Streamflow generation for the Senegal River basin

Item Type Thesis-Reproduction (electronic); text

Authors N'Diaye, Abdoulaye.

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 27/05/2021 08:32:01

Link to Item http://hdl.handle.net/10150/191850

STREAMFLOW GENERATION FOR THE

SENEGAL RIVER BASIN

by

Abdoulaye N'Diaye

A Thesis Submitted to the Faculty of the

DEPARTMENT OF HYDROLOGY AND WATER RESOURCES

In Partial Fulfillment of the RequirementsFor the Degree of

MASTER OF SCIENCEWITH A MAJOR IN WATER RESOURCES ADMINISTRATION

In the Graduate College

THE UNIVERSITY OF ARIZONA

1985

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment ofrequirements for an advanced degree at The University of Arizona and isdeposited in the University Library to be made available to borrowersunder rules of the Library.

Brief quotations from this thesis are allowable without specialpermission, provided that accurate acknowledgment of source is made.Requests for permission for extended quotation from or reproduction ofthis manuscript in whole or in part may be granted by the head of themajor department or the Dean of the Graduate College when in his orher judgment the proposed use of the material 's the interests ofscholarship. In all other instances, howeverk ssion must beobtained from the author.

.110031Nis.rains

SIGNED:

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

Nathan Buras(,)/471,d! /9' Kr

DateProfessor of Hydrology

DEDICATION

TO my late:

- brothers, Habib N I DIAYE andEl Hadji Amadou N I DIAYE;

- grand-father, El Hadji Omar N I DIAYE; and

- aunt Wdeye Fama LO;

who died while I was away and working on this degree.

ACKNOWLEDGMENTS

The author would like to express his gratitude to his advisor

and thesis director, Dr. Nathan Buras, for his continuous guidance,

moral support and constructive criticisms during the author's graduate

work and research; gratitude is also expressed to the other members of

the graduate committee, Dr. Thomas Maddock III, and Dr. Donald Davis,

for their moral and academic support.

Acknowledgments also go to Mr. Glen Slocum of the Senegal desk

(USAID, Washington, D.C.), to the staff of the Central Library

(Washington, D.C.), and to my friend Mamadou L. Thiam for providing most

of the necessary documentation.

The author is grateful to the African-American Institute for

their support for the work on this degree.

Special thanks to all my parents, friends and colleagues in

Senegal, particularly to my mother Fatou Gackou and to my father Sidy

N'Diaye.

Special thanks also go to various people in the United States,

particularly to my wife Margaret N'Diaye for her support each and every

day.

i v

TABLE OF CONTENTS

Page

LIST OF TABLES vii

LIST OF ILLUSTRATIONS x

ABSTRACT xii

1. INTRODUCTION 1

1.1 Subject Definition and Delineation 11.2 Physical Characteristics of the Senegal

River Basin 51.2.1. General Description 51.2.2. Climatic Conditions 111.2.3. Hydrologic Conditions 24

1.3 The OMVS and its Development Plan 371.3.1. The OMVS 371.3.2. The Development Plan 43

1.4 Summery 52

2. LITERATURE REVIEW AND MODEL SELECTION 55

2.1 Literature Review 552.1.1. Definitions 572.1.2. Model Review and Pre-Selection 60

2.2. Multivariate Lag-One Markov Model 642.2.1. Univariate Case 642.2.2. Multivariate Generating Processes 682.2.3. Limitations of the Markov Lag-One Model 74

2.3 FFGN Model 762.3.1. Theoretical Background 76

2.3.2. The Method of Construction of FFGN 822.4 Disaggregation Models 87

2.4.1. General Disaggregation Model 89

2.4.2. Disaggregation Models 90

2.4.3. Parameter Estimation 942.5 Summary and Conclusions 98

2.5.1. Selection 992.5.2. Model Definition 100

V

vi

TABLE OF CONTENTS--Continued

Page

3. MODEL APPLICATION 103

3.1 Data Analysis 1033.1.1. Normality Check 1053.1.2. Filling Missing Values 137

3.2 Testing of Model 1 1523.2.1. Methodology 1533.2.2. Results and Discussion 155

3.3 Summary and Conclusions 159

4. CONCLUSIONS AND RECOMMENDATIONS 161

APPENDIX A: RAW DATA MONTHLY FLOWS COLLECTED 165

APPENDIX B: THREE VERSIONS OF PROGRAMD DATA 1FOR MONTHLY, SEASONAL AND ANNUALFLOWS USED FOR THE NORMALITY CHECK 182

APPENDIX Bi: Program Data 1 for Monthly Flows 183APPENDIX B2: Program Data 1 for Seasonal Flows 188APPENDIX B3: Program Data 1 for Annual Flows 194

APPENDIX C: PROGRAMS FOR THE REGRESSION ANALYSIS 200

APPENDIX Cl: Program Data 2 201APPENDIX C2: Example Set-Up of Regression 204

APPENDIX D: PROGRAM DATA 3 206

APPENDIX E: HISTORIC DATA (MONTHLY AND ANNUALFLOWS 209

APPENDIX F: PROGRAM MMLO FOR THE GENERATION OFSTREAMFLOWS 215

REFERENCES 224

LIST OF TABLES

Table Page

1.1 Drainage Areas and Average Flows in theSenegal River Basin 10

1.2 Average and Extreme Annual Rainfall for SelectedStations in the Senegal River Basin 17

1.3 Some Extremes of Rainfall (mm) in West Africa 20

1.4 Monthly Average Lake Evaporation (mm) for SelectedStations in the Senegal River Basin 22

1.5 Mean Monthly Temperatures ( °C) at Selected Stationsin the Senegal River Basin 23

1.6 Annual Average and Extreme Flows for SelectedStations in the Senegal River Basin 29

1.7 Annual Peak Flows at Selected Stations forSelected Return Periods in the SenegalRiver Basin 33

1.8 Groundwater Resources of the Senegal RiverBasin 34

1.9 Diama Impoundment Characteristics at Water Levelsof 1.5 meters and 2.5 meters 45

1.10 Manantali Impoundment Characteristics 48

3.1 Basic Statistics of Monthly Flows at Galougo 110

3.2 Basic Statistics of Monthly Flows at Bakel 111

3.3 Basic Statistics of Monthly Flows at Kayes 112

3.4 Basic Statistics of Monthly Flows at Kidira 113

3.5 Basic Statistics of Seasonal Flows at all Sites 114

3.6 Basic Statistics of the Annual Flows at all Sites 115

vii

viii

LIST OF TABLES--Continued

Table Page

3.7 Statistics of the Chi-square and the Kolmogorov-Smirnov Tests for the Normality of theMonthly Flow Residuals at Galougo 125

3.8 Statistics of the Chi-square and the Kolmogorov-Smirnov Tests for the Normality of theMonthly Flow Residuals at Bakel 126

3.9 Statistics of the Chi-square and the Kolmogorov-Smirnov Tests for the Normality of theMonthly Flow Residuals at Kayes 128

3.10 Statistics of the Chi-square and the Kolmogorov-Smirnov Tests for the Normality of theMonthly Flow Residuals at Kidira 129

3.11 Statistics of the Chi-square and the Kolmogorov-Smirnov Tests for the Normality of theSeasonal Flow Residuals at all Sites 130

3.12 Statistics for the Chi-Square and the Kolmogorov-Smirnov Tests for the Normality of theAnnual Flow Residuals at All Sites

133

3.13 Summary Table of the Multiple Regression withDependent Variable GFLOW (flows atGalougo) 140

3.14 Summary Table of the Multiple Regression withDependent Variable KAFLOW (flows atKayes) 141

3.15 Summary Table of the Multiple Regression withDependent Variable KIFLOW (flows atKidira) 143

3.16 Summary Table of the Simple Regression withDependent Variable GFLOW (flows atGalougo)

146

3.17 Summary Table of the Simple Regression withDependent Variable KAFLOW (flows atKayes)

147

ix

LIST OF TABLES--Continued

Table Page

3.18 Summary Table of the Simple Regression withDependent Variable KIFLOW (flows atKidira) 148

3.19 Confidence Intervals on the Intercepts (A) andthe Regression Coefficients (B) for Kidiraand Kayes 150

3.20 Basic Statistics of the Historic and SyntheticFlows 156

3.21 Lag Zero Cross-Correlations of the Historicand Synthetic Flows 158

LIST OF ILLUSTRATIONS

Figure Page

1.1 Structure of Simulation Study, Indicating theTransformation of a Synthetic StreamflowSequence, Future Demands, and a SystemDesign and Operating Policy IntoSystem Performance 4

1.2 Senegal River Basin 6

1.3 Basin States of the Senegal River 7

1.4 Movement of the Inter-Tropical ConvergenceZone 13

1.5 Isohyetal Map of the Senegal River Basin(Rainfall in mm) 16

1.6 Monthly Rainfall Distribution for SelectedStations in the Senegal River Basin 19

1.7 Schematic Cross-Section of the Senetal RiverFlood Plain in the Middle Valley 26

1.8 Stage-Capacity and Stage-Area Curves of theLac de Guiers 28

1.9 Hydrographs for1958/59 Flood at SelectedStations of the Senegal River 31

1.10 Stage-Area Curve of the Diama Dam 46

1.11 Stage-Area and Stage-Capacity Curves of theManantali Dam 49

3.1 Periods of Measurements and Missing Values forMean Daily Flows (and Monthly Flows) at Bakel(1), Galougo (2), Kayes (3), and Kidira (4) 104

3.2 Annual Flows at Gal ougo 116

3.3 Log Transformed Annual Flows at Galougo 117

X

xi

LIST OF ILLUSTRATIONS--Continued

Figure Page

3.4 Annual Flows at Bakel 118

3.5 Log Transformed Annual Flows at Bakel 119

3.6 Annual Flows at Kayes 120

3.7 Log Transformed Annual Flows at Kayes 121

3.8 Annual Flows at Kidira 122

3.9 Log Transformed Annual Flows at Kidira 123

ABSTRACT

The Senegal River Basin is located in the Sahel, a drought-

stricken region of West Africa. Water scarcity in this basin and the

important constraint it represents for economic growth and the estab-

lishment of a desirable quality of life for the populations has been a

reality before the drought conditions arose in the region.

Aware of this situation and of the potential benefits of manag-

ing the resources of the basin, three of the four basin-states (Mali,

Mauritania, and Senegal) decided to develop the basin as a whole. The

integrated development plan includes the construction of an infrastruc-

ture to serve three purposes (irrigation, navigation, and hydropower)

for multiple objectives.

The optimal allocation of water resources in such complex con-

ditions for a fair and equitable use necessitates careful and detailed

studies to provide input to the various decision-making processes.

Among those inputs are those highly uncertain, like streamflows. To

handle this type of uncertainty, various water quality and quantity

simulation and optimization techniques use synthetic streamflows.

In this study, two models for the generation of streamflows in

the Senegal River and its tributaries are selected, adapted, and par-

tially tested for use in future studies and recommendations made for

future investigations.

xii

CHAPTER 1

INTRODUCTION

The great variability in time and space of rainfall in the

Senegal River Basin, and the lack of highly significant control on the

river and its waters have hindered economic growth and made the estab-

lishment of a desirable quality of life for the inhabitants of the basin

out of reach. Conscious of this situation since the first years of

their independence in the 60's, three of the four basin-states of the

Senegal River (Senegal, Mali, and Mauritania) decided to initiate and

sustain a common integrated development of the basin as a whole and

created on March 11, 1972 an intergovernmental agency, "L'Organisation

pour la Mise en Valeur du fleuve Senegal" (OMVS). This agency is

responsible for the conception, coordination and implementation of

projects for optimum exploitation of the resources of the Senegal River

Basin to alleviate the drought conditions in this part of the Sahel and

to further economic growth and social welfare.

1.1 Subject Definition and Delineation

The Senegal River Basin, referred to for the rest of this study

as the SRB, is, with respect to its development plan and the conditions

under which this plan is being carried, an unique experience. First,

nearly 80 percent of the basin is located on the western part of the

Sahel which owes its name and worldwide renown to the recurrent

1

2

droughts that have been striking this part of Africa for over a decade.

Furthermore, all three countries carrying the development of the basin

have very limited financial and natural resources and belong to the so-

called third world. In addition, the Senegal River waters have been

declared an international waterway in accordance with the Helsinki

rules since 1963 by all four basin-states including Guinea which is not

taking part in the development of the SRB although two of the three

main tributaries of the Senegal River originate in that country.

Another aspect of the SRB development plan is, as shown by the data

presented in the next sections, the relative importance of the surface

waters over the groundwaters in the basin. Finally the development

proposed includes the construction of two dams (Diama and Manantali)

for multiple purposes (irrigation, hydropower, navigation and salt

intrusion control), and for multiple objectives.

The undertaking of such an outstanding plan as the one proposed

by the OMVS cuts across multiple disciplines such as hydrology (en-

gineering), economics, public administration and water law. It is

therefore a complete example in the field of water resources (adminis-

tration) suggesting the use of water resource systems analysis to

attest the performance of the project undertaken. To guarantee the

success of the development envisioned it is necessary to dispose of a

water use plan for an equitable and efficient use of the resources of

the SRB among the different purposes and the different states. In its

final report, the consulting office Gannett, Fleming, Corddry and

Carpenter, Inc. (abbreviated GFC&C in this report) who conducted in 1978

3

a study to assess the environmental effects of the proposed develop-

ments in the SRB recommended strongly the development of an integrated

land and water use plan. To devise and implement such a plan,

simulation studies will have to be carried out.

As shown by Figure 1.1, simulation models of river basin system

prerequires the availability of a synthetic streamflow generating

model. Synthetic streamflow generation, also called operational

hydrology (Loucks, 1981), provides streamflow input for both water

quantity and quality models used in studies of various purposes ranging

from planning and design to implementation and operation. The present

study is primarily aimed at examining various procedures for generating

streamflows to recommend one or more multivariate generators for both

annual and monthly flows that can be used in future studies relating to

the SRB development plan.

To accomplish this objective, three categories of synthetic

streamflow generating models are reviewed in detail to select two

models that will be tested with streamflow data of the SRB (see

Chapters 2 and 3).

This report is organized as follows. The rest of this chapter

gives background information on the SRB and its river with respect to

the location, the climate, and the hydrology of the basin. A descrip-

tion of the proposed development plan and an analysis of the institu-

tion, the OMVS, responsible for its design and implementation is also

given in this chapter. The second chapter gives details on the approach

(method of analysis), and the theoretical and technical bases of the

4

Synthetic stream-flow and other hydrologic sequences

Simulationmodel of

river basin system

System-----4> Performance

t

tSystem design andoperating policy

Future demandsand economic data

Fig.1.1. Structure of Simulation Study, Indicating the Transformationof a Synthetic Streamflow Sequence, Future Demands, and aSystem Design and Operating Policy Into System Performance.

Source: Loucks (1981)

5

study. Chapter 3 comprises the analysis of the streamflow data used in

the study and the application of the model or models selected.

Finally, the last chapter is a summary of findings and recommendations.

1.2 Physical Characteristics of the Senegal River Basin

This section gives the general description (location and compo-

sition), and the atmospheric (rainfall, winds, evaporation) and

hydrologic (surface waters, groundwater, geology) conditions of the

basin and its river.

1.2.1. General Description

The Senegal River Basin. The SRB is located in west Africa (see

Figures 1.2 and 1.3) between the latitudes 10 030' and 17 °30' North and

the longitudes 7 °00' and 16 °30' West. The total area of the basin is

290,000 km 2 divided between four riparian countries: Guinea,

31,000 km2 ; Mali, 155,000 km 2 ; Mauritania, 76,000 km 2 ; and Senegal,

28,000 km 2 (Riley et al., 1978). The basin has traditionally been

divided into three geomorphological entities (GFC&C, 1978):

- the Upper Valley upstream of Bakel

- the Middle Valley between Bakel and Dagana

- the Delta between Dagana and Saint-Louis.

The topography varies markedly within the basin. The southern-

most part of the basin is the most mountainous. It is limited in

Guinea by the Fouta Djalon massif which towers at 1425 meters, and

in Mali by the plateau Mandingue which is located in the region

west of Bamako. This portion of the SRB has altitudes ranging

TE SA SS h

r•Dhi T PE•E

GDLAD•0.•aff.

:OuRS D CA.

API T ALES

• n LLE S •

di .•••

5 ' Ila•RAGES s TES

n-

•

-a

•

BASSIN DU FLEUVE SENEGALSENEGAL RIVER BASIN

Fig. 1.2. Senegal River Basin

Source: GFC&C (1978)

6

Fig. 1.3. Basin States of the Senegal River

Source: Riley et al. (1978)

7

8

between 1000 m and 800 m. The rest of the Upper Valley has

elevations varying between 800 m and 100 m down to Kayes on one

hand, and in the interval 100-0 m between Kayes and Bakel on the

other hand. The latter variation in elevation (0-100 m) is the same

for the Middle Valley and the Delta which are mostly flat

(Rochette, 1974).

The geology in the basin comprises sedimentary and metamorphic

rocks (Rochette, 1974). The dominant formations in the Upper Valley

are metamorphic rocks:

- the "Dolentes"

- the "Granite et Granodiorite postectoniques, birrimiens"

- the "Birrimien" (facies schistose)

and sedimentary rocks:

- the "Infracambrien" (sandstone and sandstone quartzite)

- the "Cambrien inferieur et Cambrien indifferencie (tillite,

limestone, bloodstone, sandstone).

In the Middle Valley the dominant formations are:

- the "Eocene moyen" (Middle Eocene) (limestone, dolomite, clay,

sandstone)

- the "Continental terminal" (sand, clay)

- the "Quarternaire superieur marin ou fluvial" (sand, gravel)

for the sedimentary rocks and the u serie d'Akjoujt et de Baker (schist

and quartzite) for the metamorphic rocks.

Finally, the Delta region is dominated by the "Quarternaire

superieur marin ou fluvial" and sand dunes.

9

The Senegal River Course. The Senegal River is the second

largest in west Africa, the Niger being the first. The river and its

tributaries run a total distance of 1800 km to drain the 290,000 km2 of

its basin. The actual Senegal River is formed by the junction at

Bafoulabe in Mali (1,060 km from the Atlantic Ocean) of two tributar-

ies: the Bafing, "black river" in Mandingo (local dialect of a good

proportion of the basin population), and the Bakoye, "white river" in

the same dialect. These two tributaries and a third one, the Faleme,

are accountable for practically all the flow of the Senegal River which

ends in the Atlantic Ocean at Saint-Louis in Senegal.

The Bafing is indeed the main tributary of the Senegal River

for the flows it contributes. It originates in the Fouta Djalon

mountains at an elevation of approximately 800 m, about 15 km away

from the town of Mamou in Guinea (Rochette, 1974). These mountains are

called, with due credit, the reservoir of western Africa, for most

rivers and streams of this part of the continent, including the Niger,

originate there. Because of the relatively high precipitation in Guinea,

the Bafing that drains 38,000 km 2 or 13 percent of the SRB contributes

380 m3 /s or about 50 percent of the annual average flow of the Senegal

River at Bakel (see Table 1.1).

The Bakoye originates at an elevation of 760 m in the mountains

of Menien (11 050' n, 9°40' W) northeast of Siguiri in Mali (Rochette,

1974). It drains 85,000 km2

or 29 percent of the total of the SRB.

The main tributary of the Bakoye, the Baoulé, flows out of the moun-

tainous region southeast of Bamako in Mali (Rochette, 1974). The

u)I

O NCO r- co in N

r T-••

W I• al 01• (0 IW OI (r) 01 0 ill 0O E--1-4I (NI

-

rsI-I (01 r-(I) twI 0 ail

I

I 000 0 0I 000 0 0

Ell • 0 0 0 0

1 OD &II ON CO 0I rn co r`i 1-- Chi (NI (N

CD 14

3-)0(/)

(0or

4- rn• N r`t

44 41 • (-I(0 (0 0

0(-1 I-4 p.

(1) (014/ >-1 8 CT rl 4140W I .4-1 0 (P (111>1 '44 Ad c c

(0 (0 (1.1 tti (1) 10MI CC) al AL., tn

10

11

junction Bakoye-Baoulé occurs approximately 120 km before the junction

Bafing-Bakoye. The Bakoye-Baoulé system supplies 23 percent (170 m 3 /s)

of the average annual streamflow of the Senegal River at Bakel (see

Table 1.1).

The Faleme joins the Senegal 50 km upstream of Bakel. It

originates at an altitude of 800 m in the doleritic plateau of the

Bowal Seguere Fougou (11 °52' N, 10 052' W) in Guinea near the common

border of Senegal-Mali-Guinea. The Faleme has a watershed of

29,000 km 2, or 10 percent of the total, and contributes 25 percent

(187 m3/s) of the average annual streamflow of the Senegal River at

Bakel (see Table 1.1).

All along its course from Bafoulabe to Saint-Louis, the Senegal

River receives other minor tributaries (see Figure 1.2) such as:

- the Kolombine in Mali, entering near Kayes

- The Karakoro in Mali, entering 23 km upstream of the confluence

of the Faleme and the Senegal

- the Gorgol in Mauritania, entering near Kaedi

These are intermittent streams which supply water mostly during the

flood season. Their total contributions are not sufficient to

compensate losses by evaporation and infiltration in the Senegal River

(GFC&C, 1978).

1.2.2. Climatic Conditions

In the SRB, rainfall and evaporation are the most important

aspects of the climate from a water resources point of view. However,

to understand the climate in the SRB, it is essential to place it in its

12

correct perspective and context. Therefore, we will first discuss the

reasons for the climate of the SRB. Then, we will discuss the charac-

teristics of the rainfall, the evaporation, and other atmospheric

factors (temperature and radiation, airflow, and humidity).

The SRB is located within the part of the tropics affected by

the Intertropical Convergence Zone (ITCZ). In this region there is an

interplay between the wind of the northern and southern hemispheres.

This explains why the climate in the SRB is a result of the conditions

in both hemispheres. The ITCZ migrates with the sun's apparent

movement with a lag of approximately one month. The average range of

movement is shown in Figure 1.4.

The ITCZ, also called the intertropical front (FIT), is respon-

sible for precipitation in the SRB as in the whole of west Africa

(Sircoulon, 1976) and in most of the tropics (GFC&C, 1978).

The two air masses involved in the interplay are (Sircoulon,

1976):

- the continental tropical air, hot and dry, called "harmattan" and

coming from north or northeast of the Sahara desert

- the marine tropical air, unstable, moist and relatively cool

resulting from the anticyclone of Saint Helens. This air mass

coming from the southwest is called "monsoon."

When an area, such as the SRB, is more under the influence of the

harmattan, meaning a southward movement of the ITCZ, the possibility of

rain is precluded. Conversely, if the monsoon is dominant or, in other

words, when the ITCZ is in its northern position there is a good chance

Bassin du fleuve SenegalSenegal river basin

Fig. 1.4. Movement of the Inter-Tropical Convergence Zone

Source: GFC&C (1978)

13

14

of rain. In conclusion, the relative movement of the ITCZ from south

to north and north to south in the range shown in Figure 1.4 corres-

ponds to the rainy season (May through October) and the dry season

(November through April) in the basin with some variation.

To complete this discussion on what causes the climate in the

SRB, other aspects are to be mentioned. The explanations above

concerning the rain are to be understood with a probabilistic approach.

Indeed, only the presence of the ITCZ does not guarantee rain; it just

increases the probability of the event. The reasons for this uncer-

tainty relate to other factors, such as influences of both upper-air

and local conditions and features such as topography and bodies of

water. Unfortunately, little is known at present on the upper-air and

local conditions in the SRB (GFC&C, 1978).

The relatively high elevations in the southern part of the

basin, particularly in the Fouta Djalon mountains, combined with the

longest presence of the ITCZ during the year, explain why this part of

the basin is the wettest.

Concerning the bodies of water, due to the lack of pertinent

data, it can only be thought, based on other experiences (Lake Victoria

and Great Lakes of Canada and USA), that they influence the local

climate and weather by acting upon the microscale weather (GFC&C,

1978).

Rainfall. Precipitation (rainfall) is the source of supply for

the river flows and the climatic element that shows the most variation

in the SRB (GFC&C, 1978). Streamflow and precipitation are closely

15

related in the basin (but not perfectly), because of antecedent soil

moisture, base flow, and rainfall intensity and distribution.

The pronounced variation in the amount and distribution of

rainfall in the basin is generally ascribed to the location and

intensity of the ITCZ. Figure 1.5 shows precipitation isohyets varying

from 2000 mm in the south to 250 mm in the north. Three pluvio-

metrical regimes are distinguished by Rochette (1974):

- the Guinean regime between isohyets 2000 and 1500

- the Sudanean regime between isohyets 1500 and 750

- the Sahelian regime between isohyets 750 and 250

For more illustrations, inter-annual variations in rainfall (average,

minimum and maximum annual values) are presented in Table 1.2 for 12

selected stations distributed in and around the SRB. Some of these

stations are also identified in Figure 1.5. Table 1.2 shows deviations

of annual extremes from the annual mean ranging from 34% (lowest at

Matam) to 326% (highest at Saint-Louis).

There are two facts that hold true at most places most of the

time. First, inter-annual variations are correlated with the magnitudes

of annual averages. Table 1.2 shows, with the exception of Kenieba,

Kayes and Nioro, that areas with lower annual average rainfall exhibit

larger inter-annual variations in rainfall. Second, for the upper and

middle valley,the annual rainfall has a Pearson type III distribution

(Pochette, 1974). GFC&C (1978) also found an asymmetrical distribution

for annual totals in the northern and central parts of the basin. They

found, for instance, that at Saint-Louis (Senegal) the mean annual

16

r— =

EC S—CU • n—•S— 0-I-) V) V)XLU 0

•r--C)

ea 4-.)V)

CUCs)(0

17

= <

Le. 0, tr,r•-4 .7 r•••

• r"C

- rs4 rr".

0 COO C CC CC CCr•I C0 40 •••• <7 •••••

• N. (NI •n•

n0 ON CS CT %0 CC .4',,C) en en rn rn

C C CD 0 COCO ON :0 nnnn

eNi r'••• CO ,O •••.? rs• 4' CCen CO f's nJD en en.1n111.

00000 0000000rr) CO eV Cr, 0 C aO

‘10 I" en eV fr) fr).n

▪

1

•

onI nNI ••n

0 VD 4, •0 .7 4: 1," in an ir% u-%N. fr. ,„0 ,g)

ON CI, ON ON 0, ON ON ON ON Cr, ON ON ON

I I I I II I I I I I I ItN n•n1 ) C CN .n•• ev

ev

-

•

C ev (..1 CV CV CV CV en en cr)on CM CT C 0 a' CT CT CT. 0% ON ON ON

.nI ...I ••=1.5 • en en ea0.1 • 0. 00 • .0 -.0 00 00C •I.s •=s -mt • ami • =I I.1 0.1 I. 6 4,1 OJ

• aI In1 snI 4n11 .1. ...1 0 O 0 0 0 0O 41 0 IC IC ‘14 CO CV TO re 4: CJL; r M Z z z z an z z CA ri)

co...4

O 0.0 ... 0

a.) co 0 E 0 0 .-7cu •..1 ce co a) b.. la co ... ch0 0

.0 C .... E >, 0 6. a=1 .... 0.) (i)CS ad • ..I CS er) ...o tO re • •• ... O ......1 h4 bIL cc ac Z Z z .4C cc

18

rainfall (365 mm) is exceeded only about 40 percent of the time, while

nearly 60 percent of the events yield below the average. This pattern

of higher probability of occurrence for below-average rainfall can be

generalized to the whole basin with the exception of the Fonta Djalon

mountains (GFC&C, 1978). The implication of these two facts is that

areas with marginal rainfall (in terms of agriculture and livestock)

will experience many dry years. It is therefore obvious that one must

plan and manage the SRB waters using a drought-conditions approach

(lean years exceeding fat years) (GFC&C, 1978).

Rainfall in the SRB is also highly seasonal. Monthly rainfall

patterns for 12 selected stations are presented in Figure 1.6. The

rainy season extends from July to September in the northern part of

the basin (Saint-Louis, Rosso, Aleg, Kiffa, Nara, Matam), from June

through October in the central part of the basin (Kayes, Nioro), from

May to October in the southern part of the basin (Bamako, Kita,

Kenieba), and throughout the year in the Fouta Djalon (Labe). Table 1.3

shows monthly variations for the wettest month (August) for several

stations.

Rainfall is generally of short duration and high intensity in

the northern part of the basin, and longer and more frequent in the

southern part (GFC&C, 1978).

Evaporation. Lake evaporation in the SRB has traditionally been

measured by Piche Evaporimeter. Readings using this method have been

found not reliable because the measuring instrument is located in a

shelter and therefore not influenced by air flows and solar radiation.

LABE (1923-1970) KENIERA (1942-1976) 1(11A (1931-1976) BAMAKO (1922-1976) 450

400 -

350

300

250

200

•-• 150

WO

1 50

0

R0550(1934-1965)VILLE (1902-1965)ST-LOUIS ALES (1930-1965) 'UEFA ( 1 922 -1965 )

JFMAMJ JASON() JEMAMJJASOND JFMAMJJASO ND J F MAMJ J A SONO

1

rr 4._ „rn1,__J EMAM J J A SONO JEM A MJ J A SONO J ENAMJ J A SONO JF MAMJJ AS 0 ND

NIORO (1922-1976)250

200

150

100

BO

0JEMAIAJ J A SONO JFMAMJJASONO

( 1696-1914KATES \\ 1920-1916jNARA (1921-1965) MATAM(1922-1975)

250

MO

190

*0

50

0

450

400

350

300

250EEEE 200

wz 150J 0-/ 100

Dn 50Z (1-w a

wztai

0

>-0 0 2502 <I

Z 200

>1= .1

150<I )-1- _1 100

50

cc 0a 2

0

250

200

150

100

50

0

19

Fig. 1.6. Monthly Rainfall Distribution for SelectedStations in the Senegal River Basin

Source: GFC&C (1978)

Table 1.3

Some Extremes of Rainfall (mm) in West Africa

Source: GFCE (1978)

LocationAnnual August

Yearsof

RecordHigh Low High Low

Mauritania

Aleg 544 89 280 16 36

Kaedi 762 205 364 23 50

Rosso 612 106 498 7 32

Selibabi 1099 350 415 84 33

Senegal

Bake]. 751 235 362 58 47

Matam 1112 255 473 69 47

Podor 793 98 310 12 60

Saint-Louis 1239 144 769 0 56

Mali

Falea 2167 919 622 249 20

Kayes 1136 361 526 54 57

Kenieba 1913 986 931 172 23

Nioro 965 398 463 74 35

Yelimane 975 416 319 90 32

Guinea

Dalaba 3161 1639 759 365 23

Mamou 2801 1207 668 250 35

Pita 2403 1439 737 201 32

20

21

According to GFC&C (1978), Senegal-Consult (1970) has compared Piche

observations with other readings from the Colorado Sunken Pan and cal-

culations using Penman's equation, and proposed a 0.8 Pan coefficient to

correct Piche readings.

Free water surface evaporation data (lake evaporation) for

several stations in the SRB is presented in Table 1.4. The values in

this table are Piche readings published by Rochette (1974). As one can

expect, evaporation increases in the SRB from the wet south (Kienieba,

Labe) to the dry north (Matam, Kayes, Rosso). Table 1.4 also shows

that the monthly variation of evaporation follows the rainfall pattern.

Radiation and Temperature. In the SRB as in most of the

tropics there is a relatively small variation in the amount of radiation

received at the troposphere (outer limit of earth's atmosphere). The

ratio between the highest and the lowest values is 1.4 with the

greatest deviation from the average of 17 percent (GFC&C, 1978). As

shown in Table 1.5, mean monthly temperature in the SRB exhibits very

little variation during the year. GFC&C (1978) explain this observation

by the fact that "the radiation received at the earth's surface in the

Senegal River Basin is determined primarily by the seasonal pattern of

cloud cover in the area."

Airflows. As seen earlier, during low-sun periods the ITCZ

moves south, exposing the SRB to the "harmattan." Conversely, the

basin is exposed to the "monsoon" with the ITCZ moving north during the

high-sun season. During the latter season, cloud cover and thickness

increase (GFC&C, 1974). Various intermediate positions exist between

AMe,111M/

-C)C)4-)

C)

is

"5

t" N. CN Or' CV 1.1^cr IN rn I"

en t

▪

r, -.1" e.r, ao enese .... •n1 • NI

IN. .....1 CO a% NI .....70 .7 tr., CC .. u",c•.) .....• mm• 4n1 t'N tNI

C0 4' ... CZ ... CCr... CT C CC C 0.-. -. -. NI NI

I.rl N. •.0 nT nIl 4.0

0 I/1 ,..1:II nn CO s.1"nI OWN IInIl •IMI

e."1n.11

• i

•

i-'• In11 11nI a=m1

e•4 CCO Le)

ItO c le‘

,t)• eV

el IN e••41 CO .1' nCnen IN e•NI en rn crnen .... ..... en en -..

CO CN en 1/1 CNI ,o0eV eV 0% •n• en enen C•4 •nI en en CNI

• %ID NCI en ••••1'▪ m m o en co

e-,1 cv en en CSI

C" 0 nI in en Len...? dmo CSI CSI lin N.CSI eV C.1 e•NI CNI CNI

cD en1'61

,nnnn

CD eV0

▪

n11 ..„1cs.

•$.1 0 CI) COCV c.)

a. 0 >•

CB...1

CC 7.0 0

CC CU E 0 -30.1 • vol el.) co u)>, c 411 4-, coto o., to to D ,...

o4 oc ,— z ce G')

22

Lc") cts• Cr)

- C.) 0.)S-

C.)- 4-) (• r)

fu)

- C.)Cd

(7)

23

N - CD 0 CC

▪

02

()4 IA Lt1 %it IA t.0 CC0,) N NNNNNNNN

> % CIN CD 01 NC) N N NiN N N

N 0% 0 01 01 N Nt.) NN eINNNNNN0

C. 0% C CD CO N N N t.12N m N NNNNNN

tr2

en C0 0 0% cs%Inn N N N N N N N

CC 0 0 0 h N e n.0• N M m N N N N

C .-t tCP N "1 0 CON en en N N N

> m Crn tf) N 1.11m NI IN en 1.1 N m

• N en C2 ter IN In ff1C..N N m m m N I fot

1.0 n••) in as co I n—tIQ N N N N en N tn

12 en "1 t.0 1.0 co co rn co cr%NNNNNNNNN

C N r rn •ttr Art Ui N GO 11-Irr, IN N N N N N N N N"2

014 RI

to) .0O g 7 In 0)U) 1Z IT 0 ... 'VCC 4.) C 2.. C .0O r: •,* to c) fa

z .4 x .1

24

the two mentioned extremes (Sircoulon, 1976). These modifications

result from local air circulations which are important only in the

Fouta Djalon and the delta regions. In the Fouta Djalon, wind direction

will depend a great deal on valley configuration and orientation, while

in the delta, normal land-sea breeze (onshore during the day and

offshore at night) will dominate on most days (GFC&C, 1978).

Humidity. Absolute humidity depends on the airflows. It is

high with the marine tropical air and low with the continental tropical

air. There is, however, an appreciable amount of moisture independent

of the movement of the air masses (GFC&C, 1978).

The relative humidity (absolute humidity over saturation level)

is high at night and low in the afternoon. The relative humidity, which

averages 20% most of the day during the dry season, affects consider-

ably the evaporation rates (GFC&C, 1978).

1.2.3. Hydrologic Conditions

This section of the general description was compiled from the

report by GFC&C (1978), and from the ORSTOM monograph by Rochette

(1974). The hydrologic conditions referred to herein include both

surface and ground waters. A description of the streams and lakes,

followed by a description of the streamflows, will be given for the

surface waters. Then, a description of the aquifers in the SRB will

end this part of the report.

Streams and Lakes. As mentioned before, three geomorphological

areas exist in the SRB: the Upper Valley, the Middle Valley and the

Delta region.

25

Deeply entrenched valleys and steep slopes combined with rapids

characterizes the water courses in the Upper Valley. The average

slopes are: 0.95% for the Bafing, 1.15% for the Bakoye, 1.24% for the

Faleme, and 0.30% for the actual Senegal River between Bafoulabe and

Bakel.

In the Middle Valley, starting at Bakel, the river valley

widens and the slope of the river bed decreases to 0.03% between Bakel

and Podor and down to 0.01% between Podor and Dagana. The Middle

Valley is a wide alluvial plain in a semi-desertic environment.

Numerous sills and bars are found here as in the Upper Valley. Because

of the small slope, the river meanders and forms an intricate system

of umarigots" (backwaters) and depressions in the floodplain. Figure

1.7 shows the major elements of the floodplain in the Middle Valley.

During the dry season the flow is confined in the river bed and gets to

the Walo zone at higher flows. The Fonde area (high banks) is

generally not inundated even during extreme floods. Finally the "dieri"

is the outer limit of the flood plain, coming after the Walo where

recession farming is practiced.

Before Vending (km 481) the Senegal River splits into two

channels: the Senegal and the Doue, into which 30 to 50 percent of the

total flow is diverted.

In the Delta region, host of the final stretch of the river, the

channel system comprises a well-defined main channel and numerous

branches. The slopes here are less than 10-4

, subjecting the river to

severe ocean water intrusion, felt sometimes as far as Boghe, located

26

r o

77)oo

rtscs-.)

LE)

Cl/-C

co

o254- L.)0 I-L

CC4.7 CC'>"J

CD 0

>)CIJ

CD(/)

I /C) 0(/) ( f)(/)

C.0

o

27

at 382 km upstream from Saint-Louis near the cost. In this part of the

basin it is also worth mentioning the different lakes fed by the river:

Lac de Guiers, Aftout-Es-Sahel, and Lac R'Kiz (all shown in Figure 1.2).

Lac de Guiers is connected to the Senegal River by the Taouey

marigot which was canalized. The lake waters are used for the

municipal water supply of Dakar and for irrigation. Figure 1.8 shows

the storage-area and stage capacity curves of the lake.

The Affout-es-Sahel comprises a series of depressions located

between the Atlantic Ocean and Nouakchott and is alimented by the river

in periods of above-average floods. The waters in this lake are

quickly lost after the rainy season.

Lake R I Kiz, also in Mauritania, which is recharged by the river

by the Laouwaja marigot, is dry most of the year.

Streamflows. Flows in the SRB have been recorded since 1903.

These records have been taken by various agencies and companies for

different purposes (railroad and highway construction, irrigation,

navigation, etc.), causing interruption and even discontinuation at some

stations. In 1974, Rochette and the ORSTOM carried out a study to

reevaluate the streamflow data.

Streamflows in the SRB vary in direct response to the climatic

changes. The flow values of eight representative stations presented in

Table 1.6 (see also Figure 1.2) show the importance of the Upper

Valley. At Bakel, limit of the Upper Valley, 75% of the drainage basin

has contributed almost 100% of the flow in the SRB. This suggests an

overall losing stream by evaporation, infiltration, and diversion in the

o

0.)

1.4 S- —C.) = 00CI L) C---4. Cs)CC (ts e--1

= Cl.) --..-cl) s—

L.)eRS

Cl.) L.)O CD Li-If 4—)

V)

28

W' NV V1111.V0 3A0EIV 11403HIle Nei 3.1.00

29

Table 1.6

Annual Average and Extreme Flows for SelectedStations in the Senegal River Basin

Station River

Source:

DrainageArea

Km2

GFCE (1978)

Average

Annual Flows

(m3/s)

min. max.

Periodof

Record

Soukoutali Bafing 27,800 380 227 584 1903-1975

Oualia Bakoye 84,700 168 29 302 1903-1975

alo go Senegal 128,400 606 246 974 1903-1975

Kayes Senegal .157,400 612 210 982 1903-1975

Kidira Falené 28,900 187 21 340 1903-1975

Bakel Senegal 218,000 751 266 1247 1903-1975

natam Senegal 253,000 776 283 1394 1903-1965

Dagana Senegal 268,000 691 292 969 1903-1965

30

Middle Valley and in the Delta Region where flows rarely surpass

10 m3 /s toward the end of the dry season (April, May).

Annual streamflows vary greatly in relation to the equally

great variability in precipitation. At Bakel, the key station in the

basin, the minimum observed annual flow is one-third of the annual

average flow.

The seasonal variation is also dictated by the rainfall distri-

bution in the basin and by the floodplain configuration. This within-

year pattern is characterized by a single extended flood wave during

the rainy season followed by a long recession until the river flow

almost ceases toward the end of the dry season. The transformation of

flood hydrographs from the Fouta Djalon mountains to Saint-Louis, due

to floodplain configuration, is illustrated by Figure 1.9.

The annual flood originates from the Bafing and Faleme tribu-

taries in response to the relatively heavy precipitations at their

headwaters in the southern part of the basin during the month of May.

A few weeks later, the rising limb of the hydrograph at Bakel starts

and its slope increases rapidly near the end of July. The peak is

usually obtained between mid-August and the end of September. A few

days after the peak, the falling limb of the hydrograph begins with a

steep slope. Occasionally, the recession is marked by small,

short-lived secondary peaks. The recession slows down after the middle

of November to continue during the next year.

To conclude this part on surface waters, flood peaks for return

periods of 10, 100, and 1000 years, based on flood frequency analysis

AM.

' •

Amman• A-,grannnrazacturinorms

viaE ce)D 'v >-co cc6D ac

3 1

E61x ,

ID\,

z• • 1,

• .27211.•azaun• .

• • • -Ar4ginApswarsaa

'yxivazrimma--zrAgrazratnrazarca'mow=

Aransarcurgaomr.srin-10:4727472

- •

• f

• sI •

7 °I

AN,

7/4. •

•

C E ;E 5 a

v .

MmomoisLit. lie3a

32

by Groupement Manantali (1977) and Rochette (1974), are presented in

Table 1.7.

SRB Aquifers. Data pertaining to the major aquifers of the SRB

is presented in Table 1.8. Only those aquifers known to be connected

with the Senegal River and its tributaries will be discussed here. They

are:

- the Senegal River Alluvium

- the Continental Terminal

- the Eocene Limestone

- the Maestrichtien

The Senegal River Alluvium is an unconfined deposit of sand,

clay and gravel. Therefore, it is permeable and recharged by the river

channel, floodplain innundation and direct rainfall infiltration.

Although the size of the resources stored is small due to its limited

area, it is a very significant source of supply for the inhabitants of

the valley. Wells tend to be shallow (2-10 meters) and easily dug by

hand. Flows up to 30 m3/hour with a drawdown of 2 m are possible with

bored wells. This aquifer extends both in Senegal and Mauritania and

has 0 to 0.1 billion m3 /year of renewable resources (recharge) and 0.3

to 0.6 billion m3/year of exploitable resources (see Table 1.8).

The next formation, the Continental Terminal, was deposited

during the end of the tertiary throughout most of Senegal and parts of

Mauritania. In the SRB, recharge of this aquifer happens both by direct

infiltration of rainfall and from the downstream reaches of the river.

This formation of sand and clay has a variable thickness, mostly

33

Table 1.7

Annual Peak Flows at Selected Stations for Selected ReturnPeriods in the Senegal River Basin

Source: GFC&C (1978)

Annual Peak Flows, m3 Is

Station/River 10—year 100—year 1000—year

Dagana/Senegal 3200 3800 4200

Bakel/Senegal 6900 8800 10100

Kidira/Faleme 2600 3300 3800

Kayes /Senegal 5400 6500 7400

Oualia/Bakoye 2300 3000 3400

Soukoutali/Bafing 3200 4000 4600

Eo

;I. 11,

•C

..- 5 ,ci- ••n cr.t: v,

:L. L1

I.2". I ...g= - .

G)0 0.

C

.5?

•r-

CO

s.-Q)

cc

(-0

34

n 7.

aC •..a- =

, Fi

Gr. It1.• . S. .Ca 0 E3.1 , ...• ....• 1... 1.• C111 No ••• •Ig N. ..• .1, 0 Z ••• 7, U •• •• U ••. E 0 .- •••• n I0-* ....+ '''' c ..... ... .< .0 < -= C 0 0 04., ••• ••• 0; 01 .

InOD GC1 1 VI '7I tr.

r. G)

, IC

N1 1

X

O 0 n••1 P.,' 0%

• •0 0) 0

4), NC 1•1 •G•

d

0 01

0VI 0 0. 0n-•a .-. .-.

Ln 0r.a

=.

0 0 e C.' Ce 0 C0 •e• 0 17, E...: c L.... 0 C...

...ni tr. c LI

4 4 .CE =

"it CCt-:tn = c.n. c ..... t.e; ='-.0 --_ 2 z -.... o

C E E CC

IT- . E 0 i. n0 dr, •n••

'ili a • iii—. =—1 ,./.:

ci0. LC Lr,•er ... r.

0 00 a 0 4+1 =

—, N N

C '-'- 0 = ..:.In

C."CS

C C 1..-,z R L' E.

c n n n œc CI n LI: c

......

52 vE c... ...C)

E E c -....- -.... ..• J , Y., 4, C ••• .0V V V 333 V t

-z -.-. .2. ,L,0., 0 0 ":,

..g. V: tf: 0:: '4

I

• V E 3.;

n•••••.rs1

•n•rs•E

•Ig<

o••n

35

cl

es,

PP:

36

between 100 and 200 meters. The aquifer is confined in some portions

and semiconfined in others. In Mauritania, the average capacity is

30 m3 /hour for 1 in drawdowl and 180 m3

/hour for the same drawdown in

Senegal. In the two countries the renewable resources of the aquifer

are 3 to 11 billion m3/year and the exploitable resources are 64 to

143 billion m3/year (see Table 1.8).

