KWAME NKRUMAH UNIVERSITY OF SCIENCE AND

TECHNOLOGY, KUMASI, GHANA

Development of Intensity Duration Frequency Curves for

Accra

By

Daniel Gondolo Glay

(BSc. Civil Engineering)

A Thesis submitted to the Department of Civil Engineering,

College of Engineering

In Partial Fulfilment of the requirements for the degree of

MASTER OF SCIENCE

Water Resources Engineering and Management

May, 2016

ii

DECLARATION

I hereby declare that this Thesis is my own work towards the MSc. and that, to the

best of my knowledge, it contains no material previously published by another person

or material which has been accepted for the award of any other degree of the

University, except where due acknowledgement has been made in the text.

Daniel Gondolo Glay …………………… ……………….

(PG9731313) Signature Date

Certified by:

Dr. F.O. K Anyemedu …………………….. ……………….

(Supervisor) Signature Date

Prof. Y. A. Tuffour ……………………… ………………….

(Head of Department) Signature Date

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ABSTRACT

The research work is about the development of Intensity-Duration-Frequency curves

for Accra. The IDF relationships is a mathematical relationships between the rainfall

intensity, duration and frequency. It is a hydrologic statistical tool that is commonly

used by engineers during planning and designing of hydraulic infrastructures such as

drainage system and water storage facilities for urban and rural areas. Maximum

rainfall depths of nine different durations for 28 years (1971 to 2006) were obtained

from the Ghana Meteorological Agency. Regression analysis were used to fill in the

missing gaps found in the data. Various literatures reviewed to find the probability

distribution that best fitted the acquired data chose the Gumbel Extreme Value Type I

(EVI) and the Log-Pearson Type III distributions. The Goodness of fit tests

(Kolmogorov-Smirnov Anderson - Darling and the Chi-Square) using the aid of the

Easy fit software finally selected and confirmed the Gumbel EVI as the best fit

distribution function for the analysis of the data. For each duration: (0.2hr, 0.4hr,

0.7hr, 1.0hr, 2.0hrs, 3.0hrs, 6.0hrs, 12.0hrs, and 24.0hrs) sample characteristics were

analyzed and used to determine the population parameters, which aided in finding the

frequency of the rainfall depths and intensities for the construction of the IDF curves

for Accra. Comparison made between the existing and the new intensities showed

more than 20% average increase in value over J. B. Dankwa’s from shorter to longer

durations (i.e. 0.2hr to 24.0hrs) and selected return periods (i.e. 5 to 100 years). This

implies that the rainfall intensities for Accra has changed as a result of climate

variability. Therefore, the relevant institution must begin to update the IDF curves for

the various regions in Ghana. Besides, the responsible institution should commence

an IDF curves development projects for those areas that presently lack the curves to

guide planners and engineers during the design of hydraulic infrastructures.

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DEDICATION

I dedicate this work to my late parents Mr. Fineboy Glay Yeebahn and Madame Mary

Wehyee my mother for jointly processing my existence on planet earth and for their

physical and personal interactions with me regarding my capacity building prior to the

commencement of the study for MSc in Water Resources Engineering and

Management:

May the souls of my late parents rest in perpetual peace

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ACKNOWLEDGEMENT

I am pleased to register herein my heartfelt thanks and appreciation above all to

Jehovah Jirah, the creator of heaven and earth for the maintenance of my life up to

this age for a successful achievement of my longtime dream.

My next thanks and appreciation goes to the present government of the Republic of

Liberia and the African Development Bank for providing the resources for my

training. Additionally to the management team of the Liberia Water and Sewer

Corporation under the dynamic leadership of Honorable Charles B. Allen Managing

Director and the UWSSP staff for ensuring the accomplishment.

More besides, my thanks and appreciation goes to Dr. F. O K Anyemedu my principal

thesis supervisor and Dr. Anornu the Co- supervisor who worked tirelessly to ensure

the timely and successful completion of my study.

With the highest respect due knowledge, and humanity, I also express my gratitude to

Dr. Richard Buahman, the program coordinator and all the lecturers of WRESP and

WREM including my course mates for all the supports accorded me during this

endeavor. My special and exclusive thanks and appreciation goes to Mr. Stephen

Asugre Jr., Mr. Collins Owusu and Mr. Sulemana Abubakari who aided me in all

aspects of my study with pieces of technical advice during my study and research

work. Moreover, Hon. Roger B. Woodson formal Managing Director of LWSC for

physically setting me on the path to this achievement.

Lastly, my gratitude goes to my darling wife Mrs. Wede Olivia Glay and all of my

children for their patience and prayers offered to God while I was away for the study.

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TABLE OF CONTENT

DECLARATION ........................................................................................................... ii

ABSTRACT ..................................................................................................................iii

DEDICATION .............................................................................................................. iv

ACKNOWLEDGEMENT ............................................................................................. v

LIST OF TABLES ......................................................................................................... x

LIST OF FIGURES ...................................................................................................... xi

LIST OF ABBREVIATION ........................................................................................ xii

LIST OF SYMBOLS .................................................................................................. xiv

CHAPTER 1: INTRODUCTION .................................................................................. 1

1.1 General Background ............................................................................................ 1

1.2 Problem Statement ........................................................................................... 3

1.3 Justification of Research ...................................................................................... 3

1.4 Primary Objective ............................................................................................. 4

1.5 Specific Objectives ........................................................................................... 5

1.6 Research Questions ............................................................................................ 5

1.7 Arrangement of Report ..................................................................................... 5

CHAPTER 2: LITERATURE REVIEW ....................................................................... 7

2.1 Introduction ...................................................................................................... 7

2.2 Brief Description of the IDF Curve and importance ........................................ 7

2.3 Characteristics description of the IDF Curve ................................................... 8

2.5 Short history of IDF Curves Development ....................................................... 9

2.6 Development of Intensity Duration Frequency Curves .................................. 11

2.6.1 Procedures in developing IDF Curves ...................................................... 11

2.6.1.1 Rainfall Data Collection and analysis ................................................ 11

2.6.1.2 Fitting Probability distribution to the rainfall Data ............................ 12

2.6.1.3 Frequency Factor ................................................................................ 12

2.6.1.4 Rainfall Intensity Analysis ................................................................. 13

2.6.1.5 Graphical development of the IDF Curves ......................................... 14

2.7 Theory of fitting Probability distribution to rainfall data ............................... 14

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2.7.1 Types of Probability Distribution for Hydrology data analysis ............... 14

2.7.1.1 Normal Distribution ........................................................................... 15

2.7.1.2 Log- Normal Distribution ................................................................... 15

2.7.1.3 Gamma Distribution ........................................................................... 16

2.7.1.4 Exponential Distribution .................................................................... 16

2.7.1.5 Pearson Type III Distribution ............................................................. 17

2.7.1.6 Log- Pearson Type III Distribution .................................................... 18

2.7.1.7 Extreme Value Distribution of Gumbel ............................................. 19

2.7.2 Fitting a probability distribution to Rainfall data ..................................... 21

2.7.3 Parameter Estimation ............................................................................... 22

2.7.3.1 Methods of Moment (MOM) ............................................................. 22

2.7.3.2 Method of Maximum Likelihood ....................................................... 23

2.7.3.3 Method of L-Moments ....................................................................... 25

2.7.4 Statistical Parameters ............................................................................... 26

2.7.4.1 Mean (Average) .................................................................................. 26

2.7.4.2 Variance .............................................................................................. 27

2.7.4.3 Skewness ............................................................................................ 27

2.8 Goodness of Fit Tests ..................................................................................... 28

2.8.1 Anderson Darling Test ............................................................................. 28

2.8.2 Kolmogorov-Smirnov (KS) Test .............................................................. 30

2.8.3 Chi-square (CS) test ................................................................................. 31

2.9 Statistical Test of Hypotheses ........................................................................ 32

2.9.1 Procedures for Testing Hypothesis ........................................................... 33

2.9.1.1 Stating the two Hypotheses ................................................................ 34

2.9.1.2 Test Statistic and the sample’s Distribution function ......................... 35

2.9.1.3 State the level of significance ............................................................. 35

2.9.1.4 Statistical Data Analysis ..................................................................... 36

2.9.1.5 Zone of Acceptance and Rejection ..................................................... 36

2.9.1.6 Make decision based on comparison .................................................. 37

2.9.2 Conclusion of the Literature Review ........................................................ 37

CHAPTER 3: RESEARCH METHODOLOGY ......................................................... 39

3.1 The Study Area .............................................................................................. 39

3.1.1 Climate ..................................................................................................... 40

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3.1.2 Vegetation ................................................................................................ 40

3.1.3 Topography and Drainage ........................................................................ 40

3.1.4 Geology and Soil ...................................................................................... 41

3.2 Research Methodology ................................................................................... 41

3.2.1 Procedure .................................................................................................. 42

3.2.2 Rainfall Data Collection ........................................................................... 42

3.2.3 Rainfall Data Processing .......................................................................... 43

3.2.4 Selection of appropriate Distribution Functions for the sample Data ...... 43

3.2.5 Fitting the selected Gumbel Distribution to the Sample Data .................. 44

3.2.6 Validation testing of the fitness of Gumbel Distribution ......................... 47

3.2.6.1 Procedure for Kolmogorov – Smirnov Test validation ...................... 48

3.2.6.2 Procedure for the Chi –Square Test ................................................... 49

CHAPTER 4: RESULTS AND DISCUSSION ........................................................... 53

4.1 Presentation of Results and Discussion .......................................................... 53

4.1.1 Analyze historic rainfall data for the determination of annual maximum

rainfall depth for the various durations. ................................................... 53

4.1.2 Selection and verification of the appropriate probability distribution that

best fit the sample data ............................................................................. 54

4.1.3 Fitting the selected Gumbel distribution to the sample data .................... 55

4.1.4 Compute rainfall intensity and developed IDF curves ............................. 59

4.1.5 Comparison of Results ............................................................................. 67

CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS ............................... 69

5.1 Conclusion ...................................................................................................... 69

5.2 Recommendations .......................................................................................... 69

REFERENCE ............................................................................................................... 70

APPENDICES ............................................................................................................. 77

Appendix 1: Raw Rainfall data obtained from GMA .............................................. 78

Appendix 2: Complete rainfall Data obtained after filling gaps .............................. 81

Appendix 3: Summarized Annual Maximum Rainfall depths extracted from all

durations ............................................................................................. 86

Appendix 4: Annual Maximum Series arranged for Different Durations ............... 87

Appendix 5: Summarized Distribution Parameters for different Durations ............ 95

Appendix 6: Kolmogorov – Smirnov test statistics for different Durations ............ 98

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Appendix 7: Critical Values table for Kolmogorov – Smirnov Test ..................... 105

Appendix 8: Chi –Square repartition tables for observed and expected

frequencies ........................................................................................ 106

Appendix 9: Chi-Square Test result for different durations .................................. 109

Appendix 10: Chi – Square Distribution table ....................................................... 112

Appendix 11: Estimates of Rainfall Intensities for different durations and return

periods. ............................................................................................. 113

Appendix 12: Table of Frequency Factor (K) for Extreme Value Type 1(EV1) ... 116

Appendix 13: J. B Dankwa Maximum Rainfall Intensities Duration Frequency for

Accra ................................................................................................ 117

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LIST OF TABLES

Table 4.1. Result of Correlation coefficient data s strength of relationship ................... 53

Table 4.2: Result of Easy Fit Test for EV1 and LP3 under duration 0.20 hr (12 min) .. 54

Table 4.3: Summary of result from the Easy Fit Tests for all durations ........................ 55

Table 4.4. AMS Analyzed for duration 0.20 hour ......................................................... 56

Table 4.5. Computed Gumbel Distribution parameters analyzed for 0.20 hr ................ 56

Table 4.6. Kolmogorov – Smirnov Test Result analyzed for duration 0.20 hr. ............. 58

Table 4.7: Chi -Square analyzed Test result analyzed for duration 0.20 hr ................... 58

Table 4.8: Summary of Chi- Square Test result ............................................................. 59

Table 4.9 Analyzed rainfall intensities for duration 0.20 hr. ......................................... 60

Table 4.10. Summarized estimates of rainfall intensity for all durations and return

periods ............................................................................................................... 60

Table 4.11 Comparison result of estimated Intensities (New) with J. B Dankwa’s for

(5yr.10yr. and 15yr) .......................................................................................... 65

Table 4.12. Comparison result of new estimated Intensities with Dankwa’s for (20yr.

25yr. and 50yr) .................................................................................................. 66

Table 4.13 Comparison result of estimated Intensities (New) with J. B Dankwa’s

Intensities for (100yrs) ...................................................................................... 67

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LIST OF FIGURES

Figure 3.1 Maps of Figure Ghana and Greater Accra .................................................... 39

Figure 4.1. IDF Curves for Accra – Log-Log graph ...................................................... 62

Figure 4.2 IDF Curves for Accra – Semi- Log .............................................................. 63

Figure 4.3 IDF Curves for Accra – Normal Scale .......................................................... 64

xii

LIST OF ABBREVIATION

Abbreviations Meaning

AD - Anderson Darling test

AMS - Annual Maximum Series

APD - Average percentage difference

B - Upper bound

CDF - Cumulative Distribution Function

CS - Chi-Square test

Cum. - Cumulative frequency

D - Ranked sample rainfall depth (mm)

E - Expected frequency

EDF - Empirical Distribution Function

EV1 - Extreme Value Type 1 Distribution

EV11 - Extreme Value Type 2 Distribution

EV111 - Extreme Value Type 3 Distribution

GEV - Generalized Extreme Value Distribution

GMA - Ghana Meteorological Agency

hr - Hour (Duration)

IDF - Intensity- Duration- Frequency

IHP - International Hydrological Programme

in - Inch

K - Frequency Factor

KS - Kolmogorov Smirnov test

df - degree of freedom

LP3 - Log-Pearson Type 3 Distribution

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mm - Millimeter

MOM - Method of Moments

MML - Method of maximum Likelihood

O - Observed frequency

OAPD - Overall average percentage difference

PDF - Probability Density Function

PDS - Partial Duration Series

PWM - Probability-Weighted Moment

Stdev. - Standard deviation

U - Reduced variable

U.S - United States

U.S.S.R - Union of Soviet Socialist Republics

xiv

LIST OF SYMBOLS

Symbols Meaning

HA - Alternative hypothesis

H0 - Null hypothesis

I - Rainfall intensity

K - Frequency factor

Kσ - Departure from variant

m - Rank

n - Number of events

p - Exceedance of probability

s - Scale parameter

T - Return period

u - Reduced variable

α - Significance level

- Gumbel mean

- Mean of reduced variable

- Sample mean

- Gumbel standard deviation

- Standard deviation of reduced variable

- Sample standard deviation

XG - Gumbel variable

X0 - Position parameter

XT - Expected Rainfall depth

1

CHAPTER 1: INTRODUCTION

1.1 General Background

Rainfall depths of various durations are analyzed extensively for the design of hydraulic

structures and the management of many water resources projects involving natural hazards

due to extreme rainfall occurrences world-wide. The most common rainfall frequency

analysis consists of developing a relationship between Intensity, Duration and the Frequency

or return period. Such relationships are known as Intensity Duration Frequency (IDF)

Relationship or equations and are usually derived using observed annual maximum series

(AMS) analyzed from recorded historical rainfall data at one or more sites of interest.

The IDF relationship is also a mathematical relationship between:

The rainfall intensity (I) measured in mm/hr. or inch/hr.

The duration (D) measured in minute or hours and

The return period (T) considered in year

The relationships are used to compute design storms used in many practical applications.

Moreover, the rainfall Intensity Duration Frequency Relationship is one of the widely and

most commonly used tools in water resources engineering for planning, designing and

operating of various engineering projects against floods (Koutsoyiannis et. al., 1998)

Degradation of water quality, property damage and potential loss of life due to flooding is

caused by extreme rainfall events. As such, historic rainfall statistics are analyzed and

utilized for design of flood control hydraulic structures, and others civil structures involving

hydrologic flows (McCuen, 1998; Prodanovic and Simonovic, 2007). Apart from changes in

the hydrologic cycle due to increase in greenhouse gases which cause variations in

precipitation, inadequate hydrologic data and improper planning of water resources projects

have also contributed to flooding and massive loss of life and properties. These impacts due

2

to extreme rainfall events have also forced engineers to analyze available rainfall data more

critically in most developing countries. Since rainfall data are mostly used for the planning

and designing of hydraulic structures, regular review and updating of rainfall statistics (such

as, rainfall Intensity–Duration and Frequency) are very necessary because climate variation

affects the urban area as rainfalls.

Moreover, most urban and peri-urban drainage systems are designed and built to last longer

than their design period which are usually high because of their high maintenance and

rehabilitation cost. Objectively, these surface run-off collection and transportation facilities

are designed to properly contain and convey storm water that could be caused by extreme

rainfall events during their service life usually ranging from 15 to 25 years for secondary

drains and 50 to 100 years for primary drains. But in view of economic, institutional

guideline as well as technical constraints, there exist some capacity limit for drainage systems

(Desramnaut, 2008). The secondary drainage systems should be built to be overwhelmed only

once every 5 to 10 years, while the major component of the drainage system must be

designed and constructed to support nearly every rainfall event (Arisz and Burrel, 2006). To

meet the criteria mentioned above, engineers need to accurately analyze and predict future

rainfall events by making use of the required analytical tools.

Analysis and development of Rainfall Intensity- Duration-Frequency curves are based on

long term rainfall records, considering the assumption which states that the distribution

parameters are stationary or constant over time. With the occurrence of climate variability

which has impact on rainfall also, the hypothesis of stationary precipitation occurrences

appears to be highly questionable. However, the computed design storms from the historic

data could only be accurate for the current period but not the entire service life of drainage

systems.

3

1.2 Problem Statement

Changes in the hydrologic cycle due to climate variability has contributed to the cause of

variations in rainfall intensities in various areas and regions world-wide. As such, increasing

intensities of rainfall along with unplanned development of urban areas usually worsen the

already critical problem of urban area. As it has been observed and reported concerning

increasing rainfall intensities, previous studies have agreed on the fact that as extreme rainfall

events are likely to become frequent, the carrying capacity of existing drainage systems will

surely be buried considerably and the urban hydraulic structures will not contain and control

the volume of surface run-off. The impact of the surface run-off may likely contribute to the

frequent structural failure, structural damage, unforeseen negative environmental effect and

increase in cost of maintenance or rehabilitation.

Now, the existing Intensity- Duration -Frequency (IDF) Relationships established by Dankwa

(1974) being used by engineers for the design and construction of hydraulic structures for

Accra is about 42 years old as at the year (2016) of the preparation of this research.

Computing expected rainfall depths and intensities for the design of hydraulic structures

based on the Dankwa (1974) IDF relationships may result in inaccurate estimates of expected

rainfall intensities due to changes in the pattern of rainfall caused by impact of climate

variability over a long period.

1.3 Justification of Research

The IPCC report of 2007 shows that during the last forty years climate scientists established

an evidence to indicate that the global average temperature is increasing. The report also

expressed that portion of this increase in temperature is due to the emission of greenhouse

gases caused by human activities (IPCC, 2007). Additionally, Scientists have also used

Global Climate Models (GCMs) to quantify the projections of future temperature changes.

4

The report informed that thus projection of increase in average temperature of the earth have

been made, the temperature increase will vary and increase with geographical location and

which will not be evenly distributed according to season. Because the reported average global

temperature increases will also cause evaporation to increase, the increase in evaporation will

contribute to the average increase in precipitation globally. Despite these scientific

projections, there is still a high degree of uncertainty regarding the spatial and temporal

distribution of those changes in rainfalls. Regarding this reported global impact due to

climate change, analysis and quantification of rainfall using historic data are mostly required

for the design and development of many urban infrastructures including roadside drainage,

urban water infrastructures such as water storage facilities and storm drainage system etc.

These hydraulic structures among many others are very expensive to construct and are mostly

vulnerable to stresses of climate variability which affect urban rainfall characteristics.

Due to the sensitive nature and environmental importance for the use of these structures,

accurate engineering analysis and design require the rainfall characteristics represented by the

IDF curves (or relationships) which express statistics on reoccurrence frequency of rainfall to

be accurate. As such, it is important that the tools utilized during the design process of the

structures provide accurate result. The Intensity Duration Frequency Curves aid in the design

and construction of infrastructures resilient to heavy and an extreme storm events to prevent

flooding and its destructive consequences like inundation of farmland, damage to life and

properties.

1.4 Primary Objective

The objective of this study is to develop Rainfall-Intensity-Duration- Frequency (IDF)

Curves for Accra.

5

1.5 Specific Objectives

In order to implement the primary objective of the research, the below listed specific

objectives will be achieved:

Analyze historic rainfall data for the determination of annual maximum rainfall

depths for various years and durations.

Select and verify the appropriate probability distribution function that best fit the

sample data.

Compute rainfall intensities, Construct new IDF curves and compare the new

intensities with the existing developed by J. B Dankwa

1.6 Research Questions

The main questions the study seeks to answer are as follow:

Had there been a change in the rainfall intensities, for specified duration and

frequency in the study area as a result of climate variation?

