Application of the Joint Probability
Approach to Ungauged Catchments for
Design Flood Estimation
By
Tanvir Mazumder
Student ID: 13762067
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
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.
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)
TABLE OF CONTENTS
ABSTRACT-----------------------------------------------------------------------------------viii
ACRONYMS AND ABBREVIATIONS---------------------------------------------------x
1.0 INTRODUCTION -------------------------------------------------------------------------1
1.1 Background --------------------------------------------------------------------------1
1.2 Objectives ---------------------------------------------------------------------------4
1.3 Thesis outline ----------------------------------------------------------------------4
2.0 DESIGN FLOOD ESTIMATION METHODS BASED ON THE JOINT
PROBABILITY APPROACH ----------------------------------------------------------6
2.1 General ------------------------------------------------------------------------------6
2.2 Flood estimation methods overview --------------------------------------------6
2.3 Rainfall-based flood estimation methods---------------------------------------7
(i) Empirical methods -----------------------------------------------------8
(ii) Continuous simulation method -------------------------------------11
(iii) Design Event Approach ---------------------------------------------12
(iv) Joint Probability Approach-----------------------------------------13
2.4 Description of the Joint Probability Approach -------------------------------15
(i) Approximate methods--------------------------------------------------15
(a) Discrete methods ------------------------------------------------------15
(b) Simulation techniques ------------------------------------------------17
(ii) Analytical methods ---------------------------------------------------18
i
(1) Methods based on U.S. Soil Conservation Service’s curve number
procedure--------------------------------------------------------------19
(2) Methods based on Eagleson’s kinematic runoff model ---------19
(3) Methods based on other types of rainfall-runoff models ------20
(4) Methods based on geomorphologic unit hydrograph ----------21
2.5 Recent research on the joint probability approach to design
flood estimation-----------------------------------------------------------------22
2.6 Index of major research works on the Joint Probability Approach-------26
2.7 Joint probability approach to ungauged catchments -----------------------32
2.8 Description of the Monte Carlo Simulation Technique to design
flood estimation of Rahman et al. (2002)------------------------------------32
2.8.1 General ---------------------------------------------------------------32
2.8.2 Event definition -----------------------------------------------------33
2.8.3 Distribution of flood-producing variables -----------------------33
2.8.3.1 Duration -----------------------------------------------------------34
2.8.3.2 Intensity -----------------------------------------------------------35
2.8.3.3 Temporal pattern -------------------------------------------------35
2.8.3.4 Initial loss ---------------------------------------------------------36
2.9 Simulation of derived flood frequency curves -----------------------------37
2.10 Proposed research ------------------------------------------------------------37
3.0 DESCRIPTION OF DATA ---------------------------------------------------------40
3.1 Pluviograph data-----------------------------------------------------40
3.2 Catchments for the validation of the new technique---------------------------42
ii
3.2 Summary ---------------------------------------------------------------------------43
4.0 METHODOLOGY IN THE PROPOSED RESEARCH ---------------------44
4.1 Steps in the proposed research ----------------------------------------------44
4.2 Rainfall analysis --------------------------------------------------------------45
4.3 Loss analysis -----------------------------------------------------45
4.4 Calibration of runoff routing model --------------------------48
4.5 Simulation of streamflow hydrograph ----------------------------------------49
4.6 Testing the hypothesis whether storm-core duration can be described
by an exponential distribution-------------------------------------------------50
4.7 Program to compute weighted average IFD table
at an ungauged catchment ------------------------------------------------------53
5.0 RESULTS --------------------------------------------------------------------------------55
5.1 Distribution of storm-core duration ------------------------------------------55
5.2 Derivation of intensity-frequency-duration
curves ---------------------------------------------------------------------------- 56
5.3 Regionalisation of the distributions of various flood
producing variables -------------------------------------------------------------56
5.3.1 Storm-core duration -----------------------------------------------------------57
5.3.2 Storm-core rainfall intensity ------------------------------------------------59
5.3.3 Storm-core temporal patterns------------------------------------------------59
5.3.4 Initial loss-----------------------------------------------------------------------60
5.3.5 Continuing loss, storage delay parameter, non-linearity parameter
and baseflow-----------------------------------------------------------------------61
iii
5.4 Derived flood frequency curves with regionalised
parameters of the input variables----------------------------------------------61
5.5 Sensitivity analyses-------------------------------------------------------------71
5.5.1 Continuing loss ---------------------------------------------------------------71
5.5.2 Catchment storage parameter (k) ----------------------------74
5.6 Comparison among Joint Probability Approach, Probabilistic
Rational Method and Quantile Regression Technique--------------------76
6.0 SUMMARY AND CONCLUSIONS--------------------------------------------------82
6.1 Summary --------------------------------------------------------------------------82
6.2 Conclusions---------------------------------------------------------84
6.3 Recommendation of further study----------------------------------------------85
REFERENCES--------------------------------------------------------------------------------86
APPENDICES
Appendix A: List of study pluviograph stations----------------------------------97
Appendix B: FORTRAN program to compute weighted average IFD table
at an ungauged catchment ----------------------------------------99
Appendix C: Distributions of storm-core durations for selected catchments-100
Appendix D: IFD tables for the study catchments----------------------------------------109
Appendix E: IFD curves for selected catchments----------------------------------------144
Appendix F: Flood frequency curves for selected test catchments--------------------148
Appendix G: A Numerical example illustrating the identification of a storm-core--154
iv
TABLES
Table 2.1: Index of previous research on the Joint Probability Approach--------------26
Table 3.1: Selected gauged catchments for validating the new technique --------------42
Table 3.2: Selected additional test catchments and relevant data ------------------------43
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-----------------------------------------------------------------48
Table 4.5: Output file for a40082-(Bremer River catchment) for rainfall analysis----48
Table 4.6: Parameter file re1s1.par (region-1, station-1) for simulation
of streamflow hydrograph ---------------------------------------------------------52
Table 4.7: Output file for re1s1.par (region-1, station-1)
for simulation of streamflow hydrograph---------------------------------------52
Table 5.2.1: IFD table for Station 76031-----------------------------------------------------58
Table 5.4.1: Radius of subregions for the Boggy Creek catchment ----------------------62
Table 5.4.2: The percentage difference between the DFFCs and observed
floods for the Boggy Creek catchment----------------------------------------64
Table 5.4.3: Radius of subregions for the Tarwin River catchment ---------------------65
v
Table 5.4.4: Percentage differences between the DFFCs and observed
floods for the Tarwin River catchment-----------------------------------------66
Table 5.4.5: Radius of subregions for the Avoca River catchment ---------------------67
Table 5.4.6: The percentage difference between the DFFCs and observed
floods for the Avoca River catchment------------------------------------------70
Table 5.6.1: Flood estimation obtained by three methods
(QRM, PRM and JPA) -----------------------------------------------------------78
Table 5.6.2:Median relative error values (%) for three methods JPA, QRT
and PRM-----------------------------------------------------------------------------80
FIGURES
Figure 2.1: Classification of design flood estimation methods------------------------------7
Figure 2.2: Flood Estimation by Design Event Approach ---------------------------------14
Figure 2.8.1: Histogram of storm-core durations dc at pluviograph
station Mildura MO (76031) ------------------------------------------------------34
Figure 2.9: Schematic diagram of Monte Carlo Simulation--------------------------------39
Figure 3.1: Locations of the selected pluviograph stations in Victoria -------------------41
Figure 3.2: Distribution of record lengths of the selected pluviograph stations --------41
Figure 4.3: Baseflow separation for the Bremer River Catchment -----------------------51
Figure 4.4: Observed vs. computed streamflow data for a selected event
for the Bremer River catchment----------------------------------------------------51
Figure 4.5: Output windows to compute weighted average IFD table ------------------54
Figure 5.1.1: Histogram of storm-core duration for Pluviograph Station 82121-------56
vi
Figure 5.2.1: Plot of IFD curves for Station 76031-----------------------------------------58
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---------------------------------------------60
Figure 5.4.1: Derived flood frequency curves for the Boggy Creek
catchment using regionalised parameters--------------------------------------62
Figure 5.4.2: Derived flood frequency curves for the Tarwin River
catchment using regionalised parameters---------------------------------------65
Figure 5.4.3: Derived flood frequency curves for the Avoca River
catchment of using regionalised parameters-----------------------------------67
Figure 5.4.4: Box plot of relative errors for the three test catchments--------------------69
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------------------------------------------------73
Figure 5.5.3: Effects on derived flood frequency curves for the Avoca
River Catchment of using different continuing loss values -----73
Figure 5.5.4: Effects on derived flood frequency curves for the
Boggy Creek Catchment of using different k values----------------------74
Figure 5.5.5: Effects on derived flood frequency curves for the
Tarwin River Catchment of using different k values----------------------75
vii
Figure 5.5.6: Effects on derived flood frequency curve for the
Avoca River Catchment of using different k values-----------------------76
Figure 5.6.1: Box plot for relative errors for JPA-----------------------------------------80
Figure 5.6.2: Box plot for relative errors for PRM ---------------------------------------81
Figure 5.6.3: Box plot for relative errors for QRT----------------------------------------82
viii
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
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
Acronyms and Abbreviations
ARI
Average Recurrence Interval
ARR
Australian Rainfall and Runoff
AEP
Annual Exceedance Probability
BOM
Bureau of Metrology
IFD
Intensity-frequency-duration
dc
Storm-core Duration
Ic
Storm-core Rainfall-Intensity
ILs
Initial Loss for Complete Storm
ILc
Initial Loss for Storm-Core
JPA
Joint Probability Approach
MCST
Monte Carlo Simulation Technique
TPc
Storm-core Temporal Pattern
URBS
Runoff-Routing Hydrologic Model
k
Storage delay parameter
m
Non-linearity parameter
PRM
Probabilistic Rational Method
QRM
Quantile Regression Technique
JPA
Joint Probability Approach
DFFC
Derived Flood Frequency Curve
xi
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
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
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
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.
•
distribution.
•
with the Joint Probability Approach.
To assess whether storm duration data in Victoria can be described by an Exponential
To develop a method to regionalize the distribution of rainfall intensity for application
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
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
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
Design flood estimation methods
Rainfall based methods
Streamflow based methods
Event based methods
Continuous simulation based methods
Design Event
Empirical
Joint Probability
Approach
methods
Approach
Partial
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
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
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:
(2.1)
QT = 0.278CTItc,TA
Where QT is the peak flow (m3/s) for an average recurrence interval (ARI) of T years, CT is the
runoff coefficient for the same ARI which is dimensionless, ItcT is the average rainfall intensity
(mm/hr) for a storm duration of tc hours with ARI of T years. Here, tc is the time of
concentration in hours and A is the catchment area (km2). 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 CT 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, 10, 20, 50 and 100 years ARIs based
on the data of 88 catchments from South-east Australia. These equations are provided below.
log Q 2 = − 4 .148 + 0 .667 log( area ) + 1 .417 log( I 12 )
+ 0 .930 log( sden ) + 1 .630 log( evap )
R2 = 0.72, Adjusted R2 = 0.70, SEE = 0.23 (6.43% of the mean logQ2)
9
( 2 .2 )
log Q5 = − 6 .513 + 0 .720 log( area ) + 1 .448 log( I 12 )
+ 0 .875 log( sden ) + 2 .439 log( evap )
( 2 . 3)
R2 = 0.75, Adjusted R2 = 0.74, SEE = 0.23 (6.06% of the mean logQ5)
log Q10 = −6.551 + 0.682 log( area ) + 1.377 log( I 12 )
+ 0.968 log( sden ) + 2.542 log( evap )
( 2 .4 )
R2 = 0.73, Adjusted R2 = 0.72, SEE = 0.23 (5.89% of the mean logQ10)
log Q20 = −6.166 + 0.735 log( area ) + 1.537 log( I 12)
+ 0.987 log( sden ) + 2.374 log( evap )
( 2.5)
R2 = 0.74, Adjusted R2 = 0.73, SEE = 0.24 (5.88% of the mean logQ20)
log Q50 = −6.663 + 0.729 log(area) + 1.266 log(I12)
+ 0.997 log(sden) + 2.597 log(evap) − 0.086 log(qsa)
(2.6)
R2 = 0.77, Adjusted R2 = 0.76, SEE = 0.23 (5.48% of the mean logQ50)
log Q100 = −6.587 + 0.726 log(area) + 1.351 log( I12)
+ 1.007 log(sden) + 2.580 log(evap) − 0.084 log(qsa )
(2.7)
R2 = 0.76, Adjusted R2 = 0.75, SEE = 0.23 (5.56% of the mean logQ100)
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,
km2); stream density (sden, km/km2), 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.
10
(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
̇
̇
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
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
Design Rainfall Depth
(ARI = Y, Duration = D
Temporal and Spatial
Patterns of Rainfall
Design Rainfall Event
Loss Parameters
Loss Model
(ARI = Y)
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
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.
15
In SCS curve number method (Soil
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
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
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 Tp = 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: Tq = n/j/m where Tq 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
(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 runoffrouting model utilised kinematic wave equations for both overland flow and channel 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
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,
(2.8)
Q = CIA
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
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
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
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
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
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
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
1972
Title
Author(s)
Dynamics of flood frequency
Eagleson, P.S
Summary
This paper assumed that storm characteristics (duration and
intensity) are independent random variables.
1973
Estimation of Design Floods and the Beran, M. A.
This paper presented a procedure that sampled the possible
Problem of Equating the Probability
ways in which a storm of given ARI could cause floods, and
of Rainfall and Runoff.
derived their Joint Probability distributions. It was found
that the derived flood frequency curves were much flatter
than the observed ones.
1974
Modelling
of
Stochastic- Laurenson, E. M.
Deterministic Hydrologic Systems.
This paper represents the most general application of total
probability theorem which is described by ‘transformation
matrix’ approach.
1982
Considerations for design flood Ahern, P.A. and
This paper mentioned that Joint Probability Approach,
estimation
which considers the outcomes of events with all possible
using
catchments Weinmann, P.E.
modelling.
combinations of input values and, if necessary, their
correlation structure, obtain better estimates of design flows.
26
1982
A
derived
distribution
flood
using
frequency Hebson, C and
Horton
order Wood, E.F.
They used Eagleson’s (1972) partial area runoff routing
model and their runoff routing model was based on the
Ratios.
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.
1993
Derived,
physically
based Muzik, I
This paper represented a modified SCS curve number
distribution of flood probabilities.
method in the Monte Carlo Simulation to obtain a derived
Extreme
distribution of peak discharge. Here the initial abstraction
Hydrological
Events:
Precipitation, Floods and Droughts.
and five-day antecedent rainfall values were assumed to be a
random variable.
1996
Process
Controls
Frequency.1.
on
Derived
Flood Sivapalan, M.,
They proposed a
Flood Bloschl, G. and
distribution of rainfall intensity and duration, which
Frequency
Gutknecht, D.
method of specifying the joint
considers IFD curves as conditional distributions, and
distribution of storm duration as marginal distribution.
2000
Derived distribution of floods based Iacobellis, V. and
They assumed that the peak direct flow is expressed as the
on the concept of partial area Fiorentino, M.
product of average runoff per unit area, u(a), and the peak
coverage with a climate appeal.
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
2000
A
derived
flood
frequency Goel, N. K., Kurothe,
They presented a stochastic rainfall model which assumes
distribution for correlated rainfall R.S., Mathur, B.S.
that rainfall intensity is, either positively or negatively,
intensity and duration.
correlated to the rainfall duration for the generation of the
and Vogel, R.M
rainfall.
2000
The Gumbel Mixed Model applied Yue, S.
He represented Gumbel distribution model. Based on this
to
model, one can obtain the Joint Probability distributions,
storm frequency analysis.
and the associated return periods of two correlated variables
if their marginal distributions can be represented by the
Gumbel distribution.
2002
Overcoming the joint probability Heneker, T.,
This paper represented the ways of overcoming the Joint
problem associated with initial loss Lambert, M., and
Probability problem by allowing design rainfall obtained
estimation in design flood estimation Kuczera, G.
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.
2002
Integration
of
Monte
Carlo Rahman, A,
This paper describes how a Monte Carlo simulation
simulation technique with URBS Carroll, D.G, and
technique can be applied with the industry-based runoff
model for design flood estimation.
routing model URBS to determine derived flood frequency
Weinmann, P.E.
curves.
28
2002
Monte Carlo Simulation of flood Rahman, A,
This technique is appropriate for the derivation of flood
frequency curves from rainfall.
Weinmann, P.E,
frequency curves in the ARI range 1 to 100 years for small
Hoang, T.M.T., and
gauged catchments.
Laurenson, E.M
2002
Monte Carlo simulation of flood Weinmann, P.E.,
This paper has highlighted some of the theoretical and
frequency curves from rainfall-the Rahman, A.,
practical limitations of the currently used Design Event
way ahead.
Hoang, T, Laurenson, Approach to rainfall based design flood estimation.
E.M., and
Nathan, R.J.
2002
The use of probability distributed Rahman, A.,
This study examines the role played by initial loss
initial
modelling in flood estimation for selected Victorian
losses
in
design
estimation.
flood Weinmann, P.E. and
Mein, R.G.
catchments. It has been shown that the variability of initial
loss in the Victorian catchments can be described by a fourparameter Beta distribution.
2003
Application
of
Monte
Carlo Charalambous, J.,
They found that the URBS-Monte Carlo Simulation
Simulation Technique with URBS Rahman, A., and
technique can easily be applied to large catchments.
Model for Design Flood Estimation Carroll. D.
Although the limited data availability introduced significant
in Large Catchments.
uncertainty in the distributions of the input variables e.g.
IFD curves.
29
2003
Joint Probability and Design Storms Kuczera, G.,
They suggested that the current revision of ARR needs to
at the Crossroads
Lambert, M.,
articulate the shortcomings of the design storm approach,
Heneker, T.,
identify calibration strategies, which gives guidance about
Jennings, S., Frost, A. its reliability in different application. They also notify that
and Coombes, P
event Joint Probability methods based on Monte Carlo
Simulation are computationally less demanding but require
specification of the probability distribution of initial
conditions.
2004
Appropriate spatial variability of Carroll, D. and
This paper describes the effects of spatial variability of
flood producing variables in the Rahman, A
flood producing variables on design flood frequency curves
Joint Probability Approach to design
in the Joint Probability Approach.
flood estimation.
2004
Regionalization of design rainfalls in Kader, F., and
This paper represents how the distribution of rainfall
Victoria, Australia for design flood Rahman, A.
intensity can be regionalized in the state of Victoria in
estimation
Australia.
by
Joint
Probability
Approach.
2004
Investigation of Sub tropical rainfall
Carroll, D., and
This paper examines the relationship between rainfall
characteristics for use in the Joint Rahman, A.
intensities of complete storm and storm core and the
Probability
application of multiplicative cascade model for temporal
Approach
to
design
flood estimation
pattern distribution is also discussed here.
30
2004
Study of fixed duration design
rainfalls
in
Australian
Rauf, A., and
This study examines the sampling properties of the rainfall
Rainfall Rahman, A.
events in the Australian Rainfall-Runoff (ARR) method and
Runoff and Joint Probability based
Joint Probability Approach to identify any systematic
design rainfalls.
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.
2004
An improved framework for the Nathan. R. J and
This paper presented a methodology that reduces the
characterisation of extreme floods Weinmann .P.E.
practical problems involved in the derivation of both
and for the assessment of dam
standards
safety.
methodology is based on the use of Monte Carlo
Simulation.
31
and
risk
based
design
estimates.
Here
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)
2.8.1 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
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 (ID) or a sub-storm duration (Id)
must satisfy one of the following conditions:
I D ≥ f1 × 2 I D
(2.9)
I d ≥ f ×2 I d
2
(2.10)
Where f1 and f2 are reduction factors, the threshold intensity 2ID is the 2 year ARI
design rainfall intensity for the selected storm duration D, and 2Id is the corresponding
intensity for the sub-storm duration d. The use of smaller values of f1 and f2 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 2Id for
that duration d, then selecting the bursts of that duration which produce the highest
ratio.
2.8.3 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
2.8.3.1 Duration
The storm-cores are selected from the hourly pluviograph data of selected stations and
analysed for storm-core duration (dc), average rainfall intensity (Ic), and temporal
patterns (TPc). Figure 2.8.1 shows a typical histogram of the frequencies of different
storm-core durations, indicating that dc 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.
0.7
0.6
Probability
0.5
0.4
Probability
0.3
0.2
0.1
0
1--10 11-- 21-- 31-- 41-- 51-- 61-20
30
40
50
60
70
7180
Storm core duration (h)
Figure 2.8.1 Histogram of storm-core durations dc at pluviograph station Mildura MO
(76031)
The exponential distribution has one parameter and its probability density function is
given by:
p(dc)=(1/β)e-dc/ β
(2.11)
where p stands for probability density, dc 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 dc values in a pluviograph station or over a region.
2.8.3.2 Intensity
34
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)
The range of storm-core duration dc 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), 13-36 h (24 h). For the
data in each class interval (except the 1 h class), a linear regression line
is fitted between log(dc) and log(IC). The slope of the fitted regression
line is used to adjust the intensities for all duration within the interval to
the representative duration.
