The main objective of this study was to investigate the statistical characteristics of point rainfall and the novelty of the work was the development of a hybrid probability distribution that can model the full spectrum of daily rainfall in the Onkaparinga catchment in South Australia. Daily rainfall data from 1960 to 2010 at 13 rainfall stations were considered. Spatial dependency among the rainfall maxima was assessed using madograms. Relatively strong and significant autocorrelation coefficients were observed for rainfall depths at finer (daily and monthly) temporal resolutions. The performance of different distribution models was examined considering their ability to regenerate various statistics such as standard deviation, skewness, frequency distribution, percentiles and extreme values. Model efficiency statistics of modelled percentiles of daily rainfall were found to be useful for optimum threshold selection in a hybrid of the gamma and generalized Pareto distributions. The hybrid and the mixed exponential distributions were found to be more efficient than any single distribution (Weibull, gamma and exponential) for simulating the full range of daily rainfall across the catchment. The outcomes from this study will assist water engineers and hydrologists to understand the spatial and temporal characteristics of point rainfall in the Onkaparinga catchment and will hopefully contribute to the improvement of rainfall modelling and downscaling techniques.

## INTRODUCTION

Rainfall is one of the key inputs in hydrological modelling. The generation of rainfall, both temporally and spatially, is a relatively recent research topic in rainfall simulation and downscaling. The distribution of rainfall amounts over time and space can be an important input to decision making tools that provide information on rainfall variability and help in understanding hydrological processes (Apaydin *et al.* 2006). Extreme rainfall has significant impacts on water resource management. According to the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4), extreme precipitation frequencies increased almost everywhere in the world over the 20th century and this trend will be continued into the 21st century (IPCC 2007). This is also confirmed by the recent assessment by the IPCC in the Special Report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation (SREX) (IPCC 2012). Several studies have investigated temporal and spatial changes in extreme rainfall (Suppiah & Hennessy 1998; Hennessy *et al.* 1999; Plummer *et al.* 1999; Alexander & Arblaster 2009; Argüeso *et al.* 2013; Rashid *et al.* 2013a, b). Plummer *et al.* (1999) and Hennessy *et al.* (1999) observed that heavy rainfall events and the number of rainy days both increased during the last century over Australia and that this increase depended on the season and region. Over the east and north portion of Australia, a significant increase in heavy rainfall has been observed in summer, while a decreasing trend was observed in south-west Western Australia (Hennessy *et al.* 1999). Suppiah & Hennessy (1998) observed an increasing trend in the 90th and 95th percentile rainfalls over most of Australia. Haylock & Nicholls (2000) reported a significant increase in total rainfall and number of rain days in the northern and southern regions of Australia. Jakob *et al.* (2011a) studied the existence of non-stationarity in daily and sub-daily extreme rainfall by examining the variation in the frequency and magnitude of rainfall extremes for durations from 6 min to 72 h using data from sites in the south-east of Australia for the period 1921 to 2005. They observed that the deviations from the long-term average tend to be larger for frequencies than for magnitudes and changes in frequency and magnitude may have opposite signs. Above average magnitudes were also found for the late 1980s for 12 min and 24 h durations. Quantile estimates derived for Sydney Observatory Hill for the period 1976 to 2005 showed significant decreases for the 6 and 72 h durations. Jakob *et al.* (2011b) observed significant spatial and seasonal variation in the frequency and magnitude of rainfall events.

Understanding the spatial pattern of extreme rainfall events such as annual maxima is of great importance in different practical applications, for example regional flood risk planning. Conventional geostatistical methods such as variograms, which assume that the marginal distribution of the underlying random field is Gaussian, are not relevant for assessing the spatial dependence of extreme rainfall events because these usually do not have a normal distribution. Cooley *et al.* (2006) introduced the madogram which links the geostatistical ideas to the measure of dependence of extremes. Vannitsem & Naveau (2007) used madograms to assess the spatial dependence among the precipitation maxima in Belgium. They observed that the degree of dependency varies with the site separation distance, the season (summer or winter) and the precipitation accumulation duration (hourly, daily and monthly).

Positively skewed probability distributions such as the Kappa, gamma, exponential, and Weibull are commonly used to model the frequency distribution of daily rainfall amounts (Liu *et al*. 2011; Chowdhury & Beecham 2012; Li *et al.* 2012b). These distributions are reasonably capable of reproducing low to moderate rainfall amounts but are generally not adequate for simulating rainfall extremes (Wilks 1999; Furrer & Katz 2008). Although simulation of extreme rainfall is a crucial challenge in rainfall modelling, only a limited number of studies in this area have been undertaken (Arnbjerg-Nielsen *et al.* 2013). Wilks (1999) reported that the mixed exponential (ME) distribution reproduced extreme rainfall (in the case of rainfall depths less than 100 mm) better than the gamma distribution. A few researchers have used parametric compound distributions in order to simulate the full range of rainfall (Vrac & Naveau 2007; Furrer & Katz 2008; Hundecha *et al.* 2009), but these were generally criticized because of limitations such as numerical instability, data sensitivity and computational demand. The non-parametric approach where rainfall is modelled by resampling of the observed historical rainfall has been used by Lall & Sharma (1996) and Rajagopalan & Lall (1999). This approach generally leads to underestimation of extreme rainfall events (Markovich 2007). Many weather generators have been developed to generate daily rainfall. For example, Liu *et al.* (2009) developed the stand-alone MODAWEC (MOnthly to DAily WEather Converter) model to generate daily rainfall from observed monthly data where the precipitation amount was generated from a modified exponential equation. Moncho *et al.* (2012) proposed an alternative model for estimating the probability of precipitation. The model was based on four parameters and they claimed that this performed better than other conventional probability distributions. Ceresetti *et al.* (2010) assessed the scaling properties of heavy point rainfall at different temporal resolutions ranging from 1 to 24 hours. They observed that for extreme point rainfall, the survival probability is hyperbolic and can be parameterized by a decay rate *α* and a lower bound *x*_{min}. Ceresetti *et al.* applied an objective method to detect the lower bound above which the distribution exhibits power law tails and they determined the power law exponent *α* of the hyperbolic tail of the distribution using a maximum likelihood estimator. A hybrid of the gamma and exponential distributions along with the generalized Pareto (GP) distribution were implemented by Furrer & Katz (2008) and Li *et al.* (2012a) respectively, in order to simulate the entire range of daily rainfall. They found that hybrid distributions can be a substantial improvement over gamma or exponential distributions for simulating extreme rainfall.

In statistical downscaling, probability distributions are generally used to simulate the rainfall (Monjo *et al.* 2013; Ribalaygua *et al.* 2013; Beecham *et al.* 2014). Assessment of the capability of statistical distributions to simulate various temporal and extreme characteristics of rainfall is necessary before using such distributions in a rainfall generator or downscaling model. The performance of a probability distribution can vary temporally. Wan *et al.* (2005) observed that the ME distribution performs well in the summer season, whereas the gamma distribution is suitable for the winter season in Canada. Li *et al.* (2012b) fitted six probability distributions to daily rainfall from 24 stations in Canada and assessed their performance based on their ability to reproduce several key statistics such as mean, standard deviation and percentiles of daily, monthly and annual rainfall. They observed that the performance of reproducing key statistics of rainfall time series is proportional to the number of parameters in the distribution function. They also identified that the three-parameter ME distribution outperformed others in simulation of rainfall amounts. The distribution of rainfall amounts in different months and seasons is also important for water resources planning and management. The Precipitation Concentration Index (PCI) (Oliver 1980) is a widely used index that has been applied for this purpose by various researchers (Michiels *et al.* 1992; Apaydin *et al.* 2006; De Luis *et al.* 2010, 2011; Ngongondo *et al.* 2011). Higher values of PCI indicate a higher concentration of rainfall and vice versa. For more explanation of PCI, please refer to De Luis *et al.* (2011). Ngongondo *et al.* (2011) found an unstable monthly rainfall regime in Malawi, with PCI values of more than 10. De Luis *et al.* (2011) studied the mean values of annual, seasonal, wet and dry periods of PCI in Spain. They found a significant change in PCI between two periods (1946–1975 and 1976–2005).