The Eocene Limestone located in Mauritania is an unconfined,

discontinuous aquifer composed mostly of limestone and dolomite. It is

partially covered by the Continental Terminal and is bordered on the

east by schist in the Precambrian Crystalline formation. Indication by

piezometers suggests that the Senegal River provides some recharge to

this aquifer that can generate flows up to 100 m 3/hour in existing

wells. Estimated from Table 1.8 the renewable and exploitable

resources for this formation are 0 to 0.4 billion m3/year and 6 to

12 billion m 3 /year, respectively.

The Maestrichtian, most important aquifer in Senegal with

respect to potential groundwater resource, is confined. It extends

throughout most of this country and is 200 to 250 m thick on the

average. The Continental Terminal and the Eocene Marl limestone

formations overlay this aquifer. The depth to the aqUifer varies from

50 to 500 m with potential well yields of 150-200 m 3/hour. It is

indicated that recharge in the SRB portion of the aquifers comes from

the river bed and the Lac de Guiers. The groundwater in the formation

moves 10 m per year, horizontally. Water 40,000 years old has been

found in the central part of the aquifer. The BRGM, responsible for

37

most of the information presented so far, gives no indication for the

renewable resources according to GFC&C (1978). Estimates from Table

1.8 indicate that the exploitable resources in the Maestrichtian are 37

to 75 billion m 3 /year.

1.3 The OMVS and its Development Plan

The physical characteristics of the SRB already discussed in

this report demonstrate that the Senegal River is truly an interna-

tional river in accordance with the Helsinki rules (International Law

Association, 1966). Development of such a river is usually furthered

best by the adoption by the riparian countries of complementary plans.

In the SRB, three of the four basin states have created an agency, the

OMVS, to devise and implement an integrated development plan.

This section presents an institutional analysis of the organiza-

tion for the development of the SRB (OMVS) and its plan.

1.3.1. The OMVS

In 1980 in Lagos (Nigeria), the heads of state of the Organiza-

tion of African Unity (OAU) declared:

We Commit ourselves individually and collectively, on behalf ofour governments and peoples, to promote the economic and socialdevelopment and integration of our economies with a view toachieving an increasing measure of self-sufficiency and self-sus-tainment, . . . expand economic and technical cooperation in foodand agriculture through increased trade, exchange of manpower andtechnology, and joint development programs at the subregional andregional levels. . . . We hold firmly to the view that these com-mitments will lead to the creation, at the national, subregionaland regional levels, of a dynamic, interdependent Africaneconomy. . . .

38

Before this Lagos Plan of Action, the three governments of

Senegal, Mali, and Mauritania were already involved in a partnership to

put to practical use the resources of the SRB by creating, supporting

and empowering the OMVS to plan and manage the integrated development

of the SRB and by establishing the river and its regime as an interna-

tional resource.

History of the OMVS. Less than five years after their indepen-

dence, the four basin states--Senegal, Mali, Mauritania and Guinea--

founded in 1963 the "Comité Inter-Etats pour le developpement du Bassin

du Fleuve Senegal." This decision resulted from the realization by the

four republics since their birth and by the prior colonial administra-

tion of the needs and potential benefits of managing the SRB waters.

In 1968 the four states changed the Comité Inter-Etats of 1963

into the "Organisation des Etats Riverains du fleuve Senegal" (OERS). In

1972, the state of Guinea withdrew its membership and the three basin

states left signed in March of the same year a treaty to form the

OMVS. The treaty was ratified in November 1972.

This change in name and the loss of one of the members in 1968

led to a reorientation of the organization from a political and

diplomatic instrument of vague and soothing nature to an institution

for development with concrete and precise objectives.

Jurisdiction and General Powers. Within the three basin states

(Senegal, Mali, Mauritania), the OMVS has jurisdiction over all matters

related to the management of the Senegal River, which was declared an

39

international waterway since 1963 in the context of the former organi-

zation (the "Comité Inter-Etats").

Article 1 of the 1972 treaty gives jurisdiction of the river

basin as a whole and without limitations to the OMVS, providing that

the organization's actions be the collective desires of the states

(USAID, 1982).

Overall Objectives. According to the USA1D (1982), OMVS has

basically three objectives:

- a more secure and better living conditions for the population of

the basin and its neighboring areas (including reduction of vul-

nerability to climatic and other external factors).

- a better balance between man and his environment in the basin

and also over the widest possible area within the three

countries.

- an acceleration of the economic growth of the three member

states and of the inter-state cooperation.

For the period 1972-1981 the OMVS has mainly been managing feasibility

studies and collecting money for the construction of the main

infrastructures proposed in its plan of development:

the Manantali dam

- the Diama dam

- the ports and waterways.

Organization of the OMVS. The OMVS is governed by three insti-

tutions: the Conference of Heads of State, the Council of Ministers,

and the High Commission.

40

Article 3 of the treaty stipulates that the "supreme authority

of the Organization" is the Conference of Heads of State. The

Conference meets once a year and can be called for extraordinary

session by any member. Its presidency is assumed by one member state

on a two-year rotational basis.

Article 8 of the same treaty defines the Council of Ministers

as the conceptual and control body of the organization. It is composed

of the three ministers in charge of water resources adminstration in

their respective countries. The presidency is also assumed by each

country on a two-year rotational basis. The president, legal represen-

tative of the Council with all national and international institutions,

is empowered to negotiate and sign treaties in the name of the OMVS.

The Council, a decision-making body, has the responsibility of defining

priorities for development projects, authorizing the budget, accepting

loans and grants, and determining member state contributions. It

reports to the Conference of Heads of State through its president. It

submits to the Conference matters on which its authority is limited or

when unanimity of decision cannot be reached. As for the Conference,

all decisions are to be made with unanimity.

The High Commission, the third permanent institution, is headed

by a high commissioner appointed by the Conference for a renewable

mandate of four years. The high commissioner implements the decisions

of the Council and reports to it. He is "responsible for the financial

operations" (Article 13). He also is entrusted with mobilizing financial

resources necessary, and is empowered to represent the OMVS in

41

international aid negotiations. The high commissioner is assisted by an

economic, a legal, and a press attaché in addition to the secretary

general.

Consultative and Study Bodies. To do its assignment, the High

Commission comprises four advisory and study bodies: the Permanent

Water Commission (PWC), the Inter-State Committee for Agricultural

Research and Development (CIERDA), the OMVS Consultative Committee

(CC), and the Financial Control Units. These four bodies also advise

the Council of Ministers (USAID, 1982).

The PWC, created by the 1972 convention, is entrusted with

"defining the principles and the conditions of allocating the Senegal

River waters between the different sectors: Industry, Agriculture,

Transport" (Article 20).

The CIERDA advises the Council on all matters relating to agri-

cultural research and development, while the CC is concerned with

financial planning. The latter is the OMVS financial coordinating

structure and the forum for debating policy and programmatic options.

Fourteen bilateral and multilateral financing sources are represented

in the CC.

Operating Structures. In addition to the consultative and study

bodies, the OMVS has an operating structure composed of the following

departments (USAID, 1982).

The Regional Infrastructure Directorate (RID) is the most tech-

nically sophisticated department. It manages three divisions: one

dedicated to the Diama dam, one to the Manantali dam, and one to the

42

future ports and waterways. It also comprises procurement, planning

and evaluation offices.

The Investment Directorate (ID) manages the financial portfolio

of OMVS which results from the contribution of nearly $800 million of a

consortium of financiers. The ID is responsible for the timely payment

of the two dams' contractors in accordance with funds released through

financing sources. It also plans the amortization of the loans granted

to OMVS by external sources.

The Human Resources Directorate (HRD) is the manpower-planning

unit. It is responsible for recruitment, monitoring of contractors'

compliance to national labor and social welfare laws, planning for

training, and classification and upward mobility programs.

Finally, the Development and Coordination Directorate (DCD) is

mainly concerned with the hydroagricultural development of the basin.

It has four divisions and one planning unit devoted to long-range

planning, analysis, evaluation and feasibility study of the agricultural

and industrial potential of the basin.

This part of the report on the institutional analysis will be

concluded with a summary of the conclusions reached by Ndiaye (1984)

when attempting an appraisal of the OMVS. The OMVS was found to be

well structured and institutionally adequate to serve the purposes for

which it was intended. This finding resulted from an evaluation of the

organization on the basis of selected criteria used in institutional

evaluation. For more details, the reader is referred to Ndiaye (1984).

43

1.3.2. The Development Plan

To start this section, we first describe the existing water

uses in the SRB. This will help the reader to better understand the

justification of the plan proposed by OMVS. The components of the so-

called plan are then discussed.

Existing Water Uses. The waters of the Senegal River are

presently used for irrigation, navigation, and municipal water supply.

Three types of irrigation systems are practiced, mostly in the

Middle Valley and in the Delta regions, in Senegal and Mauritania. The

first type is recession farming practiced in 120,000 ha of the river

flood plain after the high flows recede. The second type is practiced

on approximately 12,100 ha and consists of controlled surface flooding

on 11,000 ha in Senegal and 1,100 ha in Mauritania. Finally, 5,000 ha

of sugar cane in Senegal are irrigated by fully controlled pumping

irrigation (Riley et al., 1978).

Commercial navigation on the Senegal River is subject to stage

variations. During the period of high flows, the river is navigable as

far upstream as Kayes in Mali. When flows are low the navigability is

limited to Podor, 275 km from the Atlantic Ocean (Riley et al., 1978).

Municipal water supply uses are small. They are limited to the

small towns of Kayes in Mali; Rosso, Bogue and Kaedi in Mauritania; and

Bakel, Matam, Podor, Dagana, Richard-Toll and Saint-Louis in Senegal.

To conclude the actual uses of the Senegal River, let us

mention that there is presently no hydroelectric power production in

the SRB and little industrial use.

44

Proposed Developments. To reach its objective, the OMVS

decided to eliminate the constraints to economic growth and social

welfare which are mainly the lack of abundant water, food and capital

investment. This led the OMVS to call for an integrated plan,

comprising the construction of two dams and navigation infrastructures,

the development of nearly 300,000 ha of irrigated perimeters, and the

production of hydropower for industrial development.

The Diama Dam. The Diama dam, under construction since the

beginning of 1982, is located 27 km upstream of Saint-Louis, near the

mouth of the river.

The construction of the dam includes a closing dike, a lock, a

dam, a stopping dike, and a road embankment. The impoundment of the

dam is presented in Table 1.9 and Figure 1.10 exhibits the area-stage

curve of the dam.

Beside controlling salt intrusion from the ocean which is the

main function of the dam, Diama will serve the following purposes:

diversion of water to Lac de Guiers;

year-round source of fresh water for the irrigation of

42,000 ha;

- availability of surface water for the annual recharge of Lac

R I Kiz.

The Manantali Dam. An important element of the OMVS develop-

ment plan is the augmentation of low flows in the Senegal River year-

round. This will be done by the Manantali dam under construction and

located 1200 km upstream of Saint-Louis in the Bafing River.

Table 1.9

Diama Impoundment Characteristics at Water Levelsof 1.5 meters and 2.5 meters

Source: GFC&C (1978)

At 1.5 m IGN At 2.5 m IGN

Reservoir Length 360 km extending to 380 km extending to

Guede-Boghe area Boghe-Cascas area

Reservoir Width 0.3 to 5.0 km

0.3 to 5.0 km

Enclosed Surface Area 235 sq. km

440 sq. km

Water volume 0.25 billion

0.58 billion

cu. meters cu. meters

45

2.50

2.00

050-

o 200 400 600 44,1o

Fig. 1.10. Stage-Area Curve of the Diama Dam

Source: GFC&C (1978)

46

47

Controlled releases will allow:

- a year-round irrigation of 255,000 ha of land between the

village of Manantali, the dam site, and Saint-Louis.

a year-round flow of 100 m 3 /s in excess of irrigation demand

and other requirements to provide water depths needed for

navigation as far upstream as Kayes in Mali.

- the production of 800 giga-watt-hours/year of electric power at

Manantali dam.

To accomplish these goals, the dam will have the impoundment charac-

teristics presented in Table 1.10.

The design proposed by the Corps of Engineers, Groupement

Manantali, consists of three major components:

- a concrete structure located in the middle section of the dam

containing the hydropower plant, a series of gated spillways,

and stilling basin;

- two earthfill dams connecting the concrete gravity dam with the

cliffs on the right and left sides of the valley.

The dam construction will necessitate the relocation of 9 to

10,000 persons that live in the areas that will be inundated after the

dam is completed. Figure 1.11 exhibits the area-stage and stage-

capacity curves of the dam.

For more details on the two dams the reader is referred to the

studies by SOGREAH (1977) for Diama and by Groupement Manantali (1977)

for Manantali.

48

Table 1.10

Manantali Impoundment Characteristics

Source: GFC&C (1978)

At Spillway At Minimum Water Level

Elevation to be Allowed During

Reservoir Operation

Water Level

208.0 187.0

(meters IGN)

Corresponding

Surface Area of

Reservoir 477 275

(square kilometers)

Reservoir Water

Volume 11.3 billion 3.4 billion

(cubic meters)

49

00

ooo

o l\

w '13A 31 ti31VM

50

Agricultural Development. To increase crop production in the

SRB, the OMVS proposes in its development plan an alteration of the

present agricultural conditions. These conditions are dominated by

recession farming mentioned before and by rainfed agriculture, locally

called Dieri farming. In the first 15 years after completion of the

Manantali dam, recession farming will still be practiced in 100,000 ha

by releases of 2,500 m3 /s from the reservoir for "artificial flood"

during the transition period (GFC&C, 1978). Then, as the level of

technology of the population and other factors become more and more

suitable for modern intensive irrigation, new agricultural practices

will take place. Recession farming will greatly diminish as prime

recession lands are converted into irrigated perimeters. The changes

also include the use of "Dieri" lands primarily for grazing and the

completion of 255,000 ha of diked agricultural perimeters by the year

2028 (GFC&C, 1978). After the Manantali dam is operational and

dry-season releases begin, the production of two crops yearly will be

possible under modern intensive irrigation.

Development Related to Navigation. The year-round navigation

of the river between the Atlantic Ocean and Kayes in Mali is very

important for this land-locked country and for the development of the

basin. This will be made possible by the following alterations (GFC&C,

1978):

a) a navigation channel to Kayes with a minimum width of 55 m and a

minimum bend radius of 700 m;

51

h) sufficient flow to maintain a minimum water depth of 2 m

(300 m 3/s of flow at Kayes and 150 m 3/s at Podor);

c) development and upgrading of port facilities at Rosso, Richard-

Toll, Dagana, Podor, Boghe, Kaedi, Matam, Bakel, Ambidebi and

Kayes;

d) an entry channel between the estuary and the ocean 7 km

downstream of the Faidherbe bridge at Saint-Louis, a breakwater

into the ocean, and an approach channel into the ocean;

e) an estuarine approach channel connecting the entry channel to the

proposed harbor facilities at Saint-Louis;

f) a deep-water harbor along the left bank of the river south of

Saint-Louis to transfer goods from ocean-going to river-going

vessels;

g) modification of the Faidherbe bridge to facilitate passage of

vessels.

The planned developments mentioned above are to be updated by

the Canadian Agency for International Development (CA1D), however the

basic strategy mentioned above will remain unchanged (GFC&C, 1978).

Municipal-Industrial Development. The municipal and industrial

developments are contingent to the developments already described

(dams, agriculture, navigation). The future activities to be created

can be classified into two categories: the industries based on agricul-

tural and livestock products and the industries based on mining

activities. These industries are expected to use most of the 800 Gwh

52

per year to be produced at the Manantali dam, and the navigation

facilities.

Population in the basin is expected to increase from 241,200 in-

habitants in 1980 to 1,490,000 in 2028 due to the combined effects of

natural growth and OMVS program related growth. To accomodate this

population, adequate infrastructure in the area, such as housing, water,

waste disposal, power, transportation, etc., will be needed.

In conclusion to this section on the proposed development, it

can be inferred that the OMVS program includes the construction of

several structures to serve various purposes in three different

countries. Agricultural development will primarily benefit Senegal and

Mauritania and, to some extent, Mali. Navigation will benefit greatly

Mali and, to a different level, Senegal and Mauritania, which are

coastal countries. Hydropower for municipal and mainly industrial

development will promote economic growth in all countries. The main

attraction of the OMVS plan is the priority given to agriculture, con-

sidering the ill-satisfaction of food requirements in the basin and the

fact that presently the only significant economic activities in the

basin are based on agriculture and livestock.

1.4 Summary

Putting in the same picture the climate and hydrologic

conditions in the SRB, one finds out one of the most important

constraints to economic growth and a desirable quality of life in the

SRB. Rainfall, the lifeblood of the Senegal River, is small and highly

variable in time and space. Evaporation rates are high. Surface water

53

flows are high during the rainy season and very low during the dry

season. Groundwaters are significant but their use is limited to

drinking purposes. Water is therefore scarce in the SRB mainly because

of its temporal and spatial distribution, but also because of drought

conditions.

The general situation of the basin shows that the Senegal River

is truly an international river. This situation, if anything, makes the

scarcity of water more crucial because of the competition it causes.

Instead of competition which often leads to inefficiency, the

three basin-states of Senegal, Mali and Mauritania decided to set an

unprecedented example of international water resources management in

Africa. They created the OMVS and provided the organization with the

powers and support it needs to conceptualize, coordinate and implement

projects to alleviate the constraints to economic growth and a

desirable quality of life for the basin's population.

The organization mentioned came up with an integrated plan for

an optimal exploitation of the resources of the basin. The development

plan proposed included the construction of two dams, a power-plant and

navigation facilities for agricultural and industrial development for

the common interest of the three member-states. More than $800

million dollars from over ten financing sources will be involved for

the integrated plan.

The planning, design, and implementation of such a complex

water resources system can reveal to be very challenging. Aware of

this reality, the OMVS was structured to comprise consultative and

54

study bodies. Among these bodies, the Permanent Water Commission (PWC)

is of particular interest in this study since it is assigned to define

the principles and conditions for the allocation of the waters of the

Senegal River and its tributaries among the different sectors:

industry, agriculture, transport.

The optimal allocation of water among three purposes

(irrigation, navigation, and hydropower) for a fair and equitable use in

three different countries necessitates careful and detailed studies to

provide input to the decision-making process. Among those inputs

needed are those highly uncertain, like streamflows. To handle this

type of uncertainty, various simulation and optimization techniques use

synthetic streamflows that can be generated by various models. The

simulation and optimization models referred herein include those used

for both water quantity and quality management. A model for

streamflow generation can thus be a valuable tool for the operation of

the future SRB dams, and therefore for the OMVS in general and in

particular for its Permanent Water Commission in charge of water

allocation. In this study, the objective is to review some of these

synthetic streamflow generators for the selection, adaptation and

testing of those that can be suitable for use in studies dealing with

the SRB development plan. The selection and adaptation is done in the

next chapter and the testing in Chapter 3.

CHAPTER 2

LITERATURE REVIEW AND MODEL SELECTION

Following the procedure outlined in Chapter 1, we present in

this chapter a literature review on "operational hydrology," followed

by the description of the models that will be considered for further

investigation in the next chapter of this study. Section 2.1 gives some

definitions relating to streamflow generation, and a review of various

models for generating streamflows followed by the pre-selection of the

models presented in sections 2.2, 2.3, and 2.4. Section 2.2 gives a

description of the multivariate lag-one Markov model. Section 2.3

presents the fast fractional Gaussian noise (FFGN) model while section

2.4 describes the disaggregation models for both the temporal and

spatial cases. Finally, section 2.5 summarizes the different findings

of the preceding sections and gives the composition of the two models

proposed for the generation of streamflows in the Senegal River Basin

(SRB).

2.1 Literature Review

The description given in Chapter 1 of the Senegal River Basin

(SRB) and its proposed development, planned and to be implemented by

the OMVS, have led to think of the SRB's development plan in terms of a

complex water resources management scheme. The planning, design, and

analysis of complex systems have relied for a long time on techniques

55

56

which used the historical record of flows for a particular stream or

for streams in a region under study. A typical and common example of

such techniques is Rippl's (1883) mass curve diagram. Rippl was

concerned with the design of a dam that would store water within a

year when inflows were greater than demands for use during periods

when farmers and other users will need more water than could be

diverted from natural flows. Rippl's method provides a systematic way

of determining the minimum storage capacity of a dam required to meet

a pattern of target releases if it had been subjected to the historical

flow record and assuming a starting storage. Rippl's mass curve has

been widely used for planning and designing of within-year and over-

year storages. An important drawback of Rippl's method is the use of

the historical record for this historical flow sequence is not the only

one possible.

Operational hydrology, also called synthetic hydrology or time

series modeling, arose as a result of the dissatisfaction of planners

with techniques that use only the historical record (Jackson, 1975).

The hydrologists sought techniques that will consider, as mentioned by

Jackson (1975) and many other hydrologists, that:

1. The historical sequence is unlikely to reoccur.

2. It is unlikely that the extremes on a historical record are

the worst flood or drought possible.

Therefore, to carry a study of complex water resource system

there is a necessity to use a comprehensive approach, comprehensive in

the sense that it will be complete or at least produce outcomes that

57

are likely to occur, instead of being based on a single realization of

events. A complete theoretical predictive model is beyond the hydrolo-

gist's capabilities (Jackson, 1975) and may be impossible because there

will always be some errors (errors in sampling, errors in model identi-

fication, etc.) and biases (parameter estimation, etc.).

2.1.1. Definitions

Stochastic Processes. Most hydrologic phenomena such as

streamflow and rainfall are characterized by their variability in time

and space. When the outcome of a variable X cannot be predicted in a

deterministic manner, meaning with certainty, X is said to be a random

variable. Randomness does not mean lawlessness in hydrology for most

hydrologic time series vary through time according to probabilistic

laws. Such time series are called stochastic processes (Loucks, 1981).

A time series is an ordered sequence of observed values of the random

variable X, X 1 , X 2 .••X n at successive intervals of time ti , t2 ...tn. The

series X 1 , X2 ...X n is a single realization of the stochastic process for

it is possible that the set of values X l , X i ..X i could have been1 2' nobserved for the same random variable X during the same periods of

observations tl , t2 ...tn , due to the random nature of X.

Thus, the analysis of a stochastic process requires the

knowledge of the joint probability distribution f (X 1 , X2 ...X n ) of the

random variables X 1 , X2••.Xn• If f (X 1 , X2 ...X n ) . f(X 1 ) x f(X 2 ) ... x

f(X n ), the product of the marginal distributions, then the process is an

independent stochastic process and the series is an independent time

series. If f (X 1 , X 2 ...X n ) # f (X 1 ) x f (X 2 ) x...x f (X n ) there is a

58

serial correlation at some level. Then, the stochastic process is

serially correlated and the time series is a dependent series (Salas et

al., 1980). The difficulty in stochastic process studies is that one

can usually observe only one single realization for a finite set of

time points. Time series analysis is carried out to infer the probabil-

ity laws of the stochastic process.

To do this, a couple of assumptions are necessary among which

is stationarity (Loucks, 1981). Given the stochastic process

X(t2 ) ... X(t n ), the expected value of the process is in general

composed by E[X(t1 )], E[X(t2 )] ... E[X(t n )] and the variance by Var

Var [X(t2 )] ... Var [X(tn)]. If the stochastic process is

stationary then the random variables have the same mean, variance, and

distribution which in mathematical notation lead to (Loucks, 1981):

(1) E[X(t)] = p r t e [t1'

t2 ... tn ]

Var [X(t)] = a 2 T t E [t1' t2 ... tn ]

(2) Fx(t) [X(t)] = F x [X(t)]

In addition, if a process is strictly stationary, the joint distribution

of the random variables X(ti ), X(t2 ), ..., X(tn ) is the same as the joint

distribution of X(ti+t), X(t2+t), ..., X(t n+t) for all t; the joint dis-

tribution does not depend on t but on the time difference ti -ti (Loucks,

1981). If the stochastic process is stationary in the mean, it is

termed first order stationary. If the process is in addition stationary

in the covariance, that is, if the covariance for some lag k depends

only on the time lag but not on the time position t, Gov [X(t), X(t-k)]

. Gov (k), the stochastic process is a second-order stationary process.

59

The latter case is also termed stationary in the wide sense or weakly

stationary (Salas et al., 1980).

In general stochastic processes are not stationary at all. And

even if they are, they can lose that property due to urbanization,

deforestation, climatic shifts, etc. (Loucks, 1981). This makes time

series analysis more difficult. However, there are techniques to

overcome this difficulty. Some of these techniques will be discussed

in section 2.2 of this chapter.

Stochastic Models. To conduct a time series analysis one needs

to define completely and in specific terms a model that represents the

event under study. Defined in mathematical terms a model that

represents a stochastic process is called a "stochastic model" or "time

series model" (Salas et al., 1980). The model has a set of equations

and a set of parameters. The model can be complex or simple depending

on:

- The nature of the mathematical relationships;

- The number of parameters and their estimation;

The purpose, theoretical knowledge, and practical experience of

the modeler.

If we know the probability density function and some statistical char-

acteristics such as the mean and the variance, we can define very

simple models to produce values for the random variable. However, the

real distribution and the population statistics are never known.

The set of techniques and procedures to carry out in order to

define a model, also referred to as the generating process, is called

60

"time series modeling" (Salas et al., 1980). The generating process set

the basis to assess the predictability of future events, the reliability

of statistical descriptors and allows the formulation of steps to

follow for generating synthetic streamflows to be used in water

resources planning (Maddock, 1984). Synthetic streamflow means herein

flows that are not really occurring but that are likely to occur in

statistical terms.

There are two categories of generating processes discussed in

this chapter: the Markov processes defined in part 2.2 and the self-

similar processes described in part 2.3 of this chapter.

Finally, the terms single site or univariate, multisite or mul-

tivariate, annual and seasonal are also used in operational hydrology.

When flows are generated at one site at a time, the model is termed

univariate or single site on one hand. On the other hand, if the flows

are generated at several sites at the same time, the model is termed a

multivariate or multisite model. A single-site or multisite model can

be for annual time series or seasonal time series.

2.1.2. Model Review and Pre-Selection

Review. Since the realization that the historical sequence of

hydrological events is unlikely to occur again in the same way, many

hydrologists have been working on developing models that can be used

for time series generation in general and streamflows in particular.

The starting point in stream flow generation may be associated

with Hazen (1914) who obtained synthetic streamflows by combining the

annual flows from fourteen individual streams (Phien and Ruksasilp,

61

1981). Another pioneer of synthetic hydrology is Sudler (1927) who

tried to improve on the use of the historic sequence by entering a

series of annual events on a card and to reshuffle the deck of cards

repeatedly to obtain a longer sequence containing new combinations of

the original series (Benson and Matalas, 1967).

The next improvement was introduced by Barnes (1954) who used

a table of random numbers to synthesize a long record of annual flows

having the same mean and standard deviation as the original record, and

assuming a normal distribution. Although Barnes' method seems better

than that of Sudler, it does not consider any serial correlation

between flows (Benson and Matalas, 1967).

A group working at Harvard (Maass et al., 1962) considered two

other statistical parameters; the skew coefficient and the serial

correlation between successive flows in addition to the mean and the

standard deviation (Benson and Matalas, 1967).

The method developed by the Harvard group and the important

work of Thomas and Fiering (1962) set the basis of "synthetic

hydrology." Since the comprehensive model of Thomas and Fiering (1962),

a large number of models have been introduced. They are according to

Salas et al. (1980):

1. Autoregressive models;

2. Autoregressive and moving average models;

3. Fractional Gaussian noise models;

4. Broken-line models

5. Shot-noise models;

62

6. Disaggregation models;

7. Markov-mixture models;

8. ARMA-Markov models; and

9. General mixture models.

Several of these models are multivariate models and were

proposed for the design and operation of water resources systems by

Fiering (1964), Matalas (1967), Matalas and Wallis (1971), Mejia (1971),

Valencia and Schaake (1973), O'Connell (1974), and others (Salas et al.,

1980). Based on the works of Thomas and Fiering (1962, 1963), Fiering

(1966), Hufschmidt and Fiering (1966), Matalas (1967) proposed a multi-

variate lag-one Markov model with constant parameters. Following

Matalas' model, Young and Pisano (1968) devised a procedure for

applying Matalas' model to operational hydrology using residuals.

Pegram and James (1972) extended Matalas' model to the multilag case

with constant parameters. O'Connell (1974) extended it to the ARMA

(1,1) multivariate model with constant parameters. Salas et al. (1980)

introduced an ARMA (p,q) model that accounts for the correlation

structure in time and the lag-zero cross correlation in space.

Valencia and Schaake (1973) proposed a multivariate disaggregation

model for synthetic stream flow generation that maintains the annual

as well as seasonal covariance properties. Matalas and Wallis (1971)

developed the multivariate fractional Gaussian noise while Mejia came

in the same year with the multivariate broken line model.

Pre-selection. "Synthetic hydrology" is nowadays widely used in

water resources planning and analysis. However, a non-negligible

63

difficulty faced by the potential user is the question of what model to

choose. Justification for selecting a model is based on (Salas et al.,

1980):

- the ability of the model to maintain the statistical charac-

teristics thought to be relevant for the purpose of the study;

- the nature and amount of data available;

- the physical basis of the time series;

- the modeler's experience, theoretical knowledge and even

personal preference;

- ease of application of the model.

Pre-selected Models. Some of the models mentioned in the

review portion of this section are characterized by one or several of

the following considerations:

- cumbersome mathematics;

- lengthy computations;

- complex parameter estimations.

These important drawbacks make their use for practical purposes in

water resources studies very limited. Therefore, the following models

will be pre-selected for a detailed description in the next sections of

this chapter.

1. The multivariate lag-one Markov model of Matalas (1967) for

annual flows and its extension by Young and Pisano (1968) for

monthly flows both described in section 2.2;

64

2. The FFGN model introduced by Mandelbrot (1971) and modified

by Chi et al. (1973) for single-site annual flows presented in

section 2.3;

3. The temporal and spatial disaggregation models of Lane (1979)

described in section 2.4 of this chapter.

After the detailed description of these models, the final form

of the models that will be used with the data collected and their

exact composition for use in the context of this study will be

presented in the summary section of this chapter.

2.2 Multivariate Lag-One Markov Model

In this section, we present the Markov lag-one model of

Matalas (1967) for the generation of streamflows at several sites.

Subsection 2.2.1 describes the univariate or single-site case, and

subsection 2.2.2 presents the multivariate case of Matalas (1967)

followed by the extension of this model by Young and Pisano (1968).

2.2.1. Univariate Case

This subsection gives a definition of Markov processes and the

procedure for using it to generate synthetic streamflows.

Markov Process. A common practice in stochastic modeling of

water resources systems is to assume that the stochastic process X (t)

is a Markov process. A Markov process has the property that future

values' dependence on past values is summarized by the current value

(Loucks, 1981).

65That is, for k>0

F x[X(t+k)1X(t), X(t-1), X(t-2), ...] = F x [X(t+k)IX(t)]

The current value X(t) is called the state and if the state takes on

only discrete values we have a Markov chain (Loucks, 1981).

Markovian Generating Process. The basic model for generating

synthetic streamflow sequences is the lag-one Markov process, which is

defined as (Matalas, 1967):

1/22(X 1. - p ) = p (1) (X. - p ) + [1-px(1)] ax c.1+xxix 1+1 (2.1)

where:

- X i and X .111 are the streamflows at time points i and i+1,

respectively;

px and ax are the mean and the standard deviation of X,

respectively;

p x (1) is the lag-one serial correlation coefficient for X; and

- Ei+1

is a random component with zero mean and unit variance and

is independent of X.

ux , a x and p x(1) are unknown but may be estimated from the

historical record by'Ix , ax and ax(1). Using these estimates withequation (2.1), synthetic streamflows that resemble the historic events

in terms of these estimates can be generated as follows:

- X i , being the most recent streamflow recorded, e i+1 is randomly

selected from a population of zero mean and unit variance.

- Using equation (2.1), X i and 9+1 ; X i4.1 is generated.

66

- X i+1 assumes the role of X i and a new e i+1 is generated leading

to a new X ii.1 .

This process is repeated N times, N being the length of the synthetic

record. As N increases and approaches infinity ax,ax and ax (1) obtained

from the synthetic sequence should approach the estimates obtained

using the historic record.

The requirement to select E i+1 at random from a population of

unit variance and zero mean limits the applicability of equation (2.1)

to a weakly stationary process as defined in section 2.2.1 (Matalas,

1971.)

To maintain the skewness of the series, some modifications are

to be introduced in equation (2.1). These modifications can be of

various forms and will be related to the distribution assumed for X.

The modification used in this study is discussed in what follows.

If the lag-one Markov model is to represent a strictly

stationary process, the probability distribution of X i and X i+1 must be

considered. The assumption of strict stationarity leads to a

straight-forward procedure for generating synthetic streamflows that

will resemble the historic sequence in terms of '1 )c , ax and ax (1) and -7x

if a skewed distribution is considered. The use of this assumption is

illustrated by Matalas (1967) for the gamma and lognormal distribu-

tions. In what follows we reproduce the case for the three-parameter

lognormal distribution.

If a is the lower bound of the random variate X, its loga-

rithmic transform y = ln (x-a) will be normally distributed. The mean

67

A2Tlx , variance a2 and skew coefficient

x are related to the lower bound

a and to the mean p and variance a 2 of the random variate y byY Y

(Matalas, 1967):

= a exp [in c7; +

A2 = exp [2 (a 2 + p )] - exp [3. 2 + 2 p IY Y Y Y

exp [34] - 3 exp [4] + 2

If we let a = exp [Yp + 1/2 a 2 ] and n 2 = exp (a 2 ) - 1 Aitchison and Y Y

Brown (1957) show that:

2ay = in 02 + 1)

2 11/2ax

= ln2

no (

1 02 +1)

Yx 3/2[exp [a ,f) - 1]

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

in (n0 P x 1) (2.8)

ln + 1)

where

[ y ,,, 2 ]1/2 ] 1/3 [nO

x [, x 4.- 2- —4—

Y [ 2 ]1/2

x y- x + 1

2 -4-

ax

(2.9)

68

It is now possible to generate y instead of x using the following

equation:

(Yi + 1 = Py(1) (Yi (1 py2(1))

1/2 ay c i+1 (2.10)

where ei+1

is normally distributed with zero mean and unit variance and

is independent of y i . In terms of X, the generating process is

xi+1 = a + [exp [py

(1 - p)]] • (xi-a )P i+1

(2.11)

where

P (1) andY 6i+1 [[1-p;(1)] 1/2= exp ay 9 +1 ] (Matalas, 1967).

The importance of considering the distribution followed by the

events is hard to evaluate. In studies dealing with truncated flows,

as is the case for low flow augmentation, the probability distribution

might be quite important because interest is focused on the distribution

of the durations and volume deficits associated with flows less than

some specified values, say the mean (Matalas, 1967). This study is

interested in drought length and the range, departures from the mean.

The type of distribution to be assumed will therefore be considered and

determined in the next chapter.

2.2.2. Multivariate Generating Processes

The Model of Matalas (1967). Thus far we discussed the

generation of synthetic flows at one site. To generate synthetic flows

for more than one station, the cross correlation between the historic

flows at different stations must be considered in addition to çl x , ax,

69

x' and 'I;x(1) of the historic flows at each station (Matalas, 1967).

For a non-zero cross correlation between stations, which is the usual

case in a river basin, a multivariate generating process is necessary to

handle the dependence among different stations.

Given X (p p = 1, m, the flows pertaining to station p with

mean px(p standard deviation a

x(p skew coefficient ix (p) , and auto-

correlation x(p) (1), the lag-zero cross-correlation between station p

and station q, q = 1, m is denoted p x(1)(q) (0).

A straightforward way of generating multivariate synthetic

sequences is based on a multivariate weakly stationary generating

process that is defined as (Matalas, 1967)

Xi+1 = AX. + Be i+1

(2.12)

where X i+1 and X. are (mxl) matrices whose pth elements are (x! p)

1+1 -(p)%

px and (x(p)

- p(p) ), respectively, with i and i+1 being the time

points. ci+1, the random component, is a (mxl) matrix whose elements

are independent of x i . A and B are (mxm) coefficient matrices. These

coefficients of A and B must be defined in such a way that the model

will generate sequences that resemble the historic record in terms of

ûX (P) ' aX ( P ) , -7X

( P ) , f)'X(P) (1) and a (0), for all values of p and q.

To determine A and B the following reasoning was held by

Matalas (1967). If E denotes the mathematical expectation

E [c+1] = 0 since E [X i+1 ] = E [X i ] = 0.

70

Multiplying both sides of equation (2.12) by X, the transpose

of X. and taking the expectations

M 1 = A M

(2.13)

where Mo = E [X i X -ir ] and M 1 = E [Xi+1 X i ].

If the elements of i+1 are mutually independent with zero

mean and unit variances, E(c i+l c iiTI ) = I, the identity matrix. If both

sides of equation (2.12) are postmultiplied by X iji and the expecta-

tions taken,

Mo = AMT + BBT1

where M1 and BT are the transpose of M1 and B, respectively.

The matrix A is given by

A = M M -11 0

where M-1 is the matrix inverse of M0 , the variance-covariance0

matrix.

The matrix B is obtained by solving

BBT = MMO ml M1 m

0 mlT

(2.14)

(2.15)

(2.16)

Equation (2.16) can be solved by the techniques of principal

component analysis (Matalas, 1967). If so,

B = P 1/2 P-I

(2.17)

where x is an (mxm) diagonal matrix whose elements are the eigenvalues

71

of M o - M 1 M0 -1 M iT, and P contains the corresponding m eigenvectors

while p -1 is its inverse.

The matrix M 1 contains the lag-one serial correlation and the

lag one cross-correlation. If the latter is of no interest, the compu-

tation of M 1 is simplified according to Matalas (1967), in the following

manner.

The matrix A may be taken as a diagonal matrix whose elements

are the lag one serial correlation coefficients: the ()th element of

A is p x(p) (1). With A so defined:

mx!P ) = ^ ( P ) (1) x ( p ) + z b e (s)1+1 Px i P's 1+1s=1

( 2.18)

(q) (q) (q) m (s)xi+1 = 13x (1) x, + z bs=1 q,s

i+1

where

'

where b and bq,s are the (p, ․ )th

and (q, ․ )th

elements of B, respec-p,s

tively.

Multiplying both sides of equation (2.18) by x i (q) and taking

the expectation leads to

- ( P )( q ) (1) = A (P)(q) (0) "13 (P) (1)Px Px ( 2.20)

where i, x (P)( q ) (1) is the lag-one cross-correlation for the events

generated by equations (2.18) and (2.19). Equation (2.20) shows that for

Markov processes, the lag-one cross-correlation is the product of the

lag-zero cross-correlation and the lag-one serial correlation.

Therefore, by replacing a(l) by Tx(P)(q) (1), the multivariate

synthetic sequence generated by equation (2.12) will preserve

(2.19)

72

(p) (p) (p)(q)aX , X(1), and P x (0), but not ax

(p)(q) (1). So, if the lag-one

- (p)(q)cross-correlation is of no interest, it may be estimated by P x (1),

as defined by equation (2.20) instead of computing a x (p)(q) (1) from the

historic record.

As it was discussed for the single-site model, it might be

necessary to account for the skewness. To preserve the skew coeffi-

cient, x ( P ) the assumption is made that x ( P ) follows a 3-parameter

log-normal distribution with lower bound a (p) , so that y (p) = ln (x (p) -

a ( P ) ) is normally distributed. The relations between ax ( P ) ' ax ( P )

' x ( P ) ,

and ax ( P ) (1) for x ( P ) and aY , Y ' /Y

( P ) a ( P ) A ( P ) , and A ( P ) (1) of thePY

transform y are given by equations (2.2), (2.3), (2.4), (2.5), (2.6),

(2.7), and (2.8).

The lag-zero cross-correlation ax(p)(1) (0) can be computed from

solving (Matalas, 1967)

(P) A (CI) A ( P )( q ) (0)] - 1=

exp[ay ay Py13x (P)(q) ( 0)

[eXP [& 2(p)] 1..] 1/2 [exp [a 2(q)] - 1i1/2

and the cross correlation a (P)(q) (0) is (Maddock, 1984)Y

(2.21)

(p) (q) ^ (P)(q)ln (n n Px (0) + 1)(p)(q)13 (0) - [inEn2(p) + 11/2 [1nEn2(q) + 11/2 (2.22)Y

The y ( P ) , p 1, m, are assumed to be represented by the

multivariate weakly stationary process (Matalas, 1967).

akj

x)(. - mkir.k. = "

1J(2.25)

t= AY + B cYi+1 1 i+1

73

(2.23)

where A' and B' are determined in the manner indicated before for the

matrices A and B.

The determination of B' involves the matrix M 11' whose elements

are composed of the lag-one serial correlation of y(p) and y (q) , p,q =

1, ..., m, and the lag-one cross correlations coefficients of y (p) and

y ( q ) , p,q + 1, ..., m a (p)(q) (1). Again if the lag one cross correlationY

is of no interest it can be evaluated by (Matalas, 1967):

A (p)(q),I3 (p)(q) (1) = P (0) p7 (p)

(1)Y Y Y

(2.24)

It is then possible to obtain synthetic flows x(p) p = 1, ..., m,

by taking the antilog of the y ( P ) 's generated.

The Model by Young and Pisano. Following the introduction of

the model defined above, Young and Pisano (1968) devised a procedure

for generating operational hydrology (i.e., synthetic hydrologic data).

This procedure models the residuals

k -where xij is the flow for station k, k = 1, ..., n month j, j = 1, ...,

12, in year i, i = 1, ..., y and

Y kZ.

i=1 xljMkj

-Y

(2.26)

74

the mean flow of station k taken over the y years; a ki is the corres-

ponding standard deviation.

This transformation, called the cyclic linear transformation,

has been proved to be effective in removing the cyclic pattern of the

correlogram. The subtraction of the mean makes the assumption of sta-

tionarity, adopted for practical purposes, more valid in a physical

sense. This is true because a time series can be thought of as a sum

of a deterministic component and a stochastic one (Maddock, 1984). By

subtracting the mean, the main part of the deterministic component that

keeps the time series from being stationary is removed.

In the model proposed by Young and Pisano an effort to make

the residuals conform to a normal distribution can be made by taking

either log10 (x l.(.) or (x l. ( )1/2 , or performing no operation and finding

ij 1J

which option leads to the most nearly normal residuals. The ultimate

goal is to pick the option that leads to a minimum average skewness

(sum of the skewness at each one of the n sites divided by n). This

operation is called finding the minimum skewness transformation (MST)

(Young and Pisano, 1968).

The rest of the model is similar to the procedure developed by

Matalas (1967). For more details, the reader is referred to Young and

Pisano (1968).

2.2.3. Limitations of the Markov Lag-One Model

The following remarks inspired mostly by Salas et al. (1980)

pertain to the AR models in general.

75

The lag-one Markov model described before is a particular case

of the more general autoregressive model of order P, AR(P), of Box and

Jenkins (1970). The lag one Markov model of Thomas and Fiering (1962)

applied to the multivariate case by Matalas (1967) is an autoregressive

model of order one, AR(1). The model can preserve the mean, the

standard deviation, the skewness, the lag-zero and lag-one cross-corre-

lation, and the lag-one serial correlation of the time series. Although

the model preserves the lag one serial correlation and lag one cross-

correlation, it may not preserve the eventual long term dependence of

the historic record because the AR models have a "short memory" meaning

that the autocorrelation function decays very fast as the time lag

increases. The consequence of this limitation is that the AR(1) model

will tend to produce smaller droughts and smaller storage capacities if

long term persistence is present in the historic record.

The second limitation relates to the assumption of normality.

We've seen that if the original time series doesn't follow a normal

distribution, we need to work with its transform to ensure normality.

The drawback of using a transform is that the preservation of the

statistics of the transformed variable doesn't guarantee the

preservation of the statistics of the original variable.

Finally, the multivariate lag one Markov model requires that

the estimated lag-zero correlation matrix M o and the matrix BBT to be

consistent (a matrix is consistent if it is positive definite or positive

semidefinite, i.e., its eigenvalues are greater or equal to zero). This

inconsistency may occur when there are missing data or when the data

76

has different sample sizes. Crosby and Maddock (1970) have developed a

technique to guarantee the consistency of the matrices for the latter

case.

2.3 FFGN Model

For long-term trends, such as the range of "cumulative

departure" from the mean as suggested by Hurst (1956), the Markov

model has not been found to be satisfying (Askew et al., 1971;

Mandelbrot and Wallis, 1968; Chi et al., 1973). In addition, if the

Markov model is adjusted to fit the long-term persistence, e.g., using

an ARMA model, it becomes inadequate for the short-term properties of

the time series.

The ability of a model to preserve the range which gives an

indication of the storage required to regulate the long-run fluctua-

tions of a river system is a legitimate concern for any modeler who is

considering the use of synthetic hydrology in a study within the

context of the SRB. This basin is mostly located in a semi-arid region

under drought conditions for over a decade. This is the reason for

presenting on this part of the report the "fast fractional Gaussian

noise" model (FFGN) as introduced by Mandelbrot (1971) and modified by

Chi et al. (1973) for practical application.

