A nonparametric model for stochastic generation of daily rainfall amounts

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 12, 1343, doi: /2003wr002570, 2003 A nonparametric model for stochastic generation of daily rainfall amounts Timothy I. Harrold Research Institute for Humanity and Nature, Kyoto, Japan Ashish Sharma School of Civil and Environmental Engineering, University of New South Wales, Sydney, New South Wales, Australia Simon J. Sheather Australian Graduate School of Management, University of New South Wales, Sydney, New South Wales, Australia Received 6 August 2003; accepted 15 September 2003; published 5 December [1] This paper presents a model for stochastic generation of rainfall amounts on wet days that is nonparametric, accommodates seasonality, and reproduces important distributional and dependence properties of observed rainfall. The model uses kernel density estimation techniques, which minimize the assumptions that are made about the underlying probability density, and representation of seasonal variations is achieved through the use of a moving window approach. Four different classes of rainfall amount are considered and categorized according to the number of adjacent wet days, and the model is conditioned on the rainfall amount on the previous day. The proposed model can emulate the day-to-day features that exist in the historical rainfall record, including the lag 1 correlation structure of rainfall amounts. We link this model with long sequences of rainfall occurrence generated by the model of Harrold et al. [2003], which is designed to reproduce the longer-term variability of the observed record. The approach is applied to daily rainfall from Sydney, Australia, and the performance of the approach is demonstrated by presentation of model results at daily, seasonal, annual, and interannual timescales. INDEX TERMS: 1812 Hydrology: Drought; 1854 Hydrology: Precipitation (3354); 1869 Hydrology: Stochastic processes; KEYWORDS: nonparametric, rainfall, simulation, stochastic Citation: Harrold, T. I., A. Sharma, and S. J. Sheather, A nonparametric model for stochastic generation of daily rainfall amounts, Water Resour. Res., 39(12), 1343, doi: /2003wr002570, Introduction [2] Stochastic data generation is a statistical procedure that produces synthetic sequences of hydrologic data such as rainfall. These sequences are assumed to be as likely to occur in the future as the observed historical record, and are used in catchment water management studies as a tool for exploring the potential variability in the catchment response. However, droughts and sustained periods of high rainfall can be hard to reproduce in generated sequences. If these features are not present in the synthetic sequences generated by a stochastic model, then the generated sequences will not be of great value in catchment water management studies; the use of such sequences could lead to misrepresentation of the possible effects of climatic variability, and to suboptimal policies for system management. Unfortunately, existing methods for the generation of daily rainfall under-represent the longer-term variability that is associated with drought [Buishand, 1978; Wilks and Wilby, 1999]. [3] The aim of this study is to formulate methods of generating daily rainfall at a single location within a catchment, such that the generated sequences represent the historical record both at daily and longer timescales. As a Copyright 2003 by the American Geophysical Union /03/2003WR SWC 8-1 first part of this work, an approach for generating sequences of the rainfall occurrence state (dry or wet) was proposed [Harrold et al., 2003], the approach being structured to ensure that the variation in the number of wet days in a seasonal, annual or longer time period, is of the same form as observed in the historical rainfall record. This paper presents the second part of the work. The aim here is to generate a sequence of rainfall amounts for all the wet days in the synthetic rainfall record, such that the generated sequence of amounts adequately reproduces the distributional features, dependence features, and seasonal variations of the observed record, and such that the resulting synthetic rainfall record (occurrence + amounts) reproduces the observed features of the historic record at daily, seasonal, annual and interannual timescales. [4] Previous authors have shown that the distribution of rainfall amount is different on solitary wet days compared to days in the middle of a wet spell, and to days at the start or end of a wet spell [Buishand, 1978; Chapman, 1998]. This is due to the characteristics of the weather patterns that produce rainfall, and to the fact that rainfall events may start or end at any time of day, with only a few hours of the event extending into the first or last day of the wet spell. We consider these issues when we formulate our amounts model by considering four classes of wet day, namely

