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Journal of Hydrology: Regional Studies journal homepage: www.elsevier.com/locate/ejrh

Modelling hydrological losses for varying rainfall and moisture conditions in South Australian catchments S.H.P.W. Gamage a,∗, G.A. Hewa a, S. Beecham a,b a

School of Natural and Built Environments, University of South Australia, Adelaide, SA 5095, Australia Centre for Water Management and Reuse (CWMR), University of South Australia, Adelaide, SA 5095, Australia b

a r t i c l e

i n f o

Article history: Received 15 December 2014 Received in revised form 15 March 2015 Accepted 19 April 2015 Keywords: Hydrological losses Rainfall Storm duration Antecedent wetness

a b s t r a c t Study region: The study is based on unregulated catchments located in Mt. Lofty, Northern and Yorke regions of South Australia (SA). Study focus: Hydrological losses, which are frequently used in design ﬂood estimation, have a wide range of spatial and temporal variability. However, the current practice for many design applications is to use a single loss value. Adopting a single representative value for loss is likely to introduce a high degree of uncertainty and potential bias. This paper identiﬁes the relationships between losses and other parameters that can be incorporated in hydrological models to make reasonably accurate estimates of the losses. This paper assesses the variability of losses and identiﬁes a method that can model initial loss (IL) using the parameters total rainfall (TR), rainfall duration (D) and antecedent wetness (AW). This study is based on 1162 rainfall events from the selected catchments. New hydrological insights for the region: This paper introduces two nomographs, TR–D and TR–AW, which are implemented using k-colour maps and a central tendency method. The developed methods are then validated using the rainfall runoff model, Water Bound Network Model (WBNM). This study will yield improvements to existing loss models by utilising rainfall and antecedent data, instead of using representative values to generalise real situations. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Abbreviations: ARR, Australian Rainfall and Runoff; APIk , antecedent precipitation index; AW, antecedent wetness; CL, continuing loss; D, storm duration; DfW, Department for Water; IL, initial loss; JPA, joint probability approach; PL, proportional loss; QF, quickﬂow; RORB, Run Off Routing Burroughs; SA, South Australia; TR, total rainfall; TRV, total rainfall volume; TSV, total surface-runoff volume; WBNM, Water Bound Network Model. ∗ Corresponding author. Tel.: +61 8 830 29942. E-mail address: [email protected] (S.H.P.W. Gamage). http://dx.doi.org/10.1016/j.ejrh.2015.04.005 2214-5818/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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1. Introduction In hydrology, loss estimation is necessary to provide input for two main applications: (1) real-time ﬂood forecasting; and (2) design ﬂood estimation. Loss models adopted for real-time ﬂood forecasting are usually simple lumped models, examples include initial loss combined with continuing loss and proportional loss rate or runoff coefﬁcient. Estimating temporal variability of losses across storm events is a crucial part of in real-time ﬂood forecasting. For design ﬂood estimation, either statistical, design storm derivation or continuous simulation approaches can be used (Camici et al., 2011). Statistical methods attempt to estimate the design ﬂood by analysing the observed record of peak ﬂows. The limitations associated with the implementation of statistical methods include: (1) the need for a reasonable number of peak ﬂow observations (Mimikou and Gordios, 1989; Pandey and Nguyen, 1999); and (2) the need to assume that the ﬂood frequency behaviour is stationary over time (Sivapalan and Samuel, 2009). Design ﬂoods predicted using design storms estimate the entire ﬂood hydrograph not only the peak discharge, as derived from statistical methods. The design storm method uses rainfall–runoff models (routing models) in association with loss models for ﬂow hydrograph estimation. Design ﬂood hydrographs are needed for cases where storage is signiﬁcant (e.g. dam spillway design and ﬂood plain ﬂows) or where the duration of ﬂooding is required (Nandakumar et al., 1994). For these applications the accurate calculation of design losses is essential. Since actual losses vary from event to event, it is important to estimate the temporal variability of losses in the design ﬂood estimation, which uses design storms. Continuous simulation can also be employed for both real-time ﬂood forecasting (Berthet et al., 2009) and design ﬂood estimation methods (Boughton and Droop, 2003; Brocca et al., 2011; Paquet et al., 2013). Continuous simulation models automatically account for the antecedent conditions for major storm events, which avoid the difﬁculty of separately estimating the initial conditions which affect losses (Blazkova and Beven, 2009; Calver et al., 2009; Camici et al., 2011;). Continuous rainfall–runoff modelling is becoming increasingly popular because of increasing model capabilities in terms of predicting short-time interval ﬂows and the ready availability of computer resources. Although the continuous simulation approach has these advantages compared to eventbased approaches, it is difﬁcult to use this continuous simulation for design ﬂood estimation in rural catchments because of the complexity involved in model calibration. Continuous simulation models require long-term and complete time series for the input data (i.e. meteorological) and such requirements limit the use of these models in many parts of the world, particularly if hourly observations are required (Viviroli et al., 2009). Because of the complex structure of continuous simulation approaches, event-based approaches are more widely applied, mainly because of their simplicity (Berthet et al., 2009; Coustau et al., 2012; Massari, 2014). Event-based methods also have potential advantages in regions where the rain gauge network is more extensive than the ﬂow gauge network, or where longer records exist for precipitation (Caballero et al., 2011). Moreover, event-based rainfall–runoff models require less parameterisation (Berthet et al., 2009; Coustau et al., 2012; Massari, 2014). However, the major limitation of the eventbased method is the difﬁculty in assessing the antecedent soil moisture conditions, which can be very different from one storm event to the next (Coustau et al., 2012; Hino et al., 1988; Tramblay et al., 2010, 2012; Van Steenbergen and Willems, 2013). Therefore, there is a clear need to reduce the uncertainties associated with the initial moisture condition in event-based ﬂood forecasting and design ﬂood estimation techniques. Hydrological losses can be affected by the characteristics of rainfall, catchment geography (e.g. slope of the catchment and vegetation) (Hill and Mein, 1996) and soil moisture content. The rainfall characteristics can be represented by parameters such as rainfall volume, duration, intensity and average recurrence interval (ARI). The relationships identiﬁed between losses and rainfall can differ depending on the rainfall characteristics being used. For example, the relationship between losses and annual rainfall is different from that between losses and design rainfall intensity (Hill and Mein, 1996; Nandakumar et al., 1994). The losses can also be related to the pre-storm baseﬂow (Hill and Mein, 1996; Mein and O’Loughlin, 1991; Mein et al., 1995; Nathan et al., 2003; Siriwardena and Mein, 1996). However, baseﬂow is not a direct characteristic of a catchment but is a combination of other

