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Flood Estimation TechniquesApplication of Monte Carlo Simulation Technique with URBS Model for Design Flood Estimation of Large Catchments
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3.0 Method
3.1 Overview
The Monte Carlo Simulation Technique as proposed by Rahman et al., (2001) comprises
three principal elements:
(a) a (deterministic) hydrologic modelling framework to simulate the flood formation;
(b) the key model variables (inputs and parameters) with their probability distributions and
dependencies; and
(c) a stochastic modelling framework to synthesise the derived flood distribution from the
model or input distributions. These elements are discussed below.
3.1.1 Hydrologic Modelling Framework
The proposed hydrologic model of the flood formation process involves the same
components as the models most commonly used with the current Design Event Approach: a
runoff generation function or loss model; a runoff transfer function or runoff-routing model, as
shown in Figure 3.1. Together with a design rainfall depth or intensity, these components are
commonly referred to as the rainfall-runoff process, and a model which encompasses these
components, a rainfall-runoff model.
3.1.1.1 Runoff Generation Function
A runoff generation function or loss model is needed to partition the gross rainfall input into
effective runoff (or rainfall excess) and loss. Most of the previous derived distribution studies
(e.g. Eagleson, 1972; Russell, Kenning & Sunnell, 1979) have used an empirical infiltration
equation (such as Horton’s equation) or a more physically based equation (such as the Philip
and Green Ampt infiltration equations) to estimate the rainfall excess.
In design practice, use of simplified, lumped conceptual loss models is preferred over the
mathematical equations because of their simplicity and ability to approximate catchment
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Figure 3.1: Design Event Approach
runoff behaviour. This is particularly true for design losses which is probabilistic in nature and
for which complicated theoretical models may not be required. On this basis, the initial-
continuing loss model has been adopted in the present study. In this model, it is assumed
Runoff Generation or Loss Model
Rainfall Excess Hyetograph
Runoff Routing Model
Surface Runoff Hydrograph
Rainfall Depth or Rainfall Intensity
AEP = 1 in Y
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that no runoff is generated from a rainfall event until the cumulative rainfall exceeds the initial
loss value; for the remainder of the event, loss is assumed to occur at a constant rate.
3.1.1.2 Transfer Function
A catchment response model is needed to convert rainfall excess hyetograph produced by
the loss model into a surface runoff hydrograph. The models commonly used in previous
Joint Probability Approach studies includes the Kinematic Wave Model (e.g. Eagleson,
1972), Geomorphologic Unit Hydrograph Model (e.g. Diaz-Granados, Valdes & Bras, 1984),
Unit Hydrograph Method (e.g. Beran, 1973; Muzik, 1993), Clark’s Model (Russell, Kenning &
Sunnell, 1979), and parallel linear storages (Blöschl and Sivapalan, 1997).
In Australian flood design practice, it is common to use a semi-distributed and non-linear type
of catchment routing model, referred to as a runoff-routing model. This type of model, being
semi distributed in nature, can account for the areal variation of rainfall and losses to some
extent, and consider the non-linearity of the catchment routing response. Examples of
models in this group in common use include RORB (Laurenson and Mein, 1988), WBNM
(Boyd et al., 1987), URBS (Carroll, 2001) and XP-RAFTS. All these models are comparable
for most applications, although they differ in their capability to use more detailed data if
available. In this respect, URBS is probably the most advanced. For this reason and its
flexibility URBS has been integrated with the Monte Carlo Simulation Technique (Rahman,
Carroll & Weinmann, 2002). The URBS model is discussed further in Section 3.3.2.
A comparison and discussion of catchment response models used with the Monte Carlo
Simulation Technique is contained in Section 3.3.
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3.1.2 Key Model Variables
The major factors affecting storm runoff production are rainfall duration, rainfall intensity,
temporal pattern and areal patterns of rainfall, and storm losses. Factors affecting
hydrograph formation are the catchment response characteristics embodied in the runoff-
routing model (model type, structure and parameters) and design baseflow. Ideally, all the
variables should be treated as random variables but, for practical reasons, application of a
smaller number of random variables would be preferable, if it did not result in a significant
loss of accuracy. Given the dominant role of rainfall and loss in the flood formation process
for Australian conditions, it might be expected that the incorporation of the probabilistic
nature of these variables would result in significant reduction of bias and uncertainties in
design flood estimates. Although continuing loss (CL) is an important variable in the rainfall-
runoff process, it has not been included as a random variable in this study because Rahman
et al. (2002a) did not include CL as a random variable but recommended it be considered an
option for further study. Additionally the main objective of this study was to extend the
method of Rahman et al. (2002a) to large catchments. Thus, four variables have been
considered here for probabilistic representation: rainfall duration, rainfall intensity, rainfall
temporal pattern and initial loss. In contrast to this, the currently used Design Event
Approach treats only rainfall intensity for a given duration as a probabilistic variable. The
input variables or parameters that need to be considered in a probabilistic fashion are further
discussed in Section 3.2.
The areal distribution of rainfall over the catchment is assumed to be uniform, and the
average catchment rainfall intensity is obtained from the point rainfall intensity, using an areal
reduction factor (e.g. Siriwardena and Weinmann, 1996). The continuing loss is assumed to
be a constant; likewise, a constant baseflow is assumed, determined as the average
baseflow at the start of surface runoff generation in observed events. A single set of
parameter values for the runoff-routing model is used here; the calibration procedure allows
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the determination of a set of model parameters for a given catchment, which can be applied
with reasonable confidence.
Thus, the adopted Monte Carlo Simulation Technique considers probabilistic modelling
related to the runoff production only; the hydrograph formation part (e.g. runoff-routing)
remains entirely deterministic. It has been left to future research efforts to determine if the
probabilistic treatment of any of the above variables, kept constant in the simulation, might
further improve the flood estimates.
3.1.3 Stochastic Modelling Framework
The basic idea underlying the proposed new modelling is that the distribution of the flood
outputs can be directly determined by simulating the possible combinations of hydrologic
model inputs and parameters values. Here, we adopted a Monte Carlo simulation approach
for its relative simplicity and flexibility. The method is described below from Rahman et al.