(ii)
An exponential distribution is fitted to the partial series of the adjusted
intensities within the class interval, and design intensity values IC (ARI)
were computed for ARIs of 2, 5, 10, 20, 50 and 100 years.
(iii)
For a selected ARI, the computed Ic (ARI) values for each duration range
were used to fit a second-degree polynomial between log(dc) and log(Ic).
The adopted Monte Carlo simulation scheme starts with the generation of a dc
value from its marginal distribution. Given this dc and a randomly generated
ARI value, the rainfall intensity value Ic is then drawn from the conditional
distribution of Ic , expressed in the form of IFD curves.
2.8.3.3 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 (TPc). However, Rahman et al. (2002) adopted
historic temporal patterns instead of generated temporal patterns. Here the observed
35
temporal patterns are expressed in dimensionless form in 10 time intervals and are
drawn randomly from the sample corresponding to the generated dc 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.
2.8.3.4 Initial loss
The initial loss for a complete storm (ILs) is estimated to be the rainfall that occurs
prior to the commencement of surface runoff. The storm-core initial loss (ILc) is the
portion of ILs that occurs within the storm-core. The value of ILc can range from zero
(when surface runoff commences before the start of the storm-core) to ILs (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 ILc from ILs:
ILc = ILs[0.5 + 0.25log10(dc)]
(2.12)
This relationship gives ILc = ILs at dc = 100 hours, and ILc = 0.50 × ILs at dc = 1 hour.
The use of the ILs distribution with an adjustment factor, such as the one proposed in
Equation 2.12, is preferable to the use of ILc directly, as ILs 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 ILc.
Rahman et al. (2002b) also adopted a four-parameter Beta distribution to describe the
distribution of ILs in that the four parameters are estimated from the observed lower
limit (LL), upper limit (UL), mean and standard deviation of the ILs values.
2.9 SIMULATION OF DERIVED FLOOD FREQUENCY CURVES
36
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 2.9.
2.10 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:
H0: The at-site and regional distribution of storm-core durations in the study area can
be described by an Exponential distribution.
H1: The at-site and regional distribution of storm-core durations in the study area
cannot be described by an Exponential distribution.
37
Hypothesis 2:
H0:
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.
H1: 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,
⎡
⎢
⎢
IFDuc = ⎢⎢
⎢
⎢
⎣⎢
1
x1
+
1
1
x2
+
1
x3
⎤
⎥
⎤
⎥ ⎡ IFD
1 + IFD2 + IFD3 + ...⎥
+ ...⎥⎥ × ⎢⎢
⎥
x2
x3
⎥
⎥ ⎣⎢ x1
⎦
⎥
⎦⎥
(2.13)
Where IFDuc = Weighted average IFD value at the ungauged catchment;
IFD1 = IFD value of the nearest pluviograph station from the ungauged catchment;
IFD2 = IFD value of the 2nd nearest pluviograph station from the ungauged
catchment;
x1 = distance between the ungauged catchment and the nearest pluviograph station;
and
x2 = distance between the ungauged catchment and the 2nd nearest pluviograph
station, and so on.
ANALYSIS
Select stormcore events
DATA GENERATION
AND SIMULATION
dc
38
Storm-core
duration (dc)
Ic
Identify
component
distributions
Storm-core
rainfall
intensity (Ic)
Storm-core
temporal
pattern (TPc)
TPc
ILc
CL
Initial losses
(ILs, ILc)
Rainfall
excess
hyetograph
Design loss
analysis
Runoff
model
calibration
Baseflow
analysis
Derived flood
frequency curve
Route through runoff routing
model (m = 0.8, k from
calibration)
BF
Peak of simulated
streamflow
hydrograph
Figure 2.9: Schematic diagram of Monte Carlo Simulation (Rahman et al., 2002)
CHAPTER 3
39
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 75th percentile is 37 years. The distribution of
record lengths of the selected stations is shown in Figure 3.2.
40
76,031
77,087
80,110
80,109
80,102
82,039
80,006
81,049
79,082
79,052
79,079
81,038
81,115
81,003
88,029
81,013
82,011
82,121
82,076
83,031 83,067
83,074
82,016
88,153
79,046
87,029
87,152
89,016 89,025
83,025
83,017
86,142
84,005
83,033
87,097 86,038
87,033
90,058
90,166
84,123
84,125
84,112
84,078
86,224
86,314
85,026
85,103
85,236
85,106
85,072
Figure 3.1 Locations of the selected pluviograph stations in Victoria
30
25
Frequency
20
15
10
5
0
1--10
10--20
20--30
30--40
40--50
Record le ngth (years)
Figure 3.2 Distribution of record lengths of the selected pluviograph stations
41
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
station
Catchment area
2
Period of streamflow data
(km )
(years)
108
1974-1998
No.
Boggy Creek at
403226
Angleside
Tarwin
(25)
River
227226
127
East Branch
Avoca River at
Amphitheatre
1957-1979
(22)
408202
78
1975-1999
(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, km2); stream density (sden, km/km2), 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
3.2.
42
Table 3.2 Selected additional test catchments and relevant data
StationID
226410
227200
229218
230204
234200
401215
404207
405205
405212
405214
405229
408202
Lat
(deg)
38.33
38.54
37.67
37.47
37.81
36.87
36.61
37.41
37.1
37.15
36.64
37.78
Lon
(deg)
146.53
146.67
145.26
144.67
143.59
147.7
146.06
145.56
145.06
146.13
144.87
144.54
AREA
(km2)
88.8
215
36.3
79.1
324
471
451
108
337
368
108
629
I12:2
(mm/h)
4.18
4.9
3.98
4.35
3.6
4.5
4.11
4.5
3.74
4.19
3.5
4.02
EVAP
(mm)
1200
1216
1200
1153
1188
1000
1447
1125
1200
1100
1500
1320
SDEN
QSA
(km/km2) (Fraction)
1.48
0.01
1.22
0.1977
1.29
0.124
2.12
0.4273
1.6
0.1883
1.34
0.0061
1.07
0.1608
1.4
0.0406
1.68
0.0556
1.8
0.1518
1.05
0.45
2.24
0.4316
3.3 SUMMARY
This chapter selects study pluviograph stations from Victoria as well as test
catchments that will be used to validate the new technique. All the necessary data
were obtained and checked for the purpose of this study.
43
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.
•
pluviograph stations.
Obtain the intensity-frequency-duration (IFD) curves at the selected
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 2.13.
•
patterns at the selected pluviograph stations.
Obtain the dimensionless mass curves of the observed storm-core temporal
•
Obtain the regional distribution of initial loss in the selected study area.
•
parameters of the runoff routing model and baseflow.
•
are obtained, the simulation of the streamflow hydrographs can be carried out.
Obtain the regional average values of fixed variables continuing loss,
Once the regionalized values of all the variables for the runoff routing model
Apply the Monte Carlo Simulation technique (for ungauged catchments) to
test catchments and compare the results with the observed flood frequency
curves.
44
•
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 and validity of the Monte Carlo
Simulation technique (for ungauged catchments).
The proposed research will apply the Monte 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 pluviograph station is analysed to select storm-core
rainfall events. These storm-core events are then analysed to obtain probability
distributions of storm-core duration ( dc ), average rainfall intensity ( I c ), and temporal
patterns ( TPc ).
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 76031 (Mildura Mo) is shown in Table 4.2.
Important output files from the program mcsa5.for are listed in Table 4.3.
4.3 LOSS ANALYSIS
To identify the probability distribution of storm-core initial loss (ILc), 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 4.5
45
Table 4.1 Hourly pluviograph data for Pluviograph Station 76031(Mildura Mo)
Variable Year, Month, Day,
Rainfall Quality
Station ID
Type
Hour
(mm)
Code
76031
10
19530403190000
0.000
1
76031
10
19530403200000
0.000
1
76031
10
19530403210000
0.000
1
76031
10
19530403220000
0.020
1
76031
10
19530403230000
1.770
1
76031
10
19530404000000
2.800
1
76031
10
19530404010000
0.350
1
76031
10
19530404020000
0.140
1
76031
10
19530404030000
0.000
1
76031
10
19530404040000
0.000
1
76031
10
19530404050000
0.000
1
76031
10
19530404060000
0.000
1
76031
10
19530404070000
0.000
1
46
Table 4.2 Parameter file a76031.psa for rainfall analysis (for Station 76031)
Input
Description
a76031
Station ID
76031.dat
Data file, rainfall in mm
6
Dry period, between successive complete storm events, in hour
0.4
Reduction factor, F1
0.5
Reduction factor, F2
17.90
2
I 1 (Log-normal design rainfall intensity, 2 year ARI-1 hr
duration), mm/hr
2.85
2
0.70
I12 (Log-normal, ARI-2 year, duration-12 hr), mm/hr
2
I 72 (mm/hr)
41.50
50
6.75
50
I12 (mm/hr)
1.62
50
I 72 (mm/hr)
I1 (mm/hr)
0.00
Skewness(G)
140
Catchment area, km2
76031a.txt
Streamflow data file name
Table 4.3 Important output files from program mcsa5.for (for Station 76031)
Output
Description
a76031. 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
Table 4.4 Parameter file a40082.lan- (Bremer River catchment) for rainfall analysis
Input
Description
a40082
Station ID
130
Catchment area, km2
40082.dat
Data file, rainfall in mm
40082a.txt
Streamflow data file (hourly), streamflow
in m3/s
Table 4.5 Output file for a40082-(Bremer River catchment) for rainfall analysis
Output
Description
a40082.ssr
Starting time of surface runoff
a40082.ics
Initial loss for complete storm ( ILS )
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
S = kQ m
(4.4.1)
where S is the catchment storage in m3, k a storage delay parameter in hour, Q the
rate of outflow in m3/ 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
(i)
Select a number of rainfall and runoff events from the observed data at the
catchment outlet.
(ii)
For about two-thirds of the selected rainfall and runoff events, calibrate the
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.
(iii)
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
(i)
Arrange the NG simulated peaks in decreasing order of magnitude.
(ii)
Assign rank (m) 1 to the highest value, 2 to the next and so on.
(iii)
For each of the ranked floods, compute an ARI from the following equation:
ARI =
NG + 0.2
m − 0.4
×
1
λ
≅
NY + 0.2
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.
κ2 =∑
k
i =1
(oi − ei ) 2
ei
(4.6.1)
Where;
Where κ 2 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 oi and ei represents the observed and expected frequencies,
respectively, for the ith cell.
50
100
Q (m3/s)
10
1
Streamflow
0.1
Baseflow
61
55
49
43
37
31
25
19
13
7
1
0.01
Time Interval (h)
Figure 4.3 Baseflow separation for the Bremer River Catchment
45
40
35
Qobs
Qcom
Q (m3/s)
30
25
20
15
10
5
0
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61
Tim e Interval (h)
Figure 4.4 Observed vs. computed streamflow data for a selected event for the Bremer
River catchment ( IL = 16 mm, CL = 0.65 mm/h, k = 10 h, m = 0.8 , Start of event =
23/02/1975 at 10 am)
51
Table 4.6 Sample parameter file re1s1.par for simulation of streamflow hydrograph.
Here IL is initial loss and CL is continuing loss
Input
Description
Reg1s1
Catchment name
a
Run Sequence
20000
No of simulations
13.1
Mean durations
108
Catchment area, km2
reg1s1.ifd
IFD file name
reg1s1.tem
TP file name
0
IL-Lower Limit
89.00
IL-Upper Limit
23.00
IL-Mean
19.00
IL-SD
10
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
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 text10.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 2.13. 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 36.74 degree and longitude 146.40 degree. Here
the candidate region has a radius of 50 km.
53
Figure 4.5 Output windows to compute weighted average IFD table
54
CHAPTER 5
RESULTS
5.1 DISTRIBUTION OF STORM-CORE DURATION
The histograms of storm-core durations for all the selected 76 pluviograph stations
were obtained. Figure 5.1.1 shows a typical histogram for pluviograph Station 83033
while the other histograms are provided in Appendix C. The examination of the shape
of the histogram indicates an Exponential distribution.
A hypothesis test was then conducted to see whether an Exponential distribution is
adequate to approximate the distribution of storm-core duration in a particular
pluviograph station by adopting a Chi-squared test as discussed in Section 4.6. The
test result for Station 83033 is discussed below.
In the Chi-squared test for the Station 83033 the observed Chi-squared value was
12.91 and the critical value 7.81 (for alpha = 0.05 and degree of freedom 3) and hence
Ho was rejected i.e. the data for Station 83033 cannot be described by Exponential
distribution.
It was found that only 42% stations passed the test for regional Exponential
distribution. Haddad and Rahman (2005) also tested for the Gamma distribution but
did not find any better results. This thesis did not conduct any further investigation
with other candidate distributions as it was assumed that an exponential distribution
would be enough to approximate the distribution of storm-core duration as far as the
practical application of the Monte Carlo Simulation Technique to design flood
estimation is concerned.
55
5.2 DERIVATION OF INTENSITY-FREQUENCY-DURATION
CURVES
The intensity-frequency duration (IFD) curves of the selected 76 pluviograph stations
were obtained by using the method described in Section 4.2. The IFD table for Station
76031 is shown in Table 5.2.1. Plots of the IFD tables revealed that the obtained IFD
curves were consistent in that they do not intersect each other. IFD tables for the study
catchments are provided in Appendix D. A sample IFD plot is shown in Figure 5.2.1
for Station 76031. Each of the IFD tables was saved in a database for selection by the
developed FORTRAN program in estimating the IFD table for the selected ungauged
Frequency
catchment. IFD curves for the selected catchments are provided in Appendix E.
80
70
60
50
40
30
20
10
0
1--10 10-20
20-30
30-40
40-50
50-60
60-70
70-80
80-90
Storm-core duration (dc) (hour)
Figure 5.1.1 Histogram of storm-core duration for Pluviograph Station 83033
5.3 REGIONALISATION OF THE DISTRIBUTIONS OF VARIOUS FLOOD
PRODUCING VARIABLES
To apply the Monte Carlo Simulation Technique, we need to have distributions of
storm-core duration, intensity, temporal patterns and initial loss, and also the values of
the fixed input variables continuing loss, parameters of the runoff routing model and
baseflow. To apply the Monte Carlo Simulation technique for the ungauged
56
catchments, we need to have regionalized values of the above distributions and fixed
input variables.
5.3.1 Storm-core duration
To regionalize the distribution of rainfall duration, Victoria was provisionally divided
into four hydrometeorological zones, namely Zone 1 (South-eastern Victoria), Zone 2
(North-eastern Victoria), Zone 3 (North-western Victoria) and Zone 4 (South-western
Victoria), roughly cutting the state into quadrants along the Great Dividing Range and
North from Melbourne, as shown in Figure 5.3.1. The mean values of storm-core
durations are 14.5h, 13.1h, 10.5h and 12.7h respectively for Zones 1, 2, 3 and 4.
These regional average values of the storm-core durations were used to fit the regional
Exponential distribution and hypothesis testing was conducted against the at site
storm-core duration data to assess the viability of a regional Exponential distribution.
Using the Chi–squared test, the null hypothesis of a regional Exponential distribution
cannot be rejected (at 5% level of significance) for any of the stations in Zones 1, 2, 3
and 4.
The regional Exponential distributions with these mean storm-core duration values
were adopted in the Monte Carlo simulation for generation of streamflow
hydrographs.
57
Table 5.2.1 IFD table for Station 76031 (Intensity values are in mm/h and ARI in years)
ARI-0.1
7.298
3.732
1.158
0.219
0.088
0.05
0.032
ARI-1
9.242
4.958
1.97
0.687
0.425
0.325
0.264
ARI-1.11
9.934
5.329
2.12
0.743
0.461
0.354
0.288
ARI-1.25
10.709
5.744
2.288
0.805
0.502
0.386
0.315
ARI-2
13.801
7.397
2.958
1.055
0.664
0.514
0.422
ARI-5
19.83
10.618
4.261
1.54
0.979
0.765
0.632
ARI-10
24.392
13.053
5.245
1.907
1.218
0.954
0.791
ARI-20
28.953
15.488
6.229
2.274
1.457
1.144
0.95
1000
ARI-50
34.984
18.706
7.53
2.758
1.773
1.395
1.16
ARI-100
39.546
21.141
8.514
3.125
2.012
1.585
1.319
ARI-500
50.138
26.793
10.798
3.976
2.567
2.025
1.689
ARI-1000 ARI-1000000
54.7
100.164
29.227
53.485
11.781
21.583
4.343
7.995
2.805
5.186
2.215
4.106
1.848
3.435
ARI=1
ARI=1.11
ARI=1.25
100
ARI=2
ARI=5
10
Ic(mm/h)
Duration(hours)
1
2
6
24
48
72
100
ARI=10
ARI=20
ARI=50
1
ARI=100
ARI=500
0.1
ARI=1000
1
dc(h)
10
100
ARI=1000000
Figure 5.2.1 Plot of IFD curves for Station 76031 (ARI in years)
58
VICTORIA
AUSTRALIAN CAPITAL TERRITORY
Zone 3
Zone 2
NEW SOUTH WALES
Zone 1
Zone 4
Figure 5.3.1 Various zones in Victoria for regionalisation of storm-core duration
5.3.2 Storm-core rainfall intensity
To regionalise the distribution of storm-core rainfall intensity, Equation 2.13 was used
using 1, 2, 3, 4 or 5 pluviograph stations in a region. This means that IFD curves for
an ungauged catchment will be obtained based on the IFD curve(s) of the nearest 1, 2,
3, 4 or 5 pluviograph stations. For this purpose the developed FORTRAN program
called ucat1.for was used.
5.3.3 Storm-core temporal patterns
To form regional database of storm-core temporal patterns, the observed temporal
patterns of the nearest 1, 2, 3, 4 or 5 pluviograph stations are pooled together in such
a way that if IFD curves are obtained from 2 pluviograph stations, temporal pattern
database will also be obtained from the same 2 pluviograph stations. For this purpose
59
the developed FORTAN program called ucat1.for was used. A sample plot of the
temporal patterns is shown in Figure 5.3.2
100
90
Cumulative % rain
80
70
60
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
Period
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
5.3.4 Initial loss
To regionalise the distribution of initial loss, the results from Rahman et al. (2002b) is
adopted in that they adopted a four-parameter Beta distribution to specify the
distribution of initial loss. Based on 10 catchments data from Victoria, Rahman et al.
(2002b) obtained the regional average values of the four parameters of the Beta
distribution from the observed values of initial losses as 1mm (lower limit), 72mm
(upper limit), 25mm (mean) and 13mm (standard deviation). These values are adopted
in this study to fit regional Beta distribution for initial loss. The whole of Victoria was
considered to be a single region with respect to initial loss distribution.
60
5.3.5 Continuing loss, storage delay parameter, non-linearity parameter and
baseflow
The regional average values of continuing loss was taken from Rahman et al. (2001)
as 1.76 mm/h. The regional value of storage delay parameter (k) of the adopted runoff
routing model was obtained from the following equation, again based on the study of
Rahman et al. (2001).
log(k) = -0.837 + 1.013log(A)
(5.1)
where k is storage delay parameter in hour and A is catchment area of the ungauged
catchement in km2.
The value of the non-linearity parameter (m) of the adopted runoff routing model was
taken to be 0.8. The average baseflow was taken to be 0.53 m3/s based on the study of
Rahman et al. (2001).
It may be noted here that the regional average values of continuing loss and regional
estimation equation of storage delay parameter (k) were based on a very small sample
size and thus likely to introduce a high degree of error in design flood estimates.
5.4 DERIVED FLOOD FREQUENCY CURVES WITH REGIONALISED
PARAMETERS OF THE INPUT VARIABLES
To identify number of pluviograph stations to be included in obtaining IFD values at
an ungauged catchment, subregions were formed with one, two, three, four or five
pluviograph stations and derived flood frequency curves were obtained. Each time,
the radius of the subregion was selected in such a way that it captures one, two, three,
four or five stations. The method was applied to three selected test catchments: Boggy
Creek catchment, Tarwin River catchment and Avoca River catchment.
The resulting radius of the subregions for the Boggy Creek catchment is shown in
Table 5.4.1. The obtained derived flood frequency curves (DFFC) for the Boggy
61
Creek catchment is shown in Figure 5.4.1 which shows that there is no significant
differences in the DFFCs whether a subregion consists of one, two, three, four or five
stations. In obtaining these DFFCs, regional values for other input variables such as
storm-core duration and initial loss were obtained using the procedures mentioned in
Section 5.3.
Table 5.4.1 Radius of subregions for the Boggy Creek catchment
Subregion
Number of
Area (km2)
pluviograph stations
in the subregion
Subregion one
1
1.40
Subregion two
2
20.09
Subregion three
3
36.64
Subregion four
4
43.68
Subregion five
5
47.79
1000
DFFC - 1 station (DFFC1)
DFFC - 2 stations (DFFC2)
Q (m /s)
DFFC- 3 stations (DFFC3)
3
DFFC- 4 stations (DFFC4)
DFFC- 5 stations (DFFC5)
Observed floods (partial series)
100
10
1
10
100
ARI (years)
Figure 5.4.1 Derived flood frequency curves for the Boggy Creek catchment using
regionalised parameters
For the Boggy Creek catchment, the percentage differences between various sets of
DFFCs and observed floods are presented in Table 5.4.2, which shows that the
62
absolute differences between the DFFCs and observed floods vary from 0.07% to
16.3% over the ARIs of 2, 5, 10, 20, 50 and 100 years. The median absolute
difference is the lowest (1.35%) for Sub-region 1 that contains only one station. For
Sub-regions 2, 4 and 5, the median absolute difference is close to 5%. For Sub-region
3, the median absolute difference is the highest (12.75%).
