Copula based daily rainfall disaggregation model

Transcription

1 WATER RESOURCES RESEARCH, VOL. 47,, doi: /2011wr010519, 2011 Copula based daily rainfall disaggregation model Yeboah Gyasi Agyei 1 Received 6 February 2011; revised 27 April 2011; accepted 28 April 2011; published 20 July [1] A daily rainfall disaggregation model, which uses a copula to model the dependence structure between total depth, total duration of wet periods, and the maximum proportional depth of a wet period, is presented. The wet(1) dry(0) binary sequence is modeled by the nonrandomized Bartlett Lewis model with diurnal effect incorporated before superimposing the AR(1) depth process submodel. Unlike previous studies, the model is structured such that all wet day data available are considered in the analysis, without the need to discard any good quality daily data embedded in a month having some missing data. This increased the data size, thus improving the modeling process. Further, the daily data are classified according to the total duration of wet periods duration within the day. In this way a large proportion of the model parameters become seasonal invariant, the overriding factor being the total duration of wet periods. The potential of the developed model has been demonstrated by disaggregating both the data set used in developing the model parameters and also a 12 year continuous rainfall data set not used in the model parameterization. Gross rainfall statistics of several aggregation levels down to 6 min have been very well reproduced by the disaggregation model. The copula dependence structure and the variation of the depth process submodel parameters with the total duration of wet periods are also very well captured by the presented model. Citation: Gyasi Agyei, Y. (2011), Copula based daily rainfall disaggregation model, Water Resour. Res., 47,, doi: /2011wr Centre for Railway Engineering, Faculty of Sciences, Engineering and Health, CQUniversity, Rockhampton, Queensland, Australia. Published in 2011 by the American Geophysical Union. 1. Introduction [2] Floods and flash floods occur frequently around the world, costing world economies billions of dollars each year and causing loss of lives. Fine time scale (e.g., 6 min) rainfall data improves the planning, design, and management of hydraulic structures in urban areas. Environmental processes, such as erosion and pollutant transport, are better modeled with fine time scale rainfall data. The availability of fine time scale rainfall data will allow continuous simulation of rainfall runoff processes, which is increasingly becoming necessary in place of the use of design storm events for hydrological systems design and performance analysis. Such simulation allows issues such as the effect of land use change on runoff and pollutant generation to be addressed. However, fine time scale rainfall data are limited worldwide largely because of the high cost and low reliability of monitoring such data. Where fine time scale data exist, the generally short record length undermines their direct use for water resources projects, such as urban storm water planning, design and management. However, long records of daily rainfall for over 100 years of a dense network are available worldwide. For example, the Australian SILO Data Drill and BAWAP are facilities for extracting daily rainfall data, at 5 km spatial resolution over the entire continent dating back to 1889, from an archive of interpolated rainfall and climate surfaces. Gyasi Agyei [2005] observed that there is not much difference between the SILO Data Drill generated rainfall data and the observed rainfall data at an Erosion Control Experimental site near Blackwater, Queensland, Australia. A similar archive of rainfall data is managed by the U.S. National Climatic Data Center. Disaggregation of daily rainfall preserves the climate variability as seasonal and annual trends are maintained. Therefore, the importance of disaggregating the long records of daily rainfall into finer time scales cannot be overemphasized. [3] The extensive literature on rainfall data disaggregation, dating back two decades, demonstrates the importance of the subject to the water and environmental community. Simplified approaches [e.g., Hershenhorn and Woolhiser, 1987; Econopouly et al., 1990] are based on distribution functions of the storm event properties, namely starting times, volume, intensity and duration. The Bartlett Lewis (BL) and Neyman Scott rectangular pulse models [Rodriguez Iturbe et al., 1987, 1988] and their variants have been used to disaggregate daily rainfall [Bo et al., 1994; Gyasi Agyei and Willgoose, 1997, 1999; Gyasi Agyei, 1999, 2005; Cowpertwait et al., 1996; Glasbey et al., 1995; Koutsoyiannis and Onof, 2001; Gyasi Agyei and Mahbub, 2007]. Scaling (or scale invariant) based fractal and multifractal models fall into another group of models for rainfall disaggregation [e.g., Schertzer and Lovejoy, 1987; Güntner et al., 2001; Molnar and Burlando, 2005]. A few other approaches exist for disaggregation of daily rainfall data into finer time scales or generation of continuous rainfall data at large watershed scales. These include nonparametric models [e.g., Harrold et al., 2003], hidden Markov models [e.g., Lambert and 1of17