2 SWC 8-2 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS Table 1. Average Annual Rainfall, Average Number of Wet Days Per Year, and Average Rainfall on a Wet Day for El Niño and La Niña at Tenterfield El Niño Non-ENSO La Niña Annual rainfall, mm Number of wet days Rainfall on a wet day, mm solitary wet days, days at the start of wet spells, days at the end of wet spells, and days in the middle of wet spells, and then giving separate treatment to the distributions of amount on each class of wet day. [5] The weather patterns that produce rainfall introduce dependence structure into the record of amounts. For example, slow-moving cyclones or slow-moving frontal systems can result in several days of high rainfall at a given location. These effects produce small but significant correlations between amounts within a wet spell [Buishand, 1978]. Many existing rainfall amount models ignore this correlation structure and assume that the wet day amounts are independent [Wilks and Wilby, 1999]. The approach proposed in this paper accommodates within-spell correlations of rainfall amounts by using the amount on the previous day as a conditioning variable. Allan [1991] also notes that the weather patterns that produce rainfall are influenced by low-frequency ocean-atmosphere interactions such as the El Niño-Southern Oscillation. Such climatic variability can result in a rainfall record that exhibits complex low-frequency dependence features. This is the case for many rainfall records in Southeastern Australia, such as the records considered in this paper. The occurrence model of Harrold et al. [2003] has considered low-frequency dependence features in its formulation. An issue that is addressed in this paper is whether it is also necessary to consider such low-frequency dependence features when constructing our model for rainfall amounts on wet days. [6] This paper is organized as follows. Section 2 discusses the influence of low-frequency climatic features such as the El Niño-Southern Oscillation on the longer-term variability of rainfall records in Southeastern Australia, and considers how this longer-term variability can be reproduced in a daily rainfall model. Section 3 briefly describes the daily rainfall occurrence model of Harrold et al. [2003]. Section 4 reviews existing approaches to model the within-spell correlations of rainfall amounts. Kernel density estimation, which is used in formulating the amounts model, is described in section 5. The model for rainfall amounts is presented in section 6. Results from the amounts model, and from the combined occurrence/ amounts model, are given in section 7. Section 8 is the conclusion to the paper. An appendix is presented after the conclusion, which discusses both the selection of a key smoothing parameter that is used in the amounts model, and the influence of the El Niño-Southern Oscillation on rainfall amounts for each of four wet-day classes. 2. Incorporating Longer-Term Variability Into a Daily Rainfall Model [7] The El Niño-Southern Oscillation (ENSO) is a largescale coupled ocean-atmosphere quasi-periodic variation centered in the low latitudes of the Pacific Ocean [Allan, 1991], which occurs with a frequency of around 3 7 years. ENSO influences rainfall over Southeastern Australia. Table 1 presents the average annual rainfall at Tenterfield in northern New South Wales, Australia, for El Niño and La Niña years as classified by Allan [1997]. The differences between El Niño and La Niña years show that the rainfall record contains low-frequency features consistent with significant longer-term variability. This variability cannot be reproduced by a rainfall generation model unless that model incorporates mechanisms designed to simulate it. [8] The association between annual rainfalls and ENSO can be broken down into an occurrence component and an amount component. This is also shown in Table 1. The average number of wet days for El Niño and La Niña years are significantly different (a = 0.05). However the average rainfall on wet days is not very different for El Niño and La Niña years. This suggests that longer-term variability in climate has a more dominant effect on the distribution of the number of wet or dry days in a year, as compared to the amount of rainfall that is recorded on any given rainy day. Similar results can be obtained for several locations in Australia where long daily rainfall records are available. On the basis of these results, we argue that linking an occurrence model that reproduces longer-term variability with a simpler model for rainfall amounts on wet days will give a good overall model for daily rainfall. This will be tested later in this paper. [9] Harrold et al. [2003] propose a conditional simulation model for daily rainfall occurrence which is formulated to represent both the short-term and the longer-term variability of the observed record. The conditionality is imparted through the use of predictors specified at daily, seasonal, annual, and interannual levels. The longer-term predictors all act as conditioning variables which contain low-frequency signals that are similar to the information contained in ENSO, however they are not external to the generated rainfall record, as these variables are formed solely from previous values in the sequence. A significant advantage of this approach is that the model can be calibrated on the entire length of the historical rainfall record (not just on the period for which observed values of the exogenous variable(s) are available), and thus it can be designed to reproduce the full range of observed variability. This is very important for catchment water management studies, as noted in the introduction. This rainfall occurrence model is now briefly described. 3. A Model for Daily Rainfall Occurrence [10] The rainfall occurrence model of Harrold et al. [2003] is termed ROG( j) to denote rainfall occurrence generator, using j predictor variables. ROG( j) uses the moving window approach [Rajagopalan et al., 1996] to model seasonality. An l-day moving window is centered at the current calendar day, and all days falling within this window (from all historical years) form the local subset of data used in the model. This ensures a smooth transition of generated rainfall characteristics throughout the year, while providing sufficiently large data sets for simulation purposes. Actual simulation proceeds using nearest-neighbor resampling [Lall and Sharma, 1996], which approximates the random mechanism that produced the historical data; a