For the Tarwin River catchment, all the DFFCs show significant departures from the
observed floods as shown in Figure 5.4.2. However, DFFC for Sub-region 4 (with
four stations in the sub-region) shows the least departure. The radius of various
subregions for the Tarwin River catchment is shown in Table 5.4.3.
For the Tarwin River catchment, the percentage differences between various sets of
DFFCs and observed floods are presented in Table 5.4.4, which shows that the
absolute differences between the DFFCs and observed floods vary from 38.9% to
60.5% over the six ARIs. The median absolute difference is the lowest (40.95%) for
Sub-region 4 that contains four stations.
63
Table 5.4.2 The percentage difference between the DFFCs and observed floods for the Boggy Creek catchment.
ARI
Obs.
DFFC
%
%
DFFC
%
% diff.
DFFC
(years)
floods
(Sub-reg-
diff.
diff.
(Sub-
diff.
(abs)
(Sub
(abs)
reg-2)
3
(m /s)
1)
% diff.
% diff.
DFFC
(abs)
(Sub
reg-3)
% diff.
% diff.
DFFC
(abs)
(Sub
reg-4)
% diff.
% diff.
(abs)
reg-5)
2
30.00
29.22
2.60
2.60
29.31
2.30
2.30
34.47
-14.90
14.90
27.52
8.27
8.27
30.02
-0.07
0.07
5
46.00
45.51
1.07
1.07
42.30
8.04
8.04
50.42
-9.61
9.61
42.00
8.70
8.70
44.00
4.35
4.35
10
60.00
57.50
4.17
4.17
55.00
8.33
8.33
62.25
-3.75
3.75
54.00
10.0
10.0
56.50
5.83
5.83
20
72.00
72.92
-1.28
1.28
70.00
2.78
2.78
79.63
-10.60
10.60
67.80
5.83
5.83
67.80
5.83
5.83
50
92.00
90.70
1.41
1.41
84.00
8.70
8.70
107.00
-16.30
16.30
91.60
0.43
0.43
83.80
8.91
8.91
100
102.00
103.00
-0.98
0.98
96.00
5.88
5.88
118.50
-16.18
16.18
108.0
-5.88
5.88
104.20
-2.16
2.16
Median
1.35
6.96
12.75
7.07
5.09
64
Table 5.4.3 Radius of subregions for the Tarwin River catchment
Subregion
Number of pluviograph
Area (km2)
stations in the
subregion
1000
Subregion one
1
17.01
Subregion two
2
44.68
Subregion three
3
47.66
Subregion four
4
68.50
Subregion five
5
69.79
DFFC- 1 station (DF FC1)
3
Q (m /s)
DFFC-2 station (DFFC2)
DFFC-3 station (DFFC3)
DFFC -4 station (DFFC4)
DFFC- 5 station (DFFC5)
Observed floods - partial series
100
10
1
ARI (years)
10
100
Figure 5.4.2 Derived flood frequency curves for the Tarwin River catchment using
regionalised parameters
65
Table 5.4.4 Percentage differences between the DFFCs and observed floods for the Tarwin River catchment.
ARI
Obs.
DFFC
(years)
floods
(Sub-reg-
3
(m /s)
% diff.
% diff.
DFFC
%
% diff.
DFFC
(abs)
(Sub-
diff.
(abs)
(Sub
reg-2)
1)
% diff.
% diff.
DFFC
(abs)
(Sub
reg-3)
% diff.
% diff.
DFFC
%
%
(abs)
(Sub
diff.
diff.
reg-4)
(abs)
reg-5)
2
44.00
17.3
60.5
60.5
19.50
55.6
55.6
18.13
58.80
58.80
24.09
45.25
45.25
21.54
51.05
51.05
5
66.00
27.7
58.0
58.0
31.10
52.8
52.8
30.20
54.24
54.24
39.10
40.76
40.76
36.03
45.41
45.41
10
88.00
36.9
58.0
58.0
41.10
53.3
53.3
38.20
56.59
56.59
51.80
41.14
41.14
46.90
46.70
46.70
20
112.00
45.3
59.5
59.5
52.46
53.1
53.1
50.20
55.18
55.18
65.90
41.16
41.16
59.02
47.30
47.30
50
141.00
59.5
57.8
57.8
62.20
55.8
55.8
62.20
55.89
55.89
84.22
40.27
40.27
80.90
42.62
42.62
100
164.00
66.6
59.3
59.3
73.10
55.4
55.4
73.90
54.94
54.94
100.1
38.96
38.96
92.90
43.35
43.35
Median
58.7
54.3
55.53
40.95
46.06
66
For the Avoca River catchment, all the DFFCs also show significant departures from
the observed floods as shown in Figure 5.4.3. The radius of various subregions for the
Tarwin River catchment is shown in Table 5.4.5.
Table 5.4.5 Radius of subregions for the Avoca River catchment
Subregion
Area (km2)
Number of
pluviograph stations
in the subregion
Subregion one
1
27.18
Subregion two
2
43.35
Subregion three
3
49.94
Subregion four
4
55.72
Subregion five
5
56.50
3
Q (m /s)
100
DFFC - 1 station (DFFC1)
10
DFFC - 2 stations (DFFC2)
DFFC- 3 stations (DFFC3)
DFFC- 4 stations (DFFC4)
DFFC- 5 stations (DFFC5)
Observed floods (partial series)
1
1
ARI (years)
10
100
Figure 5.4.3 Derived flood frequency curves for the Avoca River catchment of using
regionalised parameters
67
For the Avoca River catchment, the percentage differences between various sets of
DFFCs and observed floods are presented in Table 5.4.6, which shows that the
absolute differences between the DFFCs and observed floods vary from 0.81% to
41.2% over the six ARIs. The median absolute difference is the lowest (13.3%) for
Sub-region 5 that contains five stations. However, for Sub-regions 1, 3 and 4, the
median absolute difference is very similar (15%) to that of the Sub-region 5.
Based on the results of the three test catchments, it is found that the new technique
shows a relative error in design flood estimation in the range of 0.07% to 60.5% with
a median value of 16.9% (ignoring the sign of the relative errors). The overall
distribution of relative errors for the three test catchments over the six ARIs is
presented in the box plot in Figure 5.4.4 (considering the sign of the relative errors).
The observed relative errors with the new technique seems to be acceptable as Rijal
and Rahman (2005) found that other methods currently applied for ungauged
catchments in South-east Australia show a relative error generally higher than this.
For example, Rijal and Rahman (2005) found that the Probabilistic Rational Method
and Quantile Regression Technique show a median relative error value of about 40%
and there is a 10% chance that relative error values may exceed 100% with these two
methods.
It is also found here that a sub-region consisting of four stations provide more
reasonable design flood estimates in that IFD curves and temporal pattern data at the
ungauged catchment is obtained from the nearest four pluviograph stations.
68
80.00
60.00
40.00
20.00
0.00
-20.00
Floodquantile
Figure 5.4.4 Box plot of relative errors for the three test catchments
69
Table 5.4.6 The percentage difference between the DFFCs and observed floods for the Avoca River catchment.
ARI
Obs.
DFFC
(years)
floods
(Sub-reg-
3
(m /s)
% diff.
% diff.
DFFC
%
% diff.
DFFC
(abs)
(Sub-
diff.
(abs)
(Sub
1)
% diff.
% diff.
DFFC
(abs)
(Sub
reg-3)
reg-2)
% diff.
% diff.
DFFC
%
%
(abs)
(Sub
diff.
diff.
reg-4)
reg-5)
(abs)
2
26.00
15.84
39.0
39.0
15.28
41.2
41.2
15.72
39.54
39.54
17.11
34.1
34.1
15.85
39.04
39.04
5
36.00
26.10
27.5
27.5
26.01
27.7
27.7
26.10
27.50
27.50
28.10
21.9
21.9
27.09
24.75
24.75
10
44.00
35.79
18.6
18.6
34.70
21.1
21.1
34.10
22.50
22.50
37.10
15.6
15.6
36.98
15.95
15.95
20
52.00
43.80
15.7
15.7
44.00
15.3
15.3
45.30
12.88
12.88
46.50
10.5
10.5
46.40
10.77
10.77
50
62.00
57.92
6.58
6.58
59.80
3.55
3.55
61.05
1.53
1.53
60.08
3.10
3.10
62.50
-0.81
0.81
100
69.00
66.70
3.33
3.33
75.59
-9.5
9.5
69.90
-1.30
1.30
77.90
-12.9
12.9
73.10
-5.94
5.94
Median
17.21
18.2
17.69
14.29
13.36
70
5.5 SENSITIVITY ANALYSES
This study used regional distributions of storm-core duration, intensity and temporal
pattern, which were obtained from the analyses of the observed rainfall data in the
region as discussed in Section 5.4. The distribution of initial loss was obtained from
the study of Rahman et al. (2002b). The fixed variables used in the simulation of
streamflow hydrograph were continuing loss (CL), catchment storage parameter (k),
and baseflow (BF) and fixing non linearity parameter (m) as 0.8. It would have been
ideal to have regional equations for the CL and k values as function of catchment
characteristics. However, this study did not focus on this; rather it has used regional
average values of CL based on a small number of rainfall and streamflow events, and
a provisionally developed regional equation for k, as mentioned in Section 5.3 from
the study of Rahman et al. (2001). This section investigates the impact of CL and k
values on the derived flood frequency curves (DFFC) by assigning different possible
values of these parameters.
5.5.1 Continuing loss
To examine the sensitivity of CL on the derived flood frequency curve, four
tentatively selected continuing loss values (0.1 mm/h, 0.52 mm/h, 0.80 mm/h, and
1.76 mm/h) were adopted. The results for the Boggy Creek Catchment is presented in
Figure 5.5.1, which indicates that CL value has significant effect on derived flood
frequency curve. The CL value of 1.76 mm/h gives DFFC having the best match with
the observed floods. As the CL value decreases, DFFC overestimates the observed
floods. For 10 years ARI, the derived floods increase by 25%, 30%, and 45% for
decrease in CL values by 94%, 70%, and 55%, respectively.
For the Tarwin River Catchment, the same four continuing loss values were used.
Here, the differences in derived flood frequency curves with the observed flood
frequency curve is relatively high as shown in Figure 5.5.2 with all four values of CL.
For the continuing loss value of 0.1 mm/h, the difference between the two curves is
minimum.
71
For the Avoca River Catchment, the CL value of 0.1 mm/h gives DFFC having the
best match with the observed floods as shown in Figure 5.5.3.
It is found here that DFFCs are very sensitive to CL value, thus it is very important to
derive regional prediction equation for CL with catchment characteristics or to obtain
a regional average CL value based on a large number of observed rainfall and
streamflow events in the region of interest. This is not done here as it falls beyond the
scope of this study.
1000
DFFC- 3 stations, CL=1.76 mm/h
DFFC- 3 stations, CL=0.80 mm/h
DFFC- 3 stations, CL=0.52 mm/h
DFFC- 3 stations, CL=0.10 mm/h
Observed floods (partial series)
3
Q (m /s)
100
10
1
10
100
ARI (years)
Figure 5.5.1 Effects on derived flood frequency curves for the Boggy Creek
Catchment of using different continuing loss values
72
1000
DFFC - 3 stations, CL =1.76 mm/h
DFFC - 3 stations, CL =0.80 mm/h
DFFC - 3 stations, CL =0.52 mm/h
DFFC - 3 stations, CL =0.1 mm/h
Observed floods - partial series
3
Q (m /s)
100
10
1
ARI (years)
10
100
Figure 5.5.2 Effects on derived flood frequency curves for the Tarwin River
Catchment of using different continuing loss values
1000
Q (m3/s)
100
DFFC- 3 stations, CL 0.1 mm/h
DFFC- - 3 stations, CL =0.52
mm/h
DFFC - 3 stations, CL =0.80 mm/h
10
DFFC-3 stations, CL =1.76 mm/h
Observed floods (partial series)
1
1
ARI (years)
10
100
Figure 5.5.3 Effects on derived flood frequency curves for the Avoca River
Catchment of using different continuing loss values
73
5.5.2 Catchment storage parameter (k)
The next sensitivity analysis examined the impact of storage parameter value (k) on
the derived flood frequency curve (DFFC) for the above three catchments. Four
different k values were used: the k value used in Section 5.4 plus three additional k
values, which are 50%, 75%, and 150% of original k value used in Section 5.4.
The result for the Boggy Creek Catchment is presented in Figure 5.5.4 which shows
that for a k value of 16.7h and a continuing loss value of 1.76 mm/h, there are no
significant differences between the derived flood frequency curves and the observed
floods. For k value of 12.52h, the DFFC significantly overestimates the observed
floods. For 10 years ARI, 150% increase in k value reduces derived flood estimate by
22%. Some 50% and 75% decrease in k values increase the derived floods by 74%
and 25% respectively.
1000
Q (m3/s)
100
DFFC- 3 stations, k=8.35, CL=1.76 mm/h
DFFC- 3 stations, k= 12.52, CL= 1.76 mm/h
10
DFFC- 3 stations, k= 16.7,CL= 1.76 mm/h
DFFC- 3 stations, k= 25.05,CL= 1.76 mm/h
Observed floods (partial series)
1
1
ARI (years)
10
100
Figure 5.5.4 Effects on derived flood frequency curves for the Boggy Creek
Catchment of using different k values
74
1000
3
Q (m /s)
100
DFFC - 3 stations, k =9.8, CL=1.76
mm/h
DFFC - 3 stations, k =14.7,CL=1.76
mm/h
DFFC - 3 stations, k =19.6,CL=1.76
mm/h
DFFC - 3 stations, k =29.4,CL=1.76
mm/h
Observed floods - partial series
10
1
1
ARI (years)
10
100
Figure 5.5.5 Effects on derived flood frequency curves for the Tarwin River
Catchment of using different k values
For the Tarwin River Catchment, k value of 9.8h gives the closest match with
observed floods. As the k value increases, the DFFCs decrease sharply as shown in
Figure 5.5.5. At 10 years ARI, an increase in k values of 50%, 100% and 200% from
k = 9.8h, the derived floods decrease by 28%, 43%, and 60%, respectively.
For the Avoca River catchment, k value of 6h gives the best match where DFFC
shows least difference with the observed floods. An increase in k value reduces the
DFFC sharply as shown in Figure 5.5.6. At 10 years ARI, an increase in k value from
6h to 9h (50%), 12h (100%), and 18h (200%), the derived floods decrease by 28%,
46%, and 60%, respectively. It is found that DFFC is highly sensitive to k value, thus
it is important to derive regional prediction equation for k as function of catchment
characteristics (such as catchment area and/or main stream length). This would
require calibration of runoff routing model (S=kQm) for a large number of catchments
in the region of interest based on a large number of observed rainfall and streamflow
events. This is not done here as it falls beyond the scope of this study.
75
1000
3
Q (m /s)
100
DFFC-3 stations, k =18.00, CL=1.76
mm/h
DFFC-3 stations, k =12.00, CL=1.76
mm/h
DFFC-3 stations, k =9.00, CL=1.76
mm/h
DFFC-3 stations, k =6.00, CL=1.76
mm/h
Observed floods (partial series)
10
1
1
ARI (years)
10
100
Figure 5.5.6 Effects on derived flood frequency curve for the Avoca River Catchment
of using different k values
5.6 COMPARISON AMONG JOINT PROBABILITY APPROACH,
PROBABILISTIC
RATIONAL
METHOD
AND
QUANTILE
REGRESSION TECHNIQUE
To assess the performance of the Joint Probability Approach (JPA) to ungauged
catchment developed here and two empirical methods (Probabilistic Rational Method
and Quantile Regression Technique), twelve additional test catchments from Southeast Australia were selected as mentioned in Chapter 3. These are gauged catchments
having recorded streamflow data.
76
The Probabilistic Rational Method was applied to these catchments using the runoff
coefficients from the Australian Rainfall Runoff (I. E. Aust., 1987). The Quantile
Regression Technique was based on the prediction equations developed by Rahman
(2005) as mentioned in Section 2.3. The JPA was applied to each of these test
catchments based on regionalized parameters of k, CL and BF as mentioned in Section
5.3. The IFD curve at each of the test catchments was estimated using Equation 2.13
based on the IFD curves of 4 nearby pluviograph stations for each of the test
catchments. Flood estimates obtained by the three methods are provided in Table
5.6.1.
77
Table 5.6.1 Flood estimation obtained by three methods (QRM, PRM and JPA) (Flood estimation in ML/day)
Catchment
ID
226410
Q2
OBS.
2200
Q5
QRM
1554
PRM
1373
JPA
1900
OBS.
4250
QRM
2826
Q10
PRM
2011
JPA
2937
OBS.
5000
QRM
4286
Q20
PRM
2452
JPA
3801
OBS.
5800
QRM
5089
Q50
PRM
2673
JPA
4665
OBS.
10000
Q100
QRM
8365
PRM
2967
JPA
5875
227200
3450
3217
4424
2073
11200
6195
6478
3456
17400
8528
7901
4665
20300
10908
8612
5875
26000
11201
9560
7257
229218
7100
706
2024
1900
1740
1237
2964
2764
2000
1942
3615
3542
2850
2163
3941
4147
3400
2715
4375
5270
230204
1460
1875
2206
2937
2900
3286
3230
4492
3850
5218
3939
5529
4820
6279
4294
6912
5800
6151
4767
8640
234200
3000
3105
5431
1728
4350
5916
7953
3024
8100
8648
9699
3974
12200
10831
10572
5875
17000
12776
11736
7516
401215
1700
3818
3139
3110
3250
6757
4597
5529
4200
8860
5606
7430
5100
12074
6111
9158
6050
18639
6784
12441
404207
6100
4804
5124
3628
18000
10586
7504
5702
25000
14538
9151
7430
35500
18290
9975
9417
52000
21895
11073
11923
405205
8100
1731
2839
2764
1300
3069
4158
4060
1750
4468
5071
5184
1950
5486
5527
6220
2200
7115
6136
7776
405212
6000
3563
11072
2419
15500
6840
16214
4147
19000
10011
19773
5702
21500
12644
21552
6912
26500
17085
23925
8985
405214
7200
4159
5006
3283
13000
7623
7330
5011
18500
10866
8939
6739
26000
14282
9744
8294
35500
15568
10817
9676
405229
1200
1470
1645
1728
2000
3092
2410
2937
3500
4708
2939
3974
5900
5261
3203
5356
12000
6268
3556
7257
408202
1900
1617
1845
1728
4200
3105
2703
2764
4900
4892
3296
3801
5150
5634
3593
4838.
5400
6754
3988
6393
OBS.
16000
30100
4000
6450
18100
6500
71000
2350
28000
46000
20100
5600
QRM
10061
PRM
3311
JPA
6825
13303
10666
8985
3214
4881
5875
10368
7532
5318
15482
13094
9244
22012
7569
15897
26300
12354
13478
8462
6846
9244
20824
26693
10713
18887
12068
12614
7498
3967
8640
8210
4450
8121
78
All these test catchments are gauged and the streamflow data were analysed by
graphical method to obtain design flood estimates. These estimates are referred to as
‘Observed flood estimates’ (Qobs). The difference between the observed flood
estimates (Qobs) and JPA or QRT or PRM was referred to as relative errors, i.e,
Relative Error (%) = (Qobs- QJPA)/ Qobs *100
(5.2)
Relative Error (%) = (Qobs - QQRT)/ Qobs *100
(5.3)
Relative Error (%) = (Qobs- QPRM)/ Qobs *100
(5.4)
Based on the absolute values of the relative errors, the median values of relative errors
are shown in Table 5.6.2 for the three methods, which shows that the new Joint
Probability Approach has median relative error in the range 49 to 66% which are
higher than those of the Probabilistic Rational Method (which shows median relative
errors in the range 41% to 47%) and the Quantile Regression technique (which has
median relative errors in the range 28% to 51%).
Considering the sign of the relative errors, box plots of the relative errors were
prepared for the three techniques in Figures 5.6.1, 5.6.2 and 5.6.3, which show that
the Joint Probability Approach has an overall wider error band as compared to the
Probabilistic Rational Method and Quantile Regression Technique.
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 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 in Section 5.5 that derived flood frequency curves from the Joint
Probability Approach were very sensitive to both k and continuing loss values.
