1. A Monte Carlo Framework for the Brisbane River Catchment Flood
Study
F.L.M. Diermanse
Expert Researcher, Deltares, Delft, The Netherlands
E-mail: ferdinand.diermanse@deltares.nl
D.G. Carroll
Director, Don Carroll Project Management, Brisbane, Australia
E-mail: don.carroll@optusnet.com.au
J.V.L. Beckers
Senior Advisor/Researcher, Deltares, Delft, The Netherlands
E-mail: joost.beckers@deltares.nl
R. Ayre
Project Leader, Aurecon, Brisbane, Australia
E-mail: Rob.Ayre@aurecongroup.com
J.M. Schuurmans
Project Leader/Advisor, Royal HaskoningDHV, Amersfoort, The Netherlands
E-mail: hanneke.schuurmans@rhdhv.nl
The State of Queensland initiated a comprehensive hydrologic assessment as part of the Brisbane
River Catchment Flood Study (BRCFS) in response to the devastating floods in January 2011 and
subsequent recommendations of the Queensland Floods Commission of Inquiry. The goal of the study
is to produce a set of competing methods for estimating design floods in the Brisbane River
catchment. Currently, the project is still in the initial stages. One of the proposed methods is based on
Monte Carlo Simulations (MCS). The proposed MCS framework describes probabilities, mutual
correlations and physical interactions of all relevant factors for flood risk in the Brisbane River
catchment. These factors include rainfall intensity, spatial and temporal distribution of rainfall,
antecedent moisture conditions, initial reservoir volumes, and ocean water levels. Various correlation
models are derived to describe the statistical dependencies between rainfall intensities on the one
hand and antecedent conditions, reservoir volumes and ocean water levels on the other. For spatial
and temporal modelling of rainfall intensities, an innovative method for stochastic generation of space-
time rainfall fields developed by the Bureau of Meteorology is incorporated in the framework.
Advanced sampling techniques (stratified sampling, importance sampling) are applied to increase the
computational efficiency and accuracy. The application of a Monte Carlo framework for design flood
estimation, while complex for large catchments, is necessary to (a) gain a better understanding of
flooding mechanisms and interactions and to (b) support a more robust and integrated design flood
estimation process.
1. INTRODUCTION
The Queensland Floods Commission of Inquiry Final Report (QFC, 2012) contained a
recommendation (Recommendation 2.2) that required a flood study be conducted of the Brisbane
River catchment. In accordance with this recommendation, the State of Queensland is managing the
conduct of this study in a number of separate phases. The second phase consists of a comprehensive
hydrologic assessment. The objective of this assessment is to produce a set of competing methods to
provide best estimates of a range of flood flows across the entire Brisbane River system. The study
includes the implementation of a Monte Carlo framework that can account for the variability of all
relevant factors for flood risk in the Brisbane River catchment. This paper describes the proposed set-
up of the framework and discusses some components in more detail. As the study is still in progress
the findings presented in this paper are preliminary and are subject to further review.

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2. SET-UP OF THE MONTE CARLO FRAMEWORK
The variability of the flood generating factors in the Brisbane River catchment like rainfall intensity,
spatial and temporal distribution of rainfall, antecedent moisture conditions, reservoir volumes,
reservoir operation rules and ocean water levels will be captured within the Monte Carlo Simulation
(MCS) framework. This is the main advantage of a joint probability approach like MCS over the more
traditional Design Event Approach (ARR 1987, 1999), which only accounts for the variability of the
rainfall intensity. To capture the variability of the flood generating factors, the relevant statistical
properties, including mutual correlations, need to be quantified. Furthermore, the respective influence
of these factors on flood levels needs to be assessed.
The latter is achieved using a runoff-routing mode of the Brisbane River hydrological model as
developed by Seqwater (2013). The model is based on the URBS hydrological model (Carroll, 2012a).
URBS is a rainfall-runoff-routing networked model of sub-catchments based on centroidal inflows.