Is the existing IDF curves analyzed and developed for Accra still giving reliable and

accurate intensities for engineering planning and design of hydraulic structures?

In order to address the concerned questions and provide the appropriate answers, this study

carried out a statistical analyses using updated historic (1971 to 2009) rainfall records

obtained from GMA and developed new IDF curves for the study area and made comparison

between the J. B Dankwa data sets and the new result.

1.7 Arrangement of Report

The report covers five chapters. Chapter one is the introduction which contains the Research

Background, the Problem Statement, Justification, Primary Objective, Specific Objectives

and Concerned Research Questions. Chapter two also contains related Literatures reviewed

for the study. The third chapter also entails the research methodology and covers the location

6

of the study area, Climate, Vegetation, Topography and Drainage, Geology and Soil,

Research methodology and Procedures, Fitting the Selected Gumbel Distribution to sample

data and determination of rainfall intensities for the study area. Chapter four contains results

determined from the analyses and discussions. Finally, chapter five also contains conclusions

and recommendations developed by the study.

7

CHAPTER 2: LITERATURE REVIEW

2.1 Introduction

The main purpose of this chapter is to review relevant literatures related to this study.

Additionally, this section aims at examining and selecting the appropriate probability

distribution function required for the analysis of the obtained rainfall data and the

development of the Intensity -Duration- Frequency Curves for the study area (Accra).

2.2 Brief Description of the IDF Curve and importance

Intensity Duration Frequency (IDF) Curve is a hydrologic statistical tools that describes the

various characteristics of an area rainfall intensities. The basic characteristics of a rainfall

event are the intensity, duration, total and frequency. The IDF curves contain these

characteristics and are also used to graphically express them. The IDF curves are used by

Civil Engineers as a basic hydrologic and statistical tools to analyze and quantify the amount

of rainfall for an area. The IDF curves are used to design more cost effective and durable

hydraulic structures for certain return periods (such as, 5, 10, 15, 25, 50 years, etc.). These

structures are designed to contain a defined volume of flow and withstand a certain degree of

risk at certain capacity above which the hydraulic structure may be exceeded during extreme

rainfall event greater than the chosen event.

According to Kabange – Numbi (2007), besides the use of the IDF curves for Urban

Infrastructure development, it also needed during rehabilitation planning and redesign of

inadequate or outdated existing drainage and surface run-off control systems (Kabange –

Numbi, 2007 cited in Van de Vyver H. and Demaree G.R., 2010). Moreover, the unplanned

extension of the urbanized periphery, demographic expansion, the increase in impervious

areas, and the increased needs for water couple with other environments factors would

require the IDF curves.

8

2.3 Characteristics description of the IDF Curve

Considering rainfall as an integral component of the hydrologic cycle, the characteristics

which make up the IDF Curves are described by some past researchers and others as follows:

Rainfall intensity is defined as the rate at which rain falls in millimeter or inch per

hour (Okonkwo and Mbajiorgu, 2010).

Dupont et al. (2000) defined rainfall Intensity-Duration-Frequency (IDF)

Relationships as a graphical representations of the amount of water that falls within a

given period of time. These graphs can be used to determine when an area will be

flooded and when a certain rainfall rate or a specific volume of flow will re-occur in

the future. The average rainfall intensity is used during statistical analysis and it can

be expressed in equation (2.1) as shown below:

(2.1)

Where P indicates the depth of rain (mm or inch) and D is the Duration, usually measured in

hours or minutes.

Rainfall Duration is the time interval a particular depth of rain falls. Mostly from

analysis of expected rainfall values, the high-intensity value of a storm has a shorter

duration than the low-intensity portion.

Frequency is how often a rainfall event with a selected intensity and duration may be

expected to occur (Okonkwo and Mbajiorgu, 2010). The frequency is most often

described in term of return period, (T) which is considered the average time interval

between the rainfall events that equal or exceed the design magnitude

2.4 The properties of IDF curves

The properties of the IDF curves are described below:

From a graphical appearance of lines, IDF curves are parallel decreasing lines, and

they cannot meet nor cross each other.

9

For any return period, high rainfall intensities are recorded for shorter duration. In

short, the higher the rainfall intensity, the shorter the duration.

For any selected or given duration of rainfall, one can graphically determine the

intensities of rainfall so long the frequency of occurrence is given.

2.5 Short history of IDF Curves Development

The rainfall Intensity-Duration-Frequency-Relationships is one of the widely and most

commonly used tools in water resources engineering for planning, designing and construction

of various engineering projects against floods (Nhat et al., 2006). It is also considered to be a

mathematical relationship between the rainfall intensity, the Duration, and the return Period.

The establishment of such relationships started early in 1932 (Bernard, 1932).

Since these historical years of the development of IDF relationships by some engineers and

researchers, different types of relationship have been established for several geographical

regions. Also, the regional characteristics of IDF relationships have been studied in many

countries, and maps have been developed generally to provide the rainfall intensities or

depths for various return periods and durations (IHP- VII, 2008). Several other studies

conducted have established that IDF Curves had received considerable attention in

engineering hydrology over the past decades world-wide. In addition to the development of

IDF curves, various methods based on statistics of rainfall data developed by others have also

been considered. Some of the various actors and theirs activities involving the development

of IDF curves are recorded below. For examples:

(Bernard, 1932) developed for localities restricted to his study, formula for rainfall

intensity for the return periods of 5, 10, 15, 25, 50 and 100 year, that could be applied

to rainfall duration of 120 to 6000 min. for the U.S.S.R.

10

Dub (1950) made the first attempts to construct regional IDF curves for Slovakia,

Bell (1969) developed IDF relation using a formula which enable him to compute the

depth – duration ratio for certain areas of U.S.S.R.

Samaj and Valovic (1973) provided comprehensive IDF curves study based on 68

stations covering the area of Slovakia using data mostly from the period (1931- 1960)

Dankwa (1974) developed IDF Curves for various cities and towns in the Republic of

Ghana.

Chen (1983) developed a simple method to derive a generalized rainfall

Intensity duration frequency formula for any location in the United States using three

isopluvial maps of the U.S Weather Bureau Technical Paper No.40. In the 1990’s,

some other approaches mathematically more consistent had been proposed.

Rainfall is quantified by using Isopluvial map and IDF Curves (Chow et al., 1988).

According to Stredinger et al. (1993) IDF Curves for precipitation determined

relationship between the mean intensity, the duration, or more precisely the aggregate

time, and the frequency of a rainfall event.

Koutsoyiannis et al. (1998) proposed a new generalization approach to the

formulation and construction of the intensity-duration-frequency curves using

efficient parameterization.

Mohymont et al. (2004) assessed IDF-curves for precipitation for three stations in

Central Africa and proposed more physically based models for the IDF-curves.

Nhat et al. (2006) established IDF curves for the Monsoon area of Vietnam.

Di Baldassarre et al. (2006) analyzed to test the capability of seven different depth-

duration-frequency curves characterized by two or three parameters to provide an

estimate of the design rainfall for storm durations shorter than 1 hour, when their

parameterization is performed by using data referred to longer storms.

11

Karahan et al. (2007) estimated parameters of a mathematical framework for IDF

relationship presented by Koutsoyiannis et al. (1998) using genetic algorithm

approach.

Prodanovic and Simonovic (2007) developed IDF curves for the City of London

under the changing climate. London: Water Resources report.

Endreny and Imbeah (2009) used short record satellite data to generate robust IDF

relations for precipitation in the context of the absent of short instrumental rainfall

records in Ghana.

2.6 Development of Intensity Duration Frequency Curves

2.6.1 Procedures in developing IDF Curves

There are three basic steps involved in developing Intensity Duration Frequency Curves for a

given location or area of interest. The steps are discussed in the sections below:

2.6.1.1 Rainfall Data Collection and analysis

Primarily, the first step is to obtain the historical rainfall data from the relevant institution for

the region, assess the data and extract the annual maximum rainfall depth for each duration

for each data year. Depending on the length of available rainfall record, accurate data analysis

should be conducted using at least 25 to 30 or more years of rainfall data. Preliminary

analyses including corrections, correlation coefficient and simple regression analyses on the

raw rainfall data are conducted to facilitate the rainfall gaps filling in the main record along

with a detailed descriptive statistical analysis for annual maximum rainfall values under each

duration. The average rainfall depth designated by and standard deviation (σ) as functions

of duration using the rainfall values are calculated. From the analysis two arrays are derived,

12

the mean of the rainfall depth and the standard deviation (σ) both serve respectively as a

function of duration for the depth of rain recorded.

2.6.1.2 Fitting Probability distribution to the rainfall Data

Secondly, a Probability Distribution Function (PDF) or a Cumulative Distribution Function

(CDF) is fitted to each set of the annual maximum rainfall data for each duration of the

rainfall data. This serves as one of the initial steps to selecting the appropriate distribution

function. It is also possible to relate the maximum rainfall intensity for each duration with the

corresponding return period from the cumulative distribution function. Given a return period

represented by T, the related cumulative frequency F can be calculated from the expression

(2.2) as shown below:

(2.2)

Alternatively,

( ) (2.3)

When the cumulative frequency is computed, the expected rainfall intensity can be analyzed

using one of the commonly used theoretical distribution function selected for the study (for

example, the Gumbel, Log-normal, Log-Pearson Type III, Gamma distribution, etc.) .

2.6.1.3 Frequency Factor

Before explaining the third step, frequency analysis using frequency factors will be explained

as below. The magnitude XT of the expected rainfall can be computed using equation 2.4 as

expressed below:

(2.4)

Where µ and ΔXT are the mean and departure respectively.

The departure may also be computed using equation 2.5 as expressed below where K and σ

represent the frequency factor and standard deviation respectively:

ΔXT = K σ (2.5)

13

The departure and the frequency factor K are functions of the return period and the type

of distribution function to be selected for the frequency analysis. The expression (2.4) may

therefore be rearranged as shown by equations (2.6) and (2.7) below:

(2.6)

which may be approximated by

(2.7)

Where in equation (2.7) is Standard deviation.

For a normal distribution, the frequency factor can be expressed from (2.6) as shown in

equation (2.8) below:

(2.8)

The application of equation (2.8) is similar to using the standard normal variable (Z) and the

cumulative distribution function for the standard normal distribution for any selected return

period (T) as expressed by equation (2.9) below:,

(2.9)

2.6.1.4 Rainfall Intensity Analysis

In the third step, the expected rainfall intensities for each duration are computed based on a

set of selected periods (for example: 1, 2, 5, 10, 20, 50, 100 years, etc.). This is obtained by

using Expressions (2.7) and (2.9) as indicated by equation (2.10) below:

( ) = ( )

[ ( )

( )] (2.10)

Where XT(D) and iT(D) represent the computed depth and the maximum rainfall intensity

associated with the duration D for the selected return period (T).

14

2.6.1.5 Graphical development of the IDF Curves

Finally, using the summarized computed expected rainfall intensities statistic for each

duration and the selected set of return periods, a set of graph representing the rainfall events

are plotted using Excel program. The set of graph showing the various curves representing

each return period being referred to as the IDF curves is developed for the area of study.

2.7 Theory of fitting Probability distribution to rainfall data

Most of the hydrologic events are considered as stochastic processes or processes controlled

by the law of chance. However, since there is no clearly defined deterministic hydrologic

processes to actually understand and be used for the description of occurrences, the use of

probability theory and frequency analysis are required extensively (Yevjevich, 1972). The

theories are used most often to forecast extreme hydrological occurrences such as rainstorms

and floods. By this also, the study was focused on analyzing the annual maximum rainfall

values which fall into the extreme hydrological value series category according to Chow

(1964).

Extreme hydrological value - series is very important and required most especially for the

design of various hydraulic infrastructures used for flood management. As a result, many

probability distributions have been found to be useful for hydrologic frequency analysis

(Chow, 1964). Considering the importance attached to their functions, the study has

identified few of the probability distribution functions which are most possible to fit the

extreme hydrological value series. Therefore, it is relevant to review and assess statistical

distribution functions suggested by various researchers.

2.7.1 Types of Probability Distribution for Hydrology data analysis

With new ideas about more appropriate distributions functions coming out as a result of the

occurrence of change in climate which have impact also on the hydrologic circle, further

15

research must be conducted on rainfall data to ensure that the most accurate and appropriate

methods widely accepted and available are used to estimate the required parameters for

statistical analysis on extreme rainfall values. As such, this section of the literature review has

mentioned below some of the commonly used and accepted methods of probability

distributions.

2.7.1.1 Normal Distribution

The normal distribution arises from the central limit theorem, which states that if a sequence

of random variables Xi are independently and identically distributed with mean μ and variance

σ2 , then the distribution of the total of n such random variables, Y= ∑

, tends

towards the normal distribution with mean nμ and variance nσ2

as n value increases. The

point is statically factual and cannot be changed no matter what the probability distribution

function is used for X. Random Hydrologic variables such as extreme yearly rainfall depth

computed as the total of several different events or values tend to follow the normal

distribution. Its limits considered during description of random variables are that, the value

changes over a continuous range [∞, ∞] while most of the variables have negative values, and

that it is symmetric about the mean.

2.7.1.2 Log- Normal Distribution

A random variable (X), represented by log(X) is considered to be normally distributed, only if

the value X is log normally distributed. (Chow, 1954) concluded that the distribution function

can be applicable to hydrologic variables formed as the products of other variables if X =

X1, X2, X3…Xn, then Y = log X =∑ logXi = ∑

which tends to the

normal distribution for large n value provided that the Xi are independent and evenly

distributed. This distribution function has been found to describe the distribution of hydraulic

conductivity in a porous medium (Freeze, 1975). Some of the advantages the function has

16

over the normal distribution are that, it is bounded (X ˃ 0) and also that the log transformed

value have a tendency to decrease the nonnegative skew-ness observed in hydrologic sample,

because computing the logarithms decreases a large numerical values proportionally more

than it does to small values as well. The few limits of the distribution include its two

parameters and that it needs logarithms of the sample data to be symmetric about their mean.

2.7.1.3 Gamma Distribution

The interval of time during which a number β of events take place in a Poisson process can be

determined by using the gamma distribution function, which is considered to be the

distribution of a total value of β independent and identical exponentially distributed random

variables. The distribution function is used when defining skewed hydrologic variables

without the application of log transformation to the sample data. The function has been used

to define the distribution of precipitation depth in storms. The distribution involves the

gamma function Г (β), which is given by equation (2.11) for positive integer β below:

Г (β) (β-1)! = (β-1) (β-2) (2.11)

It is generally expressed by equation (2.12) below:

Г(β) =∫

(2.12)

The two- parameter gamma distribution (parameters β and λ) has its lower bound at zero,

which is considered to be its disadvantage for application to hydrologic variables that have a

lower bound greater than zero.

2.7.1.4 Exponential Distribution

Series of hydrologic situations, for example storm rainfall events, may be considered as

Poisson processes in which the occurrences of event are independent and instantaneous at a

constant rate along a line. The interval of time between the occurrences of the event is

determined by the exponential distribution function whose parameter is considered to be the

17

average rate at which the rainfall events take place. The exponential distribution function is

used during analysis to determine the inter-arrival times of occurrence of the random shocks

into the hydrologic systems such as slugs flow of contaminated surface runoff discharging

into water body (stream) as surface flow of rain water cleans the pollutants. The advantage of

the distribution is it simplicity in estimating from observed sample data and that it lends itself

very well to theoretical probability models. The disadvantage is also that it needs every

occurrence to be absolutely independent of its neighbors, which could not be considered as an

appropriate assumption for a study.

2.7.1.5 Pearson Type III Distribution

The Pearson Type III distribution, referred to also as the a 3-parameter gamma distribution,

introduces a third parameter, considered as the lower bound ϵ, so that by the application of

the method of moments, three moments (described as, the sample mean, sample standard

deviation and skew-ness coefficient) derived from the sample can be transformed into three

specific parameters such as, ۸, β, and ϵ of the probability distribution. The distribution is very

flexible, considering that a number of different kinds of shapes as ۸, β, and ϵ vary. The

Pearson system of distributions includes seven types; they are considered solutions for f(x) in

an equation of the form shown by equation (2.13);

( )

( )( )

(2.13)

Where d is the mode of the distribution (the value of x for which f(x) is a maximum) and Co,

C1, and C2 are parameters to be analyzed. When C2 = 0, the solution of equation (2.10) is a

Pearson Type III distribution. For C1 = C2 = 0, a normal distribution is the solution of

equation (2.10). Thus, the normal distribution is a special case of the Pearson Type III

distribution, describes a non-skewed random variable.

18

In 1924 Foster adopted the Pearson Type III distribution in hydrology to define the

probability distribution of a yearly maximum flood peaks. It becomes limited when the value

of the sample data being used is positively skewed.

2.7.1.6 Log- Pearson Type III Distribution

If log X follows a Pearson Type III distribution, then X is said to follow a log-Pearson Type

III distribution. It is the acceptable probability distribution function that is mostly used to

conduct frequency analysis of yearly maximum floods in the United States (Benson, 1968).

Considering special case for example, when Log (X) is symmetric about its mean, the LP3

distribution function reduces to the Log-normal distribution.

The location of the bound ϵ in the LP3 distribution relies on the skew-ness of the data. If the

data are positively skewed, then log (X) ≥ ϵ and ϵ is a lower bound, while if the data is

negatively skewed, log X ≤ ϵ and ϵ is an upper bound.

When the log transformation is applied to values of sample data, it decreases the skew-ness of

the transformed data and may result into the provision of a new set of transformed data which

are skewed negatively from the original data which are also positively skewed. In that case,

the application of the log-Pearson Type III distribution would impose an artificial upper

bound on the data. The log-Pearson Type III distribution was established as a means of fitting

a curve to data. The purpose for its establishment has been justified by the fact that it yields

better results during many applications in past analyses conducted, particularly on analysis

using flood peak data.

The Log- Pearson Type III distribution is complicated, as it has two interacting shape

parameters (Stedinger and Griffts, 2007). Like the General Extreme Value distribution, the

LP3 uses 3 parameters: They are Location (µ), Scale (σ) and shape (g). A problem which

19

develops with LP3 is its tendency to give low upper bounds of the precipitation values, which

is not desirable (Cunnane, 1989).

Also, since 1967 the U.S Water Resource Council recommended and required the use of LP3

distributions for all hydrological data analysis in the U.S. This recommendation was

questioned recently by several papers in the U.S that have conducted series of studies on par

with other researchers, it was established that the GEV distribution is appropriate and an

acceptable distribution function, and often preferred over LP3 (Vogel, 1993).

2.7.1.7 Extreme Value Distribution of Gumbel

Extreme values are sets of observed maximum values of historic hydrologic data. A typical

example, in hydrology the yearly maximum rainfall depths recorded and reported for a given

station is considered to be the highest value for a record year. As such, the yearly maximum

rainfall depths extracted from each record per year for each duration make up a set of extreme

value for historical record and these values are used during statistical data analysis.

Distributions of the extreme values chosen from sets of samples of any probability

distribution has indicated convergent to one of three types of extreme value distributions,

described as Type I, Type II and Type III when the selected extreme values become larger.

The three limiting forms are special cases of a single distribution called the Generalized

Extreme Value (GEV). Its Cumulative probability Function is expressed in equation (2.14)

below as defined by (Hosking. 1997) when designing for extreme events:

( ) [ (

)

] (2.14)

where k, u and α represent parameters to be calculated.

20

The cases are:

when k = 0, CDF and PDF for Extreme Value I (EV1) is expressed in equations

(2.15) and (2.16) below as defined by (Hosking, 1997) when designing for extreme

events :

( ) [ (

)] (2.15)

( )

[

(

)] (2.16)

when k ˂ 0, the distribution for Extreme Value II (EV 2), equation (2.11) is

applicable for (u + α/k) ≤ x ≤ ∞, and

when k ˃ 0, the distribution for Extreme Value III (EV 3), for which expression

(2.11) is also applicable for -∞ ≤ x ≤ (u+α/k)

In the cases mentioned above, α is assumed to be positive. For the EV I distribution x is

unbounded, while for EV 2, x is bounded from below by (u+α/k), and in the case of EV3

distribution, x is similarly bounded from above. The Extreme value type I and Extreme value

type II distributions are considered as the Gumbel and Frechet distributions respectively. If a

variable x is described by the EV3 distribution, then –x is said to have a Weibull distribution.

Extreme value distributions have been widely used for hydrological data analysis. They are

considered as the standardized methods for analyzing flood frequency in Great Britain

(NERC, 1975). The Gumbel (EV1) distribution has been used in Europe to model flood flow

function and has been applied by the National Weather Service in analyzing precipitation

across the United States. Rainfall of various depths are also and mostly modelled by the

Extreme Value Type I (Chow, 1953) and the Weibull (Extreme Value Type III) distribution

models drought flows. Gumbel distribution is the most widely used distribution for IDF

analysis owing to its suitability for modelling maxima.