79
Table 5.6.2 Median relative error values (%) for three methods JPA, QRT and PRM
(based on absolute values)
ARI (years)
2
5
10
20
50
100
JPA
49.2
60.17
66.77
59.84
60.93
61.24
QRT
28.91
41.27
38.40
35.72
41.64
51.20
PRM
44.33
42.88
42.21
41.99
47.09
46.10
100.00
0.00
-100.00
-200.00
8
8
8
-300.00
Q2
Q5
Q10
Q20
Q50
Q100
Figure 5.6.1 Box plot for relative errors for JPA
80
8
200.00
8
8
8
8
Q20
Q50
100.00
0.00
-100.00
Q2
Q5
Q10
Q100
Figure 5.6.2 Box plot for relative errors for PRM
81
300.00
8
6
8
6
200.00
8
8
6
6
100.00
0.00
-100.00
Q2
Q5
Q10
Q20
Q50
Q100
Figure 5.6.3 Box plot for relative errors for QRT
The flood frequency curve for all these catchments are presented in Appendix F.
82
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 SUMMARY
Chapter 2 of this thesis reviews design flood estimation methods based on the Joint
Probability Approach/ Monte Carlo Simulation technique. This approach was first
introduced by Eagleson (1972) and has been widely researched in the last three decades.
Eagleson (1972) adopted analytical method to obtain derived flood frequency curves
from the marginal distributions of various flood-producing variables. Many other
researchers adopted the analytical approach by Eagleson (1972). But the problems with
this approach are that this has limited flexibility and this involves complicated
mathematical functions for real catchments and thus has limited practical applicability.
On the other hand, the Monte Carlo Simulation technique is an approximate form of the
Joint Probability Approach which can be applied easily in practice. In recent years in
Australia, there have been significant research and development on the Monte Carlo
Simulation technique, which has so far been tested to gauged catchments. This research
aimed to extend the Monte Carlo Simulation technique to ungauged catchments. A
research proposal was prepared in Chapter 2 in that it was assumed that the distribution
of storm-core duration can be described by an Exponential distribution and rainfall
intensity distribution at an ungauged catchment can be obtained from the rainfall intensity
distribution of the nearby ungauged catchments.
In Chapter 3, Victoria was selected as the study area, and a total of 76 pluviograph
stations were selected. The data of these selected stations were obtained from the Bureau
of Meteorology and abstracted using HYDSYS database software. A total of three
gauged catchments from Victoria were selected to apply the new technique of design
flood estimation. An additional 12 gauged catchments were also selected from Victoria to
compare the performances of the new technique with two existing techniques for
82
ungauged catchments namely, the Probabilistic Rational Method and Quantile Regression
Technique.
Chapter 4 outlines the steps in the proposed research and demonstrates the method of
rainfall analysis, loss analysis and runoff routing model calibration, which are major
components on the Monte Carlo Simulation technique. A FORTRAN program
(containing 1370 lines) was also developed in this chapter to estimate intensityfrequency-duration (IFD) curves of an ungauged catchment based on the IFD curves of
the pluviograph stations in the selected region.
Chapter 5 obtains the storm-core durations and IFD curves of the selected 76 pluviograph
stations. This also presents the regionalization of the distributions of other input variables
in the Monte Carlo Simulation technique. The developed FORTRAN program was then
applied and derived flood frequency curves were obtained for the three test catchments
with various candidate regions consisting of one, two, three, four and five pluviograph
stations. Finally, the new technique was applied to additional 12 test catchments and
results were compared with the Probabilistic Rational Method and Quantile Regression
technique.
The technique presented here is readily applicable to Victoria; however, this can easily be
extended to other states. The application of the technique involves the following steps:
1) Obtain the latitude and longitude of the ungauged catchment within Victoria for
which design flood estimation is required.
2) Run the FORTRAN program (ucat1.for) and enter the latitude and longitude of
the ungauged catchment selected in Step (1).
3) Enter the radius of the proposed region and note the number of pluviograph
stations falling within the region.
4) If the number of pluviograph stations obtained in (3) is other than 4,
increase/decrease the radius of the zone so that the final zone contains 4
pluviograph stations. This is because it was found that region with 4 pluviograph
stations provided relatively more accurate derived flood frequency curve.
83
5) Once the radius of the zone is finalised, the program computes the IFD table in
the ungauged catchment based on the selected IFD tables of the four pluviograph
stations falling within the region and prepare the database of the observed
temporal patterns.
6) Run the program mcdffc3.for to generate the derived flood frequency curve for
the ungauged catchment. In running this program, select the value of storm-core
duration as per the location of the ungauged catchment in one of the four stormcore duration zones shown in Figure 5.3.1. The parameters of the initial loss
distribution are obtained from Section 5.3.4. The values of continuing loss,
storage delay parameter (k), non-linearity parameter (m) and baseflow are
obtained from Section 5.3.5. Use 20,000 simulation runs in obtaining the derived
flood frequency curve.
6.2 CONCLUSIONS
This thesis attempts to extend the Monte Carlo Simulation technique to ungauged
catchments for design flood estimation. The new technique has been applied to Victorian
pluviograph stations and selected test catchments. Following conclusions can be made
from this study:
•
It has been found that the Monte Carlo Simulation technique can successfully be
applied to ungauged catchments. The developed FORTRAN program allows
estimation of intensity-frequency-duration curves at an ungauged catchment in
Victoria based on a region defined by the user that consists of one to five nearest
•
pluviograph stations.
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 be higher than those of the Probabilistic Rational Method (for this
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
84
relative error is that the test catchments used in this study were included in the
data set of derivation of the runoff coefficients 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 data sample. 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 can further be improved by addition of new and
improved regional estimation equations of initial loss, continuing loss and storage
delay parameter (k) as and when these are available.
6.3 RECOMMENDATION FOR FURTHER STUDY
To enhance the developed method of design flood estimation for the ungauged catchment
following research should be undertaken:
•
•
•
Develop new and improved regional estimation equations for initial loss,
continuing loss and storage delay parameter (k) of the runoff routing model.
Test the new technique to other study area.
Examine whether the climate change has affected the distributions of rainfall
duration, intensity and temporal patterns and if so its implication on the derived
•
•
flood frequency curves.
Validate the technique for a dataset which has not been used in the development
of the Probabilistic Rational Method and Quantile Regression Technique.
Results of this study show that Joint Probability Approach (JPA) although
represents the slopes of the flood frequency curves very well, there is consistent
underestimation in the flood magnitudes. Some hybrid method combining JPA
and quantile estimates can further improves design flood estimates for ungauged
catchments.
85
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APPENDIX A
LIST OF STUDY PLUVIOGRAPH
STATIONS
Table A-1 List of study pluviograph stations
ID
76031
77087
79046
79052
79079
79082
79086
80006
80102
80109
80110
81003
81013
81026
81038
81049
81114
81115
82011
82016
82039
82042
82076
82107
82121
83017
83025
83031
83033
Name
Mildura MO
Hopetoun RWC
Wartook Reservoir
Rocklands Res.
Tottington
Horsham
Avon R No. 3
Charlton PO
Pyramid Hill
Cobram
Kerang
Bendigo Prison
Dookie Ag. College
Laaenecoorie Weir
Natte Yallock
Tatura
Tatura (Theiss)
Wanalta Ck (DAEN)
Corryong
Euroa
Rutherglen
Strathbogie PO
Dartmout Res
Lake Nillahcootie
Wangaratta (Ovens
R)
Jamieson PO
Omeo
Whitfield
Woods Point
Latitude
(degree)
34.23
35.7
37.1
37.2
36.79
36.7
36.86
36.3
36.06
35.9
35.73
36.75
36.37
36.83
36.94
36.44
36.44
36.63
36.21
36.76
36.11
36.85
36.55
36.86
Longitude
(degree)
142.08
142.30
142.40
141.90
143.12
142.20
143.12
143.40
144.12
145.60
143.92
144.28
145.70
143.89
143.47
145.23
145.22
144.87
147.89
145.55
146.51
145.73
147.49
146.00
36.35
37.31
37.11
36.75
37.56
146.34
146.14
147.59
146.41
146.24
97
83067
83074
84005
84015
84078
84112
84122
84123
84125
85000
85034
85072
85103
85106
85170
85176
85237
85256
86038
86071
86074
86085
86142
86219
86224
86314
87017
87029
87031
87033
87036
87097
87104
87105
87133
87153
88023
88029
88037
88049
88153
89016
89025
89094
90058
90166
Bright Shire Council
Lake William Hovell
Buchen PO
Ensay Composite
Sarsfield East
Cann River PO
Genoa
Wroxham 2
Crooked River
Aberfeldy
Glenmaggie Weir
East Sale AMO
Yallourn SEC
Olsens Bridge
Traralgon L.V.W
Tanjil Bren PO
Noojee Eng. HMSD
Barkley River
Essendon Airport
Melb. Regional
Office
Mitcham
Narre Warren Nth
Mt St Leonard
Coranderrk
Dandenong Comp.
Koo-Wee-Rup
Blackwood
Lancefield
Laverton AMO
Little River
Macedon-Forestry
Parwan
Werribee Cattle
Yard
Werribee Sewerage
Geelong Nth
Lerderderg R No.3
Lake Eildon
Heathcote PO
Lauriston Res
Puckapunayal
Spring Ck No. 2
Lake Bolac PO
Skipton
Warrambine No. 3
Mortlake
Winchelsea
36.73
36.92
37.5
37.37
37.75
37.56
37.48
37.35
37.38
37.7
37.91
38.11
38.18
38.48
38.22
37.83
37.88
37.52
37.73
146.96
146.38
148.17
147.84
147.73
149.15
149.64
149.48
147.11
146.36
146.81
147.13
146.33
146.33
146.50
146.18
146.00
146.55
144.90
37.81
37.83
37.99
37.57
37.68
38.01
38.2
37.47
37.26
37.86
37.99
37.4
37.7
144.96
145.19
145.34
145.51
145.56
145.19
145.49
144.31
144.71
144.75
144.49
144.55
144.32
37.97
37.95
38.12
37.46
37.23
36.93
37.25
37.00
37.09
37.72
37.68
37.83
38.08
38.25
144.63
144.62
144.36
144.40
145.91
144.71
144.38
145.00
145.71
142.84
143.37
143.88
142.79
143.97
98
APPENDIX B
INPUT AND OUTPUT LISTS FOR
FORTRAN PROGRAM UCAT1.FOR
Inputs:
•
•
Latitude of the ungauged catchment in degrees.
•
Radius of the candidate zone in km.
•
directory.
•
Longitude of the ungauged catchment in degrees.
IFD tables of the pluviograph stations in the study region should be in the
Temporal patterns data base of the pluviograph stations in the study region should
be in the directory.
Outputs:
•
•
Weighted average IFD table of the ungauged catchment.
Dimensionless temporal pattern file for the ungauged catchment.
99
APPENDIX C
DISTRIBUTIONS OF STORM-CORE
DURATIONS
(Dc values are in hour)
Station 77087
Station 79079
80
70
60
70
60
50
40
30
20
10
Frequency
Frequency
100
90
80
50
40
30
20
10
0
0
1--10
11--20
21--30
31--40
1--10
41--50
11--20
21--30
120
120
100
100
80
80
F req u en cy
Frequency
41--50
51--60
61--70
Station 79082
Station 79046
60
40
60
40
20
20
0
0
1--10
11--20
21--30
31--40
41--50
51--60
61--70
1--10
Storm-core dura tion (Dc)
11--20
21--30
120
Frequency
100
80
60
40
20
0
11--20
21--30
31--40
41--50
51--60
Station 79086
140
1--10
31--40
Storm core durations (Dc)
Station 79052
Frequency
31--40
Storm-core durations (Dc)
Storm-core duration (Dc)
41--50
Storm-core durations (Dc)
51--60
61--70
100
90
80
70
60
50
40
30
20
10
0
1--10
11--20
21--30
31--40
41--50
51--60
Storm-core durations (Dc)
100
Station 80006
Station 81013
30
20
Frequency
Frequency
25
15
10
5
0
1--10
11--20
21--30
31--40
180
160
140
120
100
80
60
40
20
0
1--10
41--50
11--20
Station 80102
40
Frequency
Frequency
50
30
20
10
0
11--20
21--30
41--50
51--60
61--70
31--40
41--50
90
80
70
60
50
40
30
20
10
0
1--10
51--60
11--20
Storm-core durations (Dc)
21--30
31--40
41--50
51--60
61--70
71--80
61--70
71--80
Storm-core duration (Dc)
Station 81038
Station 80110
70
120
60
Frequency
100
Frequency
31--40
Station 81026
60
1--10
21--30
Storm-core duration (Dc)
Storm-core durations (Dc)
80
60
50
40
30
20
10
40
0
20
1--10
0
11--20
21--30
31--40
41--50
51--60
Storm-core duration (Dc)
1--10
21--30
11--20
31--40
41--50
51--60
Storm -core durations (Dc)
Frequency
Station 88023
Station 81003
120
Frequency
100
200
150
100
50
0
60
1--10 11-- 21-- 31-- 41-- 51-- 61-20
30
40
50
60
70
40
Storm-core duration (Dc)
80
20
0
1--10
11--20
21--30
31--40
41--50
51--60
61--70
Storm-core dura tion (Dc)
101
Station 88153
100
Frequency
Frequency
Station 88029
50
0
40
30
20
10
0
1--10
1--10 11--20 21--30 31--40 41--50 51--60
11--20
21--30
31--40
Storm-core duration (Dc)
Storm-core dura tion (Dc)
200
150
100
50
0
Station 89016
1--10 11-- 21-- 31-- 41-- 51-- 61-20
30
40
50
60
70
Storm-core duration
Frequency
Frequency
Station 88037
150
100
50
0
1--10 11--20 21--30 31--40 41--50 51--60
Storm-core duration (Dc)
60
Station 89025
40
20
0
1--10
11--20
21--30
31--40
Storm-core duration (Dc)
Frequency
Frequency
Station 88049
60
40
20
0
1--10 11--20 21--30 31--40 41--50 51--60
Storm-core duration (Dc)
102
Station 81013
Station 89094
Frequency
Frequency
200
40
30
20
10
0
1--10 11-20
21-30
31-- 41-40
50
51-60
61-70
150
100
50
0
1--10
Storm-core duration (Dc)
11--20
31--40
41--50
51--60
61--70
51-60
61-70
Station 81026
Station 90058
100
100
80
50
0
1--10 11-- 21-- 31-- 41-- 51-- 61-20
30
40
50
60
70
Frequency
Frequency
21--30
Storm-core duration (Dc)
60
40
20
0
1--10
Storm-core dura tion (Dc)
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Station 81038
70
60
40
20
0
60
1--10 11-20
21-30
31-- 41-40
50
51-60
Storm-core duration (Dc)
61-70
Frequency
Frequency
Station 90166
50
40
30
20
10
0
1--10
11--20 21--30 31--40 41--50 51--60 60--71 71--80
Storm-core duration (Dc)
103
Station 82016
100
100
80
80
Frequency
Frequency
Station 81049
60
40
20
60
40
20
0
0
1--10
11--20
21--30
31--40
41--50
1--10
11--20
S torm -core duration (Dc)
21--30
31--40
41--50
51--60
Storm -core duration (Dc)
Station 81114
60
Station 82039
40
30
20
10
0
1--10
11--20
21--30
31--40
41--50
51--60
Frequency
Frequency
50
Storm-core duration (Dc)
80
70
60
50
40
30
20
10
0
1--10
11--20
21--30
31--40
41--50
51--60
Storm -core duration (Dc)
Station 81115
100
Station 82042
60
120
40
100
20
0
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
51--60
Frequency
Frequency
80
80
60
40
20
0
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80 81--90
S torm-core duration (Dc)
104
Station 83017
80
70
60
50
40
30
20
10
0
Frequency
Frequency
Station 82076
1--10
160
140
120
100
80
60
40
20
0
11--20 21--30 31--40 41--50 51--60 61--70 71--80
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80 81--90
Storm -core duration (Dc)
Storm-core duration (Dc)
Station 82107
Station 83025
150
25
100
20
50
0
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80
