Application of the Joint Probability Approach to Ungauged Catchments for Design Flood Estimation

Transcription

1 Application of the Joint Probability Approach to Ungauged Catchments for Design Flood Estimation By Tanvir Mazumder Student ID: M. Eng. (Hons) Thesis School of Engineering and Industrial Design University of Western Sydney August 2005 Principal Supervisor: Dr. Ataur Rahman Associate Supervisor: Dr. Surendra Shrestha Associate Supervisor: Professor Steven Riley 1

2 Acknowledgements I would like to thank my Principal Supervisor Dr. Ataur Rahman for his assistance, guidance and valuable advice in undertaking this research. I would also like to acknowledge my Associate Supervisors Dr. Surendra Shrestha and Professor Steven Riley for their continued support and direction while undertaking this thesis. I also thank the School of Engineering and Industrial Design for supporting this research work.

3 Statement of Authentication The work presented in this thesis is, to the best of my knowledge and belief, original expect as acknowledged in the text. I hereby declare that I have not submitted this material, either in full or in part, for a degree at this or any other institution. Tanvir Mazumder (Signature)

4 TABLE OF CONTENTS ABSTRACT viii ACRONYMS AND ABBREVIATIONS x 1.0 INTRODUCTION Background Objectives Thesis outline DESIGN FLOOD ESTIMATION METHODS BASED ON THE JOINT PROBABILITY APPROACH General Flood estimation methods overview Rainfall-based flood estimation methods (i) Empirical methods (ii) Continuous simulation method (iii) Design Event Approach (iv) Joint Probability Approach Description of the Joint Probability Approach (i) Approximate methods (a) Discrete methods (b) Simulation techniques (ii) Analytical methods i

5 (1) Methods based on U.S. Soil Conservation Service s curve number procedure (2) Methods based on Eagleson s kinematic runoff model (3) Methods based on other types of rainfall-runoff models (4) Methods based on geomorphologic unit hydrograph Recent research on the joint probability approach to design flood estimation Index of major research works on the Joint Probability Approach Joint probability approach to ungauged catchments Description of the Monte Carlo Simulation Technique to design flood estimation of Rahman et al. (2002) General Event definition Distribution of flood-producing variables Duration Intensity Temporal pattern Initial loss Simulation of derived flood frequency curves Proposed research DESCRIPTION OF DATA Pluviograph data Catchments for the validation of the new technique ii

6 3.2 Summary METHODOLOGY IN THE PROPOSED RESEARCH Steps in the proposed research Rainfall analysis Loss analysis Calibration of runoff routing model Simulation of streamflow hydrograph Testing the hypothesis whether storm-core duration can be described by an exponential distribution Program to compute weighted average IFD table at an ungauged catchment RESULTS Distribution of storm-core duration Derivation of intensity-frequency-duration curves Regionalisation of the distributions of various flood producing variables Storm-core duration Storm-core rainfall intensity Storm-core temporal patterns Initial loss Continuing loss, storage delay parameter, non-linearity parameter and baseflow iii

7 5.4 Derived flood frequency curves with regionalised parameters of the input variables Sensitivity analyses Continuing loss Catchment storage parameter (k) Comparison among Joint Probability Approach, Probabilistic Rational Method and Quantile Regression Technique SUMMARY AND CONCLUSIONS Summary Conclusions Recommendation of further study REFERENCES APPENDICES Appendix A: List of study pluviograph stations Appendix B: FORTRAN program to compute weighted average IFD table at an ungauged catchment Appendix C: Distributions of storm-core durations for selected catchments-0 Appendix D: IFD tables for the study catchments Appendix E: IFD curves for selected catchments Appendix F: Flood frequency curves for selected test catchments Appendix G: A Numerical example illustrating the identification of a storm-core--154 iv

8 TABLES Table 2.1: Index of previous research on the Joint Probability Approach Table 3.1: Selected gauged catchments for validating the new technique Table 3.2: Selected additional test catchments and relevant data Table 4.1: Hourly pluviograph data for Pluviograph Station 76031(Mildura Mo) 46 Table 4.2: Parameter file a76031.psa for rainfall analysis (for Station 76031) -47 Table 4.3: Important output files from program mcsa5.for (for Station 76031) 47 Table 4.4: Parameter file a40082.lan- (Bremer River catchment) for rainfall analysis Table 4.5: Output file for a40082-(bremer River catchment) for rainfall analysis Table 4.6: Parameter file re1s1.par (region-1, station-1) for simulation of streamflow hydrograph Table 4.7: Output file for re1s1.par (region-1, station-1) for simulation of streamflow hydrograph Table 5.2.1: IFD table for Station Table 5.4.1: Radius of subregions for the Boggy Creek catchment Table 5.4.2: The percentage difference between the DFFCs and observed floods for the Boggy Creek catchment Table 5.4.3: Radius of subregions for the Tarwin River catchment v

9 Table 5.4.4: Percentage differences between the DFFCs and observed floods for the Tarwin River catchment Table 5.4.5: Radius of subregions for the Avoca River catchment Table 5.4.6: The percentage difference between the DFFCs and observed floods for the Avoca River catchment Table 5.6.1: Flood estimation obtained by three methods (QRM, PRM and JPA) Table 5.6.2:Median relative error values (%) for three methods JPA, QRT and PRM FIGURES Figure 2.1: Classification of design flood estimation methods Figure 2.2: Flood Estimation by Design Event Approach Figure 2.8.1: Histogram of storm-core durations d c at pluviograph station Mildura MO (76031) Figure 2.9: Schematic diagram of Monte Carlo Simulation Figure 3.1: Locations of the selected pluviograph stations in Victoria Figure 3.2: Distribution of record lengths of the selected pluviograph stations Figure 4.3: Baseflow separation for the Bremer River Catchment Figure 4.4: Observed vs. computed streamflow data for a selected event for the Bremer River catchment Figure 4.5: Output windows to compute weighted average IFD table Figure 5.1.1: Histogram of storm-core duration for Pluviograph Station vi

10 Figure 5.2.1: Plot of IFD curves for Station Figure 5.3.1: Various zones in Victoria for regionalisation of storm-core duration---59 Figure 5.3.2: Sample storm-core temporal pattern database for 12h durations for a region with 3 pluviograph stations for the Boggy Creek catchment Figure 5.4.1: Derived flood frequency curves for the Boggy Creek catchment using regionalised parameters Figure 5.4.2: Derived flood frequency curves for the Tarwin River catchment using regionalised parameters Figure 5.4.3: Derived flood frequency curves for the Avoca River catchment of using regionalised parameters Figure 5.4.4: Box plot of relative errors for the three test catchments Figure 5.5.1: Effects on derived flood frequency curves for the Boggy Creek Catchment of using different continuing loss values---72 Figure 5.5.2: Effects on derived flood frequency curves for the Tarwin River Catchment of using different continuing loss values Figure 5.5.3: Effects on derived flood frequency curves for the Avoca River Catchment of using different continuing loss values Figure 5.5.4: Effects on derived flood frequency curves for the Boggy Creek Catchment of using different k values Figure 5.5.5: Effects on derived flood frequency curves for the Tarwin River Catchment of using different k values vii

11 Figure 5.5.6: Effects on derived flood frequency curve for the Avoca River Catchment of using different k values Figure 5.6.1: Box plot for relative errors for JPA Figure 5.6.2: Box plot for relative errors for PRM Figure 5.6.3: Box plot for relative errors for QRT viii

12 Abstract Design flood estimation is often required in hydrologic practice. For catchments with sufficient streamflow data, design floods can be obtained using flood frequency analysis. For catchments with no or little streamflow data (ungauged catchments), design flood estimation is a difficult task. The currently recommended method in Australia for design flood estimation in ungauged catchments is known as the Probabilistic Rational Method. There are alternatives to this method such as Quantile Regression Technique or Index Flood Method. All these methods give the flood peak estimate but the full streamflow hydrograph is required for many applications. The currently recommended rainfall based flood estimation method in Australia that can estimate full streamflow hydrograph is known as the Design Event Approach. This considers the probabilistic nature of rainfall depth but ignores the probabilistic behavior of other flood producing variables such as rainfall temporal pattern and initial loss, and thus this is likely to produce probability bias in final flood estimates. Joint Probability Approach is a superior method of design flood estimation which considers the probabilistic nature of the input variables (such as rainfall temporal pattern and initial loss) in the rainfall-runoff modelling. Rahman et al. (2002) developed a simple Monte Carlo Simulation technique based on the principles of joint probability, which is applicable to gauged catchments. This thesis extends the Monte Carlo Simulation technique by Rahman et al. (2002) to ungauged catchments. The Joint Probability Approach/ Monte Carlo Simulation Technique requires identification of the distributions of the input variables to the rainfall-runoff model e.g. rainfall duration, rainfall intensity, rainfall temporal pattern, and initial loss. For gauged catchments, these probability distributions are identified from observed rainfall and/or streamflow data. For application of the Joint Probability Approach to ungauged catchments, the distributions of the input variables need to be regionalised. This thesis, in particular, investigates the regionalisation of the distribution of rainfall duration and intensity. ix

13 In this thesis, it is hypothesised that the distribution of storm duration can be described by Exponential distribution. The distribution of rainfall intensity is generally expressed in the form of intensity-frequency-duration (IFD) curves. Here, it is hypothesised that the IFD curves for ungauged catchment can be obtained from the weighted average IFD curves of an appropriate number of pluviograph stations in the vicinity of the ungauged catchment in question. The weighting factors in averaging can be obtained from the distances of the pluviograph stations from the ungauged catchment. It has been found that Exponential distribution can be used to describe the at-site and regional distribution of storm-core duration in Victoria. It has also been found that the Monte Carlo Simulation technique can successfully be applied to ungauged catchments. The independent testing of the new technique shows that the median relative error in design flood estimates by this technique ranges from 49 to 66% which was found to higher than those of the Probabilistic Rational Method (for this the median relative errors were in the range 41% to 47%) and the Quantile Regression technique (which had median relative errors in the range 28% to 51%). The possible reasons for the Joint Probability Approach of having a higher relative error is that the test catchments used in this study were included in the data set of derivation of the runoff coefficients for the Probabilistic Rational Method in the Australian Rainfall and Runoff and Quantile Regression Technique by Rahman (2005). Another reason may be that the Joint Probability Approach adopted provisionally developed regional estimation equation for storage delay parameter (k) of the runoff routing model, and regional average continuing loss value which were obtained from a very small sample of data. It was found that derived flood frequency curves from the Joint Probability Approach were very sensitive to both k and continuing loss values. The developed new technique of design flood estimation can provide the full hydrograph rather than only peak value as with the Probabilistic Rational Method and Quantile Regression Technique. The developed new technique can further be improved by addition of new and improved regional estimation equations for the initial loss, continuing loss and storage delay parameter (k) as and when these are available. x