2.3.1. Theoretical Background

The inadequacy of AR and ARMA models to represent both short-

term and long-term persistence can be overcome by decomposing the

approximating process into a rapidly varying (high frequency) component

77

(X h) and a slowly varying (low frequency) component (X

L).

X(t) = XL (t) + X

h (t)

(2.27)

where: X(t) is a standardized random variable. This is done using what

is known under self similar hydrology, which is the statistical model

using discrete fractional Gaussian noise (dfGn) or other related

fractional noise. Self similar hydrology, which has the basic feature

of possessing an exceedingly long memory, has been explored by

Mandelbrot (1965) and Mandelbrot and Wallis (1968, 1969) following the

work by Hurst, which is discussed next.

Hurst Phenomenon. Most hydrologists agree today that flows

for many streams seem to show persistence, high flows following high

flows and low flows following low flows. The short-term persistence

is pretty obvious. Hurst, after collecting nearly 100 years of

streamflow record on the Nile River investigated the long-term

persistence in the course of long-term storage capacities of reservoirs

(Buras, 1984).

Hurst and also Feller (1951) studied the range of cumulative

departures from the mean. The range R is defined as follows:

Consider a sequence of flows X 1 , X 2 ,..., X n with mean p and variance a2

.

The ith partial sum after i years is

iS 1 z (X. - 11) (2.28)1

j=1 J

Let M n and m n be the maximum and the minimum values of S 1 ,..., S n . The

range Rn is defined by

78

Rn

= Mn - m

n

Feller used the sample mean 1IT E 4 to define the adjusted range

" k=1 "

Rn = M n - m

(2.29)

(2.30)

where Mn and m are the maximum and minimum values of S

1' ... S. ...,

Sn with

S. = z (X. - z X /n)j=1 =11 k k

(2.31)

Hurst found that

ERn nH (2.32)

L a ]

where a is the standard deviation for the n annual flows and H the

Hurst coefficient which was found to range between 0.69 and 0.80. This

fact that observed series tend to give exponents greater than 0.5 is

called the Hurst phenomenon.

For independent normal random variables H is asymptotically 0.5

E(Rn) = (w/2)0.5 0.5

a n (2.33)

The empirical mean value of H = 0.73 found by Hurst is larger than the

value produced by simple AR models, which exhibit values of H tending

to 0.5 as n becomes large (Loucks, 1981).

The failure of AR processes, which belong to the Brownian

domain of motion, to explain completely the Hurst phenomenon led to

the development of the fractional Gaussian noise model by Mandelbrot

79

and Van Ness (1968) to produce flows with a specified value of H.

Brownian Motion and Gaussian Noise. The Brownian motion was

the finding of the British botanist Robert Brown who was studying

neutrally buoyant particles in a colloidal solution that move aimlessly

in all directions. A Brownian motion B(t) can be viewed as a sum of

Gaussian white noise, G(u), or, conversely, a Gaussian white noise G(t)

is the derivative of a Brownian motion (Chi et al., 1973). In mathemat-

ical terms,

tB(t) = jG(u)du (2.34)

-.

Markov Process. The Markov process M(t) defined in section

2.2.1 is closely related to white noise. If GM a white noise is a

forcing function of the dynamic system described by the differential

equation (Chi et al., 1973):

then

T dM(t) 4. m(t) = G(t)dt

JM(t) =

oe

.1 e-u/T

G(T-u)du0 T

(2.35)

(2 .36)

where T is the total period of time. It can be shown that for Markov

processes the correlation of lag k

HOrk = r1 ' r1 is the serial correlation.

80

Thus, Markov processes can reproduce faithfully the high frequency

component (rapidly fading memory) but fail to reproduce the low

frequency component (slowly decaying function).

Fractional Brownian Motions and Noises. The models discussed

so far will ultimately follow the S1/2 Law of Einstein giving H = 1/2

(Chi et al., 1973) after a transient period. To get a prolonged persis-

tence, Mandelbrot (1965) proposed an infinite memory length which led

to the development of the more general class of self-similar processes

of which the Brownian motion is a special case (Chi et al., 1973). This

class was designated by the terminology "fractional Brownian motion"

(Mandelbrot and Van Ness, 1968; Matalas and Wallis, 1971).

A fBm can be defined by an integral transform of a Brownian

process (Maddock, personal communication) and the derivative of a fB m

process is called a fractional noise denoted by X f(t) (Chi et al., 1973).

A fractional Gaussian noise (FGN) process is a sequence of normal

random variables with zero mean and unit variance (X 1 Xn) with auto-

correlations function

p x(k) = [11(+112H

- 211(12H

Ik-1121

(2.37)

where k is the lag and H the Hurst coefficient.

In 1968 Mandelbrot and Wallis proposed two FGN approximations,

Type I and Type II, for the computer generation of flows. However,

Mandelbrot (1971) found that the most economic and commonly used type

II approximation was inadequate and proposed consequently the fast

81

fractional Gaussian noise (FFGN). The FFGN of Mandelbrot modified by

Chi et al. (1973) will be presented herein for making practical compu-

tations that approximate fractional noise.

This method of Chi et al. (1973) uses an arbitrary "threshold"

value (see Mandelbrot, 1971) of 1/3 to separate the high and low

frequency components. They assume that the low-frequency components

that are expected to be important for larger lags S can be represented

by a weighed sum of N Markov processes of increasing correlation r up

to and approaching 1. The attractiveness of their approach is based on

the important feature of using unequal increments of r: closer

intervals are used for very high r so that the long-term behavior of

the time series is faithfully reproduced while very coarse intervals

are used for relatively low r so that the computational efficiency is

greatly improved. For the high-frequency components responsible for

the short-memory properties of the time series (low r) Chi et al.

represent it by one or more Markov processes to make up the deficiency

of covariance caused by neglecting the high-frequency components (those

below the threshold). Overall the method seems to have the following

advantages (Chi et al., 1973):

(1) High computational efficiency, for it requires relatively few

memory spaces and arithmetic steps in a computer.

(2) Potential for reproducing faithfully the low moments of the

historic data and the Hurst statistic.

82

(3) Strong emphasis on low-frequency components and provision of a

systematic method of generating the weighting coefficients (on

the basis of matching the covariances).

(4) Provision of flexibility in the selection of the high-frequency

components to fit the historic data.

The method's use as a practical simulation tool with the

concern of unifying theory and flexibility is presented below along

with criteria for parameter selection.

2.3.2. The Method of Construction of FFGN

As said before, the standardized series X f (t,H) (normally dis-

tributed with zero mean and unit variance) can be represented by the

sum of a low-frequency X L term and a high-frequency X h term. Chi et

al. (1973) put the separation level at r = e l = l/e = 0.368 or u = 1,

since u = -log r (see Mandelbrot, 1971). For XL (t) the effort is

focussed on large lags S in which case the covariance can be approxi-

mated by (Chi et al., 1973; Mandelbrot, 1971):

-1

C(S,H) = 1 [ $2H + 2HS2H- 1 4. 2H2H52H-2 - 2 52H 4. s2H

2 2

2H-1 2H(2H-1) 2H-2 2H-4- 2HS + S + 0(S )]2

H(2 1-l -1) 52H-2 A C L '(S H) (2 .3 8)

A weighted sum of Markov-Gauss processes is used to approximate the

low-frequency term instead of a single Markov process.

NX L (t) = E (W ) 1/2 M n

(t)n=0 n

83

(2.39)

where Wn are the weights fo the Markov-Gauss processes M(n)

(t).

Assuming that the Markov-Gauss processes are uncorrelated and

denoting by M (k)(t) the kth Markov-Gauss process, Chi et al. argue that

the serial correlation

Rk (S) = E [IA(k) (t+s) M (k)t] = r ISIk (2.40)

and the cross-correlation

Rid (S) . E[M (k)(t+s) M 0) (t)] = 0; k#j (2.41)

They therefore get the serial correlation l CL (S,H) of X L (t) by

NCL (S,H) . E[X L (t+S) X L (t)] = z W rISI

n nn=0(2.42)

Setting rn = e-u Chi et al. using the Laplace transform, show that for

the continuous case

CL (S 'H) = FO e-us 2H(1-H)(2H-1) x u 1-H dur(3-2H)

(2.43)

To avoid putting the emphasis on high- and middle-frequency, Chi et al.

replace u by BV. Incrementing v uniformly results in incrementing u

1CL(S,H) as given by eq. (2.42) is a covariance, to get the serial corre-lation one should divide CL(S,H) by the variance.

84

by unequal intervals for B>1. In this manner a great deal of emphasis

is put on the low-frequency (Chi et al., 1973). The substitution of B -v

for u leads to

c (s , H) _ 2H(1-H)(2H-1) log B •L r(3-2H)

co n+(1/2)-SB-v) x 82(H-z [exp ( 1)v

n=-- Jn-(1/2)I dv (2.44)

Chi et al. solve equation (2.44) and show that

H(2H-1) 1-H H-1 w 2(H-1)nCL (S 'H) - (B -B ) •z B exp(-SB-n) (2.45)r(3-2H) n=--

For the intermediate steps to get equation (2.44) and (2.45) the reader

is referred to Chi et al. (1973).

For practical purposes the limits of the integral cannot be --

and oe, which correspond to r=0 and r=1. Therefore, Chi et al. replace

the lower limit by n=0, which corresponds to v=0 or u=1 as said before.

For the upper limit Chi et al. argue that the procedure of Mandelbrot

is complicated and that experience has shown that for N>20 there will

not be any appreciable improvement in the accuracy of the approxima-

tion. Putting the finite limits on n Chi et al. come up with

where

NC L (S 'H) = z Wn exp(-SB-n )

n=0

- H(2H-1) (B 1-H - BH-1 ) B 2(H-1)nW n r(3-2H)

(2.46)

(2.47)

85

From the equation below

C(S,H) = 1 [is+1,2H+ 1S-11

2H 21s1212 I I

(2.48)

Chi et al. notice that the deficiency due to the approximation on the

covariance is

D(S,H) = C(S,H) - C L (S,H) (2.49)

Further, they notice that since CL (S,H) is a truncated version of C(S,H),

to represent the low-freqency component, D(S,H) should be positive and

decreasing as S increases. However, they realize that due to the

approximation in equation (2.45), it is possible that D(S,H) < 0. To

overcome this difficulty they propose to adjust B so that D(1,H) > O.

Then

D(0,H) = C(0,H) - C L (0,H)

and

=1 - z W n 0n=0

D(1,H) = C(1,H) - CL (1,H)

= 2 2H-1 - 1 - z W„ exp (-8 -n ) > 0n=0 "

(2.50)

(2.51)

Based on equation (2.50) and (2.51) Chi et al. argue that: if one elects

to use a Markov-Gauss process to make up the high-frequency deficiency

the process should have for variance D(0,H) and for lag one covariance

D(1,H); thus a Markov process can be used to represent the

86

high-frequency part since for such a process the covariance function is

completely defined if one knows its variance and lag one covariance

recalling that for a Markov process

= r (k)rk k = 0, ±1, ±2, (2.52)

1

which indicates

CH (S ' H) D(1,H) I5 I ISI >2 (2.53)

The approximation to fractional Gaussian noise then becomes

X f(t) = X (t) + XH (t)

(2.54)

where XH (t) is a single-term Markov process which covariance function

is given by equation (2.50), (2.51) and (2.53) (Chi et al., 1973). The

approximate covariance function of X f(t) is given by

COS,N) = CL(S,H) + CH (S,H) (2.55)

Finally Chi et al. remark that the accuracy of the approximation of the

FFGN can be improved in two ways:

Additional accuracy of the middle- and low-frequency part can

be obtained by adjusting the values of N and B in XL (t).

- Even though the method should be very accurate for the high-

frequency part, improvement for additional accuracy on this

component can also be obtained by using additional Markov-Gauss

processes to approximate this component instead of a single-

term Markov process.

87

To conclude this section on fast fractional Gaussian noise it

seems that the FFGN model presented herein has the potential to

reproduce faithfully both the short- and long-term properties of the

time series in general and in particular the Hurst statistic. In

addition, the method seems very flexible and easy to implement without

heavy computer time and space memory requirements. However, an

important limitation of this FFGN model is the fact that it is for the

generation of annual streamflows at one site. This difficulty can be

overcome by extending it to the multivariate cases. This can be done

in two ways. One is by generating the flows at different sites at the

same time using an algorithm similar to the ones devised by Matalas

(1967) or Young and Pisano (1968) and presented in section 2.2 of this

chapter. A second approach, preferred in this study and discussed in

more detail in the summary part of this chapter, is to combine the FFGN

model of Chi et al. (1973) with a disaggregation model discussed next.

2.4 Disaggregation Models

Even though a periodic multivariate model may be able to

reproduce all the properties of the time series such as mean, standard

deviation, skewness, and correlation structure that the analyst may be

interested in at a seasonal level, the model may not be able to

reproduce the annual characteristics, especially the annual serial

correlation structure. If such is the case, a multivariate annual

model may be used first to generate synthetic flows at several sites

and those annual flows are disaggregated into periodic flows.

88

Such techniques of disaggregation presented herein become an

important technique for modeling hydrologic time series (Loucks, 1981).

According to Salas et al. (1980), the first model of this kind was

introduced by Harms and Campbell (1967) who termed their model an

extension of the popular Thomas-Fiering model. In spite of its ability

to reproduce the desired results, it never caught on because of obvious

theoretical short-comings (Salas et al., 1980). The true beginning of

disaggregation techniques started with the first well-accepted model of

Valencia and Schaake (1973), which because of its classic form provides

a basis for all subsequent disaggregation models among which are the

model by Mejia and Rousselle (1976) and the one by Lane (1979), (Salas

et al., 1980).

Most disaggregation models have been applied to the temporal

domain although Lane (1979) applied the same technique to the spatial

domain (disaggregation of the total flow at a site on the main stream

into several partial flows at sites on its tributaries). The basic goal

of any of these applications is the preservation of the relevant

statistics at several levels. For instance, for temporal disaggrega-

tion, we would like to preserve the means, variances, probability

distributions of values and some correlations both at the annual and

monthly levels. We also might want to preserve the same statistical

characteristics in a main stream like the Senegal River and its

principal tributaries (Bafing, Bakoye, Faleme); this can be done using a

spatial disaggregation scheme.

89

This section of Chapter 2 gives a description of disaggregation

models for temporal and spatial applications.

2.4.1. General Disaggregation Model

In general, disaggregation modeling is performed generating a

time series dependent on a time series already available. The latter

independent series has previously been generated by any desired

stochastic process. This original series is referred to as the "key"

series and the dependent series generated from it is referred to as the

"subseries" (Salas et al., 1980). In this study the key series may be

generated by any of the models mentioned before, i.e., AR, ARMA, FFGN,

etc.

All disaggregation models may be represented by the equation

Y = AX + Be (2.56)_

where Y is the subseries, X is the key series, A and B are the_

parameters expressing the causal structure and e is a random series.

In general, Y, X and e are column matrices and A and B are parameter

matrices. For example, to jointly disaggregate annual values at two

stations into monthly flows: Y would have dimensions of 24x1 (12

monthly values for each station), X would have dimensions of 2x1

(1 annual value by station), e would have dimensions of 24x1, and A and

B would have dimensions of 24x2 and 24x24, respectively.

The following assumptions will be made:

- Each of the time series forming X and Y follow the normal dis-

tribution with mean zero.

90

- The random terms e are distributed normally with zero mean and

unit variance.

In some cases the first assumption can be omitted by adding a

constant C in equation (2.56). In addition some authors add the

condition of unit variance for X and Y. However, this restriction_ _

destroys largely some interesting properties of the data if it is

originally normally distributed, hampers the detection of some possible

computational errors, requires more storage memory in the computer

program (additional parameters to be stored) and causes a loss in the

feeling for the relative magnitudes of the various series (Salas et al.,

1980). Because of these multiple limitations of this restriction it

will not be used in this study.

2.4.2. Disaggregation Models

Temporal Disaggregation. The basic form of the models by

Valencia and Schaake (1973), Mejia and Rousselle (1976) and Lane (1979)

will be presented herein along with a consideration of their advantages

and drawbacks to select the one that will be used in this study.

The basic form of the model by Valencia and Schaake (1973) has

the following form:

Y = AX + Be (2.57)

For application at one site, X is the annual flow at that site and Y is

a column matrix of seasonal flows at the same site which sum to the

value of X. X and Y have zero mean and unit variance and maybe trans-_ _ _

formed values which do not indeed have the dimension of a flow.

91

However, we keep referring to them as flows for practical purposes.

is a column matrix of standard random series N (0,1). A and B are

parameter matrices that express the causal structure and are designed

to preserve the covariance between annual and seasonal flows and to

preserve the variance and covariance among seasonal flows. For W

seasons, W2 + W separate variance and covariances are maintained (Salas

et al., 1980). In addition to this advantage, the model is easy to use.

However, the moments preserved are not consistent for say the value

for the last season of the year is generated preserving all covariances

between itself and the preceding (W-1) seasons, while the moment for

the first season is generated without preservation of covariance

between itself and any preceding season (Salas et al., 1980).

Mejia and Rouselle (1976) extended the model of Valencia and

Schaake (1973) by adding a new term to preserve the seasonal covari-

ances between seasons of the present year and the seasons of the

preceding year. The model can be represented by

Y = AX + Be + CZ- _ _ (2.58)

where Z is a matrix column of seasonal values from the previous year

(as much seasonal values are desired). C is an additional parameter

matrix while the other terms are the same as in equation (2.57). This

extension doesn't correct the inconsistent causal structure. There are

more parameters and their estimation is more complicated (Salas et al.,

1980).

92

Lane (1979) introduced an approach that simplifies the model of

Mejia and Rouselle (1976) by setting to zero several parameters of the

model which are not important. This way the number of parameters to

estimate is reduced as well as the number of moments preserved (Salas

et al., 1980). The form of the model stays the same but it is

presented on a "one-season-at-a-time" basis and with only one lagged

season (Salas et al., 1980). Thus,

Y = AX+Be+CY,-T T- T- T T-1

(2.59)

where T takes the values 1, 2, ..., W for W seasons. AT , B T , and C T are

single element parameter matrices like Y T , X and e.

This model preserves the covariances between the annual value

and its seasonal values. It also preserves the variances and lag-one

covariances among the seasonal values. One disadvantage of this model

is that it is less straightforward. Another disadvantage, common to

the two other models if transformed data is used, is the fact that the

generated seasonal flows may not add to the annual values used to

generate them. This shortcoming can be taken care of either by

adjusting the annual values using the generated seasonal values or

preferably by adjusting the seasonal values so that they add to the

annual values (Salas et al., 1980).

Although the model by Lane (1979) is less clear, it takes care

of the moments consistency mentioned herein. It also requires less

parameters to estimate than the model by Mejia and Rouselle (1976).

Finally, it preserves the covariances between annual values and their

93

seasonal correspondents, the lag-one covariances among seasonal values,

and the means, variances, and skewness; this is good enough as far as

maintaining relevant statistical characteristics for the scope of this

study. Because of all these considerations the model by Lane is

selected in this study for further investigations.

Temporal disaggregation can also be performed with a multisite

approach to maintain additional correlations; these are the cross-

correlations between the seasonal values at various sites. In order to

preserve these additional correlations, which are indirectly maintained

to an extent through the cross-correlations of the key series and the

annual-seasonal correlations, additional parameters are required

because the matrices are bigger. For the model by Lane (1979) the

matrices in equation (2.59) have the following dimensions for jointly,

disaggregating annual flows at n sites into W seasonal values:

Y and Ynxl for each season T-T -T-1'

X and E, nxl for each season T

A T,B T,and C , nxn for each season T

Spatial Disaggregation. The model by Lane (1979) can be used

for spatial disaggregation.

= AX + Be +CZ_ _ (2.60)

where Y is a column matrix of substations (stations on tributaries)

flows being generated, X is a column matrix of key station (station on

main stream) flows to be disaggregated, and Z is a column matrix of

94

previous substation flows. A, B and C are parameter matrices. This

model based on the model of Mejia and Rousselle is designed to preserve

the lag-zero correlations among the substations and lag-zero correla-

tions between the key station and the substations (Salas et al., 1980).

Spatial disaggregation can be performed with several stages.

For instance, disaggregate a key station into several substations and

then consider each one of these substations as the new key station and

disaggregate it into several substations. Equation (2.60) can also be

used with several key stations.

2.4.3. Parameter Estimation

Temporal Disaagregation. The parameters of the model by Lane

(1979) for temporal disaggregation are estimated by (Salas et al.,

1980):

.

-1

[Sxx(T'T) - Sxy(T,T-1) Sy; 1 (T-1,T-1) Syx(T-1,T)] (2.61)

e T = [Syy (T,T-1) - A1 S xy (T,T-1)] Sy; 1 (T-1,T-1) (2.62)

g gT = S (T,T) - A S (T,T) - E s (T-1,T) (2.63)IT yy T xy T yy

equation (2.63) is solved as mentioned in Section 2.2.2. In the notation

Svw (i,j) for the covariance, i and j reflect the season (or lag)

associated with V and W, respectively. For example, S (T,T-1)xy

indicates the covariance matrix between the annual value series

AT = [S (T T) - SYY

(T,T-1)SYY

-1 (T-1'T-1)S

yx(T-1,T)]yx '

95

associated with the current season and the seasonal values associated

with the previous season.

The required moments are illustrated for joint disaggregation

(of annual flows at two sites. Given the seasonal series y_

i) where: iv,T

is the site (i =1,2); v=1, ... N denotes the year; and T=1, ... W for the

season and using a matrix approach to estimate all covariance elements

at the same time (Salas et al., 1980):

N1

S (T T) - j. EYY ' N-1 v=1

- (1) -Y V,T rv (1) v (2)1

1:' V,T ' " V,T j (2)Y_ V,T

(2.64)

1 1 t:I.S (T,Ti - — ,...yx N-1 v=1

_ (1)3' v.,T

(1) (2)][Xv , Xv (2.65)

.,(2)Y V,T _

1 N

SXX (T2T) - - EN-1 v=1[41) , x,(12)1 (2.66)

N1S (T,T-1) = EYY N-1 v=1

F.( 1 ) „(2) 1Li v o.-1' J v,v- 1.j (2.67)

96

N1S (T-1,T) - zyx N-1 v=1

- v (1) -" v,T-1

„(2)Jv,T-1

[ (1) (2)]X v , X v (2.68)

S (T,T-1)= ST (T-1,T)

xy yx

and S (T-1 T-1) is obtained from (2.64) for T-1 instead of T.YY '

Spatial Disaggregation. The spatial disaggregation is designed

for annual values; it therefore does not have an excessive number of

parameters. The parameter estimates are given by (Salas et al., 1980):

-1A = FS -S (1) S - 1 ST a)] • [S - S (1) S y-1 ST (1)] (2.69)

L yx yy yy xy xx xy y xy

E = Is (1) - "A s (1)1 s -1L YY xy j yy

giT = Syy -AS -EST (1)xy YY

(2.70)

( 2.71)

In these three equations the notation (1) means we are dealing with a

lag one. For example, S xy(1) is the covariance matrix between the key

station values in year v and the sub-station values to generate in year

(v-1).

For two key stations to disaggregate at four substations:

97

Yv (1)

y v (2)

=YY

1 r

v=1 Yv(3) [Y v (1) yv (2) yv (3) y v (4)] (2 .72 )-N-1

Yv(4)

xv(1)

SXX

1

v=1 xv(2)[xv (1) xv(2)] (2.73)

N-1

yv(1)

yv(2)

S =yx1 zv=1 yv(3)

yv(4)

[xv (1) xv (2)] (2.74)N-1

1S(l) = EYY N-1 "1

Yv(1)

Yv(2)

Yv(3)

Yv(4)

[Yv-1(1) Yv-1 (2) Yv-1 (3) Y(v-1) (4) ]

(2.75)

and

1S (1)=xy N-1 v=1[yv-1 (1) yv-1 (2) y

v-1(3) y

(v-1)(4) ]

98

(2.76)

As a summary to this part on disaggregation, disaggregation

models can be used to disaggregate annual flows into seasonal flows

(day, week, month, etc.) or to disaggregate flows at one site into

flows at several other sites within the same river basin. Used for

temporal applications, statistical characteristics relevant to many

studies (mean, variance, skewness, correlations and cross-correlations)

can be maintained at several levels, say annually and monthly. The

same statistics can also be maintained for a key station and each one

of the set of several substations if spatial disaggregation is used.

With the spatial case, disaggregation can be an aid to fill in missing

values of unequal record lengths or it can be a method to avoid this

task for the key series and the subseries do not have to be of the same

length. The only limitation of disaggregation models of relevance in

this study is the fact that disaggregated flows may not add up exactly

to their initial aggregates. This problem can be handled by adjusting

the disaggregated flows to the aggregate flows with the potential risk

of disturbing the preservation of the distribution of the time series.

2.5 Summary and Conclusions

Three categories of streamflow generating scheme were

presented in this chapter after preselection in section 2.1. In this

section we summarize their respective properties with a comparative

99

approach in order to make a final selection of the models that seem

adequate for this study. The selection will be followed by a

definition of the exact form under which the selected models will be

tested using streamflow data of the SRB.

2.5.1. Selection

The lag-one Markov-model were found by different hydrologists

(Salas et al., 1980; Phien and Ruksasilp, 1981; Matalas and Wallis,

1971) to preserve well in general the mean, the standard deviation, the

skewness (depending on the distribution and/or transform used), the

lag-one serial correlation, and the lag-zero and lag-one cross correla-

tion of the historic record. However, the model may not preserve the

eventual long-term persistence because it has a "short memory." It

will therefore tend to produce smaller droughts and smaller storage

capacities if long-term persistence is present in the record.

To overcome the "short memory" limitation of the AR(1) model

higher order AR(P) models or ARMA (p,q) models can be used. Reliable

parameter estimates are hard to get for higher order AR(P) processes

particularly when the data available is limited. Used in the appropri-

ate form ARMA (p,q) models can maintain the same statistics as the

Markov models plus long-term properties. However, while gaining these

long-term properties they might lose the short-term properties. In

fact Panu and Unny (1978) argue that they were developed by Box and

Jenkins (1970) primarily to have short-term properties. In addition,

the procedure to generate streamflow using ARMA (p,q) models is lengthy

and very complex.

100

A statistic of interest in addition to the mean, standard

deviation, skewness and correlations, particularly if a large proportion

of the flow is to be developed, which is the case for the SRB develop-

ment plan, is the range R of accumulative departure from the mean.

The maintenance of this long-term statistics by a model is related to

its ability to explain the Hurst phenomenon. To explain the Hurst

phenomenon, in other words maintain long-term properties, without

losing the short-term properties the FFGN model by Chi et al. (1973)

was presented. The limitation of this model is the fact that it is for

the generation of annual flows at one site.

Disaggregation models have also the ability of maintaining both

the short and long term properties of the historic record (Tao and

Delleur, 1976). This category of model can be used for spatial and

temporal disaggregation. However, before disaggregating annual flows

to monthly flows the annual flows have to be generated first. In the

same line, before disaggregating the flows at a key station into flows

at substations one has to generate the key series first.

2.5.2. Model Definition

In the SRB, the streamflow record available for the gaging

stations on the Senegal River is longer and the data is more reliable

than for the stations on the tributaries. Therefore, a multisite model

that will transfer the information available in the main stream to some

of the tributaries such as the Bafing where the main reservoir, the

Manantali dam is located, is more appropriate than a single site model

for stream flow generation.

101

Operation of the Manantali dam to meet the water demands of

the various components of the SRB development plan and the management

of this plan will necessitate knowledge on the within-year as well as

over-year storages of the Manantali dam. For this purpose, both

monthly and annual flows are needed.

To get these streamflow records (annual and monthly) two

approaches for their generation can be used. One approach would be to

generate directly monthly streamflows using the model by Young and

Pisano (1968) and get the annual record by aggregation. The problem

with this approach is that the monthly statistical characteristics may

be maintained but not the annual ones. A second approach which will be

used in this study is to generate the annual flows first and then get

the monthly flows by disaggregation to preserve both short- and long-

term properties of the historic record, i.e., at two levels (monthly and

annual).

Considering the advantages and limitations of the different

models presented in this report the following modeling schemes can be

used to generate synthetic streamflows:

Model 1:

(a) Generate annual flows using the lag-one Markov model of

Matalas (1967) for multisites.

(h) Oisaggregate the annual flows at each site into monthly flows

using the disaggregation model by Lane (1979).

Model 2:

(a) Generate annual flows at the key station.

102

(h) Disaggregate this key series of annual flows for one site into

several subseries of annual flows at other sites using the

model by Lane (1979) for spatial application.

(c) Disaggregate each one of the annual flow sequences for all

sites into monthly values using the model by Lane (1979) for

temporal application.

Both Model 1 and Model 2 use two assumptions:

- the stationarity assumption of the flows

- the assumption that the flows are normally distributed.

Only model 1 will be tested with the data collected in the next

chapter, and recommendations will be made relating to further investi-

gations using model 2 in the last chapter of this report.

CHAPTER 3

MODEL APPLICATION

The objective in this chapter is to see how model 1 can be used

for generating streamflows in the SRB and what kind of problems may

arise in the process. In section 3.1 we analyze the data to find out

what distribution to assume for the flows and how to fill-in the

missing values since model 1 requires that all the sites (or stations)

considered have the same record length at least for the annual flows.

Then in section 3.2 we illustrate the generation of the annual flows

and draw some conclusions in section 3.3.

3.1 Data Analysis

Mean monthly flows for every month were collected for the

gauging stations at Bakel, Kayes, Kidira and Galougo, all located in the

Upper Valley of the SRB. The periods of measurements and the years

with missing values are shown by Figure 3.1 for each site. The

locations of these sites are given in Figure 1.3 of Chapter 1. All the

flows referred to in this section and the next one are in m3 /s.

These raw data presented in Appendix A were compiled from the

monography of ORSTOM (Rochette, 1974) and from the hydrologic records

of the Senegalese Ministry of Hydraulics.

In order to use the data collected to run the two models

selected in Chapter 2, we analyzed the four historic records to check

the normality of the flows distribution, and to fill in the missing

103

1903-04(1)

1951-52 (2)

1973-74

1981-82(1980-81)

(4)

1951-52

(1968-69)

(1969-70 )'

(1970-71) —(1972-73) --

(1974-75)—

(1975-76r —(1980-81)- —

1981-8

(3)1903-04—

(1904-05)

(1914-15)

(1919-20)

(1924-25)

1964-65

104

Fig. 3.1. Periods of Measurements and Missing Values forMean Daily Flows (and Monthly Flows) at Bake](1), Galougo (2), Kayes (3), and Kidira (4).Dates in parentheses indicate years with missingvalues.

105

values. Model 1 presented in Chapter 2 requires that the time series

to use to generate the flows be normally distributed and stationary.

Since there is no basis to prove or disprove that the flows in the SRB

are stationary, we assume for the purpose of this study that the flow

residuals (standardized or not) are stationary without verification.

3.1.1. Normality Check

The objective here is to see whether the original data should

be assumed to follow a normal or lognormal distribution for the appli-

cation of model 1. We chose the logarithmic transform because it is

more commonly used in hydrology than other transforms such as the

square root and leads to a family of distributions, the lognormal dis-

tribution with or without an upper and/or a lower bound, that has some

physical meaning. The gamma distribution could also be considered.

This is not done in this study for reasons given in the summary part of

this section. The data used for this analysis did not include the years

of the streamflow record with missing values.

Methodology. To accomplish the objective mentioned above, we

tested the normal distribution and the lognormal distributions. The

check was performed on the monthly, seasonal, and annual standardized

residuals as defined below:

--Monthly residuals:

x(k) _ 7 (k)

ri(

jk ) =

01)(3.1)

106

where:

(k)-x.. = flow of month i, year j, and site k13

i = 1, ..., 12; j = Nk; k = 1, 2, 3, 4; -N k being

the total number of years, the mean being

Nk7.(k) . 1 (k)

N k j=1

and the standard deviation being

[ 1 islz k [x (k) _ afk))2] ]1/2

i,kN k -1 =1j

--Seasonal residuals:

We distinguished two seasons: a high flow season (July, August,

September, October, and November), and a low flow season (December,

January, February, March, April, May, and June). The formulas for the

seasonal residuals are the same as the one for the monthly residuals

with the difference that i takes two values, one for the high flow

season and two for the low flow season.

--Annual residuals:

The formulas are the same as for the monthly case with the

exception that one does not need the index i in Equations (3.1), (3.2),

and (3.3). If the standardized residuals are normally (or lognormally)

distributed, so will be the case for the unstandardized residuals and

the actual flows.

(3.2)

(3.3)

107

All the computations were done using three computer programs

written in Fortran V by the author: PROGRAM-DATA1 for monthly flows,

and two modified versions of this program for seasonal and annual

flows. All three versions of the program, presented in Appendix B,

uses subroutines of the IMSL package (IMSL, 1982) and subroutines

written in Fortran V by the author. The programs had the following

steps for each site:

Step 1: Input the monthly flows for the first version of PROGRAM-

DATA1 and then transform into seasonal and annual flows

for the two other versions using a subroutine called SUB.

DATA.

Step 2: Computation of the basic statistics (mean, standard deviation,

and the coefficients of skewness, kurtosis, and correlation) using

the subroutine SUB3.

Step 3: Computation of the residuals using Equations (3.1), (3.2), (3.3),

and the results of Step 2.

Step 4: Computation of the basic statistics of the residuals as in

Step 2.

Step 5: Normal probability plot of the residuals of the flows in Step

1 using the subroutine USPRP of the IMSL package.

Step 6: Performing a chi-square test on the normal distribution using

the IMSL subroutine GFIT.

Step 7: In this step, we perform another test on the distribution, the

Kolmogorov-Smirnov test, using the IMSL subroutine NKS1.

108

After performing these seven steps the natural logarithm of

the monthly, seasonal, or annual flows are taken and steps 1 through 7

included are repeated.

The chi-square test is commonly used with at least five to ten

observations per equiprobable cell to reach valid conclusions (Benjamin

and Cornell, 1970). To guarantee that this condition is met, two cells

(i.e., one degree of freedom) were chosen for all the sites except

Bakel for which five cells (three degrees of freedom) were chosen. The

subroutine GFIT gives the number of observations in each cell. It also

gives the chi-square statistic (CS) and the significance level for

accepting the hypothesized distribution.

The Kolmogorov-Smirnov test is based on differences between

the empirical and the hypothesized distributions. The subroutine NKS1

gives the significance level for accepting the hypothesized distribution

for one-sided and two-sided alternatives. For the two-sided alterna-

tive, of interest herein, the hypothesis of equality versus the case of

inequality is tested. The critical values, thus the significance levels,

given by this test are perturbed if sample estimates are used for the

theoretical distribution used as is the case in this study. However,

this drawback is relatively unimportant for the context in which the

test is used herein. The test is used to get an indication on what dis-

tribution to choose between the normal and the lognormal, and not to

decide the actual distribution of the flows, for both the annual and

monthly flows.

109

For both the chi-square and the Kolmogorov-Smirnov tests, the

distribution considered is accepted if the significance level is greater

or equal to 5%, value commonly used. The distribution with the highest

significance level will be preferred to the other.

Basic Statistics. The mean, standard deviation, skew coeffi-

cient, and serial correlation were estimated using the methods of

moments as presented in the next section. Tables 3.1 through 3.6

present the basic statistics (mean, standard deviation, skewness, and

correlation) of the flows and their logarithmic transforms for the

different sites. Tables 3.1 through 3.4 are for the monthly flows.

Table 3.5 and Table 3.6 are for the seasonal and annual flows, respec-

tively. The comparisons of the skewness values given in Tables 3.1 to

3.4 do not clearly show if the transformed or untransformed monthly

flows have a skew coefficient close to zero. Therefore we cannot use

this criteria to choose the distribution to assume. However, it seems

that for the annual and seasonal flows the untransformed flows have a

smaller skew coefficient suggesting the normal distribution.

Normal Probability Plot. The normal probability plot was not

conclusive for the monthly and seasonal flows. This failure to

conclude on which distribution fits best the data came from the fact

that in general one can fit a straight line equally well for both dis-

tributions. However, the normal distribution seems more adequate for

the annual flow residuals, as shown by Figures 3.2 through 3.9 for the

four sites.

110

4.0 LO 0'1 4.0•z:r d- LU Ln L.o

• • • • • • • • • •

L0 0 (N.1 LO Cs..1 0 4.0 LnL.C.) (e) C•1 0 Li-) co

• • • • • • •r--1

I

cr) 00 Cr) re) N (e)

ce) ce) re) LC) ,,z1- (e) d- LO• • • • • •

r-I

Gi" I-I d- CO 01 01 Lc') 0ct- C\J I-0 CO Q 0.1 LO Q 0-) ce)

LU• • • • • • • • • • • •

r-. h. L.0 LC) d- d- ce) C

CO d 0 0 LO 0 N. 01Ce) C ce) d- Ce) LO

• • • • • • • • • • • •

LU (.0 0 d- Q CTI re) CV LC) LC) LC)LLD cr) Q Lo •-4 CO 00 C.) CD VD

• • • • • • • • • • • •

T--1 1-1

CO LO Q 00 •-4 00 ce)00 00 0 d- CO 0 CO OD ct

• • • LC) d- co 1-4 P., • • •

Cn C ce) ( L1 re) L.0 LC) CTLO CO d-

LO 0 ce) c CncY) LC) 00 01 ('e) d- co oo Lso

• • • CO co • •cr, N. d- cr, Ln (\J LE) re)

0 CV LC) (Y)Ln

>.) Ç r- Cr) O. -I-) > L.) C _0 S-rCS (.) 0 CI) cr5 C1.1 c0 C.

cr V) 0

111

C(13 0s_S- 4-)CI) ° (0

(-)

(r)tr,

-0 0

(C-

(o 4_,-cs

co >4-) a)cr)

CC

t1:3 0

COLs)

•

c.)k.o

n-•-n

0

CO

•

CO

CO

•

ON..

01C

•

CO1-1

•

d-

C

O..1r-1

•

cnco

•

C..1En

•

CV

•ko

DCV

Cr)

•

1.11cr

•

LC)4n1-

•

1-1(0

•

LO•-n

N.Cci

•

coCO

•

COGt

•

CoC

•

I

coCO

•

cor4

•

0

•

COCV

•

CLICO

coCY)

•

co.c.)

•

re)CI'

•

01

•cf3

COCO

d-•

c,-)N.

•

r- 4'St

•

Ore)

•LO

d-

•

kroc3,

•

COCI'

•

DCO

•

d-Cr

ko

•

coCV

•

C)

•

tc\J

•

CT)CO

CO

•

Lr)CO

•.-4

COl0

•

(0in

•Cr)

d'CO

cs)C

•

dd-

e-1

inCO

•

LC)ttO

•

t-4

(11.1 • • • • • • • •

•-•

C dr 03 cr) cr) cs.CO CO 0 CI CO 03 Cr) CO •-n c\I

• • • • • • • • • • • •

V)

CS- 0(r.) +

3Gi"CT)

COCLI CO d- CD 0.1 CO N.

("001

''Zt•-1

,--4ID

. i"CV

C rCi•

L)•

N.LC)C•J

CO0)

drCO

LON.

cnJC \ I

Co •N.

•0 01

••--I

CC ,->

1.0 •--i .ezt CO t-4 1-1

QJL3")

roLC)cr:,

•

OJLs,

•('siCD C CV Cr) CY) Cr)

Ccs)

•

Ce)

•Cr) CO LC, CV

Cs.)CVcr.)

Ls) in (".1 4 CD CV 03t-i

CCr-D r-D

CI)

c:C

0_QJV)

4-,L.)0

>0 QJ

CD

Cro

-0a)

5-ro

S.ro_

112

01 0 01 LC) 00 CO C) LO co co• cr CO CV co CV Ce) cr)

• • • • • • • • • • • •

d - CO et LO LI1 grd- kso

cr kr) 01 CV I-I n• ---1 C) n-f• • • • • • • • • •

n-•nI I I I

LI) CV et 0 LC) CV et LC) CO 00 LI) 01er CO et et CO et CO CV CV CV Ce)

• • • • • • • • • • • •

1-1 ol N 11")t.c) in co CD CD

• • • • • • •er cc r-.

Lc, co ,-i r--I Ce)CO t•-n CV et CO

• • • • •Lf) et et Ce) CV

cd- co co oo co c) r-. kr, oo cf) 1.1-)CV CV CV CV CV re) CV

• • • • • • • • • • • •

• C) c) 00 in c0h . CO 01 LC) 0 CO Cel el- CI) et LI)

• • • • • • • • • • • •C•") r-I 1-1

01 CV LO 01 CD LC) 000 CO et CD Cf) 01 el- d- CO et n-1 CID

• • cf) 440 CD •-I LC) • • ••ZI• CV CV r.... co ix) ,--1 ,czr c) al ,--1 L.c)

k.r0 Lx-) co ,--1 ,--4

Lo clo co co co - 01 CO CV CD Lf)I-I

•

et CO CO e-1 01 ,--i L.C1• ,-I LI) CD CO C1.1 d- • r-I • •

CV CV r-I 0.1CO

>, DI Cl 4-) > = S-(ES la) 0 CD cl5 1:1) (IS 0_

r -D CC V) al z

QJ

113

tdo ol Lc, h, cr co Lc) oooo C.) co CV LC) Lt) Lf) d-

• . • • • • •

oo oo co co LO CY" Cc")CO 1.0 CD 03 1.0 CD N. Ls, 00 dr• • • • • • • • • •

C.) s-s) .--411111 111111

C11 N. Lf) LC) CD CD CD C‘I C\JCD k.10 CO LC) CO N. N. N. CD In 00

• • • • • • • • • • • •,--f Lr)

CON- CV CD VD LC) co zl crt cz, COCOcY) oo Lo co (..o LC) N. CD N.

• • • • • o • • •Cr L.0 CO

-

1 CV1 1

03 0) 0)cn d- 01 00 ch COCO o coL0 CV d- Cr) CO d-

• • • • • • • • • • • •

C') —I a) a) —t dr. Ls-) co —1 C..) d-

t.10 N. CV L.0 ' n-1 CD CV• • • • • • • • • • • •

CV) 1-0 CD 0 N. CO VD 0LC) CV CV 0)0303CO co CD Cy) •;:t

• CO (.0 00 • • • • • •▪ N. 0) CV d' co co c•-> cssi

•-• (.0 CV

LC) al co a) dr-Ls, co co co, c- co co N. to lap Lc) Lc)

• • • cs..1

•

o • • a • •

Cr (.0 0) CO to N. coCO I—I

>) r— 0) 0- 4-) >S-U 0 (1) ro at co ca.

Cl) CD CD D LL-

•r-

114

Lf) O Q C \1 CO Cy) Q 0

LO C\I C\J d- CY) L.0 (Y.)• • • • • • • •

CY) N. N. LC) C)N- ccCO CD) Is--

• • • • •

• •

1 1 1

Q cr, ‘o 1..r) Q cpco .4- CV CO •://-

• • • • • • • •

LO CO 01 0 C ("J 0

C \I 01 co co co c‘i• • • • • • • •

c...1 Lc) r•-•

CO LID CDLc -) C CV C \

• • • • • •

(" d- CD (Y) LC) (.0 CO

1.-i 1-1 CLI• • • • • • • •

CO Q CD Crlri in CO Cy, (.0

• 1".n • CD) • CO •C •Zt (y) LO 0/ COC \J CY) C

1--.. (.0 0 C\J r-4 (y) ,--1C \ I r.... LO 01 LO ('LI d- 0

• Cf) • CO • t.f) • d'CV d- 01 •--1 is, ri CD .-4i-i CO CO 00

3 330 30O r-- o

4- r- 4-4- 4-

-C -CCD ill 1.7) I.....

0 • ,- CU 0 .1... a)—1 = >, -J 2 -.Y

CO CO]C CC)

30 300 0

4-0 4-

-C CI -C3 Cr) Cr)0 r 0 0

- J r- --/

115

o cc) s..ro<NJ cv)

it)

co 01 1.0• • •

0 CV c")

s.r) cc) cc) co

CNJ

1-1 •ct 141 CI"• • •

1-0 t.C) (.0 CO

C•J o I-1

l0 CJ CnI (Y)• • •

.r-d- r, a,

- .-1 .-i d-

• • • .

