Developments in Soil ScienceSERIES EDITORS: A.E. Hartemink and A.B. McBratneyOn the CoverThe gure on the cover shows an unsupervised classication of topography fromSRTM30 DEM data by an iterative nested-means algorithm and a three part geo-metric signature (Iwahashi and Pike, 2007 available at http://gisstar.gsi.go.jp)Developments in Soil Science Volume 33GEOMORPHOMETRYConcepts, Software,ApplicationsEdited byTOMISLAV HENGLInstitute for Biodiversity and Ecosystem DynamicsUniversity of AmsterdamAmsterdam, The NetherlandsHANNES I. REUTERInstitute for Environment and SustainabilityDG Joint Research CentreLand Management andNatural Hazards Unit European CommissionIspra, ItalyAmsterdam Boston Heidelberg London New York OxfordParis San Diego San Francisco Singapore Sydney TokyoElsevierRadarweg 29, PO Box 211, 1000 AE Amsterdam, The NetherlandsLinacre House, Jordan Hill, Oxford OX2 8DP, UKFirst edition 2009Copyright 2009 Elsevier B.V. All rights reservedNo part of this publication may be reproduced, stored in a retrieval system or transmitted in anyform or by any means electronic, mechanical, photocopying, recording or otherwise without the priorwritten permission of the publisherPermissions may be sought directly from Elseviers Science & Technology Rights Department in Ox-ford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com. Al-ternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier materialNoticeNo responsibility is assumed by the publisher for any injury and/or damage to persons or property asa matter of products liability, negligence or otherwise, or from any use or operation of any methods,products, instructions or ideas contained in the material herein. Because of rapid advances in the med-ical sciences, in particular, independent verication of diagnoses and drug dosages should be madeBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Cataloging-in-Publication DataA catalog record for this book is available from the Library of CongressISBN: 978-0-12-374345-9ISSN: 0166-2481For information on all Elsevier publicationsvisit our website at elsevierdirect.comPrinted and bound in Hungary09 10 11 12 10 9 8 7 6 5 4 3 2 1This book is dedicated to all geographers and earth scientists, fromwhich two must be singled out for special mention: Waldo R. Tobler methodological revolutionary, conceptualiser of Analytical Cartography,and hero to numberless quantitative geographers; and Peter A. Burrough one of the founders of geoinformation science and mentor to a generationof GIS scientists.CONTENTSAuthors xiiiCo-authors xxiForeword xxiiiI. CONCEPTS 11. Geomorphometry: A Brief Guide 3R.J. Pike, I.S. Evans and T. Hengl1. What is geomorphometry? 32. The basic principles of geomorphometry 63. The history of geomorphometry 124. Geomorphometry today 245. The Baranja Hill case study 266. Summary points 29Important sources 302. Mathematical and Digital Models of the Land Surface 31T. Hengl and I.S. Evans1. Conceptual models of the land surface 312. Digital models of the land surface 393. The sampling, generation and analysis of land surfaces 484. Summary points 62Important sources 633. DEM Production Methods and Sources 65A. Nelson, H.I. Reuter and P. Gessler1. Ground survey techniques 652. Remote sensing sources 713. Frequently-used remote-sensing based DEM products 754. Summary points 82Important sources 854. Preparation of DEMs for Geomorphometric Analysis 87H.I. Reuter, T. Hengl, P. Gessler and P. Soille1. Introduction 872. Reducing errors in DEMs 963. Reduction of errors in parameters and objects 111viiviii Contents4. Summary 117Important sources 1205. Geostatistical Simulation and Error Propagation in Geomorphometry 121A.J.A.M. Temme, G.B.M. Heuvelink, J.M. Schoorl and L. Claessens1. Uncertainty in DEMs 1212. Geostatistical modelling of DEM errors 1233. Methods for error propagation analysis 1254. Error propagation: Baranja Hill 1275. Summary points 138Important sources 1396. Basic Land-Surface Parameters 141V. Olaya1. Introduction 1412. Local land-surface parameters 1423. Regional land-surface parameters 1634. Summary points 168Important sources 1697. Land-Surface Parameters and Objects in Hydrology 171S. Gruber and S. Peckham1. Hydrological modelling 1712. Flow direction and aspect 1723. Flow algorithms 1764. Contributing area/ow accumulation 1825. Land-surface parameters based on catchment area 1856. Land-surface objects based on ow-variables 1887. Deposition function 1918. Flow modelling using TIN-based elevation models 1939. Summary points 194Important sources 1948. Land-Surface Parameters Specic to Topo-Climatology 195J. Bhner and O. Antonic1. Land surface and climate 1952. Climate regionalisation approaches 1973. Topographic radiation 1984. Topographic temperature 2115. Topographic precipitation 2196. Topographic exposure to wind 2227. Summary points 225Important sources 2259. Landforms and Landform Elements in Geomorphometry 227R.A. MacMillan and P.A. Shary1. Geomorphology, landforms and geomorphometry 227Contents ix2. Approaches to landform classication 2333. Extracting and classifying specic landform elements 2374. Extracting and classifying repeating landform types 2445. Implementing extraction of landforms 2516. Summary points 252Important sources 254II. SOFTWARE 25510. Overview of Software Packages Used in Geomorphometry 257J. Wood1. Introduction 2572. The software landscape 2573. Approaches to using software 2604. Other packages for geomorphometry 2615. The future of geomorphometry software 264Important sources 26611. Geomorphometry in ESRI Packages 269H.I. Reuter and A. Nelson1. Getting started 2692. DEM preparation 2733. Extraction of land-surface parameters and objects 2784. Arc scripts 2835. Summary points and future direction 290Important sources 29112. Geomorphometry in SAGA 293V. Olaya and O. Conrad1. Getting started 2932. DEM preparation 2983. Derivation of land-surface parameters 3004. Summary points 307Important sources 30813. Geomorphometry in ILWIS 309T. Hengl, B.H.P. Maathuis and L. Wang1. About ILWIS 3092. Importing and deriving DEMs 3133. Deriving parameters and objects 3184. Summary points and future direction 329Important sources 33114. Geomorphometry in LandSerf 333J. Wood1. Introduction 333x Contents2. Visualisation of land-surface parameters and objects 3383. Scripting with LandSerf 3424. Summary points and future direction 348Important sources 34915. Geomorphometry in MicroDEM 351P. Guth1. Introduction 3512. Geomorphometric analysis in MicroDEM 3543. Summary points and future direction 366Important sources 36616. Geomorphometry in TAS GIS 367J.B. Lindsay1. Getting started 3672. Geomorphometric analysis 3703. Summary 384Important sources 38617. Geomorphometry in GRASS GIS 387J. Hoerka, H. Mitov and M. Neteler1. Getting started 3872. Deriving land-surface parameters 3933. Limitations of GRASS 4094. Summary points and future direction 410Important sources 41018. Geomorphometry in RiverTools 411S.D. Peckham1. Getting started 4112. Advanced GIS functionality 4133. Preparing DEMs for a study area 4144. Extracting land-surface parameters and objects from DEMs 4175. Visualisation tools 4256. Summary points 429Important sources 430III. APPLICATIONS 43119. Geomorphometry A Key to Landscape Mapping and Modelling 433T. Hengl and R.A. MacMillan1. Importance of DEMs 4332. Predictive modelling of environmental variables 4373. Summary points 459Important sources 460Contents xi20. Soil Mapping Applications 461E. Dobos and T. Hengl1. Soils and mapping of soils 4612. Topography and soils 4653. Case study 4714. Summary points 475Important sources 47921. Vegetation Mapping Applications 481S.D. Jelaska1. Mapping vegetation 4812. Case study 4873. Summary points 494Important sources 49622. Applications in Geomorphology 497I.S. Evans, T. Hengl and P. Gorsevski1. Geomorphological mapping from DEMs 4972. Geomorphometry in theories of uvial and glacial erosion 4983. Geomorphological change 5044. Extraction of specic landforms 5075. Case study 5116. Summary points 522Important sources 52523. Modelling Mass Movements and Landslide Susceptibility 527S. Gruber, C. Huggel and R. Pike1. Introduction 5272. Modelling the propagation of mass: the affected area and deposition 5283. Modelling landslide susceptibility 5394. Summary points 548Important sources 54924. Automated Predictive Mapping of Ecological Entities 551R.A. MacMillan, A. Torregrosa, D. Moon, R. Coup and N. Philips1. Introduction 5512. Case study: predictive ecosystem mapping 5563. Results 5714. Summary points 577Acknowledgements 578Important sources 57825. Geomorphometry and Spatial Hydrologic Modelling 579S.D. Peckham1. Introduction 5792. Spatial hydrologic modelling: processes and methods 5813. Scale issues in spatial hydrologic models 591xii Contents4. Preprocessing tools for spatial hydrologic models 5935. Case study: hydrologic response of north basin, Baranja Hill 5956. Summary points 601Important sources 60226. Applications in Meteorology 603S. Emeis and H.R. Knoche1. Meteorology and topography 6032. Sources for topographic variables in meteorological models 6113. Case studies 6124. Summary points 620Important sources 62127. Applications in Precision Agriculture 623H.I. Reuter and K.