Regionalization of the Modified Bartlett–Lewis rectangular pulse stochastic rainfall model across the Korean Peninsula

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ScienceDirect Journal of Hydro-environment Research xx (2014) 1e15 www.elsevier.com/locate/jher

Research paper

Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula Dongkyun Kim a, Hyun-Han Kwon b,*, Seung-Oh Lee a, Sooyoung Kim c,d b

a Department of Civil Engineering, Hongik University, Seoul 121-791, Republic of Korea Department of Civil Engineering, Chonbuk National University, Deokjin-dong 1ga, Deokjin-gu, Jeonju-si, Jeollabuk-do 561-756, South Korea c River and Coastal Research Division, Korea Institute of Construction Technology, Kyounggi-do 411-712, Republic of Korea d Department of Civil Engineering, Hongik University, Seoul 121-791, Republic of Korea

Received 21 April 2014; revised 29 July 2014; accepted 14 October 2014

Abstract The six parameters of the Modified BartletteLewis Rectangular Pulse (MBLRP) model were regionalized across the Korean Peninsula for all 12 calendar months. The parameters of the MBLRP model were estimated at each of the 59 rain gauges and they were spatially interpolated using the Ordinary Kriging method in order to produce maps. The parameter search space used in the parameter estimation process was repetitively narrowed through cross-validation in order to remove the impact of the multi-modality of the MBLRP model. The synthetic rainfall time series generated based on the parameter maps successfully reproduced the various statistical properties of the observed rainfall, such as mean, variance, lag-1 autocorrelation, and probability of zero rainfall at a wide range of time accumulation levels (e.g. hourly through daily). The maps representing the general rainfall characteristics, such as the average rainfall depth per rain storm, the average rain storm duration, the average number of rain cells per rain storm, and the average rain cell duration were also produced based on the estimated parameters. Lastly, some helpful tips in regionalizing the parameters of the Poisson cluster rainfall models are discussed. © 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.

Keywords: Regionalization; MBLRP; Rainfall simulator; Optimization

1. Introduction Poisson cluster rainfall models (Rodriguez Iturbe et al., 1987, 1988) generate the rainfall time series with the assumption that rain storms containing a series of rain cells with random intensity and duration occur according to a Poisson process. Due to this solid model assumption that is consistent with the observations of the physical rainfall process (Olsson and Burlando, 2002), Poisson cluster rainfall models are capable of generating rainfall time series with reasonably fine time resolution (e.g. 1 h), which * Corresponding author. E-mail addresses: [email protected] (D. Kim), hkwon@jbnu. ac.kr (H.-H. Kwon).

matches well with the observed rainfall statistics across a wide range of time scales. For this reason, they have been vastly applied to assess uncertainties related to hydrologic variables, such as flood (Moretti and Montanari, 2004), drought (Yoo et al., 2008), landslide (Bathurst et al., 2005), water resources (Fowler et al., 2005), and pesticide fate (Nolan et al., 2008a,b). The Poisson cluster rainfall models that are widely applied in practice include the NeymanScott rectangular pulse model (Rodriguez-Iturbe et al., 1987), the space-time Neyman-Scott rectangular pulse model (Cowpertwait, 1995), and the Modified BartletteLewis rectangular pulse model (Rodriguez Iturbe et al., 1988). However, the application of Poisson cluster models in practice cannot be expanded, despite its strengths, due to the

http://dx.doi.org/10.1016/j.jher.2014.10.004 1570-6443/© 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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complex parameter estimation process of the model (Onof et al., 2000). In this process, the five to six model parameters are determined such that the statistics of the synthetically generated rainfall time series, including mean, variance, autocorrelation and probability of zero rainfall, resemble those of the observed rainfall time series at a wide range of temporal accumulation levels (e.g. 1 h to 24 h). It is impossible to identify a parameter set that perfectly reproduces the observed rainfall statistics, because the number of rainfall statistics to be matched is greater than the number of the parameters. Instead, the parameter set is determined such that the discrepancy of the statistics between the observed and the synthetically generated rainfall time series can be minimized. Because the equations representing the statistics of synthetic rainfall have complex mathematical forms, the objective function to be minimized in the parameter estimation process does also. For this reason, identifying the optimal parameters in the complex five to six dimensional surface of the objective function requires execution of heuristic optimization algorithms, which create a major obstruction in active application of the Poisson cluster rainfall models. One way to overcome this problem and to expand the applicability of the Poisson cluster model is to predetermine the parameters at spatial locations in which the rainfall time series needs to be created. This type of work is called “parameter regionalization.” Several efforts have sought to regionalize the parameters of the Poisson cluster rainfall models. Hawk and Eagleson (1992) regionalized the parameters of the Modified BartletteLewis Rectangular Pulse model across the United States and generated contour maps of each parameter for each month of the year. Their maps were generated based on data from 75 randomly selected rain gauges and 4 additional gauges were used for validation. Cowpertwait et al. (1996) suggested the use of regression equations that related the parameters of the Neyman-Scott Rectangular Pulse (NSRP) model (Rodriguez-Iturbe et al., 1987) to regional properties (e.g., altitude and distance to the coast). Their analysis was performed using rainfall data from 112 sites in the United Kingdom. Onof et al. (2000) estimated the parameters of the BartletteLewis Rectangular Pulse model at three gauge locations in the United Kingdom and suggested based on the results that the regionalization has to be performed separately for each season due to the seasonal and regional variation of the rainfall statistics, of which the pattern cannot be easily captured with mathematical functions. They also indicated that the

