacilitated via the learning capabilities of appropriate neural networks [Kosko, 1992;Klir & Yuan, 1995; Nauck & Klawonn, 1997] or by genetic algorithms [Sanchezet al., 1997], is one of their great advantages over numerical solution approaches.In sections below, we further discuss this aspect of fuzzy logic.

3.4.2 Application of standard (“Mamdani”) inference rules to coral

reef growth

Coral animals are capable of rapid fixation of CaCO3 from seawater because of sym-biotic photosynthetic algae within their tissues. Thus, carbonate production of theseanimals is, in some way, related to light penetration into the shallow ocean. Demicco& Klir [2001] contrasted “if–then” rule-based fuzzy logic models of coral reef growthwith the deterministic models of Bosscher & Schlager [1992]. This example is againbriefly described here as background for Chapter 9 where this problem is used toillustrate a novel method for solving differential equations.

Figure 3.7 shows data on growth rates of the main Caribbean reef-building coralMontastrea annularis [Bosscher & Schlager, 1992, Figure 1, p. 503]. Bosscher &Schlager [1992], following Chalker [1981], fit the equation

G(z) = G(0) tanh(Io e−kz/Ik) (3.6)

to these data. Here z is water depth, G(z) is growth rate at a given depth (z), G(0) ismaximum growth rate (G at z = 0), Io is surface light intensity, Ik is saturation lightintensity, and k is the extinction coefficient given in the Beer–Lambert law,

Iz = Ioe−kz. (3.7)

In Figure 3.7, the two dotted curves are fit to the data of Equation 3.6 using differentvalues of the parameters G(0), Io, Ik , and k. Bosscher & Schlager [1992] extendedthese equations and developed a numerical model of the geologic history of coralreefs growing on the Atlantic shelf-slope break of Belize by a step-wise solution ofthe differential equation

dh(t)/dt = Gm tanh(Io exp{−k[ho + h(t)] − [so + s(t)]}/Ik). (3.8)

Here dh(t)/d(t) is the change in the height of the coral surface with time, ho is theinitial height of the surface at the start of a time step, h(t) is the growth increment


Figure 3.7 Measured growth rates of the main Caribbean reef-building coral Montastrea

annularis [from Bosscher & Schlager, 1992, Figure 1]. The two dotted lines are solutionsof Equation (3.6). The solid curve is the result of the fuzzy logic system described in the text.

in that time step, so is the initial sea level position for a time step, and s(t) is thevariation in sea level for that time step. A simulation of coral reef growth based onthis equation is shown in Figure 3.8a. The solution assumes an initial starting slope,initial values of Gm (maximum growth rate), Io (initial surface light intensity), k

(extinction coefficient), and the variable sea level curve over the last 80,000 years asshown in Figure 3.8c.

Demicco & Klir [2001] used a fuzzy logic system to model the growth ratesof Montastrea annularis based on a natural language description that captures theessence of the data: “If the water is shallow, then the coral growth rate is fast. If thewater is deep, then the coral growth rate is slow.” The input or antecedent parameterhere is water depth whereas the output or consequent variable is coral growth rate.Both of these variables can be represented by fuzzy sets (Figure 3.9). Figure 3.9ashows two membership functions for the fuzzy sets “shallow” and “deep” for theinput variable depth over the domain 0 to 50 m. Figure 3.9b shows two possiblemembership functions for the fuzzy sets “fast” and “slow” for the variable growthrate over the domain 0 to 10 mm/yr. The “Mamdani” interpretation [Mamdani &Assilian, 1975] of the if–then rules (“if the water is shallow, then the coral growth rate


Figure 3.8 Comparison of 2-dimensional models of the geologic history of coral reefs growingon the Atlantic shelf-slope break of Belize. (a) Stepwise solution of differential Equation (3.8)[Bosscher & Schlager, 1992]. The forward model solution for coral reef growth assumes aninitial starting slope, initial values of Gm (maximum growth rate), Io (initial surface lightintensity), and k (extinction coefficient). (b) Model of reef growth based on the same sea levelcurve, same starting slope, and same initial value of Gm, but with the fuzzy inference systemdescribed in the text replacing the differential equation for coral growth production. (c) Variablesea level curve of the past 80,000 years, input into both models.

is fast; if the water is deep, then the coral growth rate is slow”) is also employed inthis example (Figure 3.10). The left-hand column represents the input variable waterdepth whereas the right-hand column represents the output variable growth rate. Theupper row is the rule: “if depth is shallow, then growth rate is fast.” The secondrow is the rule: “if depth is deep, then growth rate is slow.” A value of the input


Figure 3.9 (a) Two membership functions for the fuzzy sets “shallow” and “deep” for the inputvariable depth over the domain range 0 to 50 m. (b) Two membership functions for the fuzzysets “fast” and “slow” for the variable growth rate over the domain 0 to 10 mm/y. Membershipfunctions were adjusted by hand to produce the visual “best fit” curve in Figure 3.7.

variable (10 m in Figure 3.10) is evaluated simultaneously for each water depth and atruth value= degree of membership of the input variable in each of the potential inputsets (“shallow” and “deep”) is calculated. These truth values truncate the membershipfunctions of the appropriate output variable. For each water depth, the maximum of thetwo truncated membership functions of the output variable is taken, and the centroidof the appropriate curve is taken as the “defuzzified” output value. The solid curvein Figure 3.7 shows this fuzzy inference system evaluated over the depth 0 to 50 m.

Demicco & Klir [2001] also developed a forward model of reef development basedon the fuzzy inference system of Figures 3.9 and 3.10. The results of this model arecompared with Bosscher & Schlager’s [1992] results in Figure 3.8b. In Chapter 9,Perfilieva describes a novel technique for solving Equation (3.8) on the basis of fuzzy

transformations. She then applies this solution technique to another forward modelof reef growth (Figure 9.10) that gives similar results to Figures 3.8a and 3.8b.


Figure 3.10 Standard (“Mamdani”) interpretation of the “if–then” rules (“if the water is shal-low, then coral growth rate is fast; if the water is deep, then the coral growth rate is slow”) isshown for a water depth of 10 m. The input variable is evaluated for each water depth and atruth value = degree of membership of the input variable in each of the potential input sets(“shallow” and “deep”) is calculated. These truth values truncate the membership functionsof the appropriate output variable. For each water depth, the truncated membership functionsof the output variable are summed, and the centroid of the appropriate curve is taken as the“defuzzified” output value.

It is important to note that fuzzy logic systems are very versatile and, indeed, can bemore versatile than deterministic equations. So far we have been using ordinary fuzzy

sets wherein for a given input value there is one output value. In general, although wewill not use them in this chapter, we can generalize ordinary fuzzy sets into second-

order fuzzy sets [Mendel, 2001], where the membership function does not assign toeach element of the universal set one real number but a fuzzy number (a fuzzy setdefined on the real numbers in the unit interval), or a closed interval of real numbersbetween the identified upper and lower bounds. Clearly, this approach would be war-ranted by the spread in the initial data on coral growth rates versus depth in Figure 3.7.

3.4.3 Application of self-adjusting inference rules to calculation of

exposure index

In control algorithms, where fuzzy logic was first developed, systems could self-adjust the shapes of the membership functions and set boundaries until the required


task was flawlessly performed. This aspect of fuzzy systems, commonly facilitated viathe learning capabilities of appropriate neural networks [Kosko, 1992; Klir & Yuan,1995; Nauck & Klawonn, 1997; Lin & Lee, 1996] or by genetic algorithms [Sanchezet al., 1997; Cordón et al., 2001], is one of their great advantages over numericalsolution approaches. In the two examples given above of burial depth versus porosityand coral reef growth versus water depth, the membership functions were adjusted“by hand” to fit the output function to the data. In this simple procedure, we usedthe naturalistic boundaries suggested by the data, and trial and error adjustment ofthe shapes of the membership functions to obtain the desired fit. In this section, wepresent a fuzzy logic system that relates elevation on a carbonate flat to the absoluteamount of time an area of the flat is exposed to the atmosphere (the “exposure index”of Ginsburg et al. [1977]). We use both a trial and error fit obtained “by hand” andthe adaptive neuro-fuzzy system that is included in the Fuzzy Logic Toolbox of thecommercial high-level language MATLAB©.

Figure 3.11a shows “exposure index” versus elevation around an arbitrarily desig-nated “mean tidal level of 0 m” for the carbonate tidal flats of northwestern AndrosIsland in the Bahamas (the data are from Ginsburg et al. [1977], their Figure 3, p. 8).The exposure index is the percentage of time an area at a certain elevation relative tothe mean tide level stays dry. In this microtidal setting, exposure index is a complicatedfunction of wind direction, strength, and duration as well as the astronomical tides.

The fuzzy logic systems we have used to this point in this chapter have been theMamdani fuzzy inference systems wherein the output variable is a linguistic vari-able whose states are standard fuzzy sets. In the final step of a Mamdani-type fuzzylogic system, a fuzzy set that has been generated by aggregating the appropriatetruncated membership functions of the output variable has to be “defuzzified” bysome averaging process (e.g., finding the “centroid”). Contrary to a Mamdani fuzzyinference system, an alternative approach to formalizing fuzzy inference systems,developed by Takagi & Sugeno [1985], employs a single “spike” as the output mem-bership functions. Thus, rather than integrating across the domain of the final outputfuzzy set, a Takagi–Sugeno-type fuzzy inference system employs only the weightedaverage of a few data points. In Figure 3.12 we show the Takagi–Sugeno systemused to generate the hand-adjusted curve in Figure 3.11a. Elevation on the tidal flatis divided into three triangular-shaped membership functions: low intertidal; highintertidal; and supratidal, shown from top to bottom in the left column of the figure.The exposure index in Figure 3.12 is represented by three Takagi–Sugeno “spike-like” (so-called “zero order”) membership functions: little exposure (centered on 0.1or 10%); medium exposure (centered on 0.5 or 50%); and high exposure (centeredon 0.99 or 99%).

The “if–then” rules for this system are, as in the case of coral reef growth, simpleand intuitive. They are represented by the three rows and are (from top to bottom):

(1) If the elevation is low intertidal, then the exposure index is little;


Figure 3.11 Asterisks in figures (a) and (b) are data taken from Ginsburg et al. [1977, Figure 3,p. 8] and plot measured “exposure index” versus height around “mean tidal level” for thecarbonate tidal flats of northwestern Andros Island in the Bahamas. The exposure index isthe percentage of time an area at a certain height relative to the mean tide level stays dry. Inthis microtidal setting, exposure index is a complicated function of wind direction, strength,and duration as well as the astronomical tides. (a) Output of “hand-tuned” Takagi–Sugenofuzzy logic system shown in Figure 3.12. (b) Output of MATLAB© adaptive neuro-fuzzyTakagi–Sugeno fuzzy logic system described in Figures 3.13 and 3.14.


Figure 3.12 Takagi–Sugeno (so-called 0-order) fuzzy inference system for relating heightrelative to sea level to exposure index evaluated at an elevation of −15 cm. In this case, foreach rule the output membership functions are “spikes” set at output values 0.1, 0.5, and 0.99.The truth value of the antecedent variables truncate the spikes at the appropriate values andtheir weighted average determines the output. The output of this system is the solid line inFigure 3.11a.

(2) If the elevation is high intertidal, then the exposure index is medium; and(3) If the elevation is supratidal, then the exposure index is high.

For an elevation of+15 cm shown in this example, the appropriate output “spikes” aretruncated and the simple weighted sum of all the output functions is computed. Thisfuzzy logic system was adjusted “by hand,” and does not do a particularly accuratejob in fitting the data.

The MATLAB© adaptive neuro-fuzzy system is a program that utilizes learningcapabilities of neural networks for tuning parameters of fuzzy inference systems on thebasis of given data. However, as explained above, the type of fuzzy inference systemsdealt with by this program are not the classical Mamdani type but rather the Takagi–Sugeno type. The program implements a training algorithm that employs the commonbackpropagation method based on the least-square error criterion [see Klir & Yuan,1995, AppendixA]. Figure 3.13 shows the four antecedent membership functions usedto “adjust” the output linear functions. These four membership functions correspondto the linguistic terms “subtidal,” “low intertidal,” “high intertidal,” and “supratidal”


Figure 3.13 Four antecedent membership functions for the input variable elevation definedrelative to an arbitrary datum of 0 water depth taken as “mean sea level.” These membershipfunctions were assigned by the adaptive neuro-fuzzy training routine of MATLAB©.

used to describe the relative position of the sediment surface with respect to “mean”tidal oscillations. The curve in Figure 3.11b shows the output of this fuzzy inferencesystem.

The training algorithm generates four linear output functions. These linear outputmembership functions (so-called “first-order” Takagi–Sugeno membership functions)differ from the so-called “zero-order” spikes used above as the positions of the“spikes” are linearly related to the value of the input membership function. For exam-ple, in Figure 3.14 the upper panel shows the evaluations of four if–then rules for anelevation of −15 cm whereas the lower panel shows the evaluations of four if-thenrules for an elevation of +15 cm. Note how the positions of the output membershipspikes change between the two diagrams. This is especially obvious for the upperrow in each panel. The location of the “spike” in each rule is given by one of the fourlinear formulas below:

y = 2.08 ∗ waterdepth + 75.46

y = 1.93 ∗ waterdepth + 41.57

y = 0.16 ∗ waterdepth + 94.12

y = 0.15 ∗ waterdepth + 91.48

Here y represents the position of the output spike in the domain−128.9 to 359.7. Asan example, consider the upper panel in Figure 3.14 where an elevation of −15 cmis input. The location of the spike in each of the rules is then given by:

2.08×−15+ 75.46 = 44.26


Figure 3.14 A “linear” (so-called first-order) Takagi–Sugeno machine-fit fuzzy inference sys-tem for the elevation versus exposure index on the tidal flats ofAndros Island. The right columnshows the four antecedent membership functions of the variable elevation over the domain−30to +52 cm. The upper set of panels evaluates the system for an elevation of −15 cm whereasthe lower set of panels evaluates the system for an elevation of +15 cm. Note that the output“spikes” change position as the input variables change. In effect the output spikes can “float”over the output domain (exposure index) in a fashion that is linearly related to the water depth.See text for further details.


1.93×−15+ 41.57 = 12.62

0.16×−15+ 94.12 = 91.72

0.15×−15+ 91.48 = 89.15

Each spike is truncated by the appropriate membership grade for the input: −15 cm.This input value truncates the membership function subtidal to degree 0.4, the mem-bership function low intertidal to degree 0.6, the membership function high intertidal

to degree 0.01, and the membership function supratidal to degree 0.004. The final out-put is given by aggregating the spike location times the truth value of the appropriateinput fuzzy set:

44.26× 0.4+ 12.62× 0.6+ 91.72× 0.01+ 89.15× 0.004 = 26.5498

There is a clear computational advantage in employing aTakagi–Sugeno fuzzy logicsystem. Moreover, it is well suited to optimization techniques and adaptive techniques.The main advantages of the Mamdani method are its widespread acceptance and itsintuitive nature.

3.4.4 Carbonate production as a function of depth and distance to

platform edge

Where the fuzzy logic system is composed of two antecedent variables and oneconsequent variable and standard fuzzy sets are used, the two input variables “map”to a singular value of the output variable that comprises a 3-dimensional surface.This is a particularly powerful way to envision a fuzzy logic system. Carbonatesediment production is commonly modeled as being linearly depth-dependent as inthe example of coral growth discussed above. However, the pioneering work ofBroecker & Takahashi [1966] showed that sediment production on shallow carbonateplatforms is dependent not only on shallow depths, but also on the residence timeof the water, which, practically speaking, translates into distance away from thenearest shelf margin. Figure 3.15 shows the shallow bathymetry of the Great BahamaBank northwest of Andros Island. The dotted contours on Figure 3.15 are carbonateproduction contours in kg/m2 per year, taken from Broecker & Takahashi [1966,their Figure 10, p. 1585]. Broecker & Takahashi did not make any measurementsthat allowed them to compute carbonate production at the steep edge of the platform.However, where such measurements have been made they cluster in the range of 1 to4 kg/m2/y. For this exercise, we have chosen 1 kg/m2/y as a target figure for carbonatesediment production at the margin of the Great Bahama Bank.

Our models of sediment production use a 1 km2 grid of bathymetric data(Figure 3.15), compute sediment production in each cell, and graph the results.


Figure 3.15 Bathymetry of the Great Bahama Bank northwest of Andros Island. The edge ofthe platform is taken as the−100 m sea level contour. The dotted lines are contours of carbonatesediment production in kg/m2 per year, taken from Broecker & Takahashi [1966, Figure 10,p. 1585].

The first model (Figure 3.16a) uses a simple linear interpolation of sedimentproduction with normalized distance from the bank margin:

production = −0.5× distance+ 1 (3.9)

shown in Figure 3.17a. The second model (Figure 3.16b) uses a simple linear functionof sediment production with depth shown in Figure 3.17b:

if depth < −10 m

production = 0.01111× depth+ 1.11111

else (3.10)

production = −0.07× depth+ 0.3


Figure 3.16 Models of carbonate sediment production on the Great Bahama Bank. (a) Sed-iment production modeled as a linear decrease with (normalized) distance from bank marginaccording to Equation (3.9) (also see Figure 3.17a). (b) Sediment production modeled using thepiecewise linear relationship of Equation (3.10) relating sediment production to water depth(also see Figure 3.17b). (c) Sediment production modeled using the piecewise planar relation-ships of Equation (3.11) relating sediment production to both water depth and distance frombank margin (also see Figure 3.18a). (d) Sediment production modeled using the fuzzy logicsystem described in the text (also see Figures 3.18b and 3.19, and color insert).

This function has a maximum at 10 m and more or less resembles the data set of coralgrowth rates in Figure 3.7. Neither of these models does a particularly good job inpredicting carbonate sedimentation production patterns of the Great Bahama Bank.The third model combines the two linear models:

if depth < −10 m

production = (0.011111× depth+ 1.11111)− (0.5× distance)

else (3.11)

production = (−0.05× depth)+ 0.3)− (0.5× distance)


Figure 3.17 Linear sediment production functions. (a) Graph of Equation (3.9). (b) Graph ofEquation (3.10).

Figure 3.18 Models of sediment production based on distance and water depth. (a) Graphof Equation (3.11): sediment production modeled as piecewise planes. (b) Graph of sedimentproduction according to the fuzzy logic system described in the text and Figure 3.19.

Figure 3.18a is a graph of this function and the results, applied to the Great BahamaBank, are shown in Figure 3.16c. This function is not easily altered to fit observationsand is not that transparent to someone unfamiliar with the problem.

Contrast this piecewise approach to a fuzzy logic system of the same problem.Figure 3.19 shows the two input variables (normalized distance from shelf edge anddepth) and the output variable production. Distance is characterized by two Gaussianmembership functions, near and far, whereas depth is characterized by two trapezoidalmembership functions (deep and shallow) and one triangular membership function(maximum production depth, abbreviated max on the figure). The output variableproduction comprises four membership functions, hardly-any, little, some, and lots.There are six rules to this fuzzy logic system:

(1) If distance is near and depth is deep, produce hardly any;


Figure 3.19 Membership functions comprising part of the fuzzy logic system that generatedthe solution in Figure 3.16d. The input variables are (normalized) distance (modeled withGaussian membership functions) and water depth (modeled by both trapezoidal or triangularmembership functions). Rules relating these variables are given in the text and the solution isshown in Figure 3.18b.

(2) If distance is near and depth is max, produce lots;(3) If distance is near and depth is shallow, then produce some;(4) If distance is far and depth is deep, produce hardly any;(5) If distance is far and depth is max, produce little; and(6) If distance is far and depth is shallow, produce hardly any.

Figure 3.18b is a graph of the production versus depth and distance determined by thisfuzzy logic system contrasted with the piecewise planar approximation. Figure 3.16dis the results of this fuzzy logic model applied to the Great Bahama Bank. Both thepiecewise planar model (Figures 3.16c and 3.16d) do a fairly good job in reproducingthe carbonate sediment production pattern on the Great Bahama Bank northwest ofAndros Island (Figure 3.15). We have adjusted the boundaries and shapes of the depthfunction “by hand” to tune this model. It is important to note that tuning the model


by adjusting the membership functions is relatively easy in comparison with tryingto recalculate the piecewise approximating equations.

For example, Figure 3.18b is the 3-dimensional representation of the fuzzy logicsystem described in Figure 3.19b. The [x,y] axes are water depth and distance, andproduction is [z]. Notice that this rather simple fuzzy logic system generates a rathercomplicated non-planar relationship between depth, distance, and production. Theshape of the output surface can be quickly changed by many operations, the two mostcommonly employed being: (1) varying the shapes and boundaries of the membershipfunctions; and (2) changing the rule connectors between fuzzy intersection (“and”) orfuzzy union (“or”) rules. We can, in effect, “warp” the output surface to any arbitraryshape we want by varying the shapes of the membership functions and the connectorsin the rule system. This is the strength of this technique.

So far, we have used the first approach and changed the shapes of the membershipfunctions to “tune” our models. We did this both by hand and through the neuro-adaptive program routines of MATLAB©. Figures 3.20 and 3.21 show an exampleof the second approach of changing the inferential connector to “or.” Where “or”is used instead of “and,” fuzzy union is implied. We can also use this approachto change the number of rules necessary to characterize our fuzzy logic system.Figures 3.20a and 3.20b show membership functions for the same two antecedentvariables as above, namely distance from platform margin (normalized to be between0 if a point is at the margin and 1 if it is farthest away) and water depth over the domain−100 m to 0. However, in this case for illustrative purposes we are using only twomembership functions to approximate the consequent variable sediment productionthat is described as either being “low” or “lots” (Figure 3.20c). In this case we modelthe system with only three rules:

(1) If the distance is near AND the depth is maximum, produce lots;(2) If the distance is far OR the depth is deep, produce little;(3) If the distance is far OR the depth is shallow, produce little.

Evaluation of these rules is shown in Figure 3.21. In the case where fuzzy unionis implied (i.e., where the connector is “or”) the appropriate output membershipfunctions are truncated at the highest degree of membership of either input functions.Figure 3.22 shows the surface generated by this fuzzy inference system. In this chapterwe use only standard representations of the connectors “and” and “or.” For otherrepresentations, see Section 2.3.3.

3.4.5 Permeability as a function of grain size and sorting using

fuzzy clustering

In this example, we consider the calculation of permeability in unconsolidated sed-iments as a function of the average size of the grains and the sorting of the grains.


Figure 3.20 Effect of changing the inferential connector to standard “or” (instead of “and”) ina fuzzy logic system. (a) and (b) show membership functions for the two antecedent variables,distance from platform margin (normalized to be between 0 if a point is at the margin, and 1 if itis farthest away) and water depth over the domain−100 m to 0. (c) Two membership functionsto approximate the consequent variable sediment production that is described as being either“low” or “lots.”

This is a highly nonlinear problem in real Earth systems because of a complicatedinterrelationship between the shapes of grains and the packing of grains.

The permeability of a sediment or sedimentary rock is defined by Darcy’s law(equation given in footnote 1) and, in natural systems, varies over at least ten orders ofmagnitude. It is very expensive and time consuming (and in some cases impractical) totake samples of sediment and put them in a device that directly measures permeabilitythrough an application of Darcy’s law. Interested readers can get the details in anyintroductory hydrology textbook. Instead, hydrologists and oil company engineershave spent many years seeking empirical equations to determine permeability frommore easily measurable parameters [see Freeze & Cherry, 1979]. The most obviousparameter might seem to be the porosity of the sediment, where the porosity is the


Figure 3.21 Standard (“Mamdani”) interpretation of the “if–then” rules (“if the distance isnear AND the depth is maximum, produce lots”; “if the distance is far OR the depth is deep,produce little”; and “if the distance is far OR the depth is shallow, produce little”) is shownfor relative distance of 0.02 and water depth of −30 m. The input variable is evaluated foreach water depth and a truth value = degree of membership of the input variable in each ofthe potential input sets (“shallow” and “deep”) is calculated. In the case where fuzzy unionis implied (i.e., where the connector is “or”) the appropriate output membership functions aretruncated at the highest degree of membership of either input function. These truth valuestruncate the membership functions of the appropriate output variable. For each water depth,the truncated membership functions of the output variable are again summed, and the centroidof the appropriate curve is taken as the “defuzzified” output value.

volume fraction of connected voids that allow fluid movement. However, the porosityin a room full of cubic close-packed bowling balls is exactly the same as the porosityin a room full of cubic close-packed marbles (except for the finger holes in thebowling balls). The permeability of sediment is a measure of frictional resistance toflow, which is a function of the “tortuosity” as well as the diameter of the flow path.Although bowling balls and marbles have equal porosities, flow paths through themarbles are more tortuous and, in the marbles, more fluid is in contact with solidgrain. For these reasons, the permeability of marbles is lower than the permeabilityof the bowling balls even though their effective porosities are identical. Thus, mostempirical equations designed to estimate permeability have the average grain size asone of the independent variables. The other independent variable that is commonlyused in empirical equations to calculate permeability is the dispersion in spread ofsizes (or sorting) of the sediment. The calculation of sorting is based on measuringthe masses of sediment that reside on a series of standard mesh-size sieve screens,


Figure 3.22 Output graph of the fuzzy logic system described in Figure 3.21.

assuming that these masses would follow a log-normal distribution. Sediments aredescribed as well sorted, medium sorted and poorly sorted, on the basis of whether therange of sizes varies over a factor of 2 (well sorted), within one order of magnitude(medium), or over more than one order of magnitude (poorly sorted).

It is intuitively obvious that the more poorly sorted a sediment is, the lower thepermeability will be. In a frequently cited study, Krumbein & Monk [1942] preparedabout 30 samples of various mean grain sizes and sorting, and measured the perme-ability of their artificial grain packs. They also derived the following semi-empiricalequation relating permeability to average grain size and sorting:

k = 760GM2e−1.31σ (3.12)

where k = permeability, GM = mean grain size, and σ is a parameter that describesthe range in grain sizes (or sorting) and is related to the standard deviation of the grainsize where it is modeled as a normal distribution. The sands and the mixtures of sandsthat Krumbein & Monk [1942] used to measure permeability are all rather coarselygrained. There have been a number of other studies where, instead of artificial mix-tures, the permeability of naturally occurring sediments has been measured along withgrain size, porosity, and sorting. Pryor [1973] measured these parameters in river andbeach sands that were mostly comprised of quartz, feldspar and other aluminosili-cate minerals. Enos & Sawatsky [1981] measured this same suite of parameters, pluspercentage of mud-sized material in a suite of carbonate sediments. The data from


Figure 3.23 Three approximations to the data of Krumbein & Monk [1942] (shown as aster-isks); Pyror [1973] (shown as triangles); and Enos & Sawatsky [1981] (shown as circles).(a) Semi-empirical fit using Equation (3.12). (b) Hand-fit Mamdani further described in textand in Figure 3.24. (c) Machine-derived linear Takagi–Sugeno fit using fuzzy clusters fromFigure 3.25 and antecedent membership functions shown in Figure 3.26 to characterize grainsize and sorting. Note how both of the fuzzy logic systems (b) and (c) are better fits to thenatural permeability measurements of Pryor [1973] and Enos & Sawatsky [1983] (see alsocolor insert).

Krumbein & Monk [1942] (labeled with asterisks), Pryor [1973] (labeled with dia-monds), and Enos & Sawatsky [1981] (labeled with circles) are shown on Figure 3.23aalong with the solution to the semi-empirical relationship given in Equation (3.12).

Two fuzzy logic models of the relationship between grain size, sorting, and per-meability are given in Figures 3.23b and 3.23c. Figure 3.23b shows a “hand-tuned”Mamdani fuzzy logic system whereas Figure 3.23c again relies on the MATLAB©

adaptive neuro-fuzzy inference engine. The input and output membership functionsfor the hand-tuned Mamdani fuzzy logic system are shown in Figure 3.24. The twelverules that govern this system are:

(1) If the grain size is coarse and the sediment is well sorted, then the permeabilityis very high;


Figure 3.24 Membership functions for the “hand-fit” Mamdani fuzzy logic system relatinggrain size, sorting and permeability. See text for fuzzy logic rules and Figure 3.23 for out-put solution. (a) Membership functions for grain size. (b) Membership functions for sorting.(c) Output membership functions for permeability.

(2) If the grain size is coarse and the sediment is medium, then the permeability ishigh;

(3) If the grain size is coarse and the sediment is poor, then the permeability ismoderate;

(4) If the grain size is medium and the sediment is well sorted, then the permeabilityis moderate;

(5) If the grain size is medium and the sediment is medium, then the permeabilityis bad;

(6) If the grain size is medium and the sediment is poor, then the permeability isvery poor;

(7) If the grain size is fine and the sediment is well sorted, then the permeability islow moderate;


(8) If the grain size is fine and the sediment is medium, then the permeability isvery poor;

(9) If the grain size is fine and the sediment is poor, then the permeability is verypoor;

(10) If the grain size is mud and the sediment is well sorted, then the permeability isvery poor;

(11) If the grain size is mud and the sediment is medium, then the permeability isvery poor; and

(12) If the grain size is mud and the sediment is poor, then the permeability is verypoor.

As the number of variables rises and the number of membership functions for eachvariable rises, there is usually an increase in the number of rules. One of the challengesfacing the use of fuzzy logic in geologic models is the elimination of redundant rules.One way to reduce the number of rules was demonstrated with the use of fuzzyunion among antecedent variables. Another way to keep the number of membershipfunctions down is to look for “clusters” in the data. Clustering encompasses a numberof mathematical techniques for identifying natural groupings in a data set. There aretwo basic methods of fuzzy clustering: fuzzy c-means and fuzzy equivalence relation-

based hierarchy. The details of these methods are outside of the scope of this chapterand the interested reader is directed to Klir & Yuan [1995]. Figure 3.25 shows threeviews of grain size, sorting, and permeability relationships from the data set outlinedabove. The large asterisks are the centers of three fuzzy clusters identified by thefuzzy c-means clustering algorithm of MATLAB©. Fuzzy c-means clustering is aniterative technique that starts with a pre-determined number of clusters (three in thiscase) and partitions the data so that each point belongs to each cluster to some degreespecified by a membership grade. Once fuzzy clusters have been identified in the data,they can serve as the starting points for the adaptive neuro-fuzzy inference engineof MATLAB©. Figure 3.23c shows the results. The output Takagi–Sugeno functionsare “zero-order” spikes centered at permeabilities of 10.50, 2.50, and 758.34.

3.4.6 Adding more antecedent variables: permeability revisited

Whereas the number of antecedent variables is not restricted to two, it is the only num-ber that can conveniently be plotted. For example, we could add a third and fourthantecedent variable to the permeability fuzzy logic system described above. Theamount of mud in a sediment (whether aluminosilicate clays or aragonite needles) fun-damentally affects permeability because these grains are not spherically shaped at allas is the case with “ideal” spherical silt- and sand-sized particles. Moreover, the degreeto which the mud in a sediment has been aggregated into fecal pellets by organisms isyet another variable that could be taken into account where calculating permeability.


Figure 3.25 Three fuzzy c-means clusters (asterisks) for the data of Krumbein & Monk [1942],Pryor [1973], and Enos & Sawatsky [1983] (circles). (a) View in the [x y] (grain size–sorting)plane. (b) View in the [x z] (grain size–permeability) plane. (c) View in the [y z] (sorting–permeability) plane.

Empirical and semi-empirical relationships among these variables and how they affectpermeability naturally would lend themselves to a fuzzy logic approach.

3.5 Summary and Conclusions

It is clear that fuzzy logic systems have the potential to produce very realistic geologicmodels when used as so-called “expert” systems. An expert system usually comprisesthe cooperation of a “knowledge engineer” (i.e., someone familiar with the techniquesdescribed in this chapter) and a geologist familiar with Earth systems problems. Ifthe geologist can distill the key points of the models into the types of “if A and ifB then C” propositions described above, then the knowledge engineer can translatethem into mathematically rigorous fuzzy logic systems. It is important to note that

References 101

there is only a practical limit to the number of antecedent propositions in a fuzzy logicstatement. A statement such as “if A and if B or if C then D” would map three inputvariables into a region of space. Moreover, there is currently an explosive growth in thetheory and application of fuzzy logic and other related “soft” computing techniques,opening new ways of modeling based on knowledge expressed in natural language.This method offers a distinct alternative to statistical modeling in geology. It is morecomputationally efficient and more intuitive for geologists than complicated modelsthat solve coupled sets of differential equations.

References

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Broecker, W. A., & Takahashi, T. [1966], “Calcium carbonate precipitation on the BahamaBanks.” Journal of Geophysical Research, 71(6), 1575–1602.

Chalker, B. E. [1981], “Simulating light-saturation curves for photosynthesis and calcificationby reef-building corals.” Marine Biology, 63(2), 135–141.

Cordón, O., Herrera, F., Hoffmann, F., & Magdalena, L. [2001], Genetic Fuzzy Systems:

Evolutionary Tuning and Learning of Fuzzy Knowledge Bases. World Scientific, Singapore.Demicco, R. V., & Klir, G. J. [2001], “Stratigraphic simulations using fuzzy logic to model

sediment dispersal.” Journal of Petroleum Science and Engineering, 31(2–4), 135–155.Enos, P., & Sawatsky, L. H. [1981], “Pore networks in Holocene carbonate sediments.” Journal

of Sedimentary Petrology, 51(3), 961–985.Ernst, W. G. (ed.) [2000], Earth Systems. Cambridge University Press, Cambridge, UK.Freeze, R. A., & Cherry, J. A. [1979], Groundwater. Prentice-Hall, Englewood Cliffs, NJ.Ginsburg, R. N., Hardie, L. A., Bricker, O. P., Garrett, P., & Wanless, H. R. [1977], “Exposure

index: a quantitative approach to defining position within the tidal zone.” In: L. A. Hardie(ed.), Sedimentation on the Modern Carbonate Tidal Flats of Northwest Andros Island,

Bahamas. The Johns Hopkins University Press, Baltimore, MD.Goldhammer, R. K. [1997], “Compaction and decompaction algorithms for sedimentary

carbonates.” Journal of Sedimentary Research, 67(3), 26–35.Hamblin, W. K., & Christiansen, E. H. [2001], Earth’s Dynamic Systems (9th edition). Prentice-

Hall, Upper Saddle River, NJ.Klir, G. J. [2001], Facets of Systems Science (2nd edition). Kluwer/Plenum, New York.Klir, G. J., & Yuan, B. [1995], Fuzzy Sets and Fuzzy Logic—Theory and Applications. Prentice-

Hall, Upper Saddle River, NJ.Knauss, J. A. [1978], Introduction to Physical Oceanography. Prentice-Hall, Englewood

Cliffs, NJ.Kosko, B. [1992], Neural Networks and Fuzzy Systems. Prentice-Hall, Englewood Cliffs, NJ.Krumbein, W. C., & Monk, G. D. [1942], “Permeability as a function of the size parameters of

unconsolidated sand.” American Institute of Mining and Metallurgical Engineers, Technical

Publication No. 1492.Kump, L. R., Kasting, J. F., & Crane, R. G. [1999], The Earth System. Prentice-Hall, Upper

Saddle River, NJ.


Lin, C.-T., & Lee, C. S. G. [1996], Neural Fuzzy Systems: A Neuro Fuzzy Synergism to

Intelligent Systems. Prentice-Hall, Upper Saddle River, NJ.Mamdani, E. H., & Assilian, S. [1975], “An experiment in linguistic synthesis with fuzzy logic

controller.” International Journal of Man–Machine Studies, 7(1), 1–13.Mendel, J. M. [2001], Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New

Directions. Prentice-Hall, Upper Saddle River, NJ.Nauck, D., & Klawonn, F. [1997], Foundations of Neuro-Fuzzy Systems. John Wiley, NewYork.Nordlund, U. [1996], “Formalizing geological knowledge—with an example of modeling

stratigraphy using fuzzy logic.” Journal of Sedimentary Research, 66(4), 689–712.Press, F., & Siever, R. [2001], Understanding Earth (3rd edition). W. H. Freeman, New York.Pryor, W. A. [1973], “Permeability–porosity patterns and variations in some Holocene

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Sedimentary Environments: Processes, Facies and Stratigraphy. Blackwell, Oxford.Sanchez, E., Shibata, T., & Zadeh, L. A. (eds.) [1997], Genetic Algorithms and Fuzzy Logic

Systems: Soft Computing Perspectives. World Scientific, Singapore.Skinner, B. J., & Porter, S. C. [1999], The Blue Planet: an Introduction to Earth System Science.

John Wiley, New York.Stanley, S. M. [1999], Earth System History. W. H. Freeman, New York.Streckeisen, A. [1974], “Classification and nomenclature of plutonic rocks.” Geologische

Rundschau, 63(2), 773–786.Takagi, T., & Sugeno, H. [1985], “Fuzzy identification of systems and its application for

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Weaver, W. [1948], “Science and complexity.” American Scientist, 36(4), 536–544.Zadeh, L. A. [1965], “Fuzzy sets.” Information and Control, 8(3), 338–353.Zadeh, L.A. [1975–76], “The concept of a linguistic variable and its application to approximate

reasoning.” Information Sciences, 8(3), 199–249, 301–357, 9(1), 43–80.

Chapter 4 Fuzzy Logic in Geological Sciences:A Literature Review

Robert V. Demicco

4.1 Introduction 1034.2 A Sketch of Soft Computing 105

References to soft computing 107

4.3 Fuzzy Logic in Geology: A Literature Review 1074.3.1 Surface hydrology 108

References to surface hydrology 109

4.3.2 Subsurface hydrology 110

References to subsurface hydrology 111

4.3.3 Groundwater risk assessment 111

References to groundwater risk assessment 112

4.3.4 Geotechnical engineering 112

References to geotechnical engineering 113

4.3.5 Hydrocarbon exploration 113

References to hydrocarbon exploration 114

4.3.6 Seismology 115

References to seismology 115

4.3.7 Soil science and landscape development 116

References to soil science and landscape development 117

4.3.8 Deposition of sediment 117

References to deposition of sediment 118

4.3.9 Miscellaneous applications 119

References to miscellaneous applications 119

4.4 Concluding Note: Quo Vadis 120

4.1 Introduction

This chapter focuses on recent literature in geologically oriented journals that dealswith applications of fuzzy logic in various branches of geological sciences, sometimesin the broader context of soft computing.1

1Soft computing is briefly introduced in Section 4.2.

103

104 4 Fuzzy Logic in Geological Sciences: A Literature Review

Geology is the application of chemical, physical, and biological principles to thestudy of the lithosphere. The trend in the Earth sciences over the last 10 yearshas been to view the Earth as a system and treat the hydrosphere, atmosphere,biosphere, and lithosphere as interconnected subsystems. This approach is inter-disciplinary and has been largely fueled by concern about the Earth’s present andpast environments, with a growing realization that what happens in any one ofthe Earth’s “spheres” has impact on the others. Thus, most modern problems ofinterest to geologists involve systems with large numbers of components and richinteractions among the components that are usually nonlinear and non-random.Such problems of organized complexity (also see Section 3.1) typify geologic sys-tems and are exemplified by the Earth systems that operate at the surface of theEarth. In response to the recognition that geology deals with the realm of orga-nized complexity, there has been a recent explosive growth in the theory andapplication of fuzzy logic and other related soft computing techniques in the Earthsciences.

Geology, of all of the natural sciences, most readily lends itself to analysis and mod-eling by soft computing techniques in general and fuzzy logic in particular. Fuzzylogic, in the broad sense (see Section 2.1), has at least a ten-year history (and agrowing body) of refereed works where it has been successfully applied to manyareas of geological research. There are a number of reasons for this. First, geologicknowledge is routinely expressed in natural language and, indeed, much of this bookis a direct outgrowth of this fact. Second, geology is primarily a field science thatbegan as an outgrowth of the mineral extraction industry. As such, the variables thatgeologists have routinely measured for hundreds of years are continua that commonlyvary over many orders of magnitude. Currently, most of these naturally continuousvariables are, more often than not, broken up into arbitrary “pigeon holes” by geol-ogists seeking to “classify.” Fuzzy sets and fuzzy logic offer a much more naturalway of describing geological variables. Third, because geological research is fieldbased, it is commonly carried out over fairly broad regions of tens to hundreds ofsquare kilometers in size. And, where subsurface data are added, the volume of atypical rock body being studied, even on a modest scale, for minerals, gas, ground-water, or information about conditions on the ancient Earth, ranges up to 1000 km3.Direct sampling of the entire rock or sediment body is clearly prohibitive, and somuch of the three-dimensional distribution of rock properties is measured over a rel-atively tiny percentage of the study area and inferred over the entire volume. Theinference is clearly a source of uncertainty and is commonly based on “expert knowl-edge” from a variety of sources to model the disposition of rock properties. Fourth,remote sensing has always been used in geological data gathering. The most commonof these methods are exploration seismic surveys and geophysical measurements ofboreholes. In the former, the differential impedance of rocks to artificial vibrationalenergy produces what amounts to an “ultra sound” of subsurface rock disposition,whereas in the latter, the gamma rays output of rocks and the electrical resistivity of

4.2 A Sketch of Soft Computing 105

rocks to an induced potential are measured as proxies for rock type, permeability, etc.There is also a growing reliance on satellite (and other) remote sensing imagery ofsurface features of the Earth in a number of geological fields. Such imagery is thebasis of geographical informational systems. All of these remote-sensing techniquesrequire correspondences between the proxy measures and the desired rock, sedi-ment, or soil properties to be established. These correspondences are commonly notone-to-one.

The remainder of this chapter is divided into three sections. Section 4.2 givesa short synopsis of how fuzzy logic relates to the broader area of soft computing.This information should be of use to introduce areas of soft computing other thanfuzzy logic, mentioned in some of the papers in the following literature review andbibliography. Section 4.3 comprises a literature review and annotated bibliographywherein we have divided the current literature on fuzzy logic into a number of broadercategories. Please note that the papers described in each section are listed at the end ofthat section rather than being amalgamated into a final list. The list of papers in eachcategory is by no means exhaustive and, as such, the papers cited in each category donot comprise a complete bibliography of materials. However, they represent recentrefereed papers in major Earth science journals that should be available in even modestuniversity libraries. Finally, Section 4.4 is a brief postscript where we note some ofthe trends that seem to have developed over the last dozen years across the differentareas where fuzzy logic has been applied to Earth sciences. We hope that this literaturereview, when coupled with the references cited in the individual chapters, will serveas an introduction into the literature.

4.2 A Sketch of Soft Computing

Soft computing comprises five areas: (1) fuzzy logic; (2) rough sets [Pawlak, 1991];(3) neural networks [Beale & Jackson, 1990]; (4) evolutionary computation [Bäck,1996]; and (5) probabilistic reasoning based on precise (classical) probabilities and,more importantly, imprecise probabilities [Walley, 1991]. There is much synergyamong these areas, with fuzzy logic playing a lead role.

Rough sets and their combinations with fuzzy sets are briefly introduced inSection 2.9. Foundations of the theory of rough sets are presented in Pawlak [1991].Combinations of rough sets and fuzzy sets (fuzzy rough sets and rough fuzzy sets)are examined in two papers by Dubois & Prade [1990, 1992].

Neural networks are computational structures that were inspired by the architectureof the natural networks of neurons in the brain [Klir & Yuan, 1995]. A neural networkcomprises many interconnected computational nodes called neurons. Each neuronhas several inputs and one output, which are connected to other neurons or to theenvironment. The output value of each neuron is at each time uniquely determinedby its input values. A connection between any two neurons or a neuron with the


environment has a “strength” expressed by a number known as a weight. Each neuronthus receives inputs whose values are real numbers. On the basis of the sum ofthese inputs, a nonlinear activation function produces an output value of the neuron.The basic capability of a neural network is to learn patterns from examples. This isaccomplished by adjusting the weights of the interconnections among the neuronsaccording to a learning algorithm.

There are various kinds of learning algorithms. It was recognized since the early1990s that a connection between fuzzy systems and neural networks is beneficialfor both areas [Kosko, 1992]. First, neural networks are capable of implementingfuzzy systems. The implementation is convenient and efficient, and it adds to fuzzysystems the capabilities of learning and adaptability. Second, neural networks werefound very useful for constructing and tuning fuzzy inference rules (membershipfunctions, operations, etc.) of fuzzy systems in various application contexts, as wellas for solving, via their learning capabilities, some inverse problems associated withfuzzy systems [Klir & Yuan, 1995]. On the other hand, neural networks are mademore flexible and robust by fuzzification. Hybrid combinations of fuzzy systems andneural networks, which are usually called neuro-fuzzy systems, have been studiedquite extensively during the last decade or so [Lin & Lee, 1996; Nauck et al., 1997;Rutkowska, 2002].

As the name suggests, evolutionary computation is another biologically inspiredcomputational structure most commonly used for optimization. In this case, thefeatures to be optimized are parameterized into a vector of real numbers. In ageneration step, the vector first gives rise to a large number of new vectors eachof which has been altered by processes analogous to processes that alter DNAstrands during replication: spot mutations (changes in individual number), tran-scription errors, recombination, etc. Next, the fitness value of each of the newlygenerated vectors is determined by criteria relative to the optimization task beingperformed. Only those offspring vectors that improve the performance surviveto give rise to the next generation. The algorithm repeats generation steps untilthe fitness value satisfies some criterion. A mutually beneficial connection, whichwas recognized recently, is the one between fuzzy systems and the area of evo-lutionary computation [Sanchez et al., 1997; Cordón et al., 2001]. On the onehand, the various components of evolutionary computation can be fuzzified, whichenhances their efficiency, flexibility, and robustness. On the other hand, evolution-ary computation is a powerful tool of learning and tuning inference rules employedin fuzzy systems. The term genetic fuzzy system was recently suggested for ahybrid system involving fuzzy systems and evolutionary computation [Cordón et al.,2001].

Reasoning with imprecise probabilities of various kinds (interval-valued, fuzzy,etc.) [Walley, 1991] and dealing with information represented by imprecise probabil-ities [Klir & Wierman, 1999] is a rapidly growing area of soft computing (see Klir,[2003] for an overview).

4.3 Fuzzy Logic in Geology: A Literature Review 107

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Beale, R., & Jackson, T. [1990], Neural Computing: An Introduction. Adam Hilger, New York.Cordón, O., Herrera, F., Hoffmann, F., & Magdalena, L. [2001], Genetic Fuzzy Systems:

Evolutionary Tuning and Learning of Fuzzy Knowledge Bases. World Scientific, Singapore.Dubois, D., & Prade, H. [1990], “Rough fuzzy sets and fuzzy rough sets.” International Journal

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Systematic Organization of Information in Fuzzy Systems. IOS Press, Amsterdam.Klir, G. J., & Wierman, M. J. [1999], Uncertainty-Based Information: Elements of Gener-

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Klir, G. J., & Yuan, B. [1995], Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall, Upper Saddle River, NJ. (See especially Appendix A.)

Kosko, B. [1992], Neural Networks and Fuzzy Systems. Prentice-Hall, Englewood Cliffs, NJ.Lin, C.-T., & Lee, C. S. G. [1996], Neural Fuzzy Systems: A Neuro Fuzzy Synergism to

Intelligent Systems. Prentice-Hall, Upper Saddle River, NJ.Nauck, D., Klawonn, F., & Kruse, R. [1997], Foundations of Neuro-Fuzzy Systems. John Wiley,

New York.Pawlak, Z. [1991], Rough Sets. Kluwer, Boston.Rutkowska, D. [2002], Neuro-Fuzzy Architectures and Hybrid Learning. Physica-Verlag/

Springer-Verlag, Heidelberg and New York.Sanchez, E., Shibata, T., & Zadeh, L. A. (eds.) [1997], Genetic Algorithms and Fuzzy Logic

Systems: Soft Computing Perspectives. World Scientific, Singapore.Walley, P. [1991], Statistical Reasoning with Imprecise Probabilities. Chapman and Hall,

London.

4.3 Fuzzy Logic in Geology: A Literature Review

The categories chosen are briefly described below. It should come as no surprise thatgeotechnical engineering and areas closely allied to engineering, surface hydrol-

ogy, subsurface hydrology, and hydrocarbon exploration, have seen early andextensive use of fuzzy logic. A literature review of surface and subsurface hydrologyis also found in Section 6.2. A subcategory of subsurface hydrology contains papersdescribing management-oriented models for groundwater risk assessment based onfuzzy logic. These will be treated in a separate category for convenience. Explorationgeophysicists in hydrocarbon exploration have adapted fuzzy logic into seismic pro-cessing and evaluation. They also use many of the soft computing techniques (mostof which incorporate fuzzy logic) for the prediction of rock properties from borehole


and other data. Earthquake seismology is a discipline still very closely wedded toNewtonian mechanics. However, a growing number of geophysicists have recentlyadapted fuzzy logic into various aspects of earthquake research. In addition to thepapers mentioned here, also consult Chapter 8. Soil science and geomorphology

have seen extensive use of fuzzy logic. Models of modern and ancient deposition of

sediments have seen modest use of fuzzy logic to simulate sediment production, sed-iment erosion, sediment transportation, and sediment deposition. Chapter 5 describesuses of fuzzy logic in forward models of basin filling, whereas Chapters 9 and 10 usenovel techniques based on fuzzy logic to interpret ancient deposits. Finally, there isa scattered miscellaneous literature in the geological sciences that outlines potentialuses of fuzzy logic in specific areas not included in the categories outlined here.

4.3.1 Surface hydrology

The discharge of a stream is the volume flow per unit time through a cross-sectionof the stream at a point along the stream’s course. Obviously, reliable prediction ofdischarge, especially low flows during droughts and high flows that lead to floods, isimportant for a variety of reasons. The discharge of a river represents a complicated,highly nonlinear response of a watershed area to precipitation. Complicating factorsinclude: the topography of the watershed; the plant cover of the watershed; thestage in the annual growth cycle of the vegetation; whether the leaves are wet ordry; the intensity, duration, and location of precipitation; the physical propertiesof the soil; whether the soils are frozen or thawed, and, if thawed, their moisturecontent; etc. One of the principal areas of research on surface hydrology is to developimproved watershed response models that will allow the prediction of discharge froma watershed on the basis of precipitation falling in the catchment area of the watershed.

A number of deterministic models have been developed to try to predict dischargefrom precipitation records or forecasts, with real-time predictions as the goal. Thesemodels vary in sophistication but, as most were developed for a specific geographicarea, they are difficult to apply globally and are highly parameterized. Franks et al.[1998] employed a well-known, deterministic model to study a small watershed inBrittany, France. However, they incorporated fuzzy sets to describe the relationshipbetween slope measurements, synthetic aperture radar measurements, and the sat-uration state of the catchment area. The saturation state of the catchment area, inturn, was input into the standard deterministic model. Yu & Yang [2000] also useda pre-existing deterministic model to study rainfall/runoff for a small catchment inTaiwan. However, in addition to using conventional objective functions (such as rootmean square error and mean absolute percent error) to calibrate the model, they alsoused a fuzzy multi-objective function to calibrate the model parameters. They arguedthat the fuzzy multi-objective function led to improved modeling results. Zhu &Mackay [2001] also used deterministic watershed response models on experimental,well-instrumented and well-studied watersheds in the western US. They used fuzzy


logic-based inferences from soil maps to provide realistic ranges of soil parame-ters into the model instead of “lumped” (crisp) parameters suggested by boundarieson the maps. Xiong et al. [2001] took the results of five deterministic watershedresponse models applied to 11 catchment areas in Taiwan and combined them into asingle response model, using a first-order Takagi–Sugeno-type model.

There are a number of recent watershed response models that are entirely based onfuzzy logic and other soft computing techniques. Gautam & Holz [2001] developeda rainfall-response model based on fuzzy rules extracted by an artificial neural fuzzyinference system (ANFIS) analysis of long-term rainfall versus runoff data from awatershed in Tuscany, Italy. Chang & Chen [2001] employed a back-propagationfuzzy-neural network on long-term data from a watershed in Taiwan. Özelkan &Duckstein [2001] began with conceptual rainfall-runoff modular models of a smallsubwatershed in the heavily instrumented Walnut Gulch experimental watershed insoutheasternArizona. They then modeled uncertainty among both linear and nonlinearconceptual models with fuzzy numbers and fuzzy arithmetic operations. Finally, fora portion of the River Ouse in northern England, See & Abrahart [2001] applied datafusion techniques to the output from three separate river level forecasting models: (1)a fuzzy neural network; (2) a rule-based fuzzy logic model; and (3) an autoregressivemoving average model. In addition to the volume of a stream’s discharge, the chemicalquality of the water, especially during low flows, is important to planners, engineers,etc. Two studies that developed fuzzy logic-based approaches to water quality modelsinclude Rantitsch [2000] and Mujumdar & Sadikumar [2002]. Rantitsch [2000] usedfuzzy c-means clustering of elements measured in stream samples from the Alps ofAustria to establish four categories of background levels of various elements. Thesecategories were better able to screen out non-anomalous concentrations of metals.

Somewhat closely related to these watershed response models are studies thathave attempted to model and predict precipitation input onto watersheds (see alsoChapter 6). Pongracz et al. [1999] tried to develop a drought prediction model forthe Great Plains of the US. They developed a set of fuzzy rules that related twoinputs—(1) the Southern Ocean Oscillation (SOI as a proxy for El Niño SouthernOscillation—ENSO); and (2) the geopotential height field of the 500 hPa level overa large area of the western hemisphere—to a long-term record of droughts in eightregions of Nebraska. Stehlík & Bárdossy [2002] extended this approach to the generalstochastic prediction of a time series for precipitation over Europe. They developeda fuzzy classification of point measurements of geopotential atmospheric pressuresurfaces over a large-scale grid of Europe as an input into their model.

References to surface hydrology

Chang, F.-J., & Chen, Y.-C. [2001], “A counterpropagation fuzzy-neural network modelingapproach to real time streamflow prediction.” Journal of Hydrology, 245(1–4), 153–164.


Franks, S. W., Gineste, P., Beven, K. J., & Merot, P. [1998], “On constraining the predictionsof a distributed model: the incorporation of fuzzy estimates of saturated areas into thecalibration process.” Water Resources Research, 34(4), 787–797.

Gautam, D. K., & Holz, K. P. [2001], “Rainfall-runoff modeling using adaptive neuro-fuzzysystems.” Journal of Hydroinformatics, 3(1), 3–10.

Mujumdar, P. P., & Sasikumar, K. [2002], “A fuzzy risk approach for seasonal water qualitymanagement of a river system.” Water Resources Research, 38(1), 1–9.

Özelkan, E. C., & Duckstein, L. [2001], “Fuzzy conceptual rainfall-runoff models.” Journal

of Hydrology, 253(1–2), 41–68.Pongracz, R., Bogardi, I., & Duckstein, L. [1999], Application of fuzzy rule-based modeling

technique to regional drought. Journal of Hydrology, 224(3–4), 100–114.Rantitsch, G. [2000], “Application of fuzzy clusters to quantify lithological background con-

centrations in stream-sediment geochemistry.” Journal of Geochemical Exploration, 71(1),73–82.

See, L., & Abrahart, R. J. [2001], “Multi-model data fusion for hydrological forecasting.”Computers & Geosciences, 27(8), 987–994.

Stehlík, J., & Bárdossy, A. [2002], “Multivariate stochastic downscaling model for gener-ating daily precipitation series based on atmospheric circulation.” Journal of Hydrology,256(1–2), 120–141.

Xiong, L., Shamseldin, A. Y., & O’Connor, K. M. [2001], “A non-linear combination of theforecasts of rainfall-runoff models by the first-order Takagi–Sugeno fuzzy system.” Journal

of Hydrology, 245(3–4), 196–217.Yu, P.-S., & Yang, T.-C. [2000], “Fuzzy multi-objective function for rainfall-runoff model

calibration.” Journal of Hydrology, 238(1–2), 1–14.Zhu, A. X., & Mackay, D. S. [2001], “Effects of spatial detail of soil information on watershed

modeling.” Journal of Hydrology, 248(1–4), 54–77.

4.3.2 Subsurface hydrology

Groundwater flow has been most commonly modeled by the piecewise solution ofthe diffusion equation, either by finite difference approximations or by finite elements.Solute transport in groundwater systems has likewise been modeled by finite differ-ence or finite element approximations of the advective-dispersive equation. Thesetypes of models, which are built around the empirical Darcy’s law (see Section 3.2),are very commonly applied to problems of groundwater wellfield development and,where pollutants are dissolved in the groundwater, problems of developing remedia-tion plans to deal with the contaminant plume. There is a growing recognition that suchmodels (although quite common) may be inadequate [Konikow & Ewing, 1999]. Thisinadequacy is due to the inherent imprecision of knowledge of the three-dimensionaldistribution of hydraulic conductivity and, more fundamentally, the general fuzzynature of the variables (such as hydraulic conductivity, storativity, specific yield, etc.)themselves.


In an important series of papers, Dou and his coworkers [Dou et al., 1995, 1997a,b]developed a series of groundwater flow models for steady-state flow, transient ground-water flow, and nonreactive solute transport via groundwater flow. These models usefuzzy numbers to capture the uncertainty in the variables in the differential equa-tions of flow and fuzzy arithmetic techniques for solution of the equations. Schulz &Huwe [1997] developed a similar model for flow in the unsaturated zone (the surface-most soil zone where hydraulic conductivities vary with the state of pore saturation).Recently, Duo et al. [1999] developed a fuzzy rule-based model for solute transportin the unsaturated zone. This change from using fuzzy numbers to represent impre-cise variables in traditional finite difference solutions to models entirely based onfuzzy rules has mirrored the similar development of surface flow models (see alsoBárdossy & Duckstein [1995]).