21

In Canada currently, the EVI is the most appropriate distribution function being used for the

analysis of precipitation along with the method of moments established by Environment

Canada (EC). Recent U.S research have been conducted to find out the usefulness of General

extreme value distribution considering the Canadian circumstances. The study conducted

from Saskatchewan (Nazemi et al., 2011) for the city of Saskatoon, found that the General

Extreme Value model is appropriate, however further studies may be required to establish the

proper use of a particular parameter referred to as the shape parameter because its value

affects the output greatly.

Consequently, having reviewed and described the various hydrological distributions functions

being used for frequency analysis of hydrologic data, the study considered the Gumbel EVI

and the Log-Pearson Type III distribution functions as the distributions that could best fit the

sample data used for the study based on the recommendations made by past researchers for

being widely used and effective for frequency analysis of hydrological data.

However, the two recommended distributions will undergo Easy fit tests prior to the final

selection of the best fitted distribution between the two. The best fit distribution selected from

the Easy fit tests will further be verified and validated for being the appropriate and best fit

function required for parameters estimation leading to frequency analysis by using the

Kolmogorov-Smirnov and Chi-Square Tests.

2.7.2 Fitting a probability distribution to Rainfall data

Probability distribution is once more defined as a function representing the probability of

occurrence of random variables. As such, when a probability distribution function is fitted to

a set of rainfall data which is considered to be a random variable, it reveals the required

distribution of the sample data. Regarding this, a great deal of statistical information related

to the sample can be summarized in the distribution function and it is related parameters.

22

Therefore, fitting the distribution function to the data can be obtained by the following

methods for the estimation of the associated parameters:

Method of moment (MOM)

Method of Maximum Likelihood (MML)

Method of L- Moment

2.7.3 Parameter Estimation

When high stream flows, low flows or random variables of extreme rainfall depths are

defined by some distribution function, it becomes the duty of the engineer in charge to

compute the required parameters of the assumed distribution so that the data needed for other

analyses can be calculated using the best fit distribution function. For instance, the normal

distribution has two parameters, µ (Mean) and σ2 (Variance). Making selection of the best

distribution functions is mostly depended on testing the sample by using probability plots and

moment ratios, the physical origins of the sample and past experience. Many types of general

methods for computing the needed parameters for a distribution are available. A simple

applicable method is the method of moments which uses the available sample to compute an

estimate so that the theoretical moments of the distribution of X is exactly equal to the

corresponding sample moments. However, the following sections will briefly discuss the

various methods of statistical parameters estimation:

2.7.3.1 Methods of Moment (MOM)

It is difficult to trace back who introduced the Method of Moments, hence research showed

that Johan Bernoulli (1667- 1748) was among the first researchers who used the method in

his work. He mentioned that with the MOM, the moments of a distribution function in terms

of its parameters are set equal to the moments of the observed sample. He reported that

analytical expressions can be derived quite easily, but the estimators can be biased and not

23

effective. He further mentioned that the moment estimators however, can be very well used

as a starting estimation in an iteration process. The central moments of distribution are

expressed by equation (2.17) shown below:

( ) ∫( ) (2.17)

The sample mean is a natural estimator for the population mean (µ). The higher sample

moments are reasonable estimator but they are not unbiased. Unbiased estimator are often

used.

However, finding theoretical moments is not easy for all probability distributions. When

Karl Pearson developed the method of moments in 1902, he took into consideration that a

better parameters estimate for probability distribution functions are those for which the

moments of the probability density function measured about the origin are equivalent to the

corresponding moment of the sample data (Chow et. al., 1988). Being one of the oldest and

the most useful methods of parameter estimation, the method of moments uses relations

between the central moments and parameters of the distribution (Aksoy, 2000). The method

of moments is a straight forward statistical technique that is mostly used for parameter

estimation. Due to its ease of use and its widespread acceptance, the method of moments is

considered to be a sound choice for use in IDF analysis.

2.7.3.2 Method of Maximum Likelihood

Also with the MML, it is difficult to say who discovered the method, although Daniel

Bernoulli (1700- 1782) was one of the first researchers who reported it (Kendall, 1961). The

Maximum Likelihood method provides the relative likelihood of the observations, as a

function of the parameters θ expressed by equation (2.18) below:

( ) = ( ) (2.18)

24

With this method one chooses that value of θ for which the likelihood function is maximized.

The researcher reported that the ML- method gives asymptotically unbiased parameter

estimations and of all the unbiased estimators it has the smallest mean square error. The

variances approach asymptotically as shown by equation (2.19):

( )=E( ( )

) (2.19)

Furthermore, these estimators are invariant, consistent and sufficient. For more details

description of the estimator, refer to Hald (1952). Analytical equations for the parameter

estimators are sometimes difficult to derive. In those cases, numerical optimization routine

have to be used to determine the maximum of the likelihood function, which can also be quite

difficult since the optimum of the likelihood function can be extremely flat for large sample

sizes. Optimization of the likelihood function may also be hampered by the presence of local

maxima. Moreover:

MML is (usually) straightforward to implement,

Maximum Likelihood estimators (MLEs) may not exist, and when they do, they may

not be hampered or give a bias error (Koch, 1991).

MLE may give inadmissible results (Lundgren, 1988)

The likelihood function can be used for much more than just finding MLE

ML is adaptable to more complex situations, because the MLE satisfies a very

convenient invariance property.

The MML is extremely useful since it is often quite straightforward to evaluate from the

MLE and the observed information. Nonetheless it is an approximation, and should only be

trusted for large values of n (though the quality of the approximation will vary from model to

model).

25

If the size of the sample is large, then there seems to be a little bit of doubt about the

maximum Likelihood Estimator being a good choice. It should be emphasized, however, that

the properties above are asymptotic (large n), and better estimator may be available when

sample size are small. R. A. Fisher who derived the estimator in 1922 reasoned that the best

value of a parameter of a probability distribution should be that value which maximizes the

likelihood or joint probability of occurrence of the observed sample (Chow et. al., 1988). The

method is the most theoretically suitable method used for fitting probability distributions to

data in the sense that it provides from its application the most efficient parameter estimates.

2.7.3.3 Method of L-Moments

Hosking (1990) introduced the L-Moments. They have become the popular tool for solving

various problems related to parameters estimation and probability distribution function

identification. It can be shown that L-Moments are linear function of probability weighted

moments (PWMs) and hence for certain applications, such as the calculation of distribution

parameters, it serves the identical purposes (Hosking, 1986). In other events, however, L-

Moments have significant advantages over the PWMs, notably their ability to summarize a

statistical distribution in a more meaningful way.

Since L-Moment estimators are linear functions of the ordered data values, they are virtually

unbiased and have relatively small sampling variance. L-Moment ratio estimator also have

small bias and variance, especially during comparison with the classical coefficients of

skewness and kurtosis. Moreover, estimators of L-moments are relatively sensitive to

outliers. The application of L-Moments for the estimation of parameters are primarily done

on the basis of linear combinations of data that have been properly organized in ascending

order (Millington et. al., 2011). The simplicity together with the robustness of this method

against outliers is the reason for its common use (Hosking, 1992).

26

Generally, the method of moments is simple and suitable to apply during practical hydrology

analysis than the method of Maximum likelihood analysis (Chow et. al., 1988).

Consequently, the study employed the method of moments (MOM) during the parameter

estimation the for selected, verified and confirmed distribution function the Gumbel EVI.

2.7.4 Statistical Parameters

The various Probability distribution functions are usually characterized by their respective

expected properties or parameters during application. These properties, adequately describe

and summarize a set of data. Moments are very useful tools in the describing hydrologic

parameters such as mean, standard deviation, Variance (σ2) etc. which are considered

members of the family of moments. These parameters are very important for describing a set

of observations on a random variable, such as extreme rainfall depth mostly consider during

analysis. A moment can be referenced to any point on the measurement axis; however, the

origin and the mean are the two most common reference points. Even though most data

computation may require only two moments in some statistical studies, it is important to take

note of the following three moments for they are mostly used:

The mean (µ), is the first moment of values measured about the origin.

The variance (σ2), is the second moment of values measured about the mean.

The skew (g), is the third moment of values measured about the mean.

2.7.4.1 Mean (Average)

In hydrologic statistical analysis the mean which is also considered as the first moment that is

measured from a point known as the origin along the horizontal axis. It is also described as

the average value of all observed random variables. More importantly and in most cases, the

mean of a population is denoted by µ, while the mean of a sample is indicated by . For a

continuous random variable, it is computed as expressed by equation (2.20):

27

( or μ) = ∫ ( )

(2.20)

Even though the mean represents a parameter of neither a population nor sample, it still does

not absolutely describe the characteristics of a random variable.

2.7.4.2 Variance

The variance is considered the second moment which is measured from the position of the

mean. These symbols, S2

and σ2

are used to denote the variances of the sample and population

respectively. The units of measurement of the variance are also taken as the square of the

units of the random variable. For a continuous random variable, the variance is expressed as

shown by equation (2.21):

(S2

or σ2) = ∫ ( )

2 ( ) (2.21)

During statistical analysis variance is considered as a very important and useful parameter of

a sample because it is needed by most statistical methods to determine some level of

measurement from the actual value. Generally, variance indicates how closed the values of

a population or sample is to the overall average or the mean. If the observed values of a

sample is equaled to the mean, the variance of the sample would be equaled to zero. Even

though the variance is used in other aspects of hydrologic statistical analysis, its use as a

descriptor is limited because of its units; specifically, the units of the variance are not the

same as those of either the random variable or the mean.

2.7.4.3 Skewness

The skew is the third moment measured about the mean. Mathematically, the skew is

expressed as indicated by equation (2.22) below for a continuous random variable:

(g or 𝛾) = ∫ ( )

3 ( )

(2.22)

28

where g is the sample skew and 𝛾 is the skew of the population.

Skew has units of the cube of the random variable. It is a measure of symmetry. A symmetric

distribution will have a skew of zero, while a non-symmetric distribution will have a positive

or negative skew depending on the location of the tail of the distribution. If the more extreme

tail of the distribution is to the right, the skew is positive; the skew is negative when the more

extreme tail is to the left of the mean (McCuen, 1941a).

2.8 Goodness of Fit Tests

This set of Tests referred to as Goodness of fit tests, is applied in hydrological statistic during

frequency analysis to help in determining the most suitable probability distribution function

that best fit the sample data. The tests are not conducted to select the best distribution, they

are used to verify and confirm the appropriate distribution that fits the sample. These tests

when used, the test-statistics of the sample are computed and used to verify and confirm how

well the sample fits the assumed distribution. The test results also help to distinguish the

differences in values between the observed data, and the expected values from the

distribution been tested. The goodness-of-fit tests is one of the appropriate means of

determining how well a sample data agrees with an assumed probability distribution as its

population. Goodness-of-fit tests contain graphical component and statistical methods, with

statistical methods preferred over graphical methods because of objectivity. Well-known

statistical goodness-of-fit tests reviewed by the study are discussed in the following sections:

2.8.1 Anderson Darling Test

The Anderson-Darling test is one of the components of a goodness-of-fit tests which its test

statistic is referred to as empirical distribution statistics because they are used to determine

the difference between the empirical distribution function of an observed sample and the

theoretical or assumed distribution to be examined. It is used mostly to compare an observed

29

CDF to an expected CDF. This approach adds more weight to the tail of the distribution than

Kolmogorov-Smirnov test, which has made the AD test to be stronger, and having more

weight than the KS test. Depending on the result of the test statistic obtained, the null

hypothesis is rejected if the value of statistic obtained is greater than a critical value selected

from a defined significance level (α). The value of the significance level mostly used is

α=0.05. This number is then compared with the test distributions statistic to determine if it

can be accepted or rejected. Anderson-Darling Test was developed to test the random

variable, X has a continuous cumulative distribution function, Fx(x, θ) where θ represents the

vector of one or more parameters entering into the probability distribution. However, for a

normal distribution, the vector θ = (µ, σ2). The empirical distribution function (EDF) is

defined as Fn (x) = proportion of sample < x.

The computation of Anderson-Darling test statistic is done by the following steps expressed

by equations (2.23) and (2.24) below: Calculate

Zi = F( X(i),ᶿ) where i = 1,…, n (2.23)

Then

A2 = - {∑ ( ) [

( )

]

(2.24)

where X(i) and Zi are in ascending order.

For the expression above the tested distribution, F(x, ᶿ) must be completely specified, that is,

the parameters in ᶿ must be known. When this is the case, the situation is considered as case

O. The statistic A2 was derived by Anderson and Darling and for Case O, they gave the

asymptotic distribution and a table of percentage points. Large values of A2

will indicate a

bad fit. The distribution of A2

for a finite sample rapidly approaches the asymptotic

distribution and for practical purposes, this distribution can be used for sample sizes greater

than 5.

30

The percentage points are provided in statistical tables. In order to calculate the value for A2

during the goodness–of–fit test, equations (2.23) and (2.24) are used and the results marched

with the percentage points provided in the tables.

The null hypothesis which states that the random variable X has the distribution F(x, ᶿ) is

rejected at the level α (significance level) if A2

exceeds the allowable percentage point at this

level.

2.8.2 Kolmogorov-Smirnov (KS) Test

The Kolmogorov–Smirnov (KS) Goodness of Fit test is applied mostly to verify and confirm

the population distribution and can be best utilized on much smaller samples than the chi-

square test. It is considered a non-parameter test, because it does not require a specific

population or distribution from which the observed sample data should come from as a

precondition. Its application is based on concept. Moreover, the test requires sample on at

least an ordinal scale, but it is also used for comparisons with continuous distributions. The

Kolmogorov–Smirnov goodness of fit test is a very simple test to perform by following the

steps below:

Formulate the null and alternative hypotheses in terms of the proposed PDF and its

parameters.

Let Dn, the value of the test statistic be considered as the maximum absolute

difference between the cumulative function of the sample and the cumulative function

of the probability function specified in the null hypothesis.

Select the level of significance (mostly 0.05 and 0.01 are considered).

Obtain a set of random sample from each duration starting with the first data under

0.20hr and derive the cumulative probability function for the sample data set. Next,

31

compute the cumulative probability function also for the assumed population and the

value of the test statistic Dn in the last column.

Obtain the critical value, Dα, from the established statistical table for the KS test. The

value of Dα is based on the values of α and the sample size, n.

Compare the higher value obtained for Dn from the test statistic with the Da (that is

the standard critical value). If the calculated value obtained for Dn, is more than the

standard critical value, Dα, then the null hypothesis should not be accepted but

rejected. The described steps for the test is repeated for all sample data under the

various durations.

During the application of the KS test, it is better to use as many as possible cells created on

the excel sheet. Increasing the number of cells also increases the likelihood of determining a

better result if the null hypothesis is, in fact, incorrect. This helps to minimize the chance of

making a type I error.

2.8.3 Chi-square (CS) test

The chi-square test is used to determine the difference in the test statistic between the

assumed distribution suggested by sample and a selected probability distribution. It is

considered as one of the popularly known and widely utilized one-sample analysis for

examining a population distribution. The test can also be used to verify and confirm an

assumed population distribution of a sample during frequency analysis. It should be noted

herein that the CS test is not a high power statistical test and is not very useful (Cunnane,

1989). In summary, the test only provides the means for comparing the observed frequency

distribution of a random variable with a population distribution based on a theoretical PDF.

The steps to make the chi-square test are as follows:

32

Put the observed data (O) and expected (E) values into intervals so as to determine the

frequency of both variables in each class. This can be well expressed by a histogram

of frequencies.

Rearrange the classification so that the minimum expected frequency in each class

becomes 5 or great. The classes with low frequency should be merged to this end.

Calculate the chi-square value for all intervals by the relation expressed by equation

(2.26) below:

( ) = ∑( )

(2.25)

In the equation above, v is the degree of freedom (df) and equals n-k-1, where n is the

number of intervals and k is the number of distribution parameters obtained from the sample

statistics (Sample mean and standard deviation).

Compare the value obtained to the chi-square statistics under x20.050 from the provided

table. The null hypothesis will be accepted if x2 < x

20.050 and rejected if otherwise.

The effectiveness of the test is reduced when the expected frequency in any of the cell is less

than 5. When this condition is experienced, both the expected and observed frequencies of the

appropriate cell should be joined with the values of an adjacent cell and the value of k should

be decreased to represent the number of cells actually used in the calculation of the test

statistic.

2.9 Statistical Test of Hypotheses

A statistical test is a tool which provide a baseline condition for making a quantitative

statistical decision about a selected probability distribution function in a systematic way. The

aim is to find out whether there exist enough proof to reject a conjecture or hypothesis about

a process. The conjecture is called the null hypothesis. Based on the ideas of probability and

statistical theory, the statistical test serves as an indicator of a method of involving the idea of

33

risk into the evaluation of another decisions. Field recorded Data should stand for the samples

of data, and test statistics calculated from the process using the sample data are considered

the estimators. However, during the period of making final decisions on the results, the true

population which is not known should be used. Again during the application of the empirical

method for decision making, the analyst of the data is only concerned about drawing

conclusion from a sample data the true statements about the population which each value that

is a component of the main sample were obtained. Since the population from which the

sample data was drawn is unknown, it is important to utilize the sample data to assist in

identifying the possible population. Moreover, in statistical testing, not rejecting may be a

good result if we want to continue to act as if we believe the null hypothesis is true. Or it may

be a disappointing result, probably if we not yet have enough data to prove something by

rejecting the null hypothesis. The selected population is then used to make as a basis for

computing the required parameters and other values during frequency analysis which leads to

predictions. Thus, hypothesis tests statistics together with statistical theory and information

obtained from the sample help to assess, determine and confirm the true population.

2.9.1 Procedures for Testing Hypothesis

Information about the theoretical sampling distribution of a test statistic based on the desire

statistic can be utilized to examine a formulated hypothesis. Hypothesis test is conducted to

help in determining if the formulated statement about the hypothesis is true. Statistics for

almost all hypothesis Tests have been provided for use. The steps detailed below were used to

conduct a hypothesis statistical analysis:

Formulate the statement for the null and alternative hypotheses before testing.

Choose the best and required statistical Model that will identify the test statistic and

its distribution function.

34

From the available statistic table select the significance level which determines a

measures of risks or uncertainty.

Use the provided extreme rainfall sample data and calculate the values of the test

statistic.

Get the critical value of the test statistic, to determine the zone of rejection and

acceptance.

Match the calculated values of the test statistic obtained from the computation with

the available critical value and finally conclude on the best decision to be made by

choosing one of the hypotheses mentioned in the first step.

2.9.1.1 Stating the two Hypotheses

The approach to first start with is by formulating a Statement for the null and alternative

hypotheses before starting the test. The alternative hypothesis indicates what the researcher is

determining to establish. The null hypothesis stands also for the opposite of what the

researcher is determined to establish. As such, if the aim of the researcher is to draw a

conclusion regarding a population, the hypothesis will be statements expressing that a

random variable belongs to or does not belong to a specified distribution with a defined

values of parameters of the population. Again, if the aim is to match more than two specified

parameters, like the mean of two independent samples, the hypotheses will be a formulated

statements which will show the presence or absence of differences between the two means. It

should be understood herein that the two hypotheses are comprised of statements which

contain the population distribution. Therefore, hypotheses should not only be stated in term

of statistics of the sample. Consequently, the null and alternative hypotheses should mutually

represent exclusive decisions. Finally, whenever the result obtained from test conducted on

the obtained data indicates that the null hypothesis should be accepted, the alternative is

considered to be incorrect.

35

2.9.1.2 Test Statistic and the sample’s Distribution function

The null and alternative hypotheses mentioned in section 2.9.1.1 permit equality or a

difference in result to be shown between defined populations. In order to assess the

hypotheses, it is important to determine or establish the test statistic that shows the variance

or disparity indicated by the alternative hypothesis. The specified test values is a set of

numerical values already established for commonly defined statistical theory. The test

statistic value of the sample will not remain constant, hence, its value changes as difference

data are obtained from one point to another because of variation observed in sampling. As

such, the test statistic is considered as a random variable and which has a definite sampling

distribution. Theoretical model should be considered as the baseline from which hypothesis

test should be referenced when defining the sampling distribution of the test statistic and its

parameters. However in the absence of the appropriate theoretical models, approximated

values are usually developed for use. Consequently, in order to conduct a hypothesis test, it is

necessary to identify the model that will be used to specify the test statistic, distribution

function and parameters under any condition.

2.9.1.3 State the level of significance

To state the level of significance, there are two types of errors associated with hypotheses that

one needs to know. They are as described below:

Type I error: which is rejecting Ho, when, in fact Ho is true.

Type II error: which is accepting Ho when factually, Ho is not true.

The two decisions mentioned above are incorrect and also not independent. For a provided

size of data, the value of one type of error adds up as the value of the other type of error

reduces. While the two kinds of error are very significant, the decision making process most

chooses one type, especially the type I error. The significance level which is mostly

36

considered as a basis for decision during hypothesis testing, stands for the probability of

creating Type I error and is indicated by the alpha (α) while the probability of a type II error

is indicated also by beta (β). However, choosing the significance level should be carried out

based on a critical analysis and assessment of the sample data being studied. During critical

statistical analysis of the test statistics, the selected value for alpha (α) is usually read from a

standardized table which is conventional. Mostly, 0.05 and 0.01 are frequently chosen from

the standard table. Since alpha and beta (α and β) are not self-govern, it is important to

consider the suggestion of both errors during the selection process of the level of

significance.