regionalization is dependent upon the ability to estimate the parameters. Kim et al. (2012b) regionalized the parameters of the Modified BartletteLewis Rectangular Pulse (MBLRP) model across the United States. They estimated the parameters of the MBLRP model at 3444 National Climate Data Center (NCDC) rain gauges using the ISPSO meta-heuristic optimization algorithm (Cho et al., 2011) and spatially interpolated the parameters in order to produce the map of 6 parameters for each calendar month of the year (total of 6
12 ¼ 72 maps). The validity of the values in the parameter maps were proven through the cross-validation technique at all 3444 gauge locations by comparing the rainfall statistics of the observed and synthetically generated rainfall time series. The present study has three primary novelties compared to the previous studies: First, the parameters of the MBLRP rainfall model were regionalized across the Korean Peninsula. To the authors' knowledge, most studies on stochastic rainfall generation in Korea have focused on methodology based on the Markov Chain (e.g. Moon et al., 2006). While methodology based on the Markov chain has strength in application due to its relatively simple algorithm implementation, it has drawbacks in modeling extreme values and persistence (e.g. Gregory et al., 1992). In addition, most Markov Chain based approaches are capable of generating rainfall time series only at a coarse temporal resolution (e.g. daily) which can cause significant error in modeling the responses of watershed with a short time of concentration (e.g. 1-hour to 24-hours). In this perspective, the present study is expected to stimulate research related to risk assessment of hydrologic variables in Korea by enabling easier generation of a synthetic rainfall time series that better represents the observed rainfall time series in comparison to the conventional Markov chain-based approaches. Second, the rainfall characteristics, such as average rainfall depth per storm, average storm duration, average number of rain cells per storm and average rain cell duration, were also derived from the estimated model parameters and spatially interpolated in order to produce maps. These maps can be used to understand the spatial variation of rainfall characteristics of the Korean Peninsula from a perspective that conventional statistical analysis on rainfall cannot provide. Lastly, we addressed the issues that arose during the process of regionalization and explained how these issues were resolved in the present study, which we believe will provide useful hints in regionalizing the parameters of the Poisson cluster models in other regions.

Fig. 1. Schematic of the MBLRP model. The white and gray circles represent the arrival time of storms and rain cells, respectively. Each rain cell is represented by a rectangle whose width and height represent its duration and rainfall intensity. Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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2. Modified BartletteLewis Rectangular Pulse (MBLRP) model The MBLRP model is the most essential part of the present study. The description provided by Kim et al. (2013a,b) is noted below to help the readers understand the model. In the MBLRP model (Rodriguez-Iturbe et al., 1987), rainfall time series are represented as sequences of storms comprised of rain cells (see Fig. 1). In the model, X1 [T] is a random variable that represents the storm arrival time, which is governed by a Poisson process with parameter l [1/T]. X2 [T] is a random variable that represents the duration of storm activity (i.e., the time window after the beginning of the storm within which rain cells can arrive), which varies according to an exponential distribution with parameter g [1/ T]. X3 [T] is a random variable that represents rain cell arrival time within the duration of storm activity, which is governed by a Poisson process with parameter b [1/T]. X4 [T] is a random variable that represents the duration of the rain cells, which varies according to an exponential distri-

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The estimation of the model parameters is accomplished by matching e or minimizing the discrepancy between e the statistics of the simulated and observed rainfall time series. Some commonly used statistics are the precipitation depth mean, variance, probability of zero rainfall and lag-s covariance at various time scales (Khaliq and Cunnane, 1996). According to Rodriguez-Iturbe et al. (1988) and Bo et al. (1994), the statistics of the synthetically generated rainfall time series at an accumulation interval T are as follows: i h n T; ð1Þ E YtðTÞ ¼ lmmc a1  Var

 YtðTÞ

    2v2a T k2 2v3a k2 k1  ¼  k1  2 a2 f ða  2Þða  3Þ f   2 k 2 þ k1 ðT þ nÞ3a  2 ðfT þ nÞ3a ; ða  2Þða  3Þ f ð2Þ

   k1 ðTÞ 3a 3a 3a  ðTÞ Cov Yt ; Ytþs ¼ ½Tðs  1Þ þ n þ ½Tðs þ 1Þ þ n  2ðTs þ nÞ ða  2Þða  3Þ   k2  2ðfTs þ nÞ3a  ½fTðs  1Þ þ n3a  ½fTðs þ 1Þ þ n3a ; þ 2 f a2 a3

ð3Þ

and.     f k þ f 4k2 þ 27kf þ 72f2 ln 1 Pðzero rainfallÞ ¼ exp  lT  1 þ fðk þ fÞ  fðk þ fÞðk þ 4fÞ þ fða  1Þ 4 72   a1  2 ln 3 k ln n 2 þ 1  k  f þ kf þ f þ þ ða  1Þðk þ fÞ 2 ða  1Þðk þ fÞ n þ ðk þ fÞT 2   2 k 3 k
1  k  f þ kf þ f2 þ ; f 2 2

bution with parameter h [1/T] that, in turn, is a random variable represented by a gamma distribution with parameters n [T] and a [dimensionless]. X5 [L/T] is a random variable that represents rain cell intensity, which varies according to an exponential distribution with parameter 1/m [T/ L]. From an intuitive viewpoint, l is the expected number of storms that arrive in a given period, 1/g is the expected duration of storm activity, b is the expected number of rain cells that arrive within the duration of storm activity, 1/h is the expected duration of rain cells, and m is the average rain cell intensity. Parameters n and a do not have a clear physical meaning, but the expected value and variance of h can be expressed as a/n and a/n2. Therefore, the model has six parameters: l, g, b, n, a and m. However, it is customary to use the dimensionless ratios 4 ¼ g/h and k ¼ b/h as the parameters instead of g and b.