Another type of flow modeling seeks to understand groundwater flow from firstprinciples instead of the empirically derived Darcy flow equation. Zeng et al. [2000]developed a model of flow in porous media where the degree of interconnectivenessamong pores in a porous medium was modeled with fuzzy sets.

References to subsurface hydrology

Bárdossy, A., & Duckstein, L. [1995], Fuzzy Rule-Based Modeling with Applications to

Geophysical, Biological and Engineering Systems. CRC Press, Boca Raton, FL.Dou, C., Woldt, W., Bogardi, I., & Dahab, M. [1995], “Steady state groundwater flow

simulation with imprecise parameters.” Water Resources Research, 31(11), 2709–2719.Dou, C., Woldt, W., Bogardi, I., & Dahab, M. [1997a], “Numerical solute transport simulation

using fuzzy sets approach.” Journal of Contaminant Hydrology, 27(1–2), 107–126.Dou, C., Woldt, W., Dahab, M., & Bogardi, I. [1997b], “Transient ground-water flow simulation

using a fuzzy set approach.” Ground Water, 35(2), 205–215.Dou, C., Woldt, W., & Bogardi, I. [1999], “Fuzzy rule-based approach to describe solute

transport in the unsaturated zone.” Journal of Hydrology, 220(1–2), 74–85.Konikow, L. F., & Ewing, R. C. [1999], “Is a probabilistic performance assessment enough?”

Ground Water, 37(4), 481.Schulz, K., & Huwe, B. [1997], “Water flow modeling in the unsaturated zone with imprecise

parameters using a fuzzy approach.” Journal of Hydrology, 201(1–4), 211–229.Zeng, Z., Vasseur, C., & Fayala, F. [2000], “Modeling microgeometric structures of porous

media with a predominant axis for predicting diffusive flow in capillaries.” Applied

Mathematical Modelling, 24(12), 969–986.

4.3.3 Groundwater risk assessment

Assessing the risk to human populations from contaminant pollution of ground-water from various anthropogenic sources is an important element of municipal and


agricultural wellfield planning. Aquifer vulnerability to contamination depends onsoil properties, precipitation, topography, etc., all of which can be modeled withfuzzy sets. Freissinet et al. [1999] focused on assessing non-point source pesticidepollution of aquifers. Zhou et al. [1999] and Cameron & Peloso [2001] modifiedthe commonly employed parametric assessment model DRASTIC by incorporat-ing fuzzy sets. Özdamar et al. [2000] developed assessment models for evaluatingthe potential of industrial groundwater contamination wherein the input parameterswere fuzzy.

References to groundwater risk assessment

Cameron, E., & Peloso, G. F. [2001], “An application of fuzzy logic to the assessment ofaquifer’s pollution potential.” Environmental Geology, 40(11–12), 1305–1315.

Freissinet, C., Vauclin, M., & Erlich, M. [1999], “Comparison of first-order analysis and fuzzyset approach for the evaluation of imprecision in a pesticide groundwater pollution screeningmodel.” Journal of Contaminant Transport, 37(1–2), 21–43.

Özdamar, L., Demirhan, M., Özpinar, A., & Kilanc, B. [2000], “A fuzzy areal assessmentapproach for potentially contaminated sites.” Computers & Geosciences, 26(3), 309–318.

Zhou, H., Wang, G., & Yang, Q. [1999], “A multi-objective fuzzy pattern recognition modelfor assessing groundwater vulnerability based on the DRASTIC system.” Hydrological

Sciences—Journal, 44(4), 611–618.

4.3.4 Geotechnical engineering

There are a number of models for river discharge control and flood management withfuzzy control systems. Teegavarapu & Simonovic [1999] describe a simple one-damsystem from Kentucky, USA, whereas Cheng [1999] describes an optimization modelfor simultaneous floodgate control of the Yangtze River involving the Three GorgesDam, flood basin areas along the river, and eight flood control dams on tributaries.Huang et al. [1999] developed an optimized watershed management plan for the LakeErhai basin in southern China, using a fuzzy multi-objective management program.This watershed is under intense environmental pressure from a variety of competingland uses (agricultural, scenic, light industry, etc.).

Al-Homoud & Al-Masri [1999] developed a fuzzy expert system to evaluate thefailure potential of road-cut slopes and embankments in a landslide-prone area ofJordan. Grima & Verhoef [1999] modeled rock trencher performance with fuzzylogic, whereas Hammah & Curran [1998] used a fuzzy clustering algorithm to iden-tify fracture sets encountered in exploration drilling. Ercanoglu & Gokceoglu [2002]developed a fuzzy rule-based model to assess landslide susceptibility in Turkey.Finally, Klose [2002] describes a fuzzy rule-based geophysical forecast system tointerpret rock types from their seismic characteristics.


References to geotechnical engineering

Al-Homoud, A. S., & Al-Masri, G. A. [1999], “CSEES: an expert system for analysis anddesign of cut slopes and embankments.” Environmental Geology, 39(1), 75–89.

Cheng, C. [1999], “Fuzzy optimal model for the flood control system of the upper and middlereaches of the Yangtze River.” Hydrological Sciences—Journal, 44(4), 573–582.

Ercanoglu, M., & Gokceoglu, C. [2002], “Assessment of landslide susceptibility for a landslide-prone area (north of Yenice, NW Turkey) by fuzzy approach.” Environmental Geology,41(6), 720–730.

Grima, A. M., & Verhoef, P. N. W. [1999], “Forecasting rock trencher performance using fuzzylogic.” International Journal of Rock Mechanics and Mining Sciences & Geomechanics

Abstracts, 36(4), 413–432.Hammah, R. E., & Curran, J. H. [1998], “Fuzzy cluster algorithm for the automatic identi-

fication of joint sets.” International Journal of Rock Mechanics and Mining Sciences &

Geomechanics Abstracts, 35(6), 889–905.Huang, G. H., Liu, L., Chakma, A., Wu, S. M., Wang, X. H., & Yin, Y. Y. [1999], “A hybrid

GIS-supported watershed modeling system: application to the Lake Erhai basin, China.”Hydrological Sciences—Journal, 44(4), 597–610.

Klose, C. D. [2002], “Fuzzy rule-based expert system for short-range seismic prediction.”Computers & Geosciences, 28(3), 337–386.

Teegavarapu. R. S. V., & Simonovic, S. P. [1999], “Modeling uncertainty in reservoir lossfunctions using fuzzy sets.” Water Resources Research, 35(9), 2815–2823.

4.3.5 Hydrocarbon exploration

Not surprisingly, this area has seen the greatest development of all so-called “softcomputing” technologies including neural networks, fuzzy logic, genetic algorithms,and data analysis [Sung, 1999]. Huang et al. [2001] employed a neural-fuzzy-geneticalgorithm for predicting permeability in petroleum reservoirs, whereas Finol & Jing[2002] tackled the prediction of permeabilities in shaly formations using a fuzzy rule-based model with well log responses as the input parameters. In a somewhat relatedpaper, Das Gupta [2001] applied fuzzy pattern recognition to well logs to determinelocations of coal seams. Janakiraman & Konno [2002] describe a fuzzy neuralnetwork designed to interpret rock facies in cross-borehole seismic exploration.

There have been a number of recent journal issues solely dedicated to this area andthey are an excellent means to gain access to the literature.

Computers and Geosciences [Vol. 26, No. 8, October 2000] produced an issueentitled “Applications of virtual intelligence to petroleum engineering.” This issuewas edited by Shahab Mohagheg and contained nine technical papers in additionto an introductory note. Most of the papers are dedicated to neural networks, manyof which rely heavily on fuzzy logic. In addition to the special issue cited above,numerous single papers on applications of soft computing to the Earth sciences areto be found in Computers and Geosciences.


The Journal of Petroleum Geology [Vol. 24, No. 4, October 2001] published athematic issue entitled “Field applications of intelligent computing techniques.” Thisissue was edited by Wong and Nikravesh and included five technical papers and anintroduction by the editors. Most of the papers were applications of neural networks.However, Wakefield et al. [2001] applied fuzzy logic to the biostratigraphic inter-pretation of mudstones in a North Sea oilfield and Finol et al. [2001] applied fuzzypartitioning to the classification and interpretation of remotely sensed resistivity andspontaneous potential wireline well logs.

The Journal of Petroleum Science and Engineering produced two special issuesdedicated to “Soft computing and Earth sciences” edited by Nikravesh, Aminzadeh,and Zadeh. Part one [Vol. 29, No. 3–4, May 2001] contained seven technical papersplus an introduction by the editors. Part two [Vol. 31, Nos. 2–4, November 2001]carried nine papers. These two special issues contain rather more variety of applica-tions of soft computing to the Earth sciences but still mostly deal with applicationsof neural networks to data analysis.

Nikravesh et al. [2003] is an extended version of these two theme issues.

References to hydrocarbon exploration

Das Gupta, S. P. [2001], “Application of a fuzzy pattern recognition method in boreholegeophysics.” Computers & Geosciences, 27(1), 85–89.

Finol, J. J., & Jing, X.-D. D. [2002], “Permeability prediction in shaly formations: the fuzzymodeling approach.” Geophysics, 67(3), 817–829.

Finol, J. J., Guo, Y. K., & Jing, X. D. [2001], “Fuzzy partitioning systems for electrofaciesclassification; a case study for the Maricaibo Basin.” Journal of Petroleum Geology, 24(4),441–458.

Huang, Y., Gedeon, T. D., & Wong, P. M. [2001], “An integrated neural-fuzzy-genetic-algorithm using hyper-surface membership functions to predict permeability in petroleumreservoirs.” Engineering Applications of Artificial Intelligence, 14(1), 15–21.

Janakiraman, K. K., & Konno, M. [2002], “Cross-borehole geological interpretation modelbased on a fuzzy neural network and geotomography.” Geophysics, 67(4), 1177–1183.

Mohagheg, S. (ed.) [2000], “Applications of virtual intelligence to petroleum engineering.”Computers and Geosciences, 26(8). (A special issue of the journal containing a number ofpapers dealing with soft computing.)

Nikravesh, M., Aminzadeh, F., & Zadeh, L. A. (eds.) [2001a], “Soft computing and EarthSciences: Part 1.” Journal of Petroleum Science and Engineering, 29(3–4). (A special issueof the journal containing a number of papers dealing with soft computing.)

Nikravesh, M., Aminzadeh, F., & Zadeh, L. A. (eds.) [2001b], “Soft computing and EarthSciences: Part 2.” Journal of Petroleum Science and Engineering, 31(2–4).

Nikravesh, M., Aminzadeh, F., & Zadeh, L. A. (eds.) [2003], Soft Computing and Intelligent

Data Analysis in Oil Exploration. Elsevier, Amsterdam.Sung, A. H. [1999], “Applications of soft computing in petroleum engineering.” SPIE Con-

ference on Applications and Science of Neural Networks, Fuzzy Systems, and Evolutionary

Computation II, 3812, 200–212.


Wakefield, M. I., Cook, R. J., Jackson, H., & Thompson, P. [2001], “Interpreting biostrati-graphical data using fuzzy logic; the identification of regional mudstones within the FlemingField, UK North Sea.” Journal of Petroleum Geology, 24(4), 417–440.

Wong, P. M., & Nikravesh, M. (eds.) [2001], “Field Applications of Intelligent ComputingTechniques.” Journal of Petroleum Geology, 24(4). (Aspecial issue of the journal containinga number of papers dealing with soft computing.)

4.3.6 Seismology

Deyi & Xihui [1985] presented the result of an international symposium on fuzzymathematics in earthquake research. Since the publication of this volume, applica-tions of fuzzy logic to seismology have focused on earthquake prediction, includingassessment of the magnitude of an earthquake (absolute amount of energy a seismicevent puts into the ground) and the pattern of surface disruption. Examples of theformer include Wang et al. [1997] and Wang et al. [1999]. The surface effects ofthe passage of seismic waves can vary dramatically in an urban area depending onthe material properties of the area, the type and depth to bedrock in the area, and theamplification of earthquake waves by the shape of the resonating deposits. Predictionof the ground motion at a site depends not only on these properties, but also on theproperties of the incident waves, and on their orientation. Papers by Muller et al.[1998, 1999] and by Huang & Leung [1999] developed fuzzy-based neural-networkapproaches toward models that integrated the properties of the waves and those ofthe ground to predict ground motion.

Locating the source site of a seismic wave arriving at a distant site has been one ofthe staples of seismology since its beginning. A number of deterministic models havebeen developed to make these calculations. Recently, Lin & Sanford [2001] describedan inversion technique wherein deviations between theoretical and observed arrivaltimes are assessed with fuzzy logic.

References to seismology

Deyi, F., & Xihui, L. (eds.) [1985], Fuzzy mathematics in earthquake researches, Pro-

ceedings of International Symposium on Fuzzy Mathematics in Earthquake Researches.Seismological Press, Beijing.

Huang, C., & Leung, Y. [1999], “Estimating the relationship between isoseismal area andearthquake magnitude by a hybrid fuzzy-neural-network method.” Fuzzy Sets and Systems,107(2), 131–146.

Lin, K., & Sanford, A. R. [2001], “Improving regional earthquake locations using a modifiedG matrix and fuzzy logic.” Bulletin of the Seismological Society of America, 91(1), 82–93.

Muller, S., Legrand, J.-F., Muller, J.-D., Cansi, Y., & Crusem, R. [1998], “Seismic eventsdiscrimination by neuro-fuzzy-based data merging.” Geophysical Research Letters, 25(18),3449–3452.


Muller, S., Garda, P., Muller, J.-D., & Cansi, Y. [1999], “Seismic events discrimination byneuro-fuzzy merging of signal and catalogue features.” Physics and Chemistry of the Earth

(A), 24(3), 201–206.Wang, W., Wu, G., Huang, B., Zhuang, K., Zhou, P., Jiang, C., Li, D., & Zhou, Y. [1997], “The

FAM (fuzzy associative memory) neural network model and its application in earthquakeprediction.” Acta Seismologica Sinica, 10(3), 321–328.

Wang, X.-Q., Zheng, Z., Qian, J., Yu, H.-Y., & Huang, X.-L. [1999], “Research on the fuzzyrelationship between the precursory anomalous elements and earthquake elements.” Acta

Seismologica Sinica, 12(4), 676–683.

4.3.7 Soil science and landscape development

Maps that depict the distribution of different soil types are a standard product of geo-logical and agricultural surveys throughout the world and have a variety of importantuses. Such uses include agricultural planning, conservation, input into watershedmodels, input into groundwater models, and the legal definition of wetlands. Soilsare three-dimensional mantles of weathered Earth materials with a complex bio-geochemical evolutionary history that depends on parent material, precipitation,temperature, topography, groundwater levels, etc. Traditional soil classifications andmaps produced from them ignored intergradations of soil types, both horizontally andvertically, and were based on widely spaced sampling pits. Although the actual micro-scopic structure of soils is difficult to assess, Moran & McBratney [1997] attemptedto develop a two-dimensional fuzzy model of soil element disposition.

In recent years soil science has been revolutionized by the development of geo-graphical information systems (GIS), improvements of remote sensing capabilities (topatches 10 meters or so across), and a fuzzy approach toward soil classifications andmapping [Davis & Keller, 1997; Galbraith et al., 1998; Galbraith & Bryant, 1998;Wilson & Burrough, 1999]. Examples of this approach applied to different areasinclude: Kollias et al. [1999] for an alluvial flood plain in western Greece; Triantafiliset al. [2001] for an area in New South Wales, eastern Australia; and Zhu et al. [2001]for areas in Wisconsin and Montana. In each of these studies, horizontal and verticalmeasurements of soil properties in test pits are employed to devise fuzzy soil classi-fication systems, i.e., systems wherein a point can belong to more than one soil type.In this way, intergradations among soil types are handled naturally and small areasof slightly different soil types within larger areas can be identified. These soil typesare then mapped onto remotely sensed images of an area at a resolution of blocksthat are approximately a few tens of meters on a side. Remote sensing commonlymeasures the intensity of various wavelengths of radiation off the Earth. A transferfunction (commonly fuzzy) is then developed to relate wavelength and intensity ofradiation to a soil type. Serandrei-Barbero et al. [1999] and Smith et al. [2000] usedthese techniques to interpret glacial features from the eastern Italian Alps and Greece,respectively.


Fuzzy logic has also been employed to understand geochemical aspects of soilsand recent stream sediments. Lahdenperä et al. [2001] also used fuzzy clustering toestablish relationships among glacial tills, bedrock, and elements measured in soilsof Finland. In addition, fuzzy logic has been incorporated into the revised universalsoil loss equation [Tran et al., 2002].

References to soil science and landscape development

Davis, T. J., & Keller, C. P. [1997], “Modelling and visualizing multiple spatial uncertainties.”Computers & Geosciences, 23(4), 397–408.

Galbraith, J. M., & Bryant, R. B. [1998], “A functional analysis of soil taxonomy in relationto expert system techniques.” Soil Science, 163(9), 739–747.

Galbraith, J. M., Bryant, R. B., & Ahrens, R. J. [1998], “An expert system of soil taxonomy.”Soil Science, 163(9), 748–758.

Kollias, V. J., Kalivas, D. P., & Yassoglou, N. J. [1999], “Mapping the soil resources of a recentalluvial plain in Greece using fuzzy sets in a GIS environment.” European Journal of Soil

Science, 50(2), 261–273.Lahdenperä, A.-M., Tamminen, P., & Tarvainen, T. [2001], “Relationships between geo-

chemistry of basal till and chemistry of surface soil at forested sites in Finland.” Applied

Geochemistry, 16(1), 123–136.Moran, C. J., & McBratney, A. B. [1997], “A two-dimensional fuzzy random model of soil

pore structure.” Mathematical Geology, 29(6), 755–777.Serandrei-Barbero, R., Rabagliati, R., Binaghi, E., & Rampini, A. [1999], “Glacial retreat in

the 1980s in the Breonie, Aurine and Pusteresi groups (eastern Alps, Italy) in Landsat TMimages.” Hydrological Sciences—Journal, 44(4), 279–296.

Smith, G. R., Woodward, J. C., Heywood, D. I., & Gibbard, P. L. [2000], “InterpretingPleistocene glacial features from SPOT HRV data using fuzzy techniques.” Computers &

Geosciences, 26(4), 479–490.Tran, L. T., Ridgley, M.A., Duckstein, L., & Sutherland, R. [2002], “Application of fuzzy-logic

modeling to improve the performance of the revised universal soil loss equation.” Catena,47(3), 203–226.

Triantafilis, J., Ward, W. T., Odeh, I. O. A., & McBratney, A. B. [2001], “Creation andinterpolation of continuous soil layer classes in the Lower Naomi Valley.” Soil Science

Society of America Journal, 65(2), 403–413.Wilson, J. P., & Burrough, P. A. [1999], “Dynamic modeling, geostatistics, and fuzzy clas-

sification: new sneakers for a new geography?” Annals of the Association of American

Geographers, 89(4), 736–746.Zhu, A. X., Hudson, B., Burt, J., Lubich, K., & Simonson, D. [2001], “Soil mapping using

GIS, expert knowledge, and fuzzy logic.” Soil Science Society of America Journal, 65(5),1463–1472.

4.3.8 Deposition of sediment

Modern sedimentary systems comprise volumes of the uppermost tens of meters oflithosphere and overlying hydrosphere and atmosphere where sediments accumulate


as, for example, on the delta of the Mississippi River or on the floor of DeathValley, California. Earth scientists have limited knowledge of the physical, chemical,and biological processes that control sediment accumulation in modern sedimen-tary systems. This presents a difficulty insofar as analogous ancient sedimentarysystems are where the sediments and sedimentary rocks, which hold much of thedirect, long-term evidence of the history of the biosphere, atmosphere, hydrosphere,and lithosphere of this planet, accumulated. In order to infer such important infor-mation as long- and short-term variations in geochemical cycles, ecosystems, andclimate, we seek to recover from ancient sedimentary deposits just these recordsof the physical, chemical, and biological processes that operated on those ancientsurfaces. Our poor understanding of modern depositional processes is due to: thelarge sizes of modern sedimentary systems (102–105 km2); difficulties in instrumen-tation (especially during rare events such as hurricanes and floods); restrictions ofobservations on active processes to a few hundred years; and labor-intensive datagathering. Fuzzy logic was initially introduced by Nordlund [1996, 1999] to over-come the computational difficulties of sedimentary modeling introduced by theseproblems. FUZZIM [Nordlund, 1999] is a share-ware program that replaces sed-imentary physics with common-sense rules based on hard and soft informationdeveloped by sedimentologists over the past 100 years. Edington et al. [1998],Parcell et al. [1998], and Demicco & Klir [2001] have developed other fuzzyrule-based models of ancient deposition. Urbat et al. [2000] developed a fuzzy clus-tering classification of diagenesis of deep-sea sediments due to flux of hydrothermalfluids.

References to deposition of sediment

Demicco, R. V., & Klir, G. J. [2001], “Stratigraphic simulations using fuzzy logic to modelsediment dispersal.” Journal of Petroleum Science and Engineering, 31(2–4), 135–155.

Edington, D. H., Poeter, E. P., & Cross, T. A. [1998], “FLUVSIM; a fuzzy-logic forwardmodel of fluvial systems.” Abstracts with Programs—Geological Society of America Annual

Meeting, 30, A105.Nordlund, U. [1996], “Formalizing geological knowledge—with an example of modeling

stratigraphy using fuzzy logic.” Journal of Sedimentary Research, 66(4), 689–712.Nordlund, U. [1999], “FUZZIM: forward stratigraphic modeling made simple.” Computers &

Geosciences, 25(4), 449–456.Parcell, W. C., Mancini, E. A., Benson, D. J., Chen, H., & Yang, W. [1998], “Geological

and computer modeling of 2-D and 3-D carbonate lithofacies trends in the Upper Jurassic(Oxfordian), Smackover Formation, Northeastern Gulf Coast.” Abstracts with Programs—

Geological Society of America Annual Meeting, 30, A338.Urbat, M., Dekkers, M. J., & Krumsiek, K. [2000], “Discharge of hydrothermal fluids through

sediment at the Escanaba Trough, Gorda Ridge (ODP Leg 169): assessing the effects on therock magnetic signal.” Earth and Planetary Science Letters, 176(3–4), 481–494.


4.3.9 Miscellaneous applications

There have been a number of general papers advocating the use of fuzzy logic in thegeological sciences. These include Fang [1997] and Fang & Chen [1990]. Bárdossy &Duckstein [1995] is a useful introductory text in which various applications of fuzzyrule-based modeling in biological and engineering applications as well as geosciencesare developed. Also, Tamas D. Gedeon edited a special issue of the International

Journal of Fuzzy Systems [Vol. 4, No. 1, 2002] with a number of papers on applicationsof soft computing in geology.

In addition to these general references there have been the following specific appli-cations of fuzzy logic to geological problems outside of the main areas listed above.Cagnoli [1998] suggested uses of fuzzy logic in the study of volcanoes. Van Wijk &Bouten [2000] applied a fuzzy rule-based model to simulation of latent heat fluxesof coniferous forests. Schulz et al. [1999] used fuzzy set theory to evaluate ther-modynamic parameters in aqueous chemical equilibrium calculations. Kruiver et al.[1999] used fuzzy clustering of paleomagnetic measurements on deep-sea sedimentsas a proxy for estimating orbital forcing of climate over the last 276,000 years. Finally,Pokrovsky et al. [2002] used fuzzy logic to study the meteorological impacts of airpollution in Hong Kong.

References to miscellaneous applications

Bárdossy, A., & Duckstein, L. [1995], Fuzzy Rule-Based Modeling with Applica-

tions to Geophysical, Biological and Engineering Systems. CRC Press, Boca Raton,FL.

Cagnoli, B. [1998], “Fuzzy logic in volcanology.” Episodes, 21(2), 94–96.Fang, J. H. [1997], “Fuzzy logic and geology.” Geotimes, 42, 23–26.Fang, J. H., & Chen, H. C. [1990], “Uncertainties are better handled by fuzzy arithmetic.”

American Association of Petroleum Geologists Bulletin, 74, 1228–1233.Gedeon, T. D. (ed.) [2002], “Soft computing in geology.” International Journal of Fuzzy

Systems, 4(1). (Special issue.)Kruiver, P. P., Kok, Y. S., Dekkers, M. J., Langereis, C. G., & Laj, C. [1999], “A psuedo-

Thellier relative palaeointensity record, and rock magnetic and geochemical parametersin relation to climate during the last 276 kyr in the Azores region.” Geophysical Journal

International, 136, 757–770.Pokrovsky, O. M., Kwok, R. H. F., & Ng, C. N. [2002], “Fuzzy logic approach for descrip-

tion of meteorological impacts on urban air pollution species: a Hong Kong case study.”Computers & Geosciences, 28(1), 119–127.

Schulz, K., Huwe, B., & Peiffer, S. [1999], “Parameter uncertainty in chemical equilibriumcalculations using fuzzy set theory.” Journal of Hydrology, 217, 119–134.

Van Wijk, M. T., & Bouten, W. [2000], “Analyzing latent heat fluxes of coniferous forests withfuzzy logic.” Water Resources Research, 36, 1865–1872.


4.4 Concluding Note: Quo Vadis

The papers cited in each category above by no means comprise a complete bibliogra-phy of aspects of use of fuzzy logic in the geological sciences. One final observationis in order. There seems to be a major trend in the use of fuzzy logic in the geologicalsciences. Initially, fuzzy sets were used to begin to capture the continuous nature ofgeologic data, and various techniques were developed to use fuzzy sets in previouslydeveloped deterministic models of geological phenomena. More and more, fuzzyrule-based models are beginning to supersede the older deterministic models. Thistrend will no doubt continue into the future.

Chapter 5 Applications of Fuzzy Logic toStratigraphic Modeling

Robert V. Demicco

5.1 Introduction 1215.2 Fuzzy Logic and Stratigraphic Models 1235.3 Death Valley, California 1245.4 Modeling Depositional Processes at a Delta Mouth 1335.5 Multidistributary Deltaic Deposition with Variable Wave and Long-Shore

Drift Regimes 1375.6 Future Developments 1475.7 Conclusions 148References 149

5.1 Introduction

Two-dimensional and three-dimensional computer-generated forward models of sed-imentary basin filling are increasingly important tools for research in applied andtheoretical geological sciences [see Tetzlaff & Harbaugh, 1989; Franseen et al., 1991;Slingerland et al., 1994; Wendebourg & Harbaugh, 1996; Harff et al., 1999; Harbaughet al., 1999; Syvitski & Hutton, 2001; Merriam & Davis, 2001]. These models producesynthetic stratigraphic cross-sections that are important for two reasons. First, theygive us a predictive picture of the subsurface distribution of petrophysical properties(such as porosity, permeability, seismic velocity, etc.) that are useful in petroleumexploration, secondary petroleum recovery, groundwater exploitation, groundwa-ter remediation, and other geotechnical applications. Second, synthetic stratigraphicmodels increase our theoretical understanding of how sediment accumulation variesin time and space in response to external factors (such as eustasy and tectonics)and internal factors (such as compaction, isostatic adjustments, and crustal flexuraladjustments made in response to tectonic loading and sedimentary accumulation) thatare known, or suspected, to influence sedimentation patterns. Physically reasonablealgorithms for eustasy, compaction, isostasy, and crustal flexure are common com-ponents of sedimentary models and can be modeled either deterministically or withfuzzy logic. For example, in Section 3.4.1 a fuzzy compaction algorithm is described.However, the main focus of this chapter is on using fuzzy logic to simulate sediment

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122 5 Applications of Fuzzy Logic to Stratigraphic Modeling

erosion, sediment transportation, and sediment accumulation within a forward model[Nordlund, 1996]. Wendebourg & Harbaugh [1996] collectively refer to this as the“sedimentary process simulator.” Models of sediment production based on fuzzyinference models are presented for reef growth in Section 3.4.2 and for biochemicalproduction of carbonate sediment on a shallow platform in 3.4.4.

Most forward models of basin filling focus on shallow-water shelf accumulation.For siliciclastic settings, a significant source of sediment input, i.e. a river delta, isusually an important component of the model. When we consider carbonate deposi-tional systems, we are confronted by the in situ formation of the sediments themselvesboth as reefs [Bosscher & Schlager, 1992], and as bank-interior sediments [Broecker& Takahashi, 1966; Morse et al., 1984]. In both siliciclastic and carbonate shallowmarine systems, waves, wave-induced currents, tidal currents, and storm-inducedwaves and currents lead to ever-changing patterns of sediment erosion, transportation,and accumulation. Buoyant plumes become important where deltas are an importantcomponent of a model.

Coastal oceanographic modelers have made great strides in dealing with the com-plexities introduced by the elements listed above. Acinas & Brebbia [1997] andKomar [1998] provide details and examples of various aspects of such models. Con-sider the following steps that would be necessary to model sedimentation in a coastalarea. First, we would need to develop a numerical solution (usually finite differenceor finite element) of the fundamental, dynamical, physical equations of circulationforced by the average or “fair weather” winds, waves, tides, and, if present, buoyantriver plumes. This step becomes immediately complicated if we suspect that stormevents are an important component of the coastal sedimentation regime, insofar as wewould then need a separate “storm” circulation model. The second step would be touse the results of the circulation model to calculate bed shear stress along the bottomof a circulation model. Finally, the bed shear stresses would then be used as input tosolve the temporal and spatial terms in bedload and suspended load sediment transportequations. These would, in turn, give us the desired sediment erosion, transportation,and deposition.

If we were to use the approach outlined in the preceding paragraph in a stratigraphicbasin filling model, we would confront at least three major difficulties. First, thesolutions of such models are site specific and depend on rigorous application ofboundary conditions, initial conditions, and wave and tidal forcing functions overa discrete domain. It is important to note that literally hundreds of constants in thecirculation models, shear stress calculations, and bedload transport equations wouldneed to be specified. Second, we would need to solve the flow and transport equationsat discrete time intervals over hundreds of thousands to millions of years as the basinslowly fills. Thus we would need to consider the duration of the individual time stepsof the stratigraphic model. The process-response models outlined above have validityfor durations of tens to (at most) hundreds of years. These are very short in comparisonto basin-filling models. Basin-filling models typically operate for time scales of 105 to

5.2 Fuzzy Logic and Stratigraphic Models 123

106 years. Any long-term changes that we would wish to incorporate into our model(such as a regional climate change) or natural, evolutionary changes in the model(such as changes in antecedent topography and changed boundary conditions) wouldentail essentially starting from scratch with a new circulation model. The third factoris simply the scale of the numerical computations, memory storage, etc., involved insuch a modeling project. We would have to run such a model many scores of timesto “tune” it so that we get realistic output.

For the reasons outlined in the preceding paragraph, sedimentary process simula-tors are the crudest parts of stratigraphic models. Siliciclastic sedimentary processsimulators typically either employ the diffusion equation to represent sediment dis-persal or use linear approximations of more complicated sediment dispersal. The2-dimensional code of Bosence & Waltham [1990] and Bosence et al. [1994], the“Dr. Sediment” code of Dunn [1991], the 2-dimensional alluvial architecture code ofBridge & Leeder [1979], the 3-dimensional update of that code by Mackey & Bridge[1995], and the “CYCOPATH 2D” code of Demicco [1998] all use such an approach.Models such as STRATAFORM [Nittrouer & Kravitz, 1996], SEDFLUX [Syvitski& Hutton, 2001], and the SEDSIM models [Tetzlaff & Harbaugh, 1989; Wende-bourg & Harbaugh, 1996; Merriam & Davis 2001] also use linear approximations(albeit to more realistic “physically based” circulation and sedimentation transportequations). Although these models have been successful, they can be computationallyquite complex.

5.2 Fuzzy Logic and Stratigraphic Models

In an effort to overcome the complexities described above, we have been developingfuzzy logic models of sediment production, erosion, transportation, and depositionbased on qualitatively and quantitatively defined observational rules. Nordlund [1996]and Fang [1997] suggested that fuzzy logic could be used to overcome some of thedifficulties inherent in modeling sediment dispersion. There is a wealth of obser-vational data on flow and sediment transport in the coastal zone, in river systems,on carbonate platforms, and in closed basin settings. Nordlund [1996] refers to thisas “soft” or qualitative information on sedimentary dynamics. However, we alsohave a fair amount of quantitative information on some sedimentary processes. Forexample, Section 3.4.4 describes a fuzzy inference model for the volumetric pro-duction of lime sediment per year on different areas of carbonate platforms, basedon the geochemical measurements of Broecker & Takahashi [1966] and Morse et al.[1984]. Examples of qualitative information would be “beach sands tend to be wellsorted and are coarser than offshore sands,” or “carbonate sediment is produced inan offshore carbonate ‘factory’ and is transported and deposited in tidal flats.” Suchstatements carry information, but are not easily quantified. Indeed, these types ofqualitative statement are commonly the exact kind of information that is obtained


by studies of ancient sedimentary sequences. Moreover, with the development of“seismic stratigraphy” and “sequence stratigraphy,” applied and academic geologistshave both moved into an arena where there is commonly a complex blend of “hard”and “soft” information. Hard data might include seismic (or outcrop-scale) geometricpatterns of reflectors or bedding geometries, whereas soft information would includedescription of rock types, interpretations of depositional settings, and their positionswithin “system tracts” [see Vail et al., 1977; Wilgus et al., 1989; Schlager, 1992,1999; Loucks & Sarg, 1993; Emery & Myers, 1996].

Fuzzy logic allows us to formalize and treat such information in a rigorous, math-ematical way. It also allows quantitative information to be treated in a more natural,continuous fashion. The purpose of this chapter is to present a number of simulationsof increasing complexity, where we have used fuzzy logic to model sediment dis-persal in 3-dimensional stratigraphic models wherein sea level changes, subsidence,isostasy, and crustal flexure are modeled using conventional mathematical represen-tations [Turcotte & Schubert, 1982; Angevine et al., 1990; Slingerland et al., 1994].The results, summarized here along with those of the model FLUVSIM [Edingtonet al., 1998] and the modeling of the Smackover formation described by Parcell et al.[1998], suggest that fuzzy logic may be a powerful and computationally efficientalternative technique to numerical modeling for the basis of a sedimentary processsimulator. It has the distinct advantage in that models based on fuzzy logic are robust,easily adaptable, computationally efficient, and can be easily altered internally toallow many different combinations of input parameters to be run in a sensitivityanalysis in a quick and efficient way.

The next three sections of this chapter describe three sets of models where fuzzylogic can be used to address some of the problems described above. The first set ofexamples models sedimentation in Death Valley, California. The second set of modelscompares a numerical solution and a fuzzy logic model applied to sedimentation ina distributary mouth bar of the Mississippi River. Finally, the last set of modelsillustrates deltaic sedimentation under variable wave and long shore current regimes.

5.3 Death Valley, California

Death Valley is an arid closed basin located in the southwestern United States. Thebasin is a half graben approximately 15 km across and 65 km long. The center of thebasin is a nearly flat complex of saline pans and playa mudflats nearly 100 m belowsea level. Gravel alluvial fans radiate from streams along steep mountain fronts onthe east side of the basin where the active border fault is inferred. These fans aresteep and grade out to the floor of the basin over a few kilometers. The mountainfront on the west side of the basin is gentler and alluvial fans issuing from streamson this side of the basin have a lower gradient than those on the east side, extendingnearly halfway across the basin floor. In 1991, a 175 m deep core was extracted from

5.3 Death Valley, California 125

Figure 5.1 Data from the Death Valley core [Lowenstein et al., 1999]. (a) Sedimentary envi-ronments versus age. (b) Paleotemperatures measured from fluid inclusions in the core. Linedenotes smoothed running average. (c) Paleoprecipitation inferred from the core and environsof Death Valley.

a salt pan in the central portion of the basin [Roberts & Spencer, 1995; Li et al., 1996;Lowenstein et al., 1999]. The core sampled basin floor sediments deposited duringthe last 191,000 years and these data are summarized in Figure 5.1. Figure 5.1a is aplot of thicknesses of the deposits of the four different environments found in the coreplotted against their ages as interpolated from U-Series dates [Ku et al., 1998]. Thepreserved deposits include: (1) disrupted, thin-bedded muds deposited in desiccatedplaya mudflats; (2) chemical sediments interlayered with aluminosilicate muds thatwere deposited in brine-saturated saline pans; (3) chemical sediments (principallyhalite) deposited in perennial saline lakes; and (4) fossil-rich aluminosilicate mudswith striking millimeter-thick laminae deposited in deep perennial lakes that werefresh. Figure 5.1b is a smoothed curve through paleotemperatures measured frombrine inclusions preserved in primary halite deposits, principally from saline lakesand saline pan deposits [Lowenstein et al., 1999]. Figure 5.1c is an interpreted recordof paleoprecipitation [Lowenstein, personal communication] based on a number of


proxies from the core and surrounding areas (dated lacustrine tufas, known lakeshorelines, pollen records, etc.) and the δ18O composition of the sulfate minerals inthe core [Yang et al., 1999].

Demicco & Klir [2001] described a fuzzy rule-based, three-dimensional modelof the last 190 kiloyears (ky) of sedimentation in Death Valley using the core datashown in Figure 5.1. That model is briefly described here as a starting point. Themodel was a grid 15 km across and 65 km long represented by approximately 1900active cells each 0.5× 1.0 km in size. The modern topography was the starting pointfor elevation at each cell in the model. Subsidence of the model was−0.2 m/ky alongthe edges of the model and increased to −1 m/ky along the steep eastern margin ofthe basin halfway down the axis of the basin. The model employs fuzzy “if–then”rules to model both alluvial fan input along the sides of the basin and deposition onthe basin floor.

Two fuzzy logic systems controlled deposition on the floor of the basin: one sys-tem generated the sediment type and the other generated the sediment thickness.The fuzzy logic system that determines the type of sediment deposited on the basinfloor is briefly described here. (Many other examples of a fuzzy inference approachto geologic problems are illustrated in Chapter 3.) The input variables in both ofthese fuzzy logic systems were: (1) the temperature signal (Figure 5.1b); and (2) theprecipitation signal (Figure 5.1c) determined from the core. Figure 5.2a shows theinput variable temperature here represented by two fuzzy sets, low temperatures andhigh temperatures, whereas Figure 5.2b shows the variable precipitation here alsorepresented by two fuzzy sets, low precipitation and high precipitation. The outputvariable here is the environment of deposition, here represented by four fuzzy sets:playa, saline pan, saline lake, and freshwater lake. The membership functions thatdescribe the fuzzy sets in Figure 5.2 are simple trapezoids or triangles. The “rules”governing the basin floor sedimentary environment are straightforward, make sensein terms of Figure 5.1, and are easily incorporated into a fuzzy inference system. Therules are:

(1) If temperature is low and precipitation is low, then the basin floor environmentis saline pan.

(2) If temperature is low and precipitation is high, then the basin floor environmentis saline lake.

(3) If temperature is high and precipitation is low, then the basin floor environmentis playa.

(4) If temperature is high and precipitation is high, then the basin floor environmentis lake.

The standard (so-called “Mamdani”) interpretation [Mamdani & Assilian, 1975] ofthese rules is shown in Figure 5.3 for input values of temperature = 29◦C andprecipitation= 165 mm/y. The left column represents the input variable temperature,


Figure 5.2 (a) Membership functions describing the fuzzy sets “low” and “high” for theinput variable temperature over the domain range 23 to 34◦C. (b) Membership functionsdescribing the fuzzy sets “low” and “high” for the input variable precipitation over the domain100 to 300 mm/y. (c) Membership functions describing the fuzzy sets “playa,” “saline pan,”“saline lake,” and “lake” for the output variable environments over the domain range 1 to 4.Membership functions were adjusted by hand to produce the curve in Figure 5.4a.

the center column represents the input variable precipitation, and the right columnrepresents the output variable environment. From top to bottom, the rows representrules (1) to (4) as listed above. The input variables (29◦C and 165 mm in this example)are evaluated simultaneously for each rule and a truth value= degree of membershipof the input variable in each of the potential input sets is calculated. In this case, 29◦

“fires” all four rules, generating truth values of approximately [0.2, 0.2, 0.8, 0.8] forrules one through four respectively whereas the truth values for an input value of165 mm are approximately [0.75, 0.3, 0.75, 0.3]. For each rule, the truth value of thelower of the two inputs truncates the membership function of the appropriate outputvariable. Here for example in rule 1, the truth value of 0.2 truncates the membership


Figure 5.3 Standard (“Mamdani”) interpretation of the “if–then” rules controlling the basinfloor environments for the input variables 29◦C and 165 mm/y. The input variables are evaluatedfor each pair of temperature and precipitation values input, and a truth value = degree ofmembership of the input variable in each of the potential input sets is calculated. These truthvalues truncate the membership functions of the appropriate output variable. For each pair ofinput variables, the truncated membership functions of the output variable are summed, andthe centroid of the appropriate curve is taken as the “defuzzified” output value, here 2.25.

function salt pan at 0.21. For each combination of temperature and precipitation, themaximum of the truncated membership functions of the output variable is taken, andthe centroid of the appropriate curve is taken as the “defuzzified” output value (2.25 inthis example). In Figure 5.4, the curve shows this fuzzy inference system evaluatedover the age range 0 to 190 ky for every appropriate input combination of temperatureand precipitation (Figure 5.1). This curve is compared to the environments recordedin the core (rectilinear line).

In robot control algorithms, where fuzzy logic was first developed, systems couldself-adjust the shapes of the membership functions and set boundaries until therequired task was flawlessly performed. This aspect of fuzzy systems, commonlyfacilitated via the learning capabilities of appropriate neural networks [Kosko, 1992;Klir & Yuan, 1995; Lin & Lee, 1996; Nauck et al., 1997] or by genetic algorithms

1The lower value is chosen here because the conjunction “and” implies a fuzzy intersection. If theconnector had been the conjunction “or” then a fuzzy union of the sets is implied and the higher of thevalues would be used. A range of conjunctions is available (see Sections 2.5 and 3.4.4).


Figure 5.4 (a) Direct comparison of the Mamdani fuzzy inference model of basin floor envi-ronments (curved line) with the geologic history of environments found in the Death Valleycore (rectilinear line). (b) Direct comparison of the machine-adjusted Takagi–Sugeno modelof basin floor environments (curved line) with the geologic history of environments found inthe Death Valley core (rectilinear line). The Takagi–Sugeno neuro-fuzzy inference system isdescribed in the text and in Figures 5.5 and 5.6.

[Sanchez et al., 1997; Cordón et al., 2001], is one of their great advantages overnumerical solution approaches. Here we illustrate an application of this self-adjustingcapability of fuzzy inference systems by employing the adaptive neuro-fuzzy systemthat is included in the Fuzzy Logic Toolbox of the commercial high-level languageMATLAB© to generate a fuzzy logic system for the Death Valley core data. Themodel is based on the premise that the deposits that accumulated on the floor ofDeath Valley were directly related in some way to a combination of temperature andrainfall. This is not an unreasonable interpretation for closed basin deposits [Smoot& Lowenstein, 1991] and is a prerequisite for using sedimentary records of lakes andother continental environments for research into paleoclimates.

The MATLAB© adaptive neuro-fuzzy system is a program that utilizes learningcapabilities of neural networks for tuning parameters of fuzzy inference systems onthe basis of given data. The program implements a training algorithm employingthe common backpropagation method based on the least square error criterion [seeKlir & Yuan, 1995, Appendix A].

The fuzzy logic system used in the previous example was the so-called “Mamdani”fuzzy inference system. In the next example, we use an alternative approachto formalizing fuzzy inference systems developed by Takagi & Sugeno [1985].


The so-called Takagi–Sugeno-type fuzzy logic system employs a single “spike” as theoutput membership functions. Thus, rather than integrating across the domain of thefinal output fuzzy set, a Takagi–Sugeno-type fuzzy inference system employs onlythe weighted average of a few data points. There is a clear computational advantage inemploying a Takagi–Sugeno fuzzy logic system. Moreover, the adaptive neuro-fuzzyinference engine of MATLAB© only supports Takagi–Sugeno-type output member-ship functions. Detailed examples of Takagi–Sugeno fuzzy logic systems can be foundin Section 3.4.3 and the reader is referred to this section for further details.

Figure 5.5 shows antecedent membership functions for the input variables “tem-perature” and “rainfall” used as input to the training algorithm. The training algorithm

Figure 5.5 Antecedent membership functions for the variable temperature (a) and precipitation(b) used as input to “tune” an adaptive neuro-fuzzy inference model relating temperature andrainfall to sedimentary environments.


Figure 5.6 Solution surface of the adaptive neuro-fuzzy logic model generated fromMATLAB©.

also requires two separate arrays of data for “training” and “verifying.” These dataarrays comprise triplets of temperature, rainfall, and resultant environment. The train-ing algorithm systematically adjusts the output functions and ultimately generates ninelinear output functions. Figure 5.6 is the surface generated by this fuzzy logic systemand Figure 5.4b is a direct comparison between our modeling results (curved line) andthe original depositional environment data (rectilinear line), both plotted against age.Clearly, the “trained” Takagi–Sugeno fuzzy inference system does a superior job inmodeling the time history of environments of deposition on the floor of Death Valley.

Figure 5.7 shows two synthetic stratigraphic cross-sections of the original model[Demicco & Klir, 2001] rerun with all conditions being the same except for thefuzzy logic systems that control deposition in the center of the basin. In this revisedmodel, the sedimentary environment in the basin center is controlled by the machine-developed fuzzy logic system described above. Figure 5.7a is a cross-section acrossthe basin in a west–east orientation, whereas Figure 5.7b is a cross-section along thenorth–south long-axis of the basin. As in the original model, alluvial input fromthe sides arose from the canyon locations at the heads of the main modern allu-vial fans around the basin margin. Deposition on the alluvial fan to playa mudflat


Figure 5.7 Synthetic stratigraphic cross-sections from a three-dimensional model of DeathValley sedimentation over the past 190 ky. The model uses the adaptive neuro-fuzzy inferencemodel to control the environment of the basin floor. AMamdani fuzzy inference system controlsthe thickness of the sediments deposited in the center of the basin, and two Mamdani fuzzyinference systems control the amount and caliber of the sediment deposited on alluvial fansalong the edges of the basin. (a) Cross-section perpendicular to the long axis of the valley inthe center of the model (approximate location of the Death Valley core). Orange and yellowson the margins indicate gravels and coarse sands of the alluvial fan systems which grade intolight blue playa muds toward the low floor of the basin. Basin floor sediments include: (1) lakemuds (which onlap the alluvial fans in the lower portions of the model) indicated by deep blue;(2) saline pan deposits indicated by reds; and (3) saline lake deposits denoted by yellows. (b)Cross-section parallel to the long axis of the valley through the point of maximum sedimentaccumulation. (See also color insert.)

drainage-ways was modeled by two Mamdani fuzzy logic systems. The input vari-ables to both models were distance from canyon mouth and slope of the sedimentsurface in each cell. These input variables controlled the particle size of the depositand the thickness of the alluvial deposits in each cell. In the synthetic cross-valleysection (Figure 5.7a) short steep fans on the eastern side of the basin comprise coarsergravels (orange) and contrast with the long, lower gradient fans on the western side

5.4 Modeling Depositional Processes at a Delta Mouth 133

of the basin that are generally composed of finer sediment (yellow and green). Thealluvial input into the basin ultimately leads to the deposition of playa muds in thefloor of the basin. The basin floor sediment is color-coded: deep freshwater lake andplaya mud flats are blue, saline pan is red, and saline lake is shades of yellow andorange. Playa mud flats develop in the floor of the basin when the chemical or lacus-trine sediments are minimal. In these revised models, the lakes lap up and over thealluvial fan deposits.

5.4 Modeling Depositional Processes at a Delta Mouth

A significant portion of the geologic record comprises siliciclastic sedimentary rocks(sandstones and shales) deposited in shallow marine settings. It is axiomatic that anappreciable thickness of such material requires a source of sediment, i.e., a river. Thus,deltaic deposits are common components of the geologic record even though waveand tidal current processes in the basin may redistribute the material into beachesor tidal flats [Reading & Collinson, 1996]. The Mississippi River Delta complex isone of the best-studied deltas on Earth and serves as an archetypical “river domi-nated” delta [Gould, 1970; Wright, 1978; Reading & Collinson, 1996]. The modernMississippi River Delta has three main distributary channels that are suggestive ofa “bird’s foot” in map view. Southwest Pass is the most studied distributary. Eachday, the Mississippi River delivers over 1 million tons of sediment, mostly silt andclay but with appreciable amounts of fine sand, to the Gulf of Mexico. The followingdiscussion of deposition at the mouth of Southwest Pass is based upon Gould [1970],Wright & Coleman [1971], Coleman & Wright [1975], and Wright [1978]. South-west Pass Channel shallows from a depth of approximately 12 m to approximately5 m where it reaches the mouth of the pass. Seaward of the pass mouth is a sand shoalknown as the “distributary mouth bar” that is some 8 km wide and extends some 10to 15 km seaward. Depths over the shoal are as shallow as 3 to 5 m (except wheredredging maintains a deeper ship channel) and are as much as 100 m at the distal endof the bar. The size of the sedimentary particles that comprise the bar grades fromfine sand on the bar crest to a mixture of sand, silt, and clay mid-bar, to clay at thesides and in front of the bar. The system builds out about 80 m per year, leaving asubsurface, pillow-shaped sand deposit encased in mud (a “bar finger” sand deposit).

The main processes of deposition are tied to the buoyant plume that emanatesfrom the mouth of the distributary and include bedload deposition of the fine sandand suspension settle-out of the silt and clay. The clay deposition is affected byflocculation that coagulates clay particles as they pass from fresh water to high ionicstrength seawater. The plume that issues from the distributary mouth and its associatedsedimentary processes has two basic modes: (1) low to average flows; and (2) floodflows. During times of average to low discharge, a salt-water wedge intrudes upSouthwest Pass 10 km or so and the buoyant, sediment-laden plume is out of contactwith the bottom. During the spring freshet and other floods, the salt wedge is driven


out of the channel and the fresh-water plume detaches from the bottom just seawardof the mouth bar. In this mode the bedload sands are flushed from the pass out ontothe distributary mouth bar and this process apparently accounts for the main sanddeposits of the mouth bar. The plume itself has a complicated internal circulationand is visible in satellite photographs extending up to 20 km or so off the mouth ofSouthwest Pass.

The scenario outlined above is supported by some limited oceanographic mea-surements but, as yet, no detailed numerical models of circulation and sedimentationfor Southwest Pass exist. Instead, as is typical of many coastal sedimentary studies,there is the combination of hard and soft data outlined above that is summarized inFigures 18 and 20 in Gould [1970] and particularly in Figure 20 in Wright [1978]. It isjust such hybrid data sets that fuzzy logic excels at modeling. The stratigraphic modelSEDFLUX 1.0C contains subroutines to simulate bedload and suspended load sedi-mentation from a buoyant plume [Syvitski & Hutton 2001]. However, this programassumes a fairly deep receiving basin and no appreciable wave or current action inthe basin. Application of the plume routines in SEDFLUX to Southwest Pass, usingvalues of discharge, flow velocity, and sediment load appropriate for the MississippiRiver at South Pass, produces a very laterally restricted suspension settle-out plume(Figure 5.8a) with most settle-out just seaward of the distributary mouth and extendingas much as 35 km offshore. SEDFLUX simulates bedload deposition by uniformlydistributing the bedload in front of the plume over cells in a specified depth range andis not shown here (all sediment is deposited by settle-out). The shape of the sedimentplume is dictated by the analytical routines in SEDFLUX and is based on momentumdissipation in a turbulent jet as modeled by Albertson et al. [1950]. The lateral extentof the plume is dictated by the width of the river mouth, and the average velocity at themouth of the river dictates the downstream length of the plume. The other adjustableparameters are the settling times of the sediments. The constants used in SEDFLUXare empirical and take into account the increased settling out of clay due to floccula-tion (Figure 5.9). One of the distinct advantages of SEDFLUX, however, is its strictadherence to conservation of mass insofar as the amount of sediment deposited inthe offshore plume is the amount that enters the model from the river mouth. Even acursory comparison of the sediment plume geometry given by the plume routines inSEDFLUX with Figures 18 and 20 in Gould [1970] and Figure 20 in Wright [1978]shows that it is very different than the observed distribution of sediment.

A fuzzy logic approach to this problem would start with the combined soft andhard information described above. In this case the target would be to reproduce thethickness of sand, silt, and mud and, in particular, the proper proportions of sand, silt,and mud on the distributary mouth bar over some specified time interval (e.g., 1 dayor 1 year). The targets of this simulation would be Figures 18 and 20 in Gould [1970]and Figure 20 in Wright [1978]. We have developed a Takagi–Sugeno model fromthese figures (Figures 5.8b, 5.8c). In this model, radial coordinates are employed andthe inputs for this model are the radial distance from the river mouth (Figure 5.10a)

5.4 Modeling Depositional Processes at a Delta Mouth 135

Figure 5.8 (a) Geometry and thickness of suspended sediment plume off Southwest Passproduced by the plume subroutine of SEDFLUX 1.0 C for values appropriate to the MississippiRiver at South Pass. The plume is for suspended sediment only; the bedload sediment would beevenly distributed in an arbitrary array of cells off the river mouth designated by the operator,and is not shown here. Horizontal and vertical scales are in km, contour scale is in mm/day. (b)Geometry and thickness of sediment plume off Southwest Pass produced by the fuzzy inferencemodel with the same input values; however, bedload is included in this model. Distances arein kilometers; contours are in mm/day. (c) Mixtures of grain sizes (fuzzy sets) produced by themodel: 0.9 = clean, well-sorted fine sand; 0.7 = silty sand; 0.5 = sandy silt; 0.3 = silt; and0.1 = clay. Compare with Figure 18 in Gould [1970].

and the absolute angle left and right from the centerline of the river (Figure 5.10b).Distance from the river mouth is divided into five fuzzy sets: (1) in channel; (2) veryclose; (3) close; (4) far; and (5) very far. Angular separation from the centerline isdivided into five triangular fuzzy sets: (1) far negative; (2) negative; (3) centerline;(4) positive; and (5) far positive. There are five linear output functions: (1) none (0);(2) very little (0.2); (3) little (0.4); (4) some (0.6); and lots (0.8). There are 17 rulesto the system (Table 5.1).


Figure 5.9 Contour maps of the various size fractions that make up the sediment plumedepicted in Figure 5.8a—note change of scales. Horizontal and vertical scales in km. In eachdiagram the contours are (from furthest out inward toward river mouth) 0.05 mm, 0.2 mm,0.4 mm, 0.6 mm, and 0.8 mm. Thickness and extent of the plumes for the various grain sizesare controlled by user input sediment amounts and settling rates. These size fractions wouldhave to be integrated in order to generate sediment distribution maps such as Figure 5.8c.

Figure 5.8b shows the thickness of sediment deposited from a fuzzy logic modeldeveloped for Southwest Pass Distributary Mouth Bar. In this model, mass conserva-tion is obtained in the following way. The relative volume of the sediment depositedin a given time step is computed by multiplying the area of each cell by the relativethickness given in the model. This volume is divided into the volume of sedimententering the model during a time step, adjusted by the porosity of the deposits. Therelative thickness is then multiplied by the calculated factor and mass balance isobtained. Figure 5.8c is the sediment type deposited. As the output variable we arecalculating is a fuzzy set of mixtures of sediment types, we are assured of getting datacompatible with the model data. In this case the sediment type is adjusted to match theinput targets. It is important to note that further tuning of the model is accomplished

5.5 Multidistributary Deltaic Deposition with Variable Wave 137

Figure 5.10 Membership functions used in the fuzzy model of distributary mouth bar deposi-tion off the mouth of Southwest Pass shown in Figures 5.8b and 5.8c. (a) Radial distance fromriver mouth. (b) Angular position relative to centerline of the model.

by adjusting the membership functions. Our fuzzy logic simulations compare morefavorably with Figure 20 in Wright [1978] than do those obtained from the plumemodel (Figure 5.8a).

5.5 Multidistributary Deltaic Deposition with Variable Wave

and Long-Shore Drift Regimes

Whereas the modern Mississippi River Delta has a few distributaries, older Holocenedeltas developed on the Mississippi delta plain had up to 40 distributaries [Gould,1970]. Clearly, modeling each distributary separately would require a significantamount of computation. Instead, we have built upon a hypothetical, simplified riverflood plain and delta system model inspired by Nordlund [1996] and further describedin Demicco & Klir [2001]. The original model (Figure 5.11) had a simple geometry


Table 5.1 Rules of the fuzzy logic system of sediment plume deposition off Southwest Pass.

If the radial distance is channel, then the thickness is none.If the radial distance is very close and the angular position is far negative, then the thicknessis some.If the radial distance is very close and the angular position is negative, then the thickness issome.If the radial distance is very close and the angular position is central, then the thickness is lots.If the radial distance is very close and the angular position is positive, then the thickness issome.If the radial distance is very close and the angular position is far positive, then the thicknessis some.If the radial distance is close and the angular position is far negative, then the thickness isnone.If the radial distance is close and the angular position is negative, then the thickness is little.If the radial distance is close and the angular position is central, then the thickness is some.If the radial distance is close and the angular position is positive, then the thickness is little.If the radial distance is close and the angular position is far positive, then the thickness is none.If the radial distance is far and the angular position is far negative, then the thickness is none.If the radial distance is far and the angular position is negative, then the thickness is very little.If the radial distance is far and the angular position is central, then the thickness is little.If the radial distance is far and the angular position is positive, then the thickness is very little.If the radial distance is far and the angular position is far positive, then the thickness is none.If the radial distance is very far, then the thickness is very little.

imposed on a grid of cells 125× 125, each 1 km2, and four fuzzy inference systems,two for the delta and two for the river, controlled sediment deposition in the model.In each case, one Mamdani-type fuzzy logic system controlled sediment grain sizeand one fuzzy logic system controlled thickness of sediment deposited in each cell.The membership functions of the fuzzy logic system that controlled grain size forthe subaqueous deltaic deposition is shown in Figure 5.12. This system had twoantecedent variables, water depth (Figure 5.12a) and distance from the mouth of theriver (Figure 5.12b), and one consequent or dependent variable, grain size. Grainsize (Figure 5.12c) was normalized over the interval 0 to 1 and characterized by fivetriangular membership functions: clay, sandy-clay, clayey-sand, sand, and clean sand.

