# Monte Carlo simulation for design flood estimation: a review of Australian practice.

1. IntroductionFloods are one of the worst natural disasters; not only costing Australian's millions of dollars annually, but also the loss of human lives and livelihoods. Design flood estimates are often used to quantify potential flood risks when planning or designing infrastructure and developments subject to flooding. This is usually given as a flood characteristic (peak flow, flood volume or time to peak) associated with a particular frequency, such as the annual exceedance probability (AEP). The 'design event' approach was traditionally favoured by hydrologists to estimate design floods in Australia.

Australian Rainfall and Runoff (ARR) has been consistently adopted, throughout Australia, since the release of the first edition in 1958. The document is an official guideline of design practice for flood estimation. The first edition of ARR (Nicol 1958) was an extension of an investigation into rainfall and runoff process in NSW (completed in 1949), and contained simple techniques (e.g. the rational method) which relied on a number of reference tables (as it was prior to the advent of personal computing). In the second edition of ARR (Pattison 1977), measurements were standardised to SI units and additional methods suitable for digital computers were included; however, reasonably simple design flood estimation methods were still widely adopted.

The third revision of ARR (Pilgrim 1987), which practitioners used until only recently, covered existing techniques in ARR 1977 in greater detail, along with a host of new topics (such as facets of hydraulic modelling and stormwater drainage). While more advanced techniques were introduced in ARR 1987, the simpler probabilistic rational method was still widely adopted for estimating peak flows throughout this period. Where the full flood hydrograph was required, however, the design event approach was commonly adopted, based on rainfall-runoff modelling. The design event approach adopts a single set of model inputs, to capture the 'typical' catchment response. A minor update to ARR 1987 was released in 1998, although only the section on large to extreme flood estimation was changed significantly.

The fourth version of ARR (Ball et al. 2016) was released in November 2016. The revision came about due to key issues identified in ARR 1987; including, significant recent changes to flood estimation techniques (mainly due to increasing computational power), theoretical advances and understanding, availability of additional recorded rainfall and streamflow data and the need to cover more topics (such as overland flows). For rainfall-runoff modelling, recommendations in ARR 2016 have shifted towards more holistic approaches such as joint probability approaches; for instance, quantile ensembles, event-based Monte Carlo simulation or continuous simulation techniques. In this regard, Nathan and Weinmann (2013) prepared a discussion paper on Monte Carlo Simulation technique supporting the latest ARR 2016.

The objective of this paper is to present the evolution of the Monte Carlo simulation technique for design flood estimation in Australia. It is expected that the outcomes of this review will enhance further development of the Monte Carlo simulation technique to make it readily applicable in Australian hydrologic practice.

2. Design event approach

While rainfall-based design flood estimation methods have changed throughout the years, the method predominantly adopted until recently (and recommended in ARR 1987) is the design event approach. This approach has been extensively used over the past two decades, both locally and internationally; nevertheless, during this period, there has been significant discussion and critique over some of the practical and theoretical issues of the approach. The design event approach derives complete flood hydrographs through the transformation of a statistical rainfall burst into a design flood event. The transformation assumes a probability neutral transformation --meaning that a flood of a specified AEP originates from a design rainfall burst of the same AEP. In the design event approach, rainfall depths are dependent on the burst duration and severity (i.e. AEP). But all other model inputs and parameters (such as losses) are fixed, with the mean or median values (from calibration or regionalisation studies) typically being adopted.

The basic steps of the design event approach are as follows:

(1) Estimate the design burst rainfall depth for a given AEP and a range of burst durations around the critical duration (often estimated by empirical equations). Then, depending on the catchment size select an areal reduction factor to convert these point rainfall depths to catchment average rainfall depths.

(2) Determine the burst temporal and spatial pattern for a range of standard AEPs (with uniform spatial patterns typically adopted for smaller catchments).

(3) Generate the rainfall excess hyetographs for a given rainfall depth, temporal pattern and a range of durations using a conceptual loss model.

(4) Route the rainfall excess hyetograph, for each duration, through the catchment model to generate the flood hydrograph.

(5) Extract the hydrograph corresponding to the duration with the highest peak flow (known as the critical duration); this event is taken as the design flood hydrograph (and corresponding design peak flow).

The key advantages of the approach are that it has had extensive use throughout Australia (implying that the method has been tailored to Australian conditions); it is able to make use of readily available data-sets (i.e. IFD data, temporal patterns, losses, routing parameters, and so on); it has been widely tested on Australian catchments; and, its limitations are well understood. In addition, the method is simple to use, has minimal data requirements and is computationally efficient. Since the datasets for this method are readily available, its application is quite straight forward.

Some of the disadvantages of the design event approach are due to aleatory uncertainty in hydrologic variables (being the inherent uncertainty in variables), catchment non-linearity, the critical duration concept and storm burst methodology. A more detailed discussion of each of these issues is provided below. A critical review of the design event approach is also discussed by Watt and Marsalek (2013), in the Canadian context.

2.1. Variability of model inputs

Factors affecting the runoff response are complex in nature, and are highly variable over space and time (Rahman, Weinmann, Hoang, et al. 2002). For instance, every storm is unique, with varying depths, durations, temporal and spatial patterns and antecedent wetness conditions. Therefore, the assumption that each of these processes (except for rainfall depth) can be represented by some value of central tendency (such as the mean or median value) is highly questionable and is unlikely to fully represent the catchment system being modelled. In addition to these hydrologic variables, certain water resources structures incorporate variables that represent the state of the system (such as the initial drawdown water level for reservoirs). Likewise, these variables are represented by a single value, which again is not fully representative, as their distributions are often skewed. For example, a reservoir with full supply levels prior to a storm will have a remarkably different response than an empty reservoir--with the latter scenario potentially resulting in no flood downstream of the reservoir.

In addition to these concerns, the variability of flood producing factors is also much more pronounced for more frequent events. This results in significant uncertainties for more frequent events; i.e. an equal sized flood could result from a small storm on a wet catchment or a large storm on a dry catchment (Da Ros and Borga 1997; Zehe and Bloschl 2004).

2.2. Non-linear catchment response

Rainfall-runoff relationships are complex interconnected physical relationships; as such, the representation of these relationships is non-linear and dynamic (changing with rainfall and catchment characteristics). It has been shown that the non-linearity of runoff response is less obvious in wet climates and larger events (i.e. floods over bankfull flow), where it appears that catchments act as a linear system (Zhang and Cordery 1999). Some of the implications of non-linear systems are the inconsistencies with a probability neutral assumption, the sensitivities to initial conditions (small inaccuracies can be amplified), the fact that catchment response cannot be inferred from its constituents (and only so much can be learnt about individual runoff processes), and lastly that catchment response may not be predictable from small-scale, short-term processes (Coulthard and Van De Wiel 2007).

The non-linearity of catchment response is typically represented in many runoff routing models by a single non-linearity parameter. While several studies have shown that catchment non-linearity is dynamic (i.e. the system linearity changes with rainfall and catchment characteristics) (Rahman and Goonetilleke 2001; Zhang and Cordery 1999), a single catchment model is typically assumed (i.e. a single nonlinearity parameter value). Rezaei-Sadr et al. (2012) established a relationship between the antecedent catchment wetness and the non-linearity parameter in WBNM. Rahman and Goonetilleke (2001) also show that large floods in particular are quite sensitive to the non-linearity parameter.