In this study, first we have fitted three widely used single probability distributions (gamma, exponential and Weibull) to assess their limitations in modelling rainfall extremes. Secondly, we have used a ME (Wilks 1999), a hybrid of the gamma and generalized Pareto (termed as HGP) (Furrer & Katz 2008), and a hybrid of the exponential and generalized Pareto (termed as HEP) distributions to model the full spectrum of daily rainfall in the Onkaparinga catchment in South Australia. Various statistics such as standard deviation, skewness, frequency distribution, percentiles and extreme values of observed and modelled rainfall were used to quantify the performance of the hybrid model. In addition, model efficiency statistics such as the coefficient of efficiency and the index of agreement were used. Moreover, we have discussed how an efficient threshold can be selected in the HGP modelling approach by examining various percentiles of observed and modelled rainfall. Finally, a hybrid of the Weibull and GP distributions is also proposed in this study for modelling daily rainfall frequency distributions in the Onkaparinga catchment. Spatial dependence among the rainfall maxima have been assessed for different accumulation periods (1–25 days) for summer and winter seasons. We have also characterized the annual and seasonal rainfall concentrations in the Onkaparinga catchment using PCI.

## STUDY AREA AND DATA

Daily rainfall time series were provided by the Bureau of Meteorology, Australia. Thirteen rainfall stations were selected which are spread across the Onkaparinga catchment, as shown in Figure 1. The details of the selected rainfall stations are listed in Table 1. Daily rainfall data for the period 1960 to 2010 were used for this study. These rainfall stations were previously used by Teoh (2003) who found that the temporal homogeneity of rainfall data at these stations was satisfactory. Seven of these selected stations match with the stations selected by Heneker & Cresswell (2010), who used a network of 20 stations for the assessment of potential climate change impact on the water resources across the Mount Lofty Ranges. The catchment is approximately 25 km south-east of the city of Adelaide in South Australia. About 60% of Adelaide's municipal water is supplied from the Onkaparinga catchment. This catchment is also important for its valuable contribution for meeting local irrigation demand. The Onkaparinga catchment is hydrologically very well instrumented, partly because of its importance as a water supply catchment and partly because it includes the Willunga Basin Super Science Site, which is funded by the Australian Commonwealth Government's Super Science programme for the development of scientific infrastructure. The median annual rainfall over the area is approximately 770 mm but this varies with a strong gradient from approximately 400 mm near the coast to 1,170 mm in upstream areas (Teoh 2003). The catchment has a strong seasonal rainfall variation with less rainfall in summer (December–February) and higher rainfall during winter (June–August) (Beecham *et al.* 2014).

BOM station ID | Station code | Latitude (decimal degree) | Longitude (decimal degree) | Elevation (m) |
---|---|---|---|---|

023726 | G1 | − 34.9 | 138.87 | 459 |

023750 | G2 | − 34.96 | 138.74 | 487 |

023707 | G3 | − 35.01 | 138.76 | 445 |

023720 | G4 | − 35.03 | 138.81 | 341 |

023709 | G5 | − 35.06 | 138.66 | 376 |

023713 | G6 | − 35.1 | 138.79 | 370 |

023710 | G7 | − 35.11 | 138.62 | 267 |

023730 | G8 | − 35.18 | 138.76 | 356 |

023753 | G9 | − 35.27 | 138.56 | 104 |

023704 | G10 | − 35.01 | 138.65 | 305 |

023721 | G11 | − 35.06 | 138.56 | 170 |

023722 | G12 | − 34.93 | 139.01 | 365 |

023733 | G13 | − 35.06 | 138.85 | 363 |

BOM station ID | Station code | Latitude (decimal degree) | Longitude (decimal degree) | Elevation (m) |
---|---|---|---|---|

023726 | G1 | − 34.9 | 138.87 | 459 |

023750 | G2 | − 34.96 | 138.74 | 487 |

023707 | G3 | − 35.01 | 138.76 | 445 |

023720 | G4 | − 35.03 | 138.81 | 341 |

023709 | G5 | − 35.06 | 138.66 | 376 |

023713 | G6 | − 35.1 | 138.79 | 370 |

023710 | G7 | − 35.11 | 138.62 | 267 |

023730 | G8 | − 35.18 | 138.76 | 356 |

023753 | G9 | − 35.27 | 138.56 | 104 |

023704 | G10 | − 35.01 | 138.65 | 305 |

023721 | G11 | − 35.06 | 138.56 | 170 |

023722 | G12 | − 34.93 | 139.01 | 365 |

023733 | G13 | − 35.06 | 138.85 | 363 |

## METHODOLOGY

### Statistical moments and autocorrelation

*x*(where

_{t}*x*

_{1},

*x*

_{2}, …,

*x*are the rainfall depths at a uniform time interval

_{n}*t*) with a sample size

*n*, the various statistical moments used in this study such as mean (), standard deviation (

*s*), skewness (

*g*) and kurtosis (

*k*) are defined by Equations (1)–(4), respectively (Sheskin 2004) Rainfall data may exhibit serial correlation as the data are collected over time. This can be checked by estimating the autocorrelation function (ACF) for the time series. The ACF is a measure of correlation between two values

*x*and

_{t}*x*for a lag

_{t+k}*k*, which can be defined as (Khan

*et al.*2006; Box

*et al.*2011) where

*R*is the ACF for any lag

*k*and is the mean of the time series. If the value of

*R*for any lag lies outside the interval defined by Equation (6), then the time series has significant serial correlation at that lag at the 95% confidence limit. If the lag 1 ACF is outside of this interval, it is assumed that the time series is not composed of random observations.

### Spatial dependence

*E*(.) is the expectation.

*M*and

_{i}*M*are the precipitation maxima at any station

_{j}*i*and

*j*defined as , where

*m*denotes the number of periods of duration

*t*over a time window,

*T*. The random variable

*Z*(

_{m}*t*) represents the precipitation amount.

*F*(.) denotes the marginal distribution of random variable

*M*(

*t*). This equation can be reduced to a simple estimator that does not depend on the form of margins as where

*N*is the number of pairs of rainfall maxima, , and

*F*(.) represents the empirical distribution of random variable

*M*(

_{i}*t*). Considering an isotropic and homogeneous condition in space, a relatively good estimate of the madogram can be obtained by aggregating the results for different bins (discrete intervals).

Daily precipitation maxima for each station and for different accumulation periods (1 day to 25 days) for summer (December–February) and winter (June–August) seasons were extracted from the daily rainfall series over the period 1960 to 2010 and the spatial dependence was estimated using a madogram.

### Probability distribution model

One parameter (exponential), two parameter (gamma and Weibull), three parameter (ME) and hybrid (HGP and HEP) distributions are used in this study to model the frequency distribution of daily rainfall. The generalized extreme value (GEV) distribution is used for frequency analysis. The probability density function and corresponding parameters of different single distributions used in this study are listed in Table 2.