3 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS SWC 8-3 pattern is chosen from the historical sequence that is similar to the current pattern in the generated sequence, and the successor to the chosen pattern is placed into the generated sequence. The patterns that are examined are those that exist in the predictor set, which for Sydney was chosen as (1) rainfall occurrence on the previous day, (2) a 90-day wetness index, (3) a 365-day wetness index, and (4) an 1825-day wetness index. [11] The wetness indexes are aggregate variables, based on the number of wet days in the preceding m i days, and discretized into five states ( very dry, dry, average, wet, and very wet ). The three longer-term predictors represent low frequency features in the simulated data. For the Sydney rainfall occurrence data, l was chosen as 15. Further details on the ROG( j) model are given by Harrold et al. [2003], who show that the ROG(4) model for Sydney rainfall occurrence ( ) reproduces the daily-level statistical characteristics of the historical occurrence record, smoothly reproduces seasonal features, and reproduces observed variability at seasonal, annual, and interannual timescales. 4. Approaches to Modeling Correlation in Wet Day Amounts [12] Gregory et al. [1993] suggest that reproduction of the structure of daily autocorrelation provides a crucial test for a stochastic rainfall generator. Multistate Markov chains [Haan et al., 1976; Srikanthan and McMahon, 1985; Gregory et al., 1993] are one type of rainfall model that do not assume independence of wet day amounts, and can therefore at least partially reproduce the correlation structure. Nonparametric models can also be formulated to reproduce daily autocorrelation by conditioning the simulated amounts on the value on the previous day [Sharma and Lall, 1999]. Sharma and Lall s approach is to apply nearest-neighbor resampling (as described in the previous section) using the length of the wet spell, the position in the wet spell, and the rainfall amount on the previous day as the conditioning variables. A disadvantage of this approach is that the resampled amounts, including the extreme events, are historical values. Conditional kernel density estimation, as described in the next section, forms a smooth empirical probability density estimate that can produce values that are different to the historical values. [13] The multistate Markov chain simulates both occurrence and amounts. Different ranges of rainfall amounts are grouped into classes, with the lowest class constituting zero rainfall (i.e., dry days), and transition probabilities among all possible pairs of classes are calculated. Once the class is decided, the rainfall within the class is calculated using a parametric distribution. It is the transition probabilities of a multistate Markov chain that allow the model to at least partially reproduce the correlation structure of rainfall amounts. Gregory et al. [1993] show that the daily serial correlation coefficients of area-average rainfall in southeast England are better reproduced by multistate Markov chains than by models that assume the rainfall amounts on wet days are independent. [14] In a study covering 14 locations in Australia, 6 locations in South Africa, 24 locations in North America, and 22 locations in the Pacific islands, Chapman [1998] compared the performance of commonly used parametric rainfall models against the multistate Markov chain of Srikanthan and McMahon [1985]. The Srikanthan and McMahon approach performed better than the other models for 53 out of the 66 locations tested. The Akaike Information criteria [Akaike, 1974] was used for the comparisons. This is a measure of the quality of a one-day-ahead forecast made using the model. We believe that the relatively good performance of the Srikanthan and McMahon approach is due to the ability of a multistate Markov chain to partially reproduce the structure of daily autocorrelation, and thus provide a more accurate forecast of the rainfall amount on a day to day basis. 5. Univariate and Conditional Kernel Density Estimators [15] A univariate kernel probability density estimator is written as ^f X ðx; hþ ¼ Xn i¼1 1 nh K ðx x iþ h where x i is the ith data point in a sample of size n, K( )isa kernel function that must integrate to 1, and h is the bandwidth of the kernel used in estimating the probability density function. [16] The density estimate in (1) is formed by summing kernels with bandwidth h centered at each observation x i. By using smooth kernel functions, the estimated probability density in (1) is smooth and continuous. The amount of smoothing depends on the choice of h. A large bandwidth results in an oversmoothed probability density, with subdued modes and over-enhanced tails. A small bandwidth, on the other hand, can lead to density estimates with noticeable bumps in the tails of the probability density. [17] A conditional probability density estimator formed from a bivariate density is written as: ^f ð xt jx t 1 Þ ¼ ^fðx t ; x t 1 Þ ^f m ðx t 1 Þ where ^f (x t, x t 1 ) is the bivariate density estimate formed by summing bivariate kernels centered at each of n historical observations (x i, x i 1 ), and ^f m () is the estimated marginal density. ^f (xt, x t 1 ) is formed in a similar way to the univariate kernel density estimate in (1), except its formation is in two dimensions, and two bandwidths are used, one in each coordinate direction. ^f (x t jx t 1 ) is a slice through this bivariate density function along the conditioning plane specified by x t 1. The conditional estimate is itself composed of a sum of slices, along the conditioning plane, through the n individual kernels that form the bivariate density estimate. This is illustrated in Figure 1. [18] Using the kernel density estimator in (1) with a Gaussian kernel function, the conditional density in (2) is estimated as: ^f ð xt jx t 1 Þ ¼ Xn i¼1! 1 ð ð2pþ 0:5 w i exp x t x i Þ 2 h 1 where ^f (x t jx t 1 ) is the conditional probability density estimate, h 1 is the bandwidth in the X t coordinate direction 2h 2 1 ð1þ ð2þ ð3þ