14 Acronyms and Abbreviations ARI ARR AEP BOM IFD d c I c IL s IL c JPA MCST TP c URBS k m PRM QRM JPA DFFC Average Recurrence Interval Australian Rainfall and Runoff Annual Exceedance Probability Bureau of Metrology Intensity-frequency-duration Storm-core Duration Storm-core Rainfall-Intensity Initial Loss for Complete Storm Initial Loss for Storm-Core Joint Probability Approach Monte Carlo Simulation Technique Storm-core Temporal Pattern Runoff-Routing Hydrologic Model Storage delay parameter Non-linearity parameter Probabilistic Rational Method Quantile Regression Technique Joint Probability Approach Derived Flood Frequency Curve xi

15 CHAPTER 1 INTRODUCTION 1.1 BACKGROUND Flood is the number one natural disaster on earth in terms of economic damage. Each year floods cause millions of dollars of damage across Australia. Annual spending on infrastructure requiring flood estimation in Australia is about $1 billion. The average annual cost of flood damage in Australia is estimated to be about $400 million (MRSTLG, 1999). Due to global climate change (resulting from greenhouse effects), the severity and frequency of floods and associated damage will increase significantly in the near future in Australia (similar to other parts of the world) (CSIRO, 2001, Muzik, 2002). Also, estimation of streamflow of a given recurrence interval (ARI) is often required in environmental studies. Due to its large economical and environmental relevance, estimation of design flood remains a subject of great importance and interest in flood hydrology. On numerous occasions, floods have to be estimated at locations where there are little or no recorded streamflow data (ungauged catchments). Also due to land use and global climate changes, many of the previously recorded rainfall and streamflow data may become of little relevance for a catchment of interest, and under such a situation, flood estimation techniques for ungauged catchments need to be applied. Thus, flood estimation at ungauged catchments is a major issue in hydrological and environmental design/studies, and is of great economic significance. The currently recommended methods to estimate design floods at ungauged catchments include empirical methods (black box type model) such as Probabilistic Rational Method (I. E. Aust., 1997), Index Flood Method (Hosking and Wallis, 1993; Rahman et al., 1999) and USGS Quantile Regression Method (Benson, 1962). These are limited to peak flows and are not particularly useful when estimation of complete streamflow hydrographs is required, e.g. 1

16 wetland design. The Design Event Approach that is based on design rainfalls such as RORB (Laurenson and Mein, 1997) and URBS (Carroll, 1994) can be used to estimate design hydrograph at ungauged catchments; a high degree of estimation error is associated with these techniques because of a high degree of error in transposing model parameters from gauged to ungauged catchments and due to a fundamental limitation of the Design Event Approach as discussed below. The Design Event Approach uses a probability-distributed rainfall depth with representative values of other input variables such as rainfall temporal pattern and initial loss and assumes that the resulting flood has the same frequency as the input rainfall depth. The key assumption involved in this approach is that the representative design values of the input variables/ model parameters at different steps can be defined in such a way that they are annual exceedance probability (AEP) neutral i.e. they result in a flood output that has the same AEP as the rainfall input. The success of this approach is crucially dependent on how well this assumption is satisfied. There are no definite guidelines on how to select the appropriate values of the input variables/ model parameters that are likely to convert a rainfall depth of a particular AEP to the design flood of the same AEP. There are many methods to determine an input value, the choice of which is totally dependent on various assumptions and preferences of the individual designer. Due to non-linearity of the transformation in the rainfall-runoff process, it is generally not possible to know a priori how a representative value for an input should be selected to preserve the AEP. In summary, the current Design Event Approach considers the probabilistic nature of rainfall depth but ignores the probabilistic behaviour of other input variables/ model parameters such as rainfall duration and losses. The assumption regarding the probability of the flood output i.e. that a particular AEP rainfall depth will produce a flood of the same AEP is unreasonable in many cases. The arbitrary treatment of the various flood producing variables, as done in the current Design Event Approach, is likely to lead to inconsistencies and significant bias in flood estimates for a given AEP. This results in either over-design or underdesign of flood structures both of which have important economic consequences. 2

17 A significant improvement in design flood estimates can be achieved through rigorous treatment of the probabilistic aspects of the major input variables/model parameters in the rainfall-runoff models. This can be done through a Joint Probability Approach, which is more holistic in nature that uses probability-distributed input variables/model parameters and their correlation structure to obtain probability-distributed flood output. While ARR (I. E. Aust., 1987) recommended the Design Event Approach to rainfall-based design flood estimation, it recognised the importance of considering the probabilistic nature of the flood- producing input variables. It thus recommended further investigation into the Joint Probability Approaches. More recently, Hill and Mein (1996), in a study of incompatibilities between storm temporal patterns and losses for design flood estimation, mentioned, A holistic approach will perhaps produce the next significant improvement in design flood estimation procedures. They found the error in design flood estimates as high as 40% in some Victorian catchments due to inconsistencies in design loss and temporal patterns values alone. The Joint Probability Approach is superior to the currently adopted Design Event Approach because the former can account for the probabilistic nature of the flood producing variables and their interactions in an explicit manner and eliminates the subjectivity in selecting the representative value of a flood producing variable that show a wide variability such as initial loss. Rahman et al. (1998) summarised the previous works (Eagleson, 1972; Beran, 1973; Russell et al., 1979; Diaz-Granados et al., 1984; Sivapalan et al., 1990) on the Joint Probability Approaches to flood estimation and found that most of the previous applications were limited to theoretical studies; mathematical complexity, difficulties in parameter estimation and limited flexibility generally preclude the application of these techniques to practical situations. Rahman et al. (2002) developed a Monte Carlo simulation technique for flood estimation based on the principles of joint probability that can employ many of the commonly adopted flood estimation models and design data. The new technique has enough flexibility for its adoption in practical situations and has the potential to provide more precise design flood estimates than the existing technique. The Monte Carlo Simulation technique by Rahman et al. (2002) has so far been applied to gauged catchments. This thesis proposes to extend the Monte Carlo Simulation technique to 3

18 ungauged catchments, which involves regionalisation of the distribution of various floodproducing variables and runoff routing model parameters such as rainfall duration, intensity, temporal pattern, losses, and runoff routing model parameters. This study, in particular, will focus on the regionalisation of the distribution of rainfall duration and rainfall intensity. The main objectives of the study are provided below. 1.2 OBJECTIVES This thesis deals with design flood estimation at ungauged catchments. This, in particular, attempts to develop a new design flood estimation technique for ungauged catchments based on the Joint Probability Approach. The objectives of this thesis are: To extend the Joint Probability Approach of design flood estimation to ungauged catchments. To assess whether storm duration data in Victoria can be described by an Exponential distribution. To develop a method to regionalize the distribution of rainfall intensity for application with the Joint Probability Approach. To compare the new Joint Probability Approach for ungauged catchments with two existing methods: Probabilistic Rational Method and Quantile Regression Technique. 1.3 THESIS OUTLINE This thesis consists of six chapters. The introductory Chapter 1 provides objective of the thesis and a detail background of flood estimation technique. It also presents the overall outline of the thesis. Review of rainfall based flood estimation methods are described in Chapter 2. A detail review of Joint Probability Approach for the design flood estimation is also covered in this chapter. At the end of this chapter, recent research works on the Joint Probability Approach for design flood estimation are reviewed. The research hypotheses to be examined in the thesis are provided at the end of this chapter. 4

19 In Chapter 3, the study area is selected. The data used in this study are described here. A total of 76 pluviograph stations are selected in this study. To validate the new technique of design flood estimation, three gauged catchments are selected from the study area. An additional 12 gauged catchments are also selected here to compare the performances of the new technique with two existing design flood estimation techniques for ungauged catchments. The proposed research methodology is described in Chapter 4. This includes steps in the proposed research, rainfall analysis, loss analysis, calibration of runoff routing model, simulation of streamflow hydrograph, testing the hypothesis whether storm-core duration can be described by an exponential distributions and program to compute weighted average intensity-frequency-duration (IFD) values at an ungauged catchment. A FORTRAN program is also developed in this chapter. Chapter 5 details the results of the study. At the beginning, the distributions of storm-core durations are examined. The IFD curves of the study pluviograph stations are obtained in this chapter. The IFD values at an ungauged catchment are obtained by the proposed method, and derived flood frequency curves are obtained for the three study catchments and compared with the at-site flood frequency analyses. The new method is also compared with Quantile Regression Technique and Probabilistic Rational Method. Finally, Chapter 6 contains summary and conclusions from the thesis. This also includes recommended further research. 5

20 CHAPTER 2 DESIGN FLOOD ESTIMATION METHODS BASED ON THE JOINT PROBABILITY APPROACH 2.1 GENERAL This chapter of the thesis reviews design flood estimation methods in general with a particular emphasis to rainfall-based design flood estimation methods. This presents a detail review of the design flood estimation methods based on the Joint Probability Approach. At the end of the chapter, a research hypothesis is formulated. 2.2 FLOOD ESTIMATION METHODS OVERVIEW Flood estimation methods can broadly be classified into two groups: streamflow based methods and rainfall based methods (Lumb and James, 1976, Feldman, 1979, James and Robinson, 1986, I. E. Aust., 1987). This classification is presented in Figure 2.1 Streamflow-based methods give estimates of design floods by analysing observed streamflow data at a particular location. However, its application is limited to situations where sufficiently long period of streamflow data are available and catchments conditions remain unchanged over the period of observation. Walsh et al. (1991) mentioned that streamflow data are often unreliable, particularly for large events where rating curves generally undergo large extrapolations. 6