-C. Kersebaum1. Introduction 6232. Precision Agriculture applications 6253. Summary points 635Acknowledgements 635Important sources 63628. The Future of Geomorphometry 637P. Gessler, R. Pike, R.A. MacMillan, T. Hengl and H.I. Reuter1. Peering into a crystal ball 6372. Concepts 6383. DEM data: demand and supply 6394. Exploiting LiDAR 6415. Morphometric tools 6436. Approaches and objectives 6457. Applications to come 6468. A geomorphometric atlas of the World 6489. Closing remarks 651Bibliography 653Index 695Colour Plate Section 707AUTHORSThis book is the joint effort of a number of specialists and researchers. Informationabout the authors and their afliations is given in the following section.Jrgen Bhner is a Professor of Physical Geography and head of theSAGA-GIS developer team at the Department for Earth Sciences, Uni-versity of Hamburg. He graduated from the University of Gttin-gen in geography, meteorology, bioclimatology and botany, where hegained his Ph.D. in 1993. Until 2004, he was a scientic assistant atthe Department of Physical Geography at Gttingen University, co-ordinating and participating in national and international research projects onclimatic variability, applied meteorology, remote sensing and process modelling.His Habilitation thesis in 2004 mirrors his major interests in modelling topo-climates. More recently, his major focus had been the creation of complex land-surface parameters for regional climate modelling purposes and the modellingof soil related processes (wind and water erosion, translocation and depositionprocesses).Current employer: Institut fr Geographie, University of Hamburg, GermanyContact: jboehner@geowiss.uni-hamburg.deEndre Dobos is an Associate professor at the University of Miskolc, De-partment of Physical Geography and Environmental Sciences. He dida Ph.D. in Soil Mapping with GIS and Remote Sensing at the Agron-omy Department of Purdue University, Indiana, USA, in 1998 and anM.Sc. in GIS and Environmental Survey at the Faculty of Civil En-gineering of the Technical University of Budapest in 1996. Endre haschaired the working group on Digital Soil Mapping of the European Commissionand co-chaired the Digital Soil Mapping working group of the International Unionof Soil Sciences.Current employer: University of Miskolc, Miskolc, HungaryContact: ecodobos@uni-miskolc.huStefan Emeis is a meteorologist with main emphasis on turbulent trans-port processes in the atmospheric boundary layer. Stefan did a post-doc at the Institute of Meteorology and Climate Research of the Uni-versity of Karlsruhe/Forschungszentrum Karlsruhe, and a habilita-tion in Meteorology at the University of Karlsruhe in 1994. He haslong-year experience in numerical modelling of atmospheric owsxiiixiv Authorsand chemistry over orography which includes the use of digital terrain and landuse data. The outcome of these modelling studies made contributions to the under-standing of pressure drag exerted on the atmosphere by sub-gridscale orographyand to the understanding of air pollution in mountainous areas. His present spe-cialisation is surface-based remote sensing of the atmospheric boundary layer.Current employer: Forschungszentrum Karlsruhe GmbH, Garmisch-Partenkirchen,GermanyContact: stefan.emeis@imk.fzk.deIan S. Evans is credited with a number of innovations in geomor-phometric concepts and techniques, starting with formalising a sys-tem of surface derivatives to replace many more arbitrary measures,and calculating them from DEMs. Ian has worked on a number ofprojects involving data analysis in Geography, including the specicgeomorphometry of glacial cirques and drumlins, using both manu-ally measured indices and DEM-based attributes. Ian worked in the NERC/RCAExperimental Cartography Unit from 1968 to 1970 and from then onwards in theGeography Department at Durham, from 1979, as Senior Lecturer, and from 1999,as Reader. He has held research grants from the US Army, from ESRC and fromthe Ministry of Defence and various ofces in professional organisations, includ-ing Chairman (19961997) of the British Geomorphological Research Group.Current employer: Durham University, Durham, UKContact: i.s.evans@durham.ac.ukPaul Gessler is an Associate Professor of Remote Sensing and SpatialEcology and Co-Director of the Geospatial Laboratory for Environ-mental Dynamics in the College of Natural Resources at the Universityof Idaho, USA. He completed a Ph.D. in Environmental Modelingat the Australian National University, Canberra, and remote sensing(M.Sc.) and soils (B.Sc.) degrees from the University of Wisconsin Madison. Paul pursued research in soillandscape modelling with the CSIRO Di-vision of Land & Water in Australia for seven years before starting in academia.Hes involved in a diversity of research and teaching involving the remote sens-ing, characterisation and monitoring of forest ecosystems along with wildland refuels and re hazard mapping, airborne sensor development, and soillandscapemodelling and digital soil mapping. All activities involving the analysis of com-plex terrain and an integrated element of geomorphometry.Current employer: University of Idaho, Moscow, Idaho, USAContact: paulg@uidaho.eduStephan Gruber rmly believes that topography makes happy. He cur-rently lives in Switzerland. His main research interests are the mea-surement and modelling of high-mountain cryosphere phenomena(permafrost, glaciers and snow). Geomorphometry is one of the tech-niques he uses it is often very powerful due to the dominatinginuence that topography has on surface and near-surface processes.Authors xvHe currently works at the University of Zrich, where he received his Ph.D. Pre-viously, he has done research in other places: the University of Giessen, Germany;the Arctic Centre/University of Lapland, Finland; the ITC in the Netherlands; theNational Snow and Ice Data Center in Boulder, Colorado, USA and the Universitde Savoie, France. Stephan is still contemplating a suitable land-surface parameterthat quanties the happiness caused by topography. Like in many other cases it isprobably somehow related to the rst derivative.Current employer: University of Zrich, Zrich, SwitzerlandContact: stgruber@geo.unizh.chPeter L. Guth is a Professor in the Department of Oceanography atthe United States Naval Academy. Peter was trained as a eld geol-ogist at the Massachusetts Institute of Technology, and has workedfor a number of summers in the Sheep Range of southern Nevada.He shifted his research focus to microcomputer land-surface analy-sis while teaching at the U.S. Military Academy, and has been de-veloping the freeware MicroDEM program for over 20 years. He has used Mi-croDEM to investigate software algorithms such as slope, line of sight, andviewshed computations; for looking at anomalies in digital elevation modelssuch as contour-line ghosts; for quantifying the degree of land-surface organi-sation and fabric; and for looking at geomorphometric land-surface character-istics computed for the United States and the entire world during the ShuttleRadar Topography Mission. He has also worked with Geodesy Base, a smallcompany that locates res, using web-based tools and GIS in lookout tow-ers.Current employer: U.S. Naval Academy, Annapolis, MD, USAContact: pguth@usna.eduTomislav Hengl is a GIS scientist with special interests in soil map-ping, land-surface modelling and the use of statistical techniques forspatial analysis in general. He studied at the Faculty of Forestry in Za-greb, then received a scholarship for a post-graduate study abroad.He nished his M.Sc. in 2000 and Ph.D. degree in 2003, both ofthem in the Netherlands, at the International Institute for Geoinfor-mation Science and Earth Observation and Wageningen University. He joinedthe Joint Research Centre of the European Commission, as a post-doctoral re-searcher, in June 2003. He has published several research papers that focus onthe preparation of land-surface parameters, their improvement using different l-tering techniques, and on the use of land-surface parameters in soillandscapemodelling, including lecture notes on extraction of DEM-parameters in ILWISGIS. His recent interests are development of automated predictive mappingtechniques and integration of geostatistics, geomorphometry and remote sens-ing.Current employer: Faculty of Science, University of Amsterdam, NetherlandsContact: hengl@science.uva.nlxvi AuthorsJaroslav Hoerka is an associate professor of Physical Geography andGeoecology and Head of the GIS Laboratory in the Department ofGeography and Regional Development at the University of Presov,Slovakia. He received a Ph.D. degree in Cartography and Geoinfor-matics from Comenius University, Bratislava, Slovakia in 1998. Hismain research activities have been focused on digital terrain modellingand applications, spatial interpolation and the modelling of landscape processes(water erosion, solar radiation) using GIS. He has been participating in the devel-opment of Open Source GRASS GIS since 1992. His other research areas includerenewable energies, spatial and temporal landscape changes and municipal infor-mation systems.Current employer: Department of Geography and Regional Development, Univer-sity of Presov, Presov, SlovakiaContact: hoerka@geomodel.skSven D. Jelaska is an assistant professor of Plant Ecology in the De-partment of Botany of Faculty of Science at the University of Zagreb,Croatia. His main interest is in scientic research dealing with oraand vegetation spatial distribution, including the issues of biologicaldiversity. Using the GIS, statistical methods (CCA, CART, DA, logit,etc.) and other technologies (e.g. RS, GPS, hemispherical canopy pho-tos) he integrates various types of data relevant for description and explanationof spatial distribution of biological entities. These were backbones of his M.Sc.and Ph.D. thesis, both in ecology, accepted at the Faculty of Science, Universityof Zagreb in 1999 and 2006, respectively. He managed the creation of prelimi-nary ecological network of Croatia, and co-managed the late Ecological Networkof Croatia as a part of PEEN and NATURA2000 network. He was actively involvedin project Mapping of habitats of the Republic of Croatia. As a biodiversity expert heparticipated in National report on climate change 19962003. He published over 20peer-reviewed papers on various aspects dealing with vascular ora and vegeta-tion.Current employer: Faculty of Science, University of Zagreb, Zagreb, CroatiaContact: sven@botanic.hrJohn Lindsay is a lecturer in physical geography and geocomputationat the University of Manchester. He studied geography at the Univer-sity of Western Ontario, Canada, where he completed an M.Sc. anda Ph.D. in the areas of uvial geomorphology and digital land-surfaceanalysis, respectively. Johns research area has focused on DEM pre-processing, particularly in relation to topographic depressions, andthe extraction of DEM-derived channel networks and network morphometrics.John also has considerable interest in the development of software and algorithmsfor digital land-surface analysis and is the author of TAS GIS.Current employer: Department of Geography, University of Guelph, Guelph, On-tario, CanadaContact: jlindsay@uoguelph.caAuthors xviiRobert A. MacMillan is a private sector consultant who earns his liv-ing applying geomorphometric techniques to map and model naturallandscapes. Bob has a B.Sc. in Geology from Carleton University, anM.Sc. in Soil Science from the University of Alberta and a Ph.D. inGIS and hydrological modelling from the University of Edinburgh.Bob graduated as a geologist but trained as a soil surveyor with boththe national and Alberta soil survey units in Canada. Bob spent more than 10years as an active eld soil surveyor (19751985) with experience in Alberta, On-tario, East Africa, Nova Scotia and New Brunswick. From 1980 onwards, Bobwas increasingly involved in developing and applying computer-based proce-dures for enhancing soil survey products, including statistics and geo-statistics,analysis of soil map variability and error, use of GIS to both create and ap-ply soil map information and use of DEMs to assist in the creation of maps.Bob led the rst project to use GIS for soil information in Alberta in 1985 andcreated his rst DEM in 1985 in which soil attributes from a grid soil surveywere related to terrain attributes computed from the DEM. Bob led the designeffort that