ð4Þ

where   a  lm kfm2 n k1 ¼ 2lmc m2 þ 2c ; a1 f 1   a  lmc kfm2 n k2 ¼ ; 2 a1 f 1 and. k mc ¼ 1 þ ; f 3. Methodology 3.1. Data description The Korean Meteorological Administration (KMA) runs 78 Automatic Synoptic Observation System (ASOS) gauges to

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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measure various meteorological variables such as pressure, air temperature, precipitation, wind direction/velocity, solar radiation, and surface temperature. The data observed from the ASOS gauges of the KMA goes through a strict quality control process (Korea Meteorological Services, 2012a) and, therefore, is considered to be the most accurate source of rainfall data in Korea. For this reason, it has been used in various meteorological and hydrologic studies in Korea (Korea Meteorological Services, 2012b). In the present study, we performed analysis based on rainfall data recorded at 59 ASOS gauges (Fig. 2). All of these gauges have a continuous record length greater than 20 years. The rainfall record has a temporal resolution of 1 h, and the earliest record starts from the year 1973 and the latest record ends in the year 2012. The parameter calibration of the MBLRP model is based on rainfall statistics and this threshold of record length ensures the accuracy of the rainfall statistics measured at each gauge location. Use of rainfall data with fine temporal resolution, such as 1 h, in the parameter calibration enables the model to reproduce the statistical property of the observed rainfall time series at a similar level of fine temporal resolution. This is particularly important, because watershed response, such as runoff and peak flow to the same depth of rainfall, can vary significantly depending on the temporal resolution of the input rainfall time series (Marani et al., 1997). 3.2. Rain gauge statistics For each of the 59 gauges and the calendar month of the year (i.e., January, February, …, November, December), the

mean, variance, lag-1 autocorrelation, and the probability of zero rainfall of the rainfall depth were calculated at 1-, 3-, 12and 24-hour accumulation levels, for a total of 16 statistics per month. Among these, the mean rainfall values at the 3-, 12-, and 24-hour accumulation levels were excluded in the parameter calibration process, because the mean does not vary with the accumulation level. According to Khaliq and Cunnane (1996), the models calibrated based on these 4 statistics at various accumulation levels produce rainfall time series that resemble historical observations. The reason why the rainfall time series was divided by each month was because the seasonality of rainfall statistics is difficult to capture with other approaches assuming a smooth transition between the seasons (Onof et al., 2000). Various levels of subdaily accumulation levels were considered, because the partitioning of rainfall into runoff and sub-surface flow was highly affected by the internal structure of rainfall time series within the sub-daily time frame (Marani et al., 1997). Fig. 3 shows maps of the rainfall statistics for the month of July. The maps were generated by interpolating the estimated rainfall statistics at each gauge using the Ordinary Kriging interpolation technique. What is particularly notable from these results in comparison to those of Kim et al. (2013a,b), which investigated the same statistics in the United States, is that the lag-1 auto-correlation shows a high regional tendency as well as the other 3 statistics. This seems to be due to the distinct characteristics of summer rainfall on the Korean Peninsula, which is highly affected by the East Asian Monsoon. During this rainy season, the frontal rain storm, of which the spatial scale is similar to the size of the Korean Peninsula, takes place over a long temporal duration (e.g. from 3 to 4 days to a couple of weeks). 3.3. Parameter estimation Once the 13 rainfall statistics were calculated for all of the 59 gauges for each of the months, the next step was to estimate the parameters of the MBLRP model through the calibration process. As mentioned earlier, it is impossible to identify the parameter set that perfectly reproduces the observed rainfall statistics by analytically solving equation (1) through equation (4). Instead, the parameter set is determined such that the discrepancy of the statistics between the observed and the synthetically generated rainfall time series can be minimized. In this study, the following equation was used to represent the discrepancy between the statistics:  2 n X Fk ðl; n; a; m; 4; kÞ OF ¼ wk 1  ; ð5Þ fk k¼1

Fig. 2. Location of the 59 KMA ASOS gauges in Korean Peninsula with hourly recording and periods of records longer than 20 years.

where n is the number of statistics being matched, Fk ðl; n; a; m; 4; kÞ is the kth statistic of the simulated rainfall time series, fk is the kth statistics of the observed rainfall time series, and wk is the weighing factor given to each rainfall statistic depending on the use of the synthetic rainfall time series (Kim and Olivera, 2012).

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 3. Rainfall statistics for a 1-h accumulation interval for the month of July. From top to bottom and left to right: mean (mm/hr), variance (mm2/hr2), lag-1 autocorrelation coefficient, and probability of zero rainfall.