2.3. Critical duration concept

The validity of critical burst duration theory has been questioned for many years now, but has never been comprehensively assessed in the literature. The burst duration resulting in the highest streamflow is random, as it is not only governed by the catchment response but also the storm characteristics. Therefore, using the critical burst duration, it is assumed that the flood produced from the critical duration is the only storm burst that contributes to the flood probability. In reality, however, many other combinations contribute to the total flood probability (Kuczera et al. 2006).

In order to highlight the deficiencies in the critical burst duration approach, Weinmann et al. (2002) compared the flood frequency curves derived using the traditional design event approach to those derived using observed values. By comparing the two, the authors were able to show that the floods observed in the annual maximum series related to storms of varying durations, which thereby define a marginal distribution of flood magnitude (despite the duration of the storm). The use of the critical duration concept is also likened to the assumption that 'the marginal distribution of flood magnitude is equal to the conditional distribution of flooding for the critical rainfall duration' (Weinmann et al. 2002).

Stochastically sampling the rainfall duration is arguably more theoretically sound than critical duration concept. However, while the critical duration concept may introduce some probability bias, the magnitude of this bias and significance to design flood estimates has never been studied explicitly. Even more so, much of the design data in Australia is based on the critical duration concept, which makes the approach more practically viable. Hence, the issue of whether to incorporate the critical duration concept will be based on the purpose of the study and is open to further research efforts.

2.4. Storm burst procedure

The design event approach follows a burst-based procedure, rather than storm-based, meaning that antecedent rainfall is ignored. A burst is the most intense rainfall period within a storm, for a specified duration. In some events the antecedent rainfall has the potential to generate surface/stream flow and fill any storage across the catchment. This will potentially influence the magnitude of the subsequent peak flow. Practitioners have recognised the need to account for antecedent rainfalls when using these procedures; however, this has been typically accounted for by adjusting the rainfall losses.

While the use of burst-based procedures may be due to the ease of calculation, there are three key issues. Firstly, (and historically) the burst depths and temporal patterns have been inconsistent with the losses, which take antecedent rainfall into account; this leads to rainfall losses being underestimated and design flood estimates being overestimated (Hill and Mein, 1996; Kuczera et al. 2006). More recent revisions of design inputs, however, are likely to overcome this issue (Hill et al. 2014). Secondly, ignoring the impact of antecedent rainfall on the flood response can result in peak flows being underestimated, due to an underestimation of the catchment wetness (Phillips, Lees, and Lynch 1994; Rigby and Bannigan 1996; Rigby et al. 2003). Lastly, the rainfall burst duration is usually much shorter than the storm from where it came, resulting in significant errors in the flood duration and the runoff volume. Pilgrim (1987) noted, however, that there are competing biases. Design losses are biased to underestimation, as calibrated events generally occur on a wet catchment, but this is counteracted by pre-burst rainfalls being disregarded, due to the use of rainfall bursts.

To overcome the potential underestimation of peak flows, Phillips, Lees, and Lynch (1994) proposed the use of an embedded design storm. This approach embeds the design burst (from ARR 1987) into a historical storm. Rigby and Bannigan (1996) modified the approach by embedding the design rainfall burst into a longer duration design burst. In this approach, rainfall intensities were adjusted to maintain the average design intensity for the storm. Further to this, Rigby et al. (2003) performed an in-depth analysis of the embedded design storm approach and Roso and Rigby (2006) analysed the approach for internal locations within a catchment.

3. Joint probability approach

The complex hydro-meteorological processes involved in flood generation are not adequately described using fixed values; as is the case in the traditional design event approach. Rather, the variability and dependencies of key model inputs should be considered. The use of fixed model inputs also compromises the probability neutral transformation of rainfall to runoff; potentially leading to bias in design flood estimates. In a seminal paper, Eagleson (1972) provided the foundation for a joint probability framework. In the paper, flood frequency curves are derived using a combination of rainfall and catchment characteristics, rather than streamflow records. Since then, two types of joint probability approaches have emerged; the first technique continuously simulates hydrologic time series over long periods (continuous simulation); whilst the second technique stochastically defines the boundary conditions for thousands to tens of thousands of discrete events (event-based joint probability approach).

3.1. Continuous simulation

Continuous simulation is an approach where continuous rainfall sequences are used to implicitly account for the joint probabilities of all flood producing variables. These rainfall sequences can either be long historic time series or synthetic time series, through stochastic rainfall generators. A detailed review of continuous simulation methods and computer models can be found in Boughton and Droop (2003) and Ling et al. (2015). As its name implies, the technique simulates both wet and dry periods using a long synthetic rainfall record. The flood response is simulated by using a sufficient number of years (usually thousands to tens of thousands of years) to incorporate the interactions of all key joint probabilities of model inputs and parameters. Given the ability of these models to generate continuous runoff sequences, these models are well-positioned for use in flood forecasting (Arduino, Reggiani, and Todini 2005). More recently, however, the ability to generate synthetic rainfall sequences at finer time scales, led to continuous simulation also being used for design flood estimation.

The procedures typically used for design flood estimation using continuous simulation are as follows:

(1) Synthetic hydro-meteorological records (such as rainfall or evaporation) are derived for thousands of years using a stochastic time series generator and observed time series records.

(2) Run the synthetic time series through a continuous rainfall-runoff model (with a pre-calibrated parameter set).

(3) Extract the annual maximum series from the synthetic streamflow time series, for a specified flood characteristic.

(4) Perform a statistical frequency analysis on the resulting annual maximum flow series.

Continuous simulation is an approach that is not limited by the probability neutral assumption associated with the traditional design event approach. It benefits from the ability to inherently consider the joint probability of key variables through both high and low flow periods. As such, initial catchment conditions and dependencies (such as the antecedent moisture condition, initial reservoir levels and dependence between cascaded storages) are implicitly considered. Storms of all durations are modelled, eliminating the need to select a critical storm duration. The method also obviates the need to add baseflow, as required with traditional event-based approaches. Finally, only a single parameter set needs to be calibrated across both wet and dry periods.

Despite the theoretical attraction of continuous simulation, there are several limitations. Ling et al. (2015), for instance, found that these limitations resulted in continuous simulation performing poorly when compared to other commonly used techniques. Traditionally, the key downfall of this approach was in data management and model efficiency; however, with the advent of modern technology, this task has become more routine and soon this may not be a cause for concern. Current limitations of the technique include the considerable effort required to calibrate the model (particularly problematic in larger catchments where there are significant complexities), the vast data requirements (spatially and temporally), the reliance on a synthetic rainfall generator, the lack of practical experience, and the inability to comprehensively incorporate uncertainties in the analysis. Some of these points are subsequently covered in a greater detail.

Continuous simulation relies on a stochastic rainfall generator that realistically reproduces the low to extreme rainfalls of observed rainfall records, as well as the dry periods between them; traditional approaches (such as design event approach) on the other hand are only interested in the moderate to extreme rainfall events, which makes for a much simpler model. The literature shows that methods used to generate synthetic time series at course time scales are quite realistic; however, the problem lies in reproducing observed rainfall statistics at finer temporal resolutions, such as hourly or sub-hourly data that is generally needed for design flood estimation (Boughton and Droop 2003; Kuczera et al. 2006; Srikanthan and McMahon 2001). As such, a typical approach is to generate the courser data series and subsequently disaggregate the series using daily rainfall statistics. Kuczera et al. (2006) points out that stochastic rainfall models are on the verge of practical application.