2 Kuczera, 1998], and nonlinear deterministic dynamic models [e.g., Srikanthan et al., 2005]. [4] Regardless of the approach employed, the existing subdaily time scale stochastic models are unable to preserve the day to day rainfall variability if not used in disaggregation. The need to generate subdaily time series fully consistent with the observed daily totals, while preserving the stochastic structure of multiple subdaily time scales, cannot be overemphasized. It is simply not sufficient for a model to only reproduce some basic scaling properties and statistics, such as mean, variance, autocorrelations, and the probability of an interval being dry at several time scales, but the generated rainfall series should also reflect annual trends, in particular for sites that exhibit a clear climate change. [5] Radar is increasingly becoming popular for gathering spatial temporal rainfall data, but this data gathering technology is prone to errors [e.g., Krajewski, 1987] and ground truth data are required for calibration. Also radar source rainfall data are only available for a few years and are not adequate for long term rainfall runoff simulation. [6] Copulas are functions that join multivariate distribution functions to their one dimensional marginal distribution function based on the Sklar theorem, and it is a way of studying scale free measures of dependence. A rainfall process is multidimensional, exhibiting important properties of storms including start time, duration, volume, peak intensity and time to peak, and with these properties changing from storm to storm and also depending on the time scale of interest. The pairwise dependence of these variables has traditionally been modeled using bivariate distributions such as the extreme value distribution, with the drawback being the assumption that the variables follow the same parametric univariate distributions [Genest and Favre, 2007]. This limitation is avoided by the copula models where the dependence between the variables is independent of the marginal distributions which could be different for the variables. While copula models have been available for years, particularly in the field of actuarial and financial applications, they are now quickly making ingress into hydrology. Notable pioneering applications of copulas in hydrology include De Michele et al. [2005] and Favre et al. [2004]. Recent applications include Bárdossy and Pegram [2009] who demonstrated the use of copula models for multisite daily rainfall modeling and Zhang and Singh [2007a, 2007b] who derived intensity frequencyduration curves using a copula model. Vandenberghe et al. [2010] used bivariate copulas to model the dependence structure between storm characteristics. Serinaldi [2009] studied the probability distribution of rainfall at different time scales by a bivariate copula based mixed model. Other recent applications of copulas in hydrology are Kao and Govindaraju [2008] and Evin and Favre [2008]. [7] The basic features of the daily rainfall disaggregation model presented in this paper are as follows. [8] 1. The model is structured such that all wet day data available are considered in the analysis, without the need to discard any good quality daily data within a month having some missing data. In this way a large amount of data are used thus improving the modeling process. [9] 2. Copulas are used to model the dependence structure between the daily rainfall depth, the total duration of wet periods and the maximum proportional depth of a wet period within a wet day. [10] 3. The wet dry binary sequence and the intensity model parameters are related to the total duration of wet periods within a wet day. [11] The paper is organized as follows. The data used in the paper are first described in section 2. This is followed by an overview of the disaggregation model, and then the different aspects of the model are presented in detail in section 3. A case study is presented to highlight the robustness of the disaggregation model before the summary and conclusions. 2. Data [12] The site of interest is Gregory Erosion Control Experimental field trials site [Gyasi Agyei, 2006] located between the and km marks on the Gregory railway line, central Queensland, Australia. It is situated at latitude S and longitude E. It has 12 years (January 1998 to December 2009) of 1 min time scale continuous rainfall data collected by a tipping bucket of tipping depth 0.18 mm. Figure 1 shows the location of the Gregory field trials site and the nearest two Bureau of Meteorology (BOM), Australia, rainfall stations (Emerald and Dingo ) within a 70 km radius. These two stations have 6 min time scale and 0.01 mm resolution rainfall data spanning between 1983 and 2006 for Emerald and between 1963 and 2006 for Dingo. The Emerald and Dingo data were combined into one data set and used to estimate the required model parameters for the case study site. As is typical of such high resolution rainfall data, this data set has a considerable amount of missing data. It turns out that 45% of the 2957 wet days are within months having missing data. However, every good quality wet day data was used in the analysis irrespective of missing data for the other days within the same month. The daily data were then grouped into 13 quantile classes of the total duration of wet periods (L) within a wet day as shown in Table 1, before analyzing the classes at 9 aggregation levels (0.1, 0.2, 0.5, 1, 2, 4, 8, 12, and 24 h). Daily rainfall amounts less than 0.1 mm were ignored as false events. While the quantiles were chosen arbitrarily, due consideration was given to the distribution of the class mean dry probability, and also a fair distribution of points on the log scale of L. The mean values of the L quantile classes were used to estimate some of the model parameters. The aggregation levels were carefully chosen so that 24 h (a day) is an integral multiple of each one of them. 3. The Rainfall Disaggregation Model Development and Parameter Estimation 3.1. An Overview [13] The rainfall depth distributed over a day {Y } is considered as a product of a wet (1) and dry (0) indicator submodel {Y} and a depth process submodel {A} expressed as {Y }={A}{Y}. The depth process is assumed to be lognormal such that {A} = exp{z}, where {Z}is a stationary first order autoregressive, AR(1), model given as z i ¼ Z þ Z ½z i 1 Z Šþe i ; ð1þ where m Z is the mean and r Z is the lag 1 autocorrelation coefficient within ±1. The innovation process {E} follows a normal distribution N(0, s E 2 ) with a mean value of zero and 2of17

3 Figure 1. Location of rainfall stations. Table 1. Distribution of Wet Days on the Basis of L Quantile Classes a Class ID Number of Days LQ (>) UQ ( ) LL (h) UL (h) ML (h) MR (mm) MM MI (mm) Emerald Dingo b Gregory Site c a LQ, lower L probability; UQ, upper L probability; LL, lower L; UL, upper L; ML, mean L; MR, mean R; MM, mean M; MI, mean maximum 6 min depth. b Total: 2957 days, 45% within months with missing data of other days. c Total: 863 wet days, no missing data. 3of17