21 Design flood estimation methods Rainfall based methods Streamflow based methods Event based methods Continuous simulation based methods Design Event Empirical Joint Probability Partial Approach methods Approach continuous simulation Complete continuous simulation Figure 2.1 Classification of design flood estimation methods 2.3 RAINFALL-BASED FLOOD ESTIMATION METHODS Rainfall based flood estimation techniques are commonly adopted in hydrologic practice where there is limitation of recorded streamflow data. Rainfall data generally have greater temporal and spatial coverage than the streamflow data. A rainfall runoff model can be used to generate long series of streamflow data using the available rainfall data. This method can be sub-divided into event-based methods and continuous simulation methods. The current Design Event Approach is an example of event-based methods, which is the recommended method to obtain 7

22 design floods in Australian Rainfall and Runoff (ARR) using hypothetical rainfall event and runoff routing model. Some important features of rainfall based flood estimation methods are: i) Areal extrapolation of rainfall data can be achieved more easily than the streamflow data due to greater density of rainfall stations than the streamflow gauging stations. ii) Physical features of catchments can easily be incorporated into a rainfall-runoff model, which facilitates extreme flood estimation. iii) Climate changes happen more slowly than the land use changes of a catchment which means that long period of recorded rainfall data can be used in the rainfall runoff model. Rainfall based flood estimation methods can be grouped into four types i) Empirical methods ii) Design Event Approach iii) Joint probability based methods or derived distribution methods iv) Continuous simulation methods. (i) Empirical methods James and Robinson (1986) mentioned that empirical methods use observed streamflow and rainfall data to calibrate one or several coefficients in an equation representing the rainfallrunoff process. Probabilistic Rational Method (I. E Aust., 1987) and USGS quantile regression methods (Benson, 1962) are the most common examples of this approach which is black box type methods. Because they do not incorporate the hydrological process in the system rather than attempt to optimise the design flood output by comparison with the observed rainfall and streamflow data. The application of the empirical methods for practical flood estimation is limited to peak flow estimation only and therefore not particularly useful in cases where complete stream flow hydrographs are required. These methods are widely used for ungauged catchments. 8

23 The Probabilistic Rational Method is the most commonly used approximate method in Australia. According to Australian Rainfall and Runoff (ARR) (I. E. Aust., 1997) this can be represented by: Q T = 0.278C T I tc,t A (2.1) Where Q T is the peak flow (m 3 /s) for an average recurrence interval (ARI) of T years, C T is the runoff coefficient for the same ARI which is dimensionless, I tct is the average rainfall intensity (mm/hr) for a storm duration of t c hours with ARI of T years. Here, t c is the time of concentration in hours and A is the catchment area (km 2 ). The original rational method, Q = CIA for computation of design discharge has further been rationalized in the Probabilistic Rational Method by incorporating the probabilistic nature of rainfall intensity (I) and storm loss and other variables that affect runoff generation through the use of runoff coefficient. ARR (I. E. Aust., 1997) has recommended this method for general use in small to medium sized ungauged catchments in Australia, particularly for South-east Australia. The major problem associated with this method is related to the estimation of runoff-coefficients and the time of concentration. The spatial distribution of C T is based on an assumption of geographical contiguity. Hollerbach and Rahman (2003) found little coherence in spatial distribution of the runoff coefficients for South-east Australian catchments. Rahman (2005) proposed a Quantile Regression Technique for South-east Australian catchments. He developed prediction equations for 2, 5,, 20, 50 and 0 years ARIs based on the data of 88 catchments from South-east Australia. These equations are provided below. log Q2 = log( area ) log( I12) log( sden ) log( evap ) (2.2) 2 2 R = 0.72, Adjusted R = 0.70, SEE = 0.23 (6.43% of the mean logq 2 ) 9

24 log Q 5 = log( area ) log( I12) log( sden ) log( evap ) (2.3) R 2 = 0.75, Adjusted R 2 = 0.74, SEE = 0.23 (6.06% of the mean logq 5 ) log Q = log( area) log( I12) log( sden) log( evap) (2.4) R 2 = 0.73, Adjusted R 2 = 0.72, SEE = 0.23 (5.89% of the mean logq ) log Q20 = log( area) log( I12) log( sden) log( evap) (2.5) R 2 = 0.74, Adjusted R 2 = 0.73, SEE = 0.24 (5.88% of the mean logq 20 ) logq50 = log( area) log( I12) log( sden) log( evap) 0.086log( qsa) (2.6) R 2 = 0.77, Adjusted R 2 = 0.76, SEE = 0.23 (5.48% of the mean logq 50 ) logq0 = log( area) log( I12) log( sden) log( evap) log( qsa) (2.7) R 2 = 0.76, Adjusted R 2 = 0.75, SEE = 0.23 (5.56% of the mean logq 0 ) Here, various variables are: rainfall intensity of 12-hour duration and 2-year average recurrence interval (I12, mm/h), mean annual class A pan evaporation (evap, mm); catchment area (area, km 2 ); stream density (sden, km/km 2 ), which is the length of stream lines divided by the catchment area; and fraction quaternary sediment area (qsa). The qsa is a measure of the extent of alluvial deposits and is an indicator of floodplain extent in the study area. The explanatory variables evap and I12 are determined at the catchment centroid.

25 (ii) Continuous simulation method Continuous simulation for design flood estimation is a rapidly developing field in hydrology. According to Boughton and Droop (2003) the term Continuous simulation when used in flood hydrology refers to the estimation of losses from rainfall and the generation of streamflow by simulating the wetting and drying of a catchment on daily, hourly, and occasionally, sub-hourly time steps. In this method, the uncertainty or randomness of flood variables like initial loss is avoided. One of the important characteristics of these models is the continuous use of water budget model for the catchment so that continuous antecedent condition to each storm event is known. The particular advantage of this method is that the variability of flood producing variable in the temporal sense is reflected in the time series of flood peaks. Event simulation is necessary when long-term rainfall time series is available. In 1997, Siriwardena and Weinmann (1997) prepared a review of continuous simulation approach for design flood estimation. The review covered a number of loss models and some flood hydrograph models, as well as combined systems. This review referred to earlier studies by James and Robinson (1986) and Thomas (1982). In Australia, continuous simulation method has been advanced by Boughton et al. (2000) as a project of the Co-operative Research Center for Catchments Hydrology. Weinmann et al. (2000) noted that the continuous simulation approach is conceptually the most desirable one. The advantage of continuous simulation method over the Design Event Approach according to Rahman et al. (1998) are: It eliminates the need for using synthetic storms by using actual storm records. It eliminates subjectivity in selecting antecedent conditions for the land surface since a water budget is accounted for in each time step of the simulation and thus automatically logs antecedent moisture condition (James and Robinson, 1986). It overcomes the problem of accounting for antecedent moisture conditions. It overcomes the problem of critical storm duration because it simulates the resultant flows for all storms (Lumb and James, 1976). 11

26 It handles the antecedent conditions correctly because the continuous time series of flows includes all effects of antecedent conditions. (Huber, et al. 1986). It undertakes a frequency analysis of the variable of interest (peakflow, flow volume, pollutant washoff etc) by statistically analysing the time series of model outputs, as opposed to assuming equal probability of floods and causative rainfall intensity (Huber et al. 1986). The problems associated with the application of continuous simulation are: Loss of sharp events if long time steps are used. Extensive data requirement. Significant amount of time and efforts are required in gathering the precipitation and other climatic data needed for simulation of long continuous sequences of these variables. Management of large amount of time series output (data management). Expertise required for determining parameter values which best reproduces historical hydrographs (model calibration effort). According to Kuczera and Coombes (2002), stochastic rainfall models and regional techniques are sufficiently improved now to generate long-term rainfall data in the absence of observed pluviograph data. (iii) Design Event Approach Design Event Approach is the currently recommended technique in Australia for estimation of design floods using runoff routing model (I. E. Aust., 1958, 1977, 1987, 1997). In the Design Event Approach, for a selected average recurrence interval (ARI), a number of trial rainfall durations and their corresponding average rainfall intensities are used with fixed temporal pattern, initial loss and other inputs to obtain a flood hydrograph for each duration. Beran (1973) and Ahern and Weinmann (1982), described the steps involved with the Design Event Approach as shown in Figure 2.2. Here the input parameters at different steps of computation should be selected in such that they result in a flood output of the same ARI as the rainfall 12

27 intensity input. This is generally done by considering the representative values (e.g. mean or median) of the input variables. Due to non-linearity of rainfall-runoff process and high degree of variability of input variables such as initial loss, this assumption of probability neutrality of other input variables are hardly satisfied. In short, this approach considers only the probabilistic nature of the rainfall depth but ignores the probabilistic behaviour of other input variable such as losses in rainfall runoff modelling. As a result, the Design Event Approach is likely to introduce significant probability bias in the final flood estimates and has been widely criticized (Kuczera et al., 2003; Rahman et al., 2002). This can result in either a systematic under or over design of engineering structures, both with important economic consequences (Weinmann et al., 1998). Rahman et al. (1998) made a detail critical review of this method including its limitations. In Australia, the Design Event Approach is commonly used with a runoff-routing model such as RORB (Laurenson and Mein, 1997) and URBS (Carroll, 1994). (iv) Joint Probability Approach The basic idea underlying this approach is that any design flood characteristic could result from a variety of combination of flood producing factors, rather than from a single combination, as done in the Design Event Approach. This approach was pioneered by Eagleson (1972) who used an analytical method to derive the probability of distribution of peak streamflows from an idealized V-shaped flow plane. This approach has been advanced and improved in the last two decades. Ahern and Weinmann (1982) mentioned that Joint Probability Approach which considers the outcomes of events with all possible combinations of input values and, if necessary, their correlation structure, should lead to better estimates of design flows. The method is regarded to be theoretically superior to the Design Event Approach and regarded as an attractive design method (I. E. Aust., 1987). This method is discussed in more details in the following section. 13