Name of the distribution | Probability density function | Parameters |
---|---|---|

Exponential | x = daily rainfall amount μ = scale parameter | |

Gamma | a = shape parameter b = scale parameter | |

Weibull | a = shape parameter b = scale parameter | |

GP | u = threshold σ = scale parameter k = shape parameter | |

GEV | μ = location parameter σ = scale parameter ξ = shape parameter |

Name of the distribution | Probability density function | Parameters |
---|---|---|

Exponential | x = daily rainfall amount μ = scale parameter | |

Gamma | a = shape parameter b = scale parameter | |

Weibull | a = shape parameter b = scale parameter | |

GP | u = threshold σ = scale parameter k = shape parameter | |

GEV | μ = location parameter σ = scale parameter ξ = shape parameter |

*p*is the mixing factor and

*μ*

_{1}and

*μ*

_{2}are the scale parameters of the two exponential distributions.

*f*and

*F*are the density and cumulative distribution functions of the gamma distribution. The density function of the GP distribution over a threshold

*u*is denoted by

*g*with shape and scale parameters

*k*and

*σ*, respectively.

*I*(

*.*) is the indicator function and [1 −

*F*(

*u;a,b*)] is the normalization factor. In order to make the hybrid density function continuous at the junction of two distributions, i.e. threshold (

*u*), it is necessary that

*h*(

*u*−) =

*h*(

*u*+). The shape parameter of the GP distribution is obtained from the GP distribution fitted to the rainfall data above the threshold value and is directly used in the hybrid distribution whereas the scale parameter can be estimated as

*et al.*(2012a) with probability density and cumulative distribution functions defined respectively as and where

*f*

_{E}(

*x;μ*) and

*f*

_{GP}(

*x;k,σ,θ*) are respectively the exponential density function with scale parameter

*μ*and the GP density at a location

*θ*with scale parameter

*σ*and shape parameter

*k*.

*I*is the indicator function while

*F*

_{E}and

*F*

_{GP}are the cumulative distribution functions for the exponential and GP distributions, respectively. The normalizing constant

*Z*is used to integrate the hybrid density to one. The threshold

*θ*is the junction point of the exponential and the GP distributions and can be defined as a function of the scale parameters of the two distributions.

*Z*and

*θ*are defined respectively as and

In this research, we have used the maximum likelihood estimators (MLEs) to estimate the parameters of the different distributions. The method of moments is an alternative to MLE but can be a poor estimator due to its inefficiency to estimate small values of the shape parameter (Thom 1958; Wilks 1990, 1995). In the case of the HGP distribution, application of the MLE directly to estimate parameters is difficult. Instead, we have followed the procedure suggested by Furrer & Katz (2008). First, the scale and shape parameters of the gamma distribution were estimated by the MLE method considering the entire rainfall series. After selecting a suitable threshold value (*u*), GP distribution parameters were estimated by MLE from rainfall data above *u*. Finally, the scale parameter of the GP distribution was adjusted to obtain a continuous density. In the case of the HEP distribution, the parameters were also estimated by the MLE method as for other distributions.

### Threshold selection in a hybrid distribution

Selection of an appropriate threshold in the hybrid distribution is crucial for its performance. The threshold should be neither too small nor too large (Li *et al.* 2012a). One of the techniques for identification of the threshold for the GP distribution involves the empirical assessment of diagnostic plots and requires prior experience of their interpretation (Davison & Smith 1990; Coles & Tawn 1994). In this method, the estimated parameters of the GP distribution are fitted using a range of thresholds. The threshold is then selected at a point where the estimated parameters become almost linear. Based on this method, Thompson *et al.* (2009) introduced a pragmatic automated threshold selection method. In their method, the optimum threshold was selected based on the distribution of the difference of the estimated parameters when the threshold is changed. Choulakian & Stephens (2001) used a trial and error method to estimate the optimum threshold for the GP distribution. They fitted a GP distribution considering a threshold and then assessed the performance of the distribution by checking the goodness of fit statistics including the Cramér-von Mises statistics (*W*^{2}) and the Anderson–Darling statistics (*AD ^{2}*). The threshold was then increased successively by the value of the smallest order statistics until the significant statistics (

*p*value) of the

*W*

^{2}and

*AD*

^{2}exceeded 10%. An optimum threshold for the GP distribution was selected based on the optimal bias robust estimator of the distribution parameter by Dupuis (1999). Selection of a threshold for the HGP distribution is a trade-off between the gamma and GP distribution. The threshold is generally selected manually by trial and error. Quantile plots of observed and modelled rainfall can be used as a tool to identify a suitable threshold (Furrer & Katz 2008). Selection of the threshold for the HGP distribution requires accurate interpretation of a quantile plot to achieve a satisfactory model fit. In contrast, in this study we have assessed the model efficiency looking at the different percentiles of the data considering an arbitrary threshold. An optimum threshold was then selected by identifying the percentile from which the gamma distribution starts to lose its performance compared to the GP distribution.

### Goodness of fit statistic

*F*(

*X*) and

_{i}*F*(

*Y*) are the empirical and theoretical cumulative distribution functions for a sample size

_{i}*N*.

*E*) and the index of agreement (

*d*) statistics have been used in this study, as described by Equations (20) and (21), where

*O*and are the observed data, model simulated data and observed mean, respectively (Legates & McCabe Jr 1999; Krause

_{i}, P_{i}*et al*. 2005). The range of values for

*E*varies from 1.0 (perfect fit) to and a value less than zero indicates that the mean of the observed series would be a better predictor than the model. The index of agreement (

*d*) varies between 0 (no correlation) and 1 (perfect fit) The quantile (Q–Q) plot, which is applied in this study, is a useful graphical method to compare the probability distributions of two series of data. If the observed and simulated series have identical distributions, then the plotting of their quantiles will be a straight line with a slope of 1:1, pointed through the origin (Wilk & Gnanadesikan 1968).

### Precipitation concentration index

*et al.*(2000), where P

_{i}is the monthly rainfall (mm) in any month

*i*. For seasonal PCI, the Australian seasons are considered as summer (December–February), autumn (March–May), winter (June–August) and spring (September–November). PCI values less than 10 indicate a uniform rainfall distribution; values between 11 and 15 denote a moderate rainfall distribution; values from 16 to 20 suggest an irregular distribution; and values above 20 represent a strong irregularity in precipitation concentration (Oliver 1980; De Luis

*et al*. 2011).

### Daily rainfall generation

*V*is a uniform random number and

_{t}*β*is the randomly picked first (

_{t}*β*

_{1}) or second (

*β*

_{2}) mean which is also dependent on the mixing factor,

*P*as described in Equation (9).

## RESULTS AND DISCUSSION

Statistical moments and lag 1 autocorrelation coefficients of rainfall for different temporal resolutions are listed in Table 3. Both spatial and temporal variation was observed in the statistical moments. Rainfall series for all temporal resolutions (except annual) had positive skewness, which indicates a right skewed distribution. A high right skewed distribution was observed for the finest temporal resolution (daily) and this reduced as the temporal resolution increased from daily to annual. A negative skewness was found in a few stations for the annual rainfall series. In the case of kurtosis, relatively high values were observed for daily rainfall and low values were observed for monthly, seasonal and annual rainfall series. On the whole, the rainfall series at a fine resolution (daily) display a strong right skewed distribution with a sharp peak near the mean whereas rainfall series at coarse resolutions (monthly, seasonal, annual) have less skewed distributions with relatively flat peaks. This is a consequence of the central limit theorem which states that the sum of a large number of independent and identically distributed random variables with finite variance converge to a normal distribution. Significant lag 1 autocorrelation (lag 1 ACF) was observed at all rainfall stations for daily and monthly rainfalls. Seasonally, in summer (DJF), a significant lag 1 autocorrelation was observed at four stations whereas no stations showed any significant lag 1 ACF for other seasons. For annual rainfall, three stations showed significant (at the 5% level) lag 1 ACF and six stations showed lag 1 ACF significant at the 10% level.