resulted in production of the seamless digital soils database for Al-berta (AGRASID). Since 1994 Bob has operated a commercial consulting com-pany (LandMapper Environmental Solutions Inc.) that has completed numerousprojects that used automated analysis of digital elevation models and ancillarydata sources to produce maps and models for government and private sectorclients. Bob developed the LandMapR toolkit to provide a custom, in-house, ca-pacity to analyse the land surface to describe and classify landforms, soils, eco-logical and hydrological spatial entities in an automated fashion. The LandMapRprocedures have been used to produce ecological and landform maps for mil-lions of ha in BC and Alberta and to classify hundreds of agricultural elds.The toolkit has been used by more that 50 individuals, private sector compa-nies, universities and major government organisations in Canada and internation-ally.Current employer: LandMapper Environmental Solutions Inc., Edmonton, AB,CanadaContact: bobmacm@telusplanet.netAndrew Nelson is a geographer with interests in the Multi ScaleModelling of environmental issues, Geographically Weighted Sta-tistics, Biodiversity Mapping and Analysis, Accessibility Models,Neural Networks, Population and Poverty Modelling and WatershedModelling. He has previously worked at the World Bank, UNEP,CGIAR and is currently a post-doctoral researcher at the EC JointResearch Centre in Italy. Andy has worked on hole-lling algorithms for theSRTM data, and multi-scale land-surface parameter extraction using geograph-ically weighted statistics to identify appropriate scales for environmental mod-elling.Current employer: European Commission, Directorate General JRC, Ispra, ItalyContact: andrew.nelson@jrc.itxviii AuthorsVictor Olaya Ferrero is a GIS developer with an interest in computa-tional hydrology and land-surface analysis. He studied Forest Engi-neering at the Polytechnic University of Madrid and received an M.Sc.degree in 2002. After that, he created a small company dedicated to thedevelopment of software for forest management. Victor is currentlyemployed as a Ph.D. student at the University of Extremadura, wherehe leads the development of the SEXTANTE project a GIS specially devel-oped for forest management purposes. Victor has developed several applicationscontaining land-surface parametrisation algorithms. He is also the author of theA gentle introduction to SAGA GIS, the ofcial manual of this GIS.Current employer: University of Extremadura, Plasencia, SpainContact: volaya@unex.esScott D. Peckham is a research scientist at INSTAAR, which is a re-search institute at the University of Colorado in Boulder. Scott hasbeen honoured to pursue research as a NASA Global Change Stu-dent Fellow (19901993) and a National Research Council ResearchAssociate (19951998). His research interests include physically-basedmathematical and numerical modelling, watershed-scale hydrologicsystems, coastal zone circulation, source-to-sink sediment transport, scaling analy-sis, differential geometry, theoretical geomorphology, grid-based computationalmethods, efcient computer algorithms and uvial landscape evolution models.Scott is also CEO and founder of Rivix LLC, which sells a software product forland-surface and watershed analysis called RiverTools, and is also the primary au-thor of a next-generation, spatially-distributed hydrologic model called TopoFlow.Current employer: University of Colorado at Boulder and Rivix LLC, Broomeld,CO, USAContact: peckhams@rivix.comRichard J. Pike has dedicated his entire career to land-surface quan-tication. His earliest research in continuous-surface morphometry(in 1961 on mean valley depth) was as a student of Walter F. Wood,the pioneering terrain analyst of the quantitative revolution in Amer-ican geography. Inspired by astrophysicist Ralph B. Baldwin, he sub-sequently became expert in the specic morphometry of planetaryimpact craters. Richard was educated both as a geologist (Tufts, B.Sc.; The Uni-versity of Michigan, Ph.D.) and a geographer (Clark, M.A.). He has worked forUSGS since 1968, when he organised creation of the Agencys rst DEMs and mor-phometric software. Among his many contributions are lunar surface-roughnessdata for the Apollo Roving Vehicle Project, the concept of the geometric signa-ture, co-authorship of the celebrated digital shaded-relief map of the United States,Supplementband 101 of the Zeitschrift fr Geomorphologie, and a 7000-entry an-notated bibliography.Current employer: U.S. Geological Survey, Menlo Park, CA, USAContact: rpike@usgs.govAuthors xixHannes I. Reuter is a geo-ecologist, who graduated from Potsdam Uni-versity with majors in Soil Science and GIS/remote sensing. He ob-tained his degree in soil science from the University of Hannover,while working at the Leibniz-Centre for Agricultural Landscape Re-search (ZALF) on precision farming topics. He used land-surface pa-rameters in his Ph.D. thesis to investigate relationships between relief,soil and plant growth, using a couple of ArcInfo Scripts. His interest is in improv-ing the understanding of landscape processes at different scales. He is currentlyworking on nding optimal methods for lling in data voids in the SRTM datamodel.Current employer: European Commission, Directorate General JRC, Ispra, ItalyContact: hannes.reuter@jrc.itArnaud Temme holds M.Sc. degrees in Soil Science and Geoinforma-tion Science (cum laude), both obtained at Wageningen University. In2003, he became a Ph.D. student there, under the supervision of thechairs of Soil Inventarisation and Land Evaluation. His main interestis the dynamic landscape, and the methods for studying it. His Ph.D.study area is in the foothills of the Drakensberg, South Africa, wherehe studies the evolution of a 100-ka landscape as a function of climatic changeand endogenous feedbacks. In his rst paper, he presented an algorithm for deal-ing with sinks in DEMs, within landscape evolution models, so that it would nolonger be necessary to remove the sinks before running the model. Arnaud hasa part-time Ph.D. job to enable him to pursue, simultaneously, a career in moun-taineering.Current employer: Wageningen University and Research Centre, Wageningen,The NetherlandsContact: Arnaud.Temme@wur.nlJo Wood has been a Senior Lecturer in the Department of Informa-tion Science at City University London since 2000. Between 1991 and2000, he was a lecturer in GIS at the University of Leicester, in theDepartment of Geography. He gained an M.Sc. in GIS at Leicester in1990 and then studied there for his Ph.D. in geomorphometry, whichhe was awarded in 1996. His teaching and research interests includeland-surface analysis, spatial programming with JAVA, geovisualisation and GIScience. Jo gained his Ph.D. on The Geomorphological Characterisation of DigitalElevation Models in 1996. This thesis, one of the rst studies to incorporate themulti-scale land-surface parametrisation of DEMs, won the 1996 AGI Student-of-the-Year Award. The approach suggested by the thesis was later incorporated intosome of the GIS GRASS modules and also led to the development of the land-surface analysis GIS, LandSerf, which Jo has been authoring for 9 years. He iscurrently supervising Ph.D. students in the areas of Ethno-physiography and inobject-eld representations of geographic information.Current employer: City University, London, UKContact: jwo@soi.city.ac.ukCO-AUTHORSOleg Antoni cCurrent employer: Rudjer Bokovi c Institute, Zagreb, CroatiaContact: oantonic@irb.hrLieven ClaessensCurrent employer: Wageningen University and Research Centre, Wageningen,The NetherlandsContact: lieven.claessens@wur.nlOlaf ConradCurrent employer: University of Gttingen, Gttingen, GermanyContact: oconrad@gwdg.dePeter V. GorsevskiCurrent employer: Bowling Green State University, Bowling Green, Ohio, USAContact: peterg@bgnet.bgsu.eduGerard B.M. HeuvelinkCurrent employer: Wageningen University and Research Centre, Wageningen,The NetherlandsContact: gerard.heuvelink@wur.nlChristian HuggelCurrent employer: University of Zrich, Zrich, SwitzerlandContact: chuggel@geo.unizh.chBen H.P. MaathuisCurrent employer: International Institute for Geo-Information Science and EarthObservation (ITC), Enschede, The NetherlandsContact: maathuis@itc.nlKurt C. KersebaumCurrent employer: Leibniz Centre for Agricultural Landscape and Land Use Re-search (ZALF), Mncheberg, GermanyContact: ckersebaum@zalf.dexxixxii Co-authorsHans R. KnocheCurrent employer: Institut fr Meteorologie und Klimaforschung, Garmisch-Partenkirchen, GermanyContact: hans-richard.knoche@imk.fzk.deHelena MitovCurrent employer: North Carolina State University, Raleigh, NC, USAContact: helena_mitasova@ncsu.eduMarkus NetelerCurrent employer: Fondazione Mach Centre for Alpine Ecology, 38100 Viotedel Monte Bondone, Trento, ItalyContact: neteler@cealp.itJeroen M. SchoorlCurrent employer: Wageningen University and Research Centre, Wageningen,The NetherlandsContact: jeroen.schoorl@wur.nlPeter A. SharyCurrent employer: Institute of physical, chemical and biological problems of soilscience, Moscow region, RussiaContact: p_shary@mail.ruPierre SoilleCurrent employer: European Commission, Directorate General JRC, Ispra, ItalyContact: pierre.soille@jrc.itAlicia TorregrosaCurrent employer: US Geological Survey, Menlo Park, CA, USAContact: atorregrosa@usgs.govLichun WangCurrent employer: International Institute for Geo-Information Science and EarthObservation (ITC), Enschede, The NetherlandsContact: lichun@itc.nlFOREWORDWHY GEOMORPHOMETRY?We began to think about a geomorphometry book in the summerof 2005 following a request to suggest auxiliary data that wouldassist the automated mapping of soils. The rst thing that cameto mind, of course, was Digital Elevation Models (DEMs). Thelonger we considered our response to the request, the more werealised that a substantial gap had opened between the formaldiscipline of land-surface quantication and a vast informal, andrapidly growing, community of DEM users.The practical aspects of morphometric analysis seemed to us neglected in theliterature. Apart fromWilson and Gallants Terrain Analysis: Principles and Applica-tions and Li, Zhu and Golds Digital Terrain Modeling: Principles and Methodology,few textbooks are suited both for training and for guiding an inexperienced DEMuser through the various steps, from obtaining a DEM to carrying out analyses inpackaged software. It was our experience that, although irreplaceable, Wilson andGallants book is not ideal for either purpose; not only it is primarily a compilationof research or review papers, but it relies heavily on Ian Moores TAPES software,a comprehensive package to be sure but just one of many now available. Mean-while, new parameters and algorithms for processing DEMs were circulating inthe scientic