(2001).
For each run of the combined loss and runoff-routing model, a specific set of input or
parameter values is selected by randomly drawing a value from each of the respective
distributions (for probability distributed variables) and by choosing a representative value (for
other variables). Any significant correlation between the input variables is allowed for by
using conditional probability distributions. For example, the strong correlation between
rainfall duration and intensity is allowed for by first drawing a value of duration and then a
value of intensity from the conditional distribution of rainfall intensity for that duration interval.
The results of the run (e.g. flood peaks at the catchment outlet) are then stored and the
Monte Carlo simulation process is repeated a sufficiently large number of times to fully reflect
the range of variation of input or parameter values in the generated output. The output values
of a selected flood characteristic (e.g. flood peak) can then be subjected to a frequency
analysis to determine the derived flood frequency curve for the AEP range of interest.
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The adopted Monte Carlo Simulation Technique is illustrated in Figure 3.2, and the steps
involved in the modelling process are detailed below:
1. Draw a random value of duration Di, from the identified marginal distribution of rainfall
duration.
2. Given the duration Di, draw a random value of rainfall intensity Ii(Di) from the conditional
distribution of rainfall intensity conditioned on Di.
3. Given the duration Di, draw a random temporal pattern TPi(Di) from the conditional
distribution of temporal pattern. The temporal patterns were conditioned on rainfall
duration in such a way that temporal patterns for Di in the range of 4 to 12 hours were
considered to be forming a homogeneous group, and temporal patterns over 12 hours
duration to be forming another homogenous group. This was based on the
recommendations of Rahman et al. (2002a). In the simulation, sampling of temporal
patterns was done from either of these two homogeneous groups, depending on the
generated rainfall duration.
4. Given the duration Di, draw a random value of initial loss ILi(Di) from the conditional
distribution of initial loss. The IL distribution was conditioned on D similar to Rahman et
al. (2002d) who found that initial loss for storm-core should be conditioned on storm-core
duration.
5. Run the randomly selected variables Di, Ii, TPi, and ILi (with a constant continuing loss)
through the loss model and runoff-routing models to simulate a flood hydrograph.
6. Add the baseflow to the simulated flood hydrograph and the note the flood peak Qi.
7. Repeat all the above steps N times (N in the order of 10,000 – 20,000).
8. Use the N simulated flood peaks to determine the derived flood frequency curve using
rank-order statistics (non-parametric method).
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Figure 3.2: Monte Carlo Simulation Technique
Randomly select storm durations, AEP, temporal pattern and initial loss from the conditional distributions
Initial lossILi(Di)
2.5 10
5 25
5 10
5
Storm durationsDi
1h
2h
3h
6h
12h
6h
Runoff Routing Model
Surface hydrographAEP = 1 in Y
Rainfall excess hyetograph
Derived Flood Frequency
Curve
Repeat N times
(10,000 - 20,000)
Temporal patternsTPi(Di)
Rainfall intensityIi(Di)
IF D Curve
Duration
10 mm/hr
(AEP = 1 in Y)Random
1 in 50
1 in 10
1 in 100
1 in 2
1 in 50
1 in 5
1 in 2
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3.2 Distributions of the Key Variables
3.2.1 Rainfall Event Definition
The Design Event Approach (I.E. Aust, 1987) treats rainfall intensity as a random variable,
and uses a number of trial durations with fixed temporal patterns to obtain design flood
estimates. The storm burst durations employed in this method are specified, predetermined
rather than random. In contrast, the proposed Joint Probability Approach in the form of the
Monte Carlo Simulation Technique treats all three rainfall characteristics (i.e. rainfall duration,
intensity and temporal pattern) as random variables. Thus, the new event definition has to
incorporate the random nature of these rainfall characteristics. For the purposes of the
approach a ‘complete storm’ and a ‘storm-core’ (the most intense part of the storm) are
defined as follows.
3.2.1.1 Complete Storm
A complete storm is defined in three steps, illustrated in Figure 3.3 in accordance with
Hoang et al. (1999):
1. A ‘gross’ storm is a period of rainfall starting and ending by a non-dry hour (i.e. hourly
rainfall greater than C1 mm/h), preceded and followed by at least six dry hours. This is
defined as the separation time, h = 6 hrs.
2. ‘Insignificant rainfall’ periods at the beginning or at the end of a gross storm, if any, are
then cut from the storm, the remaining part of the gross storm is named the ‘net’ storm.
(A period is defined as ‘having insignificant rainfall’ if all individual hourly rainfalls are ≤
C2 mm/hr, and average rainfall intensity during the dry period is ≤ C1 mm/hr. Therefore
C1 and C2 are used as insignificant ‘rainfall filters’).
3. The net storms, now referred to as complete storms, are then evaluated in terms of their
potential to produce significant storm runoff. This is performed by assessing their rainfall
magnitudes by comparing their average intensities with threshold intensities. A net storm
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is only selected for further analysis if the average rainfall intensity during the entire storm
duration (RFID) or during a sub-storm duration (RFId), satisfies one of the following two
conditions:
Condition 1: (RFID) ≥ F1.(2ID), where 0 < F1 <1
Condition 2: (RFIdmax) ≥ F2.(2Id), where 0 < F2 <1
where 2ID is the 2 year ARI burst intensity for the selected storm duration D, and 2Id
corresponding burst intensity for the sub-storm duration d. The values of 2ID and 2Id are
estimated from the design rainfall data in Australian Rainfall and Runoff (I.E. Aust., 1987).
In the above event definition, the use of appropriate reduction F1 and F2 allows the selection
of only those events that have the potential to produce significant storm runoff. The use of
smaller values of F1 and F2 captures a relatively larger number of events; appropriate values
need to be selected such that events of very small average intensity are not included. In this
study, the following parameter values have been adopted: F1 = 0.4, F2 = 0.5, C1 = 0.25
(mm/hr) and C2 = 1.2 (mm/hr) based on previous work (Hoang et al., 1999; Rahman et al.,
2001).