A number of studies outline approaches used to quantify the uncertainties in rainfall-runoff models; however, it is well understood that adopting Monte Carlo-based uncertainty estimation for continuous simulation is quite a challenging task. This task is made difficult as the distributions of the parameters and input variables of the model need to be specified. Continuous simulation intrinsically incorporates some estimate of the inherent uncertainties in the inputs by adopting many realisations of the stochastic input variables; however, this does not fully incorporate all sources of error and uncertainty.

Another point to be made is related to the structure and parameterisation of continuous models. These models have been developed to reflect the underlying physical processes, such as evaporation, infiltration, percolation, sub-surface flows and groundwater flows. However, imperfect understanding of these processes during high flow events and restricted data series for calibration result in many of these models being inappropriate for design flood estimation. Though research continues in this space.

3.2. Event-based joint probability approach

The event-based joint probability approach is similar to the continuous simulation approach (discussed in the previous section) in that the interactions between all joint probabilities are considered; however, the key difference is that the event-based approach only considers these interactions during key periods of interest (such as large storm events when investigating flood risk). It should be noted, however, that this approach does not simply ignore the periods in-between storm events, but rather treats them as probability distributed inputs which are randomly varied (or varied in relation to some correlation structure). This can be problematic as the distributions are derived separately, prior to this analysis.

Steps involved in the event-based joint probability approach, implemented via Monte Carlo simulation, are as follows:

(1) Identify model inputs/parameters that should be treated as stochastic variables and those that should be fixed;

(2) Define probability distributions for each of the stochastic variables and their correlations;

(3) Generate an input set (which includes fixed and stochastic inputs), with stochastic values sampled from the probability distribution defined in the last step (considering correlation structures where needed);

(4) Run the generated input set through a rainfall-runoff model and record the flood characteristics of interest;

(5) Repeat steps 3 and 4 (n) times (typically thousands of iterations); and

(6) Calculate the AEP of each event.

Being that continuous simulation and Monte Carlo simulation only differ in the definition of their boundaries they share some key benefits, for instance; (i) both are able to treat probability effects rigorously, thus accounting for natural variability; (ii) both model the antecedent conditions, albeit Monte Carlo simulation in a more simplistic manner (by adopting probability-distributed inputs); and (iii) both can model storms of all durations, removing the need to select a critical burst duration. Monte Carlo simulation, however, has several additional benefits; for instance, it is more amenable to uncertainty analysis, it is closely related to current and historic Australian flood estimation practice (therefore, much of the knowledge and data can be easily transferred across) and it is also more computationally efficient, particularly where variance reduction techniques are utilised (such as importance sampling or stratified sampling).

Although specific to the problem at hand, event-based Monte Carlo methods have several advantages over other approaches; nevertheless, there are several other considerations and downfalls to the method. For instance, the specification of probability distributions for model inputs can be quite difficult, and the determination of the correlation structure between variables, which could be linear or otherwise, also presents a major challenge. In addition, while implicitly considered in continuous simulation, Monte Carlo methods need to explicitly account for aleatory (statistical) uncertainties; which are different to epistemic uncertainties that exist in every model due to data or knowledge limitations. Lastly, consideration needs to be made to the processing time, which varies substantially depending on the problem at hand. In a simple case with a basic hydrologic model, Monte Carlo simulation (using automated processes, such as those provided in RORB or URBS) is quicker than traditional deterministic methods, due to manual post-processing; however, with more complex models (such as distributed or hydraulic models), Monte Carlo simulation becomes a more significant computational burden. For example, with a modest simulation time of 1 s and 5000 iterations the total run time will be about 1.5 h. These issues can be overcome, however, with continuous improvements being made to techniques (ensemble modelling or variance reduction methods), software (increasing efficiencies and allowing the use of graphics processing units) and technology.

The derived flood frequency distribution technique was first introduced by Eagleson (1972). It contains a stochastic rainfall model, a runoff routing model and a derived distribution technique that calculates the probability-distributed peak flows. The first runoff routing model adopted by Eagleson (1972) was a kinematic wave-based model. Similarly, following on from Eagleson's work, Shen, Koch, and Obeysekara (1990), Cadavid, Obeysekera, and Shen (1991) and Muzik (1994) also based their models on kinematic wave theory.

Later, the derived distribution approach was based on the geomorphological instantaneous unit hydrograph (GIUH) developed by Rodriguez-Iturbe, Gonzalez-Sanabria, and Bras (1982), such as those studies by Hebson and Wood (1982) and Wood and Hebson (1986). Diaz-Granados, Valdes, and Bras (1984) further incorporated the Philip's equation with the GIUH to model infiltration. However, Moughamian, McLaughlin, and Bras (1987) tested these formulations by Hebson and Wood (1982) and Diaz-Granados, Valdes, and Bras (1984) and found they performed poorly. Sivapalan, Wood, and Beven (1990) utilised a similar approach to investigate partial contributing areas, as generated through infiltration- and saturation-excess overland flow runoff mechanisms, which was further modified by Troch et al. (1994). Infiltration was also modelled using the Soil Conservation Service Curve Number (SCS-CN) method by Raines and Valdes (1993). Kurothe, Goel, and Mathur (1997) considered that the rainfall intensity and duration could be described by negatively correlated probability distributions. Lastly, following on from Kurothe, Goel, and Mathur (1997), Goel et al. (2000) allows for both negative and positive correlations between rainfall intensity and duration.

Another stream of research adopted the derived distribution approach based on the rational method (Haan and Wilson 1987). Brodie (2013) assessed their approach against an at-site flood frequency analysis (FFA) on an urban catchment and found limitations due to the simplicity of the method. Gottschalk and Weingartner (1998) treated the rational method in a probabilistic fashion, but only specified the probability distribution of the runoff coefficient (Franchini, Galeati, and Lolli 2005). Michele and Salvadori (2002) took this a step further to derive probability distributions of peak discharges (for a specific AEP), including a description of infiltration using the SCS-CN method. Franchini, Galeati, and Lolli (2005) extended this analysis further to consider the probabilistic CN parameter, as well as the antecedent moisture condition.

In another line of research, Iacobellis and Fiorentino (2000) developed the index flood (IF) model based on partial contributing areas. Fiorentino, Salvatore, and Iacobellis (2007) verified their proposed model structure against a continuous simulation approach using a distributed hydrologic model. Gioia et al. (2008) derived the two-component IF (TCIF) model, based on the assumption that different runoff mechanisms produce frequent and extreme flood events. Then, Fiorentino et al. (2011) produced regional parameter estimates for both the IF and TCIF models for prediction in ungauged Italian catchments. These two models have dominated recent literature; however, other recent applications of the analytical derived flood frequency approaches are provided by Allamano, Claps, and Laio (2009), Viglione and Bloschl (2009), and Brodie (2013).