28 Design Rainfall Depth (ARI = Y, Duration = D Temporal and Spatial Patterns of Rainfall Design Rainfall Event (ARI = Y) Loss Parameters Loss Model Rainfall Excess Hyetograph Catchment Response Parameters Catchment Response Model Surface Runoff Hydrograph Baseflow Design Flood Hydrograph (ARI = Y) Figure 2.2 Flood Estimation by Design Event Approach ( Rahman et al., 1998) 14

29 2.4. DESCRIPTION OF THE JOINT PROBABILITY APPROACH The Joint Probability Approach calculates the probability of an output by considering all possible combinations of design inputs. In this approach, flood output has a probability distribution instead of a single value. Here each input is treated as a random variable. The method of combining probability-distributed inputs to form a probability-distributed output is known as the derived distribution approach. A derived probability distribution can be found in two ways: (i) approximate methods and (ii) analytical methods. The choice of a method to compute a derived distribution from these options is influenced mainly by the level of analytical skills and the computer resources available for the task (Weinmann, 1994). (i) APPROXIMATE METHODS Approximate methods are often used in hydrology to determine derived frequency distribution. There are two categories of approximate methods: a) Discrete methods: Total probability theorem is generally used where continuous distributions of hydrologic variables are discritized. b) Simulation technique: Random samples are drawn from continuous distribution of input variables. A) Discrete methods Here discrete probability distributions are used to describe hydrologic variable, such as rainfall duration, antecedent precipitation index, soil moisture deficit, etc. even though they are really continuous ones. Many researchers e.g. Beran (1973), Laurenson (1974), Russell et al. (1979), Fontaine and Potter (1993) adopted this method. The accuracy of the approach depends on the degree of discretization. In discrete methods, the theorem of total probability is normally used to calculate flood probabilities. Fontaine and Potter (1993) make the simplest application of this. For a given flood, its exceedance probability is the sum of three terms, each being the joint probability of extreme rainfall and antecedent soil moisture. In SCS curve number method (Soil 15

30 Conservation Service, 1972), it is assumed to be represented by three curve numbers. In fact, this over-simplified assumption is one basic limitation of the proposed Joint Probability Approach. The same concept is applied by Russell et al. (1979) to a rainfall-runoff model represented by three parameters (time of concentration T, infiltration rate I and storage constant R). Russell et al. (1979) used actual storm rainfall records instead of a synthetic storm. The Clark rainfall runoff model (Clark, 1945) which provides the basis for the HEC1 model was used in which rainfall is lagged by a time-area curve and routed through linear storage. It was assumed that infiltration rate would be constant for any particular storm. Laurenson (1974) presented the most general application of total probability, which is described by transformation matrix approach. The method requires division of a design problem into a sequence of steps, each step transforms an input distribution into output distributions, which becomes the input to the next step (Laurenson, 1974). In applying the method, input, transformation relation and output should be expressed in matrix form. One particular value of the transformation matrix represents the conditional probability of obtaining an output value given a value of the input. The transformation matrix method provides a wide range of application (Laurenson, 1973; Laurenson, 1974; Ahern and Weinmann, 1982; Laurenson and Pearse, 1997) when the stochastic nature of the hydrologic system needs to be accounted for. The above examples demonstrate how the theorem of total probability can be applied for calculating design flood probabilities. If all the random variables involved in the design are independent, computation of flood probabilities becomes very simple once probabilities of those input variables are given. For the case of dependent variables, application of the theorem becomes relatively difficult. Beran (1973) presented a procedure that sampled the possible ways in which a storm of a given ARI could cause floods, and derived their joint probability distributions. The unit hydrograph method was used as catchment response model. In applying the method, smoothing of flood probability distributions may be required because of discretizing continuous distributions into class intervals. Shen et al. (1990) presented numerical integration to determine the derived distribution. They used a Poisson process for arrival of storm events, exponential distributions 16

31 for rainfall intensity and duration, Phillip s equation for infiltration capacity, and the kinematic wave equation to formulate a rainfall runoff model. The results of the study are applicable to given ranges of basin characteristics only. B) Simulation techniques A number of investigators have used simulation methods to determine derived flood frequency distributions. For example, Durrans (1995) represented a simulation procedure to determine derived flood frequency curve for regulated sites, which has been described as an integrated deterministic-stochastic approach to flood frequency analysis. It was done in the following steps: 1) Random sampling of unregulated annual flood peak and unregulated flood volume. 2) Random sampling of a dimensionless initial reservoir depth and dimensionless gate opening area. 3) Routing the inflow hydrograph through reservoir. 4) Replication of steps (1) to (3) N times to obtain N outflow hydrograph peaks. Here N is in the order of thousands. Muzik (1993) adopted a modified SCS curve number method in the Monte Carlo Simulation to obtain a derived distribution of peak discharge. The approach combines knowledge of physical processes with the theory of probability in that knowledge of the processes allows putting reasonable limits on the variable values. Here the initial abstraction and five-day antecedent rainfall values (P5) were assumed to be a random variable. The steps involved in the simulation are: (i) generation of a random value of P5; (ii) from the relationship between P5 and S obtaining the maximum potential retention S; (iii) generation of a random value of the initial abstraction Ia; (iv) generation of a random value of total rainfall P; and (v) computation of rainfall excess depth. The rainfall excess depth was then transformed deterministically by means of the unit hydrograph method into a flood hydrograph. Sivapalan et al. (1996) and Tavakkoli (1985) adopted a simulation approach to derive flood frequency curves for an Australian catchment. The method resulted in slight overestimation of flood peaks, which he mainly attributed to the runoff generation model. 17

32 Muzik and Beersing (1989) studied the transformation process of probability distributions of rainfall intensity for the case of runoff from a uniformly sloping impervious plane. Here kinematic wave and experimentally derived relations were used to compute the peak discharge. Beran (1973) adopted a simulation technique in that the sampling produced lower flood values at smaller ARIs than the expected flood following storms of that same ARI. This method is not a fully generalised simulation approach; it is a combination of the approximate method and the simulation technique. Here probability distributions of storm durations and temporal pattern were based on complete storms and obtained from the observed data but existing IFD (intensity-frequency-duration) curves based on storm bursts were adopted for rainfall depth. Bloschl and Sivapalan (1997) adopted a Monte Carlo simulation method for mapping rainfall ARIs to runoff ARIs. The simulation consisted of the following steps: (i) Draw storm durations from an exponential distribution. (ii) Draw precipitation probabilities from a uniform distribution P [0; 1] and calculate precipitation return period from T p = 1/(1-p)/m where m is the number of events per year; (iii) get rainfall intensities, p, from the IFD curve using the two previous pieces of information; and (iv) fit temporal pattern to rainfall, apply runoff coefficient to estimate rainfall excess, simulate streamflow hydrograph from the selected runoff routing model, and note the peaks. At the end, the flood peaks were ranked which allowed assignment of an ARI to each event by using plotting positions: T q = n/j/m where T q is the return period of the flood, n is the total number of events, and j is the rank. Finally it can be said that the mathematical framework of the Monte Carlo Simulation technique adopted by several previous studies provide examples of practical design flood estimation techniques based on the Joint Probability Approach. (ii) ANALYTICAL METHODS Bates (1994) and Sivapalan et al. (1996) presented examples where an analytical approach was used for deriving flood frequency distributions. A review of these studies is discussed hereunder depending on the runoff routing method adopted. 18

33 (1) Methods based on U.S. Soil Conservation Service s curve number procedure Haan and Edward (1986) derived the joint probability density function of runoff Q and maximum water abstraction S by using U. S. Soil Conservation Service (SCS) curve number method. The equation derived is strictly applicable to the SCS curve number method and it becomes much more difficult in situations where a more complex transformation between rainfall and runoff is required. Raines and Valdes (1993) modified Diaz-Granados et al., s (1984) approach where the SCS curve number procedure was used instead of Philip s ( 1957) infiltration equation to estimate runoff. Becciu et al. (1993) presented a derived distribution technique in flood estimation for ungauged catchments. Here point rainfall was described by a Poisson distribution; intensity and duration of rainfall were assumed to be mutually independent random variables. Catchments in Northern Italy showed its capability to satisfactorily reproduce the frequency distribution of the observed data. (2) Methods based on Eagleson s kinematic runoff model Eagleson s (1972) has pioneered the derived flood frequency approach by using kinematic model for runoff from an idealized V-shaped flow plane. This approach assumed that storm characteristics are independent random variable with a joint exponential probability function. He used the empirical areal reduction factors to convert point rainfall to catchment-average rainfall. The method has limited practical applicability. Generally, the number of parameters of the derived distributions is large (Wood and Hebson, 1986) and the assumption of independence between rainfall duration and intensity is not likely to be satisfied. Here runoff- flow and channel routing model utilised kinematic wave equations for both overland flow. Cadavid et al. (1991) applied a derived distribution approach to small urban catchments, which included Eagleson s rainfall model, Philip s (1957) infiltration equation, and kinematic wave 19