I I 1

LOLc) s..o

C 1-1

01 CO 1.0

1.70 C•1 Cn.11 -1 1.0

COo

corss- 1h r z

(1) ) 0r-

CO C.0V)

116

PROBABILITY PLOT FOR NORMAL DISTRIBUTION.14E+01 X**X x*****

XX

.10E+01 +

X.65E+00 + X •

XX

X.29E+00 +

XX

0B -.66E-01 +

X X

. VA -.42E+00+

X XX

O X

S• -.78E+00 + X

X

-.11E+01 +

X-.15E+01 +

.*X

-.19E+01 +

-.22E+01 *****x*******+*******+** ***** **********+************ ****** **-4-.01 .05 .10 .50 .75 .90 .95

Fig. 3.2. Annual Flows at Galougo

PROBABILITY PLOT FOR NORMAL DISTRIBUTION11E+01 + X**X*****X

XX

75E+00 + XX

XX

X36E+00 +

XX

30E-01 + X-X

XXX

42E+00 +X

4. X

81E+00 + X

12E+01 +

X

16E+01 +

X-.20E+01 +

-.24E+01 +

- .28E+01 +****x***+ ***** ****** ** * ** ***it-1-m* ** * ** * ** ****+************+.01 .05 .10 . . 75 .90 • 95

117

Fig. 3.3. Log Transformed Annual Flows at Galougo

118

PROBABILITY.20E+01 +

***

PLOT FOR NORMAL DISTRIBUTION

XXXX***

* X *.16E+01 + XX *

* ** X ** ** X *

.12E+01 + XXX ** X ** ** ** X *

.85E+00 + XXX ** ** XX ** *

0 * X *B .46E+00 + XX *S * X *E * X *R * X *V * XX *A .77E-01 +T

X **

I * XX *

*0 * X *N * XXX *S -.31E+00+ XX *

* XX ** X ** X ** X *

-.69E+00 + ** XX ** XX ** *

X *-.11E+01+ X *

* XXXX ** XX ** ** *

-.15E+01 + X ** X X ** X ** ** X *

-.19E+01 ++***X*** + + .01 .05 .10 .7*......, .50 .75 .90 .95

Fig. 3.4. Annual Flows at Bakel

119

PROBABILITY PLOT FOR NORMAL DISTRIBUTION.16E+01 + ******* ***************** ***** ***********************x*X****X

* X *x x* *

* X ** X *

.11E+01 + XXX ** X *

, * ** x ** XX *

•73E+00 + XX ** X ** XXX ** X *

X * *

.32E+00 + XX ** XX ** X *

X * *O * XX *B -.B4E-01 + XX *S * XX *E * X *R * XX *✓ * X *A

-.49E+00+ x *T * X *I * xO 4. X * *N * *S -.90E+00+ *

* X ** X ** XXXX ** X *

-.13E+01 + X ** ** ** ** x *

-.17E+01 + ** x *I. x ** ** x *

-.21E+01 + ** ** ** x ** x *

-.25E+01 ***** ****+****+*******+*******+******* ***** ***+****.********+*.01 .05 .10 -,..-• •••••n .50 • 75 .90 .95

Fig. 3.5. Log Transformed Annual Flows at Bakel

PROBABILITY PLOT FOR NORMAL DISTRIBUTION

.14E+01

.18E+01 +

+

XX

X

X*

•

* XX* XX X ** XX ** *

.99E+00 + X ** ** ** XX ** XXX *

.57E+00 + XX ** X ** X ** •

0 * *B .15E+00 + X *S * X *

*E * XX•R * XX

V * X •A -.27E+00 + X *T * X *I * XX *0 • XX *N * XX •S -.69E+00 + X •

* **

XXXX

-.11E+01 + X •X

XX

-.15E+01 : •X *

* X •* *• •

-.20E+01 + X ** *• ** •* *

-.24E+01 +X ***** **+ +.01 .05 .10 ......,-0.- .50 .75 .90 .95

120

Fig. 3.6. Annual Flows at Kayes

121

PROBABILITY PLOT.15E+01 +

•

FOR NORMAL DISTRIBUTION

X XX**

* X ** XX ** XXX X *

.98E+00 + ** X ** X ** XXXX •* XX *

.50E+00 + ** X ** X ** X ** XX *

.22E-01 + XX *•* X

* XX ** XX *

0 * XX *8 -.46E+00 + XXX *S * X *E * *R * *V * XX *A -.94E+00 + XX *T * X *I * *0 * X *N * X *S-.14E+01+ *

* X ** ** ** X *

-.19E+01 + X ** ** ** ** *

-.24E+01 + X ** ** ** ** *

-.29E+01 + ** ** ** ** *

-.33E+01 +X *** ******+********+*******+***+********+.01 .05 .10 . .25 .50 .75 .90 .95

Fig. 3.7. Log Transformed Annual Flows at Kayes

.10E+01

.13E+01 PROBABILITY PLOT FOR NORMAL DISTRIBUTION+

+ X X

X

X

x****,

X

X

.73E+00 + XX

X

.42E+00 + XXX

0 X.11E+00 +

VA —.19E+00 +

X

—.50E+00 + XX

—.80E+00 + XX

—.11E+01 +

X

—.14E+01 +

X X

* X—.17S+01 44.**X**4***** atata**+********+********+**** ***** ***** ***sib+

.01 .05 .10 .25 .50 .75 .90 .95

122

Fig. 3.8. Annual Flows at Kidira

PROBABILITY PLOT FOR NORMAL DISTRIBUTION.11E+01

XXXX X

X

.75E+00 + XX

X XX.43E+00 +

.12E+00 +

X0B -.19E+00 +

XX

VA -.51E+00 + X

S• -.82E+00 +

-*

-.11E+01 + X

-.15E+01 +

X

-.18E+01 +

X

•-.21E+01 +****X*** ***** *******+**** ***** ** ***** *+** ******* **+* ***** **+.01 .05 .10 .25 .50 .75 .90 .95

123

Fig. 3.9. Log Transformed Annual Flows at Kidira

124

Chi-Square Test. Comparing the significance levels given by

Tables 3.7 through 3.10, for the monthly flow residuals, we conclude

that for:

- Galougo, the normal and lognormal distributions were accepted

for all months with a preference for the first distribution;

- Bakel, the normal distribution was rejected for May and June

while the lognormal distribution was rejected for most of the

low flow season months;

Kayes, the two distributions were both rejected during May and

June and the lognormal was preferred overall for the other

months;

- Kidira, the normal distribution was accepted for all months

while the lognormal was rejected for May, June, March, and

April;

For the seasonal flow residuals, Table 3.11 leads to the

following:

- for Galougo and Bakel, the normal and lognormal distributions

were accepted for both seasons with a distinct preference for

the first distribution;

for Kayes, both distributions were accepted for both seasons

with an equal preference for the normal during the high flow

season and the lognormal during the low flow season;

for Kidira, both distributions were accepted during both seasons

with a preference for the normal distribution for the low flow

season.

Lc Lc)• CO en

en •

‘C) Ion torClor • 0

en • •

Ot re)v.4 en Le)• •

Leo 02 en ete-4

• •OO00

125

0 co ro Lnro we)

o• •

re)IL)

CV vno tr • en one

tot to. 0 CO on.• • 0

oe21 Lo

uoO• •

o c_D

s-ro

(/)r-

>

7710 • r-17) V)0 CUE

0 3. 0CO r--

CL)a.)r-

cts

oro

s-ro C.)

I 4—0

L.) >1

CU •,--

tor)0

onO CV tCr re) 0so. onI 0 CO 0

• • 0

o•••n V CVqr 0 CO

o•

CVo

CV or •C• Color oœ

o• •

OCV or V' Col•••I ••nI oœ

O• •

0CV ••• V el

re oœ• • 0

CV0

r-I V V•••• 0 CO 0

• • O

CV

Col V' 0LC) CO ro ro 0••••n • • CO

CV 0

el V1.0 CO v. rot

• •

O tn co en CVro Co)

• •

O re) ON enor. ,o1 r., ir) U.)

•

0 en CT en•n•n •n1 en 0

• •

C_- 0 CVlor CO rn ea'

• •e•-n

rt CV V ro•n•I •n•Io œ C°1

• •

Ori Co)•••1 11)

CV Cer0 CO CV

• •

CO Le, trei' top1n1 nn••n ••n••

• •

wol CV V ro•••• •••• C0 CO 0

(ts

S-0 0

(.34..

4' ,-in

•••••

co IA in 0) orC C r > >W0 0 ola CIJ 0>r ••••

10,- Ca)4J_-_ oFt o•••••• okoot L. rVI ro 10 CV Ul 41>_- >.-.- Cd E CUS. L. 0) C V) t.)CL) or (1,1 otn L. 10 I C

4) in'— in,— 00 L) St eaSo.- .0w .0 0) 7 0I.1•CI QI.) 0t. 0' ir L rc

in-,-- 0 4-

(7 ,r C •••• C I C 0, rV) 0 r 0- r IT 0=

I .0 •••• E On•n•• 4! MIR LI VI or or.0 0 on(..) IC or

-r.

.8., .-- --...In to ul Cu e-c c •s- > > 0)0 0 •IJ CI) 0>

.T. •In

10 Odd.. =0)4, ..... ro.) .••n• Jo.) L. .-•0'Jtn•n VI CV in alI. ..... > ro LI EQ.,Soo. L. CI) C 11) Id

Q)— 0.) or 5.- 10 IC

0) VI on in r tO L.) > 10L. .0 CI) .00) 7 • OU

MI OU OW er i••• Sin •,= VI .... O4_CT .4.• C 1.1.• C I C CT orV) 0 •••• Q.- r 01 OCs .c ••• E CM

r •IG ..It C.) V) I.- r.0 0 onLI IC ro•

%CI Co)C*1 4•4 43. LA VI C•J

• 9-4 • •

to C.', 0...:J. In V' 0 P.. Csi .44 en.44 nn1 AJ 1nI 4.4 • •

f ...

4.0o--1 7.) t/D.44 r-0 • •

to

C., LC)LO

-

efl C) fv.4 CV

-

4.4 • •

• 4.0- 144 1.4 1n1 1.4 en 01

• •

0.14.0 CO en

• eni • •

ff)LC) ) 1.0 LC)

CV 1..41

▪

•n• • •

CO

,4 1..4 1.4

‘.0 CO CO••.1

▪

4n41 • •

cn es,• -n • •

-

•N.

t0.-4 1-n • 1-t

ID •

to. tercp to at r, CV • 8

4n1

C-54 r•-n 04 .1 .

IA 0

..-4 •

co to LA c,

• 0)CO 040 •

OOO

4-, .- -In ch vl VW V) VI 01C C C C C •••• > >Q)O 0 0 0

.1-- •••,- 4,-0 4.4 01 0 >

44.. 10 pr. C CV44 .4, 4.• •..... 4.1 ...... 4.1- 4,4 ...... 4.4 S.. ••••4:6 .-1 ICI CS) o tn to er to to tfl CLI>,..... >•-.4 )0.4.- > 4-.0 >,.... (..) E a)L- s. L. L. L.at ... 11) .--. C I) •-• CL) r-, CL). WS.- g VIIC'j

CD In LA r 1.A r in In ,.... eff t.) > 10

L. .0 121 .0 CD .0 Q.) .0 01 .0 0.1 7 r 0 U05 OU OU OU OU OU cr 4- S. 4.-7 VI 4- 0 4-

CT '4 - C 4- C 9- C LI- C 4- C I C Ch..-LA Or Or Or Or 0 4- 4... 01 OC

I .C r E tol,- 44 44 C.) V) .n• 4,..0 O u)L.> IC .......

126

4-)

S-o4-

L-

E

.0Of

Cf1

• n•

•-)

oO

N141 121 • o

CV 1.0

NJCO 14' CO 1Cr 0.1

Co.1 cc.;

Lt) N. CO1.0 00 CV 4/0 Col 1-1 C )

Lb if) 00 Lbri ON at Ln

CI 1.01.r) N. Mir .4^ OO

1,1 01.7 %,1 •

CO LO 4:1 Le)ori 00 ON

Ln N.l•-•4, 1 Ci) CT,N.

CVv, 1 r•I 1.0

CV 1,1 Co) •N.

N.

r.0 00 0

C0N m

CV or Ni N. 4.10 • c,li 1. ..Cr C.) 0

CO •

4.1v) o V) oi) tto ti)C C = C C0 0 0 0 0 .I.)

r r r r 10.i.J .... +J r. 4a ...... 4.7 ...... 4-, •••• 4-,IO r I0 NI CO CI, 01 et OS In V)>._- >__- y. ..-... >_- y....S. S. S- S.. S. ar0) or 0) or QJ r 0.) r 0) r. L.

CU 1/1 1o1 or VI r 111 or LA 1••• 14U •IZ a) 1) al .0 0 .0 01 ra Q) =ro o c..,, OU OU OU OU Cr= VIc • ki.. C cs.. c 44- C 4- C tè.. C ILn Or Or Or Or Or ...2

Ir qft nL ,ft c.,..cL?

127

0.

.0

Q)o

0

V)

-1-)

V')

koCV

00..1 CV

cr,

• •

-er .0. CV 01 .1C41 CV1,.. .1

. •V7

z

co

.1

en CV 0 N. kr)• •

o C7 r-05 C•1CV 0 Ts.

• • •

r- CO 0 tr)e•-) CV CV )0 CO

•

en LC) 0 01 COen 04..I CV

• •V)

•.1

el V) 0, 0.1 hCC) 04 .1 CV• •

....en•

o CO r- os Cr)CV Cr-

• •07

•

LID (LI %CDtv) cs.i ID et CO• •

%IDen

VZ)kr)

• 0el •

kr) kir)

a

Cs.1 VD 0

-C C

CO 0 0'CT 0 0CV %.0 • 0 0Cil t0 0 01,1 • .

.1.) F. •••nVI Cil I/1 0)C C •••• > > Q)0 0 •F, CI1 O>

•••• .. 4V .• = Q)4.5. 4.1 )... ...

tO .I 40 CV V1 CL)>-. >- i.) E0.)S. L. a) C N UQ) g. Q)- L. 10 I C

CI) .n ,-- IA ,-- cCI C.) >4VS. .0W .0 0) 7 .. Oc)MI 00 Oc) 0 4-= LA 0 4-0 '1- C 4- C I C 01) 4-V) 0 4.- 0 •g" •g- 0) OC

I .0 E onft ft (...) kn.= OU)

L.) ......

CO 0 N. ce,en 0 N.

• II

COcs.1 en Cr-

co 0 N. onO.I n

g•-1 07 0C•1 ID

• ••-n

rx:)r Cr-

• •

O 07 r-0'CV Cr-

o OD I,. CTCLI Or-

1.1 CO 0 Ts.

('4 '.O r••

C•1 0 CT, 07C•1 Cs, 07

07 o TnC") Cr-

• •

en 0CT 0CV %SD • 0ch

-4 •

g.

4V-

LO• 0

kr).1 •

4.) g- ....Ln In Ln CVC C > >Q)0 0 4.) 0.1 O>

•••• IQ 0. =0).•-• .... 4.) ••••• 4.) L. .-4V-4tCI ... IQ CV IQ Q)> •••••• >...- L) E a)L. L. 0.1 C V)).)

W- 0) - L. 10 IC0.) c / ) .. th .. CCI C.) >4V

L. .00) .0W z 4.- Ou

137 Oc) Oc) 0'4- S. ..0 Lil 4.- 0 4--

0 1.F C ..F. C 1 C 014-V) O- 0 ••- •••• 0) OC

I .0 4- E al•-•-• ft ft L..) V) •••- ...= OU)L.) Y ....,

128

129

OCr 11 .4 CO- 4. 10 CI-

• •

- cr N. .1- ‘C)

•-• re)4•44 ••

▪

•I w •Cr ON• • •

f1") I-4 I1 CO It)41 4•41 ••4

• • •

• 0 N. n••I P.)- t0

• • •

f•-) .44 N. CO4-1 .4 4.0

• • •

N. N. N. Cr4.4•1 v. 0

• • •

• 0 N. 44.4• 4.41 10 gr tg)

• • •

0 N. •••1▪ co q- (v)• •

- in N. 00%.13

• • •

etO 1.4...er

• • •

4.0cr 0 N. •--I 0.. .-v I.13 et 0

• • 0•

U.r.

4-, -in in L.) ID .-.C c > > a)0 0 4., GI 0>

.,-. ..•-•

(D( CG)4./ ..... 4.) ...... 4-, S. v-ICI ...4 (D CJ VI CIS> s... >- t.) E a)L. L. GS C V) 1)

Q)- (1.1 ,4 /- I13 (C

(I) 0 1.44 0 r-- fa L.) > RS

L. .QG) .0 GI 7 •••• 0 1,

IO O U OU 0" ,4•4 fr *v..0 V) •••• 0 4..

cr 4. C 4.. C I C 07,-.V) Q' o 4.- •••• al 0=

I r ..-. E al-.• c, nk LS V) .- .1...0 0 VI13 NC 4.....

N. N. Cs1•••n

cr •

It) 1311 wit CNI

▪ • 0

v1) N.O CO• LO• •

OD w N.O ODv. 40 v.• •

- V) N. CO LC)▪ 4. v. 1,C)

. •

tO4:1*

• • •

Cs.1 oo CO▪ 00 Cr)

• •

v•.( (V) 00• •••4 4.0

• •

NI N. CO- v-•I 1‘) C

▪

O• •

cr+ in in Cs.)•-n • CV •O•

v. •

N. N.• .1 4. tr

• 0.0 • •

4-, 01

In 0 0 CV g-C0

••••

C0..-

•v-4-,(CI

>GS

r.-

> Q)0>CG)

4.) ..... 4J- .IJ S.. v.tel .--,>.-

tO NJ>-

(i) 0)(..) E I1)

L. L. a) c V)(...)/3) ...• C./ a.- I. (D IC

0) V) .• LA 4. IC 1) >(DL.113

.QC)O U

.00)0 U

7cr 4-

OUS.. .4.-

7cr 1.,- a 4- a

(nI

.1-c

0 14-chn-

i.r> Q.. 0 -i- -,- 0) OC.0 - E In

..-. 34c nE 13 V) 4. vv.-=I,-.)

0 V,IC .....•

C) al v-1Cl) Lr) 1-1 Csj Gt.('r) ('Ji . 6 •

r—I

r—i Tn VDCl) C•1 CV

00 Lf)✓-I 1-4

•

co

C) 00tr) r-4

• •

cr)a,

(.10L.r)

a)›.)

r--r0

(.0 _C 30.) 0

•1-u_

130

CD CD co(\J CD 0- r-r

t-i

0 nz1- r-- r-1 0r-1 r-1 LC) dt- 0

. CD

co r--. ci-r-•c t- co ,--1 co co tc) c.,r—I r--1 1--I I—I 4-1 • • .

C\JCV 1-1 Gi" CO 0I-1 r-1 0 OC) CD

. CDCD

C•1r--1 (\1 d- (Y) CDt--I .--1 CD 00 0

. . 0CD

V)=0

-,--4_, e•-ncar-4>.—L-s-

v)C09-

4..) -ro c\I>-.--s-

v)=0

9--4-) ---.(a Cl)>—s-

6/1C0

9-4-) -as .t->.—s-

V)C09-+) ...--.

(cs Lr)>.-s-

a) .--- a) ,— 0.) r-- cu r- a) I—.CL) ri) r v) r— V) r (J) r V) rS.- X) 0.1 -0 0.) _C) CI) _0 CL) -0 0.)(0 OU OU OC.) OL) OL)=Cr 4- = 4- = 4- = 4- C 4- C

V) Or Or Or9- 0'1- Or1

9- 4,_CC_.)

r-> CI)0>Ca)S-

ECUV) UIC> ra00S.- .1-0 4-01.1-0Er-0 tr)

0)0

=

CV .--1

arl LC) LÛ CV cl-,--1 . • .

rl

C.) r-1 r-I k./0

• •

- •

CD 0(.0 0)

• •

01 (•JCV r-,..

• a

CVC\i e-1

r-c I', (.0Ce) C•J CV

•

0LC) (V) ,--1CV (e) •

v-1

CVCD Q r-i r

1-1Lf)

CD lr0 COI-0 CD i... 00 Cr) Lil X)e-1 ri r-I ri • •

C')I

CVCO CD CDO) d- 0 co 0

ri eri CD

•

(e) CA CO CDCe) LC)

•CD

•

0 U)U) 0 U)C a c c ao o o o o,- .r- ,- • - •,-...-. 4_, ..--. 4..., --, 4-) ..--, 4-) ----..(CS ,-I (13 CV (Q (V) as cr as Lc>> .......s.CL) r-

>—S-a) I-

> -._.•s-Q) .-

>—s..C1.) 1-

> .....--S-Cl) 1-

(3) (i) 1- ci) 1-- Lo) r V) I-- U) 1-S- -CI Cl) -0 CD -C1 CI) -CI Cl) -CD Cl)fr3 OU OU 00 00 00=CT 4- C 4- C 4- C 4- = 4-- C

(r) Or- Or Or Or 0 -r-I

• r- 4,k 4k 4k-C

131

132

Finally, both distributions were accepted for all sites for the

annual residuals (see Table 3.12). Except for Kidira, where there was

no preference, the chi-square test produced a higher significance level

for the normal distribution.

For the purpose of this study, we are mainly interested in the

monthly and annual residuals. Model 1 described in the last part of

the second chapter requires that the same distribution be assumed for

both annual and monthly flows, and for all the sites. For the annual

values the normal distribution was preferred by the chi-square test for

three of the four sites considered. The same conclusion holds for the

monthly flows. Therefore, the chi-square test indicates that the dis-

tribution of the flows should be assumed to be normal in this study.

Kolmogorov -Smirnov Test. From Tables 3.7 through 3.10, we

notice for the monthly flows that:

for Galougo, the normal distribution was rejected and the

lognormal accepted for all the months;

for Bakel, the normal distribution was rejected for the month of

May and the lognormal was rejected for most of the low flow

season months.

- for Kayes, both distributions were rejected for May and June and

the lognormal preferred for the other months;

for Kidira, the lognormal distribution was preferred overall,

although it was rejected four times while the normal distribu-

tion was rejected only twice.

11 16 32 10

12 16 26 14

Table 3.12

Statistics for the Chi-Square and the Kolmogorov-Smirnov Testsfor the Normality of the Annual Flow Residuals at All Sites

Statistics

Galougo Bakel Kayes Kidira

Normal Distribution

Chi-Square

133

# of observationsin cell (1)

# of observationsin cell (2)

# of observationsin cell (3)

# of observationsin cell (4)

# of observationsin cell (5)

Chi-square statistic .04 1.87 .62 .67

Significance level .83 .76 .43 .41

Kolmogorov-Smirnov(significance level)

.00002 .05 .31 .008

Table 3.12, Continued

Statistics

Galougo Bakel Kayes Kidira

Log Normal Distribution

Chi-Square

# of observationsin cell (1)

# of observationsin cell (2)

# of observationsin cell (3)

# of observationsin cell (4)

# of observationsin cell (5)

9 15 25 10

14 12 33 14

15

18

1 8

Chi-square statistic 1.08 1.61 1.10 .67

Significance level .30 .81 .29 .41

Kolmogorov-Smirnov(significance level)

.00002 .35 .37 .10

134

135

For the seasonal flows, Table 3.11 shows that: the lognormal

distribution was preferred for Kayes and Kidira for both seasons; the

lognormal was preferred for the high flow season at Bakel while the

normal was preferred during the low flow season; and both distributions

were rejected for Gal ougo.

Finally, the lognormal distribution was preferred for the

annual flows at Kidira, Kayes, and Bakel, and both distributions were

rejected at Galougo (see Table 3.12).

In summary, on the Kolmogorov-Smirnov test, the lognormal dis-

tribution was preferred for three sites out of four for both the

monthly and annual records. For the monthly flows, the normal distri-

bution was preferred only at Bakel. For the annual flows both distri-

butions were rejected at Gal ougo.

We conclude this section on the normality check by saying that

neither the assumption of normality nor the assumption of log normality

is clearly indicated. First, the probability plot and the skew coeffi-

cient were not conclusive for the monthly flows and suggested the

normal distribution for the annual and the seasonal flows. These

results implied by the probability plot and the skewness are not

consistent for if the annual flows are normally distributed so should

be the case for the monthly flows. This latter remark raises the

question of whether or not the data collected is sufficient and/or

reliable. Another explanation could also be that the two methods

mentioned did not perform well. Second, the Kolmogorov-Smirnov test

preferred the lognormal distribution while the chi-square test chose

136

the normal distribution. The chi-square test is not recommended for

making a choice between two distributions (Benjamin and Cornell, 1970)

and lead to unreliable conclusions when the cells are not equiprobable

(different number of observations) as it was often the case in this

study (see Tables 3.7 through 3.12). In addition, the Kolmogorov-

Smirnov test was found more powerful than the chi-square by other

researchers like Afifi and Azen (1979). However, there is a problem

related to the parameter estimation of the distribution as mentioned

earlier. In spite of this difficulty, the conclusions of the

Kolmogorov-Smirnov should be preferred to those of the chi-square since

the lognormal distribution is more adequate than the normal distribu-

tion in drought conditions--more lean years than fat years, as is the

case for the SRB. In situations like this where there is no clear

indication for assuming the normal or lognormal distribution, Fiering

and Jackson (1971) suggest trying the gamma distribution if the skew

coefficient is significantly different from zero or use game and

decision theories to decide on which distribution to assume if

otherwise. The skew coefficients presented in Tables 3.1 to 3.6 are, in

general, significantly different from zero. Therefore one could try

the gamma distribution. However, since the reliability of the data

collected has become an open question, we do not see the point of

conducting any further check on the distribution and will assume the

lognormal distribution. But we recommend strongly that this be done

with more reliable data in future studies relating to this question.

137

3.1.2. Filling Missing Values

The objective herein is to bring the historic records of the

monthly flows at the four sites to the same length and to fill in the

missing values. This is a prerequisite for the generation of annual

flows using model 1. This work is done by performing a regression

analysis using the "New Regression" program of the SPSS package (Hull

and Nie, 1981).

Methodology. To find an estimate of the dependent variable y

as a function of the independent variables x i (1=1, k), the package

uses the regression equation:

y'A+ B 1 x1 ++ +82 x2Bkxk (3.4)

where y' represents the estimated value for y, A is the y intercept,

and the Bk are regression coefficients. The intercept and the coeffi-

cients are selected based on the minimization of z(y-y') 2 (least-squares

criterion).

With the stepwise alternative used in this study, the "New

Regression" program enters the independent variables one at a time in

Equation (3.4). A variable is entered at a given step if it satisfies

the significance level (SIGF) of the F ratio and the value of the

tolerance (T); i.e., if its introduction in the equation will improve or

at least maintain the fit obtained in the preceding steps. The

tolerance of an independent variable being considered for inclusion is

the proportion of the variance of that variable not explained by the

independent variables already in the regression equation, i.e., one

138

minus the squared multiple correlation of that independent variable

with the independent variables already in the equation. A variable does

not enter the equation if its multiple correlation with the variables

already in the equation is greater than one minus the specified T. A

tolerance of 0 indicates that a given variable is a perfect linear com-

bination of the other independent variables, and a tolerance of 1

indicates that the variable considered is uncorrelated to the other

variables. The F ratio is computed in a test for significance of a

regression coefficient at each step of the analysis for variables not

yet in the equation. This ratio, defined below, is for a given variable

the value that would be obtained if that variable was brought into the

equation:

F R2/k (1-R2 )/(N-k-1)

(3.5)

where R is the multiple correlation between the dependent variable and

the independent variables; k is the number of independent

variables; and N is the sample size.

A variable is entered if SIGF is less than the specified value. The

values of T and SIGF used in this study are 0.01 and 0.05, respectively.

These are the default values of the program.

To reconstitute the flows we did two sets of runs which

results and discussion are presented in the following paragraphs. The

program DATA2 written in FORTRAN V, and presented in Appendix C was

used to provide data input to the SPSS program. An example set-up of

139

the "New Regression" SPSS program is also presented in Appendix C.

Notations:BELOW = Monthly flows at Bakel

GFLOW = Monthly flows at Gal ougo

KAFLOW = Monthly flows at Kayes

KIFLOW = Monthly flows at Kidira

First Set of Run. In this set of run we tried to reconstitute

the flows at each site considering the flows at all the other three

sites.

For the reconstitution of the flows at Galougo only the flows

at Kayes, the closest site, were accepted in Equation (3.4). The

results of the regression analysis relevant for the context of this

study, and extracted from the computer printout, are presented in Table

3.13.

Table 3.14 shows the variables accepted in the equation for the

reconstitution of the monthly flows at Kayes. Except for the month of

June where only GFLOW was accepted, BELOW was generally in the

equation.

At Kidira (Table 3.15) the regressive equation included BELOW

alone for most months. However, for February and March KAFLOW was the

only variable accepted.

Second Set of Run. In this set of run we considered the recon-

stitution of the dependent variable KAFLOW, KIFLOW, or GFLOW using only

one independent variable, BFLOW. This set up result into a simple

regression analysis which is a special case of the multiple regression

analysis performed in the first set of run.

Table 3.13

Summary Table of the Multiple Regression with DependentVariable GFLOW (flows at Galougo)

Variable entered = Kaye

Month Multiple Corre- SIGF Coefficient InterceptR lation T Bi A

May .9247 .9247 .000 1.0266 3.931.000

Jun .9284 .9284 .000 .8340 36.331.000

Jul .8539 .8539 .000 .9790 12.401.000

Aug .8923 .8923 .000 .8433 99.301.000

Sep not applicable

Oct .7112 .7112 .004 .8412 3061.000

Nov .9269 .9269 .000 .9802 20.261.000

Dec .8846 .8846 .000 .9174 21.511.000

Jan .7395 .7395 .003 .7778 31.601.000

Feb .6257 .6257 .017 .6296 30.521.000

Mar .7248 .7248 .003 .6956 15.781.000

Apr .7956 .7956 .001 .8912 6.351.000

140

s- C/)

d' a) 0CC r--

4-

..o- ,-I 01 N.. CO COco c.) 0 1-1 Q-4osial

r,. C71

COal

CO0.1

VO Crl0.1 N.

• • • •

Q Q Q Q 0 0Q Q Q Q CD 0Q Q Q Q CD CD

• • • •

,--1c1" 0 01 N.. CO t.0CO N... 0 ,--t 0 LOC1..1cn

C \Jcn

COon

COcn

k.o 1-,on cn

cn cnVD VD CD CD CD CD Cr) Cr)CO CO CD CD CD CD is, l',.01 01 CD CD CD CD un V1COCO •• • CO COCQ 0.1 r-t r-I r-I r r-I r

• • • •

Cf 01 CD N. VD CD 1.11 CDcf 1.11 1--i cf LO ,...-i Cr) VD CO Cf CO CQ ct01 VD r-i Cr) VD CO CJ 01 r-i cn r, cn ol •co ol • ol • 0 CQ CO • P. up 01 00 00LO ct CN.1 CD Csd —4-4 CO 1-4 l-r) 0-1 l0 d-

• • 1 • OJ • I • t.0 . • • 1,--4 I .--1 I

3 c)3 30 0 Q Q Q Q 0 _I--I —I Li- Li-CC1 CD

u- .. u..CD CO CL1 CO CO

C •-i

141

C.Q 011`... 'JDCNI L.0cn on

• • • . • •

C r cn cLco a)

•z:C

3 3 3 3:c cp

3 3C c) cp cp_ j c) _ j c) c.- J -I -1 -.1 U--I L. --I --I

L.L. U- UU.. U.. -CO 1-$ CD

Li-CO CO CO CO CO

-C)c:CCO co

LC) CO cl" 1-1 C•./ CO COCO cl- ch 01 CM ch CMCO 4-0 01 d- CV 0.1 0CM CM cs CT) CTI r•-n al

rn 0.1LO CV01 CM01 01

• • • • • • •

0 0 0 CD 00 0 0 0 00 0 0 0 0

• • • • ••

r, 04 141 01 CV .--i CD cn COVD CJ 01 nt cn op ul r, cncn cn oo VD Iss. ch VD r, CDCM cn cn cn CA CM CM cn cn

• • • • • • •

0 0 0 CC0 0 0 LC) C10 0

.0 ,--i ,-.4

e--i ,A,-.1 1-1 1-1 C') C')i

cn

• •

CD CD CM CD VD v1 QD VD 01CQ CV ,,zr VD 01 CO CD cn 0.1CM • CO • QD • Q1 CV •0, .zt-

•0,

•01,--i

CO•CV

ILO .--4

• •141,-4

01 I

-0o CU 0) CO

0CM ch r-i 0Ul cr CV CDC) 1--4 1--i •

0.1 CV L.0 .-4• • •

CO h*-- LID ,--i CDcr 01 I-4 1.--I CA 0101 CD ,--I n-I CM •cr CV CQ I VD 0,)

• • • • ICO

142

LLC-01-1V)

CC

•.tr03 ms

r.-4 c.° ro tep 1-1 CD ct CO01 CV hn CA Ct •-•1 CVCO Ch 01 Cg CD ChCo cf Co r, CN cn cn n,

• •

▪

• •

CD CD CD 01 CD CD CD CD

CD CD CD CD CD CD CD CD

CD CD CD CD CD CD CD CD• •

•

• •

143

r-i V1 01 VD ,--i01 co hn Ch .7r.--I 01 cn ol ,-1CO as CO 1.-n 01

• • • a

hn hn CD CD CDCO CO CD CD CD00 01 CD CD CDCg CQ • •• •

LC)

c, cp

,-i r--4

'<t"

r-i

• (NI C\I

33 3 3 3 3 3 CDC) C) 0 CD 0 0 0 __I--I -IU- U-

-JLi-

--1U.-

-.JU..

-.JL.L. -1 U...

CO W CO CO co co co

CD cf 01,-.1 C\J 0.1Ci CD cf01 0)0)

• • •

CD CO COCD VD VDCD hn hn

• CD CDn-4 • •

Z-* '''''":" 21 (74 '7"Eor> co VD eV• • • • • •

1-1 C.1 1-1 1-1

C7) Cl. 4-)

C:r CD

-C)

4-)

0

LI)

•

r—

CL) 0

(a 4-)

S-ftS CU

>S..

CO 0

V)

°•r-rIZS

CC

0

,---tcc)L)Cs)

•

000

.

i--4(Y)U)Cs)

•

000

•1-1

0alis)Cri

•

000

.

0Cs)is)Cs)

•

000

er-4

tr)

01

CO00

000

Cc)CoCOCO

000

t--I

r::1-

cilCS1

•

000

.

d-cl-cr)Cs)

•

000

•t--1

CoCV

c\ICT)

•

000

.

("JCS)CV01

•

000

•1-41

CO

COCO

.

000

.

CON.coCO

•

000

•r-i

40- r-i CJ C rr) C"1 ro)CY) CO CO

CD • (\I • N. 0 0 CNJ• C c:C ('-J l-0LS) 0 wzi- • n-4 • r-4

CO co C\ co (NJ L.0 • 0 •

a) 0_c

• I

3

•

3

I •

3

I

(0 -I-) 4->•r-S-(0 •r-- 0-

4-, "--

0-JLi-CO

r-I

0-JLi-CO

0—ILi-CO

_J

t-I

U_

r-I

—J

Co

V)

-C4-)

0 O wCD

wS-(TS

S-

c:C

144

145

For this analysis Tables 3.16 through 3.18 included present the

results. For Galougo, BFLOW was not accepted in the equation for five

of the twelve months. One could force BFLOW in Equation (3.4) by using

less stringent values of SIGF and T. However, such practice will not

guarantee a good fit because of increasing error in the estimation

process. For the two other sites, Kayes and Kidira, BFLOW was accepted

for all 12 months with high significance levels in general.

Discussion. To decide what option to use to reconstitute the

data we first make the following remarks. As shown by Figure 3.1 the

year 1980-81 is missing for all four sites. Therefore, we consider the

year 1979-80 as the last year of the flow records for the rest of this

study. We do not consider reconstitution by serial correlation in this

study. In addition, Figure 3.1 shows that only Bakel has a complete

record for the period 1903-04 to 1979-80. Finally, the flows collected

for the years before 1951-52 have already undergone some reconstitution

by Rochette (1974). Thus, the flows between 1951-52 and 1979-80 will

be considered for the rest of this study because using data that has

been reconstituted twice to generate synthetic streamflows would very

likely result in a net loss of information.

In light of the remarks above we conclude that the first run is

not adequate for the purpose of this study. Although this run produces

theoretically the best regression equations, these equations cannot be

used for the direct reconstitution of all the flows at all sites. This

holds because the only variable with a complete record of flows (BELOW)

was not accepted for:

Table 3.16

Summary Table of the Simple Regressionwith Dependent Variable GFLOW (flows at Galougo)

Month Step Variable Bi T Correlation SIGFi in the and R

Equation A

May 1 BELOW .5838 1.000 .7235 .000.97

Jun 1 BELOW .6926 1.000 .7500 .00038

Jul 1 BELOW .4131 1.000 .4307 .040312

Aug 1 BELOW .6713 1.000 .7754 .000

Sep

Oct

1

1

none

none

526

Nov 1 BELOW .3727 1.000 .4817 .020256

Dec 1 BELOW .3373 1.000 .4396 .036

Jan

Feb

Mar

1

1

1

none

none

none

129

Apr 1 BELOW .3894 1.000 .4528 .035.8

146

Table 3.17

Summary Table of the Simple Regression with DependentVariable KAFLOW (flows at Kayes)

Month Step Variable Bi T Correlation SIGFin the ana

Equation A

May 1 BELOW .93834 1.000 .8428 .000.12907

Jun 1 BFLOW .73655 1.000 .7838 .00036.1648

Jul 1 BELOW .98476 1.000 .9263 .000-45.7337

Aug 1 BELOW .75396 1.000 .9548 .000289.3072

Sep 1 BELOW .62468 1.000 .9721 .000492.7174

Oct 1 BELOW .74091 1.000 .9743 .00053.6621

Nov 1 BELOW .69822 1.000 .9162 .00064.4761

Dec 1 BELOW .54478 1.000 .7808 .00080.9054

Jan 1 BELOW .58671 1.000 .7520 .00040.3570

Feb 1 BELOW .61627 1.000 .7583 .00019.0803

Mar 1 BELOW .57027 1.000 .7629 .0006.2479

Apr 1 BFLOW .51226 1.000 .7604 .0003.6990

147

Table 3.18

Summary Table of the Simple Regression with DependentVariable KIFLOW (flows at Kidira)

Month Step Variable Bi T Correlation SIGFin the and

Equation A

May 1 BELOW .03388 1.000 .4344 .027.2446

Jun 1 BELOW .21774 1.000 .8844 .000-3.017

Jul 1 BELOW .29817 1.000 .8600 .000-41.52

Aug 1 BELOW .3043 1.000 .9315 .000-33.65

Sep 1 BELOW .30514 1.000 .9499 .000-110.19

Oct 1 BELOW .36344 1.000 .9125 .000-210.86

Nov 1 BELOW .21447 1.000 .9311 .000-22.48

Dec 1 BELOW .19307 1.000 .8831 .000-9.50

Jan 1 BELOW .13692 1.000 .9156 .000-2.24

Feb 1 BELOW .09607 1.000 .8797 .000-.2275

Mar 1 BELOW .07351 1.000 .7743 .000.2759

Apr 1 BFLOW .04448 1.000 .5951 .001.9255

148

149

- GFLOW, where KAFLOW was accepted for all months;

- KAFLOW, during the month of June where GFLOW was accepted; and

- KIFLOW, for February and March where KAFLOW was accepted.

One could perform a series of reconstitutions, for example, reconsti-

tute the flow at one site for a given month, then use those reconsti-

tuted flows to reconstitute the flows at another site. However, this

type of reconstitution might just add more noise to the time series.

Therefore, we reconstitute the historic flows using the second set of

runs. Since we cannot reconstitute directly the flows at Kayes and

Kidira only, we exclude Galougo for the rest of this study.

Reconstitution. For the period 1951-52 to 1979-80, we recon-

stituted the monthly flows for Kayes and Kidira using the flows at

Bakel with the program DATA3 in Appendix D.

Among the monthly flows generated at Kayes for the period

1965-66 to 1979-80 only the value for July 1970-71 was unacceptable,

being negative, using the intercepts and regression coefficients of

Table 3.17. This value was adjusted using the upper bound of the 95%

interval confidence of the intercept A, given in Table 3.19, instead of

A itself which produces the negative value.

For Kidira more than 80 flows were reconstituted. Three of

these reconstituted values were found unacceptable because they were

negative. These values were also rectified using the same technique as

above, i.e., using the upper bound of the 95% confidence interval on A

(Table 3.19) instead of A itself.

aiC)CCU 0

•r-(4- 75

g>S.-a)

-o=

o_L o

=

01h.

.

alLC)

CV

cr,N.

•CV

I

: \ .1CV

•CS)LO

1-1 CO InCD N. ,--)

• •CO r--.. ,--1CV 3D I

I

',ZrCV

•,-i

CDCV

•

d-

Lo•

1-4

cl-Lo

•I-.1

,---1r---.

r-f

. +-1 s- -o d- •tzl- d- .z:1- CO 0) 00('si N-h. CV LO 01 CO CV LOO) 01

3 =00 I CO

•CD

•4-0

• •

Cr) CO•

0 L0•

4-I• •

o

u")

(0

-J CO I COI

Cf)

Icv cr.) I

I I

CVI

I I I

-0CUS-01 S- -0 CO ) L.0 00 cr CS)CL)

CO= .zzl- CD

.ZrCDCO

CVal

CD N. CO•Zr r••n CO

d- u-)L.0

n-1CO

Cll(f)

CCU 0

0_ 0CC)

LOCD

•

COCV

LC)CO

•

d-

co•

cv• •

<---1•

)•-n•

01CD

•

L.00

4-) a.)

-0 ca

cv.S-0

‘"-

g>S-W4-) S- C) n-1 cr) LO

c:r C

cr)C

3 000 CD

d-

N.

ce)COCV

LI

Ln

O N-LO 01 CD(0 0)03co Ix)

Lc) 1_0LO

rn-.0) CD

( ) S- -_I CO CV CV CV CV4-) •c)- -cc

• • • • • • •

CL) •

s.-ai s.-4-) 0C 4- s- -cc LO C) cr) Ln LO CV CO

t-I

a) CO_C4-)

ai

ai0

a)

o_ o= co

CO•

h.•

laLs-)

CO•

CD

CTI•

CD 01 CLI• •

cz) Lo

l0

•

r-4

LO•

co

•

re)

c:1-•

1-1•

C 4-)• r-

0ai

()r-•

"-'0 S-C-) ai

(rS S- "tD LO CO CD P CD CT CD h..> 4-S- 4-CU CL)4-) 0C C.)

Lf) CU c300

--J CO

LU•

LC)•

LC-)

Cv•

CVr-I

LO

CD

L...0• •

LS) 00 d-

d-re) I

cs,

C-) CD•

LA

cD•

1-4

aiCL)C.)

>-)co

CL) N. LA CO CO (0(00300 C) 00 CO LO COS- -C3 h. h. C) CO cD •nt- c) coCll CO CV CV LO LO LO r-4 Gt CO al 01

CU co O. 01 al 01 •-n 4.0 CO N. 4.0 CV L0 01 CVCI- 0 0 CO CD 03(0 N. N. LO

0C-) Cl) 0

= CC) • • • • • • • •

•r-4-

0 s-

S- -0 CV d- 0-1 CD CD LO N. N. Cr) LC) CD .1-

LC)CY)

C

00

CYNN-1--..

COCDCO

cf-

N.h.

.--1,--1CY)

c') CD cl-ct L.C1 1.0co CT -

CD000..i

CD014:1-

ctcl-N.

01CDCI'

,--)LO01

-J03 N. L0 CO l0 L.0 liD l0 ci- ci- cl- cl- CO• • • • • • • • • • •

..04-)

o>,ctS

C

N-D

n•-• CJ)

c:C

D- >CU L)OV) CD =

(-)W CO

"0

_rDai

LI_123 0_

150

151

With the adjustments made above, the reconstituted monthly

flows for Kayes and Kidira along with the flows record of Bakel from

which reconstitution was done are used to compute the annual flows at

all three sites. The three monthly and annual flow records thus

obtained, referred to as the historic records for the rest of this

report, are used to test model 1 and are presented in Appendix E. The

reconstitution of the data do not appear to be very reliable because of

the negative flows we had to adjust. The results of the reconstitution

could be improved if more data was available and maybe more stringent

statistical driteria (more stringent value of F ratio and Tolerance)

used during the regression analysis.

In summary, we performed two analyses on the data collected so

that it can be used to test Model 1. First, we analyzed the data to

find out what distribution to assume for the streamflows, limiting

ourselves to the normal and lognormal distributions. We concluded that

analysis by adopting the lognormal distribution for the monthly as well

as annual flows at all four sites. However, there was no clear

indication on what distribution to assume. Second and last, we carried

a regression analysis considering two cases: a multiple regression

case and a simple regression case. The multiple case for which one

would reconstitute the flows at each site using all the other sites

accepted in the regression equation was not found adequate due to the

limited nature of the data and the results provided by this method. We

therefore decided to reconstitute the flows at Kayes and Kidira using

the flows at Bakel excluding the flows at Galougo bacause for the

152

simple regression at this site, the flows at Bakel did not pass the

statistical tests to enter the regression equation. The three annual

and monthly flow records for 1951-52 to 1979-80 without missing values

will be used as the historic record in the next section. The reconsti-

tution was limited to this period because the flows before 1951-52

collected have already been reconstituted and 29 years of flows seems

sufficient for testing the model. We mentioned earlier in this section

that the historic data obtained by filling the missing values do not

seem reliable. This question on the reliability of the data also arose

on the check of the normality.

3.2 Testing of Model 1

In this section we test the first component of model 1 only,

the generation of the annual flows for the following reasons. First,

the analysis of the data in the preceding section did not clearly

indicate what distribution to assume. This difficulty led us to

question the reliability of the raw data collected. In addition, the

reconstitution of the raw data to fill in the missing values did not

produce reliable historic records for Kayes and Kidira. The distribu-

tion to assume and the reliability of the data to use to generate the

annual flows are crucial to the testing of the model. Unless these

two questions are answered, we cannot attest the capabilities of the

model for the generation of synthetic flows. We will generate the

annual flows using the first component of model 1, the multivariate

Markov lag-one, to illustrate how it can be used and leave the

generation of the monthly flows for further investigations.