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physical characteristics such as soil type and vegetation type (Hill and Mein, 1996). Soil moisture content has a direct relationship with hydrological losses as it controls the proportion of rainfall that inﬁltrates into the soil. However, soil moisture data may not be readily available. Therefore soil moisture proxies such as soil wetness index, antecedent precipitation index (APIk ), 5-day antecedent precipitation index (AW) and antecedent baseﬂow index are often used to indicate the soil moisture content (Mishra et al., 2004; Brocca et al., 2008). Different types of relationships between soil moisture content and hydrological losses can be expected due to the variety of indicators (soil moisture proxies) used to represent the soil moisture content (Brocca et al., 2008; Mishra et al., 2004). The relationship between initial loss (IL) and APIk was ﬁrst reported in the 1970s (Nandakumar et al., 1994). However, relationships are yet to be established between many other soil moisture proxies and losses. Analysis of the relationships between losses and soil moisture content should include a detailed investigation of the ﬂood formation process. The role and effects of soil moisture content on the ﬂood formation process has been analysed in small experimental catchments (Castillo et al., 2003; Goodrich et al., 1994; Merz and Bárdossy, 1998; Zehe and Blöschl, 2004), but the results obtained have been contradictory (Brocca et al., 2009b). Some investigators have argued that the initial soil moisture state of the catchment is the most important factor for determining the outcome of an event (De Michele and Salvadori, 2002; Stephenson and Freeze, 1974). In contrast, other investigators have suggested that the initial conditions of the catchment are not critical, particularly in the case of large events (Bronstert and Bárdossy, 1999; Castillo et al., 2003; Merz and Plate, 1997). In addition, the role of initial conditions in the ﬂood formation process depends on the dominant runoff mechanisms (Vertessy et al., 2000). For saturated source area runoff, correct speciﬁcation of the initial saturation deﬁcit is critical for accurate storm modelling. In the case of excess runoff, the importance of initial conditions depends on the storm intensity relative to the inﬁltration characteristics of the soil (Brocca et al., 2009b). Some studies have attempted to relate the rainfall–runoff model’s initial conditions to different external indicators of soil moisture as estimated using in situ, satellite and modelled data (Beck et al., 2009; Brocca et al., 2009a,b, 2011; Coustau et al., 2012; Graeff et al., 2012; Tramblay et al., 2010, 2011, 2012). Many studies have investigated the relationship between soil moisture and runoff, and indirectly determined the potential beneﬁt of analysing soil moisture conditions for rainfall–runoff modelling (e.g. Penna et al., 2011; Matgen et al., 2012; Graeff et al., 2012). Beck et al. (2009) and Brocca et al. (2009a, 2011) conducted rainfall–runoff modelling in Italy, Luxembourg and Australia to investigate the relationship between modelled and observed antecedent wetness conditions. A recent study (Massari, 2014) introduced a simpliﬁed continuous rainfall–runoff model, which uses satellite soil moisture data to identify the initial wetness conditions of a catchment to simulate discharge hydrographs. In Australia, commonly used loss models include the Initial Loss–Continuing Loss model (IL–CL) and the Initial Loss–Proportional Loss model (IL–PL). Australian Rainfall and Runoff (ARR) (IEAust., 1987), which is the national guide for ﬂood estimation in Australia, provides only representative single values (median or mean) for hydrological losses. Although design applications require generalised loss values for the catchment of interest, representative values can result in even greater errors if the variability of losses and other factors that affect losses are not considered. Also many event based rainfall–runoff models such as Run Off Routing Burroughs (RORB) and Water Bound Network Model (WBNM) use representative single value losses. This practice tends to underestimate the actual loss values and is likely to introduce a high degree of uncertainty and possible bias in the resulting ﬂood/ﬂow estimates (Haddad et al., 2010; Hill and Mein, 1996; Loveridge et al., 2013; Walsh, 1991; Waugh, 1991). The primary aim of this study is to improve event-based rainfall–runoff by introducing new, more effective methods that incorporate initial soil moisture conditions and substitute representative single values of losses with rainfall and antecedent data. The method introduced in this paper can be used even when satellite data are not available and it generally requires less parameterisation than continuous simulation models. This paper: (1) investigates the variability of initial loss (IL), continuing loss (CL) and proportional loss (PL) and estimates the errors associated in current single value loss estimation; (2) identiﬁes the loss patterns with respect to rainfall characteristics and antecedent moisture content; and (3) develops a simple model for estimating IL using the parameters of total rainfall (TR), storm duration (D) and antecedent wetness (AW).