4 Figure 2. Three dimensional vine copula construction. variance s 2 E =[1 r 2 Z ]s 2 Z. Rainfall data are sampled at a fixed time scale t s, 0.1 h in this paper. This implies that a daily rainfall depth R (mm) is distributed over a finite number of wet periods, n L, giving a total daily duration of wet periods L (hours) as the product of t s and n L. On the other hand, if R and L (both considered continuous random variables) are given, then n L is given as the nearest integer of (L/t s + 0.5). The distribution of the n L wet periods (1 s) within the wet day, the process {Y}, is modeled by the nonrandomized Bartlett Lewis (BL) model [Rodriguez Iturbe et al., 1987] with diurnal effect incorporated. Invoking the properties of the lognormal distribution, the mean value of {A}, (R/n L ), is related to the statistics of {Z} as ln R ¼ z þ 1 n L 2 2 z : ð2þ [14] Hence, the knowledge of s E and r z will lead to the estimation of s z, and with R/n L, m z can be evaluated. Wet period depths [r i = exp(z i ), i =1,n L ] are then sampled from equation (1) with the following two conditions: (1) Pn L r i is i¼1 within acceptable limits from R and (2) maximum of r i /R, max(r i /R), is within acceptable limits from M, the maximum proportional depth of a wet period conditioned on the values of R and L. [15] The random variable L is sampled from a copula model for a given value of R, and M is sampled from a different copula model conditioned on the values of R and L. In sections , details of the different aspects of the disaggregation model are presented Copula Modeling Introduction [16] Copula modeling is based on the Sklar theorem which says that a continuous d variate cumulative distribution function (cdf) F 12..d (x), of a random vector X consisting of random variables X i with univariate continuous marginal cdfs F i (x i ), is given as F 12::d ðx 1 ; x 2 ; ::; x d Þ ¼ C 12::d ff 1 ðx 1 Þ; F 2 ðx 2 Þ; ::; F d ðx d Þg: ð3þ C 12..d (.) is a unique d dimensional copula that maps d dimensional marginal uniform distributions into their joint (d dimensional) cdf expressed as [0,1] d [0,1]. Denoting the probability u i = F i (x i ) and F 1 i (.) as the quasi inverse (or the quantile) function of F i (.), the Sklar s inversion theorem is given as C 12::d ðu 1 ; u 2 ; ::; u d Þ ¼ F 12::d F 1 ð Þ; F 1 ð Þ; ::; F 1 ðu d Þ ð4þ 1 u 1 2 u 2 d for u i 2 [0, 1] d. The three random variables to be modeled by copulas are the total daily rainfall depth (R, mm), the total duration of wet periods (L, hours), and the maximum proportional depth of a wet period (M) within a wet day. Therefore, the ensuing is restricted to the three dimensional (3 D) copula case. For dimensions greater that 3, readers are referred to Joe [1997], Nelson [1999] and Salvadori et al. [2007]. [17] For computational tractability, particularly for higherdimensional copulas, vine copulas have been introduced [Joe, 1996; Bedford and Cooke, 2001, 2002] and were recently illustrated by Aas et al. [2009]. Vine copulas decompose the multivariate densities into bivariate linking copulas thus involving lower dimensional integrals. For dimensions greater than 2 there are several possible 2 D copula (the basic building block) constructions. Two of the vine copulas (C vine and D vine) are illustrated in Figure 2 for the 3 D case. In general a d D vine copula uses d(d 1)/2 bivariate copulas on a tree structure. In the 3 D case there are 2 trees (T1 and T2); the first tree has 3 nodes (circles) and 2 edges (lines), and the second tree has 2 nodes and 1 edge. The 2 edges in T1 become the nodes for T2, and a 2 D copula is applied between the nodes of the trees. An additional advantage is that the vine copula offers the flexibility of selecting different bivariate family copulas for each edge. In the simulation process, the starting point is the selection of the vine type, followed by the selection of the bivariate copulas for the first tree. The conditional distribution functions for T2 are computed from the partial derivative (G functions) of T1 copulas. This is then followed by the determination of the conditional independent copula for T2. The choice of the vine copula depends on the dependence structure of the data being modeled. As explained later, the D vine is the preferred option for the application presented in this paper Parameter Estimation and Simulation Procedure [18] Using the chain rule, Joe [1996] showed that the trivariate cdf is given as F 123 ðx 1 ; x 2 ; x 3 ; 12 ; 13 ; 23 Þ ¼ PX ð 1 x 1 ; X 2 x 2 ; X 3 x 3 Þ Z x2 ¼ C 13j2 F 1j2 ðx 1 jz 2 ; 12 Þ; F 3=2 ðx 3 jz 2 ; 23 Þ F2 dz 2 ; ð5þ where C ij (u i, u j ; ij ) is a bivariate copula with uniform margins and dependence parameter ij between nodes i and j, and F i j is the conditional distribution of variable i given variable j. The conditional distributions are defined as G 2j1 ½u 2 ju 1 ; 12 Š ¼ F 2j1 ðx 2 jx 1 Þ C 12 ½u 1 ; u 2 ; 12 Š 1 for the 2 D case and G 3j12 ðu 3 ju 1 ; u 2 Þ ¼ PX ð 3 x 3 jx 1 ¼ x 1 ; X 2 ¼ x 2 Þ C 13j2 G 3j2 ; G 1j2 ; 1j2 for the 3 D case. Note that for the G i j (.) function, the derivative is with respect to the second variable j. The parameter 13 2 is conditional dependent of the first and third univariate margins given the second. It needs to be underlined that the conditional distribution G i j (.) values lie within the unit interval [0,1], and is continuous and an increasing function. Equation (7) shows how the 3 D copula is decomposed into 3 separate 2 D copulas (the edges of the ð7þ 4of17