153

3.2.1. Methodology

The theoretical and technical basis for generating synthetic

flows using model 1 was described in Chapter 2. Therefore, only the

parameters estimation will be presented herein.

The following steps were performed using the program MMLO in

Appendix F. The program contains subroutines written in Fortran V and

use subroutines of the IMSL package (IMSL, 1982).

Step I: Generate n(0,1) random variables for each station (Bakel,

Kayes, Kidira).

Step 2: Read in the streamflow data (historic record for all three

stations).

Step 3: Computation of the statistical estimators of the historical

flows using the method of moments (for N=29 years).

1= z x(j) = (3.6)

j=1

-2ax 1= — z [x(j) - = S2

NJ=1

x

= [1 14z [x(j) - R] 3]/S3N j=1

(I) N-1 - 2"x

N1-1

z [x(j)-R] [x(j+1)-x]/S x (N-1)j=1

(3.7)

(3.8)

(3.9)

where:

154

[N-1 N-11/2

S2x(N-1) = E [ X (i) -31 ]2 [X(j+1)-7(]2L 1 j=1

(3. 1 0)

Step 4: Computation of the lag zero cross correlation for each pair

of stations (p,q) for the flows by

ax(p)(q)(°) = z Ex(P)(j) - R ( P ) ] [x ( q ) (j) - R ( q ) ] ( p ) a (0Lax xJ=1(3.11)

Step 5: Computation of a, n' ' a

2 ' py as defined by equations (2.5)0 y y

through (2.9) for each station.

Step 6: Computations of the lag zero cross-correlation for each pair

of stations for the log transform flows using equation 2.22.

Step 7: Computation of the lag one cross-correlation for the same

flows for each pair using Equation (2.24).

Step 8: Construct the (3x3) matrix Mo whose diagonal elements,

= A 2(i); 71 =

Y

and whose off-diagonal elements

( i )(i) A (i) A (j)M O (i ,j ) = ay (°) ay ay ; .91 # j

Step 9: Construct the (3x3) matrix M 1 whose diagonal elements

M 1 (i 'j) = y (i) (1) a 2(i) ; 71=jY

and whose off-diagonal elements

(i)(j) A (i) A (i)

M 1 (i,j) = f;y ( 1) a ; "t=jY Y

(3.12)

(3.13)

(3.14)

(3.15)

155

Steps 10 through 17: These steps consist of the construction of the

various matrices necessary for the obtention of the two

matrices A and B used in equation 2.12 as described in section

2.2.2 (see equations 2.13 to 2.17)

Step 18: In this step we compute, for each station i, the initial value

of the log-transformed flows with zero mean using

37 1 " ) = in (a x (i) (last)) - Tly (i) (3.16)

where a. is the upper bound of the distribution and x(last) is the lasti i

flow of the historic record for station i.

Step 19: We generate the log transformed flows with zero mean 31i 0)

for 1=1,2,3 and j=2, ..., 291 using equation (2.23). Then we

obtain the corresponding synthetic annual flows for 290

years, using:

x. a" ) - exp [Y. (i) + p (i lJ J Y

(3.17)

Step 20: We compute the statistics of the synthetic annual flows as in

step 3.

3.2.2. Results and Discussion

Table 3.20 gives the statistics of the historic and synthetic

flows and the difference in percent between the two. The difference in

percent is defined as the positive difference between the two values of

the statistic considered for the historic and the synthetic flows at a

given station divided by the biggest value of the two. The mean and

156

4-)

0)• :-.10 L()

u•)Cr)

CJ f",

CD_Y 4-

C`J

•

'D

•

N. •-n LI) 0Lf)

C) .D

Q)o

o

1

"-4-) LC) 004-) 0.1

f•J0•-t

0•-i

CO LC) r-tC\J

s_s-o

(Ve)•

(r)•

4-)

(11os-

0-

C

Lt)-o 0s-ro 4-)

-{=3Crt;)

d-coc•J

(r)

c.)

0r-U-

U

Lf)0 CO

C\J00

ftSto

4-) >w •r- ••-•i=)

04-)Cll

4-) -C(i)

4-)

>"(V) f_10

o

Q)s-

4—4—•

Cr) Cr) cc C1.1 0 01 C)0N.

COL •)

N.t•O

C•JLf)

LC1 •-4

t-4 C•J es.)C)

C•JCD

•r-4-3

a)

ro>,

s-

-c)

.--W

-1Cro

Lf)al>-•ro

r1:$S-

••--0• 1'. cri

Lr)

cri

cri

•r--0

CO CO Y

157

the standard deviation were well preserved; they are at most 10% and

9% off, respectively, for all sites. The autocorrelations are at most

20% off. However, the skew coefficients are more than 30% off for

all stations. The nonpreservation of the skew coefficient and

therefore of the distribution was anticipated since the distribution

assumed, the lognormal distribution, was not clearly indicated during

the check on the normality in the preceding section. Furthermore, the

skewness indicated by the values in Table 3.20 suggest that the

historic flows are negatively skewed for all three stations which is

not consistent with the indications of the skew coefficients of the raw

data in Table 3.6 for which the flows at Kayes and Kidira are

negatively skewed while the flows at Bakel are positively skewed. This

discrepancy between the raw data and the historic data raises, once

again, the question of the reliability of the raw data (with missing

values) and/or of the historic data (reconstituted). Another explana-

tion of this situation could simply be that the two sets of record have

a different length; the historic data covers the period 1951-1980 while

the raw data includes flows before 1951 and sometimes as far back as

1903. This leads to the questions of stationarity and/or long-term

persistence of the flows. Both questions are out of the scope of this

study.

The lag-zero cross-correlation of the historic and synthetic

flows presented in Table 3.21 show that the model is capable of main-

taining the cross-correlation.

Table 3.21

Lag Zero Cross-Correlations of theHistoric and Synthetic Flows

Pair of

Historic

SyntheticStations

Flows

Flows

Bakel-Kayes .9947 .9942

Bakel-Kidira .9532 .9610

Kayes-Kidira .9334 .9406

158

159

The following conclusions seem reasonable. The first component

of model 1, the multivariate Markov lag-one model preserves the mean,

the standard deviation, the autocorrelation, and the lag-zero cross-

correlations. The skew coefficient was not preserved perhaps because

the data collected is not sufficient or reliable to allow the main-

tenance of the distribution. The nonpreservation of the distribution

can also be approached with considerations about the stationarity

and/or the long-term persistence of the flows in the SRB.

3.3 Summary and Conclusions

In this chapter we did two things. First, we performed an

analysis of the data to know what distribution to assume and to fill in

the missing values so that we can test model 1 described in the

preceding chapter. Because we failed to clearly assume a distribution

and fill in the missing values while maintaining the characteristics of

the raw data, we did not test the model completely. Both failures

seem to be explained by a limitation on the data collected, its

reliability. The partial test of model 1 seems to indicate that the

generation of annual flows which preserves the historic means, standard

deviations, autocorrelations, lag-zero cross correlation and perhaps the

historic skew coefficient is possible provided that: reliable data is

available, the distribution to assume is clearly indicated, and that the

eventual missing values are successfully reconstituted. If the distri-

bution is still not preserved we suggest the verification of the

stationarity assumption and/or the type of persistence (long-term or

short-term) of relevance. Then one would pursue the testing of the

160

model by generating monthly flows and verify if the statistical charac-

teristics of the monthly flows (as those mentioned above for the annual

flows) are maintained. These monthly flows would be generated by dis-

aggregating the annual flows as indicated in Chapter 2. One should

also verify if the correlation structures between annual and monthly

flows and among monthly flows for each site are maintained, and if the

cross correlation structure of the monthly flows between pairs of

sites is maintained.

CHAPTER 4

CONCLUSIONS AND RECOMMENDATIONS

In Chapter 1, we concluded that the planning, design, and

implementation of the SRB development plan can reveal to be very chal-

lenging. One of the most important constraints to economic growth and

the establishment of a desirable quality of life in the basin is the

scarcity of water because of rainfall (the lifeblood of the Senegal

River) variations in time and in space combined with significant losses

by evaporation and infiltration, and severe drought conditions, particu-

larly during the last 10 or 15 years. Aware of this situation and the

potential benefits of managing the resources of the basin together,

three of the four basin states (Senegal, Mali, and Mauritania) created

an international agency, the OMVS. Responsible for the development of

the basin as a whole, empowered and supported by the three member

states, the organization proposed a plan, actually under implementa-

tion, including the construction of two dams for multiple purposes

(irrigation, hydropower, navigation, and salt intrusion control), and

multiple objectives (see subsection 1.3.1).

The undertaking of such a complex project requires detailed

studies for the management of both water quantity and water quality.

Among other methods for assessing the performance of such a complex

water resources system, systems analysis can provide useful information

161

162

for the decision-making processes. The approach mentioned often uses

simulation models that prerequire the availability of synthetic stream-

flows. This is why the focus of this study was to select, adapt, and

eventually test models that could be used for generating synthetic

flows.

In Chapter 2, we conducted a detailed review of several models

and selected at the end of the chapter two models (model 1 and model

2) which components are discussed in detail throughout the chapter.

The two models were selected on the realization that the range of

cumulative departures of the flows from the mean may or may not be of

interest for the case of the SRB. This question related to the Hurst

phenomenon leads to the question of whether or not the flows in the

SRB exhibit a long-term persistence or not. If long-term persistence

is evidenced and is of relevance for the purpose of the study for which

the flows are being generated, then the modeler should consider the use

of model 2. Otherwise, model 1 may be adequate. Answering this

question is out of the scope of this study. However, considering the

persisting drought conditions in the SRB for the last decade, it is

strongly advised that this aspect be investigated in further studies.

Finally, in Chapter 3 we attempted the testing of model 1. The

model was tested partially and just for illustrative purposes. The

difficulty of testing the model resulted from the failure of the data

to clearly indicate the distribution to assume and to allow a reconsti-

tution of the missing flows. The importance of the type of distribu-

tion to assume depends on the purpose for which the synthetic flows

163

generated will be used. For instance, if the simulation model using

the generated flows is not sensitive to the type of distribution

considered, one can assume either the normal, the lognormal, or the

gamma distribution. Otherwise decisions on what distribution to assume

can be made through techniques such as game and decision theories to

evaluate the risk associated with the assumption of one distribution or

another. The two problems faced when testing model 1 could be

overcome with a good data set. It is therefore recommended that more

data be collected to test model 1. If uncertainty on what distribution

to assume remains we recommend that methods of analysis as the one

mentioned above (sensitivity analysis) be used to decide on what distri-

bution to use. It is also recommended that the stationarity assumption

be verified.

The final word of this report relates to a recent development

concerning the SRB project. In Chapter 1, we mentioned that the

strategy proposed by the OMVS for the agricultural development is to

replace gradually the actual recession farming system by a modern

intensive irrigation system. At the time we were to conclude this

report we have been informed that the OMVS is reconsidering the rate at

which this change should be implemented. As anticipated since the

beginning of this study, the need for studies to refine the plan is

already felt while the dams are under construction. To thoroughly deal

with this question, simulation models embodying the various constraints

and purposes of the project are necessary along with other approaches.

164

As said before, synthetic streamflow models can be a valuable tool in

studies of this kind providing further justifications for the investiga-

tions recommended in this report and to be carried in the future.

APPENDIX A

RAW DATAMONTHLY FLOWS COLLECTED

165

year month GALOUGO BAK EL K AYES K IIIIRA

1903-04 i .00000 10.00000 9.10000 .00000

1903-04 2 .00000 120.00000 108.00000 .00000

1903-04 3 •00000 746.00000 526.00000 .00000

1903-04 4 .00000 1937.00000 1794.00000 .00000

1903-04 5 .00000 2759.00000 2318.00000 .00000

1903-04 6 .00000 1060.00000 810.00000 .00000

1903-04 7 .00000 476.00000 280.00000 .00000

1903-04 8 •00000 202.00000 160.00000 .00000

1903-04 9 .00000 124.00000 90.00000 .00000

1903-04 0 .00000 74.00000 48.00000 .00000

1903-04 1 .00000 40.00000 16.00000 .00000

1903-04 2 .00000 15.00000 5.00000 .00000

1904-05 1 .00000 10.00000 .00000 .00000

1904-05 2 .00000 29.00000 .00000 .00000

1904-05 3 .00000 682.00000 .00000

1904-05 4 .00000 2626.00000 .00000 .00000

1904-05 5 .00000 3187.00000 .00000 .00000

1904-05 6 .00000 1113.00000 .00000 .00000

1904-05 7 .00000 583.00000 .00000 .00000

1904-05 8 .00000 272.00000 .00000 .00000

1904-05 9 .00000 144.00000 .00000 .00000

1904-05 Ito .00000 86.00000 .00000 .00000

1904-05 II .00000 50.00000 .00000 .00000

1904-05 12 •00000 22.00000 .00000 .00000

1905-06 1 •00000 10.00000 15.00000 .00000

1905-06 2 .00000 235.00000 210.00000 .00000

1905-06 3 .00000 919.00000 862.00000 .00000

1905-06 4 •00000 2740.00000 2448.00000 .00000

1905-04 5 .00000 2284.00000 1900.00000 .00000

1905-06 6 .00000 2381.00000 1776.00000 .00000

1 905-06 7 .00000 1077.00000 725.00000 .00000

1905-06 8 .00000 375.00000 280.00000 .00000

1905-06 9 .00000 192.00000 155.00000 .00000

1905-04 10 .00000 113.00000 92.00000 .00000

1905-06 11 .00000 64.00000 44.00000 .00000

1905-06 12 .00000 31,c4)000 17.00000 .00000

1904,07 1 .00000 15.000) 9.10000 .00000

1906-07 2 .0000n 143.001700 100.00000 .00000

1906-07 3 .00000 11..71.00464) 1050.00000 .00000

1906-07 4 .00000 5831.00000 3783.00000 .00000

1904-07 5 .00000 4186.00000 3279.00000 .00000

1904-07 6 .00000 1607.00000 1198.00000 .00000

1906-07 7 .00000 825.00000 520.00000 .00000

1906-07 8 .00000 465.00000 230.00000 .00000

1904-07 9 .00000 250.00000 130.00000 .00000

1904-07 10 .00000 140.00000 75.00000 .00000

1906-07 11 .00000 80.00000 35.00000 .00000

1906-07 12 .00000 40.00000 13.00000 .00000

1907-08 1 .00000 10.00000 9.10000 .00000

1907-08 2 .00000 120.00000 108.00000 .00000

1907-08 3 .00000 403.00000 296.00000 .00000

1907-08 4 .00000 905.00000 813.00000 .00000

1907-08 5 .00000 2194.00000 1823.00000 .00000

1907-08 6 .00000 1282.00000 968.00000 .00000

1907-08 7 .00000 613.00000 433.00000 .00000

1907-08 8 .00000 340.00000 200.00000 .00000

1 907-08 • .00000 185.00000 112.00000 .00000

1907-08 10 .00000 110.00000 64.00000 .00000

1907-018 11 .00000 62.00000 20.00000 .00000

1 907-08 12 .00000 28.00000 9.00000 .00000

166

1908-09 I . 00000 10.00000 9.10000 .00000

1908-09 , .00000 91.00000 120.00000 .00000

1908-09 3 .00000 799.00000 823.00000 .00000

1908-09 4 .00000 2195.00000 2352.00000 .00000

1908-09 5 .000,0 3691.00000 3156.00000 .00000

1908-09 6 . 00000 1395.00000 121•. 00000 .00000

1908 -09 7 . 00000 500.00000 473.00000 .00000

1908-09 e .00000 235.00000 250.00000 .00000

1908-09 9 .00000 130.00000 140.00000 .00000

1908-09 10 .00000 75.00000 80.00000 .00000

1908-09 11 .00000 42.00000 38.00000 .00000

1908-09 12 .000*0 18.00000 14.00000 .00000

1909-10 1 .00000 10.00000 15.00000 .00000

1909-10 2 .00000 286.00000 377.00000 .00000

*909-10 3 .00000 949.00000 921.00000 .00000

1909-10 4 .00000 2967.00000 3024.00000 .00000

1909-10 5 .00000 4144.00000 3279.00000 .00000

1909-10 6 .00000 1296.00000 1114.00000 .00000

1909-10 7 .00000 590.00000 703.00000 .00000

1909-10 a .00000 255.00000 290.00000 .00000

1909-10 9 .00000 140.00000 165.00000 .00000

1909-10 10 .00000 83.00000 96.00000 .00000

1909-10 11 .00000 46.00000 47.00000 .00000

1909-10 12 .00000 20.00000 18.00000 .00000

1910-11 1 .00000 10.00000 9.10000 .00000

1910-11 2 .00000 120.00000 108.00000 .00000

1910-11 3 .00000 590.00000 503.00000 .00000

1910-11 4 .00000 2134.00000 1951.00000 .00000

1910-11 5 .00000 3004.00000 2692.00000 .00000

1910-11 6 .00000 1221.00000 1023.00000 .00000

1910-11 7 .00000 472.00000 410.00000 .00000

1910-11 e .00000 215.00000 190.00000 .00000

1910-11 9 .00000 120.00000 106.00000 .00000

1910-11 10 .00000 70.00000 60.00000 .00000

1910-11 11 .00000 38.00000 26.00000 .00000

1910-11 12 .00000 16.00000 9.00000

1911 - 12 1 .00000 10.00000 9.10000 .00000

1911-12 2 .00000 120.00000 109.00000 .00000

1911-12 3 .00000 590.00000 423.00000 .00000

1911-12 4 .00000 1455.00000 1418.00000 .00000

1911-12 5 .00000 2439.00000 2041.00000 .00000

1911-12 6 .00000 930.00000 756.00000 .00000

1911-12 7 .00000 431.00000 342.00000 .00000

1911 - 12 8 .00000 220.00000 170.00000 .00000

1911 - 12 9 .00000 125.00000 96.00000 .00000

1911 - 12 10 .00000 72.00000 59.00000 .00000

1911-12 11 .00000 38.00000 22.00000 .00000

1911-12 12 .00000 16.00000 7.00000 .00000

1912- 13 1 .00000 10.00000 7.00000 .00000

1912- 13 2 .00000 120.00000 60.00000 .00000

1912-13 3 .00000 590.00000 491.00000 .00000

1912-13 4 .00000 1425.051000 1326.00000 .00000

1912-13 5 .00000 2348.00000 2057.00000 .00000

1912-13 6 .00000 1305.00000 978.00000 .00000

1912-13 7 .00000 436.00000 298.00000 .00000

1912-13 8 .00000 230.00000 140.00000 .00000

1912-13 9 .00000 135.00000 78.00000 .00000

1912-13 10 .00000 78.00000 42.00000 .00000

1912-13 11 .00000 43.00000 16.00000 .00000

1912-13 12 .00000 18.00000 4.00000 .00000

167

1913-14 1 .00400 10.00010 7.00000 .00000

1913-14 2 .00000 120.00000 90.00000 .00000

1913-14 3 .00000 333.00000 214.00000 .00000

1913-14 4 .00000 704.00000 532.00000 .00000

1913-14 .00000 919.00000 747.00000 .00000

1913-14 6 .00000 680.00000 472.00000 .00000

1913-14 7 .00000 251.00000 215.00000 .00000

1913-14 . 00000 121. 00000 115.00000 .00000

1913-14 9 .00000 64.00000 65.00000 .00000

1913-14 0 .00000 30.00000 35.00000 .00000

1913-14 1 .00000 10.00000 13.00000 .00000

1913-14 2 .00000 4.00000 3.00000 .00000

1914-15 I .00000 10.00000 .00000 .00000

1914-15 2 .00000 120.00000 .00000 .00000

1914-15 3 .00000 590.00000 .00000 .00000

1914-15 4 .00000 1323.00000 .00000 .00000

/914-15 5 .00000 1423.00000 .00000 .00000

1914-15 6 .00000 1035.00000 .00000 .00000

1914-15 7 .00000 360.00000 .00000 .00000

1914-15 8 .00000 200.00000 .00000 .00000

1914-15 9 •00000 115.00000 .00000 .00000

1914-15 10 .00000 70.00000 .00000 .00000

1914-15 11 .00000 40.00000 .00000 .00000

1914-15 12 .00000 16.00000 .00000 .00000

1915-16 1 .00000 10.00000 9.10000 .00000

1915-16 2 .00000 90.00000 108.00000 .00000

1915-16 3 .00000 636.00000 597.00000 .00000

1915-16 4 .00000 1896.00000 1931.00000 .00000

1915-16 5 .00000 2442.00000 2049.00000 .00000

1915-16 6 .00000 1261.00000 1008.00000 .00000

1915-16 7 .00000 350.00000 382.00000 .00000

1915-16 0 .00000 190.00000 220.00000 .00000

1915-16 9 .00000 105.00000 125.00000 .00000

1915-16 10 .00000 62.00000 70.00000 .00000

1915-16 11 .00000 34.00000 32.00000 . 00000

1915-16 12 .00000 12.00000 12.00000 . 00000,

1916-.1. I .00000 5.00000 9.10000 .00000

1916-17 2 .00000 4.00000 108.00000 .00000

1916-.7 3 .00000 726.00000 811.00000 .00000

1916-17 4 .00000 1782.00000 1883.00000 .00000

1916-17 5 .00000 3223.00000 2723.00000 .00000

1916-17 6 .00000 1664.00000 1239.00000 .00000

1916-17 7 .00000 400.00000 303.00000 .00000

191 4- 17 8 .00000 210.00000 215.00000 .00000

1916-17 9 .00000 120.00000 120.00000 .00000

1916-17 10 .00000 70.00000 68.00000 .00000

191 6- 17 11 .00000 38.00000 31.00000 .00000

1916- 17 12 .00000 16.00000 11.00000 .00000

1917-18

1917-is1

2

.00000 10.00000 9.10000 .00000

1917-18 3

.00000 20.00000 100.00000 .00000

1917-18 4

.00000 293.00000 243.00000 .00000

1917-18 5

.00000 2130.00000 1903.00000 .00000

1917-18 6

.00000

.00000

3393.00000

1195.00000.•00000

2626.00000

19 1 7-18

1917-18

7

e.000X) 330.00000 295.00000

.00000

.00000

1917-18

.00000 185.00000 190.00000

1917-18

9

*0

.00000 100.00000 102.00000

1917-19

1917-18

11

12

.00000

.00000

58.00000

32.00000

54.00000

24.00000

.00000

.00000.00000 1 1.00000 0.00000 .00000

168

1918-19 .00003 10.00000 9.10000

1918-19 2 .00000 200.00000 190.00000

191B-19 3 .00000 836.00000 730.00000 .00000

1918-19 4 .00000 3447.00000 2580.00000 .00000

191 8-19 5 .00000 5216.00000 3180.00000 .00000

1918-19 6 .00000 2573.00000 1723.00000 .00000

1918-19 7 .00000 645.00000 540.00000 .00000

1918-19 B .00000 335.00400 300.00000 .00000

1918-19 9 .00000 202.00000 170.00000 .00000

1918-19 i0 .00000 122.00000 100.00000 .00000

19 1B-19 .1 .00000 65.00000 48.00000 .00000

1918-19 .2 .00000 30.00000 19.00000 .00000

1919-20 1 .00000 10.00000 .00000 .00000

1919-2( 2 .00000 140.00000 .00000 .00000

1919-2C 3 .00000 404.00000 .00000 .00000

1919-20 4 .00000 1704.00000 .00000 .00000

1919-20 5 .00000 2261.00000 .00000 .00000

1919-20 6 .00000 1026.00000 .00000 .00000

1919-20 7 .00000 356.00000 .00000 .00000

1919-20 8 .00000 210.00000 .00000 .00000

1919-20 9 .00000 115.00000 .00000 .00000

1919-20 10 .00000 70.00000 .00000 .00000

1919-20 11 .00000 38.00000 .00000 .00000

1919-20 12 .00000 15.00000 .00000 .00000

1920-21 1 .00000 10.00000 9.10000 .00000

1920-21 z om000 ' 120.00000 108.00000 .00000

1920-21 3 .00000 540.00000 376.00000 .00000

1920-21 4 .00000 2535.00000 2041.00000

1920-21 5 .00000 4252.00000 3128.00000 .00000

1920-21 6 .00000 1311.00000 998.00000 .00000

1920-21 7 .00000 596.00040 310.00000 .00000

1920-21 e .00000 290.00000 180.00000 .00000

1920-21 9 .00600 160.00000 102.00000 .00000

1920-21 0 .00000 95.00000 56.00000 .00000

1920-21 1 ..0000 52.00000 24.00000 .00000

1920-21 2 .00000 .-., 3. 00(00 8.00000 .00000

1921-22. :

.00000 10.00000 9.10000 .00000

1921-22 .00000 120.00000 108.00000 .00000

1921 .... 3 .00000 396.00000 295.00000 .00000

1921-2.. 4 .021000 1201.00000 1189.00000 .00000

1921-22 ', .00000 2100.00000 1908.00000 .00000

1921-22 6 .00000 736.00000 646.00000 .00000

1921-22 7 .00000 270.00000 270.00000 .00000

1921-22 8 .00000 150.00000 155.00000 .00000

1921-22 9 .00000 90.00000 85.00000 .00000

1921-22 .0 .00400 50.00000 48.00000 .00000

1921-22 11 .00000 26.00000 19.00000 .00000

1921-22 12 .00000 10.00000 6.00000 .00000

1922-.A 1 .00000 10.00000 9.10000 .00000

192- 23 2 .00000 40.00000 108.00000 .00000

1922-23 3 .00000 402.00000 490.00000 .00000

1922-23 4 .00000 3213.00000 2724.00000 .00000

1922-23 5 .00000 6746.00000 4525.00000 .00000

1922-23 6 .00000 2778.00000 2100.00000 .00000

1922-23 7 .00000 778.00000 612.00000 .00000

1922-23 8 .v0.000 316.00400 237.00000 .00000

1922-23 9 .00000 158.00000 136.00000 .00000

1922-23 10 .00000 95.00000 78.00000 .00000

1922-23 11 .00000 53.00000 37.00000 .00000

1922-23 12 .00000 • 23.00000 14.00000 .00000

169

1o13-24 1 000000 i0.0,A00 9.1.000 .00000

1923-24 2 .00000 90.00000 108.00000 .00000

1923-24 3 .0,.,00,, e.28.00000 450.00000 .00000

1923-24 4 •00000 1808.00000 1635.00000 .00000

1923-24 5 . (K 0000 3764.0'000 2960.00000 .00000

1923-24 6 .00000 1463.00000 1055.00000 .00000

1923-24 7 .00000 741.00000 625.00000 .00000

1923-24 8 .00000 272.00000 300.00000 .00000

1923-24 9 .00000 138.00000 170.00000 .00000

1923-24 10 .00000 80.00000 100.00000 .00000

1923-24 11 .00000 44.00000 48.00000 .00000

19231'24 12 .00000 19.00000 20.0000 .00000

1924-25 1 .00000 10.00000 .00000 .00000

1924-25 2 .00000 144.00000 .00000 .00000

1924-25 3 .00000 1385.00000 .00000 .00000

1924-25 4 .00000 3973.00000 .00000 .00000

1924-25 5 .00000 5300.00000 .00000 .00000

1924-25 6 .00000 2463.00000 .00000 .00000

1924-25 7 .00000 796.00000 .00000 .00000

1924-25 8 -.00000 384.00000 .00000 .00000

1924-25 9 .00000 210.00100 .00000 .00000

1924-25 10 .00000 125.00000 .00000 .00000

1924-25 11 .00000 70.00000 .00000 .00000

1924-25 12 .00000 32.00000 .00000 .00000

1925-26 1 .00000 14.00000 9.10000 .00000

1925-26 2 .00000 101.00000 108.00000 .00000

1925-26 3 .00000 397.00000 555.00000 .00000

1925-26 4 .00000 2280.00000 2063.00000 .00000

1925,26 5 .00000 3275.00000 2812.00000 .00000

1925-26 6 . 00000 2506.00000 2204.00000 .00000

1925-26 7 .00000 765.00000 638.00000 .00000

1925-26 8 .00000 325.00000 270.00000 .00000

1925-26 9 .00000 185.00000 155.00000 .00000

1925-26 10 .00000 110.00000 90.00000 .00000

1925-26 11 .000.0 65.00000 43.00000 .00000

1925-26 .7 .00000 30.00000 16.00000 .00000

1926-27 1 .00000 1 0 .00000 9.10000 .000001926-27 2 . .,...),0 1 4.) .. + n )000 108.00000 .000001926-27 3 000, 507.. n0./00 555.00000 .000001926-27 4 .00000 1607.00000 1505.00000 .000001926-27 5 .00000 1741.00000 1536.00000 .000001926-27 6 .00000 973.00000 900.00000 .000001926-27 7 .00000 715.00000 420.00000 .000001926-27 8 .00000 270.00000 210.00000 .000001926-27 9 .00000 130.00000 120.00000 .000001926-27 10 .00000 76.00000 70.00000 .000001926-27 11 .00000 43.00000 30.00000 .000001926-27 12 .00000 18.00000 10.00000 .000001927-28 1 .00000 10.00000 9.10000 .000001927-28 2 .00000 120.00000 108.00000 .000001927-28 3 .00000 777.00000 578.00000 .000001927-28 4 .00000 2800.00000 2371.00000 .000001927-28 5 .00000 4745.00000 3440.00000 .000001927-28 6 .00000 2743.00000 2130.00000 .000001977-28 .00000 878.06000 761.00000 .000001927-28 .00000 3130.00000 310.00000 .000001927-28 9 .00000 205.00000 175.00000 .000001927-28 10 .00000 120.00000 105.00000 .000001927-28 11 .00000 70.00000 50.00000 .000001927-28 12 .00000 32.00000 20.00000 .00000

170

1928-29 1 .00000 10.00000. 9.10000 .00000

1928-29 2 .00000 50.00000 108.00000 .00000

1928-29 3 .00000 351.00000 .00000341.00000

1928-29 4 .0004.) 2973.00000 2534.00000 .00000

1928-29 5 .80000 4568.00000 3439.00000 .00000

1928-29 6 .00000 1679.00000 1442.00000 .00000

1928-29 7 .00000 696.00000 660.00000 .00000

1928-29 8 .00000 240.00000 275.00000 .00000

1928-29 9 .00000 130.00000 155.00000 .00000

1928-29 10 .00000 77.00000 90.00000 .00000

19.13-29 11 .000(0 39.0400 ) 44.00000 .00000

1928-29 12 .00000 15.......00• 17.00000 .00000

1929-30 1 . 05.5000 10.0(5000 9.10000 .00000

1929-30 2 .00000 300.00000 305.00000 .00000

1929-30 3 .00000 964. 00(0 660.00000 .00000

1929-30 • .00000 2948.00000 2383.00000 .00000

1929-30 5 .00000 4399.00000 3294.00000 .00000

1929-30 6 .00000 1340.00000 1147.00000 .00000

1929-30 7 .00000 434.00000 481.00000 .00000

1929-30 8 .00000 217.00000 235.00000 .00000

1929-30 9 .00000 123.00000 130.00000 .00000

1929-30 10 .00000 71.00000 75.00000 .00000

1929-30 11 .00000 38.00.00 35.00000 .00000

1929-30 12 .00000 15.00000 13.00000 .00000

1930-31 1 .00000 10.00000 9.10000 .00000

1930-31 2 .00000 170.00000 135.00000 .00000

1930-31 3 .00000 649.00000 491.00000 .00000

1930-31 4 .00000 262 1 .00000 2005.00000 .00000

1930-31 5 .00000 3412.00000 2477.00000 .00000

1930-31 6 .00000 1929.00000 1432.00000 .00000

1930-31 7 .00000 605.00000 478.00000 .00000

1930-31 8 .00000 290.00000 215.00000 .00000

1930-31 9 .00000 167.00000 122.00000 .00000

1930-31 10 .00000 97.00000 68.00000 .00000

1930-31 11 .00000 58.00000 32.00000 .00000

1930-31 12 .00000 25.00000 11.00000 .00000

1931-32 I .00000 10.00000 9.10000 .00000

1931-32 2 .00000 170.00000 140.00000 .00000

1931-32 3 .00000 940.00000 707.00000 .00000

1931-32 4 .00000 1735.00000 1465.00000 .00000

1931-32 5 .00000 2715. 00000 2400. 00000 .00000

1931-32 6 .00000 2119.00000 1641.00000 .00000

1931-32 7 .00000 550.00000 414.00000 .00000

1931-32 0 . 04)000 270.00000 225.00000 .00000

1931-32 9 . 000110 1 rni. 04.1090 127. 00000 .00000

1931-32 10 .00000 0.. n ...4)0 77.00000 .00000

1931-32 11 000000 •‘,....05000 34. 00000 .00000

193E-32 12 • 04)49) .:2. 00000 1.... 80000 .00000

1932-33 I .00000 10.00000 9.10000 .000001932-33 2 .00000 130.00000 .145.00000 .000001932-33 3 .004) 10 /80.00000 992.00000 .000001932-33 • .00000 2780.00000 2792.00000 .000001932-33 5 .00000 3181.00000 2840.00000 .000001932-33 6 .00000 1369.00000 .000001147.00000

1932-33 7 .00000 445.00000 485.00000 .000001932-33 8 .00000 227.00000 245.00000 .000001932-33 9 .00000 130.00000 138.00000 .000001932-33 10 .00000 75.00000 80.00000 .000001932-33 11 .00000 43.00000 .0000037.00000

1932-33 12 .00000 19.00000 14.00000 .00000

171

1933-34 - 1 .00000 10.00000 9.10000 .00000

1933-34 2 .0('000 153.00000 250.00000 .00040

1933-34 3 .0000) 1087.000,0 1258.00000 .00000

1933-34 4 .00000 3302.00000 2873.00000 .00000

1933-34 5 .00000 3571.00000 2652.000(10 .00000

1933-34 6 .00000 1066.00000 884.00000 .00000

1933-34 7 .00000 386.00000 342.00000 .00000

1933-34 a .00000 200.00000 200.00000 .00000

1933-34 9 .00000 115.00000 112.00000 .00000

1933-34 10 .00000 68.00000 64.00000 .00000

t933-34 ti .00000 35.00000 28.00000 .00000

1933-34 12 .00000 12.00000 9.00000 .00000

1934-35 1 .00000 10.00000 9.10000 .00000

1934-35 2 .00000 20 . ,,,00..0 108.00000 .00000

1934-35 3 .00000 270.00000 249.00000 .00000

1934-35 4 .00000 2339.00000 2103.00000 .00000

1934-35 5 .00000 3496.00000 2559.00000 .00000

1934-35 6 .00000 133t,. 00000 1104.00000 .00000

1934-35 .00000 440.00000 380.00000 .00000

1934-35 .000Q., :::20.900)0 190.00)00 .00000

1934-35 9 .1)001,0 130.0,7,000 110.0000'0 .00000

1934-35 10 .00000 75.00000 60.00000 .00000

1934-35 11 .00000 43.00000 31.00000 .00000

1914-35 12 .0.000 39. 04,11.,1) 8. 0000) .00000

1935-36 1 .00000 10.00000 9.10000 .00000

1935-36 2 .00000 120.00000 109.00000 .00000

1935-36 3 .00000 896.00000 889.00000 .00000

1935-36 4 .00000 4269.00000 3730.00000 .00000

1935-36 5 .00000 4971.00000 3610.00000 .00000

1935-36 6 .00000 2487.00000 1883.00000 .00000

1935-36 7 .00000 630.00000 491.00000 .00000

1935-36 0 .00000 265.00000 230.00000 .00000

1935-36 9 . 00000 152.00000 130.00000 .00000

1935-36 10 .00000 88.00000 75.00000 .00000

1935-36 11 .00000 50.00000 34.00000 .00000

1935-36 12 .00000 20.00000 13.00000 .00000

1936-37 1 .000(10 10. 00000 9.10000 .00000

1936-37 2 .00000 85.00000 108.00000 .00000

1936-37 3 .00000 599.00000 499.000C* .00000

1936-37 4 .00000 4593.00000 3803.00000 .00000

1936-37 5 .00000 5825.00000 4204.00000 .00000

1936-37 6 .00000 2261.00000 1727.00000 .00000

1936-37 7 .00000 707.00000 528.00000 .00000

1936-37 8 .00000 334.00000 260.00000 .00000

1936-37 9 .00000 172.00000 148.00000 .00000

1936-37 10 .000.* 105.00000 86.00000 .00000

1936-37 It .00000 62.00000 40.00000 .00000

1936-37 12 .00000 25.1.0000 15.00000 .00000

1937-38 1 .00000 10.00000 9.10000 .00000

1937-38 2 .00000 120.00000 1061.00000 .00000

1937-38 3 .00000 397.00000 308.00000 .00000

1937-38 4 .00000 1748.00000 1573.00000 .00000

1937-38 5 .0000o 3100.00007 2327.00000 .00000

1937-38 6 .00000 1339.110001 969.0000) .00000

1937-38 7 .0000.0 504.00000 365.00000 .00000

1937-38 a .00000 230.00000 180.00000 .00000

1937-38 9 .01000 130.00000 100.00000 .00000

1937-38 10 .00000 75.00000 56.00000 .00000

1937-38 11 410,0.0 24.00000 .00)00

1937-38 12 .00000 17.0000) 8.00000 .00000

172

1938-39 1 .00000 10.00000 9.10000 .00000

1938-39 2 .00000 12fl.00000 109.00000 .00000

1938-39 3 .00000 479.00000 437.00000 .00000

1938-39 4 .00000 I 826.00000 1801.00000 .00000

1938-39 5 .00000 3995.00000 3199.00000 .00000

1938-39 6 .00000 187,.00000 1329.00000 .00000

1938-39 7 .00000 800.00000 579.00000 .00000

1938-39 13 .00000 270.00000 230.00000 .00000

1938-39 9 .00000 150.001700 130.00000 .00000

1938-39 10 .00000 88.00000 75.00000 .00000

1938-39 11 . 0041,10 48.00000 34.00000 .00000

1938-39 12 .00000 21.00000 13.00000 .00000

1939-40 1 .00000 2.30000 9.10000 .00000

1939-40 , .00000 28.00000 108.00000 .00000

1939-40 3 .00000 362.00000 281.00000 .00000

1939-40 4 .00000 1935.00000 1711.00000 .00000

1939-40 5 .00000 2089.00000 1554.00000 .00000

1939-40 6 .00000 1377.00000 1004.00000 .00000

1939-40 7 .00000 435.00000 320.00000 .00000

1939-40 e .00000 220.00000 160.00000 .00000

1939-40 9 .00000 125.00000 70.00000 .00000

*939-40 10 .00000 72.00000 50.00000 .00000

1939-40 11 .00000 40.00000 20.00000 .00000

1939-40 12 .00000 16.00000 6.00000 .00000

1940-41 1 .00000 10.00000 9.10000 .00000

1940-41 2 .00000 50.00000 108.00000 .00000

1940-41 3 .00000 210.00000 193.00000 .00000

1940-41 4 .00000 1316.00000 997.00000 .00000

1940-41 5 .00'4" 1096.00000 .00000

1940-41 6 .00000 1254.00000 931.00000 .00000

1940-41 7 .0000O 529.00000 375.00000 .00000

1940-41 e .00000 200.00000 190.00000 .00000

1940-41 9 .00000 I.:0.00000 105.00000 .00000

1940-41 10 ,00000 68.00000 60.00000 .00000

1940-41 11 .00000 30.00000 26.00000 .00000

1940-41 12 .r...XWQ 15.00000 8.00000 .00000

1941-42 1 .00000 10.00000 9.10000 .00000

1941-42 2 .00000 120.00000 108.001300 .00000

1941-42 3 .00000 339.00000 304.00000 .00000

1941-42 4 .00000 1158.00000 907.00000 .00000

1941-42 5 .00000 2115.00000 1631.00000 .00000

1941-42 6 .00000 740.00000 535.00000 .00000

1941-42 7 .00000 247.00000 220.00000 .00000

1941-42 e .00000 130.00000 125.00000 .00000

1941-42 9 .00000 75.00000 70.00000 .00000

1941-42 10 .00000 44.00000 38.00000 .00000

1941-42 11 .00000 19.00000 14.00000 .00000

1941-42 12 .00000 6.50000 3.50000 .00000

1942-43 1 .00000 10.00000 9.10000 .00000

1942-43 2 .00000 120.00000 109.00000 .00000

1942-43 3 .00000 385.00000 370.00000 .00000

1942-43 4 .00000 1896.00000 1672.00000 .00000

1942-43 5 .00000 1715.00000 1269.00000 .00000

1947-43 .., .00 707 539.00000 447.00000 .00000

1942-43 7 .00000 266.00000 260.00000 .00000

1942-43 8 .00000 140.00000 150.00000 .00000

1942-43 9 .00000 60.00000 84.00000 .00000

1942-43 10 .00000 43.00000 44.00000 .00000

1942-43 11 .00000 22 .00000 15.00000 .000,00

1942-43 12 .00000 8.00000 5.00000 .00000

173

1943-44 . 00000 10. 00000 . 7.10000 .00000

1943-44 12.l0 10,3.

