montecarlo hydro graph

Flood Estimation TechniquesApplication of Monte Carlo Simulation Technique with URBS Model for Design Flood Estimation of Large Catchments

32

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


33

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


34

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.


35

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


36

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.


37

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).


38

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


39

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


40

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).


41

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


42

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


43

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.


44

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


45

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).


46

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


47

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.


48

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


49

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).


50

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);


51

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


52

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


53

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


54

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.


55

• 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


56

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).


58

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.


59

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.


61

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


64

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


66

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)

montecarlo hydro graph

Documents

rainfallrunoff model

urbs model

runoffrouting model

bythe loss model

rainfallrunoff process

rainfall event

initialcontinuing loss

clarks model russell