Each storage component is conceptually represented as a non-linear storage. The URBS model setup
can readily represent a networked cascade catchment structure. Seqwater adopted the ‘split’ model to
simulate runoff routing. This model splits the conceptual non-linear storage into sub-catchment routing
and channel routing components. Key routing variables used by URBS are: stream length, catchment
area, fraction urbanised (various degrees) and fraction forested and, optionally, channel roughness
and slope. Backwater effects can be approximated within URBS using multivariate or dependent rating
relationships.
Seqwater divided the Brisbane River catchment into seven distinct sub-catchment models based on
review of topography, drainage patterns, and major dam locations. Seqwater also considered the key
locations of interest for real time flood operations of the dams and the best use of available data
including water level gauges. The seven sub-catchments are further subdivided into sub-areas (539 in
total, see Table 1). The primary reason for introducing the sub-areas is to be able to properly model
spatial variability of rainfall in the catchment.
Table 1The seven sub-catchments used to model the Brisbane River catchment
Sub-catchment Size (km2
) #sub-areas
Stanley River to Somerset Dam 1,324 76
Upper Brisbane River to Wivenhoe Dam 5,645 99
Lockyer Creek to O’Reilly’s Weir 2,964 138
Bremer River to Walloon 634 42
Warrill Creek to Amberley 902 56
Purga Creek to Loamside 209 19
Lower Brisbane River to the river mouth 1,855 109
In the MCS framework for the Brisbane River catchment, event rainfall is characterised by hourly
rainfall time series for each of the 539 sub-areas. These are used as input for the URBS hydrological
model. The URBS model provides several methods to compute excess rainfall from total rainfall. For
the Brisbane River catchment model, the initial loss-continuing loss model was adopted by Seqwater.
This means the net rainfall, i.e. the portion of the rainfall that generates surface runoff, is derived by
first subtracting the initial loss from the event rainfall and subsequently subtracting the continuing loss
from the “remaining” rainfall. Parameters IL (initial loss), CL (continuing loss),  (channel lag
parameter) and  (catchment lag parameter) were derived per event, based on model calibration by
Seqwater (2013).
Figure 1 shows the computational scheme of the proposed setup of the MCS framework. The
procedure illustrated will be carried out separately for each river location/gauge of interest.
Approximately twenty different locations located within the catchment have been identified for
assessment in the Brisbane River catchment study. The components of Figure 1 can be summarised
as follows:
1. Generate N synthetic flood events, characterized by samples of the duration, the annual
exceedance probability (AEP) of the rainfall intensity, initial losses in the sub-catchments,

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reservoir volumes and ocean water levels. In the sampling process, mutual correlations are
taken into account. The choice of N is determined by the sampling method and the desired
computational accuracy.
2. Use the available IFD curves of the Bureau of Meteorology (Green et al, 2012) and CRC-
FORGE correction factors for more extreme events to derive rainfall intensities from the
sampled AEP of step 1 for each of the 539 sub-areas of the URBS model.
3. Apply an Areal Reduction Factor (ARF) to the rainfall intensities of step 2 (ARR, 2013b) and
subsequently derive the catchment average rainfall depth. The catchment in this case refers to
the upstream catchment of the river gauge/location under consideration. Note an alternative
approach for steps 2 and 3 will also be investigated in which catchment averaged IFD curves
are derived according to the method of ARR (2013b).
4. For each synthetic event, sample one of the synthetic storm patterns as generated by the BoM
stochastic space-time simulation model and reported in SKM (2013) and scale the rainfall
intensities of these patterns in such a way that the catchment average total rainfall depth of
each event is in accordance with step 3.
5. Prepare input files for the URBS hydrological model for the 539 sub-areas, based on the
samples of step 1 and the scaled synthetic rainfall field of step 4.
6. Simulate the N synthetic events with the URBS hydrological model.
7. Derive (N) maximum water levels, discharges and runoff volumes at the catchment outlet.
8. Apply MCS post-processing to derive water levels, peak discharges and runoff volumes for a
set of pre-defined Average Return Intervals (ARI) or Annual Exceedance Probabilities (AEP).
