Efficient simulation of a space-time Neyman-Scott rainfall model

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1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2006wr004986, 2006 Efficient simulation of a space-time Neyman-Scott rainfall model M. Leonard, 1 A. V. Metcalfe, 2 and M. F. Lambert 1 Received 21 February 2006; revised 6 July 2006; accepted 27 July 2006; published 9 November [1] Existing space-time Neyman-Scott models characterize rainfall as the arrival of rain cells, clustered in time and independently distributed in space. Each cell is described as a cylinder having a random intensity (height) and random radial coverage. With this formulation it is possible to have cells with centers lying outside of a target simulation region yet having radii large enough to cover points within the region. To avoid significant boundary effects, it is necessary to include these points in the simulation. However, this can introduce inefficiency into the algorithm that is computationally restrictive. To overcome this, an efficient method is derived and demonstrated to improve computational performance. Citation: Leonard, M., A. V. Metcalfe, and M. F. Lambert (2006), Efficient simulation of a space-time Neyman-Scott rainfall model, Water Resour. Res., 42,, doi: /2006wr Introduction [2] There is an extensive amount of literature on the development of clustered rainfall models at a single point [Onof et al., 2000]. The Neyman-Scott rectangular pulse model is one type of clustered process and has been demonstrated to successfully reproduce a wide range of rainfall statistics at various locations. The general structure of the model consists of a set of random variables to describe the arrival of storms in time, the displacement of storm cells relative to the storm origin, the lifetime of storm cells and the intensity of storm cells. Where cells overlap in time, the total rainfall intensity is the sum of the rainfall contributions from each cell. Cowpertwait [1995] introduced a spatial extension to the Neyman-Scott point model by specifying random variables for the position and radial coverage of each cell in space. Where cells overlap in space, as for the temporal case, cell depths are aggregated. While this spatial description of rainfall is highly idealized (for example, it does not include advection), it is (1) one of only a few rainfall models that characterizes rainfall continuously in time and space and (2) can be solely calibrated to rain gauge data. Wheater et al. [2005] discuss an alternative space-time rainfall model, having a Bartlett-Lewis temporal process and calibrated using radar data. [3] A simulation proceeds by sampling, from calibrated distributions, the arrival of storms, number of rain cells and cell properties over some target region. However, with this formulation it is possible to have rain cells with centers lying outside of a target region, yet having radii large enough to cover points within the region. This introduces a boundary effect that significantly reduces the simulated rainfall depth at points within the region if these cells are ignored. It is essential to account for this boundary effect 1 School of Civil and Environmental Engineering, University of Adelaide, Adelaide, Australia. 2 School of Mathematical Sciences, University of Adelaide, Adelaide, Australia. Copyright 2006 by the American Geophysical Union /06/2006WR because the size of a rain cell is often of a similar magnitude to the size of the target region, meaning that all points within the region (and not just those near the outer perimeter) will be affected. One approach for avoiding this is to wrap the effects of rain cells across opposing points on the boundary of the region. This approach is efficient and effective with the exception that spurious cross correlations will be observed for points separated by large distances. An alternative approach is to mitigate the boundary effect by implementing a buffer around the target region. This approach is demonstrated to substantially inhibit the computational efficiency of the model, hence an algorithm is proposed that avoids the use of a buffer region. This algorithm directly simulates the number of cells that occur outside of the target region yet are known to intersect it. Whereas the buffer algorithm is approximate due to the finite size of the buffer, the direct algorithm is exact with respect to reproducing rainfall depths within the region. The method is not specific to the Neyman-Scott model as it could be applied to similar cluster-based models having the same spatial structure. 2. Algorithm for Simulation of Cells Using a Buffer Region [4] Consider a target region, of fixed radius, r t,asin Figure 1. For any given point within this region it is desired to model the rainfall process using the space-time Neyman- Scott model. According to this model, cells associated with a particular storm arrive within the region as a Poisson process having a spatial rate 8p r t 2, where 8 is the rate parameter. [5] It is however also possible to have cells arrive outside of the target region, referred to as an outer region, yet having a cell radius, R c, greater than the distance from the circumference of the target region to the center of the cell, R xy. Consider therefore a buffer, having radius, r t + r, where r is defined relative to the target region. It is possible to also simulate cells over this region according to a Poisson process with the same spatial rate parameter, 8. Cells that intersect the target region are accepted, otherwise cells that lie solely outside are rejected. 1of5