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2. Catchment selection and data The criteria used in selecting the catchments include catchment regulation, size, land-use type and available rainfall and streamﬂow record lengths. The selected catchments were unregulated and in the small to medium size range with no major land-use changes. The deﬁnition of small to medium size catchments is arbitrary and is considered here to have an upper limit of 1000 km2 in area (Haddad et al., 2010). Both rainfall and streamﬂow record lengths of the catchments were sufﬁcient to provide a robust analysis. According to several studies (Boni et al., 2007; Jingyi and Hall, 2004; Kumar and Chatterjee, 2005), the record length of data should be at least 10 years for adequate empirical analysis. Six catchments with hourly rainfall and streamﬂow data were selected for this study: Scott Bottom (A5030502), Mt. Pleasant (A5040512), Yaldara (A5050502), Rhynie (A5060500), Penrice (A5050517) and Spalding (A5070501). A location map of the selected catchments is given in Fig. 1, and details of the geographic, climatic and meteorological data for each of the catchments are provided in Table 1. 3. Methodology 3.1. Quantifying losses This paper focuses on the event based conceptual models of initial loss–continuing loss (IL–CL) and initial loss–proportional loss (IL–PL) for quantifying losses. The process of calculating losses involves: (1) extracting events; (2) baseﬂow separation; and (3) calculating IL, CL and PL components. For this study, 1162 rainfall events, which have the potential to produce signiﬁcant runoff, were selected from the six catchments over a 25 year observation period. The criteria described by Rahman et al. (2001) were adopted for selecting suitable rainfall events. According to the Rahman et al. (2001) method, A ‘gross’ storm is a period of rainfall starting and ending with a non-dry hour (i.e. hourly rainfall >0.25 mm/h), preceded and followed by at least 6 ‘dry hours’. Any period of ‘insigniﬁcant rainfall’ at the beginning or end of a gross storm (referred to as a ‘dry period’) is then cut off from the gross storm to produce the ‘net’ storm of duration D (a period is dry if all hourly rainfall totals in the period are ≤1.2 mm and the average rainfall intensity during the period is ≤0.25 mm/h). A net storm is only selected for further analysis if the average rainfall intensity during the entire storm duration (RFID ) or during a sub-storm duration (RFId ), satisﬁes the condition RFID ≤ F1 × 2 ID or RFId ≥ F2 × 2 Id , where 2 ID is the 2 year ARI intensity for the selected storm duration D and 2 Id the corresponding intensity for the sub-storm duration d. F1and F2 are reduction factors deﬁned in Rahman et al. (2001). In this study, the values of 2 ID and 2 Id are estimated from the design rainfall data. The use of smaller values of F1 and F2 captures a relatively larger number of events. As appropriate values need to be selected to exclude events with very small average intensities, values of 0.4 and 0.5 were assigned for F1 and F2, respectively. Table 1 Basic catchment attributes of the selected catchments.

Station no River Area (km2 ) Elevation at gauging station (m) Elevation at upper stream (m)b Average slope of the catchmentc Annual average rainfall (mm)a Rainfall and streamﬂow record length (years) Mean annual evaporation a

Scott Bottom

Mt. Pleasant

Yaldara

Penrice

Rhynie

Spalding

A5030502 Scott 26.8 205 423 0.024 830 19

A5040512 Torrens 26 415 484 0.010 553 21

A5050502 North Para 384 145 286 0.007 468 25

A5050517 North Para 118 285 435 0.007 487 30

A5060500 Wakeﬁeld 417 202 427 0.009 433 48

A5070501 Hutt 280 270 443 0.005 421 37

1.38

1.69

1.69

1.69

3.36

3.36

Calculated based on data provided by the DfW. Calculated using GIS. Calculated as the difference between the highest point and the lowest point of the catchment/distance between these two points. b c

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Fig. 1. Location map of the study catchments.

Then the streamﬂow and rainfall time series were synchronised (plotted) and corresponding streamﬂow events were extracted for the selected rainfall events using the HYDSTRA (KISTERS, 2008) software. For hydrological loss estimations, baseﬂow should be separated from the original streamﬂow data set. The Lyne and Hollick algorithm (Nathan and McMahon, 1990) in the HYDSTRA (KISTERS, 2008) program was used for baseﬂow separation. Nathan and McMahon (1990) compared this method of baseﬂow separation with several other rigorous algorithms and concluded that the Lyne and Hollick

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Fig. 2. Graphical representation of conventional IL–CL model.

algorithm was simpler and produced as good results as the alternatives. It also shows agreement with the most recent baseﬂow separation method proposed by Eckhardt (2008). The Lyne and Hollick algorithm is given in Eq. (1). qf (i) =∝ qf (i−1) + (q(i) − q(i−1) )(1+∝)/2