2+3. IFD for the upstream catchment1. Sampling (N)
- rainfall duration
- AEP of intensity
- initial loss
- reservoir volume
- surge
3. catchment rainfall
-spatial averaging
- areal reduction factor
2. IFD
rainfall intensity
for all URBS
model subareas
4 synthetic events
- random pattern selection
- scaling to catchment rainfall
6 URBS runs
periods of 7 days
5 URBS input
539 subareas
- rainfall time series
- initial losses
- continous losses
- ocean water levels
0.1 0.0010.01
AEP
8 post-processing
- importance sampling
- stratified sampling
7 URBS output
- max h&q&V at output
- for each simulation
N discharges
Q
Figure 1 Schematic overview of the proposed Monte Carlo Simulation framework
The computation scheme of Figure 1 provides a broad outline of what is required for the BRCFS-
hydrology phase: a joint probability approach for the derivation of design flows and volumes, taking
into account spatial and temporal variation of rainfall over the Brisbane River catchment and variability
of initial losses, reservoir volumes and ocean water levels. The Monte Carlo framework is
implemented in Delft-FEWS (Werner et al., 2013). Delft-FEWS is a component-based modeling
framework that incorporates a wide range of general data handling utilities and open interfaces to
many hydrological and hydraulic models. FEWS is mainly used for flow forecasting, e.g. by the
Environment Agency (UK) and the National Weather Service (US). Currently it is also implemented by
Deltares and BoM for flood forecasting in Australia. The component based generic set-up of FEWS
makes the framework relatively easy to transfer to other catchments in Australia.

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3. SAMPLING METHODS
The choice of sampling method is crucial for the BRCFS-hydrology phase. There are several
candidate methods, each with their own advantages and disadvantages, some of which will be
discussed in this section. We emphasise that a final choice of method has not yet been made.
Currently, there are two published MCS methods for estimating design floods in Australia: the
Cooperative Research Centre – Catchment Hydrology (CRC-CH) method (Rahman et al, 2001; 2002)
and the Total Probability Theorem (TPT) method (see e.g. ARR, 2013a). The TPT and CRC-CH
methods differ in their sampling schemes (stratified sampling versus crude Monte Carlo), but also in
the type of rainfall statistics that are used. The TPT method is based on storm bursts statistics, the
CRC-CH method is usually applied to storm event statistics. This is reflected in a number of other
differences between the two methods:
 The CRC-CH method takes the storm duration as a random variable; the TPT method
considers a fixed set of potentially critical durations and takes the maximum design water
levels over the considered durations, which means the duration is not a random variable in the
TPT approach.
 The CRC-CH method uses a parameter  which represents the number of events per year.
The TPT method uses rainfall statistics that are defined in terms of exceedance probabilities
per year.
 As a consequence of the difference in rainfall statistics (burst- versus event-based), the
distributions of other random variables differ between TPT and CRC-CH method. Specifically,
the probability distributions of initial losses, initial reservoir storage levels and temporal
patterns should be consistent with the rainfall statistics.
The CRC-CH method is referred to as conceptually superior by Mirfenderesk et al (2013).
Furthermore, the CRC-CH methodology considers whole storm events and is therefore more suitable
for volume sensitive applications, compared to the TPT method (Carroll, 2012b). On the other hand,
the TPT method has some practical advantages over the CRC-CH. First of all, the nation-wide
available IFD curves from the Bureau of Meteorology (Green et al, 2012) can be applied directly in the
TPT method, since these IFD curves were derived using storm bursts. Should the event-based CRC-
CH method be applied, these curves need to be transformed into IFD curves based on events, which
means an empirical relationship is required that will introduce additional uncertainties and, potentially,
errors. A better alternative for the CRC-CH approach would be to derive IFD curves for events directly
from rainfall data, but this a costly and time-consuming process and may be limited by the availability
of continuous rainfall data.