5 Table 2. Kendall s Tau and Spearman Rank Correlation Coefficient Number of Kendall s Tau Spearman s Month Wet Days R L R M L M R L R M L M Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec vine diagrams) using the chain rule. The D vine copula parameters are estimated by the following procedure. [19] 1. Generate three vectors of normalized ranks from the data u 1, u 2, u 3. [20] 2. Fit a copula to u 1 and u 2 to estimate the parameter 12. [21] 3. Fit a copula to u 2 and u 3 to estimate the parameter 23. [22] 4. Generate the series v 12 using the fitted copula to u 1 and u 2 as v 12 = G 1 2 [u 1 u 2 ; 12 ]. [23] 5. Generate the series v 32 using the fitted copula to u 2 and u 3 as v 32 = G 3 2 [u 3 u 2 ; 23 ]. [24] 6. Fit a copula to v 12 and v 32 to estimate the parameter [25] The normalized ranks used are defined as u ij = r ij /(m +1) where m is the number of data triplets, and r ij is the jth rank of variable i [e.g., Genest and Favre, 2007]. [26] During simulation, inversion of the G i j (.) function is required; where analytical expression of the inverse function G 1 i j(.) is not available, numerical inversion is employed. The general simulation procedure based on the Sklar s theorem and the probability integral transform [e.g., Salvadori et al., 2007] is as follows. [27] 1. First generate w 1, w 2, and w 3 from independent uniform U(0,1). [28] 2. Set u 1 = w 1. [29] 3. Calculate the conditional probability u 2 = G [w 2 u 1 ; 12 ]. [30] 4. Calculate the conditional probability v 12 = G 1 2 [u 1 u 2 ; 12 ]. [31] 5. Calculate the conditional probability v 32 = G [w 3 v 12 ; 13 2 ]. [32] 6. Calculate the conditional probability u 3 = G [v 32 u 2 ; 23 ] Bivariate Copula Selection [33] A basic requirement for copula modeling is that the random variables should be continuous and increasing functions. Therefore, the discontinuities in the rainfall data as a result of the sampling, measuring, or discretization process should first be removed. By so doing all ties, or duplications, will be removed. For each wet day a value Ra is drawn from a uniform distribution Ra U( 0.005,0.005) for 0.01 mm rainfall depth resolution. Then a uniformly distributed random noise ra i U( 0.005,0.005) is added to each of the 6 min rainfall depths before they are uniformly scaled to ensure that the total amount added to the daily depth matches Ra. For the variable L the added value La is drawn from U( 0.05,.05), for 0.1 h (6 min) time scale resolution. Vandenberghe et al. [2010] applied a similar treatment to a 10 min rainfall data from Belgium. [34] For the disaggregation process the daily rainfall depth (R) is always known and as such it is the leading variable. Table 2 shows the Kendall s Tau and Spearman s rank correlation coefficients between the variables for the months for the Emerald Dingo data set used to estimate the model parameters. There is a strong positive correlation between R and L but a weak negative one between R and M. The very strong negative correlation between L and M suggests the D vine construction as a better choice, calling for a copula model between R and L and between L and M on T1. Relating to the D vine diagram in Figure 2, variables D, L and M are 1, 2 and 3, respectively. [35] A pairwise scatterplot of the empirical normalized rank distributions shown in Figure 3a for R L and Figure 3b for L M gives an idea of the type of copula function to be used. The dependence structures for R L and L M are nearly elliptical with positive dependence between R and L and negative between L and M. Maximum proportion of the simulation time scale was used instead of maximum depth in order to remove scale effects, and also because it has a stronger dependence on R and L. [36] The serial independence test of paired random variables R, L and M was carried out using procedures outlined by Genest and Remillard [2004] and Kojadinovic and Holmes [2009] and as implemented in the R copula package [Kojadinovic and Figure 3. Dependence structure of the T1 and T2 vine copula variables for December, Emerald Dingo. 5of17

6 Table 3. The p Values of the Cramer von Mises Sn Statistics and the Calibrated Parameters of the Selected Bivariate Copulas a p Values Copula Parameters Month R L NC L M NC v 12 v 32 CL R L NC L M NC v 12 v 32 CL Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average a NC, normal copula; CL, Clayton copula. Yan, 2010]. All p values of the serial independence test obtained were less than 0.005, suggesting that the random variables are dependent. Therefore, the following single parameter bivariate copulas having different tail behavior were considered: Archimedean copulas (Frank, Clayton, Gumbel Hougaard, Ali Milhail Haq), elliptical copulas (normal and t with different degrees of freedom) and the Plackett copula. [37] The R copula package function gofcopula based on the Cramer von Mises Sn statistic [Scaillet, 2005; Remillard and Scaillet, 2009; Kojadinovic and Yan, 2010] was used to assess the goodness of fit of the copulas considered. The multiplier option was used with the number of multiplier iterations set to In all cases the maximum likelihood method was used to estimate the parameters. Table 3 presents the bivariate copulas with the highest p values. The elliptical copulas (normal and t) generally performed better than the other copulas considered. At the 5% significance level, both normal and t 15 copulas for R L cannot be rejected for all months except for November which is acceptable at the 1% significance level. In the end, the normal copula (NC) was selected to model the dependence between R and L because it had the highest monthly average value and the highest p value for November. Similarly, the elliptical copulas outperformed the other copulas for the dependence between L and M, with 9 months acceptable at the 5% significance level, 2 months at the 1% significance level, with January p value being below 1%. Based on the p values, the NC was again selected. The T2 random variables v 12 and v 32 were generated from the G(.) function of NC selected for R L and L M dependence. The Clayton copula (CL) emerged as the best copula for T2 with 9 months passing at the 5% significance level, and the remaining months at 1% (Table 3). Finally, the parameters of the T1 and T2 copulas were jointly calibrated by maximizing the sum of the three log likelihoods, and the results are presented in Table Marginal Distributions Selection [38] Figure 4 presents an example histogram for the random variables for December. Five two parameter family distributions (Pareto, Gumbel, Weibull, gamma, lognormal) were selected for testing for the marginal distributions of the random variables R, L and M. Here too the maximum likelihood method was used to estimate the parameters. Table 4 presents p values of the Anderson Darling goodness of fit test, adopted to select suitable marginal distributions, for the best three distributions. For the random variable R, the lognormal distribution outperformed the others with all monthly p values in excess of 5% suggesting that it cannot be rejected at the 5% significance level. The lognormal distribution, having all p values above 0.1, was selected for the random variable L. All monthly p values being greater than 0.05, the lognormal distribution was also selected for the random variable M. In Table 5 are presented the calibrated parameters of the selected marginal distributions The Nonrandomized Bartlett Lewis Model [39] The nonrandomized Bartlett Lewis (BL) model [Rodriguez Iturbe et al., 1987] is used to generate the indicator function of 1 for a wet and 0 for a dry binary sequence, the process {Y}, for a value of n L calculated from R and L. Below are the details of the BL model. [40] 1. Storms arrive in a Poisson process of rate l(h 1 ). [41] 2. Following the storm origins are cell arrivals also governed by a Poisson process of rate b(h 1 ), one cell arriving at each storm origin. [42] 3. Cell arrivals associated with a storm cease after an exponentially distributed time with rate g(h 1 ). [43] 4. The duration of the rectangular pulse associated with each cell follows an exponential distribution with rate h(h 1 ). Figure 4. Histogram of rainfall variables for December, Emerald Dingo. 6of17