The new delta model has user-adjustable lateral dimensions and starting topog-raphy. Moreover, the new model employs Takagi–Sugeno type fuzzy logic systemsand incorporates two new elements: basin energy and long-shore drift regime. Basinenergy is intended to generally model the wave climate in the basin, with 0 represent-ing low wave energy and 1 representing high wave energy. Long-shore drift regimealso varies between 0 and 1. In a 0 long shore drift setting, long-term wave approachis parallel to the shore, whereas a drift regime setting of 1 indicates substantialobliqueness in the general direction of wave approach and standing long shore drift

Figure 5.11 Outputs from basic delta model described in Demicco & Klir [2001] with sinu-soidal sea level oscillation of 10 ky and 10 m height. (a) through (d) are isometric views ofhypothetical delta simulation at different time steps. Dark blues represent finest floodplainmuds (0 on colorbar) and deepest marine muds whereas reds denote clean sands in the river

Continued


[Komar, 1998]. In addition, the positions of the distributary mouths grow seawardin an expanding fractal pattern and mass conservation is accomplished by the sameartifice as is used in the single distributary model.

Figure 5.13 conceptually illustrates the delta sedimentation patterns under end-member conditions of basin energy and drift regime. Figure 5.13a illustrates a con-dition of low wave energy and parallel wave approach (i.e., no long-shore drift) andis essentially the model described by Demicco & Klir [2001]. Figure 5.13b illustratesthe case where basin wave energy is still low but there is “eastward” directed long-shore drift. In Figure 5.13c, basin wave energy is high but wave approach is parallel tothe shore. In this case, sediment is “trapped” in the littoral zone and distributed alongbeaches that flank the delta mouth. Finally, Figure 5.13d illustrates the case wherethere is strong wave energy in the receiving basin and a markedly preferred directionof wave approach, resulting in strong “eastward” directed long-shore currents.

In this new model, five antecedent (input) variables control the geometry of theoffshore sediment plume: basin energy, drift regime, angular position (theta), radialdistance, and depth (Figure 5.14). Relative basin energy and drift regime both haverange and domain of 0 to 1 (Figures 5.14a, 5.14b). Angular position (Figure 5.14c) andradial distance (Figure 5.14d) are measured in polar coordinates from a distributarymouth position.Angular trend of the shoreline is represented by angular measurementsin radians from the river mouth position and is divided into three fuzzy sets: low,intermediate, and high. Low angles are “easterly,” intermediate angles are arrangedon either side of a perpendicular line seaward from the delta mouth, and high anglesare “westerly.” Distance from the river mouth is measured radially from a distributary

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Figure 5.11 Continued. and at the river mouth (0.9 on colorbar). Deltaic dispersion conevaries from sands (0.7) through muddy sands (0.5) to sandy muds (0.3) of the deeper shelf.(a) Sediment surface at start of simulation. (b) Sediment surface at end of 20,000 years afterone sea level rise and fall. (c) Sediment surface at the end of 28,000 years during sea levelfall. (d) Sediment surface at the end of 30,000 years at the end of sea level fall. Note howthe river deposits of abandoned channels sink into the floodplain surface. Also note how theriver “meanders” across the floodplain as it seeks the lowest path to the shoreline. Syntheticstratigraphic cross-sections generated by the model perpendicular to shore (e) and parallel toshore (f) produced at the end of the model run. In the parallel section, marine deltaic deposits(red sands through yellow silts to light blue marine mud) alternate with the dark blue fluvialmud and the aggrading channel deposits (vertical red bars flanked by yellow levee deposits). Inparticular, note the four avulsion events preserved in the topmost fluvial section in the shore-parallel. The fluvial to marine cycles, ten meters or so thick, are dictated by the external sealevel driver and are thus known as “allocycles.” The other, smaller scale cycles are driven byavulsion and kilometer-scale shift of the mouth of the delta between time steps. These kindsof cycle, driven by internal variability in sedimentation, are called autocycles. (See also colorinsert.)


Figure 5.12 (a) Membership functions for the fuzzy sets “shallow” and “deep” for the inputvariable depth over the domain range 0 to −300 m for delta model in Demicco & Klir [2001].(b) Membership functions for the fuzzy sets “at-source,” “near-source,” and “far-from-source”for the input variable location over the domain 0 to 250 km for the delta model in Demicco &Klir [2001]. (c) Membership functions and their designations for the output variable grain sizeover a normalized domain for the delta model in Demicco & Klir [2001].

mouth bar and has been normalized to a range of 0 to 1 by dividing radial distancesby the maximum radial distance in the model for a given time step. Radial distanceis divided into three fuzzy sets (Figure 5.14d) at source, near source, and far fromsource. Depth range (Figure 5.14e) in the model is also divided into three fuzzy sets:shoreline, shelf, and basin. Sediment thickness deposited at each cell is divided into


Figure 5.13 Matrix showing sediment plume produced under various regimes of relativebasin energy and long-shore drift. Drift regime increases from right to left whereas basinenergy regime increases from top to bottom. (a) Relative basin energy= 0, relative long-shoredrift strength = 0. This is essentially the model described in Demicco & Klir [2001] and inFigures 5.11 and 5.12. (b) Relative basin energy= 0, relative long-shore drift strength= 1. (c)Relative basin energy= 1, relative long-shore drift strength= 0. (d) Relative basin energy= 1,relative long-shore drift strength = 1. (See also color insert.)

five Takagi–Sugeno linear membership functions: background (0.01), very little (0.1),little (0.33), some (0.66), and lots (1.0). Sediment type deposited at each cell is alsodivided into five Takagi–Sugeno linear functions.

There are a total of 57 rules in this system. A typical rule reads: “If the basinenergy is high and the drift regime is low and the angular location is low and theradial distance is at source and the depth is shoreline, then the thickness is lots.”In this case, if every combination of input variables were allowed, there would be108 rules. However, it is clear from the context that some of the rules would beirrelevant and can therefore be dropped (or “pruned”) from the system. For example,it would never arise that a point that was at the source (at the distributary) would have


Figure 5.14 Input membership functions for deltaic fuzzy model with variable basin waveenergy and long-shore drift regimes. (a) Relative basin energy. (b) Relative long-shore driftregime strength. (c) Angular relationship to model centerline. (d) Radial distance from rivermouth. (e) Water depth.


a water depth of anything but shoreline, so rules with shelf or basin water depths “atthe source” were ignored. Another way to reduce rules is to use the standard fuzzy

complement of membership functions. In fuzzy logic, the adjective “not” implies thefuzzy complement of the fuzzy set it modifies. Consider another rule from the deltasystem: “if the basin energy is high and the drift regime is low and the angular locationis not intermediate and the radial distance is near source and the depth is shoreline,then the thickness is some.” For any angular position, not intermediate = 1 minusintermediate. Thus the membership function “not intermediate” covers the low andhigh options of the angular position.

Two model runs each simulating 50,000 years of sedimentation at 200-year timesteps are presented here over a 200 × 200 km2 grid. In both cases, sea level is heldconstant. In both models, the subsidence is a maximum in the center of the modeland falls off to the edges of the model but isostatic compensation of the sedimentsaccording to the formulas given by Turcotte & Schubert [1982] are incorporated.Tectonic “subsidence” remains constant through the simulation time. In the simulationshown in Figure 5.15, basin energy and long-shore drift regime are initially assignedinput values of 0.1 and 0.1. During the course of the model run, however, a randomfluctuation in the second decimal point of the input values for basin energy and long-shore drift regime is allowed at each time step. In the simulation shown in Figure 5.16,the basin energy input value and the long-shore drift input value are 0.85 and 0.85,respectively, and are held constant at each time step. (It is important to note that inthese simulations maximum long-shore drift direction is to the “west.”) Finally, in bothmodels, there are random (in time) avulsions of the river upstream of the model. Afteran avulsion, the river enters the upstream end of the model at the lowest point. Theinterplay of subsidence with floodplain aggradation history and location of the priorchannel belts will determine this low point. From the lowest point at the upstream endof the model the river finds the lowest set of adjacent cells to reach the shoreline. Thecolors in Figures 5.15 and 5.16 reflect the caliber of the sediments that are depositedon the surface (Figures 5.15a and 5.16a) in each cell at each time step. The dark bluerepresents the finest-grained flood plain mud with lighter blues representing offshoremarine mud. The red through light blue hues indicate environments of depositiondominated by clean, coarse sand (red) through finer sands (represented by yellow) tocoarse silt (greenish blues). These environments include: (1) the distributary mouthbars (here amalgamated into a continuous sheet); (2) the river and distributary systemthat feeds the deltaic system; and (3) the river’s levee–crevasse–splay system.

Note that on sediment surfaces at the ends of the runs (Figures 5.15a and 5.16a)the channel belt widens toward the shore and the distributary portion of the deltais not a single cell. This is because, at every avulsion, as the delta re-establishesitself the distributary mouth bars are added to shoreward nodes in a random, fractalpattern. This fractal growth pattern is generated by a “diffusive” mechanism [Witten& Sander, 1983]. Also note the differential buildup of an offshore delta platform inthe case of low long-shore drift versus high long-shore drift.


Figure 5.15 (a) Isometric view of hypothetical delta simulation at end of run 1. Dark bluesrepresent finest floodplain muds and deepest marine muds whereas reds denote clean sands atthe river mouth. The deltaic dispersion cone varies from sands through muddy sands to sandymuds of the deeper adjacent basin. Horizontal and lateral scale in km, vertical scale in m. Sealevel held constant throughout the simulation. Basin energy and long-shore drift regime areboth low in this simulation with a bit of second-order “noise” (random variation in the seconddecimal place). The river widens into a broad distributary system and there are approximately10 distributaries at this point in the simulation. Synthetic stratigraphic cross-sections: (b) viewperpendicular to shore and (c) parallel to shore, produced at the end of the model run. In theparallel section, marine deltaic deposits (red sands through yellow silts to light blue marinemud) grade up to dark blue fluvial mud and the aggrading channel deposits (vertical orangebars flanked by green levee deposits). In particular, note the avulsion events preserved in thetopmost fluvial section in the shore-parallel. The smaller, 10 m thick scale cycles are driven byavulsion and kilometer-scale shifts of the distributary mouths of the delta between time steps.These kinds of cycle, driven by internal variability in sedimentation, are called autocycles.(See also color insert.)


Figure 5.16 (a) Isometric view of hypothetical delta simulation at end of run 2. Dark bluesrepresent finest floodplain muds and deepest marine muds whereas reds denote clean sands atthe river mouth. The deltaic dispersion cone varies from sands through muddy sands to sandymuds of the deeper adjacent basin. Horizontal and lateral scale in km, vertical scale in m. Sealevel held constant throughout the simulation. Basin energy and long-shore drift regime areboth high in this simulation with a bit of second order “noise” (random variation in the seconddecimal place). The river widens into a broad distributary system and there are approximately10 distributaries at this point in the simulation. Synthetic stratigraphic cross-sections: viewperpendicular to shore (b) and parallel to shore (c), produced at the end of the model run. Inthe parallel section, note how the strong long-shore drift has skewed the deposits to the west.Overall, marine deltaic deposits (red sands through yellow silts to light blue marine mud)grade up to dark blue fluvial mud at the left side of the cross-section. Thickness here is, in part,generated by increased isostatic adjustments due to increased sediment loads on this side ofthe model. A starved shelf system obtains on the right (“east”) side of the model. The smaller,10 m thick scale cycles are driven by avulsion and kilometer-scale shifts of the distributarymouths of the delta between time steps. These kinds of cycle, driven by internal variability insedimentation, are called autocycles. (See also color insert.)

5.6 Future Developments 147

Figures 5.15b, 5.15c, 5.16b, and 5.16c show synthetic cross-sections through thedeltaic deposit at the end of the model runs. The upper panel in each diagram is a shoreperpendicular view whereas the lower panel is a shore parallel view out in the basin:colors have the same meanings as in the oblique surface views. Figure 5.15 showsthe case with low long-shore drift and wave energy whereas Figure 5.16 shows themarked effects of strong preferential long-shore drift. There are a number of scales ofcycles in both figures. The largest scale of cycles (tens of meters) is due to the seawardprogradation of the river system resulting in a cap of fluvial deposits. The next smallercycles are autocycles due to the avulsion of the river. Finally, there are meter-scalecycles due to the change in the position and number of distributaries as the deltabuilds out. This alters the position of the delta by as much as a few kilometers in eachtime step and produces the small-scale cycles. The strong long-shore drift and strongwave energy have clearly strongly influenced the deposits depicted in Figure 5.16.Here, isostatic compensation has produced thick deposits on the western side of thebasin and “starved basin” conditions on the east.

5.6 Future Developments

We hope that the examples of this chapter demonstrate that fuzzy logic systems arevery versatile and, indeed, can be more versatile than classical mechanics equations.There are a number of areas that warrant further research and development of appli-cations of fuzzy logic to computer-generated stratigraphic simulations. These areoutlined below.

The applications discussed in this chapter restrict the use of fuzzy sets to modelsediment dispersal within the various models illustrated. There is no reason whyfuzzy logic systems could not be used in other parts of these simulations. For exam-ple, compaction routines developed in Chapter 3 could easily be adapted for use inthese models. Indeed, compaction routines employed in many models are essentiallyinstantaneous and are solved at designated time steps. It is clear from modern coastalareas that compaction has a definite time lag component that is not accounted forin deterministic equations. Realistic time delay factors could easily be inserted in afuzzy logic compaction routine.

So far we have been using standard fuzzy sets wherein for a given input value thereis one degree of membership in the unit interval [0, 1]. It is worthwhile to explorethe use of various nonstandard fuzzy sets (as introduced in Section 2.9), in particularfuzzy sets of type 2 [Mendel, 2001]. Clearly, fuzzy sets of type 2 would be warrantedby the spread of the initial data on coral growth rates (Figure 3.7) used in the sedimentproduction function of Section 3.4.2.

Another area of fuzzy logic stratigraphic modeling that warrants further investiga-tion is in rule “pruning.” In general, the number of rules equals the product of the


number of membership functions of the antecedent variables. However, as we see inthe delta example discussed in Section 5.5 above, not all the rules make sense in thecontext of the model. Although automated procedures have been developed for rulepruning, these have yet to be employed in stratigraphic simulation models.

Finally, we are just beginning to turn our attention to automating the entire modelingprocess. In such a system, we would input the desired geologic information, a cross-section or a seismic line, or a series of core holes, and ask the machine to come up withforward models that predict unknown deposits. As a tentative first step, Chapter 10describes the extraction of a sea level signal using fuzzy linguistic rules from outcropdata. Clearly, one of the major inputs to a depositional model is the initial sea levelsignal, and an automated example of sea level extraction based on fuzzy expert rulesis demonstrated in that chapter.

5.7 Conclusions

There are a number of distinct advantages in employing fuzzy inference systems tomodel sediment dispersal in stratigraphic models. First, fuzzy sets describe systemsin “natural language” and provide the tools to rigorously quantify “soft” infor-mation. Second, fuzzy inference systems are more computationally efficient thanfinite-element or finite-difference models, and can even run faster than a simplelinear interpolation scheme. Last, and most importantly, the shapes of the mem-bership functions can easily be changed by small increments, thereby allowingrapid “sensitivity analysis” of the effects of changing the boundaries of the fuzzysets. In robot control algorithms, where fuzzy logic was first developed, systemscould self-adjust the shapes of the membership functions and set boundaries untilthe required task was flawlessly performed. This aspect of fuzzy systems, com-monly facilitated via the learning capabilities of appropriate neural networks [Kosko,1992; Klir & Yuan, 1995; Nauck et al., 1997] or by genetic algorithms [Sanchezet al., 1998; Cordón et al., 2001], is one of their great advantages over numeri-cal solution approaches. Fuzzy logic models hold the potential to accurately modelsubsurface distribution of sedimentary facies (not just water-depths of deposition)in terms of the natural variables of geology. As exploration moves further intouse of 3-dimensional seismic data gathering, the utility of easy-to-use, flexible 3-dimensional forward models is obvious. Such models could be used to producesynthetic seismic sections. Moreover, the “learning ability” of fuzzy logic systemscoupled with neural networks offers the long-term possibility of self-tuning sed-imentary models that can match 3-dimensional seismic subsurface information ina “nonhuman” expert system. This method offers an alternative to the statisticalmodeling of subsurface geology. It is more computationally efficient and more intu-itive for geologists than complicated models that solve coupled sets of differentialequations.

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Chapter 6 Fuzzy Logic in Hydrology and WaterResources

Istvan Bogardi, Andras Bardossy, Lucien Duckstein, and Rita Pongracz

6.1 Introduction 1546.2 Overview 1546.3 Fuzzy Rule-Based Hydroclimatic Modeling 157

6.3.1 Selection of the input and output base variables 158

6.3.2 Definition of fuzzy sets 161

6.3.3 Definition of the training and validation data sets 164

6.3.4 Rule construction 166

6.3.5 Validation procedure 169

6.3.6 Assessment of the fuzzy rule system 170

6.3.7 Evaluating the fuzzy rule-based model 172

6.4 Application Examples for Nebraska, Arizona, Germany, andHungary 1736.4.1 Long-term statistical forecasting of drought index in Nebraska

and Hungary 173

6.4.2 Long-term statistical forecasting of precipitation in Hungary,

Arizona, and Germany 178

6.5 Discussion and Conclusions 184References 187

Abstract

From the early application of fuzzy logic to hydrology a large amount of research has beenpursued and at present, fuzzy logic has more and more become a practical tool in hydrologicanalysis and water resources decision making. In this chapter the main areas of applications arehighlighted. Then, one major area of hydrology, namely, hydro-climatic modeling of hydro-logical extremes (i.e., droughts and intensive precipitation) is selected to describe in details themethodology using fuzzy rules of inference (or in other words the fuzzy rule-based modelingtechnique). Results over four regions—Arizona, Nebraska, Germany and Hungary—and underthree different climates—semiarid, dry and wet continental—suggest that fuzzy rule-basedapproach can be used successfully to predict the statistical properties of monthly precipitationand drought index from the joint forcing of macrocirculation patterns and ENSO information.

153

154 6 Fuzzy Logic in Hydrology and Water Resources

6.1 Introduction

Hydrology and water resources commonly involve a system of concepts, principles,and methods for dealing with modes of reasoning that are approximate rather thanexact. In other words, hydrology is hampered by uncertainties caused by nature (e.g.,climate), limited data, and imprecise modeling. For instance, aquifer parameters areobtained from a few locations that represent a small fraction of the total volume.Definition of system boundaries and initial conditions also introduce uncertainty.Future stresses on the system are also imprecisely known. The stochastic approachof uncertainty analysis considers aquifer properties as random variables with knowndistributions. Thus, the outputs from a stochastic model are also characterized bythe statistical moments or the full probability density function. However, despite thetheoretical development of the stochastic approach, its practical application is ratherlimited, especially if a point process model needs to be upscaled.

From the early application of fuzzy logic to hydrology [Bogardi et al., 1983], alarge amount of research has been pursued and, at present, fuzzy logic has becomea practical tool in hydrologic analysis and water resources decision making. In thischapter, the main areas of applications are highlighted and one major area is selectedto describe details of the methodology and examples of application results.

6.2 Overview

In contrast with or in complement to a probabilistic approach, fuzzy logic also allowsus to consider the treatment of imprecision (or vagueness) in hydrology. For example,consider the following statements: “runoff increases with higher antecedent mois-ture,” or “the water supply is less than the demand in midsummer,” or “transmissivityis greater near the foothills,” or “salt water intrusion may occur if pumpage gets closeto capacity,” or “high intensity rain.” How can decisions be made under these con-ditions? Also, is a trade-off possible between imprecision and other criteria, such ascost or risk?

As an example, let us consider a deficit or shortage in water supply as h = D−Q,where D, Q denote a given demand and supply, respectively. If we simply define afailure F as the event h > 0, then we have an ordinary set inclusion. If, however, wefollow Duckstein et al. [1988a] and define a failure incident I ∗ as an ordered pairdefined on the set of real numbers R: I ∗={(h, F (h)): h∈R;F(h) is the membershipgrade function of h in F with values in [0,1]}, then I ∗ is a fuzzy set. Such a definitionmakes it possible to accept an imprecise definition of a failure: incident I ∗(h) may stillrepresent only a fair supply condition for a 10% surplus. Thus any value of deficit h

belongs to the set F of supply failures with a non-negative membership function F(h).Stream discharge is another example of imprecision (or vagueness) in hydrology.

Discharge curves corresponding to time-invariant hydrological conditions are often

6.2 Overview 155

based on a few data points where the underlying discharge measurements may beinaccurate. Critical hydrologic decisions must therefore be made over such parts ofthe discharge curve where no data points or only imprecise data points are available[Chow et al., 1988]. Fuzzy regression may be used to express the uncertainty in dis-charge curves. Modeling of various relationships between variables describing waterquantity and quality provides other examples. For instance, the dissolved chemicalconcentration of contaminants such as phosphorus stemming from non-point sourcesis commonly related to the peak flow or total volume of runoff events [Wetzel, 1975].The number of such measurements is often not sufficient to perform standard sta-tistical regression analysis. Similarly, sediment transport relationships may use riverflow quantities to estimate suspended sediment concentration or bed-load. Since theriver regime may change quite fast, often there is only enough time to obtain just afew observation points during important sediment transport events. In addition, thesediment measurements are themselves quite imprecise.

In groundwater hydrology several examples can be mentioned. Aquifer parametersare estimated using expensive field tests, and time (or budget) constraints may leadto the availability of a relatively small number of data points. Further, dispersioncoefficients (which are quite difficult to measure directly) may be estimated indi-rectly from other aquifer parameters [Fried, 1975]. Even though calculated pollutionconcentration may be quite sensitive to the dispersion coefficient, the inaccuracyinvolved is rarely accounted for. Fuzzy regression may also be helpful in this regard.Rate constants for dissolved oxygen models may be estimated from average waterdepth and flow velocity [Biswas, 1981]. However, measured rate constants may beavailable at only a few points; thus the use of statistical regression analysis is notjustified. Flow in fractured rocks is strongly related to the geologic properties of thematerial. Parameters governing such flow can be estimated indirectly from geologicquantities which, however, are often difficult and time consuming to measure. As aresult, the relationships may have to be modeled from just a few data points [Bogardiet al., 1982]. Environmental health risk analyses use dose–response relationshipsbased on a few animal experiments over a dose region which considerably exceedsdoses occurring in contaminated groundwater. Health risk estimated from such uncer-tain relationships may be the basis of environmental regulation. Fuzzy logic offersa possibility to express health risk under uncertainty and to select cost-effective riskreduction alternatives [Bardossy et al., 1991a].

Main domains of fuzzy logic applications in hydrology include:

1. Fuzzy regression, which is useful when it is known that a causal relation exists,but only very few data points are available [Bardossy et al., 1990, 1991b; Ozelkan& Duckstein, 2000].

2. Hydrologic forecasting, for instance to embed short-term flood forecasting intomedium-term forecasting. Kalman filtering is used for the short-term component,


while fuzzy logic operates on the medium term, leading to a complete real-timeforecasting system [Kojiri, 1988].

3. Hydrological modeling, where traditional rainfall runoff models can be replacedby fuzzy rule systems with similar performance [Hundecha et al. 2001].

4. Fuzzy set geostatistics allows us to use imprecise and possibly indirect measure-ments and small datasets in spatial statistical analysis [Bardossy et al., 1988,1990].

5. Incorporation of spatial variability into groundwater flow and transport modeling

with fuzzy logic [Dou et al., 1995, 1997a; Woldt et al., 1997]. In this approach,the imprecision of hydraulic parameters is embedded directly into the govern-ing differential equations as fuzzy numbers [Dou et al., 1995, 1997a]. Then thesystem of finite difference equations is solved using fuzzy set theory methods.This fuzzy modeling technique can handle imprecise parameters in a direct waywithout generating a large number of realizations (which is the common featureof the stochastic approach).

6. Regional water resources management aims at selecting among alternative man-agement schemes under small data sets and imprecisely known or modeledobjectives [Bogardi et al., 1982; Nachtnebel et al., 1986; Bardossy et al., 1989].

7. Multicriterion decision making (MCDM) under uncertainty is essential whenwater resources systems face multiple and conflicting criteria (objectives), e.g.,economic efficiency and environmental preservation, and the criteria corre-sponding to alternative systems are imprecisely known [Duckstein et al., 1988b;Bardossy et al., 1992; Bogardi et al., 1996]. These criteria are defined as fuzzynumbers and MCDM is performed in a fuzzy logic framework.

8. Fuzzy risk analysis considers uncertainty in any or all elements of risk analysis:exposure or load, resistance or capacity, and consequence [Bogardi et al., 1989;Duckstein & Bogardi, 1991]. The uncertainties are defined as fuzzy numbers, sothe risk is also obtained as a fuzzy number. In a risk management framework,management options are evaluated to identify the best option, say in a risk–costtrade-off formulation [Lee et al., 1994, 1995; Stansbury et al., 1999; Mujumdar &Sasikumar, 2002].

9. Reservoir operation planning may apply fuzzy logic to derive operation rules[Simonovic, 1992; Shrestha et al., 1996]. Operation rules are generated on thebasis of economic development criteria such as hydropower; municipal; indus-trial and irrigation demands; flood control and navigation; and environmentalcriteria such as water quality for fish and wildlife preservation, recreationalneeds, and downstream flow regulation. Split sampling of historical data (meandaily time series of flow, lake level, demands, and releases) is used to train andthen validate the fuzzy logic model. Such models appear to be easy to construct,apply, and extend to a complex system of reservoirs [Teegavarapu & Simonovic,1999].

6.3 Fuzzy Rule-Based Hydroclimatic Modeling 157

10. Climatic modeling of hydrological extremes has applied fuzzy logic to describea stochastic linkage between large-scale climatic forcing and local hydrologi-cal variables [Pesti et al., 1996; Pongracz et al., 2001]. Large-scale climaticforcing may include atmospheric circulation patterns (weather types) and seasurface temperature (SST) indicating El Niño or La Niña in the tropical Pacificregion. Local hydrological variables may include, among others, daily precip-itation, temperature, evaporation, wind velocity, and drought indices. This lastapplication is the subject of the remainder of this chapter.

6.3 Fuzzy Rule-Based Hydroclimatic Modeling

Fuzzy rule-based modeling shows much potential in cases when a causal relationshipis well established but difficult to calculate under real-life conditions, when data arescarce and imprecise, or when a given input vector has several contradictory responseswhich may be true to varying degrees. These features are often present in hydrologyand water resources.

Fuzzy rule-based modeling may be considered as an extension of fuzzy logic.The primary difference is that fuzzy logic is traditionally used for system controlwith feedback, whereas fuzzy rule-based modeling is employed to simulate pro-cesses, usually without a feedback mechanism [Sugeno & Yasukawa, 1993; Wang &Mendel, 1992; DeCampos & Moral, 1993]. The advent of fuzzy rule-based modelingis a recent development that currently exists without an extensive base of scientificapplications such as that enjoyed by fuzzy logic adherents in engineering disciplines.Consequently, the use of this approach to enhance the modeling of hydrologicalprocesses is relatively new [Bardossy & Duckstein, 1995].

Fuzzy rule-based modeling has been used in several areas of hydrology, including:

● classification of spatial hydrometeorological events [Bardossy et al., 1995];● climatic modeling of flooding [Bogardi et al., 1995];● modeling of groundwater flow and transport [Bardossy & Disse, 1993; Dou et al.,

1997b, 1999; Woldt et al., 1997];● modeling regional-scale nitrate leaching using available soil and cultivation data

[Bardossy et al., 2003; Haberlandt et al., 2002];● forecasting pollutants transport in surface waters [Di Natale et al., 2000];● hydroclimatic modeling of hydrological extremes, i.e., droughts and intensive

precipitation.

We use the hydroclimatic modeling of hydrological extremes to describe a typicalfuzzy rule-based approach as applied in hydrology. An application of fuzzy rulesof inference (or, in other words, the fuzzy rule-based (FRB) modeling technique)is illustrated by several examples. Basic definitions and the main characteristics of


knowledge-based fuzzy systems are presented in Chapter 2. Definition of fuzzy rulesis provided in Section 2.7. In the case of FRB modeling, experts are substitutedfor observed data and several conditional, unqualified fuzzy propositions are used(see Section 2.5).

Because of several difficulties experienced in traditional statistical analysis, FRBmodeling can be used for estimating different hydroclimatological variables. Themain advantages of the FRB approach are that it has a relatively simple structure andrequires neither independency nor long data sets. In the following, an FRB technique,called a weighted counting algorithm [Bardossy & Duckstein, 1995], is adopted toestimate a drought index. The weighted counting algorithm is applied to a subset ofthe data (known as a training set). Results are then composed to a validation subsetof the data. It is described in a step-by-step manner so that the same steps can be usedin other similar cases.

6.3.1 Selection of the input and output base variables

First, the output base variable (so-called response) that the FRB model aims to calcu-late is selected. Here we consider agricultural drought events represented by varioustypes of drought indices (further described below) as the output variable. Next, inputbase variables (so-called premises) are selected in this case on the basis of both phys-ical reasoning and statistical analysis. Drought events (e.g., over the Great Plains ofNorth America) are strongly related to large-scale climatic forcings such as: (1) con-tinental scale atmospheric circulation patterns; and (2) climate oscillations presentboth in the ocean and in the atmosphere (e.g., ENSO with El Niño and La Niña events[Glantz et al., 1991], and see below).

Large-scale atmospheric circulation patterns (CPs) can be represented by either thesea surface pressure field or daily geopotential height fields (e.g., 500 hPa, 700 hPalevel) above a continental-size area containing the study region [NCAR, 1966]. Toovercome the time-scale difference between monthly drought index and daily CP, theeffects of CP on droughts are represented by the monthly empirical relative frequen-cies of daily CP types. The CP types can be identified by a combined multivariatetechnique [Wilks, 1995], namely, principal component analysis and cluster analy-sis using the k-means method [MacQueen, 1967]. Details of this methodology arepresented in Matyasovszky et al. [1993]. Other studies in Europe are based on thesemi-objective CP classification system of Hess and Brezowsky [1952, 1977], or onfuzzy rule-based classification [Bardossy et al.,1995, 2002].

The importance of ENSO effects on weather anomalies and crop production inthe U.S. Midwest was shown by many researchers, and is summarized by Carlsonet al. [1996]. The association of drought with ENSO has been demonstrated for thewhole United States [Piechota & Dracup, 1996], but the correlations are not strongenough to predict drought from any ENSO index alone. The ENSO phenomena are


represented by the time series of SOI (Southern Oscillation Index), which is one ofthe most commonly used indices in ENSO research [NOAA, 2001a]. SOI is definedas the monthly pressure difference between Tahiti and Darwin [Clarke & Li, 1995].Positive and negative SOI values refer to La Niña and El Niño episodes, respectively.

After consideration of physical causality, possible statistical relationships betweenthe selected response and the premises are analyzed. In this case, the correlationcoefficients between the monthly relative frequencies of CP types and lagged droughtindex (DI) and between the monthly relative frequencies of CP types and SOI aresmaller than 0.2, and mostly not significant. On the other hand, the empirical fre-quency distributions of CP types during the five drought categories (as defined inTable 6.1) are different at the 0.01 significance level. Figure 6.1 shows the frequen-cies of CP types during the two most extreme DI intervals: very dry and very wetconditions. The frequencies of CP types during the three ENSO phases (as definedin Table 6.2) are also significantly different. The correlation coefficients between DIand lagged SOI reach 0.39 and are significant (Figure 6.2). Both directions of laghave been evaluated since simultaneous, lag, and pre-lag teleconnections of climate

Table 6.1 Categories defined on drought index (DI).

DI intervals Drought categories

DI < −3 very dry−3 ≤ DI < −1 dry−1 ≤ DI ≤ +1 normal+1 < DI ≤ +3 wetDI > +3 very wet

Figure 6.1 Empirical relative frequency distributions of CP types during extreme droughtconditions in climate division 8.


Table 6.2 Categories defined on SOI.

SOI intervals ENSO phases

SOI ≤ −1 El Niño−1 < SOI < +1 neutralSOI ≥ +1 La Niña

Figure 6.2 Correlation coefficients between drought and the lagged SOI.

variables may be related to ENSO [Wright, 1985]. The conditional frequency distri-butions of DI during El Niño and La Niña periods (Figure 6.3) are also significantlydifferent.

These statistical analyses reinforce earlier findings (e.g., [Piechota & Dracup,1996]) that, despite the strong teleconnection between ENSO and droughts, droughtshave occurred under various phases of ENSO [Carlson et al., 1996]. Thus, for instancein the Great Plains, the partial signals of ENSO and CP on drought are weaker thanin other regions. Also, CP and ENSO are evidently interdependent since they bothrepresent parts of the complex climate system. Thus, the more traditional stochasticapproach to regress SOI and the frequencies of CP types with a drought index doesnot work (see below).

In summary, the SOI and the monthly frequency distribution of CP types constitutethe input base variables, forcing functions, or premises. Next, the question arises as tohow many prior monthly premises should be considered to predict the drought index.There is no strict rule for this case; here we use a selection based on the correlationanalysis between SOI with different lag periods and the drought. For the CP types,none of the prior months has any significant correlation; thus, only the simultaneousfrequency distributions of CP types represent the first type of premises (X1, . . . , X6).


Figure 6.3 Empirical relative frequency distributions of drought conditions during El Niñoand La Niña.

For SOI, as already shown on Figure 6.2, the lag correlations are significant up tothe prior six months. The highest correlation between DI and SOI occurs for a lagof 6 months in the case of Nebraska, but then the correlation coefficients weaken.However, another local maximum correlation can be seen at the−4 month lag period(Figure 6.2). Furthermore, theoretically, as an annual cycle is considered, no lagperiods beyond six months in either direction are taken into account. On the basisof these findings, we used four lagged periods (0,−2,−4, and −6 months) of highcorrelations as SOI-type premises (X7, X8, X9, X10). Note the trade-off between theincreasing number of premises and the length of the data set.

6.3.2 Definition of fuzzy sets

Fuzzy sets (basic characteristics of which are described in Chapter 2) are definedfor each variable involved in the model. Here, all fuzzy sets are fuzzy members(Chapter 2) with triangular membership functions. Each fuzzy number can be rep-resented by a triple 〈a, b, c〉 of real numbers, where b defines its core and the openinterval (a, c) defines its support. Definitions of fuzzy numbers are based on theranges of premises and the response. A fuzzy partition is applied to each variable(SOI values, CP relative frequencies, DI values).

Fuzzy numbers defined on premises

The entire range of possible premise values is divided into several overlapping classeseach forming a fuzzy number. The more fuzzy numbers we define, the better estima-tion can be expected for the values of DI. However, as we are going to use subsets of


the data to define and validate our FRB system, if too many fuzzy numbers are definedon the premises the validation set might contain too many observations that have neveroccurred in the training set. As a compromise, all premises (relative frequencies of CPtypes, and lagged SOI time series) are divided into five regions, namely for monthlyCP occurrence: very rare A1, rare A2, medium A3, frequent A4, and very frequent A5

(Figure 6.4). Then, for SOI: strong El Niño A1, weak El Niño A2, neutral A3, weakLa Niña A4, and strong La Niña phases A5 (Figure 6.5). Various CP types occur withdifferent frequencies, so for the sake of comparability the highest monthly frequency

Figure 6.4 Fuzzy numbers defined on the monthly relative frequency of a given CP type.

Figure 6.5 Fuzzy numbers defined on SOI.


Table 6.3 Monthly maximum relative frequencies (max)for daily CP types and their proportions.

CP1 CP2 CP3 CP4 CP5 CP6

max 0.77 0.50 0.93 0.74 0.94 0.60¾·max 0.58 0.38 0.70 0.56 0.70 0.45½·max 0.38 0.25 0.47 0.37 0.47 0.30¼·max 0.19 0.13 0.23 0.19 0.23 0.15

Table 6.4 Values of premise membership functions for the data array April 1946.

A1i A2i A3i A4i A5i

i Xi1 Very rare Rare Medium Frequent Very frequent

1 0.30 0 0.44 0.56 0 02 0 1.00 0 0 0 03 0.20 0.14 0.86 0 0 04 0.07 0.64 0.36 0 0 05 0.03 0.86 0.14 0 0 06 0.40 0 0 0.33 0.67 0

Strong Weak Weak StrongEl Niño El Niño Neutral La Niña La Niña

7 −1.04 0 0.69 0.31 0 08 0.31 0 0 0.79 0.21 09 0.60 0 0 0.60 0.40 0

10 0.25 0 0 0.83 0.17 0

that ever occurred in the data set is defined as the maximum of the given CP-typepremise (the present case is included in Table 6.3).

As an example, inApril 1946 (representing a data array) the occurrences of CPtypesare: CP1: 9, CP2: 0, CP3: 6, CP4: 2, CP5:1, and CP6: 12 days. Thus, for that monthX1 = 0.30 (relative frequency of CP1), X2 = 0, X3 = 0.20, X4 = 0.07, X5 =

0.03, X6 = 0.40, X7 = −1.04 (simultaneous SOI), X8 = 0.31 (SOI −2 monthsbefore), X9 = 0.60 (SOI −4 months before), X10 = 0.25 (SOI −6 months before).The corresponding membership functions are given in Table 6.4. For example, therelative frequency of CP1 X1,1 = 0.30 possesses membership values (different from0) in both fuzzy sets “Rare monthly CP occurrence” and “Medium monthly CP occur-rence,” 0.44 and 0.56, respectively; or the relative frequency of CP5 X5,1 = 0.03has membership values (different from 0) in both fuzzy sets “Very rare monthly CPoccurrence” and “Rare monthly CP occurrence,” 0.86 and 0.14, respectively.


Figure 6.6 Fuzzy numbers defined on PMDI.

Fuzzy numbers defined on the response

Drought Index (DI) as the response (Y ) is considered in the present example for theeight climate divisions of Nebraska established by NOAA [2001b] and the spatialaverage of the entire state. Different types of fuzzy numbers can be defined on therange of DI from extremely dry (large negative DI values) to extremely wet (largepositive DI values) conditions. As the total number of fuzzy numbers increases (7, 8,11, 12, 17, 18), the accuracy of the FRB model improves. So the last option waschosen with 18 fuzzy numbers: B1, . . . , B18 (Figure 6.6). This fuzzy partition offersa proper representation of the wide range of DI, and the data set is able to provideseveral arrays in each interval.

For the example of April 1946, values of the DI membership function are given inTable 6.5 for the eight climate divisions and the spatial average. Note that in othercases linear partitioning may not be applied to the response, e.g., if it follows a skewedfrequency distribution, as precipitation does. In this case, a different partitioning isselected; namely, the 10th percentiles are assigned to the core of the fuzzy sets(Figure 6.7).

6.3.3 Definition of the training and validation data sets

The entire data set {Xij ;Yj }i=1,...,k;j=1,...,n contains k(= 6 + 4 = 10 in the presentexample) premises Xi and n observations on the premises and the response Y . Theentire time series is split into two parts: a training set τ (2/3 of the entire period)and a validation set ν (1/3 of the entire period). The training set is used to learnthe fuzzy rules, so it must be long enough to provide valuable model outputs. Thevalidation set is applied to validate the rules derived from the training set, namely,how correctly they estimate the observed response. Different partitions of the data setcan be used to check the sensitivity of results to this operation; in the present examplecase, the results are not sensitive to the selection of partitions. Besides the continuouspartitioning, it is possible to select every third data point for the validation procedure,while the other two are parts of the training set.

Table 6.5 Values of drought response membership functions for the data array April 1946 in different regions of Nebraska.

Drought Membership values

B1 B2 . . . B6 B7 B8 B9 B10 . . . B17 B18Div. Y1 Extreme dry Dry 7 . . . Dry 3 Dry 2 Dry 1 Normal Wet 1 . . . Wet 8 Extreme wet

1 −1.16 0 0 . . . 0 0.16 0.84 0 0 . . . 0 02 −1.71 0 0 . . . 0 0.71 0.29 0 0 . . . 0 03 −1.31 0 0 . . . 0 0.31 0.69 0 0 . . . 0 05 −2.47 0 0 . . . 0.47 0.53 0 0 0 . . . 0 06 −1.47 0 0 . . . 0 0.47 0.53 0 0 . . . 0 07 −2.08 0 0 . . . 0.08 0.92 0 0 0 . . . 0 08 −2.29 0 0 . . . 0.29 0.71 0 0 0 . . . 0 09 −1.76 0 0 . . . 0 0.76 0.24 0 0 . . . 0 0NE −1.84 0 0 . . . 0 0.84 0.16 0 0 . . . 0 0


Figure 6.7 Fuzzy numbers defined on the response variable based on the 10th percentiles ofthe precipitation time series in Hungary.

6.3.4 Rule construction

Fuzzy rules are constructed using the training set τ : {Xij ;Yj }i=1,...,k;j=1,...,nt(where

nt < n, number of observations in the time series of the training set) by applying thefollowing steps.

Determine the highest values of all membership functions

for each data array

First, values of membership functions are calculated for each observed premise and theresponse: Ali (Xij ) (for li = 1, . . . , 5; i = 1, . . . , k) and Bl(Yj ). Then, the maximumvalues of membership functions are selected. Thus, each Xij data array within thedata set (j = 1, . . . , nt ) possesses a value Mij :

Mij = maxli=1,...,5

(Ali (Xij )),

and also each response Yj possesses a value M0j :

M0j = maxl=1,...,18

(Bl(Yj )).

Table 6.6 shows these selected maximum values for the data array, April 1946.

Combined effect of fuzzy numbers (using operator AND)

Since we have more than one premise, the effects of premises should be com-bined. The two most commonly used operators for fuzzy numbers are AND and


Table 6.6 Maximum membership function values and weights for the data array April 1946.

Maximum value Name of thei Name Mi1 fuzzy number

1 CP1 0.56 Medium2 CP2 1.00 Very rare3 CP3 0.86 Rare4 CP4 0.64 Very rare5 CP5 0.86 Very rare6 CP6 0.67 Frequent

7 SOI 0.69 Weak El Niño8 SOI (−2) 0.79 Normal9 SOI (−4) 0.60 Normal

10 SOI (−6) 0.83 Normal

DOF1 = 0.049

Response Maximum Name of the Weight of rule 1variable Location value M01 fuzzy number ω1 = DOF1 ·M01

DI div. 1 W-Ne 0.84 dry 1 0.041DI div. 2 N-Ne 0.71 dry 2 0.035DI div. 3 NE-Ne 0.69 dry 1 0.034DI div. 5 Central-Ne 0.53 dry 2 0.026DI div. 6 E-Ne 0.53 dry 1 0.026DI div. 7 SW-Ne 0.92 dry 2 0.045DI div. 8 S-Central Ne 0.71 dry 2 0.035DI div. 9 SE-Ne 0.76 dry 2 0.037DI/NE Nebraska 0.84 dry 2 0.041

OR [Zimmermann, 1985]. In the present model we use only the operator AND to addthe effects of different premises. So a rule looks like this:

IF (X1j is Al1 AND X2j is Al2 AND . . . AND X10j is Al10) THEN Yj is Bl .

The combined effect of all premises is represented here by the product of mem-bership functions called degree of fulfillment (DOF), which indicates the degree ofapplicability of the rule within the FRB system. Thus, the DOF of the j th set of dataarray (DOFj ) is calculated as DOFj =

∏ki=1 Mij . For the data array of April 1946,

we obtain

DOF1 = 0.56 · 1.00 · 0.86 · 0.64 · 0.86 · 0.67 · 0.69 · 0.79 · 0.60 · 0.83 = 0.049.


In the very beginning, the fuzzy rule system is empty, containing no rules at all—thefirst rule is derived from the first observed values. In the present case, this first rule forthe entire state of Nebraska, and for Northern, Central, Southwestern, South-central,and Southeastern Nebraska, is as follows:

Rule (1)

IF

(Medium CP1 occurrence) AND (Very rare CP2 occurrence) AND (Rare

CP3 occurrence) AND (Very rare CP4 occurrence) AND (Very rare CP5occurrence) AND (Frequent CP6 occurrence) AND (Weak El Niño in the actualmonth) AND (Neutral phase 2 month before) AND (Neutral phase 4 month

before) AND (Neutral phase 6 month before)

THEN

(Dry2 drought condition)

whereas for Western, Northeastern, and Eastern Nebraska, it is:

Rule (2)

IF

(Medium CP1 occurrence) AND (Very rare CP2 occurrence) AND (Rare CP3occurrence) AND (Very rare CP4 occurrence) AND (Very rare CP5 occurrence)AND (Frequent CP6 occurrence)AND (Weak El Niño in the actual month)AND

(Neutral phase 2 month before) AND (Neutral phase 4 month before) AND

(Neutral phase 6 month before)

THEN

(Dry1 drought condition)

The rule system will grow as more and more rules are added on the basis of theobserved data array. If a rule derived from a given set of data arrays is not includedin the rule system yet, then it should be added to the rule system.


Assign a weight to each rule

Weights indicate the proportion of the training data sets explained by a given (mth)rule. They are calculated as the sum of the products of DOFj and the value ofmembership function of the response variable (M0j ):

ωm =

nt∑

j=1

DOFj ·M0j .

For the data point April 1946, the weights of rule (1) or (2), depending on the areaconsidered, are shown in Table 6.6.

If the first rule (1) or (2) appears in more data arrays, the individual weights aresummed.

After proceeding throughout the entire training set, all derived rules will possess aweight that will be used in the validation procedure when the estimated values of theresponse are calculated during the defuzzification step.

6.3.5 Validation procedure

Fuzzy rules derived from the training set τ are validated using the validation data setν: {Xij ;Yj }i=1,...,k;j=nt+1,...,n in the following steps.

Calculate all possible DOF for each data array

All values of membership functions are calculated for each premise, so we have allAli (Xij ) (for li = 1, . . . , 5; i = 1, . . . , k) values. Since the fuzzy sets are definedas overlapping intervals, all the data array will fall into two different fuzzy sets of agiven premise (Figures 6.4, 6.5). Thus, theoretically, there are 2k possible rules, butmost of them are either impossible or did not occur in the training set (the maximumnumber of rules is determined by the length of the training set, nt , which is much lessthan 2k). Therefore, only a few existing rules will be taken into account in specifyingthe response output. As an example, for a data array from the validation set (July1966), the possible membership values are calculated in the western Nebraska region(Table 6.7). The total number of potentially applicable fuzzy rules is 210 = 1024 inthe present study.

Combine the fuzzy responses: defuzzification

At this time, the application of each rule provides a fuzzy response. The defuzzificationprocess combines the fuzzy responses as the weighted linear combination and arrives


Table 6.7 Membership function values for the data array July 1966 (western Nebraska).

A1i A2i A3i A4i A5i

i Xi124 Very rare Rare Medium Frequent Very frequent

1 0.13 0.33 0.67 0 0 02 0.07 0.48 0.52 0 0 03 0.23 0.03 0.97 0 0 04 0.32 0 0.26 0.74 0 05 0.22 0.03 0.97 0 0 06 0.03 0.79 0.21 0 0 0

Strong Weak Weak StrongEl Niño El Niño Neutral La Niña La Niña

7 −0.24 0 0.16 0.84 0 08 −0.65 0 0.43 0.57 0 09 −1.77 0.18 0.82 0 0 0

10 −1.33 0 0.89 0.11 0 0

at a crisp (a real number) estimated response. The center of gravity can be commonlyused to obtain the estimated value of the response variable (Yj ):

Yj =

∑

m∈τ

DOFm · ωm · Bm(2)

∑

m∈τ

DOFm · ωm

,

where Bm(2) is the core of the consequent fuzzy set Bm (when the membership value

equals 1) defined on DI.In our example, for data array July 1966 five rules are applicable out of the 1024

possible fuzzy rules (Table 6.8). So the estimation for DI in western Nebraska at July1966 is:

Y124=10−5·(0.096·0.64·(−1)+ 0.25·0.14·1+ 3.59·0.23·3+ 1.01·0.15·1+ 9.12·0.30·(−3))

10−5·(0.096·0.64+ 0.25·0.14+ 3.59·0.23+ 1.01·0.15+ 9.12·0.30)

= −5.393.73 = −1.45.

6.3.6 Assessment of the fuzzy rule system

A more automated method can be used to obtain the entire fuzzy rule system,namely, simulated annealing using the Metropolis–Hastings algorithm [Chib &


Table 6.8 Characteristics of the applied rules for the data array July 1966 (western Nebraska).(VR = very rare, R = rare, M = medium, wE = weak El Niño, N = neutral.)

WeightApplied (mth) rule DOFm [10−5] ωm [10−2] Bm

(2)

VR, R, VR, R, R, R, N, wE, wE, N→ dry1 0.96 6.4 −1VR, R, VR, M, R, VR, wE, N, wE, N→ wet1 2.51 1.4 1VR, R, R, R, R, VR, wE, wE, sE, wE→ wet3 35.91 2.3 3R, VR, VR, N, R, R, wE, N, wE, wE→ wet1 10.12 1.5 1R, R, R, R, R, R, wE, wE, wE, wE→ dry3 91.20 3.0 −3

Greenberg 1995]. In the first step, the performance P of the rule system is definedusing the estimated and observed response values:

P =∑

j

F(Yj , Yj ).

Typically, F can be chosen as an lp measure:

F(Yj , Yj ) =∣∣Yj − Yj

∣∣p .

Other performance functions such as a likelihood-type measure, a geometric distance,or a performance related to proportional errors can also be formulated. Once one has ameasure of performance an automatic assessment of the rules can be established. Thismeans that the goal is to find the rule system for which the performance P reaches itsminimal value. Since the number of possible different rule systems is very large, thereis no possibility to try out each possible rule combination to find the best. Therefore,discrete optimization methods have to be used to find “good” rule systems. Geneticalgorithm or simulated annealing are possible candidates for this task.

Here, simulated annealing is summarized as a tool for finding the rule system R

with optimal performance P(R). The algorithm is as follows:

1. The possible fuzzy sets for the arguments Ali and the responses Bl are defined.2. An initial rule system R is generated at random.3. The performance of the rule system P(R) is calculated.4. An initial so-called annealing temperature ta is selected.5. A rule I of the rule system is picked at random.6. An argument or a response of this rule is chosen at random.7. If argument h ≤ k is chosen, an index 1 ≤ h∗ ≤ lh is chosen at random and a

new rule system R∗, with ki,h∗ replacing ki,h, is considered.8. If response h > k is chosen, an index 1 ≤ h∗ ≤ lh is chosen at random and a

new rule system R∗, with li = h∗ replacing li , is considered.


9. In both cases the performance of the new rule system P(R∗) is evaluated.10. If P(R∗) < P (R) then R∗ replaces R.11. If P(R∗) ≥ P(R) then the quantity

π = exp

(P(R)− P(R)∗

ta

)

is calculated. With the probability π , the rule system R∗ replaces R (negativechanges).

12. Steps 5–11 are repeated NN times.13. The annealing temperature ta is reduced.14. Steps 12–13 are repeated until the proportion of positive changes becomes less

than a threshold ε > 0.

The above algorithm yields a rule system with “optimal” performance. However, therules obtained might reflect some specific features corresponding to a small numberof cases in the data set. To avoid rules that are derived from too few cases, the perfor-mance function is modified. The insufficient generality of a rule can be recognized onthe number of cases to which it is applied. As an alternative, the degree of fulfillmentof the rules can also be considered. In order to ensure the transferability of the rules,the performance of the rule system is modified by taking the sum of the DOFs intoaccount.

P ′(R) = P(R)∏

i

⎡⎢⎢⎣1+

⎛⎜⎜⎝

v′ −∑

t

v′i(x1(t), . . . , xJ (t))

v′

⎞⎟⎟⎠+

⎤⎥⎥⎦ (6.1)

Here (·)+ is the positive part function

x+ =

{x if x ≥ 0

0 if x < 0

v′ is the desired lower limit for the applicability of the rules, in this case expressedby the sum of DOFs. If the sum of DOFs for a rule is less than v′, then a penalty isapplied. If P ′ is used in the optimization procedure, then rules that are based on a fewcases and are seldom used are penalized. The degree of penalty depends on the gradeto which the desired limit v′ exceeds the actual sum of DOFs for a selected rule.

6.3.7 Evaluating the fuzzy rule-based model

The FRB model must be evaluated in terms of how well it reproduces the statisticalproperties and the actual time series of the consequences in the validation set. In order

6.4 Application Examples 173

to fulfill the evaluation procedure, it is possible to compare various statistical param-eters of the observed and the calculated time series, e.g., mean, standard deviation,quartiles, deciles, etc. The strength of the linear relationship between the observedand the FRM-modeled time series can be represented by the correlation coefficient;in the optimal case it should be equal to 1.

Graphical comparison includes the plot of the observed and FRB-modeled timeseries, or the scatterplot diagrams of the observed and calculated data. Statisticaldistributions of the two time series can be compared by using the relative frequencydistribution histograms or the empirical probability functions.

Furthermore, several types of error terms are available [Wilks, 1995] to describethe reproduction of observed data by the FRB model. Only some of them are listedhere. Definition of the mean error (ME) is written as follows:

ME =1

n·

n∑

j=1

(Yj − Yj ).

The mean absolute error (MAE) can be defined as follows:

MAE =1

n·

n∑

j=1

∣∣Yj − Yj

∣∣

Finally, the root-mean squared error (RMSE) is the most often used error term; it canbe defined as follows:

RMSE =

√√√√1

n·

n∑

j=1

(Yj − Yj )2

In the next section of this chapter several examples of these evaluation forms areshown in order to illustrate the applicability of FRB modeling and the goodness-of-fitbetween the observed and modeled time series.

6.4 Application Examples for Nebraska, Arizona, Germany,

and Hungary

6.4.1 Long-term statistical forecasting of drought index in Nebraska

and Hungary

Drought indices serve as common tools to measure the intensity and spatial extentof droughts. One of the most commonly used climatic drought indices in the U.S.is the Palmer Drought Severity Index (PDSI) [Palmer, 1965]. The PDSI is based onthe principle of a balance between moisture supply and demand when man-made


Figure 6.8 Drought conditions observed in Nebraska and Hungary.

changes are not considered. This index indicates the severity of a wet or dry spell—the greater the absolute value, the more severe the dry or the wet spell. The PDSI wasmodified by the National Weather Service Climate Analysis Center to obtain anotherindex (modified PDSI or PMDI), which is more sensitive to the transition periodsbetween dry and wet conditions [Heddinghause & Sabol, 1991]. For the examplepresented here, the modified Palmer index is considered. However, the methodologyis applicable to any other drought indices such as the Standardized Precipitation Index[McKee et al., 1993] or the Bhalme–Mooley drought index [Bogardi et al., 1994].A long-term historical data set of monthly PMDI values exists for climatic divisionsaround the U.S. [Guttman & Quayle, 1996]. The data set of the monthly PMDI startsin 1895 [NOAA, 2001b]. Drought events occur in the case of negative PMDI valueswhereas positive values imply wet conditions.

The observed PMDI in Nebraska and PDSI in Hungary (Figure 6.8) indicate thehigh variability and persistence of droughts. Drought indices are evaluated duringthe summer season (May–August). Drought is a normal part of the climate in bothHungary and Nebraska, and is different from other natural hazards. Drought is a slow-onset, insidious hazard that is often well established before it is recognized as a threat,taking months or years to develop. Very severe drought occurred at both locations insummer 2000. Although both locations have mainly continental climate, Hungariansummers may be interrupted occasionally by oceanic and Mediterranean influences.The climate of the different regions within both locations varies considerably: thewestern part of Nebraska is in general colder and drier than the eastern part, whereasin Hungary the opposite is true.

A historical data set of monthly PDSI values exists for 16 climatic stations inHungary (Mika, 2000). The question arises whether the monthly PDSI values arehomogeneous. Figure 6.9 shows the cumulative frequency distribution of PMDI forSzarvas (located in the eastern part of Hungary) for two periods: the training set of


Figure 6.9 Cumulative frequency distributions of monthly PDSI for Szarvas, located in easternHungary.

Figure 6.10 Distributions of PMDI for the training and validation sets in climate division 1,Nebraska.

1881–1960 and the validation set of 1961–1990. The two frequency distributions aredifferent at the 0.01 significance level, using the two-sample Kolmogorov–Smirnovtest [Wilks, 1995], which indicates a drier climate during this latter period. The otherstations in Hungary generally behave similarly, with some rare exceptions.

For Nebraska, Figure 6.10 shows the cumulative frequency distribution of PMDIin climate division 1 for two periods: the training sets of 1946–62, 1978–94 and thevalidation set of 1963–77. The two frequency distributions are different at the 0.1, butthe same at the 0.05 significance level, using the two-sample Kolmogorov–Smirnovtest. The other divisions in Nebraska behave similarly.


Figure 6.11 Cumulative frequency distribution of PMDI time series (1946–1994) in south-central Nebraska (climate division 8).