2 LEONARD ET AL.: TECHNICAL NOTE Figure 1. Schematic diagram of cells generated inside and outside the target region. [6] The arrival of cells according to a Poisson distribution requires their cells to be uniformly distributed over the region. Efficiently simulating uniformly over a circular region can be achieved using a transformation to polar coordinates. The equations for simulating a point (x,y) uniformly within a circular region of radius r t + r are, pffiffiffiffi x ¼ ðr t þ rþ p U ffiffiffiffi cos Q ð1þ y ¼ ðr t þ rþ U sin Q where Q Uniform[0,2p] and U Uniform[0,1]. [7] To illustrate the impact of a buffer region, the Arno Basin case study given by Cowpertwait et al. [2002] is used as an example. A circular target region of radius r t =65km encompasses all of the rain gauges within this region. Cell radii are distributed exponentially with parameter f. For the two example months of January and July reported values of this parameter for the Arno Basin are f = km 1 and f = km 1 respectively. An exponential distribution of cell radii having f = km 1 is capable of producing cells with large radii, for example, the 0.99 quantile gives a cell radius of 103 km. Even though such cells have a low probability of occurrence, they need to be included in a simulation otherwise the simulated rainfall will be substantially lower than the observed. For this reason, the buffer region needs to be sufficiently large to accommodate cells that occur with large radii up to an equally large distance. [8] Consider different sizes for a buffer region, r, set to quantiles 0.0, 0.5, 0.8, 0.95, 0.99 and of the exponential distribution of cell radii. The 0.0 quantile gives the case of having no buffer. Figure 2 shows the simulated proportion of mean rainfall depth for each buffer size for the two example months, where increasing buffer sizes give increasing proportions. The nondimensional quantity f(r t + r) is used to standardize the cell size with respect to the size of the region. In order to obtain accurate estimates of the simulated proportion, the statistic was estimated from an arbitrarily long simulation of 120,000 years length (limited only because of computational requirements). The point located at the center of the 65 km target region was used for 2of5 the comparison, while points closer to the perimeter will demonstrate an even greater reduction in the simulated proportion. [9] Figure 2 gives a theoretical comparison for a point at the center of the region and shows that the proportion will reduce significantly in the event of either smaller target regions or larger expected cell radii (smaller f). The proportion, denoted m, can be obtained theoretically as the ratio of the number of cells landing within the target and buffer region that overlap the center point to the number of cells that overlap the center point for a region extending infinitely, given as m ¼ Z 2p Z rtþr Z 2p Z 1 ¼ 1 e f ð rþrt 8qe fq dq dq 8qe fq dq dq Þ ð1 þ fr þ fr t Þ ð2þ [10] Figure 2 shows that the size of the buffer region strongly affects the simulated statistics within the region, and that large buffer sizes are necessary to avoid reduction in the simulated statistics. For example, in January, the proportion of rainfall is 0.79 for the case having no buffer, and for a buffer set to the 0.99 quantile (r = 103 km), the proportion rainfall is By comparing the two months, Figure 2 also shows that the proportion of simulated rainfall varies with each month due to the variation in the parameter f. [11] The computational inefficiency introduced by the buffer is proportional to the ratio of the area of the target region to this same region with the additional buffer. Assuming, that the 0.99 quantile is used to set the buffer size, the total simulation region is approximately 6.5 times larger for January and 3 times larger for July. For the Arno Basin case study, an average month has a total simulation region 4 times larger than the target region. This inefficiency is important given that the computational requirements for Figure 2. Proportion reduction in simulated rainfall depth, simulations for January and July compared with theoretical result.

3 LEONARD ET AL.: TECHNICAL NOTE simulating the model are intensive. In order to address this problem, the following section develops an algorithm for directly simulating the number, location and radius of cells that land outside of the target region yet partially cover the region. 3. Algorithm for Direct Simulation of Cells [12] Consider a target region, of fixed radius, r t,asin Figure 1. Cells arrive within this region according to a Poisson process with spatial rate parameter, 8. Consider also the same Poisson process occurring over an outer region that extends infinitely, with the exception that it is desired to retain only those cells that intersect the target region. This occurs when the cell radius, R c, is greater than the distance from the circumference of the target region to the center of the cell, R xy. The motivation then is to derive three distributions for the number, location and radius of cells, where each distribution is conditioned on the event that R c > R xy. Equation (3) defines the discrete distribution of the number of cells in the outer region, N o, that intersect with the target region. Having defined the distribution for the number of cells, it is necessary to define the properties for each given cell. The continuous distribution of cell centers, R xy, is given in equation (4) for distances defined relative to the circumference of the target region. This distribution is conditioned on the event R c > r xy, as it is more likely for cells close to the target region to intersect it. The continuous distribution for cell radii is defined in equation (5). This distribution also depends upon the event R c > r xy, since the cell radius