3.2.1.2 Storm-Core
The available IFD information in Australian Rainfall and Runoff (I.E. Aust., 1987) is not based
on the generation of complete storms but on periods of intense rainfall within complete
storms, called bursts. If this existing information is to be used with the proposed new
approach, it is more useful to undertake the design rainfall analysis in terms of storm bursts.
However, as the duration of the bursts in ARR87 analysis were predetermined rather than
random, it is necessary to consider a new storm burst definition that will produce randomly
distributed storm burst durations. These newly defined storm bursts are referred to as storm-
cores (Rahman et al., 1998).
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For each complete storm, a single storm-core can be defined as the ‘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 index rainfall intensity 2Id for the relevant duration d, then
selecting the burst of that duration which produces the highest ratio.
For example, in Figure 3.4 (Rahman et al., 2001) the storm-core has a duration of 3 hours.
For that duration the ratio with 2I3 is 4.0, compared to a value of 1.4 for 2I1 (duration of 1hour)
which is the most intense rainfall burst within the complete storm.
Figure 3.3: Rainfall Events: Complete Storms and Storm-Cores (Rahman et al., 2001)
0
1
2
3
4
5
6
7
8
9
10
1 5 9 13 17 21 25 29 33 37 41 45Time (h)
Rai
nfal
l Int
ensi
ty (m
m/h
r)
Storm-core
Start of Storm
End of net storm
End of gross storm
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Figure 3.4: Identification of a Storm-Core (Rahman et al., 2001)
3.2.1.3 Storm-Core Duration
Using the definition of a rainfall event above, the distributions of storm-core duration (Dc) are
determined from the rainfall (ALERT or pluviograph) stations located across the catchments
of interest. A storm analysis is conducted from which the mean, standard deviation and
skewness of the observed Dc values for these stations are determined. Rahman et al. (2001)
obtained storm-core distributions for 29 pluviograph stations of varying record length (at least
20 years) in Victoria. The distributions of (Dc) were examined and an exponential distribution
was found to approximate the distribution. This implies that, at a particular station, there are
many more short duration storm-cores, than longer duration ones, and that the number of
storms reduces exponentially with duration. The exponential distribution has one parameter
and its probability density function is given by:
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6
Time (h)
Rai
nfal
l Int
ensi
ty (m
m/h
r)
1-hr relative intensity = 28/20 = 1.43-hr relative intensity = 20/5 = 4.0
Storm-core
3-hr average intensity = 20
2I3 = 5
2I1= 20
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β
β/1
)( cDc eDp −= Equation 3.1
where p () stands for probability density, Dc is the storm-core duration and is β the
parameter of the exponential distribution.
The parameter β can be taken as the mean of the observed Dc values. The exponential
distribution has a skewness of 2, and its mean and standard deviation are equal. For the
purposes of this study an exponential distribution of storm-core duration (Dc) has been
adopted.
3.2.2 Storm-Core Rainfall Intensity
In practice, the conditional distribution of rainfall intensity is expressed in the form of the
intensity-frequency-duration (IFD) curves, where rainfall intensity is plotted as a function of
rainfall duration and frequency. In the Joint Probability Approach adopted here, the IFD
curves for storm-core rainfall intensity have been developed in a number of steps, as
described below.
3.2.2.1 Development of Storm-Core IFD Curves
As expected, Rahman et al. (2001) observed that there was a strong relationship between
storm-core rainfall intensity (Ic) and duration (Dc). The strong relationship between Dc and Ic
means that the distribution of Ic needs to be conditioned on Dc. The procedure adopted to
develop storm-core IFD curves is outlined below from Rahman et al. (2002a).
1. The range of storm-core duration (Dc) is divided into a number of class intervals (with a
representative or mid-point duration for each class). An example is given in Table 3.1.
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2. For the data in each class interval (except the 1hr class), a linear regression line was
fitted between log(Dc) and log(Ic). The slope of the fitted regression line was used to
adjust the intensities for all durations within the interval to the representative duration.
3. The adjusted intensity values in a duration class interval form a partial series. An
exponential distribution is fitted to the partial series II (I=1,…,M), where M is the number
of data points in a class. Quantiles are obtained from the following equation:
)ln()( 0 TITI λβ+= Equation 3.2
where I0 is the smallest value in the series; β � � �Ii/M-I0; λ =M/N; N is the number of years
of data; and T is the average recurrence interval (ARI) in years. Rahman et al. (2001) found
that an exponential distribution better fitted the partial series rainfall intensity data than the
other candidate distributions and recommended the adoption of an exponential distribution,
which has been followed here.
Adopting the fitted distribution, design rainfall intensity values Ic(T) for the given duration
interval are computed for ARIs of 2, 5, 10, 20, 50 and 100 years.
4. For a selected ARI, the computed Ic(T) values for each duration range are used to fit a
second degree polynomial between log(Dc) and log(Ic) using the equation below.
cDbDaI ccc ++= ))(log())(log()log( 2 Equation 3.3
where a, b and c are constants.
Table 3.1: An Example of Class Intervals and Representative Points for Storm-Core Duration(Dc) for Developing IFD Curves
Class interval(hours)
Representative duration(hours)
1 12 - 3 2
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4 - 12 613 - 36 2437 - 96 48
3.2.2.2 Preparation of IFD Table
The adopted Monte Carlo Simulation Technique begins 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 will then be drawn from the conditional distribution of Ic, expressed in the
form of IFD curves. This requires the definition of a continuous distribution function, ideally in
the form of a functional relationship between Dc, Ic, and ARI. However, as it is difficult to
derive a functional relationship that suits different conditions, an IFD table is used with an
interpolation procedure to generate Ic values for any given combination of Dc and ARI.
Equation 3.2 and Equation 3.3 are the basis of the IFD table, used for data generation in the
adopted Monte Carlo Simulation Technique. In an IFD table, Ic values are tabulated for Dc
values of 1, 2, 6, 24, 48, 72 and 100 hours, and ARIs of 0.1, 1, 1.11, 1.25, 2, 5, 10, 20, 50,
100, 500, 1,000 and 1,000,000 years. A linear interpolation function in the log domain is used
between the tabulated values of Dc and ARI.