7 Table 4. The p Values of the Anderson Darling Test Statistic of the Marginal Distributions Month R L M Weibull Gamma L norm Weibull Gamma L norm Weibull Gamma L norm Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average [44] 5. The rectangular pulses of cells are allowed to overlap with cells of the same storm and across cells of different storms. [45] Defining the dimensionless parameters = b/h and = g/h, the probability that an interval of duration h his dry, P(h), is given as [Rodriguez Iturbe et al., 1987] Ph ð Þ ¼ exp½ ðh þ u T CÞŠ; ð8þ where C = G þ P½þe ð Þh Š þ and functions u T and G P are defined as u T = 1 h 1 + h 1R 0 1 dn R 0 1 dtn 1 t 1 [1 (1 nt)e n(1 t) ], and G P = h 1 e R 0 1 t 1 (1 t)e t dt. Both functions u T and G P are solved by numerical integration. [46] The dry probability (DP) of several aggregation levels are used to estimate the BL model parameters l, b, g and h. A major advancement in this paper is that each good quality wet day data available is considered in the analysis as explained in section 2. Previously, the model was calibrated on a monthly basis discarding good quality daily data within a month having some missing data. For the Emerald Dingo data set, 45% of the good quality data were embedded in a month having some missing data. Figure 5a shows a plot of monthly DP of the considered time scale aggregation levels, noting that the 24 h DP on a wet day is zero. The slight differences in the monthly DP curves are due to the differences in the monthly mean values of random variable L. This is exemplified in Figure 5b where the simulation time scale (0.1 h) DP is linearly related to the mean value of L for the 12 months. For the simulation time scale the theoretical value of DP is given as (1 L/24) as plotted in Figure 5b. Therefore, the BL model was fitted to the mean values for each aggregation time scale without regard to the month by minimizing the objective function, J: J ¼ X8 k¼1 ½P ok ðh k Þ P ak ðh k ÞŠ 2 ; ð9þ where P ok (h k ) and P ak (h k ) are the observed and analytical DPs for the aggregation time scale h k (0.1, 0.2, 0.5, 1, 2, 4, 8, 12 h). The shuffled complex evolution global probabilistic search strategy over a hypercube [Duan et al., 1992] as implemented in NLFIT Bayesian nonlinear regression software [Kuczera, 1994] was used to calibrate the parameters presented in Table 6. This specific optimization method was chosen due to experience and its ability to identify the BL model parameters [e.g., Gyasi Agyei and Mahbub, 2007]. The perfect fit of the BL model to the data is shown in Figure 5c. [47] As observed by Gyasi Agyei [2001], diurnal effect plays an important role in rainfall modeling, particularly in the subtropical and tropical climates of the case study region. In this paper diurnal effect is captured by the time of occurrence of the first storm, or the beginning of the rainfall for a wet day. For a given value of L (hours), the first storm can only occur within 0 to (24 L) h labeled as the available start time. The distribution of rainfall start time of all wet days within each of the L quantile classes (Table 1) was analyzed. Figure 6a shows the frequency distribution of the L quantile class 8 (4.38 to 6.05 h). Start time within the first hour stands out for all classes so it was selected as the first option. While start times within the remaining hours (24 L 1) show some variation, two options were identified as first and second halves of (24 L 1) h. Therefore, the diurnal effect was modeled by a trinomial model: option 1 is that it will start within 0 to 1 h; option 2 within [(24 L 1)/2 + 1] to [24 L] h, and option 3 within 1 to [(24 L 1)/2 + 1] h. Options 1 and 2 probability variation with the mean wet duration of the L quantile classes were fitted with a polynomial trend as shown in Figures 6b and 6c. The polynomial trend probabilities are given as op1 ¼ 0:1105 þ 0:0373L op2 ¼ 0:6988 0:0632L þ 0:0015L 2 op3 ¼ 1 op1 op2 ð10þ Table 5. Parameter Values of the Selected Marginal Distributions R L M Month meanlog sdlog meanlog sdlog meanlog sdlog Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average of17

