A comparison of three methods for downscaling daily precipitation in the Punjab region

Transcription

1 HYDROLOGICAL PROCESSES Hydrol. Process. 25, (2011) Published online 6 April 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: /hyp.8083 A comparison of three methods for downscaling daily precipitation in the Punjab region Deepashree Raje 1 * and P. P. Mujumdar 2,3 1 Center for Climate Change Research, Indian Institute of Tropical Meteorology, Pune , India 2 Department of Civil Engineering, Indian Institute of Science, Bangalore , India 3 Divecha Center for Climate Change, Indian Institute of Science, Bangalore , India Abstract: Many downscaling techniques have been developed in the past few years for projection of station-scale hydrological variables from large-scale atmospheric variables simulated by general circulation models (GCMs) to assess the hydrological impacts of climate change. This article compares the performances of three downscaling methods, viz. conditional random field (CRF), K-nearest neighbour (KNN) and support vector machine (SVM) methods in downscaling precipitation in the Punjab region of India, belonging to the monsoon regime. The CRF model is a recently developed method for downscaling hydrological variables in a probabilistic framework, while the SVM model is a popular machine learning tool useful in terms of its ability to generalize and capture nonlinear relationships between predictors and predictand. The KNN model is an analogue-type method that queries days similar to a given feature vector from the training data and classifies future days by random sampling from a weighted set of K closest training examples. The models are applied for downscaling monsoon (June to September) daily precipitation at six locations in Punjab. Model performances with respect to reproduction of various statistics such as dry and wet spell length distributions, daily rainfall distribution, and intersite correlations are examined. It is found that the CRF and KNN models perform slightly better than the SVM model in reproducing most daily rainfall statistics. These models are then used to project future precipitation at the six locations. Output from the Canadian global climate model (CGCM3) GCM for three scenarios, viz. A1B, A2, and B1 is used for projection of future precipitation. The projections show a change in probability density functions of daily rainfall amount and changes in the wet and dry spell distributions of daily precipitation. Copyright 2011 John Wiley & Sons, Ltd. KEY WORDS downscaling; comparison; precipitation; Punjab; monsoon Received 17 September 2010; Accepted 24 February 2011 INTRODUCTION Many studies have been conducted in the past decade on use of various downscaling techniques for projection of hydrological variables at river basin scale under climate change impacts in different parts of the globe. Climate impact studies typically use general circulation models (GCMs) for simulating current and future time series of climate variables for the entire globe. The models account for various internal and external atmospheric forcings, including various scenarios for increases in greenhouse gases and socio-economic changes. However, the resolution provided by typical GCMs is inadequate for the purpose of hydrological impact assessment. In addition, GCMs have limited skill in resolving subgridscale features such as clouds, convection, and topography (Xu, 1999). Hence, large-scale climatic variables are projected or downscaled to a finer resolution station-scale variable. In statistical downscaling (SD), a statistical or empirical relationship is derived between the large-scale * Correspondence to: Deepashree Raje, Center for Climate Change Research, Indian Institute of Tropical Meteorology, Pune , India. deepashree@tropmet.res.in atmospheric variables simulated by the GCM (called predictors) and the regional climate variables (called predictands). SD uses area-average climate variables from GCMs (such as mean sea level pressure, air temperature or specific humidity) and relates them to point-scale variables, such as station precipitation or streamflow. Implicit assumptions in SD are (Wilby and Wigley, 1997; von Storch et al., 2000) that (1) the predictors are variables of relevance and are realistically modelled by the GCM, (2) the predictors employed fully represent the climate change signal, and (3) the relationship is valid under altered climate conditions. Wilby and Wigley (1997) presented a comprehensive study that compared empirical transfer functions, weather generators, and circulation classification schemes over North America. They downscaled daily precipitation for six locations over North America, spanning different climate regimes, and compared 14 measures of skill, emphasizing daily statistics such as wet and dry spell length, 95th percentile values, wet-wet day probabilities, and several measures of standard deviation. More recently, Frías et al. (2006) studied two SD methods, namely canonical correlation analysis and an analogue search method, to estimate past and future precipitation in the Iberian and Scandinavian Peninsulas, using climate records from a coupled Copyright 2011 John Wiley & Sons, Ltd.