literature; an update and summary of the eld seemed increasinglyappealing. Richard Pike later told us that he (and others) had pondered a geomor-phometry text for many years. We also discovered that there is quite some disorderin the eld. A major problem is the absence of standards for extracting descrip-tive measures (parameters) and surface features (objects) from DEMs. Manyusers are confused by the fact that values of even basic parameters such as slopegradient may vary depending on the mathematical model by which they are cal-culated, size of the search window, the grid resolution. . . although the measuresthemselves might appear quite stable. Serious issues also exist over operationalprinciples, for example, pre- and postprocessing of DEMs: should unwanted de-pressions (sinks, or pits) be ltered out, or not? which algorithms should be usedto propagate DEMerror through subsequent analyses? should DEMs be smoothedprior to their morphometric application or not, and if so, by howmuch? These andother questions got us thinking about many aspects of land-surface quantication.In November 2005, we prepared the initial draft of a Table of Contents andimmediately agreed on three things: the book should be (1) practical, (2) compre-hensive, and (3) a fully integrated volume rather than an ad hoc compilation ofxxiiixxiv ForewordFIGURE 1 Participants in the rst meeting of the authors, Plasencia, Spain, 1822 May 2006.papers. We also knew that our goals would be more likely achieved in collabo-ration with a number of co-authors. Initially, we invited ten colleagues to join usbut the number slowly grew, along with interest in the book. Our third objectiveposed difculties how to synchronise the output of well over a dozen authors?To solve this problem, we launched an online editorial system that allowed usto exchange documents and data sets with all the authors, thereby encouragingtransparent discussion among everyone in the group. It became clear that therewould be many iterations before the chapters were nalised and authors sent intheir last word.Our action leader at JRC, Luca Montanarella, soon recognised the importanceof this project and supported us in organising the rst authors meeting, whichwas kindly hosted by Victor Olaya and Juan Carlos Gimenez of the Universidad deExtremadura in Plasencia, Spain. At this meeting, we found ourselves convincedof the effectiveness of a group approach to the writing; enthusiasm for the bookwas overwhelming. In response to last-minute invitations, Paul Gessler and IanEvans joined the group (Paul took less than 24 hours to decide to make the 12,000kilometre trip from the western U.S., even though the meeting would convene injust 4 days) and immediately provided useful feedback.It was Ian Evans who rocked the boat by opening a discussion on some of theelds terminology. First to be scrutinised, and heavily criticised, was terrain.Gradually we began to see the problems arising from its use and elected to adoptless ambiguous language. We understand that whatever our arguments, the wideruser community will not readily abandon terrain and terrain analysis in favour ofour preferred land surface and geomorphometry (indeed, there is not 100%agreementamong this books authors), but we hope that the reader will at least agree to thinkalong with us. The Plasencia meeting further revealed that most authors were inForeword xxvFIGURE 2 Geomorphometrists are easily recognised by their obsession with shape explaining a morphometric algorithm often requires much use of the hands.favour of pricing the book at a non-commercial rate, thereby opening it up to thewidest possible readership yet without jeopardising its scientic and technicalcontent.The meeting also led us to suspect a gender gap in the eld. Despite theirmany contributions over the years, women geomorphometrists were absent atPlasencia. We hasten to add that we invited several women colleagues to join us,but only four were able to participate in preparing this rst edition. We look for-ward to an improved balance in the next, and succeeding, editions of this bookand take encouragement from Peter Shary, who reported from the 2006 NanjingSymposium on Terrain Analysis and Digital Terrain Modelling that the number ofyounger women now working with DEMs (at least in Asia) is clearly on the rise.During nal editing of the books initial draft we decided to prepare a state-of-the-art gallery of land-surface parameters and objects, to assist less experiencedreaders in applying DEMs to their best advantage, and then to support an inde-pendent Web site to encourage further evolution of the Geomorphometry Project.You are nowinvited to visit this site, post comments on it, evaluate software scriptsand packages, upload announcements of events or jobs, and eventually post yourown articles. The oor is open to all.WHAT CAN YOU FIND IN THIS BOOK?The volume is organised in three sections: theoretical (concepts), technical (soft-ware), and discipline-specic (applications). Most of the latter are in the environ-mental and Earth sciences, so that the book might best be compared with that ofWilson and Gallant (2000). Our book differs, however, in that it offers technical de-tails on a variety of software packages and more instruction on how to carry outsimilar data analyses yourself.This book is more about the surface properties that can be extracted froma DEM than about creating the DEM itself. To appreciate our chosen operationalxxvi Forewordfocus, a basic acquaintance with geographical information systems (GIS) (Bur-rough, 1986) and (geo)statistics (Goovaerts, 1997) will be helpful. Readers whorequire added technical information on DEMs and how to generate them shouldconsult the books by Li et al. (2005) Digital Terrain Modeling: Principles and Method-ology and Maune (2001) Digital Elevation Model Technologies and Applications: TheDEM Users Manual.Each of the books three sections consists of nine or ten chapters that followa logical sequence from data processing to extraction of land-surface parametersand objects from DEMs. Many chapters overlap in both content and examples,illustrating not only the many types of land-surface parameters, but also their vari-ants differing parameter values calculated from an identical DEM by differentsoftware. Links to external sources and important literature can be found at theend of each chapter, and well over 100 text boxes ag (important) remarks through-out the book. All major types of land-surface parameters and objects, together witha quick reference to their signicance and interpretation, are listed in the galleryof parameters and objects available on the Geomorphometry Web Site. A list ofreferences and an index are provided at the end of the book.Part I: ConceptsThe books opening Chapter 1 will rst orient you to the eld of geomorphometry,its basic concepts and principles, and major applications. This introduction is fol-lowed by a historical review of the discipline, from before the rst contour lines tothe computer programs by which early DEMs were processed. You will also nda detailed description of the Baranja Hill case study, which is used to demonstratealgorithms and applications throughout the book.Chapter 2 in Part I is a mathematical introduction to modelling the land sur-face. Following a discussion of the most important model properties, includingsurface-specicity, is a list of mathematical models and data structures to representtopography and its intrinsic attributes, such as scale dependence, multi-fractality,and the t of a model to the true land surface. Special attention is accorded formu-las for calculating rst- and second-order surface derivatives.The most common sources of digital elevation data are reviewed in Chapter 3.Each DEM source is described in terms of the equipment or hardware used to col-lect elevation data, as well as the advantages and disadvantages of postprocessingin converting the raw data into a DEM. Also compared are such key character-istics of the different sources as cost per km2, typical footprints, postprocessingrequirements, and data accuracy and precision.Chapter 4 is devoted to techniques for improving the quality of DEMs prior togeomorphometric analysis. Included are algorithms to: detect artefacts, systematicerrors, and noise in DEMs; deal with missing values (voids), water bodies, andtree-canopy distortion (e.g. in SRTM data); and lter out spurious DEM depres-sions. The chapter closes with a discussion of simulation techniques to minimiseDEM error.Ageostatistical technique to model uncertainty in DEMs and analyse its impacton the calculation of land-surface parameters (slope, wetness index, soil redis-Foreword xxviitribution) is introduced in Chapter 5. The focus is on propagation of DEM errorthrough subsequent analyses using the sequential Gaussian simulation.Chapter 6 is an overview of basic morphometric parameters, measures de-rived directly from DEMs without added special input. The measures range fromlocal land-surface parameters (slope, aspect, solar aspect, curvature) to regionalparameters (catchment area, slope length, relative relief) and statistical parameterssuch as terrain roughness, complexity, and anisotropy. Each measure is illustratedby the Baranja Hill test site.Following in Chapter 7 are hydrological land-surface parameters for quan-tifying water ow and allied surface processes. This overview will guide youthrough the key concepts behind DEM-based ow modelling, again, illustratedby our Baranja Hill case study. Methods for parameterising the physics involvedin moving mass (water, sediment, ice) over an irregular surface (topography) areexplained, as well as related parameters and objects derived from modelled ow.Chapter 8 contains an extensive review of solar radiation models and ap-proaches to quantifying exposure of the land surface to climatic inuences. Firstdiscussed are algorithms by which incoming solar radiation may be estimatedfrom DEMs. Topo-climatic modelling is then extended to the estimation of land-surface temperature, precipitation, snow-cover, and exposure to wind and the owof cold air.The nal Chapter 9 in Part I introduces landform types and elements andtheir relation to continuous topography versus specic geomorphic features. Nextdescribed are techniques for extracting landform classes, either from a list of pre-dened geomorphic types or by automated extraction of generic surface facetsfrom DEMs. An extensive comparison of approaches to landform classicationhighlights the value of geomorphometric standards and data-systems that couldwin wide (international) acceptance.Part II: SoftwareChapter 10 opens the middle third of the book with a general inventory andprospect of all packaged computer programs suited to geomorphometry (of whichwe are aware), including software