8 Figure 5. (a) Mean monthly dry probability, (b) simulation time scale dry probability versus mean wet duration, and (c) observed and fitted site mean dry probability, Emerald Dingo. and noting that the probabilities vary between 0 and 1. With these three probabilities, an option i is sampled from a trinomial discrete distribution, and then the first storm location, S i, is sampled from a uniform distribution within the corresponding option s time interval limits, i.e., S 1 U(0, 1), S 2 U {[(24 L 1)/2 + 1], [24 L]}, S 3 U{[1, [(24 L 1)/2 + 1]}. The remaining storm arrivals are generated as explained above. Hence, the BL model with diurnal effect has nine parameters AR(1) Submodel Parameter Estimation [48] The AR(1) model for the simulation time scale depth process, equation (1), requires estimation of the statistics s E and r Z for each wet day. These statistics were estimated for each wet day and the mean and the standard deviation for each of the 13 L quantile classes were calculated separating the months. The monthly class mean values did not display any apparent seasonal variation. Therefore, all the monthly data were combined (Figures 7a and 7b) for the determination of the L quantile class distributions. Figures 7c and 7d show the histogram of s E and r Z for the L quantile class 8. For ease of distribution fitting, r Z values were transformed as r Z t = (1 r Z ) in order to make all values positive and the distribution positive (right) skewed as depicted in Figure 7e, basically a reflection along the axis r Z = 1 of Figure 7d. A similar analysis carried out in section was used to select a common distribution for the 13 L quantile classes. The Weibull distribution emerged as the best for s E, with all p values in excess of 0.18, that is passing at the 5% significance level. For the statistic r Z t, the lognormal distribution was the best with 11 L quantile classes p values greater than 0.05, but the first L quantile class failed at the 1% significance level. However, the p value for this class for the normal distribution was Figures 7f 7i depict the variation of the distribution parameters with the mean wet duration, L, which were fitted to a polynomial trend of the form presents the fitted polynomial parameter values. For the t first L quantile class, r Z is sampled from the normal distribution N(m, s 2 ) with the parameters derived as ¼ exp * þ *2 2 ð12þ 2 ¼ exp 2* þ * 2 exp * 2 1 ; where m* (meanlog) and s* (sdlog) are obtained from the fitted polynomial curves (Figures 7f and 7g). The statistic r Z is obtained as (1 r Z t ) Summary of the Simulation Procedure [49] The summary of the simulation procedure of the disaggregation model is as follows. [50] 1. For a given daily rainfall value R, the probability w 1 = u 1 = F R (R) is obtained from R s marginal distribution F R (.). [51] 2. Sample w 2 and w 3 from independent uniform distribution. [52] 3. The conditional probabilities u 2 and u 3 are obtained using steps 3 6 of the general simulation procedure in section [53] 4. Calculate the quantile L from L s marginal distribution F L (.): L = F 1 L(u 2 ). [54] 5. Calculate the quantile M from M s marginal distribution F M (.): M = F 1 M(u 3 ). [55] 6. L is used to estimate the diurnal trinomial discrete distribution parameters (equation (10)). [56] 7. L is used to estimate the depth process parameters (equation (11)). [57] 8. The distribution of the n L = nearest integer [(L/t s + 0.5)] wet periods (1 s) within the wet day is simulated by the pp ¼ a þ b log 10 ðlþþc log 10 ðlþ 2 þ d log 10 ðlþ 3 ; ð11þ where pp is the distribution parameter of interest, and a, b, c and d are the polynomial constants to be evaluated. Table 7 Table 6. Calibrated BL Model Parameters l (h 1 ) b (h 1 ) g (h 1 ) h (h 1 ) of17

9 Figure 6. (a) Distribution of start times for class 8 ( h), (b) probability of falling within the first hour, and (c) probability of falling within the second half for the 13 classes, Emerald Dingo. BL model which is run repeatedly and terminates when the simulated n* L is within a set limit of ±2.5% of n L. [58] 9. Simulation of n* L depths [r i = exp(z i ), i = 1, n* L ]is repeatedly carried out, terminating when two conditions, P n L* r i is within a set limit of ±5% of R and max(r i /R) is within i¼1 a set limit of ±2.5% of M, are met. [59] 10. The depths r i are then proportionally adjusted such that Pn L* r i = R. i¼1 [60] The partial derivative G(.) functions for the Normal and Clayton copulas are given in Appendix A. 4. Results and Discussion [61] The model is examined for the reproduction of rainfall statistics of the Emerald Dingo data used to estimate the model parameters and the Gregory case study site. For the assessment of the model performance of the Gregory site data, the 6 min simulation time scale data of depth resolution of 0.01 mm is resampled with the 0.18 mm tipping bucket (depth resolution of the Gregory data). Where wet period depths are significantly lower than the resampling resolution, the numerical value of L could be reduced. For example, consider daily rainfall depth R = 3.6 mm, lasting for L = 30 min (5 wet periods) with period depths of r i = (0.10, 0.14, 2.00, 1.26 and 0.10 mm). Resampling with the 0.18 mm tipping bucket will produce depths of r i * = (0.00, 0.18, 1.98, 1.26, 0.18) reducing L by 1/5. The relevant statistics of the resampled data were compared with the observed ones at the case study site. One hundred twenty simulation runs were carried out for both data sets, estimating the mean of each statistic at all aggregation levels. For each statistic, the 120 values were ranked and the 4th and 117th values were used to define the 95% prediction limits Dependence Structure [62] Figures 8a, 8b, and 8c compare the observed and one simulation run results of the bivariate dependence structure of R, L and M for December (Emerald Dingo). Also shown in Figures 8d, 8e, and 8f are the covariance structure of the actual values of the rainfall random variables. It is evident that the copula model captures the dependence structure very well, and this was the case for all the simulation runs. This is supported by the reproduction of the covariance structure of the Gregory site rainfall as depicted in Figure 9 for December. The QQ plots for L and M shown in Figure 10 indicate that the marginal distributions are very well predicted, but with slight overestimation of L values less than 1 h, and M values less than However, the majority of the observed values are within the prediction limits of the 120 simulation runs Dry Probability [63] The reproduction of DP is analyzed using the 13 L quantile classes. Figure 11 presents the simulation results with and without the diurnal submodel. The error bars represent the simulation prediction limits. In Figure 11 is also shown the limiting case of DP ( p =1 h k /24, n L = 1), the uppermost curve. As can be seen in Figure 11a, without the diurnal model component, DP is overestimated for L greater than 10.5 h and underestimated for L less than this threshold. The limiting case of DP at h k = 0.1 h (simulation time scale) given as p =1 L/24 is preserved with or without the diurnal component as this is a basic property of the rainfall process. [64] For longer wet period durations, if the storm start time is further from the 0 h the wet periods will have the tendency to be generated closer together, with reduced dry periods within the storm resulting in a higher DP as the aggregation time scale increases. Incorporation of the diurnal effects forces the start time closer to the 0 h thus creating the opportunity for the wet periods to be spread out resulting in a decrease in DP with aggregation time scale. The converse is true for shorter wet period durations. Figure 11b indicates that the simulation of the DPs significantly improved with the incorporation of the diurnal submodel. In general the prediction interval of the DP values is very small. [65] Averaging the DPs for the aggregation time scales without regard to the wet period duration may give deceptive results, as results with and without diurnal effect may match the observed mean values very well (Figure 11c). The simpler Poisson model with two parameters [Rodriguez Iturbe et al., 1987] was initially tested. While the analytical fit was perfect, the simulation results (even with the diurnal submodel) tend to overestimate the DP because of the tendency to cluster the wet periods together with limited dry periods within. Figure 12 demonstrates a very good reproduction of the DPs 9of17