2 3576 D. RAJE AND P. P. MUJUMDAR climate model simulation. They found that the differences in downscaling winter precipitation arose largely due to the use of different predictors than downscaling method. Haylock et al. (2006) compared six statistical and two dynamical downscaling models with regard to their ability to downscale seven seasonal indices of heavy precipitation for two station networks in northwest and southeast England. They found that models based on nonlinear artificial neural networks were best at modelling the interannual variability of the indices; however, they underestimated extremes. Segui et al. (2010) compared three SD techniques: a weather regimes method, a quantile-mapping method, and an anomaly method, to force a distributed hydrological model to simulate the French Mediterranean basins. Their study showed that the three methods produced similar anomalies of the mean annual precipitation, but there were important differences, mainly in terms of spatial patterns. However, there are only a few studies which compare the relative performance of downscaling methods, especially for regions in the monsoon regime. Various studies demonstrate that the GCM simulations for the 20th and 21st centuries do not represent the real temporal evolution of large-scale weather states in the past (cf Widmann and Bretherton, 2000; Maraun et al, 2010). Hence, a time series of the downscaled variable cannot be used directly for performance evaluation, but statistical characteristics of the downscaled variable can be compared. This article evaluates the performances of three downscaling models in downscaling precipitation with respect to reproduction of various statistics such as means, standard deviations, dry and wet spells, and cumulative distribution functions (CDFs), and presents a comparison of results. In the conditional random field (CRF) model (Raje and Mujumdar, 2009), the precipitation sequence and atmospheric variables are represented as a CRF to downscale to precipitation in a probabilistic framework. As it models the conditional distribution of the precipitation sequence given the observed climate data, the method does not need assumptions about distributions of climate variables. The CRF model can also use high-dimensional features derived from observed climatic data and can hence handle complex dependencies in time between climate variables and downscaled variable. Support vector machine (SVM) uses the structural risk minimization (SRM) principle that minimizes an upper bound on the expected risk (Tripathi et al., 2006). The SVM regression aims to find a function that has at most ε deviation from the actually obtained target precipitation for all the training data and is at the same time as flat as possible (Smola and Scholkopf, 2004). The SVM algorithm depends only on dot products between input data patterns (climate variables). Using a kernel function mapping on training data to make the algorithm nonlinear, it is sufficient to know the kernel function rather than the mapped function explicitly, and a linear solution in the mapped higher dimensional feature space corresponds to a nonlinear solution in the original lower dimensional input space. These features lead to a good generalization ability of the SVM in the downscaling problem, in capturing nonlinear regression relationships between predictors and predictand. The K-nearest neighbour (KNN) algorithm is an analogue-type method that queries days similar to a given feature vector from the training data. The model identifies a subset of K days similar to the feature day. These K days are then weighted using a bisquare weight function and randomly sampled to generate hydrological projections (Gangopadhyay et al., 2005). The models are used to downscale precipitation at six sites in the Punjab region in India. Output from the Canadian global climate model (CGCM3) GCM for three scenarios is used for projection of future precipitation. Projected results from the CRF method are compared to projections from the KNN method to examine hydrological implications for the region. This article is organized as follows. The Section Data and Input to the Models gives details of the case study region, data used, and inputs to the models. The Section Downscaling Methods provides an introduction to the three downscaling models and application of the models for downscaling. A comparison of model performances and future projections are presented in the Section Results and Discussion and the conclusions are presented in the Section Concluding Remarks. DATA AND INPUT TO THE MODELS The Punjab state of India covers an area of km 2 with a population of around 24 million (Census of India, 2001). Five rivers, tributaries of the Indus River, namely the Beas, Chenab, Jhelum, Ravi, and Sutlej, flow through the region, which is now divided between India and Pakistan. Figure 1 shows the Punjab state physical map with rivers and topography. Most of Punjab is a fertile, alluvial plain with an extensive irrigation canal system. The southwest of the state is semi-arid, eventually merging into the Thar Desert. The Siwalik Hills extend along the northeastern part of the state at the foot of the Himalayas. The region is ideal for wheat growing; rice, sugar cane, fruits, and vegetables are also grown (Lall, 2009). There are three seasons, namely summer, monsoon, and winter. The rainy season or monsoon lasts for 4 months from June to September, during which most of the annual rainfall occurs. Daily precipitation in the Punjab region is downscaled in this study at six locations. Only monsoon rainfall is downscaled because there is negligible rainfall during the remaining part of the year. Appropriate choice of predictor variables for downscaling is important in the downscaling process. Ideally, predictor variables chosen should be reliably simulated by GCMs, readily available from archives of GCM outputs, and strongly correlated with the surface variables of interest (Wilby et al., 1999; Wetterhall et al., 2005). The mechanism of monsoon precipitation is linked to the location of low pressure systems, the corresponding circulation pattern, air mass transport, and atmospheric