Rainfall temporal | Statistical | Rainfall station | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

resolution | moments | G1 | G2 | G3 | G4 | G5 | G6 | G7 | G8 | G9 | G10 | G11 | G12 | G13 |

Daily | Mean | 2.69 | 3.05 | 2.98 | 2.31 | 2.47 | 2.32 | 2.32 | 2.49 | 1.88 | 2.37 | 1.83 | 1.59 | 2 |

Std. dev. | 7.73 | 7.99 | 8.06 | 6.03 | 5.96 | 6.41 | 5.98 | 6.64 | 5.28 | 6.29 | 4.78 | 4.67 | 5.32 | |

Skewness | 5.45 | 4.72 | 4.81 | 4.91 | 4.3 | 5.54 | 4.77 | 5.11 | 5.41 | 4.58 | 5.12 | 6.41 | 5.75 | |

Kurtosis | 49.247 | 33.985 | 33.89 | 38.51 | 28.67 | 53.44 | 36.84 | 41.7 | 51.95 | 30.44 | 40.91 | 70.57 | 60.83 | |

ACF(lag 1) | 0.17* | 0.22* | 0.18* | 0.26* | 0.27* | 0.14* | 0.19* | 0.17* | 0.13* | 0.2* | 0.22* | 0.24* | 0.26* | |

Monthly | Mean | 72.35 | 87.06 | 81.35 | 66.90 | 74.67 | 62.62 | 65.40 | 68.84 | 51.63 | 69.30 | 55.51 | 47.21 | 60.78 |

Std. dev. | 60.78 | 70.05 | 65.15 | 53.91 | 55.00 | 48.43 | 50.07 | 50.89 | 39.35 | 52.49 | 41.41 | 38.75 | 45.94 | |

Skewness | 1.07 | 0.98 | 1.13 | 1.13 | 0.83 | 0.94 | 0.94 | 0.90 | 0.82 | 0.83 | 0.86 | 1.27 | 1.02 | |

Kurtosis | 0.84 | 0.59 | 1.40 | 1.68 | 0.40 | 0.82 | 0.93 | 0.62 | 0.16 | 0.30 | 0.47 | 2.35 | 1.08 | |

ACF(lag 1) | 0.45* | 0.43* | 0.46* | 0.41* | 0.44* | 0.43* | 0.43* | 0.41* | 0.44* | 0.42* | 0.41* | 0.41* | 0.37* | |

DJF | Mean | 86.45 | 109.71 | 101.47 | 87.93 | 95.06 | 84.52 | 80.56 | 91.80 | 66.32 | 89.57 | 76.47 | 75.37 | 85.75 |

Std. dev. | 46.57 | 51.34 | 46.07 | 46.65 | 43.26 | 45.67 | 37.82 | 44.87 | 35.27 | 41.04 | 38.33 | 53.24 | 44.05 | |

Skewness | 0.67 | 0.53 | 0.58 | 0.86 | 0.53 | 0.92 | 0.70 | 0.87 | 1.11 | 0.56 | 0.65 | 1.73 | 0.95 | |

Kurtosis | 0.41 | − 0.28 | − 0.23 | 0.61 | − 0.14 | 0.53 | 0.07 | 0.75 | 2.26 | 0.29 | − 0.27 | 4.27 | 0.64 | |

ACF(lag 1) | 0.19 | 0.27* | 0.05 | 0.27* | 0.23** | 0.23** | 0.19 | 0.19 | 0.18 | 0.13 | 0.14 | − 0.02 | 0.11 | |

MAM | Mean | 187.88 | 242.80 | 224.18 | 182.17 | 219.62 | 170.83 | 196.50 | 189.37 | 149.04 | 205.48 | 165.99 | 118.92 | 167.03 |

Std. dev. | 91.98 | 113.05 | 96.80 | 90.50 | 94.79 | 74.75 | 89.37 | 73.79 | 63.44 | 83.38 | 70.42 | 56.18 | 72.56 | |

Skewness | 0.69 | 0.39 | 0.51 | 0.95 | 0.75 | 0.37 | 1.24 | 0.57 | 0.84 | 0.22 | 0.53 | 0.48 | 0.75 | |

Kurtosis | 1.20 | 0.79 | 1.55 | 2.14 | 1.29 | 0.95 | 3.82 | 1.22 | 1.46 | 0.47 | 1.16 | 0.88 | 1.26 | |

ACF(lag 1) | 0.01 | − 0.04 | − 0.07 | 0.06 | − 0.07 | 0.01 | − 0.10 | − 0.13 | − 0.16 | − 0.06 | − 0.07 | − 0.12 | 0.01 | |

JJA | Mean | 380.61 | 442.70 | 418.47 | 336.77 | 365.29 | 309.33 | 323.24 | 340.54 | 259.60 | 338.32 | 267.00 | 224.27 | 294.91 |

Std. dev. | 123.76 | 138.47 | 137.52 | 97.61 | 95.09 | 105.39 | 84.74 | 91.89 | 62.99 | 94.21 | 74.44 | 79.03 | 83.87 | |

Skewness | 0.39 | 0.14 | 0.84 | 0.57 | 0.42 | 0.15 | 0.35 | 0.48 | 0.14 | 0.56 | 0.49 | − 0.21 | 0.33 | |

Kurtosis | 0.55 | 0.55 | 1.32 | 1.00 | 0.26 | 1.90 | 0.40 | 0.73 | − 0.71 | 0.57 | 0.33 | 0.16 | 0.11 | |

ACF(lag 1) | − 0.15 | − 0.05 | 0.14 | − 0.12 | 0.02 | 0.07 | 0.04 | − 0.10 | − 0.08 | 0.02 | 0.01 | 0.09 | − 0.12 | |

SON | Mean | 213.31 | 249.55 | 232.06 | 195.95 | 216.08 | 186.75 | 184.57 | 204.41 | 144.55 | 198.17 | 156.66 | 147.97 | 181.66 |

Std. dev. | 89.92 | 105.20 | 89.87 | 81.96 | 80.53 | 65.28 | 69.15 | 68.90 | 57.28 | 83.95 | 63.20 | 60.51 | 64.26 | |

Std. dev. | 0.83 | 1.00 | 0.73 | 0.43 | 0.69 | 0.60 | 0.52 | 0.19 | 1.07 | 0.86 | 0.75 | 0.40 | 0.52 | |

Kurtosis | 0.58 | 1.09 | 0.88 | 0.63 | 0.68 | 0.59 | 0.65 | − 0.07 | 1.43 | 0.73 | 0.53 | 0.75 | 0.53 | |

ACF(lag 1) | − 0.15 | − 0.14 | − 0.11 | − 0.01 | − 0.06 | − 0.06 | − 0.03 | − 0.06 | − 0.11 | − 0.08 | 0.03 | − 0.19 | − 0.04 | |

Annual | Mean | 868.25 | 1,044.75 | 976.19 | 802.82 | 896.05 | 751.42 | 784.86 | 826.11 | 619.50 | 831.55 | 666.11 | 566.53 | 729.35 |

Std. dev. | 221.70 | 238.48 | 202.15 | 191.39 | 170.00 | 171.94 | 165.60 | 145.86 | 124.86 | 163.12 | 138.94 | 158.63 | 149.22 | |

Skewness | − 0.04 | 0.13 | − 0.13 | 0.16 | 0.17 | − 0.21 | 0.49 | − 0.29 | 0.56 | − 0.06 | 0.03 | − 0.37 | 0.20 | |