Each of the aforementioned studies are based on the same key framework (being the analytical derived distribution approach) and largely sought to improve the catchment response mechanisms or the statistical rainfall model. This method is advantageous as the effect of the inputs on the resulting AEP can be clearly distinguished in the final set of equations. Nevertheless, it suffers from being mathematically complex, hence is impractical to use in many real-world scenarios (Rahman et al. 1998). Furthermore, assumptions that are required to simplify the problem also cause the method to perform poorly; simplifications are largely made to the rainfall model, meaning that the spatial and temporal distribution of rainfall is typically ignored. Therefore, the method is only feasible when the catchment response model and the stochastic elements of the model, specifically the rainfall and antecedent conditions, are sufficiently simple to warrant its use.

The most commonly used joint probability approach, both in research and practice, is the approximate solution. This is highlighted by the number of studies on the topic. Approximate solutions are mainly used due to their ease in application, particularly when it is too difficult to apply analytical methods. Stochastic simulation is a special variant of the approximate solution, which is a branch of experimental mathematics concerned with experiments on random numbers (Rahman et al. 1998), also known as Monte Carlo simulation.

One of the earlier applications is by Laurenson (1974) who adopted a transition matrix to solve total probability theory in rainfall-runoff modelling. Another early application is by Beran (1973), whom stochastically sampled both the storm depth and duration to estimate flood frequencies. Tavakkoli (1985) went on further to consider the dependence of key inputs (Sivapalan, Bloschl, and Gutknecht 1996). Muzik (1993) included the stochastic nature of antecedent moisture conditions using the SCS-CN method to model runoff generation processes. Early application to water storage structures was performed by Durrans (1995), for which the technique proved to perform well. Bloschl and Sivapalan (1997) derived a more practically applicable Monte Carlo simulation technique.

Since these early applications of Monte Carlo simulation techniques, there have been significant increases in the processing power of personal computers; with high-performance computing often available for use these days. These advancements have given rise to Monte Carlo simulation techniques being researched much more extensively and being adopted into more practical applications, both nationally and internationally. Likely one of the most cited papers of Monte Carlo simulation is that by Rahman, Weinmann, Hoang, et al. (2002). In the study, the authors use a non-linear routing scheme coupled with a lumped loss model (initial loss-continuing loss model, herein known as IL-CL) within a Monte Carlo simulation framework. Stochastic rainfall characteristics are adopted (rainfall intensity, duration and temporal distribution), along with the antecedent moisture condition of the catchment, being the initial loss (IL) parameter, while all other inputs are fixed. Through comparison to at-site FFA for three Victorian catchments (Australia), the Monte Carlo simulation method was found to perform remarkably well.

Around the same period as the study by Rahman, Weinmann, Hoang, et al. (2002), Loukas (2002) describes a slightly simpler framework, where the Nash model (using linear reservoir routing) is coupled with a power relationship to describe infiltration. In line with critical duration concept, the duration was fixed. While rainfall depths and temporal patterns were treated as stochastic inputs, all other parameters were fixed. Albeit temporal patterns were treated in a simplistic manner--using a triangular form. The application of the approach on eight British Columbia catchments show that the method produced reliable, robust estimates of peak flows and flood volumes. At the same time, another study by Nathan, Weinmann, and Hill (2002) derived a more practically applicable Monte Carlo framework in line with traditional guidelines on flood estimation and dam safety management. Similar to Loukas (2002) critical duration concept was adopted--in line with design inputs and guidelines. However, in addition to treating the rainfall depths and temporal patterns as stochastic inputs, Nathan, Weinmann, and Hill (2002) also considered seasonality, initial loss and reservoir operations (specifically drawdown and spillway condition) as stochastic variables. The expected probability quantiles were then estimated using the Total Probability Theorem. By estimating floods for events rarer than the 1 in 20 year event, the authors were able to demonstrate the benefits of combining a Monte Carlo framework with traditional guidelines.

In order to apply Monte Carlo simulation to ungauged or poorly gauged catchments, Aronica and Candela (2007) derived a methodology where calibration of the rainfall-runoff model is not required. The authors adopted a stochastic rainfall generator to model rainfall, and the instantaneous unit hydrograph coupled with the SCS-CN method to model the catchment response. The rainfall depths were stochastically generated using a two-component extreme value distribution (describing the effect of two different meteorologic factors) for a fixed duration, and regionalised through a cluster analysis. The antecedent moisture condition was also stochastically varied according to a discrete probability distribution. The application of the model on six Sicilian catchments (Italy) resulted in reasonably accurate estimates of observed floods.

At the start of the new millennium, a range of research efforts began to focus on bridging the gap between event-based Monte Carlo simulation and continuous simulation methods. The SCHADEX method (Simulation Climato-Hydrologique pour l'Appreciation des Debits Extremes) (Paquet, Gailhard, and Garcon 2006), for instance, uses a continuous rainfall-runoff model with synthetic rainfall events that are placed within historical rainfall data series; thus, overcoming the need for continuous rainfall generation models. Although the soil moisture conditions are not explicitly considered as stochastic variables, the wetting and drying sequences are described by a continuous rainfall-runoff model based on historic records. Garavaglia et al. (2010) went further to produce more robust method for the derivation of synthetic rainfall events based on underlying weather patterns--namely the Multi-Exponential Weather Patterns (MEWP) distribution. By combining the SCADEX method with the MEWP distribution (Garavaglia et al. 2010), Paquet et al. (2013) found good results for the River Tarn in the South of France.

In a similar space, Kjeldsen, Svensson, and Jones (2010) adopted an event-based rainfall-runoff model with synthetic rainfall events that are temporally related to one another through statistical distributions. Initially, a stochastic rainfall generator is used to create a partial series of individual events using stochastically sampled rainfall durations, intensities and inter-event arrival times from respective probability distributions. Following this, runoff hydrographs are calculated using the PDM, with additional stochastic parameters; specifically, the initial flow and soil moisture deficit at the start and end of the event. In addition, dependencies and seasonal variation of key parameters were also considered. Peak flow estimates from the river Blyth in England were found to produce reasonably accurate estimates as compared to observed flood events.

Svensson, Kjeldsen, and Jones (2013) extended the work of Kjeldsen, Svensson, and Jones (2010); one of the changes resulted in the original triangular profile (representing the temporal distribution of rainfall) being replaced by a stochastic double peaked triangle (reflecting two rainfall bursts). The revised technique was applied and compared to at-site FFA estimates for four catchments in England; overall more frequent events were reasonably accurate, but inaccuracies were found in the rarer events. Although it should be noted that both methods (including the at-site FFA) are prone to errors for infrequent events due to the rarity of such events in observed records.

In other studies, Saghafian, Golian, and Ghasemi (2014) considered the joint probabilities of rainfall intensities on catchment sub-areas; Kottegoda, Natale, and Raiteri (2014) derived stochastic rainfall hyetographs for use in Monte Carlo simulation; Li et al. (2016) incorporated more detailed analysis of seasonality, particularly for the estimation of soil moisture; and several more have coupled hydrologic model outputs with hydraulic/ hydrodynamic models (Aronica et al. 2012; Kalyanapu et al. 2011; Yu, Qin, and Larsen 2012).