10 Figure 7. Variation of daily (a) lag 1 autocorrelation (r z ); (b) standard deviation (s E ) with wet duration; L quantile class 8 histogram for (c) s E, (d) r z, and (e) the transform (1 r z ); observed and fitted lognormal parameters (f) meanlog and (g) sdlog to r z ; and observed and fitted Weibull parameters (h) shape and (i) scale to s E, Emerald Dingo. for the Gregory site, although there is a slight overestimation after an aggregation time scale of 1 h AR(1) Parameters [66] The simulation of the AR(1) submodel parameters was initially tested by using the observed daily total duration of wet periods of the Emerald Dingo data. Simulation results shown in Figure 13 match the observed very well with the class mean values indistinguishable. This justifies the choice of the marginal distributions. Figures 14a and 14b compare the class mean of the observed and the 120 simulation runs (complete disaggregation model) of the AR(1) submodel parameters. The slight underestimations of both the lag 1 autocorrelations and the standard deviations are due to the slight overestimation of the L values, particularly under 1 h, causing a shift to the right. Comparison of the observed and simulated AR(1) parameters for June (winter) Table 7. Calibrated AR(1) Model Parameters Coefficients Parameter a b c d meanlog r z sdlog r z shape s E scale s E of 17

11 Figure 8. (a f) Comparison of the dependence and covariance structure of the observed (open circles) and one simulation run (pluses) for December, Emerald Dingo. Figure 9. (a c) Comparison of the covariance structure of the observed (open circles) and one simulation run (pluses) for December, Gregory. 11 of 17

12 Figure 10. limits. QQ plot for random variables L and M, Emerald Dingo. Limits indicate the 95% prediction Figure 11. Dry probability distribution of (top to bottom) classes 1, 4, 6, 8, 9, 10, 11, 12, and 13: (a) no diurnal effect, (b) with diurnal effect, and (c) mean of all data, Emerald Dingo. For Figures 11a and 11b, error bars indicate the 95% prediction limits. 12 of 17

13 good, bearing in mind that the case study site data were not used in the development of the model parameters. Hence, this quality of disaggregated data for a site which has only daily rainfall data is more than acceptable for hydrological modeling. Figure 12. Dry probability distribution of (top to bottom) classes 1, 3, and 4 with diurnal effect, Gregory Site. Error bars are the 95% prediction limits. and December (summer) months for one simulation run in Figures 14c 14f demonstrates the satisfactory performance of the AR(1) submodel, the overriding factor being the total wet periods duration Variance and Intensity Frequency Duration [67] Preservation of monthly rainfall statistics of variance and the intensity frequency duration (IFD) curves by the disaggregation model is demonstrated in Figures 15, 16, 17, and 18 (R 0.1 mm). Nearly all the observed values of these statistics are within the prediction limits, and the means are very close to the observed. The extreme value trends, shown for the 20 year (Emerald Dingo) and 10 year (Gregory) frequencies are typical of the other IFD curves not shown in this paper. [68] As the number of years of daily data to be disaggregated increases, the prediction intervals are expected to narrow. Twelve years of data is very short and the overall performance of the developed disaggregation model is very 5. Summary and Conclusions [69] A model for disaggregating daily rainfall data into fine time scale is presented in this paper. It contributes to the use of existing long records of daily rainfall data for continuous hydrologic simulations. The wet(1) dry(0) sequence is modeled first by the nonrandomized Bartlett Lewis model with incorporated diurnal effect before superimposing the AR(1) depth process model. Unlike previous simulation efforts where quality daily data embedded in a month with some missing daily data were discarded, the presented model is structured to use all available quality daily data. Copula theory which is gaining ground in hydrology is used to model the dependence structure between daily rainfall depth (R), the total duration of wet periods (L) in a wet day, and the maximum proportional depth of a wet period (M). These random variables have been identified as important features of the daily rainfall process. [70] Most of the model parameters are related to the total wet period duration within a wet day. Limited fine time scale rainfall data near the site of interest are used to derive the model parameters. This has been demonstrated for a case study site where data from two nearby sites were combined to obtain the region specific model parameters. One hundred twenty simulation runs were carried out to verify the performance of the disaggregation model. The results indicated that gross rainfall statistics (variance, dry probability, and several intensity frequency duration curves), and the dependence structure of daily rainfall random variables (R, L and M), are very well preserved by the disaggregation model. The variation of the depth process submodel parameters with the total daily duration of wet periods is also very well captured by the presented model. It is expected that the prediction interval of the statistics will be narrowed with an increase in the number of years of daily data disaggregated. Figure 13. Comparison of observed and simulated lag 1 autocorrelation (r z ) and standard deviation (s E ) with wet duration, Emerald Dingo. 13 of 17