not demonstrated in this book. The remain-ing chapters illustrate eight well-known packages currently available for land-surface analysis, ranging from commercial (ArcGIS) to medium-cost (RiverTools)and freely-available (including open-source) (SAGA, GRASS, ILWIS, LandSerf, TAS,MicroDEM) software. Five chapters are authored by the originators of the software,and three by later developers or expert users; each chapter follows a commonstructure: Description of the software, its origins and target users, and how to acquirethe package and install it. Using the software package for the rst time what it can, or cannot do;where and how to get support. How to import and display DEMs, using our Baranja Hill case study. Which land-surface parameters and object-parameters can be derived fromthe package, and how they are calculated.xxviii Foreword How particular land-surface parameters and objects can be interpreted andapplied. Summary of strong and weak points of the software, any known bugs, andhow the package may be expected to evolve.We intend that each chapter serve a dual purpose, as a user manual and asa review of scientic information. For readers requiring further support, linksto original user guides, mailing lists, and technical documentation and where todownload them are given in each chapter.Part III: ApplicationsThe nal section of the book exemplies the role of geomorphometry in geo- andenvironmental sciences ranging from soil and vegetation mapping, hydrologicaland climatic modelling, to geomorphology and precision agriculture. Chapter 19introduces the role of digital land-surface analysis in creating maps and modelsacross a broad spectrum of disciplines. It explains why DEM analysis has becomeso essential for quantifying and understanding the natural landscape. The chapterreviews basic concepts underlying the many uses of geomorphometry as well ashow these applications incorporate automated mapping and modelling. It alsodescribes some of the mathematical, statistical, and empirical methods by whichpredictive scenarios have been modelled using land-surface data.Subsequent chapters of Part III describe specic cases of automated DEManalysis in various disciplines. These examples are not necessarily all-encompass-ing, but illustrate some of the many different approaches to using geomorphom-etry to generate and interpret spatial information. Each of the next eight chaptersfollows a common structure: Introduction to state-of-the-art applications, explaining the importance ofgeomorphometry in this eld and reviewing recent research. Guided analysis of an example, usually the Baranja Hill case study, includingan interpretation of the results. Summary of opportunities and limitations as well as suggestions for futureresearch.In considering the prospect for geomorphometry, the books closing chapterpeers into a crystal ball what breakthroughs might emerge from future advancesin technology? Which concepts, applications, and societal needs are likely to drivethe discipline? How dramatic an increase in detail and accuracy can be expectedof future DEMs? The chapter also includes a proposal for the design and op-eration of a geomorphometric atlas of the world that could provide a referencedata-repository for most applications of DEM-derived information.CLOSING THOUGHTS AND ACKNOWLEDGEMENTSThis book is intended primarily for (a) universities and research institutes wheregraduate or post-graduate courses are conducted in geography and other envi-Foreword xxixronmental and geo-sciences, and (b) GIS specialists and project teams involved inmapping, modelling, and managing natural resources at various spatial scales. Webelieve, moreover, that it will prove its worth as a tutorial and reference source toanyone involved in the analysis of DEMs.It is not our intention that this volume deliver an exhaustive synthesis of geo-morphometry. A reader with a background in civil engineering, for example, willquickly note applications and technical areas that are under-represented or absent.This does not mean that we did not think it worthwhile to include them, but ratherthat other books are better suited to the task. Nonetheless, we hope that a diversereadership will come to regard our book as a worthwhile source of information onthe methods and applications of modern geomorphometry. We offer the book notso much as a stand-alone achievement, but rather as part of an initiative to pro-mote development of the science so that not only researchers in geomorphometry,but also the wider community of DEM users, will apply it wisely. We offer ourapologies if we have inadvertently and unintentionally omitted anyones contri-butions to geomorphometry.We wish to thank our science reviewers, Bodo Bookhagen (Stanford Univer-sity, School of Earth Sciences, Stanford, CA, USA), Peter Burrough (University ofUtrecht, The Netherlands), Ian S. Evans (Durham University, Durham, UK), PeterFisher (City University, London, UK), John Gallant (CSIRO Land and Water, Can-berra, Australia), Gerard B.M. Heuvelink (Wageningen University and ResearchCentre, Wageningen, The Netherlands), Robert A. MacMillan (LandMapper Envi-ronmental Solutions Inc., Edmonton, AB, Canada), Richard Pike (U.S. GeologicalSurvey, Menlo Park, CA, USA), David Tarboton (Utah State University, Logan, UT,USA), Stephen Wise (University of Shefeld, Shefeld, UK), and Ole Wendroth (Uni-versity of Kentucky, Kentucky, US). Their numerous comments and suggestionsfor improving and extending various chapters have been invaluable in bringingthis project to a successful conclusion.We are especially grateful to Richard Pike and Ian S. Evans (two fathers of mod-ern geomorphometry) for providing the support and encouragement during thelast phases of line-editing. We are also grateful to Roko Mra (the Croatian StateGeodetic Department) for organising a licence to use the Baranja Hill datasets.Last, but not least, we thank JRC colleagues Nicola Lugeri for cross-checking over1000 references, Nadine Bhr for her tipsntricks of graphical editing, PierangelloPrincipalli and Alessandro Piedepalumbo for their professional-quality printing andbinding of v1.0 and v2.0 of the book, our secretary Grazia Faber for providingcontinual remedy for the inevitable bureaucratic headaches, and many other col-leagues within JRC and farther aeld who have supported us in this endeavour.Every effort has been made to trace copyright holders. We apologize for anyunintentional omissions and would be pleased to add an acknowledgment in fu-ture editions.Tomislav Hengl and Hannes I. ReuterIspra (VA), July 2007CHAPTER1Geomorphometry: A Brief GuideR.J. Pike, I.S. Evans and T. Henglbasic denitions the land surface land-surface parameters and objects digital elevation models (DEMs) basic principles of geomorphometry froma GIS perspective inputs/outputs, data structures & algorithms historyof geomorphometry geomorphometry today data set used in this book1. WHAT IS GEOMORPHOMETRY?Geomorphometry is the science of quantitative land-surface analysis (Pike, 1995, 2000a;Rasemann et al., 2004). It is a modern, analytical-cartographic approach to rep-resenting bare-earth topography by the computer manipulation of terrain height(Tobler, 1976, 2000). Geomorphometry is an interdisciplinary eld that has evolvedfrom mathematics, the Earth sciences, and most recently computer science(Figure 1). Although geomorphometry1has been regarded as an activity withinmore established elds, ranging from geography and geomorphology to soil sci-ence and military engineering, it is no longer just a collection of numerical tech-niques but a discipline in its own right (Pike, 1995).It is well to keep in mind the two overarching modes of geomorphometricanalysis rst distinguished by Evans (1972): specic, addressing discrete surfacefeatures (i.e. landforms), and general, treating the continuous land surface. Themorphometry of landforms per se, by or without the use of digital data, is morecorrectly considered part of quantitative geomorphology (Thorn, 1988; Scheidegger,1991; Leopold et al., 1995; Rhoads and Thorn, 1996). Geomorphometry in this bookis primarily the computer characterisation and analysis of continuous topography.A ne-scale counterpart of geomorphometry in manufacturing is industrial surfacemetrology (Thomas, 1999; Pike, 2000b).The ground beneath our feet is universally understood to be the interface be-tween soil or bare rock and the atmosphere. Just what to call this surface and itsscience of measurement, however, is less obvious. Numerical representation of the1The term, distinguished from morphometry in other sciences (e.g. biology), dates back at least to Neuenschwander(1944) and Tricart (1947).Developments in Soil Science, Volume 33 2009 Elsevier B.V.ISSN 0166-2481, DOI: 10.1016/S0166-2481(08)00001-9. All rights reserved.34 R. J. Pike et al.FIGURE 1 Geomorphometry and its relation to source and end-user disciplines. Modied afterPike (1995).land surface is known variously as terrain modelling (Li et al., 2005), terrain analysis(Wilson and Gallant, 2000), or the science of topography (Mark and Smith, 2004).2Quantitative descriptors, or measures, of land-surface form have been referred toas topographic attributes or properties (Wilson and Gallant, 2000), land-form parame-ters (Speight, 1968), morphometric variables (Shary et al., 2002), terrain information(Martinoni, 2002), terrain attributes (Pennock, 2003), and geomorphometric attributes(Schmidt and Dikau, 1999).REMARK 1. Geomorphometry is the science of topographic quantication; itsoperational focus is the extraction of land-surface parameters and objects fromdigital elevation models (DEMs).Despite widespread usage, as a technical term terrain is imprecise. Terrainmeans different things to different specialists; it is associated not only with landform, hydrographic features, soil, vegetation, and geology but also (like topogra-phy) with the socio-economic aspects of an area (Li et al., 2005). Terrain3also cansignify an area of ground, a region. . . unrelated to shape of the land surface. Themuch used terrain analysis (Moore et al., 1991a; Wilson and Gallant, 2000) is con-fusing (unless preceded by quantitative), because it has long denoted qualitative(manual) stereoscopic photo- or image-interpretation (Way, 1973). Nor does themore precise digital terrain modelling (Weibel and Heller, 1991) escape ambiguity, asterrain modelling can infer measurement or display of surface heights, unspeciedquantication of topography, or any digital processing of Earth-surface features.2The most frequent equivalents of geomorphometry in Googles online database appear to be surface or terrain modelling,terrain analysis and digital terrain modelling (Pike, 2002).3Terrain is from the Latin terrenum, which might be translated as of the