Kurtosis | 1.43 | 1.79 | 0.08 | 0.41 | 0.33 | 0.42 | 0.20 | 0.10 | 0.62 | 0.05 | − 0.32 | 3.96 | 0.57 | |

ACF(lag 1) | − 0.24** | − 0.24** | − 0.22** | − 0.08 | − 0.22** | − 0.24** | − 0.12 | − 0.3* | − 0.23** | − 0.26* | − 0.17 | − 0.14 | − 0.25* |

Rainfall temporal | Statistical | Rainfall station | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

resolution | moments | G1 | G2 | G3 | G4 | G5 | G6 | G7 | G8 | G9 | G10 | G11 | G12 | G13 |

Daily | Mean | 2.69 | 3.05 | 2.98 | 2.31 | 2.47 | 2.32 | 2.32 | 2.49 | 1.88 | 2.37 | 1.83 | 1.59 | 2 |

Std. dev. | 7.73 | 7.99 | 8.06 | 6.03 | 5.96 | 6.41 | 5.98 | 6.64 | 5.28 | 6.29 | 4.78 | 4.67 | 5.32 | |

Skewness | 5.45 | 4.72 | 4.81 | 4.91 | 4.3 | 5.54 | 4.77 | 5.11 | 5.41 | 4.58 | 5.12 | 6.41 | 5.75 | |

Kurtosis | 49.247 | 33.985 | 33.89 | 38.51 | 28.67 | 53.44 | 36.84 | 41.7 | 51.95 | 30.44 | 40.91 | 70.57 | 60.83 | |

ACF(lag 1) | 0.17* | 0.22* | 0.18* | 0.26* | 0.27* | 0.14* | 0.19* | 0.17* | 0.13* | 0.2* | 0.22* | 0.24* | 0.26* | |

Monthly | Mean | 72.35 | 87.06 | 81.35 | 66.90 | 74.67 | 62.62 | 65.40 | 68.84 | 51.63 | 69.30 | 55.51 | 47.21 | 60.78 |

Std. dev. | 60.78 | 70.05 | 65.15 | 53.91 | 55.00 | 48.43 | 50.07 | 50.89 | 39.35 | 52.49 | 41.41 | 38.75 | 45.94 | |

Skewness | 1.07 | 0.98 | 1.13 | 1.13 | 0.83 | 0.94 | 0.94 | 0.90 | 0.82 | 0.83 | 0.86 | 1.27 | 1.02 | |

Kurtosis | 0.84 | 0.59 | 1.40 | 1.68 | 0.40 | 0.82 | 0.93 | 0.62 | 0.16 | 0.30 | 0.47 | 2.35 | 1.08 | |

ACF(lag 1) | 0.45* | 0.43* | 0.46* | 0.41* | 0.44* | 0.43* | 0.43* | 0.41* | 0.44* | 0.42* | 0.41* | 0.41* | 0.37* | |

DJF | Mean | 86.45 | 109.71 | 101.47 | 87.93 | 95.06 | 84.52 | 80.56 | 91.80 | 66.32 | 89.57 | 76.47 | 75.37 | 85.75 |

Std. dev. | 46.57 | 51.34 | 46.07 | 46.65 | 43.26 | 45.67 | 37.82 | 44.87 | 35.27 | 41.04 | 38.33 | 53.24 | 44.05 | |

Skewness | 0.67 | 0.53 | 0.58 | 0.86 | 0.53 | 0.92 | 0.70 | 0.87 | 1.11 | 0.56 | 0.65 | 1.73 | 0.95 | |

Kurtosis | 0.41 | − 0.28 | − 0.23 | 0.61 | − 0.14 | 0.53 | 0.07 | 0.75 | 2.26 | 0.29 | − 0.27 | 4.27 | 0.64 | |

ACF(lag 1) | 0.19 | 0.27* | 0.05 | 0.27* | 0.23** | 0.23** | 0.19 | 0.19 | 0.18 | 0.13 | 0.14 | − 0.02 | 0.11 | |

MAM | Mean | 187.88 | 242.80 | 224.18 | 182.17 | 219.62 | 170.83 | 196.50 | 189.37 | 149.04 | 205.48 | 165.99 | 118.92 | 167.03 |

Std. dev. | 91.98 | 113.05 | 96.80 | 90.50 | 94.79 | 74.75 | 89.37 | 73.79 | 63.44 | 83.38 | 70.42 | 56.18 | 72.56 | |

Skewness | 0.69 | 0.39 | 0.51 | 0.95 | 0.75 | 0.37 | 1.24 | 0.57 | 0.84 | 0.22 | 0.53 | 0.48 | 0.75 | |

Kurtosis | 1.20 | 0.79 | 1.55 | 2.14 | 1.29 | 0.95 | 3.82 | 1.22 | 1.46 | 0.47 | 1.16 | 0.88 | 1.26 | |

ACF(lag 1) | 0.01 | − 0.04 | − 0.07 | 0.06 | − 0.07 | 0.01 | − 0.10 | − 0.13 | − 0.16 | − 0.06 | − 0.07 | − 0.12 | 0.01 | |

JJA | Mean | 380.61 | 442.70 | 418.47 | 336.77 | 365.29 | 309.33 | 323.24 | 340.54 | 259.60 | 338.32 | 267.00 | 224.27 | 294.91 |

Std. dev. | 123.76 | 138.47 | 137.52 | 97.61 | 95.09 | 105.39 | 84.74 | 91.89 | 62.99 | 94.21 | 74.44 | 79.03 | 83.87 | |

Skewness | 0.39 | 0.14 | 0.84 | 0.57 | 0.42 | 0.15 | 0.35 | 0.48 | 0.14 | 0.56 | 0.49 | − 0.21 | 0.33 | |

Kurtosis | 0.55 | 0.55 | 1.32 | 1.00 | 0.26 | 1.90 | 0.40 | 0.73 | − 0.71 | 0.57 | 0.33 | 0.16 | 0.11 | |

ACF(lag 1) | − 0.15 | − 0.05 | 0.14 | − 0.12 | 0.02 | 0.07 | 0.04 | − 0.10 | − 0.08 | 0.02 | 0.01 | 0.09 | − 0.12 | |

SON | Mean | 213.31 | 249.55 | 232.06 | 195.95 | 216.08 | 186.75 | 184.57 | 204.41 | 144.55 | 198.17 | 156.66 | 147.97 | 181.66 |

Std. dev. | 89.92 | 105.20 | 89.87 | 81.96 | 80.53 | 65.28 | 69.15 | 68.90 | 57.28 | 83.95 | 63.20 | 60.51 | 64.26 | |

Std. dev. | 0.83 | 1.00 | 0.73 | 0.43 | 0.69 | 0.60 | 0.52 | 0.19 | 1.07 | 0.86 | 0.75 | 0.40 | 0.52 | |

Kurtosis | 0.58 | 1.09 | 0.88 | 0.63 | 0.68 | 0.59 | 0.65 | − 0.07 | 1.43 | 0.73 | 0.53 | 0.75 | 0.53 | |

ACF(lag 1) | − 0.15 | − 0.14 | − 0.11 | − 0.01 | − 0.06 | − 0.06 | − 0.03 | − 0.06 | − 0.11 | − 0.08 | 0.03 | − 0.19 | − 0.04 | |

Annual | Mean | 868.25 | 1,044.75 | 976.19 | 802.82 | 896.05 | 751.42 | 784.86 | 826.11 | 619.50 | 831.55 | 666.11 | 566.53 | 729.35 |

Std. dev. | 221.70 | 238.48 | 202.15 | 191.39 | 170.00 | 171.94 | 165.60 | 145.86 | 124.86 | 163.12 | 138.94 | 158.63 | 149.22 | |

Skewness | − 0.04 | 0.13 | − 0.13 | 0.16 | 0.17 | − 0.21 | 0.49 | − 0.29 | 0.56 | − 0.06 | 0.03 | − 0.37 | 0.20 | |

Kurtosis | 1.43 | 1.79 | 0.08 | 0.41 | 0.33 | 0.42 | 0.20 | 0.10 | 0.62 | 0.05 | − 0.32 | 3.96 | 0.57 | |

ACF(lag 1) | − 0.24** | − 0.24** | − 0.22** | − 0.08 | − 0.22** | − 0.24** | − 0.12 | − 0.3* | − 0.23** | − 0.26* | − 0.17 | − 0.14 | − 0.25* |

‘*’ and ‘**’ indicate significance at the 0.05 and 0.01 levels, respectively.