4 SWC 8-4 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS [21] Application of the univariate and conditional kernel density estimation techniques required a choice of bandwidth to be made. Several methods for estimating an optimal bandwidth have been proposed in the statistical literature. A relatively simple bandwidth choice suggested for use with samples having a high coefficient of skewness is used here: h ¼ lrn 1=5 ð4þ Figure 1. Illustration of the conditional probability density function ^f (x t jx t 1 ) (after Sharma et al. [1997]). (see Figure 1), w i is the weight associated with each kernel slice that constitutes the conditional probability density, w i ¼ P n j¼1 exp exp! ð x t 1 x i 1 Þ 2 2h 2 2 x 2!; t 1 x j 1 h 2 is the bandwidth in the X t 1 coordinate direction, and x i is the conditional mean associated with each kernel slice. [19] The specification of w i ensures that simulation proceeds with more emphasis given to the observed data points (x i, x i 1 ) that lie close to the conditioning plane (x t 1 ), and less emphasis given to the data points that lie further away. The derivation of equation (3) is presented by Sharma and O Neill [2002] and Sharma et al. [1997], except a bivariate kernel that is scaled by h 1 and h 2 is presented here. Sharma and O Neill used the full covariance matrix to scale the kernel, in a process known as sphering [Fukunaga, 1972, p. 175]. A sphered kernel will intercept the conditioning plane (x t 1 ) at an angle, which can result in kernel slices that have a negative conditional mean. 6. Model for Amounts [20] The model for rainfall amounts is called RAG to denote rainfall amount generator. The one-predictor RAG(1) model, described here, uses conditional kernel density estimation to provide an empirical estimate of the distributional features of the observed data. The model is conditioned on the rainfall amount on the previous day, so that it can reproduce the short-term dependence structure of the observed amounts, and seasonality is modeled using the moving window approach that is described in section 3. The model is applied separately to solitary wet days, days at the start of wet spells, days at the end of wet spells, and days in the middle of wet spells, with the amounts on each class of wet day labeled as class 0, class 1a, class 1b, and class 2, respectively. An l-day moving window is used in the model. (Note that for this data set, we chose l = 31 as a compromise between forming a smoothly varying seasonal pattern, and having a width small enough to capture any rapidly changing seasonal effects.) A threshold of 0.3 mm is used to define a wet day. 2h 2 2 where R is the sample interquartile range of the data, and l a factor that depends on the sample coefficient of skewness. A value of l = 0.3 was chosen in the present study based on the average coefficient of skewness noted for the Sydney daily rainfall amounts. This skewness was stable across seasons and across the wet-day classes, with an average value of about 4. The derivation of equation (4) is discussed in Appendix A to this paper. For the conditional kernel, both h 1 and h 2 are calculated using equation (4), and are estimated based on the respective samples denoting the X t and X t 1 coordinate directions. However the values of R and n used for calculation h 1 of are conditional to the value of x t 1. Details of these conditional calculations are also given in the appendix to the paper. [22] For class 0 and class 1a amounts, the amount on the previous day is always zero, and the univariate kernel density of equation (1) is used. Simulation of class 0 and class 1a amounts (x t ) proceeds as follows: [23] 1. Form a seasonal subsample X of n class 0 (or class 1a) amounts x i from the historical record, using a moving window centered on the calendar day corresponding to x t. [24] 2. Estimate h from X using equation (4). [25] 3. Pick an x i value from X with probability 1/n. [26] 4. Select x t as a random variate from the kernel centered on x i : x t ¼ x i þ hw t where W t is a random variate from a normal distribution with mean of 0 and variance of 1. [27] Equation (5) can lead to values for the rainfall amounts that are less than the threshold amount. To deal with this problem, we use a variable kernel and boundary renormalization (as described by Sharma and O Neill [2002]) near the threshold amount value of 0.3 mm. The bandwidth used in step 4 of the algorithm is reduced depending on how far x i is from 0.3. The modified step 4 is as follows. [28] 4a. Estimate a transformed bandwidth h 0 such that h 0 ¼ h if F Nxi;h ð 2 Þðx t < 0:3Þ a ¼ h 0 if F Nxi;h ð 2 Þðx t < 0:3Þ > a; such that F Nxi;h ð 02 Þðx t < 0:3Þ¼a ¼ 0 if x i ¼ 0:3 where F N(m,s 2) is the cumulative probability of a normal distribution with mean m and variance s 2. [29] 4b. Sample a new value of x t as x t = x i + h 0 W t. [30] 4c. Repeat step 4b if the sampled x t is less than [31] A threshold probability a equal to 0.06 has been used here (after Sharma and O Neill [2002]). If an amount less than 0.25 mm is simulated (as happens in less than 6% of all cases for kernels lying close to the zero-flow bound- ð5þ