(1)

where qf(i) is the ﬁltered quickﬂow for the ith sampling instant qf(i−1) is the ﬁltered quickﬂow for the previous sampling instant q(i) is the original streamﬂow for the ith sampling instant q(i−1) is the original streamﬂow for the previous sampling instant ˛ is a ﬁlter parameter. In this study, the sampling interval was considered as 60 min. The ﬁltering factor was considered as 0.925, as recommended by Nathan and McMahon (1990). Having separated baseﬂow, the IL, CL and PL were calculated for the selected events using an author-developed programme. The basic principle behind the developed computer code is as follows. The IL is deﬁned as the amount of rainfall that occurs before the start of the runoff; whereas CL is deﬁned as the average rate of loss in mm/h over the remaining period of the rainfall event (illustrated in Fig. 2). The IL can be expressed using Eqs. (1) and (2) respectively. IL =

n

Ii

(2)

i=1

where n is the duration in hours of the storm burst that ends before runoff begins, and Ii is hourly rainfall in mm. The Total Rainfall (TR) can be expressed according to Eq. (3). Quickﬂow (QF) is calculated by subtracting the baseﬂow from total stream ﬂow. Eq. (3) then can be rearranged as in Eq. (4) to calculate the CL. TR = IL + CL × t + QF

(3)

TR − IL − QF CF = t

(4)

where TR, IL and QF are in mm, CL is in mm/h and t is the time (in hours) elapsed between the start of the surface runoff and end of the rainfall event.

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Quickﬂow (QF) calculated using Eq. (1) was in the units of m3 /s. In order to substitute values of quickﬂow into Eq. (4), quickﬂow values need to be converted into mm using Eq. (5). QF =

(qf (i) ) ×

1000t A

(5)

where t is streamﬂow duration in seconds and A is catchment area in m2 . The PL, which is assumed to be a ﬁxed proportion of the storm rainfall, was estimated using Eq. (6). PL = 1 −

TSV TRV − IL(A × 10−3 )

(6)

where TSV is the total surface-runoff volume, TRV is the total rainfall volume and A is catchment area in m2 TSV and TRV were calculated using Eqs. (7) and (8), respectively. TSV =

n

t × qf (i)

(7)

i=1

where t is the duration of the streamﬂow event. TRV =

n i=1

ri ×

A 1000

(8)

where ri is the hourly rainfall in mm during the ith hour, A is the catchment area in m2 , and n is the rainfall duration. 3.2. Analysing loss variability with rainfall characteristics and antecedent wetness As discussed earlier, there are a number of parameters that affect losses. However, it is impractical to incorporate all those parameters into a model as this would both increase the model complexity and limit the usability of the model. In this study, the variability of the losses (IL, CL and PL) with parameters total rainfall, rainfall duration and ﬁve-day antecedent wetness was investigated. The total rainfall volume of the event expressed in terms of the average depth of total rainfall (TR) in mm over the catchment, and rainfall duration (D) in hours. AW is deﬁned as the total rainfall that the catchment receives in the ﬁve days prior to the start of the selected event. A bivariate analysis was performed to investigate the relationship between the loss components and the selected variables. The loss components (IL, CL and PL) were considered as dependent variables (Y) and the TR, D and AW were considered as independent variables (X). Scatter plots were constructed to visually demonstrate the results. If both Y and X variables are random and if Y shows signiﬁcant variability for a given value of X, then the Random X model can be used for analysing variability. The Random X model shows the mean value of Y against a given value of X. In this analysis, the variability of the losses with D and AW were investigated using Random X models. Multiple-regression was carried out to estimate the combined effect of the independent variables (TR, AW and D) on the dependent variable (IL, CL). However, as the multiple-regression analysis was not accurate enough to enable IL to be estimated, further analysis of IL was carried out as described below: 1) The relationship between IL and TR was further assessed by calculating IL over TR (IL/TR). 2) Contour and dot maps were used to examine the effect of the variables on each other. On these maps each loss value was mapped in the third dimension as it improves the interpretation of the loss distribution patterns. The possibility of forming either a contour map or a dot map, and the patterns of distribution of each loss component compared to two other independent variables were investigated. 3) The k-coloured dot maps were developed by plotting the observed IL values against (AW and TR), (TR and D) and (AW and D). In order to identify clusters, all the observed IL values were ranked and

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Table 2 Loss aggregations. Aggregation No.

1

2

3

4

5

6

7

8

Range of loss values (mm)

0–5

6–10

11–15

16–20

21–25

26–30

31–40

41–50

aggregated to 8 arrays, as shown in Table 2. Then, the aggregations were plotted against various other combinations of the independent variables. These combinations included (IL/D and TR), (TR/D and D) and (TR/D and AW). 4) Finally, two nomographs were introduced to describe the IL distribution patterns. Nomographs were used to describe a “map” of the variables (IL, AW and TR) in an efﬁcient and simpliﬁed way. In this way, it was possible to identify whether the IL distribution has a pattern, or the map is in some sense random. In this study, two-dimensional nomographs were introduced to describe how two variables change with IL. Therefore the variable sets (TR and AW) and (TR and D) were selected to develop TR–AW and TR–D nomographs, respectively. A total of 1162 events were used to develop these nomographs and 100 randomly selected events were used for validation purposes. The methods followed to develop the nomographs are now described. 3.2.1. TR–D nomograph When developing the TR–D nomograph, ﬁrst TR, D and IL were recorded for each event. Then the rate of IL (ILrate) was calculated using Eq. (9) and this was multiplied by 100 and rounded to the nearest decimal place in order to enlarge the plotted area for better identiﬁcation of the patterns. Finally, the values of (ILrate) × 100 were graphically presented using a Cartesian grid with TR (X-axis) and D (Y-axis). Clusters were then identiﬁed and marked on the nomograph. ILrate =