earth.Geomorphometry: A Brief Guide 5Additionally, in many countries (e.g. France, Spain, Russia, Slovakia) relief4issynonymous with morphology of the land surface (King et al., 1999). This usageis less evident in Anglophone regions (e.g. Great Britain, North America), whererelief, usually prexed by relative or local, has come to denote the difference be-tween maximal and minimal elevation within an area (Partsch, 1911; Smith, 1953;Evans, 1979), low and high relief indicating small and large elevation contrastsrespectively.5To minimise confusion, the authors of this book have agreed to consistentlyuse geomorphometry to denote the scientic discipline and land surface6to indicatethe principal object of study. Digital representation of the land surface thus willbe referred to as a digital land surface model (DLSM), a specic type of digital surfacemodel (DSM) that is more or less equivalent to the widely-accepted term digitalelevation model7(DEM).An area of interest may have several DSMs, for example, surface models show-ing slope gradient or other height derivative, the tree canopy, buildings, or a geo-logical substrate. DSMs from laser altimetry (LiDAR, light detection and ranging)data can show more than one return surface depending on how deep the rayspenetrate. Multiple DLSMs are usually less common but can include DEMs fromdifferent sources or gridded at different resolutions, as well as elevation arraysstructured differently from square-grid DEMs (Wilson and Gallant, 2000). Objectsof the built environment are of course not part of the land surface and must beremoved to create a true bare-earth DLSM.Digital elevation model (DEM) has become the favoured term for the datamost commonly input to geomorphometry, ever since the U.S. Geological Sur-vey (USGS) rst began distribution of 3-arc-second DEMs in 1974 (Allder et al.,1982). Even elevation is not unique as it can also mean surface uplift (e.g. the Hi-malayas have an elevation of 5 mm/year). However, the alternative terms are lesssatisfactory: height is relative to a nearby low point, and altitude commonly refersto vertical distance between sea level and an aircraft, satellite, or spacecraft. Thusdigital height model and altitude matrix (Evans, 1972) are avoided here.REMARK 2. The usual input to geomorphometric analysis is a square-grid rep-resentation of the land surface: a digital elevation (or land surface) model (DEMor DLSM).In this book, DEM refers to a gridded set of points in Cartesian space attributedwith elevation values that approximate Earths ground surface (e.g. Figure 5, be-low). Thus, contour data or other types of sampled elevations, such as a triangulararray, are not DEMs as the term is used here. DEM implies that elevation isavailable continuously at each grid location, at a given resolution. See Chapter 2for a detailed treatment of topography and elevation models.4fren. Topographie, germ. Relief, russ. , span. Relieve.5This quantity is also known as reliefenergie (Gutersohn, 1932), particularly in Germany and Japan.6fren. Surface terrestre, germ. Gelnde, russ. , span. Topografa. A term that became widely knownthrough the morphometric work of Hammond (1964).7fren. Modle numrique de terrain, germ. Digitales Gelnde Model, russ. , span. Modelo deelevacin digital.6 R. J. Pike et al.Finally, we dene parameter and object, the two DEM-derived entities funda-mental to modern geomorphometry (see, e.g., Mark and Smith, 2004). A land-surface parameter8is a descriptive measure of surface form (e.g. slope, aspect, wet-ness index); it is arrayed in a continuous eld of values, usually as a raster imageor map, for the same referent area as its source DEM. A land-surface object9is a dis-crete spatial feature (e.g. watershed line, cirque, alluvial fan, drainage network),best represented on a vector map consisting of points, lines, and/or polygons ex-tracted from the square-grid DEM.It is also important to distinguish parameters per se, which describe the landsurface at a point or local sample area, from quantitative attributes that describeobjects. For example, slope gradient at a given point refers only to its x, y location,whereas the volume of, say, a doline (limestone sink) applies to the entire areaoccupied by that surface depression; slope is a land-surface parameter, while de-pression volume over an area is an attribute of a land-surface object. Each of thesequantities can be obtained from a DEM by a series of mathematical operations, ormorphometric algorithms.2. THE BASIC PRINCIPLES OF GEOMORPHOMETRY2.1 Inputs and outputsThe fundamental operation in geomorphometry is extraction of parameters and ob-jects from DEMs (Figure 2). DEMs, i.e. digital land-surface models, are the primaryinput to morphometric analysis. In GIS (geographic information system) terms,a DEM is simply a raster or a vector map showing the height of the land sur-face above mean sea level or some other referent horizon (see further Section 2 inChapter 2).Geomorphometry commonly is implemented in ve steps (Figure 2):1. Sampling the land surface (height measurements).2. Generating a surface model from the sampled heights.3. Correcting errors and artefacts in the surface model.4. Deriving land-surface parameters and objects.5. Applications of the resulting parameters and objects.Land-surface parameters and objects can be grouped according to variouscriteria. Parameters commonly are distinguished as primary or secondary, de-pending on whether they derive directly from a DEM or additional processingsteps/inputs are required (Wilson and Gallant, 2000). In this book, we will followa somewhat different classication that reects the purpose and type of analysis.Three main groups of land-surface parameters and objects are identied: Basic morphometric parameters and objects (see Chapter 6); Parameters and objects specic to hydrology (see Chapter 7); Parameters and objects specic to climate and meteorology (see Chapter 8);8fren. Paramtre de la surface terrestre, germ. Reliefparameter, russ. , span. Variable del terreno.9fren. Object de la surface terrestre, germ. Reliefobjeckt, russ. , span. Elemento del terreno.Geomorphometry: A Brief Guide 7FIGURE 2 The operational focus of geomorphometry is extraction of land-surface parametersand objects from DEMs.Basic parameters and objects describe local morphology of the land surface(e.g. slope gradient, aspect and curvature). Hydrological or ow-accumulation pa-rameters and objects reect potential movement of material over the land surface(e.g. indices of erosion or mass movement). The third group of parameters andobjects is often calculated by adjusting climatic or meteorological quantities to theinuence of surface relief.A special group of land-surface objects geomorphological units, land ele-ments and landforms receives its own chapter (Chapter 9). A landform is a dis-crete morphologic feature such as a watershed, sand dune, or drumlin that is a functionally interrelated part of the land surface formed by a specicgeomorphological process or group of processes. Each landformmay be composedof several landform elements, smaller divisions of the land surface that have rela-tively constant morphometric properties.REMARK 3. A landform element is a division of the land surface, at a givenscale or spatial resolution, bounded by topographic discontinuities and having(relatively) uniform morphometry.8 R. J. Pike et al.Recognition of landforms and less exactly dened tracts, commonly referredto as land-surface types, from the analysis of DEMs is increasingly important. Manyareas of the Earths surface are homogeneous overall or structured in a distinctiveway at a particular scale (e.g. a dune eld) and need to be so delineated (Iwa-hashi and Pike, 2007). In the special case of landforms extracted as membershipsby a fuzzy classication algorithm, such forms can be considered to partake ofa particular land-surface object instead of directly mapping, say, a stream chan-nel, we can obtain a membership value10to that landform.2.2 The raster data structureMany land-surface representations, such as the background topography seen invideo games and animated lms, are modelled by mass-produced surface heightsarrayed in some variant of the surface-specic triangulated irregular network (TIN)model (Blow, 2000; Hormann, 1969; see Chapter 2, Section 2.1). Most geomorpho-metric applications, however, use the square-grid DEM model. To be able to applythe techniques of geomorphometry effectively, it is essential to be familiar with theconcept of a raster GIS and its unique properties.Although the raster structure has a number of disadvantages, including a rec-tangular data array regardless of the morphology of the study area, large data-storage requirements, and under- and over-sampling of different parts of a diversestudy area, it will remain the most popular format for spatial modelling in theforeseeable future. This structure is especially advantageous to geomorphometrybecause most of its technical properties are controlled automatically by a singlemeasure: spatial resolution, grid size or cell size,11expressed as a constant x, y spac-ing (usually in metres) (Hengl, 2006).In addition to grid resolution, we also need to know the coordinates of at leastone grid intersection (usually marking the lower left-hand corner of the entireDEM array) and the number of rows and columns, whereupon we should be ableto dene the entire map (Figure 3). This of course assumes that the map is projectedinto an orthogonal system where all grid nodes are of exactly equal size and orientedtoward cartographic North.Accordingly, the small 66-pixel DEMin Figure 5 (see below) can also be codedin an ASCII le as an array of heights:ncols 6nrows 6xllcorner 0yllcorner 0cellsize 10.00nodata_value -3276710 16 23 16 9 614 11 18 11 18 1919 15 13 21 23 2520 20 19 14 38 4524 20 20 28 18 4923 24 34 38 45 5110Such a value has been designated by the rather clumsy term channelness.11Cell size is a more appropriate term than grid size because grid size can also imply size of the whole grid.Geomorphometry: A Brief Guide 9FIGURE 3 An orthogonal raster map can be dened by just ve parameters: (a & b) number ofrows and columns; (c & d) coordinates of the lower left corner and (e) cell size.where ncols is number of columns, nrows is number of rows, xllcorner is thewestern edge of the map, yllcorner is the southern edge of the map, cellsizeis grid resolution in metres, nodata_value is the arbitrary value used to maskout locations outside the area of interest and 10, 16, 23, 16, 9, 6 are the elevationvalues in the (rst) row. This is the standard format for ASCII grid les used byESRI Inc. for its ArcInfo and ArcGIS software. It is necessary to dene the initialpoint of the grid system correctly: there is a difference in x, y location of half thecellsize, depending