3 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3577 Figure 1. Physical map of Punjab state water content. Hence, local-scale precipitation can be related to atmospheric circulation or pressure patterns and wind velocities, specific humidity, and temperature. In this study, large-scale atmospheric predictors chosen were mean sea level pressure, surface-specific humidity, specific humidity at 850 hpa, surface temperature at 2 m, surface U-wind (zonal), and surface V-wind (meridional). These were based on an initial screening of predictor variables through a correlation analysis with daily monsoon precipitation at one of the downscaling locations. Predictor data over an area from 25 N to35 N and from 70 E to 80 E are obtained from the National Center for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data for years (Kalnay et al., 1996). Gridded daily precipitation data (at 1 by 1 resolution, interpolated from station data) from 74Ð5 E to76ð5 E and from 30Ð5 N to 31Ð5 N for years (Rajeevan et al., 2006a,b) were obtained from the India Meteorological Department (IMD) and used as predictand for training the models. Rain-gauge data in India are collected and maintained by the IMD. The 1 by 1 gridded daily rainfall data set was released by IMD for based on these data (Rajeevan et al., 2006a,b). This data set was constructed by interpolating station data using Shepard s interpolation scheme, which is essentially an inverse-distance weighting scheme. Suprit and Shankar (2008) tested three gridded data sets including the Climate Research Unit s (CRU, University of East Anglia) CRU TS 2Ð0 data set, IMD, and the Tropical Rainfall Measuring Mission (TRMM, 2006) data set on the west coast region of India and found that the IMD data set gives the best discharge estimates when used for forcing a hydrological model for the Mandovi River basin in west India. However, they found that the discharge was underestimated by the IMD rainfall, the error being larger in years with high discharge and therefore high rainfall. Thus, the IMD data were unable to capture the large interannual variability inherent in the west-coast monsoon rainfall and the simulated discharge had a much lower variance than observed. This was found attributable to the coarse resolution of the data set, for which the interpolation algorithm ignores the effect of elevation and the need to ensure continuity which led to the use of fewer rain gauges than were actually available for some basins. Thus, for analysis of extremes or flood risk, it would be preferable to use station or gauge data where available. However, in the absence of station data for the region, IMD data were considered adequate for the purpose of this study. Daily atmospheric variables from the CGCM3 model developed by the Canadian Centre for Climate Modeling and Analysis for years for three scenarios, viz. A1B, A2, and B1 were used for prediction (Nakicenovic et al., 2000). The grid locations where downscaling to daily precipitation is performed are shown in Table I. Figure 2 shows the locations of NCEP and GCM grid points, at which large-scale climate variables are simulated. Large-scale atmospheric data have to be preprocessed before using them for training the downscaling models. The locations of NCEP/NCAR grid points and CGCM3 grid points do not match; hence interpolation is necessary before using GCM outputs for prediction. A linear Table I. Locations for downscaling precipitation Location Latitude ( N) Longitude ( E)

4 3578 D. RAJE AND P. P. MUJUMDAR presented in Table II. As the components are obtained by projection in the principal directions (in addition to the period being slightly different from baseline), it is seen that the means are slightly different for NCEP and CGCM3 data. The following sections explain the terminology, training, and testing of the downscaling models. Figure 2. NCEP and CGCM3 grid point locations interpolation is performed to obtain CGCM3 variable values at the respective NCEP grid points. Standardization (Wilby et al., 2004) is then performed to reduce systematic biases in the mean and variances of GCM predictors relative to the observations or NCEP/NCAR data. This involves subtraction of mean and division by standard deviation of the predictor variable for a predefined baseline period for both NCEP and GCM outputs for the 20th century. The period is used as a baseline in this study. Each standardized variable is further mean-centred with zero mean and unit standard deviation by applying normalization. The number of predictor variables from all NCEP grid points is very large and training the models with these would be