The equations representing the statistics of synthetic rainfall, which is Fk ðl; n; a; m; 4; kÞ in equation (5) or equation (1) through equation (4), have highly non-linear mathematical forms and, therefore, the objective function, or OF in equation (5), needs to be minimized in the parameter estimation process. For this reason, the conventional slope-based optimization algorithm usually fails to find the optimal parameter set (l; n; a; m; 4; k) that minimizes the OF value. In this study, we used a meta-heuristic optimization algorithm (ISPSO, Cho et al., 2011), which was proven to successfully identify the parameters of the MBLRP model. In this particular study, the identical weight factor (wk in equation (5)) was applied for all of the types of statistics despite the fact that not all of the rainfall statistics have the same relative importance (Kim and Olivera, 2012) and the parameters determined with it will reflect this assumption.

readers understand the following: rain gauges A and B should be located close to each other. Despite the fact that the gauges were at different locations, for the sake of simplicity, it was assumed that both were affected identically by the storm depicted in Fig. 4. In the figure, the storm at gauge A is represented by two rain cells, while the one at gauge B is represented by six rain cells. The rain cells in A have a longer duration and greater rainfall depth than those in B, but the overall results are the same. In other words, a storm e or a precipitation time series if we refer to a longer period e can be represented by different parameter sets. The existence of different and equally-correct solutions for the parameter vector is represented as a multiple minima of the objective function in the parameter space and it is referred to as multi-modality (Gyasi-Agyei, 1999; Onof et al., 2000). The spatial interpolation of the parameter values between gauges A and B might yield incorrect estimates because of this multi-modality.

3.4. Multi-modality and repetitive cross validation Once the parameter set was estimated for each of the gauges, each parameter was spatially interpolated to produce a map. However, the multi-modality of the objective function (Gyasi-Agyei, 1999; Onof et al., 2000) causes a problem in the interpolation process. In this study, we presented a description of multi-modality by Kim et al. (2012) in order to help the

Fig. 4. Rainfall event modeled with two different parameter sets.

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 5. Example of cross-validation and the identification of the new parameter search space.

In order to address this multi-modality problem, we invented the method of repetitive cross -validation in this study. In this method, the parameter set corresponding to the global minimum of the objective function was first identified for all of the gauges using the Isolated Speciation-based Particle Swarm Optimization (ISPSO) method (Cho et al., 2011) within a given parameter space; then, the values were “cross-validated”. In crossvalidation, the parameter value at the gauge was compared to the value that was obtained by the spatial interpolation of the parameter values of the neighboring gauges with the assumption that the gauge being compared did not exist. Fig. 5 explains this in greater detail. In Fig. 5, gauges A, B, D, and E have parameter values of 2, 4, 7, and 6, which are relatively similar to each other in comparison to the parameter value of gauge C, which is 25. In this situation, it can be assumed that the first 4 values were estimated within the same or “major” cluster of the parameter space, except for the latter. The plot on the right side of Fig. 5 shows the results of the cross-validation corresponding to this situation. The x-axis of the plot represents estimated parameter values and the y-axis of the plot represents cross-validated parameter values. In the plot, the point corresponding to gauge C has a cross-validated value or the y-coordinate that is affected only by the parameter values of A, B, D, and E falls within the major cluster of the parameter space. The upper and the lower boundary of the cross-validated values exclude the parameter value originally estimated at gauge C, which was estimated at a different mode of the objective function compared to the rest. Then, the upper and the lower boundaries obtained through cross-validation were used as the new range in the next iteration of the parameter estimation using ISPSO. This repetitive process of parameter estimation using ISPSO and cross-validation to assess the new parameter range in the next iteration of parameter estimation was termed as “repetitive cross-validation.” In this study, 3 repetitive cross-validations yielded the parameter values estimated at similar modes for all 6 of the parameters. More details on this subject are provided in Section 4.1. 3.5. Spatial interpolation Both for the process of repetitive cross-validation and the process of generating final parameter maps, the method of

spatial interpolation is required. This study employed the Ordinary Kriging interpolation method (Journel and Huijbregts, 1978). The Ordinary Kriging was used since no prior knowledge of the spatial trend of the parameter values being interpolated is required. Other interpolation methods, such as Simple or Universal Kriging, require knowledge of the constant or variable spatial trend of the data, respectively, so significantly more exploration of the data is required, while higher accuracy is not always guaranteed in comparison to the Ordinary Kriging. A total of 72 monthly parameter maps (i.e., 6 parameters
12 months) were generated. The variograms used for the interpolation of the model parameters were assumed to have a spherical shape and their properties were determined so that the sum of the square of the residuals between the model and sample variogram was minimized. 4. Results 4.1. Parameter maps Fig. 6 shows the parameter maps for the month of July. All of the maps except for the map of parameter a show very smooth spatial variation. Even though the map of parameter a shows relatively rougher spatial variation, the map still has smoother variation than the parameter maps of the contiguous United States (Kim et al., 2013a,b). This is primarily due to the fact that the spatial trend of the lag-1 auto-correlation of the Korean Peninsula is clearer than that of the United States. According to Islam et al. (1990), the parameter a, which governs the duration of rain cells along with parameter n, is closely related to the autocorrelation of rainfall depth. Therefore, smoother spatial variation of lag-1 autocorrelation leads to smoother spatial pattern of the parameter a. In addition, repetitive cross-validation contributed greatly to the production of parameter maps with smoother spatial variation than those corresponding to the contiguous United States (Kim et al., 2013a,b). The maps of the parameters became significantly smoother when the multimodality of the estimated parameters was reduced through repetitive crossvalidation. Fig. 7 shows the maps of parameter m produced at different levels of repetitive cross-validation. The range of