It should be noted here that Ic values for ARIs less than 1 year and greater than 100 years
are of less direct interest in the development of derived flood frequency curves for design
flood estimation up to the limit of the 100 year ARI. However, these extrapolated values are
required to cover the range that might arise in the Monte Carlo Simulation. The part of the
developed IFD curves for ARIs of 100 to 1,000,000 years is subject to very large estimation
errors from rainfall data records of limited lengths (in this study less than 30 years). Where
the interest is on rare to extreme floods (ARI greater than 100 years), this part of the curves
needs to be adjusted using design rainfall data from some appropriate regionalisation
approach, for example the CRC FORGE method (Nandakumar et al., 1997; Weinmann,
Nandakumar & Siriwardena, 1999).
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3.2.3 Storm-Core Rainfall Temporal Pattern
A rainfall temporal pattern is a dimensionless representation of the variation of rainfall depth
over the duration of the rainfall event. Following the procedure of Hoang (2001), in this study
the time distribution of rainfall has been characterised by a dimensionless mass curve, (i.e. a
graph of dimensionless cumulative rainfall depth versus dimensionless storm time with 10
equal time increments).
Rahman et al. (2001) found that temporal patterns of rainfall depth for storm-cores (TPc) are
not dependent on season and total storm depth. This means that dimensionless temporal
patterns from different seasons and for different rainfall depths can be pooled. However, the
patterns were found to be dependent on storm duration, yielding two groups: (1) up to 12
hours duration; and (2) greater than 12 hours duration.
As the rainfall data used in the analysis was only defined at hourly intervals, the minimum
storm-core duration used in the temporal pattern analysis was 4 hours. Storms with less
than 4-hour duration are assumed to have the same patterns as the observed 4 to 12 hour
storms.
Design temporal patterns for storm-cores (TPc) could be generated by the ‘multiplicative
cascade model’ applied by Hoang (2001). However, in the present Monte Carlo Simulation
Technique, historic temporal patterns have been used directly instead of generated temporal
patterns similar to Rahman et al. (2001). That is, observed temporal patterns (in
dimensionless form) were drawn randomly from the sample corresponding to the generated
Dc value.
3.2.4 Storm-Core Initial Loss
The initial loss (ILs) for a complete storm is estimated to be the rainfall that occurs prior to the
commencement of surface runoff (following the approach adopted by Hill et al., 1996), as
shown in Figure 3.5. The storm-core initial loss (ILc) is the portion of ILs that occurs within the
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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 storm event). In computing loss, a surface runoff threshold value of 0.01 mm/hr has been
used, similar to Hill and Mein (1996); it is considered that surface runoff commences when
the surface runoff threshold has been exceeded.
Catchment average rainfall is used in the computation of losses in the cases where more
than one pluviograph station is available within the catchment. To enable the calculation of
average rainfall, an access database program ‘Rain Converter.mdb’ was developed (Morris,
2003). The program takes ‘Thiessen Polygon Weightings’ calculated by GIS and computes
catchment average rainfall.
3.2.4.1 Fitting a Theoretical Distribution to Initial Loss Data
Based on the results of Rahman et al. (2001) and Hill and Mein (1996), the relationship
between ILc, ILs, and Dc was expressed by the following empirical equation:
)](10log25.05.0[ cDsILcIL += Equation 3.4
This relationship gives ILc = 0.5. ILs at Dc = 1 hour, and ILs = ILc at Dc = 100 hours. It might be
noted here that the use of ILs distribution (with a adjustment factor) as proposed in Equation
3.4 is preferable to the use of ILc directly as ILs is more readily determined from the data and
can probably be derived using existing design loss data (e.g. Hill & Mein, 1996). Rahman et
al. (2001) found the distributions of ILs for the study catchments in Victoria were positively
skewed, and a four-parameter Beta distribution was used to approximate distribution of ILs.
The Beta distribution was adopted for this study.
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Figure 3.5: Initial Loss for Complete Storm (ILs) and Initial Loss for Storm-Core (ILc).(Rahman et al., 2001)
The four-parameter Beta distribution is detailed below (Benjamin and Cornell, 1970):
111 )()(
)(1
)( −−−− −−
−= rtr
tY ybayabB
yf Equation 3.5
bya ≤≤ and t > r
where fY(y) is the probability density, a, b, t and r are parameters and B is the beta function
defined below:
)!1()!1()!1(
−−−−=
trtr
B Equation 3.6
Time
Rai
nfal
l/Str
eam
flow
Flood Hydrograph
Storm-Core
IL c
IL s
Start of surface runoff
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The mean and variance of the Beta distribution are given by:
)( abtr
aY −+=µ Equation 3.7
)1()()(
2
22
+−−=
ttrtrab
Yσ Equation 3.8
The parameters of the Beta distribution r, t can thus be determined from known values of a,
b, Yµ and Yσ , that is, the lower and upper limits, mean and standard deviation respectively
of the observed loss values at a site.
There are a number of possible alternatives to the selected Beta distribution to describe the
variability of initial loss, e.g. the Gamma, Exponential and Truncated Normal Distributions.
The Beta distribution was adopted for its flexibility and because its parameters lend
themselves readily to physical interpretation (Rahman, Weinmann & Mein, 2002d).
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3.3 Catchment Response Models
As discussed in Section 3.1.1.2, a transfer function or catchment response model is needed
to convert the rainfall excess hyetograph produced by the loss model into a surface runoff
hydrograph. The process is also referred to as hydrograph formation and involves
transforming the runoff from different parts of the catchment into a flood hydrograph, by
means of runoff-routing (typically an Australian method) or unit hydrograph (typically a North
American method).
The following section discusses these two methods and how they have been integrated with
the Monte Carlo Simulation Technique.
3.3.1 Runoff Routing Model
Rainfall runoff modelling methods in Australia involves the analysis and selection of rainfall
intensity or rainfall depth for a given AEP. The rainfall depth is combined with a runoff
generation or loss model to produce a rainfall excess hyetograph, which is the runoff from a
storm event. The rainfall excess hyetograph is then routed through a catchment response or
runoff-routing model to produce a flood discharge or flood estimate at the outlet of the
catchment.