IL D

(9)

where D is rainfall duration in hours. 3.2.2. TR–AW Nomograph When developing the TR–AW nomograph, ﬁrst TR, AW and IL were recorded for each event. Secondly, the IL values were aggregated as shown in Table 2. Thirdly, the aggregation numbers that represent IL values were marked on a Cartesian grid with TR (X-axis) and AW (Y-axis). The aggregations numbered 1–5, which are described in Table 2, were used in developing this nomograph. Other aggregations (numbered 6–8) were excluded due to the limited number of events found in this range. Unlike the TR–D nomograph, clusters could not be easily identiﬁed in the TR–AW nomograph. Therefore, to identify clusters (or a distribution of the IL values with respect to TR and AW), the centre value and the dispersion of IL in each aggregation were estimated. The centre values (mean centres) for each aggregation were estimated using Eqs. (10) and (11). X¯ =

n X

i

i=1

Y¯ =

n

n Y

i

i=1

n

(10)

(11)

where (Xi , Yi ), i = 1,2,. . .,n are the coordinates of a given set of n points for each aggregation. The dispersion of IL for each aggregation was determined using the standard deviation. The standard distance (SD) is directly related to the standard deviation (Burt et al., 2009). Therefore, the SD of each aggregation was calculated by using the coordinates of each point, as shown in Eq. (12).

SD =

x2 + y2

(12)

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Table 3 Scenarios considered in validating the TR–AW nomograph.

1 2

IL values

CL values

Using single representative value recommended losses Taking TR and D into consideration

Single representative value (median) Selecting representative values from the TR–D nomograph (Fig. 9) Selecting representative values from the TR–AW nomograph (Fig. 10)

Single representative value (median) Two different representative values for low and high rainfalla Two different representative values for low and high rainfalla

Taking TR and AW into consideration

3

a

Scenario

The threshold between the higher and lower rainfall events were selected based on the median value of the TR.

where x2

n =

i=1

¯ 2 (Xi − X) n

n and

y2

=

i=1

2 (Yi − Y¯ )

n

Finally, the IL distribution that corresponded to each aggregation was marked on the Cartesian grid as a circle with a radius equal to the standard distance from the mean centre of the distribution. 3.3. Validating results using WBNM In this paper, an independent catchment, the Torrens Gumeracha Weir catchment (A5040500), was used to validate the two nomographs. The Water Bound Network Model (WBNM) was used to compare the peak discharge of an event when changing the representative values of losses, as suggested by this research. WBNM is a ﬂood hydrograph model developed by Boyd et al. (1996), which calculates ﬂood runoff from rainfall hyetographs. WBNM model has wide acceptance in Australia and have been used for analysing ﬂoods (Boyd and Bodhinayake, 2006; Boyd et al., 2002; Ryan and Boyd, 2002; Van Drie et al., 2001). This model requires users to input values for IL and CL parameters. When using this model, common practice is to use recommended single representative values. For example, recommended values for the humid zone of SA are 10 mm for IL and 2.5 mm/h for CL in winter, and 25 mm for IL and 4 mm/h for CL in summer (IEAust., 1987). As WBNM allows users to deﬁne IL and CL values it was selected to validate the results for this study. In this study, the WBNM model was run for 50 randomly selected rainfall events in the test catchment. Observed (measured) steamﬂow events corresponding to each of these rainfall events were also recorded for comparison. For each rainfall event the model was run three times, each time changing the values for IL and CL, as shown in Table 3. The peak ﬂows at the end of the catchment for each scenario were recorded. In scenario 1, losses are represented by single values (median of losses). In scenarios 2 and 3, the developed nomographs were used to calculate losses. Observed (measured) streamﬂow data for selected events were then compared with each WBNM output by calculating the error magnitudes, using Eq. (13).

PDmod − PDobs × 100% PD

|error| =

(13)

obs

where PDmod is the modelled peak discharge and PDobs is the observed peak discharge. The scenario 2 output was used to demonstrate how much improvement can be achieved by incorporating rainfall characteristics TR and D when representing losses. Scenario 3 was used to demonstrate how much further improvement could be achieved by considering both TR and AW when selecting representative values for losses. To determine the representative value for CL in scenarios 2 and 3, the losses were ranked against total rainfall and divided into two categories: low rainfall events and high rainfall events. The threshold between the higher and lower rainfall events was selected based on the median value of the TR. Then two different sets of representative values (median) for CL were introduced for each category.

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4. Results and discussion The numbers of events extracted for the six selected catchments over the 25 year observation period are listed in Table 4. This study investigates the variability of loss components (IL, CL and PL) with the rainfall characteristics (TR and D) and antecedent moisture content. AW is used as a substitute for the antecedent moisture content, because AW is an easily measurable parameter that has a direct relationship with antecedent moisture content. The commonly used Antecedent Precipitation Index (APIk ) was also investigated in this study but it was found that IL has a stronger correlation with AW than that with APIk . As TR, D and AW are easily measurable parameters, this should increase the useability of the developed model. When analysing the variability of losses with the selected parameters, it was found that the variability of IL and CL changed considerably with TR. Fig. 3 shows the variability of IL, CL and PL with TR for the catchment A5040512. The other catchments also exhibit similar trends. The selected events were ranked in ascending order of TR values and X-axis of Fig. 3 indicates the rank of events. The threshold between the higher and lower rainfall events were selected based on the median value of the TR. Fig. 3 shows that the higher the TR in an event, the greater the variation in IL and CL. The summary statistics for the higher and lower rainfall events are shown in Table 5. Consequently, it is clear that the ARR (IEAust., 1987) recommendation of constant values for the humid zone of SA: 10 mm for IL and 2.5 mm/h for CL in winter; and 25 mm for IL and 4 mm/h for CL in summer (IEAust., 1987) are perhaps no longer valid. This is an important ﬁnding for applications where the mean or median values of IL are used as an input parameter (e.g. WBMN model). For such design applications, it is important to identify the TR of the event before assigning a representative value for the IL or CL parameters. A simple graphical method, such as that shown in Fig. 3 is not sufﬁcient to illustrate the variability of losses with AW and D. The main reason for this is the weak linear correlations between each loss component and the parameters D and AW. It has been found that the correlation coefﬁcients of the losses with AW and D are less than 0.5. The low correlation coefﬁcient occurs because of the large variation of losses across the individual units of independent parameters (D and AW). For an instance, the variance of IL in the study catchment is 53 mm2 when the AW value is zero. Therefore, the Random X model was used to explore the relationship between each loss component and parameters AW and D. The Random X model shows the mean of each loss component for a given range of independent values. While the Random X model shown in Figs. 3(c) and 4(a) is intended to explain the variability of each loss component with the AW, Fig. 3(d)–(f) shows their variability with D. According to Fig. 4(a), (b), (d) and (e), both the IL and CL components change markedly with D and AW. Random X models can be used to identify the variation of each component with other independent variables. However Table 4 Selected events. Catchment name