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 6. The MBLRP model parameters for the month of July. From the top to bottom and left to right: l (1/hr), n (hr), a, m (mm/hr), f, and k.

parameter estimation for each level of repetitive crossvalidation is given in Table 1. The top-left map shows the map when the range of parameter estimation was not adjusted with cross-validation. While the map shows the smooth transition of the parameter values over the space, it does not reflect the observational evidence taking into consideration that parameter m has the physical meaning of average rain cell intensity and that such abrupt variation of rain cell characteristics over the space is not physically possible. The remaining 3 maps show that the spatial trend of parameter m became significantly more discernible as the parameter space was narrowed through repetitive crossvalidation.

4.2. Validity of the parameter maps The primary purpose of the parameter maps is to enable the MBLRP model to generate a rainfall time series that resembles the statistics of the observed rainfall. If the rainfall time series generated from the MBLRP model based on the parameter values read from the maps has similar statistics to the observed ones, then the validity of the parameter maps is verified. In this context, the following approach was used for the parameter map verification. First, the parameter values were crossvalidated at a given gauge location. Then, the crossvalidated parameter values were put into equation (1) through equation (4) in order to obtain the rainfall statistics.

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 7. Impact of repetitive cross-validation in order to narrow down the parameter estimation range. The maps of the parameter m (mm/hr) after (a) 0, (b) 1, (c) 2, and (d) 3 repetitive cross-validations are shown.

Lastly, the calculated rainfall statistics were compared to the statistics of the observed rainfall time series. Fig. 8 shows the results of the rainfall statistics comparison. The x-axis and yaxis of each plot represent the observed rainfall statistics and the ones calculated based on the cross-validated parameters, respectively. The 1st through the 4th columns of Fig. 8 show the plots corresponding to the mean, standard deviation, lag-1 auto-correlation, and probability of zero rainfall, respectively. The 1st through 4th rows of Fig. 8 show the plots corresponding to temporal accumulation levels at 1, 3, 12, and 24 h, respectively. In general, a good fit was observed for the mean and standard deviation at all of the accumulation levels. The lag-1 auto-correlation and probability of zero rainfall of the simulated rainfall was concentrated within a range that was narrower than that of the observed statistics. Kim and Olivera (2012) provided a good answer as to how this measure of error

introduced when reproducing the lag-1 auto-correlation and the probability of zero rainfall will impact the accuracy of hydrologic studies. For example, the relative importance of the lag-1 auto-correlation and probability of zero rainfall to the mean rainfall while modeling floods in a highly impervious watershed (Curve Number ¼ 90) were both 8%. In addition, the relative importance of the lag-1 auto-correlation and probability of zero rainfall to the mean rainfall while modeling the runoff volume in the same watershed were 17% and 21%, respectively. Fig. 9 shows the plots comparing the empirical cumulative density function (CDF) of the observed rainfall time series at hourly accumulation level and that of the synthetic rainfall time series generated by the MBLRPM model for the six selected locations of Korea (See Fig. 2). The hourly rainfall records of the four wettest months (June, July, August, and

Table 1 Range of each parameter used for parameter estimation narrowed down through repetitive cross-validation. Number of XV

l

y

А

m

F

k

0 1 2 3

0.00001e0.05 0.00698e0.0187 0.00773e0.0131 0.00880e0.0128

2e6 2.24e5.85 3.93e4.79 4.22e4.73

3e300 5.84e297 25.9e71.1 26.7e35.6

0.5e250 5.15e66.8 9.46e21.2 10.4e15.1

0e0.4 0.00409e0.132 0.0278e0.0614 0.0321e0.0433

0.01e0.999 0.0988e0.953 0.273e0.625 0.420e0.624

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 8. Comparison of the observed rainfall statistics (x-axis) and statistics of the synthetically generated rainfall time series (y-axis) based on cross-validated parameter values. Each row corresponds to a different level of time accumulation and each column corresponds to a different type of rainfall statistic.

September) of Korea with non-zero depth values were used to generate the plots. The plots indicate the good match between the observed and synthetic rainfall time series. The twosample KolmogroveSmirnov test (KeS test) was used to check the similarity of the two distributions at all 59 gaging stations. For all 59 cases, the null hypothesis stating that the distributions of the two time series are from the same distribution was not rejected. Considering that the probability of zero rainfall values are well reproduced with the MBLRP model, this result means that the MBLRP model is capable of generating synthetic rainfall time series of which distribution is similar to that of the observed rainfall time series.