must be larger than the distance r xy. PN o ¼ n o jr c > R xy f Rxy f Rc r xy jr c > r xy r c jr c > r xy [13] All other properties of the cell, the intensity, duration and starting time are independent of the location of the cell within the outer region Number of Outer Cells Intersecting Target [14] The following derivation relies on the Poisson distribution as the limit of a large number of Bernoulli trials. Consider an element, as in Figure 1, having radial increment, dr, elemental angle, dq, and total radius, r + r t, with cells arriving over that element according to a Poisson process having rate, 8. The probability of one cell landing inside the element is proportional to the elemental area, (r + r t )drdq, multiplied by the spatial rate. The probability of more than one cell landing in the element is of the order (drdq) 2 and becomes zero in the limit. If a cell radius is distributed exponentially with parameter f, then the probability that a cell, landing at distance r, intersects the target region is the survivor function, e fr. Thus the probability that a cell lands in the element and extends to the target region is given as ð3þ ð4þ ð5þ pr ðþ/8ðrþr t Þdrdq e fr : ð6þ [15] The proportionality factor required for equation (6) is obtained from the Poisson distribution as e 8(r+rt)drdq. This factor is omitted from the resulting derivation for notational convenience as the probability in equation (6) is linearized with respect to q when taking the limit dq! 0, since this gives e 8(r+rt)drdq! 1. It is clear from equation (6), that for a fixed elemental angle dq, the associated probability will vary with respect to the radius. Alternatively, for an arbitrary constant, p(r) =p o, on rearranging equation (6), the elemental angle can be made to vary for a given radius to ensure that each element maintains this fixed probability, given as p o dqðþ¼ r : ð7þ 8ðr þ r t Þdr efr [16] The set of elements, varying across all radii in the outer region and having elemental angles for a given radius specified by equation (7), define a set of Bernoulli trials. Each trial has a fixed probability, p o, that a rain cell lands in the elemental area at radius r and also intersects the target region. At a given radius the number of Bernoulli trials is given by 2p/dq(r), and the total number of trials, N tot, can be obtained by integrating over all radii in the outer region, Z 1 82pðr þ r t Þ N tot ¼ e fr dr r¼0 p o ¼ 82p r t p o f þ 1 f 2 : [17] A set of Bernoulli trials of size N tot, from which n o will be successful is distributed according to a binomial distribution with PN ð o ¼ n o Þ ¼ N tot n o p no o ð8þ ð1 p o Þ Ntotno : ð9þ [18] The Poisson distribution is derived as the limit to the binomial distribution as N tot!1and is specified with a rate parameter, v = p o N tot. Consequently, the number of cells in the outer region that have radius large enough to reach the target region is Poisson distributed with parameter v, n no PN o ¼ n o jr c > R xy ¼ n o! en ; ð10þ where the parameter v = 82p ( r t f þ 1 Þ is obtained from f equation (8) Cell Center Conditioned on Intersecting Target [19] The probability that a cell intersecting the target comes from an element with radial increment, dr, and elemental angle, dq, is the ratio of the expected number of cells with center in the element that intersect the target, to the expected number of outer cells that intersect the target. This is defined as ¼ Z 1 r xy¼0 PR xy ¼ r xy jr c > R xy 8 r xy þ r t e fr xy dr xy dq Z 2p : ð11þ 8 r xy þ r t e fr xy dr xy dq 3of5

4 LEONARD ET AL.: TECHNICAL NOTE Figure 3. Comparison of direct algorithm and buffer algorithm for proportion of memory usage and simulation run time. Results are reported for a range of sizes of the target region using different parameters across 12 months. [20] Evaluating the denominator for all angles 0 < q 2p, and all radii (0 < r xy 1), and taking the limit drdq!0 this becomes the continuous distribution, f Rxy r xy jr c > r xy ¼ kfe fr xy þ ð1 kþf 2 r xy e frxy ; ð12þ where the resulting expression in equation (12) represents a mixture of an exponential distribution, Exp[f], and a gamma distribution, Gamma[2, 1 f ], and where the mixture ratio k = f r t /(f r t +1) depends upon the radius of the target region and the parameter for cell radii. For larger target regions and smaller expected cell radii, the distribution in equation (12) will tend toward an exponential distribution. The simulation of a point within the outer region requires a random angle, q, which can be independently sampled from a uniform distribution Q Uniform[0, 2p], giving the coordinates of the point as x ¼ R xy þ r t cos Q ð13þ y ¼ R xy þ r t sin Q: 3.3. Cell Radius Conditioned on Location [21] Given that a cell has landed in an element at a distance r xy from the edge of the target region, for the cell to intersect the region, its radius must be greater than the distance r xy. Given that the cell radius is exponentially distributed and that r xy is the constant defined by equation (12), the remaining distance R c r xy is also exponential, based on the standard properties of this distribution. The distribution of the cell radius is therefore given as f Rc r c jr c > r xy ¼ fe fðr cr xy Þ : ð14þ [22] Simulation of the direct algorithm proceeds as previously outlined for the arrival or storms and occurrence of 4of5 cells over the target region. For the cells that land outside the region yet intersect it, the number of cells is sampled using equation (10), their location is sampled using equation (12), and their radius is sampled using equation (14). All other cells properties such as the starting time, lifetime and intensity are independent of the location of the cell. 4. Results