computationally very expensive. Principal component analysis (PCA) is hence applied to the predictor variables to reduce and effectively summarize the information from all variables from selected grid points. The first five PCs accounting for more than 70% of the variance were taken as predictors to train the model. The coefficients of the PCA (empirical orthogonal functions) were applied to the standardized and normalized (with respect to NCEP) CGCM3 data to get their projections in the principal directions. Further details of the procedures used for preprocessing predictors are given in Raje and Mujumdar (2009). The mean values for the period for the first five principal components (PCs), which are used as predictors for the NCEP and CGCM3 20th century run, are DOWNSCALING METHODS CRF model CRFs belong to a class of models called undirected graphical models (Lafferty et al., 2001). CRFs have been successfully applied to a variety of domains such as text processing, computer vision, image processing and bioinformatics, protein structure prediction, motion tracking, voice recognition, activity recognition, and information extraction (Kumar and Hebert, 2003; Liao et al, 2007; Vail et al., 2007). In the CRF-downscaling model (Raje and Mujumdar, 2009), daily precipitation sequence at a site (y D <y 1,y 2,...,y T >) and the observed daily atmospheric variable sequence (x D <x 1,x 2,...,x T >) for time t D 1toT are represented as a linear-chain CRF. The conditional distribution of the precipitation sequence y is given by p yjx D 1 { T } K Z x exp k f k y t,y t 1, x 1 td1 KD1 where Z x D { T } exp K td1 KD1 kf k y t,y t 1, x is y a normalization function summed over all possible output sequences y D <y 1,y 2,...,y T >, Df k g2< K is a parameter vector, and f k y t,y t 1, x K kd1 is a set of realvalued feature functions for features k D 1toK, defined on pairs of consecutive precipitation values and the entire sequence of atmospheric data as shown in Figure 3. Each feature function can depend on observations from any time step. The feature functions are fixed beforehand y 1 y 2 y t-1 y t y T x=x 1,x 2,..x T f 1 (y t-1,y t,x) f 2 (y t-1,y t,x) f K (y t-1,y t,x) Figure 3. Feature functions defined on a linear-chain CRF Table II. Comparison of mean values of predictors from NCEP and CGCM3 20th century data for Predictor variable 1 Predictor variable 2 Predictor variable 3 Predictor variable 4 Predictor variable 5 NCEP ( ) mean Ð CGCM3 20C3M ( ) mean Ð

5 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3579 and are constructed to express some characteristic of the actual (observed) distribution which should hold in the model distribution. The general form for their definition is f i y, x Dhy D iig x and f ij y t,y t 1 x D hy t D iihy t 1 D jig x where hði is the indicator function which is equal to 1 only when the given condition is true and 0 otherwise, and g(x) is a function (observation feature) of the observed climate variables. There are many different types of observation features, including the features that pick out specific dimensions of the input data. In general, observation features may be overlapping and non-independent, and may be complex functions that involve multiple parts of the observations. In the present CRF model, different feature functions that have been used include intercept, transition, raw observation, product, sum, difference, and threshold features. For the case study, training of the CRF-downscaling model was performed using maximum likelihood (Raje and Mujumdar, 2009). Here, the parameter values were chosen to maximize the regularized log likelihood l D f k g, which imposes a penalty on weight vectors whose norm is too large, given as l D ( N log p ( y i jx i ) D id1 y i t,y i t 1, x i ) N id1 T td1 kd1 K k f k N log Z ( x i ) id1 k kd1 2 k where each x i Dfx i 1, x i 2,...,x i T g is a sequence of inputs for time t D 1toT and y i Dfy i 1,y i 2,...,y i T g is a sequence of the desired predictions, i is the data set number, and N is the number of independent identically distributed (iid) training data sets D Dfx i, y i g N id1 available. 2 is a regularization parameter that determines the strength of the penalty for weights whose norm is too large. It is a free parameter and the accuracy of the model has been found to be insensitive to its chosen value for changes up to a factor of 10 (Sutton and McCallum, 2006). The CRF model was trained using sets of PCs of NCEP atmospheric variables and IMD daily precipitation data to estimate optimum parameters. Precipitation was classified into ten classes including one class for zero precipitation, for numerical computation. A limited-memory version of BFGS (lbfgs) (Nocedal and Wright, 1999) was used for optimization. For prediction, the maximum a posteriori or most probable precipitation sequence is computed for a given observation (climate) sequence using the Viterbi algorithm (Rabiner, 1989). In practice, the unnormalized probabilities computed in the forward backward algorithm used for computations of likelihood and gradient, and the Viterbi algorithm, are very large, which overflow the computer. Hence, the implementation of the algorithms is in log space. SVM model The foundations of SVM have been developed by Vapnik (1995), initially for optical character recognition. SVM learning has now evolved into an active area of research. The SVM model has been used (Tripathi et al., 2006; Anandhi