3.3. Australian practice

Australian research into the practical application of design flood estimation within a Monte Carlo framework has been around for two decades; however, it is only recently that Australian guidelines (eds. Ball et al. 2016) recommended that the joint probability of key factors are to be considered explicitly in rainfall-runoff modelling. The guidelines recommend one of two approaches, being an ensemble approach or Monte Carlo approach. The ensemble approach simulates a discrete number of events (10-20), each with a unique rainfall temporal pattern (or alternatively IL value) with all other inputs fixed. The approach is more simplistic than a full Monte Carlo simulation, but attempts to capture the variability of the main flood-producing factor (only a single variable). Alternatively, Monte Carlo simulation considers the stochasticity and joint probability between key flood-producing factors, while keeping all other factors fixed. In line with critical duration concept, the recommended approach suggests taking an envelope of model results to determine the rainfall duration that results in the maximum value--rather than treating the duration as a stochastic variable. More guidance can be found in Chapter 4 of ARR (Nathan and Ling 2016; Nathan and Weinmann 2016), resulting from an earlier discussion paper by Nathan and Weinmann (2013); however, other reviews of previous applications and theory are also available. Rahman et al. (1998), for instance, provided an early review of joint probability analyses, but after years of practical application Nathan and Hill (2011) provided an overview of the practical implementation of Monte Carlo methods.

Monte Carlo simulation originally required some degree of programming knowledge (or at least advanced excel skills); however, these techniques have recently become more accessible to every practitioner with a range of tools, techniques and software packages available. For example, the ensemble approach recommended in ARR 2016 is a simple extension of the traditional approaches. Where a full Monte Carlo approach is required, other simplifying techniques have included the use of look-up tables (e.g. Sih et al. 2012), which also overcomes the computational burden of 2D hydraulic models that can take hours or days to run. Tools that have been released, such as the joint probability modelling tool for estuarine environments (Westra, Leonard, and Zheng 2016) or the RORB temporal pattern extractor tool (Laurenson, Mein, and Nathan 2010) enable more complex techniques to be implemented with ease. Further to this, a few software packages now enable practitioners to readily employ Monte Carlo simulation, including RORB (i.e. Nathan, Mein, and Weinmann 2006) and URBS (Carroll 2001). The availability of these tools, techniques and software packages have further promoted the application of Monte Carlo techniques within Australian design flood estimation practice.

Following the millennium, Monte Carlo simulation surged in Australia with a myriad research projects aimed at improving knowledge for the practical implementation of design flood estimation techniques. An early method, proposed by Rahman et al. (2001) (also outlined by Weinmann et al. 2002), considered four key flood-producing factors to be stochastic variables; namely, the rainfall depth, event duration, temporal distribution of rainfall and initial rainfall loss (IL). Rainfall depths were extracted from IFD data, the IL and event duration were randomly sampled from probability distributions and the rainfall temporal patterns were sampled from a pool of historic events. The method was used to estimate the peak design flood for three small, simple Victorian catchments (78-127 [km.sup.2]) using a single non-linear storage routing model (Rahman, Weinmann, Hoang, et al. 2002). While the method showed promise over traditional approaches, it was limited by the simple routing scheme employed which is unrealistic in real-world scenarios. Therefore, the methodology was later extended to adopt semi-distributed runoff routing models (such as RORB and URBS) and tested, with success, on several catchments along the east coast of Australia (Carroll and Malone 2008; Charalambous, Rahman, and Carroll 2013; Pate and Rahman 2010; Rahman, Smith, and Stathos 2005; Rahman et al. 2006).

These early studies made no assumption of the critical duration by randomly sampling rainfall durations; however, this also meant that standard Australian design information (such as design IFD rainfalls) could not be adopted. For this reason, Nathan, Weinmann and Hill (2003) proposed a Monte Carlo technique based on critical duration concept (further details in Nathan and Weinmann 2004, 2013). Like Rahman et al. (2001), the rainfall depth, temporal distribution of rainfall and IL were treated as stochastic variables; however, given the study sought to estimate the reservoir outflow up to the probable maximum flood, additional factors were considered including seasonality, initial reservoir level and the spillway condition (i.e. the likelihood of blockage). Rainfall depths are extracted from IFD data for fixed durations, the season is extracted from a discrete distribution (based on seasonal frequencies of past floods), the IL and initial reservoir levels are randomly sampled from a conditional probability distribution (based on the season) and the spillway condition is sampled from a discrete distribution (based on historic likelihood of blockage) This methodology was tested on a much larger hypothetical catchment (1,500 [km.sup.2]) with a storage reservoir to estimate a range of possible outflows (rather than a single estimate). The approach showed promise as a way to include current design inputs in a Monte Carlo approach.

Since this early research there have been numerous practical applications of design flood estimation within a Monte Carlo framework. While much of the literature has been covered here not everything could be included, with many projects having never been openly published. In some earlier applications, joint probability techniques were used to simulate the dependence between the initial reservoir levels of two or more cascaded storages. For instance, Nazarov (1999) investigated the impact of storage dependence (between Lake St Clair and Lake King William at the head of the Derwent River, Tasmania) on the flood capacity of the spillway using a cascaded storage equation. The authors found that the dependence between initial reservoir levels had a direct impact on the AEP of critical levels. Following on from this, Smythe and Jeffries (2002) adopted a simpler single storage equation to approximate the dependencies between four cascading storages on the Pieman River (Tasmania). In their study, the authors assumed a storage level based on the joint probability analysis of the upstream dam which was then used to calculate inflows to the downstream storage. The cascaded and single storage equations were later compared by Nandakumar and Green (2003) for a cascade of two storage reservoirs in NSW (Keepit and Split Rock Dams). They noted small biases between the two approaches, but overall recommended the simpler single storage equation due to efficiency gains and only minor reductions in accuracy.

While the dependence between initial reservoir levels is important, more recent studies have also considered the variability in other key flood-producing factors. Mittiga et al. (2006, 2007), for instance, considered the uncertainty in extrapolated rainfall depths up to the PMP and the catchment routing parameter ([k.sub.c] in RORB). Confidence limits derived from 100 sets of 10,000 Monte Carlo simulations highlighted that routing parameter uncertainties dominate the variability for more frequent events, while the rainfall depth becomes more important with increasing rarity. In more detailed hydrologic study of the Hume Dam, Nandakumar et al. (2011) considered the variability of rainfall depths, seasonality, temporal distribution of rainfall, initial moisture content of the catchment and the initial reservoir level for two dams (including their dependence). In a similar study of Lake Rowallan in Tasmania, Robinson et al. (2012) investigated the effect of stochastic inputs on outflows. They found that the initial moisture content of the catchment had negligible effect on the outflows, the temporal distribution of rainfall had a noticeable effect on outflows and that the initial reservoir level had the biggest impact on outflows (particularly in more frequent events). In another study, rather than the flood capacity of the spillway being assessed, Cohen et al. (2015) considered the warning times of a high consequence dam with stochastic variables including the rainfall depth, initial moisture of the catchment, temporal distribution of rainfall and initial reservoir level. For this purpose, they found that the initial reservoir level had the largest effect on trigger times for shorter events, but the temporal distribution of rainfall became dominant for longer events. From this sample of studies, it is clear that while a number of variables can directly impact results, the initial reservoir level and temporal pattern of rainfall have consistently proved to be an important consideration when assessing dam safety and operation.