Results for Nebraska

First, examples of the results obtained for Nebraska are given. The distributions ofthe calculated PMDI reproduce the empirical distributions (Figure 6.11). Figure 6.12compares the observed and estimated (calculated) time series for climate division 8in the case of using the FRB modeling technique (upper panel) and the multivariateregression (MR) model (lower panel). The FRB model performs almost perfectlyover the training set, and quite well over the entire period, whereas the MR techniqueis not able to reproduce the variability of the observed PMDI time series, not even inthe training period. The calculated values of the FRB model are not exactly the sameas the observed drought index during the validation period. This is evident and wasexpected, since droughts are triggered by a large number of atmospheric, hydrologic,agricultural, and other phenomena in addition to the two types of premises this modelconsiders. Another reason is that during the “learning” process huge and persistentnegative (years 1954–57) and positive (1992–94) peaks must be “assimilated.” Themodel did learn all the peaks, which is necessary in order to apply the FRB model tothe entire range of PMDI.

Results for Hungary

The FRB approach leads to similar promising results in Hungary. Being located incentral Europe, CPs are represented by the daily time series of the Hess–Brezowsky[1952, 1977] CP types. Here, the original 29 CP types have been aggregated into sixlarger classes according to: (1) the three main circulations (zonal classes (4 types),


Figure 6.12 Observed and estimated PMDI time series for climate division 8, NE.

half-meridional classes (7 types), and meridional classes (18 types)); (2) cyclonic andanticyclonic CP types; and (3) the combination of (1) and (2). The frequency distri-butions of the calculated drought index, PDSI, reproduce the empirical distributions(Figure 6.13).

This performance of the approach, both in Nebraska and in Hungary, is evenmore noteworthy if we consider the much larger difference between the empiricaldistributions in the training set (from which the rules are derived) and the validationset; that is, the model works under non-stationary climatic conditions. We concludethat the FRB technique is able to reproduce the variability of droughts influenced byCP and ENSO both in Nebraska and Hungary.


Figure 6.13 Cumulative frequency distribution of PDSI time series (1881–1990) in south-western Hungary (for station Pécs).

6.4.2 Long-term statistical forecasting of precipitation in Hungary,

Arizona, and Germany

Results for Hungary

Although there is a considerable variability in average monthly precipitation inHungary, the frequency distributions of monthly precipitation for selected clustersof months do not change significantly. Namely, for Keszthely, three clusters ofmonths can be separated: January to March, May to August, and September toNovember (Figure 6.14). Note that April and December do not pertain to anyclusters.

Here the FRB models contain the CP premises without any lag becauseregional precipitation does not exhibit any correlation with the relative fre-quency of CP classes for prior months. This is evident physically since themean residence time of a water vapor particle in the atmosphere is not morethan 10–15 days. Thus the simultaneous frequency distributions of the six CPclasses represent the first type of premises. For SOI, we selected two previ-ous monthly values representing the highest correlation with monthly precip-itation (for instance, in the case of station Keszthely, −1 and −5 monthsSOI).

The following example rule indicates how it reflects typical weather condi-tions leading to “extremely high” precipitation for station Keszthely and season


Figure 6.14 Cumulative frequency distributions of monthly precipitation at Keszthely for thethree clusters.

May–August in Hungary:

IF

((Very rare zonal–cyclonic CP class occurrence) AND

(Very rare halfmeridional–cyclonic CP class occurrence) AND

(Very frequent meridional–cyclonic CP class occurrence) AND

(Very rare zonal–anticyclonic CP class occurrence) AND

(Rare halfmeridional–anticyclonic CP class occurrence) AND

(Rare meridional–anticyclonic CP class occurrence) in the actual month) AND

(Strong El Niño 1 month before) AND

(Neutral phase 5 months before)

THEN

(Extremely high monthly precipitation).

This rule can be physically expected because much precipitation results from merid-ional transport dominance together with strong cyclonic activity above the Europeancontinent.


Table 6.9 Standard deviation of observed and estimated time series using differentfuzzy rule-based models for station Keszthely in western Hungary. (Model 1: cyclonic–anticyclonic CP+ 2 SOI; Model 2: zonal–half-meridional–meridional CP+ 2 SOI;Model 3: cyclonic–anticyclonic dominancy and zonality (6 CP classes)+ 2 SOI; Model4: only 6 CP classes; Model 5: only 2 SOI.)

Season Jan.–March May–Aug. Sept.–Nov.

Observed time series 23.6 41.5 36.8

Model 1 11.3 10.1 14.7Model 2 15.8 23.6 16.7Model 3 22.0 35.1 30.0

Model 4 18.7 28.5 25.5Model 5 4.1 7.3 4.4

The results of the FRB models using five fuzzy sets for each premise and 11 for theresponse are summarized by providing the standard deviations (Table 6.9) of observedand estimated monthly precipitation for all seasons for the climate station Keszthelylocated in the western part of the country. It is evident from Table 6.9 that the FRBmodel using 6 + 2 premises reproduces the empirical standard deviation quite well.The frequency distributions of the calculated precipitation reproduce the empiricaldistributions (Figure 6.15).

The results are quite sensitive to the selection of the number of premises. Withfewer premises the reproduction of standard deviation becomes worse (Table 6.9). Forinstance, if only either cyclonic–anticyclonic or zonal–halfmeridional–meridional CPclasses are used, the model does not reproduce the empirical frequency distributions(Figure 6.16). The necessity of considering both cyclonic–anticyclonic dominanceand zonality can be explained easily. Zonal or meridional airflow determines themain circulation characteristics, whereas the major synoptic phenomena, cyclonesand anticyclones, are mainly responsible for the regional precipitation.

Results for Arizona

In Arizona monthly precipitation data from eight stations, distributed quite evenlyover the state, are used with the FRB model. Daily observed 500 hPa level geopotentialheight data describe the atmospheric CPs defined at 35 grid points on a diamond gridover the southwestern U.S. For the classification of CP, an automated, nonhierarchicalmethod, namely, PCA coupled with k-means clustering technique, is used. Twentyprincipal components are maintained to explain 97–98% of the variance. There aresix, seven, seven, and eight types for winter (January–March), spring (April–June),summer (July–September) and fall (October–December), respectively.


Figure 6.15 Cumulative frequency distributions of monthly precipitation time series inwestern Hungary (Keszthely) using six CP classes and two SOI as premises.


Figure 6.16 Cumulative frequency distributions of estimated monthly precipitation time seriescompared to observed precipitation in western Hungary (Keszthely) using two (cyclonic–anticyclonic) or three (zonal–half-meridional–meridional) CP classes and two SOI as premises.

Concerning the other type of premises, lag correlations between monthly precip-itation and SOI are higher than in Hungary; e.g., in Grand Canyon the statisticallysignificant (80%) correlations, <0.19, correspond to lags of −2,−5,−9, and −10months. Thus, SOI is represented with these four monthly values as premises at thisstation.

The number of years chosen for validation is up to 8 years. Figure 6.17 shows theresults of the FRB model for December at Grand Canyon, compared with the resultsof a multiple regression model. It is clear that the FRB model provides a better fit forboth the calibration and validation period.

Results for Germany

Twelve CPs have been defined by using 500 hPa geopotential height anomalies overthe North Atlantic/European region. The defining process of fuzzy rules is basedon the position of high and low air pressure anomalies [Bardossy et al., 1995]. Thefuzzy rules are obtained automatically by using an optimization of the performanceof the classification, which is measured by rainfall frequencies and rainfall amountsconditioned on the CP. In this application wet and dry CPs were defined. Here, thetraining and validation periods were selected as 1980–89 and 1970–79, respectively.

Figures 6.18 and 6.19 show the distributions of the mean normalized 500 hPageopotential height anomalies for the wettest and the driest CPs, respectively. CP 1(Figure 6.19) is the most frequent CP and dominates during the whole year, havingthe average annual frequency of about 40%. At the same time it is the driest CP withthe lowest precipitation probability (25.6%), lowest mean wet-day amount (1.1 mm),


Figure 6.17 Comparison of the results of fuzzy rule-based and multiple regression modelswith observed monthly precipitation, Grand Canyon, December.

and lowest wetness index (0.63). The map shows that CP 1 is characterized by apronounced high-pressure anomaly east of the British Isles, which causes a weakair movement and transport of dry air masses from northeastern Europe to centralEurope. Figure 6.18 shows CP 3. CP 3 is a typical wet CP which has the second high-est precipitation probability (66.9%), highest mean wet-day amount (2.9 mm), andhighest wetness index (1.65). Figure 6.18 shows that CP 3 is characterized by a typ-ical negative air pressure anomaly north of the British Isles and a positive anomalyover the eastern Atlantic. This distribution of air pressure anomalies causes a typ-ical west cyclonic transport of wet, ocean air mass from the northern Atlantic tocentral Europe. All the maps of air pressure anomalies produced by the automatedclassification method show physically realistic results.

Wetness indices (defined as the ratio of precipitation contribution and occurrencefrequency) for every CP, in every season (and for the annual cycle), were computedfor the nine stations in Germany (Figure 6.20). For instance, in the case of stationStuttgart (indicated by ‘∗’ on the maps), the wettest CP is CP 3 and the driest CP isCP 1. Figure 6.20 shows that both the above-described CPs have the same (wet ordry) character at all stations. Also, in the case of other CPs, it holds that wet CPsare wet and dry CPs are dry for almost all stations simultaneously. The spatiallyhomogeneous character of wetness index patterns is reflected in the relatively lowvalues of the coefficient of variation calculated from all stations and for a selectedCP (ranging to 0.08 for CP 9 and 0.29 for CP 2). The negative correlation (−0.47)between the average wetness index and the coefficient of variation indicates that


Figure 6.18 Mean normalized distributions of 500 hPa geopotential height of a wet CP, aver-aged over 1970–79. (CPs are precipitation-optimized in 1980–89 with 500 hPa data and ninestations in Germany.) (See also color insert.)

the spatial variability of the wetness index is greater for dry CPs. This is causedby the fact that the wet CPs have a west or north cyclonic character which causesprecipitation events covering large areas. However, for dry CPs, local precipitationevents (especially convective rainfalls in summer) are typical.

6.5 Discussion and Conclusions

The fuzzy rule-based approach was used successfully over four regions—Arizona,Nebraska, Germany, and Hungary—and under three different climates—semiarid,dry, and wet continental—to predict the statistical properties of monthly precipitationand drought index from the joint forcing of CP and ENSO. Because of the weak-ness and nonlinearity of this relationship over these regions, traditional methods offorecasting have limited possibilities. The FRB technique is able to reproduce theobserved time series even for the validation period if the relationship is relativelystronger, as in Arizona. The reproduction of the time series is relatively weaker inNebraska and even more in Hungary, located very far from the ENSO region. The

6.5 Discussion and Conclusions 185

Figure 6.19 Mean normalized distributions of 500 hPa geopotential height of a dry CP, aver-aged over 1970–79. (CPs are precipitation-optimized in 1980–89 with 500 hPa data and ninestations in Germany.) (See also color insert.)

possibility of using this technique for real-time forecasting is thus variable. On theother hand, in every case, the observed frequency distributions of both precipitationand drought index are correctly reproduced.

The fuzzy rule-based technique has the potential to generate time series of regionaldrought indices and/or precipitation under climate change scenarios. The main ideais to use, instead of the historical CP and ENSO data, results of general circulationmodels (GCM) with the established fuzzy rule-based linkage. Several GCMs areable to reproduce features of present atmospheric general circulation patterns quitecorrectly (e.g., [Mearns et al., 1999]). Also, recently, GCM-produced ENSO indiceshave become available [Meehl and Washington, 1996; Timmermann et al., 1999].

The following conclusions can be drawn from the experience gained at the fourremote regions under different climates:

1. In every case, the fuzzy rule-based approach reproduces the statistical propertiesof monthly precipitation and drought index.

2. The best results require consideration of the joint forcing of CP and ENSO infor-mation. Separate use of either the relative frequencies of CP types as premises

Figure 6.20 Spatial variability of yearly wetness index for one wet (top) and one dry (bottom)CP for nine stations in Germany, 1970–79. (CPs are precipitation-optimized in 1980–89 with500 hPa data and nine stations in Germany.)

References 187

or the lagged SOI shows that neither formulation can reproduce the empiricalfrequency distributions.

3. Statistical measures of dependence between CP, ENSO, and precipitation/droughtindex are relatively weak, precluding the use of other techniques such as multi-variate regression.

4. In every case, the calculated time series reproduces the observed time series forthe calibration period.

5. In Arizona the calculated precipitation time series, and in Nebraska the calcu-lated drought index, reproduce fairly well the observed time series, even for thevalidation periods.

6. In Hungary the observed time series for the validation periods are not reproduced.7. All the maps of air pressure anomalies produced by the automated classification

method show physically realistic results. All wet and dry CPs provide a spatiallyhomogeneous character of wetness index in the German stations.

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Chapter 7 Formal Concept Analysis in Geology

Radim Belohlávek

7.1 Introduction 1927.1.1 Directly observable data: objects and their attributes 192

7.1.2 Analysis: discovery of hidden attribute dependencies and natural

concepts 192

7.2 Formal Concept Analysis: What and Why 1937.2.1 Origins 193

7.2.2 Informal outline 193

7.2.3 Discovering natural concepts hidden in the data 194

7.2.4 Hierarchy of discovered concepts 196

7.2.5 Attribute dependencies 197

7.2.6 Fuzziness and similarity issues 198

7.3 Formal Concept Analysis of Fuzzy Data: a Guided Tour 1997.3.1 Fuzzy context and fuzzy concepts: input data and hidden

concepts 199

7.3.2 Fuzzy concept lattices: hierarchy of hidden concepts 203

7.3.3 Attribute implications 204

7.4 Similarity and Logical Precision 2077.4.1 Similarity relations 207

7.4.2 Similarity of objects and similarity of attributes 209

7.4.3 Similarity of concepts 211

7.4.4 Compatible similarities and factorization 212

7.4.5 Similarity of concept lattices 214

7.4.6 Logical precision 215

7.5 Formal Concept Analysis Demonstrated: Examples 217Acknowledgments 236References 236

191

192 7 Formal Concept Analysis in Geology

7.1 Introduction

7.1.1 Directly observable data: objects and their attributes

When humans formulate their knowledge about some domain of interest, they usu-ally recognize objects and (their) properties. Objects and attributes (properties) are,indeed, primary phenomena when one observes the physical world. When expertsbegin to explore an unknown area of interest, their first step is to identify relevantobjects and their attributes. Then, experts identify what objects have which attributes.With this object–attribute knowledge at hand, experts can start further investigationssuch as various kinds of relationships between attributes and a natural classificationscheme. Attributes can be useful in devising suitable criteria according to which theobjects relevant to the domain may be naturally classified. Furthermore, it is oftenfound that, in order to get an insightful view into the domain of interest, one needsto establish a reasonable conceptual system, i.e., a collection of concepts (specific tothe domain) with basic relationships between the concepts.

7.1.2 Analysis: discovery of hidden attribute dependencies and

natural concepts

What was outlined above is the more true for biological sciences, geological sciences,etc. Let us illustrate the above general description by an example. Suppose we arriveat a new territory with completely unknown living organisms (or suppose we are inour world but do not know anything about the organisms living here) and supposewe want to know more about the organisms. In other words, we want to be able to domore than just recognize and distinguish the organisms we encounter. The objects ofour domain are thus the living organisms (or a suitable collection of them). We maynow select several attributes of these organisms that seem to some extent relevant andelementary (directly observable). So far, our knowledge is limited to the knowledgeof what objects have what attributes. Note that this knowledge is naturally depictedin the form of a (two-dimensional) table with rows corresponding to objects andcolumns corresponding to attributes. A data entry corresponding to a table cell, whichis the intersection of the xth row (row corresponding to object x) and the yth column(column corresponding to attribute y), contains the value of attribute y on object x.This value may be a numerical value (if the attribute is quantifiable, e.g., the weightin kilograms), a logical value (in case of qualitative attributes like “hard”), or someother value (e.g. where the object was found). Typical of geological and biologicalsciences is the fact that some attributes are commonly fuzzy (e.g. “hard”) in the sensethat an attribute applies to an object only to a certain degree.

To get a deeper insight into the object–attribute data, one naturally asks what are thedependencies and relationships among the attributes that can be read from the table.

7.2 Formal Concept Analysis: What and Why 193

For instance, there might be dependencies, not visible on the surface, that tell us thata certain combination of certain attributes determines to some extent another attributeor attributes. As an example, consider a dependency “if x lives in the water then x

has . . .” One wishes to have an automatic procedure for obtaining the dependenciesfrom the data. Another natural question relates to the fact that it is almost impossible tocommunicate knowledge without a conceptual system that is appropriate for a givendomain. That is, one looks for what natural concepts are hidden in the data and what isthe hierarchical structure of these concepts. For instance, one expects that some naturalconcepts like “a flying predator” are in some way hidden in the present object–attributedata, and that some natural hierarchy of those concepts is in the data as well.

7.2 Formal Concept Analysis: What and Why

7.2.1 Origins

The kind of data analysis outlined in the previous section is the main objective offormal concept analysis. The roots of formal concept analysis go to the paper byWille [1982]. In this paper, he outlined his program of “restructuring lattice theory.”The main aim was to develop a lattice theory close to the original motivations of thetheory of ordering. This is best illustrated by the following quotation from the paper:

“The approach to lattice theory outlined in this paper is based on an attempt to reinvigorate thegeneral view of order. For this purpose we go back to the origin of lattice concept in nineteenth-century attempts to formalize logic, where the study of hierarchies of concepts played a centralrole [cf. Schröder, 1890–95] . . .. In set-theoretical language, this gives rise to lattices whoseelements correspond to the concepts . . . and whose order comes from the hierarchy of concepts.”

The theory that resulted from this endeavor is called the theory of concept lattices.The part dealing with applications to the analysis of object–attribute data is knownas formal concept analysis. The basic reference is Ganter & Wille [1999], where onecan also find an extensive list of publications related to both theory and applications;see also Ganter [1994], Ganter et al. [1987], and Wille [1992]. Extension of conceptlattices and formal concept analysis to the case of fuzzy data (which is the tool we areinterested in) can be found in Belohlávek [2002] and Pollandt [1997]; the first paperon this topic is Burusco & Fuentes-Gonzáles [1994].

7.2.2 Informal outline

The rest of this section is devoted to an informal outline of the conceptual frameworkof formal concept analysis. The basic notion which serves to represent the inputobject–attribute knowledge is that of a formal context. Formal context consists of a


Table 7.1 Input data in the tabular form. I (xi , yj ) is the degree towhich attribute yj applies to object xi .

y1 … yj … yl

x1...

......

xi … … I (xi , yj ) … …...

...

xk

...

set X of objects, a set Y of attributes, and a relation I between objects and attributes.The set X represents objects that are relevant to our domain of interest, i.e., objects towhich we restrict our attention. Likewise, Y contains relevant attributes. We restrictourselves to the case where attributes are qualitative, i.e., they either apply or don’tapply or apply only to a certain truth degree. An example of such an attribute is “tobe found in North America.” Provided North America is sharply delineated, this isan example of a so-called crisp attribute; each object x either was found in NorthAmerica (the attribute takes logical value 1 on this object) or was not (the attributetakes logical value 0 on this object). The attribute “hard” is an example of a typicalfuzzy attribute; the fact that an object is hard may be assigned a truth degree, say, 0.7if the object is more or less hard. Therefore, the input data specify for each object x

from X and each attribute y from Y to which extent the attribute y applies to the objectx. This is naturally done by specifying the truth degree I (x, y) of the fact “y appliesto x.” The degree I (x, y) is supplied by an expert’s observation of the domain. Inpractice, the sets X and Y are of course finite. The input data can thus be put into atable specifying the values I (x, y) for each x from X and y from Y . Let the elementsof X be x1, . . . , xk , and the elements of Y be y1, . . . , yl . Then the input data, i.e., thetriple 〈X, Y, I 〉, can be represented by a table (Table 7.1).

Having specified the input data 〈X, Y, I 〉, one is interested in the analysis of thesedata. In this preliminary outline, we focus on the discovery of natural concepts hiddenin the data, the discovery of attribute dependencies in the form of attribute implica-tions, and the measurement of similarity of attributes and similarity of the discoveredconcepts.

7.2.3 Discovering natural concepts hidden in the data

First of all, one needs to say what is to be understood by a (formal) concept. In formalconcept analysis, a concept is understood according to a longstanding tradition of


Port-Royal logic [Arnauld & Nicole, 1662]; see also Höfler [1906]. A concept isdetermined by its extent and its intent. The extent of a concept is the collection of allobjects that are covered by the concept. The intent of a concept is the collection ofall attributes covered by the concept. For instance, consider the concept DOG. Theextent of DOG consists of all dogs while the intent of DOG consists of all attributesthat apply to dogs (like “to be a mammal,” “to bark,” etc.). Therefore, a concept informal concept analysis is understood to be a pair 〈A, B〉 consisting of a collectionA of objects (extent) and a collection B of attributes (intent). In order to qualify as aconcept, the pair 〈A, B〉 has to satisfy some constraints. Note that this is extremelyimportant and makes formal concept analysis what it is. If there were no constraint,the whole thing would be useless; in this case, each pair 〈A, B〉 would be a conceptand so “being a concept” would carry no information. The constraint being used informal concept analysis is very simple and can be described verbally as follows: Apair 〈A, B〉 is called a (formal) concept if

B is the collection of all attributes shared by all objects from A

and

A is the collection of all objects sharing all the attributes from B.

Therefore, given the input data 〈X, Y, I 〉, there might be several pairs 〈A, B〉 whereA is a subcollection of X and B is a subcollection of Y that satisfy the definition ofa formal concept. These formal concepts are hidden in the input data in that theirpresence is not obvious by just looking at the table. A procedure that takes 〈X, Y, I 〉

as its input and generates the list of formal concepts that are hidden in 〈X, Y, I 〉 maybe considered as performing a discovery of natural concepts that “exist in the data.”Formal concepts that arise from such a procedure are, in a sense, meaningful clustersof objects and attributes (where “meaningful” is to be considered with respect to(w.r.t.) the conceptual interpretation). We denote the collection of all formal conceptshidden in 〈X, Y, I 〉 by B (X, Y, I ). Let us illustrate the notion of a concept by a simpleexample.

Example 7.1

A simple example illustrates these notions. Let X = {x1, x2, x3}, Y = {y1, y2, y3, y4}

and consider a binary relation I given by Table 7.2. That is, x1 has attributes y1 andy2 but does not have attributes y3 and y4. Although this is a very simple example,we will interpret xi and yj as follows. Let x1, x2, x3 be some geological objects, sayminerals; let y1 mean “was found in North America,” y2 mean “was found in SouthAmerica,” y3 mean “was found in Asia,” y4 mean “was found in Europe.” However,note that “object” is just a technical term. In our example, xi refers to a whole groupof minerals of the same sort (one particular mineral cannot be found in both North


Table 7.2 Input data to Example 7.1.

y1 y2 y3 y4

x1 1 1 0 0x2 0 1 1 0x3 0 0 1 1

Table 7.3 Concepts from Example 7.1.

x1 x2 x3 y1 y2 y3 y4

c1 1 1 1 0 0 0 0c2 0 1 1 0 0 1 0c3 0 0 1 0 0 1 1c4 1 1 0 0 1 0 0c5 0 1 0 0 1 1 0c6 1 0 0 1 1 0 0c7 0 0 0 1 1 1 1

America and South America). There are seven concepts hidden in these data; theyare listed in Table 7.3. For instance, concept c4 is a pair 〈A4, B4〉 with the extentA4 = {x1, x2} and the intent B4 = {y2}. That is, c4 covers mineral x1 and mineralx2, and covers the attribute “found in South America.” We comment further on theconcepts below.

7.2.4 Hierarchy of discovered concepts

The next issue is the hierarchy of discovered concepts. Hierarchy of concepts w.r.t.their generality is a basic relation that accompanies concepts. For instance, the conceptDOG is a subconcept of the concept MAMMAL, MAMMAL is a superconcept ofDOG. We denote the conceptual hierarchy by ≤ and write 〈A1, B1〉 ≤ 〈A2, B2〉

to denote that the concept 〈A1, B1〉 is a subconcept of the concept 〈A2, B2〉. Beingmore general means covering more objects (or, which is equivalent, less attributes).Therefore, it is only natural to define ≤ by

〈A1, B1〉 ≤ 〈A2, B2〉 if and only if A1 is a subcollection of A2 or,equivalently, if and only if B2 is a subcollection of B1.

It is easily seen that hierarchy defined in this way is a partial order, i.e., it is reflexive(c ≤ c), antisymmetric (c1 ≤ c2 and c2 ≤ c1 imply c1 = c2), and transitive (c1 ≤ c2

and c2 ≤ c3 imply c1 ≤ c3). Moreover, for each collection of formal concepts


Figure 7.1 Hierarchy of hidden formal concepts from Example 7.1.

from B (X, Y, I ), there exist both their direct superconcept (generalization) and theirdirect subconcept (specialization) in B (X, Y, I ) (see the next section). Therefore, ≤obeys the laws naturally required for a complete conceptual system. The hierarchicalstructure of the collection B (X, Y, I ) w.r.t. the hierarchy order ≤ is easily depictedby a so-called Hasse diagram. The next example serves to illustrate this.

Example 7.2

Consider the input data from Example 7.1. The hierarchical structure of B (X, Y, I )

is depicted in Figure 7.1. Concept c1 is the most general concept; its extent containsall objects (x1, x2, x3). On the other hand, c7 is the empty concept; its extent does notcontain any object. Between c1 and c7 there are five concepts. For instance, conceptsc3 and c5 have the verbal descriptions “to be found in Asia and in Europe” and “to befound in South America and in Asia,” respectively. Concept c2 is the join of c3 andc5 and, therefore, c2 is the direct generalization of c3 and c5. Indeed, the intent of c2

is {y3}, which means that the verbal description of c2 is “to be found in Asia.”

7.2.5 Attribute dependencies

Attribute dependencies, as approached in formal concept analysis, are expressed byimplications of the form

attributes y1, . . . z1 imply attributes y2, . . . , z2,

written formally {y1, . . . , z1} ⇒ {y2, . . . , z2}. This implication means that eachobject that has all of the attributes y1, . . . , z1 has also all of the attributes y2, . . . , z2;


it is this sense in which the implication is valid in given input data 〈X, Y, I 〉. Attributeimplications reveal other kinds of knowledge hidden in the input data. The analysisof attributes may reveal that some attributes are only simple combinations of others.We are surely interested in all implications that are valid in the input data. However,listing of all implications can yield a huge number of implications. Moreover, someof them are trivial (such as {y1} ⇒ {y1}), some of them follow (in a natural but pre-cisely defined sense) from other implications. Formal concept analysis has means togenerate only basic implications (in the sense that all other implications follow fromthe basic ones). The next example illustrates the basics of attribute implications.

Example 7.3

Consider again the input data of Example 7.1. We can see that there are, for instance,the attribute implications {y4} ⇒ {y3} and {y1} ⇒ {y2} which are true in the inputdata. Implication {y4} ⇒ {y3} says that each mineral found in Europe was also foundin Asia, and {y1} ⇒ {y2} says that each mineral found in North America was alsofound in South America. On the contrary, implication {y3} ⇒ {y4} is not true in thedata since x3 has the attribute y3 but does not have the attribute y4.

7.2.6 Fuzziness and similarity issues

The example we used to demonstrate the basic notions of formal concept analysis wasspecific in that the attributes were crisp. However, most empirical attributes are fuzzy.What was described above also applies if attributes are fuzzy; we only need correctlyto interpret the verbal description (see next section). In the case that the input datacontains fuzzy attributes, an important phenomenon is that of similarity. Similaritycan be basically considered on three levels: similarity of attributes (and similarity ofobjects); similarity of concepts; and similarity of the conceptual structures.

Similarity is a graded notion. Objects x and y may be more similar than objectsx and z are. Simplifying reality using similarity is the very nature of how humanscope with the complexity of the outer world. Basically, the simplification is done byidentifying objects that are “very similar.” The process of identification of elementsis known as factorization. It is the usual formal model of what people call abstraction.What “very similar” means depends on how coarse the factorization is and how muchabstraction is required. Fuzzy logic has natural means to model factorization w.r.t.graded similarity.

Intuitively, we consider two attributes similar if they apply to each object of thedomain of discourse approximately to the same extent. This makes it possible to reducethe input data, i.e., to identify attributes that are “very much” similar. Distinguishingvery similar attributes would lead to overly detailed and extensive analysis.

From the input (fuzzy) data one can generate the list of all formal concepts hiddenin the data and the hierarchical structure of the concepts. A natural question is that of

7.3 Formal Concept Analysis of Fuzzy Data: a Guided Tour 199

how the similarity of the formal concepts can be measured. Intuitively, we considertwo concepts similar if they apply to all objects to approximately the same extent.If one finds one does not need that level of discernibility which is represented bythe generated structure of concepts, one may wish to simplify the conceptual systemby identifying concepts that are sufficiently similar. Doing so, one obtains a simplerconceptual system that is (for the given purposes) sufficient.

Finally, a natural problem is the similarity of two conceptual systems. A conceptualsystem is a system of basic abstract units (concepts) which allows efficient commu-nication. Given two systems, an immediate question is to what extent are the twoequivalent in that each concept of one of them may be described by concepts in theother one.

Formal concept analysis of fuzzy data has means for naturally modeling all of thethree levels of similarity described above. Basically, it answers the questions of (a)how to measure similarity, and (b) how to simplify (factorize) by similarity.

7.3 Formal Concept Analysis of Fuzzy Data: a Guided Tour

This section presents basic notions and results of formal concept analysis of fuzzydata. Theorems are presented without proofs, which can be found in Belohlávek[2002b].

7.3.1 Fuzzy context and fuzzy concepts: input data and hidden

concepts

Let X be a set of objects and Y be a set of attributes to which we restrict our attention.Let L be a (suitable) set of truth degrees. Furthermore, let I be a binary fuzzy relationwith truth degrees in L; that is, I assigns to each x ∈ X and each y ∈ Y a truth degreeI (x, y) ∈ L. The degree I (x, y) is interpreted as the truth degree to which object x

has attribute y.

Definition 7.1

The above triple 〈X, Y, I 〉 is called a fuzzy context.Mostly, L is taken to be some subset of [0, 1]. As we will see, we need operations

on L that correspond to logical connectives. That is, L should be equipped with acouple of operations corresponding to conjunction, implication, etc. We will providea general structure for this purpose (L will be equipped with a structure of so-calledcomplete residuated lattice) and then show particular examples of this structure thatare most commonly used in applications.

Complete residuated lattices are the basic structures of truth values used in fuzzylogic in the narrow sense [Goguen, 1967, 1968–69; Höhle, 1995, 1996; Hájek, 1998].


The reader can find basic information about lattices and partially ordered sets in Davey& Priestley [1990]. A complete residuated lattice [Ward & Dilworth, 1939] is analgebra L = 〈L,∧,∨,⊗,→, 0, 1〉 such that: (1) 〈L,∧,∨, 0, 1〉 is a complete latticewith the least element 0 and the greatest element 1; (2) 〈L,⊗, 1〉 is a commutativemonoid, i.e., ⊗ is commutative, associative, and x⊗ 1 = x holds for each x ∈ L;and (3) ⊗,→ are binary operations which form an adjoint pair, i.e., x⊗ y ≤ z ifand only if x ≤ y → z holds for all x, y, z ∈ L. The operation ⊗ corresponds toconjunction;→ corresponds to implication.

The most studied and applied set of truth values is the real interval[0, 1]. Each left-continuous t-norm ⊗ induces a complete residuated lattice〈[0, 1], min, max,⊗,→, 0, 1〉 where → is given by a → b = max{c | a⊗ c ≤

b} (and conversely, each residuated lattice on [0, 1] is induced in this way bysome left-continuous t-norm); for details and more information see Belohlávek[2002b], Hájek [1998]. The most popular t-norms are: (1) the Łukasiewicz t-norm(a⊗ b = max(a + b − 1, 0), a → b = min(1 − a + b, 1)); (2) the Gödel t-norm (a⊗ b = min(a, b), a → b = 1 if a ≤ b and a → b = b else); and (3)the product t-norm (a⊗ b = a · b, a → b = 1 if a ≤ b and a → b = b/a

else). They are all continuous. On the other hand, any continuous t-norm can becomposed in a simple way out of these three; see Belohlávek [2002b] or Hájek[1998]. Another important set of truth values is the set {a0 = 0, a1, . . . , an = 1},a0 < · · · < an, where the ordering determines the complete lattice structure. Twot-norms are often considered: (1) ak ⊗ al = amax(k+l−n,0) and the corresponding →given by ak → al = amin(n−k+l,n) (Łukasiewicz); and (2) ak ⊗ al = amin(k,l) and thecorresponding→ given by ak → al = 1 if k ≤ l and al else (Gödel). A special caseof the latter algebras is the Boolean algebra 2 of classical logic with the support set{0, 1}. It may be easily verified that the only t-norm on {0, 1} is the classical conjunc-tion operation∧, i.e., a∧b = 1 if and only if a = 1 and b = 1, which implies that theonly residuum operation is the classical implication operation→, i.e., a → b = 0 ifand only if a = 1 and b = 0. Note that each of the preceding residuated lattices iscomplete.

In the following, L always denotes some complete residuated lattice. However,there will be no substantial loss if one assumes that L is [0, 1] or some finite subchainof [0, 1].

A fuzzy set (or L-set) A in a universe set X is a mapping A : X → L. The valueA(x) ∈ L is interpreted as the truth value of the statement “the element x belongsto A.” The set of all fuzzy sets in X is denoted by LX. For A1, A2 ∈ LX we writeA1 ⊆ A2 if and only if A1(x) ≤ A2(x) for all x ∈ X. Similarly, a binary fuzzy relationR between X and Y is a mapping R : X × Y → L. Particularly, {a1/x1, . . . , an/xn}

denotes a fuzzy set A with A(x1) = a1, . . . , A(xn) = an, and A(x) = 0 for x �= xi

(i = 1, . . . , n).Coming back to the notion of a fuzzy context, we want to formalize the notion of

a concept. According to Port-Royal, a concept consists of a collection of objects (its


extent) and a collection of attributes (its intent). If the attributes in the context arefuzzy, both extent and intent are assumed to be fuzzy sets. Consider, for example,the concept hard mineral. Clearly, its extent is a fuzzy set rather than a crisp set.Following the verbal definition, we need to define two operators, ↑ and ↓. The intendedmeaning of ↑ and ↓ is the following. For a fuzzy set A of objects (i.e., A ∈ LX),A↑ is the fuzzy set of all attributes (i.e., A↑ ∈ LY ) shared by all objects from A;for a fuzzy set B of attributes (i.e., B ∈ LY ), B↓ is the fuzzy set of all objects (i.e.,B↓ ∈ LX) sharing all attributes from B. The basic semantic rules of fuzzy logic givethe following. For a fuzzy context 〈X, Y, I 〉, A ∈ LX and B ∈ LY , A↑ and B↓ are afuzzy set in Y and a fuzzy set in X, respectively, defined by

A↑I (y) =∧

x∈X

A(x)→ I (x, y) (7.1)

B↓I (x) =∧

y∈Y

B(y)→ I (x, y) (7.2)

for each y ∈ Y and x ∈ X. Therefore, (7.1) and (7.2) define mappings ↑I : LX → LY ,↓I : LY → LX. If I is obvious, we write only ↑ and ↓ instead of ↑I and ↓I .

Example 7.4

Let X = {x1, x2, x3}, Y = {y1, y2}, L = [0, 1], I (x1, y1) = 1, I (x1, y2) = 0.3,I (x2, y1) = 0.8, I (x2, y2) = 0.9, I (x3, y1) = 0, I (x3, y2) = 0.1. Consider A =

{0.5/x1, 1/x2, 0/x3}, B = {0.7/y1, 0.3/y2}. We want to determine A↑I and B↓I . Forinstance, with the Łukasiewicz structure on [0, 1], we have

A↑I (y1) =∧

x∈X

A(x)→ I (x, y1)

= [A(x1)→ I (x1, y1)] ∧ [A(x2)→ I (x2, y1)] ∧ [A(x3)→ I (x3, y1)]

= [0.5 → 1] ∧ [1 → 0.8] ∧ [0 → 0]

= 1 ∧ 0.8 ∧ 1 = 0.8

A↑I (y2) =∧

x∈X

A(x)→ I (x, y2)

= [A(x1)→ I (x1, y2)] ∧ [A(x2)→ I (x2, y2)] ∧ [A(x3)→ I (x3, y2)]

= [0.5 → 0.3] ∧ [1 → 0.9] ∧ [0 → 0.1]

= 0.8 ∧ 0.9 ∧ 1 = 0.8


B↓I (x1) =∧

y∈Y

B(y)→ I (x1, y)

= [B(y1)→ I (x1, y1)] ∧ [B(y2)→ I (x1, y2)]

= [0.7 → 1] ∧ [0.3 → 0.3]

= 1 ∧ 1 = 1

B↓I (x2) =∧

y∈Y

B(y)→ I (x2, y)

= [B(y1)→ I (x2, y1)] ∧ [B(y2)→ I (x2, y2)]

= [0.7 → 0.8] ∧ [0.3 → 0.7]

= 1 ∧ 1 = 1

B↓I (x3) =∧

y∈Y

B(y)→ I (x3, y)

= [B(y1)→ I (x3, y1)] ∧ [B(y2)→ I (x3, y2)]

= [0.7 → 0] ∧ [0.3 → 0.1]

= 0.3 ∧ 0.8 = 0.3.

Changing the structure on [0, 1] changes the operators ↑I and ↓I . For instance, withthe Gödel structure on [0, 1], we get A↑I (y1) = 0.8, A↑I (y2) = 0.3, B↓I (x1) = 1,B↓I (x2) = 1, B↓I (x3) = 0.

With these definitions, the verbal definition of a concept may be formalized asfollows.

Definition 7.2

A fuzzy concept in a fuzzy context 〈X, Y, I 〉 is each pair 〈A, B〉 of a fuzzy set A ∈ LX

of objects and a fuzzy set B ∈ LY of attributes such that A↑ = B and B↓ = A.Indeed, this definition just makes formal the verbal constraints that have to be

fulfilled by the extent A and the intent B of a concept consisting of A and B.The set of all fuzzy concepts in a given fuzzy context 〈X, Y, I 〉 will be denoted by

B (X, Y, I ). That is, we have

B (X, Y, I ) = {〈A, B〉 ∈ LX × LY | A↑I = B, B↓I = A}.

B (X, Y, I ) will be called a fuzzy concept lattice determined by a fuzzy context〈X, Y, I 〉 (the term concept “lattice” will be justified later). We write only “con-text,” “concept,” and “concept lattice” if L is obvious. On the other hand, if we want


to emphasize the structure L of truth values, we write “L-context,” “L-concept,” and“L-concept lattice.” It is important to note that if L = 2, i.e., the structure of truthvalues is the two-element set L = {0, 1}, then the notions of an L-context, L-concept,and L-concept lattice become the notions of a (crisp) context, (crisp) concept, and a(crisp) fuzzy concept lattice developed by Wille [1982].

Example 7.5

For L = 2 (i.e., L = {0, 1}, thus we have 0 and 1 as the only truth values), wehave that A↑(y) = 1 (i.e., y belongs to A↑) if and only if for each x ∈ X such thatA(x) = 1 we have I (x, y) = 1, or, in other words, each x ∈ A is in relation I withy, which is the meaning of A↑ in the crisp case. The situation for B↓ is symmetric.

Fuzzy concepts may be viewed as maximal rectangles in the object–attribute tablecorresponding to a fuzzy context. Although this is especially appealing in the crispcase, i.e., L = 2, we will demonstrate this alternative view of concepts in general. Abinary fuzzy relation I ′ between X and Y is called a rectangular relation if and only ifthere are A ∈ LX, B ∈ LY such that I ′(x, y) = A(x)⊗B(y) (for all x ∈ X, y ∈ Y ),written I ′ = A⊗B. In this case, the pair 〈A, B〉 is called a rectangle. A rectangle〈A, B〉 is said to be contained in a binary fuzzy relation I ′′ if A⊗B is contained inI ′′, i.e., if A⊗B ⊆ I ′′. There is a naturally defined ordering ≤ defined on the setof all rectangles by 〈A1, B1〉 ≤ 〈A2, B2〉 if and only if for all x ∈ X, y ∈ Y wehave A1(x) ≤ A2(x) and B1(y) ≤ B2(y). The following theorem, if interpreted forL = 2, says that concepts are just maximal rectangles of I which are filled with 1s(if we consider the two-valued relation I as a matrix-table of 0s and 1s).

Theorem 7.1

For a fuzzy context 〈X, Y, I 〉 and A ∈ LX, B ∈ LY , we have that 〈A, B〉 is a fuzzy

concept iff it is a maximal rectangle contained in I .

7.3.2 Fuzzy concept lattices: hierarchy of hidden concepts

We are now going to investigate the set of all fuzzy concepts with its hierarchicalstructure. A thorough treatment on this topic can be found in Belohlávek [2003]. Atthis moment, we confine ourselves to a special case: we concentrate on the “crisp”hierarchy of fuzzy concepts. The subconcept–superconcept relation ≤ on the setB (X, Y, I ) of all fuzzy concepts in a fuzzy context 〈X, Y, I 〉 (i.e., the conceptualhierarchy) is naturally defined by

〈A1, B1〉 ≤ 〈A2, B2〉 iff A1 ⊆ A2 (iff B1 ⊇ B2). (7.3)

That is, the fuzzy concept 〈A1, B1〉 is a subconcept of the fuzzy concept 〈A2, B2〉

if the extent A2 is greater than the extent A1. This means that the degree to which


any object x belongs to A2 is at least as high as the degree to which x belongs toA1. Equivalently, 〈A1, B1〉 is a subconcept of 〈A2, B2〉 if the intent B1 is greater thanthe intent B2. Saying that 〈A1, B1〉 is a subconcept of 〈A2, B2〉 may be equivalentlyexpressed by saying that 〈A1, B1〉 is more special than 〈A2, B2〉, or that 〈A2, B2〉 ismore general than 〈A1, B1〉, or that 〈A2, B2〉 is a superconcept of 〈A1, B1〉.

The structure of B (X, Y, I ) w.r.t. ≤ and its characterization is the subject of thefollowing theorem [Belohlávek 2001, 2003].

Theorem 7.2 (main theorem of fuzzy concept lattices, crisp order)Let 〈X, Y, I 〉 be an L-context. (1) B (X, Y, I ) is a complete lattice in which infima

and suprema can be described as follows:

∧

j∈J

⟨Aj , Bj

⟩=

⟨⋂

j∈J

Aj , (⋂

j∈J

Aj )↑

⟩=

⟨⋂

j∈J

Aj , (⋃

j∈J

Bj )↓↑

⟩(7.4)

∨

j∈J

⟨Aj , Bj

⟩=

⟨(⋂

j∈J

Bj )↓,⋂

j∈J

Bj

⟩=

⟨(⋃

j∈J

Aj )↑↓,⋂

j∈J

Bj

⟩. (7.5)

(2) Moreover, a complete lattice V = 〈V,≤〉 is isomorphic to B (X, Y, I ) iff there

are mappings γ : X × L → V , μ : Y × L → V , such that γ (X × L) is supremally

dense in V, μ(Y × L) is infimally dense in V, and a⊗ b ≤ I (x, y) is equivalent to

γ (x, a) ≤ μ(y, b) for all x ∈ X, y ∈ Y , a, b ∈ L.

Recall that the fact that B (X, Y, I ) is a complete lattice is a very natural one.Recall [Birkhoff, 1967] that a complete lattice is a partially ordered set V where,to each subset V ′ of V , there exists the infimum of V ′ as well as the supremum.Now the supremum of V ′ is the least one of all elements of V which are greaterthan each element of V ′. That is, under the conceptual interpretation, if V ′ is a setof concepts then the supremum of V ′ is a concept which can be thought of as thedirect generalization of all concepts from V ′. Dually, the infimum of V ′ is the conceptwhich can be thought of as the direct common specialization of the concepts from V ′.

7.3.3 Attribute implications

Attribute implications represent another useful set of information that can be extractedfrom the input data (fuzzy context). The basic idea is this. An attribute implication

consists of a pair comprising a fuzzy set A of attributes and a fuzzy set B of attributes.Such an attribute implication will be briefly denoted by A⇒ B (note that we use→to denote residuum (implication operation) in structures of truth values and alsoto denote attribute implications—however, there is no danger of confusion). Now,given the input data, i.e., a fuzzy context 〈X, Y, I 〉, an attribute implication may


be true in 〈X, Y, I 〉 to a certain degree. The degree A⇒〈X,Y,I 〉 B to which A⇒ B

is true in 〈X, Y, I 〉 is verbally described as the degree to which “for each x ∈ X: if x

has all the attributes from A then x has also all the attributes from B.” Formally, wehave

A⇒〈X,Y,I 〉 B =∧

x∈X

(∧

y∈Y

(A(y)→ I (x, y))→∧

y∈Y

(B(y)→ I (x, y))).

In a similar way, one can consider the degree to which an attribute implication is truein B (X, Y, I ). We first elaborate on the details. Then we go to the notion of entailmentof attribute implications and to the notion of a base of attribute implications. Recallthat for fuzzy sets C, D ∈ LU , the degree S(C, D) to which C is contained in D isdefined by S(C, D) =

∧x∈U (C(x)→ D(x)).

Definition 7.3

For a fuzzy context 〈X, Y, I 〉, a fuzzy set M ∈ LY , and an attribute implicationA⇒ B, we put

|= (M, A⇒ B) = S(A, M)→ S(B, M)

and call |= (M, A⇒ B) the degree to which A⇒ B is true in M .Definition 7.3 has a natural interpretation: |= (M, A⇒ B) is the truth degree to

which it is true that whenever A is contained in M then B is as well.We are going to define the notion of validity of a collection of attribute implications

in a collection of fuzzy sets of attributes. |= is an L-relation between the set LY of all L-sets of attributes and the set LY ×LY of all L-attribute implications. As established byBelohlávek [1999] (see also Ore [1944]), this relation induces an L-Galois connection〈∧, ∨〉 between LY and LY × LY , i.e., for M ∈ LLY

and I ∈ LLY×LYthe L-sets

M∧ ∈ LLY×LYand I∨ ∈ LLY

are given by

M∧(A⇒ B) =∧

M∈LLY

M(M)→ |= (M, A⇒ B)

I∨(M) =∧

(A⇒B)∈LLY×LY

I(A⇒ B)→ |= (M, A⇒ B).

Therefore, M∧(A⇒ B) is the truth degree to which A⇒ B is true in each M fromM, and I∨(M) is the truth degree to which each implication from I is true in M .

Particularly, we will be interested in {{1/x}↑ | x ∈ X}∧ and Int(X, Y, I )∧:{{1/x}↑ | x ∈ X}∧(A⇒ B) is the truth degree to which A⇒ B is true in each{1/x}↑ (i.e., the intent of the elementary fuzzy concept 〈{1/x}↑↓, {1/x}↑〉) andInt(X, Y, I )∧(A⇒ B) is the truth degree to which A⇒ B is true in all intentsof B (X, Y, I ). For the sake of brevity we denote for M ∈ LLY

and A⇒ B the


degree M∧(A⇒ B) by A⇒M B and, furthermore, if M = {{1/x}↑ | x ∈ X} wewrite A⇒〈X,Y,I 〉 B instead of A⇒M B. This makes good sense: since {1/x}↑(y) =

I (x, y), A⇒〈X,Y,I 〉 B is the degree to which it is true that, for each x ∈ X, if x has allthe attributes from A then x has all the attributes of B; i.e., we get the notion of a valid-ity of an attribute implication in a fuzzy context. For M = 〈D | 〈C, D〉 ∈ B (X, Y, I )〉

we denote A⇒M B by A⇒B(X,Y,I ) B. That is, A⇒B(X,Y,I ) B is the degreeto which it is true that if A is contained in an intent D of some fuzzy concept〈C, D〉 from the fuzzy concept lattice B (X, Y, I ), then B is contained in D aswell.

Theorem 7.3

For any fuzzy context 〈X, Y, I 〉 and any attribute implication A⇒ B we have

A⇒〈X,Y,I 〉 B = A⇒B(X,Y,I ) B,

i.e., for any implication A⇒ B, the degree to which A⇒ B is valid in the fuzzy

context 〈X, Y, I 〉 equals the degree to which A⇒ B is true in B (X, Y, I ).

Since A⇒〈X,Y,I 〉 B and A⇒B(X,Y,I ) B coincide, we denote both of them simplyby A⇒I B or even by A⇒ B. The following assertions list some basic rules of thecalculus of attribute implications.

Theorem 7.4

For any fuzzy context 〈X, Y, I 〉 and any attribute implication A⇒ B, the truth degrees

A⇒ B, S(A↓, B↓), and S(B, A↓↑) are equal.

Theorem 7.5

For each fuzzy context 〈X, Y, I 〉 we have

A⇒I A = 1, (A⇒I B)⊗(B ⇒I C) ≤ (A⇒I C) (7.6)

S(A1, A2)⊗ S(B2, B1)⊗(A1 ⇒ B1) ≤ (A2 ⇒ B2). (7.7)

Thus, (7.6) says that A always implies A, and that if A implies B and B implies C

then A implies C; (7.7) says that if A1 implies B1 and if A1 is contained in A2 andB1 contains B2, then A2 implies B2.

These rules indicate that some of the attribute implications “follow” from otherimplications. Therefore, it is not necessary to list all the attribute implications.Rather, it is desirable to have only a relatively small “base” of attribute implicationsfrom which all other implications follow. We now formalize these intuitiveconsiderations.

We say that an attribute implication A⇒ B follows from a set {Aj ⇒ Bj | j ∈ J }

of attribute implications A, B, Aj , Bj ∈ LY if, for each fuzzy set D ∈ LY , wehave that A⇒M B = 1 (i.e., the truth degree to which A⇒ B is true in {M} is 1)

7.4 Similarity and Logical Precision 207

whenever, for each j ∈ J , we have Aj ⇒M Bj = 1. For a fuzzy context 〈X, Y, I 〉

we say that a set I = {Aj ⇒ Bj | j ∈ J } of attribute implications which aretrue in degree 1 in 〈X, Y, I 〉 forms a base for 〈X, Y, I 〉 if for each attribute impli-cation A⇒ B which is true in degree 1 in 〈X, Y, I 〉 we have that A⇒ B followsfrom I. A base I for 〈X, Y, I 〉 is called irredundant if no A⇒ B ∈ I followsfrom I − {A⇒ B}. Therefore, an irredundant base provides complete irredun-dant information about the attribute implications. Further information about attributeimplications (including algorithms) can be found in Ganter & Wille [1999] and inPollandt [1997].

7.4 Similarity and Logical Precision

7.4.1 Similarity relations

The similarity phenomenon plays a crucial role in the way humans regard the world.In fact, similarities are induced by the very nature of human perception. Gradualsimilarity of concepts is one of the fundamental preconditions for powerful humanreasoning and communication. The similarity phenomenon is thus one of the mostimportant ones accompanying conceptual structures.

In fuzzy set theory, the similarity phenomenon is approached via so-called simi-larity relations (fuzzy equivalence relations); see Chapter 2. For a structure L of truthvalues, a similarity relation (fuzzy equivalence) [Zadeh, 1971] on a set U is a binaryfuzzy relation E : U × U → L on a universe U satisfying the following propertiesfor all x, y, z ∈ U :

E(x, x) = 1 (7.8)

E(x, y) = E(y, x) (7.9)

E(x, y)⊗E(y, z) ≤ E(x, z). (7.10)

Properties (7.8), (7.9), and (7.10) are called reflexivity, symmetry, and transitivity,respectively. A similarity class of x ∈ U is the fuzzy set [x]E ∈ LU given by[x]E(y) = E(x, y) for each y ∈ U , i.e., it is a collection of elements similar tox. A fuzzy set A ∈ LU is said to be compatible w.r.t. E if for every x, y ∈ U wehave A(x)⊗E(x, y) ≤ A(y). Verbally, this condition says that, with each elementx, A contains all the elements similar to x. It is easily seen that in the crisp case,i.e., L = {0, 1}, similarity relations are equivalence relations. For the study of thesimilarity phenomenon, the crisp case is a degenerate one and non-interesting—two elements x and y may be “fully similar” (E(x, y) = 1) or “fully dissimilar”(E(x, y) = 0).


Example 7.6

For the three basic structures on [0, 1], i.e., Łukasiewicz, Gödel, and product,transitivity translates to

max(0, E(x, y)+ E(y, z)− 1) ≤ E(x, z)

min(E(x, y), E(y, z)) ≤ E(x, z)

E(x, y) · E(y, z) ≤ E(x, z),

respectively. Reflexivity and symmetry conditions are the same for each of the threestructures.

Note that transitivity expresses a condition which can be formulated in words as “ifx and y are similar and if y and z are similar then x and z are similar.” For example,if E(x, y) = 0.8 (x and y are similar in degree 0.8) and E(y, z) = 0.8 (y and z aresimilar in degree 0.8) then x and z have to be similar at least in degree 0.8⊗ 0.8.Thus, in the case of the product structure, transitivity forces E(x, z) ≥ 0.8⊗0.8 = 0.64.

To model the equivalence (or closeness) of truth values we have at our disposal theso called biresiduum (or biimplication) [Pavelka, 1979] operation↔ defined by

a ↔ b = (a → b) ∧ (b → a).

The following lemma will be useful in our considerations.

Lemma 7.1

Let E be a similarity on U , S = {Ai ∈ LU | i ∈ I } be a family of fuzzy sets. (1) E is

the largest similarity relation compatible with all [x]E . (2) The relation ES defined by

ES(x, y) =∧

i∈I

(Ai(x)↔ Ai(y)) (7.11)

is the largest similarity relation compatible with all Ai ∈ S. Moreover, Ai(x) = 1implies [x]ES

⊆ Ai .

Notice that for the crisp case (i.e., L = {0, 1}), ES is a crisp equivalence relation—two elements of the universe are equivalent if and only if there is no set of the familywhich separates them.

Remark 7.1

Lemma 7.1 has a very natural interpretation. Each fuzzy set Ai in U represents someproperty of elements of U ; Ai(x) is the degree to which the element x has the propertyrepresented by Ai . The degree Ai(x) ↔ Ai(y) is the truth degree to which “x hasthe property Ai if and only if y has the property Ai .” Therefore, ES(x, y) is thetruth degree of “for each property A from S: x has A if and only if y has A.” If


S is the system of all relevant properties, ES(x, y) is the truth degree to which x andy have the same (relevant) properties.

Example 7.7

For the three basic structures on the real unit interval [0, 1], i.e., Łukasiewicz, Gödel,and product, we have

ES(x, y) = infi∈I{1− |Ai(x)− Ai(y)|} (Łukasiewicz)

ES(x, y) =

{1 if ∀i : Ai(x) = Ai(y)

inf i∈I {min(Ai(x), Ai(y))} otherwise(Gödel)

ES(x, y) = infi∈I{min(Ai(x)/Ai(y), Ai(y)/Ai(x))} (product)

where we put 0/0 = 1 and a/0 = ∞ for a �= 0.

Example 7.8

We illustrate ES . Let U = {x, y, z}, S = {A, B} where A = {1/x, 0.5/y, 0.1/z},B = {0.9/x, 0.4/y, 0.1/z}. For the Łukasiewicz structure on [0, 1] we get

ES(x, y) = [A(x)↔ A(y)] ∧ [B(x)↔ B(y)]

= [1 ↔ 0.5] ∧ [0.9 ↔ 0.4] = 0.5 ∧ 0.5 = 0.5,

and ES(y, z) = 0.6, ES(x, z) = 0.1. Note that, for the Gödel structure, we getES(x, y) = 0.4, ES(y, z) = 0.1, ES(x, z) = 0.1; thus we see that ES depends onthe structure on L.

In the following we consider the problem of similarities on three levels: similarityof objects (and attributes), similarity of attributes, and similarity of concept lattices.The proofs and further results can be found in Belohlávek [2000a].

7.4.2 Similarity of objects and similarity of attributes

First, let us propose a way to measure similarity of objects and similarity of attributesof a given fuzzy context. This similarity is induced by the structure of fuzzy conceptsdetermined by the fuzzy context. It turns out that these similarities may be determineddirectly from the fuzzy context; this is relevant from the computational point of view.

Lemma 7.1 can be directly applied to our problem of measuring similarity of objectsand attributes. We are given objects (elements of X) and their observed attributes(elements of Y ). A natural question is that of the similarity relation on the objectsand attributes. The given fuzzy context gives rise to a complete lattice of all fuzzyconcepts hidden in the fuzzy context. As mentioned, the fuzzy concept lattice maybe used for the conceptual classification of objects and attributes. It seems therefore


reasonable to use the induced conceptual structure B (X, Y, I ) to define similarityrelations on X and on Y .

Consider the problem of similarity of objects. Informally, two objects x1, x2 ∈ X

are similar if they cannot be separated by any concept, or more precisely, if for eachconcept c it holds that x1 belongs to the extent of c if and only if x2 belongs to theextent of c. This leads to the following definition of a relation EX

B(X,Y,I )∈ LX×X:

EXB(X,Y,I )(x1, x2) =

∧

〈A,B〉∈B(X,Y,I )

(A(x1)↔ A(x2)

). (7.12)

The relation EXB(X,Y,I )

will be called induced (by B (X, Y, I )) similarity on X. ByLemma 7.1 we immediately get the following statement.

Theorem 7.6

The relation EXB(X,Y,I )

is the largest similarity relation on X compatible with the

extents of all concepts of B (X, Y, I ).