1943 - 44 3 .00,1. 3.1.f.,...0000

1943-44 4 .00000 1867.00000 1705.00.;,00 •00000

1943-44 5 .00000 251.00000 2269.0)000 .00200

1943-44 6 .00000 1601.00000 1252.00000 .00000

1943-44 7 .00000 443. 7.1c.00000 .0000

1943-44 a .0(.7.-x) 195.00000 175.00009 . nre-snel

1943-44 9 .00000 110.00000 9e. 000 .., .70000

1943-44 10 .00000 65.00000 55 .00000 .00002

1943-44 11 .00000 35.20000 23.20000 .00000

1943-44 12 .00000 14.00000 7.000)0 .00000

1944-45 1 .00000 10.00.000 9.10000 .20000

1944-45 2 .00000 120.00000 137.00200 .00000

1944-45 3 .00000 225.00000 220.00000 .00000

1944-45 4 .00000 814.00000 768.00000 .00000

1944-45 5 .00000 1444.00000 1283.00000 .00000

1944-45 6 .00000 663.00000 484.00000 .00000

1944-45 7 .00000 339.00000 260.00000 .00000

1944-45 a .00000 160.00000 140.00000 .00000

1944-45 9 .00000 95.00000 78.00000 .00000

1944-45 10 .00600 55.00000 43.00000 .00000

1944-45 11 .00000 28.00000 21.00000 .00000

1944-45 12 .00000 10.00000 4.50000 .00000

1945-46 1 .00000 10.00000 9.10000 .00000

1945-46 2 .00000 120.00000 108.00000 .00000

1945-46 3 .00000 396.00000 300.00000 .00000

1945-46 4 .00000 3260.00000 2545.00000 .00000

1945-46 5 .00000 4738.00000 3473.00000 .00000

1945-46 6 .00000 1909.00000 1439.00000 .00000

1945-46 7 .00000 464.00000 460.00000 .00000

1945-46 8 .00000 195.00000 255.00000 .00000

1945-46 9 .00000 110.00000 145.00000 .00000

1945-46 10 .00000 65.00000 85.00000 .00000

1945-46 11 .00000 35.00000 39.00000 .00000

1945-46 12 .0000, 14.00001 1 15.00000 .00000

1946-47 1 .00000 10.00000 9.10000 .00000

1946-47 2 .00000 120.00000 109.00000 .00000

1946-47 3 .00000 362.00000 415.00000 .00000

1946-47 4 .00000 2505.00000 2251.00000 .00000

1946-47 5 .00000 3024.00000 2397.00000 .00000

1946-47 6 .00000 1819.00000 1439.00000 .00000

1946-47 7 .00000 580.00000 456.00000 .00000

1946-47 a .00000 23E1.00000 210.00000 .00000

1946-47 9 .60000 130.00000 118.00000 .00000

1946-47 10 .00000 75.00000 67.00000 .00000

1946-47 11 .00000 41.00000 30.00000 .00000

1946-47 12 .00000 17.00000 10.00000 .00000

1947-48 1 .00000 10.00000 9.10000 .00000

1947-48 2 .00000 120.00000 108.00000 .00000

1947-48 3 .00000 343.00000 255.00000 .00000

1947-48 4 .00000 1860.00000 1362.00000 .00000

1947-48 5 .00000 3363.00000 2157.00000 .00000

.947-48 .00000 1509.00000 779.00000 .00000

1947-48 7 .00000 397.00000 300.00000 .00000

1947-48 .00000 180.00000 170.00000 .00000

1947-48 9 .00000 105.00000 95.00000 .00000

1947-48 10 .00000 60.00000 53.00000 .00000

1947-48 11 .00000 32.00000 22.00000 .00000

1947-48 12 .00000 12.00000 7.00000 .00000

174

1948-49 1 .00000 5.00000 9.10000 .00000

1948-49 2 .00000 31.00000 109.00000 .00000

1948-49 3 .00000 591.00000 544.00000 .00000

1948-49 4 .00000 1836.00000 1656.00000 .00000

1948-49 5 .00000 2656.00000 2264.00000 .00000

1948-49 6 .00000 961.00000 864.00000 .00000

1948-49 7 .00000 398.00000 400.00000 .00000

.000001948-49 .00000 168.00000 =20.00000

1948-49 9 . 00000 105.00,..0, 124.00000 .00000

1948-49 10 .00000 60.00,0 70.00000 .00000

1948-49 11 .0(X,00 32.00000 32.00000 .00000

1948-49 12 .•...0 hl 11..000(0 11.00000 .00000

1949-50 1 ..AK 00 5.00000 4.00000 .00000

1949-50 .00000 9 • 00000 20.00000 .00000

1949-50 3 .00000 325.00000 331.00000 .00000

1949-50 .00000 2052.00000 2027.00000 .00000

1949-50 .00000 1912.00000 1487.00000 .00000

1949-50 809.00000 621.00000 .00000

1949-50 7 .00000 216.00000 285.00000 .00000

1949-50 8 .00000 123.00000 165.00000 .00000

1949-50 9 .00000 73.00000 93.00000 .00000

1949-50 10 .00000 42.00000 52.00000 .00000

1949-50 11 .00000 20.00000 21..00000 .00000

1949-50 12 .00000 10.00000

5.00060

7.00000 .00000

1950-51 1 .00000 2.80000 .00000

1954-51 2 .00000 3.00000 18.70000 .00000

1950-51 3 .00000 545.00000 466.00000 .00000

1950-51 4 .00000 2914.00000 2224.00000 .00000

1950-51 5 .00000 5891.00000 3856.00000 .00000

1950-51 à .00000 3071.00000 2492.00000 .00000

1950-51 7 .00000 778.00000 642.00000 .00000

1950-51 9 .00000 304.00000 242.00000 .00000

1950-51 9 .00000 153.00000 125.00000 .00000

1950-51 10 .00000 86.00000 62.00000 .00000

1950-51 II .00000 43.00000 30.00000 .00000

1950-51 12 .00000. 13.00000 10.50000 .00000

1951-52 1 10.30000 4.00000 6.50000 .30000

1951-52 2 63.00000 57.00000 69.40000 5.80000

1951-52 3 379.00000 387.00000 356.00000 64.00000

1951-52 4 1373.00000 1418.00000 1368.00000 327.00000

1951-52 5 2001.00000 2331.00000 1999.00000 613.00000

1951-52 6 2592.00000 3581.00000 2625.00000 1341.00000

1951-52 7 1131.00000 1455.00000 1147.00000 301.00000

1951-52 8 338.00000 423.00000 346.00000 76.00000

1951-52 9 185.04000 214.00000 190.00000 33.10000

1951-52 10 109.00000 125.00000 119.00000 14.60000

1951-52 11 55.00000 64.00000 55.30000 6.10000

1951-52 12 24.00000 27.00000 23.10000 2.20000

1952 -53 1 ,., 7rn.n.. , 5%1,14100 7.70000 .70000

1152-53 -2 38.00000 22.00000 32.20000 .20000

1952-53 3 505.00000 524.00000 473.00000 133.00000

1952-53 4 1283.00000 1395.00000 1259.00000 401.00000

1952-53 5 2094.00000 2421.00000 2180.00000 792.00000

1952-53 6 1868.00000 3126.00000 2006.00000 1096.00000

1952-53 7 407.00000 597.00000 431.00000 132.00000

1952-53 8 200.00000 246.00000 199.00000 48.90000

1952-53 9 102.00000 134.00000 108.001000 23.10000

1952-53 10 49.00000 71.00000 55.20000 9.00000

1952-53 11 31.00000 37.00000 27.30000 4.60000

1952-53 12 13.30000 17.00000 11.80000 1.110000

175

1953-54 1 2.80000 3.00000 3.90000 .40000

1953-54 2 169.00000 101.00000 149.00000 9.70000

1953-54 3 B33.00000 788.00000 831.00000 144.00000

1953-54 4 1427.00000 1547.00000 1432.00000 357.00000

1953-54 5 2237.00000 2926.00000 2409.00000 725.00000

1953-54 6 1003.00000 1236.00000 1023.00000 214.00000

1953-54 7 356.00000 464.00000 389.00000 70.00000

1953-54 a 1E16.00000 219.00000 201.00000 28.00000

1953-54 9 122.00000 140.00000 133.00000 13.50000

1953-54 0 63.00000 81.00000 72.30000 8.30000

1953-54 1 33.00000 41.00000 32.20000 3.60000

1953-54 2 13.40000 13.00000 9.50000 1.40000

1954-55 1 ZB . 00000 12.00000 18.00000 .30000

2 954-55 2 234.00000 253.00000 224.00000 60.00000

1954-55 3 1024.00000 963.00000 949.00000 253.00000

1954-55 4 3456.00000 3987.00000 3610.00000 1123.00000

1954-55 5 3031.00000 4419.00000 3214.00000 1199.00000

1954-55 6 1308.00000 1655.00000 1343.00000 289.00000

1954-55 7 570.r0000 681.00000 554.00000 126.00000

1954-53 8 331.00000 396.00000 330.00000 60.00000

1954-5t 9 170.00000 197.00000 171.00000 29.10000

1954-51 10 96.00000 I16.00, 0K. 95.50000 13.60000

1954-52 II 53.00000 68.00000 52.10000 7.20000

1954-55 12 39.00000 42.00000 32.50000 3.00000

1755-56 1 40.00000 32.00000 38.60000 1.80000

1955-56 2 197.00000 207.00000 194.00000 43.40000

1955-56 3 642o00000 612.00000 606.00000 180.00000

1955-56 4 2948.00000 3563.00000 2931.00000 1222.00000

1955-56 5 3174.00000 4004.00000 3232.00000 1032.00000

1955-56 6 1772.00000 2615.00000 1909.00000 572.00000

1955-56 7 630.00000 770.00000 631.00000 126.00000

1955-56 8 291.00000 347.00000 298.00000 55.00000

1955-56 9 170.00000 203.00000 176.00000 27.90000

1955-56 10 100.00000 119.00000 105.00000 13.90000

1955-56 11 56.00000 69.00000 54.30000 7.40000

1955-56 12 25.00000 34.00000 17.20000 3.50000

1956-57 1 12.50000 13.00000 7.50000 1.30000

1956-57 2 6E1.00000 40.00000 48.00000 10.60000

1956-57 3 461.00000 493.00000 436.00000 137.00000

1956-57 4 2279.00000 2210.00000 2191.00000 601.00000

1956-57 5 3240.00000 5237.00000 34138.00000 1780.00000

1 956-57 6 1704.00000 2159.00000 1750.00000 368.00000

1956-57 7 504.00000 634.00000 503.00000 97.00000

1956-57 8 232.00000 285.00000 234.00000 42.60000

1956-57 9 134.00000 163.00000 136.00000 20.00000

1956-57 10 77.00000 99.00000 76.70000 9.00000

1956-37 11 40.00000 60.00000 39.40000 4.60000

1956-57 12 17.60000 24.00000 13.90000 2.00000

1957-58 1 6.40000 8.00000 5.40000 .90000

1957-58 2 208.00000 215.00000 199.00000 48.60000

1957-58 3 539.00000 608.00000 525.00000 122.00000

1957-58 4 2612.00000 2668.00000 2562.00000 735.00000

1957-58 5 3145.00000 4227.00000 3295.00000 1141.00000

1957-58 6 2438.00000 2904.00300 2451.00000 500.00000

1957-58 7 761.00000 935.00000 752.00000 129.00000

1957-58 e 293.00000 351.00000 295.00000 52.00000

1957-58 9 166.00000 197.00000 168.00000 23.80000

1957-58 10 94.00000 118.00000 98.00000 10.70000

I957-5E II 50.00000 67.00000 48.40000 4.90000

1 957-5E 12 23.20000 32.00000 1E1.70000 2.20000

176

1955-59 1 18.40000 18.00000 12.00000 1.20000

*958-59 2 167.00000 175.00000 162.00000 22.60000

1958-59 3 519.00000 568.00000 479.00000 131.00000

t 958-59 4 3760.00000 3985.00000 3625.00000 990.00000

1958-59 5 2913.00000 4028.00000 3025.00000 795.00000

*958-59 6 1541.00000 1916.00000 1563.00000 370.00000

1958-59 7 672.00000 785.00000 643.00000 143.00000

1958-59 8 355.00000 444.00000 350.00000 80.00000

1958-59 9 192.00000 237.00000 191.00000 28.60000

1958-59 10 110.00000 139.00000 110.00000 12.90000

1955-59 11 63.00000 84.00000 61.00000 7.80000

1958-59 12 28.00000 40.00000 23.00000 3.00000

1959-60 1 26.00000 19.00000 17.30000 1.20000

1959-60 2 184.00000 164.00000 161.00000 13.80000

1959-60 3 460.00000 583.00000 435.00000 73.00000

1959-60 4 2077.00000 2434.00000 2159.00000 855.00000

1959-60 5 21536.00000 4047.00000 2987.00000 1118.00000

1959-60 6 915.00000 1242.00000 928.00000 242.00000

1959-60 7 380.00000 487.00000 377.00000 71.00000

1759-60 8 179.00000 223.00000 181.00000 33.00000

1959-60 9 104.00000 126.00000 106.00000 16.20000

1959-60 10 58.00000 76.00000 55.00000 8.10000

1959-60 11 29.00000 42.00000 24.30000 4.10000

1959-60 12 11.80000 17.00000 9.00000 1.90000

1960-61 1 3.90000 5.00000 3.10000 1.10000

1960-61 2 89.00000 82.00000 75.00000 7.10000

1960-61 3 839.00000 789.00000 726.00000 191.00000

1960-61 • 1414.00000 1790.00000 1446.00000 551.00000

1960-61 5 2006.00000 2508.00000 2133.00000 625.00000

1960-61 6 1042.00000 1301.00000 1045.00000 250.00000

1960-61 7 397.00000 504.00000 402.00000 70.00000

1960-61 8 171.00000 213.00000 177.00000 30.20000

1960- ,.1

1960-61

9 93.00000 120.00000 98.80000 14.80000

1960-61

10 56.00000 75.00000 54.50000 0.10000

1960-61

11 29o00000 41.00000 25.20000 3.80000

1961-62

12 10.80000 16.000001.800000.00000

1 4.14400 3.50000 2.70000 .500001961-62

1961-62

2 98.00000 102.00000 77.50000 34.70000

1961-62

3

4

400.00000 781.00000 713.00000 188.00000

1961-62 5

2470.00000 2956.00000 2760.00000 706.00000

1961-62

3130.00000 5201.00000 3723.00000 1709.00000

1961-62

6 1447.00000 1360.00000 1051.00000 -209.00000

1961-62

7 484.00000 458.00000 373.00000 61.00000

1961-62

8 207.00000 207.00000 174.00000 23.00000

1961-62

9 127.00000 121.00000 97.00000 11.20000

1961-62

10 70.0000051.30000 4.10000

74.00000

11 35.00000 21.10000 2.6000040.00000

1961-62

1962-63

12 16.70000 12.00000 5.70000 .90000

1962-63

1 10.50000 2.70000 2.70000 .20000

1962-63

2 64.30000 85.00000 80.30000 22.40000

1962-63

3 2741.00000 511.00000 436.00000 122.00000

1962-63

4 1370.00000 2220.00000 1927.00000 746.00000

1762-63

5 2350.00000 3632.00000 260/.00000 1245.00000

1962-63

6 2680.00000 1313.00000 324.000001620.00000

1962-63

7 643.00000 594.00000 478.00000 110.00000

1962-63

8 246.00000 262.00000 218.00000 35.40000

1962-43

9 149.00000 138.00000 117.00000 16.90000

1962-63

10 82.10000 86.00000 64.00000 7.90000

1962-63

11 39.9000027.30000 3.90000

43.00000

1963-64

12 17.30000 18.00000 9.20000 1.70000

1963-64

1 8.64000 8.00000 5.70000 .40000

1963-64

2 93.80000 7.00000 10.10000 .90000

1963-64

3 523.00000 473.00000 370.00000 170.00000

1963-64

4 2310.00000 1620.00000 1279.00000 524.000005 3000.00000 2772.00000 2306.00000 744.00000

177

1963-64 6 1990.00000 1988.00000 1792.00000 395.00000

1965-64 7 593.00000 636.00000 516.00000 83.00000

1963-64 8 267.00000 230.00000 197.00000 30.50000

1963-64 9 165.00000 129.00000 107.00000 13.90000

1963-64 10 108.00000 72.00000 52.00000 6.30000

1963-64 1 53.70000 36.00000 22.30000 2.80000

1963-64 .2 22.40000 13.80000 6.30000 .80000

1964-65 I 14.50000 3.20000 2.90000 .10000

1964-65 2 94.70000 171.00000 29.00000 40.90000

1964-65 3 399.00000 602.00000 519.00000 180.00000

1964-6$ 4 1040.00000 1973.00000 2100.00000 714.00000

1964-6$ 5 1810.00000 5680.00000 4135.00000 1005.00000

1964-65 6 718.00000 3999.00000 1462.00000 329.00000

1964-65 7 253.00000 580.00000 453.00000 93.00000

1964-65 8 149.00000 285.00000 227.00000 44.20000

1964-65 9 76.30000 166.00000 136.00000 19.60000

1964-65 0 39.30000 105.00000 78.00000 9.90000

1964-65 1 16.50000 58.00000 32.00000 5.00000

1964-65 2 4.31000 26.00000 11.50000 2.10000

1965-66 1 1.86000 9.18000 .00000 .60100

1965-64 2 44.00000 84.10000 .00000 18.90000

1965-66 3 601.00000 513.00000 .00000 94.40000

1965-66 4 1420.00000 3270.00000 .00000 1120.00000

1965-66 5 2870.00000 3340.00000 .00000 1310.00000

1965-66 6 1800.00000 2050.00000 .00000 439.00000

1965-66. 7 727.00000 641.00000 .00000 135.00000

1965-66 e 248.00000 290.00000 .00000 63.80000

1965-66 9 133.00000 171.00000 .00000 25.00000

1965-66 0 74.70000 104.00000 .00000 10.00000

1965-66 1 33.500.10 57.50000 .00000 5.00000

1965-66 2 14.10000 28.20000 .00000 2.00000

1966-67 1 3.52000 10.60000 .00000 1.00000

1966-67 2' 35.30000 76.00000 .00000 26.00000

1966-67 3 281.00000 367.00000 .00000 74.90000

1966-67 4 1910.00000 1380.00000 .00000 496.00000

1966-67 5 2270.00000 2830.00000 .00000 898.00000

1966-67 6 660.00000 3900.00000 .00000 1620.00000

1966-67 7 238.00000 053.00000 .00000 231.00000

1966-67 e 124.00000 321.00000 .00000 95.10000

1966-67 9 64.70000 174.00000 .00000 14.70000

1966-67 10 35.90000 105.00000 .00000 5.27000

1966-67 11 15.70000 61.60000 .00000 2.40000

1966-67 12 3.38000 27.50000 .00000 1.74000

1967-68 1 1.56000 11.00000 .00000 1.23000

(967-68 2 24.80000 89.40000 .00000 20.30000

1967-68 3 455.00000 560.00000 .00000 123.00000

1967-62 4 2290.00000 2410.00000 .00000 513.00000

1967-68 5 2170.00000 5830.00000 .00000 1550.00000

1967-68 6 610.00000 2800.00000 .00000 774.00000

1967-68 7 206.00000 764.00000 .00000 147.00000

1967-68 e 107.00000 346.00000 .00000 35.10000

1967-68 9 52.80000 211.00000 .00000 27.30000

1967-68 10 28.30000 134.00000 .00000 13.60000

1967-68 II 9.09000 77.70000 .00000 5.63000

1967-68 12 1.53000 36.30000 .00000 4.20000

1968-69 1 .38400 16.60000 .00000 3.33000

1968-69 2 91.10000 76.00000 .00000 4.73000

1968-69 3 305.00000 421.00000 .00000 73.90000

1968-69 4 776.00000 1010.00000 .00000 144.00000

1968-69 5 1020.00000 1800.00000 .00000 379.00000

1968-69 6 469.00000 853.00000 .00000 166.00000

1968-69 7 196.00000 301.00000 .00000 .00000

1968-69 8 111.00000 169.00000 .00000 .00000

1968-69 9 50.80000 93.30000 .00000 7.48000

1969-69 10 26.00000 54.80000 .00000 4.65000

1968-69 11 8.44000 27.101100 .00000 1.96000

1968-69 12 1.44000 8.01000 .00000

178

1969-70 1 .3190) 2.60600 .00000 .00000

1969-70 7 139.0000) 71.40000 .00000 13.70000

1969-70 3 359.00000 683.00000 .00000 148.00000

1969-70 4 1610.00000 1650.00000 •00000 491.00000

1969-70 5 1190.0k 000 3150.00000 .00000 664.00000

1969-70 6 429.00000 2040.00000 .00000 483.00000

1969-70 7 164.0(0(0 947.00000 .00000 164.00000

1969-7( 9 72.10000 308.00000 .00000 43.20000

1969-7C 9 36.00000 157.00000 .00000 .00000

1969-7C 10 18.90000 92.90000 .00000 9.85000

1969-7( 11 5.80000 50.50000 .00000 5.40000

1969-7( 12 .76900 24.50000 .00000 1.63000

1970-71 i 3.40000 5.47000 .00000 .45000

1970-71 2 103.00000 29.60000 .00000 .21000

1970-71 3 733.00000 29.70000 .00000 .00000

1970-71 4 2724.00000 2250.00000 .00000 742.00000

4970-71 5 3589.00000 2500.00000 .00000 568.00000

1970-71 6 1030.00000 791.00000 .00000 106.00000

1970-71 7 380.00000 2134.00000 .00000 40.80000

1970-71 171.00000 144.00000 .00000 14.20000

1970-71 9 96.00000 85.00000 .00000 9.86000

1970-71 10 52.00000 52.60000 .00000 4.73000

1970-71 It 25.00000 27.70000 .00000 1.64000

1970-71 12 7.50000 10.20000 .00000 1.38000

1971-72 t 5.40000 3.90000 .00000 .00000

1971-72 2 101.00000 2.90000 .00000 6.75000

1971-72 3 505.00000 481.00000 .00000 81.10000

1971-72 4 1894.00000 2530.00000 .00000 033.000400

1971-72 5 2512.00000 2740.00000 .00000 682.00000

1971-72 6 1320.00000 810.00000 .00000 142.00000

1971-72 7 486.00000 261.00000 .00000 39.10000

1971-72 a 217.00000 131.00000 .00000 14.90000

1971-72 9 116.00000 75.60000 .00000 7.64000

1971-72 10 66.00000 46.40000 .00000 3.80000

1971-72 11 30.00000 20.80000 .00000 1.01000

1971-72 12 10.70000 3.18000 .00000 .01500

1972-73 1 3.30000 .90000 .00000 .00000

1972-73 2 6.60000 43.0004)0 .00000 3.76000

1972-73 3 382.00000 291.00000 .00000 40.30000

1972-73 4 1266.00000 795.00000 .00000 193.00000

1972-73 5 2179.00000 1060.00000 .00000 169.00000

1972-73 6 1790.00000 499.00000 .00000 .00000

1972-73 7 509.00000 218.00000 .00000 .00000

1972-73 191.00000 106.00000 .00000 .00000

1972-73 9 103.00000 54.70000 .00000 .00000

1972-73 10 53.00000 27.00000 .00000 .00000

1972-73 11 24.00000 9.20000 .00000 .00000

1972-73 12 7.90000 18.30000 .00000 .00000

1973-74 I 2.80000 .33900 .00000 .00000

1973-74 2 162.00000 126.00000 .00000 8.56000

1973-74 3 568.00000 327.00000 .00000 4.02000

1973-74 4 2059.00000 1670.00000 .00000 458.00000

1973-74 5 3995.00000 1360.00000 .00000 314.00000

1973-74 6 1403.00000 497.00000 .00000 50.00000

1773- 74 446.0000C 120.00000 .00000 13.40000

1973-74 8 226.00000 72.60000 .00000 7.37000

1973-74 9 " 123.00000 39.60000 .00000 3.00000

1973-74 10 66.00000 18.60000. .00000 1.00000

1973-74 II 33.50000 7.04000 .00000 .10000

1973-74 12 10.00000 1.18000 .00000 .00000

179

180

1974-75 1 .00000

1974-75 2 .00000 3.00000

1974-75 3 .00000 739.00000

1974-75 4 ..00») 3236.00000

1974-75 5 .00000 3138.00000

1974-75 6 .00000 1321.00000

1974-75 7 .00000 371.00000

1974-75.00000 143.00000

1974-75 9.01,X0' 59.7000)

1974-75 10 .00,-.K.) 34.10000

1974-75 11.r.)000, 14.4(000

1974-75.

12 .00000 4.20000

1975-76 1 .00000 1.23500

1975-76 2 .00000 .31700

1975-76

1975-76

1975-76

3

4

5

•00000

.00000

.0000n

552.00000

1586.0,0000

3281. 00000

1975-76 6 •00000 1158.00000

1975-76

1975-76

1975-76

7

8

9

.0000n

.00000

.00000

3132.00000

149.00000

60.70000

1975-76 10 .00000 34.40000

1975-76 11 .00000 14.09000

1975-76

1976-77

1976-77

12

1

2

.00000

.00000

.00000

2.53000

40.00000

57.00000

1976-77 3 .00000 302.00000

1976-77

1976-77

4

5

.00000

.00000

547.00000

485.00000

1976-77 6 .00000 487.00000

1976-77 7 .00000 420.00000

2976-77

1976-77

8

9

.00000

.00000

225.00000

165.00000

1979-77 10 .00000 124.00000

1976-77 11 .00000 93.00000

1976-77 12 .00000 69.00000

1977-78 1 .00000 1.38000

1977-78 2 .00000 1.68000

1977-78 3 .00000 230.00000

1977-78 4 -.00000 841.00000

1977-78 5 .00000 1728.00000

1977-78 6 .00000 752.00000

1977-78 7 .00000 211.00000

1977-78 8 .00000 61.00000

1977-78 9 .00000 32.20000

1977-78 10 .00000 12.00000

1977-78 11 .00000 3.24000

1977-78 12 .00000 .90000

1978-79 1 .00000 .70600

1970-79 2 .00000 7.79000

1978-79 3 .00000 359.00000

1970-79 4 .00000 1764.00000

1970-79 5 .00000 1892.00000

1970-79 6 .00000 1314.00000

1978-79 7 .00000 462.00000

1970-79 8 153.00000.00000

1970-79 9 .00000 67.90000

1970-79 10 .00000 31.50000

1970-79 11 .00000 9.65000

1970-75 12 .00000 2.85000

.00000 .00000

.00000 .00000

.00000 222.00000

•oo0o0 899.00000

.00000 seg509.00000

.00000

226.00000

.00000

43.40000

.00000

13.900(10

.00000

7.10000

.0000,1

3.56n00

.00000

1.39000

.00000 .23000

.00000 .00000

.00000 .00000

.00000

109.00000

.00n00

318.00000

.00000

/27.00000

....00:0000000 194.00000

0

47.000

0

17.00000

00

.00000 7.31000

.00000 3.34000

.54600.00000

.00000 :0000000060

.00000

.00000

2.64000.00000

109.50000

.00000 250.00000.00000

173.80000

188.80000.00000

.00000 120.00000

.00000 9.13000

23.60000.00000

.00000 4.35000

.00000 .98300

.00000 .00700

.00000 .00000

.0000 0 .38400

.00000

23.60000

.00000 125.00000

.00000 410.00000

.00000 184.00000

.00000 31.00000

.00000

.170001

.00000 3.76000

.00000 1.04000

.00000

.00600

.00000 .00000

.00000 .00000

.00000 .00000

.00000 36.60000

.00000 590.00000

.00000 431.00000

.00000 292.00000

.00000 95.60000

.00000 20.70000

.00000 9.73000

.00000

3.73000

.00000 .50200

.00000 .00000

1979-80 1 .00000 1.68000 .00000 .00000

1979-80 2 .00000 42.30000 .00000 1.43000

1979-00 3 .00000 300.00000 .00000 50.00000

1979-80 4 .00000 991.00000 .00000 187.00000

1979-80 .00000 1263.00000 .00000 312.00000

1979-80 6 .00000 573.00000 .00000 121.00000

1979-90 7 .00000 293.00000 .00000 34.60000

1979-80 8 .00000 98.00000 .00000 10.60000

1979-80 9 .00000 43.20000 .00000 3.66000

1979-80 10 .00000 17.30000 .00000 .25500

1979-80 11 .00000 4.20000 .00000 . 0 '..,300

1979-80 1, .00000 L 4300.)3000 . 00000 . 00000

1900-81 I o3 ,000 .00000 .00000 .00000

1980-81 2 . 00000 .00000 .00000 .00000

1900-81 3 .00000 .00000 .00000 .00000

1980-81 4 .00000 .00000 .00000

1980-81 5 .00000 .00000 .00000 .00000

1980-81 6 .00000 .00000 .00000 .00000

1980-81 7 .00000 .00000 .00000 .00000

1990-81 8 .00000 .00000 .00000 .00000

1980-81 9 .00000 .00000 .00000 .00000

1900-81 10 .00000 .00000 .00000 .00000

1980-91 11 .00000 . 00000 .0000o .00000

1990-81 12 .00000 .00000 .00000 .00000

1981-82 1 .00000 .32900 .00000 .00000

1981-82 2 .00000 30.40000 .00000 .00000

1981-82 3 .00000 457.00000 .00000 71.90000

1981-82 4 .00000 1921.00000 .00000 468.00000

1981-82 5 .00000 • 1748.00000 .00000 477.00000

1981-82 6 .00000 457.00000 .00000 97.90000

1991-82 7 .00000 229.00000 .00000 24.30000

1991-82 9 .00000 83.50000 .00000 9.93000

1981-82 9 .00000 40.80000 .00000 2.49000

1991-82 10 .00000 18.70000 .00000 .35400

1981-82 11 .00000 4.66000 .00000 .00000

1981-82 12 .00000 1.26000 .00000 .00000

181

APPENDIX B

THREE VERSIONS OF PROGRAM DATA 1 FOR MONTHLY,SEASONAL AND ANNUAL FLOWS USED FOR THE NORMALITY CHECK

182

APPENDIX Bi:

Program Data 1 for Monthly Flows

183

184

1 4.y..41ç poocrAm ANALY7ES THE CAT/. TO CHFCK Frc p.CP"ALITy ANC clturN

2 •A4ITY. THE CHFCm WILI Fr rFr r(PmF0 UN Thr PrthTFLT I.ESICUALS.3 +7.-ITç MODK is orp“ tY MASIER,S OEGPFF THESIS4 LIST nF vArIARLFÇ

,:: mriou(1.J.K).0.1A,- rrNTpLT FIOVE OPTAINFO Fi- P1 THE VAILY FLCVE

h RcLnwEI.J.0),OFEllUALS

9144MI:gtr41:13F TrANSFCRPqn FLJVS7

OC N(I)KNUMAFE fir YEAS FCP EITF I

IJ C YPAO(LrI).YEAP L JF PECOKD FOP E/ TF I.

11 C SITE(11.NAmF OF THE SITES1 1 N0mf(II=NLm8EP Cr CAY5 I IN A GIVEN MONTH

23 2 xmEANFP.1=mFAN CF tONTH M

14C U(1.m).STP,OARD DEVIATION OF MONTH M

15 g u(o.m),CKFwNESS rc PONTH M-g

15 g U(1.m)=AuTLCCFPFLA1ION COFFFTrIFNT OF MONTH pi

17 ro. THE rTI- , rp VAFIAPLEE SEF IFSL SvArotiIINIS USEn IN THIS PROGRAm

19 P90G94m DATi3ITNFU1,OPIPLT,TAFr5=INPUTOAPEe.OUTru1l

1 9 PEAL DFLOV(4.100,12).MFLCV(411C0.12),PFLOV(4,100,12),LF104(4,100,

23 •12).PmEAN(I2),Ut3,121 11,0(700),COMP(2).CS,0.MFA.SD.

21 •PITF(61,CELLEF21,TEMP1(10C,12),TEMP2(1C3.12),TEMP3E1UC.11,22 •TENP4(100.1),YE1C),11,X1(100.4 1

23 INTEGER N(4),N0m(12/rm,Z,TY,IFm.TNCD(90),1.J,m,N1.1,

24 44,41.10I5TOOPT,IPF0E3E1C0/,IND,FELX,V,R4

25 CHAPACTEF*7 EITE(4),YEAP(I0C,4)

76 C-EAPACTFP*3 MONE/7)

27 CHAPACTER*30 VAPP VAR2,VAP3.VAP4

2P COMMGN/CNF/MEA.3O

79 EYTFPNAL NORm

30 C INPUT NAME OF SITES1 1 00 16 T=1,4

52 9E8DF5=1 I STT((I)

33 1 FOPmAT f5X,A71

34 16 CONT1N1IF

35 , INPUT TI r NUmPER OF YFAPS FOP EACH SITE

36 RFA0(5,*) INII/riz1,4)

27 C IN9UT THE NurcEP rIF DAYS TN EACw MONTH AND THF MONTH

39 ' 01 8 1=1,12'IQ RFAn (5=2) PONIII,NCm(I)

ZO ;FOPMAT (5X.A3.1Y.I7)41 rrn NTINIIF42 C INPUT THE NUMPEP JF EQUIPPBAALF CELLS FOR EACH SITE FOP TF, E CHI —SCuARrD

43 cc.Ar(5,*) (K3(i),I.1,4144 C INPUT Ti-IF FLCW DATA45 OJ 10 Iw1,4

46 V=0

47 DO 11 L.10(I)

4 4 9rAn(501 YFAP(I.1)43 F0RmAT E5F , A7I

50 11 CONTINUF

51 Or 13 L•IrP(I)

57 RFAn 150.1 IMFICJI1.10(100.1,17/

53 13 CONTINUE

54 C POINTING THE MONTNLY FLCWE

55 VARI.IMEAN mlINTHlY FL065 1

VAF7.,PESIOUALS OF MONTHLY FLOWS ,

5 CALL SrP1(6A11,SITt,8F10 61 ,8ON.YFAR , I , N 1COMPUTATION OF MEAN,ETANOAPC rEVIATIUN, COEFF OF SKEWNESS,KUPTOSIE,AND CUPRELATION CrFFF FOE Ti-IF MONTHLY 'FLOWS

FFLY=NCTIDO 70 0.1,12

DC 71 J.1.NEI1TEmP1(.11,0).PFLOVOI , J ,0 /TEMP3(.1.1).TFPP1(JoK)

71 CONTINUECALL S1P3(TEMP3,DrEA1,05131,DC01 , 0S 0 F 1 pFE 1 Y/XNEAN(0).DMEA1 •11(1.00.rS01U(2.K1=DSKE1U(3,1().DC01

70 rnNTENvrCALL SUB2(vAR1.SITE,m0N,xmEANN1 I pi1COMPUTATION OF THE RESIDUALS

DO 3CDr 31 V.1.12PFLO10(1,1,K).(mFLOw11, ,, K1 —xmFANIK 1) /U (1,K1

31 CnNTINuE3)CONTINUE

PRINTING OF ifir RESIDUALSCALL SUBI(VAR2.SITE,RFLOW=MON.YEARFI , N )

COMPUTATION OF:THF BASIC STATISTICS OF THE RESIDUALSFELY.N(I100 72 0.1,12

DC 73 J.I.N(1)TEmP2(J,0).PFLOW(I , J. 0 ).TENP4E5,11•TEMP21.1, 0 1

Y1.11, 11A1EMP4(J.1)73 CENTINUE

CALL SUB3(TEMP4IDM6A2,0502,DCO2,DS 0 E 2 oFELY/YmEAN(0).DMEA7U(1,,K)KDSD?U(2.1().DSNF2P(3.1().rCC2

PROdABILITY PLOT JE EACH MONTHLY' RESIDUALS USING IMSLN1.17.N( I)N2AN(I/IlIsT=1IOPT=0

55575 95960618263646566t768Sc70717273747576777Q7960f1P2P314851687I- 99953clc2935495Qh97GP59

185

1C) CALL "SPDP(TFrP 4 .7,00,N2,1('T5T,1097.6r,IFP)1 101 C CHI S0LJAPEC TEST1C2 'qA=YNFAN(K)1C3 SO.U(I,r11C4 14.r3(I)105 TOF.c106 CALL GFIT(Ncpr,N4,1FrP4,2,CrLLS,r0rP,CS,IDF,°,TER)107 oPINT 6901CA 4.Q3 r0PrA7(1x,r CrI-tcuAREC TF:T ".1 )ILQ PPINT 7L1,(CELLEW,J=1,A4)11 1 700 F)PPAT Ilkt"CLANTS CF LASFFVATILMS IN CELLS"./41X4 F(F1C45o2T111 4- ),I43211 .7(F3C4:42T)4//*)112 ppINT 7C1A(CCFP(J)AJA10(4)

113 771 FOANAT (lxCENpoNrNTS CF EN1-SOUARFC STATISTIC",/,1X,114 A °(C1C.5,2X),/,32Y,7(FlO.5,2X),,/,)115 PAINT 7C2,CS,C.10F1/5 702 ; ) 4.AT (1W!CSA",4Y,F1C.5./.1X1"0.",5Y.F10.5A/Ir1Ag"ILJA . ,117 4 3Y.T5,1Y)115 C KOLM9GOPCV S!'IFAlli TEST11Q CALL srPr(x)123 CALL NKs1(Nr;m.x.7.rniF.Irr)121 • opINT 7 0 1122 790 FO7mATI//,1X," xumccoRtiv—smIRNry TEST "f/1123 ''RINT FOOA(PnIrWAJA1,6)1 7 4 ROC FfirmAi(1x,e(F10.5,2Y.)./)125 r ARINTING /PC rONTH PLOTTED126 "PINT 9.C.rON(r1127 .0 rOPMAT f3GY,A3,////)1?P 72 CONTIP4F129 C PRTNTINC, CF TF STATS CF THr RrSIDUALS110 CALL SV:32(VAR2.SITL,r0N.xMIAN,U .1)131 v.V.E113? TF (V.F0.1) INFN133 DO 6C , ..1, N(I1114 ru ti K.1.12135 MFLCA(I.JrN)AL0G(PFLOW(IpJAK))116 61 CONTINUE137 60 CoNTINUE11° VAR1.1LOG TRANSFORrED reNTFLY FLOWS'119 vAp7.IRFSICUALS OF TPANSF0FrED fLORS 1

143 GO TO 5141 FLSF142 END IF143 10 CONTINUF144 STOP145END

186

1 SUBROUTINE NORM(XFP)2 COMMON/ONE/NEA,SD

REAL MEA, SDT* (X ..-MEA)/SD

5 P..5*ERECI—T*.7071068)6 P.!ETURN7 END

123456

SUBROUTINE SUB1(A,B.C,O,E.11,E)REAL C(4,100.12)INTEGER 11,1,JoR,E(1C0)CHARACTER*30 ACHAR4CTER*3 0( 12)CHARACTER*? E(100,4),B(4)

7 PRINT 100,A,B(I1)II 130 FORMAT(50X,A30,/p50Y," FOP" rA81//)9

101PRINT 101,( 0 (K )#0, 1,6)

13 F0RMAT(//.2*,"1YEAR ".2Y,61A10,8X),,)

D1 10 J*11pFII)12 PRINT 1021 E(J,I1 ) 41(C(Ilo.),K )00, 1,6 )13 102 F0RMAT(IX,A7,2X,6(F15.5.3X),/)14 10 CONTINUE15 PRINT 111,(0(K), ( *7,12)16 111 FORMAT(//r2X."YEAR")2X,6(A10.8X)IP/)17 DO 11 J*1,F(I1)18 PRINT 112,EIJIII)r(C(IIIJ.<),K*7,12)19 112 F0RM4T(1XoA7,2X,6(F15.5,3%),/)20 11 CONTINUE2122

RETURNEND

1

187

SUBROUTINE Sj82(AArBB,C2,00.FE,0G)

2 REAL 00(12).EE(3.12)3

4 /HTAFRIETE7 B8(4)

5 CHARACTER*3 00(12)

;'' 4igtCHÔ:liott(GG)8 10C FORMAT(55X." BASIC STATS OF "o/150X.A30,/,50%, " FOP ",A7r1X,//11)

9 PRINT 301.((8),K*1.6)

10 301 FORM4T(8X,6(i10,8Y)./)

11 PRINT 302,(DD(K),K*1,6 ).(EF(1,1().**1.6 )

12 1.(EE(20().K.1.6 )01LE(3.1().K*1.6 )

13 312 FORMAT (lx," MEAN "f3Y,6(F15.5.3Y),//111X)"5OFV",3X,6(F15.5,3Y)14

15•.04iX571S,7E(17$3%5(FIW5,3X).//,1X," CORP ".3Y.6(F15.5,3X),/.1

V3 311 FORMAT(8%.6(410.6).;)PRINT 312.(3)(K),K*7,12).(EE(11,K).10 , 7,12)

18 *,(EE(20K),K*7.12/1(EE(3,K).0.7.12)

19 312 FORMAT (lx." MEAN "f3X,6(F15.5,3Y).//.1*,"SDF 1d".3Y.6(F15.5.3X)

20 +,//olx." SKE4 "Jp3X.6(F15.5,3X),//.1%," CORP ".3Y,61F15.5.3X1,/,/

21 RETURN

22 END

I SUBROJTINE SUB3 .(8.B.CPCPEr0FLX)

2 REAL AlDELX.1)

3 INTEGER DEL%4 B*0.0

5 C*0.0

6 D*0.0

7 E*0.0

8 G*0.0

9 H*(1.0

10 S*.0.0

iiC COMPUTATION OF MEAN

.00 10 I*1,DELX

13 1103+A(I.1)

14 10CONTINUE

15 B*B/DELX

16 C COMPUTATION OF STANDARD DFVIATION

17 DO 15 1.1,DELX

18 G.G+((A(I.1)-8)**2)

19 15 CONTINUE

20 G*G/DELX

21 C*SORT(G)

22 C COMPUTATIUN OF CORRELATION

23 DO 20 I*1.DELX-1

24 H.H4(A(I.1)-..1)*(4(I*111)—B)

25 20 CONTINUE

26 D*11/((DELX-1)*(C**2))

27 : COMPUTATION OF THE SKEW COEFFICIENT

28 DO 25 1.1,DELK

29 S*S4((A(I,1 )-8)**3)

30 25 CONTINUE

31 . E*S/(DELX*(C**3))32ii RETURN

END

1 SU6kOUTINE S)RT(XNC)2 INTEGER XN0(100,1).TEMR

3 INTEGER I.J.M4 DO 32 J*1.99

5 M.100.-J6 DO 22 1.1tti

7 IF (XN0(111 1).LTOING(1+1, 1)) GO 10 228 TEMP*XN0(1. 1)9 wm0(1, 1) , x4J(14.1, 1)10 X40(I+1. 1)*TEMP

11 22 CONTINUE

12 3i CONTINUE •

13 RETURN

14 END

APPENDIX B2:

Program Data 1 for Seasonal Flows

188

189

1 C +THIS PROGRAM ANALYZES THE DATA FOR NORMALITY AND STATION

2 C +ARITY THE CHECK WILL BE PERFORMED ON THE SEASONAL RESIDUALS3 q +THIS WORK IS DONE MY MASTER'S DEGREE THESIS4 .:4 LIST OF VARIABLES5 MFLOW(I.J.K)=MEAN MONTHLY FLOWS OBTAINED FROM THE DAILY FLOWS

6 r.... RFLOW(I,J.K)=RESIDUALS7 C LFLOW(1,30()=LOG TRANSFORMED FLOWS9 c motw3t.moNTH J9 C N(I)=NUMBER OF YEARS FOR SITE I10 C YEAR(L,I)=YEAR L OF RECORD FOR SITE I.11 C SITE(I)=NAME OF THE SITES .12 C NDM((I)=NUMBER OF DAYS I IN A GIVEN MONTH13 C XMEAN(M)=MEAN OF MONTH M14 C U(101)=STANDARD DEVIATION OF MONTH M15 C U(2,M)=SKEWNESS OF MONTH M16 C U(3,M)=AUTOCORRELATION COEFFICIENT OF MONTH M17 C FOR THE OTHER VARIABLES SEE IMSL SUBROUTINES USED IN THIS PROGRAM

18 PROGRAM DATAI(INFUT.OUTPUT,TAPE5=INPUT,TAPE 6=OUTPUT )

19 REAL DFLOW(4,100.12).MFLOW(4.100.12) , RFLOW( 4 . 100 . 12) .LFLOW(4 . 100,

20 +12),XMEAN(12),U(3,12) .WK(200),COMP(2) , CS , OsMEA , SD ,

21 +PDIF(6).CELLS(2),TEMP1(100.12),TEMP2(100.12).TEMP3(100.1).22 +TEMP4(10011).X(100,1),X1(100,4 )'SFLOW(4.100,12),AFLOW(4,100.12)23 INTEGER N(4),NDM(12).M.Z,IX.IER , INCD(80) , I.J.K.N 1 . 1— ,

24 +N2,IDIST'IOPT,IDF,K3(100).IND,FELX,V,K425 CHARACTER*7 SITE(4),YEAR(100.4)26 CHARACTER*30 MON(12)27 CHARACTER*30 VARI.VAR24VAR3.VAR428 COMMON/ONE/MEA,SD29 EXTERNAL NORM30 C INPUT NAME OF SITES31 DO 16 1=1,432 READ(511 ) SITE(I)33 t FORMAT (5X.A7)34 16 CONTINUE35 C INPUT THE NUMBER OF YEARS FOR EACH SITE36 READ(51*) (N(I),I=1.4)37 C INPUT THE NUMBER OF EQUIPRBABLE CELLS FOR EACH SITE FOR

THE CHI—SQUARED

38 READ(500) (K3(I).I=1,4)39 C INPUT THE THE SEASON40 DO 91 1=1,241 READ(5,4) MON(I)42 4 FORMAT (5X.A16)43 91 CONTINUE44 C INPUT THE FLOW DATA45 DO ro 1=1.446 V=047 DO 11 L=1.N(I)48 READ(5.3) YEAR(LII)49 3 FORMAT (5X.A7)50 11 CONTINUE51 DO 13 L=IIN(I)w•-n...I4 READ (5,*) (MFLOW(I.LIK),K=1 , 12)53 13 CONTINUE54 C COMPUTATION OF THE SEASONAL AND ANNUAL FLOWS55 CALL DATA(MFLOW.SFLOW.AFLOW.I.N)

56 C PRINTING THE SEASONAL FLOWS57 VAR1='MEAN SEASONAL FLOW ,

58 VAR2=,RESIDUALS or SEASONAL FLOW ,

59 5 .CALL SU81(VAR1,SITE,SFLOW,MON.YEAR,I,N)60 C COMPUTATION OF MEAN.STANDARD DEVIATION, COEFF OF SKEWNESS,61 C KURTOSIS,AND CORRELATION COEFF FOR THE SEASONAL FLOW62 FELX=N(I)63 DO 70 K=1,264 DO 71 J=1.N(I)65 TEMP1(i,K)=SFLOW(I,3.K)66 TEMP3(1.1)=TEMP1(J.K)67 71 CONTINUE68 CALL 5UB3(TEMP3,DMEA1,DSDI.DC01,DSKEI,FELX)69 XMEAN(K)=DMEA170 i U(1.14)=DSD171 U(2,K)=DSKE172 U(3,K)=DC0173 70 CONTINUE74 CALL SUB2(VAR1,SITE,MON,XMEAN,U ,I)75 C COMPUTATION OF THE RESIDUALS76 DO 30 J=1,N(1) .._ .

77 DO 31 K=1,278 RFLOW(I,J,K)=(SFLOW(I,J.K)—XMEAN(K))/U(1,1079 71 CONTINUE80 70 CONTINUE81 C PRINTING OF THE RESIDUALS82 CALL SUB1(VAR2'SITE.RFLOW.MON,YEAR.I.N)63 C COMPUTATION or THF BASIC STATISTICS OF THE RESIDUALS84 FELX=N(I)65 no 72 K=1,:.86 DO 73 J=1.N(I)87 TEMP2(J.K)=RFLOW(I,J.K)88 TEMP4(J.1)=TEMP2(J.K.69 X(J'1)=TEMF4(J,1)90 73 CONTINUE91 CALL SUB3(TEMP4,DME42.DSD2.0CO2,DSKE2,FELX)92 XME4N(K)=DMEA293 U(1,K)=D5D294 U(2,K)=DSKE295 U(3,K)=00O296 C PROBABILITY PLOT OF EACH SEASONALRESIDUALS USING IMSL

190

97 N1=1

98 Z=N(I)

99 N2=N(I)

100 IDIST=I

101 IOPT=0

102 CALL USPRP(TEMP4.Z,N1,N2,IDIST.IOPT,WR,IER)

103 C CHI SOUARED TEST104 MEA=xMEAN(K)

105 S0=U(1,K)

106 K4=K3(I)

107 IDF=0108 CALL GFIT(NORm.144,TEMP4,Z.CELLS,COMP.CS,IDF,Q.IER)

109 PRINT 690

110 670 FORmAT(1X," CHI-SQUARED TEST "./)

111 PRINT 700.(CELL5(3),3=1,144)

112 700 FORMAT (1X,"COUNTS OF OBSERVATIONS IN CELLS.!. lx, 81F10.5.2X

113 + ).1,32X ,7(F10.5.2X),//.)