Catchment number

Data collection period

No. of events selected

Scott Bottom Mt. Pleasant Yaldara Penrice Rhynie Spalding

A5030502 A5040512 A5050502 A5050517 A5060500 A5070501

1991–2010 1989–2011 1985–2011 1986–2011 1985–2011 1992–2011

200 227 200 185 208 142

Table 5 Summary statistics for the loss components, based on changes in TR. Mean

IL (mm) CL (mm/h) PL a

Median

Variance

Highera

Lower

Higher

Lower

Higher

Lower

7.90 0.37 0.64

17.20 2.10 0.73

8.10 0.25 0.75

16.39 0.94 0.86

13.70 0.16 0.07

80.10 10.71 0.07

The threshold between the higher and lower rainfall events were selected based on the median value of the TR.

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Fig. 3. (a) Variability of IL with total rainfall, (b) variability of CL with total rainfall, (c) variability of PL with total rainfall. The threshold between the higher and lower rainfall events were selected based on the median value of the TR.

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Fig. 4. Random X models for: (a) IL vs AW; (b) CL vs AW; (c) PL vs AW; (d) IL vs D; (e) CL vs D; (f) PL vs D.

the variation of CL with D cannot be identiﬁed just by using the Random X model and this requires further investigation, which is beyond the scope of this paper. As shown in Fig. 4(c) and (f) PL is almost constant throughout the range of AW and D values considered. The analysis up to this point suggests that the IL and CL are highly dependent on the rainfall characteristics (TR and D) and AW. Also the mean is biased by the high outliers (high loss values) and can cause underestimated design ﬂood quantiles. Therefore it is inaccurate to use single representative values (mean or median) for either IL or CL. However, using a single representative value for the PL is reasonable as the variation with both rainfall characteristics and AW are small. As shown in Fig. 3(c), the variation of the PL with TR is quite low. In fact, in this particular catchment the variance of PL is as small as 0.07 with the presence of high outliers. Not only is PL independent of some parameters that have effects on other loss components, it also shows much less sensitivity to the high outliers. Therefore, it can be inferred that the PL model is more suitable than the CL model. This also supports the ﬁndings of Goyen (2000) who evaluated the performance of PL for New South Wales, Australia. However, it seems that IL and CL needs further investigation before it can be efﬁciently incorporated into design applications. The following discussion examines the distribution patterns of the IL values that can be used in design applications.

4.1. Identiﬁed distribution patterns of IL with respect to TR, D and AW Before analysing the distribution patterns of IL with respect to the rainfall characteristics (TR and D) and AW conditions, it is useful to determine the combined effects of TR, D and AW on IL. Hence a multiple-regression was carried out. Table 6 presents the results for the stepwise regression procedure. The R2 value of 0.7 implies that the variables TR, AW and D explain 70% of the variation in IL. It is clear that rainfall and antecedent conditions can only explain up to 18% of the variation in CL and parameters. Also for the CL, the adjusted R2 value does not improve much with incorporating either second or third variables. Therefore, the variables TR, D and AW cannot be used to represent CL accurately. The variables TR and D are inter-correlated. However, the correlation between TR and D is not linear. In the regression analysis for IL, it was noted that incorporating TR and AW will improve the adjusted R2

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Table 6 Regression statistics for IL and CL with variables TR, AW and D. Model scenarios

R2

Adjusted R2

Standard error of estimate

IL vs TR IL vs TR, AW IL vs TR, AW and D CL vs TR CL vs TR, AW CL vs TR, AW and D

0.44 0.59 0.70 0.17 0.18 0.18

0.43 0.58 0.70 0.17 0.18 0.18

7.06 4.98 4.08 2.40 2.30 4.05

value from 0.44 to 0.59 and the adjusted R2 value increased up to 0.7 when D was incorporated. The best regression model identiﬁed for IL is given in Eq. (14). IL = 0.53TR − 1.5AW0.1 − 0.0004D2

(14)

Further investigation showed that the regression model for IL did not produce accurate results. Also the residual plots of the developed regression models indicated that the regression rule of homoscedacity was violated. The residual bands for the variable TR increased as TR increased and had a conical distribution. Any signiﬁcant departure from a rectangular pattern residual plot is usually evidence of some violation of the regression rules. Hence, a regression model that considers all three independent variables was not recommended for calculating IL. Therefore, a better approach for describing IL and incorporating the predictor variables into models for estimating losses was required. In this study, graphical methods are investigated for this purpose and these are now described. 4.1.1. IL as a proportion of TR In this study the relationship between IL and TR is further analysed by calculating IL over TR (IL/TR) and IL over D (IL/D). The distribution of IL/TR ratio for 100 randomly selected events is presented in Fig. 5 in which X-axis presents the rank of the events organised in ascending order. Fig. 5 shows that the relationship between TR and IL/TR is highly variable. Therefore, IL values were aggregated as shown in Table 7, to better represent the relationship. According to Table 7, the percentage of IL/TR is 40–60% for most of the events and the IL/TR ratio decreases with an increase in TR. The IL/TR ratio also becomes lower, approaching 28% for higher loss values. However, since the events with IL values over 60 mm are relatively fewer, the value of the IL/TR ratio is less accurate.