34 model for runoff routing. Their model did not show good fits, particularly for higher ARI floods. (3) Methods based on other types of rainfall-runoff models In 1986, Bevan (1986) adopted a Joint Probability Approach to flood estimation that combined the topographically base TOPMODEL with a routing model based on catchment width function. In this study, he found that the proportion of saturated area of flood increased with increasing ARIs. Haan and Wilson (1987) mentioned a methodology for computing runoff frequencies based on the Joint Probability Approach. According to them the derived distribution of peak flows was based on the Rational Method, Q = CIA (2.8) The probability distribution of runoff coefficient (C) and I were described by Beta and Extreme Value Type I distributions respectively. They used numerical integration to obtain derived distribution under the assumption of independence of C and I. They found that consideration of runoff coefficient as a random variable provided larger peak flows than that obtained assuming C as a constant, particularly at higher ARIs. Schakke et al. (1967) mentioned that C may be larger for storms with greater ARIs which is also been recognized in ARR (I. E. Aust., 1987). Haan and Wilson (1987) demonstrated the appropriateness of the Joint Probability Approach and suggested further study on this approach. Sivapalan et al. (1996) illustrated the use of intensity frequency duration (IFD) curves in the derived distribution procedure, which would help to unify the theoretical research on derived flood frequency with traditional design practice. They utilised the derived flood frequency methodology to investigate the link between process control and flood frequency. Sivapalan et al. (1996) proposed a method of specifying the joint distribution of rainfall intensity and duration, which considers IFD curves as conditional distributions, and distribution of storm duration as marginal distribution. They specified the joint distribution of rainfall intensity and 20

35 duration by multiplying IFD curves (conditional distributions) with marginal distributions of duration. They identified that temporal pattern, multiple storms and the nonlinear dependence of runoff coefficients on event rainfall depth are the major factors controlling the shape of flood frequency curve. Bloschl and Sivapalan (1997) investigated the effects of various flood-producing factors (runoff coefficients, antecedent conditions, storm durations and temporal pattern) on flood frequency curve in a derived distribution frame work. They mentioned that the case of independent intensity-duration gives vastly steeper flood frequency curves than the case of dependent intensity-duration. They argued that this non-linearity might be the reason that flood frequency curves tend to be much steeper than rainfall frequency curves. It might be noted that the different slopes and shapes of rainfall and flood frequency curves have been observed for many catchments. (4) Methods based on geomorphologic unit hydrograph Hebson and Wood (1982) and Diaz-Granados et al. (1984) have extended Eagleson s (1972) rainfall-runoff model by means of the geomorphologic unit hydrograph (GUH) theory proposed by Rodriguez-Iturbe and Valdes (1979). The GUH theory assumes that rainfall excess is generated uniformly throughout the catchment area. Their procedure was tested on two Appalachian Mountain catchments and the results compared well with the observed streamflow data. Wood and Hebson (1986) adopted the scaling of rainfall duration by a characteristics basin time, which is a function of basin size. In deriving the joint probability distribution they assumed a uniform rainfall intensity over the excess storm duration and independence between average areal storm depth and excess storm duration. Diaz-Granados et al. (1984) presented an infiltration excess runoff generation model based on Phillip s (1957) representation of the infiltration process. They tested their procedure against the sample flood frequency distributions for arid and wet climates and achieved good and reasonable fits, respectively. Moughamian et al. (1987) examined the performance of the derived flood frequency models of 21

36 Hebson and Wood (1982) and Diaz-Granados et al. (1984) on three catchments and found both models performed poorly in every catchment when compared to sample distribution. Sivapalan et al. (1990) developed a flood frequency model that includes runoff generation on partial areas by Hortonian equation and integrated the partial area model with GUH based runoff routing model. For catchment in humid conditions, Sivapalan et al. (1990) found that different runoff generation processes dominate different ARIs of the flood frequency distribution. Torch et al. (1994) applied a model similar to that developed by Sivapalan et al. (1990) to study the relative importance to hydrologic controls of large floods in a small basin. The catchment was situated in Pennsylvania. Here the channel routing model was expressed in terms of the basin s width function. 2.5 RECENT RESEARCH ON THE JOINT PROBABILITY APPROACH TO DESIGN FLOOD ESTIMATION A review of the most recent works is presented below with a particular focus on the results of the studies in relation to practical applicability of the Joint Probability Approach to design flood estimation. Kuczera et al. (2000) represented KinDog kinematics model which was used to route the rainfall to the catchments outlet. This is based on the Field-william kinematic model. It conceptualises rainfall excess as Hortonian overland flow routed through a non-linear storage into the channel. Two more analytical attempts to estimate the flood probability distribution with the derived distribution methodology are by Iacobellis and Fiorentino (2000) and by Goel et al. (2000). Iacobellis and Fiorentino (2000) assumed that the peak direct flow is expressed as the product of average runoff per unit area, u(a), and the peak contributing area, a. They assumed that the probability distribution of u(a) is conditional and is related to the probability distribution of the rainfall depth occurring in a duration equal to the characteristic response time. 22

37 Goel et al. (2000) used a stochastic rainfall model which assumes that rainfall intensity is, either positively or negatively, correlated to the rainfall duration for the generation of the rainfall. Here Rainfall runoff processes were modelled using an φ-index infiltration model and a triangular geomorphoclimatic instantaneous unit hydrograph model. Yue (2000) represented Gumbel distribution model in derived flood frequency analysis. Based on this model, one can obtain the Joint Probability distributions, and the associated return periods of two correlated variables if their marginal distributions can be represented by the Gumbel distribution. Weinmann et al. (2002) highlighted some of the theoretical and practical limitations of the currently used Design Event Approach to rainfall based design flood estimation. They noted that Monte Carlo simulation has the advantage that it can utilise some of the models and design data used with the Design Event Approach, which would allow it to be more readily applied to flood estimation in practical situations. Rahman et al. (2002) presented a more holistic approach of design flood estimation based on the principle of Joint Probability Approach. This Monte Carlo simulation technique based on the Joint Probability Approach offers a theoretically superior method of design flood estimation as it allows explicitly for the effects of inherent variability in the flood producing factors and correlations between them. Rahman et al. (2002c) presented a study illustrating how Monte Carlo simulation technique can be integrated with industry-based model such as URBS. It was found that the integrated URBS-Monte Carlo Technique can be used to obtain more precise flood estimates for small to large catchments. Rahman et al. (2002b) examines the variability of initial losses and specification of its probability distribution for use in the Joint Probability Approach. It was found that the use of a mean value instead of the probability distribution of initial losses reduces flood magnitudes significantly, particularly at smaller average recurrence intervals. 23

38 Heneker et al. (2002) represented the ways of overcoming the Joint Probability problem by allowing design rainfall obtained from ARR to be directly converted into rainfall excess. They employed a continuous simulation approach using calibrated stochastic point rainfall, stochastic evaporation and water balance models to determine rainfall excess exceedance probabilities for various durations. Charalambous et al. (2003) extended the URBS-Monte Carlo Simulation technique to two large catchments in Queensland. They found that the URBS-Monte Carlo Simulation technique can easily be applied to large catchments. Although the limited data availability in their application introduced significant uncertainty in the distributions of the input variables e.g. IFD curves. Kuczera et al. (2003) suggested that the current revision of ARR needs to articulate the shortcomings of the design storm approach, identify calibration strategies, which gives guidance about its reliability in different application. They also notify that event Joint Probability methods based on Monte Carlo Simulation are computationally less demanding but require specification of the probability distribution of initial conditions. Kader and Rahman (2004) applied the Joint Probability Approach to design flood estimation for ungauged catchments. They attempted to find out how the distribution of rainfall intensity can be regionalised in the state of Victoria in Australia. They examined the regional relationship between two types of design rainfalls, Australian Rainfall Runoff (ARR) and Joint Probability Approach (JP). They found that the regionally predicted JP IFD values to be linearly correlated with the corresponding ARR IFD values. They also found that ARR IFD values are generally higher than the corresponding JP IFD values. The developed regional relationship between JP IFD and ARR IFD values did not produce satisfactory derived flood frequency curves for the ungauged catchment. Rauf and Rahman (2004) examined the sampling properties of rainfall events for constructing intensity frequency duration (IFD) curves in ARR method and Joint Probability Approach. To examine how frequently the same rainfall spell can appear in the data series across various durations in the Victorian state a term commonality was used which measured the frequency 24

39 of repetition of the storm event of a duration in the storm events of subsequent longer duration. They found for 91 stations in Victoria that about 50% storm burst events share common rainfall spells in ARR method implying that many data points across various durations are not independent. Carroll and Rahman (2004) investigated the subtropical rainfall characteristics for use in the Joint Probability Approach to design flood estimation in South-east Queensland. It was found that the complete storm durations in South-East Queensland can be approximated by an exponential distribution but the storm core durations are better approximated by the Gamma distribution. They also discussed the application of Multiplicative-cascade model for temporal pattern distribution. It was suggested that a regional temporal pattern distribution can be used to generate temporal pattern for either complete storm or storm-core in South-east Queensland. Rahman and Carroll (2004) examined the effects of spatial variability of the flood producing variable on derived flood frequency curves in the Joint Probability Approach. It was found that a spatial variation of 20% in mean rainfall duration from sub-catchment to sub-catchment would have little effect on the derived flood frequency curve and it is not necessary to consider different parameters of the initial loss distribution for various sub catchments in the Monte Carlo Simulation for medium to large catchments. 25

40 2.6 INDEX OF MAJOR RESEARCH WORKS ON THE JOINT PROBABILITY APPROACH Table 2.1 has been compiled to show a number of significant technical reports and papers that have played a major role in the past on the research and development of the Joint Probability Approach to design flood estimation. Table 2.1 Index of previous research on the Joint Probability Approach to design flood estimation Year Title Author(s) Summary 1972 Dynamics of flood frequency Eagleson, P.S This paper assumed that storm characteristics (duration and intensity) are independent random variables Estimation of Design Floods and the Problem of Equating the Probability of Rainfall and Runoff. Beran, M. A. This paper presented a procedure that sampled the possible ways in which a storm of given ARI could cause floods, and derived their Joint Probability distributions. It was found that the derived flood frequency curves were much flatter than the observed ones Modelling of Stochastic- Laurenson, E. M. This paper represents the most general application of total Deterministic Hydrologic Systems. probability theorem which is described by transformation matrix approach Considerations for design flood estimation using catchments modelling. Ahern, P.A. and Weinmann, P.E. This paper mentioned that Joint Probability Approach, which considers the outcomes of events with all possible combinations of input values and, if necessary, their correlation structure, obtain better estimates of design flows. 26