Several flood studies have also incorporated joint probability analyses due to uncertainties in model inputs. Within the early 2010s a flood study was commissioned for the Hawkesbury-Nepean valley due to a government initiative to reduce the flood risk for surrounding communities (Babister, Retallick, Loveridge, et al. 2015; Babister, Retallick, Varga, et al. 2015; Babister, Retallick, Varga, Loveridge, et al. 2015; Babister, Retallick, Loveridge, Testoni, et al. 2016). Their study primarily considered mitigation strategies such as gate operations, pre-release and dam raising, but was also used to inform emergency management strategies. Therefore, unlike most previous studies, many other flood characteristics were of interest; for instance, the rate of rise or recession, time to reach a certain trigger height (such as bridge levels), time of inundation (above a certain level), whether a trigger height is reached or the flow exceeds a certain threshold. For these purposes, stochastic variables included the rainfall depth, spatial and temporal distributions of rainfall (disregarding any dependence between the two), initial moisture content of the catchment, pre-burst rainfall, initial reservoir level and the timing of tributary response. Synthetic events were run through a hydrologic model to derive flows, followed by a 1D hydraulic model (given its speed) to derive flood levels. By comparing simulated and historic events, the model proved to be effective at matching the variability and magnitude of flood characteristics. Overall, the authors found that the Monte Carlo framework is a powerful tool for considering the entire range of possible flood events, including the optimisation of management strategies.

So far, many studies have considered temperate catchments with storage reservoirs; however, given the diversity throughout Australia other factors may need to be considered. In a quite unique Australian study, Stephens et al. (2016) included snowmelt in their assessment of the annual flood risk for two catchments in the Snowy Mountains (south-east NSW). They found that while snowmelt had an appreciable impact on the flood risk in large events, it had minimal effect on the flood risk for the PMP AEP.

While snow cover is limited in Australia, we are surrounded by ocean; therefore, many studies have investigated the joint probability between extreme rainfall events and sea levels (e.g. Aijaz et al. 2011; Need, Lambert, and Metcalfe 2006; Westra 2011). One larger project not only determined the dependence along the entire coastline of Australia, but also created an efficient joint probability framework and a tool that implements the more complex aspects (see Westra, Leonard, and Zheng 2016). It was found that the correlation between extreme rainfall and storm surge varied throughout Australia and is influenced by the storm burst duration and lag between the extreme rainfall event and storm surge event (Westra 2011; Zheng, Westra, and Sisson 2013; Zheng et al. 2014). Though it is important to note that both very small and large catchments are unlikely to be affected by the coincidence of extreme rainfall and storm surge events due to the time taken for the peak flow to reach the outlet. In the joint probability framework, a table of flood levels corresponding to different exceedance probabilities for extreme rainfall and storm surge is estimated, typically using a hydraulic model. This framework was tested in the Nambucca catchment using a 2D hydraulic model (TUFLOW), with the authors noting that although computationally expensive, the framework produced good results (Leonard et al. 2015).

In other circumstances, it might be important to consider the coincidence of flows between tributaries. For example, Sih et al. (2012) conducted a thorough flood study of the Nerang River catchment (south-east QLD) to assess the floodplain risks of the lower Nerang catchment. As with previous studies a range of key flood-producing factors were taken as stochastic model inputs, including the rainfall depth, temporal and spatial distribution of rainfall (assuming independence), initial moisture content of the catchment and initial reservoir level. However, given the complexities of the floodplain, the coincidence between the two main tributaries and sea level were also considered using a multidimensional table of levels (from hydraulic model results). While the authors found the method to be computationally expensive, the framework showed promise for future investigations.

Several studies to this point have considered the spatial and temporal distribution of rainfall but assumed they were independent. This simplified assumption was overcome by Seed, Srikanthan, and Menabde (1999), who proposed a space-time stochastic model of rainfall that uses a multiplicative cascade model in combination with a hierarchy of ARMA models to distribute rainfall first in space then time. Several investigations have shown that these rain fields match the statistical structure of observed events (Seed 2004; Seed, Srikanthan, and Menabde 1999; Seed et al. 2014). Jordan, Nathan, and Seed (2015) tested these patterns for estimating design floods within a Monte Carlo framework against simpler approaches on the Stanley River catchment (south-east QLD). The authors found that dam inflows were insensitive to the spatial distribution of rainfall (including space-time coincidence), if the temporal distribution of rainfall was stochastically sampled. However, it should be noted that these results will not necessarily generalise to other catchments.

Finally, the Brisbane River flood study--the most comprehensive Australian flood study to date--arose due to a flood inquiry following the devastating floods of 2010/2011 (e.g. Diermanse et al. 2014). There were many facets to this study, bringing together many of the elements already discussed and more. Three Monte Carlo frameworks were initially tested, including methods that adopt the critical duration concept and others that treat rainfall duration as a stochastic variable (Aurecon et al. 2015). Biases were noted between the approaches due to differences in the treatment of IFD data; however, the Monte Carlo framework that assumes a critical duration was adopted due to its ability to adopt current Australian design information. The adopted Monte Carlo framework was applied separately to 23 sub-catchments with the rainfall depth, space-time rain fields, initial moisture of the catchment and initial reservoir level treated as stochastic variables; with Ryan et al. (2015) also considering stochastic tidal levels using a 1-dimensional hydraulic model. But even more-so the dependency between key factors was also considered, including the spatial and temporal correlation of rainfall, coincidence of extreme rainfall events and storm surge events, correlation between extreme rainfall and initial reservoir level (remaining uncorrelated for frequent events), mutual dependence between the initial moisture of the catchment and initial reservoir level and the dependence between initial moisture contents of different sub-catchments. Given the need for hydraulic assessment of the floodplain within a Monte Carlo framework, Ryan et al. (2015) found that simpler 1D models show more potential with a selective subset being run through a 2D model for greater detail. Nathan et al. (2016) took this further and suggested more effort be spent on the transformation of rainfall to flow (opposed to flow to depth), given that aleatory uncertainties have a bigger impact on flows (than depths). Other notable publications include those by Diermanse et al. (2016), Ayre et al. (2015), Barton et al. (2015), Diermanse et al. (2015), Jordan et al. (2014) and Seed et al. (2014).

3.4. Probabilistic model inputs for Australian catchments

Monte Carlo simulation for flood estimation requires the expected variability of key model inputs to be defined using data from the site of interest or, when this is not possible, using regional data from similar locations. Various model inputs have been investigated for two decades now, however, it is only in the past 15 years or so that the probabilistic nature of these inputs has been studied in all respects and applied within full Monte Carlo frameworks. The most important input to any rainfall-runoff model, including those within a Monte Carlo framework, is the rainfall depth, which is often described using design IFD rainfall data--relating the rainfall depth to the event duration and severity (AEP). Other model inputs can also play an important role in the transformation of rainfall to runoff; however, their importance in any particular study is usually dependent on several factors, including the purpose of the study (i.e. whether the peak flow, volume, timing or other characteristic is of interest) and catchment features (i.e. the presence of large storages or downstream tidal boundaries).