2.9.1.4 Statistical Data Analysis

At this stage, the provided extreme rainfall sample data are used to compute the values of the

test result. The sample data are often utilized to calculate the required parameters which help

in the process of determining the appropriate distribution function the sample is drawn from.

2.9.1.5 Zone of Acceptance and Rejection

The rejection zone comprises numerical values of the defined test statistic that may

doubtfully take place when the null hypothesis is true. It is located in one or both tails of the

distribution. The position of the rejection zone relies on the formulated statement of the null

hypothesis. Meanwhile, the acceptance zone contains values of the test statistic that may

occur if the null hypothesis is true. The critical value of the test statistic is used to describe

the separation between the two zones. The critical value relies on the following conditions

during decision making:

The formulated statement of the null hypothesis,

The selected distribution of the test statistic,

The chosen significance level and

37

The Characters or nature of the sample.

Finally, decision making should always be determined from the nature of the problem to be

tested and not based on statistical values.

2.9.1.6 Make decision based on comparison

A decision to accept the null hypothesis should be based on a making comparison of the

calculated test results with the defined critical value. The null hypothesis is rejected in favor

of the alternative hypothesis if only the result is found in the rejection zone. Let it be

considered that the rejection of the null hypothesis indicates the acceptance of the alternative

hypothesis HA.

2.9.2 Conclusion of the Literature Review

Having gone through the various steps and processes of the literature review phase of the

study, it can be concluded for the benefit of the research which leads to the development of

the IDF curves for the study area that during the exercise, the Gumbel Extreme Value Type I

(EV I) and Log – Pearson Type III (LP3) among many distribution functions were

highlighted and recommended as the best distribution functions for IDF analysis and

development by past researchers. However, it was established that the recommended

distribution function currently being used for Rainfall analysis by Environmental Canada is

the Gumbel Extreme Value Type I (EVI) Distribution Function coupled with Method of

Moment (MOM) for parameters estimation. Meanwhile, the Log–Pearson Type III (LP3)

Distribution function was also recommended purposely for used in the United States. Even

though past researchers have recommended the use of the probability distribution functions,

the study did assess, verify and validate the fitness of each to the sample data for final

selection of the one needed for the estimation of parameters required for the frequency

analysis. The Easy fit Software was used to initiate the selection process while the

38

Kolmogorov-Smirnov and Chi-Square Good-ness of fit tests were carried out to finally verify

and confirm the fitness of the selected probability distribution function.

39

CHAPTER 3: RESEARCH METHODOLOGY

3.1 The Study Area

Accra is the capital city of the Republic of Ghana. It is located within the Greater Accra

Region of Ghana. Besides being the political capital, it also serves as the administrative

capital of the metropolitan assembly. The metro shares common boundaries with La-dade

Kotokpon municipal from the east and Ga west municipal, Ga central municipal and Ga south

municipal assembly from the west. The Accra metropolis administrative area has a total land

space of 200 km2. The figure 3.1 shows the general map of Ghana on the left with an

extracted map of Greater Accra on the right within which Accra is located. (Ghana

Districts.com, 2006a)

Figure 3.1 Maps of Figure Ghana and Greater Accra

40

3.1.1 Climate

The Accra Metropolitan Assembly is situated within the Savannah zone.

Accra experiences two rainy seasons annually with an average rainfall depth of

approximately of 730mm, which primarily falls during the two rainy seasons. The first rain

season starts in May of each year while the next season commences in the middle of July and

ends in October every year. Regarding temperature, there usually occurs a very little variation

throughout the year. The monthly average temperature fluctuates between 24.7o C in August

to 28.0o C in March with an annual average of 26.8

o C. The relative humidity is generally

high varying from 65% in the mid-afternoon to 95% at night. (Ghana Districts.com, 2006b)

3.1.2 Vegetation

The Metropolitan has three major vegetation zones which include shrub land, grassland and

coastal lands. The shrub land is mostly found in the north towards the Aburi Hill and the

western outskirts of Accra. (Ghana Districts.com, 2006c).

3.1.3 Topography and Drainage

The Accra Metropolitan drainage catchment area extends from the eastern boundary of the

Nyanyamu catchment on the west of greater Accra regional boundary to Laboi east of Tema.

Densu River Catchment and Sakumo Lagoon

This is the largest of all the four coastal basins within the study area. The total drainage area

is about 2500km2. It is divided into two sections above and below the Weija dam. (Ghana

Districts.com, 2006d)

The northern section of the basin, which extends inland along the the Densu River and its

tributaries 100 km, is hilly with the highest point reaching 230m above sea level.

The southern section of the basin is low lying land comprising the Sakumo lagoon and

Pandros salt pans. The rest of the catchment are:

41

Korlie-Chemu Catchment covers about 250 Km2

Kpeshie Catchment covers about 110 Km2

Songo-Mokwe Catchment covers about 50 Km2

3.1.4 Geology and Soil

The geological formation of Accra comprises Dehomeyan, Precambrian, Dehomeyan,

Schists, including many others while the Togo series is comprised of mainly quartzite,

phillites, phylitones and quartz breccias. Meanwhile, there exists other formations apart from

the few mentioned in the study area.

The four major groups of soils described below are found in the study area:

Drift materials resulting from deposits by wind-blown erosion;

Alluvial and marine motted clays of comparatively recent origin derived from

underlying shale;

Residual clays and gravel derived from weathered quartzites, gneiss and schist rocks,

and

Lateritic sandy clay soils derived from weathered Accraian sandstone bedrock

formation

Pockets of alluvial black cotton soils are mainly found in several low lying poorly drained

areas. (Ghana Districts.com, 2006e)

3.2 Research Methodology

The research methodology mainly include rainfall data collection; rainfall data processing;

selection of probability distribution; frequency analysis and development of the IDF curves.

The methods and procedures used in the study are discussed in the steps below.

42

3.2.1 Procedure

The procedure used to conduct the study is briefly outlined in Figure 3.2:

Figure 3.2. Flow chart showing procedures for methodology

3.2.2 Rainfall Data Collection

Extreme annual rainfall data was collected from the Ghana Meteorological Agency (GMA).

The rainfall depths were recorded in nine (9) durations of time series in minutes and hours as

12min, 24min, 42min, 1 hour, 2 hours, 3 hours, 6 hours, 12 hours and 24 hours. The time

series data covered the period of twenty eight (28) years beginning from January of 1971 to

December of 2009. The data was obtained for the IDF curves development. Appendix 1.0

Fitting of Probability

Distribution to data Testing the Fitness

of Distribution

Function

Frequency Analysis

Filling of

Gaps Recording of

AMS

Rainfall data collection

Correlation Coefficient and

Regression Analysis

Development of New IDF Curves

Comparison of New intensities with the

Existing

Rainfall Data Processing

Selection of Probability Distribution

Function (Easy – Fit Test)

Determination of

Rainfall Intensities

43

contains the entire raw rainfall data collected. The data were observed to have (299) missing

values.

3.2.3 Rainfall Data Processing

In order to have a complete and accurate rainfall record for the period January 1971 to

December 2009 free of gaps, a simple regression analysis using excel was conducted to fill

the 299 gaps observed. During the process, the strength of the relationship determined

between rainfall values of the neighboring durations was assessed and results from the

process recorded. This was followed by using linear equation derived from the regression

analysis to fill in the missing data. The computation of the missing values for variable X and

Y to fill the blank cells in the missing data sheet was done in the Excel spread sheet. This was

applied to all the nine (9) durations. The process started with 0.20 hr and 0.40 hr durations

and continued for all paired columns. Table 4.1 contains details of analyzed correlation

coefficients representing strength of the relationship between values of paired durations in red

color. The paired neighboring durations were arranged systematically in groups of (0.20 Vs

0.40), (0.40 Vs 0.70), (0.70 Vs 1.0), (1.0 Vs 2.0), (2.0 Vs 3.0), (3.0 Vs 6.0), (6.0 Vs 12.0),

and (12.0 Vs 24.0). Finally, the maximum annual extreme rainfall values for each year under

the respective durations were orderly recorded and arranged for another level of data analysis

after completing the rainfall gaps filling exercise. Appendixes 1 and (2) contain details of raw

data with gaps and filled gaps as it was obtained from GMA.

3.2.4 Selection of appropriate Distribution Functions for the sample Data

Past researchers have recommended the use of two probability distribution functions

(Gumbel–EVI and Log-Pearson Type III). The function which best fits the extreme rainfall

data was selected for this research on the basis of application of the Easy–Fit- Software

which displayed results to aid in the selection process. As discussed earlier, the Easy Fit is a

statistical tool which helps in the quick selection process of the appropriate probability

44

Distribution function which best fit the hydrologic data. Its application also helps to reduce

the manual data processing time. The Easy Fit software makes use of the various tests which

include Kolmogorov-Smirnov, Anderson Darling, and Chi–Squared Tests as its components

to help compare the tests statistics for each distribution function (EV1 and LP3) for the

identification and selection of the most appropriate distribution. The following steps were

used during the computer aided testing processes:

The recorded maximum rainfall values for each duration was loaded into the Easy –

Fit software. The Gumbel (EVI) and Log – Pearson Type 3 distributions were selected

among the many functions to help determine the best fit function for the rainfall data.

The maximum rainfall values recorded under duration 12 minutes (0.20 hr) in

Appendix 3 for the 28 years were the first to be applied and the process was repeated

for the rest of the durations. See Table 4.2 for typical result obtained for 0.20 hr

duration. The test statistic obtained from the exercise for the various durations were

summarized in Tables 4.3.

The detail of ranks gathered from the respective Easy Fit Tests (Kolmogorov

Smirnov, Anderson Darling and Chi –Squared Tests) were recorded under each

Probability Distribution Function. After processing the extreme rainfall data for each

duration by using the Easy Fit method, the test results were assessed and compiled for

comparison and final selection of the best fit distribution function. That is, the results

for EV1 and LP3. Table 4.3 contains result for the distribution functions that scored

the rank of 1(one) and the rank of 2 (two) during the Easy Fit tests on the maximum

annual rainfall data set for Accra.

3.2.5 Fitting the selected Gumbel Distribution to the Sample Data

After the Gumbel Distribution was selected from the tests as the appropriate distribution

function for the performance of a frequency analysis, it was fitted to the rainfall data to

45

estimate the various parameters of the Gumbel distribution required from the sample data.

Thereafter, a Kolmogorov – Smirnov and Chi- Square Goodness of fit tests were separately

conducted using AMS to assess the validity of the fitness per duration. Additionally, fitting a

hydrologic distribution function to an extreme rainfall data is to truly ascertain if the rainfall

data (sample data) was truly drawn from population with a specified distribution function like

the Gumbel. Moreover, fitting was conducted in order to also prove the hypothesis that “the

extreme rainfall events are drawn from a specific distribution” (Gumbel distribution).

The steps below described the processes used for estimating the required parameters leading

to the performance of frequency analysis and fitting of the Gumbel distribution to the data:

The maximum annual rainfall depths recorded were ranked under each duration from

the highest value to the lowest (that is, in a descending order) using the excel program.

The rank (m) was assigned to each rainfall value in the column, starting with the

highest value having the numerical rank of 1 and the lowest value (last rainfall value) a

rank (m = 28). See Table 4.4 for 0.20 hr duration result.

The probability of exceedence (P) was computed for each row. The Gringorten plotting

position formula was considered for the computation of exceedance probability (P) on

the basis of average frequency value analyzed among four selected frequency formulas

representing Hazen William, Blom, Gringorten and Cunname.

The Weibull plotting position formula could not be used because it gave an exact

and low value when applied as compared with the other formulas. Additionally, the

Gringorten formula was considered for the analysis of the probability of exceedence

(P) for this study on the basis of recommendation made by past researchers. Chow et

al. (1988) stated that for data distributed according to the Gumbel distribution, the

Gringorten formula is the best. The expression (3.1) below which represents the

46

Gringorten formula was used to compute the probability of exceedence (P) as

expressed by equation (3.1) below:

=

(3.1)

Where (m) represents the rank and (n) is the total sample size (28) for this study, P, is the

probability of exceedence also expressed as below in (3.2):

P = P(X ≥ x) (3.2)

Also note that P = 1 – F(x) and expressed in (3.3) below:

F(X) = P(X < x) (3.3)

The reduced variable (u) was computed from the result of the Probability of

Exceedence (p) value as expressed by equation (3.4) below:

( ( )) (3.4)

Using excel program, the values for the sample mean and standard deviation

represented by (µs) and (σs), were computed respectively.

The position and scale parameters represented by (Xo) and (S), were computed

respectively using the expressions (3.5) and (3.6) below:

(3.5)

(3.6)

Where µN is the mean of the reduced variable and σN standard deviation of reduced variable.

Gumbel mean (µG) and standard deviation (σG) were computed using the expressions

(3.7) and (3.8) below :

(3.7)

(3.8)

47

Note: The parameters described in the equations above were applied to each rank and

duration respectively to compute the required statistic, and Table 4.5 shows the detail.

The Gumbel variable (XG) which is the expected rainfall depth for each rank and

duration was computed using the expression (3.9) below. Record of the results is

shown in Table 4.5.

(3.9)

Where D represents the ranked sample rainfall depth (mm) for each historical year.

The Gumbel’s variable (XG) is considered as the expected rainfall depth. Analysis conducted

for duration 0.20 hr Annual Maximum Series (AMS) and estimated parameters are recorded

in Tables 4.4 and 4.5. Additionally, results for the rest of the durations are presented in

Appendixes 4 and 5.

3.2.6 Validation testing of the fitness of Gumbel Distribution

The null hypothesis (Ho) drawn for this test is that “the annual maximum rainfall data used

for the study is drawn from a Gumbel Distribution”. This implies that the data (sample)

should come from a population that is characterized by Gumbel Distribution function. As

such, at the completion of the validation testing, the null hypothesis should either be accepted

or rejected. In order to accomplish this, the Kolmogorov – Smirnov and Chi – Square tests

were conducted to verify and confirm the stated null hypothesis. The significance level under

which the tests were conducted was generally chosen as 0.05 (5%) for the both tests. The

significance level is the level of probability at which null hypothesis is accepted or rejected

depending on the output of test statistic. When the result is true and the null hypothesis is

rejected, an error is made. Hence, such error is considered to be Type One Error. Tables 4.6

and 4.7 contain results for Kolmogorov – Smirnov and Chi- Square test results for duration

0.20 hr. Table 4.9 also represents a summarized results of the Chi – Square test conducted for

48

the various durations. The steps used to conduct the Chi – Square Test has earlier been

discussed under section 2.8.3 of the literature review.

3.2.6.1 Procedure for Kolmogorov – Smirnov Test validation

The following procedures were used to compute the test statistics required for the validation

and confirmation of the selected probability distribution function (Gumbel EVI) for each set

of sample rainfall data for every duration:

The set of maximum annual rainfall depths under each duration were determined and

logically arranged the intervals starting from the least rainfall depth in the data set.

The interval values were recorded in column 1.0. Table 4.6 represents detail for

duration 0.20 hr statistic;

The derived value of the upper boundary in column 2 labelled (B) determined from

column 1.0 was recorded.

The number of the observed frequency of rainfall depth obtained from sample data

were recorded in column 3 labelled (O) using the interval arranged in column (1) as

a guide.

The number of cumulative frequency (Cum.) of the rainfall depth for each row was

computed and result recorded in column 4.0.

The ratio of cumulative frequency to sample size (n) was computed and the result

recorded in column 5.0 for each set of interval using the expression (3.10) below:

( )= (

) (3.10)

The reduced variable (U) was computed using the expression (3.11) below and result

recorded in Column 6 for each set of interval:

(3.11)

49

Where U is the reduced variable, B is the upper boundary, xo is the position parameter and S

is scale parameter.

The Gumbel cumulative probability distribution was computed using the expression

(3.12) below and result recorded in Column 7:

( ) (3.12)

Ft (u) can simply be computed as: ( )=exp( ( ))

The Kolmogorov-Smirnov differences (DN) was computed using expression (3.13)

below and the result recorded in Column 8:

| ( ) ( ) (3.13)

This exercise was repeated for all of durations starting with duration 0.20 hr to 24.0 hr. See

Table 4.6 for detail of the statistic recorded from the exercise for duration 0.20 hr.

3.2.6.2 Procedure for the Chi –Square Test

The following Chi –Square Test procedures were used to compute the test statistics required

for the validation and confirmation of the selected probability distribution function (Gumbel

EVI) for each set of sample rainfall data for every duration starting with duration 0.20hr: The

steps are as follows:

The observed data was recorded in column labelled (O) and expected (E) values into

intervals so as to determine the frequency of both variables in each class. This can be

well expressed by a histogram of frequencies.

The classification was rearranged so that the minimum expected frequency in each

class becomes 5 or great. The classes with low frequency should be merged to this

end.

The chi-square value was calculated for all intervals by the relation expressed by

equation (3.14) below:

50

( ) = ∑( )

(3.14)

In the equation above, v is the degree of freedom (df) and equals n-k-1, where n is the

number of intervals and k is the number of distribution parameters obtained from the sample

statistics (Sample mean and standard deviation).

The value obtained was compared with the chi-square statistics under x20.050 from the

provided Appendix 10. The null hypothesis will be accepted if x2 < x

20.050 and rejected

if otherwise.

The effectiveness of the test is reduced when the expected frequency in any of the cell is less

than 5. When this condition is experienced, both the expected and observed frequencies of the

appropriate cell should be joined with the values of an adjacent cell and the value of k should

be decreased to represent the number of cells actually used in the calculation of the test

statistic.

At the end of the tests, the highest Kolmogorov difference (Dn) and sum of Chi-Square result

((O - E)2/E) for the different durations were recorded and compared with the critical values

developed for both tests. The decision about the hypothesis is made on the basis of the

highest value of the difference recorded in the test statistic table for Kolmogorov – Smirnov

while that of the Chi – Square is made on the basis of the sum obtained for (O –E)2/E and that

of the critical value under the 5% significance level using the computed degree of freedom

(df) as a guide. If the respective values obtained from the test statistic for each duration is less

than the value of the actual critical value, then the null hypothesis is accepted. If not the null

hypothesis is rejected. The critical value is determined based on the level of significance

(0.05 or 0.01) selected and the sample size for the KS test while the degree of freedom (df) is

additionally used for the Chi - Square test.

51

For this study, the critical value for the KS test was derived by interpolation because the

actual critical value for the sample size of 28 falls between 25 and 30 at 5% significance level

while the value for CS Test was read from a special table. See Appendixes 7 and 10. The

Kolmogorov – Smirnov and Chi –Square validation testing processes were repeated for the

rest of the durations. See Tables 4.7 and 4.8 for analyzed result for duration 0.20hr and

summary of the test.

3.2.7 Determination of Rainfall Intensity

On the basis of the Kolmogorov-Smirnov and Chi-Square validity tests which confirmed the

use of the Gumbel Probability distribution (EVI) function for hydrological data analysis, the

expected rainfall depths were then computed for all the durations and the selected return

periods, using equation (3.16). The determination of frequency factors (K) for each selected

return period were obtained from factor table prepared by Kendall (1959). Chow (1953)

confirmed that frequency factor for Gumbel Distribution function can be calculated using the

expressions (3.14) and (3.15) below:

√

( [ (

)]) (3.14)

Note: when T = 1, the expression below is applicable

√

[ ( )

] (3.15)

But the limitation of the above approach is that the determination of frequency factor (K)

depends only on the selected return period (T). Besides, the equations mentioned above,

Kendall (1959) derived and provided frequency factors in a tabular form to be used on

statistical analysis related to the Gumbel distribution. The determination of the frequency

factor developed by Kendall (1959) depends also on two parameters; the sample size and

return period. On the basis of the two parameters, the Kendall’s frequency factor was

considered and derived by interpolation processes using provided standardized numerical

52

values (factors) in tabular form as a guide. See Appendix 12 for Kendall’s frequency factors

table. Consequently, the study further derived the frequency factors (K) and computed the

expected rainfall depths and intensities for each duration and selected return periods. The

equations (3.16) and (3.17) below were used. The next chapter will show results for the

rainfall intensities statistic recorded. See Table (4.10) for statistics obtained from the

application of the method under duration 0.20 hr.

( ) (3.16)

(3.17)

Where XT is the estimated or expected rainfall depth in (mm), µG and σG are the Gumbel

mean and standard deviation respectively and I, represents the expected rainfall intensity and

also Hr. represents the durations under which the rain falls. Details of the expected rainfall

depths and intensities statistics for the rest of the durations are provided in Appendixes 11.0.