Daily rainfall maxima for each station and for different accumulation periods (1 to 25 days) for summer (December–February) and winter (June–August) seasons were extracted from the daily rainfall series over the period 1960 to 2010 and their spatial dependence was assessed using madograms. Figure 2 shows the variation of the madogram as a function of site separation distance for different accumulation periods. During summer, the spatial dependence of the maxima decreases with an increase in site separation distance. In the winter season, a cyclic madogram was observed which reveals that there is a periodic pattern in the spatial dependence of the rainfall maxima. In winter (JJA) the madogram significantly decreased for durations of 1 to 5 days. The changes for 5 to 25 days were less than the changes for 1 to 5 days. In summer (DJF), the madograms for 15 to 25 days were more or less identical. In general summer rainfall showed stronger spatial dependence than the winter rainfall. The spatial dependency of multi-day extremes compared to single day extremes was stronger in winter whereas in summer the spatial dependence did not vary much for single day and multi-day extremes. It was observed that in general increasing the accumulation period (from 1 to 25 days) strengthened the spatial dependence.

### Probability distribution

#### Single and two component distribution

In this study, commonly used probability distributions such as exponential, gamma and Weibull have been fitted to daily rainfall for 13 rainfall stations. Table 4 lists the summary of goodness of fit statistics, including KS-statistics and MAE. It was observed that the gamma and Weibull had better capability to reproduce the empirical cumulative distribution of daily rainfall. The exponential distribution had less skill compared to the gamma and Weibull. However, all three distributions failed to simulate the high rainfall depths, as shown in Figure 3. The upper tails of the distribution were not heavy enough and continuously underestimated the extreme rainfall.

KS-statistics | MAE-cdf | |||||
---|---|---|---|---|---|---|

Exponential | Gamma | Weibull | Exponential | Gamma | Weibull | |

Min | 0.3793 | 0.0621 | 0.0719 | 0.0876 | 0.0217 | 0.0171 |

Mean | 0.4189 | 0.0867 | 0.0831 | 0.1389 | 0.0285 | 0.0217 |

Max | 0.4800 | 0.1239 | 0.1037 | 0.1820 | 0.0337 | 0.0264 |

Stand. Dev. | 0.0291 | 0.0148 | 0.0099 | 0.0313 | 0.0043 | 0.0031 |

KS-statistics | MAE-cdf | |||||
---|---|---|---|---|---|---|

Exponential | Gamma | Weibull | Exponential | Gamma | Weibull | |

Min | 0.3793 | 0.0621 | 0.0719 | 0.0876 | 0.0217 | 0.0171 |

Mean | 0.4189 | 0.0867 | 0.0831 | 0.1389 | 0.0285 | 0.0217 |

Max | 0.4800 | 0.1239 | 0.1037 | 0.1820 | 0.0337 | 0.0264 |

Stand. Dev. | 0.0291 | 0.0148 | 0.0099 | 0.0313 | 0.0043 | 0.0031 |

A two component compound ME distribution was fitted to the daily rainfall. Figure 4 shows the QQ plot of observed and modelled daily rainfall for the ME distribution. Although the ME distribution showed an improvement in reproducing the higher rainfall, compared to the single distributions, it had an over-predicting tendency. This was also the case when compared with the observed annual total rainfall for different average recurrence intervals (ARIs) as shown in Figure 12.

The scale and shape parameters obtained from the gamma distribution fitted to daily non-zero rainfall from all 13 rainfall stations were spatially interpolated using an Inverse Weighted Distance technique to understand the spatial variability of rainfall through the study area, as shown in Figure 5. The spatial distribution of gamma parameters in any area is a useful tool for understanding the spatial variability of rainfall. This was applied by Husak *et al.* (2006) for drought monitoring applications in Africa. Interpretation of the gamma distribution is not as straightforward as the normal distribution where any single parameter such as the mean or standard deviation can be directly used to understand the characteristics of the distribution. Both the shape and scale parameters need to be interpreted together to acquire information from the gamma distribution. For example, areas with the same shape parameter values, but with different scale values, have different density functions. The shape and scale parameters of the gamma distribution are not independent. The product of the estimates of shape and scale is equal to the mean of the non-zero rainfall observations. The variance of the gamma distribution is the estimated shape parameter multiplied by the square of the scale parameter. So, in other words, the square root of the shape parameter is approximately equal to the mean divided by the variance. A larger shape parameter value within a low mean rainfall area will have low variance. A smaller shape parameter value in an area with high mean rainfall will indicate high variance. It is apparent that as the shape and scale parameters are related to each other through the mean rainfall, the area with minimal rainfall amounts is described by either a relatively large shape or scale parameter but not large values of both parameters. In this study we have found that the minimal rainfall can be described by the larger shape parameter. The term shape-dominant rainfall refers to the locations where larger shape parameters exist and the term scale-dominated rainfall refers to the locations with larger scale parameters. A shape-dominated regime describes a pattern where the rainfall tends to be symmetrically distributed, indicating that drier-than-average events are as common as wetter-than-average events. On the other hand, scale-dominated rainfall describes locations where the variance is quite large in comparison to the mean.

It has been observed in this study that the area near the coast (downstream portion of the catchment) has a shape-dominated rainfall pattern, which indicates that the area has a symmetrically distributed rainfall pattern with fewer extreme events. In contrast, the upstream (north-east) portion of the catchment shows a scale-dominated rainfall region, which indicates more variability in the rainfall with more extreme events.

From Figure 5, it is observed that the areas that experience a large amount of rainfall are described by large scale or small shape parameter values in the gamma distribution. The mean daily rainfall and mean number of wet days (>0.5 mm) per year were spatially distributed, with these being lower in the downstream areas near the coast and higher in the upper catchment areas (Figure 6).

#### Hybrid distribution

Application of the hybrid distribution (HGP) improved the performance of the model for fitting the higher quantiles of the rainfall amounts, as shown in Figure 7 (only rainfall stations G1, G3 and G7 are shown here).