14 Figure 14. Comparison of the variation of lag 1 (a) autocorrelation (r z ) and (b) standard deviation (s E ) with wet duration (class mean for all 120 simulations with very small prediction interval) and one simulation run for (c, d) June (winter month) and (e, f) December (summer month), Emerald Dingo. Figure 15. Comparison of observed (triangles) and simulated (open circles) for variance, Emerald Dingo. The error bars define the 95% prediction limits, and the aggregation time scales are indicated on the plots. 14 of 17

15 Figure 16. Comparison of observed (triangles) and simulated (open circles) intensity frequency duration, Emerald Dingo. The error bars define the 95% prediction limits, and the durations are indicated on the plots for the 20 year frequency. Figure 17. Comparison of observed (triangles) and simulated (open circles) for variance, Gregory Site. The error bars define the 95% prediction limits, and the aggregation time scales are indicated on the plots. 15 of 17

16 Figure 18. Comparison of observed (triangles) and simulated (open circles) intensity frequency duration, Gregory Site. The error bars define the 95% prediction limits, and the durations are indicated on the plots for the 10 year frequency. [71] However, the diurnal submodel of the BL model requires improvements. Also metaelliptic copulas [e.g., Genest et al., 2007] may be investigated for improvements of the copula component of the disaggregation model. While the number of model parameters appears to be large, ongoing research work indicates that the parameters related to the total daily duration of wet periods could be similar for a large region such as the Australian continent. Fourier harmonic series are also being investigated for fitting to the copula parameters in order to reduce the number of parameters. An extension of this paper to spatial daily rainfall disaggregation is also being investigated. Appendix A: Copula Partial Derivative G(.) Functions A1. Normal Copula [72] The normal copula is G 1j2 ½u 1 ju 2 ; 12 Š ¼ 1 ðu 1 Þ 12 1 ðu 2 Þ7 4 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ Š ¼ 1 ðu 1 Þ G 1 1j2 u 1ju 2 ; 12 þ 12 1 ðu 2 i Þ ; where is the standard univariate normal distribution and 1 the inverse of it. A2. Clayton Copula [73] The Clayton copula is G 1j2 ½u 1 ju 2 ; 12 G 1 1j2 u 1ju 2 ; Š ¼ u u 12 1 þ u " 12 # 1 1þ 12 ½ Š ¼ u 1 u 1þ12 2 u 12 2 þ 1 [74] Acknowledgments. Continuing financial support by Queensland Rail and QRNational to the author through the HEFRAIL Project is gratefully acknowledged. We would like to thank George Kuczera for providing the NLFIT software. Comments by an associate editor and two anonymous reviewers were very constructive. 12 : References Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009), Pair copula constructions of multiple dependence, Insur. Math. Econ., 44, , doi: /j.insmatheco Bárdossy, A., and G. G. S. Pegram (2009), Copula based multisite model for daily precipitation simulation, Hydrol. Earth Syst. Sci., 13, , doi: /hess Bedford, T., and R. M. Cooke (2001), Probability density decomposition for conditionally dependent random variables modeled by vines, Ann. Math. Artif. Intell., 32, , doi: /a: Bedford, T., and R. M. Cooke (2002), Vines A new graphical model for dependent random variables, Ann. Stat., 30, , doi: / aos/ Bo, Z., S. Islam, and E. A. B. Eltahir (1994), Aggregation disaggregation properties of a stochastic rainfall model, Water Resour. Res., 30(12), , doi: /94wr Cowpertwait, P. S. P., P. E. O Connell, A. V. Metcalfe, and J. A. Mawdsley (1996), Stochastic point process modelling of rainfall. I. Single site fitting and validation, J. Hydrol., 175, 17 46, doi: /s (96) De Michele, C., G. Salvadori, M. Canossi, A. Petaccia, and R. Rosso (2005), Bivariate statistical approach to check adequacy of dam spillway, J. Hydrol. Eng., 10, 50 57, doi: /(asce) (2005)10:1(50). Duan, Q., S. Sorooshian, and V. Gupta (1992), Effective and efficient global optimization for conceptual rainfall runoff models, Water Resour. Res., 28(4), , doi: /91wr Econopouly, T. W., D. R. Davis, and D. A. Woolhiser (1990), Parameter transferability for a daily rainfall disaggregation model, J. Hydrol., 118, , doi: / (90)90259-z. Evin, G., and A. C. Favre (2008), A new rainfall model based on the Neyman Scott process using cubic copulas, Water Resour. Res., 44, W03433, doi: /2007wr Favre, A. C., S. E. Adlouni, L. Perreault, N. Thiémonge, and B. Bobée (2004), Multivariate hydrological frequency analysis using copulas, Water Resour. Res., 40, W01101, doi: /2003wr Genest, C., and A. C. Favre (2007), Everything you always wanted to know about copula modelling but were afraid to ask, J. Hydrol. Eng., 12, , doi: /(asce) (2007)12:4(347). Genest, C., and B. Remillard (2004), Tests of independence and randomness based on the empirical copula process, Test, 13(2), , doi: /bf Genest, C., A. C. Favre, J. Béliveau, and C. Jacques (2007), Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data, Water Resour. Res., 43, W09401, doi: /2006wr Glasbey, C. A., G. Cooper, and M. B. McGechan (1995), Disaggregation of daily rainfall by conditional simulation from a point process model, J. Hydrol., 165, 1 9, doi: / (94) Güntner, A., J. Olsson, A. Calver, and B. Gannon (2001), Cascade based disaggregation of continuous rainfall time series: The influence of climate, Hydrol. Earth Syst. Sci., 5(2), , doi: /hess of 17