Additionally, summarized estimated rainfall intensities for the development of the Intensity

Duration Frequency Curves for all durations and selected return periods are provided in

Tables 4.10 in the next chapter. In order to represent the analyzed values in a graphical form

called the IDF Curves, the Microsoft Excel program was used on the data recorded in Table

4.10. The analyzed values were plotted in Log-Log, Semi – Log and Normal scales as shown

in Figures (4.1), (4.2), and (4.3) respectively.

53

CHAPTER 4: RESULTS AND DISCUSSION

4.1 Presentation of Results and Discussion

This section presents the related results and discussions made towards achieving the study

objectives arising from the data processing and analysis.

4.1.1 Analyze historic rainfall data for the determination of annual maximum rainfall depth

for the various durations.

Table 4.1 contains the result of the correlation coefficient analysis. It was evident by the

values obtained from the analysis that the paired rainfall data under the various durations are

strongly correlated. Appendix 3.0 also contains the analyzed and determined annual

maximum rainfall depths for the various years and durations which serves as a fulfillment of

the first specific objective of the study.

Table 4.1. Result of Correlation coefficient data s strength of relationship

Duration,

(hour)

Duration (hour)

0.2 0.4 0.7 1 2 3 6 12 24

0.2 1.000 0.813 0.643 0.416 0.418 0.419 0.395 0.374 0.425

0.4 0.813 1.000 0.858 0.585 0.500 0.474 0.430 0.459 0.427

0.7 0.643 0.858 1.000 0.794 0.627 0.535 0.429 0.368 0.267

1 0.416 0.585 0.794 1.000 0.843 0.718 0.609 0.502 0.363

2 0.418 0.500 0.627 0.843 1.000 0.943 0.878 0.808 0.740

3 0.419 0.474 0.535 0.718 0.943 1.000 0.973 0.930 0.874

6 0.395 0.430 0.429 0.609 0.878 0.973 1.000 0.982 0.925

12 0.374 0.459 0.368 0.502 0.808 0.930 0.982 1.000 0.970

24 0.425 0.427 0.267 0.363 0.740 0.874 0.925 0.970 1.000

54

4.1.2 Selection and verification of the appropriate probability distribution that best fit the

sample data

The results gathered and summarized from the application of the Easy Fit software to assess

and select the best fit distribution showed that the Gumbel EVI distribution is the most

appropriate distribution. From the result in Table 4.3 showed that EVI under Kolmogorov-

Smirnov scored seven (7) rank of one (1) and under Anderson Darling EVI scored nine (9)

ranks of one. The rank of one (1) represents a better fit while 2 is less satisfactory fit. Hence,

Gumbel (EVI) distribution scored a better than that of Log-Pearson Type3 (LP3). Finally,

the study selected Gumbel (EVI) distribution as the best fit. This selection facilitated the

frequency analysis processes for the estimation of the required parameters. Moreover, the

Gumbel Extreme Value 1(EV 1) was considered as the specific distribution function for the

population from which the maximum annual rainfall data used for the study was drawn from.

Tables 4.2 and 4.3 contain results of the selection exercises.

Table 4.2: Result of Easy Fit Test for EV1 and LP3 under duration 0.20 hr (12 min)

Duration Distribution

Rank

Kolmogorov

Smirnov Anderson Darling Chi-Square

0.20 Hr EV 1 1 1 2

LP 3 2 2 1

55

Table 4.3: Summary of result from the Easy Fit Tests for all durations

Duration

(hour)

Rank

Kolmogorov Smirnov Anderson Darling Chi- Squared

EV1 LP3 EV1 LP3 EV1 LP3

0.20 1 2 1 2 2 1

0.40 1 2 1 2 1 2

0.70 1 2 1 2 2 1

1.00 1 2 1 2 2 1

2.00 1 2 1 2 1 2

3.00 2 1 1 2 2 1

6.00 2 1 1 2 1 2

12.00 1 2 1 2 2 1

24.00 1 2 1 2 2 1

As shown in the table, the rank of 1 indicates a best fit than that of 2

4.1.3 Fitting the selected Gumbel distribution to the sample data

In order to perform the frequency analysis, the selected Gumbel distribution was fitted to the

annual maximum rainfall data to estimate various parameters of the distribution required

from the sample data. Table 4.4 and 4.5 show the results of estimated parameters for duration

0.20hr. Appendixes 4 and 5 contain the results for the rest of the durations.

56

Table 4.4. AMS Analyzed for duration 0.20 hour

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel

Variable, XG

(mm)

2004 47 1 0.0199 3.9063 71.98

1991 39 2 0.0555 2.8634 57.31

1980 30 3.8 0.1195 2.0616 43.18

2002 30 3.8 0.1195 2.0616 43.18

2003 30 3.8 0.1195 2.0616 43.18

2007 30 3.8 0.1195 2.0616 43.18

2008 30 3.8 0.1195 2.0616 43.18

1974 28 8.67 0.2927 1.0606 34.78

1975 28 8.67 0.2927 1.0606 34.78

2005 28 8.67 0.2927 1.0606 34.78

1992 26 11 0.3755 0.7532 30.82

1973 25 12.5 0.4289 0.5796 28.71

1996 25 12.5 0.4289 0.5796 28.71

1993 24 14 0.4822 0.4182 26.68

1972 23 15 0.5178 0.3156 25.02

2001 22 16 0.5533 0.2157 23.38

2000 21 17 0.5889 0.1177 21.75

1976 20 18 0.6245 0.0208 20.13

1995 18 19 0.6600 -0.0759 17.51

2006 17 20 0.6956 -0.1734 15.89

1971 16 21.5 0.7489 -0.3236 13.93

2009 16 21.5 0.7489 -0.3236 13.93

1977 15 23.75 0.8289 -0.5686 11.36

1978 15 23.75 0.8289 -0.5686 11.36

1994 15 23.75 0.8289 -0.5686 11.36

1998 15 23.75 0.8289 -0.5686 11.36

1979 14 27.5 0.9623 -1.1873 6.41

1997 14 27.5 0.9623 -1.1873 6.41

Table 4.5. Computed Gumbel Distribution parameters analyzed for 0.20 hr

Parameter Description Value (mm)

Sample Mean (µs) 23.61

Sample Standard Deviation (σs) 8.05

Position Parameter (xo) 19.56

Scale Parameter (S) 6.40

Gumbel Mean (µG) 23.25

Gumbel Standard Deviation (σG) 8.20

Mean of Reduced Variable (µN) 0.63

Standard Deviation of Reduced Variable (σN) 1.26

57

4.1.4 Validity Testing Results

The Tables 4.6 and 4.7 below show recorded results under duration 0.20hr for the validation

tests which confirmed the Gumbel distribution as the best fit distribution for the sample data

used for the study. The rest of the results are recorded under Appendixes 6, 8 and 9. All of the

results under the Kolmogorov – Smirnov confirmed Gumbel (EVI) as the best fit distribution

while 55.56% of the Chi-Square test accepted the Gumbel as the appropriate distribution.

This also fulfilled the accomplishment of the second specific objective.

The derived critical value for the KS test is 0.252. Besides, the highest Kolmogorov

difference (Dn) computed and recorded under column 8 of Table 4.6 for duration 0.20 hr is

0.106. However, since 0.106 is less than the critical value 0.252, it implies that that all the

data passed the KS test and so the null hypothesis was accepted. This implied that the rainfall

data was drawn from a population with Gumbel Distribution. The test result was successful

for the rest of the duration.

Regarding the test statistics obtained from Chi-Square tests conducted to further verify and

confirm the EVI as the best fit probability distribution function from which the sample was

drawn, the CS test result recorded from the Easy fit test statistic was considered very poor

under EVI and LP3 as recorded in Table 4.3. However due to its simplicity in analyzing test

statistic, the CS test was used to still counter check results determined from KS test. In

summary, the result analyzed indicates that 55.56% of the test statistics compiled from all the

durations accepted the null hypothesis while 44.44% rejected the null hypothesis as

summarized and recorded in Table 4.8. Thus the study did not find any statistical theory to

refer to for conclusion but it was reasoned that Chi-Square test result is satisfactory and it

supports the KS test for the verification and confirmation of the EVI being the best fit

distribution by virtue of the percentages of scores (55.56% accepted and 44.44% rejected).

Additionally, visual and statistical assessment of the result showed that some values which

58

rejected the null hypothesis are very close to the zone of acceptance. This also fulfilled the

accomplishment of the second specific objective.

Table 4.6. Kolmogorov – Smirnov Test Result analyzed for duration 0.20 hr.

Range B Frequency

Fo(x) U Ft

(x) Dn = Ft(x) - Fo(x)

O Cum

10 – 15 15.00 6 6 0.214 -0.713 0.130 0.084

16 – 20 20.00 5 11 0.393 0.069 0.393 0.000

21 – 25 25.00 6 17 0.607 0.850 0.652 0.045

26 – 30 30.00 9 26 0.929 1.632 0.822 0.106

31 – 35 35.00 0 26 0.929 2.414 0.914 0.014

36 – 40 40.00 1 27 0.964 3.196 0.960 0.004

41 – 45 45.00 0 27 0.964 3.978 0.981 0.017

46 – 50 50.00 1 28 1.000 4.760 0.991 0.009 0.106 is less than 0.252. Therefore, null hypothesis is accepted

Table 4.7: Chi -Square analyzed Test result analyzed for duration 0.20 hr

5.41< 5.991 X2 value obtained at 5% significant level, hence null hypothesis is accepted

Interval

(mm)

O E O – E (O - E)2 (O - E)

2 / E

No of Frequency

5 – 20 11 10 1 1 0.10

20 – 25 6 3 3 9 3.00

25 – 55 11 13 -2 4 0.31

55 – 60 0 1 -1 1 1.00

60 – 75 0 1 -1 1 1.00

Total 5.41

59

Table 4.8: Summary of Chi- Square Test result

Duration

(hr)

Chi-Square

Test result

(O – E)2/E

DF X2 at 5%

Decision of hypothesis

based on test result

0.20

5.1

2

5.991

5.10<5.991, null

hypothesis is accepted.

0.40 5.256 2 5.991 5.256<5.991, null

hypothesis is accepted

0.70 10.275 6 12.592 10.275<12.592, null

Hypothesis is accepted

1.0 4.667 1 3.841 4.667>3.841, null

hypothesis is rejected

2.0 11.57 7 14.067 11.57<14.067, null

hypothesis is accepted

3.0 7.574 4 9.488 7.574<9.488, null

hypothesis is accepted

6.0 8.099 1 3.841 8.099>3.841, null

hypothesis is rejected

12.0 4.99 1 3.841 4.99>3.841, null

hypothesis is rejected

24.0 6.251 1 3.841 6.251>3.841, null

hypothesis is rejected

DF: degree of freedom, O: Observed frequency, E: Expected frequency and X2 CS result 5% significant level

4.1.4 Compute rainfall intensity and developed IDF curves

Table 4.9 contains details of the expected rainfall depths (XT) and rainfall intensities

computed using the Kendall (1959) frequency factor table in Appendix 12 for duration 0.20hr

and selected return periods. Details for the rest are recorded under Appendix 11. Equation

3.16 was used to finally compute the expected rainfall depths (XT) while equation (3.17) was

used to calculate the intensities. The results discussed above fulfilled the accomplishment of a

portion of the third specific objective of the study.

60

Meanwhile, Table 4.10 contains summarized estimates of the expected rainfall intensities for

all durations and selected return periods used by this study to develop the new IDF curves for

Accra. Figures 4.1, 4.2 and 4.3 are graphical representations of the statistics used to develop

the new IDF curves for the study area (Accra). Figures, 4.1 is Log-Log graph, 4.2 semi-log

graph and 4.3 normal scale graph of the curves. Consequently, these values and graphs totally

accomplished the third specific and primary objectives of the study.

Finally, the set of rainfall statistic used by J. B Dankwa for the development of the IDF curve

for Accra is also recorded in table under Appendix 13 for reference. Source: J. B Dankwa

(1974) GMA

Table 4.9 Analyzed rainfall intensities for duration 0.20 hr.

Return

Period

(Year)

Duration

(hr)

Frequency

Factor (K)

Gumbel

Mean

(µG) mm

Gumbel Rainfall

Depth

XT (mm)

Intensity

I = XT

(mm)/hr

Stdev.

(σG)mm

5 0.20 0.875 23.2525 8.2016 30.43 152.14

10 0.20 1.5546 23.2525 8.2016 36.00 180.01

15 0.20 1.9384 23.2525 8.2016 39.15 195.75

20 0.20 2.2068 23.2525 8.2016 41.35 206.76

25 0.20 2.4134 23.2525 8.2016 43.05 215.23

50 0.20 3.0508 23.2525 8.2016 48.27 241.37

100 0.20 3.6834 23.2525 8.2016 53.46 267.31

Table 4.10. Summarized estimates of rainfall intensity for all durations and return periods

Rainfall

Duration

(hr)

RETURN PERIODS

(Years)

5 10 15 20 25 50 100

0.20 152.14 180.01 195.75 206.76 215.23 241.37 267.31

0.40 115.93 136.53 148.17 156.30 162.56 181.89 201.06

0.70 84.45 98.84 106.97 112.65 117.03 130.53 143.93

1.00 74.27 87.15 94.43 99.51 103.43 115.51 127.50

2.00 46.62 55.60 60.67 64.22 66.95 75.38 83.74

3.00 36.22 43.61 47.79 50.71 52.95 59.89 66.77

6.00 19.74 24.01 26.42 28.11 29.40 33.41 37.38

12.00 10.92 13.51 14.97 15.99 16.78 19.21 21.62

24.00 5.54 6.85 7.59 8.10 8.50 9.73 10.95

61

The rainfall intensities estimated for the graphical representation of the new IDF curves for

the study area conformed to the general characteristics and properties of IDF relationship.

Additionally, the annual maximum rainfall depths determined or recorded under Appendix 3

were used for selecting the appropriate probability distribution, estimating the required

parameters for the selected distribution (Gumbel), testing the fitness of the selected

distribution and frequency analysis. Besides, the values of the estimated rainfall intensities in

Table 4.10 gradually increased from a shorter return period to higher return period for all

durations indicating an intense rainfall. Moreover, the curves representing the various return

periods, intensities and durations did not intersect nor meet each other at any point along.

Thus, they run parallel to each other. Figures 4.1, 4.2, and 4.3 represent the different plots.

The durations and selected interval of occurrence (Return Periods) remain the same as

established in the past by J. B Dankwa (1974).

62

Figure 4.1. IDF Curves for Accra – Log-Log graph

I = 64.214x-0.695

I = 76.31x-0.682

I = 83.135x-0.677

I = 87.906x-0.674

I = 91.577x-0.672

I = 102.9x-0.666

I = 104.45x-0.665

1

10

100

1000

0.10 1.00 10.00 100.00

RA

INFA

LL IN

TEN

SITY

mm

/hr)

RAINFALL DURATION (HOUR)

5 Years

10 Years

15 Years

20 Years

25 Years

50 Years

100 Years

Return Period

63

Figure 4.2 IDF Curves for Accra – Semi- Log

I = 64.214x-0.695

I = 76.31x-0.682

I = 83.135x-0.677

I = 87.906x-0.674

I = 91.577x-0.672

I = 102.9x-0.666

I = 104.45x-0.665

0

50

100

150

200

250

300

350

0.10 1.00 10.00 100.00

RA

INFA

LL

IN

TE

NS

ITY

(m

m/h

r)

RAINFALL DURATION (HOUR)

5 Years

10 Years

15 Years

20 Years

25 Years

50 Years

100 Years

Return Period

64

Figure 4.3 IDF Curves for Accra – Normal Scale

I = 64.214x-0.695

I = 76.31x-0.682

I = 83.135x-0.677

I = 87.906x-0.674

I = 91.577x-0.672

I = 102.9x-0.666

I = 104.45x-0.665

0

50

100

150

200

250

300

350

0.00 5.00 10.00 15.00 20.00 25.00 30.00

RA

INFA

LL IN

TEN

SITY

(m

m/h

r)

RAINFALL DURATION (HOUR)

5 Years

10 Years

15 Years

20 Years

25 Years

50 Years

100 Years

Return Period

65

Table 4.11 Comparison result of estimated Intensities (New) with J. B Dankwa’s for (5yr.10yr. and 15yr)

Duration

(hr)

Return Periods

5 Years 10 Years 15 years

J B. Dankwa

(mm/hr)

New

Result

(mm/hr)

Diff

(%)

J B. Dankwa

(mm/hr)

New Result

(mm/hr)

Diff

(%)

J B. Dankwa

(mm/hr)

New

Result

(mm/hr)

Diff

(%)

0.20 127.00 152.14 19.80 140.97 180.01 27.70 149.01 195.75 31.37

0.40 99.10 115.93 16.98 116.84 136.53 16.85 121.92 148.17 21.53

0.70 74.40 84.45 13.50 85.6 98.84 15.47 90.17 106.97 18.63

1.00 62.50 74.27 18.84 71.88 87.15 21.25 75.94 94.43 24.34

2.00 37.85 46.62 23.16 44.5 55.60 24.94 47.28 60.67 28.33

3.00 29.21 36.22 24.01 33.02 43.61 32.08 34.71 47.79 37.68

6.00 15.75 19.74 25.35 19.56 24.01 22.75 21.00 26.42 25.81

12.00 8.64 10.92 26.35 10.67 13.51 26.59 11.52 14.97 29.95

24.00 4.32 5.54 28.22 5.33 6.85 28.50 5.75 7.59 31.96

Average 21.80 24.01 27.75

Note: The intensities values in red color were interpolated due to the absence of data for years 15 and 20 in the original J. B. Dankwa results for

Accra (1974)

66

Table 4.12. Comparison result of new estimated Intensities with Dankwa’s for (20yr. 25yr. and 50yr)

Duration,

(hr)

Return Periods

20 Years 25 Years 50 years

J B. Dankwa

(mm/hr)

New Result

(mm/hr)

Diff

(%)

J B. Dankwa

(mm/hr)

New Result

(mm/hr)

Diff

(%)

J B. Dankwa

(mm/hr)

New

Result

(mm/hr)

Diff

(%)

0.20 157.06 206.76 31.64 165.10 215.23 30.36 180.34 241.37 33.84

0.40 126.99 156.30 23.08 132.08 162.56 23.08 147.32 181.89 23.46

0.70 94.74 112.65 18.91 99.31 117.03 17.84 109.98 130.53 18.68

1.00 80.01 99.51 24.38 84.07 103.43 23.03 92.96 115.51 24.26

2.00 50.05 64.22 28.32 52.83 66.95 26.73 59.18 75.38 27.37

3.00 36.41 50.71 39.27 38.10 52.95 38.99 43.67 59.89 37.13

6.00 22.44 28.11 25.25 23.88 29.40 23.13 27.18 33.41 22.91

12.00 12.36 15.99 29.39 13.21 16.78 27.03 15.24 19.21 26.05

24.00 6.18 8.10 31.14 6.60 8.50 28.83 7.62 9.73 27.69

Average 27.93 26.56 26.82

Note: The intensities values in red color were interpolated due to the absence of data for years 15 and 20 in the original J. B. Dankwa results for

Accra (1974)

67

Table 4.13 Comparison result of estimated Intensities (New) with J. B Dankwa’s

Intensities for (100yrs)

4.1.5 Comparison of Results

Visual and statistical comparison conducted between the new and the existing

maximum rainfall intensities analyzed by J. B. Dankwa for the study area were

recorded in Tables 4.11, 4.12 and 4.13. It showed that the average percentage

difference between the new and existing intensities which shows increase in the

rainfall varies from the shorter to the longer durations under all return periods.

Additionally, the computed values also indicate more than 20% average increase in

intensities over the existing rainfall intensities derived by J. B. Dankwa since 1974

from shorter to longer durations (i. e., from 0.20hr to 24.0hrs) and for all return

periods. Similarly, the results are shown in Tables 4.11, 4.12 and 4.13. Thus the

increase in percentage is not uniform, but it fluctuates along with the increase in

duration for all the return periods. Finally, it was observed from the comparison made

between the new estimated rainfall intensities and that of the J. B Dankwa’s values for

Duration,

(hr)

Return Period

100 Years

J B. Dankwa

(mm/hr)

New Result

(mm/hr)

Diff

(%)

0.20 196.85 267.31 35.79

0.40 162.56 201.06 23.68

0.70 120.40 143.93 19.54

1.00 101.85 127.50 25.18

2.00 65.53 83.74 27.79

3.00 48.26 66.77 38.35

6.00 28.70 37.38 30.24

12.00 17.02 21.62 27.03

24.00 8.64 10.95 26.71

Average 28.26

68

the study area showed that an increase in the rainfall intensities has occurred over the

existing values derived since 1974.

Consequently, it is believed that rainfall in the study area is becoming more intense

for all durations and return periods. This change in intensities could be attributed to

climate variability.