5 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS SWC 8-5 Figure 2. Julian day. Historical mean daily rainfall for Sydney on class 0, 1a, 1b, and 2 wet days as a function of ary), a new value is sampled from the same kernel until an acceptable value results. This approach recognizes that the variability associated with low-rainfall values is significantly smaller than that associated with higher-rainfall values. While the proposed approach introduces bias into the simulated values lying close to the boundary, this bias is considerably less than would be the case if a variable kernel was not used, and the results presented below show that the effect of this bias on the overall model performance is negligible. After simulation, we round the generated rainfall amounts to the nearest 0.1 mm, because the historical amounts are also rounded to the nearest 0.1 mm. [32] For class 1b and class 2 amounts, the rainfall on the previous day is considered, and the conditional kernel density estimator of equation (3) is used. The simulation of class 1b and class 2 amounts proceeds as follows. [33] 1. Form a seasonal subsample of class 1b (or class 2) amounts (x i ) from the historical record. Also identify all corresponding x i 1 values. These form n pairs of amounts (x i, x i 1 ). Call this seasonal subsample X. [34] 2. Estimate h 2 from the marginal distribution of x i 1 using equation (4); [35] 3. Estimate the weights w i for the kernel slices that are associated with each data pair (x i, x i 1 ). [36] 4. Estimate h 1 from X using equation (4), with values of R and n chosen conditional to the value of x t 1 (cf. section 9.1); [37] 5. Pick a data pair (x i, x i 1 ) from X with probability w i. The bivariate kernel centered on this data point intersects the conditioning plane specified by x t 1, giving a kernel slice centered at x i. (Parameters x i, h 1, and w i give the center, spread, and relative area of each individual kernel slice at the conditioning plane specified by x t 1 ). [38] 6. Select x t as a random variate from the kernel slice centered on x i : x t ¼ x i þ h 1 W t ð6þ where W t is a random variate from a normal distribution with mean of 0 and variance of 1. [39] The use of a variable kernel and boundary renormalization modifies step 7 of the above procedure near the threshold amount value of 0.3 mm, in a way similar to the unconditional case that has already been described. Note that equation (6) is identical to equation (5), with the only differences here being that the x i and h 1 values are selected conditionally. 7. Results [40] The differences between class 0, class 1a, class 1b, and class 2 amounts are illustrated in Figure 2, which shows the historical mean daily amounts for each class for Sydney ( ), as they vary with time of year. Note that the calculations for each day shown in Figures 2 and 3 use a 31-day moving window. [41] We combined the ROG(4) model for occurrences of Harrold et al. [2003] with RAG(1) for amounts to give a combined model for rainfall generation. Results for this model for Sydney daily rainfall ( ) are shown in Figures 3 6. [42] Figure 3 shows daily statistics of class 2 rainfall amounts for the RAG(1) model. The distribution of each statistic from 100 generated sequences is shown by the 5th percentile, median, and 95th percentile lines (each of the 100 sequences generated by the RAG(1) model is of the same length as the historical sequence). Superimposed on each graph are the historical values (circles). It can be seen that the historical values vary smoothly with time of year, and so do the generated values, with RAG(1) adequately reproducing the historical values. The reproduction of the lag 1 correlation structure that is shown in Figure 3 is better than that reported for a multistate Markov chain by Gregory et al. [1993]. [43] The results for class 0, class 1a, and class 1b amounts also reproduced the corresponding historical daily level

6 SWC 8-6 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS Figure 3. RAG(1): Statistics of daily rainfall for Sydney on class 2 wet days as a function of Julian day. (a) Mean daily rainfall. (b) Standard deviation of daily rainfall. (c) Skew of daily rainfall. (d) Lag 1 correlation of daily rainfall. statistics. The only exception to this was that simulated mean class 1b rainfalls were slightly too high. It was found that the high rainfalls occurred because the model overestimates the amounts on the days before class 1b days. When the class 1b amounts are conditionally simulated, the positive correlation structure causes the class 1b amounts to be similarly overestimated. However this does not significantly alter the overall model performance. We found that the seasonal and annual simulated rainfall totals generated by the model adequately reproduced the observed values. Note that this result also shows that the bias introduced by the handling of generated values below the wet-day threshold of 0.3 is negligible. [44] About 115 days in the Sydney rainfall record have rainfalls of greater than 100 mm. Amounts over 100 mm usually occur on class 2 wet days, but can occur on class 1b, or (less commonly) on class 1a wet days. Amounts over 100 mm do not occur on class 0 wet days. Six days in the Sydney record have amounts over 200 mm. One of these is a class 1b wet day (234 mm on 9 November 1984) and the rest are class 2 wet days. The two wettest days are 280 mm on 28 March 1942 and 328 mm on 6 August 1986, corresponding to Julian days 88 and 219, respectively. The increase in skewness (Figure 3c) that occurs around