41 1982 A derived flood frequency distribution using Horton order Ratios Derived, physically based distribution of flood probabilities. Extreme Hydrological Events: Precipitation, Floods and Droughts Process Controls on Flood Frequency.1. Derived Flood Frequency Hebson, C and Wood, E.F. Muzik, I Sivapalan, M., Bloschl, G. and Gutknecht, D. They used Eagleson s (1972) partial area runoff routing model and their runoff routing model was based on the third-order geomorphologic unit hydrograph model. Their procedure was tested on two Appalachian Mountain catchments and the results compared well with the observed streamflow data. This paper represented a modified SCS curve number method in the Monte Carlo Simulation to obtain a derived distribution of peak discharge. Here the initial abstraction and five-day antecedent rainfall values were assumed to be a random variable. They proposed a method of specifying the joint distribution of rainfall intensity and duration, which considers IFD curves as conditional distributions, and distribution of storm duration as marginal distribution Derived distribution of floods based on the concept of partial area coverage with a climate appeal. Iacobellis, V. and Fiorentino, M. They assumed that the peak direct flow is expressed as the product of average runoff per unit area, u(a), and the peak contributing area, a. They assumed that the probability distribution of u(a) is conditional and is related to the probability distribution of the rainfall depth occurring in a duration equal to the characteristic response time. 27

42 2000 A derived flood frequency distribution for correlated rainfall intensity and duration The Gumbel Mixed Model applied to storm frequency analysis Overcoming the joint probability problem associated with initial loss estimation in design flood estimation Goel, N. K., Kurothe, R.S., Mathur, B.S. and Vogel, R.M Yue, S. Heneker, T., Lambert, M., and Kuczera, G. They presented a stochastic rainfall model which assumes that rainfall intensity is, either positively or negatively, correlated to the rainfall duration for the generation of the rainfall. He represented Gumbel distribution model. Based on this model, one can obtain the Joint Probability distributions, and the associated return periods of two correlated variables if their marginal distributions can be represented by the Gumbel distribution. This paper represented the ways of overcoming the Joint Probability problem by allowing design rainfall obtained from ARR to be directly converted into rainfall excess. They employed a continuous simulation approach using calibrated stochastic point rainfall, stochastic evaporation and water balance models to determine rainfall excess exceedance probabilities for various durations Integration of Monte Carlo simulation technique with URBS model for design flood estimation. Rahman, A, Carroll, D.G, and Weinmann, P.E. This paper describes how a Monte Carlo simulation technique can be applied with the industry-based runoff routing model URBS to determine derived flood frequency curves. 28

43 2002 Monte Carlo Simulation of flood frequency curves from rainfall Monte Carlo simulation of flood frequency curves from rainfall-the way ahead. Rahman, A, Weinmann, P.E, Hoang, T.M.T., and Laurenson, E.M Weinmann, P.E., Rahman, A., Hoang, T, Laurenson, E.M., and Nathan, R.J. This technique is appropriate for the derivation of flood frequency curves in the ARI range 1 to 0 years for small gauged catchments. This paper has highlighted some of the theoretical and practical limitations of the currently used Design Event Approach to rainfall based design flood estimation The use of probability distributed initial losses in design flood estimation. Rahman, A., Weinmann, P.E. and Mein, R.G. This study examines the role played by initial loss modelling in flood estimation for selected Victorian catchments. It has been shown that the variability of initial loss in the Victorian catchments can be described by a fourparameter Beta distribution Application of Monte Carlo Simulation Technique with URBS Model for Design Flood Estimation in Large Catchments. Charalambous, J., Rahman, A., and Carroll. D. They found that the URBS-Monte Carlo Simulation technique can easily be applied to large catchments. Although the limited data availability introduced significant uncertainty in the distributions of the input variables e.g. IFD curves. 29

44 2003 Joint Probability and Design Storms at the Crossroads Kuczera, G., Lambert, M., Heneker, T., Jennings, S., Frost, A. and Coombes, P They suggested that the current revision of ARR needs to articulate the shortcomings of the design storm approach, identify calibration strategies, which gives guidance about its reliability in different application. They also notify that event Joint Probability methods based on Monte Carlo Simulation are computationally less demanding but require specification of the probability distribution of initial conditions Appropriate spatial variability of flood producing variables in the Joint Probability Approach to design flood estimation Regionalization of design rainfalls in Victoria, Australia for design flood estimation by Joint Probability Approach Investigation of Sub tropical rainfall characteristics for use in the Joint Probability Approach to design flood estimation Carroll, D. and Rahman, A Kader, F., and Rahman, A. Carroll, D., and Rahman, A. This paper describes the effects of spatial variability of flood producing variables on design flood frequency curves in the Joint Probability Approach. This paper represents how the distribution of rainfall intensity can be regionalized in the state of Victoria in Australia. This paper examines the relationship between rainfall intensities of complete storm and storm core and the application of multiplicative cascade model for temporal pattern distribution is also discussed here. 30

45 2004 Study of fixed duration design rainfalls in Australian Rainfall Runoff and Joint Probability based design rainfalls An improved framework for the characterisation of extreme floods and for the assessment of dam safety. Rauf, A., and Rahman, A. Nathan. R. J and Weinmann.P.E. This study examines the sampling properties of the rainfall events in the Australian Rainfall-Runoff (ARR) method and Joint Probability Approach to identify any systematic differences between them. It has been found that ARR design rainfall estimates are generally higher than the Joint Probability based estimates, however these differences vary with location, duration and ARI. This paper presented a methodology that reduces the practical problems involved in the derivation of both standards and risk based design estimates. Here methodology is based on the use of Monte Carlo Simulation. 31

46 2.7 JOINT PROBABILITY APPROACH TO UNGAUGED CATCHMENTS Estimation of hydrologic variables at ungauged sites is perhaps among the oldest challenges for the hydrologic practitioner. In fact, flood and environmental flow estimation at catchments with no streamflow data is a common problem faced by practicing engineers and environmental scientists In Australia, of the 12 drainage divisions, seven do not have a stream with 20 or more years of data (Vogel et al., 1993). Thus flood estimation at ungauged catchments is a major issue in hydrological and environmental design in Australia. The currently recommended methods to estimate design floods at ungauged catchments include empirical methods such as Probabilistic Rational Method (I. E. Aust., 1987), Index Flood Method (Hosking and Walls, 1993; Rahman et al., 1999) and USGS Quantile Regression Method (Benson, 1962). Generally, these provide only the flood peak estimate. Design flood estimation based on Joint Probability Approach using Monte Carlo simulation technique (Rahman et al., 2002) has shown potential to become a practical tool for estimation of design flood for small catchments. The application of this technique to ungauged catchments will require regionalisation of the parameters of the input variables in a region such as rainfall duration, intensity and initial loss. The proposed research intends to adopt Monte Carlo Simulation Approach of Rahman et al. (2002) to ungauged catchments and hence this will be discussed in more details in the next section. 2.8 DESCRIPTION OF THE MONTE CARLO SIMULATION TECHNIQUE TO DESIGN FLOOD ESTIMATION OF RAHMAN ET AL. (2002) General The Design Event Approach treats rainfall intensity as a random variable, and uses a number of trial rainfall burst durations with fixed temporal patterns to obtain design flood estimates. In contrast, Monte Carlo Simulation Approach by Rahman et al. (2002) requires rainfall events to provide random duration unlike the rainfall bursts of predetermined durations used in the Design Event Approach. For the purposes of the proposed method, two types of rainfall events were defined. These are as follows. 32

47 2.8.2 Event definition A complete storm is a period of significant rainfall that is separated from previous and subsequent rainfall events by a dry period. Here a period is defined a dry if it lasts at least 6 h. A complete storm is considered to be significant if it has the potential to produce significant runoff. This is assessed by comparing its average rainfall intensity with a threshold intensity. Thus, for a complete storm, the average rainfall intensity during the entire storm duration (I D ) or a sub-storm duration (I d ) must satisfy one of the following conditions: 2 I f1 I (2.9) D D I d f 2 I 2 d (2.) Where f 1 and f 2 are reduction factors, the threshold intensity 2 I D is the 2 year ARI design rainfall intensity for the selected storm duration D, and 2 I d is the corresponding intensity for the sub-storm duration d. The use of smaller values of f 1 and f 2 captures a relatively larger number of events. A numerical example illustrating the identification of a storm-core has been given in Appendix G. For each complete storm, a single storm-core can be identified, defined as the most intense rainfall burst within a complete storm. It is found by calculating the average intensities of all possible storm-bursts, and the ratio with an rainfall intensity 2 I d for that duration d, then selecting the bursts of that duration which produce the highest ratio Distribution of flood-producing variables In the Monte Carlo Simulation Approach by Rahman et al. (2002), four variables were considered for probabilistic representation. These were rainfall duration, rainfall intensity, rainfall temporal pattern and initial loss. 33

48 Duration The storm-cores are selected from the hourly pluviograph data of selected stations and analysed for storm-core duration (d c ), average rainfall intensity (I c ), and temporal patterns (TP c ). Figure shows a typical histogram of the frequencies of different storm-core durations, indicating that d c values are approximately exponentially distributed. This implies that, at a particular station, there are many more short duration storm-cores than longer duration ones and that number of storms reduce exponentially with duration Probability Probability Storm core duration (h) Figure Histogram of storm-core durations d c at pluviograph station Mildura MO (76031) The exponential distribution has one parameter and its probability density function is given by: p(d c )=(1/β)e -dc/ β (2.11) where p stands for probability density, d c is the storm-core duration and β is the parameter of the exponential distribution. The parameter β can be taken as the mean of the observed d c values in a pluviograph station or over a region Intensity 34