69

CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusion

After a thorough literature review and statistical data analyses conducted on the

extreme rainfall values obtained for the study area, it was observed that the IDF

curves developed for the study shows a general increase in trend of the rainfall

intensities. The analysis also showed that Gumbel, (Extreme Value Type 1) is the

most appropriate probability distribution that best fitted the rainfall data used for the

study. Furthermore, the new rainfall intensities computed showed more than 20%

average increase in value over the existing from shorter to longer durations (i. e., from

0.20hr to 24.0hrs) and for all return periods (i.e., from 5years). This implies that there

is now much more intense rainfall being experienced in the study area compare to the

rainfall intensities determined by J. B. Dankwa.

5.2 Recommendations

As a result of the outcome of the study, the following recommendations were

developed for consideration:

That systematic monitoring of the various rain gauges and regular data

recording should be encouraged to reduce the number of gaps in future

records.

That the existing data management system be assessed and improve if

necessary for easy access.

That IDF curves for the remaining 9 regions in Ghana should be updated for

now and thereafter GMA should encourage a periodic update of the IDF

curves for the various regions.

70

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APPENDICES

78

Appendix 1: Raw Rainfall data obtained from GMA

Rainfall depth (mm)

Station: Accra

DURATION 0.2 hr 0.4 hr 0.7 hr 1.0 hr 2.0 hrs 3.0 hrs 6.0 hrs 12. hrs 24.0 hrs

Year Month

1971 1 16 21 23 25 34 34 34 34 47

1972 2 13 19 22 25

1972 3 14

26 27

1972 4 23 41 56 78 87 89 95 96 96

1972 5

22 28 33 33 33 36

1972 6

22 32 37 38 43 61 62 66

1972 9 13 15

1972 10

33 37 37 38 38 38

1973 3 14 23 24 26 31 32 32

1973 5 19 23 29 33 36 36 36 36

1973 6 25 45 59 66 67 107

1973 7

21

31 39 72 73 73

1973 9 13 20 34 36 48 52 54 55 72

1974 3

17 22

1974 5 14 24 29 30 37 47 49 49 49

1974 6 28 45 69 74 84 86 88 88 88

1974 7

24 32 37 45 45 45 45 45

1974 11

21 26 43 45 45 45 45 45

1975 2

24 26 27 30

1975 3 24 38 42 42 42 42 42 42 42

1975 5 13 22 33 37 37 37 37 37

1975 6 28 45 63 65 82 85 86 93 95

1975 7

26 32 36 46 46

1975 9 23 32 35 36 39 40 40 40 40

1975 11

25 32 33 34 35 38 38

1975 12 14

44 53 55 55 55 55 55

1976 4 17 25 26 28 33 45 47 47 47

1976 6 15 20 24 27 33 37 43 47 47

1976 10 20 30 35 43 71 75 75 75 75

1977 4 15 30 56 65 65 65 65 65 65

1977 10 14 24 27 30 38 47 75 103 103

1978 4

23

26 30 33 33 42

1978 5

34 40 68 74 77 77

1978 9 15 16

1979 3

15

1979 4 13 16 21

1979 6

16 21 32 49 49 49 49 49

1979 9

22 22

1980 4 12 25 35 53 70 85 85 85 85

1980 5

59

1980 6 15 19 25 30 52 53 57 58 58

79

1991 4 24 28 33 60 80 81 85 85 85

1991 5

54 55 57 57 57 60 87 87

1991 6

30 31

1991 7

31 73 98 124 133 138

1991 10

29 29 29

1992 3 15 15

1992 4

27 27 38 40 40 40 40 40

1992 5

37 37 62 62 65 69 69 69

1992 6

43 43 57 64 67 67

1993 1 18 18

1993 6 17 20 29 29 39 49 59 61 61

1993 9

42 48 48 48 48 48 48

1993 10

15

1993 11

20 30 30 35 35 35

1993 12

29 45 66 67 67 67 67 67

1994 5

28 33 33 37 38 38 44

1994 6

23 40 63 67 88 99 100 100

1995 2 13 16 20

1995 3 13 18 25 27 29 33 40

1995 4 18 25 32 34 35 36

1995 5

25 35 37 40 45 45 45 45

1995 6 13 35 41 41 41 41 41 41 41

1995 7

30 42 107 170 207 259 262

1995 11 18 26 44 45 45 47 48

1996 3

23 25 29 31 32 32

1996 5 25 31 33 33 33 33 47 51

1996 6 25 32 58 63 69 72 75 76 76

1997 3 14 19 25

1997 4

18 30 35 42

1997 5

20 30

1997 6 12 20 25 32

1998 5 15 18 28 47 47

1998 10 12 17 30 50 50

2000 10

15 24 27 33 36 36 36

2000 11 21 23 24

2001 3 16 18 22

2001 9 22 31 36 37 39 40 40 40 40

2001 12

16

2002 1 18 28 35 94 110 120 121 122 122

2002 3 17 24 31 33 35 35 35 35

2002 4 13 17 25 29 33 36 36 36

2002 5

21 25 28 28 35 35 35

2002 6 30 45 48 55 90 99 104 109 126

2003 4 30 36 40 46 94 110 115 120 120

2003 7

20 28 31 32 34 35 35

2003 9

25 26

2003 10 17 28 52 64 68 69 76 77 77

2003 12 12 15

80

2004 5 21 33 33 33 33 33 33 33

2004 6 47 62 71 77 92 95 97 98 98

2004 9 15 23 32 44 45 45 46 46 46

2005 3 19 21 24 44 56 57 60 60 60

2005 6 25 60 67 67 68 69 69 71 72

2005 8 12 16

2005 10 15 32 35 36 40 44 44 44 44

2005 11 28 47 55 57

2005 12

20 31

2006 5 17 40 50 51 52 54

54

2006 6 16 30 39 39 43 62 68 68 68

2007 3 15 27 48 58 59 63 63 63 63

2007 4 30 36 41 53 54 56

57 57

2007 5 27 34 43 46 50

2007 6 12

22 28 29 31 35 37

2007 9 25 31 37 40 43

2007 10 13 15

2008 3 25 38 40 41

2008 4 18 21 37 46 61 66 69

2008 5 16 33 53 98 149 151 151 151 151

2008 6 15 29 32

2008 7 14 22 34 40 47 64 64 73 83

2008 10 30 41 42 43 57 57 59 79 126

2008 11 22 26 42 45 48 49 49 67

2008 12 30 41 42 43 57 57 59 79 126

2009 3 13 20 23

2009 4 13 18 23 26 32 35 36 36 36

2009 6 16 36 38 45 61

81

Appendix 2: Complete rainfall Data obtained after filling gaps

Station: Accra

Duration 0.2 hr 0.4 hr 0.70 hr 1.0hr 2.0 hrs 3.0 hr 6.0 hr 12.0 hr 24.0 hr

Year Month

1971 1 16 21 23 25 34 34 34 34 47

Max 16 21 23 25 34 34 34 34 47

1972 2 13 19 22 25 29 32 34 32 35

1972 3 14 22 30 37 26 27 28 25 28

1972 4 23 41 56 78 87 89 95 96 96

1972 5 14 22 28 33 33 33 36 34 37

1972 6 14 22 32 37 38 43 61 62 66

1972 9 13 15 22 29 34 37 39 38 41

1972 10 23 33 37 37 38 38 38 37 40

Max 23 41 56 78 87 89 95 96 96

1973 3 14 23 24 26 31 32 32 30 33

1973 5 19 23 29 33 36 36 36 36 39

1973 6 25 45 59 66 67 107 115 130 134

1973 7 8 14 21 27 31 39 72 73 73

1973 9 13 20 34 36 48 52 54 55 72

Max 25 45 59 66 67 107 115 130 134

1974 3 11 17 22 29 34 37 39 38 41

1974 5 14 24 29 30 37 47 49 49 49

1974 6 28 45 69 74 84 86 88 88 88

1974 7 16 24 32 37 45 45 45 45 45

1974 11 14 21 26 43 45 45 45 45 45

Max 28 45 69 74 84 86 88 88 88

1975 2 11 17 24 26 27 30 32 30 33

1975 3 24 38 42 42 42 42 42 42 42

1975 5 13 22 33 37 37 37 37 37 40

1975 6 28 45 63 65 82 85 86 93 95

1975 7 4 9 16 22 26 32 36 46 46

1975 9 23 32 35 36 39 40 40 40 40

1975 11 17 25 32 33 34 35 38 38 41

1975 12 14 22 44 53 55 55 55 55 55

Max 28 45 63 65 82 85 86 93 95

82

1976 4 17 25 26 28 33 45 47 47 47

1976 6 15 20 24 27 33 37 43 47 47

1976 10 20 30 35 43 71 75 75 75 75

Max 20 30 35 43 71 75 75 75 75

1977 4 15 30 56 65 65 65 65 65 65

1977 10 14 24 27 30 38 47 75 103 103

Max 15 30 56 65 65 65 75 103 103

1978 4 10 16 23 30 26 30 33 33 42

1978 5 12 19 27 34 40 68 74 77 77

1978 9 15 16 24 31 37 41 44 44 47

Max 15 19 27 34 40 68 74 77 77

1979 3 9 15 22 29 34 37 39 38 41

1979 4 13 16 21 27 32 35 37 36 39

1979 6 10 16 21 32 49 49 49 49 49

1979 9 14 22 22 29 34 37 39 38 41

Max 14 22 22 32 49 49 49 49 49

1980 4 12 25 35 53 70 85 85 85 85

1980 5 30 43 51 59 70 78 84 92 95

1980 6 15 19 25 30 52 53 57 58 58

Max 30 43 51 59 70 85 85 92 95

1991 4 24 28 33 60 80 81 85 85 85

1991 5 39 54 55 57 57 57 60 87 87

1991 6 27 39 30 31 37 41 44 44 47

1991 7 11 17 24 31 73 98 124 133 138

1991 10 10 16 23 29 29 29 31 28 31

Max 39 54 55 60 80 98 124 133 138

1992 3 15 15 22 29 34 37 39 38 41

1992 4 18 27 27 38 40 40 40 40 40

1992 5 26 37 37 62 62 65 69 69 69

1992 6 19 28 36 43 43 57 64 67 67

Max 26 37 37 62 62 65 69 69 69

1993 1 18 18 26 33 40 44 47 48 51

1993 6 17 20 29 29 39 49 59 61 61

83

1993 9 24 34 42 48 48 48 48 48 48

1993 10 9 15 22 29 34 37 39 38 41

1993 11 8 13 20 30 30 35 35 35 38

1993 12 20 29 45 66 67 67 67 67 67

Max 24 34 45 66 67 67 67 67 67

1994 5 13 20 28 33 33 37 38 38 44

1994 6 15 23 40 63 67 88 99 100 100

Max 15 23 40 63 67 88 99 100 100

1995 2 13 16 20 26 31 34 36 34 37

1995 3 13 18 25 27 29 33 40 39 42

1995 4 18 25 32 34 35 36 38 37 40

1995 5 17 25 35 37 40 45 45 45 45

1995 6 13 35 41 41 41 41 41 41 41

1995 7 14 22 30 42 107 170 207 259 262

1995 11 18 26 44 45 45 47 48 49 52

Max 18 35 44 45 107 170 207 259 262

1996 3 10 16 23 25 29 31 32 32 35

1996 5 25 31 33 33 33 33 47 51 54

1996 6 25 32 58 63 69 72 75 76 76

Max 25 32 58 63 69 72 75 76 76

1997 3 14 19 25 32 38 42 45 45 48

1997 4 11 18 30 35 42 46 49 50 53

1997 5 13 20 30 37 44 49 52 54 57

1997 6 12 20 25 32 38 42 45 45 48

Max 14 20 30 37 44 49 52 54 57

1998 5 15 18 28 47 47 52 56 58 61

1998 10 12 17 30 50 50 55 59 62 65

Max 15 18 30 50 50 55 59 62 65

2000 10 9 15 24 27 33 36 36 36 39

2000 11 21 23 24 31 37 41 44 44 47

Max 21 23 24 31 37 41 44 44 47

2001 3 16 18 22 29 34 37 39 38 41

2001 9 22 31 36 37 39 40 40 40 40

84

2001 12 10 16 24 31 37 41 44 44 47

Max 22 31 36 37 39 41 44 44 47

2002 1 18 28 35 94 110 120 121 122 122

2002 3 17 24 31 33 35 35 35 35 38

2002 4 13 17 25 29 33 36 36 36 39

2002 5 14 21 25 28 28 35 35 35 38

2002 6 30 45 48 55 90 99 104 109 126

Max 30 45 48 94 110 120 121 122 126

2003 4 30 36 40 46 94 110 115 120 120

2003 7 13 20 28 31 32 34 35 35 38

2003 9 4 8 15 21 25 26 27 23 26

2003 10 17 28 52 64 68 69 76 77 77

2003 12 12 15 22 29 34 37 39 38 41

Max 30 36 52 64 94 110 115 120 120

2004 5 21 33 33 33 33 33 33 33 36

2004 6 47 62 71 77 92 95 97 98 98

2004 9 15 23 32 44 45 45 46 46 46

Max 47 62 71 77 92 95 97 98 98

2005 3 19 21 24 44 56 57 60 60 60

2005 6 25 60 67 67 68 69 69 71 72

2005 8 12 16 24 31 37 41 44 44 47

2005 10 15 32 35 36 40 44 44 44 44

2005 11 28 47 55 57 67 75 80 87 91

2005 12 13 20 31 38 45 50 53 55 58

Max 28 60 67 67 68 75 80 87 91

2006 5 17 40 50 51 52 54 58 54 57

2006 6 16 30 39 39 43 62 68 68 68

Max 17 40 50 51 52 62 68 68 68

2007 3 15 27 48 58 59 63 63 63 63

2007 4 30 36 41 53 54 56 60 57 57

2007 5 27 34 43 46 50 55 59 62 65

2007 6 12 19 22 28 29 31 35 37 40

2007 9 25 31 37 40 43 47 50 51 54

85

2007 10 13 15 22 29 34 37 39 38 41

Max 30 36 48 58 59 63 63 63 65

2008 3 25 38 40 41 48 53 57 60 63

2008 4 18 21 37 46 61 66 69 74 77

2008 5 16 33 53 98 149 151 151 151 151

2008 6 15 29 32 40 47 52 56 58

2008 7 14 22 34 40 47 64 64 73 83

2008 10 30 41 42 43 57 57 59 79 126

2008 11 22 26 42 45 48 49 49 67 70

2008 12 30 41 42 43 57 57 59 79 126

Max 30 41 53 98 149 151 151 151 151

2009 3 13 20 23 30 35 38 40 39 42

2009 4 13 18 23 26 32 35 36 36 36

2009 6 16 36 38 45 61 68 73 79 82

Max 16 36 38 45 61 68 73 79 82

Note: The rainfall depths in red color represent values analyzed to fill existing gaps in record obtained from

GMA.

86

Appendix 3: Summarized Annual Maximum Rainfall depths extracted from all

durations

Duration 0.2 hr 0.4 hr 0.70 hr 1.0 hr 2.0 hrs 3.0 hrs 6.0 hrs 12.0

hrs

24.0

hrs

Year Maximum Rainfall depth (mm)