Frequency
Frequency
200
15
10
5
Storm-core duration (Dc)
0
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Station 82121
120
Station 83031
80
140
60
120
40
20
0
1--10
11--20
21--30
31--40
Storm-core duration (Dc)
41--50
Frequency
Frequency
100
100
80
60
40
20
0
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80
Storm-core duration (Dc)
105
Station 84005
80
70
60
50
40
30
20
10
0
100
80
Frequency
Frequency
Station 83033
60
40
20
0
1--10
21--30
31--40
41--50
Storm-core duration (Dc)
Station 83067
Station 84015
100
100
80
80
60
40
20
51--60
60
40
20
0
0
1--10
11--20
21--30
31--40
41--50
51--60
61--70
1--10
11--20
21--30
31--40
Storm-core duration (Dc)
Storm-core duration (Dc)
Station 83074
Station 84078
35
120
30
100
25
Frequency
Frequency
11--20
Storm -core duration (Dc)
Frequency
Frequency
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80 81--90
20
15
10
41--50
80
60
40
20
5
0
0
1--10
11--20 21--30 31--40 41--50 51--60 61--70 71--80
Storm-core duration
1--10
11--20
21--30
31--40
41--50
51--60
61--70
Storm-core duration (Dc)
106
Station 84112
Station 84125
70
20
15
Frequency
Frequency
60
10
5
50
40
30
20
10
0
0
11--20
21--30
31--40
1--10
41--50
21--30
31--40
51--60
41--50
Storm-core duration (Dc)
Station 84122
Station 85000
70
35
60
30
50
25
40
30
20
61--70
20
15
10
10
5
0
0
1--10
11--20
21--30
31--40
41--50
51--60
61--70
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Storm-core duration (Dc)
Station 84123
Station 85026
30
51--60
60
50
Frequency
25
Frequency
11--20
Storm -core duration (Dc)
Frequency
Frequency
1--10
20
15
10
40
30
20
10
5
0
0
1--10
11--20
21--30
31--40
Storm-core duration (Dc)
41--50
1--10
11--20
21--30
31--40
41--50
51--60
Storm-core duration (Dc)
107
Station 85034
Station 85106
140
100
Frequency
Frequency
120
80
60
40
20
0
1--10
11--20
21--30
31--40
41--50
51-60
80
70
60
50
40
30
20
10
0
61--70
1--10
11--20
Storm-core duration (Dc)
21--30
31--40
41--50
51--60
61--70
Storm-core duration (Dc)
160
140
120
100
80
60
40
20
0
Station 85170
70
60
1--10
11--20
21--30
31--40
41--50
51--60
61--70
Frequency
Frequency
Station 85072
Storm-core duration (Dc)
50
40
30
20
10
0
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Station 85103
100
Station 85176
60
40
20
0
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
51--60
Frequency
Frequency
80
80
70
60
50
40
30
20
10
0
1--10
11--20
21--30
31--40
41--50
51--60
61--70
Storm-core dura tion (Dc)
108
Station 86074
70
120
60
100
50
Frequency
Frequency
Station 85237
40
30
20
60
40
20
10
0
0
1--10
11--20 21--30 31--40 41--50 51--60 61--70 71--80
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Storm-core duration (Dc)
Station 85256
Station 82011
60
120
50
100
Frequency
Frequency
80
40
30
20
10
51--60
80
60
40
20
0
0
1--10
11--20
21--30
31--40
41--50
51--60
Storm-core duration (Dc)
1--10
11--20
21--30
31--40
41--50
51--60
Storm-core duration (Dc)
Station 86038
120
Frequency
100
80
60
40
20
0
1--10
11--20
21--30
31--40
41--50
Storm-core duration (Dc)
Station 86071
500
Frequency
400
300
200
100
0
1--10 11--20 21--30 31--40 41--50 51--60 61--70 71--80
Storm-core duration (Dc)
109
APPENDIX D: IFD TABLES
(Intensity values are in mm/h and ARI in years)
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.978
3.585
1.127
0.219
0.09
0.052
0.033
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.308
4.641
1.703
0.293
0.099
0.049
0.027
77087
ARI-1
8.488
4.584
1.849
0.663
0.417
0.323
0.265
ARI1.11
9.238
5.037
2.047
0.731
0.457
0.351
0.286
ARI1.25
10.076
5.544
2.269
0.807
0.501
0.383
0.311
ARI-2
13.421
7.561
3.153
1.111
0.676
0.51
0.407
ARI-5
19.944
11.488
4.875
1.703
1.019
0.757
0.597
ARI-10
24.878
14.458
6.178
2.15
1.278
0.945
0.74
ARI-20
29.813
17.426
7.481
2.598
1.537
1.132
0.884
ARI-50
36.337
21.35
9.203
3.19
1.88
1.38
1.074
ARI-100
41.273
24.318
10.506
3.637
2.139
1.567
1.218
ARI-500
52.732
31.209
13.531
4.677
2.741
2.003
1.553
ARI1000
57.667
34.177
14.834
5.125
3
2.19
1.697
ARI1000000
106.852
63.751
27.817
9.586
5.583
4.06
3.132
79046
ARI-1
11.248
6.948
3.191
1.164
0.696
0.513
0.4
ARI1.11
12.007
7.344
3.353
1.236
0.747
0.556
0.438
ARI1.25
12.856
7.787
3.534
1.315
0.805
0.605
0.48
ARI-2
16.248
9.548
4.257
1.632
1.037
0.801
0.653
ARI-5
22.874
12.972
5.661
2.246
1.489
1.19
1
ARI-10
27.891
15.557
6.721
2.71
1.832
1.486
1.267
ARI-20
32.911
18.14
7.781
3.172
2.175
1.784
1.537
ARI-50
39.549
21.553
9.18
3.783
2.629
2.178
1.895
ARI-100
44.572
24.133
10.239
4.244
2.973
2.477
2.168
ARI-500
56.238
30.122
12.695
5.315
3.771
3.173
2.802
ARI1000
61.263
32.701
13.753
5.777
4.114
3.473
3.076
ARI1000000
111.35
58.392
24.291
10.37
7.54
6.468
5.815
110
Duration(hours)
1
2
6
24
48
72
100
Station
82011
ARI-0.1
9.332
5.14
1.773
0.376
0.159
0.093
0.06
ARI-1
12.116
7.093
3.167
1.234
0.794
0.62
0.51
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
9.791
5.042
1.623
0.337
0.145
0.087
0.057
ARI1.11
12.705
7.487
3.357
1.3
0.83
0.644
0.527
ARI1.25
13.36
7.926
3.569
1.373
0.871
0.672
0.546
ARI-2
15.948
9.665
4.412
1.668
1.032
0.782
0.625
ARI-5
20.925
13.026
6.053
2.241
1.349
0.999
0.781
ARI-10
24.66
15.554
7.292
2.675
1.588
1.164
0.901
ARI-20
28.382
18.076
8.532
3.108
1.828
1.329
1.022
ARI-50
33.291
21.404
10.17
3.681
2.145
1.548
1.182
ARI-100
36.998
23.918
11.409
4.115
2.384
1.714
1.304
ARI-500
45.593
29.751
14.285
5.122
2.941
2.099
1.586
ARI1000
49.292
32.262
15.523
5.555
3.181
2.265
1.708
ARI1000000
86.104
57.257
27.866
9.877
5.572
3.921
2.923
82016
ARI-1
12.526
6.712
2.749
1.054
0.701
0.564
0.478
ARI1.11
13.529
7.216
2.935
1.117
0.739
0.594
0.503
ARI1.25
14.651
7.779
3.142
1.186
0.783
0.627
0.53
ARI-2
19.124
10.02
3.968
1.464
0.955
0.761
0.641
ARI-5
27.842
14.379
5.571
2.004
1.292
1.022
0.856
ARI-10
34.434
17.672
6.782
2.411
1.547
1.22
1.019
ARI-20
41.026
20.963
7.992
2.819
1.801
1.418
1.182
ARI-50
49.739
25.313
9.591
3.358
2.138
1.679
1.398
ARI-100
56.329
28.602
10.799
3.765
2.392
1.877
1.561
ARI-500
71.631
36.239
13.606
4.711
2.983
2.336
1.94
ARI1000
78.221
39.528
14.814
5.118
3.238
2.534
2.103
ARI1000000
143.893
72.298
26.854
9.176
5.773
4.504
3.729
111
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.597
4.18
1.486
0.346
0.156
0.096
0.065
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
10.291
5.886
2.115
0.456
0.191
0.112
0.071
82039
ARI-1
10.436
6.013
2.741
1.187
0.833
0.691
0.601
ARI1.11
11.106
6.408
2.916
1.252
0.873
0.72
0.623
ARI1.25
11.856
6.849
3.112
1.325
0.917
0.753
0.649
ARI-2
14.847
8.611
3.893
1.614
1.094
0.885
0.752
ARI-5
20.676
12.042
5.411
2.177
1.44
1.145
0.957
ARI-10
25.085
14.638
6.558
2.602
1.702
1.343
1.115
ARI-20
29.493
17.232
7.704
3.027
1.964
1.541
1.273
ARI-50
35.321
20.662
9.219
3.589
2.31
1.803
1.483
ARI-100
39.73
23.257
10.365
4.013
2.573
2.002
1.642
ARI-500
49.967
29.28
13.024
4.999
3.182
2.464
2.012
ARI1000
54.375
31.875
14.169
5.424
3.445
2.663
2.171
ARI1000000
98.311
57.726
25.58
9.652
6.061
4.648
3.762
ARI1.11
15.011
9.006
4.094
1.571
0.989
0.758
0.612
ARI1.25
16.102
9.642
4.373
1.676
1.055
0.808
0.653
ARI-2
20.448
12.175
5.486
2.095
1.318
1.011
0.818
ARI-5
28.91
17.099
7.649
2.91
1.832
1.406
1.139
ARI-10
35.307
20.82
9.282
3.526
2.221
1.706
1.382
ARI-20
41.702
24.537
10.915
4.142
2.609
2.005
1.626
ARI-50
50.153
29.45
13.072
4.956
3.123
2.401
1.948
ARI-100
56.546
33.166
14.703
5.572
3.512
2.701
2.192
ARI-500
71.388
41.791
18.49
7.002
4.415
3.397
2.758
ARI1000
77.779
45.505
20.12
7.618
4.804
3.697
3.002
ARI1000000
141.47
82.514
36.367
13.755
8.681
6.684
5.432
82042
ARI-1
14.035
8.436
3.843
1.477
0.93
0.713
0.576
112
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
9.912
5.764
2.064
0.421
0.168
0.095
0.058
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.643
5.459
2.083
0.409
0.153
0.081
0.047
82076
ARI-1
12.026
7.333
3.37
1.278
0.791
0.598
0.478
ARI1.11
12.618
7.712
3.549
1.343
0.829
0.626
0.499
ARI1.25
13.28
8.134
3.749
1.416
0.872
0.657
0.522
ARI-2
15.918
9.821
4.546
1.706
1.041
0.779
0.616
ARI-5
21.059
13.107
6.1
2.271
1.372
1.019
0.799
ARI-10
24.945
15.592
7.276
2.698
1.622
1.2
0.938
ARI-20
28.831
18.077
8.452
3.125
1.872
1.381
1.076
ARI-50
33.968
21.362
10.006
3.689
2.202
1.62
1.26
ARI-100
37.853
23.846
11.182
4.116
2.452
1.801
1.398
ARI-500
46.873
29.614
13.911
5.108
3.033
2.222
1.721
ARI1000
50.758
32.098
15.087
5.535
3.283
2.403
1.86
ARI1000000
89.468
56.853
26.803
9.79
5.774
4.208
3.244
ARI1.11
14.328
8.893
4.236
1.703
1.092
0.844
0.687
ARI1.25
15.338
9.455
4.48
1.807
1.164
0.904
0.738
ARI-2
19.354
11.684
5.446
2.217
1.452
1.144
0.947
ARI-5
27.161
15.989
7.311
3.011
2.015
1.613
1.357
ARI-10
33.057
19.23
8.714
3.609
2.44
1.97
1.67
ARI-20
38.949
22.464
10.113
4.206
2.864
2.327
1.984
ARI-50
46.736
26.732
11.96
4.995
3.426
2.799
2.399
ARI-100
52.625
29.959
13.356
5.592
3.851
3.156
2.714
ARI-500
66.295
37.444
16.593
6.976
4.837
3.986
3.446
ARI1000
72.182
40.666
17.987
7.571
5.261
4.343
3.762
ARI1000000
130.841
72.754
31.866
13.505
9.493
7.909
6.91
82107
ARI-1
13.424
8.388
4.016
1.611
1.027
0.791
0.64
113
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.725
4.39
1.554
0.326
0.135
0.078
0.049
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.112
5.496
2.266
0.463
0.172
0.09
0.052
82121
ARI-1
10.132
6.044
2.752
1.079
0.691
0.537
0.439
ARI1.11
10.769
6.437
2.932
1.145
0.731
0.566
0.461
ARI1.25
11.482
6.876
3.133
1.218
0.775
0.598
0.487
ARI-2
14.326
8.627
3.935
1.512
0.95
0.727
0.587
ARI-5
19.873
12.037
5.497
2.083
1.293
0.98
0.784
ARI-10
24.069
14.617
6.677
2.515
1.552
1.172
0.934
ARI-20
28.265
17.196
7.857
2.947
1.811
1.363
1.084
ARI-50
33.813
20.604
9.416
3.518
2.153
1.616
1.282
ARI-100
38.01
23.183
10.595
3.949
2.412
1.808
1.432
ARI-500
47.755
29.17
13.333
4.951
3.014
2.253
1.78
ARI1000
51.952
31.748
14.512
5.383
3.273
2.445
1.93
ARI1000000
93.779
57.441
26.261
9.683
5.855
4.357
3.426
ARI1.11
13.393
8.713
4.313
1.709
1.059
0.796
0.63
ARI1.25
14.356
9.219
4.511
1.79
1.117
0.845
0.673
ARI-2
18.216
11.229
5.299
2.112
1.349
1.041
0.846
ARI-5
25.79
15.117
6.825
2.737
1.8
1.425
1.187
ARI-10
31.544
18.043
7.974
3.209
2.14
1.716
1.446
ARI-20
37.31
20.963
9.12
3.679
2.481
2.006
1.706
ARI-50
44.944
24.816
10.634
4.301
2.93
2.391
2.052
ARI-100
50.725
27.728
11.778
4.771
3.27
2.682
2.313
ARI-500
64.161
34.482
14.432
5.862
4.059
3.359
2.921
ARI1000
69.951
37.389
15.575
6.332
4.399
3.65
3.184
ARI1000000
127.708
66.333
26.955
11.01
7.783
6.555
5.801
83017
ARI-1
12.534
8.258
4.134
1.636
1.007
0.752
0.592
114
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.102
4.461
1.582
0.365
0.164
0.1
0.067
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
9.692
5.391
1.863
0.387
0.16
0.092
0.058
83025
ARI-1
10.41
6.014
2.736
1.169
0.812
0.668
0.576
ARI1.11
11.028
6.382
2.903
1.235
0.853
0.7
0.602
ARI1.25
11.72
6.792
3.09
1.308
0.9
0.736
0.631
ARI-2
14.477
8.432
3.835
1.6
1.087
0.88
0.748
ARI-5
19.849
11.629
5.287
2.168
1.45
1.161
0.977
ARI-10
23.911
14.048
6.386
2.598
1.725
1.374
1.151
ARI-20
27.973
16.467
7.485
3.028
2
1.588
1.325
ARI-50
33.342
19.664
8.938
3.596
2.364
1.869
1.555
ARI-100
37.403
22.083
10.037
4.026
2.639
2.083
1.729
ARI-500
46.832
27.7
12.589
5.024
3.277
2.578
2.133
ARI1000
50.893
30.119
13.688
5.454
3.552
2.791
2.308
ARI1000000
91.359
54.227
24.643
9.736
6.292
4.916
4.044
ARI1.11
13.994
8.183
3.686
1.478
0.973
0.771
0.643
ARI1.25
14.826
8.685
3.92
1.574
1.036
0.821
0.684
ARI-2
18.146
10.687
4.852
1.954
1.286
1.018
0.848
ARI-5
24.62
14.588
6.669
2.695
1.773
1.402
1.166
ARI-10
29.519
17.54
8.043
3.255
2.141
1.693
1.407
ARI-20
34.418
20.491
9.417
3.816
2.509
1.984
1.648
ARI-50
40.894
24.392
11.233
4.557
2.996
2.368
1.967
ARI-100
45.793
27.343
12.606
5.117
3.364
2.658
2.208
ARI-500
57.169
34.195
15.796
6.418
4.22
3.333
2.768
ARI1000
62.069
37.146
17.169
6.978
4.588
3.624
3.009
ARI1000000
110.897
66.553
30.858
12.562
8.258
6.52
5.411
83031
ARI-1
13.25
7.734
3.477
1.393
0.917
0.727
0.606
115
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.406
5.526
2.244
0.475
0.184
0.1
0.059
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
9.712
5.674
2.108
0.474
0.203
0.12
0.077
83033
ARI-1
10.692
7.304
3.825
1.568
0.973
0.729
0.573
ARI1.11
11.266
7.672
4.027
1.676
1.054
0.797
0.632
ARI1.25
11.908
8.084
4.252
1.797
1.144
0.873
0.699
ARI-2
14.466
9.732
5.152
2.278
1.507
1.181
0.968
ARI-5
19.446
12.949
6.906
3.211
2.215
1.789
1.508
ARI-10
23.21
15.385
8.232
3.915
2.752
2.253
1.922
ARI-20
26.974
17.823
9.558
4.618
3.29
2.718
2.339
ARI-50
31.948
21.045
11.31
5.546
4.001
3.335
2.893
ARI-100
35.711
23.484
12.635
6.247
4.538
3.803
3.313
ARI-500
44.447
29.146
15.712
7.876
5.788
4.889
4.292
ARI1000
48.209
31.585
17.037
8.577
6.326
5.358
4.714
ARI1000000
85.699
55.891
30.24
15.56
11.691
10.033
8.932
ARI1.11
13.723
8.206
3.866
1.668
1.147
0.934
0.797
ARI1.25
14.696
8.736
4.099
1.775
1.227
1.004
0.861
ARI-2
18.565
10.84
5.02
2.199
1.546
1.283
1.113
ARI-5
26.079
14.911
6.804
3.019
2.166
1.825
1.609
ARI-10
31.752
17.979
8.148
3.638
2.634
2.236
1.985
ARI-20
37.42
21.043
9.489
4.255
3.102
2.647
2.361
ARI-50
44.908
25.089
11.261
5.071
3.72
3.19
2.859
ARI-100
50.571
28.147
12.6
5.687
4.187
3.601
3.236
ARI-500
63.716
35.244
15.708
7.118
5.271
4.555
4.111
ARI1000
69.376
38.299
17.045
7.734
5.738
4.965
4.488
ARI1000000
125.773
68.734
30.372
13.868
10.39
9.06
8.249
83067
ARI-1
12.852
7.731
3.658
1.573
1.075
0.872
0.741
116
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
11.285
6.692
2.466
0.519
0.21
0.119
0.074
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.013
4.8
1.881
0.464
0.211
0.129
0.085
83074
ARI-1
14.075
8.718
4.125
1.636
1.038
0.798
0.645
ARI1.11
14.629
9.045
4.287
1.717
1.099
0.849
0.691
ARI1.25
15.248
9.412
4.469
1.808
1.166
0.907
0.741
ARI-2
17.704
10.876
5.195
2.167
1.435
1.135
0.943
ARI-5
22.461
13.739
6.617
2.865
1.954
1.579
1.336
ARI-10
26.045
15.908
7.696
3.392
2.345
1.913
1.633
ARI-20
29.621
18.077
8.776
3.917
2.735
2.247
1.93
ARI-50
34.343
20.947
10.206
4.611
3.25
2.687
2.322
ARI-100
37.91
23.118
11.287
5.136
3.639
3.02
2.618
ARI-500
46.188
28.16
13.8
6.354
4.541
3.792
3.305
ARI1000
49.751
30.332
14.883
6.878
4.93
4.124
3.601
ARI1000000
85.227
51.977
25.676
12.099
8.797
7.432
6.545
ARI1.11
10.664
6.707
3.405
1.6
1.144
0.952
0.826
ARI1.25
11.478
7.233
3.684
1.738
1.245
1.038
0.901
ARI-2
14.721
9.314
4.781
2.284
1.648
1.38
1.203
ARI-5
21.024
13.33
6.889
3.339
2.433
2.051
1.799
ARI-10
25.786
16.352
8.471
4.135
3.028
2.56
2.253
ARI-20
30.545
19.367
10.049
4.928
3.622
3.071
2.708
ARI-50
36.834
23.348
12.129
5.976
4.408
3.746
3.311
ARI-100
41.591
26.357
13.701
6.768
5.002
4.257
3.768
ARI-500
52.635
33.338
17.347
8.606
6.383
5.445
4.829
ARI1000
57.39
36.343
18.916
9.397
6.978
5.957
5.287
ARI1000000
104.778
66.275
34.537
17.278
12.903
11.062
9.853
84005
ARI-1
9.934
6.234
3.154
1.476
1.053
0.876
0.76
117
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.866
4.513
1.845
0.4
0.157
0.087
0.052
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.124
5.038
1.934
0.407
0.161
0.09
0.054
84015
ARI-1
9.339
6.368
3.308
1.331
0.815
0.605
0.472
ARI1.11
9.99
6.762
3.504
1.427
0.885
0.664
0.523
ARI1.25
10.717
7.204
3.723
1.536
0.965
0.73
0.58
ARI-2
13.618
8.964
4.599
1.966
1.281
0.996
0.812
ARI-5
19.271
12.397
6.308
2.802