In Australian flood design practice, it is common to use a semi-distributed and non-linear type
of catchment routing model, referred to as a runoff-routing model. For hydrograph formation,
different categories of runoff-routing models can be distinguished according to how they deal
with particular aspects of representing catchment characteristics in the model, such as:
1. lumped or spatially distributed representation of the catchment’s runoff-routing
characteristics and, for distributed (or semi-distributed) representation, the method of
catchment sub-division (topographically-based or isochronal lines);
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2. combined or separate modelling of the routing response of different catchment elements
(overland flow, streamflow, natural and artificial storages);
3. adoption of a linear or non-linear form of relationship between storage and discharge;
and
4. the ability of the model to deal with special features of the catchment or drainage
network, such as modifications to natural flow characteristics in parts of the catchment,
flow diversion points and various flow control structures.
3.3.1.1 Role of Storage-Discharge Relationship
All the runoff-routing models use either a hydraulic or hydrologic routing method to represent
the modifying effect that a particular routing element (e.g. an overland flow path, stream
reach, a reservoir or other storage) has on the input hydrograph. These routing methods are
based on the simultaneous solution of two equations:
1. The continuity equation expressing the principle of conservation of mass:
StQI ∆=∆− )( Equation 3.9
that is, the difference between inflow (I) to the routing element and the outflow (Q) from it
over a time difference ( ∆ t) is equal to the change in storage within the element ( ∆ S).
2. An equation that relates the discharge to the characteristics of the routing element. In the
widely used storage routing methods this is the storage-discharge (S-Q) relationship:
)(QfS = Equation 3.10
The problem of parameter determination for a runoff-routing model can be seen as finding
the set of parameters that defines the storage-discharge relationships of all the routing
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elements used to represent the catchment’s routing response. This set of parameter values
must adequately reflect the influence of different catchment factors and their variations, and
its determination should be based on a sound understanding of the basis of the adopted S-Q
relationship for each of the routing elements. In the following section different forms of S-Q
relationships are introduced, and their hydraulic basis explained.
3.3.1.2 Storage-Discharge Relationships
For a particular routing element, say a river reach, the relationship between storage and
discharge in the reach may vary considerably for different flow magnitudes, reflecting the
varying influence of the factors that determine flow and storage in the river or floodplain at
different water levels. This is shown conceptually in Figure 3.6.
Figure 3.6: Example of Actual S-Q relationship
It is generally not possible to accurately identify the form of the actual S-Q relationship for the
complete range of flow magnitudes from either the flood observations or from hydraulic
calculations. Different runoff-routing models use different forms of equations to represent the
actual S-Q relationship in a simplified fashion.
Range of observed flood events
Discharge Q
Sto
rage
S
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The simplest form is the linear relationship for a concentrated storage element, where the
volume stored in the element is linearly related to the outflow from the element:
KQS = Equation 3.11
The well-known Muskingum flood routing method uses a linear S-Q relationship for a
distributed storage element, where the volume stored in the element is related to both the
inflow and the outflow from the element:
))1(( QXXIkS −+= Equation 3.12
and the parameter X indicates the relative influence of the inflow (I) and outflow (Q) on the
storage in the reach.
In Australia, the most commonly applied runoff-routing models fall into the category of semi-
distributed models (node-link models) that use a topographic division into subcatchments,
and a network of routing elements that are typically characterised by a non-linear power
function relationship between storage and discharge:
mkQS = Equation 3.13
where S is the catchment storage in m3h/s, k is the non-linear routing coefficient, Q is the rate
of outflow in m3/s and m indicates the degree of non-linearity of the S-Q relationship (typically
0.8).
This is the form adopted in most routing elements of the RORB, URBS and WBNM models,
and in the overland flow component of the XP-RAFTs model. The URBS model uses a non-
linear form of the Muskingum equation for the stream routing elements:
nQXXIkS ))1(( −+= Equation 3.14
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3.3.1.3 Network Runoff Routing Model
Some models use a lumped representation of the overland flow and streamflow components,
while other models treat these separately. All the commonly used modelling packages make
provision for more detailed representation of natural or artificial storages.
In most applications in Australia a network runoff-routing model is employed for design flood
estimation. The network model arranges storages to represent the natural drainage network
of the catchment. The distributed nature of the catchment storage is simulated by a series of
concentrated storages for the main stream and major tributaries. Advantages and
disadvantages of the network runoff-routing model are discussed below.
Advantages
• Non-linear catchment response can be modelled.
• Spatial distribution of catchment storage can be realistically modelled.
• The effect of significant storages (such as reservoirs or large floodplains) on a catchment
can be modelled.
• Variations or changes in catchment characteristics can be modelled.
• The separate effects of more than one significant stream can be modelled.
• Hydrographs can be estimated at more than one location in the model.
• Spatial variations in rainfall and losses can be taken into account.
Disadvantages
• Considerable expertise is required for the valid operation of models and interpretation of
their results.
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• The power law form non-linearity assumption may not be valid for large floods on natural
catchments with large floodplains when linearity may be approximated.
• The use of a relatively complex model provides a false sense of security in the user.
• Where models have more than one parameter to be evaluated, interaction between the
parameters often occurs.
• The effects of data errors may be greater with complex models than simple models.
• As with other methods of flood estimation, accurate estimation of losses to determine the
rainfall excess is necessary even if the best, and most complex of models is applied.
The most commonly used network runoff-routing models in Australia include RORB, WBNM,
URBS, XP-RAFTS. All these models are comparable for most applications, but differ in their
capability to use more detailed data. Presently URBS is the only model which has been
integrated with the Monte Carlo Simulation Technique (Rahman et al., 2002b). A discussion
of the URBS model is contained in Section 3.3.2.
3.3.2 URBS
3.3.2.1 Overview
URBS (Carroll, 2001) is an event-based runoff-routing model suitable for integrated
catchment management and flood forecasting and is a versatile runoff-routing networked
model of subcatchments based on centroidal inflows. The URBS model originated from the
WT42 model developed by the Queensland Department of Primary Industries – Water
Resources. The model, written in C programming language, is available under a number of
operating systems including DOS and Microsoft WINDOWS.