1971 16 21 23 25 34 34 34 34 47

1972 23 41 56 78 87 89 95 96 96

1973 25 45 59 66 67 107 115 130 134

1974 28 45 69 74 84 86 88 88 88

1975 28 45 63 65 82 85 86 93 95

1976 20 30 35 43 71 75 75 75 75

1977 15 30 56 65 65 65 75 103 103

1978 15 19 27 34 40 68 74 77 77

1979 14 22 22 32 49 49 49 49 49

1980 30 43 51 59 70 85 85 92 95

1991 39 54 55 60 80 98 124 133 138

1992 26 37 37 62 62 65 69 69 69

1993 24 34 45 66 67 67 67 67 67

1994 15 23 40 63 67 88 99 100 100

1995 18 35 44 45 107 170 207 259 262

1996 25 32 58 63 69 72 75 76 76

1997 14 20 30 37 44 49 52 54 57

1998 15 18 30 50 50 55 59 62 65

2000 21 23 24 31 37 41 44 44 47

2001 22 31 36 37 39 41 44 44 47

2002 30 45 48 94 110 120 121 122 126

2003 30 36 52 64 94 110 115 120 120

2004 47 62 71 77 92 95 97 98 98

2005 28 60 67 67 68 75 80 87 91

2006 17 40 50 51 52 62 68 68 68

2007 30 36 48 58 59 63 63 63 65

2008 30 41 53 98 149 151 151 151 151

2009 16 36 38 45 61 68 73 79 82

87

Appendix 4: Annual Maximum Series arranged for Different Durations

Duration: 0.40 hr

Ranked

Year

Rainfall

Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

2004 62 1 0.0199 3.9063 98.93

2005 60 2 0.0555 2.8634 87.07

1991 54 3 0.0910 2.3491 76.21

1973 45 4.75 0.1533 1.7935 61.96

1974 45 4.75 0.1533 1.7935 61.96

1975 45 4.75 0.1533 1.7935 61.96

2002 45 4.75 0.1533 1.7935 61.96

1980 43 8 0.2688 1.1611 53.98

1972 41 9.5 0.3222 0.9445 49.93

2008 41 9.5 0.3222 0.9445 49.93

2006 40 11 0.3755 0.7532 47.12

1992 37 12 0.4111 0.6358 43.01

2003 36 13.67 0.4705 0.4529 40.28

2007 36 13.67 0.4705 0.4529 40.28

2009 36 13.67 0.4705 0.4529 40.28

1995 35 16 0.5533 0.2157 37.04

1993 34 17 0.5889 0.1177 35.11

1996 32 18 0.6245 0.0208 32.20

2001 31 19 0.6600 -0.0759 30.28

1976 30 20.5 0.7134 -0.2228 27.89

1977 30 20.5 0.7134 -0.2228 27.89

1994 23 22.5 0.7845 -0.4284 18.95

2000 23 22.5 0.7845 -0.4284 18.95

1979 22 24 0.8378 -0.5984 16.34

1971 21 25 0.8734 -0.7260 14.14

1997 20 26 0.9090 -0.8740 11.74

1978 19 27 0.9445 -1.0619 8.96

1998 18 28 0.9801 -1.3651 5.09

88

Duration: 0.70 hr

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

2004 71 1 0.0199 3.9063 116.16

1974 69 2 0.0555 2.8634 102.10

2005 67 3 0.0910 2.3491 94.16

1975 63 4 0.1266 1.9998 86.12

1973 59 5 0.1622 1.7320 79.02

1996 58 6 0.1977 1.5128 75.49

1972 56 7 0.2333 1.3256 71.33

1977 56 8 0.2688 1.1611 69.42

1991 55 9 0.3044 1.0134 66.72

2008 53 10 0.3400 0.8783 63.15

2003 52 11 0.3755 0.7532 60.71

1980 51 12 0.4111 0.6358 58.35

2006 50 13 0.4467 0.5246 56.07

2002 48 14.5 0.5000 0.3665 52.24

2007 48 14.5 0.5000 0.3665 52.24

1993 45 16 0.5533 0.2157 47.49

1995 44 17 0.5889 0.1177 45.36

1994 40 18 0.6245 0.0208 40.24

2009 38 19 0.6600 -0.0759 37.12

1992 37 20 0.6956 -0.1734 35.00

2001 36 21 0.7312 -0.2728 32.85

1976 35 22 0.7667 -0.3753 30.66

1997 30 23.5 0.8201 -0.5395 23.76

1998 30 23.5 0.8201 -0.5395 23.76

1978 27 25 0.8734 -0.7260 18.61

2000 24 26 0.9090 -0.8740 13.90

1971 23 27 0.9445 -1.0619 10.72

1979 22 28 0.9801 -1.3651 6.22

89

Duration: 1.0 hr

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

2008 98 1 0.0199 3.9063 155.72

2002 94 2 0.0555 2.8634 136.31

1972 78 3 0.0910 2.3491 112.71

2004 77 4 0.1266 1.9998 106.55

1974 74 5 0.1622 1.7320 99.59

2005 67 6 0.1977 1.5128 89.35

1973 66 7.5 0.2511 1.2410 84.34

1993 66 7.5 0.2511 1.2410 84.34

1975 65 9.5 0.3222 0.9445 78.96

1977 65 9.5 0.3222 0.9445 78.96

2003 64 11 0.3755 0.7532 75.13

1994 63 12.5 0.4289 0.5796 71.56

1996 63 12.5 0.4289 0.5796 71.56

1992 62 14 0.4822 0.4182 68.18

1991 60 15 0.5178 0.3156 64.66

1980 59 16 0.5533 0.2157 62.19

2007 58 17 0.5889 0.1177 59.74

2006 51 18 0.6245 0.0208 51.31

1998 50 19 0.6600 -0.0759 48.88

1995 45 20.5 0.7134 -0.2228 41.71

2009 45 20.5 0.7134 -0.2228 41.71

1976 43 22 0.7667 -0.3753 37.45

1997 37 23.5 0.8201 -0.5395 29.03

2001 37 23.5 0.8201 -0.5395 29.03

1978 34 25 0.8734 -0.7260 23.27

1979 32 26 0.9090 -0.8740 19.10

2000 31 27 0.9445 -1.0619 15.31

1971 25 28 0.9801 -1.3651 4.83

90

Duration: 2.0 hr

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

2008 149 1 0.0199 3.9063 229.53

2002 110 2 0.0555 2.8634 169.03

1995 107 3 0.0910 2.3491 155.43

2003 94 4 0.1266 1.9998 135.23

2004 92 5 0.1622 1.7320 127.71

1972 87 6 0.1977 1.5128 118.19

1974 84 7 0.2333 1.3256 111.33

1975 82 8 0.2688 1.1611 105.94

1991 80 9 0.3044 1.0134 100.89

1976 71 10 0.3400 0.8783 89.11

1980 70 11 0.3755 0.7532 85.53

1996 69 12 0.4111 0.6358 82.11

2005 68 13 0.4467 0.5246 78.82

1973 67 14.67 0.5060 0.3491 74.20

1993 67 14.67 0.5060 0.3491 74.20

1994 67 14.67 0.5060 0.3491 74.20

1977 65 17 0.5889 0.1177 67.43

1992 62 18 0.6245 0.0208 62.43

2009 61 19 0.6600 -0.0759 59.44

2007 59 20 0.6956 -0.1734 55.43

2006 52 21 0.7312 -0.2728 46.38

1998 50 22 0.7667 -0.3753 42.26

1979 49 23 0.8023 -0.4830 39.04

1997 44 24 0.8378 -0.5984 31.66

1978 40 25 0.8734 -0.7260 25.03

2001 39 26 0.9090 -0.8740 20.98

2000 37 27 0.9445 -1.0619 15.11

1971 34 28 0.9801 -1.3651 5.86

91

Duration: 3.0 hr

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

1995 170 1 0.0199 3.9063 269.37

2008 151 2 0.0555 2.8634 223.84

2002 120 3 0.0910 2.3491 179.76

2003 110 4 0.1266 1.9998 160.87

1973 107 5 0.1622 1.7320 151.06

1991 98 6 0.1977 1.5128 136.48

2004 95 7 0.2333 1.3256 128.72

1972 89 8 0.2688 1.1611 118.54

1994 88 9 0.3044 1.0134 113.78

1974 86 10 0.3400 0.8783 108.35

1975 85 11.5 0.3933 0.6937 102.65

1980 85 11.5 0.3933 0.6937 102.65

1976 75 13.5 0.4644 0.4709 86.98

2005 75 13.5 0.4644 0.4709 86.98

1996 72 15 0.5178 0.3156 80.03

1978 68 16.5 0.5711 0.1665 72.24

2009 68 16.5 0.5711 0.1665 72.24

1993 67 18 0.6245 0.0208 67.53

1977 65 19.5 0.6778 -0.1245 61.83

1992 65 19.5 0.6778 -0.1245 61.83

2007 63 21 0.7312 -0.2728 56.06

2006 62 22 0.7667 -0.3753 52.45

1998 55 23 0.8023 -0.4830 42.71

1979 49 24.5 0.8556 -0.6603 32.20

1997 49 24.5 0.8556 -0.6603 32.20

2000 41 26.5 0.9267 -0.9608 16.56

2001 41 26.5 0.9267 -0.9608 16.56

1971 34 28 0.9801 -1.3651 -0.73

92

Duration: 6.0 hr

Ranked

Year

Rainfall Depth,

d(mm)

Rank

(m)

Execeedence

Probability, P

Reduced

Variable, µ

Gumbel Variable,

XG (mm)

1995 207 1 0.0199 3.9063 321.78

2008 151 2 0.0555 2.8634 235.13

1991 124 3 0.0910 2.3491 193.02

2002 121 4 0.1266 1.9998 179.76

1973 115 5.5 0.1799 1.6176 162.53

2003 115 5.5 0.1799 1.6176 162.53

1994 99 7 0.2333 1.3256 137.95

2004 97 8 0.2688 1.1611 131.12

1972 95 9 0.3044 1.0134 124.78

1974 88 10 0.3400 0.8783 113.81

1975 86 11 0.3755 0.7532 108.13

1980 85 12 0.4111 0.6358 103.68

2005 80 13 0.4467 0.5246 95.42

1976 75 14.67 0.5060 0.3491 85.26

1977 75 14.67 0.5060 0.3491 85.26

1996 75 14.67 0.5060 0.3491 85.26

1978 74 17 0.5889 0.1177 77.46

2009 73 18 0.6245 0.0208 73.61

1992 69 19 0.6600 -0.0759 66.77

2006 68 20 0.6956 -0.1734 62.90

1993 67 21 0.7312 -0.2728 58.99

2007 63 22 0.7667 -0.3753 51.97

1998 59 23 0.8023 -0.4830 44.81

1997 52 24 0.8378 -0.5984 34.42

1979 49 25 0.8734 -0.7260 27.67

2000 44 26.5 0.9267 -0.9608 15.77

2001 44 26.5 0.9267 -0.9608 15.77

1971 34 28 0.9801 -1.3651 -6.11

93

Duration: 12.0 hr

Ranked Rainfall Depth, Rank Execeedence Reduced Gumbel Variable,

Year d(mm) (m) Probability, P Variable, µ XG (mm)

1995 259 1 0.0199 3.9063 398.31

2008 151 2 0.0555 2.8634 253.12

1991 133 3 0.0910 2.3491 216.78

1973 130 4 0.1266 1.9998 201.32

2002 122 5 0.1622 1.7320 183.77

2003 120 6 0.1977 1.5128 173.9503

1977 103 7 0.2333 1.3256 150.28

1994 100 8 0.2688 1.1611 141.41

2004 98 9 0.3044 1.0134 134.14

1972 96 10 0.3400 0.8783 127.33

1975 93 11 0.3755 0.7532 119.86

1980 92 12 0.4111 0.6358 114.68

1974 88 13 0.4467 0.5246 106.71

2005 87 14 0.4822 0.4182 102.26

2009 79 15 0.5178 0.3156 90.16

1978 77 16 0.5533 0.2157 84.69

1996 76 17 0.5889 0.1177 80.20

1976 75 18 0.6245 0.0208 75.74

1992 69 19 0.6600 -0.0759 66.29

2006 68 20 0.6956 -0.1734 61.82

1993 67 21 0.7312 -0.2728 57.27

2007 63 22 0.7667 -0.3753 49.61

1998 62 23 0.8023 -0.4830 44.80

1997 54 24 0.8378 -0.5984 32.23

1979 49 25 0.8734 -0.7260 23.11

2000 44 26.5 0.9267 -0.9608 9.68

2001 44 26.5 0.9267 -0.9608 9.68

1971 34 28 0.9801 -1.3651 -14.69

94

Duration: 24.0 hr

Ranked Rainfall Depth, Rank Execeedence Reduced Gumbel Variable,

Year d(mm) (m) Probability, P Variable, µ XG (mm)

1995 262 1 0.0199 3.9063 402.74

2008 151 2 0.0555 2.8634 254.17

1991 138 3 0.0910 2.3491 222.64

1973 134 4 0.1266 1.9998 205.96

2002 126 5 0.1622 1.7320 188.40

2003 120 6 0.1977 1.5128 174.50

1977 103 7 0.2333 1.3256 150.76

1994 100 8 0.2688 1.1611 141.83

2004 98 9 0.3044 1.0134 134.51

1972 96 10 0.3400 0.8783 127.65

1980 95 11.5 0.3933 0.6937 120.43

1975 95 11.5 0.3933 0.6937 119.99

2005 91 13 0.4467 0.5246 109.62

1974 88 14 0.4822 0.4182 103.07

2009 82 15 0.5178 0.3156 93.55

1978 77 16 0.5533 0.2157 84.77

1996 76 17 0.5889 0.1177 80.24

1976 75 18 0.6245 0.0208 75.75

1992 69 19 0.6600 -0.0759 66.26

2006 68 20 0.6956 -0.1734 61.75

1993 67 21 0.7312 -0.2728 57.17

1998 65 22.5 0.7845 -0.4284 49.65

2007 65 22.5 0.7845 -0.4284 49.65

1997 57 24 0.8378 -0.5984 34.98

1979 49 25 0.8734 -0.7260 22.84

1971 47 26.67 0.9328 -0.9932 11.22

2000 47 26.67 0.9328 -0.9932 10.99

2001 47 26.67 0.9328 -0.9932 10.99

95

Appendix 5: Summarized Distribution Parameters for different Durations

Duration: 0.40 hr

Duration: 0.70 hr

Parameter Description Value (mm)

Sample Mean (µs) 45.96

Sample Standard Deviation (σs) 14.25

Position Parameter (xo) 39.47

Scale Parameter (S) 11.56

Gumbel Mean (µG) 46.14

Gumbel Standard Deviation (σG) 14.83

Mean of Reduced Variable (µN) 0.56

Standard Deviation of Reduced Variable (σN) 1.23

Duration: 1.0 hr

Parameter Description Value (mm)

Sample Mean (µs) 35.86

Sample Standard Deviation (σs) 11.80

Position Parameter (xo) 30.31

Scale Parameter (S) 9.45

Gumbel Mean (µG) 35.76

Gumbel Standard Deviation (σG) 12.13

Mean of Reduced Variable (µN) 0.59

Standard Deviation of Reduced Variable (σN) 1.25

Parameter Description Value (mm)

Sample Mean (µs) 57.46

Sample Standard Deviation (σs) 18.20

Position Parameter (xo) 49.16

Scale Parameter (S) 14.78

Gumbel Mean (µG) 57.69

Gumbel Standard Deviation (σG) 18.95

Mean of Reduced Variable (µN) 0.56

Standard Deviation of Reduced Variable (σN) 1.23

96

Duration: 2.0 hr

Duration: 3.0 hr

Parameter Description Value

Sample Mean (µs) 79.75

Sample Standard Deviation (σs) 31.32

Position Parameter (xo) 65.44

Scale Parameter (S) 25.44

Gumbel Mean (µG) 80.12

Gumbel Standard Deviation (σG) 32.63

Mean of Reduced Variable (µN) 0.56

Standard Deviation of Reduced Variable (σN) 1.23

Duration: 6.0 hr

Parameter Value (mm)

Sample Mean (µs) 85.14

Sample Standard Deviation (σs) 36.15

Position Parameter (xo) 68.52

Scale Parameter (S) 29.38

Gumbel Mean (µG) 85.48

Gumbel Standard Deviation (σG) 37.68

Mean of Reduced Variable (µN) 0.57

Standard Deviation of Reduced Variable (σN) 1.23

Parameter Description Value (mm)

Sample Mean (µs) 69.86

Sample Standard Deviation (σs) 25.39

Position Parameter (xo) 58.20

Scale Parameter (S) 20.62

Gumbel Mean (µG) 70.10

Gumbel Standard Deviation (σG) 26.44

Mean of Reduced Variable (µN) 0.57

Standard Deviation of Reduced Variable (σN) 1.23

97

Duration: 12.0 hr

Parameter Description Value (mm)

Sample Mean (µs) 90.46

Sample Standard Deviation (σs) 43.9303

Position Parameter (xo) 70.39

Scale Parameter (S) 35.66

Gumbel Mean (µG) 90.98

Gumbel Standard Deviation (σG) 45.74

Mean of Reduced Variable (µN) 0.56

Standard Deviation of Reduced Variable (σN) 1.23

Duration: 24.0 hr

Parameter Description Value (mm)

Sample Mean (µs) 92.41

Sample Standard Deviation (σs) 43.77

Position Parameter (xo) 71.75

Scale Parameter (S) 36.03

Gumbel Mean (µG) 92.54

Gumbel Standard Deviation (σG) 46.21

Mean of Reduced Variable (µN) 0.57

Standard Deviation of Reduced Variable (σN) 1.22

98

Appendix 6: Kolmogorov – Smirnov test statistics for different Durations

Duration 0.40 hr

Range B

Frequency Fo(X) U Ft (X)

Dn = Ft(X) - Fo(X) O Cum

16 – 20 20 3 3 0.1071 -1.090 0.051 0.056

21 – 25 25 4 7 0.2500 -0.561 0.173 0.0767

26 – 30 30 2 9 0.3214 -0.032 0.356 0.0346

31 – 35 35 4 13 0.4643 0.497 0.544 0.0798

36 – 40 40 5 18 0.6429 1.025 0.699 0.0558

41 – 45 45 7 25 0.8929 1.554 0.809 0.0834

46 - 50 50 0 25 0.8929 2.083 0.883 0.0100

51 – 55 55 1 26 0.9286 2.612 0.929 -0.0007

56 – 60 60 1 27 0.9643 3.141 0.958 0.0066

61 - 65 65 1 28 1.0000 3.670 0.975 0.0252

Dn value 0.0834 is less than 0.252, therefore null hypothesis is accepted

Duration 0.70 hr

Range B

Frequency Fo(X) U Ft (X)

Dn = Ft(X) - Fo(X) O Cum

21 – 25 25 3 3 0.1071 -1.2513 0.030 0.0768

26 – 30 30 3 6 0.2143 -0.8188 0.104 0.1108

31 – 35 35 1 7 0.2500 -0.3863 0.230 0.0204

36 – 40 40 4 11 0.3929 0.0462 0.385 0.0080

41 – 45 45 2 13 0.4643 0.4787 0.538 0.0739

46 – 50 50 3 16 0.5714 0.9112 0.669 0.0975

51 – 55 55 4 20 0.7143 1.3437 0.770 0.0561

56 – 60 60 4 24 0.8571 1.7762 0.844 0.0129

61 – 65 65 1 25 0.8929 2.2087 0.896 0.0031

66 – 70 70 2 27 0.9643 2.6413 0.931 0.0331

71 – 75 75 1 28 1.0000 3.0738 0.955 0.0452

Dn value 0.1108 is less than 0.252, therefore null hypothesis is accepted

99

Duration 1.0 hr

Range B

Frequency Fo(X) U Ft (X)

Dn = Ft(X) -

Fo(X) O Cum

21 – 25 25 1 1 0.0357 -1.6351 0.006 0.0298

26 – 30 30 0 1 0.0357 -1.2967 0.026 0.0099

31 – 35 35 3 4 0.1429 -0.9584 0.074 0.0691

36 – 40 40 2 6 0.2143 -0.6200 0.156 0.0584

41 – 45 45 3 9 0.3214 -0.2816 0.266 0.0557

46 – 50 50 1 10 0.3571 0.0567 0.389 0.0316

51 – 55 55 1 11 0.3929 0.3951 0.510 0.1170

56 – 60 60 3 14 0.5000 0.7335 0.619 0.1186

61 – 65 65 6 20 0.7143 1.0718 0.710 0.0042

66 – 70 70 3 23 0.8214 1.4102 0.783 0.0380

71 – 75 75 1 24 0.8571 1.7485 0.840 0.0169

76 – 80 80 2 26 0.9286 2.0869 0.883 0.0453

81 -85 85 0 26 0.9286 2.4253 0.915 0.0132

86 – 90 90 0 26 0.9286 2.7636 0.939 0.0103

91 – 95 95 1 27 0.9643 3.1020 0.956 0.0082

96 – 100 100 1 28 1.0000 3.4404 0.968 0.0315

Dn value 0.1186 is less than 0.252, therefore null hypothesis is accepted

100

Duration 2.0 hr

Range B

Frequency Fo(X) U Ft (X)

Dn = Ft(X) -

Fo(X) O Cum

31 – 35 35 1 1 0.0357 -1.1253 0.046 0.0102

36 – 40 40 3 4 0.1429 -0.8827 0.089 0.0537

41 – 45 45 1 5 0.1786 -0.6402 0.150 0.0285

46 – 50 50 2 7 0.2500 -0.3977 0.226 0.0243

51 – 55 55 1 8 0.2857 -0.1551 0.311 0.0253

56 – 60 60 2 10 0.3571 0.0874 0.400 0.0429

61 – 65 65 3 13 0.4643 0.3299 0.487 0.0230

66 – 70 70 6 19 0.6786 0.5725 0.569 0.1097

71 – 75 75 1 20 0.7143 0.8150 0.642 0.0719

76 – 80 80 1 21 0.7500 1.0576 0.707 0.0434

81 – 85 85 1 22 0.7857 1.3001 0.761 0.0242

86 – 90 90 1 23 0.8214 1.5426 0.807 0.0139

91 – 95 95 2 25 0.8929 1.7852 0.846 0.0473

96 - 100 100 0 25 0.8929 2.0277 0.877 0.0162

101 - 105 105 0 25 0.8929 2.2702 0.902 0.0090

106 - 110 110 2 27 0.9643 2.5128 0.922 0.0421

111 -115 115 0 27 0.9643 2.7553 0.938 0.0259

116 - 120 120 0 27 0.9643 2.9978 0.951 0.0130

121 -125 125 0 27 0.9643 3.2404 0.962 0.0027

126 -130 130 0 27 0.9643 3.4829 0.970 0.0055

131 - 135 135 0 27 0.9643 3.7254 0.976 0.0119

136 - 140 140 0 27 0.9643 3.9680 0.981 0.0170

141 - 145 145 0 27 0.9643 4.2105 0.985 0.0210

146 - 150 150 1 28 1.0000 4.4530 0.988 0.0116

Dn value 0.1097 is less than 0.252, therefore null hypothesis is accepted

101

Duration 3.0 hr

Range

B Frequency

Fo(X) U Ft (X) Dn = Ft(X) -

Fo(X) O Cum

31 - 40 40 1 1 0.0357 -0.9999 0.066 0.0303

41 - 50 50 4 5 0.1786 -0.6068 0.160 0.0189

51 - 60 60 1 6 0.2143 -0.2137 0.290 0.0756

61 - 70 70 7 13 0.4643 0.1793 0.434 0.0308

71 - 80 80 3 16 0.5714 0.5724 0.569 0.0026

81 - 90 90 5 21 0.7500 0.9655 0.683 0.0667

91 - 100 100 2 23 0.8214 1.3586 0.773 0.0481

101 - 110 110 2 25 0.8929 1.7517 0.841 0.0521

111 - 120 120 1 26 0.9286 2.1448 0.890 0.0391

121 - 130 130 0 26 0.9286 2.5379 0.924 0.0046

131 - 140 140 0 26 0.9286 2.9310 0.948 0.0195

141 - 150 150 0 26 0.9286 3.3240 0.965 0.0361

151 - 160 160 1 27 0.9643 3.7171 0.976 0.0117

161 - 170 170 1 28 1.0000 4.1102 0.984 0.0163

Dn value 0.0756 is less than 0.252, therefore null hypothesis is accepted

102

Duration 6.0 hr

Range B Frequency

Fo(X) U Ft (X) Dn = Ft(X) - Fo(X) O Cum

31 – 40 40 1 1 0.0357 -0.9707 0.071 0.0357

41 - 50 50 3 4 0.1429 -0.6304 0.153 0.0100

51 – 60 60 2 6 0.2143 -0.2900 0.263 0.0485

61 – 70 70 4 10 0.3571 0.0503 0.386 0.0292

71 – 80 80 6 16 0.5714 0.3907 0.508 0.0631

81 – 90 90 3 19 0.6786 0.7310 0.618 0.0607

91 – 100 100 3 22 0.7857 1.0713 0.710 0.0758

101 - 110 110 0 22 0.7857 1.4117 0.784 0.0020

111 - 120 120 2 24 0.8571 1.7520 0.841 0.0164

121 - 130 130 2 26 0.9286 2.0923 0.884 0.0447

131 - 140 140 0 26 0.9286 2.4327 0.916 0.0126

141 - 150 150 0 26 0.9286 2.7730 0.939 0.0109

151 - 160 160 1 27 0.9643 3.1134 0.957 0.0078

161 - 170 170 0 27 0.9643 3.4537 0.969 0.0046

171 - 180 180 0 27 0.9643 3.7940 0.978 0.0135

181 - 190 190 0 27 0.9643 4.1344 0.984 0.0198

191 - 200 200 0 27 0.9643 4.4747 0.989 0.0244

201 - 210 210 1 28 1.0000 4.8151 0.992 0.0081

Dn value 0.0758 is less than 0.252, therefore null hypothesis is accepted

103

Duration 12.0 hr

Range B

Frequency Fo(X) U Ft (X) Dn = Ft(X) - Fo(X)