et al., 2008) as a downscaling technique for predicting subdivisional precipitation of different regions in India. SVM uses the SRM principle which minimizes an upper bound on the expected risk, as opposed to empirical risk minimization that minimizes the error on the training data. The SVM regression aims to find a function f x that has at most ε deviation from the actually obtained targets y i for all the training data and is at the same time as flat as possible (Smola and Scholkopf, 2004). The SV formulation of the optimization problem and construction of the Langrange function lead to conditions where the SV algorithm depends only on dot products between input data patterns x i. To make the algorithm nonlinear, the training data are mapped by a function, 8, to a feature space. Using a kernel function mapping, it is sufficient to know the kernel function rather than the mapped function 8 explicitly. The Karush Kuhn Tucker (KKT) conditions also ensure that all samples lying within the ε-tube (or with lesser deviation than ε) are redundant in construction of the fitted function f x. These features lead to a good generalization ability of the SVM in the downscaling problem, in capturing nonlinear regression relationships between predictors and predictand. Given training data f x 1,y 1, x 2,y 2... x n,y n ; x 2< k,y 2<gfor time t D 1tonwhere k is the number of predictor variables, the support vector regression equation can be given as (Smola, 1996) y D f x D n w i ð K x i,x j C b id1 3 where K x i,x j is the inner product kernel function and w i are the corresponding weights used in the regression. b is a constant called the bias. Figure 4 shows the architecture of an SVM. Here support vectors are those training data points lying on the surface of the ε-deviation from fitted function tube, which do not vanish as a result of the KKT optimization conditions and are used to determine prediction for a test data point or test vector. There are several possibilities for the choice of kernel function, including linear, polynomial, sigmoid, splines, and radial basis function (RBF). The kernel function used has to be defined in accordance with Mercer s theorem, i.e. functions should correspond to a dot product in some Output: Σ w i k(x,x i ) + b w 1 w 2 w l K(x,x 1 ) K(x,x 2 ) K(x,x l ) x 1 x 2 x l x Test vector Figure 4. Architecture of the SV regression machine Support vectors

6 3580 D. RAJE AND P. P. MUJUMDAR other feature space. The RBF kernels have localized and finite responses across the entire range of predictors. Moreover, the RBF is computationally simpler than a polynomial kernel, which has more parameters. Hence, in this study the RBF kernel is used, which is given as ( K x i,x j D exp kx i x j k 2 ) 4 where is the width of the RBF kernel, which can be adjusted to control the expressivity of RBF. This width controls the smoothness of the derived function (Smola et al., 1998). A large kernel width acts as a low-pass filter in frequency domain, attenuating higher order frequencies and thus resulting in a smooth function, while an RBF kernel with small kernel width retains most of the higher order frequencies leading to an approximation with a complex function (Smola, 1996). The least square support vector machine (LS-SVM) has been used in this study to downscale precipitation. The LS-SVM provides a computational advantage over standard SVM (Suykens and Vandewalle, 1999). LS-SVM is a least squares version of SVM, where the solution to the optimization problem is found by solving a set of linear equations instead of a convex quadratic programming for classical SVMs. The LS-SVM optimization problem for function estimation is formulated by minimizing the cost function given as w,e D 1 2 wt w C 1 n 2 C 5 subject to the equality constraints id1 y i Oy i D e i for i D 1,...,n 6 where is the cost function, w are the parameters of the regression, C is a positive real constant which serves as a penalty parameter for large model errors and Oy i is the actual model output. Developing LS-SVM with RBF kernel involves selection of RBF kernel width and the penalty parameter C. The linear correlation coefficient, or R-value, obtained is used as an index to assess the performance of the model for fixing these parameters. In this study, grid search procedure (Gestel et al., 2004) is used to find the optimum range for the parameters C and, giving highest R-values for testing. The range of values used in this study was C D 1 to 500 and D 1 to 100, with the final values used in the model being C D 100 and D 10. As the regression equation gives a continuous range of values for precipitation, daily precipitation amounts computed as below 1 mm were taken to be dry days. KNN model The KNN algorithm is described for use in a stochastic weather generator by Lall and Sharma (1996), Rajagopalan and Lall (1998), Buishand and Brandsma (2001), and Yates et al. (2003). Gangopadhyay et al. e 2 i (2005) used the KNN model for downscaling local-scale temperature and precipitation in the United States. The KNN algorithm searches for analogues of a feature vector based on similarity criteria in the observed time series. The steps involved in prediction are as follows, slightly modified from the methodology of Gangopadhyay et al. (2005): 1. The climate variables for the chosen feature day are projected on principal directions obtained from PCA of training climate data. A feature vector is compiled of these projections (x t ) of climate variables projected on the PC space of training data, referred to as predictor variables henceforth. The feature vector ( EF t f ) consists of values for all predictor variables for the feature day t, for which analogues in terms of KNNs are sought. EF t f D [xt 1 xt 2,...,xt n ] 7 where xj t is the jth predictor variable on day t and n is the total number of predictors (equal to 5, the number of PCs retained). 