Julian day 219 is due to the 328 mm value. This also affects the standard deviations (Figure 3b). Note that extreme values affect the results up to 15 days on either side of the event, because a 31-day moving window is being used. [45] Figure 4 shows how the combined ROG(4)/RAG(1) model reproduces the variability of Sydney rainfall at several timescales. Figures 4a and 4b show the standard deviation of rainfall per season and per year, respectively. Figure 4c is a plot of standard deviation of rainfall totals versus timescale in years. Standard deviations are shown on this plot for annual rainfall, biannual rainfall, triannual rainfall, 4-yearly rainfall, 5-yearly rainfall, 6-yearly rainfall, and 7-yearly rainfall. The standard deviation of annual rainfall is identical to the plot shown in Figure 4b. The other standard deviations are an indicator of the very-longterm variability of the sequences. Historical values of the standard deviations are shown by lines (Figures 4a and 4c) or by a solid circle (Figure 4b). The box plots show the distribution of each statistic from 100 generated sequences, with 5th percentile and 95th percentile values forming the whiskers. The seasonal level standard deviations are reproduced adequately, however the annual and longer-term standard deviations are slightly overrepresented. [46] The overrepresentation of longer-term variability is unexpected, since the rainfall occurrence model (ROG(4)) adequately represents the variability of wet days per year

7 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS SWC 8-7 Figure 4. Combined ROG(4)/RAG(1): Variability of Sydney rainfall totals at several timescales. (a) Standard deviation of seasonal rainfall. (b) Standard deviation of annual rainfall. (c) Standard deviation of rainfall totals at very long timescales. [Harrold et al., 2003], and previous studies found that annual variability was underrepresented by daily rainfall models [Wilks and Wilby, 1999]. Upon investigation, we found that the overrepresentation of variability was linked to complex low-frequency features in the historical rainfall record, relating to differences between the wet day classes. In particular, the apparent cause of over-representation of annual and longer-term standard deviations is that the historical class 2 daily amount is lower in wet years than in dry years (see Table A2 and the discussion in section A2). Without the use of an annual-level predictor, the RAG(1) model is insensitive to this feature of the historical record. To overcome this problem, the amounts model was conditioned on the 365-day wetness index (very dry, dry, average, wet, or very wet; as described in section 3), based on the number of wet days over the past 365 days. The resulting two-predictor model, which is denoted as RAG(2), is formulated in the same way as the one-predictor model described above, except the observations x i are separated into five data sets according to the historical values of the 365-day wetness index. The values of the wetness index in the generated sequence determine which data set to use in the simulation for a particular day. The combined ROG(4)/RAG(2) model gives the results shown in Figures 5 8. The daily-level results for this model are similar to those shown in Figure 3, and the model also reproduced the observed seasonal and annual mean rainfalls. However, Figure 5 shows that the use of the annual-level predictor in the amounts model improves the representation of the annual-level and longer-term variability significantly. Figure 6 shows that the distribution of annual rainfall amounts for Sydney is well-represented by the combined ROG(4)/RAG(2) model. Figure 7 shows 5% n-day rainfall totals, which represent drought; and Figure 8 shows 95% n-day rainfall totals, which represent sustained periods of high rainfall, for n = 7, 30, 90, 365, Figures 7 and 8 show that the model adequately reproduces n-day rainfall totals that represent both drought conditions and very wet conditions. [47] We also applied the combined ROG(4)/RAG(1) model and the combined ROG(4)/RAG(2) model to Melbourne daily rainfall ( ). The results of this work are reported by Harrold [2002] and are similar to the results for Sydney. We also tried using a 90-day wetness index instead of the 365-day wetness index in the RAG(2) amounts model for Melbourne, but this did not improve the reproduction of longer-term variability. 8. Conclusion [48] The model for stochastic generation of rainfall amounts that is presented in this paper is an improvement over existing models. It can reproduce the day-to-day features that exist in the historical rainfall record, including the lag 1 correlation structure, and differences in the distribution of amounts between solitary wet days, days at the start or end of a wet spell, and days in the middle of a

8 SWC 8-8 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS Figure 5. Combined ROG(4)/RAG(2): Variability of Sydney rainfall totals at several timescales. (a) Standard deviation of seasonal rainfall. (b) Standard deviation of annual rainfall. (c) Standard deviation of rainfall totals at very long timescales. wet spell. Seasonal variations in the rainfall record are also smoothly reproduced. Existing amounts models are not designed to reproduce all of these features. In addition, the use of an annual level predictor in the proposed model allows complex low-frequency features of the historical record to be reproduced. All of this is achieved within a nonparametric framework that minimizes the assumptions made in simulating the rainfall amounts. Figure 6. Combined ROG(4)/RAG(2): Distribution of annual rainfall amounts for Sydney.