O Cum

31 - 40 40 1 1 0.0357 -0.8522 0.096 0.0602

41 - 50 50 3 4 0.1429 -0.5718 0.170 0.0272

51 - 60 60 1 5 0.1786 -0.2914 0.262 0.0837

61 - 70 70 5 10 0.3571 -0.0110 0.364 0.0067

71 - 80 80 4 14 0.5000 0.2694 0.466 0.0341

81 - 90 90 2 16 0.5714 0.5498 0.562 0.0099

91 - 100 100 5 21 0.7500 0.8302 0.647 0.1034

101 - 110 110 1 22 0.7857 1.1106 0.719 0.0663

111 - 120 120 1 23 0.8214 1.3910 0.780 0.0417

121 - 130 130 2 25 0.8929 1.6714 0.829 0.0642

131 - 140 140 1 26 0.9286 1.9518 0.868 0.0610

141 - 150 150 0 26 0.9286 2.2322 0.898 0.0303

151 - 160 160 1 27 0.9643 2.5126 0.922 0.0421

161 - 170 170 0 27 0.9643 2.7930 0.941 0.0237

171 - 180 180 0 27 0.9643 3.0734 0.955 0.0095

181 - 190 190 0 27 0.9643 3.3538 0.966 0.0014

191 - 200 200 0 27 0.9643 3.6342 0.974 0.0097

201 - 210 210 0 27 0.9643 3.9146 0.980 0.0160

211 - 220 220 0 27 0.9643 4.1950 0.985 0.0208

221 - 230 230 0 27 0.9643 4.4754 0.989 0.0244

231 - 240 240 0 27 0.9643 4.7558 0.991 0.0271

241 - 250 250 0 27 0.9643 5.0362 0.994 0.0292

251 - 260 260 1 28 1.0000 5.3166 0.995 0.0049

Dn value 0.1034 is less than 0.252, therefore null hypothesis is accepted

104

Duration 24.0 hr

Range B

Frequency

Fo(X)

U

Ft (X)

Dn = Ft(X) - Fo(X) Observed Cum

41 - 50 50 4 4 0.1429 -0.6036 0.161 0.0178

51 - 60 60 1 5 0.1786 -0.3261 0.250 0.0716

61 - 70 70 5 10 0.3571 -0.0485 0.350 0.0071

71 - 80 80 3 13 0.4643 0.2291 0.451 0.0128

81 - 90 90 2 15 0.5357 0.5066 0.547 0.0117

91 - 100 100 6 21 0.7500 0.7842 0.633 0.1165

101 - 110 110 1 22 0.7857 1.0617 0.708 0.0781

111 - 120 120 1 23 0.8214 1.3393 0.769 0.0519

121 - 130 130 1 24 0.8571 1.6168 0.820 0.0372

131 - 140 140 2 26 0.9286 1.8944 0.860 0.0682

141 - 150 150 0 26 0.9286 2.1719 0.892 0.0363

151 - 160 160 1 27 0.9643 2.4495 0.917 0.0470

161 - 170 170 0 27 0.9643 2.7271 0.937 0.0276

171 - 180 180 0 27 0.9643 3.0046 0.952 0.0126

181 - 190 190 0 27 0.9643 3.2822 0.963 0.0011

191 - 200 200 0 27 0.9643 3.5597 0.972 0.0077

201 - 210 210 0 27 0.9643 3.8373 0.979 0.0144

211 - 220 220 0 27 0.9643 4.1148 0.984 0.0195

221 - 230 230 0 27 0.9643 4.3924 0.988 0.0234

231 - 240 240 0 27 0.9643 4.6699 0.991 0.0264

241 - 250 250 0 27 0.9643 4.9475 0.993 0.0286

251 - 260 260 0 27 0.9643 5.2250 0.995 0.0303

261 - 270 270 1 28 1.0000 5.5026 0.996 0.0041

Dn value 0.1165 is less than 0.252, therefore null hypothesis is accepted

105

Appendix 7: Critical Values table for Kolmogorov – Smirnov Test

Sample

Size Level of Significance

(n) α = 0.20 α = 0.15 α = 0.10 α = 0.05 α = 0.01

1 0.900 0.925 0.950 0.975 0.995

2 0.684 0.726 0.776 0.842 0.929

3 0.565 0.597 0.642 0.708 0.828

4 0.494 0.525 0.564 0.624 0.733

5 0.446 0.474 0.510 0.565 0.669

6 0.410 0.436 0.470 0.521 0.618

7 0.381 0.405 0.438 0.486 0.577

8 0.358 0.381 0.411 0.457 0.543

9 0.339 0.360 0.388 0.432 0.514

10 0.322 0.342 0.368 0.410 0.490

11 0.307 0.326 0.352 0.391 0.468

12 0.295 0.313 0.338 0.375 0.450

13 0.284 0.302 0.325 0.361 0.433

14 0.274 0.292 0.314 0.349 0.418

15 0.266 0.283 0.304 0.338 0.404

16 0.258 0.274 0.295 0.328 0.392

17 0.250 0.266 0.286 0.318 0.381

18 0.244 0.259 0.278 0.309 0.371

19 0.237 0.252 0.272 0.301 0.363

20 0.231 0.246 0.264 0.294 0.356

25 0.210 0.220 0.240 0.270 0.320

30 0.190 0.200 0.220 0.240 0.290

35 0.180 0.190 0.210 0.230 0.270

> 35

(Source: MCcuen, 1941b)

4/√𝑛 /√𝑛 /√𝑛 36/√𝑛 63/√𝑛

106

Appendix 8: Chi –Square repartition tables for observed and expected frequencies

Duration 0.20 hr Duration 0.40 hr

Interval Observed Expected

(mm) Frequency Frequency

5 to 20 3 7

0 2- 25 4 0

25 – 30 2 2

30 – 35 4 2

35 – 40 5 2

40 -45 7 4

45 – 50 0 3

50 -55 1 1

55 – 60 1 0

60 – 65 1 4

65 – 70 0 0

70 – 75 0 0

75 – 80 0 1

80 – 85 0 0

85 -90 0 1

90 - 100 0 1 Total 28 28

Duration 0.70 hr Duration 1.0 hr

Interval Observed Expected

(mm) Frequency Frequency

5 to 15 0 8

15 – 20 6 2

20 – 25 5 3

25 – 30 6 4

30 – 35 9 4

35 – 40 0 0

40 – 45 1 5

45 – 50 0 0

50 – 55 1 0

55 – 60 0 1

60 -65 0 0

65 – 70 0 0

70 -75 0 1

Total 28 28

Interval observed Expected

(mm) Frequency Frequency

5 -10 0 1

10 - 15 0 2

15 - 20 0 1

20 - 25 3 2

25 - 30 3 0

30 - 35 1 3

35 - 40 4 1

40 - 45 2 1

45 - 50 3 2

50 - 55 4 2

55 - 60 4 2

60 - 65 1 2

65 - 70 2 2

70 - 75 1 1

75 - 80 0 2

80 - 85 0 0 85 -90 0 1

90 - 95 0 1

95 - 100 0 0

100 - 105 0 1

105 - 120 0 1

Total 28 28

Interval observed Expected

(mm) Frequency Frequency

< 20 1 3

20 – 25 0 1

25 – 30 3 2

30 – 35 2 0

35 - 40 3 1

40 -45 1 2

45 – 50 1 1

50 – 55 3 1

55 – 60 6 1

60 – 65 3 2

65 – 70 1 1

70 – 75 2 2

75 - 80 0 3

80 – 85 1 2

85 - 90 1 1

90 – 95 0 0

95 - 100 0 1

100 - 105 0 0

105 - 110 0 1

110 -115 0 1

115 - 120 0 0

120 - 160 0 2

Total 28 28

107

Duration 2.0 hr Duration 3.0 hr

Interval Observed Expected

(mm) Frequency Frequency

< 30 0 3

30 - 35 1 2

35 - 40 0 0

40 - 45 2 1

45 - 50 2 0

50 - 55 1 1

55 - 60 0 1

60 - 65 4 2

65 - 70 3 1

70 - 75 3 2

75 - 80 0 0

80 - 85 2 1

85 - 90 3 2

90 - 95 1 0

95 - 100 1 0

100 - 105 0 2

105 - 110 2 1

110 - 115 0 1

115 - 120 1 1

> 120 2 7

Total 28 28

Interval observed Expected

(mm) Frequency Frequency

< 20 0 2

20 - 25 0 1

25 - 30 0 1

30 - 35 1 1

35 - 40 3 1

40 -45 1 1

45 - 50 2 1

50 -55 1 0

55 - 60 1 2

60 - 65 3 1

65 - 70 6 1

70 - 75 1 3

75 - 80 1 1

80 - 85 2 1

85 -90 1 2

90 - 95 2 0

95 -100 0 0

100 - 105 0 1

105 - 110 2 1

110 - 115 0 1

115 - 120 0 1

120 - 125 0 0

125 - 130 0 1

130 - 135 0 0

135 - 140 0 1

140 - 145 0 0

145 - 230 1 3

Total 28 28

108

Duration 6.0 hr Duration 12.0 hr

Interval Observed Expected

(mm) Frequency Frequency

< 30 0 4

30 - 35 1 1

35 - 40 0 0

40 - 45 2 1

45 - 50 1 1

50 - 55 1 0

55 - 60 0 1

60 - 65 2 1

65 - 70 3 1

70 - 75 1 0

75 - 80 3 1

80 - 85 0 2

85 - 90 2 0

90 - 95 2 1

95 - 100 3 0

100 - 105 1 1

105 - 110 0 1

110 - 115 0 1

115 - 120 1 1

120 - 125 1 0

125 - 130 1 1

> 130 3 9

Total 28 28

Duration 24.0 hr

Interval Observed Expected

(mm) Frequency Frequency

< 30 0 4

30 - 35 1 1

35 - 40 0 0

40 - 45 2 1

45 - 50 1 0

50 - 55 1 1

55 - 60 1 1

60 - 65 1 1

65 - 70 3 2

70 - 75 5 1

75 - 80 1 1

80 - 85 1 0

85 - 90 2 3

90 - 95 1 0

95 - 100 2 0

100 - 105 0 1

105 - 110 0 1

110 - 115 2 1

115 - 120 0 0

120 - 125 2 1

125 -130 0 0

> 130 2 8

Total 28 28

Interval observed Expected

(mm) Frequency Frequency

10 to 35 0 5

35- 40 0 0

40 - 45 0 0

45 - 50 4 2

50 - 55 0 0

55 - 60 1 1

60 - 65 2 1

65 - 70 3 1

70 - 75 1 0

75 - 80 2 1

80 - 85 1 2

85 - 90 1 0

90 - 95 3 1

95 - 100 3 0

100 - 105 1 1

105 - 110 0 1

110 - 115 0 0

115 - 120 1 1

120 - 125 0 1

125 - 130 1 1

> 130 4 9

Total 28 28

109

Appendix 9: Chi-Square Test result for different durations

Duration 0.40 hr

5.26 < 5.991 at 5% significance level, hence null hypothesis is accepted

Duration 0.70 hr

10.28 < 12.592 at 5% significance level, hence null hypothesis is accepted

Duration 1.0 hr

Interval O E O - E (O - E)2 (O - E)

2 / E

(mm) No of Frequency

< 60 14 12 2 4 0.33

60 - 100 14 12 2 4 0.33

100 - 110 0 1 -1 1 1.00

110 - 120 0 3 -3 9 3.00

Total 4.67 4.67 > 3.841 at 5% significance level, hence null hypothesis is rejected

Interval O E O - E (O - E)2 (O - E)

2 / E

(mm) No of Frequency

5 to 40 18 13 5 25 1.92

40 - 65 10 12 -2 4 0.33

65 - 80 0 1 -1 1 1.00

80 - 90 0 1 -1 1 1.00

90 - 100 0 1 -1 1 1.00

Total 5.26

Interval

(mm) O E O – E (O - E)

2 (O - E)

2 / E

No of Frequency

5 to 50 16 13 3 9 0.69

50 -55 4 2 2 4 2.00

55 -60 4 2 2 4 2.00

60 -70 3 4 -1 1 0.25

70 - 80 1 3 -2 4 1.33

80 - 90 0 1 -1 1 1.00

90 - 95 0 1 -1 1 1.00

95 - 105 0 1 -1 1 1.00

105 - 120 0 1 -1 1 1.00

Total 10.28

110

Duration 2.0 hr

11.57 < 14.067 at 5% significance level, hence null hypothesis is accepted

Duration 3.0 hr

7.57 < 9.488 at 5% significance level, hence null hypothesis is accepted

Duration 6.0 hr

Interval O E O – E (O - E)2 (O - E)2 / E

(mm) No of Frequency

< 55 5 7 -2 4 0.57 55 - 70 5 4 1 1 0.25 70 - 110 14 8 6 36 4.50

> 110 4 9 -5 25 2.78

Total 8.10 8.10 > 3.841 at 5% significance level, hence null hypothesis is rejected

Interval O E O - E (O - E)2 (O - E)

2 / E

(mm) No of Frequency

< 45 5 7 -2 4 0.57

45 -60 4 3 1 1 0.33

60 - 75 10 5 5 25 5.00

75 - 85 3 2 1 1 0.50

85 - 90 1 2 -1 1 0.50

90 - 115 4 3 1 1 0.33

115 - 120 0 1 -1 1 1.00

120 - 130 0 1 -1 1 1.00

130 - 140 0 1 -1 1 1.00

140 - 230 1 3 -2 4 1.33

Total 11.57

Interval O E O - E (O - E)2 (O - E)

2 / E

(mm) No of Frequency

< 30 0 3 -3 9 3.00

30 - 35 1 2 -1 1 0.50

35 - 85 7 9 -2 4 0.44

85 - 90 3 2 1 1 0.50

90 - 105 2 2 0 0 0.00

105 - 115 2 2 0 0 0.00

115 - 270 3 8 -5 25 3.13

Total 7.57

111

Duration 12.0 hr

Interval O E O - E (O - E)2

(O - E)2 /

E

(mm) No of Frequency

< 75 11 10 1 1 0.10

75 - 95 7 4 3 9 2.25

95 - 115 4 3 1 1 0.33

> 115 6 11 -5 25 2.27

Total 4.96 4.96 > 3.841 at 5% significance level, hence null hypothesis rejected

Duration 24.0 hr

Interval O E O - E (O - E)2 (O - E)

2 / E

(mm) No of Frequencies

10 to 95 18 14 4 16 1.14

95 - 110 4 2 2 4 2.00

110 - 130 2 3 -1 1 0.33

> 130 4 9 -5 25 2.78

Total 6.25

6.25 > 3.841 at 5% significance level, hence null hypothesis is rejected

112

Appendix 10: Chi – Square Distribution table

df is degree of freedom

113

Appendix 11: Estimates of Rainfall Intensities for different durations and return

periods.

Duration 0.40 hr

Return

Period

Year

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean (µG)

Gumbel

Stdev.(σG

)

Rainfall

Depth XT

(mm)

Intensity

I = XT

(mm)/Hr

5 0.40 0.875 35.7627 12.1252 46.37 115.93

10 0.40 1.5546 35.7627 12.1252 54.61 136.53

15 0.40 1.9384 35.7627 12.1252 59.27 148.17

20 0.40 2.2068 35.7627 12.1252 62.52 156.30

25 0.40 2.4134 35.7627 12.1252 65.03 162.56

50 0.40 3.0508 35.7627 12.1252 72.75 181.89

100 0.40 3.6834 35.7627 12.1252 80.42 201.06

Duration 0.70 hr

Return

Period

Year

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean (µG )

Gumbel

Stdev. (σG

)

Rainfall

Depth

XT (mm)

Intensity

I = XT

(mm)/Hr

5 0.70 0.8750 46.1386 14.8262 59.11 84.45

10 0.70 1.5546 46.1386 14.8262 69.19 98.84

15 0.70 1.9384 46.1386 14.8262 74.88 106.97

20 0.70 2.2068 46.1386 14.8262 78.86 112.65

25 0.70 2.4134 46.1386 14.8262 81.92 117.03

50 0.70 3.0508 46.1386 14.8262 91.37 130.53

100 0.70 3.6834 46.1386 14.8262 100.75 143.93

114

Duration 1.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean

(µG)

Gumbel

Stdev.(σG)

Rainfall

Depth

XT (mm)

Intensity

I = XT (mm)/Hr

5 1.00 0.875 57.6911 18.9515 74.27 74.27

10 1.00 1.5546 57.6911 18.9515 87.15 87.15

15 1.00 1.9384 57.6911 18.9515 94.43 94.43

20 1.00 2.2068 57.6911 18.9515 99.51 99.51

25 1.00 2.4134 57.6911 18.9515 103.43 103.43

50 1.00 3.0508 57.6911 18.9515 115.51 115.51

100 1.00 3.6834 57.6911 18.9515 127.50 127.50

Duration 2.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean(µG)

Gumbel

Stdev(σG)

Rainfall

Depth

XT

(mm)

Intensity

I = XT (mm)/Hr

5 2.00 0.875 70.0972 26.4395 93.23 46.62

10 2.00 1.5546 70.0972 26.4395 111.20 55.60

15 2.00 1.9384 70.0972 26.4395 121.35 60.67

20 2.00 2.2068 70.0972 26.4395 128.44 64.22

25 2.00 2.4134 70.0972 26.4395 133.91 66.95

50 2.00 3.0508 70.0972 26.4395 150.76 75.38

100 2.00 3.6834 70.0972 26.4395 167.48 83.74

Duration 3.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean(µG)

Gumbel

Stdev(σG)

Rainfall

Depth

XT

(mm)

Intensity

I = XT (mm)/Hr

5 3.00 0.875 80.12 32.6263 108.67 36.22

10 3.00 1.5546 80.12 32.6263 130.84 43.61

15 3.00 1.9384 80.12 32.6263 143.36 47.79

20 3.00 2.2068 80.12 32.6263 152.12 50.71

25 3.00 2.4134 80.12 32.6263 158.86 52.95

50 3.00 3.0508 80.12 32.6263 179.66 59.89

100 3.00 3.6834 80.12 32.6263 200.30 66.77

115

Duration 6.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean(µ)

Gumbel

Stdev(σG)

Rainfall

Depth

XT (mm)

Intensity

I = XT (mm)/Hr

5 6.00 0.875 85.4802 37.683 118.45 19.74

10 6.00 1.5546 85.4802 37.683 144.06 24.01

15 6.00 1.9384 85.4802 37.683 158.52 26.42

20 6.00 2.2068 85.4802 37.683 168.64 28.11

25 6.00 2.4134 85.4802 37.683 176.42 29.40

50 6.00 3.0508 85.4802 37.683 200.44 33.41

100 6.00 3.6834 85.4802 37.683 224.28 37.38

Duration 12.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor(K)

Gumbel

Mean

(µG )

Gumbel

Stdev(σG)

Rainfall

Depth

XT (mm)

Intensity

I = XT (mm)/Hr

5 12.00 0.875 90.9770 45.7386 131.00 10.92

10 12.00 1.5546 90.9770 45.7386 162.08 13.51

15 12.00 1.9384 90.9770 45.7386 179.64 14.97

20 12.00 2.2068 90.9770 45.7386 191.91 15.99

25 12.00 2.4134 90.9770 45.7386 201.36 16.78

50 12.00 3.0508 90.9770 45.7386 230.52 19.21

100 12.00 3.6834 90.9770 45.7386 259.45 21.62

Duration 24.0 hr

Return

Periods

Years

Duration

Hr

Frequency

Factor (K)

Gumbel

Mean(µG)

Gumbel

Stdev(σG)

Rainfall

Depth

XT (mm)

Intensity

I = XT (mm)/Hr

5 24.00 0.875 92.5431 46.207 132.97 5.54

10 24.00 1.5546 92.5431 46.207 164.38 6.85

15 24.00 1.9384 92.5431 46.207 182.11 7.59

20 24.00 2.2068 92.5431 46.207 194.51 8.10

25 24.00 2.4134 92.5431 46.207 204.06 8.50

50 24.00 3.0508 92.5431 46.207 233.51 9.73

100 24.00 3.6834 92.5431 46.207 262.74 10.95

116

Appendix 12: Table of Frequency Factor (K) for Extreme Value Type 1(EV1)

Sample Size Return Period (years)

N 5 10 15 20 25 50 100

15 0.967 1.703 2.117 2.410 2.632 3.321 4.005

20 0.919 1.625 2.023 2.302 2.517 3.179 3.836

25 0.888 1.575 1.963 2.235 2.444 3.088 3.729

30 0.866 1.541 1.922 2.188 2.393 3.026 3.653

35 0.851 1.516 1.891 2.152 2.354 2.979 3.598

40 0.838 1.495 1.866 2.126 2.326 2.943 3.554

45 0.829 1.478 1.847 2.104 2.303 2.913 3.520

50 0.820 1.466 1.831 2.086 2.283 2.889 3.491

75 0.792 1.423 1.780 2.029 2.220 2.812 4.400

100 0.779 1.401 1.752 1.998 2.187 2.770 3.349

0.719 1.305 1.635 1.866 2.044 2.592 3.137

(Source: Kendall, 1959)

∞

117

Appendix 13: J. B Dankwa Maximum Rainfall Intensities Duration Frequency for

Accra

Rainfall Duration

Hours

RETURN PERIODS

(YEARS)

5 10 25 50 100

0.20 127.00 140.97 165.10 180.34 196.85

0.40 99.10 116.84 132.08 147.32 162.56

0.70 74.40 85.60 99.31 109.98 120.40

1.00 62.50 71.88 84.07 92.96 101.85

2.00 37.85 44.50 52.83 59.18 65.53

3.00 29.21 33.02 38.10 43.67 48.26

6.00 15.75 19.56 23.88 27.18 28.70

12.00 8.64 10.67 13.21 15.24 17.02

24.00 4.32 5.33 6.60 7.62 8.64