1.9
1.522
1.275
ARI-10
23.545
14.994
7.601
3.433
2.369
1.922
1.63
ARI-20
27.82
17.591
8.893
4.064
2.838
2.323
1.986
ARI-50
33.469
21.025
10.602
4.897
3.458
2.855
2.46
ARI-100
37.743
23.622
11.895
5.527
3.927
3.257
2.819
ARI-500
47.666
29.652
14.897
6.99
5.018
4.194
3.655
ARI1000
51.939
32.249
16.19
7.62
5.487
4.597
4.015
ARI1000000
94.526
58.131
29.074
13.895
10.168
8.621
7.614
ARI1.11
10.611
7
3.52
1.407
0.872
0.655
0.517
ARI1.25
11.205
7.442
3.774
1.52
0.943
0.709
0.56
ARI-2
13.586
9.198
4.781
1.966
1.228
0.925
0.732
ARI-5
18.249
12.603
6.723
2.829
1.78
1.345
1.066
ARI-10
21.784
15.172
8.185
3.48
2.197
1.663
1.319
ARI-20
25.321
17.737
9.644
4.13
2.614
1.98
1.573
ARI-50
29.999
21.127
11.569
4.988
3.165
2.401
1.908
ARI-100
33.538
23.69
13.024
5.636
3.582
2.718
2.161
ARI-500
41.759
29.638
16.4
7.142
4.549
3.456
2.75
ARI1000
45.299
32.199
17.853
7.79
4.965
3.774
3.003
ARI1000000
80.591
57.714
32.328
14.246
9.115
6.94
5.53
84078
ARI-1
10.082
6.603
3.291
1.306
0.808
0.606
0.479
118
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.039
5.445
2.355
0.556
0.23
0.131
0.08
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
11.419
6.442
2.466
0.669
0.335
0.221
0.157
84112
ARI-1
10.202
7.125
3.961
1.83
1.227
0.968
0.796
ARI1.11
10.771
7.462
4.152
1.966
1.349
1.081
0.903
ARI1.25
11.406
7.84
4.366
2.118
1.484
1.208
1.024
ARI-2
13.927
9.36
5.224
2.717
2.027
1.727
1.525
ARI-5
18.817
12.345
6.909
3.871
3.089
2.762
2.549
ARI-10
22.507
14.612
8.187
4.739
3.895
3.556
3.346
ARI-20
26.193
16.882
9.466
5.605
4.701
4.356
4.154
ARI-50
31.064
19.885
11.157
6.747
5.767
5.417
5.232
ARI-100
34.748
22.158
12.438
7.61
6.574
6.223
6.052
ARI-500
43.297
27.437
15.411
9.612
8.449
8.098
7.967
ARI1000
46.979
29.712
16.691
10.474
9.257
8.908
8.794
ARI1000000
83.661
52.386
29.456
19.055
17.309
16.99
17.081
ARI1.11
14.332
9.599
5.324
2.741
2.034
1.727
1.52
ARI1.25
14.715
10.142
5.778
2.979
2.183
1.831
1.593
ARI-2
16.295
12.265
7.59
3.933
2.771
2.242
1.882
ARI-5
19.492
16.29
11.128
5.808
3.909
3.034
2.442
ARI-10
21.953
19.284
13.809
7.235
4.766
3.627
2.861
ARI-20
24.432
22.255
16.493
8.666
5.622
4.217
3.279
ARI-50
27.724
26.159
20.045
10.563
6.75
4.994
3.829
ARI-100
30.22
29.101
22.734
12
7.602
5.58
4.244
ARI-500
36.03
35.91
28.984
15.343
9.58
6.936
5.205
ARI1000
38.535
38.835
31.677
16.785
10.43
7.52
5.618
ARI1000000
63.545
67.896
58.542
31.174
18.899
13.317
9.723
84123
ARI-1
13.996
9.108
4.917
2.529
1.902
1.633
1.455
119
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.094
5.888
2.369
0.369
0.108
0.048
0.024
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.736
6.473
2.806
0.52
0.172
0.083
0.044
84125
ARI-1
10.825
7.873
4.038
1.307
0.66
0.427
0.294
ARI1.11
11.443
8.217
4.203
1.398
0.725
0.477
0.334
ARI1.25
12.131
8.604
4.389
1.498
0.796
0.534
0.381
ARI-2
14.853
10.172
5.145
1.888
1.077
0.762
0.57
ARI-5
20.11
13.273
6.652
2.63
1.615
1.205
0.947
ARI-10
24.069
15.634
7.802
3.184
2.017
1.541
1.237
ARI-20
28.021
18.001
8.957
3.734
2.417
1.876
1.528
ARI-50
33.24
21.135
10.487
4.46
2.944
2.318
1.914
ARI-100
37.185
23.509
11.645
5.007
3.342
2.653
2.207
ARI-500
46.339
29.024
14.339
6.276
4.265
3.43
2.888
ARI1000
50.281
31.401
15.499
6.822
4.661
3.764
3.181
ARI1000000
89.541
55.098
27.077
12.253
8.608
7.096
6.112
ARI1.11
11.277
8.607
4.725
1.642
0.854
0.561
0.39
ARI1.25
11.964
9.034
4.924
1.724
0.906
0.6
0.421
ARI-2
14.709
10.737
5.716
2.05
1.115
0.758
0.545
ARI-5
20.078
14.046
7.26
2.679
1.519
1.067
0.793
ARI-10
24.147
16.543
8.426
3.153
1.825
1.303
0.983
ARI-20
28.22
19.038
9.592
3.625
2.13
1.539
1.174
ARI-50
33.608
22.332
11.132
4.249
2.534
1.852
1.428
ARI-100
37.686
24.823
12.298
4.721
2.839
2.089
1.622
ARI-500
47.159
30.604
15.003
5.815
3.547
2.64
2.072
ARI1000
51.24
33.093
16.168
6.285
3.852
2.878
2.266
ARI1000000
91.931
57.884
27.774
10.973
6.892
5.248
4.209
85000
ARI-1
10.663
8.224
4.547
1.569
0.807
0.525
0.363
120
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.96
4.876
1.95
0.48
0.215
0.13
0.085
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.976
4.378
1.703
0.361
0.143
0.079
0.048
85026
ARI-1
10.203
6.53
3.334
1.517
1.05
0.853
0.724
ARI1.11
10.968
7.004
3.565
1.618
1.118
0.908
0.771
ARI1.25
11.822
7.532
3.822
1.73
1.195
0.971
0.824
ARI-2
15.222
9.63
4.844
2.178
1.502
1.22
1.035
ARI-5
21.832
13.7
6.826
3.05
2.102
1.707
1.45
ARI-10
26.827
16.77
8.32
3.709
2.557
2.077
1.765
ARI-20
31.819
19.837
9.812
4.367
3.011
2.447
2.08
ARI-50
38.416
23.889
11.784
5.237
3.612
2.937
2.498
ARI-100
43.405
26.953
13.274
5.896
4.066
3.307
2.813
ARI-500
54.988
34.064
16.734
7.424
5.122
4.168
3.547
ARI1000
59.976
37.126
18.223
8.082
5.577
4.538
3.864
ARI1000000
109.68
67.632
33.061
14.639
10.109
8.234
7.016
ARI1.11
10.032
6.663
3.411
1.412
0.895
0.682
0.546
ARI1.25
10.813
7.209
3.713
1.549
0.985
0.753
0.604
ARI-2
13.925
9.382
4.913
2.094
1.346
1.035
0.834
ARI-5
19.984
13.61
7.249
3.155
2.049
1.585
1.283
ARI-10
24.564
16.806
9.015
3.957
2.581
2.001
1.623
ARI-20
29.142
20
10.779
4.759
3.113
2.417
1.963
ARI-50
35.194
24.222
13.112
5.819
3.816
2.967
2.413
ARI-100
39.772
27.415
14.876
6.621
4.348
3.383
2.753
ARI-500
50.399
34.828
18.971
8.483
5.583
4.349
3.543
ARI1000
54.976
38.021
20.734
9.285
6.115
4.766
3.884
ARI1000000
100.583
69.833
38.308
17.276
11.415
8.913
7.274
85034
ARI-1
9.334
6.175
3.141
1.29
0.814
0.619
0.494
121
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.181
4.082
1.637
0.33
0.123
0.065
0.038
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.413
4.476
1.869
0.365
0.129
0.066
0.037
85072
ARI-1
7.932
5.486
2.83
1.071
0.623
0.445
0.336
ARI1.11
8.433
5.832
3.017
1.152
0.674
0.484
0.367
ARI1.25
8.994
6.22
3.226
1.242
0.731
0.528
0.401
ARI-2
11.233
7.766
4.059
1.601
0.96
0.701
0.54
ARI-5
15.596
10.782
5.684
2.301
1.406
1.041
0.81
ARI-10
18.896
13.063
6.913
2.83
1.744
1.298
1.016
ARI-20
22.197
15.345
8.141
3.359
2.081
1.556
1.222
ARI-50
26.559
18.362
9.765
4.058
2.528
1.897
1.494
ARI-100
29.86
20.644
10.993
4.586
2.866
2.155
1.701
ARI-500
37.523
25.943
13.845
5.814
3.65
2.754
2.18
ARI1000
40.823
28.225
15.073
6.342
3.988
3.011
2.386
ARI1000000
73.714
50.969
27.313
11.61
7.356
5.583
4.444
ARI1.11
9.65
6.601
3.347
1.241
0.714
0.508
0.382
ARI1.25
10.484
7.051
3.519
1.303
0.756
0.541
0.41
ARI-2
13.834
8.837
4.202
1.551
0.92
0.673
0.52
ARI-5
20.433
12.293
5.523
2.032
1.237
0.926
0.733
ARI-10
25.457
14.895
6.519
2.394
1.475
1.117
0.895
ARI-20
30.497
17.491
7.513
2.755
1.713
1.308
1.056
ARI-50
37.174
20.917
8.824
3.232
2.027
1.559
1.268
ARI-100
42.233
23.506
9.816
3.593
2.263
1.749
1.429
ARI-500
53.995
29.513
12.115
4.429
2.813
2.189
1.802
ARI1000
59.066
32.099
13.105
4.789
3.049
2.379
1.963
ARI1000000
109.664
57.845
22.961
8.374
5.401
4.265
3.561
85103
ARI-1
8.907
6.196
3.193
1.185
0.677
0.478
0.357
122
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.356
4.929
2.039
0.434
0.167
0.09
0.053
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.432
3.958
1.499
0.31
0.122
0.067
0.041
85106
ARI-1
10.258
7.129
3.825
1.61
1.01
0.761
0.601
ARI1.11
10.685
7.425
3.985
1.679
1.055
0.795
0.629
ARI1.25
11.163
7.755
4.164
1.757
1.105
0.833
0.66
ARI-2
13.064
9.072
4.877
2.067
1.304
0.987
0.783
ARI-5
16.76
11.635
6.264
2.671
1.693
1.284
1.022
ARI-10
19.551
13.572
7.314
3.127
1.986
1.51
1.203
ARI-20
22.34
15.507
8.362
3.583
2.28
1.734
1.383
ARI-50
26.023
18.064
9.748
4.186
2.668
2.032
1.622
ARI-100
28.809
19.998
10.797
4.642
2.961
2.257
1.802
ARI-500
35.274
24.488
13.23
5.701
3.642
2.778
2.221
ARI1000
38.057
26.421
14.278
6.157
3.935
3.003
2.402
ARI1000000
65.786
45.68
24.722
10.699
6.856
5.242
4.199
ARI1.11
10.283
6.372
2.989
1.153
0.717
0.543
0.434
ARI1.25
11.103
6.84
3.183
1.219
0.756
0.572
0.456
ARI-2
14.386
8.702
3.954
1.481
0.912
0.688
0.548
ARI-5
20.813
12.322
5.444
1.989
1.215
0.913
0.726
ARI-10
25.688
15.056
6.566
2.372
1.444
1.084
0.861
ARI-20
30.57
17.788
7.685
2.755
1.672
1.254
0.996
ARI-50
37.03
21.398
9.162
3.26
1.974
1.48
1.174
ARI-100
41.919
24.127
10.279
3.641
2.203
1.65
1.309
ARI-500
53.279
30.463
12.869
4.527
2.733
2.046
1.623
ARI1000
58.173
33.191
13.983
4.908
2.961
2.216
1.757
ARI1000000
106.975
60.369
25.082
8.703
5.234
3.913
3.102
85170
ARI-1
9.55
5.954
2.815
1.094
0.682
0.517
0.413
123
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.179
5.882
2.613
0.566
0.214
0.113
0.065
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.484
5.067
1.906
0.419
0.175
0.101
0.064
85176
ARI-1
10.536
7.83
4.387
1.746
1.017
0.724
0.542
ARI1.11
11.329
8.271
4.571
1.833
1.083
0.779
0.59
ARI1.25
12.215
8.764
4.777
1.929
1.156
0.841
0.644
ARI-2
15.753
10.715
5.604
2.316
1.449
1.092
0.864
ARI-5
22.655
14.487
7.224
3.07
2.02
1.586
1.306
ARI-10
27.879
17.327
8.453
3.64
2.453
1.963
1.646
ARI-20
33.105
20.16
9.683
4.21
2.885
2.341
1.989
ARI-50
40.015
23.9
11.311
4.964
3.458
2.843
2.446
ARI-100
45.244
26.726
12.543
5.533
3.891
3.223
2.793
ARI-500
57.386
33.283
15.404
6.857
4.897
4.107
3.603
ARI1000
62.616
36.105
16.636
7.426
5.33
4.489
3.952
ARI1000000
114.746
64.21
28.923
13.104
9.65
8.295
7.449
ARI1.11
11.17
7.151
3.535
1.459
0.939
0.726
0.59
ARI1.25
11.609
7.487
3.731
1.545
0.994
0.767
0.622
ARI-2
13.359
8.825
4.512
1.89
1.211
0.931
0.751
ARI-5
16.765
11.425
6.038
2.564
1.634
1.248
0.999
ARI-10
19.339
13.387
7.194
3.074
1.954
1.486
1.185
ARI-20
21.912
15.347
8.35
3.585
2.274
1.725
1.371
ARI-50
25.312
17.937
9.879
4.26
2.696
2.039
1.617
ARI-100
27.883
19.894
11.036
4.771
3.015
2.277
1.802
ARI-500
33.851
24.438
13.723
5.959
3.756
2.828
2.231
ARI1000
36.421
26.394
14.881
6.47
4.075
3.066
2.416
ARI1000000
62.025
45.877
26.42
11.571
7.254
5.429
4.254
85237
ARI-1
10.778
6.85
3.36
1.382
0.89
0.689
0.561
124
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.405
5.195
2.003
0.431
0.174
0.098
0.06
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.085
4.214
1.558
0.329
0.134
0.076
0.047
85256
ARI-1
10.164
6.582
3.247
1.289
0.802
0.605
0.48
ARI1.11
10.893
7.055
3.468
1.362
0.84
0.63
0.497
ARI1.25
11.704
7.58
3.712
1.442
0.882
0.658
0.517
ARI-2
14.901
9.643
4.671
1.76
1.053
0.773
0.6
ARI-5
21.054
13.594
6.502
2.374
1.389
1.005
0.769
ARI-10
25.679
16.554
7.872
2.835
1.643
1.182
0.899
ARI-20
30.293
19.504
9.235
3.296
1.898
1.359
1.031
ARI-50
36.383
23.393
11.031
3.903
2.236
1.595
1.206
ARI-100
40.985
26.33
12.387
4.363
2.491
1.774
1.338
ARI-500
51.662
33.14
15.531
5.428
3.084
2.189
1.647
ARI1000
56.258
36.07
16.883
5.887
3.34
2.368
1.78
ARI1000000
102.031
65.239
30.339
10.454
5.887
4.153
3.109
ARI1.11
9.231
5.84
2.815
1.113
0.698
0.531
0.425
ARI1.25
9.702
6.183
3.01
1.202
0.757
0.577
0.462
ARI-2
11.589
7.547
3.786
1.558
0.992
0.76
0.612
ARI-5
15.28
10.199
5.291
2.249
1.449
1.117
0.903
ARI-10
18.078
12.204
6.427
2.771
1.795
1.387
1.123
ARI-20
20.878
14.207
7.563
3.293
2.141
1.657
1.342
ARI-50
24.581
16.855
9.063
3.984
2.598
2.013
1.633
ARI-100
27.383
18.857
10.198
4.506
2.944
2.283
1.853
ARI-500
33.891
23.505
12.831
5.718
3.747
2.91
2.363
ARI1000
36.694
25.507
13.965
6.24
4.093
3.179
2.583
ARI1000000
64.636
45.451
25.265
11.444
7.539
5.867
4.772
86038
ARI-1
8.811
5.534
2.641
1.034
0.645
0.49
0.391
125
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.157
4.281
1.56
0.31
0.12
0.066
0.039
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.503
4.383
1.87
0.429
0.174
0.098
0.059
86071
ARI-1
10.255
6.322
2.938
1.118
0.69
0.52
0.414
ARI1.11
11.037
6.759
3.139
1.214
0.761
0.58
0.466
ARI1.25
11.91
7.248
3.363
1.321
0.839
0.647
0.525
ARI-2
15.386
9.202
4.262
1.746
1.155
0.916
0.763
ARI-5
22.149
13.017
6.016
2.57
1.769
1.444
1.236
ARI-10
27.26
15.905
7.344
3.192
2.233
1.846
1.597
ARI-20
32.369
18.794
8.672
3.813
2.698
2.248
1.96
ARI-50
39.122
22.612
10.427
4.634
3.312
2.78
2.44
ARI-100
44.229
25.501
11.756
5.255
3.776
3.182
2.804
ARI-500
56.086
32.21
14.84
6.697
4.855
4.118
3.65
ARI1000
61.193
35.099
16.168
7.317
5.32
4.521
4.015
ARI1000000
112.077
63.893
29.406
13.501
9.949
8.541
7.655
ARI1.11
9.288
6.53
3.575
1.549
0.987
0.751
0.599
ARI1.25
9.855
6.925
3.79
1.642
1.047
0.797
0.635
ARI-2
12.116
8.501
4.646
2.013
1.284
0.978
0.78
ARI-5
16.528
11.571
6.311
2.734
1.746
1.331
1.063
ARI-10
19.867
13.892
7.569
3.278
2.095
1.597
1.276
ARI-20
23.206
16.214
8.827
3.823
2.444
1.864
1.49
ARI-50
27.62
19.281
10.489
4.542
2.905
2.217
1.772
ARI-100
30.96
21.602
11.745
5.087
3.253
2.483
1.985
ARI-500
38.714
26.989
14.663
6.35
4.063
3.102
2.481
ARI1000
42.054
29.309
15.919
6.894
4.412
3.369
2.695
ARI1000000
75.341
52.43
28.437
12.313
7.885
6.024
4.822
86074
ARI-1
8.781
6.176
3.383
1.465
0.934
0.71
0.566
126
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.261
5.133
2.036
0.47
0.199
0.116
0.073
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.613
5.217
2.269
0.527
0.214
0.12
0.073
86142
ARI-1
11.935
7.642
3.875
1.726
1.175
0.945
0.794
ARI1.11
12.672
8.1
4.098
1.822
1.24
0.996
0.837
ARI1.25
13.496
8.61
4.347
1.929
1.312
1.054
0.886
ARI-2
16.778
10.644
5.336
2.356
1.601
1.286
1.081
ARI-5
23.17
14.595
7.256
3.187
2.165
1.74
1.464
ARI-10
28.002
17.578
8.706
3.815
2.592
2.084
1.754
ARI-20
32.833
20.559
10.153
4.443
3.019
2.428
2.045
ARI-50
39.218
24.497
12.066
5.273
3.584
2.883
2.429
ARI-100
44.048
27.475
13.512
5.901
4.011
3.228
2.72
ARI-500
55.261
34.388
16.868
7.358
5.003
4.028
3.395
ARI1000
60.09
37.365
18.313
7.986
5.43
4.372
3.686
ARI1000000
108.21
67.02
32.709
14.238
9.687
7.807
6.588
ARI1.11
10.329
7.3
4.025
1.755
1.121
0.853
0.681
ARI1.25
10.99
7.718
4.237
1.856
1.194
0.914
0.732
ARI-2
13.614
9.377
5.084
2.261
1.485
1.155
0.94
ARI-5
18.699
12.596
6.731
3.051
2.053
1.628
1.35
ARI-10
22.531
15.023
7.976
3.647
2.483
1.987
1.661
ARI-20
26.357
17.447
9.221
4.244
2.914
2.347
1.974
ARI-50
31.409
20.647
10.866
5.033
3.483
2.823
2.388
ARI-100
35.227
23.067
12.111
5.629
3.914
3.183
2.702
ARI-500
44.087
28.682
14.999
7.014
4.914
4.021
3.431
ARI1000
47.901
31.099
16.243
7.61
5.345
4.381
3.746
ARI1000000
85.886
55.177
28.636
13.553
9.639
7.978
6.884
86219
ARI-1
9.737
6.926
3.835
1.664
1.055
0.8
0.634
127
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.002
4.496
1.604
0.356
0.154
0.092
0.059
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.775
4.06
1.488
0.3
0.117
0.065
0.039
86224
ARI-1
12.469
7.099
3.2
1.385
0.977
0.814
0.711
ARI1.11
13.073
7.418
3.335
1.446
1.023
0.855
0.748
ARI1.25
13.746
7.773
3.486
1.514
1.075
0.9
0.789
ARI-2
16.428
9.187
4.086
1.785
1.28
1.081
0.955
ARI-5
21.64
11.931
5.252
2.313
1.68
1.433
1.279
ARI-10
25.574
14.001
6.132
2.712
1.982
1.699
1.524
ARI-20
29.503
16.068
7.01
3.11
2.285
1.966
1.77
ARI-50
34.694
18.797
8.171
3.636
2.684
2.319
2.095
ARI-100
38.618
20.861
9.048
4.034
2.986
2.585
2.341
ARI-500
47.724
25.649
11.084
4.958
3.688
3.205
2.912
ARI1000
51.645
27.71
11.961
5.355
3.99
3.472
3.158
ARI1000000
90.697
48.237
20.691
9.316
6.999
6.131
5.613
ARI1.11
9.792
6.153
2.881
1.063
0.635
0.468
0.364
ARI1.25
10.43
6.551
3.066
1.132
0.677
0.498
0.388
ARI-2
12.972
8.139
3.806
1.405
0.841
0.619
0.482
ARI-5
17.928
11.233
5.246
1.937
1.16
0.855
0.666
ARI-10
21.677
13.572
6.335
2.339
1.4
1.033
0.805
ARI-20
25.425
15.911
7.422
2.74
1.641
1.211
0.944
ARI-50
30.38
19.002
8.86
3.271
1.959
1.445
1.127
ARI-100
34.128
21.34
9.947
3.672
2.2
1.623
1.266
ARI-500
42.831
26.768