From the computational point of view, the foregoing definition leads to the follow-ing algorithm for computing the similarity relation EX

B(X,Y,I ). Take a fuzzy context,

generate all the fuzzy concepts of B (X, Y, I ), and determine the similarity of eachpair 〈x1, x2〉 ∈ X × X by (7.12). The fuzzy concept lattice may, however, be quiteextensive. This poses the question whether the computational cost can be reduced. An(exact) solution which significantly reduces the computational costs follows. Definea relation EX

〈X,Y,I 〉 ∈ LX×X by

EX〈X,Y,I 〉(x1, x2) =

∧

y∈Y

(I (x1, y)↔ I (x2, y)). (7.13)

EX〈X,Y,I 〉(x1, x2) may be obtained from the L-context 〈X, Y, I 〉 computing |Y | times

the operation ↔. Using Lemma 7.1 (put X = X, I = Y , Ai(x) = I (x, y)), we getthe following theorem.

Theorem 7.7

The relation EX〈X,Y,I 〉 is the largest similarity relation on X compatible with all

I (_, y) ∈ LX, y ∈ Y .

The following theorem solves the problem of finding an efficient procedure forcomputing the similarity relation EX

B(X,Y,I ).

Theorem 7.8

Let 〈X, Y, I 〉 be a fuzzy context. Then, for the similarity relations defined by (7.13)

and (7.12), we have

EXB(X,Y,I ) = EX

〈X,Y,I 〉. (7.14)


Hence, the computation ofEXB(X,Y,I )

may be reduced to the computation ofEX〈X,Y,I 〉,

which is much simpler. In a completely analogous way we may get the results for thesimilarity relations on Y .

7.4.3 Similarity of concepts

The next level on which the similarity phenomenon will be considered is the level ofconcepts. Observe first the following fact, which follows from Lemma 7.1.

Lemma 7.2

For any universe U , the relation E on LU given for any A1, A2 ∈ LU by

E(A1, A2) =∧

x∈U

(A1(x)↔ A2(x))

is the largest similarity relation on LU such that A1(x)⊗E(A1, A2) ≤ A2(x) holds

for each x ∈ U , A1, A2 ∈ LU .

E(A1, A2) is thus the truth degree to which it is true that x belongs to A1 if andonly if x belongs to A2. In the following, it will be clear what universe U the relationE concerns.

Example 7.9

Let again U = {x, y, z}, S = {A, B}, where A = {1/x, 0.5/y, 0.1/z}, B =

{0.9/x, 0.4/y, 0.1/z}. For the Łukasiewicz structure on [0, 1] we get

E(A, B) = [A(x)↔ B(x)] ∧ [A(y)↔ B(y)] ∧ [A(z)↔ B(z)]

= [1 ↔ 0.9] ∧ [0.5 ↔ 0.4] ∧ [0.1 ↔ 0.1] = 0.9 ∧ 0.9 ∧ 1 = 0.9.

For the Gödel structure we analogously have E(A, B) = 0.4, while for the productstructure we have E(A, B) = 0.8.

Consider first the relations EExt and EInt on B (X, Y, I ); call them induced

similarity by extents and induced similarity by intents, respectively:

EExt(〈A1, B1〉, 〈A2, B2〉) = E(A1, A2) =∧

x∈X

(A1(x)↔ A2(x))

EInt(〈A1, B1〉, 〈A2, B2〉) = E(B1, B2) =∧

y∈Y

(B1(y)↔ B2(y)).

Lemma 7.2 gives immediately the following statement.


Theorem 7.9

EExt and EInt are the largest similarity relations on B (X, Y, I ) such that A1(x)⊗

EExt(〈A1, B1〉, 〈A2, B2〉) ≤ A2(x) and B1(y)⊗ EInt(〈A1, B1〉, 〈A2, B2〉) ≤ B2(y)

hold for every x ∈ X, y ∈ Y , 〈A1, B1〉, 〈A2, B2〉 ∈ B (X, Y, I ).

The following theorem answers the question of how the relations EExt and EInt

are related.

Theorem 7.10

For any fuzzy context 〈X, Y, I 〉 we have EExt = EInt.

We can therefore write E instead of EExt and EInt and call it the induced similarity

on concepts.

7.4.4 Compatible similarities and factorization

The primary importance of similarity relations in human reasoning is the reduc-

tion of the complexity of the world at a reasonable price. The complexity isreduced by considering the “collections of similar elements of concern” ratherthan the particular elements themselves [Zadeh, 1997]. This is known in gen-eral system theory as the abstraction process by factorization: moving from agiven level of abstraction (distinguishability) one level up where the elements arecollections of elements of the lower level. Instead of the original system onetherefore considers the “system modulo similarity.” The price paid is the loss ofprecision.

Our concern in the following is the reduction of the complexity of the concept

lattice by factorization modulo similarity. The concept lattice of a given contextrepresents the overall conceptual structure, which can be considerably intricate. Togain insight one has to look for methods of reducing the complexity of the structure.In the two-valued (crisp) case, considerable attention has been paid to this problem[Ganter & Wille, 1999]. In the fuzzy case, one would expect to find methods forgradual reduction of the complexity. The idea is to factorize the concept lattice by anappropriate α-cut αE of the similarity E (note that αE = {〈c1, c2〉 | α ≤ E(c1, c2)}),controlling thus the complexity by α ∈ L. Clearly, the lower α ∈ L, the coarser thefactorization.

The process of factorization of a system consists of two steps. First, specificationof the elements, and, secondly, specification of the structure of the factor system.Since both of the steps are non-standard in our case, we will describe them in moredetail. In general, algebraic systems can be factorized by congruences, i.e., equiva-lences compatible with the structure of the system. We deal with conceptual structuresthat are complete lattices. The α-cut αE is clearly a tolerance relation (i.e., reflexiveand symmetric), not transitive in general. In general, factorization of algebras bycompatible tolerances is not possible. Surprisingly, Czédli [1982] showed a way to


factorize lattices by compatible tolerance relations. The construction has then beenused for the factorization of ordinary concept lattices [Wille, 1985]. In the follow-ing, we describe the construction of the factor lattice of a fuzzy concept lattice by acompatible tolerance relation. Let 〈X, Y, I 〉 be a fuzzy context. A tolerance relationT on B (X, Y, I ) is said to be compatible if it is preserved under arbitrary supremaand infima, i.e., if 〈cj , c

′j 〉 ∈ T , j ∈ J , implies both 〈

∨j∈J cj ,

∨j∈J c′j 〉 ∈ T

and 〈∧

j∈J cj ,∧

j∈J c′j 〉 ∈ T for any cj , c′j ∈ B (X, Y, I ), j ∈ J . For a com-

patible tolerance relation T on B (X, Y, I ) denote cT =∧〈c,c′〉∈T c′ and cT =∨

〈c,c′〉∈T c′. Call [c]T = [cT , (cT )T ] = {c′ ∈ B (X, Y, I ) | cT ≤ c′ ≤ (cT )T }

a block of T and denote B (X, Y, I )/T = {[c]T | c ∈ B (X, Y, I )} the set of allblocks. Introduce a relation ≤T on B (X, Y, I )/T by [c]T ≤T [c

′]T if and only if∧[c]T ≤

∧[c′]T (if and only if

∨[c]T ≤

∨[c′]T ). The justification of the con-

struction is given by the following statement which follows immediately from Wille[1985].

Theorem 7.11

(1) B (X, Y, I )/T is the set of all maximal tolerance blocks, i.e., B (X, Y, I )/T =

{B ⊆ B (X, Y, I ) | (B × B ⊆ T ) & ((∀B ′ ⊃ B)B ′ × B ′ �⊆ T )}. (2)〈B (X, Y, I )/T ,≤T 〉 is a complete lattice (factor lattice) where suprema and infima

are described by

∨

j∈J

[cj ]T = [∨

j∈J

cj ]T and∧

j∈J

[cj ]T = [(∧

j∈J

cj )T ]T (7.15)

for every cj ∈ B (X, Y, I ), j ∈ J .

Substituting (7.4) and (7.5) into (7.15), we get a more concrete description of thelattice operations.

Coming back to the induced similarityE on B (X, Y, I ), the ultimate question is thatof the compatibility of the α-cuts of E. We call a similarity relation F on B (X, Y, I )

compatible if αE is a compatible tolerance relation on B (X, Y, I ) for each α ∈ L.Notice that for the two-valued (crisp) case the situation is completely uninteresting.Namely, as one may easily check, the only cases are 0E = B (X, Y, I )×B (X, Y, I )

and 1E = idB(X,Y,I ) = {〈c, c〉 | c ∈ B (X, Y, I )}. In the first case, B (X, Y, I )/0E =

{B (X, Y, I )}, i.e., the factor lattice collapses into a one-element lattice, while in thesecond case, B (X, Y, I )/1E = {{〈A, B〉} | 〈A, B〉 ∈ B (X, Y, I )}, i.e., B (X, Y, I )

and B (X, Y, I )/1E are isomorphic.Note that we need not confine ourselves to the induced similarity E. On

the other hand, taking into account only similarity relations F satisfyingA(x)⊗F(〈A, B〉, 〈A′, B ′〉) ≤ A′(x) (which is quite natural—it reads “object belong-ing to the extent of some concept belongs also to the extent of any similar concept”)


for each x ∈ X, Theorem 7.9 tells us that E provides the most extensive reduction:for any other F and each α ∈ L, αE is coarser than αF .

Theorem 7.12

The induced similarity E on B (X, Y, I ) is compatible. If α ∈ L is ⊗-idempotent

(i.e., α⊗α = α) then αE is, moreover, transitive, i.e., a congruence relation on

B (X, Y, I ).

Remark 7.2

Theorem 7.12 and the construction described above yield a method for factorizingany fuzzy concept lattice B (X, Y, I ) by any α-cut αE of the induced similarity E. Itis worth noticing that the similarity E is defined “internally,” i.e., it is not suppliedfrom the outside.

Remark 7.3

If L is the Gödel algebra on [0, 1], i.e., ⊗ is min, then each α-cut of E is indeed acongruence relation.

7.4.5 Similarity of concept lattices

Finally, we consider similarity of concept lattices. A natural way to define the simi-larity degree of two concept lattices over the sets X and Y is based on the followingintuition. Concept lattices B (X, Y, I1) and B (X, Y, I2) are similar if and only if foreach concept c1 ∈ B (X, Y, I1) there is a concept c2 ∈ B (X, Y, I2) such that c1 andc2 are similar and, conversely, for each concept c2 ∈ B (X, Y, I2) there is a conceptc1 ∈ B (X, Y, I1) such that c1 and c2 are similar. In the following we write B1 and B2

instead of B (X, Y, I1) and B (X, Y, I2), respectively. According to how the similarityof concepts is measured, we distinguish two rules for the definition of the similaritydegree of two concept lattices:

E∗(B (X, Y, I1), B (X, Y, I2)) (7.16)

=∧

〈A1,B1〉∈B1

∨

〈A2,B2〉∈B2

E∗(〈A1, B1〉, 〈A2, B2〉)∧

∧

〈A2,B2〉∈B2

∨

〈A1,B1〉∈B1

E∗(〈A1, B1〉, 〈A2, B2〉),

∗ ∈ {Ext, Int}, where we put

EExt(〈A1, B1〉, 〈A2, B2〉) = E(A1, A2)

EInt(〈A1, B1〉, 〈A2, B2〉) = E(B1, B2).


We shall see that all of the above relations are, in fact, similarity relations. Themeaning of the similarity relation E defined on the set of all contexts is:

E(〈X, Y, I1〉, 〈X, Y, I2〉) (7.17)

= E(I1, I2) =∧

〈x,y〉∈X×Y

I1(x, y)↔ I2(x, y).

Lemma 7.2 immediately gives the result that E is a similarity relation. The mainresult showing the relationships between the relations introduced above is containedin the following theorem.

Theorem 7.13

For every pair of fuzzy contexts 〈X, Y, I1〉, 〈X, Y, I2〉 we have

E(〈X, Y, I1〉, 〈X, Y, I2〉) ≤ EExt(B1, B2)

and

E(〈X, Y, I1〉, 〈X, Y, I2〉) ≤ EInt(B1, B2).

Moreover, EExt and EInt are similarity relations on {B (X, Y, I ) | I ∈ LX×Y }.

7.4.6 Logical precision

Consider a structure L of truth values. The set L is the set of all possible truth valueswhich we have at our disposal for logical modeling of our knowledge. It can beconsidered as representing “logical discernibility.” Consider for example the two-element Boolean algebra. Then the level of discernibility is low—we can discernonly fully true statements from fully false statements. An n-element chain of truthvalues offers more—we can discern n logical “levels.” Very loosely, using more truthvalues means more logical precision (in the above sense). From the point of view oflogical modeling, it is natural to be able to change the set of truth values (in order toincrease or decrease the logical discernibility) so that the structural properties of themodel remain preserved.

An important role is played by the structure of the set of truth values. Considertwo structures L1 and L2 of truth values such that there is a surjective mappingh : L1 → L2, i.e., h(L1) = L2. If h preserves the structure of the sets of truthvalues, then the change from L2 to L1 can be considered as an increase of logicalprecision and, conversely, change from L1 to L2 can be considered as a decrease oflogical precision. The requirement of preserving the structure of truth values may be,from the algebraic point of view, seen as fulfilled if h is a homomorphism [Grätzer,


1968]. In our case, h is a homomorphism if and only if the following conditions aresatisfied:

h(a ∨ b) = h(a) ∨ h(b)

h(a ∧ b) = h(a) ∧ h(b)

h(a⊗ b) = h(a)⊗h(b)

h(a → b) = h(a)→ h(b)

h(0) = 0

h(1) = 1.

In the following we will suppose that all the homomorphisms under considerationwill be

∧-preserving, i.e., for each K ⊆ L1 we have h(

∧k∈K k) =

∧k∈K h(k).

Given two structures L1 and L2 of truth values and a homomorphism h : L1 → L2,we define for each L1-fuzzy set A in X (A ∈ LX

1 ) the corresponding L2-fuzzy seth(A) ∈ LX

2 by (h(A)) (x) = h(A(x)) for all x ∈ X. The following two statementsshow how the systematic change of the set of truth values (i.e., increase or decreaseof logical precision) influences the structure of the respective concepts [Belohlávek,2002a].

Lemma 7.3

Let L1, L2 be complete residuated lattices and h : L1 → L2 be an onto homo-

morphism. Let 〈X, Y, I 〉 be an L1-context. Then for C ∈ LX2 , D ∈ LY

2 , the

following holds: 〈C, D〉 ∈ B (X, Y, h(I )) iff there are A ∈ LX1 , B ∈ LY

1 such

that 〈A, B〉 ∈ B (X, Y, I ), h(A) = C, and h(B) = D.

A lattice homomorphism h : V1 → V2 between two complete lattices V1 andV2 is called complete if for each K ⊆ V1 we have h(

∧k∈K) =

∧k∈K h(k) and

h(∨

k∈K) =∨

k∈K h(k).

Theorem 7.14

Under the conditions of the preceding lemma, the mapping h∗ as defined by

h∗(A, B) = 〈h(A), h(B)〉

is a complete homomorphism of B (X, Y, I ) onto B (X, Y, h(I )).

Remark 7.4

The foregoing theorem is relevant from the application point of view. Suppose wehave a concept with truth values from L1. A further analysis on the level of L1 maybe (for various reasons, e.g. computational ones) “too precise.” We can then skip to

7.5 Formal Concept Analysis Demonstrated: Examples 217

a level of L2 = h(L1) which is appropriate. Due to the theorem, the structure of theconcepts changes systematically, i.e., the structure of concepts in L1 is in a systematicway more precise than that one in L2.

There is an important consequence of Theorem 7.14: the fuzzy concept latticeB (X, Y, h(I )) can be thought of as if obtained from B (X, Y, I ) by factorization, i.e.,by the process of abstraction. Namely, the mapping h∗ of B (X, Y, I ) to B (X, Y, h(I ))

induces an equivalence relation θh∗ on B (X, Y, I ) by

〈〈A1, B1〉, 〈A2, B2〉〉 ∈ θh∗ iff h∗(A1, B1) = h∗(A2, B2).

That is, we can consider a so-called factor set B (X, Y, I )/θh∗ of B (X, Y, I )

modulo θh∗ . The elements of B (X, Y, I )/θh∗ are classes [〈A, B〉]θh∗of pairwise

θh∗ -equivalent fuzzy concepts, i.e.,

[〈A, B〉]θh∗= {〈A′, B ′〉 ∈ B (X, Y, I ) | 〈〈A, B〉, 〈A′, B ′〉〉 ∈ θh∗}.

Moreover, θh∗ is a complete congruence on B (X, Y, I ). This means that one candefine operations on B (X, Y, I )/θh∗ in such a way that it becomes a complete lattice:we put

[〈A1, B1〉]θh∗∧ [〈A2, B2〉]θh∗

= [〈A1, B1〉 ∧ 〈A2, B2〉]θh∗

[〈A1, B1〉]θh∗∨ [〈A2, B2〉]θh∗

= [〈A1, B1〉 ∨ 〈A2, B2〉]θh∗∧

i∈I

[〈Ai, Bi〉]θh∗= [∧

i∈I

〈Ai, Bi〉]θh∗

∨

i∈I

[〈Ai, Bi〉]θh∗= [∨

i∈I

〈Ai, Bi〉]θh∗.

The following theorem follows from elementary facts of general algebra.

Theorem 7.15

B (X, Y, I )/θh∗ equipped with the above-defined operations is isomorphic to

B (X, Y, h(I )).

This means that B (X, Y, I )/θh∗ and B (X, Y, h(I )) differ only in relabeling theirelements. That is, B (X, Y, h(I )) may be seen as if obtained from B (X, Y, I ) byabstraction.

7.5 Formal Concept Analysis Demonstrated: Examples

Example 7.10

Our main purpose is to illustrate the methods described above. The example is takenfrom paleontology and is a simplified version of a larger example examined in a


Figure 7.2 Fossils from Example 7.10.

forthcoming paper by Belohlávek and Košt’ák. The input data are hypothetical fossiloutlines depicted in Figure 7.2.

Each of the fossils consists of a body and a spine. Intuitively, the fossils belongto some general category C (covering them all). Suppose we have no previous


knowledge about what natural categories (subcategories of C) exist. A first approxi-mation of natural subcategories of C may be automatically extracted from the inputdata using formal concept analysis.

The first step is to write down an appropriate context from the input data. That is,we have to identify objects and (fuzzy) attributes. We naturally take the nine fossilsfor the objects; we assign numbers 1–9 to them, according to Figure 7.2. Now wehave to identify suitable (fuzzy) attributes. Needless to say, identification of attributesof the fossils is an arbitrary process. What makes the fossils look different here arebasically two features: first, the size of the spine and, second, the shape of the body.Concerning the first feature, we identify two attributes, “spine small” and “spinebig.” These attributes are naturally fuzzy ones. Concerning the shape of the body,the shape goes from circle-shaped to very much oval-shaped. The key feature isthus the ratio length:width. We identify two attributes, “oval-shaped” (length:widthbig) and“circle-shaped” (length:width small). The attributes and their meaning aresummarized in Table 7.4.

We now have to take an appropriate set of truth values and equip it with an appro-priate structure.We take L = {0, 1

2 , 1} and will consider two structures defined on L,the Łukasiewicz one and the Gödel one. The context is given by Table 7.5. Therefore,

Table 7.4 Attributes of fossils and their meaning.

Abbreviation Meaning

ss has a small spinesb has a big spinecs is circle-shapedos is oval-shaped

Table 7.5 Fuzzy context given by fossils and their properties.

Spine small Spine big Circle-shaped Oval-shapedss sb cs os

fossil 1 1 0 0 1fossil 2 1 0 0 1fossil 3 1 0 0 1fossil 4 1 0 1

2 1fossil 5 0 1 1 1

2

fossil 6 0 1 1 12

fossil 7 12

12 1 0

fossil 8 12

12 1 0

fossil 9 1 0 1 0


Figure 7.3 Fuzzy concept lattice of the context in Table 7.5 (Łukasiewicz structure).

the set X of objects contains nine elements (denoted by 1,…,9); the set Y of attributescontains four elements (denoted ss, sb, cs, os).

The corresponding fuzzy concept lattices are depicted in Figure 7.3 (Łukasiewiczstructure on L) and Figure 7.4 (Gödel structure on L).

To gain more insight, the elements (i.e., concepts) of the lattice are identified inTable 7.6 (Łukasiewicz structure) and Table 7.7 (Gödel structure). Table 7.8 showsthe similarity relation EX

B(X,Y,I )on the fossils.

We now illustrate reduction of a fuzzy concept lattice by decrease of logical pre-cision (see Section 7.4). Consider the Gödel structure L on L and the mappingh : L→ {0, 1} defined by

h :

0 �→ 012 �→ 1

1 �→ 1

We can verify that h is a (∧

-preserving) homomorphism of L onto 2. The context〈X, Y, h(I )〉 is depicted in Table 7.9.


Figure 7.4 Fuzzy concept lattice of the context in Table 7.5 (Gödel structure).

According toTheorems 7.14 and 7.15, the mappingh∗ sending 〈A, B〉 ∈ B(X, Y, I)

to 〈h(A), h(B)〉 is a complete homomorphism of B (X, Y, I ) onto B (X, Y, h(I )). Thismapping induces a congruence θh∗ on B (X, Y, I ) so that two fuzzy concepts 〈A1, B1〉

and 〈A2, B2〉 from B (X, Y, I ) belong to the same class of θh∗ (are θh∗ -congruent) iffh(A1) = h(A2) and h(B1) = h(B2). The classes of θh∗ are depicted in Figure 7.5. Theelements of B (X, Y, h(I )) corresponding to the classes of θh∗ are listed in Table 7.10.The lattice B (X, Y, h(I )) (which is isomorphic to B (X, Y, I )/θh∗ ) is depicted inFigure 7.6.

Consider now the Łukasiewicz structure on {0, 12 , 1}. We illustrate the factorization

by similarity. Consider the α-cut of the induced similarity E on B (X, Y, I ) for α = 12 ,

i.e., consider12 E. The tolerance blocks in B (X, Y, I ) (which are, in fact, complete

sublattices) are depicted in Figure 7.7. Note that each block is a maximal subset ofL-concepts which are similar in the degree of at least 1

2 . The corresponding factor

lattice B (X, Y, I )/12 E is depicted in Figure 7.6.

Afew remarks on this example follow. We can identify apparently natural concepts.Consider, e.g., the Łukasiewicz structure. Concept no. 14 is naturally described as “afossil with a small spine and an oval shape.” Concept no. 26 is “a fossil with a circleshape which has a rather small spine.” Concept no. 1 is an example of an (empirically)empty concept (its extent is an empty set). Concept no. 17 (and also all the conceptsbetween 1 and 17, i.e., 2, 3, 4, 5, 6, 7, 8, 10, 11, 12), however, do not contain anyobject in degree 1. One may thus wish to consider them empty concepts as well.

Table 7.6 Fuzzy concepts of the context of Table 7.5 (Łukasiewicz structure).

No. Extent Intent

1 2 3 4 5 6 7 8 9 ss sb cs os

1. 0 0 0 0 0 0 0 0 0 1 1 1 1

2. 0 0 0 12 0 0 0 0 0 1 1

2 1 1

3. 0 0 0 0 12

12 0 0 0 1

2 1 1 1

4. 0 0 0 0 0 0 12

12 0 1 1 1 1

2

5. 12

12

12

12 0 0 0 0 0 1 1

212 1

6. 0 0 0 12

12

12 0 0 0 1

212 1 1

7. 0 0 0 12 0 0 1

212

12 1 1

2 1 12

8. 0 0 0 0 12

12

12

12 0 1

2 1 1 12

9. 12

12

12 1 0 0 0 0 0 1 0 1

2 1

10. 12

12

12

12

12

12 0 0 0 1

212

12 1

11. 12

12

12

12 0 0 1

212

12 1 1

212

12

12. 0 0 0 12

12

12

12

12

12

12

12 1 1

2

13. 0 0 0 0 1 1 12

12 0 0 1 1 1

214. 1 1 1 1 0 0 0 0 0 1 0 0 1

15. 12

12

12 1 1

212 0 0 0 1

2 0 12 1

16. 12

12

12 1 0 0 1

212

12 1 0 1

212

17. 12

12

12

12

12

12

12

12

12

12

12

12

12

18. 0 0 0 12 0 0 1

212 1 1 0 1 0

19. 0 0 0 12 1 1 1

212

12 0 1

2 1 12

20. 0 0 0 12

12

12 1 1 1

212

12 1 0

21. 1 1 1 1 12

12 0 0 0 1

2 0 0 1

22. 1 1 1 1 0 0 12

12

12 1 0 0 1

2

23. 12

12

12 1 1

212

12

12

12

12 0 1

212

24. 12

12

12 1 0 0 1

212 1 1 0 1

2 0

25. 12

12

12

12 1 1 1

212

12 0 1

212

12

26. 12

12

12

12

12

12 1 1 1

212

12

12 0

27. 0 0 0 12

12

12 1 1 1 1

2 0 1 0

28. 0 0 0 12 1 1 1 1 1

2 0 12 1 0

29. 1 1 1 1 12

12

12

12

12

12 0 0 1

2

30. 1 1 1 1 0 0 12

12 1 1 0 0 0

31. 12

12

12 1 1 1 1

212

12 0 0 1

212

32. 12

12

12 1 1

212 1 1 1 1

2 0 12 0

33. 12

12

12

12 1 1 1 1 1

2 0 12

12 0

34. 0 0 0 12 1 1 1 1 1 0 0 1 0

35. 1 1 1 1 1 1 12

12

12 0 0 0 1

2

36. 1 1 1 1 12

12 1 1 1 1

2 0 0 0

37. 12

12

12 1 1 1 1 1 1 0 0 1

2 0

38. 1 1 1 1 1 1 1 1 1 0 0 0 0


Table 7.7 Fuzzy concepts of the context of Table 7.5 (Gödel structure).

No. Extent Intent

1 2 3 4 5 6 7 8 9 ss sb cs os

1. 0 0 0 0 0 0 0 0 0 1 1 1 12. 0 0 0 1

2 0 0 0 0 0 1 0 1 13. 0 0 0 0 0 0 1

212 0 1 1 1 0

4. 0 0 0 0 12

12 0 0 0 0 1 1 1

5. 0 0 0 1 0 0 0 0 0 1 0 12 1

6. 0 0 0 12 0 0 1

212 1 1 0 1 0

7. 0 0 0 12

12

12 0 0 0 0 0 1 1

8. 0 0 0 0 0 0 1 1 0 12

12 1 0

9. 0 0 0 0 1 1 0 0 0 0 1 1 12

10. 0 0 0 0 1 1 12

12 0 0 1 1 0

11. 1 1 1 1 0 0 0 0 0 1 0 0 112. 0 0 0 1 0 0 1

212 1 1 0 1

2 0

13. 0 0 0 1 12

12 0 0 0 0 0 1

2 114. 0 0 0 1

2 0 0 1 1 1 12 0 1 1

2

15. 0 0 0 12 1 1 0 0 0 0 0 1 1

216. 0 0 0 0 1 1 1 1 0 0 1

2 1 017. 1 1 1 1 0 0 1

212 1 1 0 0 0

18. 1 1 1 1 12

12 0 0 0 0 0 0 1

19. 0 0 0 1 0 0 1 1 1 12 0 1

2 0

20. 0 0 0 1 1 1 0 0 0 0 0 12

12

21. 0 0 0 12 1 1 1 1 1 0 0 1 0

22. 1 1 1 1 0 0 1 1 1 12 0 0 0

23. 1 1 1 1 1 1 0 0 0 0 0 0 12

24. 0 0 0 1 1 1 1 1 1 0 0 12 0

25. 1 1 1 1 1 1 1 1 1 0 0 0 0

This is possible if one goes from the original fuzzy concept lattice B (X, Y, I ) to thelattice which results from B (X, Y, I ) by factorization modulo the induced similarity12 E. Indeed, as we can see from Figure 7.3, the concepts between 1 and 17 form one

block of12 E and can thus be considered one concept in the factor lattice. The same

applies to other similar concepts of B (X, Y, I ). We thus see that the factorization

modulo12 E makes the concept lattice smaller in that similar concepts which need not

be distinguished are put together.


Table 7.8 Similarity EXB(X,Y,I )

on fossils from Figure 7.2 and Table 7.5.

fos. 1 fos. 2 fos. 3 fos. 4 fos. 5 fos. 6 fos. 7 fos. 8 fos. 9

fossil 1 1 1 1 12 0 0 0 0 0

fossil 2 1 1 12 0 0 0 0 0

fossil 3 1 12 0 0 0 0 0

fossil 4 1 0 0 0 0 0fossil 5 1 1 1

212 0

fossil 6 1 1 12 0

fossil 7 1 1 12

fossil 8 1 12

fossil 9 1

Table 7.9 Context 〈X, Y, h(I )〉 corresponding to 〈X, Y, I 〉 from Table 7.5.

Spine small Spine big Circle-shaped Oval-shapedss sb cs os

fossil 1 1 0 0 1fossil 2 1 0 0 1fossil 3 1 0 0 1fossil 4 1 0 1 1fossil 5 0 1 1 1fossil 6 0 1 1 1fossil 7 1 1 1 0fossil 8 1 1 1 0fossil 9 1 0 1 0

Note that both B (X, Y, h(I )) obtained from B (X, Y, I ) (Gödel structure) and

B (X, Y, I )/12 E obtained from B (X, Y, I ) (Łukasiewicz structure) are isomorphic.

Note also that, taking the Gödel structure,12 E is just the partition induced by the

homomorphism h∗ induced above, i.e., B (X, Y, I )/12 E again is isomorphic to the

lattice in Figure 7.6.

Example 7.11

The next example illustrates attribute implications. Table 7.11 shows the input data.There are 17 objects (denoted by 1–17) and 36 attributes (denoted by a–J). In thiscase, the structure of truth values is the two-element Boolean algebra, i.e., only


Figure 7.5 Classes of the congruence relation induced by h.

truth values 0 (false) and 1 (true) are employed. The objects are extinct cephalopods(identified by paleontologists) as listed in Table 7.12. The organisms possess attributeswhich are listed in Table 7.13. In view of our purpose we do not comment on theappropriateness of these attributes from the paleontological point of view. Belohlávek& Košt’ák (in preparation) contains more information and details.

There is a relatively large number of attributes, obviously more than a humanmind can grasp at once. Therefore, it is desirable to get information that gives us anadditional insight. A suitable one is in the form of attribute dependencies. Tables 7.14,7.15, and 7.16 show an irredundant basis of all attribute implications that are validin the input data (that is, no one of the implications follows from the others). Theimplications are to be read as follows. Implications correspond to rows in the table(in order to fit the page, tables are split into two parts: one corresponding to attributesa–r and one corresponding to attributes s–J; therefore, each row is split into twoparts). “A” denotes that the corresponding attribute belongs to the antecedent of theimplication, “C” denotes that the attribute belongs to the consequent. Thus, the firstrow says that the implication {J}⇒{p} is true in the input data; the eighth row saysthat the implication {A,B}⇒{l} is true in the input data, etc.


Table 7.10 Concepts of B (X, Y, h(I )).

No. Extent Intent

1 2 3 4 5 6 7 8 9 ss sb cs os

1. 0 0 0 0 0 0 0 0 0 1 1 1 12. 0 0 0 1 0 0 0 0 0 1 0 1 13. 0 0 0 0 0 0 1 1 0 1 1 1 04. 0 0 0 0 1 1 0 0 0 0 1 1 15. 1 1 1 1 0 0 0 0 0 1 0 0 16. 0 0 0 1 0 0 1 1 1 1 0 1 07. 0 0 0 1 1 1 0 0 0 0 0 1 18. 0 0 0 0 1 1 1 1 0 0 1 1 09. 1 1 1 1 0 0 1 1 1 1 0 0 0

10. 1 1 1 1 1 1 0 0 0 0 0 0 111. 0 0 0 1 1 1 1 1 1 0 0 1 012. 1 1 1 1 1 1 1 1 1 0 0 0 0

Figure 7.6 Lattice B (X, Y, h(I )). Factor lattice B (X, Y, I )/12 E.

Computation of concept lattices and attribute implications is too extensive to bedone by hand and, therefore, requires the use of a computer. In our examples weused a software tool that is being developed jointly in the Department of ComputerScience, Palacký University, Olomouc (Czech Republic), and in the Department ofComputer Science, Technical University of Ostrava (Czech Republic).

Figure 7.7 Blocks of the tolerance relation12 E on the concept lattice of Figure 7.3.


Table 7.11 Fuzzy context given by fossils and their properties.

a b c d e f g h i j k l m n o p q r s t u v w x y z A B C D E F G H I J

1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 02 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 13 1 1 1 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 14 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 15 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 16 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 17 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 08 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 19 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1

10 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 111 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 012 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 113 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 014 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 115 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 016 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 017 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1

Table 7.12 Objects of the context ofTable 7.11.

1 Actinocamax verus antefragilis2 Praectinocamax primus3 Praectinocamax plenus4 Praectinocamax plenus cf. strehlensis5 Praectinocamax triangulus6 Praectinocamax aff triangulus7 Praectinocamax sozhenzis8 Praectinocamax contractus9 Praectinocamax planus

10 Praectinocamax coronatus11 Praectinocamax matesovae12 Praectinocamax medwedicicus13 Praectinocamax sp.114 Praectinocamax sp.215 Goniocamax intermedius16 Goniocamax surensis17 Goniocamax volgensis


Table 7.13 Attributes of the context ofTable 7.11.

a rostra bigb rostra mediumc rostra smalld cigar shape in dorsoventral viewe lanceolat in dorsoventral viewf little lanceolat in dorsoventral viewg subcylindric in dorsoventral viewh conic in dorsoventral viewi cigar shape in lateral viewj lanceolat in lateral viewk little lanceolat in lateral viewl subcylindric in lateral viewm conic in lateral viewn flat lateralo flat dorsalp flat ventralq alveolar fracture highly conicr alveolar fracture lowly conics pseudoalveol flatt pseudoalveol deepu cut of alveolar fracture ovalv cut of alveolar fracture oval-triangularw cut of alveolar fracture triangularx cut alveolar fracture circle-shapedy conellaez joinA dorsolateral lineB dorsolateral listelC rostrum granulationD rostrum granulation partlyE rostrum striationF rostrum striation partlyG vessel engramH vessel engram rarelyI mucroJ ventral line


Table 7.14 Minimal base of implications of the context of Table 7.11 (first part).

a b c d e f g h i j k l m n o p q r

1 . . . . . . . . . . . . . . . C . .2 . . . . . . . . . . . . . . . C . .3 . . . . . . . . . . . . . . . . . .4 . . . . . . . . . . . . . . . . . .5 . . . . . . . . . . . . . . . C . .6 . . . . . . . . . . . A . . . . . .7 . . . . . . . . . . . C . . . . . .8 . . . . . . . . . . . C . . . . . .9 . . . . . . . . . . . C . . . . . .

10 . . . . . . . . . . . A . . . . . .11 . . . . . . . . . . . C . . . . . .12 . . . . . . . . . . . . . . . . . .13 . . . . . . . . . . . . . . . . . .14 . . . . . A . . . . . . . . . . . .15 . . . . . C . . . . . . . . . . . .16 . . . A . . . . . . . . . . . . . .17 . . . A . . . . . . . C . . . A . .18 . . . A . . . . . . . . . . . . . .19 . . . A . . . . . . . . . . . . . .20 . . C . . . . . . . . . . . . . A .21 . . A . . . . . . . . C . . . A . .22 . . C . . . . . . . . . . . . . . .23 . . A . . . . . . . . . . . . . . .24 . . . . . . . . . . . . . . . . . .25 . . A . . . . . . . . . . . . . . .26 . . A . . C . . . . . . . . . . . .27 . . . . . . . . C . . . . . . . . .28 . . . . . . . . . . . . . A . . . .29 . . . C . . . . A . . . . . . . C .30 . . . A . . . . . . . . . C . . A .31 . . C A . . . . . . . . . . . . . .32 . . A A . C . . . . . A . . . . . .33 . . C . . . . . . . . . . . . . . .34 . . C A . . . . . . . . . . . . . .35 . . A C . . . . . . . . . . . . . .

s t u v w x y z A B C D E F G H I J

1 . . . . . . . . . . . . . . . . . A2 . . . . . . . . . A . . . . . . . .3 . . . . . . . . . C . . . . . . A .4 . . . A . . . . . C . . . . . . . .

continued


Table 7.14 Continued

5 . A . . . . . . . . . . . . . . . .6 . . . . . . . . . C . . . . . . . .7 . . . . . . . . . . . . . A . . . .8 . . . . . . . . A A . . . . . . . .9 . . . . . . . . . . . . . . . A C .

10 . . . . . . . . C . . . . . . C A .11 . A . . . . . . . A . . . . . . . .12 . A . C . . . . . . . . . A . . . .13 A . . . . . . . . . . . . . . C . .14 . . . . . . . . . . . . . C . . . .15 . A . . . . . . A . . . . A . . . .16 . . . . . . . . C . . . . . . . . .17 . . . . . . . . . . . . . . . . . .18 . C . A . . . . . . . . . . . . . .19 . A . C . . . . . . . . . . . . . .20 . . . . . . . . C . . . . . . . . .21 . . . . . . . . . . . . . . . . . .22 . . . A . . . . . . . . . . . A . .23 . . . A . . . . A . . . . . . C . .24 . . . C . . . . . . . . . A . A . .25 . A . . . . . . A . . . . . . C . .26 . A . . . . . . . . . . . A . . . .27 . . . . . A . . . . . . . . . . . .28 . . . . . C . . . . . . . . . . . .29 . . . . . . . . . . . . . . . . . .30 . . . . . . . . . . . . . . . . . .31 . . . . . . . . . . . . . A . . . .32 . . . . . . . . . . . . . . . . . .33 A A . . . . . . . . . . . . . . . .34 . . . . . . . . . . . . . . . A . .35 A . . . . . . . . . . . . . . . . .

Table 7.15 Minimal base of implications of the context of Table 7.11 (second part).


36 . . A . . A . . . . . . . . . . . .37 . A . . . . . . . . . . . . . C . .38 . C . . . . . . . . . . . . . . . .39 . . . . . . . . . . . . . . . . . .40 . C . . . . . . . . . C . . . . . .41 . . . . . . . . . . . . . . . . . .42 . C . . . . . . . . A C . . . . . .

continued




43 . C . . . . A . . . . . . . . . . .44 . . . . . . A . . . . . . . . . . .45 . . . . . . A . . . . . . . . . . .46 . . . . . . A . . . . . . . . . . .47 . . . . . . C . . . . . A . . . . .48 . . . . . . A . . . . . C . . . . .49 . . . . . . A . . . . C . . . . . .50 . . . . . . A . . . . A . . . . . .51 . C . . A . . . . . . . . . . . . .52 . . . . C . . . . . . . . . . . . .53 . . . . C . . . . . . . . . . . . .54 . . . . A . . . . . . . . . . . . .55 . . . . C . . . . . . . . . . . . A56 . . . . A . . . . . . . . . . . . C57 . . . . . . . . . . . . . . . . . A58 . A . . C . . . . . . . . . . . . .59 . . . . A . . . . . . . . . . . . .60 . . . . A . . . . . . . . . . . . .61 . . . . C . . . . . . . . . . . . .62 . . . . A . . . . . . A . . . . . .63 . . . . A . . . . . . . . . . . . .64 . . . . A . . . . . . . . . . . . .65 . A . . C . . . . . . . . . . . . .66 . . . . . . . . . . . . . . C . . .67 . . . . . . . . . . . . . . . . . .68 . . . . C . . . . . . . . . A . . .69 . . . . A . . . . . . . . . . . . .70 . . . . C . . . . . A . . . . . . .


36 C A . . . . . . A . . . . . . . . .37 . . . . . . . . . . . . . . . . . .38 . . A . . . . . . . . . . . . . . .39 . A A . . . . . . . . . . . . . . C40 . . . . . . . . C . . . A . . . . .41 . A . . . . . . . . . . A . . C . .42 . . . . . . . . C . . . . . . . . .43 . . . . . . . . . . . . . . . . . .44 . . . . . . . . C . . . . . . . . A45 . . . . . . . . A . . . . . . . . C



46 . . A . . . . . C . . . . . . . . .47 . C C . . . . . . . . . . . . . . .48 . A A . . . . . . . . . . . . . . .49 . . . . . . . . . A . . . . . . . .50 . A . C . . . . . . . . . . . . . .51 . . . . . . . . . C . . . . . . . .52 . . . . . . . . . . . . . . . . A A53 . . A . . . . . . A . . . . . . . .54 . . C A . . . . . . . . . . . . . C55 . . C . . . . . . . . . . . . . . C56 . . . A . . . . . . . . . . . . A .57 . . . A . . . . . . . . . . . . C .58 . . . . . . . . . . . . . A . . . A59 . . C . . . . . . . . . . A . . . C60 . . C . . . . . . . . . A . . . . .61 . . . . . . . . . . . . A . . . . A62 . . . A . . . . . . . . . C . . . .63 . A . . . . . . . . . . . . . . . C64 . A . . . . . . A . . . . . . C . .65 A . . . . . . . . . . . C . . . . .66 . . . . . . . A . . . . . . . . . .67 . . . . . . A C . . . . . . . . . .68 . C . . . . . . . . . . C . . . . .69 . A A . . . C . A . . . . . . . . .70 . . . . . . . . . . . . . . . . . A

Table 7.16 Minimal base of implications of the context of Table 7.11 (third part).


71 . . . . A . . . . . A . . . . . . .72 . . . . . . . . . . C A . . . . . A73 . . . . . . . . . . A . . . . . . C74 . . . . . . . . . A . . . . . . . C75 . . . . . . . . . A . . . . . . . .76 . C . A . . . . . . . . . . . . . .77 . A . A . . C . . . . . . . . . . .78 . A . C . . . . . . . . . . . . . .79 . A A . . . . . . . C . . . . . . .80 . . C . . . . . . . A . . . . . . .

continued




81 . . C . . . . . . . . . . . . . . .82 . . A . . . . . . . . . . . . . . .83 . . C . . . . . . . A . . . . . . .84 . . . . . . . . . . A . . . . . . .85 . . C . . . A . . . . . . . . . . .86 . A . . . A C . . . . . . . . . . .87 . . A . . C A . . . . . . . . . . .88 . . A . A . . . . . C . . . . . . .89 . . A . C . . . . . . . . . . . . .90 . A C . C . . . . . . . . . . . . .91 . . A . . . . . . . . . . . . . . A92 A . . . . . . . . . . C . . . . . .93 A . . . . . . . . . . . . . . . . .94 C . . . . A . . . . . . . . . . . .95 A . . . . C . . . . . . . . . . . .96 A . . . . C . . . . . . . . . . . .97 A C A . . . . . . . . . . . . . . .98 A A C . C . . . . . . . . . . . . .99 A C . . . . . . . . . . . . . . . .

100 C . A . A . . . . . . . . . . . . .101 A . . . . . . . . . . . . . . . . .102 C . C . . . . . . . . . . . . . . .103 . . A . A . . . . . . . . . . . . .104 C . . . . . . . . . . A . . . . A .105 C . . . A . A . . . . . . . . . . .106 . . . C . . . . . A A . . . . . . .107 A . . A . . . . . C . . . . . . . .108 . . . . . . . C . . . . . . . . . .109 . . . . . . . . . . . . . . . . . .110 . . . . . . . . . . . . . . . . . .111 . . . . . C . A C . . . . . . . . .112 . . . . . . . C . . . . . . . . . .113 . . . . . A . . . . . . C . . . A .


71 . . . . . . . . . . . . . . . A . C72 . . . . . . . . . . . . . . . . . .73 . . A . . . . . . . . . . . . . . A74 . . . . . . . . . . . . . . . . . .75 . . . C . . . . . . . . . . . . A .76 . . . . . . . . . . . . . . . . . A

continued



77 . . . . . . . . . . . . . . . . . .78 . A . A . . . . A . . . . . . . . A79 . . . . . . . . A . . . . . . . . .80 . . . A . . . . . . . . . . . . . .81 . . . A . . . . . . . . A . . . . .82 . . . C . . . . . . . . A . . A . .83 . A . . . . . . . . . . . . . . . .84 . A . A . . . . . . . . C . . . . .85 . . . . . . . . . . . . . A . . . .86 . . . . . . . . . . . . . . . . . .87 . . . . . . . . . . . . . . . . . .88 . . . . . . . . . . . . . . . . . .89 . . . . . . . . . . . . . . . . . A90 . . . . . . . . A . . . . A . . . .91 . . . . . . . . . . . . . C . . . .92 . . . . . . . . C . . . . . . . . .93 . A . . . . . . . . . . . . . . . C94 . . . . . . . . . . . . . . . . . A95 . . . . . . . . . . . . . A . . . C96 . . . A . . . . . . . . . . . . . .97 . . . . . . . . . . . . . . . . . .98 . . . . . . . . . . . . . . . . . .99 . . . . . . . . . . . . . . . A . .

100 . . . . . . . . . . . . A . . . . .101 . . A . . . . . . . . . C . . . . .102 . C . . A . . . . . . . . . . . . .103 . A . . C . . . . . . . . . . . . .104 . . C . . . . . . . . . . . . . . .105 . . . . . . . . . . . . . . . . . .106 . . . . . . . . . . . . . . . . . .107 . . . . . . . . . . . . . . . . . .108 . . . . . . . . . . . . . . A . . .109 . . . . . . . . . . . A . . C . . .110 . . . . . . . . . . A C . . . . . .111 . . . . . . . . . . . . . . . . . .112 A . . . . . . . . . . . . . . . . A113 . . . . . . . . . . C . . . . . . .


Acknowledgments

The work has been supported by the project Kontakt ME 468 of the MŠMT of theCzech Republic as the international supplement to the NSF project “Stratigraphicsimulation using fuzzy logic to model sediment dispersal” and partly by GACR201/99/P060 and GACR 201/02/P076. The author thanks Dr Martin Košt’ák forproviding him with paleontological data.

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Chapter 8 Fuzzy Logic and Earthquake Research

Chongfu Huang

8.1 Introduction 2398.2 Basic Terminology in Earthquake Research 242

8.2.1 Earthquake and seismology 242

8.2.2 Earthquake engineering 244

8.3 Fuzzy Logic in Earthquake Prediction 2458.3.1 Direct method of fuzzy pattern recognition in earthquake

prediction 245

8.3.2 Fuzzy pattern for earthquake prediction based on seismicity

indices 247

8.4 Fuzzy Logic in Earthquake Engineering 2498.4.1 Fuzzy earthquake intensity 250

8.4.2 Estimating earthquake damage with fuzzy logic 255

8.5 Hybrid Fuzzy Neural Networks with Information Diffusion Method 2598.5.1 Fuzzy relationships given by the information diffusion method 260

8.5.2 Pattern smoothing 262

8.5.3 Learning relationships by BP neural networks 264

8.5.4 An application 264

8.6 Conclusion and Discussion 269Appendix 8.A: Modified Mercalli Intensity Scale Used in China 270Acknowledgments 271References 271

8.1 Introduction

An earthquake is a sudden, rapid shaking of the Earth caused by the breaking andshifting of rock beneath the Earth’s surface. Ground shaking from earthquakes can col-lapse buildings and bridges; disrupt gas, electric, and phone services; and sometimestrigger landslides, avalanches, flash floods, fires, and tsunamis.

No story will ever be written that will tell the awfulness of a few hours follow-ing a terrible earthquake such as the San Francisco earthquake, 1906 (April 18at 5:15 a.m., measuring 8.25 on the Richter scale), or the Richter magnitude 7.8

239

240 8 Fuzzy Logic and Earthquake Research

Tangshan earthquake on July 28, 1976, that destroyed almost all of the city and killed242,000 people (investigated by the China Academy of Building Research [1986]).No pen of the most powerful descriptive power could ever place on paper the impres-sion of any one of the hundreds of thousands who felt the mighty Earth tremble. Nopen can record the sufferings of those who were crushed to death or buried in theruins that encompassed them in the instant after a destructive earthquake.

Although the world’s largest earthquakes do have a clear spatial pattern, we are notable to predict individual earthquakes. Some Chinese seismologists claimed that theHaicheng earthquake (February 4, 1975, with Richter magnitude 7.3) was success-fully predicted. However, many scientists deny it (see Geller et al. [1997]). Stormsapproach, fires spread, floods migrate downstream after large rainfalls, but earth-quakes can turn a perfectly normal day into disaster in seconds. There is a growingpossibility that in the near future we will be able to correctly predict earthquakes thathave obvious precursors. Unfortunately, most earthquakes do not have obvious pre-cursors such as changes in land elevation, changes in groundwater levels, widespreadreports of peculiar animal behavior, and foreshocks. A contemporary problem facingseismologists is to explore more powerful tools for monitoring seismic activity andanalyzing collected geophysical data and other related information.

Seismologists have developed several approaches based on fuzzy logic for ana-lyzing earthquake data [Feng et al., 1996; Junji & Feng, 1995]. Furthermore, anew branch of seismology called fuzzy seismology has been proposed by Feng et al.[1992b]. Fuzzy seismology includes the following ten fuzzy methods applied in earth-quake prediction: (1) fuzzy pattern recognition; (2) fuzzy clustering analysis; (3) fuzzyinformation retrieval; (4) fuzzy similarity choice; (5) fuzzy multi-factorial evaluation;(6) fuzzy reasoning; (7) fuzzy self-similarity analysis and fuzzy fractal dimension;(8) gray fuzzy prediction; (9) fuzzy neural network; and (10) fuzzy analyzing andprocessing software systems.

Lin and Sanford [2001] incorporated a fuzzy logic algorithm into the locationprogram SEISMOS to increase stability in locating regional earthquakes. This tech-nique was converted into a computer subroutine as an initial hypocenter estimator andincorporated into SEISMOS in early 1994 at the New Mexico Institute of Mining andTechnology [Lin, 1994]. The Fuzzy/SEISMOS combination has proved to be veryeffective in locating earthquakes.

The current success rate in earthquake prediction is very low. Even if we might oneday successfully predict any earthquake, it is impossible to move the buildings or theproperty within buildings away from the area of a predicted earthquake. We are ableto mitigate earthquake disasters by considering earthquake loads when designing andconstructing buildings. This discipline is a natural branch of civil engineering, calledearthquake engineering.

The use of fuzzy sets in civil engineering in the USA was alluded to in 1971[Brown & Leonards, 1971], but little serious work occurred until the advent ofBlockley’s 1975 and 1977 papers. These UK studies had a profound effect on workin the USA where concern had existed about the inclusion of semantic wisdom,

8.1 Introduction 241

as opposed to countable information, into building safety and design reliabilityconsiderations [Brown, 1985]. An immediate application is the assessment of seismicdamage [Yao, 1980]. This led to the discovery by English-speaking workers of theexisting fuzzy set literature in Japan [Tokyo Institute of Technology, 1975–1985] and,in particular, to applications of fuzzy set theory to earthquake engineering in China[Feng et al., 1982; Liu, 1982; Liu & Dong, 1982; Tian, 1983; Wang, F., 1983; Wang,1984; Huang & Liu, 1985; Liu et al., 1985; He & Guo, 1988; Huang & Xiu, 1988].

In the early 1980s, some scientific researchers working in earthquake-resistantdesign came in touch with fuzzy logic and it quickly became a powerful tool to incor-porate the copious imprecise knowledge in this field quantitatively and in a morescientific fashion. For example, Wang [1982] suggested a fuzzy method for deter-mining the value of certain parameters used in structure design, so as to avoid theproblem of increasing or decreasing the design parameters by a factor of 2. Chiang andDong [1987] suggested that when available data on structural parameters are crudeand do not support a rigorous probabilistic model, the fuzzy set approach should beconsidered in view of its simplicity. To select the best design strategy, Wu and Wang[1988] developed a fuzzy approach to reasonably relate the condition of construction,the environment of the structure, the probability of foundation damage, the arrange-ment of horizontal and vertical planes of the structure, and so on. Subramaniam et al.[1996] presented one of the first experimental applications of fuzzy control to a build-ing structure, showing the feasibility of the implementation of fuzzy logic to highlynonlinear problems.

For efficient fuzzy information processing relevant in earthquake engineering,Huang [1997] suggested the principle of information diffusion to deal optimally withrecognizing the underlying relationships from a small sample. Properties of informa-tion diffusion estimators on probability density functions confirm that the principle istrue. The simplest diffusion function is the 1-dimensional linear information distribu-tion [Liu & Huang, 1990]. Huang [2000] demonstrated that the work efficiency of themethod of information distribution is about 28% higher than the histogram method.In other words, we can use a 28% smaller sample size and obtain the same accuracyby the method of information distribution. If we need a sample of 30 observationsfor the histogram method, a sample with 22 observations can give an estimate bythe method of information distribution with the same accuracy. However, we nevereliminate the imprecision. Huang [1998b] studied the possibility of using the methodof information distribution to calculate a fuzzy probability distribution to representthe imprecision.

In this chapter we review some contemporary methodologies of fuzzy logic forearthquake research. The focus is on applications of fuzzy logic developed by seis-mologists rather than civil engineers.

The chapter is organized as follows. To help the readers who are unfamiliar withearthquake research, Section 2 introduces basic terminology of seismology andearthquake engineering. Section 3 briefly presents two fuzzy methods for earth-quake prediction. Section 4 reviews the studies of fuzzy earthquake intensity and


the estimation of earthquake damage with fuzzy logic. In Section 5, we give oneexample of a hybrid fuzzy neural network that estimates the relationship betweenisoseismal area and earthquake magnitude. The chapter is then summarized with aconclusion in Section 6.

8.2 Basic Terminology in Earthquake Research

8.2.1 Earthquake and seismology

An earthquake is the vibration of the Earth produced by the rapid release of energy.This energy radiates in all directions from its source, the focus, in the form of waves.Just as the impact of the stone sets water waves in motion, an earthquake generatesseismic waves that radiate throughout the Earth. Even though the energy dissipatesrapidly with increasing distance from the focus, instruments located throughout theworld can record the event.

Although an earthquake is a natural phenomenon, seismology is the branch of Earthscience concerned with the study of natural earthquakes, man-made earthquakes, andrelated phenomena such as underground nuclear bomb testing.

Early attempts to establish the size or strength of earthquakes relied heavily onsubjective description. There was an obvious problem with this method becausepeople’s accounts vary. Thus it was difficult to make an accurate determination ofthe quake’s absolute strength. In 1902, a fairly reliable intensity scale based on theamount of damage caused to various types of structure was developed by GiuseppeMercalli [Tarbuck & Lutgens, 1991]. A modified form of this tool is presently usedin China (see Appendix 8.A).

By definition, the earthquake intensity (also called seismic intensity) is a measureof the effects of a quake at a particular location. It is important to note that earthquakeintensity depends not only on the absolute strength of the earthquake, but also on otherfactors. These factors include the distance from the epicenter (the vertical projectionof the focus on the surface), the nature of the surface materials, and building design.The intensity at the epicenter is called epicentral intensity.

In 1935, Charles Richter of the California Institute of Technology introduced theconcept of earthquake magnitude when attempting to rank earthquakes of southernCalifornia [Berlin, 1980]. The earthquake magnitude is basically a relative scale[Kasahara, 1981]. It defines a standard size of earthquake and rates other earthquakesin a relative manner by their maximum amplitude of ground motion under identicalobservational conditions. This is evident from Richter’s definition:

M = log[A(�)/A0(�)] = log A(�)− log A0(�), (8.1)

where � is an epicentral distance, and A0 and A denote the maximum trace ampli-tudes, on a specified seismograph, of the standard event and the one to be measured,

8.2 Basic Terminology in Earthquake Research 243

respectively. The standard earthquake, i.e., M = 0 (= log 1) in Richter’s formula, issuch as to give the maximum trace amplitude of 0.001 mm on a Wood–Anderson-typeseismograph at � = 100 km.

The intensity scale differs from the Richter magnitude scale in that the effects ofany one earthquake vary greatly from place to place, so there may be many intensityvalues measured from one earthquake. Each earthquake, on the other hand, shouldhave just one magnitude, although different methods of estimating it will yield slightlydifferent values.

Earthquake prediction [Aki, 1995] is usually defined as the specification of thetime, location, and magnitude of a future earthquake within stated limits.

Earthquake activity is called seismicity and is usually described by statistical indicescalled seismicity indices. In China, seismicity indices are calculated each month fromthe data for the last 100 events before the end of the month in question. The followingthree indices are commonly used to quantify earthquake activity in an area: (1) the b

value; (2) the η value; (3) the c value.

1. The b value is a basic characteristic of the seismicity rate in an area, which canbe calculated by using the earthquake data set collected in the studied area. The b

value is a parameter in the Gutenberg–Richter law (Gutenberg & Richter, 1944],

log10 N = a − bM, (8.2)

that determines the correlation between the magnitude M of earthquakes and theirrelative numbers N .

2. Let Mi be the ith magnitude of the given data with size n, M0 is the minimummagnitude within the activity, and Xi = Mi −M0. Then, the η value is defined as

η =(1

n

n∑

i=1

X2i

)/(1

n

n∑

i=1

Xi

)2. (8.3)

It is a degree of deviation from the Gutenberg–Richter law.3. The c value is another deviation degree that is defined by using the Kullback–

Leibler directed divergence (Kullback, 1959]:

c =

n∑

i=1

p(Xi) ln[p(Xi)/q(Xi)] (8.4)

p(Xi) = ni/N

q(Xi) = B exp(−BXi)�M

B = b ln 10

Xi = Mi −M0,


where �M is the length of an earthquake range, ni and N denote the numberof earthquake range from Mi to Mi +�M and the total number of earthquakes,respectively, and M0 is the minimum magnitude within the activity.

8.2.2 Earthquake engineering

Earthquake engineering can be defined as the branch of civil engineering devotedto mitigating earthquake hazards [Committee on Earthquake Engineering, ResearchCommission on Engineering and Technical Systems, National Research Council,1982]. In this broad sense, earthquake engineering covers the investigation and solu-tion of the problems created by damaging earthquakes, and consequently the workinvolved in the practical application of these solutions, i.e., in planning, designing,constructing, and managing earthquake-resistant structures and facilities.