114 PRINT 701,(COMP(3),J=1,1(4)

115 701 FORMAT (1X,"COmPONENTs OF CHI-SQUARED STATISTIC"./.1X,

116 + 8(F10.5,2X),/,32x,7(F10.5,2X),//,)

117 PRINT 702,CS.0.I 0F

118 7C2 FORMAT (1X,"CS= ,4X,F10.5./,1X,"0= - ,MX,F10.5,/,1X,"IDF=,

119 + 3X,I15,IX)

120 C KOLMOGOROV SMIRNOV TEST121 CALL SORT(X)

122 CALL NKS1(NORM,X,Z,PDIF,IER)

123 PRINT 790

124 790 FORMAT(//,1X," KOLMOGOROV-SMIRNOV TEST ",/)

125 PRINT 800.(PDIF(J),J=1.6)

126 BOO FORMAT(1X,6(F10.5,2X,),/)

127 C PRINTING THE MONTH PLOTTED128 PRINT 90,MON(K)

129 90 cORMAT (I0X,A30,///)

130 72 CONTINUE131 C PRINTING OF THE STATS OF THE RESIDUALS132 ' CALL SUB2(VAR2,SITE,MON.XMEAN,U ,I)

133 V=V+1 _

134 IF (V.E0.1) THEN135 DO 60 3=1, N(I)

136 DO 61 K=1,2

137 SFLOW(I,J.1()=LOG(SFLOW(1,J,K))

138 61 CONTINUE139 60 CONTINUE140 VAR1=,LOG TRANSFORMED SEASONAL FLOWS/141 VAR2=,RESIDUALS OF TRANSFORMED FLOWS ,

142 SO TO 5

143 ELSE144 END IF145 10 CONTINUE146 STOP .

147 END

191

1 SUBROUTINE NORM(X,P)

2 COMMON/ONE/MEA,SD

3 REAL MEA,SD4 T., (X-MEA7/SD5 P=.5*ERFC(—T‘.7071068)

6 RETURN

7 END

.4 SUBROUTINE SUB1(A,B,C,D,E,I1,F)REAL C(4,100.12)5 INTEGER I1,L,J,K,F(100)

4 CHARACTER*30 A

5 CHARACTER*30 0(12)

6 CHARACTER*7 E(100,4).B(4)

7 PRINT 100,A1B(I1)8 100 PORMAT(50X.A30,/,50X," FOR " .AS,//)

9 PRINT 101,(D(K )0(-1,2)

10 101 FORMAT(//,2X," YEAR ".2X12(416,SX),/)

11 DO 10 3=1,F(I1)

12 PRINT 102. E(J.I1 ) .(C(Il.J.K ),K=1,2 )

13 102 FORMAT(1X.A7,2X,2(F15.5.9X)./)

14 10 CONTINUE

15 11 CONTINUE

16 RETURN

17 END

1 SUBROUTINE SUB2(AAIBB.CC,DD,EE,GG)

2 REAL DD(12),EE(3.12)INTEGER GG

! CHARACTER*7 BB(4)

gCHARACTER*30 CC(12)CHARACTER*30 AA

7 PRINT 100,AAIBB(GG)B 100 FORMAT(5.5X," BASIC STATS OF "si,50X,A30,/,50X. " FOR ",A7.1)(7//,)

9 PRINT 301,(CC(K),K=1.2)

10 301 FORMAT(8X,2(1416.8X),/)

11 PRINT 302,(DD(K),K=1,2 ),(EE(1110,10.1,2 )

12 +.(EE(2,K),K=1,2 ),(EE(3,K),K61,2 )

13 302 FORMAT (1)(." MEAN ",50(1 , 2(F15.5,3X),//v1X,"SDEV",9X,2(F15.513X)

14 +.//,1X," SKEW "19X,2(F15.5,3X),//v1X," CORR ",9X,2(F15.5,3X),/,)

15 RETURN

16 END

192

1 SUBROUTINE SUB3(A,B,C,D,E,DELX)

2 REAL A(DELX,1)

3 INTEGER DELX

4 B=0.0

5 C=0.0

6 D=0.0

7 E=0.0B G=0.0

9 H=0.0

10 S=0.0

11 C COMPUTATION OF MEAN12 DO 10 I=1,DELX

13 B=B+A(I,1)

14 10 CONTINUE15 B=B/DELX

16 C COMPUTATION OF STANDARD DEVIATION17 DO 15 I=1.DELX

18 G=G+((A(I,1)-B)**2)

19 15 CONTINUE20 G=G/DELX

21 C=SORT(G)

--).-. C COMPUTATION OF CORRELATION...

23 DO 20 I=1,DELX-1

24 H=H+(A(I.1)-B)*(A(I+1,1)-B)

-,.,- 20 CONTINUE.,

26 DH/( (DELX-1)*(C*4 ,2))

27 C COMPUTATION OF THE SKEW COEFFICIENT28 DO 25 I=1.DELX

29 S=S+((A(1,1)-B)**3)

30 ,..., CONTINUE31 E=S/(DELX*(C*4.3))

32 RETURN33 END

1 SUBROUTINE SORT(XN0)

2 INTEGER XNO(100,1),TEMP

3 INTEGER 1,3,M

g DO 32 3=1.99M=100-3

6 DO 22 I=1,M

7 IF (XNO(I. 1).LT.XNO(I+1, 1)) GO TO 22

8 TEMP=XNO(I, 1)

9 XNO(I, 1)=XNO(I+1, 1)

10 XNO(I+1, 1)=TEMP

11 22 CONTINUE12 32 CONTINUE13 RETURN14 END

193

1 SUBROUTINE DATA(MF,SF,AF,I1,7)

3C THIS SUBROUTINE TRANSFORMS MONTHLY FLOWS INTO SEASONALC AND ANNUAL FLOWS

4 C VARIABLE LISTC MF(I.J,K)=MEAN MONTHLY FLOWS FOR SITE 11.YEARJ MONTH K

g C SF(I,J,K)=SEASONAL FLOW FOR SITE I1,YEAR J REASON K

7 C AF(I,J)=ANNUAL FLOW FOR YEAR Je c I1=SITE NUMBER

9 C DM(K)=NUMBER OF DAYS IN A MONTH 1410 C Z(11)=NUMBER OF YEARS FOR A SITE II

11 REAL DF(4,100,12),MF(4,100,12),SF(4,100,12),AF(4,100,12)

12 INTEGER DM(12),Z(4),I1

13 C INITIALIZATION

14 DO 1 3=1,Z(I1)

15 AF(I1,2,1)=0.0

16 DO 2 K=1,12

17 SF(I1,3,K)=0.0

18 2 CONTINUE

19 1 CONTINUE

20 C COMPUTATION OF ANNUAL FLOWS

21 DO 10 3=1.Z(I1)

22 AF(I1,3,1)=0.0

23 DO 11 K=1.12

24 AF(I1,J,1)=Ar(11,3,1)+MF(I1,J,K)

2 41 CONTINUE

26 AF(11,3,1)=(AF(I1,3,1))/12

27 10 CONTINUE

28 C COMPUTATION OF THE SEASONAL.FLOWS

29 C HIGH FLOW SEASON(SEASONI=JUL,AUG,SEP,OCT,NOV)

30 DO 20 3=1,Z(I1)

31 SF(I1,3,1)=0.0

32 DO 21 K=3,7

33 SF(I1.3,1)=SF(I1,3,1)+MF(I1,3,K)

34 21 CONTINUE

35 SE(I1,3,1)=(SF(11,3.1))/5

36 20 CONTINUE

37 C LOW FLOW SEASON(SEASON2=DEC,JAN,FEB,MAR,APR,MAY)

38 DO 30 3=1,Z(I1)

39 SF(I1,3,2)=0.0

40 DO 31 K=8,12

41 SF(I1,3,2)=SF(11,3,2)+MF(II,J,K)

42 31 CONTINUE

43 DO 32 K=1.2

44 SF(I1,3,2)=SF(I1,3,2)+MF(I1,J,K)

45 32 CONTINUE

46 SE(11,3,2)=(SF(II,3,2))/7

47 30 CONTINUE

48 RETURN

49 END

APPENDIX B3:

Program Data 1 for Annual Flows

194

195

1 C +THIS PROGRAM ANALYZES THE DATA FOR NORMALITY ANC STATION

2 C +AR1TY THE CHECK WILL RE PERFORMED ON THE ANNUAL RESIDUALS

3 C +THIS WORK IS DONE my mAsTER,S DEGREE THESIS

4 C LIST OF VARIABLES

5 C mFLOw(I,J,K)=mEAN MONTHLY FLOWS OBTAINED FROM THE DAILY FLOwS

6 C RFLOw(1,J,K)=RESIDUALS

7 C LFLOw(I,J,m)=LOG TRANSFORMED FLOWS

e C moN«J)=MONTH J

9 C N(I)=NUMBER OF YEARS FOR s;TE 1

10 C yEAR(L,I)=yEAR L OF REcoRD FOR SITE I.

11 C SITE(I)=NAME OF THE SITES

12 C NDm«I)=NUmBER OF DAYS 1 IN A GIVEN MONTH

13 C xmEAN(m)=mEAN OF MONTH M

14 C U(1M)=STANDARD DEVIATION OF MONTH M

IS C U(2,m)=SKEwNESS OF MONTH M

16 C U(3,m)=AuTOCoRRELATIoN COEFFICIENT OF MONTH M

17 c FOR THE OTHER VARIABLES SEE ImSL SUBROUTINES USED IN THIS PROGRAM .

le PROGRAM DATAI(INpuT,OUTRUT,TARE5=INPuT,TAPE6=OUTRUT)

19 REAL DFLOW(4,100,12),MFLOW(4,100 , 121,RFLOW(4,100 , 12) , LFLOw(4 , 100 ,

20 +12),xmEAN(12),U(3,12) ,WK(200),COmP(2),CS,O,MEA,SD ,

21 +FDIF(6),CELLS(2),TEMpl(100,12),TEMP2(100,12),TEMP3(100,1),

22 +TEmp4(100,1),X(100,1),X1(100,4 ),5FLOW(4,100,12),AFLOW(4,1 10,12)

23 INTEGER N(4),NDm(12),M,Z,IX,IER,INCD(80),I,3,K.N1,L,

24 +N2,IDIST,101.1,1DF,K3(100),IND,FELx,v,K4

25 CHARACTER=7 SITE(4),yEAR(100,4) .k.

26 cHARACTER=30 MON(12)

27 CHARACTER=30 vAR1,VAR2,vAR3,VAR4.

2e COmmoNioNEMEA,SD

29 EXTERNAL NORM

30 C INPUT NAME OF SITES

31 DO 16 1=1,4

32 READ(5,1 ) SITE(I)

33 1 FORMAT (5X,A7)

34 16 CONTINUE

35 C INPUT THE NUMBER OF )(EARS FOR EACH SITE

36 READ(5,1)-(N(1),1=1,4)

37 C INPUT THE NUMBER OF EQuipRBABLE CELLS FOR EACH SITE FOR THE 04I-S0uARED

38 READ(5,1) (K3(I),I=1,4)

39 C INPUT THE THE SEASON

40 DO 91 1=1,2

41 READ(5,4) MON(I)

42 c. FORMAT (5X,A16)

43 91 CONTINUE

44 C INPUT THE FLOW DATA

4n Do 10 1=1,4

46 V=0

47 DO 11 L=1,N(I)

48 READ(5,3) YEAR(L,I)

49 3 FORMAT (5X,A7)

50 11 CONTINUE

51 DO 13 L=1,N(I)

52 READ (5,4) (MFLOW(I,L,K),K=1,12)

53 13 CONTINUE

54 C COMPUTATION OF THE SEASONAL AND ANNUAL FLOWS

55 CALL DATA(MFLOW,SFLOW,AFLOW,I,N)

56 C PRINTING THE ANNUAL FLOWS

57 vAR1=,MEAN ANNUAL FLOW ,

58 vAR2=,RESIDUALS OF ANNUAL FLOW ,

59 5 CALL SUB1(VAR1,SITE,AFLOW , MON , YEAR , I , N )

60 C COMPUTATION OF MEAN,STANDARD DEVIATION, COEFF OF SKEWNESS,

61 C KuRTOSIS,AND CORRELATION COEFF FOR THE ANNUAL FLOW

62 FELX=N(I)

63 DO 70 K=1,1

64 DO 71 3=1,N(I)

65 TEmpi(J,K)=AFLOW(I,J,K)

66 TEmP3(.1,1)=TEmP1(3,K)

67 71 CONTINUE

68 CALL SUB3(TEMP3,DMEA1,DSD1 , DC0 1, DSKElcFELX )

69 XMEAN(K)=DMEA1

70 U(1,K)=DSD1

71 U(2,K)=DSKE1

72 U(3,K)=DC01

73 70 CONTINUE

74 CALL SUB2(vARI,SITE , MoN , xMEAN , U ,i)

75 C COMPUTATION OF THE RESIDUALs

76 DO 30 3=1,N(I)

77 DO 31 K=1,I7E1 RFLOW(I.J,K)0(AFLOW(I,J,K)7XMEAN(14»/U(1,K)79 31 CONTINUE80 30 CONTINUE

81 C PRINTING OF THE RESIDUALS82 CALL SUB1(VAR2,SITE,RFL0W , mON , yEAR.I.N )

83 C COMPUTATION OF THE BASIC STATISTICS OF THE RESIDUALS

84 FELX=N(I) .85 DO 72 K-1,186 DO 73 3=1,N(1)87 TEMFT(J,K)=RFLOW(I,J,K)ea TEMP4(J.1)=TEMP2(3,K)89 X(3,1)=TEMP4(3,1)90 73 CONTINUE91 CALL SUB3(TEMP4,DMEA2.DSD2 , DCO2, DSKE2, FELX )

92 XMEAN(K)=DMEA293 U(1,K)=DSD294 . U(2,K)=DsKE295 U(3,K)=Dc0296 C PROBABILITY PLOT OF EACH SEASONALRESIDUALS USING IMSL

196

97 N1=198 Z=N(I)99 N2=N(I)100 IDIST=1101 IOPT=0102 CALL USPRP(TEMP4,7,N1442,IDIST , I0PT , WK , IER)103 C CHI SOUARED TEST104 MEA=XMEAN(K)105 SD=U(1,K)106 K4=K3(I)107 IDF=0100 CALL GFIT(NORM,K4,TEMP4,Z,CELLS.COMP,CS , IDF.Q , IER)109 PRINT 690110 690 FORMAT(1X." CHI-SQUARED TEST ".1)111 PRINT 700,(CELLS(3),3=1,K4)112 700 ctIRMAT (1)WCOUNTS OF OBSERVATIONS IN CELLS",!, IX, 8(F10.5,2X

113 + ),/,32X ,7(F10.5,2X),//,)114 PRINT 701,(COMP(J)LJ-1,K4)115 701 FORMAT (1X,"COMPONLNTS OF CHI-SQUARED STATISTIC",/,1X,116 + 8(F10.5,2X),/,32X,7(F10.5,2X),//,)117 PRINT 702_,CS,Q.ADF118 702 FORMAT (1X,"CS= ,4X,F10.5,/,1X,"0=",5X,F10.5,/,1X."IDF=" ,

119 + 3X,I521X)120 C KOLMOGOROV SMIRNO" TEST121 CALL SORT(X)122 CALL NKS1(NORM,X,Z.PDIF,IER)123 PRINT 790124 790 FORMAT(//,1X," KOLMOGOROV-SMIRNOV TEST ",/)125 _ PRINT 800.(PDIF(J).3=1.6)126 800 FORMAT(1X,6(F10.5,2X,),/)127 C PRINTING THE MONTH PLOTTED128 PRINT 90,MON(K)129 90 FORMAT (10X.A30.////130 72 CONTINUE131 C PRINTING OF THE STATS OF THE RESIDUALS132 CALL SUB2(VAR2.07TE,MON,XMEAN.0 .1)133 V=V+1134 'F (V.E0.1) THEN135 . 1 60 3=1. N(I)136 DO 61 K=1,1137 • AFLOW(I,J.K)=LOG(AFLOW(I.J.K))138 61 CONTINUE139 60 CONTINUE140 VAR1='L0G TRANSFORMED ANNUAL FLOW'141 VAR2='RESIDUALS OF TRANSFORMEDFLOWS?142 GO TO .,143 ELSE144 END IF145 10 CONTINUE146 STOP147 END

1 SUBROUTINE NORM(X,P)2 COMMON/ONE/MEA.SD3 REAL MEA,SD4 T=(X-MEA)/SD5 P=.5*ERFC(-T*.7°71.068)6 RETURN7 END

4.3

SUBROUTINE SUB1(A,B,C,D,E,I1,F)REAL C(4,100,12)INTEGER I1,L,J,K,F(100)

4 CHARACTER*30 A5 CHARACTER*30 D(12)6 CHARACTER*7 E(100.4),B(4)7 PRINT 100.A,B(I1)B 100 FORMAT(50X,A30,/,50X," FOR " ,A8,//)9 PRINT 101,(D(( ),K=1,1)

10 101 FORMAT(//,2X," YEAR ",2X,1(A16,8X),/)11 DO 10 J=1,F(I1)12 PRINT 102. E(J.I1 ) ,(C(I1,J,K ),K=1.1 )13 102 FORMAT(1X,A7,2X,1(F15.5,9X),/)14 10 CONTINUE15 11 CONTINUE16 RETURN17 END

1 ' SUBROUTINE SUB2(AA,BB,CC,DD,EE,GG)

2 REAL DD(12),EE(3,12)

3 INTEGER GB

.I.,CHARACTER*7 BB(4)

6CHARACTER*30 CC(12)CHARACTER*30 AA

7 PRINT 100,AA,BB(GG)B 100 FORMAT(55X," BASIC STATS OF "./.50X,A30,/,50X. " FOR ",A7,1X,//,)

9 PRINT 301,(CC(K),K1,1)

10 301 FORMAT(BX.14A16,8X),/)

11 PRINT 302,(DD(K),K1.1 )•(EE(1,K),K=1,1 )

12 +,(EE(2,K),K=1,1 ),(EE(3,1(),K=1,1 )

13 302 FORMAT (1X," MEAN ",9X,1(F15.5,3X),//.1X,"SDEV",9X,1(F15.5,3X)

14 +,//,1X," SKEW ",9X,1(F15.5,3X),//,1X," CORR ",9X,1(F15.5,3X),/,)

15 RETURN16 END

197

123456

SUBROUTINE SUB3(A,B,C,D,E,DELX)REAL A(DELX,1)INTEGER DELXB=0.0C=0.0D=0.0

7 E=0.08 G=0.09 H=0.010 S=0.011 C COMPUTATION OF MEAN12 DO 10 I=1,DELX13 B=B+A(I,1)14 10 CONTINUE15 B=B/DELX16 C COMPUTATION OF STANDARD DEVIATION17 DO 15 I=1,DELX18 15=13+((A(I,1)-B)+*2)19 15 CONTINUE20 G=G/DELX21 C=SORT(G).....,

Cal. C COMPUTATION OF CORRELATION23 DO 20 I=1,DELX-1

24 H=H+(A(I,1)-B)*(A(I+1,1)-B)

25 20 CONTINUE26 D=H/((DELX-1)*(C**2))

27 C COMPUTATION OF THE SKEW COEFFICIENT28 DO 25 I=1,DELX

29 5=5+((A(I,I)-B)**3)

30 25 CONTINUE31 E=S/(DELX*(C**3))

32 RETURN33 END

IA SUBROUTINE SORT(XN0)

5 INTEGER XNO(100,1),TEMPINTEGER I.J,M .

4 DO 32 3=1,99

5 M=100-J

6 DO 22 I=1,M

7 IF (XNO(I, 1).LT.XNO(I+1, 1)) GO TO 22

B TEMP=XNO(I, 1)

9 XNO(I, 1)=XN0(I+1, 1)

10 XNO(I+1, 1)=TEMP

11 22 CONTINUE12 32 CONTINUE13 RETURN14 END

198

199

I SUBROUTINE DATA(MF,SF,AF,I1Z)

2 C THIS SUBROUTINE TRANSFORMS MONTHLY FL0 6)5 INTn SEASONAL

3 C AND ANNUAL FLOWS

4 C VARIABLE LIST

5 C MF(13K)=MEAN MONTHLY FLOWS FOR SITE Il,YEARJ MONTH K

6 C SF(I,J,K)=SEASONAL FLOW FOR SITE 11,YEAR J SEASON K

7 C AF(I,J)=ANNUAL FLOW FOR YEAR J8 C I1=SITE NUMBER

9 C DM(K)=NUMBER OF DAYS IN A MONTH K

10 C 2(11)=NUMBER OF YEARS FOR A SITE II

11 REAL DF(4,100,12),MF(4,100,12),SF(4,100,12),AF(4,100,12)

12 INTEGER DM(12),Z(4),I1

13 C INITIALIZATION

14 DO 1 3=1,2(I1)

15 AF(I1,J,1)=0.0

16 DO 2 K=1,12

17 SF(II,3,K)=0.0

18 2 CONTINUE

19 i CONTINUE

20C COMPUTATION OF ANNUAL FLOWS

21 DO 10 3=1,2(II)

22 AF(I1,J,1)=0.0

23 DO 11 K=1.12

24 AF(I1,3,1)=AF(11,3,1)+MF(11,J,K)

25 .11 CONTINUE

26 AF(I1,3,1)=(AF(I1,3,1))/12

27 10 CONTINUE _

28 C COMPUTATION OF THE SEASONAL.FLOWS

29 C HIGH FLOW SEASON(SEASONI=JUL,AUG,SEP,OCT,NOV)

30 DO 20 3=1,2(I1)

31 SF(11,J,1)=0.0

32 DO 21 K=3,7

33 SF(I1,J,1)=SF(II,J,1)+MF(I1,J,K)

34 21 CONTINUE

35 SF(II,3.1)=ISF(I1.3,11)/5

36 20 CONTINUE

37 C LOW FLOW SEASON(SEASON2=DEC,JAN,FEB,MAR,APR,MAY)

38 DO 30 3=1,Z(I1)

39 SF(I1,J,2)=0.0

40 DO 31 K=8,12

41 SF.(11,3,2)=SF(I1,3,2)+MF(I1,3,K)

42 31 CONTINUE

43 DO 32 K=1,2

44 SF(I1,3,2)=SF(I1,J,2)+MF(I1,3,K)

45 32 CONTINUE

46 SF(I1,J,2)=(SF(I1,J,2))/7

47 30 CONTINUE

48 RETURN

49 END

APPENDIX C

PROGRAMS FOR THE REGRESSION ANALYSIS

200

APPENDIX Cl:

Program Data 2

201

202

1 C THIS FROGRAM IS AT OF THE MUTIPLE REGRE5SION ANALYSIS

.: C +THIS wORK IS DO! .2E AETER-S DEGREE THESIS

3 C ALL MISSING VALUE= ARE REPLACED EY ZEROS

4 C LIST OF VARIAELFS

5C DFLOW(I,J,K)=F,AW DATA,MEAN DAILY FLOW FOR SITE 1,YEAR J,MONTH K

6 C MFLOW(I,J,K)=MEAN MONTHLY FLOWS OBTAINED FROM THE DAILY FLOWS

7 C RFLOW(I,J,K)=RESIDUALS

8 C SFLOW(1,J.K)=19FAN SEASONAL FLOW OF SITE I YEAR J SEASON K

9 C LFLOW(I,J,K)=LOG TRANSFORMED FLOWSIo C NDMi(1)=NUMEER OF DAYS I IN A GIVEN MONTH

11 C MUN(J)=SEASON J

12 C N(I)=NUMBER OF YEARS FOR SITE I

13 C YEAR(L,I)=YEAR L OF RECORD FOR SITE I.

14 C SITE(I)=NAME OF THE SITES

15 C XMEAN(M)=NEAN OF SEASON M .

16 C U(1,M)=STANDARD DEVIATION OF SEASON M

17 C U(2,M)=SKEWNESS OF SEASON Mle C U(3,M)=4UT000RRELATION COEFFICIENT OF SEASON M

19 C FOR THE OTHER VARIABLES SEE IMSL SUBROUTINES USED IN THIS PROGRAM

20 PROGRAM DATA2(1NPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT)

21 REAL DFLOW(4,100,12),MFLOW(4'100,12),RFLOW(4,100,12),LFLOW(4,100,

22 +12)•XMEAN(12),U(3,12) ,WV(200),COMP(2),CS,O,MEA,SD,

23 +PDIF(6),CELLS(2),TENP1(100,12),TEMP2(100,12),TEMP3(100,1),

24 +TEMF4(100,1),X(1(,0,1),X1(100,4 ),SFLOW(4,100,12),AFLOW(4,100,12),

25 +TFLOW(4,10(',12)

26 INTEGER N(4),NDm(12),m,z,ix,IER,INCri(eo),I,J,K,NI,L, •

27 +N2,IPIST,IOPT,IDF,K3(100),IND,FELX,V,K4

28 CHARACTER*7 SITE(4),YEAR(100,4)

29 CHARACTER*30 MON(12)

30 CHARACTER*70 VAR1,VAR2,VAR3,VAR4

31 C INPUT NAME OF SITES

32 DO 16 I=1,4

33 READ(5,1 ) SITE(I)

34 1 FORMAT (5X,A7)

35 16 CONTINUE •

36 C INPUT THE NUMBER OF YEARS FOR EACH SITE

37 READ(5,*) (N(1),I=1,4)

3U C INPUT THE FLOW DATA

39 DO 10 1=1,4

40 DO 11 L=1,N(I)

41 READ(53) YEAR(L,I)

42 3 FORMAT (5X,A7)

43 11 CONTINUE

44 DO 13 L=1,N(I)

45 READ (5,*) (MFLOW(I,L,K),K=1,12)

46 13 CONTINUE

47 C COMPUTATION OF 'SEASONAL AND ANNUAL FLOWS48 CALL DATA(MFLOW'SFLOW,AFLOW,I'N)

49 10 CONTINUE

50 C PRINTING THE MONTHLY FLOWS

51 DO 60 J=1,N(2)

52 DO 61 K=1,12

53 IF (3.LE.4e) THEN

54 TFLOW(1,J,K)=0.0

55 TFLOW(2,J,K)=MFLOW(2,J,K)

56575859606162636465666768697071727374

TFLOW(3,J, ()=NFLOW(3,J,K)TFLOW(4,J,K1=0.0ELSEJJ=J-48

TFLOW(1 , J , K)=MFLOW(1,3J,10TFLOW(2 , J , K)=MFLOW(2,J,K)TFLOW(3 , 3 , K)=MFLOW(3,3,1))TFLOW(4 , J , K)=MFLOW(4,JJ,K)

END IF61 CONTINUE60 CONTINUE

DO 70 J=1,N'2)•DO 71 K=1,12

WRITE(6,300) YEAR(3,2),3 ,K,(TFLOW(I,J,K).1=1,4.300 FORMAT (1X,A7,1X,12,1X,12,1X,4(F15.5,1X),/,)71 CONTINUE70 CONTINUE

STOPEND

203

1 SUBROUTINE DATA(MF,SF,AF,I1,Z)C THIS SUBROUTINE TRANSFORMS M0NTHI Y FLOWS INTO SEASONAL

5 C AND ANNUAL FLOWS4 C VARIABLE LIST5 C MFII,J,K)=MEAN MONTHLY FLOWS FOR SITE I1,YEARJ MONTH K

6 C SF(I,J,Y)=SEASONAL FLOW FOR SITE 11,YEAR J SEASON K

7 C AF(I,J)=ANNUAL FLOW FOR YEAR J

8 C I1=SITE NUMBER9 C DM(K)=NUMBER OF DAYS IN A MONTH K

10 C Z(11)=NUMBER OF YEARS FOR A SITE II.11 REAL DF(43100,12),MF(4,100,12),SF(4,100,12),AF(4 , 100,12)

12 INTEGER DM(12),Z(4),I1

13 C INITIALIZATION14 DO 1 J=1,1(11)

15 AF(I1,J,1)=0.0

16 DO 2 K=1,12

17 SF(11,3,K)=0.0

18 2 CONTINUE19 1 CONTINUE2 0 C COMPUTATION OF ANNUAL FLOWS2 1 DO 10 3=1,Z(I1)

22 AF(I1,3,1)=0.0

2 13DO 11 K=1.12

24 AF(I1,3.1)=AF(I1,3,1)+MF(I1,3,K)

25 11 CONTINUE26 AF(I1,3.1)=(AF(I1,3,1))/12

2710 CONTINUE28 C COMPUTATION OF THE SEASONAL FLOWS29 C HIGH FLOW SEASON(SEASON1=JUL,AUG,SEP , OCT,NOV)

30 DO 20 3=1.Z(I1)

31 SF(I1.3,1)=0.0

32 DO 21 K=3.7

33 SF(II,J,1)=SF(I1,3,1)+MF(II,J,K)

34 21 CONTINUE35 SF(11.3.1)=(SF(I1..J,1))/5

36 20 CONTINUE37 C LOW FLOW SEASON(SEASON2=DEC.JAN,FEB.MAR,APR,MAY)

38 DO 30 3=1,Z(11)

39 SF(I1,3,2)=0.0

40 DO 31 14=8.12

41 SF(11,3,2)=SF(II,J.2)+MF(I1.J,K)42 71 CONTINUE43 DO 32 K=1,2

44 SF(I1-1,2)=SF(I1,3,2)+MF(II,3,K)

45 32 CONTINUE46 SF(I1,3,2)=(SF(I1,3,2))/7

47 30 CONTINUE46 RETURN- END

APPENDIX C2:

Example Set-up of Regression

204

nmis

Is= sss

=

'T

warX'".

a .WI,ut .1...

...1

Z2

an.02mca

ret sz0 I

0sasos

a z on ua < AIn pi. ON. Y m0 . -V 0 Y W V0 MZ .m Y N.414

SIC..Z

02M>

0 . vo00

IX .. -4 Z cm

Ism s..Z Z-40 ,MI .

...WOT >

M Mmmas mor s..rn .Z .00 0Z. .. m. X 0 WM y n

or n i. Os -0,44.0 It 0 404. . • 2M MM 2 n •o= taxanz« < a. z o i

s z * <<zs< nmc ssmmonm s m ....,a, sno. 0 M 4.1 O nmmia. 0MY TTMMZaso a M ZioX / m m

WT T IC FT MMNYV r- DO .m00 m . 0 vmz . Z .NM . CN0. 244v 00CC . M C. X 0 miv 0 0 . ...241., 20N CC m 0 .1.2 C N .ta 0 rn n w Z m ZN . . am. 2 .4. .

c4 0 • .. am 0m. 4,4 . . Z vmZ al cmV p a 0 MW aT 14440 0 0.. 4% rn -Ia %A , . , MCCM 0.0 . a M n m

0 N N 0 T NTWOX 0.0% T o4 ZM Re M

IL 0 N MMeln af MyYm Me 0 ssr e so s es secweve ma tsauwva la s's 0 I r•

•A 0 rn rn 0 • • . • • • I. 2t K - •c n z re veas 0s.4 s 0 •-• z aM 2 0 Im Al W44444004 . 2 IT n n

T OMMAOMN<OZ g sage ..... ms< a m.cz z a nm MWTYTM1WTm 0 n YMC 0 M xM a wiANTam n CAN IA O VW a m W n m Y X n

.1' m -.4f.-. nA =M000mMy 0M1 m lc MMY X 00 MC7CMTMn VY 0C.YMMX ZM n n a ca..., o m

Z = ms=s0ave2sz ma n -csra 2 0 C T-CO a p0 m aN n MT .... M 0.2mn am0 mrsm m . .mm . . .

Z M. v.m.Zr.... Mmw•Zola ioT •••nnn•n••n•• Ca *MT 0 0

1 TN MONm1MMAy7 MDZMCM M MAA T To m m M

Z 0 0we •=NaCv nZ.,m. w mm• O >ZM TI 0Al TA.7 MT CDTT YX7a0 M flo a M0T N a 1

T MT mT n TN n XM0o.n VA Ml Z • z71 -4 A z

0 om./1 TnMM.= r.-CZNi a xt 7./ n4 mmn M / mTa ro

a !7 n .2m3-r- K .. 0 1P2N0.6.,PV 0 . 440.• .2 Omv4CmK .2* T • •.n -IIIIIII r- • . MA A

, awn- CY •AA Tv-DX Z D 11,TA .7

= TTT ...tee XX, r.0 Z 0 1. sr Z C Y0. n

.. 0ZA m sae* 0C1Z Z . yPww.. 2 > .4cO .8N up. Ton C T.-. I.- MN/70/mM N a 0•Z n

!TN My Woow M M y vat r

Z MZ2 OD a-4n* 4. W m•TM MY. z 0 ..otnin. • Va A MT7 tol

C vr• r- mg 1.1- n m r .. 0sa M.Yam C XanT M TC m

L 010, so YMCM a in na., M

in mT C OM•0 T ear Z0 SP W 4-4, 34m > M Ow. nA ... • a 71474p. 0 m •• .1,4 .a mM . W>Z> . ... N

.4 v. M s., T •• KM,.01A to,, Z T,-Z. n1 .1Z. m-1 n worm o sn0 op m ac.cm saZ . s • • • m 00 Tm

a M CZTM in •-• .1V , 10 T7mC NmC) M WZZ-. m • 0

Z M 00 ao4C7 M VIC• 0 O X n n ••

SC im AMP... 7 Mn y AP1TA /

• c% Z /. J. ...... . aTOT N ••a Z MM nM n

T S. 0,-.0 TI ••

M COE 0 m0 -. 2m M

C >00. .• .0. 0X •• n• C. X X N4r, n• .M K v0 M MIC N C. -4.m .

205

APPENDIX D

PROGRAM DATA 3

206

207

1 O THIS PROGRAM FILLS IN THE MISSING VALUES OF THE MONTHLY FLOWS2 C FOR ( AYES AND VIDIPA USING THE DATA STORED IN A FILE BY3 C PROGRAM 1ATA2(USED TO INPUT DATA FOR THE REGRESSION WITH 5P55)4 PROGRAM DATA3(INPUT,OUTPUT,TAPE5=INPUT,TAPE8,TAPE9)5 C LIST OF VARIABLES6 C MFLOW=MONTHLY FLOWS7 C AFLOW=ANNUAL FLOWS8 C A=INTEPCEPTS A9 C P=REGRESSION COEFFICIENTS

10 C UBA=UPPER BOUND OF A11 REAL MFLOW(3,10n,12),AFLOW(3,100),A43,12),B(3,12),UBA(3 , 12)12 CHARACTER*7 SITE(3),YEAR(100)13 C INPUT NAME OF SITES14 READ(5,2) (SITE(I),I=1,3)15 2 FORMAT(5X,3(A7,1X))16 C INPUT THE INTERCEPTS AND THE REGRESSION COEFFICIENTS17 DO 13 1=1,318 DO 14 3=1,1219 A(I,3)=0.020 B(I,3)=0.021 UBA(I,J)=0.022 14 CONTINUE23 13 CONTINUE24 DO 15 1=1,22- 11=1412g DO 16 3=1,1227 READ(5,*) B(II,J),A(II,J)20 16 CONTINUE29 15 CONTINUE30 C PRINTING THE INTECEPTS AND THE REGRRESSION COEFFICIENTS31 DO 50 1=1,232 II=I+133 WRITE(8,102) SITE(II),(A(II,3),3=1,12)34 102 FORMAT(1X,"INTERCEPTS FOR",1X,A7,/, 6(F15.5,1X),/,1X,6(F15.5,1X)35 +,//,)36 WR1TE(8,98) SITE(II),(B(II.3),3=1,12)37 98 FORMAT(1X,"REGRESSION COEFFICIENTS FOR,1X,A7./,6(F10.5,1X),/,38 +6(F10.5,1X),///,)39 50 CONTINUE40 -C INPUT THE UPPER BOUND OF A41 DO 51 1=1,242 II=I+143 DO 52 3=1,1244 READ(5,*) UBA(II,J)45 52 CONTINUE46 51 CONTINUE47 C PRINTING THE UPPER BOUND OF A48 DO 53 1=1,249 II=I+150 WRITE(8,202) SITE(II),(UBA(II,3),3=1,12)51 202 FORMAT(1X,"INTERBOUND FOR".1X,A7,/, 6(F15.5,1X),/.1X,6(F15.5,1X)52 +,//,)53 53 CONTINUE54 -C INPUT THE MONTHLY FLOWS FOR 1903-04 TO81-8255 DO 10 1=1.79

56 DO 11 K=1,1257 READ(9,1) YEAR(3),M,N,MFLOW(1,3,K),MFLOW(2,3,K),MFLOW(3,3,K)58 1 FORMAT(1X,A7, 1X, I2, 1X, I2, 17X,3(F15....,, 1)() , /, )59 11 CONTINUE60 i0 CONTINUE61 PRINTING THE MONTHLY FLOWS USED TO FILL IN MISSING VALUES62 WRITE(8,100) (SITE(I),I=1,3)63 100 FORMAT(20X,"MONTHLY FLOWS",/,20X,"NOT FILLED-IN",//,23X,3(A7,9X))64 DO 40 3=1,7965 DO 41 K=1,1266 WRITE(8,101) YEAR(J),j ,K,(MFLOW(I.3,K),I=1,3)67 101 FORMAT(1X,A7,1X,I2,1X,I2,1X,3(F15.5,1X))68 41 CONTINUE69 i0 CONTINUE70 COMPUTATION OF THE MISSING VALUES FOR 1903-04 TO 81-8271 DO 20 1..1,272 II=I+173 DO 21 3=1,7974 DO 22 K=1,1275 IF (MFLOW(II,3,K).E0.0) THEN76 MFLOW(II,3,K)=A(II.K)+MFLOW(1 ,3,K)4.8(11,1077 ELSE78 END IF79 C ADJUSTEMENT OF THE NEGATIVE FLOWS80 IF(MFLOW(II,3,K).LT.0) THEN81 MFLOW(II,J,K)=UBA(II,K)+MFLOW(1,J,K).1.8(II,K)82 ELSE .83 END IF84. 22 CONTINUE

21 CONTINUE86 20 CONTINUE87 C PRINTING THE MONTHLY FLOWS RECONSTITUTED88 WRITE(8,103) (SITE(I),I=1,3)89 103 FORMAT(30X,"MONTHLY FLOWS",/,30K," FILLED-IN",//,23X.3(A7.9X))90 DO 60 3=49.77

208

91 33=3-4892 DO 61 K-1,1293 WRITE(8,104) YEAR(J),..73,K,(MFLOW(I,3,K),1 n1,3)94 104 FORMAT(IX,A7,1X,I2,1X,I2,1X.3(F15.5,1X))95 61 CONTINUE96 60 CONTINUE97 C COMPUTATION OF THE ANNUAL FLOWS FOR 1951-52 TO 79-9098 DO 30 161,399 DO 31 3..49,77100 AFLOW(I,3)=0.0101 DO 32 K=1,12102 AFLOW(I,J)=AFLOW(I,3)+MFLOW(I,3,10103 32 CONTINUE104 AFLOW(I.3)=(AFLOW(I,J))/12105 31 CONTINUE .106 30 CONTINUE107 C PRINTING THE ANNUAL FLOWS (COMPUTED FROM THE RECONSTITUTED DATA)108 WRITE(8,105) (SITE(I),I=1,3)109 105 FORMAT(//////,30X,"ANNUAL FLOWS",//,15X,3(A7,9X))110 DO 70 3=49,77111 WRITE (8,106) YEAR(J)1(AFLOW(I,J),I=1,3)112 106 FORMAT (1X,A7,1X,3(F15.5,1X))

113 7) CONTINUE114 STOP .115 END.