12.471
4.603
2.758
2.035
1.588
ARI1000
46.579
29.106
13.558
5.004
2.998
2.213
1.726
ARI1000000
83.931
52.4
24.387
8.999
5.394
3.982
3.107
86314
ARI-1
9.222
5.796
2.714
1.002
0.599
0.441
0.343
128
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.39
4.556
1.756
0.38
0.154
0.087
0.054
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.98
3.795
1.316
0.294
0.13
0.079
0.052
87029
ARI-1
10.717
6.76
3.26
1.3
0.822
0.628
0.506
ARI1.11
11.457
7.184
3.44
1.367
0.864
0.661
0.532
ARI1.25
12.285
7.657
3.642
1.441
0.911
0.697
0.562
ARI-2
15.59
9.543
4.446
1.738
1.098
0.842
0.68
ARI-5
22.036
13.216
6.013
2.318
1.463
1.124
0.91
ARI-10
26.914
15.993
7.199
2.756
1.739
1.337
1.083
ARI-20
31.792
18.769
8.385
3.195
2.015
1.549
1.257
ARI-50
38.242
22.439
9.953
3.775
2.38
1.83
1.485
ARI-100
43.12
25.214
11.139
4.213
2.656
2.043
1.658
ARI-500
54.449
31.658
13.893
5.232
3.296
2.536
2.06
ARI1000
59.328
34.433
15.079
5.671
3.572
2.749
2.233
ARI1000000
107.955
62.087
26.897
10.044
6.321
4.866
3.955
ARI1.11
10.264
5.976
2.722
1.142
0.779
0.633
0.539
ARI1.25
11.082
6.493
2.968
1.237
0.838
0.676
0.573
ARI-2
14.346
8.551
3.949
1.619
1.073
0.853
0.712
ARI-5
20.714
12.558
5.857
2.36
1.532
1.198
0.986
ARI-10
25.533
15.588
7.3
2.921
1.88
1.461
1.194
ARI-20
30.353
18.617
8.742
3.482
2.228
1.723
1.402
ARI-50
36.725
22.62
10.649
4.222
2.688
2.07
1.679
ARI-100
41.546
25.649
12.09
4.783
3.035
2.333
1.888
ARI-500
52.739
32.679
15.438
6.083
3.843
2.943
2.374
ARI1000
57.56
35.707
16.879
6.644
4.191
3.206
2.583
ARI1000000
105.605
65.881
31.243
12.226
7.658
5.827
4.671
87031
ARI-1
9.534
5.514
2.501
1.056
0.726
0.593
0.509
129
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.112
4.154
1.514
0.322
0.132
0.076
0.047
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.036
4.492
1.784
0.384
0.153
0.085
0.051
87033
ARI-1
9.471
5.727
2.645
1.043
0.667
0.516
0.421
ARI1.11
10.182
6.108
2.805
1.111
0.715
0.557
0.457
ARI1.25
10.977
6.534
2.984
1.187
0.77
0.603
0.497
ARI-2
14.151
8.231
3.696
1.491
0.988
0.787
0.659
ARI-5
20.342
11.533
5.084
2.082
1.413
1.147
0.978
ARI-10
25.026
14.028
6.134
2.529
1.735
1.42
1.221
ARI-20
29.711
16.522
7.183
2.976
2.057
1.694
1.465
ARI-50
35.905
19.818
8.57
3.567
2.482
2.056
1.787
ARI-100
40.591
22.311
9.619
4.014
2.804
2.33
2.031
ARI-500
51.471
28.097
12.054
5.051
3.552
2.966
2.599
ARI1000
56.157
30.589
13.103
5.498
3.874
3.24
2.843
ARI1000000
102.859
55.42
23.554
9.949
7.084
5.973
5.282
ARI1.11
9.396
6.318
3.208
1.253
0.756
0.556
0.431
ARI1.25
10.066
6.777
3.446
1.345
0.811
0.596
0.462
ARI-2
12.739
8.609
4.39
1.713
1.03
0.756
0.584
ARI-5
17.95
12.176
6.229
2.428
1.456
1.066
0.823
ARI-10
21.892
14.873
7.619
2.968
1.778
1.301
1.003
ARI-20
25.834
17.57
9.008
3.509
2.1
1.536
1.183
ARI-50
31.045
21.134
10.844
4.223
2.526
1.846
1.422
ARI-100
34.987
23.83
12.233
4.763
2.848
2.081
1.602
ARI-500
44.14
30.089
15.456
6.017
3.596
2.626
2.021
ARI1000
48.082
32.785
16.845
6.557
3.918
2.861
2.201
ARI1000000
87.368
59.648
30.679
11.938
7.127
5.2
3.998
87036
ARI-1
8.797
5.907
2.996
1.171
0.707
0.521
0.404
130
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.997
3.83
1.265
0.239
0.093
0.052
0.032
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.619
3.9
1.435
0.306
0.126
0.072
0.045
79082
ARI-1
9.187
5.283
2.261
0.815
0.499
0.377
0.302
ARI1.11
9.69
5.58
2.399
0.873
0.538
0.409
0.328
ARI1.25
10.251
5.912
2.554
0.938
0.582
0.444
0.358
ARI-2
12.488
7.237
3.169
1.197
0.757
0.584
0.476
ARI-5
16.845
9.816
4.365
1.701
1.098
0.859
0.708
ARI-10
20.139
11.765
5.269
2.082
1.356
1.067
0.884
ARI-20
23.431
13.715
6.173
2.462
1.614
1.275
1.06
ARI-50
27.782
16.291
7.366
2.965
1.955
1.55
1.293
ARI-100
31.074
18.239
8.269
3.346
2.213
1.759
1.47
ARI-500
38.715
22.763
10.365
4.229
2.812
2.243
1.88
ARI1000
42.005
24.712
11.268
4.609
3.071
2.452
2.057
ARI1000000
74.794
44.125
20.261
8.399
5.643
4.531
3.82
ARI1.11
10.776
6.44
2.928
1.137
0.723
0.558
0.454
ARI1.25
11.681
6.956
3.137
1.2
0.756
0.58
0.47
ARI-2
15.296
9.011
3.965
1.453
0.891
0.672
0.536
ARI-5
22.354
13.011
5.571
1.942
1.155
0.854
0.669
ARI-10
27.697
16.035
6.781
2.312
1.355
0.993
0.772
ARI-20
33.043
19.058
7.989
2.681
1.556
1.133
0.876
ARI-50
40.111
23.053
9.585
3.168
1.822
1.318
1.014
ARI-100
45.459
26.075
10.791
3.537
2.023
1.458
1.119
ARI-500
57.879
33.091
13.59
4.392
2.49
1.785
1.362
ARI1000
63.229
36.113
14.795
4.76
2.691
1.926
1.468
ARI1000000
116.55
66.22
26.798
8.429
4.697
3.33
2.518
79086
ARI-1
9.966
5.979
2.741
1.08
0.693
0.538
0.44
131
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.451
4.067
1.57
0.321
0.123
0.067
0.04
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
5.463
3.275
1.31
0.343
0.163
0.103
0.07
87097
ARI-1
9.732
6.161
3.061
1.322
0.885
0.704
0.587
ARI1.11
10.475
6.587
3.268
1.434
0.974
0.783
0.659
ARI1.25
11.306
7.063
3.501
1.559
1.074
0.872
0.741
ARI-2
14.618
8.961
4.425
2.054
1.473
1.232
1.075
ARI-5
21.072
12.66
6.225
3.015
2.254
1.943
1.744
ARI-10
25.953
15.458
7.585
3.74
2.845
2.485
2.258
ARI-20
30.833
18.256
8.945
4.465
3.438
3.029
2.774
ARI-50
37.285
21.954
10.742
5.423
4.221
3.75
3.46
ARI-100
42.165
24.751
12.102
6.147
4.814
4.296
3.98
ARI-500
53.495
31.246
15.258
7.827
6.191
5.565
5.19
ARI1000
58.375
34.043
16.617
8.551
6.784
6.112
5.712
ARI1000000
107.005
61.919
30.161
15.761
12.696
11.568
10.923
ARI1.11
8.34
5.177
2.612
1.248
0.909
0.768
0.675
ARI1.25
9.012
5.587
2.804
1.324
0.957
0.803
0.703
ARI-2
11.697
7.22
3.567
1.625
1.145
0.946
0.816
ARI-5
16.942
10.4
5.044
2.208
1.515
1.229
1.044
ARI-10
20.914
12.804
6.158
2.648
1.795
1.445
1.219
ARI-20
24.889
15.207
7.269
3.087
2.076
1.662
1.395
ARI-50
30.145
18.382
8.737
3.667
2.447
1.949
1.628
ARI-100
34.123
20.783
9.846
4.105
2.727
2.166
1.805
ARI-500
43.36
26.359
12.421
5.123
3.379
2.671
2.217
ARI1000
47.339
28.76
13.529
5.561
3.66
2.889
2.395
ARI1000000
87.004
52.684
24.567
9.925
6.458
5.06
4.169
87104
ARI-1
7.739
4.81
2.44
1.18
0.867
0.736
0.651
132
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.057
3.962
1.293
0.317
0.158
0.105
0.076
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.717
3.796
1.336
0.28
0.116
0.067
0.042
87105
ARI-1
10.294
5.287
2.187
0.972
0.735
0.649
0.6
ARI1.11
11.274
5.727
2.34
1.033
0.782
0.692
0.641
ARI1.25
12.374
6.218
2.51
1.102
0.835
0.74
0.686
ARI-2
16.792
8.17
3.182
1.372
1.042
0.928
0.866
ARI-5
25.485
11.953
4.473
1.891
1.442
1.292
1.214
ARI-10
32.097
14.805
5.44
2.281
1.741
1.565
1.475
ARI-20
38.726
17.654
6.403
2.668
2.04
1.837
1.736
ARI-50
47.507
21.415
7.673
3.179
2.433
2.196
2.079
ARI-100
54.157
24.258
8.632
3.565
2.731
2.466
2.339
ARI-500
69.617
30.855
10.854
4.459
3.42
3.095
2.941
ARI1000
76.28
33.695
11.81
4.844
3.716
3.365
3.2
ARI1000000
142.752
61.982
21.323
8.672
6.667
6.056
5.781
ARI1.11
9.127
5.581
2.579
0.987
0.614
0.466
0.373
ARI1.25
9.612
5.947
2.779
1.064
0.658
0.497
0.396
ARI-2
11.536
7.402
3.577
1.371
0.834
0.62
0.486
ARI-5
15.261
10.218
5.138
1.971
1.175
0.858
0.661
ARI-10
18.069
12.341
6.32
2.425
1.433
1.038
0.793
ARI-20
20.871
14.459
7.504
2.881
1.69
1.217
0.924
ARI-50
24.571
17.256
9.07
3.483
2.031
1.454
1.097
ARI-100
27.368
19.371
10.255
3.938
2.288
1.632
1.228
ARI-500
33.858
24.276
13.007
4.997
2.885
2.047
1.532
ARI1000
36.652
26.388
14.192
5.453
3.142
2.226
1.663
ARI1000000
64.48
47.419
26.011
9.998
5.705
4.004
2.965
87133
ARI-1
8.692
5.252
2.401
0.918
0.574
0.438
0.352
133
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.522
4.012
1.452
0.389
0.198
0.133
0.096
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.61
5.337
2.032
0.418
0.163
0.089
0.054
87153
ARI-1
12.624
7.509
3.595
1.652
1.193
1.006
0.885
ARI1.11
13.489
8.095
3.889
1.763
1.255
1.047
0.912
ARI1.25
14.456
8.749
4.218
1.887
1.325
1.094
0.944
ARI-2
18.317
11.361
5.531
2.381
1.605
1.286
1.079
ARI-5
25.841
16.454
8.093
3.339
2.154
1.669
1.358
ARI-10
31.532
20.308
10.031
4.062
2.569
1.962
1.576
ARI-20
37.223
24.163
11.969
4.784
2.985
2.257
1.795
ARI-50
44.745
29.258
14.532
5.737
3.536
2.648
2.088
ARI-100
50.435
33.113
16.471
6.458
3.952
2.944
2.31
ARI-500
63.647
42.065
20.973
8.132
4.919
3.633
2.829
ARI1000
69.337
45.92
22.912
8.853
5.336
3.93
3.052
ARI1000000
126.038
84.344
42.239
16.032
9.487
6.894
5.287
ARI1.11
13.321
8.4
3.992
1.526
0.934
0.699
0.552
ARI1.25
14.001
8.804
4.166
1.584
0.968
0.723
0.57
ARI-2
16.701
10.411
4.86
1.817
1.101
0.819
0.643
ARI-5
21.933
13.531
6.21
2.271
1.36
1.005
0.785
ARI-10
25.873
15.884
7.23
2.614
1.556
1.145
0.892
ARI-20
29.804
18.235
8.25
2.957
1.752
1.285
0.999
ARI-50
34.992
21.339
9.597
3.41
2.011
1.471
1.14
ARI-100
38.913
23.685
10.616
3.753
2.206
1.611
1.247
ARI-500
48.006
29.13
12.981
4.549
2.661
1.937
1.495
ARI1000
51.92
31.473
14
4.892
2.856
2.077
1.601
ARI1000000
90.882
54.817
24.146
8.306
4.806
3.475
2.666
88023
ARI-1
12.712
8.038
3.836
1.474
0.905
0.678
0.536
134
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.85
4.264
1.607
0.317
0.119
0.064
0.038
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.378
4.574
1.573
0.338
0.145
0.086
0.056
88029
ARI-1
8.743
5.664
2.72
0.996
0.583
0.422
0.323
ARI1.11
9.594
6.198
2.955
1.066
0.618
0.444
0.338
ARI1.25
10.544
6.795
3.216
1.143
0.656
0.469
0.355
ARI-2
14.334
9.175
4.257
1.452
0.811
0.569
0.424
ARI-5
21.717
13.809
6.277
2.051
1.113
0.766
0.561
ARI-10
27.3
17.312
7.803
2.504
1.343
0.917
0.667
ARI-20
32.883
20.815
9.329
2.957
1.572
1.068
0.773
ARI-50
40.263
25.446
11.344
3.555
1.876
1.267
0.913
ARI-100
45.845
28.948
12.868
4.007
2.106
1.419
1.02
ARI-500
58.806
37.081
16.407
5.057
2.639
1.77
1.267
ARI1000
64.388
40.583
17.93
5.51
2.869
1.921
1.374
ARI1000000
120.014
75.486
33.113
10.016
5.16
3.431
2.439
ARI1.11
11.838
6.728
2.913
1.122
0.726
0.571
0.473
ARI1.25
12.854
7.281
3.129
1.19
0.765
0.599
0.494
ARI-2
16.911
9.484
3.991
1.464
0.921
0.71
0.579
ARI-5
24.833
13.777
5.664
1.997
1.225
0.93
0.748
ARI-10
30.832
17.023
6.927
2.399
1.456
1.098
0.878
ARI-20
36.833
20.269
8.189
2.8
1.687
1.265
1.007
ARI-50
44.768
24.559
9.855
3.331
1.992
1.487
1.18
ARI-100
50.772
27.804
11.115
3.732
2.223
1.656
1.31
ARI-500
64.715
35.339
14.04
4.664
2.759
2.046
1.613
ARI1000
70.72
38.584
15.299
5.065
2.99
2.215
1.744
ARI1000000
130.579
70.919
27.844
9.06
5.293
3.894
3.049
88037
ARI-1
10.93
6.234
2.719
1.06
0.692
0.546
0.454
135
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.014
5.208
1.964
0.344
0.116
0.058
0.031
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.93
4.557
1.721
0.426
0.197
0.123
0.083
88049
ARI-1
11.887
7.564
3.443
1.127
0.612
0.422
0.309
ARI1.11
12.862
8.063
3.608
1.17
0.636
0.439
0.322
ARI1.25
13.956
8.619
3.793
1.219
0.663
0.458
0.337
ARI-2
18.343
10.827
4.526
1.412
0.768
0.533
0.395
ARI-5
26.969
15.101
5.943
1.79
0.973
0.679
0.506
ARI-10
33.531
18.321
7.01
2.075
1.127
0.788
0.59
ARI-20
40.113
21.534
8.073
2.361
1.281
0.897
0.672
ARI-50
48.833
25.776
9.476
2.737
1.484
1.041
0.781
ARI-100
55.439
28.981
10.535
3.022
1.638
1.149
0.864
ARI-500
70.8
36.416
12.991
3.683
1.995
1.401
1.055
ARI1000
77.423
39.616
14.048
3.968
2.148
1.509
1.137
ARI1000000
143.513
71.48
24.563
6.803
3.677
2.587
1.955
ARI1.11
11.375
6.87
3.276
1.426
0.982
0.8
0.683
ARI1.25
12.025
7.264
3.452
1.486
1.015
0.822
0.698
ARI-2
14.622
8.836
4.151
1.724
1.146
0.911
0.761
ARI-5
19.688
11.896
5.503
2.183
1.403
1.09
0.892
ARI-10
23.523
14.209
6.522
2.529
1.598
1.227
0.994
ARI-20
27.359
16.521
7.538
2.874
1.794
1.366
1.097
ARI-50
32.431
19.576
8.879
3.329
2.052
1.55
1.236
ARI-100
36.269
21.887
9.892
3.672
2.248
1.689
1.341
ARI-500
45.181
27.251
12.243
4.469
2.703
2.014
1.588
ARI1000
49.02
29.561
13.254
4.813
2.899
2.155
1.694
ARI1000000
87.285
52.577
23.324
8.228
4.853
3.556
2.761
88153
ARI-1
10.793
6.517
3.119
1.372
0.953
0.781
0.669
136
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.757
3.942
1.37
0.253
0.094
0.05
0.029
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
5.957
3.891
1.503
0.279
0.098
0.05
0.028
89016
ARI-1
8.402
5.147
2.282
0.767
0.433
0.307
0.231
ARI1.11
9.052
5.528
2.45
0.829
0.471
0.336
0.254
ARI1.25
9.778
5.953
2.637
0.899
0.514
0.368
0.28
ARI-2
12.672
7.649
3.383
1.176
0.685
0.498
0.383
ARI-5
18.31
10.952
4.836
1.715
1.019
0.751
0.586
ARI-10
22.573
13.45
5.935
2.122
1.271
0.943
0.74
ARI-20
26.835
15.948
7.033
2.53
1.524
1.135
0.894
ARI-50
32.468
19.249
8.485
3.068
1.858
1.388
1.098
ARI-100
36.73
21.746
9.583
3.476
2.111
1.58
1.252
ARI-500
46.624
27.544
12.133
4.421
2.697
2.027
1.61
ARI1000
50.884
30.04
13.23
4.829
2.95
2.219
1.765
ARI1000000
93.346
54.921
24.172
8.887
5.468
4.134
3.304
ARI1.11
8.459
5.899
2.954
0.999
0.532
0.358
0.256
ARI1.25
8.99
6.302
3.174
1.077
0.574
0.386
0.276
ARI-2
11.111
7.909
4.054
1.39
0.74
0.497
0.354
ARI-5
15.249
11.036
5.764
1.998
1.064
0.713
0.507
ARI-10
18.38
13.4
7.057
2.458
1.309
0.877
0.623
ARI-20
21.513
15.762
8.349
2.918
1.554
1.041
0.74
ARI-50
25.654
18.883
10.056
3.526
1.879
1.258
0.893
ARI-100
28.787
21.244
11.347
3.986
2.124
1.422
1.009
ARI-500
36.062
26.725
14.344
5.054
2.693
1.803
1.279
ARI1000
39.195
29.086
15.635
5.514
2.939
1.967
1.395
ARI1000000
70.421
52.606
28.496
10.099
5.383
3.601
2.552
89025
ARI-1
7.984
5.538
2.756
0.929
0.495
0.333
0.239
137
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.479
5.573
2.059
0.329
0.103
0.049
0.025
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
5.214
3.568
1.466
0.288
0.104
0.053
0.03
89094
ARI-1
11.857
7.74
3.69
1.294
0.73
0.515
0.385
ARI1.11
12.304
7.972
3.801
1.362
0.784
0.561
0.425
ARI1.25
12.802
8.233
3.927
1.437
0.844
0.612
0.47
ARI-2
14.776
9.276
4.427
1.735
1.085
0.824
0.659
ARI-5
18.59
11.321
5.405
2.307
1.56
1.254
1.057
ARI-10
21.458
12.871
6.145
2.736
1.923
1.589
1.374
ARI-20
24.317
14.422
6.885
3.162
2.287
1.93
1.701
ARI-50
28.089
16.472
7.862
3.723
2.77
2.386
2.142
ARI-100
30.938
18.024
8.601
4.147
3.136
2.734
2.482
ARI-500
37.543
21.626
10.317
5.128
3.988
3.549
3.283
ARI1000
40.384
23.177
11.056
5.55
4.356
3.902
3.631
ARI1000000
68.662
38.633
18.414
9.743
8.03
7.453
7.165
ARI1.11
8.369
5.673
2.832
1.027
0.584
0.412
0.308
ARI1.25
8.981
6.033
2.98
1.075
0.611
0.432
0.323
ARI-2
11.419
7.465
3.574
1.265
0.719
0.509
0.382
ARI-5
16.165
10.249
4.734
1.636
0.927
0.658
0.495
ARI-10
19.751
12.353
5.613
1.918
1.084
0.769
0.579
ARI-20
23.336
14.456
6.492
2.199
1.241
0.88
0.663
ARI-50
28.072
17.234
7.655
2.571
1.449
1.027
0.773
ARI-100
31.654
19.335
8.534
2.852
1.605
1.137
0.857
ARI-500
39.969
24.213
10.577
3.506
1.969
1.394
1.05
ARI1000
43.549
26.314
11.457
3.787
2.125
1.505
1.133
ARI1000000
79.223
47.244
20.227
6.592
3.684
2.604
1.959
90058
ARI-1
7.821
5.351
2.699
0.984
0.56
0.395
0.294
138
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
4.897
3.823
1.745
0.327
0.106
0.05
0.026
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.521
3.718
1.165
0.249
0.112
0.069
0.047
80006
ARI-1
6.659
5.278
2.989
1.027
0.52
0.334
0.227
ARI1.11
7.261
5.654
3.143
1.073
0.545
0.352
0.241
ARI1.25
7.934
6.072
3.315
1.125
0.574
0.371
0.255
ARI-2
10.627
7.733
4.004
1.331
0.686
0.449
0.313
ARI-5
15.894
10.949
5.348
1.738
0.903
0.599
0.423
ARI-10
19.886
13.372
6.365
2.046
1.068
0.711
0.505
ARI-20
23.881
15.792
7.383
2.355
1.232
0.823
0.587
ARI-50
29.167
18.987
8.728
2.763
1.448
0.971
0.694
ARI-100
33.166
21.403
9.746
3.073
1.612