4.3. Maps of rainfall characteristics The MBLRP model conceptualizes the rainfall into the sequences of rain storms composed of the sequences of the rain cells with random duration and intensity. Based on this conceptualization, some interesting rainfall characteristics, such as average rain depth per storm, average storm duration, average number of rain cells per storm, and average rain cell duration, can be calculated using the MBLRP parameters. The following equations developed by Hawk and Eagleson (1992) were used to calculate these rainfall characteristics:

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 9. Cumulative density function (CDF) of the observed rainfall time series and synthetic rainfall time series generated by the MBLRP model. The hourly rainfall records of the four wettest months (June, July, August, and September) of Korea with non-zero depth values were used to generate the plots.

  n k Average rainfall depth per storm ½L ¼ m 1þ a f Average duration of storm duration ½T ¼

n fa

Average number of rain cells per storm ¼ 1 þ Average rain cell duration ½T ¼

n a

k f

These rainfall characteristics were calculated at all gauges and were spatially interpolated using the Ordinary Kriging technique in order to produce a surface map. Fig. 10 shows the maps for the month of July, during which the Korean Peninsula experiences the most frequent occurrences of floodrelated disasters among all calendar months. In general, characteristics related to rain storms indicated a spatial pattern varying in the latitudinal direction. The average rain depth per storm and the average storm duration were greater in the northern area than they were in the southern area meaning that once it rains, it rains longer and harder in the north than in the

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 10. Storm and rain cell characteristics for the month of July according to the MBLRP model parameters. (a) Average rainfall depth per storm (mm), (b) average storm duration (hr), (c) average number of rain cells per storm, and (d) average rain cell duration (hr) are shown.

south. The regional characteristics of the rain cells showed a different spatial pattern. The average number of rain cells per storm was greater for the eastern area than the western area and the average rain cell duration was greater for the western area than the eastern area. This clear spatial pattern of the latitudinal variation of rain storms and the longitudinal variation of rain cells is very interesting considering that no rainfall statistics for the Korean Peninsula (see Fig. 3) have such patterns. This indicates that the MBLRP model factorizes the rainfall process in a unique manner compared to typical statistical analysis of rainfall, which is the subsidiary advantage of modeling and regionalizing rainfall based on conceptualization of the Poisson cluster model. 5. Discussion The rainfall characteristics of Korea are unique and they are characterized in particular by a long spell of rainy weather during the early summer season. These unique Korean rainfall characteristics caused several issues while conducting the parameter regionalization that did not occur in a similar study performed in the continental United States (Kim et al., 2013a,b). In this study, we focused on discussing these issues and how they were resolved. 5.1. Determination of the appropriate parameter ranges In the previous section, we learned that defining the correct parameter range in the parameter estimation process is

important for reducing the problems associated with the multimodality of the MLBRP model, which helps to obtain maps that reflect the “true” spatial variability of the parameters. In addition, it should be noted that different parameter ranges have to be used for the parameter estimation of seasons with different rainfall characteristics. This finding is consistent with that of Onof et al. (2000). In order to verify this, we estimated the parameters using ISPSO based on the same parameter range for different months with the distinct rainfall characteristics in this study. In the case when the initial parameter range (2nd row of Table 1) was used, the mean of the objective function values (equation (5)) estimated at all 59 of the gauges for the months of April, July, and October were 1.34, 0.15, and 0.26, respectively. In the meantime, when the parameter range that was narrowed down through 3 repetitive cross-validations for the month of July (4th row of Table 1) was used for the parameter estimation process for not only July, but also for the remaining two months, the mean of the objective function values for April, July, and October were 157.8, 0.101, and 36.49 respectively. Considering that a greater value of the objective function eventually means poorer performance of the model, it should be noted that the parameter estimation process and the corresponding model performance of the MBLRP model can be significantly degraded if a parameter range that does not take into consideration the seasonality of the rainfall characteristics is used for the model calibration. Based on this result, the following remarks should be emphasized when regionalizing the parameters of the Poisson cluster models: (1) The parameter range to be used in the

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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parameter estimation process should be selected distinctively for different seasons with different rainfall characteristics; and (2) The “correct” parameter range that can eliminate the negative effects of the multi-modality of the MBLRP model can be obtained through the repetitive cross-validation technique. Fig. 11 shows the variation of the mean of the objective function values estimated at 59 ASOS gauges with regards to tightening the parameter estimation range through the repetitive cross-validation processes. The upper line shows the average objective function values for the dry season (October to April) and the lower line shows the one for the wet season (May to September). It is noteworthy that the sacrifice of the MBLRP model performance as indicated by the increasing average objective function value due to the tightening parameter estimation range for regionalization is minimal for the wet season. This indicates that cross-validation is an effective tool for obtaining the appropriate parameter range in order to reduce the problems related to the multi-modality especially for the wet season. The line for the dry season indicates that the objective function value ranged between 3.5 and 4.5. Even though these values were greater than those corresponding to the wet season, they were still significantly lower than the case in which the parameter range of the wet season was used for the model calibration. This result also indicates the importance of applying the appropriate parameter range for model calibration. In addition, the average objective function value consistently increased for the dry season with the tightening parameter estimation range. This indicates that, if the MBLRP model is to be used at a single site with no spatial or regional consideration for the dry season, a wider parameter range can be used for model calibration instead of the one tightened up for the parameter regionalization. 5.2. Seasonality of the MBLRP model parameters Cowpertwait et al. (1996) previously performed a parameter regionalization study in Great Britain. They suggested that a mathematical periodic function could be used to describe the seasonality of a Poisson cluster rainfall model. We also sought to find the periodicity of the MBLRP model parameters in this study, but we could not identify a seasonal pattern that was

Fig. 11. Mean of the objective function values for the wet and dry seasons with a parameter estimation range narrowed through cross-validation.