and Discussion [23] To demonstrate that the bias of the proposed algorithm is practically negligible, 100 replicates of 1,000,000 years length were simulated for the Arno Basin case study. The resulting median proportion of rainfall was , with a standard deviation of , which arises because of the finite simulation length. This proportion is not statistically significantly different from a value of 1.0 at the 10% level. [24] To illustrate the efficiency of the direct simulation algorithm a comparison was conducted with the buffer algorithm for the Arno Basin case study. Results for the direct algorithm are presented in Figure 3 as a proportion of the requirements for the buffer algorithm for a range of values of the nondimensional quantity fr t. The results were obtained using values of f reported for the 12 months of the year for the Arno Basin case study and for a range of target regions r t = {20, 40, 60, 80, 100} km. For each month, the radius of the buffer region, r, was specified as the constant corresponding to the 0.99 quantile of the cell radii for that month. The results are reported for two variables, the memory usage and simulation time. Because the results of the direct algorithm are reported as ratios of the buffer algorithm, they can loosely be considered independent of the computational platform. The efficiency is largely independent of the number of simulation years, and a length of 300 years was used in order to measure run times with sufficient accuracy. It is important to note that the measured efficiencies depend to some extent on the details of implementation in the computer code, since there is often a tradeoff between memory usage and computational time. Therefore Figure 3 should be regarded as an indication of the order magnitude of the efficiency and as a qualitative indication of the trend in efficiency with respect to various attributes. [25] Figure 3 shows that the proportion of memory usage across the various months and target regions respectively follow two curves, with scatter attributed to random variation of the simulating process and the variability of parameters other than f for each month. To interpret Figure 3, consider a parameter value of f = 0.1. For for a small radius of the target region, r t = 20 km, the buffer region is comparatively large to the target region, hence the direct simulation method is considerably more efficient. The direct method gives a proportion of memory usage at 0.25 and a proportion of duration at approximately 0.6. For a target region having a large radius of r t = 100 km, the buffer region is not as comparatively large, hence the efficiency of the direct method is reduced. At this radius, the proportion of memory usage is 0.4 and the proportion of duration is approximately 0.8. For this latter case, it is important to note that even a small reduction in the duration of the simulation is significant since a simulation will always maximize the available resources, for example by increasing the simulation length or by computing additional replicates. Similar

5 LEONARD ET AL.: TECHNICAL NOTE comparisons could be made from Figure 3 by considering the variation with respect to the parameter f with respect to a fixed target region. [26] It would be unlikely for the space-time Neyman- Scott model to be applied to regions with radius larger than 100 km due to the requirement that the observed rainfall must be homogeneous over this region. Furthermore, it would be expected that the majority of applications of this model would be to urban catchments that are considerably smaller than r t = 100 km. Therefore Figure 3 demonstrates that the direct simulation algorithm significantly reduces the computational requirement of this model. 5. Conclusions [27] The need to simulate cells outside of a region of interest was demonstrated in order to avoid significant boundary effects. An efficient method was derived for directly simulating these cells without the use of a buffer region. The performance of the algorithm varies with respect to the parameter values and the size of the target region, but in all cases was demonstrated to provide considerable improvement. The significance of the direct method is due to the computationally intensive nature of the model and the desire to increase simulation lengths and the number of replicates of the model. [28] Acknowledgments. The authors acknowledge the Australian Research Council for funding this project through a Discovery Project grant. The authors also wish to thank the three anonymous reviewers for their helpful suggestions that have significantly improved this paper. References Cowpertwait, P. (1995), A generalized spatial-temporal model of rainfall based on a clustered point process, Proc. R. Soc. London, Ser. A, 450(1938), Cowpertwait, P., C. G. Kilsby, and P. E. O Connell (2002), A space-time Neyman-Scott model of rainfall: Empirical analysis of extremes, Water Resour. Res., 38(8), 1131, doi: /2001wr Onof, C., R. E. Chandler, A. Kakou, P. Northrop, H. S. Wheater, and V. Isham (2000), Rainfall modelling using Poisson-cluster processes: A review of developments, Stochastic Environ. Res. Risk Assess., 14(6), , doi: /s Wheater, H. S., R. E. Chandler, C. Onof, V. S. Isham, E. Bellone, C. Yang, D. Lekkas, G. Lourmas, and M. L. Segond (2005), Spatial-temporal rainfall modelling for flood risk estimation, Stochastic Environ. Res. Risk Assess., 19(6), , doi: /s M. F. Lambert and M. Leonard, School of Civil and Environmental Engineering, University of Adelaide, Adelaide, SA 5005, Australia. (mleonard@civeng.adelaide.edu.au) A. V. Metcalfe, School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia. 5of5

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