1.083
0.776
ARI-500
42.457
27.008
12.109
3.791
1.993
1.342
0.964
ARI1000
46.459
29.421
13.127
4.101
2.157
1.454
1.045
ARI1000000
86.358
53.46
23.27
7.186
3.789
2.564
1.852
ARI1.11
10.057
5.381
2.195
0.835
0.552
0.443
0.374
ARI1.25
10.587
5.755
2.377
0.899
0.587
0.466
0.39
ARI-2
12.702
7.239
3.105
1.153
0.726
0.559
0.455
ARI-5
16.827
10.113
4.525
1.649
0.997
0.744
0.587
ARI-10
19.947
12.278
5.601
2.024
1.203
0.884
0.688
ARI-20
23.067
14.439
6.677
2.399
1.408
1.025
0.789
ARI-50
27.191
17.293
8.101
2.896
1.68
1.211
0.924
ARI-100
30.311
19.45
9.178
3.271
1.886
1.351
1.026
ARI-500
37.554
24.456
11.679
4.143
2.363
1.679
1.263
ARI1000
40.673
26.611
12.756
4.519
2.568
1.82
1.366
ARI1000000
71.758
48.074
23.496
8.262
4.617
3.225
2.388
80102
ARI-1
9.584
5.046
2.032
0.778
0.521
0.422
0.361
139
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.967
4.008
1.408
0.279
0.11
0.061
0.037
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.666
3.719
1.255
0.24
0.093
0.052
0.031
80109
ARI-1
10.547
6.372
2.846
1.016
0.604
0.445
0.347
ARI1.11
11.306
6.845
3.047
1.072
0.629
0.459
0.355
ARI1.25
12.154
7.373
3.272
1.133
0.657
0.475
0.365
ARI-2
15.528
9.48
4.167
1.378
0.77
0.542
0.407
ARI-5
22.086
13.586
5.909
1.852
0.991
0.677
0.494
ARI-10
27.038
16.692
7.224
2.21
1.158
0.781
0.562
ARI-20
31.987
19.799
8.54
2.567
1.327
0.885
0.632
ARI-50
38.525
23.905
10.278
3.039
1.549
1.024
0.725
ARI-100
43.469
27.012
11.593
3.395
1.718
1.13
0.795
ARI-500
54.945
34.227
14.646
4.223
2.109
1.374
0.96
ARI1000
59.886
37.334
15.961
4.579
2.278
1.48
1.031
ARI1000000
109.121
68.303
29.062
8.128
3.96
2.535
1.742
ARI1.11
9.502
5.541
2.413
0.88
0.541
0.409
0.327
ARI1.25
10.192
5.944
2.591
0.948
0.584
0.442
0.354
ARI-2
12.947
7.55
3.3
1.219
0.756
0.575
0.462
ARI-5
18.32
10.679
4.682
1.746
1.091
0.834
0.674
ARI-10
22.386
13.046
5.727
2.145
1.344
1.03
0.834
ARI-20
26.452
15.413
6.771
2.543
1.597
1.226
0.994
ARI-50
31.827
18.541
8.151
3.07
1.932
1.485
1.205
ARI-100
35.894
20.908
9.195
3.468
2.185
1.681
1.365
ARI-500
45.336
26.402
11.619
4.393
2.773
2.137
1.737
ARI1000
49.403
28.769
12.663
4.791
3.027
2.333
1.897
ARI1000000
89.931
52.351
23.064
8.76
5.551
4.287
3.494
80110
ARI-1
8.885
5.181
2.253
0.82
0.502
0.379
0.303
140
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.061
4.281
1.596
0.328
0.129
0.071
0.043
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.635
4.374
1.548
0.318
0.129
0.073
0.045
81003
ARI-1
8.706
5.512
2.634
1.012
0.621
0.465
0.367
ARI1.11
9.322
5.886
2.809
1.083
0.667
0.501
0.397
ARI1.25
10.011
6.304
3.005
1.163
0.719
0.542
0.43
ARI-2
12.762
7.968
3.781
1.479
0.926
0.704
0.564
ARI-5
18.131
11.202
5.286
2.093
1.328
1.021
0.826
ARI-10
22.196
13.645
6.421
2.556
1.633
1.262
1.026
ARI-20
26.261
16.087
7.554
3.019
1.938
1.502
1.225
ARI-50
31.637
19.313
9.051
3.631
2.341
1.821
1.49
ARI-100
35.704
21.752
10.183
4.093
2.646
2.062
1.69
ARI-500
45.148
27.416
12.81
5.167
3.353
2.622
2.156
ARI1000
49.215
29.855
13.942
5.629
3.658
2.863
2.356
ARI1000000
89.755
54.154
25.209
10.234
6.696
5.268
4.356
ARI1.11
11.676
6.725
2.969
1.169
0.764
0.604
0.503
ARI1.25
12.55
7.155
3.137
1.245
0.823
0.656
0.55
ARI-2
16.04
8.861
3.805
1.547
1.058
0.866
0.744
ARI-5
22.848
12.167
5.098
2.132
1.516
1.278
1.131
ARI-10
28.001
14.659
6.073
2.573
1.862
1.592
1.428
ARI-20
33.155
17.147
7.047
3.013
2.208
1.907
1.726
ARI-50
39.969
20.433
8.332
3.594
2.666
2.325
2.123
ARI-100
45.125
22.917
9.303
4.033
3.013
2.641
2.425
ARI-500
57.098
28.681
11.558
5.052
3.817
3.376
3.126
ARI1000
62.255
31.163
12.529
5.49
4.163
3.693
3.429
ARI1000000
113.654
55.881
22.197
9.858
7.617
6.856
6.454
81013
ARI-1
10.893
6.34
2.818
1.1
0.712
0.558
0.46
141
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.072
4.476
1.489
0.278
0.106
0.058
0.035
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.978
4.435
1.499
0.291
0.114
0.064
0.039
81026
ARI-1
9.887
5.759
2.45
0.836
0.489
0.357
0.277
ARI1.11
10.537
6.155
2.618
0.886
0.515
0.374
0.289
ARI1.25
11.263
6.597
2.806
0.942
0.543
0.393
0.302
ARI-2
14.158
8.363
3.556
1.166
0.658
0.468
0.355
ARI-5
19.795
11.805
5.019
1.602
0.882
0.617
0.46
ARI-10
24.057
14.408
6.125
1.931
1.051
0.729
0.539
ARI-20
28.318
17.011
7.232
2.261
1.221
0.842
0.62
ARI-50
33.95
20.453
8.694
2.696
1.445
0.991
0.726
ARI-100
38.209
23.056
9.8
3.025
1.615
1.104
0.807
ARI-500
48.099
29.1
12.369
3.789
2.009
1.367
0.994
ARI1000
52.358
31.704
13.475
4.118
2.178
1.48
1.075
ARI1000000
94.799
57.647
24.5
7.398
3.87
2.609
1.88
ARI1.11
10.792
6.272
2.703
0.965
0.584
0.438
0.347
ARI1.25
11.444
6.643
2.858
1.019
0.617
0.462
0.366
ARI-2
14.045
8.125
3.479
1.234
0.745
0.557
0.441
ARI-5
19.114
11.014
4.689
1.652
0.995
0.743
0.588
ARI-10
22.949
13.199
5.604
1.969
1.184
0.884
0.699
ARI-20
26.783
15.384
6.519
2.286
1.373
1.024
0.81
ARI-50
31.852
18.273
7.729
2.704
1.624
1.21
0.956
ARI-100
35.686
20.459
8.645
3.021
1.813
1.351
1.067
ARI-500
44.589
25.533
10.77
3.756
2.252
1.677
1.324
ARI1000
48.423
27.718
11.686
4.073
2.441
1.818
1.435
ARI1000000
86.633
49.497
20.809
7.23
4.326
3.219
2.54
81038
ARI-1
10.209
5.939
2.564
0.917
0.556
0.416
0.33
142
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
7.8
4.142
1.372
0.285
0.12
0.071
0.046
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.382
4.489
1.467
0.286
0.115
0.065
0.041
81049
ARI-1
10.067
5.615
2.366
0.885
0.566
0.442
0.364
ARI1.11
10.924
6.086
2.555
0.948
0.603
0.469
0.386
ARI1.25
11.883
6.613
2.766
1.019
0.645
0.5
0.41
ARI-2
15.709
8.714
3.609
1.3
0.811
0.623
0.506
ARI-5
23.168
12.811
5.249
1.848
1.135
0.862
0.694
ARI-10
28.811
15.91
6.49
2.263
1.38
1.044
0.837
ARI-20
34.454
19.01
7.73
2.677
1.625
1.226
0.98
ARI-50
41.914
23.107
9.37
3.225
1.95
1.466
1.17
ARI-100
47.557
26.206
10.61
3.639
2.195
1.648
1.313
ARI-500
60.66
33.402
13.49
4.601
2.765
2.07
1.646
ARI1000
66.303
36.502
14.73
5.015
3.01
2.252
1.789
ARI1000000
122.542
67.389
27.089
9.142
5.455
4.066
3.219
ARI1.11
11.208
6.282
2.618
0.934
0.576
0.438
0.353
ARI1.25
11.961
6.694
2.785
0.993
0.612
0.466
0.375
ARI-2
14.966
8.338
3.453
1.229
0.758
0.577
0.465
ARI-5
20.824
11.544
4.755
1.688
1.042
0.794
0.641
ARI-10
25.255
13.969
5.74
2.036
1.257
0.959
0.774
ARI-20
29.686
16.394
6.724
2.383
1.472
1.123
0.907
ARI-50
35.545
19.6
8.026
2.843
1.756
1.34
1.083
ARI-100
39.976
22.025
9.011
3.19
1.971
1.504
1.216
ARI-500
50.266
27.655
11.298
3.997
2.47
1.885
1.525
ARI1000
54.697
30.08
12.282
4.345
2.685
2.05
1.658
ARI1000000
98.861
54.246
22.097
7.809
4.826
3.686
2.983
81114
ARI-1
10.535
5.913
2.468
0.881
0.543
0.413
0.332
143
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
8.382
4.489
1.467
0.286
0.115
0.065
0.041
Station
Duration(hours)
1
2
6
24
48
72
100
ARI-0.1
6.859
4.128
1.458
0.259
0.092
0.048
0.027
81114
ARI-1
10.535
5.913
2.468
0.881
0.543
0.413
0.332
ARI1.11
11.208
6.282
2.618
0.934
0.576
0.438
0.353
ARI1.25
11.961
6.694
2.785
0.993
0.612
0.466
0.375
ARI-2
14.966
8.338
3.453
1.229
0.758
0.577
0.465
ARI-5
20.824
11.544
4.755
1.688
1.042
0.794
0.641
ARI-10
25.255
13.969
5.74
2.036
1.257
0.959
0.774
ARI-20
29.686
16.394
6.724
2.383
1.472
1.123
0.907
ARI-50
35.545
19.6
8.026
2.843
1.756
1.34
1.083
ARI-100
39.976
22.025
9.011
3.19
1.971
1.504
1.216
ARI-500
50.266
27.655
11.298
3.997
2.47
1.885
1.525
ARI1000
54.697
30.08
12.282
4.345
2.685
2.05
1.658
ARI1000000
98.861
54.246
22.097
7.809
4.826
3.686
2.983
ARI1.11
9.867
6.064
2.669
0.87
0.479
0.334
0.248
ARI1.25
10.877
6.608
2.873
0.931
0.514
0.359
0.268
ARI-2
14.914
8.771
3.683
1.177
0.653
0.459
0.345
ARI-5
22.806
12.969
5.255
1.655
0.923
0.655
0.495
ARI-10
28.786
16.137
6.441
2.017
1.128
0.802
0.609
ARI-20
34.768
19.304
7.626
2.379
1.333
0.95
0.723
ARI-50
42.68
23.487
9.193
2.857
1.603
1.146
0.873
ARI-100
48.667
26.65
10.377
3.219
1.808
1.293
0.987
ARI-500
62.57
33.992
13.126
4.06
2.284
1.637
1.252
ARI1000
68.558
37.154
14.31
4.422
2.488
1.785
1.366
ARI1000000
128.248
68.655
26.105
8.028
4.529
3.258
2.502
81115
ARI-1
8.965
5.576
2.486
0.815
0.448
0.312
0.231
144
APPENDIX E
INTENSITY- FREQUENCY-DURATION
CURVES (ARI in years)
ARI-1
Station 77087
ARI-1.11
1000
Station 82016
ARI-1
1000
ARI-1.11
ARI-1.25
100
ARI-2
ARI-5
ARI-5
ARI-10
10
ARI-1.25
ARI-2
Ic (mm/h)
Ic (mm/h)
100
ARI-20
ARI-10
10
ARI-20
ARI-50
1
ARI-100
0.1
ARI-50
1
ARI-100
ARI-500
1
10
100
dc(h)
ARI-1000
ARI-500
0.1
1
ARI-1000000
Station 82011
ARI-1
1000
dc (h)
10
100
Ic (mm/h)
ARI-2
ARI-1.11
ARI-5
ARI-10
10
ARI-20
ARI-1.25
100
Ic (mm/h)
100
ARI-2
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-50
1
ARI-100
ARI-100
1
ARI-1000
1
dc (h)
10
100
Station 82039
ARI-500
0.1
ARI-500
0.1
10
dc (h)
ARI-1000000
ARI-1.11
100
ARI-1
ARI-1.11
1000
ARI-1.25
ARI-1.25
100
ARI-2
ARI-10
10
ARI-20
ARI-2
100
ARI-5
Ic (mm/h)
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
ARI-50
1
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
1
10
dc (h)
100
ARI-1000
0.1
ARI-1000
1
ARI-1000000
ARI-1.11
dc (h)
10
100
Station 82107
ARI-1
Station 79046
1000
ARI-1.11
ARI-1.25
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
ARI-5
Ic (mm/h)
Ic (mm/h)
ARI-2
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
1
10
dc (h)
100
ARI-1000
ARI-1000000
ARI-1000000
ARI-1
1000
ARI-1.25
100
ARI-1000
ARI-1000000
Station 82076
ARI-1
1000
ARI-1000000
ARI-1
Station 82042
1000
ARI-1.11
ARI-1.25
ARI-1000
0.1
1
dc (h)
10
100
ARI-1000
ARI-1000000
145
ARI-1.11
ARI-1
Station 83067
ARI-1
Station 82121
1000
ARI-1.11
1000
ARI-1.25
ARI-1.25
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
Ic (mm/h)
ARI-2
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
ARI-500
0.1
ARI-1000
1
dc (h)
10
100
1
10
dc (h)
ARI-1000000
ARI-1.11
ARI-1
Station 83074
ARI-1.11
1000
ARI-1.25
ARI-1.25
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
ARI-5
Ic (mm/h)
ARI-2
100
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
1
dc (h)
10
100
ARI-500
0.1
ARI-1000
1
dc (h) 10
100
ARI-1000000
Station 83031
Station 84005
ARI-1
ARI-1.25
ARI-2
ARI-10
10
ARI-20
ARI-5
ARI-10
10
ARI-20
ARI-50
ARI-50
1
ARI-2
100
Ic (mm/h)
Ic (mm/h)
ARI-5
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
1
10
100
0.1
ARI-1000
ARI-1.11
1000
dc (h)
10
100
ARI-1.11
1000
ARI-1.25
ARI-1.25
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
Ic (mm/h)
ARI-2
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
1
dc (h)
10
100
ARI-1000
ARI-1000000
ARI-1000000
ARI-1
Station 84015
ARI-1
Station 83033
ARI-1000
1
ARI-1000000
dc (h)
ARI-1000000
ARI-1.11
ARI-1.25
100
ARI-1000
ARI-1
1000
ARI-1.11
1000
ARI-1000
ARI-1000000
ARI-1
Station 83025
1000
100
ARI-500
0.1
ARI-1000
1
dc (h)
10
100
ARI-1000000
146
ARI-1
Station 84078
ARI-1.11
1000
ARI-0.1
Station 79079
ARI-1
1000
ARI-1.11
ARI-1.25
Ic (mm/h)
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-1.25
100
Ic (mm/h)
ARI-2
ARI-2
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
ARI-500
0.1
ARI-1000
1
dc (h)
10
100
ARI-1000
1
ARI-1000000
ARI-1
Station 84112
ARI-1.11
1000
10
dc (h)
100
Station 79052
ARI-1
ARI-1.11
1000
ARI-1.25
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-1.25
ARI-2
100
ARI-5
Ic (mm/h)
ARI-2
100
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
ARI-1000
1
dc (h)
10
100
ARI-500
ARI-1000
0.1
1
ARI-1000000
100
ARI-1.11
ARI-2
100
ARI-1.11
1000
ARI-1.25
ARI-5
ARI-20
ARI-50
1
ARI-100
Ic (mm/h)
10
ARI-2
100
ARI-10
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-500
ARI-100
ARI-1000
0.1
dc (h)
10
100
ARI-500
ARI-1000000
0.1
1
dc (h)
10
100
ARI-1.11
ARI-1
ARI-1.11
1000
ARI-1.25
ARI-1.25
ARI-2
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
Ic (mm/h)
Ic (mm/h)
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
ARI-1000
dc (h)
10
100
ARI-1000000
ARI-1000
ARI-1000000
Station 85034
ARI-1
Station 85000
1000
1
ARI-1000000
ARI-1
Station 85026
ARI-1.25
Ic (mm/h)
10
dc (h)
ARI-1
Station 84125
1000
1
ARI-1000000
ARI-500
0.1
1
10
dc (h)
100
ARI-1000
ARI-1000000
147
ARI-0.1
Station 85072
ARI-1
1000
ARI-1
Station 85176
ARI-1.11
1000
ARI-1.11
ARI-1.25
ARI-1.25
ARI-2
Ic (mm/h)
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
Ic (mm/h)
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
0.1
ARI-1000
1
dc (h)
10
100
1
ARI-1000000
ARI-1.11
1000
10
100
ARI-1.25
ARI-5
ARI-10
10
ARI-20
ARI-2
100
Ic (mm/h)
Ic (mm/h)
100
ARI-5
ARI-10
10
ARI-20
ARI-50
ARI-50
1
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
1
10
100
dc (h)
ARI-1000
0.1
ARI-1000
1
ARI-1000000
ARI-1.11
1000
dc (h)
10
100
ARI-1
1000
ARI-1.11
ARI-1.25
ARI-10
10
ARI-20
Ic (mm/h)
ARI-5
Ic (mm/h)
ARI-1.25
100
ARI-2
100
ARI-2
ARI-5
ARI-10
10
ARI-20
ARI-50
ARI-50
1
1
ARI-100
ARI-100
ARI-500
ARI-500
0.1
ARI-1000
0.1
1
dc (h)
10
100
ARI-1000
1
ARI-1000000
ARI-1.11
1000
dc (h)
10
100
ARI-1.11
1000
ARI-1.25
ARI-1.25
ARI-2
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-2
100
Ic (mm/h)
Ic (mm/h)
100
ARI-5
ARI-10
10
ARI-20
ARI-50
1
ARI-100
ARI-100
ARI-500
0.1
ARI-1000
1
dc (h)
10
100
ARI-1000000
ARI-1000000
ARI-1
Station 86038
ARI-1
Station 85170
ARI-1000000
ARI-0.1
Station 85256
ARI-1
Station 85106
ARI-1000000
ARI-1.11
1000
ARI-1.25
ARI-2
ARI-1000
ARI-1
Station 85237
ARI-1
Station 85103
dc (h)
ARI-500
0.1
ARI-1000
1
10
dc (h)
100
ARI-1000000
148
APPENDIX F
DERIVED FLOOD FREQUENCY CURVES OF THE
TWELVE TEST CATCHMENTS
DFFC for 222202
310
260
Q (m3/s)
210
160
110
60
10
1
10
100
ARI (years)
149
DFFC for 223202
210
Q (m3/s)
160
110
60
10
1
10
100
ARI (years)
DFFC-for 226204
260
Q (m3/s)
210
160
110
60
10
1
10
100
ARI (years)
150
DFFC for 226410
100
90
80
Q (m3/s)
70
60
50
40
30
20
10
1
10
100
ARI (years)
DFFC-for 227200
150
130
Q (m3/s)
110
90
70
50
30
10
1
10
100
ARI (years)
151
DFFC-for 229218
70
60
Q (m3/s)
50
40
30
20
10
1
10
100
ARI (years)
DFFC-for 230204
110
Q (m3/s)
90
70
50
30
10
1
10
100
ARI (years)
152
DFFC-for 234200
105
Q (m3/s)
85
65
45
25
5
1
10
100
ARI (years)
DFFC-for 235211
85
75
Q (m3/s)
65
55
45
35
25
15
5
1
10
100
ARI (years)
153
DFFC-for 237205
100
90
80
Q (m3/s)
70
60
50
40
30
20
10
0
1
10
100
ARI (years)
DFFC for 238229
100
90
80
Q (m3/s)
70
60
50
40
30
20
10
0
1
10
100
ARI (years)
154
APPENDIX G
A NUMERICAL EXAMPLE ILLUSTRATING THE
IDENTIFICATION OF A STORM-CORE
Consider the complete storm given in Figure G1. It has 5 hour duration. The stormcore from this complete storm is identified below.
18
16
14
I (mm/h)
12
10
8
6
4
2
0
1
2
3
4
5
Period
Figure G1. A plot of complete storm
Total rainfall = 1.5+5.2+8.9+15.4+1.2
= 32.2 mm.
Duration = 5 hour
Rainfall intensity = 6.44 mm/hour
From AUSIFD 2I5=12.8 mm/h
P
P
CHECK:
Ist Crieteria:
B
B
I D ≥ f1 × 2 I D
155
2nd Crieteria:
I d ≥ f ×2 I d
2
P
P
B
Here, f1 = 0.4 and f2 = 0.5
B
B
I D ≥ f1 × 2 I D
B
Ist Crieteria:
B
So, f1 x 2ID= 0.4 x 12.8 = 5.12 mm/hour
P
B
B
P
B
B
Where 6.44 > 5.12 Accept the complete storm.
Now find the storm-core, as shown in Table G1. Here 3 hour duration gives the
highest ratio, thus, the storm-core consists of 5.2mm, 8.9mm and 15.4mm rainfalls, as
plotted in Figure G2.
Table G1 Identification of storm-core
Storm-core duration
A
1
2
3
4
5
Storm-core intensity (mm/h)
B
15.40
12.15
9.83
7.75
6.44
2ID (ARR design rainfall), mm/h
C
39.40
24.40
18.40
15.00
12.80
Ratio
D (=B/C)
0.39
0.50
0.53
0.52
0.50
18
16
14
I (mm/h)
12
10
8
6
4
2
0
1
2
3
Period
Figure G2. Identified storm-core
156