on whether the rst coordinate is at the lower left-handcorner of the lower left-hand grid cell (llcorner) or at the centre of that cell(llcenter).REMARK 4. The principal advantage of a raster GIS over other spatial datastructures is that a single measure the cell or pixel size automaticallycontrols most technical properties.2.3 Geomorphometric algorithmsPerforming morphometric operations within a raster GIS usually involves calcu-lating intermediate quantities (over the same grid of interest) which are then usedto compute the nal output. Most morphometric algorithms work through theneighbourhood operation a procedure that moves a small regular matrix of cells(variously termed a sub-grid or lter window) over the entire map from the upperleft to the lower right corner and repeats a mathematical formula at each place-ment of this sampling grid.Neighbouring pixels in a sampling window are commonly dened in relationto a central pixel, i.e. the location for which a parameter or an object membershipis derived. In principle, there are several ways to designate neighbouring pixels,most commonly either by an identier or by their position relative to the central10 R. J. Pike et al.FIGURE 4 The common designation of neighbours in 33 and 55 window environments:(a) by unique identiers (as implemented in ILWIS GIS), (b) by row and column separation(in pixels) from the central pixel (as implemented in the ArcInfo GIS).pixel (Figure 4). The latter (e.g. implemented by the DOCELL command in ArcInfo)is the more widely used because it can readily pinpoint almost any of the neigh-bouring cells anywhere on the map [Figure 4(b)].Computing a DEM derivative can be simple repetition of a given formula overthe area of interest. Consider a very small DEM of just 66 pixels. You could zoominto these values (elevations) and derive the desired parameter on a pocket calcu-lator (Figure 5). For example, using a 33 sampling window, slope gradient at thecentral pixel can be derived as the average change in elevation. Three steps arerequired; rst, the difference in relative elevation is calculated in x and y direc-tions, whereupon slope gradient is obtained as the average of the two quadratics(Figure 5). By the EvansYoung method12(Pennock et al., 1987), slope gradient iscalculated (see further Chapter 6):G = zNB3+zNB6+zNB9zNB1zNB4zNB76 sH = zNB1+zNB2+zNB3zNB7zNB8zNB96 s12Often, one land-surface parameter can be calculated by several different formulas or approaches; we caution that theresults can differ substantially!Geomorphometry: A Brief Guide 11FIGURE 5 Numerical example showing slope tangent (in %) extracted from a DEM using a 33window.where G is the rst derivative in the x direction (df /dx), H is the rst derivative inthe y direction (df /dy), zNB5 is the (central) cell for which the nal value of slopeis desired, zNB1,2,3,4,6,7,8,9 are the eight neighbouring cells, and s is pixel size inmetres (Figure 5). The slope gradient as a tangent is nally computed as:SLOPE =

H2+G2Note that the example in Figure 5 shows values of slope gradient for rows andcolumns at the edge of the map, although we did not actually have the necessaryelevation values outside the map area. Normally, a neighbourhood operation ispossible only at a grid location surrounded by its eight immediate neighbours.Because keeping to this practice loses the outermost rows and columns, the ex-pedient solution illustrated in this example is to estimate missing neighbours byduplicating cells at the edges of the DEM and tolerating the (usually) modest errorin the nal result. By so doing, the output map retains exactly the same size as theinput map.REMARK 5. Because most land-surface parameters vary with spatial scale, orcan be calculated by different algorithms and sampling grids, no map computedfrom a DEM is denitive.12 R. J. Pike et al.Adjustments such as these differ among software packages, so that almost al-ways some small differences will be found in outputs from exactly the same math-ematical formulas. To avoid confusion, in referring to various types of generalland-surface parameters and objects we will consistently specify (1) the algorithm(reference), (2) size of the sampling window and (3) the cell size. The exampleabove would be slope (land-surface parameter type) calculated by the EvansYoungmethod (Pennock et al., 1987) (variant) in a 33 window environment (sub-variant)using a 10 m DEM (cell size). The rounding factor also can be important becausesome intermediate quantities require high precision (many decimal places), whileothers must never equal zero or take a negative value.Finally, in Figure 5 we can see that the pixel with highest slope, 125%, is atlocation row = 5, column = 5 and the lowest slope, 5%, is at location row = 6,column = 1. Of course, in a GIS map the heights are rarely represented as num-bers but rather by colour or greyscale legends.3. THE HISTORY OF GEOMORPHOMETRYBefore exploring data, algorithms and applications in detail, it is well to step backand consider the evolution of geomorphometry, from the pioneering work of Ger-man geographers and French and English mathematicians to results from recentSpace Shuttle and planetary-exploration missions. While its ultimate origins maybe lost in antiquity, geomorphometry as we know it today began to evolve asa scientic eld with the discoveries of Barnab Brisson (17771828), Carl Gauss(17771855), Alexander von Humboldt (17691859), and others, reaching maturityonly after development of the digital computer in the mid- to late-20th century.REMARK 6. Geomorphometry evolved from a mix of mathematics, computerprocessing, civil and military engineering, and the Earth sciences especiallygeomorphology.The earliest geomorphometry was a minor sub-activity of exploration, naturalphilosophy, and physical geography especially geomorphology; today it is in-extricably linked with geoinformatics, various branches of engineering, and mostof the Earth and environmental sciences (Figure 1). In the following sections wewill briey describe the approaches and concepts of pre-DEM morphometry aswell as analytical methods applied to contemporary data. Additional backgroundis available in Gutersohn (1932), Neuenschwander (1944), Zakrzewska (1963), Ku-gler (1964), Hormann (1969), Zavoianu (1985), Krcho (2001), and Pike (1995, 2002).3.1 Hypsometry and planimetric formGeomorphometry began with the systematic measurement of elevation above sealevel, i.e. land surveying almost certainly in ancient Egypt.13Height measure-ment by cast shadows is ascribed to the Greek philosopher Thales of Miletus13Land surveying that focuses on measurement of terrain height is often referred to as hypsometry, from the Greeko height.Geomorphometry: A Brief Guide 13(ca. 624546 B.C.). The concept of the elevation contour to describe topographydates to 1584 when the Dutch surveyor Pieter Bruinz drew lines of equal depth inthe River Spaarne; but this was an unpublished manuscript (Imhof, 1982). In 1725Marsigli published a map of depth contours in the Golfe du Lion, i.e. the open sea.In 1737 (published in 1752) Buache mapped the depth of the Canal de la Manche(English Channel), and in 1791 Dupain-Triel published a rather crude contour mapof France (Robinson, 1982, pp. 87101/210215).In 1774, British mathematician Charles Hutton was asked to summarise theheight measurements made by Charles Mason,14an astronomer who wanted to es-timate the mass of Earth. Hutton used a pen to connect points of the same heighton the Scottish mountain Schiehallion, developing the isohypse (or isoline) con-cept. This has proved very effective in representing topography and is one of themost important innovations in the history of mapping by virtue of its convenience,exactness, and ease of perception (Robinson, 1982). DeLuc, Maskelyne, Roy, Wol-laston, and von Humboldt were among many early investigators who used thebarometer invented by Evangelista Torricelli (16081647) and developed by BlaisePascal (16231662) to measure elevation; see also Cajori (1929) and de Dainville(1970).With the spread of precise surveying in late 18th- and early 19th-century Eu-rope, illustrations ranking mountain-top elevations and the lengths of rivers beganto appear in atlases.15Mountain heights and groupings were studied qualita-tively, often by military engineers (von Sonklar, 1873), as orography, their heightsand derived parameters as orometry (Figure 6). Early 19th-century German geog-raphers such as von Humboldt (recently cited in Pike, 2002, and Rasemann etal., 2004) compared summit heights in different ranges. Von Sonklar (1873), andearlier regional monographs, went further and considered the elevations of sum-mits, ridges, passes and valleys as well as relative heights, gradients and volumes.Orometry with emphasis on mean slope, mean elevation and volume, planimet-ric form, relative relief, and drainage density became a favoured dissertationtopic for scores of European geographers (Neuenschwander, 1944). The overar-ching charter of geomorphometry was nicely captured many years ago by theGerman geographer Alfred Hettner (18591941), when he wrote in a brief consid-eration and critique of 19th-century orometry: But it is more important to enquirewhether we cannot express the entire character of a landscape numerically (Hettner,1928, p. 160; republished in 1972).Before the wider availability of contour maps in the mid-19th century,16mostquantitative analyses of topography were of broad-scale linear features: riversand coasts. The concavity of longitudinal river proles, adequately determinedfrom spot heights, came to be represented by exponential and parabolic equations(Chorley et al., 1964, 23). Carl Ritter (17791859) introduced indices of Kstenen-twicklungen (Coastal Development) to distinguish intricate coastlines such as fjordsfrom simpler ones such as long beaches. Some indices were more descriptive than14This is the same Charles Mason who, with Jeremiah Dixon, surveyed the MasonDixon Line in the USA between 1763and 1767.15Tufte (1990, p. 77) reproduces just such a detailed 1864 diagram from J.H. Colton.16Because early topographic maps represented relief by hachures, not contours, analysis of slope required detailed eldsurvey and thus was rare.14 R. J. Pike et al.FIGURE 6 Two landmarks of early geomorphometry from Germany and Austria, arguably thecradle of geomorphometry. The brief 19-page chapter on orometrie in von Sonklars 1873textbook (left) presented twelve quantitative measures of mountain morphology, whichstimulated much publication on land-surface characterisation. One of the best summarytreatments of early geomorphometry (including criticism of Sonklar!) was a much longer andwider-ranging chapter in Pencks 1894 textbook (right). Photos by R. Pike.others; the ratio of an islands area to the square of its