Over the past two decades, design IFD rainfall data was provided within ARR 1987 for durations ranging from 5 min to 72 h and Average Recurrence Intervals (ARIs) between 1 year and 100 years. Since the release of this IFD data, practitioners have noted key gaps, for instance, that the rainfall depths are based on the burst within a complete storm event and that the rainfall depths do not extend to very frequent or rare events. Several research efforts have since filled these gaps. The Cooperative Research Centre--FOcused Rainfall Growth Estimation (CRC-FORGE) method arose following the work of Nandakumar et al. (1997) to estimate design rainfall depths for very rare long-duration events. The CRC-FORGE method has since been applied to all states and territories in Australia (Durrant and Bowman 2004; Gamble, Turner, and Smythe 1998; Hargraves 2004; Hill et al. 2000; Nandakumar et al. 1997, 2012). Jordan et al. (2005) went further to provide estimates of very rare short-duration rainfall depths in southern Australia. Similarly, Haddad, Rahman, and Green (2011) applied the Generalised Least Squares Regression (GLSR) and L-moments methods to estimate very rare short-duration rainfall depths.

Storm-based IFD rainfall data may also be required in situations where the total rainfall volume is of interest. Several studies have derived storm IFD rainfall data, including those by Carroll (2008) for south-east QLD and Caballero et al. (2011) who regionalised data across NSW. These methods are limited by the event definition adopted, as inconsistencies between design inputs may arise without industry accepted standards for the event definition. In an alternative approach, Minty and Meighen (1999) transformed design burst IFD rainfall into storm IFD rainfall through the analysis of pre-burst rainfall depths for long-duration storms in southern Australia. More recently, Loveridge et al. (2016) applied a similar transformation for NSW catchments using the storm to burst rainfall ratio (from historic events). However, the simplistic conversion of burst IFD data to storm IFD data is limited by the fundamental differences between burst and storm events. Leonard, Ball, and Lambert (2011), for instance, highlighted the coincidence of extreme rainfall bursts, as found in the design IFD rainfall. This essential duplication of events across multiple durations, however, does not hold true for complete storm events, which are only included once (only the single rarest internal burst period of any duration is included). While storm-based IFD rainfall would be useful there are implicit difficulties in their derivation, particularly with event definition and the lack of unique events (both in space and time).

A comprehensive regionalisation of design IFD rainfalls across Australia was recently completed by the BoM (Green et al. 2015; Green, Johnson, et al., 2016), for 1 min to 168 h and from 12 Exceedances per Year (EY) to an AEP of 1 in 2000 years (noting that very rare rainfalls are only provided for durations over a day). A quality controlled pluviograph database (Green, Xuereb, and Siriwardena 2011) was used to extract Annual Maximum Series (AMS). Each AMS was then fitted to a GEV distribution to derive L-moment statistics (Green et al. 2012), with sub-daily statistics inferred using a Bayesian GLSR (Johnson, Haddad, et al., 2012). Each stations sampling uncertainty was reduced using a region of influence scheme (Johnson, Xuereb, et al., 2012) and then the final data was gridded using thin plate smoothing splines within ANUSPLIN (The et al. 2014). This approach was used to derive the frequent to infrequent design IFD rainfalls, however, as with the ARR 1987 IFD data, these estimates required extending to very frequent and rare rainfalls. The rare design rainfall estimates were derived in a similar fashion, but with the minimum data length increased to 60 years and the region of influence increased to a minimum of 2000 station years (Green, Beesley, et al., 2016). The very frequent design rainfalls were extended using the Partial Duration Series (PDS), opposed to the AMS, (Xuereb and Green 2012) fitted with L-moments to the generalised Pareto distribution (The et al. 2015).

Jolly, Velasco-Forero, and Green (2015) compared the reviewed design IFD rainfall data (Green, Johnson, et al., 2016) to ARR 1987 estimates for Tasmania. It was found that the additional density and length of rainfall stations, along with advanced statistical techniques (i.e. elevation enhanced regionalisation schemes) led to more representative design rainfall estimates, particularly in topographically complex areas. In another study, Stensmyr and Babister (2015) compared areal IFD rainfall estimates for the Brisbane River and Hawkesbury-Nepean River catchments to the newly revised design IFD rainfalls with Areal Reduction Factors (ARFs) applied. The authors noted that while there were substantial differences in the methodologies, both produced remarkably similar results. A similar analysis by Scorah, Lang, et al. (2015), found consistent bias when using the AWAP database to derive areal IFD rainfalls. Other comparisons have also been made to alternative IFD methodologies by Beesley and Green (2015) and CRC-FORGE estimates by Green, Beesley, et al. (2016).

Following rainfall depths, the temporal distribution of rainfall has received the greatest attention in Australian literature. Early research efforts extracted observed patterns of rainfall for the estimation of the PMP or PMPDF; for instance, Nathan (1992) extracted a set of observed temporal patterns for rare, long-duration storms in southern Australia. Further to this, Meighen and Minty (1998), Walland et al. (2003), Green et al. (2003, 2005) also extracted rainfall temporal patterns for various catchments in south-east Australia and tropical Australia. Jordan et al. (2005) later extracted patterns for rare, short-duration events (30 min to 12 h). Several other regional studies have extracted and regionalised observed temporal patterns of rainfall; for instance, Caballero and Rahman (2013) adopted an Inverse Distance Weighting (IDW) scheme to regionalise temporal patterns, Herron et al. (2011) pooled 50 regional patterns per duration within the greater Melbourne area and similarly, Cunningham and Muncaster (2011) pooled temporal patterns within the Melbourne region. In a more unique approach, Scorah, Lang, et al. (2015) utilised the BoM's AWAP data-set to extract long-duration temporal patterns of rainfall.

While the use of observed temporal patterns of rainfall is relatively simple, it is not able to consider future climate conditions or events that have not yet been observed. For this reason, many studies have sought to generate synthetic temporal patterns of rainfall that reflect the statistical characteristics of observed events. Hoang (2001) proposed a multiplicative cascade model to generate temporal patterns of rainfall for each season, duration and depth. Their model was later used by Carroll and Rahman (2004) to estimate design floods in sub-tropical regions of Australia. Varga, Ball, and Babister (2009) used conditional random walk theory to develop synthetic rainfall temporal patterns, which were applied by Ball and Ara (2014) to estimate the variability of design floods in Centennial Park in Sydney.

Most recently, an Australia-wide study was conducted to regionalise temporal patterns of rainfall (Babister, Retallick, Loveridge, Testoni, and Podger 2016). Initially, the study derived a comprehensive Australian events database (with over 100,000 complete storm events) to store events with associated storm and burst information (Loveridge et al. 2015a). This events database was used to define ten temporal patterns of rainfall for each of the 12 fixed regions, four AEP bins (over 1 EY) and 24 burst durations (10 min to 168 h) (Testoni et al. 2016). Loveridge, Babister, and Retallick (2015b) found that once the variability in temporal patterns is considered, the next most important impact is the initial state of the catchment. The study also noted that the rainfall temporal patterns and losses were interdependent when using the IL-CL model (Loveridge and Babister, 2016); specifically, that events with higher internal variability have higher volumes of lost rainfall (due to continuing losses), front-mass events are more sensitive to initial conditions and back-mass events tend to have the highest peak flows.