2. For each time, the Euclidean distance between the feature vector and the PCs of training data from time i D 1, 2,...,Tis computed. The distance d i of feature day t from training day i is given as 1/2 n d i D xj t pi j 2 8 jd1 3. The distances d i are sorted in ascending order and only the first K neighbours are retained. The choice of K is based on the square root of all possible candidates, i.e. K D p T (Rajagopalan and Lall, 1998; Yates et al., 2003). 4. A weight w i (0 <w i < 1) is assigned to each of the K neighbours using a bisquare weight function (Huber, 2003) as [ ( ) ] 2 2 di 1 w i D [ K 1 id1 d K ( di d K ) 2 ] 2 9 where d K is the sorted distance of neighbour K. 5. A neighbour is randomly selected from the K neighbours as an analogue for feature day t. For this, a uniform random number u ¾ U[0,1] is first generated. If u ½ w 1, then the day corresponding to distance d 1 is selected. If u w K, then the day corresponding to distance d K is selected. For w 1 <u<w K, the day corresponding to that d i is selected for which u is closest to w i. 6. The neighbour day for each feature day obtained using the KNN algorithm was used to select the dailyobserved precipitation value for that location. This constituted the downscaled precipitation series for each of the locations used in this study.

7 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3581 Table III. Statistics of model-computed versus observed daily monsoon rainfall in independent testing for years Location Mean (observed) Mean (CRF computed) Mean (KNN computed) Mean (SVM computed) S.D. (observed) S.D. (CRF computed) S.D. (KNN computed) S.D. (SVM computed) Ð82 7Ð59 8Ð75 8Ð Ð32 7Ð11 11Ð55 10Ð Ð80 11Ð54 14Ð24 12Ð Ð90 7Ð92 11Ð38 10Ð Ð86 11Ð40 11Ð30 10Ð Ð66 11Ð65 10Ð66 10Ð19 S.D., standard deviation. Figure 5. Percentage error in computed means and standard deviations in testing for years , using the three downscaling methods RESULTS AND DISCUSSION All downscaling methods were used with the same set of predictors for training, namely, the first five PCs of the NCEP climate variables: mean sea level pressure, surface-specific humidity, specific humidity at 850 hpa, surface temperature at 2m, surface U-wind (eastwards), and surface V-wind (northwards). In the SVM, they are directly used in fitting the regression equation, while in the CRF model features are defined (such as their direct values, differences, etc.) based on these variables. In the KNN model they are used to identify nearest neighbours for a given day. In this study, NCEP data from the first 30 years ( ) are chosen for calibrating the models and the remaining NCEP data ( ) are used for validation. The three models were tested for the period for reproduction of various daily precipitation statistics. Table III shows a comparison of computed and observed statistics of daily rainfall at the six locations for the testing period, using each of the three models. Figure 5 shows these results as a percentage error in computed means and standard deviations in testing for years , using the three downscaling methods. It is seen that the SVM model overestimates mean daily precipitation significantly at four of six locations and the KNN model underestimates the mean at three of six locations and overestimates it at one location. The CRF model shows the least error in the mean of the three methods. The standard deviation is consistently underestimated by the CRF and SVM models, while the KNN model also underestimates it in three of six locations. Figure 6 shows a box-plot comparison of modelcomputed versus observed daily monsoon rainfall for the six locations. The monsoon season is taken from June to September, consisting of 122 days per year. The length of the whiskers is 1Ð5 times the interquartile range, and the outliers are shown as red crosses. Mean precipitation at each location is shown as a black cross mark. It is seen that all models perform reasonably well in reproducing the mean, median, and interquartile ranges for the data, with the exception of location 1. This location shows very low (almost zero) median rainfall, and it is seen that only the KNN model reproduces the spread of data well here. In this context, the choice and applicability of a downscaling model would appear to be dependent on the rainfall characteristics of a region. For a location which is characterized by a higher number of dry days, an analogue approach like KNN gives better results. It is not clear whether this is a fundamental weakness of the other methods. If so, it also remains to be seen whether it is a better option to model the occurrence and amount of precipitation separately, as in Markov Chain and generalized linear model methods. The performance of the models for precipitation amount (without including dry days) is hence tested separately and is shown in Figure 7. It is seen that the CRF and KNN models are able to reproduce the mean daily precipitation amounts well at most locations (including dryer locations 1 and 2),