URBS has a similar catchment discretisation to that of the RORB model (Laurenson and
Mein, 1997). An important feature of the URBS model is the ability to split the hydrograph
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routing into catchment and channel components. The URBS model is used extensively
throughout Australia for flood forecasting and design event modelling. Further information on
the URBS model is contained in Carroll (2001).
Two routing models are available to describe catchment and channel storage routing
behaviour:
1. The ‘Basic’ model is a RORB-like model (Mein, Laurenson & McMahon, 1974) and
assumes that catchment and channel storage for each sub-catchment is lumped together
and represented as a single non-linear reservoir.
2. The ‘Split’ model separates the channel and catchment storage components of each sub-
catchment for routing purposes.
The Split Model is the most versatile of the two, and may also be less demanding in terms of
the number of subcatchments required to adequately define the catchment. Irrespective of
the model used, each storage component is conceptually represented as a non-linear
reservoir and Muskingum routing is used for channel routing (Carroll 1996; 2001).
3.3.2.2 Split Model
This study has used the Split model. The Split model identifies the catchment and channel
routing in each sub-catchment and calculates their effects separately. First, the rainfall on a
sub-catchment is routed to the creek channel. This inflow to the sub-catchment into the
channel is assumed to occur at the centroid of the sub-catchment. The lag of the sub-
catchment storage is assumed to be proportional to the square root of the sub-catchment
area. Next, the inflow is routed along a reach using non-linear Muskingum method, with a
lag time proportional to the length (derivative) of the reach (Carroll 1996; 2001).
The Split model is similar to the Watershed Bounded Network Model or WBNM model (Boyd
et al., 1987) except the WBNM model assumes the channel storage is proportional to sub-
catchment area rather than channel length.
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3.3.2.2.1 Catchment Routing
The storage discharge relationship for this reservoir which models catchment storage is
given as:
mQU
FAS
++= 2
2
)1()1(β
Equation 3.15
where S = catchment storage (m3h/s), β = catchment lag parameter, A = area of sub-
catchment (km2), U = fraction urbanisation of sub-catchment, F = fraction of sub-catchment
forested, m = catchment non-linearity parameter.
The non-linearity catchment routing parameter (m) is typically between 0.6 and 0.8. It is
noted that the effects of urbanisation and forestation are applied to the catchment routing
components. Therefore, through flows are unaffected by local sub-catchment urbanisation or
forestation. Accordingly, this model is more suitable for large creeks and rivers where the
main channel hydraulic properties are largely unaffected by the extent of catchment
urbanisation or forestation (Carroll 1996; 2001).
3.3.2.2.2 Channel Routing
Channel routing which is based on the non-linear Muskingum model as is given as:
ndu
c
QXXQS
nLfS ))1(( −+= α Equation 3.16
where S = channel storage (m3h/s), α = channel lag parameter, f = reach lag parameter, n =
Manning’s n or channel roughness, L = length of reach (km), Sc = channel slope (m/m), Qu =
inflow at upstream end of reach (includes catchment inflow, Qd = outflow at downstream end
of the channel reach (m3/s), X = Muskingum translation parameter, n = Muskingum non-
linearity parameter (exponent).
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It is noted that setting Muskingum n = m, X = 0 and β = 0, reduces the Split Model to the
Basic model, or the simplest form of the RORB model. Setting β = 0 and n = 1, the model
reduces to the Muskingum model. Setting Muskingum n to a value other than 1, assumes
that the non-linear Muskingum model, which allows the model to vary lag with flows; a value
less than 1 implies a decrease in lag with increasing flow, whereas a value greater than 1
implies the opposite (Carroll 1996; 2001).
3.3.2.2.3 Calibration
Calibration of the Split model is best done by first matching the output of a rigorous hydraulic
model to the Muskingum model to establish the value α (e.g. Della & McGarry, 1993) or by
estimating the channel celerity in km/hr, α is the inverse of the average wave speed (km/hr)
when n, the channel linearity parameter (exponent) is assumed to be 1 and stream length
alone is used to characterise the routing process. Once α and n have been calibrated, β
and m are calibrated by matching recorded events.
3.3.2.2.4 Input and Outputs
The input to the model includes catchment specification, recorded river flows or heights,
rainfall data and rating curves if river height data are either input or required as output. The
catchment specification contains the subcatchment descriptions and river reach details used
to characterise the hydrology of the river catchment being studied.
Outputs generated from the model include screen plots of observed and modelled
hydrographs and rainfall hyetographs. A series of output files are generated containing
depths of excess rainfall, observed and modelled river discharges and heights (if rating
tables are present), sediment washoff and traffic disruption.
The model has been used successfully for both design and operational flood hydrology in
Australia. The URBS model is one of the models used by the Australian Bureau of
Meteorology.
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3.3.3 Calibration of Runoff Routing Method
In conceptual hydrologic models, such as commonly used runoff-routing models, the
complex physical processes that determine the actual catchment response are represented
by relatively simple equations with a small number of parameters. As indicated above, the
relationship of these parameters with physical characteristics may be quite complex and
difficult to establish in an individual case. In this situation, a small number of parameters can
be inferred from observed model inputs and outputs by a process called model calibration. In
the calibration of runoff-routing models, use is made of the large amount of catchment
information that is embodied in flood hydrographs.
The URBS models used in the study were calibrated Split models provided by the
Queensland Bureau of Meteorology. Therefore no change of key calibration parameters such
as catchment linearity = m, α = channel lag parameter and β = catchment lag parameter
has been undertaken. The URBS catchment definitions files or vector files for the two
catchments studied are contained in Appendix F.
3.4 Monte Carlo Simulation
‘Monte Carlo Simulation’ refers to a mathematical technique that is used to determine the
outputs from a model represented by a complex set of equations that cannot be readily
solved analytically. In this study, the Monte Carlo Simulation approach is used to generate a
sample of NG (Number Generated) different runoff events from NG different combinations of
rainfall and loss inputs. For each event, a set of values of Dc, Ic, TPc, and ILc is generated to
define the rainfall excess hyetograph, which is then routed through a calibrated runoff-routing
model to produce a corresponding streamflow hydrograph. A large number of hydrographs
(in the order of thousands 10,000 to 20,000) is typically generated and the resulting flood
peaks are extracted and subjected to a frequency analysis to obtain the derived flood
frequency curve. The Monte Carlo Simulation Technique adopted in this study is similar to
Rahman et al. (2001) and is summarised below.