A second advantage of the TPT method is the efficient stratified sampling scheme, which is not
incorporated in the CRC-CH approach because the latter uses variable event durations. The CRC-CH
method is therefore considered unsuitable for extreme floods (Mirfenderesk et al, 2013). This is for
instance demonstrated in the paper of Rahman et al (2002), which showed a significant variability of
estimated peak discharges for high Average Return Intervals (ARI). This could be resolved by
increasing the number of samples. However, that would lead to millions of hydrological model runs to
reduce the variability to a desired level for high ARIs, making the method untenable. For this reason,
we trialled an alternative approach for the crude Monte Carlo sampling scheme as commonly applied
in the CRC-CH method. The alternative approach incorporates “importance sampling” into the CRC-
CH method. Importance sampling (Engelund & Rackwitz, 1993; Koopman et al, 2009) is a Monte
Carlo sampling technique in which the percentage of extreme events can be increased by adapting
the sampling distribution functions. Importance sampling can potentially strongly reduce the variability
in estimates of probabilities of extremes. The applicability of the method was tested for a hypothetical
catchment, similar to the example used by Rahman et al (2002). Preliminary results are presented in
this paper with details to be published in a later separate paper.
Both the crude Monte Carlo and importance sampling approaches were applied using N=20,000
simulated storm events, representing a series of 4,000 years (5 storms per year on average). The
procedure was repeated 10 times to assess the variability in the Monte Carlo estimates, and the
results are shown in Figure 2. Two disadvantages of the crude Monte Carlo sampling method are
clearly demonstrated in this graph:
 the variation in estimated design discharges increases with increasing value of ARI; and

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 for ARIs that are larger than the number of simulated years (4,000 in this case) there are no
estimates available.
If importance sampling is applied, the variance in estimated peak discharges decreases dramatically
and estimates are available for a much wider range of ARIs. This clearly demonstrates that the CRC-
CH can also be applied for extremely high ARIs (low AEPs) without requiring millions of model
simulations. This creates opportunities for using the CRC-CH method in applications were probabilities
of extremes are relevant, such as the BRCFS project or dam design studies.
10
0
10
2
10
4
10
6
10
8
0
200
400
600
800
1000
1200
1400
1600
1800
2000
ARI (years)
discharge(m3/s)
crude Monte Carlo
importance sampling
Figure 2 Average Return Intervals (ARI) and corresponding peak discharges as estimated from
10 different MC runs with crude Monte Carlo and 10 different MC runs with Monte Carlo with
importance sampling; 20,000 samples.
A further advantage of TPT over CRC-CH is that the methodology builds on the existing Design Event
Analysis (DEA) and will therefore be more readily taken up by the industry’s practitioners
(Mirfenderesk et al, 2013). A disadvantage of the TPT approach is that the concept of a single critical
duration may introduce a probability bias, for example if in reality several durations contribute to the
exceedance probability of a flood level. For larger catchments that consist of multiple sub-catchments,
each having their own runoff characteristics, it is unlikely that only a single duration is critical.
4. CORRELATIONS
Statistical interdependence between flood forcing factors can be relevant for flood hazard assessment
since a correlation between these factors may significantly increase the probability of high discharges
and flood levels. This means the relevant correlations need to be taken into account in the sampling
procedure (step 1 in Figure 2-1). As an example, this section describes the approach for simulating
statistical dependence between initial losses of the different sub-catchments. Information on initial
losses in the Brisbane River Catchment is available from Seqwater (2013). In their study, initial losses
for 48 events were derived from model calibrations. Analyses were carried out for seven sub-
catchments, in which initial losses were assumed uniform over the entire sub-catchment. The resulting
information, a 48 by 7 matrix of initial losses, was used to derive correlations between sub-catchment
initial losses. Table 2 shows the correlations are significantly higher than 0.

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Table 2 Derived correlations of initial losses between the seven sub-catchments.
Area Stanley
Upper
Brisbane Lockyer Bremer Warrill Purga
Lower
Brisbane
Stanley 1.00 0.42 0.51 0.56 0.49 0.47 0.62
Upper 0.42 1.00 0.81 0.64 0.71 0.64 0.52
Lockyer 0.51 0.81 1.00 0.81 0.71 0.56 0.60
Bremer 0.56 0.64 0.81 1.00 0.87 0.60 0.68
Warrill 0.49 0.71 0.71 0.87 1.00 0.64 0.65
Purga 0.47 0.64 0.56 0.60 0.64 1.00 0.89
Lower 0.62 0.52 0.60 0.68 0.65 0.89 1.00
In the Monte Carlo simulations, the various correlations are incorporated using a Gaussian Copula
approach (see e.g. Diermanse and Geerse, 2012). This method requires the n-by n correlation matrix,
C, as input, where n is the number of mutually correlated variables. As proven by Fang et al (2002), C
should be taken equal to sin(/2), where  is Kendall’s rank correlation matrix. The procedure to
generate correlated samples is as follows:
1. Derive a matrix P for which: PP’ = C, through Cholesky decomposition of correlation matrix C
(see, e.g. Strang, 1982). Note: P’ is the transpose of matrix P.