9 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS SWC 8-9 Figure 7. Combined ROG(4)/RAG(2): 5th percentile n-day rainfall totals for Sydney. [49] We link the proposed rainfall amount model with long sequences of rainfall occurrence generated by the model of Harrold et al. [2003] and show that the longerterm variability present in the historical rainfall record can be reproduced by this combined model. The resulting generated sequences provide a better representation of the variability associated with droughts and sustained wet periods than was previously possible. Such features are of great interest in catchment management studies, and the generated sequences can be used in catchment studies to enable better quantification of the uncertainty in the catchment response that is due to climatic variability. [50] The combined model is composed of two parts, which are a multipredictor rainfall occurrence generator (ROG(4)), and a two-predictor rainfall amount generator (RAG(2)) for amounts on wet days. In both parts of the model, the use of the moving window approach [Rajagopalan et al., 1996] ensures accurate representation of the seasonal variations present in the rainfall time series. ROG(4) resamples from a seasonal subset of the historical record of rainfall occurrence, conditional to the values of a set of multiple predictors. The predictors are formed solely from previous values in the sequence, and represent short-term, seasonal, annual, and interannual features of the rainfall sequence. The use of these multiple predictors in the resampling model produces generated occurrence sequences that closely reproduce the historical longer-term variability. The model for generation of amounts (RAG(2)) works with a seasonal subset of the historical record of rainfall amounts, and is conditioned on both daily-level and annual-level predictors. A smoothed empirical estimate of the conditional probability distribution of amounts is formed, and amounts are generated from this empirical distribution. Separate RAG models are applied to class 0, class 1a, class 1b, and class 2 wet days, where each wet day is assigned to a class based on the number of adjacent wet days, and class 1a and class 1b refer to wet days at the start and at the end of a wet spell, respectively. The use of these various classes is important since the probability density function of the rainfall varies between each class. [51] An important conclusion to be made from this research is that a substantial portion of the longer-term variability in rainfall is accounted for by the variability in the rainfall occurrence process. The use of a rainfall occurrence model that is capable of simulating low-frequency variability is the key to ensuring that the overall daily rainfall model can reproduce the observed statistics at seasonal, annual, and longer timescales. The subsequent simulation of amounts on each wet day is relatively simple. Additionally, the use of categorical aggregate variables to represent long-term features in the record has significant benefits, and ensures an accurate representation of lowfrequency features in the simulations. In our application of the combined rainfall occurrence and rainfall amounts model to Sydney rainfall, we show that the simulations reproduce statistics at both a daily level and at longer timescales. This would not have been possible without the use of categorical aggregate variables. [52] A feature of the amounts model is that the behavior of the model for extreme values is essentially assumptionfree (except for the issue of bandwidth choice). The shape

10 SWC 8-10 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS Figure 8. Combined ROG(4)/RAG(2): 95th percentile n-day rainfall totals for Sydney. of the kernel probability density function is determined by the observed data, each kernel representing a scaled probability density centered at individual data points. The generated extreme values are sampled from a kernel associated with an observed extreme value, but the smooth nature of the kernel function ensures that the new value is a unique realization that has not been recorded in the observed record. The extent to which the generated values are different from the observations depends on the bandwidth of the kernel density function. While the approach used in the current study uses a relatively simple formulation for the bandwidth, variants that assign different bandwidths depending on the local variance of the data exist [e.g., Sharma et al., 1998]. It should, however, be noted that all nonparametric approaches are data based. If observations are not representative of the population being modeled, or if the observations are not sufficient in number, then the sequences that are generated will be limited in their representation of the process. Having said this, the same limitation exists when adopting a parametric approach. The difference is that a nonrepresentative record leads to poor parameter and distribution choices when using a parametric alternative, while the impact is directly on the estimated probability distribution in a nonparametric approach. [53] The approaches for stochastic data generation outlined here assume that the observed data is stationary, i.e., that the effects of climate change and anthropogenic change on the observed record are negligible. Under this assumption, the aim of the stochastic analysis is to produce multiple synthetic sequences that represent the climate variability contained in the observed record. Note that these synthetic sequences would form a useful base-case for comparing the hydrologic effects of climate variability against the hydrologic effects of climate change, if projected synthetic rainfall under climate change conditions was available from a climate change study. Appendix A A1. Bandwidth Selection for Kernel Density Estimation [54] As noted in section 5, the choice of the bandwidth is the key to an accurate estimate of the probability density. Some possible bandwidth selection rules are now discussed. A1.1. Univariate Case [55] A useful choice of univariate bandwidth h is one that minimizes the mean integrated square error (MISE): Z MISE ^f ¼ E n o 2dx fðþ ^f x X ðx; hþ ða1þ where f (x) is the underlying ( population) probability density and ^f x (x; h) is the kernel density estimate of f (x). [56] If f (x) is normally distributed, the choice of bandwidth that minimizes MISE is given by the Gaussian reference bandwidth [Silverman, 1986, p. 45]; expressed in terms of the interquartile range R, this is: h ref ¼ 0:79Rn 1=5 ða2þ