When different thresholds were applied to the HGP distribution, it was observed that for comparatively lower thresholds, the higher quantiles of the observed rainfall series were overestimated, as shown in Figure 7. For the HGP distribution, first a threshold value of 10 mm was selected for all stations. In order to further check the performance of each model in reproducing the observed rainfall, various percentiles of daily rainfall such as the 5th, 10th, 30th, 50th, 70th, 80th, 90th, 95th and 99th percentile values were estimated and plotted in Figure 8 and this shows that single distributions (exponential, gamma and Weibull) are not able to simulate percentile values of daily rainfall. For the lower percentiles (10th to 70th percentile) most of the models overestimated the observed rainfall whereas the hybrid model performed well for higher percentiles (90th percentile and above) of rainfall, as shown in Figure 8. The goodness of fit statistics for different distribution models are given in Table 5. For the HGP distribution, since rainfall amounts that are less than the threshold value (10 mm) have been modelled by the gamma distribution, there is no difference in the performance of the hybrid and gamma distributions for lower percentiles (less than 10 mm) as shown in Figure 8 and Table 5. According to the efficiency statistics in Table 5, the Weibull distribution performed better than the gamma distribution in reproducing the lower percentiles of daily rainfall (below the threshold of the hybrid distribution) in all rainfall stations (station G3 is shown here). Therefore, using a hybrid distribution consisting of the Weibull distribution for lower percentiles and the GP distribution for higher percentiles of daily rainfall will provide a better overall efficiency than the hybrid of the gamma and GP distributions for modelling the full spectrum of daily rainfall.

Coefficient of efficiency, (E) | Index of agreement (d) | |||||||
---|---|---|---|---|---|---|---|---|

Rainfall percentile | Gamma | Exponential | Weibull | Hybrid | Gamma | Exponential | Weibull | Hybrid |

5th | − 0.48 | − 0.47 | − 0.87 | − 0.48 | 0.57 | 0.58 | 0.45 | 0.57 |

10th | 0.39 | − 0.22 | 0.48 | 0.39 | 0.75 | 0.51 | 0.79 | 0.75 |

20th | 0.07 | − 0.29 | 0.44 | 0.07 | 0.61 | 0.49 | 0.76 | 0.61 |

30th | − 0.04 | − 0.29 | 0.28 | − 0.04 | 0.55 | 0.47 | 0.67 | 0.55 |

50th | 0.18 | 0.07 | 0.39 | 0.11 | 0.60 | 0.56 | 0.69 | 0.58 |

70th | 0.58 | 0.55 | 0.68 | 0.27 | 0.77 | 0.75 | 0.83 | 0.65 |

80th | 0.81 | 0.78 | 0.86 | 0.84 | 0.90 | 0.88 | 0.92 | 0.91 |

90th | 0.68 | 0.60 | 0.75 | 0.85 | 0.82 | 0.78 | 0.87 | 0.92 |

95th | 0.44 | 0.33 | 0.59 | 0.87 | 0.70 | 0.65 | 0.78 | 0.93 |

99th | 0.19 | 0.06 | 0.42 | 0.79 | 0.59 | 0.54 | 0.70 | 0.89 |

Coefficient of efficiency, (E) | Index of agreement (d) | |||||||
---|---|---|---|---|---|---|---|---|

Rainfall percentile | Gamma | Exponential | Weibull | Hybrid | Gamma | Exponential | Weibull | Hybrid |

5th | − 0.48 | − 0.47 | − 0.87 | − 0.48 | 0.57 | 0.58 | 0.45 | 0.57 |

10th | 0.39 | − 0.22 | 0.48 | 0.39 | 0.75 | 0.51 | 0.79 | 0.75 |

20th | 0.07 | − 0.29 | 0.44 | 0.07 | 0.61 | 0.49 | 0.76 | 0.61 |

30th | − 0.04 | − 0.29 | 0.28 | − 0.04 | 0.55 | 0.47 | 0.67 | 0.55 |

50th | 0.18 | 0.07 | 0.39 | 0.11 | 0.60 | 0.56 | 0.69 | 0.58 |

70th | 0.58 | 0.55 | 0.68 | 0.27 | 0.77 | 0.75 | 0.83 | 0.65 |

80th | 0.81 | 0.78 | 0.86 | 0.84 | 0.90 | 0.88 | 0.92 | 0.91 |

90th | 0.68 | 0.60 | 0.75 | 0.85 | 0.82 | 0.78 | 0.87 | 0.92 |

95th | 0.44 | 0.33 | 0.59 | 0.87 | 0.70 | 0.65 | 0.78 | 0.93 |

99th | 0.19 | 0.06 | 0.42 | 0.79 | 0.59 | 0.54 | 0.70 | 0.89 |

The quantile plot in Figure 7 indicates that 10 mm is a suitable threshold for the HGP model. This value, which is equal to the 68th percentile of daily rainfall, is used for the HGP distribution at rainfall station G3. The scatter plot (Figure 8) and goodness of fit statistics (Table 5) both show that the gamma distribution performed better than the hybrid distribution for the 50th, 70th and 80th percentiles of daily rainfall. Therefore a new threshold value of 25 mm, which is equal to the 90th percentile of daily rainfall, was selected and goodness of fit statistics show that the overall performance of the HGP distribution has been improved. It was observed that for most of the stations the thresholds were around the 90th percentile of the wet day daily rainfall. Selection of the threshold value for the HGP model based on the quantile plots might be misleading. Therefore, the efficiency statistics and the scatter plot of the observed and modelled percentiles of rainfall have also been considered for optimum selection of the threshold value in the HGP model.

The HGP distribution exhibits a poorer performance compared to other single distributions (for example, the gamma) when the daily rainfall is aggregated to estimate the seasonal and annual rainfall totals, which is an interesting finding. It is evident in Figure 8 that the lower percentiles (for example, daily rainfall less than 15 mm for station G3) of daily rainfall are overestimated by the gamma distribution and even by the HGP distribution. On the other hand, the higher percentiles of daily rainfall (daily rainfall greater than 15 mm for station G3) are under-predicted by the gamma distribution, which overestimated the lower percentile rainfalls. In contrast, for the HGP distribution, the overestimated lower percentile rainfalls are not counterbalanced by a noticeable underestimation of the higher percentile rainfalls. As a result, the annual total rainfall is always over-predicted by the HGP distribution, more so than by a single distribution. From Table 6, it is clear that both the HGP and gamma distributions overestimate the total rainfall (below the threshold value of 15 mm) over the 51 year period by 2,572 mm. This is counterbalanced by the underestimated total rainfall (1,973 mm) above the threshold in the case of the gamma distribution, whereas the HGP distribution is not able to counterbalance this, and instead again overestimates the total rainfall. Finally, considering the daily rainfall of station G3 over the period 1960 to 2010, the HGP and the gamma distributions overestimated the 51 year total rainfall by 2,793 mm and 599 mm, respectively. This over-prediction varies from year to year due to the variability of the total amount of rainfall above and below the threshold. For example, for station G3 in the year 1960, the HGP distribution over-predicted the annual total rainfall by 64.27 mm, whereas the gamma distribution underestimated the annual total rainfall by 4.96 mm. In the following year (1961), annual rainfall is overestimated by 73.44 mm and 57.88 mm by the HGP and gamma distributions, respectively. So the choice of rainfall distribution could be varied depending on the required temporal resolution (daily, monthly, seasonal and annual) of model output data. The underestimation of the annual total rainfall by the HGP distribution might be due to the fitting procedure. Fitting a gamma distribution for lower rainfall percentiles (below the threshold, *u*) rather for the total rainfall series and then fitting a GP distribution for the higher rainfall percentiles (above the threshold, *u*) would reduce the limitation of the HGP distribution. While the performance of the HGP distribution for annual and seasonal rainfall simulation is not satisfactory compared to the gamma and Weibull distributions, the HGP distribution can reproduce the full range of observed daily rainfall in the Onkaparinga catchment.