2. Sample values u1,…,un from the standard normal distribution function; store the results in an
1xn vector u.
3. Compute: u*= uP’.
The resulting vector u* is a sample of n correlated standard normally distributed random variables.
The u*-values can subsequently be translated to samples of “real-world” variables such as initial
losses. This translation is done by means of conservation of probability of non-exceedance:
   *i i iu F x  (1)
Where  is the standard normal distribution function, ui* is the i
th
component of vector u*, xi is the
“real-world” realisation of the i
th
random variable and Fi is the marginal distribution function of xi.
Figure 3 shows samples of initial losses for Stanley, Upper Brisbane, Lockyer and Bremer sub-
catchments, that were generated according to this procedure. For each subcatchment, 1000 values for
the initial losses were sampled and this set of 1000 samples is shown in each sub-plot. It can be seen
from Figure 3 that, for example, the sampled losses for Lockyer and Bremer sub-catchments are
stronger correlated than the sampled losses for Lockyer and Stanley sub-catchments, which is in
accordance with Table 2. The fact that this procedure provides simultaneous samples of mutually
correlated variables makes it a powerful tool for Monte Carlo analysis, because this means the
samples provide realistic spatial patterns of initial losses.
The Gaussian Copula is asymptotically independent, which means there is no increased statistical
dependence in the tail ends of the distributions. This makes the model unsuitable for situations where
asymptotic dependence of random variables is present and significantly influences the flood risk. This
might also be the case for the initial losses of the sub-catchments of the Brisbane River catchment, as
extremely wet antecedent soil conditions in the sub-catchments are likely to coincide. In that case,
correlation models that model asymptotic dependence are required, such as the student-t copula
(Fang et al, 2002), the threshold-excess logistic model (Zheng et al, 2013) or the Gumbel-Hougard
Copula (Diermanse and Geerse, 2012). However, the available data set on initial losses is too small to
provide evidence for asymptotic dependence, which is why the Gaussian copula is considered to be
suitable. A sensitivity analysis will be carried out to assess the potential influence of asymptotic
dependence in initial losses on the derived flood levels in the Brisbane River catchment. For this
purpose, additional Monte Carlo simulations will be carried out with the student-t copula and resulting
flood levels will be compared with the flood levels as derived with the Gaussian Copula.

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Figure 3 Correlation plot of simulated initial losses (IL) for four sub-catchments (Stanley, Upper
Brisbane, Lockyer and Bremer)
5. SUMMARY AND CONCLUSIONS
The objective of the Comprehensive Hydrologic Assessment of the Brisbane River Catchment Flood
Study is to produce a set of competing methods for estimating design floods in the Brisbane River
catchment. One of the approaches considered is a joint probability approach, based on Monte Carlo
Simulations. This approach can account for the variability of all relevant factors for flood risk in the
Brisbane River catchment. This paper described the set-up of the Monte Carlo framework and
discussed some components in more detail, such as sampling techniques and models for statistical
interdependencies. The authors wish to stress that this work is still in progress and that all results are
preliminary. Nevertheless, the application of a Monte Carlo framework for design flood estimation,
while complex for large catchments, is considered necessary to (a) gain a better understanding of
flooding mechanisms and interactions and to (b) support a more robust and integrative design flood
estimation process.

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6. ACKNOWLEDGMENTS
The authors want to express their gratitude for the valuable comments on the work described in this
paper by representatives of the Queensland Government, by members of the Technical Working
Group and by members of the Independent Panel of Experts, all of whom are involved in the Brisbane
River Catchment Flood Study.
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