8 3582 D. RAJE AND P. P. MUJUMDAR 120 Location Location Location Location Location Location Daily rainfall Obs CRF SVM KNN 0 Obs CRF SVM KNN 0 Obs CRF SVM KNN 0 Obs CRF SVM KNN 0 Obs CRF SVM KNN 0 Obs CRF SVM KNN Figure 6. Box-plots of model-computed versus observed daily monsoon rainfall for the six locations, in testing for years Mean precipitation is shown by a cross. Outliers are shown as red crosses Figure 7. Precipitation amount statistics for wet days in testing for years while the SVM model overestimates mean precipitation at most locations. Standard deviation of precipitation amount is usually underestimated. Figure 8 shows a comparison of the CDFs for observed versus computed rainfall at three locations. It is seen that the KNN and CRF models give the best fits to the CDF of rainfall in independent testing, whereas the SVM model is unable to reproduce it well. The fits are similar at the remaining two locations. To check the performance of the methods in reproduction of extreme precipitation values, a plot of the quantiles of observed versus computed distributions against each other was used. Figure 9 shows the quantile-quantile plot in log scale at all locations for years At all locations, it is seen that the CRF and KNN models follow the 45 line better than the SVM model for amounts above 1 mm rainfall, suggesting the CRF and KNN models are closer to the observed rainfall distribution. There are outliers for all methods for low precipitation amounts, but not for high amounts, which shows that high amounts are better simulated than low amounts. A general observation is that for all locations, the KNN-computed distribution is more dispersed than the observed distribution, while the SVMand CNN-computed distributions are less dispersed than the observed distribution. In the case of the CRF model, discretization of precipitation means that a less dispersed distribution below 1 mm precipitation is unavoidable. Figure 10 shows the probability mass functions (PMFs) of dry and wet spell lengths for monsoon rainfall at two locations and a comparison with model-computed PMFs for the testing years The other locations show similar fits for the PMFs. It can be seen from the figure that CRF and SVM models are able to model the PMF of wet spell length reasonably well, but not the KNN model, whereas CRF and KNN are able to model dry spell length PMF well, but not the SVM model. It is likely that the SVM and KNN models underperform in these statistics, as they do not consider temporal climate variable changes, whereas these are incorporated in the CRF model through feature functions. The statistics for mean wet and dry spell lengths computed at all the locations are presented in Tables IV and V. It is seen that the SVM model underestimates the mean dry spell length, whereas the CRF model underestimates the mean wet spell length, while the KNN model performs well for both means. A potential reason that the KNN model performs well is that the distributions of climatic variables may be sharply different for wet versus dry days. Figure 11 shows a scatter plot of intersite correlations computed by the models versus actual correlations between the sites, for testing years KNN and SVM models are seen to be able to capture the intersite correlations better than the CRF model, although there is no feature in any of the models by which intersite correlations are explicitly preserved. To evaluate biases for the current climatology, the downscaling models were used to project precipitation

9 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3583 Figure 8. CDFs of model-computed versus observed daily monsoon rainfall at four locations in independent testing for years Probability of dry days is the point where the CDFs intersect the y-axis from the 20th century simulations of CGCM3 model (20 CM3). Table VI presents downscaled means for the three downscaling methodologies using 20C3M simulations versus observed means at all locations. It is seen that there exists a bias in the downscaled precipitation values, which is largest for the SVM results and least for KNN results, and these biases need to be suitably handled in projection of results. For this purpose, the downscaling models could be fit to the control run of CGCM3 directly and then used for future projections using this same relation, as suggested by the reviewer, which would however ignore potential useful forcings and feedbacks captured from actual NCEP reanalysis data. Another way would be to perform a correction in projections using the bias information, such as through a scaling factor or additively. Overall, it is seen from testing results that most statistics are consistently well reproduced by the CRF and KNN models. Various statistics including means and distributions of daily rainfall, interquartile range, quantile-quantile mapping, and dry and wet spell length distributions are reliably reproduced by the CRF and KNN models but not as well by the SVM. As the CRF and KNN models were found to perform relatively better than SVM, these two models were used for projection of daily rainfall at the six locations, using output from the CGCM3 GCM for the chosen three scenarios for years The projected results were also corrected for mean rainfall bias exhibited for years , as projected in Table VI, by adding the bias amount to the model results. Table VII shows