APPENDIX E

HISTORIC DATA(MONTHLY AND ANNUAL FLOWS)

209

year monthBAKU. KAYES KIDIRA

1951-521951-521951-521951-52

1234

4.0000057.00000

387.000001418.00000

6.5000069.40000

356.000001368.00000

.300005.80000

64.00000327.00000

1951-521951-52

56

2331.000003581.00000

1999.000002625.00000

693.000001341.00000

1951-52 7 1455.00000 1147.00000 301.000001951-52 e 423.00000 346.00000 76.000001951-52 9 214.00000 190.00000 . 33.900001951-152 10 123.00000 119.00000 14.600001951-521951-52

1112

64.0000027.00000

55.3000023.10000

6.900002.20000

1952-53 1 5.00000 7.70000 .700001952-53 2 22.00000 32.20000 .200001952-53 3 524.00000 473.00000 133.000001952-53 4 2395.00000 1259.00000 401.000001952-53 5 2421.00000 2180.00000 792.000001952-53 6 3126.00000 2006.00000 1096.000001952-53 7 597.00000 431.00000 132.000002952-531952-53

e9

246.00000134.00000

199.00000108.00000

48.9000023.10000

1952-53 10 71.00000 55.20000 9.000001952-53 11 37.00000 27.30000 4.600001952-153 12 17.00000 11.80000 1.900001953-54 1 3.00000 3.90000 .400001953-541953-541953-54

234

101.00000788.000001547.00000

149.00000831.000001432.00000

9.70000144.00000357.00000

1953-541953-541953-54

567

2926.000001236.00000464.00000

2409.000001023.00000389.00000

725.00000214.0000070.00000

1953-34 e 219.00000 201.00000 28.000001953-54 9 140.00000 133.00000 13.500001953-54 10 81.00000 72.30000 8.300001953-54 11 41.00000 32.20000 3.600001953-54 12 13.00000 9.50000 1.400001954-551954-55 1 12.00000

253.0000018.00000

224.00000.30000

60.000001954-35 § 963.00000 949.00000 253.000001954-35 4 3987.00000 3610.00000 1123.000001954-55 5 4419.00000 3214.00000 1189.0000019154-55/954-55

67

1655.00000681.00000

1343.00000554.00000

289.00000126.00000

1954-55 8 396.00000 330.00000 60.000001954-55 9 197.00000 171.00000 29.1000019154-35 10 116.00000 95.50000 13.600001954-55 11 68.00000 52.10000 7.200001954-55 12 42.00000 32.50000 3.000001955-56 1 32.00000 38.60000 1.800001955-56 2 207.00000 194.00000 43.400001955-56 3 612.00000 606.00000 180.000001955-56 4 3563.00000 2931.00000 1222.000001955-56 5 4004.00000 3232.00000 1032.000001955-56 6 2615.00000 1909.00000 572.000002955-56 7 770.00000 631.00000 126.0000019=5-56 8 347.00000 298.00000 55.000001955-56 9 203.00000 176.00000 27.900001955-56 10 119.00000 105.00000 13.800001955-56 11 69.00000 54.30000 7.400001955-56 12 34.00000 17.20000 3.500001956-571956-57 4 13.00000

40.000007.5000048.00000

1.3000010.60000

1956-57 5 495.00000 436.00000 137.000001956-571956-57

45

2210.000005237.00000

2191.000003488.000w

601.000001780.00000

1956-57 6 2159.00000 1750.00000 368.000001956-57 7 634.00000 503.00000 97.000001956-57 e 285.00000 234.00000 42.600001956-57 9 163.00000 136.00000 20.000001956-57 10 99.00000 76.70000 9.000001956-57 11 60.00000 39.40000 4.6000019156-57 12 24.00000 11.90000 2.000001957-581957-58 1 8.00000

215.000000.40000

199.00000.90000

48.600001957-58 S 608.00000 525.00000 122.000001957-58 4 2668.00000 2562.00000 7315.000001957-58 5 4227.00000 3295.00000 1141.000001957-58 6 2904.00000 2451.00000 500.0000019117-58 7 935.00000 752.00000 129.000001957-58 e 351.00000 295.00000 52.000001957-58 9 197.00000 168.00000 23.800001957-58 10 118.00000 98.00000 10.700001957-58 11 67.00000 48.40000 4.900001957-58 12 32.00000 18.70000 2.200001958-391958-59 1 18.00000

175.0000012.00000

162.000001.20000

32.6000019158-59 S 568.00000 479.00000 131.0000019158-59 4 3985.00000 3625.00000 990.0000019158-59 5 4028.00000 3025.00000 795.000001958-59 6 1916.00000 1563.00000 370.000001938-59 7 785.00000 643.00000 143.00000

210

1958-59 8 444.00000 350.00000 80.000001958-59 q 237.00000 191.00000 28.600001958-59 10 139.00000 110.00000 12.900001958-59 11 84.00000 61.00000 7.800001958-59 12 40.00000 23.00000 3.000001959-60 1 19.00000 17.30000 1.200001959-60 2 164.00000 161.00000 13.800001959-60 3 583.00000 435.00000 73.000001959-60 4 2434.00000 2159.00000 855.000001959-60 5 4047.00000 2987.00000 1118.000001959-60 §, 1242.00000 928.00000 242.000001959-60 489.00000 377.00000 71.000001959-60 6 223.00000 181.00000 33.800001959-60 9 126.00000 106.00000 16.200001959-60 10 76.00000 55.00000 8.100001959-60 1) 42.00000 24.30000 4.100001959-60 12 17.00000 9.00000 1.900001960-61 1 5.00000 3.10000 1.100001960-61 2 82.00000 75.00000 7.100001960-61 3 789.00000 726.00000 191.000001960-61 4 1790.00000 1446.00000 551.000001960-61 5 250E1.00000 2133.00000 625.000001960-61 6 1301.00000 1045.00000 250.000001960-61 7 504.00000 402.00000 70.000001960-61 8 213.00000 177.00000 30.200001960-61 9 120.00000 98.80000 14.800001960-61 10 75.00000 54.50000 8.100001960-61 11 41.00000 25.20000 3.800001960-61 12 16.00000 8.00000 1.800001961-62 1 3.50000 2.90000 .500001961-62 2 102.00000 77.50000 34.700001961-62 3 781.00000 713.00000 188.000001961-62

! 2956.00000 2768.00000 706.000001961-62 5201.00000 3723.00000 1709.000001961-62 a: 1360.00000 1051.00000 209.000001961-62 7 458.00000 373.00000 61.000001961-62 e 207.00000 174.00000 25.000001961-62 5 121.00000 97.00000 11.200001961-62 10 74.00000 51.30000 6.100001961-62 11 40.00000 21.10000 2.600001961-62 12 12.00000 5.70000 .900001962-63 1 2.70000 2.70000 .200001962-63 2 85.00000 80.50000 22.400001962-63 3 511.00000 456.00000 122.000001962-63 4 2220.00000 1927.00000 746.000001962-63 3632.00000 2609.00000 1245.000001962-63 a' 1620.00000 1313.00000 324.000001962-63 7 594.00000 478.00000 110.000001962-63 8 262.00000 218.00000 35.400001962-63 9 138.00000 117.00000 16.900001962-63 10 86.00000 64.00000 7.900001962-63 11 43.00000 27.30000 3.900001962-63 12 18.00000 9.20000 1.700001963-64 1 8.00000 5.70000 .400001963-64 2 7.00000 10.10000 .900001963-64 3 473.00000 370.00000 170.000001963-64 4 1620.00000 1279.00000 524.000001963-64 5 2772.00000 2306.00000 746.000001963-64 6 1988.00000 1792.00000 395.000001963-64 7 636.00000 516.00000 83.00000

1963-64 8 230.00000 197.00000 30.500001963-64 9 129.00000 127.00000 13.800001963-64 10 72.00000 52.00000 6.300001963-64 11 36.00000 22.30000 2.800001963-64 12 13.80000 6.30000 .800001964-65 I 3.20000 2.90000 .100001964-65 2 171.00000 28.00000 40.900001964-65 3 602.00000 519.00000 1130.000001964-65 4 1973.00000 2100.00000 714.000001964-65 5 5680.00000 4135.00000 1805.000001964-65 6 1989.00000 1462.00000 329.000001964-65 7 580.00000 453.00000 93.000001964-65 8 285.00000 227.00000 44.200001964-65 9 166.00000 136.00000 19.600001964-65 10 105.00000 78.00000 9.900001964-65 II 58.00000 32.00000 5.000001964-65 12 26.00000 11.50000 2.100001965-66 1 9.18000 8.74303 .601001965-66 2 84.10000 98.10866 18.900001965-66 3 513.00000 459.44818 94.400001965-66 4 3270.00000 2754.75640 1120.000001965-661965-66

56

5340.000002050.00000

3828.508601572.52760

1310.00000439.000001965-66 7 649.00000 517.62088 135.00000

1965-66 8 290.00000 238.89160 63.800001965-66 9 171.00000 140.68441 25.000001965-66 10 104.00000 83.17238 10.000001965-66 11 57.50000 39.03843 5.000001965-66 12 28.20000 18.14473 2.00000

211

1966-67 1 10.60000 10.07547 1.000001966-67 2 76.00000 92.14260 26.000001966-67 3 367.00000 315.67322 74.900001966-67 4 1380.00000 1329.77200 496.000001966-67 5 2830.00000 2260.56180 898.000001966-67 6 3900.00000 2943.21110 1620.000001966-67 7 853.00000 660.05776 231.000001966-67 9 321.00000 255.77978 95.100001966-67 9 174.00000 142.44454 14.700001966-67 10 105.00000 83.78865 5.270001966-67 11 61.60000 41.37653 2.400001966-67 12 27.50000 17.78615 1.740001967-68 1 11.00000 10.45081 1.230001967-68 2 89.40000 102.01237 20.300001967-68 3 560.00000 505.73190 123.000001967-68 4 2410.00000 2106.35080 513.000001967-68 3 5830.00000 4134.60180 1550.000001967-68 6 2800.00000 2128.21010 774.000001967-68 7 764.00000 597.91618 147.000001967-68 8 346.00000 269.39928 35.100001967-68 9 211.00000 164.15281 27.300001967-68 10 134.00000 101.66048 13.600001967-68 11 77.70000 50.55788 5.650001967-68 12 36.30000 22.29404 4.200001968-69 1 16.60000 15.70551 3.330001968-69 2 76.00000 92.14260 4.730001968-69 3 421.00000 368.85026 73.900001968-69 4 1010.00000 1050.80680 144.000001968-69 5 1800.00000 1617.14140 379.000001968-69 6 853.00000 685.65833 166.000001968-69 7 301.00000 274.64032 42.07547

1968-69 el 169.00000 172.97322 23.128831968-69 9 93.30000 95.09704 7.480001968-69 10 54.80000 52.85190 4.650001968-69 11 27.10000 21.70222 1.960001968-69 12 8.01000 7.80220 .285001969-70 1 4.60000 2.56875 .332691969-70 2 71.40000 88.75447 13.700001969-70 3 683.00000 626.85738 148.000001969-70 4 1650.00000 1533.34120 491.000001969-70 5 3150.00000 2460.45940 664.000001969-70 6 2040.00000 1565.11850 483.000001969-70 7 947.00000 725.69044 164.000001969-70 8 308.00000 248.69764 43.200001969-70 9 157.00000 132.47047 19.256441969-70 10 92.90000 76.33178 9.850001969-70 11 50.50000 35.04654 5.400001969-70 12 24.50000 16.24937 1.630001970-711970-71 4 5.47000

29.600005.26179

57.96668.45000.21800

1970-71 5 29.70000 50.05737 6.065651970-71 4 2250.00000 1985.71720 742.000001970-71 5 2500.00000 2054.41740 568.000001970-71 6 791.00000 639.72191 106.000001970-71 7 284.00000 262.77058, 40.800001970-71 8 - 144.00000 159.35372 14.200001970-71 9 85.00000 90.22735 9.860001970-71 10 52.60000 51.49610 4.730001970-71 11 27.70000 22.04438 1.640001970-71 12 10.20000 8.92405 1.380001971-721971-72 4 3.90000

4.900003.78860

38.30080.37673

6.750001971-72 5 481.00000 427.93586 81.100001971-72 4 2530.00000 2196.82600 833.000001971-72 5 2740.00000 2204.34060 682.000001971-72 6 810.00000 653.79920 142.000001971-72 7 261.00000 246.71152 39.100001971-72 8 131.00000 152.27158 14.900001971-72 9 75.60000 84.71228 7.640001971-72 10 46.40000 47.67523 3.800001971-72 11 20.80000 18.10952 1.010001971-72 12 3.18000 5.32799 .015001972-731972-73 .1, .90000

43.00000.97358

67.83645.27509

3.760001972-73 S 291.00000 240.83146 40.300001972-73 4 795.00000 888.70540 193.000001972-73 5 1060.00000 1154.87820 169.000001972-73 6 499.00000 423.37619 93.576561972-73 7 218.00000 216.68806 24.274461972-73 8 106.00000 138.65208 10.965421972-73 9 54.70000 72.45004 5.249521972-73 10 27.00000 35.71959 2.366391972-73 11 9.20000 11.49438 .952191972-73 12 18.30000 13.07336 1.739481973-74 1 .33900 .44717 .256091973-74 2 126.00000 128.97010 8.560001973-74 3 327.00000 276.28282 4.020001973-74 4 1670.00000 1548.42040 458.000001973-74 5 1360.00000 1342.28220 314.000001973-74 6 497.00000 421.89437 50.000001973-74 7 180.00000 190.15570 13.40000

212

1973-74 e 72.60000 120.45643 7.370001973-74 9 39.60000 63.59072 3.000001973-74 10 18.60000 30.54292 1.000001973-74 11 7.04000 10.26260 .100001973-74 12 1.18000 4.30347 .977991974-75 1 .60000 .69207 .264931974-75 2 3.00000 38.37445 3.343221974-75 3 739.00000 682.00394 222.000001974-75 4 3236.00000 2729.12176 899.000001974-75 5 3138.00000 2452.96324 589.000001974-75 6 1321.00000 1032.40421 226.000001974-75 7 371.00000 323.51572 43.400001974-75 e 143.00000 158.80894 13.900001974-75 9 59.70000 75.38359 7.100001974-75 10 34.10000 40.09511 3.560001974-75 11 14.40000 14.45979 1.390001974-75 12 4.20000 5.85049 .230001975-76 1 1.23500 1.28792 .286441975-76 2 .31700 36.39829 2.759021975-76 3 552.00000 497.85382 109.000001975-76 4 1586.00000 1485.08776 318.000001975-76 5 3281.00000 2542.29248 927.000001975-76 6 1158.00000 911.63588 194.000001975-76 7 382.00000 331.19614 47.000001975-76 e 149.00000 162.07762 17.000001975-76 9 60.70000 75.97030 7.310001975-76 10 34.40000 40.27999 3.340001975-76 11 14.09000 14.28300 .546001975-76 12 2.53000 4.99502 .006001976-77 1 40.00000 37.66267 1.599801976-77 2 57.00000 78.14815 2.640001976-77 3 302.00000 251.66382 109.500001976-77 4 547.00000 701.72332 258.000001976-77 5 485.00000 795.68720 173.800001976-77 6 487.00000 414.48527 188.800001976-77 7 420.00000 357.72850 120.000001976-77 e 225.00000 203.48090 23.600001976-77 9 165.00000 137.16415 9.130001976-77 10 124.00000 95.49778 4.350001976-77 11 93.00000 59.2E1301 .983001976-77 12 69.00000 39.04494 .007001977-78 1 1.38000 1.42398 .291351977-78 2 1.68000 37.40220 .384001977-78 3 230.00000 180.76110 23.600001977-78 4 841.00000 923.38756 125.000001977-78 5 1728.00000 1572.16444 410.000001977-78 6 752.00000 610.82642 184.000001977-78 7 211.00000 211:80052 31.000001977-78 e 61.00000 114.13698 9.170001977-78 9 32.20000 59.24906 3.760001977-78 10 12.80000 26.96E156 1.040001977-78 11 3.24000 8.09557 .006001977-78 12 .90000 4.16003 .965531978-79 1 .70600 .79154 .268521978-79 2 7.79000 41.90252 4.386191978-79 3 359.00000 307.79514 36.600001978-79 4 1764.00000 1619.29264 590.000001978-79 5 1892.00000 1674.61196 431.000001978-79 6 1314.00000 1027.21784 292.000001978-79 7 462.00000 387.05374 95.60000

1978-79 8 153.00000 164.25674 20.700001978-79 9 67.90000 80.19461 9.750001978-79 10 31.50000 38.49281 3.930001978-79 11 9.65000 11.75101 .502001978-79 12 2.85000 5.15894 1.052271979-80 1 1.68000 1.70548 .301521979-80 2 42.30000 67.32087 1.430001979-80 3 308.00000 257.57236 50.000001979-80 4 991.00000 1036.48156 187.000001979-80 5 1263.00000 1281.68824 312.000001979-80 6 573.00000 - 478.20353 121.000001979-80 7 293.00000 269.05456 34.600001979-80 8 98.00000 134.29384 10.600001979-80 9 43.20000 65.70287 3.660001979-80 10 17.30000 29.74177 .555001979-80 11 4.20000 8.64303 .003001979-80 12 1.43000 4.43153 .98911

213

BAKEL

ANNUAL FLOWS

KAYES KIDIRA1951-52 840.50000 692.02500 238.808331952-53 716.25000 565.85000 220.200001953-54 629.91667 557.07500 131.241671954-55 1065.75000 882.75833 262.766671955-56 1047.91667 849.34167 273.733331956-57 951.58333 743.62500 256.091671957-58 1027.50000 868.12500 230.841671958-59 1034.91667 853.66667 216.258331959-60 788.50000 619.96667 203.175001960-61 620.33333 516.13333 146.158331961-62 942.95833 754.79167 246.166671962-63 767.64167 608.47500 219.616671963-64 665.40000 555.28333 164.458331964-65 969035000 765.36667 270.233331965-66 1047.16500 813.30374 268.641751966-67 842.14167 679.38913 288.842501967-68 1105.7E1333 849.44487 267.865001968-69 402.48417 371.28098 70.878281969-70 764.74167 625.96550 170.280761970-71 517.43917 448.99654 124.611971971-72 592.14833 506.64993 150.974311972-73 260.17500 272.05657 45.454931973-74 358.27992 344.80074 71.723671974-75 755.33333 629.47278 167.432351975-76 601.77267 508.61318 135.520621976-77 251.16667 264.29748 74.367481977-78 322.93333 312.53137 65.768071978-79 505.36633 446.54329 123.815751979-80 303.00917 302.90331 60.17822

214

APPENDIX F

PROGRAM MMLO FOR THEGENERATION OF STREAMFLOWS

215

216

1 C THIS PROGRAM IS AN APPLICATION OF MULTIVARIAT E GENERATING PROCESS.? C SYNTHETIC STREAM FLOW ARE GENERATED FOR A MULTISITE SYSTEM OF 3

.; C DIFFERENT GAGING STATION Xl, X2, X3, USING THE FIRST ORDER AUTO4 C REGRESSIVE MODEL WITH LOGNORMALLY RANDOM NUMBERS. FROM 29 YEARS

g C OF RECORD, 290 YEARS ARE GENERATED FOR EACH STATION.C THIS WORK IS DONE FOR MY THESIS (TESTING OF MODEL!)

7 PROGRAM MMLO(INPUT,OUTPUT,TAPEB,TAPE9)

13 REAL DEMP1(30,2),DEMP2(30,2),DEMP3(30,2),D1(3),D2(3),P1(3,3),

9 +m0(3,3),M1(3,3).BBT(3,3),M1T(3,3),P(3,3),PM1(3,3),P2(3,3),

10 +B(3,3),DHALF(3,3),PDHALF(3,3),Y(300,3),Y1( 3,1)0(2(3,1),

11 +M0MI(3.3),WKAREA(30),MA(3,3),MB(3,3),WK(24),AAA(3),D(3),

12 +X(300,3),B1(3,1),R1(3,1),MEANY(3),S1(300,2),S2(300,2),S3(300,2),

13 +u(300,3),U1,112.U3,SMEAN(3),RK(3),V(300.3),DFLOW(30,3),R(400)

14 INTEGER FELX,NR,K0(104,IA,IDGT,IER,L,M,IB,IC,IJOB,12,30BN,SELX+.N1,N215

16 CHARACTER+7 YEAR(30)

17 DOUBLE PRECISION DSEED

18 DATA DEMP1/604.0/,DEMP2/60=0/,DEMP3/60=0/,D1/3=0/,02/3.0/,

19 +M0/9=0/,M1/9=0/,BBT/9.00/,M1T/94.0/,P/9.60/,PM1/9=0/,

20 +B/9=0/,DHALF/9=0/,PDHALF/9=0/,Y/900*0/,Y1/34,0/..Y213=0/,

21 +MOM1/9,00/,MA/9=0/,MB/9=0/,AAA/3=0/.D/3.0/,

22 +X/900=0/,81/34.0/,R1/3=0/,MEANY/3.00/,S1/600.1,0/,S2/600*0/,

23 +53/600=0/0.1/900=0/,U1/1=0/,U2/14,0/,U3/1=0/.SMEAN/3=0/,RK/3410/.

24 +v/900+0/,DFLOW/90 *0/1R/400+0/,P1/94.0/,P2/9=0/

i:C RANDOM NUMBER GENERATION

NR=400

27 DSEED=123457.D0

28 CALL GGNML (DSEED,NR,R)

29 DO 4 1=2,291

30 U(I21)=R(1+100)

31 4 CONTINUE

32 CALL GGNML(DSEED,NR,R)

33 DO 7 1=2,291

34 U(I,2)=R(I+100) .

35 7 CONTINUE

36 CALL GGNML(DSEED,NR,R)

37 DO 8 1=2,291

38 U(I,3)=R(I+100)

39 8 CONTINUE

40 PRINT 98

41 98 FORMAT(//,1X,"GENERATED RANDOM NUMBERS ",//,16X,"STATION1",9X,

42 +.STATION2"113X,"STATION3", //,)

43 DO 2 1=2,291

44 PRINT 99,I,(U(1,3),3=1,3)

45 99 FORMAT (1X,13,4X.3(F15.5,4X),/,)

46 2 CONTINUE

47 C INPUT HISTORIC ANNUAL FLOWS48 PRINT 100

49 100 FORMAT (1X," ANNUAL FLOW",//,1X,"YEAR".10X,"STATION1",6X,"

50 +STATION2",6X,"STATION3",//,)

51 READ(B,103)w,

..J. 103 FORMAT(360(/))

53 DO 12 1=1,29

54 READ(8,102) YEAR(I),(DFLOW(I,3) , 3=1,3)...

..i -.1 102 FORMAT( 1X,A7.1X.3(F15.5,1)())

217

56 PRINT 101.YEAR(I).(DFLOW(I,J),3=1.3)57 101 FORMAT(1X,A7,1X.3(F15.5,1X))58 12 CONTINUE59 C COMPUTATION OF THE MEAN. STANDARD DEVIATION. AUTOCORRELATION AND60 C SKEWNESS FOR THE 29 YEARS RECORDS 6OR 51CH STAT9ON W9 8 X(3)61 FELX=2962 DO 20 J=1.2963 DEMP1(3,2)=DFLOW(3,1)64 DEMP2(.3,2)=DFLOW(3.2)65 DEMP3(J,2)=DFLOW(J.3)66 20 CONTINUE67 CALL SUB1(DEMP1,DMEA1,DSDI,DC01,DSKE1,FELX)68 CALL SURICDEMP2,DMEA2.DSD2.0CO2.DSKE2.FELX)69 CALL SUB1(DEMP3,DMEA3.DSD3,DCO3,DS(E3,FELX)70 'PRINT 20071 200 FORMAT(///,1X,"ESTIMATED STATS FROM DATA",//,13X,"MEAN",10X, "STD-72 +DEVIA",2X,"AUTOCOR",4X,"51(EW",//)73 PRINT 201,DMEAI.DSD1.DC01,DSKE174 201 FORMAT (1X,"STATION1",4X,2(F9.1,2X),2(F9.5,2X),//)75 PRINT 202,0MEA2,DSD2,DCO2,DSKE276 202 FORMAT (1X,"STATION2",4X,2(F9.1.2X),2(F9.5.2X). 1/)77 PRINT 203,DMEA3,DSD3,DCO3,DSKE3__78 203 FORMAT (1)."STATION3",4x,2(F9.1.2Xi.ZiF9.5.2xi,/i)79 C COMPUTATION OF THE CROSS CORRELATION FOR EACH PAIR OF STATIONSBO C USING THE HISTORIC RECORDS FOR A ZERO LAG81 FELX=2982 CALL 5U82(DEMPI,DEMP2.DMEA1,0MEA2,0501,0S02.CRCORI,FELX)83 CALL SUB2(DEMPI,DEMP3,DMEA1,DMEA3,DSD1,0S03,CRCOR2,FELX)84 CALL SUB2(DEMP2.DEMP3.0MEA2.DMEA3.DSO2.0503.CRCOR4,FELX)85 PRINT 30086 300 FORMAT (///.1X."HISTORIC CROSS CORRELA1ION".//.1X."STATION",4X,"C87 +ROSS CORR",/)88 PRINT 301,CRCOR1.CRCOR2, CRCOR489 301 FORMAT (4X."1-2",6X,F9.5,/,4X,"1-3,6X,F9.5,/,90 +4X,"2-3".6X,F9.5,/)91 C COMPUTE OF A,ETA.MU OF Y.SIGMA,OF Y.R0 OF Y92 CALL SUB3(DSKEI,DBD1,DME41,ETAI,SIGY1,DMEAY1,AAA1,DCORY1,DC01)93 CALL SUB3(DSKE2,DSD2,DMEA2.ETA2,SIGY2,DMEAY2.AAA27DCORY2,DCO2)94 CALL SUB3(DSKE3.DSD303MEA3,ETA3.SIGY3.DMEAY3.AAA3,DCORY3,DCO3)95 PRINT 40096 400 FORMAT(///.1X,"ESTIMATED STATS OF THE TRANSFORM Y",//,20X."ETA",1397 +x,"STDDEVIA".8X,"MEAN" 10X."COEFF A",13X,"CORRELATION",//)98 PRINT 401.ETA1,SIGYI.DMEAY1,AAA1,DCORY199 401 FORMAT(1X,"STATION1",3X,5(F15.5,4X),//)

100 PRINT 402.ETA2.SIGY2.DMEAY2,AAA2,DCORY2101 402 FORMAT(1X,"STATION2"..3X,5(F15.5,4X),//)102 PRINT 403,ETA3,SIGY3.DMEAY3.AAA3.DCORY3103 403 FORMAT(IX,"STATION3",3X,5(F15.5,4X),//)104 C COMPUTATION OF THE CROSS—CORRE FOR THE TRANSFORM Y105 C FOR A LAG ZERO106 CCY012=(LOG(ETAlsETA2*CRCOR1+1))/((LOG(ETA14.02+1))*(LOWETA2s*2+1)107 +7)**0.5108 CCY013=(LOWETAI*ETA3*CRCOR2+1))/((LOG(ETA1*.02+1))*(LOG(ETA3,00,2+1)109 +)).,*0.5 .110 CCY023=(LOG(ETA2sETA3*CRCOR4+1))/((LOS(ETA2**2+1))+(LOG(ETA3**2+1)111 +))**0.5112 PRINT 405

218

113 405 FORMAT(///,1X,"CROSS CORR OF Y FORR LAG ZERD",//.1X,114 +"STATION",4X,"CRO9S COR",/)115 PRINT 406,CCY012,CCY013, CCY023116 406 FORMAT (4X , "1-2",6X,F9.5,/,4X,"1-3",6X,F9.5,/,117 +4X,"2-3,6X,F9.5,/)118 C COMPUTE OF THE CROSS CORR FOR THE TRANSFORM Y FOR A LAG ONE119 CCY112=DCORYI*CCY012120 CCY113=DCORY1*CCY013121 CCY123=DCORY2*CCY023122 CCY121=DCORY2*CCY012123 CCY131=DCORY3*CCY013124 CCY132=DCORY3*CCY023125 PRINT 407126 407 FORMAT(///,1X,"CROSS CORR OF Y FEAR LAG ONE",//,1X,127 +"STATION",4X,"CROSS COR",/)128 PRINT 408,CCY112,CCY113, CCY123, CCY121,CCY131129 + .CCY132130 408 FORMAT (4X,"1-2",6X,F9.5,/,4X,"1-3",6X,F9.5,/,131 +4X."2-3".6X,F9.5,/,4X,132 +"2-1,6X,F9.5,/, 4X,3-1",6X,F9.5,/,4X,"3-2",6X,F9.5,/,)133 C CONSTRUCTION OF THE MATRICES MO(I.J) FOR YHE CASE OF THREE SITES134 M0(1,1)=SIGYI**2135 M0(1.2)=CCY012*SIGYI*SIGY2136 M0(1.3)=CCY013*SIGYI*SIGY3137 M0(2,1)=M0(1,2)138 M0(2,2)=SI5Y2**2139 M0(2,3)=CCY023*SIGY2*SIGY3140 M0 (3,1)=M0(1.3)141 M0(3,2)=M0(2,3)142 MO(3,3)=SIGY3**2143 PRINT 499144 499 FORMAT (///,1WMATRICE MO",//)145 DO 30 1=1,3146 PRINT 500,(M0(I,3),J=1,3)147 500 FORMAT (//,1X,3(F9.5,2X),/,)148 30 CONTINUE149 C CONSTRUCTION OF THE MATRICE MI(I.J)150 M1(1,1)=DCORY1*(SIGY1**2)151 MI(1.2)=CCY112*SIGY1*SIGY2152 H1(1,3)=CCY113*SIGYI*SIGY3153 MI(2,1)=CCY121*SIGY2*SIGY1154 MI(2,2)=DCORY2*(SIGY2**2)155 M1(2,3)=CCY123*SIGY2*SIGY3156 MI(3,1)=CCY131*SIGY3*SIGY1157 MI(3,2)=CCY132*SIGY3*SIGY2158 M1(3,3)=DCORY3*(SIGY3**2)159 PRINT 493160 495 FORMAT (///,1X,"M4TRICE MI",//)161 DO 31 1=1,3162 PRINT 496,(M1(I,J),3=1.3)163 496 FORMAT (//,1X,3(F9.5,2X),/,)164 31 CONTINUE165 C CONSTRUCTION OF THE INVERSE OF MO.(MOMI)166 N=3167 IA=316e IDGT=12169 CALL LINV2F(MO,N,IA,MOM1,IDGT,W4(AREA.IER)

219

170 PRINT 700171 700 FORM4T(1X,"INVERSE MATRICE OF MO",/,)172 DO 70 I=1,3173 PRINT 701,(MOM1(I,3),3=1,3)174 701 FORMAT (/,1X,3(F15.5,2X))175 70 CONTINUE176 C COMPUTATION OF THE MATRICE A=Ml*MOMI,(MA)177 L=3178 M=3179 N=3180 IA=3181 IB=3182 IC=3183 CALL VMULFF(M1,M0M1,L,M,N,IA,IB,MA,IC,IER)184 PRINT 710185 710 FORMAT (/,1X,"MATRICE A",/)186 DO BO 1=1,3187 PRINT 711,(MA(I,3),2=1,3)188 711 FORMAT (/,1X,3(E15-5,2X))189 80 CONTINUE190 C COMPUTE THE TRANSPOSE OF M1,(MIT)191 DO 90 1=1,3192 DO 91 J=1,3193 M1T(I,J)=M1(3,I)194 91 CONTINUE195 90 CONTINUE196 PRINT 720197 720 FORMAT (/,1X,"TRANSPOSE OF MI",/)198 DO 92 1=1,3199 PRINT 7214(M1T(1,2),3=1,3)200 721 FORMAT(/,1)(.3(F9..4,2X))201 92 CONTINUE202 C COMPUTE THE MATRICE A*M1T,(MB)203 CALL VMULFF(MA,M1T,L,M,N,IA,IB,MB,IC,IER)204 C COMPUTE THE MATRICE BBT205 DO 93 1=1,3206 DO 94 3=1,3207 BBT(I,J)=MO(I,J)-MB(I,J)208 94 CONTINUE209 93 CONTINUE210 PRINT 730211 730 FORMAT(1X,"MATRICE BBT",/)212 DO 95 1=1,3213 PRINT 731,(BBT(I,3),J=1,3)214 731 FORMAT(1)(,3(F9.5,2X))215 95 CONTINUE216 C COMPUTE THE MATRICE B BY TRIANBULARISATION METHOD217 N=3218 J08N=12219 II=3220 CALL EIGRS(BBT,N,JOBN,D,P,1Z,WK,IER)221 CALL EISRS(MO,N,3OBN,D1,P1,IZ,WK,IER)222 CALL EISRS(MB.N,JOBN,02,P2,IZ,WK,IER)223 PRINT 740224 740 FORMAT (//,1X,"EISENVECTORS",47X,"EIGENVALUES" , / , )225 DO 15 1=1,3226 PRINT 741,(P(I,J),J=1,3),D(I)

220

227 741 FORMAT(/,IX,228 15 CONTINUE229 PRINT 742230 742 FORMAT (//,231 DO 67 1=1,3232 PRINT 743,233 743 FORMATU,IX,234 67 CONTINUE235 PRINT 744236 744 FORMAT (//,237 DO 68 1=1,3238 PRINT 745,239 745 FORMAT(/,1X,240 68 CONTINUE

3(F9.5,..2X),12X,F9.5,2X)

IX,"EIGENVECTORS",47X,"EIGENVALUES",/,)

(P1(1,3),3=1,31,01(1)3(F9.5,2X),12X,F9.5,2X 1

IX,"EIGENVECTORS",47X,"EIGENVALUES",/,)

(P2(1,3),3=1,3),D2S1)3(F9.5,2X),12X,F9.5.2X1

241 C COMPUTE THE INVERSE OF MATRICE P (PM1)242 N=3243 IDGT=12244 CALL LINV2F(P,N,IA,PM1,IDGT,WKAREA,IER)245 PRINT 750246 750 FORMOT(/11)(1"INVERSE OF MATRICE P")247 DO ?..., 1=1,3248 PRINT 751,(PM1(I,3),J=1,3)249 751 FORMAT(/FIX,3(F9.5.2X))250 25 CONTINUE251 C FORMING THE MATRICE DHALF252 DO 35 1=1,3253 DO 36 3=1,3254 IF(J.EO./) THEN255 DHALF(I,3)=(13(3))**0.5256 ELSE257 — DHALF(I,3)=0.0258 END IF259 36 CONTINUE260 35 CONTINUE261 C FORMING THE MATRICE PRODUCT P*DHALF262 CALL VMULFF(P,DHALF,L,M,N,IA,IBIPDHALF,IC,IER)263 C FORMING THE MATRICE PRODUCT PDHALF*PM1,(13)264 CALL VMULFF(PDHALF,PM1,LIM,N,IA,I8,8,IC,IER)265 PRINT 760266 760 FORMAT (/,1)(s"MATRICE 13",//)267 DO 45 1=1.3268 PRINT 761v(8(1,3)73=1,3)269 761 FORMAT(/,1X,3(F9.5,2X))270 45 CONTINUE271 C COMPUTE INITIAL VALUE OF Y272 AAA(1)=AAA1273 AAA(2)=AAA2274 AAA(3)=AAA3275 MEANY(1)=DMEAY1276 MEANY(2)+DMEAY2277 MEANY(3)=DMEAY3278 Y(1,1)=LOG(AAA(1)—DEMPI(29,2))—MEANY(1)279 Y(1,2)=LOG(AAA(2)—DEMP2(29,2))—MEANY(2)280 Y(1,3)=LOG(AAA(3)—DEMP3(29,2))•—MEANY(3)281 C COMPUTE THE SYNTHETIC FLOW (290 YEARS) —282Ni-1283 DO 55 K=2,291

221

284 DO 56 3=1,3285 B1(3,N1)=Y(K-1,3)286 RI(J,N1)=U(K,3)287 56 CONTINUE288 CALL VMULFF(MA,B1,1_,M,N1,IA,18,Y1,1C,IER)289 CALL VMULFF(g ,R1,L,M,N1,IA,IP,Y2,IC.IER)290 DO 57 KI=1,3291 Y(K,K1)=Y1(K1,1)+Y2((1,1)292 X(K,K1)=AAA(K1)-EXP(Y(K,KI)+MEANY(K1))293 IF (X((,)(1).LT.0.0) THEN294 X(K,K1)=0.0295 ELSE296 END IF29729g i; CRYINUÉE-299 PRINT 770300 770 FORMAT(///,IX,"SYNTHETIC ANNUAL FLOW OF THE TRANSFORM Y",/,1X,301 +"YEAR",7X,"STATIONI",4X,"STATION2",4X,"STATION3",/)302 DO 65 K=2,291303 PRINT 771,K-1,(Y(K,K1),K1=1.3)304 771 FORMAT(1X,I3,4X.3(F9.5,3X))305 S5 _CTINUE_

-306 PRINT 780307 780 FORMAT(///.1X,"SYNTHETIC ANNUAL FLOW ,FINAL FORM",/,IX,"YEAR",9X,308 +"STATION1",12X,"STATION2",.. X,"STATION3",/)309 DO 75 K=2,29I310 PRINT 781,K-1,(X(K,K1),K1=1,3)311 781 FORMAT(1X,I3,4X,3(F15.5,3X))312 75 CONTINUE313li LOMPVTATIDN_K THEJSAN, STANDARD_DEVIATION,_AUTOCORRELATIONLANa

-314 S EWNESS OM SYNTHETIC FLOW FOR EACH STATION315 SELX=290316 DO 3 1=1,290317 S1(I,2)=X(I+1,1)318 S2(1,2)=X(I+1,2)319.53(I.2)=X(I+1,3)320 3 CONTINUE32, CALL SUB1(S1,SMEA1,SSD1,SCOI,SSKEI,SELX)322 CALL SUB1(52,SMEA2,SSD2,SCO2,SSKE2,SELX)323 CALL SUB1(53,SMEA3,SSD3,SCO3,SSKE3,SELX)324 PRINT 900 •325 900 FORMAT(///,"ESTIMATED STATS FROM SYNTHETIC FLOW",//,13X,"MEAN",7X,326 +"STD DEVIA" .._5X,"AUTOCOR",4X,"SKEW",//)327 PRINT 901,SmEA1,SSD1,SC01,SSKE1328 901 FORMAT (1X,"STATION1",4X,2(F9.1,2X),2(F9.5,2X),//)329 PRINT 9021SMEA2,SSD2,SCO2,SSKE2330 9-02 FORMAT (1X,"STATION2",4X,2(F9.1,0) -,2(q.5.2 -X),/i)331 PRINT 903,SMEA3,SSD3,5CO3,5SKE3332 903 FORMAT (1X,"STATION3",4X.2(F9.1.2X) 12(F9.5,2X). 1/)333 C COMPUTATION OF THE LAG ZERO CROSS CORRELATION334 C USING THE SYNTHETIC FLOWS335 FELX=290336 CALL SUB2( Si. S2,SMEA1,SMEA2,SSDI,SSD2,CPC012,FELX)337 CALL 6UE12( Si, S3,SMEAI,SMEA3,SSD1,SSD3,CRC013,FELX)338 CALL SUB24 S2, S3,SMEA2,SMEA3,SSD2,5SD3,CRCO23,FELX)339 PRINT 305340 305 FORMAT (///,1X,"SYNTHETIC CROSS CORRELATION",//,IX,"STATION",4X,"C341 +ROSS CORR"./)342 PRINT 306,C7C012,CRCO13,CR6323343 306 FORMAT (4X,'I-2".6X,F9.5,/,4X,"1-3",6X,F9.5,/,344 +4X,"2-3",6X,F9.5,/)345 STOP346 END

1234567

SUBROUTINE SUB1(A,B,C.D,E,DELX)REAL A(DELX,2)INTEGER DELxB=0.0C=0.0D=0.0E=0.0

e 8=0.09 H=0.010 S=0.011 C COMPUTATION OF MEAN12 DO 10 I=1,DELX,3 B=B+A(I,2,44 10 CONTINUE45 B=B/DELX16 C COMPUTATION OF STANDARD DEVIATION17 DO 15 I=1,DELX1 8 G=G+C(ACI,2>-8)+4,2)19 15 CONTINUE20 G=G/DELX21 C=SORTXG7-...,a. .,. C COMPUTATION OF CORRELATION23 DO 20 I=1,DELX-124 H=H+(A(I,2)—B)*(A(I+1,2)—B)25 20 CONTINUE26 D=H/((DELX )*(C 4.4,2))27 C COMPUTATION OF THE SKEW COEFFICIENT28 DO 25 I=1,DELX29 S=S+C(ACI,27—B)**3Y30 25 CONTINUE31 E=S/(DELX*(C.i3))32 RETURN33 END

222

223

1 SUBROUTINE SUB3(GAX,SIGX,MUX,ETA,SIGY,MUY,AAA,ROY,ROX)2 REAL MUY,MUX

3 ETA=0.0

g SIGY=0.0

6MUY=0.0AAA=0.0

7 ROY=0.0B ETA=(GAX/2.4((GAX**2)/4+1)**0.5)**(1.0/3.0)-(ABS(GAX/2-

9 +((GAX**2)/4+I)**0.5))**(1.0/3.0)

10 SIGY=(LOG(ETA**2+1))**0.5

11 MUY=LOG(((SIGX**2)/((ETA**2)*(ETA**2+1)))**0.5)

12 AAA=MUX- (SIGX/ETA)

13 ROY=(LOG(CETA**2)*ROX+1))/(LO(ETA**2+1))

14 RETURN

1Z END

1 SUBROUTINE SUB2(AA,BB,CC,DD,EE,FF,HH,MELX)

2 REAL AA(MELX,2),BB(MELX,2)

3 INTEGER MELX2 YI=0.0

gHH=0.0

DO 10 I=1,MELX7 Y1=Y14.(AA(I,2)-CC)*(013(1,2)-DD)B 10 CONTINUE9 HH=Y1/(MELX*EE*FF)

10 RETURN

11 END

REFERENCES

AFIFI, A.A., and AZEN, S.P., Statistical Analysis, A Computer Oriented Approach, Academic Press, New York, 1979.

AITCHISON, J., and BROWN, J.A.C., The Log-normal Distribution, CambridgeUniversity Press, London, England, 1957.

ASKEW, A.J., YEN, W.G., HALL, W.A., A Comparative Study of CriticalDrought Simulation, Water Res. Research, Feb. 1971.

BENJAMIN, J.R., and CORNELL, C.A., Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill Book Company, 1970.

BENSON, M.A., and MATALAS, N.C., Synthetic hydrology based on regionalstatistical parameters, Water Resour. Res., 3(4), 931-945, 1967.

BOX, G.E.P., and JENKINS, G.M., Time Series Analysis--Forecasting andControl. (Revised edition), Holden-Day, 575 pp, 1970.

BURAS, N., class notes, WRA 643, 1984 (unpublished).

CHI, M., NEAL, E., and YOUNG, G.K., Practical application of fractionalBrownian motion and noise to synthetic hydrology, Water Resour. Res., 9(6), 1973.

CROSBY, D.S., and MADDOCK, T., III, Estimating coefficients of a flowgenerator for monotone samples of data. Water Resour. Res.,6(4), 1079-1086, 1970.

EINSTEIN, A., Eine neue Bestimmung der Molekuldimension, Ann. Phys., 19,289-306, 1905.

FELLER, W.F., The asymptotic distribution of the range of sums of inde-pendent random variables, Ann. Math. Statist., 22, 427, 1951.

FIERING, M.B., Multivariate techniques for synthetic hydrology. J. Hydr. Div. ASCE, 90, HY5, 43-60, 1964.

FIERING, M.B., Streamflow Synthesis, pp. 66ff, Harvard University Press,Cambridge, Mass., 1967.

FIERING, M.B., and JACKSON, B.B., Synthetic streamflows, Water ResourcesMonograph, American Geophysical Union, Washington, D.C., 1971.

224

225

GANNETT, FLEMING, CORDDRY and CARPENTER, INC. (GFC&C), in associationwith ORGATEC, "Assessment of Environmental Effects of ProposedDevelopments in the Senegal River Basin," Dakar, Senegal, 1978.

GROUPEMENT MANANTALI, "Manantali Dam Study," 1977.

HULL, C.H., NIE, H.N., "SPSS Update 7-9, New Procedures and Facilitiesfor Releases 7-9", McGraw-Hill Book Company, U.S.A., 1981.

HURST, H.E., Long-term storage capacity of reservoirs, Trans. Amer. Soc. Civil Eng., 116, 770, 1951.

HURST, H.E., Methods of using long-term storage in reservoirs, Proc. Inst. Civil Eng., 5, part 1, 519, 1956.

HURST, H.E., BLACK, R.P., and SIMAIKA, Y.N., Long Term Storage, an Exper-imental Study, Constable, London, 1965.

IMSL, LIB-0009, Red manual, Vol. 1, Edition 9, 7500 Bellaire Blvd.,Houston, Texas, 77036-5085, June 1982.

INTERNATIONAL LAW ASSOCIATION, "International Rivers--The HelsinkiRules," International Law Association, 1956 (reprinted in Kniper,1971).

JACKSON, B.B., Birth-Death Models for Differential Persistence, Water Resources Research, Feb. 1975.

JACKSON, B.B., The use of streamflow models in planning, Water Resour. Res., 11(1), 1975.

LANE, W.L., Applied stochastic techniques (LAST computer package), usermanual. Division of Planning Technical Services, Bureau ofReclamation, Denver, Colorado, December 1979.

"Le Fleuve Senegal et son Environment," Marches Tropicaux, 17 Avril,1981.

LOUCKS, D.P., STEDINGER, J.R., and HAITH, D.A., Water Resources Systems Planning and Analysis, Prentice-Hall, Inc., Englewood Cliffs, NewJerwey 0/632, 1981.

MADDOCK, T., class notes, HYD 643, 1984 (unpublished).

MANDELBROT, B.B., Une classe de processus stochastiques homothetiques asoi; Application a la loi climatologique de H. E. Hurst, C. R.Acad. Sci. Paris, 260, 3274, 1965.

MANDELBROT, B.B., A fast fractional Gaussian noise generator, Water Resour. Res., 7(3), 543-554, 1971.

226

MANDELBROT, B.B., and VAN NESS, J.W., Fractional Brownian motions,fractional noises, and applications, SIAM Rev., 10(4), 422, 1968.

MANDELBROT, B.B., and WALLIS, J.R., Noah, Joseph, and operationalhydrology, Water Resour. Res., 4(5), 909-918, 1968.

MANDELBROT, B.B., and WALLIS, J.R., Computer experiments with fractionalGaussian noises--Part 1: Averages and variances. Water Resour. Res., 1, 228-241, 1969.

MATALAS, N.C., Mathematical assessment of synthetic hydrology. WaterResour. Res., 3(4), 937-945, 1967.

MATALAS, N.C., and WALLIS, J.R. Statistical properties of multivariatefractional noise processes, Water Resour. Res., 7(6), 1460-1468,1971.

MEJIA, J.M., On the generation ofmultivariate sequences exhibiting theHurst phenomenon and some flood frequency analyses. Ph.D.Dissertation, Colorado State University, Fort Collins, Colorado,1971.

MEJIA, J.M., and ROUSSELLE, J., Disaggregation models in hydrologyrevisited. Water Resour. Res., 12(2), 185-186, 1976.

NDIAYE, ABDOULAYE, Institutional Analysis and Appraisal of the OMVS,1984, unpublished paper.

NIE, H.N., HULL, C.H., JENKINS, J.G., STEINBRENNER, K., BENT, H.D., Sta-tistical Package for the Social Sciences, 2nd ed., McGraw-HillBook Company, U.S.A., 1975.

O'CONNELL, P.E., Stochastic modeling of long-term persistence instreamflow sequences. Ph.D. Thesis, Imperial College, Universityof London, 1974.

PANU, U.S., UNNY, T.E., RAGADE, R.K., A feature prediction model insynthetic hydrology based on concepts of pattern recognition,Water Resour. Res., 14(2), 335-344, 1978.

PEGRAM, G.C.S., and JAMES, W., Multilag multivariate autoregressivemodel for the generation of operational hydrology. Water Resour. Res., 8(4), 1074-1076, 1972.

PHIEN, H.N., RUKSASILP, W., A Review of Single-Site Models for MonthlyStreamflow Generation, J. Hydrology, 52, 1-12, 1981.

PHIEN, H.N., KHAN, Ayub. M., Comparison of Two Autoregressive Models forMonthly Streamflow Generation, Water. Res. Bulletin, Dec. 1981.

227

PHIEN, H.M., ARBHABHIRAMA, A., SUTABUTR, P., Range Analysis for ReservoirStorage with Independent Flows, J. Hydrology, 47, 53-64, 1980.

RILEY, J.P. ANDERSON, J.C., BISHOP, A.B., BOWLES, D.S. and KEITH, J.E.,Cost Allocation Alternatives for the Senegal River DevelopmentProgram, Utah Water Research Laboratory, College of Engineering,Utah State University, Logan, Utah, 84322, Water ResourcesPlanning Series, UWRL/P-78/06, August 1978.

RIPPL, W., The capacity of storage reservoirs for water supply, Proc. Inst. Civil Eng., 71, 1883.

ROCHETTE, C., Le Bassin du Fleuve Senegal, Monographies Hydrologiques,ORSTOM, Paris, France, 1974.

SALAS, J.D., DELLEUR, J.W., YEVJEVICH, V., and LANE, W.L., Applied Modeling of Hydrologic Time Series, Water Resources Publications,Littleton, Colorado, 1980.

SIRCOULON, J., Les Donnees Hydrologiques de la Secheresse Recente enAfrique Intertropicale comparaison avec les Secheresses "1913" et"1940", Cahiers ORSTOM Sec. Hydrol. Vol. XIII, No. 2, NumeroSpecial Secheresse, pp. 75-174.

SOGREAH, Diama Dam Study, Grenoble, France, 1977.

TAO, P.C., and DELLEUR, J.W., Seasonal and nonseasonal ARIMA models inhydrology. Proc. Am. Soc-Civil Engrs., J. of Hydr. Div., 102,HY10, 1541-1559, 1976.

THOMAS, H.A., Jr., and FIERING, M.B., "Mathematical synthesis ofstreamflow sequences for the analysis of river basins bysimulation," in Design of Water Resource Systems, A. Maass, M.M.Hufschmidt, R. Dorfman, H.A. Thomas, Jr., S.A. Marglin, and G.M.Fair (eds.), Harvard University Press, Cambridge, Mass., 1962.

USAID/OMVS, Integrated Development Project #625-0621, "InstitutionalAnalysis," Vol. III, Section 4, USAID/RBDO, Oct. 1982, Dakar,Senegal.

VALENCIA, D.R., and SCHAAKE, J.C., Jr., Disaggregation processes instochastic hydrology, Water Resour. Res., 9(3), 580-585, 1973.

YOUNG, G.D., and PISANO, W.C., Operational hydrology using residuals, J.Hydr. Div. ASCE, 94, HY4, 909-923, 1968.