49 In practice, the conditional distribution of rainfall intensity is expressed in the form of intensity-frequency-distribution (IFD) curves, where rainfall intensity is plotted as a function of rainfall duration and frequency. The IFD curves for storm-core rainfall intensity were developed in a number of steps, as described below from Rahman et al. (2002). (i) (ii) (iii) The range of storm-core duration d c is divided into a number of class intervals (with a representative or midpoint for each class). For example 2-3 h (representation duration 2 h), 4-12 h (6 h), h (24 h). For the data in each class interval (except the 1 h class), a linear regression line is fitted between log(d c ) and log(i C ). The slope of the fitted regression line is used to adjust the intensities for all duration within the interval to the representative duration. An exponential distribution is fitted to the partial series of the adjusted intensities within the class interval, and design intensity values I C (ARI) were computed for ARIs of 2, 5,, 20, 50 and 0 years. For a selected ARI, the computed I c (ARI) values for each duration range were used to fit a second-degree polynomial between log(d c ) and log(i c ). The adopted Monte Carlo simulation scheme starts with the generation of a d c value from its marginal distribution. Given this d c and a randomly generated ARI value, the rainfall intensity value I c is then drawn from the conditional distribution of I c, expressed in the form of IFD curves Temporal pattern A rainfall temporal pattern is a dimensionless representation of the variation of rainfall intensity over the duration of rainfall event. The time distribution of rainfall during a storm are characterised by a dimensionless mass curve, i.e. a graph of dimensionless cumulative rainfall depth versus dimensionless storm time. The temporal pattern generation model applied by Hoang (2001) could generate design temporal patterns for storm-cores (TP c ). However, Rahman et al. (2002) adopted historic temporal patterns instead of generated temporal patterns. Here the observed 35

50 temporal patterns are expressed in dimensionless form in time intervals and are drawn randomly from the sample corresponding to the generated d c value during the simulation of streamflow hydrograph. The observed temporal patterns are expressed in two groups: one up to 12 hours duration, and the other greater than 12 hours duration. Storms with less than 4 hours durations are assumed to have the same temporal patterns as the observed 4 to 12 hours storms Initial loss The initial loss for a complete storm (IL s ) is estimated to be the rainfall that occurs prior to the commencement of surface runoff. The storm-core initial loss (IL c ) is the portion of IL s that occurs within the storm-core. The value of IL c can range from zero (when surface runoff commences before the start of the storm-core) to IL s (when the start of the storm-core coincides with the start of the complete storm event). Rahman et al. (2002b) proposed following equation to estimate IL c from IL s : IL c = IL s [ log (d c )] (2.12) This relationship gives IL c = IL s at d c = 0 hours, and IL c = 0.50 IL s at d c = 1 hour. The use of the IL s distribution with an adjustment factor, such as the one proposed in Equation 2.12, is preferable to the use of IL c directly, as IL s is more readily determined from data and can probably be derived using existing design loss data. However, Equation 2.12 tends to slightly underestimate observed values of IL c. Rahman et al. (2002b) also adopted a four-parameter Beta distribution to describe the distribution of IL s in that the four parameters are estimated from the observed lower limit (LL), upper limit (UL), mean and standard deviation of the IL s values. 2.9 SIMULATION OF DERIVED FLOOD FREQUENCY CURVES 36

51 This is the main component of the modelling framework for simulating flood frequency curves, known as Monte Carlo Simulation Technique. The basic principle of this technique involves simulation of a large number of flood events from a large number of rainfall events based on a wide range of likely combinations of floodproducing variables. The flood peaks from the simulation are subjected to flood frequency analysis to derive flood frequency curves. The overall procedure is illustrated in Figure PROPOSED RESEARCH From the literature review, it has been found that the Monte Carlo Simulation technique of design flood estimation is based on a sounder probabilistic formulation than the currently recommended method Design Event Approach. The proposed research is aimed to extend the Monte Carlo Simulation technique of Rahman et al. (2002) to ungauged catchments. This requires the regionalization of the distributions of rainfall duration, rainfall intensity, rainfall temporal pattern and initial loss. This also requires identification of regional average values of the fixed variables in the runoff routing model such as continuing loss, runoff routing model parameters and baseflow. The proposed research, in particular, is aimed at investigating the regionalisation of the distribution of rainfall duration and intensity. In this research, IFD curves are generated from the historical pluviograph data. Here, the ARR IFD curves are not used in the Monte Carlo Simulation, although the ARR IFD values of standard durations and ARIs are used as threshold values in selecting the storm-core events from the pluviograph data. The following hypotheses were tested in this research: Hypothesis 1: H 0 : The at-site and regional distribution of storm-core durations in the study area can be described by an Exponential distribution. H 1 : The at-site and regional distribution of storm-core durations in the study area cannot be described by an Exponential distribution. 37

52 Hypothesis 2: H 0 : The IFD curves at an ungauged catchment can be obtained from the weighted average IFD curves of an appropriate number of pluviograph stations in the vicinity of the ungauged catchment. H 1 : The IFD curves at an ungauged catchment cannot be obtained from the weighted average IFD curves of an appropriate number of pluviograph stations in the vicinity of the ungauged catchment. The weighting factors in obtaining the average IFD values can be obtained from the distances of the pluviograph stations from the ungauged catchment. That is, 1 = IFD IFD IFD IFD uc x x x x 1 x2 x3 (2.13) Where IFD uc = Weighted average IFD value at the ungauged catchment; IFD 1 = IFD value of the nearest pluviograph station from the ungauged catchment; IFD 2 = IFD value of the 2 nd catchment; and nearest pluviograph station from the ungauged x 1 = distance between the ungauged catchment and the nearest pluviograph station; x 2 = distance between the ungauged catchment and the 2 nd nearest pluviograph station, and so on. A NALYSIS Select stormcore events DATA GENERATION AND SIMULATION d c 38

53 Storm-core duration (d c ) I c Identify component distributions Storm-core rainfall intensity (I c ) TP c Storm-core temporal pattern (TP c ) IL c CL Initial losses (IL s, IL c ) Design loss analysis Rainfall excess hyetograph Runoff model calibration Route through runoff routing model (m = 0.8, k from calibration) Baseflow analysis BF Derived flood frequency curve Peak of simulated streamflow hydrograph Figure 2.9: Schematic diagram of Monte Carlo Simulation (Rahman et al., 2002) CHAPTER 3 39

54 DESCRIPTION OF DATA 3.1 PLUVIOGRAPH DATA To generate flood frequency curves using the Joint Probability based Monte Carlo Simulation technique two sets of data are required: (i) Time series pluviograph data to identify probability distributions of rainfall variables, e. g. duration, intensity and temporal pattern. (ii) Time series pluviograph data on the catchment and streamflow data at the catchments outlet location to identify probability distribution of initial loss. In addition, rainfall and streamflow data for a number of selected events are needed to calibrate the parameters of the adopted runoff routing model. In this study, Victoria is selected as the study area. A total of 76 pluviograph stations are selected from Victoria having a reasonably long record lengths. The locations of the selected pluviograph stations are shown in Figure 3.1. The names of the selected pluviograph stations are shown in Appendix A. The average record length of the selected pluviograph stations is 30 years, the range is 7 years to 128 years and the 75 th percentile is 37 years. The distribution of record lengths of the selected stations is shown in Figure

55 76,031 77,087 80,1 80,9 80,2 82,039 82,011 80,006 81,013 82,121 81,049 82,076 81,115 79,082 79,079 81,003 82,016 83,031 83,067 81,038 88,029 83,074 79,046 88,153 83,025 79,052 87,029 83,017 87,152 84,125 83,033 84,005 86,142 89,016 89,025 87,097 86,038 84,078 90,058 90,166 87,033 86,224 86,314 85,026 85,072 85,3 85,236 85,6 84,112 84,123 Figure 3.1 Locations of the selected pluviograph stations in Victoria Frequency Record length (years) Figure 3.2 Distribution of record lengths of the selected pluviograph stations 41

56 3.2 CATCHMENTS FOR THE VALIDATION OF THE NEW TECHNIQUE Three gauged catchments are selected from Victoria for validating the derived flood frequency curves obtained from the new technique. Some important characteristics of these catchments are shown in Table 3.1. Table 3.1 Selected gauged catchments for validating the new technique Catchment Streamflow Catchment area Period of streamflow data station (km 2 ) (years) No. Boggy Creek at Angleside (25) Tarwin River East Branch (22) Avoca River at Amphitheatre (25) An additional 12 gauged catchments were selected from Victoria (listed in Table 3.2) to compare the performances of the new technique with two other design flood estimation methods: the Probabilistic Rational Method and Quantile Regression Technique. To apply the quantile regression technique, data of the four flood producing variables were obtained: rainfall intensity of 12-hour duration and 2-year average recurrence interval (I12, mm/h), mean annual class A pan evaporation (evap, mm); catchment area (area, km 2 ); stream density (sden, km/km 2 ), which is the length of stream lines divided by the catchment area; and fraction quaternary sediment area (qsa). The qsa is a measure of the extent of alluvial deposits and is an indicator of floodplain extent in the study area. The explanatory variables evap and I12 are determined at the catchment centroid. These data for the selected 12 catchments are provided in Table

57 Table 3.2 Selected additional test catchments and relevant data StationID Lat Lon AREA I12:2 EVAP SDEN QSA (deg) (deg) (km2) (mm/h) (mm) (km/km2) (Fraction) SUMMARY This chapter selects study pluviograph stations from Victoria as well as test catchm ents that will be used to validate the new technique. All the necessary data were obtained and checked for the purpose of this study. 43

58 CHAPTER 4 METHODOLOGY IN THE PROPOSED RESEARCH 4.1 STEPS IN THE PROPOSED RESEARCH The proposed research aims to extend the Monte Carlo Simulation Technique of Rahman et al. (2002) to ungauged catchments. This will involve the following steps: Obtain the at-site distributions of storm-core duration at the selected pluviograph stations. Obtain the regional distributions of storm-core durations for Victoria. Obtain the intensity-frequency-duration (IFD) curves at the selected pluviograph stations. Develop a FORTRAN program to compute IFD curves at an ungauged catchment from the IFD curves of the nearby pluviograph stations according to the proposed Equation Obtain the dimensionless mass curves of the observed storm-core temporal patterns at the selected pluviograph stations. Obtain the regional distribution of initial loss in the selected study area. Obtain the regional average values of fixed variables continuing loss, parameters of the runoff routing model and baseflow. Once the regionalized values of all the variables for the runoff routing model are obtained, the simulation of the streamflow hydrographs can be carried out. Apply the Monte Carlo Simulation technique (for ungauged catchments) to test catchments and compare the results with the observed flood frequency curves. 44