However, most earthquake engineers and seismologists consider earthquake engi-neering as a scientific discipline that is more concerned with estimation of earthquake

loading and earthquake damage assessment. Therefore, as seismologists, earthquakeengineers also study the seismic (elastic) waves generated by an earthquake. Earth-quake engineers pay much attention to predicting the intensity of shaking at a givensite for an earthquake of given magnitude. This intensity is called site intensity andis measured by macro-seismic intensity (as a modified Mercalli intensity) or groundacceleration. Earthquake engineers can then change the site intensity into earthquakeloading for earthquake-resistant design.

Earthquake engineers, in the course of their work, are faced with many uncertaintiesand must use sound engineering judgments to develop safe solutions to challengingproblems.

In earthquake engineering there are two terms that are frequently used beside theterm “intensity” (defined by Appendix 8.A). These two terms are “building damageindex” and “peak ground acceleration.”

1. The building damage index is a measure of damage done to a building during anearthquake. If a building is without any damage after an earthquake, the value ofthe damage index of this building is 0. If a building is totally collapsed after anearthquake, the value of the damage index is 1. The building damage index is thusin the unit interval [0, 1]. We usually use the set given in Equation (8.5) as theuniverse of discourse of the building damage index, where the step length is 0.1.

U = {u1, u2, u3, . . . , u11} = {0, 0.1, 0.2, . . . , 1} (8.5)

2. The peak ground acceleration (PGA) is the largest acceleration recorded by a par-ticular instrumentation station during an earthquake. PGA is what is experiencedby a particle on the ground. PGA is measured in terms of the galileo, g(cm/s2).The PGA of the vertical component will usually be different from the PGA of the

8.3 Fuzzy Logic in Earthquake Prediction 245

horizontal component. Both values of PGA are dependent on the distance fromthe source, but for short distances the PGA of the vertical component may actuallyexceed the PGA of the horizontal component. In general usage, however, PGAmeans the PGA of the horizontal component.

8.3 Fuzzy Logic in Earthquake Prediction

Although the leading seismological authorities of each era have generally concludedthat earthquake prediction is not feasible [Geller et al., 1997], we believe that earth-quake prediction is worthy of study. Feng et al. [1992a] used “fuzzy recognition”to analyze seismic precursors observed in China and Japan. Their result was that allearthquakes of M ≥ 6 that occurred in the given area during the given period hadcommon precursors. The longest precursory time was about 5 months, and the short-est precursory time was about 2 months. Feng et al. [1992a] and Junji & Feng [1995]used “multi-story fuzzy multifactorial evaluation” (advanced by Wang, P. Z. [1983])to assess and examine the potential strengths of 15 induced earthquakes due to waterreservoirs in Canada, China, India, Greece, USA, and Zambia with magnitudes 2.0–6.5. The result shows that the multi-story evaluation method can be used to assess thepotential strength of an induced earthquake more effectively and more accurately.

In this section, we review two methods of earthquake prediction with fuzzy patternrecognition.

8.3.1 Direct method of fuzzy pattern recognition in earthquake

prediction

General method

According to the method of fuzzy set theory, the direct method of fuzzy patternrecognition may in principle include the following three steps:

1. Choosing characteristics. For each considered object u (e.g., seismic belt, radoncontent) we choose key characteristics relevant to earthquake prediction.

2. Constructing the membership function. If the object u may belong to several fuzzysets Ai (i = 1, 2, . . . , n), then we must construct their membership functions.

3. Recognizing and judging the object. According to some principle of belongingnessthe element u can be judged, and the kind of set to which the object u belongs canbe determined. Two principles of belongingness are usually used. They are:

(a) The principle of maximal belongingness: If we have a fuzzy set Ai that satisfies

Ai(u) = max{A1(u), A2(u), . . . , An(u)} (8.6)


then it can be considered that the object u relatively belongs to Ai ; otherwise,if we have an object ui which satisfies

A(ui) = max{A(u1), A(u2), . . . , A(un)} (8.7)

then it can be considered that among all objects u1, u2, . . . , un the object ui

belongs to fuzzy set A with maximum weight.

(b) The principle of threshold-value. After defining a threshold-value λ ∈ [0, 1]and taking μmax = max{A1(u), A2(u), . . . , An(u)}, if μmax < λ, then wereject recognition of u; if μmax ≥ λ, then we recognize the object. If we haveA1(u), A2(u), . . . , An(u) ≥ λ, then object u belongs to A1 ∪A2 ∪ · · · ∪An.Otherwise, if we have a series of objects u1, u2, . . . , um that satisfies A(ui) ≥

λ (i = 1, 2, . . . , m), then all these objects can belong to the same fuzzy setA. The threshold-value λ can be determined empirically and is usually takenas λ > μ∗ = 0.5, where μ∗ is called the “fuzzy boundary point.”

Fuzzy analysis and recognition of earthquake precursors

The direct method of fuzzy pattern recognition has been used to analyze and recognizeseismic precursors (seismic belt, seismic gap, seismicity quiescence, self-similarity ofearthquake sequence, etc.) and nonseismic precursors (radon content in groundwater,water level, tilt, tide, volume strain, levelling, earth resistivity, geomagnetic field,etc.) observed in China and in Japan [Feng & Ichikawa, 1989; Feng et al., 1989].

The basic technique of the direct method of fuzzy pattern recognition is to constructthe suitable membership function for each single precursor and the total membershipfunction for several precursors. For the majority of the observational curves of singleprecursors y = f (t), the membership function of a single precursor is representedby the following analytical expression [Junji & Feng, 1995]:

μ(yi) =(

1+a

|k(yi)| · |r(yi)|

)−1(8.8)

where yi denotes the precursor data in the ith time interval, |k(yi)| and |r(yi)| are theabsolute value of slope and coefficient of correlation of curvey = f (t), respectively, ais an empirical constant, and μ(yi) is the grade of membership of the precursor data yi

being anomalistic. The total membership function of n precursors can be expressed by

μ(y1i, y2i, . . . , yni) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∨nj=1 μj (yji), if there are m or more precursors

whose memberships are largerthan 0.5 in the same time interval,shorter than k days

0, otherwise, (8.9)

8.3 Fuzzy Logic in Earthquake Prediction 247

where yji denotes the j th precursor data in the ith time interval; μj (yji) is the gradeof membership of the precursor data yji being anomalistic; the sign

∨denotes the

disjunction, i.e., max; the parameters n, m, k must be determined empirically for thedifferent kinds of precursors in the different regions. For the Tokai area of Japan,Junji and Feng [1995] have found n = 9, m = 3, and k = 15 in the case of differentkinds of precursors, and n = 30, m = 6, and k = 5 in the case of single volume strainprecursors at different stations. For the Beijing–Tianjin–Tangshan area of China, theyhave n = 14, m = 5, and k = 15 in the case of single radon content precursors at dif-ferent stations (all precursors are radon content but are recorded at different stations).The formula (8.9) with different values of n, m, k can be called the precursor patternfor medium–short-term M ≥ M0 earthquake prediction. The above values of n, m,k for Japan and China correspond to M0 = 6.0 and in this case formula (8.9) can bewritten as

μ(y1i, y2i, . . . , yni) =

n∨

j=1

μj (yji) = μ1(y1i)∨

μ2(y2i)∨· · ·∨

μn(yni)

(8.10)

The results given by Junji & Feng [1995] show that common precursors occurredbefore all earthquakes of M ≥ 6 took place in the given area and the given period.The longest precursory time (i.e., the beginning time of a precursor anomaly) wasabout 5 months and the shortest precursory time was about 2 months. Therefore theyconsider the precursors recognized by this means as medium–short-term earthquakeprecursors.

Zheng and Feng [1989] also used “fuzzy recognition” to analyze recorded ground-water levels in 33 wells near Tangshan before and after the earthquake. They foundpreviously unknown anomalies in the correlations between water levels in differentwells beginning three hours before the quake. The anomalies were distinct from typ-ical fluctuations during the 5-day period analyzed, and increased in strength as theearthquake approached.

8.3.2 Fuzzy pattern for earthquake prediction based on seismicity

indices

We can separately study anomalies in the seismicity indices b, η, c before largeearthquakes in a given region by choosing their threshold values b∗, η∗, and c∗. Forexample, Junji and Feng [1995] studied the variations of b, η, c for the Songpanregion (in Sichuan Province, China) and found b∗ = 0.73, η∗ = 1.75, c∗ = 0.87.Their prediction check results, however, were not satisfactory.

To improve the prediction in the seismicity indices b, η, c, Junji and Feng [1995]established two fuzzy patterns for one-year and half-year earthquake prediction,respectively.


First, they defined three membership functions on domains of indices b, η, c,respectively. In standard fuzzy terms, these functions would be written as μ1(b),μ2(η), μ3(c), representing the grades of membership of the indices being anomalistic.However, in their papers, the functions are written as μb(ti), μη(ti), μc(tj ), andwithout any indication of the forms of the functions.

Second, they discovered two numbers μ∗, μ∗∗ that can help us to judge whether anearthquake would occur. Different regions have different μ∗ and μ∗∗. They givetwo patterns for one-year and half-year earthquake prediction with μ∗ and μ∗∗,respectively, as follows:

Pattern A

μ(bti , ηti , ctj ) = μ1(bti )∧

μ2(ηti )∧

μ3(ctj ) ≥ μ∗

⇒ an earthquake more than M1 would occur within one year (8.11)

(for Songpan M1 = 5.4)

where j = i or i + 1 or i + 2.Pattern A indicates that if b and η are anomalistic (more than some degree) during

time interval ti , and c is anomalistic (also more than some degree) during this timeinterval or the next interval ti+1 or the following interval ti+2, then, within one yearfollowing the time interval when c is anomalistic (ti or ti+1 or ti+2), an earthquakemore than M1 would occur. For the Songpan region, M1 = 5.4.

Pattern B

μ(bti , ηti , ctj ) = μ1(bti )∧

μ2(ηti )∧

μ3(ctj ) ≥ μ∗∗

⇒ an earthquake more than M2 would occur within half a year(8.12)

where also j = i or i + 1 or i + 2.Pattern B indicates that if b and η are anomalistic (more than some degree) during

time interval ti , and c is anomalistic (also more than some degree) during this timeinterval or the next interval ti+1 or the following interval ti+2, then, within half a yearfollowing the time interval when c is anomalistic (ti or ti+1 or ti+2), an earthquakemore than M2 would occur.

Table 8.1 shows the result for the Songpan region where, for earthquake M ≥

5.4, the correct rates of prediction are 91% and 68% for Pattern A and Pattern B,respectively. The correct rates are higher than those of other methods in which theseismicity indices b, η, c are used separately.

8.4 Fuzzy Logic in Earthquake Engineering 249

Table 8.1 The check results of fuzzy comprehensive prediction of M ≥ 5.4 earthquakes inthe Songpan region, using b, η, c.

Pattern A Pattern BActual earthquake

period and M Prediction period Check Prediction period Check

1970 II (5.5; 5.5) 1970 II–1971 I B 1970 II B1972 II (5.6; 5.5) 1972 II–1973 I B 1972 II B1973 II (6.5) 1973 II–1974 I A 1973 II A1974 I (5.7) 1974 I–II B 1974 I A1974 II (5.7) 1974 I–II B 1974 I D1976 II (7.2) 1976 II–1977 I A 1976 II A1981 I (6.9) 1981 I–II A 1981 I A

1984 II C1985 II (5.4) 1984 II–1985 I A 1985 I A1987 I (6.2)∗ D D

1988II C1989 I (5.4) 1988 II–1989 I A 1989 I A1989 II (6.3) 1989 II–1990 I A 1989 II A

I, from January to June; II, from July to December; A, accurate prediction; B, basically accurate prediction;C, false prediction; D, no prediction.* This event was at the north boundary of Songpan region.Source: Junji & Feng [1995].

8.4 Fuzzy Logic in Earthquake Engineering

The main tasks of earthquake engineering are to estimate site intensity and to esti-mate earthquake damage. Due to the fuzziness of the earthquake intensity, there isa number of different sources that define and evaluate earthquake intensity. Due tothe complicated nature of building damage caused by earthquakes, only five gradelevels of damage to buildings are often used: (1) intact; (2) slight damage; (3) mod-erate damage; (4) severe damage; and (5) collapse. These grades comprise fuzzysubsets in the universe of discourse of average building damage index [0, 1]. Liuand Wang [Liu et al., 1985; Wang et al., 1986] used set-valued statistics to outlinea procedure to take into account the number of buildings damaged at each level soas to devise an urban disaster mitigation strategy. In order to take into account allthe factors that might affect urban development in a seismic area, they developed afuzzy dynamic analysis procedure. Xiu and Huang [1989] employed the method ofinformation distribution [Liu & Huang, 1990] and fuzzy inference [Huang & Shi,2002] to estimate the earthquake damage to brick-column single-story factory build-ings. The result indicates that the fuzzy model can attain minimum error in damageprediction.


8.4.1 Fuzzy earthquake intensity

The macro-seismic intensity scale is a subjective measure of the effect of the groundshaking and is not a precise engineering measurement (Appendix 8.A). This is truedespite a great deal of effort since the 1942 study of Gutenberg and Richter to take theaverage values of peak ground acceleration from the seismic instrumental records indamaged areas as a precise physical reference. Correlations have not been successfulbecause of the great deviation of both values in different locations.

From the point of view of fuzzy mathematics, Feng et al. [1982] first describedseismic intensities of VI to XII as normal functions, using different grades ofbuilding damage as base variables. More than 700 field investigations of Chineseearthquakes provided data to shape the functions. Liu et al. [1983] formally pro-posed the name “fuzzy intensity” and defined it as fuzzy subsets in the universeof discourse U with the base variables coming from the average damage index[0, 1] of buildings, on the basis of numerous earthquake field investigations car-ried out in China in recent decades. A more detailed description of this workis given by Tian [1983], in which 1364 data points are used. Wang, F. [1983]studied the relation between epicentral intensity and magnitude. He expressedepicentral intensity as fuzzy subsets of magnitude by using normal configura-tions as membership function with shapes determined from earthquake records inChina. He obtained better results than those obtained by statistical methods ofcorrelation.

Wang [1984] outlined a two-stage procedure for intensity evaluation. He firstdivided intensity evaluating factors into four categories, each of which was con-sidered as a factor subset. These were the subset U1 of damage indexes of buildings,subset U2 of peak values of ground motion and some response spectra, subset U3 ofearthquake characteristics, and subset U4 of human reactions and geologic condition.All the factors in subsets U1 and U2 were described by a normal configuration. At thefirst stage, the fuzzy intensity vector in each category was determined by inferencerules, and then the second stage was carried out by treating such intensity vector as afactor vector combined with a weight factor.

Liu [1982] was the first scientist to introduce the concept of linguistic variable infuzzy set theory to study fuzzy earthquake intensity. He established a fuzzy model toinfer PGA from earthquake intensity, both of which were defined as fuzzy subsets inthe universes of discrete acceleration and building damage index, respectively. Liuused the model to check the 1980 Intensity Scale of China. The result is satisfactory.

In Liu’s model, earthquake intensities V–X are defined by Table 8.2, whereu1, u2, . . . , u12 are values of building damage index. For example, fuzzy intensityX1 (i.e., intensity level V) is a fuzzy subset in the universes of building damage indexwith membership function

X1 = 1/u1 + 0.90/u2 + 0.83/u3 + 0.50/u4.


Table 8.2 Fuzzy earthquake intensities V–X with the universe U .

General building X1 X2 X3 X4 X5 X6

Damage Damage index (V) (VI) (VII) (VIII) (IX) (X)

Intact u1 < 0.03 1 0.270.03 ≤ u2 < 0.10 0.90 0.67 0.42

Slight damage 0.10 ≤ u3 < 0.16 0.83 0.88 0.81 0.400.16 ≤ u4 < 0.23 0.50 0.81 0.96 0.500.23 ≤ u5 < 0.30 0.71 1 0.61

Moderate damage 0.30 ≤ u6 < 0.36 0.54 1 0.690.36 ≤ u7 < 0.43 0.37 0.79 0.810.43 ≤ u8 < 0.50 0.52 0.92 0.27

Severe damage 0.50 ≤ u9 < 0.56 0.37 1 0.42 0.300.56 ≤ u10 < 0.63 0.81 0.60 0.500.63 ≤ u11 < 0.70 0.60 0.77 0.70

Collapse 0.70 ≤ u12 0.98 1

u1, u2, . . . , u12 are the elements of the universe U .Source: Liu [1982].

Liu also defined 12 fuzzy PGAs by Table 8.3, where v1, v2, . . . , v16 are the tradi-tional PGAs expressed in g (galileo or cm/s2). For example, fuzzy PGA Y2 is a fuzzysubset in the universes of PGA with membership function

Y2 = 0.75/v1 + 0.90/v2 + 0.75/v3 + 0.45/v4 + 0.25/v5.

With the fuzzy sets in Tables 8.2 and 8.3, Liu [1982] collected 117 observationswhere intensity and acceleration were changed into an information matrix M , shownin Equation (8.13). Let (I, v) be an observation with intensity I and acceleration v. IfI fires fuzzy subset Xi (i.e., I is just Xi) and v fires fuzzy subset Yj (i.e., μyj

(v) > 0),then (I, v) is regarded as an observation that belongs to fuzzy subset Xi × Yj . Theactual number of observations belonging to Xi × Yj is assigned to the element of M

in Xi row and Yj column.

M =

X1

X2

X3

X4

X5

X6

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12⎛⎜⎜⎜⎜⎜⎜⎝

4 2 0 1 0 0 0 0 0 0 0 04 10 10 2 1 1 2 2 0 0 0 00 5 16 14 12 9 5 7 1 2 0 00 0 1 0 1 1 0 2 1 0 0 00 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎠

.

(8.13)


Table 8.3 Fuzzy PGA Y with the universe V .

Acceleration Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12

(g) v1 < 25 1 0.75 0.5025 ≤ v2 < 50 0.85 0.90 0.85 0.2550 ≤ v3 < 75 0.50 0.75 1 0.45 0.2575 ≤ v4 < 100 0.30 0.45 0.85 0.75 0.40 0.25

100 ≤ v5 < 130 0.25 0.50 0.90 0.70 0.45 0.20130 ≤ v6 < 160 0.30 0.75 0.85 0.75 0.40160 ≤ v7 < 190 0.45 0.70 0.90 0.65 0.25190 ≤ v8 < 230 0.25 0.45 0.75 0.80 0.45 0.25230 ≤ v9 < 270 0.25 0.45 0.65 0.75 0.40270 ≤ v10 < 315 0.25 0.40 0.90 0.70 0.30315 ≤ v11 < 360 0.20 0.75 0.85 0.50 0.25360 ≤ v12 < 420 0.45 0.70 0.85 0.45420 ≤ v13 < 500 0.25 0.40 1 0.75 0.30500 ≤ v14 < 600 0.25 0.85 0.90 0.50600 ≤ v15 < 700 0.50 0.75 0.85700 ≤ v16 (g) 0.30 0.45 1

v1, v2, . . . , v16 are the elements of the universe V .Source: Liu [1982].

Then, with his experience, from M Liu chose the following five elements:

E = {(1, 2), (2, 3), (3, 5), (4, 8), (5, 12)} (8.14)

to make the following five fuzzy rules:

g1�= If X is X1 then Y is Y2,




g5�= If X is X5 then Y is Y12.

Let

Ri = Xi × Yi, (8.15)

where the membership function of fuzzy set Ri is

Ri(u, v) = minu∈U,v∈V

{Xi(u), Yi(v)}, (8.16)


Table 8.4 Fuzzy relation between intensity I and acceleration v.

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16

V 0.75 0.90 0.83 0.83 0.70 0.81 0.70 0.45 0.50 0.50 0.50 0.45 0.25 0 0 0VI 0.75 0.85 0.88 0.85 0.70 0.81 0.70 0.45 0.61 0.61 0.61 0.45 0.25 0 0 0VII 0.75 0.81 0.81 0.81 0.70 0.85 0.70 0.45 0.75 0.79 0.75 0.45 0.30 0.37 0.37 0.37VIII 0.50 0.61 0.61 0.61 0.70 0.79 0.70 0.45 0.75 0.90 0.75 0.45 0.30 0.50 0.60 0.60IX 0 0 0.25 0.37 0.37 0.37 0.37 0.45 0.60 0.60 0.60 0.45 0.30 0.50 0.85 0.98

and

R = R1 ∪ R2 ∪ R3 ∪ R4 ∪ R5. (8.17)

Hence, Liu [1982] obtained a fuzzy relation between building damage index u andacceleration v. Here we omit the fuzzy relation because it is very large. Then, withthe max–min fuzzy composition rule, from a given intensity I he obtained a fuzzyPGA, Q, defined in the universe of discourse of acceleration, V = {v1, v2, . . . , v16}.For example, if I = V , then

I = V = X1 = 1/u1 + 0.90/u2 + 0.83/u3 + 0.50/u4

μQ(vj ) = max{min[μX1(ui), μR(ui, vj )]}, j = 1, 2, . . . , 16

Q = 0.75/v1 + 0.90/v2 + 0.83/v3 + 0.83/v4 + 0.70/v5 + 0.81/v6 + 0.70/v7

+ 0.45/v8 + 0.50/v9 + 0.50/v10 + 0.50/v11 + 0.45/v12 + 0.25/v13.

Therefore, he obtained a fuzzy relation between intensity (not building damage index)and acceleration, shown in Table 8.4.

Finally, using the so-called “max” principle, he defuzzified Q into a crisp result.That is, let

Q = μQ(v1)/v1 + μQ(v2)/v2 + · · · + μQ(v16)/v16

If

μQ(vk) = max{μQ(v1), μQ(v2), . . . , μQ(v16)},

then Q’s crisp result is

v = vk.

Hence, we can estimate acceleration from a given intensity. All results from the fuzzymodel are shown in column 4 of Table 8.5.

To judge whether the Q obtained by using the above fuzzy inference methodis reliable, Liu introduced two indexes: (1) maximum possibility E�(Q); and


Table 8.5 1980 Intensity Scale of China and results from Liu’s [1982] model.

1980 Intensity Scale of China Fuzzy inference

Intensity Damage index PGA PGA Error E�(Q) EC(Q)

V — 31 (22–44) 37.5 20% 0.77 0.62VI 0–0.1 63 (45–89) 62.5 0 0.78 0.62VII 0.11–0.30 125 (90–177) 145 16% 0.78 0.63VIII 0.31–0.50 250 (178–353) 292.5 17% 0.70 0.51IX 0.51–0.70 500 (354–707) 700 40% 0.43 0.27X 0.71–0.90 1000 (708–1414) — — — —

(2) minimum possibility EC(Q), which are defined by the formulas

E�(Q) =

12∑

i=1

pi sup(Q ∩ Yi) (8.18)

EC(Q) = 1− E�(Q), (8.19)

where Q is the complement of Q, and

pi =1

117

6∑

k=1

mki, (8.20)

mki is the value of the element in the kth-row and ith-column of the informationmatrix (8.13). pi is the probability of the evidence qi that can fire Yi . The probabilitydistribution is

P = {p1, p2, . . . , p12}

= {0.068, 0.145, 0.231, 0.145, 0.120, 0.094, 0.060, 0.094, 0.017, 0.017, 0, 0.009}

For example,

p1 =1

117

6∑

k=1

mk1 =4+ 4+ 0+ 0+ 0+ 0

117= 0.068.

In Liu’s estimation, the larger E�(Q) and EC(Q) are, the more reliable Q is.Liu’s E�(Q) and EC(Q) are given in columns 6 and 7 of Table 8.5, respectively.

For example, V’s row of Table 8.4 is a fuzzy PGA, denoted by Q1; that is:

Q1 = 0.75/v1 + 0.90/v2 + · · · + 0.25/v13.


Hence

E�(Q1) =

12∑

i=1

pi sup(Q1 ∩ Yi) = 0.77

EC(Q1) = 1− E�(Q1) = 0.62.

It is interesting to note that, in Table 8.5, the error of fuzzy inference is lower ifthe E�(Q) and EC(Q) are higher. Therefore, this fuzzy model provides a way ofchecking the 1980 Intensity Scale of China. For example, from Table 8.5 we knowthat the PGAs assigned by the 1980 Intensity Scale of China for intensities VI andVII are correct. Among the 117 data serving for Liu’s model, much of the data hasintensities VI and VII. Hence, the results for these intensities are more reliable. Thus,in this model, the 1980 Intensity Scale of China has been proven in theory for thefirst time.

8.4.2 Estimating earthquake damage with fuzzy logic

There has been a number of efforts to estimate earthquake damage using mechanicsmethods with probability approaches. However, only a few of these have been directedtowards determining structural damage states with respect to earthquakes. Currentmethods emphasize stochastic damage evaluation, and calculations of the reliabilityof structural systems with uncertain parameters under various seismic excitations arepresented. The reliability functions of structural systems are determined analyticallyby introducing approximate probability density functions of damage. This practiceverifies that the approximate probability density functions are only applicable to thereliability analysis of uncertain structural systems when we can collect sufficient datafrom the building and earthquake load.

In the current seismic code of China (GBJ11-89), a definite three-level seismicfortification requirement is proposed, that is, “Do not be damaged under a minorearthquake. Be repairable under a moderate earthquake. Do not collapse under amajor earthquake.” The relative relations between the three-level intensity and thebasic intensity are summed up as shown in Table 8.6.

Owing to the complicated nature of building damage caused by earthquakes, onlyfive grade levels of damage to buildings are often used: (1) intact; (2) slight damage;(3) moderate damage; (4) severe damage; and (5) collapse. These grades comprisefuzzy subsets in the universe of discourse.

In the current seismic code used in China, there are five grades to measure earth-quake damage to engineered structures: intact; slight damage; moderate damage;severe damage; and collapse. In fact, earthquake damage grade is a fuzzy concept,because it is difficult to completely separate the different damage grades by some


Table 8.6 Relationship of three-level seismic levels with the basic intensity (the designreference period is 50 years).

Seismic levels Minor earthquake Moderate earthquake Major earthquake

Exceedance 0.632 0.10 0.02–0.03probability

Relationship 1.55 degrees lower equals the basic about 1 degreewith basic than basic intensity higher thanintensity intensity basic intensity

Design do not be be repairable do not collapserequirement damaged

specific values of structural response. We denote Ai as the ith fuzzy seismic damagegrade; that is to say, the universe set of the seismic damage grade is

D = {A1, A2, A3, A4, A5}

= {intact, slight damage, moderate damage, severe damage, collapse}

For a specific type of building, the main task of estimating earthquake damage isto identify the relationship between the damage grade and the earthquake load. Forthe brick-column, single-story factory building, Xiu and Huang [1989] suggesteda typical model with fuzzy logic. In their model, the earthquake loads are the siteintensities VI, VII, VIII, IX, and X, and the damage grades were defined as the fuzzysubsets in the universes of building damage index as the following:

A1 = intact = 1/u1 + 0.7/u2 + 0.2/u3

A2 = slight damage = 0.2/u1 + 0.7/u2 + 1/u3 + 0.7/u4 + 0.2/u5

A3 = moderate damage = 0.2/u3 + 0.7/u4 + 1/u5 + 0.7/u6 + 0.2/u7

A4 = severe damage = 0.2/u5 + 0.7/u6 + 1/u7 + 0.7/u8 + 0.2/u9

A5 = collapse = 0.2/u7 + 0.7/u8 + 1/u9 + 0.7/u10 + 0.2/u11

where the values of building damage index, u1, u2, . . . , u11, are given in (8.5).The relationship in their model is a regression result obtained from the data

recording the damage caused by the Tangshan earthquake, shown in column 3 ofTable 8.7.

To change the given sample into a fuzzy relationship between intensity and damage(with building parameters such as building height H , distance d between two brickcolumns, and so on), we first calculate the dynamic response x, shown in column 4of Table 8.7. We then obtain 18 data with dynamic response x and real damage D.With these data and the method of information distribution, we construct a primary


Table 8.7 Earthquake damage to brick-column single-story factory buildings in area withintensity VIII during the Tangshan earthquake, 1976.

No. Name of workshop Real Dynamic Fuzzydamage D response x conclusion

1 Washing workshop of TMF A3 1.2481 A22 #28 machining workshop of TMF A1 0.926054 A13 Metalforming workshop of TMF A1 0.536433 A14 Finished products storehouse of TMF A1 0.94601 A15 Coil workshop of TGEW A1 0.57222 A16 Spare parts storehouse of TGEW A1 1.1049 A27 Wooden molds storehouse of TGEW A1 0.243632 A18 Spray paint workshop of TGEW A2 1.27026 A29 Machining workshop of TDEW A1 3.59285 A5

10 Metalforming workshop of TDTF A1 0.74636 A111 Stamping workshop of TDTF A1 1.41304 A312 Machining workshop of TDMF A4 0.830951 A113 Machining workshop of TAEF A5 1.59838 A514 Assembling workshop of TAEF A4 1.59838 A515 Repairing workshop of TMPM A4 1.98074 A516 Preserving workshop of TSCF A3 1.45437 A317 Stamping workshop of TLVSF A2 1.10519 A218 Oil filling workshop of TITW A2 0.914276 A1

TAEF—Tanggu Auto Engine Factory;TDEW—Tianjin Diesel Engine Works;TDMF—Tianjin Dongfeng Metalforming Factory;TDTF—Tianjin Dongfanghong Tractor Factory;TGEW—Tianjin Generating Equipment Works;TITW—Tianjin Instrument Transformer Works;TLVSF—Tianjin Low-Voltage Switch Factory;TMF—Tianjin Machinery Factory;TMPM—Tianjin Medium Plate Mill;TSCF—Tianjin Second Cable Factory.

information distribution matrix, shown in Table 8.8. The element of the ith row andj th column is denoted by qij . For example, q23 = 0.42, q24 = 0.29, and so forth.

For the sake of clarity, here we use only two places following the decimal point.In fact, the real value of every element stored in the computer has nine digits. Someof the zero entries in the matrix are only approximately equal to zero. Non-entry isreally zero.

Let

R1 = (r1ij )6×11

R2 = (r2ij )6×11,


Table 8.8 Primary information distribution matrix QVIII of damage cases in VIII zone duringthe Tangshan earthquake, 1976.

Dynamic Damage indexresponse 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.96 0.64 0.47 0.21 0.20 0.27 0.21 0.09 0.03 0.02 0.00 0.001.12 0.39 0.47 0.42 0.29 0.14 0.05 0.01 0.00 0.00 0.00 0.001.28 0.13 0.28 0.41 0.44 0.35 0.20 0.06 0.00 0.00 0.00 0.001.44 0.30 0.21 0.12 0.23 0.32 0.23 0.06 0.00 0.00 0.00 0.001.60 0.00 0.00 0.01 0.02 0.10 0.26 0.41 0.48 0.49 0.25 0.101.76 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 8.9 Fuzzy relationship matrix RVIII based on damage cases in VIII zone during theTangshan earthquake, 1976.

Dynamic Damage indexresponse 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.96 1.00 0.73 0.33 0.31 0.42 0.32 0.14 0.04 0.03 0.00 0.001.12 0.61 0.99 0.91 0.62 0.30 0.11 0.03 0.00 0.00 0.00 0.001.28 0.19 0.59 0.93 1.00 0.81 0.46 0.13 0.00 0.00 0.00 0.001.44 0.46 0.45 0.29 0.52 0.91 0.70 0.16 0.00 0.00 0.00 0.001.60 0.00 0.00 0.01 0.05 0.21 0.52 0.84 0.99 1.00 0.51 0.151.76 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

where

r1ij =

qij

max{q1j , q2j , . . . , q6j }, j = 1, 2, . . . , 11

r2ij =

qij

max{qi1, qi2, . . . , qi11}, i = 1, 2, . . . , 6,

then

R = R1

∧R2 (i.e., rij = min{r1

ij , r2ij })

is the fuzzy relationship from x to D, shown in Table 8.9.The fuzzy relationship matrix is a fuzzy set defined in universe V × U , where

V = {v1, v2, . . . , v5} = {0.96, 1.12, 1.28, 1.44, 1.60, 1.76},

is the universe of the dynamic response and U is the universe of damage index asgiven in (8.5).

8.5 Hybrid Fuzzy Neural Networks 259

By means of the max–min model for fuzzy inference, we can obtain a fuzzy con-clusion D from a response x. We choose the closest damage grade A to be the lastconclusion. Experientially, Xiu & Huang [1989] employed the method of informationdistribution [Liu & Huang, 1990] to change a crisp value x into a fuzzy set μX(v)

with universe V . The fuzzy set has the following membership function:

μX(v) =

{1− | x − v | /�v, if |x − v| ≤ �v

0, otherwise,(8.21)

where �v, the step length of v, is v2 − v1 = 0.16. For example, for a given inputx = 1.2481, Xiu and Huang changed it into a fuzzy input

X = 0.72/v2 + 0.26/v3 = 0.72/1.12+ 0.26/1.28.

They then obtained a fuzzy conclusion

D = 0.61/u1 + 0.74/u2 + 0.74/u3 + 0.62/u4 + 0.3/u5 + 0.26/u6 + 0.13/u7

= 0.61/0+ 0.74/0.1+ 0.74/0.2+ 0.62/0.30+ 0.3/0.4+ 0.26/0.5+ 0.13/0.6.

Because A2 is the closest grade to the D among all damage grades, we choose A2 tobe the final conclusion.

The last column of Table 8.7 gives the conclusion from the fuzzy inference. Thisresult is better than any other result from a traditional method (such as the method ofleast squares).

8.5 Hybrid Fuzzy Neural Networks with Information

Diffusion Method

The studies of earthquake prediction and earthquake engineering often involve findingnonlinear functions based on data that are scanty, incomplete, and contradictory. Inthis section, we estimate the relationship between isoseismal area and earthquakemagnitude as an example to introduce a new approach.

In the mid-1970s, some researchers [e.g., Howell & Schultz, 1975; Gupta & Nuttli,1976] were active in the search for the following (or similar) expression relatingintensity I to magnitude M and hypocentral distance � (in kilometers):

I = aM − b log10 �+ c (8.22)

where a, b, and c are empirical constants. Since there is a 60% probability that anobserved intensity is more than one degree greater or smaller than its predicted value[Lomnitz & Rosenblueth, 1976], a more appropriate expression [Huang & Liu, 1985]


relating isoseismal area S (in square kilometers), intensity I , and magnitude M wasdeveloped:

log10 S(I) = a + bM (8.23)

where a and b are empirical constants.However, several studies [Liu et al., 1987] have demonstrated that the linear rela-

tionship does not fit the seismicity of any region. In the western United States ofAmerica, many destructive earthquakes are controlled by one huge fault, the SanAndreas fault, and the relationship between isoseismal area and earthquake mag-nitude is approximately linear. Nevertheless, in regions where earthquakes are notcontrolled by one single fault, the linear relationship generally fails.

In principle, if there were many observations recording historical earthquakes,researchers could use powerful statistical tools such as regression analysis [Bollingeret al., 1993; Cavallini & Rebez, 1996; Fukushima et al., 1995] to reveal the non-linear relationship. However, destructive earthquakes are infrequent events withvery small probability of occurrence. Thus, we do not have enough data to employthe traditional statistical tools to estimate the relationship between isoseismal areaand earthquake magnitude. Furthermore, seismotectonic structures are very com-plex and the relationship between isoseismal area and earthquake magnitude isstrongly nonlinear. Since we do not know which nonlinear function would bestdescribe the relationship, it is profitable to employ neural networks to search forthe mapping from the input, which is earthquake magnitude, to the output, whichis isoseismal area by observations. However, in the real world, the observationsare strongly scattered and contradictory patterns do occur. Since neural informationprocessing models largely assume that the learning patterns for training a neuralnetwork are compatible, a neural network approach to this problem does not con-verge because the adjustments of weights and thresholds do not know where toturn because of the ambiguity brought forth by the contradictory patterns. Hence,to estimate the relationship, we need to handle the fuzziness and granularity of theobservations.

Hybrid fuzzy neural networks [Huang & Leung, 1999] based on the normal diffu-sion function [Huang, 1997] can be used to estimate fuzzy relationships with scanty,incomplete, and contradictory data.

8.5.1 Fuzzy relationships given by the information diffusion method

To facilitate our discussion, we first give some basic concepts and definitions.Unless stated otherwise, we assume that we are given a sample of n real-valuedobservations, xi (i = 1, 2, . . . , n), which have two components, earthquake magni-tude mi and isoseismal area Si , whose underlying relationship is to be estimated.


An observation is also called a pattern. A given sample is represented asfollows:

X = {x1, x2, . . . , xn} = {(m1, S1), (m2, S2), . . . , (mn, Sn)}.

To reduce scattering in the sample, we generally consider logarithmic isoseismal area,g = log10 S, instead and the given sample can be rewritten as

X = {x1, x2, . . . , xn} = {(m1, g1), (m2, g2), . . . , (mn, gn)}. (8.24)

In general, fuzzy subsets used to construct fuzzy rules and fuzzy relationshipsare derived from “expert” information, and different choices of fuzzy subsets leadto different results. During development, fuzzy subsets can be adjusted (“turned”)according to the results. However, this approach in which first impressions arestrongest does not fit incomplete data problems where there is insufficient informationto support any first impressions.

The usual method is to look for evidence to support fuzzy relationships. The trac-ing curve of sample-data clusters is regarded as an input–output fuzzy relationshipfunction. Cluster analysis techniques can be developed to choose reasonable fuzzysubsets. When a sample is small or scattered, there does not exist any tracing curve ofclusters. The clustering approach in small data sets can be replaced by an informationdiffusion method [Huang, 1997; Huang & Shi, 2002] which helps us to change anobservation into a fuzzy subset to fill the gap caused by incomplete data.

Let M = {m1, m2, . . . , mn} be a given sample with the universe of discourse U .A mapping from M × U to [0, 1]

μ : M × U → [0, 1]

is called information diffusion of M on U if it satisfies:

(1) ∀mi ∈ M, if u0 = mi, then μ(mi, u0) = supu∈U μ(mi, u)

(2) ∀mi ∈ M, μ(mi, u) is a convex function about u.

μ(mi, u) is called an information diffusion function of M on U . When U is discrete,the function can also be written as μ(mi, uj ).

Let X be a given sample which can be used to estimate the real-valued relation R

by the operator γ . If the estimator is calculated by using the information distributionfunction, the estimator is called the information diffusion estimator.

The principle of information diffusion, which has been justified [Huang, 1997,1998a, 2000; Huang & Shi, 2002], asserts that there must be some reasonable infor-mation diffusion functions to improve the non-diffusion estimator if and only if X

is incomplete. The principle is obvious when we understand the fuzziness of incom-plete data. And, by the methods of classical mathematics, we have justified that theprinciple is efficient for estimating probability density functions with small samples.


On the basis of similarities of information and molecules, and with the help of themolecular diffusion theory, we obtain, by solving the partial differential equation, anormal diffusion function

μ(mi, u) = exp

[−

(u−mi)2

2h2

](8.25)

where h is called the normal diffusion coefficient, which can be simply calculated[Huang, 1997] by

h =

⎧⎪⎪⎨⎪⎪⎩

1.6987(b − a)/(n− 1), for 1 < n ≤ 51.4456(b − a)/(n− 1), for 6 ≤ n ≤ 71.4230(b − a)/(n− 1), for 8 ≤ n ≤ 91.4208(b − a)/(n− 1), for 10 ≤ n.

(8.26)

Using the normal diffusion function, we can change any input–output observation(mi, gi) ∈ X in (8.24) into two fuzzy subsets

Ai =

∫

U

μ(mi, u)/u =

∫

U

exp

[−

(u−mi)2

2h2m

]/u (8.27)

and

Bi =

∫

V

μ(gi, v)/v =

∫

V

exp

[−

(v − gi)2

2h2g

]/v. (8.28)

Obviously, (mi, gi) means

Ai → Bi . (8.29)

In order to preserve more information, we employ the correlation-product encoding[Kosko, 1992] to produce a fuzzy relationship based on Ai → Bi instead of thecorrelation-minimum encoding in the Mamdani–Togai model. Therefore,

Ri(u, v) = Ai(u)Bi(v); u ∈ U, v ∈ V. (8.30)

So, we can get n fuzzy relationships from n historical earthquake observations.

8.5.2 Pattern smoothing

Suppose we have n observations (m1, g1), (m2, g2), . . . , (mn, gn). Using the infor-mation diffusion technique, we can get n fuzzy if-then rules A1 → B1, A2 →

B2, . . . , An → Bn.


Now, if a crisp input value m0 (the antecedent) is known, then one needs a way toinfer the consequent g0 from m0 and Ri .

In practical calculation, U is generally discrete, so that m0 is not just equal to somevalue in U . We can employ the information distribution formula in (8.31) to get afuzzy subset as an input:

m0(uj ) =

{1− |m0 − uj |/�, if |m0 − uj | ≤ �

0, if |m0 − uj | > �(8.31)

where � = u2 − u1. A fuzzy consequent g0∼

from m0∼

and Ri is then obtained as

g0(v) =∑

u

m0(u)Ri(u, v). (8.32)

When we defuzzify g0∼

into a crisp output valueg0, there is no operator that can avoidsystem error. In order to get direct crisp output values, we only need to change themagnitude component into fuzzy subsets. In other words, for the sake of defuzzifyingeasily, it is unnecessary to change the logarithmic isoseismal area component intofuzzy subsets when we construct the relationships Ri . That is, for an observation(mi, gi), we employ Equation (8.27) to change input mi into a fuzzy subset Ai withmembership function Ai(u), but employ Equation (8.33) instead of (8.28) to changeoutput gi into a fuzzy subset Bi with membership function Bi(v); that is, a singleton.

Bi(v) =

{1, if v = gi

0, if v �= gi .(8.33)

In this case, Equation (8.30) is changed into (8.34):

Ri(u, v) =

{Ai(u), if v = gi

0, if v �= gi .(8.34)

Therefore

g0(v) =

{∑u m0(u)Ai(u), if v = gi

0, if v �= gi .(8.35)

Let

Wi =∑

u

m0(u)Ai(u). (8.36)

(In fact, Wi is the possibility (weight) that consequent g0 may be gi .) Then, to integrateall results coming from R1, R2, . . . , Rn, the relevant output value g0 becomes

g0 =

(n∑

i=1

Wigi

)/( n∑

i=1

Wi

). (8.37)


The procedure comprising (8.31)–(8.37) is called “information-diffusion approximatereasoning” (IDAR).

Any observation (mi, gi) of the sample can thus be changed into a new pattern(mi, gi) via information-diffusion approximate reasoning. The new patterns must besmoother.

8.5.3 Learning relationships by BP neural networks

Using neural networks for automatic learning by examples has been a commonapproach employed in the construction of artificial systems. Backpropagation (BP)neural networks [Pao, 1989; Rumelhart & McClelland, 1973], a class of feedforwardneural networks, are models commonly used for learning and reasoning.

However, neural information processing models generally assume that the patternsused for training a neural network are compatible. Because the new patterns that resultfrom information-diffusion approximate reasoning are smoother, they are compati-ble. Therefore, we can employ the BP neural networks to learn relationships fromthe new patterns. We integrate the information-diffusion approximate reasoning andconventional BP neural network into a hybrid model (HM) depicted in Figure 8.1,called a hybrid fuzzy neural network.

In the hybrid model, observations (m1, g1), (m2, g2), . . . , (mn, gn) are firstchanged, via the information-diffusion technique, into new patterns (m1, g1),(m2, g2), . . . , (mn, gn). Then a conventional BP neural network is employed to learnthe relationship between isoseismal area g and earthquake magnitude m.

8.5.4 An application

In Yunnan province of China, there is a data set of strong earthquakes consisting of25 records from 1913 to 1976 with magnitude and isoseismal area, the latter surveyedfrom the region with intensity I ≥ VII. The magnitude and the isoseismal area areshown in the M column and the SI≥VII column of Table 8.10, respectively. The givensample with magnitude m and logarithmic isoseismal area g = log10 S is shown inEquation (8.38):

X = {x1, x2, . . . , x25}

= {(m1, g1), (m2, g2), . . . , (m25, g25)}

= {(6.5, 3.455), (6.5, 3.545), (7, 3.677), (5.75, 2.892), (7, 3.414),

(7, 3.219), (6.25, 3.530), (6.25, 3.129), (5.75, 2.279), (6, 1.944), (8.38)

(5.8, 1.672), (6, 3.554), (6.2, 2.652), (6.1, 2.865), (5.1, 1.279),


Figure 8.1 Architecture of the hybrid model (HM) integrating information-diffusion approx-imate reasoning (IDAR) and a conventional BP neural network.

(6.5, 3.231), (5.4, 2.417), (6.4, 2.606), (7.7, 3.913), (5.5, 2.000),

(6.7, 2.326), (5.5, 1.255), (6.8, 2.301), (7.1, 2.923), (5.7, 1.996)}.

Linear regression methods have conventionally been used to estimate the relationshipbetween g and m. Regressing g on m with data in the given sample, we obtain theregression line shown in (8.39), whose r2 = 0.503 is relatively small.

g = −2.60767+ 0.8521531m (8.39)

To establish a relationship by the neural network approach, we constructed a con-ventional BP neural network with one node in the input layer, 15 nodes in the hiddenlayer, and one node in the output layer. Setting the momentum rate η = 0.9 andlearning rate α = 0.7, we directly used X in (8.38) to train the BP network. After600,000 iterations, the normalized system error is 0.015594.

The results obtained by the regression and neural network methods are compared inFigure 8.2. Apparently, the BP curve does not quite adequately capture the observedrelationship. Neither does the regression method.


Figure 8.2 Relationship between earthquake magnitude and logarithmic isoseismal area,estimated by linear regression and BP network.

A careful examination of X shows that it is, in fact, relatively small and datacontained are incomplete and sometimes contradictory. Therefore, any observationin X can be regarded as a piece of fuzzy information that represents partially therelationship between the two variables. Furthermore, an observation can producea simple fuzzy relationship via the diffusion method. The integration of all fuzzyrelationships will, in turn, produce a more appropriate description.

For the analysis of data in Table 8.10, the procedure for our proposed approach isas follows.

First, let the discrete universe of discourse of earthquake magnitudes be

U = {u1, u2, . . . , u30} = {5.010, 5.106, . . . , 7.790} (8.40)

where the step length is 0.096.Then, from Xm = {mi |i = 1, 2, . . . , 25}, using Equation (8.26), we obtain the

normal diffusion coefficient as

hm = 1.4208(7.7− 5.1)/(25− 1) = 0.15392. (8.41)


Table 8.10 Magnitudes and isoseismal areas.

Date M SI≥VII Date M SI≥VII

1913.12.21 6.5 2848 1965.7.3 6.1 7331917.7.31 6.5 3506 1966.1.31 5.1 191925.3.16 7 4758 1966.2.5 6.5 17031930.5.15 5.75 779 1966.9.19 5.4 2611941.5.16 7 2593 1966.9.28 6.4 4041941.12.26 7 1656 1970.1.5 7.7 81761951.12.21 6.25 3385 1970.2.7 5.5 1001952.6.19 6.25 1345 1971.4.28 6.7 2121952.12.28 5.75 190 1973.3.22 5.5 181955.6.7 6 88 1973.8.6 6.8 2001961.6.12 5.8 47 1974.5.11 7.1 8371961.6.27 6 3582 1976.2.16 5.7 991962.6.24 6.2 449

M—Magnitude measured on the Richter scale;S—Isoseismal area measured in square kilometers.

Applying the normal information diffusion formula (8.25), observation mi can bechanged into a fuzzy subset as

Ai =

∫

U

mi(uj )/uj =

∫

U

exp

[−

(uj −mi)2

2h2m

]/uj . (8.42)

Therefore, an observation xi = (mi, gi) can be transformed into a single-columnfuzzy relationship matrix:

Ri =

u1

u2

·

·

·

u30

gi⎛⎜⎜⎜⎜⎜⎜⎝

mi(u1)

mi(u2)

·

·

·

mi(u30)

⎞⎟⎟⎟⎟⎟⎟⎠

. (8.43)

When magnitude m0 is given, we can change it into a fuzzy subset on U through theinformation distribution formula in (8.31). Employing the weight formula in (8.36),we calculate the relevant logarithmic isoseismal area g0. Let m0 = mi ∈ Xm; we canobtain a new sample as follows:

X = {x1, x2, . . . , x25}

= {(m1, g1), (m2, g2), . . . , (m25, g25)}

= {(6.5, 3.113), (6.5, 3.113), (7, 3.184), (5.75, 2.231), (7, 3.184),


(7, 3.184), (6.25, 3.028), (6.25, 3.028), (5.75, 2.231), (6, 2.684), (8.44)

(5.8, 2.300), (6, 2.684), (6.2, 2.979), (6.1, 2.858), (5.1, 1.444),

(6.5, 3.113), (5.4, 1.931), (6.4, 3.120), (7.7, 3.912), (5.5, 1.963),

(6.7, 2.871), (5.5, 1.963), (6.8, 2.884), (7.1, 3.231), (5.7, 2.168)}.

This is used to train a conventional BP neural network [Rumelhart et al., 1986;Pao, 1989] with one unit in the input layer, one hidden layer with 15 units, and oneunit in the output layer. Let the momentum rate be η = 0.9 and the learning rate beα = 0.7. We use X in (8.44) to train the BP neural network. After 153,780 iterations,the normalized system error is 0.00001. Using the hybrid model consisting of theinformation-diffusion approximate reasoning method and a BP neural network, wecan get a better estimator, depicted in Figure 8.3.

Figure 8.3 Relationship between earthquake magnitude and logarithmic isoseismal area,estimated by the hybrid model (HM) consisting of the IDAR method and a BP network.

8.6 Conclusion and Discussion 269

For comparison, the average sums of squared errors ǫ of the three estimators: g′,the linear-regression estimator (LR); g′′, the BP neural network estimator (BP); andg, the hybrid-model (HM); are computed as follows:

⎧⎪⎪⎨⎪⎪⎩

ǫLR= 1

25

∑25i=1(gi − g′i)

2 = 0.273425

ǫBP= 1

25

∑25i=1(gi − g′′i )2 = 0.2102895

ǫHM= 1

25

∑25i=1(gi − gi)

2 = 0.1993096

(8.45)

Obviously, the hybrid model is much better than the linear-regression model andthe conventional BP model, because the HM curve represents the nonlinearity of therelationship between earthquake magnitude and logarithmic isoseismal area, and thecurve is more stable than the BP curve.

8.6 Conclusion and Discussion

This chapter surveys some contemporary efforts in which the methodologies of fuzzylogic have been applied to earthquake research. Taxonomies of the types and scopeof applications of fuzzy logic in seismology, as well as earthquake engineering, aredescribed. Methodological needs to enable efficacious assessment of the effects ofpotential earthquakes with the aid of a few historical earthquake records are discussed.

In the past 30 years, scientists and engineers in seismology and earthquakeengineering have introduced many methods of fuzzy logic and developed severalapproaches based on fuzzy logic for earthquake research and the analysis of com-plex systems. Early investigations of fuzzy logic for earthquake research focused onfuzzy concepts in earthquake prediction, earthquake engineering, and earthquake-resistant design. The most important results based on fuzzy logic were producedfrom 1985 through 1995. More recent studies focus on developing hybrid approachesintegrating fuzzy logic, neurocomputing, evolutionary computing, and probabilisticcomputing.

Civil engineers in the USA first introduced fuzzy logic into earthquake research.Seismologists and earthquake engineers in China and Japan developed the tradi-tional fuzzy methods and suggested some new fuzzy methods to promote earthquakeresearch.

After reviewing the contemporary efforts relevant to the development of somemethodologies of fuzzy logic for earthquake research, we can come to the followingconclusions.

Fuzzy seismology, as a new branch of seismology, has provided some tools toanalyze seismic information for improving earthquake prediction. There have beencommonly accepted approaches to quantify some fuzzy concepts in earthquakeresearch such as earthquake intensity and the grade of earthquake damage. The


data in earthquake research are often scanty, incomplete, and contradictory. We canemploy the methods of information diffusion to analyze these data and obtain a fuzzyrelationship among the factors.

There are many applications of fuzzy logic in earthquake-resistant design.After reviewing the effectiveness of the fuzzy approach to earthquake research, we

believe that it is important to note the following points.We are not able to predict individual earthquakes. We do not regard fuzzy logic as

a technique to resolve this problem. Rather, we regard it as a method that can enhanceother techniques to promote earthquake prediction.

Fuzzy logic has provided powerful tools to quantify some inherently fuzzy conceptsin earthquake research. In this field, however, fuzzy methods are still dependent onexpert knowledge. Most people are interested in transforming the expert knowledgeinto a quantified model. A few pay attention to finding new knowledge from data.We shall pay more attention to finding knowledge from seismic data and earthquake-disaster data, which are often scanty, incomplete, and contradictory.

It is important to note that neither the classical models nor the fuzzy models governthe physical processes in nature. Scientists propose them as a compensation for theirown limitations in understanding the processes concerned.

With the development of soft computing and computational intelligence, one daywe shall see that we have no choice but to use fuzzy logic for analyzing seismic dataand earthquake-disaster data when they are scanty, incomplete, or contradictory.

Acknowledgments

The author is especially indebted to Professor George J. Klir of BinghamtonUniversity—SUNY who found the value of this research and recommended the chap-ter for publication. The project is sponsored by the Scientific Research Foundationfor Returned Overseas Chinese Scholars, State Education Ministry. This research wasdone at the Key Laboratory of Environmental Change and Natural Disaster, Ministryof Education of China.

APPENDIX 8.A: Modified Mercalli Intensity Scale Used in China

I. People do not feel any Earth movement.II. A few people might notice movement if they are at rest and/or on the upper

floors of tall buildings.III. Many people indoors feel movement. Hanging objects swing back and forth.

People outdoors might not realize that an earthquake is occurring.

References 271

IV. Most people indoors feel movement. Hanging objects swing. Dishes, windows,and doors rattle. The earthquake feels like a heavy truck hitting the walls. Afew people outdoors may feel movement. Parked cars rock.

V. Almost everyone feels movement. Sleeping people are awakened. Doors swingopen or close. Dishes are broken. Pictures on the wall move. Small objectsmove or are turned over. Trees might shake. Liquids might spill out of opencontainers.

VI. Everyone feels movement. People have trouble walking. Objects fall fromshelves. Pictures fall off walls. Furniture moves. Plaster in walls might crack.Trees and bushes shake. Damage is slight in poorly built buildings. No structuraldamage.

VII. People have difficulty standing. Drivers feel their cars shaking. Some furniturebreaks. Loose bricks fall from buildings. Damage is slight to moderate inwell-built buildings; considerable in poorly built buildings.

VIII. Drivers have trouble steering. Houses that are not bolted down might shift ontheir foundations. Tall structures such as towers and chimneys might twist andfall. Well-built buildings suffer slight damage. Poorly built structures suffersevere damage. Tree branches break. Hillsides might crack if the ground iswet. Water levels in wells might change.

IX. Well-built buildings suffer considerable damage. Houses that are not bolteddown move off their foundations. Some underground pipes are broken. Theground cracks. Reservoirs suffer serious damage.

X. Most buildings and their foundations are destroyed. Some bridges aredestroyed. Dams are seriously damaged. Large landslides occur. Water isthrown on the banks of canals, rivers, lakes. The ground cracks in large areas.Railroad tracks are bent slightly.

XI. Most buildings collapse. Some bridges are destroyed. Large cracks appear inthe ground. Underground pipelines are destroyed. Railroad tracks are badlybent.

XII. Almost everything is destroyed. Objects are thrown into the air. The groundmoves in waves or ripples. Large amounts of rock may move.

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Chapter 9 Fuzzy Transform: Application to theReef Growth Problem

Irina Perfilieva

9.1 Introduction 2759.2 Preliminaries 2769.3 Fuzzy Partition of the Universe 2779.4 F-Transform 2809.5 Inverse F-Transform 2839.6 Approximate Solution to the Cauchy Problem 287

9.6.1 The generalized Euler method 288

9.6.2 The generalized Euler–Cauchy method 292

9.7 Reef Growth Model and Sea Level Extraction 2949.8 Conclusions 297Acknowledgment 299References 300

9.1 Introduction

Fuzzy logic provides a basis for the approximate description of different dependen-cies. Fuzzy logic’s ability to produce smooth descriptions has especially attractedresearchers from various areas, and many of their results are unified by thecommon name—the universal approximation. These results have also proved theapproximation property of so-called fuzzy models. Generally speaking, each modeldiffers from the others by the choice of logical operations (see Novák et al.[1999]).

Unfortunately, the technique of producing fuzzy approximation models has notbeen followed by other methods. We know from the theory of numerical methods thatapproximation models can replace original complex functions in some computations,e.g., in solving differential equations, etc. This chapter is a new contribution to thisarea. We call this type of technique numerical methods based on fuzzy approximation

models.In this chapter, we show how it is possible to obtain an approximate value of a

definite integral as well as an approximate solution to an ordinary differential equation

275

276 9 Fuzzy Transform: Application to the Reef Growth Problem

on the basis of a certain class of fuzzy approximation models. We then apply thistechnique to the solution of a differential equation modeling reef growth (see alsoChapter 3). Furthermore, we use it to model ancient sea level variations (see alsoChapter 10).

9.2 Preliminaries

We confine ourselves to models that can be represented by functions of one variable.Generalization to the case of two and more variables is straightforward.