The initial state of a catchment can have a noticeable impact on flood estimates and many catchments lack sufficient data to calibrate these models. Therefore, a myriad of studies have investigated the variability of IL values to regionalise losses throughout different parts of Australia. Some earlier studies include those by Rahman, Weinmann, and Mein (2002) who described the IL for Victorian catchments using a 4-parameter Beta distribution and Nathan, Weinmann, and Hill (2002) who described the IL values for south-east Australia using a non-parametric distribution (following the work of Hill et al., 1997). Many others, however, have investigated the variability of losses, including studies by Loveridge and Rahman (2014), Gamage, Hewa, and Beecham (2013) and Tularam and Ilahee (2007). A more recent extensive study was conducted by Hill et al. (2014) using 38 catchments throughout Australia. Through a comparison of individual non-parametric IL distributions, the authors found that the shape of the distribution remained reasonably constant, while the median loss of the distribution varied substantially between locations (Hill et al. 2015). Four regional distributions were derived for each of the inland and coastal sub-regions of the GSAM and GTSMR regions. Although the authors investigated the correlation of IL with other key factors (such as AEP, antecedent rainfall, etc.) no dependencies were found.

Most design data in Australia is based on burst periods within complete storm events. However, situations arise where the entire storm response is relevant, in these circumstances pre-burst rainfall estimates have been adopted. Minty and Meighen (1999), for instance, derived pre-burst rainfall depths and temporal distributions for rare, long-duration rainfall bursts in southeast Australia. Jordan et al. (2005) took this further to consider the pre-burst rainfall for short-duration events throughout Australia. In a more recent study, Loveridge et al. Babister, Stensmyr, et al. (2015) regionalised pre-burst rainfall depths and distributions throughout Australia. They found that pre-burst depths were negligible for many locations, but needed to be considered along the eastern seaboard and across parts of northern Australia. The pre-burst to burst rainfall ratios taken from storms within the Australian events database (Loveridge, Babister, and Retallick 2015a) were then regionalised using a region of influence scheme (with three similarity criteria). Unlike the IL distributions, pre-burst distributions varied substantially from point to point making regionalisation difficult. Previous studies have only found dependence between the pre-burst to burst rainfall ratio and duration but not burst rarity (BoM 1999; Scorah, Hill, et al., 2015); however, an extensive Australia-wide study by Loveridge et al. Babister, Stensmyr, et al. (2015) highlighted that the dependency between burst severity (AEP) and the pre-burst to burst rainfall ratio is location dependent.

Where the critical duration concept is not assumed, the event duration is treated as a stochastic variable. Distributions of rainfall duration have been derived and regionalised in several studies; such as Hoang et al. (1999) who derived distributions for burst and storm events in Victoria, Haddad and Rahman (2005) who undertook a similar study in Victoria for the burst rainfall duration, and Caballero and Rahman (2011) who regionalised the storm rainfall duration for NSW. A handful of studies have also treated the inter-event arrival time (being the time between events) as a random variable. For instance, Caballero and Rahman (2014b) regionalised inter-event arrival times throughout NSW. This was followed by a similar study of the east coast of NSW by Loveridge et al. (2016). Similar to other event characteristics, the complete storm duration and inter-event arrival times are also limited by the event definition adopted in each study (due to a lack of consistency).

Other model inputs that have been considered as stochastic variables include continuing losses (e.g. Ilahee and Imteaz 2009) and routing parameters (e.g. Pate and Rahman 2010); however, many were covered within the previous section, such as the spatial pattern of rainfall (e.g. Scorah, Lang, et al. 2015 and Jordan, Nathan, and Seed 2015), space-time distribution of rainfall (e.g. Seed et al. 2014 and Seed , 2004), the dependence between seal level and extreme rainfall (e.g. Aijaz et al. 2011; Need, Lambert, and Metcalfe 2006; Zheng et al. 2014), initial storage levels (e.g. Cohen et al. 2015; Mittiga et al. 2006; Nandakumar et al. 2011; Nathan, Weinmann, and Hill 2002) and coincidence of tributary inflows (e.g. Sih et al. 2012).

4. Conclusion

Australian practice is progressively moving away from the traditional design event approach, as defined in ARR 1987, and towards holistic approaches. For instance, the national flood estimation guidelines (ARR 2016) have recently recommended the use of joint probability approaches (such as Monte Carlo simulation). This shift has come about due to key limitations of traditional methods and the increasing ability to employ more sophisticated methods for design flood estimation.

Joint probability approaches have come a long way over the past two decades in Australia. Practitioners are now able to apply these approaches without great difficulty, as a range of techniques of varying complexity are now available which are suited to different study requirements. The most basic ensemble approach can be used throughout Australia with regionalised design data to provide a better understanding of the likely variability in design floods. However, a full Monte Carlo simulation is able to consider the entire range of expected floods and is particularly useful where more complex joint probability interactions exist. Although these situations often require study specific input distributions and dependence structures to be derived.

Although Monte Carlo simulation has been recommended for use in ARR 2016, practitioners may be hesitant to adopt these methods. This may be due to unfamiliarity with the theory, difficulty in application and/or increases in set-up and processing times. The ARR 2016 revision projects were undertaken to fill many of the gaps in hydrologic knowledge and input data. However, existing software (used in the application of the method) only considers the joint interaction between initial losses and temporal patterns. While this may be suitable for many cases, more complex situations can require the joint probability of other variables to be considered. In such circumstances, tailored solutions and specialist knowledge is required. Therefore, further development of specialised software is prudent, so that the broader community are able to utilise more sophisticated Monte Carlo simulation techniques.

https://doi.org/10.1080/13241583.2018.1453979

Acknowledgements

The authors gratefully appreciate the detailed and valuable comments of Dr. Rory Nathan, whose contribution increased the overall quality of this paper. Thanks also to Mark Babister for his useful comments regarding ARR guidance.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Melanie Loveridge is a hydrologist at WaterNSW. She is currently developing a hydrological forecasting system for the real-time operation of Warragamba Dam. Her PhD research focused on design inputs for flood estimation within a Monte Carlo framework. She later used Monte Carlo simulation in a practical assessment of flood mitigation options to reduce flood risk in the Hawkesbury-Nepean valley and in the development of rainfall temporal patterns for Australian Rainfall and Runoff. Melanie has research interests in stochastic simulation, operational forecasting, hydrology, flood estimation, catchment modelling and requirements analysis.

Ataur Rahman is an associate professor in Water and Environmental Engineering in Western Sydney University. He obtained his BSc Eng. degree from

Khulna University of Engineering and Technology in Bangladesh, MSc (Hydrology) from National University of Ireland and PhD in Hydrology from Monash University. His research interest includes flood hydrology, urban hydrology and environmental risk assessment. He received 'The G. N. Alexander Medal' from the Institution of Engineers Australia in 2002. He has published over 360 research papers in water and environmental engineering. He served as Project 5 Leader (Regional flood methods in Australia) in the 4th edition of Australian Rainfall and Runoff.

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ARTICLE HISTORY

Received 17 July 2016

Accepted 13 March 2018

Melanie Loveridge and Ataur Rahman

School of computing, Engineering and Mathematics, Western Sydney university, Sydney, Australia

CONTACT Melanie Loveridge [mail] melanie.a.loveridge@gmail.com

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Author: | Loveridge, Melanie; Rahman, Ataur |
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Publication: | Australian Journal of Water Resources |

Article Type: | Report |

Geographic Code: | 8AUST |

Date: | Apr 1, 2018 |

Words: | 16085 |

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