Fig. 5. Variation of IL/TR ratio with total rainfall.

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Table 7 Summary statistics of IL/TR as calculated for the study catchments. Aggregation No.

1

2

3

4

5

6

IL (mm) Mean TR Mean IL IL/TR*100

0–10 7.12 4.27 59.93

10–20 15.42 8.80 57.05

20–30 24.94 11.95 47.93

30–40 34.37 16.01 46.59

40–60 61.44 24.95 40.62

>60 82.48 44.5 27.64

4.1.2. Contour, k-coloured and dot maps In order to provide a better representation of the distribution, IL was described with respect to at least two variables, rather than examining one variable at a time. The ﬁrst approach used to identify the distribution pattern with respect to two other parameters was the use of k-colour maps with contour patterns. Fig. 6 shows the k-colour map for the IL distribution with respect to AW and TR. It appears that no contour can be formed with this distribution and similar observations were made for the other parameter combinations as well. One of the problems with k-colour maps is that it is difﬁcult to obtain an accurate visual impression of the degree of similarity because too many colours can lead to unnecessary complexity. Therefore data aggregation was performed to minimise the complexity. The aggregation was done carefully as too few aggregations can mask the real variation displayed by the observed data. In this study, the data was aggregated into 8 arrays (Table 2), minimising the inevitable loss of information during the aggregation. Fig. 7(a)–(c) shows the distribution of IL with respect to parameter pairs: (1) AW and TR; (2) TR and D; and (3) AW and D, respectively. These pairs are three

Fig. 6. k-colour map to describe IL with AW and TR.

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Fig. 7. k-colour map with aggregation: (a) IL with AW and TR; (b) IL with TR and D; and (c) IL with AW and D.

of the many combinations tested. It is evident from Fig. 7 that even with the aggregations it is still difﬁcult to visualise a pattern and determining the distribution of IL with respect to these parameters is fairly complex. Although further aggregation might solve the problem of complexity, it can (as previously stated) cause inevitable loss of information during the aggregation process. Therefore, instead of further aggregation, the parameters were transposed to other combinations and the IL patterns were then investigated. The other combinations that were investigated included (IL/D and TR), (TR/D and D) and (TR/D and AW). However, a pattern could only be found on the map where IL

Fig. 8. Distribution of IL with IL rate and TR.

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Fig. 9. TR–D nomograph for South Australian catchments. Derived from 1162 events in six study catchments. The IL rate is considered to be same for each region.

aggregations were plotted against IL rate (IL/D) and TR, as shown in Fig. 8. The patterns found for the other two combinations are just as complex as the distributions shown in Fig. 7. The purpose of this study is to ﬁnd a method that can predict or estimate IL values. Therefore the beneﬁts of a distribution such as that shown in Fig. 8 are quite limited for design applications because IL itself appears on one axis. In order to be effective in design applications, both axes should be represented by readily measured parameters. However, this distribution provided useful information for developing the next approach, which was to investigate the distribution patterns of IL rate. As a result, a TR–D nomograph was developed. 4.1.3. TR–D nomograph The TR–D nomograph (see Fig. 9) shows clear clusters for IL rate with respect to TR and D, as most of the ILrate values that fell between 0 and 10 mm/h were in one region (Region 1) and the values in the range 11–20 mm/h were clustered in the next region (Region 2), and so on. Therefore, the TR–D nomograph can be used to determine IL if TR and D values are known. 4.1.4. AW–TR nomograph As the distribution of IL, with respect to TR and AW, was complex and no pattern could be found from the k-coloured map, a AW–TR nomograph was constructed using a central tendency method. The application of the AW–TR nomograph for ascertaining the distribution of IL with respect to AW and TR, is shown in Fig. 10. In Fig. 10, the red colour cells indicate the mean centres of each aggregation and the radius of each circle represents the standard distance. It should be noted that the both measures are very sensitive to extreme observations. The circles drawn around each centre represent the distribution of losses with respect to AW and TR. The overlapping sections indicate the variability of losses in the same range of

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Fig. 10. AW–TR nomograph for South Australian catchments. Derived from 1162 events in six study catchments. The coloured regions represent different IL ranges. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