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3.4.1 Number of runoff events generated
The number of separate events to be generated depends on the range of ARIs of interest,
the degree of accuracy required, the number of probability-distributed variables involved and
the degree of correlation between them. For the study catchments it was found that at least
6000 to 12000 rainfall events have to be generated to produce relatively stable estimates of
the derived flood frequency curve in the ARI range from 1 to 100 years.
If the purpose of the Monte Carlo Simulation was to estimate flood events in the extreme
range, or if more independent random variables were involved, the required number of
generated events would increase by orders of magnitude. It would then be desirable to apply
more efficient Monte Carlo Simulation methods, such as importance sampling (e.g.
Thompson, Stedinger & Heath, 1997).
The number of partial series flood events to be generated (NG) is obtained from the following
equation:
NYNG .λ= Equation 3.17
where λ is the average number of storm-core events per year, and NY is the number of
years of data to be generated. As an example, for λ equal to 5, a total of 10,000 data points
have to be generated to simulate 2000 years of data.
3.4.2 Steps in Simulation
To simplify the Monte Carlo simulation, a total of NG runoff events or stochastic events are
first generated and stored as individual data files for use in the simulation. Each event is
defined by random values of rainfall duration and ARI, which define the average rainfall
intensity, a random temporal pattern, and a random value of initial loss. These values are
generated from the distributions identified in Section 3.1.2.
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As a first step, values of storm-core duration, Dc, are generated from an assumed
exponential distribution. This has one parameter estimated as the observed mean Dc value,
obtained from the pluviograph data from the catchment of interest.
In the second step, a random value of storm-core rainfall intensity (Ic) for each given value of
Dc is generated, using the IFD table described in Section 3.2.2.1. First a random ARI value is
selected from the following equation (after Stedinger et al., 1993, equation 18.6.3b):
)1ln(1AEP
ARI−−= Equation 3.18
where AEP is the annual exceedance probability, obtained from a uniform distribution U
(0,1). Since the primary aim is to develop derived flood frequency curves in the range of
annual exceedance probabilities of say 1 in 100 to 1 in 2, the interval U (0,1) is too wide.
However, to cover a sufficiently wide range of rainfall intensities that might be of interest in
the simulation, U was limited the range 10-6 ≤ U ≤ 1 – e- λ . As an example, for an average
annual number of storm-core events λ equal to 5, this results in 10-6 ≤ U ≤ 0.993; in terms
of ARI (years) this is equivalent to 106 ≥ ARI ≥ 0.2. For the given Dc and ARI values, an Ic
value is then read from the IFD table for the site of interest, using linear interpolation with
respect to both log(Dc) and log (ARI) in accordance with Rahman et al., (2001).
In the third step to generate a temporal pattern, the adopted simulation method randomly
selects a historic temporal pattern recorded at the site of interest depending on the
previously generated Dc value (refer to Section 3.1.3). The procedure is repeated NG times
to sample NG temporal patterns.
In the fourth step, storm-core initial loss values are derived by first generating a storm initial
loss value from the Beta distribution fitted to the ILs data from the observed events at the site
of interest. The generated ILs value is then converted to a storm-core loss ILc, using
Equation 3.4. The procedure is repeated NG times to generate NG values of Ic.
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In addition to these four stochastic inputs, the simulation of streamflow hydrographs requires
the following fixed inputs: (a) catchment area in km2 (for calculation of areal reduction factor)
and (b) an estimate of continuing loss (CL) in mm/hr. Examples of the parameter files, and
how they were applied to generate the NG runoff events for the two study catchments are
detailed in Section 5.0.
Finally, with the above fixed and stochastic inputs, each generated rainfall event can be
converted to an input runoff hydrograph for the catchment and then routed through the URBS
model to obtain a simulated flood hydrograph at the catchment outlet. The peak of each of
the NG simulated hydrographs is stored for later analysis to determine a derived flood
frequency curve. Given the parameters of the storm-core distributions Dc, Ic, TPc and ILc, the
generation of data files from these distributions takes five minutes for 10,000 events, and the
simulation of these data files takes about 1-1.5 hours for 10,000 events on a Pentium III 733
MHz personal computer.
3.5 Flood Frequency Analysis
3.5.1 Derived Flood Frequency Curve
A non-parametric frequency analysis method is used to construct a derived flood frequency
curve from the set of NG simulated flood peaks. 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 outlined in Rahman et al, (2001):
1. Arrange the NG simulated peaks in decreasing order of magnitude and assign rank 1 to
the highest value, 2 to the next one and so on.
2. For each of the ranked floods, compute an ARI from the following equation:
4.02.01
.4.02.0
−+≅
−+=
mNY
mNG
ARIλ
Equation 3.19
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where NG is the number of simulated peaks, m is the rank, and λ the average number of
storm-core events per year at the catchment of interest, and NY is the number of years of
simulated data.
3. Prepare a plot of ARI versus flood peaks, that is a plot of the empirical flood frequency
curve defined by the simulated flood peaks.
4. Compute flood quantiles for selected ARIs by fitting a smooth curve through neighbouring
points. (Given the large number of data points, logarithmic interpolation between the two
neighbouring data points, without any smoothing, has been adopted in this study.)
3.6 Applied Method and Programs
Flood modelling and estimation is more an applied science than a pure science. As such the
methodology, programs and calculative steps used in applying the Monte Carlo Simulation
Technique need to be recognised and understood. Specifically, there are quite a number of
stepwise flood estimation procedural requirements as well as a number of programs that the
new approach requires the operator to master. To illustrate the inherent complexities of the
method in practical application, a process-program flowchart is provided in Figure 3.7. All
programs used in the study were either developed by (Rahman et al., 1998, 2001, 2002) or
Carroll (1996, 2001) for the application of Monte Carlo Simulation and URBS components
respectively.