clear enough to be expressed with a simple mathematical formula. Fig. 12 shows the seasonal variation of each of the six parameters for months from April to October in three different cities in Korea (see Fig. 2 for the location of the gauges). The plots for the remaining dry months (November to March of the next year) were not provided, because the parameter estimation process was not as successful as it was for the wet months, so some parameters estimated in these months had values of which the magnitude differed by several degrees. The problems associated with the parameter estimation in the dry months will be discussed in the next section of this article. Parameter k shows a clear pattern of declination during the months of June, July, and August for Seoul and Daejeon. However, such a pattern was not observed in Busan, which is considered to have a different climatology in comparison to the remaining two cities, because of its geographical location at the southern edge of Korea. Parameter n showed an opposite pattern with greater values during the wet months. Parameter l for Seoul showed a sinusoidal pattern with greater values during the wettest months (July and August). No other clear seasonal pattern was observed except for these three parameters. This finding leads to the conclusion that expressing the seasonality of the MBLRP model parameters with a mathematical periodic function is a challenging task. First, this is due to the fact that the periodicity of each of the parameters is not smooth enough to be expressed using a sinusoidal function, which is partly due to the fact that the conversion between the wet and dry season in Korean Peninsula is extremely abrupt. Second, each of the MBLRP model parameters explains only a limited portion of the conceptualized rainfall process. In other words, accumulated rainfall with a clear seasonal tendency is the reflection of a complex combination of storm and rain cell arrival rate, storm and rain cell duration, and rain cell intensity. While the rainfall processes are further factorized into the parameters representing the individual portions of the rainfall processes, the clarity of the seasonality in accumulated rainfall can be lost. For example, the average rain cell intensity m (plot [E] in Fig. 13) in Seoul does not

Fig. 12. Variation in the mean of objective function values (Equation (5)) calculated at 59 ASOS gauges during the MBLRP parameter estimation process. The variation in monthly precipitation on the Korean Peninsula is also shown.

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Fig. 13. Seasonal variation of the MBLRP model parameters for the Seoul, Daejeon, and Busan ASOS gauges.

reflect the clear seasonal tendency of the accumulated rainfall, because it is greatest during the months of May and September, while the rainfall depth is greatest during the months of July and August. However, the storm arrival rate l (plot [A] in Fig. 13) in Seoul has a clear seasonal tendency reflecting that of accumulated rainfall, compensating for the vague seasonal tendency of parameter m. Lastly, Fig. 13 indicates that the seasonal pattern of the parameters, if there is any, is very region-specific. While Seoul and Daejeon are only 100 km apart, the seasonal patterns of Seoul and Daejeon are not similar enough to each other to be treated as having a similar tendency. This result suggests the need to apply the distinct parameter sets of the periodic function in order to express seasonal tendencies for different geographical locations. 5.3. Applicability of MBLRP model for dry winter seasons in Korea One of the most distinct characteristics of the climate of the Korean Peninsula is that two thirds of the annual rainfall is concentrated during the summer season. This conversely means that the amount of precipitation during the remaining seasons is relatively small. The regionalization of the MBLRP model for the wet seasons could be performed with no particular problems and the produced map showed similar performance, as shown in Fig. 8. However, the parameter set that can match the observed rainfall statistics for the dry winter season was not easily identified even when a wide range of parameter search space was given to the ISPSO. The median of the objective function values at all gauges for the months of December, January, and February were 7.82, 7.82, and 7.11, respectively. These median values were significantly greater than those of the summer seasons, which were 0.0435, 0.0677, and 0.0786 for the months of June, July, and August,

respectively. This observation leads to the assumption that the fundamental structure of the MBLRP model may not be adequate to model the winter precipitation of the Korean Peninsula. We investigated the statistical properties of these winter rainfall values in order to clarify the limitation of the MBLRP model in this study. When the rainfall has the following statistical properties, the MBLRP model may not perform well: (1) The mean at the hourly accumulation level is less than 0.05 cm (or 36 cm per 30 days); (2) The variance at the hourly accumulation level is less than 0.5 cm2; (3) The Lag-1 autocorrelation at the hourly accumulation level is less than 0.01; and (4) The probability of zero rainfall at the hourly accumulation level is greater than 0.99. 5.4. Importance of determining the appropriate no rainfall threshold The ASOS rainfall data of the KMA that were used in this study had recordings of rainfall depth as small as 0.1 mm. While this small amount of rainfall does not make a substantial difference in hydrological analysis, especially in modeling floods, it significantly changes the value of the probability of zero rainfall. Considering that the MBLRP model conceptualizes the rain cell as a temporally invariant rectangle with height determined by one parameter's exponential distribution, this high value of the probability of zero rainfall reflecting the trivial rainfall intensity can exert negative impacts on the parameter estimation process of the MBLRP model. The first row of Table 2 shows the average of the hourly rainfall statistics and the objective function value (equation (5)) of the original rainfall time series observed at 59 of the ASOS gauges. The median of the objective function value was 0.103, which was relatively high in comparison to similar analysis performed in the continental United States (see Kim et al., 2013). Considering that lower objective

Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004

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Table 2 Variation of the rainfall statistics and the objective function with regard to rainfall depth filtering threshold. Threshold

Mean (mm)

Var (mm2)