TR and AW values. This nomograph can be used to predict the IL value of an event if TR and AW values are known. When implementing the AW–TR nomograph for different hydrological regions, it is also very important to use a large number of data sets. If data are aggregated into fewer and larger units, it may mask the important associations between variables or it may overemphasise other associations. Finally, if correct aggregations are not used, there can be substantial variations in the value of most statistics. For example, the correlation coefﬁcient can be changed dramatically by the aggregations, especially if there are fewer data points or if there are fewer aggregations. In fact, sometimes correlation coefﬁcients can be changed from negative to positive values. The applicability of developed nomographs was validated using 100 randomly selected events. For the test events, IL values were calculated, aggregated and plotted on the nomographs shown in Figs 9 and 10. The probability of an event that does not fall into the correct range was calculated to be 0.03 for AW–TR nomograph and 0.05 for the TR–D nomograph. Also the standard error of estimate for IL when using the regression equation and the AW–TR and TR–D nomographs were calculated as 4.08, 2.44 and 2.85, respectively. Therefore the introduced nomographs produce better results than the regression model. Further validation of this method using the WBNM model will be presented in Section 4.2. Although these nomographs are only valid for the study catchments, and perhaps for other hydrologically similar catchments, both nomographs can be generalised by extending this work to other catchments and calculating Z scores. However, Generalising these nomographs by including different hydrological regions is subject to further research. Both the TR–D and the AW–TR nomograph can help to overcome the problems associated with using representative single values for a wide range of events. A loss model that can represent

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Fig. 11. Percentage of error of estimated losses compared to observed losses. Scenario 1: when using median values, scenario 2: when using TR–D nomograph, scenario 3: when using TR–AW nomograph to represent IL.

seasonal climatic variation is required for IL calculations. The nomographs provide seasonal representation to a certain degree because the parameters (TR, D and AW) change with the seasons. It should be noted, however, that both these methods have some shortcomings. Firstly, they are sensitive to the choice of boundaries (outliers) and secondly, they are not independent of scale.

Fig. 12. Observed peak ﬂow discharge and modelled peak discharge for Scenario 1 (when using median values as losses), scenario 2 (when IL is derived from the TR–D nomograph) and scenario 3 (when IL is derived from the TR–AW nomograph).

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4.2. Validation of introduced nomographs using WBNM Error associated with current practice (use of single representative loss values) and the improvements that can be made to design applications (e.g. RR models) by selecting IL values using the TR–AW and TR–D nomographs are then tested using the WBNM model output and observed data. Based on the 50 test events, it was found that the peak ﬂow estimation of the WBNM model can be improved by using the TR–D nomograph and using two different sets of representative CL values instead of using a single representative value. The results can also be further improved by using the AW–TR nomograph, in which IL is represented as a function of TR and AW. Fig. 11 shows the percentage error reduced by introducing scenario 2 (representation based on TR and D) and scenario 3 (representation based on TR and AW), instead of using scenario 1 (representation based on a single value) for each event. Fig. 12 shows the WBNM results for the peak discharge of 50 test events and how the peak discharge varies in each scenario compared to the observed peak discharge. Therefore, it can be concluded that the signiﬁcant improvements in the output of design applications can be achieved by improving representative loss values. It should be noted that the demonstrated improvements are achieved by introducing different sets of IL values with respect to different rainfall and antecedent conditions. In addition, improvements can be made by incorporating the probability distribution of losses (Gamage et al., 2013). 5. Conclusion This paper has investigated the variability of hydrological losses with the characteristics of rainfall and antecedent wetness conditions. The effects of parameters TR, D and AW on loss components IL, CL and PL were investigated. It was found that 70% of IL could be explained by the variables TR, D and AW. The inaccuracies that can occur through using a simple representative value (mean or median) of losses in design applications were discussed. It was recommended to use IL values as a function of TR, D and AW rather than using simple representative values. Two nomographs were introduced to determine the IL when a minimum of two independent variables are available. Both nomographs were tested for the selected region and the possibility of generalising these methods was also discussed. The percentage improvements that can be achieved by representing IL as a function of TR, D and AW were also estimated using WBNM model. The results presented in this paper should be useful for improving existing event based loss models. This paper will also encourage practitioners to utilise multiple data sets to estimate losses, instead of using hypothetical or representative values to generalise real situations. Conﬂict of interest The authors declare that there are no conﬂicts of interest. References Beck, H.E., de Jeu, R.A., Schellekens, J., van Dijk, A.I., Bruijnzeel, L., 2009. Improving curve number based storm runoff estimates using soil moisture proxies. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2 (4), 250–259. Berthet, L., Andréassian, V., Perrin, C., Javelle, P., 2009. How crucial is it to account for the antecedent moisture conditions in ﬂood forecasting? Comparison of event-based and continuous approaches on 178 catchments. Hydrol. Earth Syst. Sci. 13, 819–831, http://dx.doi.org/10.5194/hess-13-819-2009. Blazkova, S., Beven, K., 2009. A limits of acceptability approach to model evaluation and uncertainty estimation in ﬂood frequency by continuous simulation: Skalka catchment, Czech Republic. Water Resour. Res. 45, W00B16, http://dx.doi.org/10.1029/2007WR006726. Boni, G., Ferraris, L., Giannoni, F., Roth, G., Rudari, R., 2007. Flood probability analysis for un-gauged watersheds by means of a simple distributed hydrologic model. Adv. Water Resour. 30 (10), 2135–2144. Boughton, W., Droop, O., 2003. Continuous simulation for design ﬂood estimation – a review. Environ. Model. Softw. 18 (4), 309–318. Boyd, M.J., Bodhinayake, N., 2006. WBNM runoff routing parameters for South and Eastern Australia. Aust. J. Water Resour. 10 (1), 35–48. Boyd, M.J., Rigby, T., Van Drie, R., Schymitzek, I., 2002. WBNM 2000 For Flood Studies on Natural and Urban Catchments. Mathematical Models of Small Watershed Hydrology and Applications. Water Resources Publications, Chelsea, MI, USA, pp. 225–258.

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