The flowchart represents the Monte Carlo Simulation Technique and has been termed the
methodology matrix. The methodology has been termed a matrix, as the process is stepwise
and outputs of a particular step are required as inputs for the next step.
The first column of the matrix represents the theory and background which have been
previously described. The second column, represents the programs, processes and analyses
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which are involved when applying the Monte Carlo Simulation Technique in practice. The
third column represents the inputs/outputs for each step and their stepwise application.
The methodology matrix highlights that the Monte Carlo Simulation Technique is quite a
complex method, and at the moment a high degree of flood modelling experience and
expertise in flood hydrology, and flood estimation is required to complete a full simulation
from start to finish. In addition, there a number of sub-steps and intuitive decision-making
processes that require the skill-set and level of understanding that only an experienced
practitioner possesses to adequately deal with the new approach on a practical and applied
level.
What this means in the short-term, is that the new approach can be only be applied by
experienced flood modellers and practitioners, whose cumulative hydrologic, hydraulic and
flood estimation experience can overcome any unforseen obstacles or shortfalls in the
process. This is particularly relevant to data limitations, which have the potential to short-
circuit the entire application of the Monte Carlo Simulation Technique as an alternative
method of flood estimation. A further discussion of the application of the new approach can
be found in Section 5.0.
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Figure 3.7: Monte Carlo Simulation Technique – Methodology Matrix
Step
1. Data Preperation
2. Storm Analysis
3. Loss Analysis
4. Temporal Pattern Analysis
5. Monte Carlo Simulation (Generation of 10,000 rainfall excess hyetographs)
6. Surface Hydrograph Formation with runoff routing model URBS
7. Non-parametric Flood Frequency Analysis (FFA) and compare to Design Event Approach and Observed Partial
Series
Process and
Programs
Data collection and reviewManual review of rainfall data and stream
gauging data
URBS Rainfall Converter/HYDSYS
Use Fortran program’mcsa11.exe ’
(Monte Carlo Storm Analysis)
Calculate catchment average rainfall using ’Rain Converter.mdb’
Then use catchment average rainfall with streamflow file and Fortran program
’Lossca.exe’(Monte Carlo Loss Analysis)
Pools temporal patterns from storm analysis of pluviographs into two groups
with Fortran program’Tpana1.exe’(Monte Carlo Temporal Pattern Analysis)
Use parameter file and outputs from previous progams above with program
’Rainurbs.exe’(Monte Carlo Simulation)
Use ’for loop’ with ’URBS32.exe’ and URBS vector file for study catchment of
interest
Use spreadsheet for FFARun Design Event Approach
Use program ’FREQ.exe’ for observed partial series analysis
Inputs/Outputs
Rainfall data (ALERT or Pluviograph) in hourly format quality checked and
compatible with storm analysis programConcurrent stream guaging data in hourly
format compatible with loss analysis program
The mean D c t o develop the exponential distribution of storm-core duration
Conditional probability distribution of I c /D c in the form of IFD curves
Dimensionalised Temporal Patterns TP c
Four parameters: Mean, standard deviation, upper limit and lower limit of IL s
to fit Beta-distribution of IL s .
For a given D c, IL c is then determined
using the empircal relationship with IL s .
TP c up to 12 hrs
TP c greater than 12 hrs
10,000 rainfall excess hyetographs
’URBSlog.csv’ is an output file which stores the 10,000 flood peaks/volumes at
the outlet and other key locations within the study catchment
Derived flood frequency curveObserved Partial Series
Design Event Approach results
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3.7 Data Management
3.7.1 Overview
Data management is a key part of applying the Monte Carlo Simulation Technique. As
indicated in Section 3.6, the present approach is stepwise and therefore excellent ‘house-
keeping’ of files and organisation of directory structures underpins successful application of
the approach.
In addition, the technique in its present form produces a high number of output files
automatically during each analysis. These output files require interpretation, modification and
in some cases analysis for the approach to continue and be completed fully. These steps are
detailed below.
3.7.2 Filename Nomenclature
Filename nomenclature is extremely important when applying the approach, particularly with
the quantity and variety of data input files and outputs. The input and output file naming
system employed for storm analysis, loss analysis and temporal pattern analysis are
outlined in the following tables.
Table 3.2: Input and Output Nomenclature for Storm Analysis (using program ‘mcsa11.exe’)Input Description Output Description.psa Parameter file
For example:(a2500.psa)
.dit Duration, intensity and total rainfall for complete storm
.dcs Duration of complete storm
.cdr Storm-core duration
.cdi Storm-core duration and intensity
.stc Starting time of storm-core
.pcr Sum of pre storm-core rain
.scs Starting time of complete storm
.etc End time of complete storm
.tpo Output file for temporal pattern analysis
.ney Number of events per year
.mcd Mean of storm-core duration
.oqn Input file to generate IFD table
.rln Record length for a site used in subroutine iana to generate.oqn file
.slt List of .cdi sites used to generate .oqn file in the subroutine
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Input Description Output Descriptioniana
.ifd IFD table output from subroutine ifdt
.stf Statistics of fit for IFD analysis from subroutine ifdt
Table 3.3: Input and Output Nomenclature for Temporal Pattern (using program ‘tpana1.exe’)Input Description Output Description.tpo Parameter file
For example(john.tpo)
.tpc TPc up to 12 hours duration and greater than 12 hoursduration
tpl12.dat TPc up to 12 hourstpgt12.dat TPc greater than 12 hours durationobsrain.dat Observed rainfall details for each temporal pattern
Table 3.4: Input and Output Nomenclature for Loss Analysis (using program ‘losssca.exe’)Input Description Output Description.lan Parameter file
For example(anorthjla1.lan)
.ssr Starting time of surface runoff
.ics Initial loss for complete storm (ILs)
.isc Initial loss for storm-core (ILc)
.mls Month vs. ILs
.mlc Month vs. ILc
.psc Concurrent pluvio and streamflow data for loss analysis
.acs API of complete storm event
.asc API for storm-core
.slp ILs statistics (lower limit, upper limit, mean and standarddeviation)
.clp ILc statistics (lower limit, upper limit, mean and standarddeviation)