59 Compare the results obtained from the Monte Carlo Simulation technique (for ungauged catchments) with other design flood estimation techniques for ungauged catchments i.e. the Probabilistic Rational Method and Quantile Regression Technique. Make conclusions regarding the applicability an d validity of the Monte Carlo Simulation technique (for ung auged catchments). The proposed research will apply the Mont e Carlo Simulation technique of Rahman et al. (2002), and hence major steps of this technique are described below. 4.2 RAINFALL ANALYSIS In Applying the Monte Carlo Simulation Technique of Rahman et al. (2002), hourly pluviograph data at a given pluvio graph station is analysed to select storm-core rainfall events. These storm-core e vents are then analysed to obtain probability distributions of storm-core duration ( d ), average rainfall intensi ty ( I c ), and temporal patterns ( TP ). c c For rainfall analysis, a FORTRAN program called mcsa5.for is used (Rahman et al., 2002). This program requires hourly pluviograph data (sample shown in Table 4.1). The input to the program is given via a parameter file e.g. a76031.psa. An example of parameter file for Pluviograph Station (Mildura Mo) is shown in Table 4.2. Important output files from the program mcsa5.for are listed in Table LOSS ANALYSIS To identify the probability distribution of storm-core initial loss (IL c ), a FORTRAN program called Losssca.for (Rahman et al., 2002) is used. The basic data input to this program are hourly streamflow and rainfall data. The required input to the program is given through a parameter file, with extension lan, e.g.a40082.lan. The parameter file for the Bremer River catchment (Station 40082) is shown in Table 4.4. Important output files from the program losssca.for are listed in Table

60 Table 4.1 Hourly pluviograph data for Pluviograph Station 76031(Mildura Mo) Station ID Variable Type Year, Month, Day, Hour Rainfall (mm) Quality Code 46

61 Table 4.2 Parameter file a76031.psa for rainfall analysis (for Station 76031) Input a dat Description Station ID Data file, rainfall in mm 6 Dry period, between successive complete storm events, in hour 0.4 Reduction factor, F1 0.5 Reduction factor, F I (Log-normal design rainfall intensity, 2 year 1 hr 1 duration), mm/hr I 12 (Log-normal, 2 year, duration-12 hr), mm/hr I 72 (mm/hr) I 1 ( mm/hr) I12 (mm/hr) 1.62 (mm/hr) 50 I Skewness(G) 140 Catchment area, km a.txt Streamflow data file name Table 4.3 Important output files from program mcsa5.for (for Station 76031) Output Description a dit Duration, intensity and total rainfall for complete storm a76031.dcs Duration of complete storm a76031.cdr Storm-core duration a76031.tpo Output file for temporal pattern analysis a76031.ifd IFD table 47

62 Tab le 4.4 Parameter file a40082.lan- (Bremer River catchment) for rainfall analysis Input Description a40082 Station ID 130 Catchment area, km dat Data file, rainfall in mm 40082a.txt Streamflow data file (hourly), streamflow in m 3 /s Table 4.5 Output file for a40082-(bremer River catchment) for rainfall analysis Output a40082.ssr Description Starting time of surface runoff a40082.ics Initial loss for complete storm ( IL S ) a40082.slp ILS statistics (lower limit, upper limit, mean and standard deviation) 4.4 CALIBRATION OF RUNOFF ROUTING MODEL To determine flood frequency curve for a station, runoff routing model needs to be calibrated for the station. In Monte Carlo Simulation technique of Rahman et al. (2002), the adopted runoff routing model uses a storage discharge relationship of the form m S = kq (4.4.1) 3 where S is the catchment storage in m, k a storage delay parameter in hour, Q the rate of outflow in m 3 / s and m is non-linearity parameter (taken as 0.8). The objective of model calibration is to determine a value of k that results in satisfactory fit for a range of recorded rainfall and runoff events at the catchment outlet. The following strategy (Rahman et al., 2002) has been found to be useful in calibration: 48

63 (i) Select a number of rainfall and runoff events from the observed data at the (i i) For about two-thirds of the selected rainfall and runoff events, calibrate the (iii) catchment outlet. model for an appropriate value of k. At first, change the initial and continuing loss value to match the rising limb of the computed and observed hydrographs and to obtain a volume balance. When these are achieved, provisionally fix the initial and continuing loss value but change the k value to match the peak. From the values of k obtained above, select a global k value for all events. (iv) Finally, use the selected k value with the remaining observed rainfall and runoff events to validate the calibrated model. A FORTRAN program called cali4.for (Rahman et al., 2002) is used for the runoff routing model calibration. Figure 4.3 and Figure 4.4 show baseflow separation and observed vs. computed streamflow data for the Bremer River catchment, respectively. 4.5 SIMULATION OF STREAMFLOW HYDROGRAPH To simulate the streamflow hydrograph a FORTRAN program called mcdffc3.for (Rahman et al., 2002) is used. The required input to the program is given through a parameter file e.g.re1s1.par. An example of parameter file is shown in Table 4.6. Important output files from the program mcdffc3.for are listed in Table 4.7. In this study, 20,000 simulation runs were adopted to sufficiently reflect all the possible variability and combinations of the various flood-producing variables that could occur in real flood generation process. The set of simulated flood peaks (NG), obtained as above, is used to construct a derived flood frequency curve. As these flood peaks are obtained from a partial series of storm-core rainfall events, they also form a partial series. Construction of the derived flood frequency curve from the generated partial series of flood peaks involves the following steps as per Rahman et al. (2002): 49

64 (i) (ii) (iii) Arrange the NG simulated peaks in decreasing order of magnitude. Assign rank (m) 1 to the highest value, 2 to the next and so on. For each of the ranked floods, compute an ARI from the following equation: ARI = NG m λ NY m 0.4 (4.5.1) where NG is the number of simulated peaks, m is the rank, λ is the average number of storm-core events per year for the catchment of interest, and NY is the number of years of simulated flood data. (iv) Prepare a plot of ARI versus flood peaks, i.e. a plot of the empirical flood frequency curve defined by the simulated flood peaks. 4.6 TESTING THE HYPOTHESIS WHETHER STORM-CORE DURATION CAN BE DESCRIBED BY AN EXPONENTIAL DISTRIBUTION Previous study on a smaller number of pluviograph stations in Victoria (Rahman et al., 2002) indicated that probability distribution of storm-core durations can be approximated by an Exponential distribution. To test the statistical hypothesis that storm-core duration data in one of the selected 76 pluviograph stations follow an Exponential distribution, Chi-squared test was applied at 5% level of significance. The Chi-squared test is based on the Chi-squared statistic, which is related to the weighted sum of squared differences between the observed and theoretical frequencies. The Chi squared test is given by the following equation. κ k 2 2 ( oi ) = i= 1 ei e i (4.6.1) Where; Where 2 κ is a value of a random variable whose sampling distribution is approximated very closely by the chi-squared distribution with ν = k 1 degrees of freedom. The symbols o i and e i represents the observed and expected frequencies, respectively, for the ith cell. 50

65 0 Q (m3/s) Streamflow Baseflow Time Interval ( h) Figure 4.3 Baseflow separation for the Bremer River Catchment Qobs Qcom s) Q (m3/ Time Interval (h) Figure 4.4 Observed vs. computed streamflow data for a selected event for the Bremer River catchment ( IL = 16 mm, CL = mm/h, k = h, m = 0. 8, Start of event = 23/02/1975 at am) 51

66 Table 4.6 Sample parameter file re1s1.par for simulation of streamflow hydrograph. Here IL is initial loss and CL is continuing loss Input Reg1s1 Description Catchment name a Run Sequence No of simulations 13.1 Mean durations 8 Catchment area, km 2 reg1s1.ifd IFD file name reg1s1.tem TP file name 0 IL-Lower Limit IL-Upper Limit IL-Mean IL-SD TP intervals 1.90 CL 0.8 m 21 k 1.16 Baseflow 2000 Length of simulation Table 4.7 Sample output file for re1s1.par for simulation of streamflow hydrograph Output Description areg1s1.gdc Generated storm-core durations areg1s1.gic Generated storm-core rainfall intensities areg1s1.glc Generated storm-core loss values areg1s1.ffc Derived flood frequency curves 52

67 4.7 PROGRAM TO COMPUTE WEIGHTED AVERAGE IFD TABLE AT AN UNGAUGED CATCHMENT In this study, a FORTRAN program has been developed called ucat1.for to compute the IFD table at an ungauged catchment from the IFD tables of the pluviograph stations in the vicinity of the ungauged catchment. The program contains 1320 lines of codes and is given in Appendix B. The program ucat1.for operates as follows: (i) At first, the IFD tables of all the selected 76 pluviograph stations have been stored in a data file called text.dat. (ii) When the program executes, it asks the user to enter latitude and longitude of the ungauged catchment. (iii) Then it calculates the distances between the ungauged catchment and all the 76 pluviograph stations. (iv) After that, these distances are sorted in a data file called text11.srt. (v) In the next step, the program asks the user to enter the radius of proposed region (km). Depending on the radius of the candidate region, the program finds out how many stations fall within that region and stores the names of these stations in a data file called text12.reg. The program at this moment allows selection of maximum five pluviograph stations in a region. (vi) To calculate the average IFD table, the program takes only those station s IFD table which falls in the proposed region. The relevant IFD tables are stored in a file called text14.ilt. (vii) Finally, depending on the number of stations in a region, the program calls the appropriate subroutine and calculates the IFD table at the ungauged catchment using Equation The final IFD table for the ungauged catchment is called regional. ifd. As an example, Figure 4.5 shows an output window of the program ucat1.for to compute the IFD table for latitude degree and longitude degree. Here the candidate region has a radius of 50 km. 53

68 Figure 4.5 Output windows to compute weighted average IFD table 54