Suppose that we are given data and are concerned with construction of a representa-tive model. By this we mean the construction of a formula which represents (preciselyor approximately) the given data. For example, if our data comprise a collection ofpairs (xi, yi), i = 1, . . . , n, then the model can be given by a formula F(x) suchthat the function y = F(x) interpolates or approximates the data. In the first case wehave

yi = F(xi), i = 1, . . . , n

while in the second case, the equality may not hold, but the function y = F(x)

“nicely” approximates (in the sense of some criterion) the data. We choose interpola-tion when our data are precise, otherwise we look for an approximation. Let us stressthat in both cases the model may not be unique. It depends on the primary choiceof a class of formulas (F(x) in our notation) which is chosen to represent the dataas well as the approximation criterion. With this in mind, we come to the followingformulation of the problem investigated in this chapter.

We are given data which are not precise (e.g., the data contain errors of mea-

surement, or comprise numeric or linguistic expert estimations, etc.). Our goal is to

construct an approximating model of the given data, represented by a formula from

a certain class and such that a certain criterion is satisfied.

Moreover, having an approximating model at our disposal, we would like to apply itin further investigations which may be connected with the original data. In this chapterwe will demonstrate how this technique works in solving differential equations wheresome parameters are replaced by their approximating models.

The structure of the chapter is as follows. In Section 9.3, the concepts of a fuzzypartition and uniform fuzzy partition of the universe are introduced. In Sections 9.4and 9.5, the technique of direct and inverse fuzzy transforms (F-transforms hereafter)is introduced and approximating properties of the inverse F-transform are established.Section 9.6 presents the technique of approximate solvability of ordinary differentialequations. Section 9.7 is an example of an application.

9.3 Fuzzy Partition of the Universe 277

9.3 Fuzzy Partition of the Universe

We take an interval [a, b] as a universe. That is, all (real-valued) functions consideredin this chapter have this interval as a common domain. Let us introduce fuzzy sets(given by their membership functions) which are subsets of the universe [a, b] andwhich form a fuzzy partition of the universe.

Definition 9.1

Let x1, . . . , xn be fixed nodes within [a, b], such that x1 = a, xn = b, and n ≥

2. We say that fuzzy sets A1, . . . , An identified with their membership functionsA1(x), . . . , An(x) defined on [a, b], form a fuzzy partition of [a, b] if they fulfill thefollowing conditions:

1. Ak : [a, b] → [0, 1], Ak(xk) = 1.2. Ak(x) = 0 if x �∈ (xk−1, xk+1), where x−1 = a, xn+1 = b.3. Ak(x) is continuous.4. Ak(x) monotonically increases on [xk−1, xk] and monotonically decreases on[xk, xk+1].

5.∑n

k=1 Ak(x) = 1, for all x.

The membership functions A1(x), . . . , An(x) are called basic functions.Figure 9.1 shows a fuzzy partition of the interval [1, 4] by fuzzy sets with triangular-

shaped membership functions. The following formulas give the formal representation

Figure 9.1 An example of a fuzzy partition of [1, 4] by triangular membership functions.


of such triangular membership functions:

A1(x) =

⎧⎨⎩

1− (x−x1)h1

, x ∈ [x1, x2]

0, otherwise

Ak(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(x−xk−1)

hk−1, x ∈ [xk−1, xk]

1− (x−xk)hk

, x ∈ [xk, xk+1]

0, otherwise

An(x) =

⎧⎨⎩

(x−xn−1)

hn−1, x ∈ [xn−1, xn]

0, otherwise

where k = 2, . . . , n− 1, and hk = xk+1 − xk .Moreover, we say that a fuzzy partition is uniform if the nodes x1, . . . , xn are

equidistant, i.e., xk = a+h(k−1), k = 1, . . . , n, where h = (b−a)/(n−1), n ≥ 2,and two additional properties are met:

6. Ak(xk − x) = Ak(xk + x), for all x, k = 2, . . . , n− 1, n > 2.7. Ak+1(x) = Ak(x − h), for all x, k = 2, . . . , n− 2, n > 2.

In the case of a uniform fuzzy partition, h is the length of the support of A1 or An

while 2h is the length of support of the other basic functions Ak , k = 2, . . . , n − 1.Figure 9.2 shows a uniform partition of the interval [1, 4] by sinusoidal-shaped basicfunctions. Their formal expressions are given below.

A1(x) =

⎧⎨⎩

0.5(cos πh(x − x1)+ 1), x ∈ [x1, x2]

0, otherwise

Ak(x) =

⎧⎨⎩

0.5(cos πh(x − xk)+ 1), x ∈ [xk−1, xk+1]

0, otherwise

where k = 2, . . . n− 1, and

An(x) =

⎧⎨⎩

0.5(cos πh(x − xn)+ 1), x ∈ [xn−1, xn]

0, otherwise.

For the sake of simplicity, here we consider only uniform partitions, though someresults remain true in the general case. We will point out when this occurs.

9.3 Fuzzy Partition of the Universe 279

Figure 9.2 An example of a uniform fuzzy partition of [1, 4] by sinusoidal membershipfunctions.

The following lemma shows that, in the case of a uniform partition, the definiteintegral of a basic function does not depend on its concrete shape. This property willbe further used to simplify a direct F-transform.

Lemma 9.1

Let the uniform partition of [a, b] be given by basic functions A1(x), . . . , An(x). Then

∫ x2

x1

A1(x)dx =

∫ xn

xn−1

An(x)dx =h

2(9.1)

and for k = 2, . . . , n− 1∫ xk+1

xk−1

Ak(x)dx = h (9.2)

where h is the length of the support of A1.

Proof

Obviously,∫ x3

x1

A2(x)dx = · · · =

∫ xn

xn−2

An−1(x)dx.

Therefore, to prove (9.2) it is sufficient to estimate∫ h

−h

A(x)dx


where A(x) = A2(x + a + h) and x ∈ [−h, h]. Based on properties 5 and 7 of basicfunctions, we can deduce that

1− A(x) = A(x + h), x ∈ [−h, 0].

Then

∫ h

0A(x)dx =

∫ 0

−h

A(x + h)dx = h−

∫ 0

−h

A(x)dx

which implies (9.2). Equation (9.1) follows immediately from the symmetry of basicfunctions (property 6). �

9.4 F-Transform

In this section we introduce the technique of two F-transforms: direct and inverse. Thedirect F-transform takes the original function (which should be at least integrable)and converts it into an n-dimensional vector. The inverse F-transform converts then-dimensional vector into a special function which approximates the original one.The advantage of the direct F-transform is that it produces a simple and uniquerepresentation of the original function which enables us to use the former instead ofthe latter in complex computations. After finishing the computations, the result canbe brought back into the space of ordinary functions by the inverse F-transform. Tobe sure that this can be done we need to prove a number of theorems.

The following definition [see also Perfilieva & Chaldeeva, 2001] introduces thedirect F-transform (or fuzzy transform) of a given function.

Definition 9.2

Let f (x) be any continuous (real-valued) function on [a, b] and A1(x), . . . , An(x)

be basic functions which form a fuzzy partition of [a, b]. We say that the n-tuple ofreal numbers [F1, . . . , Fn] is the F-transform of f w.r.t. A1, . . . , An if

Fk =

∫ b

af (x)Ak(x)dx∫ b

aAk(x)dx

. (9.3)

Suppose that the basic functions A1, . . . , An are fixed. Denote the F-transform off w.r.t. A1, . . . , An by Fn[f ]. Then, according to Definition 9.2, we can write

Fn[f ] = [F1, . . . , Fn]. (9.4)

The elements F1, . . . , Fn are called components of the F-transform.

9.4 F-Transform 281

If the partition of [a, b] by A1, . . . , An is uniform, then the expression (9.2) forcomponents of the F-transform may be simplified on the basis of Lemma 9.1:

F1 =2

h

∫ x2

x1

f (x)A1(x)dx

Fn =2

h

∫ xn

xn−1

f (x)An(x)dx

Fk =1

h

∫ xk+1

xk−1

f (x)Ak(x)dx k = 2, . . . , n− 1.

Remark 9.1

Even in the case where f (x) is known only at some nodes x1, . . . , xl ∈ [a, b], theF-transform components of f w.r.t. A1, . . . , An can be computed as follows:

Fk =

∑lj=1 f (xj )Ak(xj )∑l

j=1 Ak(xj )

where 1 ≤ k ≤ n and n < l.It is easy to see that if a fuzzy partition (and therefore, basic functions) is fixed,

then the F-transform as a mapping from C[a, b] (the set of all continuous functionson [a, b]) to R

n is linear, so that

Fn[αf + βg] = αFn[f ] + βFn[g]

for α, β ∈ R and functions f, g ∈ C[a, b].One may say that we lose information by using an F-transform instead of the

original function. However, we can investigate this problem by asking the followingquestion: how fully is the original function f represented by its F-transform? Firstof all, we will try to estimate each component Fk , k = 1, . . . , n, using differentassumptions about the smoothness of f .

Lemma 9.2

Let f (x) be any continuous function on [a, b] and A1(x), . . . , An(x) be basic func-

tions which form a uniform fuzzy partition of [a, b]. Then for each k = 2, . . . , n− 1,

there exist two constants ck1 ∈ [xk−1, xk] and ck2 ∈ [xk, xk+1] such that

Fk =1

h

∫ ck2

ck1

f (x)dx

and for k = 1 (k = n) there exists c ∈ [x1, x2] (c ∈ [xn−1, xn]) such that

F1 =2

h

∫ c

x1

f (x)dx (Fn =2

h

∫ xn

c

f (x)dx).


Proof

The proof can be easily obtained from the second mean-value theorem. �

Therefore, by Lemma 9.2 we can say that Fk is a mean value of f within theinterval [ck1, ck2] and thus it accumulates the information about function f withinthis interval. We can evaluate Fk more precisely if the function f is twice continuouslydifferentiable.

Lemma 9.3

Let the conditions of Lemma 9.2 be fulfilled, but function f be twice continuously

differentiable in (a, b). Then for each k = 1, . . . , n

Fk = f (xk)+O(h2). (9.5)

Proof

The proof will be given for one fixed value of k which lies between 2 and n− 1. Theother two cases k = 1 and k = n are considered analogously. We apply the trapezoidformula with nodes xk−1, xk, xk+1 to the numerical computation of the integral

1

h

∫ xk+1

xk−1

f (x)Ak(x)dx

and obtain

Fk =1

h

∫ xk+1

xk−1

f (x)Ak(x)dx

=1

h·h

2(f (xk−1)Ak(xk−1)+ 2f (xk)Ak(xk)+ f (xk+1)Ak(xk+1))+O(h2)

= f (xk)+O(h2). �

Using (9.5) and applying again the trapezoid formula with nodes x1, . . . , xn to thenumerical computation of the integral

∫ xk

x1

f (x)dx

we easily come to the following corollary.

Corollary 9.1 (Computation of definite integral)Let the conditions of Lemma 9.3 be fulfilled and the F-transform of f be given by

(9.4). Then for each k = 2, . . . , n− 1∫ xk

x1

f (x)dx = h( 12F1 + F2 + · · · + Fk−1 +

12Fk)+O(h2). (9.6)

9.5 Inverse F-Transform 283

Moreover, for any continuous function f (x) the integral∫ b

af (x)dx can be computed

precisely:

∫ b

a

f (x)dx = h( 12F1 + F2 + · · · + Fn−1 +

12Fn). (9.7)

Returning to the problem of losing information by dealing with an F-transforminstead of its original, we can say that an F-transform preserves mean values offunction f over 2h long subintervals with an accuracy of up to h2.

9.5 Inverse F-Transform

A reasonable question is the following: can we reconstruct the original function fromits F-transform? The answer is clear: in general not precisely, because we are losinginformation when changing to the direct F-transform. However, the function whichcan be reconstructed (by the inverse F-transform) approximates the original one insuch a way that a universal convergence can be established. Moreover, the inverseF-transform fulfills the best approximation criterion which can be called the piecewiseintegral least-square criterion.

Definition 9.3

Let Fn[f ] = [F1, . . . , Fn] be the F-transform of a function f (x) w.r.t. A1, . . . , An.The function

fF,n(x) =

n∑

k=1

FkAk(x) (9.8)

will be called the inversion formula or the inverse F-transform.The lemma below shows that the sequence of functions fF,n uniformly converges

to f .

Lemma 9.4

Let f (x) be any continuous function on [a, b] and let {(A(n)1 , . . . , A

(n)n )n} be a

sequence of uniform fuzzy partitions of [a, b], one for each n. Let {fF,n(x)}

be the sequence of inverse F-transforms, each with respect to the given n-tuple

A(n)1 , . . . , A

(n)n . Then for any ε > 0 there exists nε such that for each n > nε and for

all x ∈ [a, b]

|f (x)− fF,n(x)| < ε. (9.9)


Proof

Note that the function f is uniformly continuous on [a, b]; i.e., for each ε > 0there exists δ = δ(ε) > 0 such that for all x1, x2 ∈ [a, b] |x1 − x2| < δ implies|f (x1) − f (x2)| < ε. To prove our lemma we choose some ε > 0 and find therespective δ. Let nε ≥ 2 be such that h = (b − a)/(nε − 1) ≤ δ/2. We willshow that with n ≥ nε (9.9) holds true. Let F1n, . . . , Fnn be the components of thesingle F-transform of f w.r.t. basic functions A

(n)1 , . . . , A

(n)n , n ≥ nε. Then for all

t ∈ [xk, xk+1], k = 1, . . . , n− 1, we can evaluate

|f (t)− Fkn| = |f (t)−1

h

∫ xk+1

xk−1

f (x)A(n)k (x)dx|

≤1

h

∫ xk+1

xk−1

|f (t)− f (x)|A(n)k (x)dx < ε

and analogously

|f (t)− Fk+1,n| < ε.

Therefore,

|f (t)−

n∑

k=1

FknA(n)k (t)| ≤

n∑

k=1

A(n)k (t)|f (t)− Fkn| < ε(A

(n)k (t)+ A

(n)k+1(t)) = ε.

Because argument t has been chosen arbitrarily, this proves the requiredinequality. �

The following corollary reformulates Lemma 9.4.

Corollary 9.2

Let the assumptions of Lemma 9.4 be fulfilled. Then the sequence of inverse

F-transforms {fF,n} uniformly converges to f .

To illustrate this fact we choose two functions with different behavior, sin(1/x)

and sin x (see Figures 9.3, 9.4), and consider different values of n. As we see below,the greater the value of n, the closer the approximating curve approaches the originalfunction.

Since approximation by the inverse F-transform converges uniformly to the originalfunction, we are interested in the criterion that distinguishes it among other functionsfrom the certain class. As discussed at the beginning, this criterion guarantees us thatthe inverse F-transform is the best approximation in the sense we explain below.

Let A1(x), . . . , An(x) be basic functions which form a fuzzy partition of [a, b],and let f (x) be an integrable function on [a, b]. By FT we denote the class of

9.5 Inverse F-Transform 285

Figure 9.3 sin(1/x) and its inverse transformations based on triangular-shaped basic functions:n is the number of nodes used to approximate the function.

approximating functions represented by the formula

n∑

i=1

ciAi(x) (9.10)

where c1, . . . , cn are arbitrary real coefficients.Let the following piecewise integral least-square criterion

�(c1, . . . , cn) =

∫ b

a

(n∑

i=1

(f (x)− ci)2Ai(x)

)dx (9.11)

characterize the closeness between f (x) and a function from FT . Then the compo-nents F1, . . . , Fn of the F-transform minimize (9.11) and therefore determine the bestapproximation of f (x) in FT . We leave this fact unproved because the proof is atechnical exercise.


Figure 9.4 sin(x) and its inverse transformation based on sinusoidal-shaped basic functions(n = 20).

It is worth noticing that, so far, we have not specified any concrete shape for the basicfunctions. Thus, a natural question arises concerning the influence of different shapesof basic functions on the quality of the approximation. We can say the following. Theproperties of an approximating function are determined by the respective propertiesof basic functions. For example, if basic functions are of triangular shape then theapproximating function will be piecewise linear. If we choose sinusoidal-shaped basicfunctions, then the approximating function will be, of course, smoother. In any case,whatever we want to obtain from the approximating function is required from the basicfunctions as well. The following lemma shows how the difference between any twoapproximations of a given function by inverse F-transformations can be estimated.As can be seen, it depends on the character of smoothness of the original functionexpressed by its modulus of continuity (see below).

Lemma 9.5

Let f ′F,n(x) and f ′′F,n(x) be two inverse F-transformations of the same function f (x)

w.r.t. n-tuples of different basic functions, n ≥ 2. Then

|f ′F,n(x)− f ′′F,n(x)| ≤ 2ω(f, 2h)

where ω is the modulus of continuity of f (x):

ω(f, 2h) = max|δ|≤2h

maxx∈[a,b]

|f (x + δ)− f (x)|.

We illustrate Lemma 9.5 by considering two different inverse F-transforms offunctions sin(1/x) and sin x. One is based on triangular-shaped basic functions and

9.6 Approximate Solution to the Cauchy Problem 287

Figure 9.5 sin(1/x) and its inverse transforms based on triangular and sinusoidal-shaped basicfunctions.

Figure 9.6 sin x and its inverse transforms based on triangular and sinusoidal-shaped basicfunctions.

one is based on sinusoidal-shaped basic functions (see Figures 9.5, 9.6). Becausesin(1/x) has a modulus of continuity greater than sin x, the approximation of thelatter with the same value of n looks nicer (both approximations practically coincidewith the original function).

9.6 Approximate Solution to the Cauchy Problem

In this section we show how the approximation models based on the F-transform canbe used in applications. In general, we mean by this that if the original function isreplaced by an approximation model as described above, then a certain simplificationof complex computations can be achieved. For demonstration, we will consider the


Cauchy problem{

y′(x) = f (x, y)

y(x1) = y1(9.12)

and show how it can be approximately solved in the interval [x1, xn] if F-transformis applied to both sides of the differential equation. Let us stress that in this sectionwe need a uniform fuzzy partition of [x1, xn].

9.6.1 The generalized Euler method

Suppose that we are given the Cauchy problem (9.12) where the functions y(x) andf (x, y(x)) on [x1, xn] are sufficiently smooth. Let us choose some uniform fuzzypartition of interval [x1, xn] with parameter h = (xn − x1)/(n − 1), n ≥ 2, andapply the direct F-transform to both parts of the differential equation. In this way wetransfer the original Cauchy problem to the space of fuzzy units, solve it in the newspace, and then transfer it back by the inverse F-transform. We describe the sequenceof steps which leads to the solution. The justification is proved in Theorem 9.1.

Before we apply the direct F-transform to both parts of the differential equation,we replace y′(x) by its approximation (y(x + h)− y(x))/h so that

y(x + h) = y(x)+ hy′(x)+O(h2). (9.13)

Denote y1(x) = y(x + h) as a new function and apply the direct F-transform to bothparts of (9.13). By the linearity of F-transform and Lemma 9.3 we obtain from (9.13)the expression for F-transform components of the respective functions

Fn[y′] =

1

h(Fn[y1] − Fn[y])+O(h2). (9.14)

Here Fn[y′] = [Y ′1, . . . , Y

′n−1], Fn[y] = [Y1, . . . , Yn−1], and Fn[y1] = [Y11, . . . ,

Y1n−1]. Note that these vectors are one component shorter than in Definition 9.2because the function y1(x) may not be defined on [xn−1, xn], xn = b. It is notdifficult to prove that

Y11 = Y2 +O(h2)

Y1k = Yk+1, k = 2, . . . , n− 1.

Indeed, for values k = 2, . . . , n− 2

Y1k =1

h

∫ xk+1

xk−1

y(x + h)Ak(x)dx =1

h

∫ xk+2

xk

y(t)Ak+1(t)dt = Yk+1.


For the values k = 1, n− 1 the proof is analogous. Therefore, Equation (9.14) givesus the way to compute components of the F-transform of y′ via components of theF-transform of y:

Y ′k =1

h(Yk+1 − Yk)+O(h2), k = 1, . . . , n− 1. (9.15)

Let us introduce the (n− 1)× n matrix

D =1

h

⎛⎜⎜⎜⎝

−1 1 0 · · · 0 00 −1 1 · · · 0 0...

0 0 0 · · · −1 1

⎞⎟⎟⎟⎠ (9.16)

so that equality (9.15) can be rewritten (up to O(h2)) as matrix equality

Fn[y′] = DFn[y] (9.17)

where Fn[y′] = [Y ′1, . . . , Y

′n−1]

T and Fn[y] = [Y1, . . . , Yn]T.

Coming back to the Cauchy problem (9.12) and applying F-transform to both sidesof the differential equation, we will obtain the following system of linear equationswith respect to the unknown Fn[y]:

DFn[y] = Fn[f ] (9.18)

where Fn[f ] = [F1, . . . , Fn−1]T is the F-transform of f (x, y) as the function of x

w.r.t. the chosen basic functions A1, . . . , An. The last component Fn is not present inFn[f ] due to the preservation of dimensionality.

Note that system (9.18) does not include the initial condition of (9.12). For this, letus complete matrix D by adding the first row

Dc =1

h

⎛⎜⎜⎜⎝

1 0 0 · · · 0 0−1 1 0 · · · 0 0

...

0 0 0 · · · −1 1

⎞⎟⎟⎟⎠

so that Dc is an n × n nonsingular matrix. Analogously, let us complete the vectorFn[f ] by the first component y1/h so that

F cn [f ] = [

y1

h, F1, . . . , Fn−1]

T.

Then, the transformed Cauchy problem can be fully represented by the followinglinear system of equations with respect to the unknown Fn[y]:

DcFn[y] = F cn [f ]. (9.19)


The solution of (9.19) is given by the formula

Fn[y] = (Dc)−1F cn [f ] (9.20)

which, in fact, is the generalized Euler method. To make sure, we compute the inversematrix

(Dc)−1 = h

⎛⎜⎜⎜⎝

1 0 0 · · · 0 01 1 0 · · · 0 0...

1 1 1 · · · 1 1

⎞⎟⎟⎟⎠

and rewrite (9.20) componentwise:

Y1 = y1

Y2 = y1 + hF1

Y3 = y1 + hF1 + hF2

...

Yn = y1 + hF1 + · · · + hFn−1

or, in a more concise way,

Y1 = y1 (9.21)

Yk+1 = Yk + hFk, k = 1, . . . , n− 1.

Formulas (9.21) can be applied to the computation of Y2, . . . , Yn provided that theway of computing F1, . . . , Fn−1 is known. However, it cannot be done directlyusing formulas (9.3) because the expression for function f (x, y) includes also theunknown function y. Therefore, we have to get around this difficulty. The followingapproximation

Fk =

∫ b

af (x, Yk)Ak(x)dx∫ b

aAk(x)dx

(9.22)

for Fk , k = 1, . . . , n − 1, is suggested. The theorem given below provides thejustification.

Theorem 9.1

Let the Cauchy problem (9.12) with twice differentiable parameters be transformed

by applying F-transform w.r.t. basic functions A1, . . . , An to both sides of a given


differential equation. Then the components of the F-transform of y w.r.t. the same

basic functions can be found approximately from the following system of equations

Y1 = y1 (9.23)

Yk+1 = Yk + hFk, k = 1, . . . , n− 1, (9.24)

where Fk is given by (9.22). The local approximation error is of the order h2.

Proof

It has been shown that the system of linear equations (9.19) represents the F-transformof the Cauchy problem (9.12) up to O(h2). Therefore, to prove the theorem it issufficient to show that for each k = 1, . . . , n− 1, the order of the difference Fk − Fk

is h2. Let us denote y(xk) = yk . First, we estimate (using the trapezoid formula) theintermediate difference

f (xk, yk)− Fk =1

h

∫ xk+1

xk−1

(f (xk, yk)− f (x, Yk))Ak(x)dx

=1

h·h

2· 2(f (xk, yk)− f (xk, Yk))+O(h2) =

∂f

∂y(xk, y)(yk − Yk)+O(h2)

where y ∈ [yk, Yk]. By Lemma 9.3, we have

yk − Yk = O(h2)

which, when substituted into the expression above, leads to the estimation

f (xk, yk)− Fk = O(h2).

Again, by Lemma 9.3, we get

f (xk, yk)− Fk = O(h2)

which together with the previous estimation proves that

Fk − Fk = O(h2).

This completes the proof. �

Corollary 9.3

The generalized Euler method for (9.12) is given by the recursive scheme (9.23)–

(9.24) with the local error O(h2). The approximate solution to (9.12) can be found


Figure 9.7 Precise solution (gray line) and approximate solution (black line) of the Cauchyproblem obtained by the generalized Euler method (n = 10).

by taking the inverse F-transform

yY,n(x) =

n∑

k=1

YkAk(x)

where A1, . . . , An are fixed basic functions.

Let us illustrate the generalized Euler method for the Cauchy problem (9.12)given by

{y′(x) = x2 − y

y(x1) = 1.

Figure 9.7 shows the precise solution (gray line) and the approximate one obtainedby the generalized Euler method. The global error of the approximate solutionwith n = 10 nodes has the order 10−1, which corresponds to the theoreticalestimation.

9.6.2 The generalized Euler–Cauchy method

The generalized Euler method for the Cauchy problem has the same disadvantage as itsclassical prototype, namely, it is not sufficiently precise. Therefore, we will constructthe generalization of the more advanced method known as the Euler–Cauchy method.Recall that its classical prototype belongs to the family of the Runge–Kutta methods.


The following scheme provides formulas for the computation of components of theF-transform of the unknown function y(x) w.r.t. some basic functions A1, . . . , An:

Y1 = y1 (9.25)

Y ∗k+1 = Yk + hFk (9.26)

Yk+1 = Yk +h

2(Fk + F ∗k+1), k = 1, . . . , n− 1, (9.27)

where

Fk =

∫ b

af (x, Yk)Ak(x)dx∫ b

aAk(x)dx

F ∗k+1 =

∫ b

af (x, Y ∗k+1)Ak+1(x)dx∫ b

aAk+1(x)dx

.

This method computes the approximate coordinates [Y1, . . . , Yn] of the direct F-transform of the function y(x). The inverse F-transform

yY,n(x) =

n∑

k=1

YkAk(x)

approximates the solution y(x) of the Cauchy problem.It can be proved that the generalized Euler–Cauchy method (9.25)–(9.27) has a

local error of order h3.Let us illustrate the generalized Euler–Cauchy method for the Cauchy problem

considered above with the same number of nodes n = 10. Figure 9.8 shows theprecise solution (gray line) and the approximate one obtained by the generalizedEuler–Cauchy method. The global error of the approximate solution with n = 10nodes is of the order 10−2 which again corresponds to the theoretical estimation.

Remark 9.2

The demonstrated generalized Runge–Kutta methods can be applied to the Cauchyproblem where y(x) and f (x, y) are vector functions, i.e., to the system of ordinarydifferential equations with initial values.

Remark 9.3

The demonstrated generalized Runge–Kutta methods can be applied to the Cauchyproblem even if the function f (x, y) is only partially given at a finite number of nodesor by a description using fuzzy “IF–THEN” rules. Formalization and realization ofthis description in the form of the inverse F-transform can be preliminarily constructed(see Perfilieva [2001a,b, 2002]).


Figure 9.8 Precise solution (gray line) and approximate solution (black line) of the Cauchyproblem obtained by the generalized Euler–Cauchy method (n = 10).

Remark 9.4

The generalized Runge–Kutta methods based on F-transformation can also be appliedto the Cauchy problem when the initial value y1 is not known precisely. For example,y1 may be a fuzzy number.

9.7 Reef Growth Model and Sea Level Extraction

We are going to apply the generalized Euler method based on F-transforms tocomputer-based modeling of carbonate sedimentation. There are well-justified rea-sons for the use of fuzzy-based modeling: the available data are imprecise and to agreat extent averaged in nature (this is especially true for data related to the past);geological processes are very slow and the changes can be described qualitativelyrather than quantitatively; geological processes are locally non-homogeneous and aredependent on a specific place while, at the same time, homogeneous in that they obeyuniversal laws.

All these reasons argue in favor of using a fuzzy-based approach which is suffi-ciently robust and computationally efficient (see also Chapters 3 and 10). Though thefuzzy approach is mainly associated with linguistic characterization of vague (andthus, qualitatively expressed) events, we here demonstrate another technique relatedto fuzziness on one side and to classical analysis and numeric methods on the other.

We investigated the following two problems which use the reef growth model (9.28)of Bosschler & Schlager [1992] and Demicco & Klir [2001]. In the first problem,we use our technique to model Belize reef growth based on a differential equation(9.28) which characterizes the process. In the second problem, we use a measured

9.7 Reef Growth Model and Sea Level Extraction 295

stratigraphic section with well-defined third-order and fourth-order cycles and applythe reef growth model to back-calculate a sea level record for the section.

In the Bosschler and Schlager [1992] model, the growth rate of corals dependslargely upon the amount of light available for photosynthesis. As light decreases withwater depth, so does reef growth. The following differential equation characterizesthe process of reef growth under changing sea level regime:

dh(t)

dt= Gm tanh

(I0

Ik

exp(−k[h0 + h(t)− (s0 + s(t))])

)(9.28)

where

● h(t) is the growth increment,● h0 the initial height,● Gm the maximal growth rate,● I0 the surface light intensity,● Ik the saturating light intensity,● k the extinction coefficient,● s0 the initial sea level position, and● s(t) the sea level variation.

Note that sea level is included in (9.28) as a parameter. We first apply our techniqueof solving ordinary differential equations to Equation (9.28) and obtain a model ofthe Belize barrier reef growth over the past 80,000 years. The parameters are takenfrom Bosscher and Schlager [1992] and relate to the carbonate sediment productionpattern on the Atlantic shelf-slope break of Belize. The sea level curve s(t) of thepast 80,000 years was reconstructed from the numeric data [Bosscher and Schlager,1992] related to the pattern by the use of F-transforms (see Figure 9.9).

Figure 9.10 is a graph of carbonate production versus depth and distance, deter-mined by solving differential equation (9.28) using the technique of the generalizedEuler method (9.23)–(9.24) described above. This graph is similar to that shown inBosscher and Schlager [1992] (their Figure 8), which they obtained by numericalsolution of (9.28) using the fourth-order Runge–Kutta method. It is also similar to thegraph presented in Demicco and Klir [2001] (their Figure 2d; see also Section 3.4.2,Figure 3.8) obtained using a fuzzy linguistic model of the same problem.

Our second exercise is the inverse problem to that considered above: using astratigraphic measured section where we are given the growth increment h(t), finds(t), the sea level history. The problem formulation and the empirical data have beenprovided by Demicco (personal communication).

There are some difficulties in obtaining a solution to the inverse problem consideredabove. They are caused by the way in which the growth increment h(t) is given. Wealso recognize that intertidal carbonate cycles are strictly not totally aggradational(although subtidal cycles may be), so that this exercise is a starting point to illustrate


Figure 9.9 The sea level curve reconstructed by F-transforms from numeric data.

Figure 9.10 Carbonate production on the Atlantic shelf-slope break of Belize. The horizontalaxis is the distance from the sea shore and the vertical axis is the water depth.

9.8 Conclusions 297

potential applications. Below we describe the construction of a mathematical modelof h(t).

The input data have been obtained empirically from a vertical measured sectionof an Upper Cambrian limestone from western Maryland (see Chapter 10 for moredetails). The section comprises a sequence of thicknesses of eight types of rocks. Eachtype corresponds to a certain water depth and, therefore, relatively determines theancient sea level. Moreover, if we apply the Bosscher and Schlager [1992] model whenrelative sea level rises, the growth rate significantly decreases. Since sea level risesup and then goes down repeatedly, the total sequence of thicknesses of types of rockscan be divided into cycles characterizing one period of sea level rise and, particularly,its fall. The division into cycles cannot be determined by our mathematical model andis performed by either an expert or by a program implementing expert actions (seeChapter 10 for details). Thus, a mathematical model of h(t) can be constructed froma sequence of thicknesses of various types of rocks divided into respective cyclesplus information about the correspondence between a rock type and a water depth.Examples of the input data are given in Table 9.1.

Given this, we can solve equation (9.28) with respect to unknown s(t) for eachcycle and in this way obtain the data that characterize the decreasing portion of eachcycle in the sea level history. The result is depicted in Figure 9.11.

Let us stress that the data in Figure 9.11 characterizing sea level is artificially joinedby a continuous curve. To obtain a mathematical model of the sea level data obtainedabove, we use F-transforms with sinusoidal-shaped fuzzy sets. The results are shownin Figures 9.12 and 9.13. Let us remark that the curve in Figure 9.12 is the third-order sea level history insofar as it is showing trends in the fourth-order cycle data.Moreover, although the values of sea level are negative, the relative magnitude oftens of meters is of the right order as judged from sedimentologic inference.

9.8 Conclusions

This chapter is a contribution to a new area that can be called numerical methods onthe basis of fuzzy approximating models. Fuzzy basic functions have been introducedand two kinds of the so-called F-transform of an original function w.r.t. chosen basicfunctions have been presented. The convergence of the approximation models thatare obtained by the inverse F-transform has been demonstrated. Finally, the abilityof the new models to be used in numerical methods instead of the original functionhas been demonstrated on examples of solving ordinary differential equations. Thefollowing facts make this technique attractive:

● computation simplicity;● good accuracy, comparable with analogous numerical methods;● stability with respect to changes in initial data.


Table 9.1 Example of input data.

Rock Rock Cycle Cycletype thickness number thickness

4 1.2 1 1.21 0.4 1 1.63 0.1 1 1.74 5.5 1 7.23 0.1 2 0.14 3.0 2 3.12 0.7 2 3.84 0.3 2 4.1...

......

...

4 3.1 44 3.13 0.6 44 3.74 1.4 44 5.15 0.6 44 5.73 0.9 44 6.65 0.8 44 7.43 1.0 44 8.45 3.1 44 11.56 1.1 44 12.63 1.2 45 1.22 0.8 45 2.05 2.4 45 4.44 2.0 45 6.43 2.7 45 9.1

Figure 9.11 The sea level data.

9.8 Conclusions 299

Figure 9.12 Sea level curve obtained using F-transform.

Figure 9.13 Sea level data and curve obtained using F-transform.

We have applied numeric methods on the basis of F-transform to the solution ofordinary differential equations which are encountered in geological practice. Theresults are convincing and confirm that, in the case of averaged initial data, theproposed methodology has advantages over other methods. In further investigation wewould like to extend this methodology to the solution of partial differential equations.

Acknowledgment

This paper has been supported by grant IAA118730 of the GAAC CR and projectME468 of the MŠMT of the Czech Republic as the international supplement tothe project NSF “Stratigraphic Simulation Using Fuzzy Logic to Model Sediment


Dispersal.” The author wishes to thank her PhD student Martina Danková and diplomastudent Dagmar Plšková for realization of all the computations in Matematica.

References

Bosscher, H., & Schlager, W. [1992], “Computer simulation of reef growth.” Sedimentology,39, 503–512.

Demicco, R. V., & Klir, G. J. [2001], “Stratigraphic simulations using fuzzy logic to modelsediment dispersal.” Journal of Petroleum Science and Engineering, 31, 135–155.

Novák, V., Perfilieva, I., & Mockor, J. [1999], Mathematical Principles of Fuzzy Logic. Kluwer,Boston and Dordrecht.

Perfilieva, I. [2001a], “Logical approximation: general approach to the construction ofapproximating formulas.” Proceedings of EUSFLAT’2001, Leicester, UK.

Perfilieva, I. [2001b], “Neural nets and normal forms from fuzzy logic point of view.” Neural

Network World, 11, 627–638.Perfilieva, I. [2001c], “Normal forms for fuzzy logic functions and their approximation ability.”

Fuzzy Sets and Systems, 124, 371–384.Perfilieva, I. [2002], “Logical approximation.” Soft Computing, 7(2), 73–78.Perfilieva, I., & Chaldeeva, E. [2001], “Fuzzy transformation.” Proceedings of IFSA’2001

World Congress, Vancouver, Canada.

Chapter 10 Ancient Sea Level Estimation

Vilem Novák

10.1 Introduction 30110.2 Special Fuzzy Logic Techniques 303

10.2.1 Outline of the methods 303

10.2.2 The theory of evaluating linguistic expressions 305

10.2.3 Linguistic description and logical deduction 314

10.2.4 Fuzzy transform 319

10.3 Automatic Determination of Rock Sequences 32210.3.1 Geological characterization 322

10.3.2 General rules and the fuzzy algorithm 325

10.3.3 Results of tests 326

10.4 Sea Level Estimation 32810.5 Conclusion 335Acknowledgment 335References 335

10.1 Introduction

This chapter presents an application of fuzzy logic to the estimation of ancient sealevel changes. The main idea is to utilize the geologist’s expert knowledge expressedin natural language.

The initial situation is as follows. We are given the data about rocks found ina vertical section (such as an oil well, or a large outcrop). The total thickness ofthe section is about 250 meters. Up to eight rock types (mostly limestone) may bedistinguished in the section and the thickness of each rock type is given. The oldestrocks are at the bottom (at 0 meters) and the youngest rocks are on the top.

Our principal goal is to estimate the behavior of ancient sea level on the basis of therock type data. This goal can be solved on the basis of thickness of certain shallowing-upwards sequences of rocks deposited in the given vertical section. Let us note thatthe “accommodation potential” of a sedimentary deposit is a complicated sum ofsubsidence, sea level change, and sediment deposition. For a first cut, we assume that

301

302 10 Ancient Sea Level Estimation

subsidence is constant and negative. We also assume that sediment production can“fill up the holes” quickly.

Thus, the above goal splits into two tasks. The first task is to determine sequencesof rocks deposited during one cycle of sea level rise and fall. The second task isto use those determined sequences to estimate the sea level changes. This can bedone because the sequence thickness roughly corresponds to the maximum ancientsea level position in the given time period. We have used fuzzy logic in the broadersense1 to solve both tasks.

To solve the first task, a special algorithm has been developed and written in BorlandPASCAL 7.0. The algorithm is based on a geologist’s description of the way he/shedetermines the hierarchy of rock sequences. The description is in natural languageand its specific feature is the use of vague predicates for determination of the characterof thickness of rocks and their sequences.

The second task is solved using two basic techniques. The first technique isF-transform, described in Chapter 9. We benefit especially from its ability to filter thedata. Hence, this method is well suited for determining ancient sea level fluctuationssince the input data are far from being precise. However, the geologist may have stillmore specific information, which is difficult (or impossible) to include in the algo-rithm of F-transform. Therefore, it would be useful to be able to use this informationto directly affect the method of sea level estimation.

Such a possibility can be accomplished when the information about the sequencethicknesses is provided using a linguistic description. This is a set of linguisticallyspecified IF–THEN rules characterizing the sea level position in the respective timeperiod. Linguistic description is an efficient tool, which embraces the geologist’sspecific knowledge and thus allows a more realistic model of the actual sea levelposition to be obtained. In this chapter, we describe a method in which such a linguisticdescription can be learned from the data, and also how sea level position can bededuced from it.

All the algorithms use routines from the software package LFLC 1.5 developedat the University of Ostrava, Czech Republic. Moreover, a new routine for “smoothlogical deduction” has been developed and successfully applied.

The chapter is divided into four sections. Section 10.2 describes special fuzzytechniques used for solution of the above tasks. We start with an outline of the theoryof evaluating linguistic expressions and their semantics since these concepts playan essential role in the learning and deduction methods used later. Then the logicaldeduction and the F-transform methods are described. In Section 10.3, the algorithmfor determining rock sequences is described. In Section 10.4, we present the results,

1The following classification of fuzzy logic has been generally agreed: fuzzy logic in the narrow sense(FLn), which is a special many-valued logic providing tools for modeling the vagueness phenomenon;and that in the broader sense (FLb), which is an extension of FLn by some aspects of natural languagesemantics to enable modeling of natural human reasoning. More about the general theory of fuzzy logiccan be found in Novák & Perfilieva [2000].

10.2 Special Fuzzy Logic Techniques 303

an estimation of the sea level fluctuation based on three methods: (1) F-transform; (2)logical deduction on the basis of linguistic descriptions learned from the informationabout sequence thicknesses; and (3) logical deduction where the linguistic descriptionis learned from the F-transformed data.

10.2 Special Fuzzy Logic Techniques

This section describes special fuzzy techniques used for solving the two tasks definedabove. The three techniques described below are the theory of evaluating linguis-tic expressions, fuzzy logic deduction based on linguistic descriptions, and fuzzytransform.

10.2.1 Outline of the methods

This subsection informally describes the methods explained in more detail below. Itis intended for those who are more interested in the results of our methods for rocksequence determination and ancient sea level estimation presented below than in themethods themselves. Such readers can skip the rest and continue with Section 10.3.On the other hand, the reader interested in the details of the methods can skip thissubsection.

The essential constituent of our methods are the so-called evaluating linguistic

expressions and the model of their semantics. These are expressions such as “verydeep,” “more or less thick,” “very roughly small,” “about 200,” “shallow,” etc. Ingeneral, we can say that these expressions characterize linguistically some value ornumber. They form a small but very important and often used part of natural language.Fuzzy logic has been able to offer a mathematical model that works well in describingthe semantics of evaluating expressions. Such a model is described in the next section.Using it, we can solve various important problems.

The first problem is to imitate a geologist’s reasoning when determining rocksequences derived from the measured stratigraphic sections. The rules he/she usescontain a lot of vague evaluating expressions and thus it is fairly difficult to developan algorithm mimicking a human specialist. In Section 10.3, we describe a possiblealgorithm for determination of rock sequences based on the theory of semantics ofevaluating expressions which does this job with almost 90% success.

Evaluating linguistic expressions are used also in the so-called fuzzy IF–THEN

rules. For example,

C := IF sequence is very thin THEN sea level change is rather small (10.1)

In this example, “very thin” and “rather small” are evaluating expressions, “sequence”is an independent variable, and “sea level change” is a dependent variable.


We usually disregard their real meaning and denote them simply by X and Y ,respectively.

A set of fuzzy IF–THEN rules forms a linguistic description of some situation. Inthis chapter, we use linguistic descriptions to characterize ancient sea level positionestimated on the basis of the rock sequences determined using the algorithm describedin Section 10.3. Therefore, we face the problem, how can a linguistic description befound?

Avery simple but effective method can be used to learn a linguistic description fromthe data. We suppose that the data are organized into lines consisting of measuredvalues for all variables (for simplicity we assume only independent variable X anddependent variable Y ). Then we must specify linguistic context for both variablesX and Y . In our case, this means that we specify intervals in which all values ofthe variables can fall. Limits of each interval characterize the smallest and highest

possible values. For the given value we find a typical evaluating expression. If wedo it for both values in the given data line, we obtain one fuzzy IF–THEN rule.Repeating this procedure for the whole data set, we derive a linguistic descriptioncharacterizing the same situation as is characterized by the data. In our case, thedata consist of numbers of time cycles during which each rock sequence has beendeposited (variable X) and thicknesses of the rock sequences (variable Y ). Hence,the obtained linguistic description characterizes ancient sea level changes. Of course,the linguistic description may contain redundant and also contradictory rules. Theseare partly reduced automatically but otherwise have to be elaborated directly by thegeologist. Surprisingly, this is a potential advantage of the linguistic descriptions sincethey make it possible to include additional special knowledge not contained in the data.

The linguistic descriptions serve to derive the output given the input. A procedurecalled a logical deduction in fuzzy logic in the broader sense is used in this chapter.Roughly speaking, the given input value is classified so that the most suitable ruleis fired. From the logical point of view this means that the rule of modus ponens isapplied: given the fact A and the implication A ⇒ B, we conclude the fact B. Forexample, if we know that “the sequence is very thin” and also the rule (10.1), thenwe conclude that “sea level change is rather small.”

A powerful technique based on fuzzy logic is fuzzy transform (F-transform),described in detail in Chapter 9. This is a general technique which enables us tofilter the data and find a good fit with their trend.

In this chapter, we use F-transform in two ways. First, it is used for the estimationof ancient sea level on the basis of the determined rock sequences. Second, the nicesmoothness properties of F-transform are utilized to improve the logical deduction.The resulting method, called smooth logical deduction, is better suited for approxi-mating data on the basis of linguistic expressions than the pure deduction mentionedabove. The linguistic descriptions learned from rock sequences (determined by analgorithm based on evaluating expressions), with possible expert modification, areapplied to the estimation of ancient sea level changes.


10.2.2 The theory of evaluating linguistic expressions

Specific to applications of fuzzy logic and also important for dealing with the first taskspecified above, namely determination of rock sequences, are the so-called evaluating

linguistic expressions [see Novák, 2001; Novák et al., 1999]. These are expressionssuch as “very large, extremely deep, roughly one thousand, rather thick,” etc. Suchexpressions are used by people practically every time they want to characterize length,age, depth, thickness, and many other kinds of measurable objects. It is important tonote that these terms are vague in nature and never express precise numbers. Let usremark that the use of imprecise numbers and values is very typical in geology. Evenconcrete numbers used by geologists are always imprecise (i.e., fuzzy) and should betreated as such.

Components of evaluating linguistic expressions

The basic components of all evaluating expressions are atomic evaluating expres-

sions. They comprise any of the adjectives “small,” “medium,” or “big.” Let us stressthat “small,” “medium,” or “big” should be taken as canonical and can be replaced byany other cases such as “thin,” “thick,” “old,” “new,” etc. Atomic expressions are alsofuzzy quantities, namely “approximately x0.” These are linguistic expressions char-acterizing some element x0 on an ordered set. Examples of fuzzy quantities are one

million, the value x0, etc. As mentioned, quantities in common human understandingare almost always understood as imprecise; for example, “one million” never meansthe number 1,000,000, but “something close to it.”

Atomic evaluating expressions usually form pairs of antonyms, i.e., the pairs

〈nominal adjective〉 − 〈antonym〉. (10.2)

When completed by a middle term, such as “medium,” “average,” etc., they formthe so-called basic linguistic trichotomy. Thus, in geology, the pair of antonyms canbe “thin–thick,” “old–young,” “shallow–deep,” and the basic linguistic trichotomycan be “thin–medium thick–thick.” Note that natural language is quite rich in thebasic pairs (10.2) but the middle term has mostly the form “medium” or “average”completed by the corresponding adjective (e.g., “deep”).

Simple evaluating expressions are expressions of the form

〈linguistic hedge〉〈atomic evaluating expression〉. (10.3)

Examples of simple evaluating expressions are very thin, more or less medium,

roughly thick, about two thousand, approximately z, etc.Linguistic hedges are special adjectives modifying the meaning of the adjec-

tives before which they stand. Examples of them are “very, extremely, roughly,


approximately,” etc. In general, we speak about linguistic hedges with either a nar-

rowing effect (very, significantly, etc.) or with a widening effect (more or less, roughly,etc.). A specific case arises when no linguistic hedge is present. We will consider thiscase as the presence of an empty linguistic hedge and handle it in the same way as theother simple expressions. Hence, the pure atomic expressions “small,” “medium,”and “big” are also simple evaluating expressions.

The concept of linguistic hedge and the first outline of a possible theory of itssemantics was introduced by Zadeh [1975]. His theory has been further analyzedfrom the linguistic point of view by Lakoff [1973] and further elaborated, first inNovák [1989] and elsewhere. In this section, a novel theory is presented, whichconforms with the linguistic analysis.

In fuzzy logic, we will use several linguistic hedges with precisely defined seman-tics (see below) forming simple evaluating expressions with the following naturalorder according to their meaning:

“extremely” 〈atomic expression〉“significantly” 〈atomic expression〉“very” 〈atomic expression〉(empty hedge) 〈atomic expression〉“more or less” 〈atomic expression〉“roughly” 〈atomic expression〉“quite roughly” 〈atomic expression〉“very roughly” 〈atomic expression〉

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10.4)

where 〈atomic expression〉 is one of “small,” “medium,” or “big,” with the caveatthat hedges with narrowing effect cannot be used together with “medium” (e.g., theexpression “very medium” has no meaning). We will denote evaluating linguisticexpressions by script letters A, B, . . ..

Evaluating linguistic expressions can be joined by the connectives “and” and “or,”thus forming compound evaluating linguistic expressions. In general, they take theform

C := A 〈connective〉 B, (10.5)

where A, B are evaluating linguistic expressions and 〈connective〉 is either “and” or“or.”2 Examples of compound evaluating expressions are “very deep or deep,” “thickor thin,” etc.

The evaluating expressions are often assigned to nouns, characterizing sizes ofthe objects (or their parts) as their more specific properties. This leads to the

2One of the possibilities is the connective “but,” which always assumes that the second expression hasa negative form, e.g., “small but not very small.” The meaning of “but” is, in this case, the same as “and.”However, the theory of negation of evaluating expressions requires a deeper linguistic analysis and thusis not considered in this chapter. A discussion of the mathematical model of linguistic negation and otherproblems can be found in Novák [1992].


concept of evaluating linguistic predication. In general, these are expressions of theform

〈noun〉 is A, (10.6)

where A is an evaluating linguistic expression. If A is simple, then (10.6) is calledthe simple evaluating predication. Examples of simple evaluating predications are,e.g., “rock thickness is very big,” “the rock deposit is roughly small,” “the sea levelincrease is very high,” etc.

To finish this section, we have to mention a very important class of linguisticexpressions forming conditional clauses

C := IF 〈noun〉 is A THEN 〈noun〉 is B (10.7)

where A, B are evaluating expressions. Clauses (10.7) are in fuzzy logic called fuzzy

IF–THEN rules and stand behind most of its successful applications.

Semantics of evaluating linguistic expressions

Fuzzy logic offers a sophisticated theory of the semantics of evaluating linguisticexpressions and predications. To explain its fundamentals, we have to start withsome general remarks.

The general model of the semantics of linguistic expressions is based on the distinc-tion between their intension and extension in the sense introduced by Carnap [1947].First, we start with the concept of a possible world. This can be understood as a stateof the world at the given time moment and place. Alternatively, we can understand italso as a particular context in which the given linguistic expression is used. Then wecan distinguish the following.

Intension of a linguistic expression, sentence, or of a concept, can be identifiedwith the property denoted by it. An intension may lead to different truth values invarious possible worlds but it is invariant with respect to them. This means that, givena concept, it has just one intension which does not change when the possible worldis changed. For example, the expression “deep” is the name of an intension, being acertain property of depth, which in a concrete context (i.e., possible world) providesinformation about the size of depth.

Extension is a class of elements determined by an intension, which fall into themeaning of a linguistic expression in a given possible world. Thus, it depends on theparticular context of use and changes whenever the possible world (context, time,place) is changed. Our example, “deep,” may mean 1 cm when a beetle needs to crossa puddle, 3 m in a small lake, but 3 km or more in the ocean.

Expressions A of natural language are names of intensions. Let us remark that a lotof convincing arguments have been made that the meaning of expressions of naturallanguage cannot be identified with their extensions (see, e.g., Gallin [1975]).


To formalize the theory of the meaning of evaluating linguistic expressions, we willstart again with the concept of possible world. For the case of evaluating expressions,it is useful to understand possible worlds as closed intervals of real numbers. Thus,the set of possible worlds is the set of triples of numbers

W = {〈vL, vS, vR〉 | vL, vS, vR ∈ [0,∞) and vL < vS < vR}.

The numbers vL, vS, vR represent left limit, center, and right limit of possible valueswhich may fall into the meaning of the expressions of concern, respectively (seebelow).

Let v ∈ R be some value (R is the set of all real numbers). We say that v belongs

to the possible world w = 〈vL, vS, vR〉 if v ∈ [vL, vR]. In this case we usually writesimply v ∈ w. In the theory of evaluating expressions, we commonly replace theterm possible world by the term linguistic context (or simply, context).

Let w1, . . . , wn be an n-tuple of possible worlds. Then their Cartesian product isdefined by

w1 × · · · × wn = [v1,L, v1,R] × · · · × [vn,L, vn,R].

Let V be a set of elements which may fall into the meaning of the linguisticexpression. In our case, we usually put V = R. Then intension is formally a functionfrom the set of all possible worlds into the set of all fuzzy sets in the universe V , i.e.,

A : W → F(V ). (10.8)

The extension in the given possible world w ∈ W is a fuzzy set

A(w) ⊂∼

V,

i.e., it is a functional value of the intension (a function) A in the given possible world.Recall that the fuzzy set A(w) is itself a function

A(w) : V → [0, 1].

For example, let us consider the expression “small.” This is a property which hassome intension Sm : W → F(R). Then in each possible world (context) w ∈ W , themeaning of “small” is a certain fuzzy set of real numbers. The general principles ofhow that particular fuzzy set can be determined are described below.

A difficulty in the notation arises at this point. Let v ∈ V be some element.Since A(w) is a function, then its value at the point v must be written as A(w)(v),which is somewhat awkward. To overcome this inconvenience, we will usually(but not always!) write Aw instead of A(w) in the case that w is a possibleworld.

Our goal now is to characterize both intension and all the extensions of the eval-uating linguistic expressions. The main idea of the corresponding mathematical


model is the following. Take the expression “small.” Then, given a possible worldw = 〈vL, vS, vR〉, the value vL and all values “close” to it are small. We can imaginethis situation as that of an “observer” who stands in the position vL and looks overthe universe so that all small values fall inside the horizon of his/her view. Similarly,“big” means that the observer stands in vR and looks back. Finally, “medium” meansthat the observer stands in vS and looks at “both sides.”

Mathematically, we model this idea using three of linear functionsL, R : W×R →

[0, 1] defined in each possible world w ∈ W by

Lw(x) =

(vS − x

vS − vL

)∗(left horizon) (10.9)

Rw(x) =

(x − vS

vR − vS

)∗(right horizon) (10.10)

and a middle horizon function

Mw(x) = ¬Lw(x) ∧ ¬Rw(x) =

(x − vL

vS − vL

)∗∧

(vR − x

vR − vS

)∗(10.11)

where¬ is the negation operation defined by¬a = 1−a, a ∈ [0, 1]. The star used in(10.9), (10.10), (10.11) means cut of the values to the interval [0, 1]; i.e., if Lw(x) < 0then we put Lw(x) = 0 and if Lw(x) > 1 then we put Lw(x) = 1. Similarly for Rw

and Mw.Furthermore, we consider a class of abstract hedges, being functions

νa,b,c(y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1, c ≤ y

1− (c − y)2

(c − b)(c − a), b ≤ y < c

(y − a)2

(b − a)(c − a), a ≤ y < b

0, y < a

where a, b, c are special parameters fulfilling a, b ∈ (−∞, 1), c ∈ (0.5, 1], anda < b < c. The function ν can be also seen as a deformation of the horizon.

The set of all abstract hedges νa,b,c will be denoted by Hf . To simplify the notation,we will often write only ν ∈ Hf , understanding that v is, in fact, determined by theparameters a, b, c.3

3Functions ν ∈ Hf are quadratic. Their simpler, but less adequate, form is linear. On the other hand, amore complicated form for them is not necessary.


We will now define three special classes of intensions (recall that by F(R) wedenote the set of all fuzzy sets on R):

(i) Intensions of type Small:

Sm = {Smν : W → F(R) | Smν,w(x) = ν(Lw(x)), ν ∈ Hf}.

(ii) Intensions of type Medium:

Me = {Meν : W → F(R) | Meν,w(x) = ν(Mw(x)), ν ∈ Hf}.

(iii) Intensions of type Big:

Bi = {Biν : W → F(R) | Biν,w(x) = ν(Rw(x)), ν ∈ Hf}.

We are now ready to define the meaning of simple evaluating linguistic expressions.Let A be such an expression. Then its meaning is identified with its intension, whichis a function

Int(A) ∈ Sm∪Me∪Bi . (10.12)

Furthermore, for each possible world w ∈ W , the extension of the evaluatingexpression A in w ∈ W is

Extw(A) = Int(A)(w).

This means that if w = 〈vL, vS, vR〉 is a possible world then the extension Extw(A)

of the evaluating expression A in w is a fuzzy set Extw(A) ⊂∼[vL, vR].

Let us remark that the meaning of fuzzy quantities is modeled similarly to that ofexpressions of type “medium.” For simplicity, we have omitted details in this chapter.The interested reader should consult Mareš [1994].

If we rewrite the above formulas in more detail, we obtain:

Int(〈linguistic hedge〉 small) = Smν ∈ Sm

Int(〈linguistic hedge〉medium) = Meν ∈ Me

Int(〈linguistic hedge〉 big) = Biν ∈ Bi

where Smν , Meν , and Biν are the functions defined above. Using them, we can derivefor each possible world the explicit formulas for the corresponding extensions.

Let the parameters of the linguistic hedge ν be a, b, c ∈ [0, 1]. Put K1 = (c−b)(c−

a), K2 = (b−a)(c−a). Then the extension in each possible world w = 〈vL, vS, vR〉


is given by one of the following formulas:

Smν,w =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, x ∈ [vL, cSm], cSm = cvL + (1− c)vS

1− (x − cSm)2

K1(vS − vL)2 , x ∈ (cSm, bSm], bSm = bvL + (1− b)vS

(aSm − x)2

K2(vS − vL)2 , x ∈ (bSm, aSm), aSm = avL + (1− a)vS

0, x ≥ aSm

Meν,w =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, x ∈ [cLMe, c

RMe], cL

Me = cvS + (1− c)vL

cRMe = cvS + (1− c)vR,

1−(cL

Me − x)2

K1(vS − vL)2 , x ∈ [bLMe, c

LMe), bL

Me = bvS + (1− b)vL

1−(x − cR

Me)2

K1(vR − vS)2 x ∈ (cRMe, b

RMe], bR

Me = bvS + (1− b)vR

(x − aLMe)

2

K2(vS − vL)2 , x ∈ (aLMe, b

LMe), aL

Me = avS + (1− a)vL

(aRMe − x)2

K2(vR − vS)2 x ∈ (bRMe, a

RMe), aR

Me = avS + (1− a)vR

0 x ≤ aLMe, x ≥ aR

Me

Biν,w =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, x ∈ [cBi, vR], cBi = cvR + (1− c)vS

1− (cBi − x)2

K1(vR − vS)2 , x ∈ [bBi, cBi), bBi = bvR + (1− b)vS

(x − aBi)2

K2(vR − vS)2 , x ∈ (aBi, bBi), aBi = avR + (1− a)vS

0, x ≤ aBi .