Total rainfall (mm) | |||||
---|---|---|---|---|---|

Rainfall category | Observed | Hybrid | Gamma | Observed-Hybrid | Observed-Gamma |

≤15 mm | 19,427 | 21,999 | 21,999 | 2,572 | 2,572 |

>15 mm | 30,012 | 30,233 | 28,039 | 221 | − 1,973 |

Total | 2,793 | 599 |

Total rainfall (mm) | |||||
---|---|---|---|---|---|

Rainfall category | Observed | Hybrid | Gamma | Observed-Hybrid | Observed-Gamma |

≤15 mm | 19,427 | 21,999 | 21,999 | 2,572 | 2,572 |

>15 mm | 30,012 | 30,233 | 28,039 | 221 | − 1,973 |

Total | 2,793 | 599 |

The higher percentiles of daily rainfall were reasonably simulated by the HEP distribution compared to the single distributions (gamma, exponential and Weibull). Figure 9 shows the QQ plot of daily observed and simulated rainfall for the HEP model. The model also performed better than the single distributions in simulating the annual maximum (AM) for different ARIs, as shown in Figure 11.

Figure 10 shows the monthly variation of the standard deviation and skewness of daily rainfall over the period 1960 to 2010. The standard deviation and skewness of daily rainfall (considering only wet day rainfall) were estimated for each month and for each year over the period 1960 to 2010, and then the average for each month was estimated separately. No single distribution was able to reproduce the observed standard deviation and skewness of daily rainfall for all months. However, in general the hybrid (HGP and HEP) and ME distributions performed better than the single distributions (gamma, exponential and Weibull).

A frequency analysis has been performed for rainfall extremes (annual maxima) of daily observed and modelled rainfall obtained for each year and each rainfall station by fitting the GEV distribution. Figure 11 shows the annual maxima of daily rainfall amounts for different ARIs. The rainfall modelled by the hybrid (HGP and HEP) and the ME distributions can reproduce the observed ARI rainfall more accurately compared to the values obtained by other single distributions such as the gamma, exponential and Weibull. The ME distribution model appeared to be the best for reproducing the observed frequency of AM daily rainfall at 11 of the 13 stations. The HGP model showed the best result for the other two rainfall stations. The annual total rainfalls for different ARIs were overestimated by the ME distribution model at 12 stations. At nine stations the hybrid (HGP and HEP) and single distribution (gamma, exponential and Weibull) models showed similar performances in reproducing the frequency of the annual rainfall, as shown in Figure 12. In general, the hybrid distribution (HGP and HEP) did not show a significant improvement in reproducing the annual rainfall. However, at stations G1, G4, G7 and G13, the HGP distribution provided the best estimation of annual rainfall compared to the other distribution models. The best performing model varies from one station to another in reproducing extreme rainfall (annual total and annual maxima) for different ARIs.

### Precipitation concentration index

PCI is a key index that provides information on how the rainfall amount is distributed within a specific period of time. For example, a lower PCI on an annual scale indicates that the rainfall total is uniform over each month of the year and vice versa. PCI < 10 indicates a uniform rainfall concentration which means that the rainfall amount is uniformly distributed over a period of time. On the other hand, higher PCI values indicate higher percentages of total rainfall occurring in only a few rainy days which has the potential to cause floods and/or droughts. So, higher PCI values suggest an increased likelihood of extreme events. PCI could therefore be a useful decision tool for sustainable water resources management. In this study, we have calculated the PCI for annual and seasonal scales. The spatial and inter-annual variation of PCI is shown in Figure 13. The median values of annual and seasonal PCI were quite homogeneous throughout the study area. However, inter-annual variability was observed for both annual and seasonal PCI. Mean annual and seasonal (summer and autumn) PCI indicate a moderate precipitation concentration, whereas winter and spring seasonal PCI values show a uniform precipitation concentration. The PCI analysis indicates that the period from December to May is more susceptible to extreme events than the period June to November. Inter-annual variability of PCI is found to be higher for seasonal and lower for annual temporal resolutions. Summer PCI shows the highest variability, while winter PCI exhibits the lowest variability as shown in Figure 14. This could be because the Adelaide region has winter dominant rainfall and the sparse nature of summer rainfall may lead to increased variability. Since the summer season exhibits moderate PCI values with high inter-annual variability, sustainable management of water resources in summer may be more challenging with increased vulnerability and reduced security of supply, which in turn may require more careful planning.

## CONCLUSION

In this study, the limitation of applying widely used single distributions such as the gamma, exponential and Weibull to model the entire range (low to high) of daily rainfall time series has been demonstrated. Instead, hybrid (HGP and HEP) and ME distributions were fitted to observed daily rainfall data and this was found to reproduce the full spectrum of rainfall as demonstrated by several statistics including the standard deviation, skewness, frequency distribution, percentiles and extreme values. The quantile plot, MAE of CDF and KS-statistics were used to identify the best fit probability distribution for daily rainfall.

The study shows that the gamma and Weibull distributions exhibit better performance in reproducing the rainfall compared to the exponential distribution. But none of these models reproduced the extreme rainfall depths satisfactorily. The spatial distribution of the shape and scale parameters of the gamma distribution provided important information on the characteristics of rainfall in the study area. Downstream regions near the coast of the Onkaparinga catchment displayed a shape-dominant rainfall (less variability and fewer extreme events). In contrast, the upstream regions of the catchment were characterized by scale-dominated rainfall (more variability and more extreme events). Spatial dependency among the rainfall maxima varied with site separation distance, seasons and rainfall accumulation periods. The spatial dependency of multi-day extremes compared to single day extremes was stronger in winter whereas in summer the spatial dependence did not vary much for single day and multi-day extremes.

The quantile plots clearly indicate that the HGP and HEP model is able to simulate the higher percentiles of daily rainfall well. The ME distribution model over-predicted the higher percentiles of daily rainfall. The HGP model performed less satisfactorily for annual and seasonal rainfall totals. This might be due to the fitting procedure for the HGP distribution. Application of the hybrid (HGP and HEG) and ME distributions for daily rainfall modelling was able to successfully model extreme events (annual maxima of daily rainfall) compared to the single distribution models. The best performing model varied from one station to another in terms of the ability to reproduce extreme rainfall for different ARIs. However, in general the ME model appeared to be the best for reproducing the observed frequency of AM daily rainfall. In general, the hybrid distributions (HGP and HEP) did not show significant improvement in terms of reproducing the annual rainfall for different ARIs. However, at stations G1, G4, G7 and G13 the HGP distribution provided the best estimation of annual rainfall compared to the other distribution models. The ME distribution overestimated the annual rainfall for all stations except station G5.

Examining the model efficiency to simulate the observed percentiles of daily rainfall depth is useful for identifying an optimum threshold for the HGP distribution. This study shows that the hybrid of the Weibull and GP distributions is a better model than the hybrid of the gamma and GP distributions, at least for the Onkaparinga catchment.

The mean of the annual and seasonal PCIs was found to be approximately the same for all rainfall stations. However, inter-annual variability was observed in both the annual and seasonal rainfall totals. These values represent a moderate rainfall concentration in almost all stations throughout the year and for all seasons except winter. The summer PCI shows the highest variability whereas the winter PCI exhibits the lowest variability. So it can be inferred that sustainable management of water resources in the summer season is more challenging than in other seasons. This may help inform water resource investment strategies.

Finally, it is expected that the incorporation of the hybrid and the ME distributions in daily rainfall modelling and downscaling will improve the efficiency of models for simulating the entire range of the daily rainfall time series. Moreover, the spatial distribution of the gamma parameters and variability of PCI will assist in developing more sustainable water resource management strategies. The outcomes of this research will also be useful for water resource investment decisions.

## ACKNOWLEDGEMENTS

This study was funded by the South Australian Government's Goyder Institute for Water Research through Grant C.1.1. The Goyder Institute also provided additional scholarship funding for the first author. The researchers are also grateful to the Australian Bureau of Meteorology for providing meteorological data.