perimeter, for example, com-bined coastal sinuosity with compactness, whereas the ratio of its area to area ofthe smallest circumscribed circle was only an inverse measure of elongation, notcircularity as claimed.The impossibility of agreeing on a denitive length for a section of coastlineeventually led to Richardsons (1961) establishment of a scaling relation betweenstep length (i.e. measurement resolution) and estimated line length, and later thefractal concepts (Mandelbrot, 1967, 1977) of self-similarity and non-Euclidean form.As Mandelbrots (1967) title implies, these widely applied scaling concepts werermly rooted in coastal geomorphometry.17Once contour maps were more available, relief analysis ourished. Measure-ment of highest and lowest points within a sample area (commonly a square orcircle) quantied the vertical dimension as relief (Reliefenergie in German), whichdeveloped from the need to express relative height (Gutersohn, 1932). Partsch17Much further evidence could have been found in Volkov (1950), not cited by Mandelbrot (see also Maling, 1989, pages277303, and pages 6683 citing the 1894 measurements of A. Penck on the Istrian coast).Geomorphometry: A Brief Guide 15(1911) used elevation range per 55 km square to produce what probably is therst quantitative map of local (relative) relief. Other denitions expressed relief fora hillslope (ridge crest to valley oor) or for a uvial drainage basin: catchmentor watershed relief (Sherman, 1932). Attempts to dene relief as the separa-tion between an upper relief envelope or summit surface and a valley or streamlinesurface (reviewed in Rasemann et al., 2004) were less successful because of scalevariations. Working for the U.S. Army, W.F. Wood (19141971) quantied the de-pendency of relief upon area by statistical analysis of 213 samples measured onU.S. contour maps (Wood and Snell, 1957).Geographers and later geomorphologists planimetered the areas enclosed bycontours to generate plots of elevation versus area. Estimates for the entire globeby Murray (1888) were rough but sufcient to establish the bimodality of Earthselevations, peaking near 0 and 4600 m, which posed numerous questions forgeologists and geophysicists. This hypsographic curve could be cumulated and inte-grated for comparative studies of regions (de Martonne, 1941). The histograms ofde Martonne (1941) are misleading, however, because he used two class intervalswith the same linear vertical scale.The dimensionless hypsometric integral, rst applied to landforms (cirques) byImamura (1937) and to regions by Pguy (1942), approaches zero where a fewhigh points rise above a plain, and 1.0 where most surface heights cluster near themaximum. Although this device is useful morphologically and in geomorphology,hydrologic and other applications often require retention of landform dimensions.Strahler (1952) popularised an integral of the hypsometric curve, which later wasproven identical to a simpler measure as well as the approximate reciprocal of ele-vation skewness18(Pike and Wilson, 1971). Pguy (1948) called further for a moreconventional statistical approach and proposed the standard deviation of eleva-tion as a measure of relief because of instability of the maximum. He asserted:Like all adult science, the geography of the second half of this century will be called tomake more and more continuous appeal to mathematical methods (Pguy, 1948, p. 5).Clarke (1966) critically reviewed hypsometry, clinometry and altimetric analy-sis, which had often been used in the search for old erosion (planation) surfacesover the prior 40 years. He showed that several types of clinographic curves, go-ing back to the earliest examples by Sebastian Finsterwalder and Carl Peucker in1890, can be misleading in their attempts to plot average slope gradient againstelevation.3.2 Drainage topology and slope frequencyIn 1859, Alfred Cayley published On contour lines and slope lines, which laid outthe mathematical foundation of geomorphometry.19In this extraordinary paper,the land surface is considered in the gravitational eld, and thus certain lines andpoints are more signicant than others. Cayley dened slope lines as being alwaysat right angles to contours. On a smooth, single-valued surface, all slope lines runfrom summits to pits (ultimately the ocean), except those joining summits (ridge18See further Figure 4 in Chapter 28.19He was preceded by even earlier French mathematicians and geometers (Pike, 2002).16 R. J. Pike et al.lines) and those joining pits (course lines). Passes are the lowest points on the former,and pales are the highest points on the latter. Each pass and pale is located at theintersection of a ridge line and a course line.James Clerk Maxwell (1870) further noted that each territory dened by thesespecial lines was part of both a hill whose lines of slope run down from the samesummit, and a dale whose slope lines run down to the same pit. Hills are boundedby course lines, and dales by ridge lines. These pioneering semantics remained ne-glected until their rediscovery by Warntz (1966, 1975) and Mark (1979). They havesince been again rediscovered by the engineering-metrology community (Scott,2004).Fluvial geomorphometry evolved from concepts of stream frequency (and itsreciprocal, drainage density) and stream order, notably in the pioneering work ofLudwig Neumann and Heinrich Gravelius (Neuenschwander, 1944). The quan-titative study of rivers and river networks initially was dominated by hydraulicengineers rather than geographers or geomorphologists, the work of Horton (1932,1945) on network topology and related geometric attributes of drainage basinsbeing especially inuential. His revolutionary 1945 synthesis of hydrology and ge-omorphology rapidly evolved into the sub-eld of drainage network analysis inthe 1950s and 1960s (Shreve, 1974), which grew to such an extent that elaborationof stream-order topology overshadowed geometric analysis of the land surface.Many geomorphological studies from the 1960s through the 1980s sought torelate hillslopes to streams (see later section) and in so doing exhaustively pa-rameterised the shape and relief of individual drainage basins (Zavoianu, 1985;Gardiner, 1990). The drainage basin is Earths dominant land-surface object andits analysis is, strictly speaking, a branch of specic geomorphometry. However,uvial networks occupy so high a fraction of Earths surface that the analysis ofdistributed drainage systems has come to dominate the more process-oriented im-plementations of general geomorphometry (Rodrguez-Iturbe and Rinaldo, 1997).Statistical analysis of large samples of slopes began with Strahlers (1950)work in southern California, leading to the Columbia School of quantitative anddynamic uvial geomorphology (Morisawa, 1985). Strahler measured maximumslope down a hillside prole (ow-line) and mean (overall) gradient, and relatedboth to the gradient and topological order of the stream below. Tricart and Muslin(1951) advocated measuring large samples of 100 to 200 slope gradients from crestto foot on maps, in degrees rather than percentage; histograms for a homogeneoussample area tended to be symmetric and conspicuously peaked. Adapting a tech-nique from structural geology, Chapman (1952) added a third dimension to slopeanalysis by treating planar surfaces as poles to the plane. He constructed radialplots of slope gradient against aspect (calculated from a gridded sample of points)to visually interpret asymmetry and lineation, an approach subsequently incorpo-rated in the MicroDEM package (Guth et al., 1987).The adoption of frequency distributions and statistical tests represented con-siderable progress and was promoted by Chorley (1957, 1966) for both drainagebasins and individual slope segments. Tricart (1965) critically reviewed slope anduvial morphometry, asserting that scale cannot be ignored if river proles andchannel incision are to be related to slope processes (Schumm, 1956). Yet despiteGeomorphometry: A Brief Guide 17such advances, the more dominant view among geologists and geographers in theearly- to mid-1950s remained: mathematical analyses of topographic maps. . . are te-dious, time-consuming, and do not always yield results commensurate with the amount oftime required for their preparation (Thornbury, 1954, p. 529).20Hormann (1969) brought a more distributed context to topographic analy-sis by devising a Triangulated Irregular Network (TIN), linking selected pointson divides, drainage lines and breaks in slope to interrelate height, slope gradi-ent, and aspect. Rather than individual data points, Hormann plotted averagesover intervals, but also was able to consider valley length, depth, gradient, anddirection. Criticised by one German colleague as excessively coarse and mech-anistic, Hormanns TIN model was successfully developed in North America(Peucker and Douglas, 1975). Its surface-specic vector structure, complemen-tary to the raster square-grid model, has since become a staple of both geo-morphometry and GIS packages (Jones et al., 1990; Weibel and Brandli, 1995;Tucker et al., 2001).Slopes had been proled in the eld (down lines of maximum gradient) in the19th century (Tylor, 1875), but early geomorphometricians calculated slope fromthe contour spacing on maps21(as illustrated in Figure 7). As geomorphologistsgrew dissatised with the inadequacies of contour maps, eld measurement ofgradients and proles became widespread in the 1950s. Slope proling developedespecially in Britain where many contours were interpolated yet photogrammetrywas regarded as inadequate by the ofcial mapping agency. Slope proles weresurveyed either in variable-length segments or with a xed 1.52 m frame (Young,1964, 1972; Pitty, 1969)22; still, a truly random sample of sinuous lines from a roughsurface proved elusive. One motive for plotting frequency distributions of slopegradient was to discover characteristic slope angles, and upper and lower limit-ing angles relevant to slope processes (Young, 1972, pp. 163167). Parsons (1988)reviewed further developments in slope proling and slope evolution.Local shape of the land surface is largely a function of curvature, or change ofslope, a second derivative of elevation (Minr and Evans, 2008). Its importance inboth prole and plan for hydrology and soils has long been recognised (Figure 7)and it forms the basis of a generic nine-fold (33) classication into elementaryforms that are convex, straight or concave in plan, and in prole (Richter, 1962).This appealing taxonomy is useful, but precisely what constitutes a straight (i.e.planar) slope must be dened operationally; e.g. Dikau (1989) used a 600 m radiusof curvature as the threshold of convexity and concavity (see further Figure 7 inChapter 9).The breaks and inections of slope that delimit elementary forms or