Lag1

Prob0

Obj

0 mm 0.5 mm 1 mm 2 mm

0.3734 0.369 0.3536 0.3246

4.187 4.3181 4.317 4.289

0.5407 0.5534 0.5464 0.5277

0.8774 0.9211 0.9382 0.9561

0.103 0.072 0.069 0.060

function values indicate better performance of the parameter estimation process, this result indicates that the parameter estimation process was not as successful as the study performed in the continental United States. In order to resolve this problem, the rainfall time series was filtered out with the threshold values of 0.5 mm/h, 1 mm/h, and 2 mm/h. Then, the parameters of the MBLRP model were estimated for each of the time series with different intensity threshold values. The last 3 rows of Table 2 show the change in the median of hourly rainfall statistics and the objective function values according to change in the filtering threshold. The change in probability of zero rainfall with regard to the filtering rainfall threshold value was significant and the median objective function value became smaller with an increase in the filtering rainfall threshold. In this study, we adopted a filtering threshold value of 2.0 mm in order to enhance the accuracy of the MBLRP model parameters. Using this threshold introduced slight bias into the generated rainfall time series, but it was expected that this bias would be compensated for by reducing the error that was introduced by the misestimated parameters. 6. Summary and conclusions Seventy-two parameter maps (¼ 6 parameters
12 months) of the MBLRP stochastic rainfall generation model were generated for the Korean Peninsula. These maps allowed for the implementation of the MBLRP model at any location on the Korean Peninsula without having to calibrate and validate the model each time. The parameters were estimated at 59 KMA ASOS rain gauges by matching 13 rainfall statistics (i.e., mean, variance, probability of zero rainfall and lag-1 autocorrelation coefficient) at different rainfall accumulation levels (i.e., 1, 3, 12 and 24 h). The parameters obtained at the gauges were then interpolated using the Ordinary Kriging technique in order to generate surface maps. The parameter maps were validated through crossvalidation in order to assess their ability to reproduce rainfall statistics when used in the MBLRP model. The results of validation indicated that the values read from the parameter maps successfully reproduced rainfall statistics including mean, variance, lag-1 autocorrelation, and probability of zero rainfall within the practical range of the temporal accumulation level (1 he24 h). We also produced maps of rainfall characteristics based on the estimated parameter values, such as the average storm duration, average rain cell duration, average number of rain cells per storm, and average rain depth per storm. These maps provided the opportunity to examine the rainfall characteristics

of Korea from a perspective that typical rainfall statistical analysis cannot provide. In addition, we suggested some important notes to be considered while estimating the parameters of the MBLRP model and regionalizing them in this study. First, the multimodality of the MBLRP model should be carefully addressed in regionalization studies of Poisson cluster rainfall models. Otherwise, the parameters estimated at different modes of the parameter space cause problems during spatial interpolation. In this study, we suggested the technique of “repetitive crossvalidation” in order to resolve this issue. In this approach, the search parameter space was tightened up through successive cross-validations. The parameter maps showed significantly smoother spatial tendency as the levels of the repetitive crossvalidation were increased. Second, we suggested in this study that the MBLRP model may not be adequate for modeling precipitation during the snowy season in Korea, which has a small precipitation depth and a low lag-1 autocorrelation close to 0. The discrepancy in the rainfall statistics between the observed and simulated rainfall time series was significant for the snowy season. Lastly, we discovered in this study that trivial rainfall depth values, such as those less than 0.1 mm, can greatly affect rainfall statistics, particularly the probability of zero rainfall, which subsequently exerted negative impacts on the parameter estimation process. Therefore, the filtering threshold value should be selected carefully based on the type of hydrologic modeling study being conducted. Acknowledgement This research was supported through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT (Grant ID: NRF-2013R1A1A1011676). References Bathurst, J.C., Moretti, G., El-Hames, A., Moaven-Hashemi, A., Burton, A., 2005. Scenario modeling of basin-scale, shallow landslide sediment yield, Valsassina, Italian Southern Alps. Nat. Hazards ?Earth Syst. Sci. 5, 189e202. Bo, Z., Islam, S., Eltahir, E.A.B., 1994. Aggregation-disaggregation properties of a stochastic rainfall model. Water Res. Res. 30 (12), 3423e3435. Cho, H., Kim, D., Olivera, F., Guikema, S., 2011. Enhanced speciation in particle swarm optimization for multi-modal problems. Eur. J. Operat. Res. 213, 15e23. Cowpertwait, P.S.P., 1995. A generalized spatialetemporal model of rainfall based on a clustered point process. Proc. Royal Soc. London 450, 163e175. Cowpertwait, P.S.P., O'Connell, P.E., Metclafe, A.V., Mawdsley, J.A., 1996. Stochastic point process modeling of rainfall, II, regionalization and disaggregation. J. Hydrol. 175 (1e4), 47e65. Fowler, H.J., Kilsby, C.G., O'Connell, P.E., Burton, A., 2005. A weather-type conditioned multi-site stochastic rainfall model for the generation of scenarios of climatic variability and change. J. Hydrol. 308 (1e4), 50e66. Gregory, J.M., Wigley, T.M.L., Jones, P.D., 1992. Determining and interpreting the order of a 2-state MarkoveChain e application to models of daily precipitation. Water Res. Res. 28 (5), 1443e1446. Gyasi-Agyei, Y., 1999. Identification of regional parameters of a stochastic model for rainfall disaggregation. Journal of Hydrology 223 (3e4), 148e163.

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Please cite this article in press as: Kim, D., et al., Regionalization of the Modified BartletteLewis rectangular pulse stochastic rainfall model across the Korean Peninsula, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.10.004