11 HARROLD ET AL.: SIMULATION OF DAILY RAINFALL AMOUNTS SWC 8-11 Scott [1979] suggests that a correction factor be applied to adjust for skewness in the sample. Silverman [1986, p. 47] presents a graph that shows appropriate skewness correction factors that can be applied to equation (A2), for a lognormal parent distribution of given skewness. Applying this correction factor, we obtain: h rot ¼ lrn 1=5 ða3þ where l is the skewness correction factor for a given sample value of skewness and rot denotes rule of thumb. [57] For sample skewness of 4, which is the average value for the data considered in this paper, the value of l given by Silverman s method is 0.3. Note that this bandwidth selection rule was tested against the LSCV method [Silverman, 1986, p. 48], the SJ method [Sheather and Jones, 1991], and a Gaussian reference bandwidth, and found to perform as well as, or better than, these alternatives. A1.2. Conditional Case [58] Rules for calculating appropriate bandwidths for conditional densities (such as, ^f (x t jx t 1 ) shown in Figure 1) are not well developed. Here, we apply equation (A3) to calculate bandwidths for use with this conditional density, except the values of R and n that are used are chosen conditionally. This is consistent with the principle given in section 5, which states that simulation should proceed with more emphasis given to the observed data points lying closer to the conditioning plane, and lesser emphasis given to the data points that lie further away. In this proposed rule, the interquartile range R can vary with x t 1, and the effective number of data points n eff depends on the weights associated with the kernel slices that constitute the conditional probability density (i.e., the w i values in equation (3)). For a data set with positive skew, this rule leads to increasing values of the bandwidth h 1 as x t 1 increases. [59] For simplicity, we consider three broad classes for x t 1 : low, medium, and high. R values are calculated for each of these data ranges, and h 1 is calculated separately for each range: h low ¼ l R low n 1=5 eff ða4aþ Table A2. Average Rainfall Amount on Class 0, Class 1, and Class 2 Wet Days at Tenterfield a Class El Niño Non-ENSO La Niña all a Values are in mm/d. remarkably stable, so that it was appropriate to use l =0.3 consistently. For the conditional simulation of amounts (x t ) for the class 1b and class 2 rainfall, we allocated x t 1 to one of three broad rainfall ranges: low rainfall was defined as 0.3 to 4 mm, medium rainfall was defined as mm, and high rainfall was defined as anything over 15 mm. R values were calculated for each of these data ranges. h 1 was calculated separately for each range using equation (A4). This specification of h 1 gave the results presented in this paper. Results obtained using a single data range for the rainfall, and using n eff = n, were also adequate; however the chosen specification gave better results, and this way of defining h 1 is closer to the theoretically optimum bandwidth specification. A2. Long-Term Variability and Wet Day Classes [60] The association between annual rainfalls and ENSO at Tenterfield (Table 1), can be broken down into wet day classes based on the number of adjacent wet days. This is shown in Tables A1 and A2. There are differences between the classes. For example, the average number of class 0 wet days in La Niña years is about the same as in El Niño years, but the average rainfall amount on a class 0 wet day in La Niña years is slightly higher than in El Niño years. Also note that the average number of class 2 wet days in La Niña years is much higher than in El Niño years, but the relationship for the average rainfall amount on a class 2 wet day is reversed; the wet day amounts in La Niña years are slightly lower than in El Niño years. These complex features of the historical data cannot be modeled unless the classes are modeled separately, and an annual-level predictor is introduced to our model. h med ¼ l R med n 1=5 eff ða4bþ [61] Acknowledgments. The authors gratefully acknowledge the Australian Research Council and the NSW Department of Land and Water Conservation for funding this research. Support was also received from the Japan Society for the Promotion of Science. We thank our reviewers for constructive comments on the draft paper. h high ¼ l R high n 1=5 eff ða4cþ where n eff = 1/max(w i ), and l is calculated using Silverman s method. We found that the skewness in the samples was Table A1. Average Number of Class 0, Class 1, and Class 2 Wet Days for El Niño and La Niña at Tenterfield Class El Niño Non-ENSO La Niña All References Akaike, H., A new look at the statistical model identification, IEEE Trans. Automat. Control, 19, , Allan, R. J., Australasia, in Teleconnections Linking Worldwide Climate Anomalies, edited by H. Glantz, R. W. Katz, and N. Nicholls, pp , Cambridge Univ. Press, New York, Allan, R., El Niño/La Niña year classification, in Twelve Month Australian Rainfall Relative to Historical Records ( poster), edited by N. R. Flood and A. Peacock, Resour. Sci. Cent., Queensland Dep. of Nat. Resour., Indooroopilly, Queensland, Australia, Buishand, T. A., Some remarks on the use of daily rainfall models, J. Hydrol., 36, , Chapman, T. G., Stochastic modelling of daily rainfall: The impact of adjoining wet days on the distribution of rainfall amounts, Environ. Modell. 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