The meaning of the parameters aSm, bSm, cSm for small and likewise for medium

and big is clear from Figure 10.1.


Figure 10.1 Scheme of the construction of the extension of evaluating linguistic expressions inthe given possible world w. The νa,b,c function is turned over to the left of the y-axis. Dashedlines represent the horizon functions Lw, Mw, Rw . Each of the parameters a, b, c is projectedonto them and results in the corresponding parameters aA, bA, cA for A := Sm, Me, Bi.Note that cA determines edge of kernel, aA edge of support, and bA determines inflexionpoint.

Remark 10.1

If A is an evaluating linguistic predication (10.6), then we put its intension equal tothe intension of the evaluating expression inside (10.6).

We will say that an abstract hedge ν1 ∈ Hf is sharper than ν2 ∈ Hf ,

ν1 < ν2, if 〈a2, b2, c2〉 < 〈a1, b1, c1〉. (10.13)

Note that the ordering of hedges at the same time induces ordering of simple evaluatingexpressions. Using it, the natural ordering introduced in (10.14) can be modeled infuzzy logic.

We now select the following adverbs: “extremely (Ex), significantly (Si), very (Ve),

more or less (ML), roughly (Ro), quite roughly (QR), very roughly (VR).”

Furthermore, we choose certain numbers a0, b0, c0 ∈ [0, 1] and assign the abstracthedge νa0,b0,c0 ∈ Hf to the “empty hedge.” Then we choose three abstract hedgesνEx, νSi, νV e ∈ Hf , for which

νEx < νSi < νV e < νa0,b0,c0

holds, and four abstract hedges νML, νRo, νQR, νV R ∈ Hf , for which

νa0,b0,c0 < νML < νRo < νQR < νV R


Table 10.1 Empirically estimated possible values of parameters a, b, c of some linguistichedges. They follow the natural ordering of simple evaluating expressions in (10.4).

Linguistic hedge a b c

Extremely 0.5 0.75 0.95Significantly 0.47 0.6 0.8Very 0.35 0.58 0.83empty 0.27 0.5 0.8Rather 0.4 0.5 0.8More or less 0.23 0.45 0.76Roughly 0.2 0.4 0.7Quite roughly 0.15 0.32 0.65Very roughly 0.09 0.2 0.6

holds. Finally we assign these hedges to the words selected above and understandthe former as the meaning of the latter. This procedure allows us to construct themeaning of each evaluating expression (i.e., its intension) as well as its extension ineach possible world.

Note that we can also define other kinds of modifiers not belonging to the abovegroup, for example “rather.” This should be assigned the abstract hedge νa,b,c witha > a0, b ≤ b0, and c < c0. Thus the above theory encompasses a large class oflinguistic hedges.

The empirically estimated values of the parameters a, b, c of the various linguistichedges are given in Table 10.1.

Shapes of the membership functions of extensions of the experimentally determinedpure evaluating expressions are depicted in Figure 10.2.

Finding a suitable expression

The theory described above enables us to master the meaning of evaluating linguisticexpressions and apply it to a great variety of problems. In our case, we have used itto develop an algorithm for determination of rock sequences.

To gain insight into practical application of the theory, imagine a possible world, sayw = 〈1, 40, 100〉, and a given value x = 7. How can we linguistically characterizesize of x with respect to w? One may state quite naturally that this value is small or,to be more precise, even very small. The question arises whether we can mimic thisempirical procedure. Certain general requirements can be formulated enabling us todevelop a satisfactory algorithm. This has been described in Dvorák & Novák [1993]and is called the “Suit” procedure.


Figure 10.2 Membership functions of extensions of selected evaluating expressions fromTable 10.1. The order of curves corresponds to the natural ordering from (10.4), i.e., for “small”the extensions go from “extremely” to “very roughly,” for “big” in the opposite direction, andfor “medium” from “rather” to “very roughly.”

Let us consider a possible world w = 〈vL, vS, vR〉 and a value u ∈ w. Then Suitis a function

Suit : R×W → S (10.14)

where S is the set of all simple evaluating linguistic expressions. In practice,Suit(u, w) is such an evaluating expression A, that the value u is typical for itsextension Extw(A).

The general idea of finding the functional value of Suit(u, w), given the possibleworld w and a value u ∈ w, is to choose that evaluating expression A ∈ S whoseextension Extw(A) in the possible world w is the sharpest (smallest) one in the senseof the ordering (10.13), provided that the membership degree Extw(A)(u) is non-zeroand maximal.

We do not give more details here and refer to the software LFLC (Linguistic FuzzyLogic Controller), versions 1.5 and 2000, developed in the University of Ostrava andin which the Suit procedure is successfully implemented.4

10.2.3 Linguistic description and logical deduction

Linguistic description is a powerful tool that enables us to characterize in a reasonable(but not too detailed) form a situation requiring decision, the behavior of a system,

4The demo version of LFLC 2000 can be found in www.ac030.osu.cz where also information onhow to order it is provided.


a control strategy, the character of a data set, etc. Linguistic description can beunderstood as a slightly formalized form of a wide class of descriptions provided bypeople freely using natural language. Moreover, on the basis of linguistic descriptionand of some concrete observation (in the given possible world), people are able toderive conclusions and take appropriate actions. The power of fuzzy logic consistsin its ability to model both the meaning of linguistic description as well as deductionbased on it, provided that an observation is given.

Interpretation of linguistic description

The linguistic description is a set of fuzzy IF–THEN rules

R := {R1, R2, . . . ,Rm} (10.15)

where each Ri is a conditional clause of the general form (10.7). Since, in practice, weare usually not interested in the concrete 〈noun〉 in the linguistic predications (10.6),we usually replace it by some variable. Thus, the fuzzy IF–THEN rules in fuzzy logicRi take the form

Ri := IF X is Ai THEN Y is Bi . (10.16)

The part before THEN is called the antecedent and the part after it is called thesuccedent. Let us also remark that the rules may have more than one variable in theantecedent. For simplicity of explanation, we do not elaborate on this case here.

Let the fuzzy IF–THEN rule R have the form (10.7). Let the meaning of thepredication “〈noun〉 is A” be

Int(〈noun〉 is A) = EvA

from (10.12)5 and similarly EvS for “〈noun〉 is B” (the subscript A stands for“antecedent” and S for “succedent”). Then the intension (meaning) of the fuzzyIF–THEN rule (10.7) is

Int(R) := EvA ⇒⇒⇒ EvS (10.17)

where⇒⇒⇒ is an implication connective interpreted by some implication operation→.There are good reasons to take→ as a Łukasiewicz implication defined by

a → b = min(1, 1− a + b), a, b ∈ [0, 1]. (10.18)

Given a couple of possible worlds w, w′ ∈ W , then the extension of (10.7) is

Ext〈w,w′〉(R) := EvA,w → EvS,w′ (10.19)

5In accordance with Remark 10.1, we semantically do not distinguish evaluating linguistic predicationfrom the linguistic expression inside it.


where EvA,w is the extension of the linguistic predication “〈noun〉 is A” in the pos-sible world w and, similarly, EvS,w′ is the extension of the linguistic predication“〈noun〉 is B” in the possible world w′.

The extension (10.19) is a fuzzy relation Ext〈w,w′〉(R) ⊂∼

w × w′ defined by

Ext〈w,w′〉(R)(v, v′) = EvA,w(v)→ EvS,w′(v′), v ∈ w, v′ ∈ w′. (10.20)

Defuzzification

The defuzzification operation plays an important role in fuzzy logic applications. Thereason is that fuzzy logic works with fuzzy sets but, at the end, we usually need somespecific number as an output. Therefore, a certain procedure which transforms a fuzzyset into a number is needed. In the literature [e.g., Klir & Yuan, 1995; Novák, 1989;Novák, et al., 1999], many kinds of defuzzification operations have been described.To deal with evaluating expressions, we employ the Least of Maxima (LOM), First

of Maxima (FOM), and Center of Gravity (COG) methods.In our case, the first two can be computed for the respective expressions of the type

small and big as follows. Let w = 〈vL, vS, vR〉 be a possible world. Then

LOM(Smν(w) = cvL + (1− c)vS (10.21)

FOM(Biν(w) = cvR + (1− c)vS (10.22)

where c ∈ (0.5, 1] is the parameter of the corresponding linguistic hedge ν in (10.21)and (10.22). The COG method is defined by

COG(Ev(w)) =

∫w

v Evw(v)dv∫w

Evw(v)dv. (10.23)

The following special defuzzification operation DEE (Defuzzification of Evaluat-ing Expressions) works very well for evaluating linguistic expressions: let Ev be themeaning (intension) of some evaluating expression and w ∈ W be a possible world.Then the defuzzification operation

DEE(Ev(w)) =

⎧⎪⎨⎪⎩

LOM(Sm(w)), if Ev ∈ Sm

COG(Ev(w)), if Ev ∈ Me

FOM(Bi(w)), if Ev ∈ Bi

⎫⎪⎬⎪⎭

. (10.24)

Thus, the result of the defuzzification operation depends on the type of evaluatingexpression to be defuzzified. A schematic picture demonstrating the behavior of theDEE defuzzification method is given in Figure 10.3.


Figure 10.3 Scheme of the Defuzzification of Evaluating Expressions (DEE) method.

Logical deduction

The way people make inferences on the basis of linguistic description can be explainedwith an example. Let us consider a linguistic description that consists of two rules:

R1 := IF X is small THEN Y is big

R2 := IF X is big THEN Y is small.

Furthermore, let the linguistic context (possible worlds) for the variables X, Y bew = w′ = 〈0, 0.5, 1〉. Then small values are some values around 0.3 (and smaller)and big ones some values around 0.7 (and bigger). We know from the linguisticdescription that small input values correspond to big output ones and vice versa.Therefore, given the input, e.g., X = 0.3, then we expect the result Y ≈ 0.7 due tothe rule R1 since we evaluate the input value as being small, and thus, in this casethe output should be big. Similarly, for X = 0.75 we expect the result Y ≈ 0.25 dueto the rule R2.

We expect the formal theory of logical deduction to give analogous results. Let R

be a linguistic description (10.15). In fuzzy logic, its meaning can be represented bya set of intensions (10.17)

Int(R1) = EvA,1 ⇒⇒⇒ EvS,1

. . . . . . . . . . . . . . . . . . . . . . . . . .

Int(Rm) = EvA,m ⇒⇒⇒ EvS,m

⎫⎬⎭ (10.25)

The logical deduction may proceed, provided that an observation Ev equal to someof the antecedents EvA,i from (10.25) is given. Then the modus ponens rule can be


applied

EvA,i, EvA,i ⇒⇒⇒ EvS,i

EvS,i

(10.26)

giving as the output the expression EvS,i .Our problem, however, consists in the fact that we are given only an observation

u ∈ w in a possible world w. Because we work in the space of evaluating expressions,we transform u into the most specific antecedent EvA,i0 in the sense of the ordering(10.13) such that the membership degree of u in its extension EvA,i0,w(u) is nonzeroand maximal. This can be justified by the empirical observation that, given the possibleworld, each value in it can be classified by some evaluating expression (in fact, theyexist in natural language just for this purpose). Furthermore, since the expressionsare more or less specific, the most specific one gives the most precise information.Consequently, this expression (if it exists in the given linguistic description) is usedas an argument in the modus ponens (10.26). We say that the corresponding rule Ri0

“fired.”Logical deduction determines a function fR given by

fR(u) = DEE(EvA,i0,w(u)→ EvS,i0,w′) (10.27)

where i0 is the number of the rule which fired on the basis of modus ponens and themethod described above.

The resulting function fR is, in general, only partially continuous. This is nota problem in decision-making or even in fuzzy control, but can be a drawback indata approximation. On the other hand, logical deduction provides an efficient andhuman-like behavior. Moreover, the linguistic description in the form consideredabove is natural and easily understood by people. Let us note that the fuzzy transformdescribed below enables us to overcome the non-continuity of the logical deductionand, at the same time, to preserve its main characteristics.

Learning linguistic description from the data

The Suit function (10.14) can be effectively used for learning linguistic descriptionon the basis of given data [see Belohlávek & Novák, 2002].

Consider a pair of possible worlds w, w′ and the data

(u1, v1)

. . . . . . . .(uN , vN )

⎫⎬⎭ , (10.28)


such that each pair (uj , vj ) ∈ w×w′. We suppose that these data can be characterizedusing some linguistic description (10.15) consisting of rules of the form

Ri := IF X is Ai THEN Y is Bi (10.29)

where Ai, Bi are evaluating linguistic expressions such that for each pair (uj , vj ) ofthe data

Ai = Suit(uj , w) and Bi = Suit(vj , w′) (10.30)

hold true. Hence, each data item leads to some fuzzy IF–THEN rule (10.29) formedusing (10.30). Of course, it may happen that two different data items lead to thesame IF–THEN rule. Moreover, some learned rules may turn out to be superfluouswhen used in logical deduction. Hence, the following learning procedure leads to thelinguistic description which has the ability to approximate the data (10.30).

1. Repeat the learning procedure (10.30) for all j = 1, . . . , N and generate thelinguistic description {R1, . . . ,RN }.

2. Reduce the learned linguistic description as follows:

(a) Replace all the sets of identical rules by one rule only.(b) Let Ri and Rk be two learned rules such that their succedents are identical.

Let the antecedent of Ri be wider (in the sense of the ordering (10.13)) thanthat of Rk . Then exclude the rule Rk .

(c) Let Ri and Rk be two learned rules such that their antecedents are identical.Let the succedent of Ri be sharper (in the sense of (10.13)) than that of Rk .Then exclude the latter rule.

The result of the learning procedure is a linguistic description of the form (10.15).We employ this procedure to model ancient sea level position—see below.

Let us remark, however, that the linguistic description learned may still need someexpert modification. The reason for such discrepancies arise from the character ofthe data and vagueness of the antecedents of the rules. If the values ui in (10.28) arefairly close, and the values vi vary, then the learning procedure may lead to rules thathave the same antecedent but different succedents. Therefore, some of the rules haveto be deleted (or modified) on the basis of special expert knowledge. This problemshould be the subject of further investigation.

10.2.4 Fuzzy transform

The fuzzy transform technique developed by I. Perfilieva is described in Chapter 9(see also Perfilieva [2003]). We briefly review it with respect to our methods andnotation.


The F-transform is applied to some continuous function f (x) defined on an intervalof real numbers [vL, vR] ⊂ R. From the point of view of the explanation in previoussections, this interval is our possible world (the context of use). As above, we willwrite x ∈ w in the meaning x ∈ [vL, vR].

The interval w is divided into a set of equidistant nodes x0,k = vL + h(k − 1),k = 1, . . . , n, where

h = (vr − vL)/(n− 1), n ≥ 2, (10.31)

is the fixed distance between each neighboring set of nodes.

Direct F-transform

In the direct F-transform we next define n basic functions, which cover w and takethe role of fuzzy points (or granules), dividing w into n vague areas. For us, thebasic functions will be fuzzy numbers Fnν,x0 , where ν is a linguistic hedge and x0

is the central point around which the fuzzy number is defined. Thus, the fuzzy num-ber is an extension of the linguistic expression “approximately x0.” The consideredmembership function is the following:

Fnν,x0 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, x ∈ [cLx0

, cRx0], cL

x0= x0 − (1− c)h

cRx0= x0 + (1− c)h

1−(cL

x0− x)2

K1h2 , x ∈ [bL

x0, cL

x0), bL

x0= x0 − (1− b)h

1−(x − cR

x0)2

K1h2 , x ∈ (cR

x0, bR

x0], bR

x0= x0 + (1− b)h

(x − aLx0

)2

K2h2 , x ∈ (aL

x0, bL

x0), aL

x0= x0 − (1− a)h

(aRx0− x)2

K2h2 , x ∈ (bR

x0, aR

x0), aR

x0= x0 + (1− a)h

0, x ≤ aLx0

, x ≥ aRx0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (10.32)

Note that Fnν,x0(x0) = 1 and Fnν,x0(x0 ± h) = 0. Thus, one fuzzy number isspread over three neighboring nodes x0 − h, x0, x0 + h. Furthermore,

n∑

k=1

Fnν,x0k(x) = 1


holds for each x ∈ w. Consequently, each x ∈ w is covered by exactly two neigh-boring fuzzy numbers Fnν,x0k

, Fnν,x0k+1 . This means that x0k ≤ x ≤ x0k+1 andFnν,x0k

(x) + Fnν,x0k+1(x) = 1. This property is used below for smooth logicaldeduction.

Using the fuzzy numbers as basic functions, we transform the values of the givenfunction f (x) into an n-tuple of real numbers [F1, . . . , Fn]. In reality (and also inthe application described below) the function f (x) is known only at some pointsx1, . . . , xN . Thus, the input data form a set of couples

(x1, f (x1))

. . . . . . . . . . . .(xN , f (xN ))

⎫⎬⎭ . (10.33)

Then the direct F-transform is defined by

Fk =

∑Nj=1 f (xj ) Fnν,x0k

(xj )∑N

j=1 Fnν,x0k(xj )

, k = 1, . . . , n. (10.34)

Inverse F-transform

After realizing the direct transform, we can transform [F1, . . . , Fn] back to obtain afunction

fF,n(x) =

n∑

k=1

Fk · Fnν,x0k(xj ). (10.35)

The result of the application of F-transform to (10.33) is filtered data

(x1, fF,n(x1))

. . . . . . . . . . . . . .(xN , fF,n(xN ))

⎫⎬⎭ . (10.36)

It can be proved that if n increases then fF,n(xj ) converges to f (xj ). Let us alsoremark that the F-transform has very nice filtering properties and is easy to compute.Therefore, we have used it to solve of the sea level determination problem.

Smooth logical deduction

Recall that logical deduction gives a partially continuous function fR defined in(10.27). This drawback can be overcome by joining it with the F-transform. The ideais very simple.

Let a linguistic description R (10.15) be given. Furthermore, consider the linguisticcontext (possible world) w and let the value u0 ∈ w be given. On the basis of R andu0, the result of logical deduction at the point u0 is fR(u0), by (10.27). Then the


function fR(x) may take the role of the above-considered function f (x) to be filteredusing the F-transform.

First we choose some smoothing number, which is the number of nodes n, andcompute the distance h in (10.31). Since only one data item is to be filtered, namely(u0, fR(u0)), only two numbers Fk, Fk+1 in (10.34) must be computed for somenodes x0k , x0k+1 such that x0k ≤ u0 ≤ x0k+1. To do this we have to choosea step r > 0, compute a sequence of values u1 = x0k−1, u2 = u1 + r, u3 =

u1 + 2r, . . . , up = x0k+2 laying between the nodes x0k−1 and x0k+2, and generate asequence of auxiliary data

(u1, fR(u1))

. . . . . . . . . . . . .(up, fR(up))

⎫⎬⎭ (10.37)

where fR(uj ), j = 1, . . . , p is a result of the logical deduction (10.27) based on thelinguistic description R and the input value uj .

Now, the smooth logical deduction consists of two steps. First we compute thenumbers Fk, Fk+1 according to

Fk =

∑p

j=1 fR(uj ) · Fnν,x0k(uj )

∑p

j=1 Fnν,x0k(uj )

(10.38)

(note that the summation in (10.38) is sufficient only for values x0k−1 ≤ uj ≤ x0k+1

in the case of Fk and x0k ≤ uj ≤ x0k+2 in the case of Fk+1 since the membershipfunction of the corresponding fuzzy number is zero otherwise). Second, we computethe resulting smoothed output using

fR,n(u0) = Fk · Fnν,x0k(u0)+ Fk+1 · Fnν,x0k+1(u0). (10.39)

The scheme of the smooth logical deduction is depicted in Figure 10.4.As can be seen, the algorithm is more complicated than the simple logical deduction

(10.27). However, its advantage lies in both its ability to mimic human reasoningbased on linguistic description and its continuity.

10.3 Automatic Determination of Rock Sequences

10.3.1 Geological characterization

We tested this procedure on two detailed, measured stratigraphic sections from LowerPaleozoic carbonates. The first case is from the Lower Ordovician El Paso Group fromwest Texas described in Goldhammer et al. [1993] and given in Appendix C of thatpaper. The second is an approximately 250 meter thick composite section of the Upper

10.3 Automatic Determination of Rock Sequences 323

Figure 10.4 Scheme of the smooth logical deduction.

Cambrian Conococheague Limestone from western Maryland. Details of the sedi-mentology of the Conococheague Limestone can be found in Demicco [1985]. Bothsections comprise hundreds of individual measured rock units distributed among eightrock types: (1) stromatolites; (2) dolomitic mudstones; (3) thrombolitic bioherms (thatcontain sponges in the Ordovician section); (4) cross-stratified grainstones; (5) wavybedded “ribbon rocks”; (6) mud cracked cryptomicrobial laminites; (7) mud crackedplanar laminites; and (8) breccias (in the Cambrian Conococheague Limestone) oraeolian sandstones (in the El Paso Group). Rock types 1 and 2 represent deepestsubtidal settings. Rock types 3 and 4 represent shallow subtidal shoals and shallowsubtidal bioherms (these may be lateral equivalents of each other in terms of absolutedepth). Rock types 5, 6, and 7 were deposited on lowest intertidal mixed sand andmud flats, high intertidal mud flats covered with cyanobacterial mats, and supratidalplaya-like mudflats, respectively. Rock type 8 in both cases represents desiccated tocompletely disrupted sabkhas.

Lower Paleozoic carbonates are, in general, divisible into two orders of cycles:fourth-order cycles 1 to 10 meters thick, and third-order cycles tens up to one hun-dred meters thick [Goldhammer et al., 1993; Kerans & Tinker, 1997]. Systematicvariations in the thickness and rock types that comprise the fourth-order cycles (i.e.,the “stacking pattern”) define the third-order cycles which are directly analogous toseismic sequences in that they record long-term (one to a few tens of millions ofyears) variations in rates of sea level rise and fall. Third-order cycles start with expo-sure breccias (or, in the case of the El Paso Group, aeolian flats) that are followedby fourth-order cycles deposited on tidal flats that systematically become thickerupward as accommodation increases during rapid third-order sea level rise. Whenaccommodation potential is highest, cycles are thickest (up to 10 meters) and are


capped either by thick thrombolitic bioherms or by shoal deposits. (Note that this isa slightly different way of defining subtidal cycles than was given by Goldhammeret al. [1993].) Finally, as the rate of third-order sea level rise slows, cycles becomethinner upward and once again contain more intertidal and supratidal flat deposits,culminating in exposure breccias or wind-swept sabkhas at the slowest point of third-order sea level rise (the next sequence boundary). Figure 10.5 shows 75 m of sectionillustrating changes in stacking patterns of the Conococheague Limestone from a zoneof maximum accommodation through thinning intertidal cycles to a breccia markinga sequence boundary. Goldhammer et al. [1993] described similar stacking patternsin the El Paso Group of west Texas.

Figure 10.5 Illustration of changes in stacking patterns of the Conococheague Limestone froma zone of maximum accommodation through thinning intertidal cycles to a breccia marking asequence boundary.


10.3.2 General rules and the fuzzy algorithm

Our point of departure is to develop a series of “IF–THEN rules” specified by ageologist that capture the elements of fourth-order cycles and how they are organizedinto third-order cycles.

1. The lower the number of a rock type, the deeper the water it was deposited in;however, there may be errors and exceptions.

2. The higher the number of a rock type, the shallower the water it was deposited in.3. Rock types 6, 7, and 8 were deposited at or slightly above ancient sea level.

Furthermore, more detailed linguistic rules for interpreting “accommodationpotential” in limestone are to be used:

A1. If rock types 6, 7, or 8 are “far above” rock types 1, 2, 3 then sequences usuallystart with rock type 3 and end with 4 or 5.

A2. If rock types 6, 7, or 8 are “not far above” rock types 1, 2, 3 then sequencesusually start with rock type 3 and end in rock type 6 or 7, rarely 8.

A3. Sequences are usually 1 to 10 m thick.A4. If most sequences end in rock type 4, or if the majority of the section is rock

types 3 and 4, then this is a zone of maximum accommodation.A5. If most sequences end in rock types 6 or 7 and sequences are becoming thinner

upward then this is a zone of decreasing accommodation.A6. If most sequences end in rock types 6 and some with 7, or if the majority of the

section is rock type 5, then this is a zone of increasing accommodation.A7. If the section is between a zone of decreasing accommodation and a zone of

increasing accommodation, then this is a zone of minimal accommodation andshould correspond to rock type 8.

On the basis of the above principles, a special algorithm for determination of rocksequences has been developed. Its main feature is the use of the evaluating linguisticexpressions for branching inside the algorithm.

The input data have the following form:

Rock type Rock thickness

9 9.99 9.9...

...


The algorithm for determination of rock sequences is based especially on rulesA1–A3. Its global description is the following:

1. Find all potential ends of sequences. These should be rocks of type 6, 7, or 8 ifthey are followed by rock type 1, 2, or 3 (rule A2). If it happens that the given rockhas a lower number followed again by 6, 7, or 8 and it is too thin then it is ignored.

2. Check whether the obtained sequences are sufficiently thick (rule A3). If the givensequence is too thin then it is joined with the following one, provided that theresulting sequence does not become too thick.

3. If the given sequence is too thick then it is further divided using rule A1. For this,check all rock types 4 and mark them as ends of a new sequence provided that thenew sequence is not too thin; mark the new sequence only if it is sufficiently thick.

The linguistic expressions in the previous algorithm marked in italics are evaluatingexpressions which are modeled using the theory described above. More concretely,

● too thin means: the thickness is significantly small or smaller,● too thick means: thickness is very big or bigger,● sufficiently thick means: the thickness is medium.

The given input value—rock or sequence thickness—is evaluated by means ofthe assignment of the proper evaluating expressions using the Suit (10.14) pro-cedure. However, to do this properly, first the linguistic context (possible world)w = 〈vL, vS, vR〉 must be specified. This depends on the specified data—on thecharacter of the area where they have been obtained.

10.3.3 Results of tests

The two sets of data we tested are denoted as D1 [Demicco, 1985] and G1[Goldhammer et al., 1993], respectively. The measuring units are in meters.

The linguistic context for characterization of rocks and their sequences is thefollowing:

Data Context of Possible world Typical valuethickness

D1 rock w = 〈0, 4.8, 12〉 too thin = 0.6sequence w = 〈0, 9.6, 24〉 normal = 9.8

G1 rock w = 〈0, 2, 5〉 too thin = 0.2sequence w = 〈0, 2, 5〉 normal = 2


The typical value is the value representing what is too thin or normal in the givencontext. This means that any other value around them (or smaller in the case of “toothin”) is evaluated accordingly.

The results of sequence determination have been compared with the geologist’sexpert judgment. Let us stress that he/she also utilizes other, quite special experience,whereas the algorithm is nondeterministic and allows ambiguous solutions.

Data G1

These data are from the Lower Ordovician El Paso Group observed in the FranklinMountains of west Texas [see Goldhammer et al., 1993]. The total number of rockunits in this data set is 272 and comprises a significant proportion of the measuredsection shown in Appendix C of Goldhammer et al. [1993]. Geologists determined99 rock sequences and our algorithm determined 97 sequences. Of these, 87 arecoincident with the geologist’s solution, an 88% agreement.

The first discrepancy between the geologist’s solution and our algorithm occurswith sequence No. 43 (numbered by the algorithm). This consists of the rocks 3, 5,3, 5, 3, 5. The corresponding data are summarized in Table 10.2. One can see fromit that our algorithm checked all the geologist’s sequences as well. However, sinceall the rocks of type 5 are too thin (thickness = 0.1 m) and, at the same time theconsidered sequence G43 (geologist’s numbering) is rather thin (thickness= 0.7 m),and even the sequence G43+G44 is still quite roughly thin (thickness= 1.3 m), bothsequences were neglected by the algorithm and joined into one larger sequence No.43. The other discrepancies are similar.

However, also vice versa, our algorithm has marked three sequences which thegeologist has neglected. The corresponding data are in Table 10.3.

In all cases, the rock thickness, though small, does not correspond to the description“too thin,” and thus the algorithm decided to mark one more sequence in comparisonwith the geologist’s point of view. There are two other similar cases in the G1 data set.

Table 10.2 Sequence No. 43 (determined by the algorithm) from the point of view ofthe geologist.

Rock Rock Rock Sequence Sequence Geologist’stype thickness character thickness character number

3 0.6 Sm 0.6 G435 0.1 SiSm 0.7 RaSm G433 0.5 Sm 1.2 G445 0.1 SiSm 1.3 QRSm G443 0.5 Sm 1.8 G455 0.1 SiSm 1.9 RaMe G45


Table 10.3 Geologist’s sequence No. G66 divided into sequences Nos. 63 and 64(numbering by the algorithm).

Rock Rock Rock Sequence Sequence Algorithmtype thickness character thickness character number

3 0.3 VeSm 0.3 634 0.9 MLSm 1.2 RoSm 633 0.3 VeSm 0.3 645 0.5 Sm 0.8 RaSm 64

Let us stress that we can hardly expect the above discrepancies to be removed usingthe criteria specified above. In other words, full agreement with the geologist can beobtained only by using other information not included in the algorithm as of now.

Data D1

These data have been obtained in western Maryland from the Upper Cambrian depositConococheague Limestone. The total number of rock units in this data set is 226. Thegeologist determined 44 sequences while our algorithm determined 47 sequences. Ofthese, 36 are coincident with the geologist’s solution, which means an 81% agreement.The character of discrepancies is the same as is seen in section G1.

10.4 Sea Level Estimation

Ancient sea level is estimated on the basis of the stacking pattern of sequences deter-mined in the previous step. Each determined sequence corresponds to a certain timeperiod which, for simplicity, is assumed to be constant. The determined sequencethickness then corresponds in a uniform way to the sea level in the given time period.

The input data for sea level estimation are derived from the data sets given in theprevious section and contain the sequence number (which coincides with the timeperiod) and the sequence thickness determined in the previous step. These data areinput to further analysis. Its result is the following file, summarizing total informationabout ancient sea level. Part of the data G1 is in Table 10.4. The first column con-tains sequence number and the second one its thickness (these have been determinedusing the algorithm described above). Three “approximation” columns contain theestimated sea level using the three methods described above. These methods yielddifferent sea level estimations from the standard “Accommodation Plots” of the data[see Figure 14 in Goldhammer et al., 1993]. The third column is obtained using fuzzyF-transform.

10.4 Sea Level Estimation 329

Table 10.4 Part of the G1 data.

Input values Approximation

Sequence Sequence F-transform LD from LD fromnumber thickness seq. thickness F-transform

1 8.4 5.1 4.2 5.02 3.6 5.1 4.2 5.03 3.8 5.1 4.2 5.04 1.5 5.4 4.2 5.05 3.0 5.5 4.2 5.06 11.3 5.5 4.2 5.0...

......

......

92 0.9 2.1 2.3 2.393 2.9 2.1 2.4 2.494 6.3 2.8 2.6 2.595 1.8 4.5 2.7 2.696 8.5 5.1 3.3 3.597 4.0 5.1 3.8 4.2

Figure 10.6 Estimation of ancient sea level. Source: data G1, automatically determined rocksequences. Estimation obtained using F-transform.


Figure 10.7 Estimation of ancient sea level. Source: data G1, automatically determined rocksequences used for learning of linguistic description with fuzzy numbers in the antecedent.Estimation obtained using the learned linguistic description and smooth logical deduction.

The fourth column is computed using the learned linguistic description of ancientsea level behavior on the basis of the determined thickness of sequences. The fifthcolumn is the same but the linguistic description of the ancient sea level behavioris learned on the basis of the F-transformed sequence thicknesses. The advantage ofthe latter two methods comes from the fact that we have the linguistic descriptionof third-order and fourth-order sequence significance at our disposal. Hence, it ispossible to modify the results on the basis of some geologist’s additional knowledgeor information.

Some of the obtained results are graphically depicted in Figures 10.6–10.12. InFigure 10.6, the result of approximating the ancient sea level from the data G1 usingF-transform (third column of the above data) is presented. The following Figure 10.7shows the estimation of sea level using the linguistic description. The descriptionhas been learned from the original data (first and second columns). The antecedenthas been learned using fuzzy numbers; the succedent contains general evaluatingexpressions.

The linguistic context of the time slice (antecedent) is w1 = [1.0, 97.0] and changeof sea level (succedent) is w2 = [0.0, 5.0] (this coincides with the context for

10.4 Sea Level Estimation 331

Figure 10.8 Estimation of ancient sea level. Source: data G1, automatically determinedrock sequences used for learning of linguistic description with evaluating expressions in theantecedent. Estimation obtained using the learned linguistic description and smooth logicaldeduction.

thickness of sequences). The learned linguistic description has the form given inTable 10.5.

The expressions in the tables are written briefly, using “shorts” in correspondencewith Section 10.2. The expression “Ap5” means “approximately 5” since we workwith fuzzy numbers only. Some contradictory rules have been manually deleted (theoriginal learned description had 67 rules). For comparison, the same approximationusing the learned linguistic description, which has in its antecedent only evaluatingexpressions, is presented in Figure 10.8.

The linguistic description has the form given in Table 10.6 (after manual deletionof some contradictory rules; the original description had 25 rules).

Figure 10.9 shows the result of estimation of the sea level, again using thelearned linguistic description in which the third column has been used instead of thesecond one. Similarly, Figure 10.10 contains the same but using a linguistic descrip-tion with evaluating expressions only. These descriptions have not been manuallymodified.

Figure 10.9 Estimation of ancient sea level. Source: data G1, automatically determined rocksequences. These have been first filtered using F-transform and the result has been used forlearning of linguistic description with fuzzy numbers in the antecedent. Estimation obtainedusing the learned linguistic description and smooth logical deduction.

Figure 10.10 Estimation of ancient sea level using the same procedure as in Figure 10.9, butthe linguistic description is formed only of evaluating expressions.

Figure 10.11 Estimation of ancient sea level for data D1 using the same procedure as inFigure 10.9.

Figure 10.12 The same estimation as in Figure 10.11, but the linguistic description has beenmodified manually on the basis of additional knowledge.


Table 10.5 Learned linguistic description for the G1 data.

Rule No. Time slice⇒⇒⇒ sea level[1.0, 97.0]⇒⇒⇒ [0.0, 5.0]

1 Ro1 ⇒⇒⇒ MLBi2 Ro2.9 ⇒⇒⇒ VRSm3 Ro5 ⇒⇒⇒ QRBi4 Ro7 ⇒⇒⇒ Ap55 Ro9 ⇒⇒⇒ VRBi6 Ro9 ⇒⇒⇒ Ap57 Ro11 ⇒⇒⇒ RaMe...

... ⇒⇒⇒...

46 Ro87 ⇒⇒⇒ MLSm47 Ro89 ⇒⇒⇒ Sm48 Ro91 ⇒⇒⇒ MLSm49 Ro93 ⇒⇒⇒ QRBi50 Ro95 ⇒⇒⇒ MLMe51 Ro97 ⇒⇒⇒ Bi

Table 10.6 Manually modified linguistic description containingevaluating expressions only.

Rule No. Time slice⇒⇒⇒ sea level[1.0, 97.0]⇒⇒⇒ [0.0, 5.0]

1 RaSm ⇒⇒⇒ MLBi2 RoSm ⇒⇒⇒ RaMe3 QRSm ⇒⇒⇒ RoBi4 VRSm ⇒⇒⇒ VRSm5 MLMe ⇒⇒⇒ MLMe6 Bi ⇒⇒⇒ RoBi7 VeBi ⇒⇒⇒ VRBi8 SiBi ⇒⇒⇒ Sm9 ExBi ⇒⇒⇒ QRBi

10 VR97 ⇒⇒⇒ Bi

An example of the power of linguistic description is shown in Figures 10.11 and10.12, which show the sea level estimation for the data D1. The first figure showsthe estimation based on the original learned linguistic description. The second figureshows the same after expert modification of the description. One may see that specialknowledge can be well included in the linguistic descriptions.

10.5 Conclusion 335

10.5 Conclusion

This chapter describes some fuzzy techniques useful for application in geology,namely the theory of evaluating linguistic expressions, linguistic descriptions andlogical deduction based on them, and the fuzzy transform (F-transform) from I.Perfilieva (Chapter 9). These techniques have been applied to the determinationof rock sequences. On the basis of rock sequences ancient sea level position andfluctuations can be estimated in variety of ways.

Our results demonstrate that fuzzy techniques provide a strong tool capable ofsolving interesting geologic problems. We see our results as a strong hint supportingfurther development. For example, sea level estimation should be improved whenother influences are included, such as subsidence and sediment deposition. It is clearthat a lot of geological information has not been included, which thus made us unableto estimate the time periods during which the changes took place. It is also clear thatextraction of sea level signals is the first step in any attempt to “inverse model” asedimentary deposit. Thus, we have opened an extensive area for further research infuzzy logic and its applications to geology.

Acknowledgment

This chapter has been supported by project ME468 of the MŠMT of the Czech Repub-lic as the international supplement to project NSF “Stratigraphic Simulation UsingFuzzy Logic to Model Sediment Dispersal.”

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Boston and Dordrecht.Perfilieva, I. [2003], “Fuzzy transform.” In: Dubois, D. et al. (eds.) Rough and Fuzzy

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reasoning I, II, III,” Information Sciences, 8, 199–257, 301–357; 9, 43–80.

Absolute quantifiers 44Abstraction process by factorization 212Accommodation potential 325Aggregating operations 25, 27Aggregation operations, basic classes

28–29Algebraic product 23Algebraic sum 24α-cuts 15–16, 30, 38Ancient sea level estimation 301–337

fuzzy logic techniques 303–322if–then rules 303–304overview 301–303presentation of results 328–334

Ancient sedimentary sequences 124Andros Island 88–92Antecedent membership functions 130

for burial depth 74–76Antecedent variables 99Approximate reasoning 36, 44–46Approximation models 287Arithmetic mean 27Arithmetic operations on fuzzy intervals

28–31Arizona, long-term statistical forecasting of

precipitation 180–182Artificial neural fuzzy inference system

(ANFIS) analysis 109Atmospheric circulation patterns (CPs) 158Atomic evaluating expressions 305Attribute implications 204–207, 224Automation, modeling process 148Averaging operations 25–28

requirements 26

Backpropagation (BP) neural networks264

Basic functions 277Basic linguistic trichotomy 305Basin energy 138, 142, 145–146

Basin floor sedimentary environment 126Basin wave energy 143Beer–Lambert law 78Binary fuzzy relations 33Binary relations 31, 36Biresiduum 208Boolean algebra 25Bounded difference 24Bounded sum 24Building damage index 244, 250, 256Burial depth, antecedent membership

functions for 74–76

Carbonate production versus depth anddistance 295

Carbonate sediment productionas function of depth and distance to

platform edge 88–93if–then rules 91–93

Cartesian product 31–32Cauchy problem, approximate solution

287–289Center of area method 52Centroid method 52Characteristic function

definition 66of crisp sets 18

China 112seismic code 255

Circulation pattern (CP) 162, 177,183–187

Classical measure theory 1Classical set theory 1

and fuzzy set theory 17Climatic modeling of hydrological extremes

157Coastal oceanographic modelers 122Combinations of variables 70Commutativity requirements 23Compaction curves, application of standard

inference rules 74–78

339

Index

340 Index

Compatibility relations 38Compatible similarities 212–214Compatible tolerance relations 213Complements 21–22

classes 22requirements 21

Complexity 2managing 4

Compositional rule of inference 45–46Compositions 33–34, 52

properties 34Computational cost 4Computational theory of perceptions

(CTP) xiComputer technology, emergence of 64Computers and Geosciences 113Computing with words and perceptions

(CWP) xConcept lattices, similarity of 214–215Conditional and qualified fuzzy

propositions 44Conditional and unqualified fuzzy

propositions 43–44Congruence relation 225Congruences 212Connected binary relations 34Conococheague limestone 323–324, 328Constrained fuzzy arithmetic 31Coral reef growth, standard inference rules

78–82Core 15Credibility 2Crisp sets 12

and geology 66–67characteristic functions 18grain size 69granite 70tidal range 67versus fuzzy sets, representation of grain

size 12Cutworthy generalization 33Cutworthy property 17CYCOPATH 2D code 123Cylindric closures 32–33Cylindric extensions 32–33

Darcy’s Law 66, 94Data sets, training 164Death Valley, California 118, 124–133Decomposition 35

DEE (Defuzzification of EvaluatingExpressions) 316–317

Defuzzification 52–53, 77, 83, 128,169–170, 316

Degree of fulfillment (DOF) 167,169–170, 172

Degree of membership 4, 13, 31, 38–39, 70Delta sedimentation patterns 140Delta simulation 145–146Depositional processes, modeling 133–137Deterministic systems 2Direct F-transform 200, 320–321Direct methods 54–55Disorganized complexity 65“Dr. Sediment” code 123Drastic intersection 24Drastic union 24Drought, long-term record 109Drought index (DI) 159, 161, 164, 170

long-term statistical forecasting173–177

Drought response membership functions,Nebraska 165

Earth sciences 104, 114overview 63–102

Earthquake damage 249assessment 244brick-column single-story factory

buildings 257estimation 255

Earthquake engineering 244–245fuzzy information processing in 241fuzzy logic 249–259

Earthquake intensity 242, 249–255Earthquake loading estimation 244Earthquake magnitude 242

and logarithmic isoseismal area 266,268

Earthquake precursors 246–247Earthquake prediction 240, 243

based on seismicity indices, fuzzy patternfor 247–248

fuzzy logic 245–248fuzzy pattern recognition 245–247

Earthquake research 239–274basic terminology 242–245

Earthquake-resistant design 241El Niño Southern Oscillation (ENSO) 109,

158–161, 177, 184–185, 187

Index 341

Environmental health risk analyses 155Epicenter 242Epicentral intensity 242Evaluating linguistic expressions

finding a suitable expression 313–314semantics 307–313theory 305

Evaluating linguistic predication 307Evolutionary computation 106Expert systems 100, 112Exposure index versus height around mean

tidal level 84Extension principle 17–18

Factorization 212–214Factorization modulo 223Factorization modulo similarity 212Fitness value 106FLUVSIM 124Formal concept analysis 191–237

attribute dependencies 197–198attribute implications 204–207, 224directly observable data: objects and their

attributes 192discovering natural concepts hidden in

data 194–196discovery of hidden attribute

dependencies and natural concepts192–193

examples 217–235extent and intent 195fuzziness and similarity issues 198–199fuzzy data 199–207hierarchy of discovered concepts

196–197hierarchy of hidden concepts 203–204informal outline 193–194input data and hidden concepts 199–203origins 193

F-transform 275–300, 304components 280–281direct 280, 320–321inverse 280, 283–287, 292, 321technique 319–322

Fuzzified function 17FUZZIM 118Fuzzy algorithm 325–326Fuzzy approximation models 275

overview 276Fuzzy arithmetic 31, 109

Fuzzy clusteringbasic methods 99c-means 99–100sorting using 93–96

Fuzzy compaction algorithm 121Fuzzy compatibility, examples 39Fuzzy compatibility relations 38Fuzzy complements

involutive 22membership functions 144

Fuzzy concept lattices 203–204, 220–221,223

crisp order 204Fuzzy concepts 199–203Fuzzy contexts 199–203, 206, 215Fuzzy data 4Fuzzy earthquake intensity 250–255Fuzzy equivalence, examples 39Fuzzy equivalence relations 38Fuzzy fact 46Fuzzy implication 43–44, 46Fuzzy inference rules 73Fuzzy inference system 126, 129

output values 77Fuzzy information processing in earthquake

engineering 241Fuzzy intervals 49Fuzzy logic 38–46

broad sense 3–4, 12emergence of 6engineering applications 5impact on mathematics and logic 6narrow sense 3, 11output graph 96principal sources for further study

57–59role of 5scientific applications 5specialized tutorial 11–61systems 73–100use of term 3, 11

Fuzzy Logic Toolbox 83, 129Fuzzy measures 1Fuzzy models 275Fuzzy monotone measures 1Fuzzy numbers 29, 44, 82, 109

combined effect 166defined on monthly relative frequency of

given CP type 162defined on PMDI 164

342 Index

Fuzzy numbers (continued )

defined on premises 161–163defined on response variable 166defined on SOI 162

Fuzzy operations, construction 53–55Fuzzy partial ordering 38, 40Fuzzy partition 277

uniform 278universe 277

Fuzzy patternfor earthquake prediction based on

seismicity indices 247–248recognition in earthquake prediction

245–247Fuzzy power set 14Fuzzy predicates 41Fuzzy probability 41Fuzzy propability qualifier 42Fuzzy propositions 3–4, 38–39

and fuzzy sets 41Fuzzy quantifiers 41–42, 44Fuzzy regression 155Fuzzy relation equations 34

applications 36solving 35–36

Fuzzy relations 31–38antisymmetric 38between building damage index and

acceleration 253diagram and matrix representation 37information diffusion method 260–262reflexive 37single set 36–38symmetric 37

Fuzzy relationship matrix 258Fuzzy risk analysis 156Fuzzy rule 46Fuzzy rule-based fuzzy logic model 109Fuzzy rule-based geophysical forecast

system 112Fuzzy rule-based hydrology modeling 157Fuzzy rule-based models

evaluation 172–173future trends 120

Fuzzy rule system assessment 170–172Fuzzy seismology 240Fuzzy set theory 1, 3–4, 12, 53

and classical set theory 17emergence of 6engineering applications 5

impact on mathematics and logic 6principal sources for further study

57–59role of 5scientific applications 5

Fuzzy sets 3–4, 200and fuzzy propositions 41basic characteristics 16basic concepts 14–18construction 53–55definition 161–164geostatistics 156in geology 68level 2 56level k(k > 2) 56operations on 19type 2 56type k(k > 2) 56versus crisp set, representation of grain

size 12see also specific types

Fuzzy systems 49–53knowledge-based 50literature 53model-based 50

Fuzzy transform see F-transformFuzzy transformations 81Fuzzy transitivity 38Fuzzy truth qualifiers 42Fuzzy truth values 41

Gaussian membership functions 91General circulation models (GCM) 185Generalized Euler method 288–292Generalized Euler–Cauchy method

292–294Generalized means 27Generalized modus ponens 45Generalized Runge–Kutta methods

293–294Genetic algorithms 83Genetic fuzzy system 106Geographical information systems (GIS)

105, 116Geological sciences, literature review

103–120Geology

fuzzy set in 68literature review 107–119

Index 343

miscellaneous applications of fuzzy logic119

Geometric mean 27Geomorphology 108Geotechnical engineering 107

literature review 112–113Germany, long-term statistical forecasting of

precipitation 182–184Grain size 68

crisp set 69versus fuzzy set representation 12

permeability as function of 93–96sorting and permeability 93–96

Granitecrisp set 70fuzzy set representation 70

Granul 49Granulation 49

of systems variables 4Great Bahama Bank 88–92Groundwater flow 156Groundwater flow modeling 110–111Groundwater risk assessment 107

literature review 111–112Gutenberg–Richter Law 243

Haicheng earthquake 240Harmonic mean 27Height 15Homomorphisms 216Hungary, long-term statistical forecasting of

precipitation 178–180Hybrid fuzzy neural networks 259–269Hybrid fuzzy systems 50Hybrid model (HM) 269Hydrocarbon exploration 107

literature review 113–115Hydroclimatic modeling 157–173Hydrological extremes, climatic modeling of

157Hydrological forecasting 155–156Hydrological modeling 156Hydrology 153–190

application 155–157fuzzy rule-based modeling 157overview 154–157

If–then rules 73–74, 100–101, 142, 168,293, 302–304, 307, 315, 325

carbonate sediment production 91–93

Death Valley 126–127Mamdani fuzzy logic system 97–99sediment plume deposition off Southwest

Pass 138standard (“Mamdani”) interpretation

74–82, 95, 126, 128Takagi–Sugeno system 83–85

Igneous rock, classification 70Indirect methods 54–55Inferential connector, effect of changing 94Information-diffusion approximate

reasoning (IDAR) 264–265, 268Information diffusion estimator 261Information diffusion method 259–269

fuzzy relationships 260–262Information distribution formula 263, 267Input base variables 158–161Input membership functions 143Input-variable fuzzy sets 73Interactions 64International Union of Geological Sciences

70, 72Intersections 22–25

examples 23–24Interval-valued fuzzy sets 55Intuitionistic fuzzy sets 56Inverse F-transform 280, 283–287, 292,

321Inverse function 17Inverse transformation 286–287Inverses 33–34Inversion formula 283Involutive fuzzy complements 22

Joins 33–34Journal of Petroleum Geology 114Journal of Petroleum Science and

Engineering 114

Knowledge acquisition 53Knowledge-based fuzzy systems 50Kullback–Leibler directed divergence 243

L-fuzzy sets 56–57La Niña 158–161Landscape development, literature review

116–117Learned linguistic description 334Learning algorithm 106

344 Index

Learning relationships by BP neuralnetworks 264

Least of Maxima (LOM) method 316Level set 15LFLC (Linguistic Fuzzy Logic Controller)

314Linear regression methods 265Linguistic description 304, 314–319

learning from data 318–319power of 334

Linguistic expressions 41–42components of evaluating 305theory of evaluating 305see also Evaluating linguistic expressions

Linguistic hedges 305–306Linguistic terms 49, 53–54

membership functions 85–86Linguistic variable 49Linkages 64Logical deduction 304, 317–318Logical precision 215–217Long-shore drift regimes 138, 142–143,

145–147

Macro-seismic intensity scale 250Mamdani fuzzy inference model of basin

floor environments 129Mamdani fuzzy inference systems 83Mamdani fuzzy logic system 83, 97–99,

132, 138see also If–then rules

Mamdani inference rules 78–82Mamdani interpretation of if–then rules

74–82, 95, 126, 128Mamdani method 88Mamdani–Togai model 262Mathematical models and physical

reality 3Mathematics as core of system science 64MATLAB 83–85, 93, 97, 99, 129–131Max-min transitivity 38Mean absolute error (MAE) 173Mean error (ME) 173Membership degree 4, 13, 31, 38–39, 70Membership functions 12, 37, 39, 53, 68,

70, 126, 277antecedent 74–76, 130constructing 54–55, 245distributary mouth bar deposition 137examples 13

extensions of selected evaluatingexpressions 314

fuzzy complement of 144fuzzy sets “shallow” and “deep” for input

variable depth 141input and output 97linguistic terms 85–86modified 20–21notations 13possible shapes 14triangular-shaped 78variable porosity 74–76

Metropolis–Hastings algorithm 170Minimal solution 35Mississippi River Delta 118, 124, 133, 137Model-based fuzzy systems 50Modeling process, automation 148Modified membership function 20–21Modified Mercalli intensity scale 270–271Modifiers 19–21

types 20Modus ponens 45Modus tollens 45Monotone measures 1–3Montastrea annularis 78–79Multicriterion decision making (MCDM)

under uncertainty 156Multidistributary deltaic depostion

137–147

National Weather Service Climate AnalysisCenter 174

n-dimensional relations 31–33Nebraska, drought response membership

functions 165Necessity measure 47–48Nested family 16Neural networks 57, 83, 105–106, 109,

128–129Neuro-fuzzy sytems 106Newtonian mechanics 65Nondeterministic systems 2Nonlinear activation function 106Nonstandard fuzzy intersections 25Nonstandard fuzzy sets 13, 55–57, 70Nonstandard fuzzy unions 25Normal diffusion coefficient 262Normal information diffusion formula 267Numerical methods based on fuzzy

approximation models 275

Index 345

Numerical modeling, fuzzy logic asalternative technique 73

Optimization 106Ordinary fuzzy sets 82Organized complexity 104Organized simplicity 65Output base variables 158–161Output-variable fuzzy sets 73

Pattern smoothing 262–264PDSI (Palmer Drought Severity Index)

174–175Peak ground acceleration (PGA) 244–245,

250membership function 251

Perennial lakes 125Permeability 66

as function of grain size 93–96calculation of 93–96grain size, sorting and 93–96

Permeability and average grain size 96Petroleum engineering, virtual intelligence

in 113Plutonic igneous rocks, classification 72Porosity, membership functions 74–76Possibility measures 47–48Possibility theory 46–49

standard fuzzy-set interpretation 48Precipitation

input onto watersheds 109long-term statistical forecasting

178–184Primary information distribution matrix

258Principle of information diffusion 261Principle of maximal belongingness 245Principle of threshold-value 246Principles of belongingness 245Probability and uncertainty 1Probability-qualified form 42Probability theory 1–2Projections 32–33Propositional forms, basic types 41–44

Rainfall versus runoff data 109Reef growth model 294–297Reflexivity 207–208Regional water resources management 156Relational join 34

Remote sensing 104, 116Reservoir operation planning 156Rock properties 104Rock sequences

automatic determination 322–329,331–332

determination of 305results of tests 326–328

Root-mean squared error (RMSE) 173Rough fuzzy sets 56Rough sets 105Rule-based models see Fuzzy rule-based

modelsRule construction 166–169Runge–Kutta methods 292

Saline lakes 125Saline pans 125San Francisco earthquake 239Satellite remote sensing imagery 105Scalar cardinality 15Scientific knowledge, organizing 2Sea level

curve 296data 298–299extraction 294–297see also Ancient sea level estimation

Second-order fuzzy sets 82SEDFLUX 134–135Sediment

accumulation 122deposition 108, 117–118erosion 121–122plume 135–136, 142production 122transportation 122

Sedimentary basin filling,computer-generated forward models121

Sedimentary particles 12SEDSIM 123Seismic code, China 255Seismic damage grade 256Seismic intensity 242Seismic levels and basic intensity 256Seismic stratigraphy 124Seismicity 243Seismicity indices 243

fuzzy pattern for earthquake predictionbased on 247–248

346 Index

Seismology 108literature review 115–116

SEISMOS 240Self-adjusting inference rules, calculation of

exposure index 82–88Semantic level 43Sequence number 328Sequence stratigraphy 124Sigma count 15Siliciclastic sedimentary process simulators

123Siliciclastic sedimentary rocks 133Similarity 207–215

class 207of attributes 209–211of concept lattices 214–215of concepts 211of objects 209–211relations 207–215

Simple evaluating expressions 305Simple evaluating predication 307Simulated annealing 171Single-column fuzzy relationship matrix

267Sinusoidal membership functions 279Sinusoidal shape 286Site intensity 244, 249Smackover formation 124Smooth logical deduction 302, 304,

322–323Soft computing 103–107, 114

applications 119areas 105

SOI (Southern Ocean Oscillation) 109,159, 161

Soil science 108literature review 116–117

Solution set of fuzzy relation equations35–36

Sortinggrain size and permeability 93–96using fuzzy clustering 93–96

Southern Ocean Oscillation (SOI) 109,159, 161

Southwest Pass 135, 137Southwest Pass Distributary Mouth Bar

136Standard averaging operation 27Standard composition 33–34Standard fuzzy arithmetic 30–31

Standard fuzzy complement 22Standard fuzzy intersection 23Standard fuzzy sets 13, 68

interpretation of possibility theory 48Standard inference rules

application to compaction curves74–78

coral reef growth 78–82Standard (“Mamdani”) interpretation of

if–then rules 74–82, 95, 126, 128STRATAFORM 123Stratigraphic basin filling models 122–123Stratigraphic models 123–124Stratigraphic simulations, future

developments 147Stream sediments 117Strength of relation 31Strong α-cut 15–16Subconcept–superconcept relation 203Subnormal 15Subset 14–15Subsurface hydrology 107

literature review 110–111Support 15Supremum 16–17Surface hydrology 107

literature review 108–110Symmetric averaging operations 27Symmetry 207–208Syntactic level 43Synthetic stratigraphic cross-sections of

Death Valley sedimentation 132System science

mathematics as core of 64roots of 64

Systems thinking 64

Takagi–Sugeno first-order membershipfunctions 86–87

Takagi–Sugeno fuzzy inference system 83,85, 131

Takagi–Sugeno fuzzy logic system 88,130, 138

Takagi–Sugeno linear functions 142Takagi–Sugeno linear membership functions

142Takagi–Sugeno system, if–then rules

83–85Tangshan earthquake 240, 256, 258Theory of monotone measures 1

Index 347

Tidal range 49–50Tolerance relations 213, 227Training data sets 164Transitivity 207–208Transport modeling 156Trapezoid formula 282Triangular conorms (t-conorms) 23Triangular membership functions 91,

277–278Triangular norms (t-norms) 23Triangular-shaped membership functions

78Truth assignment function 39Truth-qualified form 42Truth values 76, 81, 95, 127, 203, 208, 215

Uncertainty 1–2multicriterion decision making (MCDM)

under 156Uncertainty and probability 1Unconditional and qualified propositions

42–43Unconditional and unqualified propositions

42Understanding Earth 63Understanding the Earth System 63

Uniform fuzzy partition 278–279Unions 22–25, 52

examples 23–24Unique maximum solution of fuzzy relation

equations 35Universal set 15

Vagueness 4Validation data sets 164, 169Virtual intelligence in petroleum engineering

113

Water quality models 109Water resources 153–190

overview 154–157Watershed response models 109Weighted average 54Weighted generalized means 27Weights 28

calculation of 169Wetness index 183, 186Wood–Anderson-type seismograph 243

Yager class of fuzzy complements 22Yager class of fuzzy intersections 24Yangtze River 112

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Fuzzy Logic in Geologyfa.mie.sut.ac.ir/Downloads/AcademicStaff/5/Courses...book is to make researchers in fuzzy logic aware of the emerging opportunities for the application of their