a comparison of current and future projections for mean daily precipitation at the six locations, showing raw as well as bias-corrected results. Figure 12(a) and (b) shows this as changes in future mean daily precipitation, depicted as a bar chart for the six locations for raw and bias-corrected projections. It is seen that after bias correction, many projections, especially for the CRF model, show a reversal in sign of change. This is due to high bias as compared to mean precipitation value. The projection results show a diverse picture for the raw projections. Figure 12(a) shows an increase in mean daily precipitation for the majority of the combinations of downscaling method and scenarios for locations 1, 3, and 6, a decrease for locations 4 and 5, and conflicting projections for location 2. The KNN model shows an increase in mean daily precipitation at

10 3584 D. RAJE AND P. P. MUJUMDAR 10 3 Quantile-quantile plot of daily rainfall for years for location Quantile-quantile plot of daily rainfall for years forlocation 2 Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Quantile-quantile plot of daily rainfall for years for location Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Quantile-quantile plot of daily rainfall for years for location Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Quantile-quantile plot of daily rainfall for years for location Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Quantile-quantile plot of daily rainfall for years for location Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Computed Daily Rainfall CRF Computed SVM Computed KNN Computed Observed Daily Rainfall Figure 9. Quantile-quantile plot on log scale of daily rainfall obtained in testing for years using the three downscaling methods most locations for all scenarios, while the CRF model shows a decrease in mean daily precipitation for four of the six locations. However Figure 12(b) shows that after bias correction, projection results show a consistent increase in mean daily precipitation at all locations for the majority of the combinations of downscaling method and scenarios, except a decrease projected for B1 scenario by the CRF model for locations 1, 3, and 4. Figure 13 shows a comparison of kernel density estimates of the PDF of current and projected daily precipitation amount for two locations. A comparison of the present and future predicted PDFs shows a change in shape for years The CRF model shows an increase in lower precipitation amounts and a decrease in medium and higher amounts, while the KNN model projects an increase in medium precipitation amounts, for the chosen GCM. Hence, moderately high precipitation events are projected to increase by the KNN model while they are projected to decrease by the CRF model. The other locations show similar results for changes in the shapes of PDFs. It is also seen that the KNN model projections vary only slightly with choice of scenario, as seen by the almost overlapping dotted lines in the graph. The result that the type of downscaling methodology results in larger differences than the choice of a scenario is unexpected given the large differences in

11 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3585 Figure 10. PMFs of dry and wet spell lengths for monsoon rainfall at two locations for years Table IV. Mean dry spell length (days) for years Table V. Mean wet spell length (days) for years Location Observed CRF computed KNN computed SVM computed Location Observed CRF computed KNN computed SVM computed 1 7Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð greenhouse emissions, etc. projected by each scenario. The KNN model is only sensitive to changes in distributions of climatic variables hence if climate change results in similar joint climatic distributions for two scenarios, the downscaled rainfall results will have similar distributions. On the other hand, the CRF model is also sensitive to the temporal changes in climatic variables, as it uses temporal features as well as point values in defining rainfall distribution. This may make the model very sensitive to small changes between scenarios, leading to different projections. Figure 14 shows projections for the PMFs of dry and wet spell lengths using the two models at location 1. It is seen that the CRF model shows an increase, but KNN model projects a decrease, in probability of short duration wet spells (<2 days), with both models showing an increase in probability of longer duration wet spells (>7 days). Similarly, short-duration dry spells are also projected to increase by the CRF model but a decrease is projected by the KNN model. Other locations show similar general results for these PMFs. These results show, unusually, that differences arising

12 3586 D. RAJE AND P. P. MUJUMDAR Figure 11. Daily rainfall intersite correlations, model-computed versus observed in independent testing for years There are 15 computed values for each model from different scenarios are small as compared to those from the downscaling methods used. The reasons for these discrepancies in projection of wet/dry spells, as discussed earlier, are likely to be methodological rather than physical. The three downscaling methods chosen for this study belong to three different types of downscaling approaches, namely transfer function (SVM), stochastic (CRF), and analogue (KNN). The SVM regression can capture nonlinear regression relationships between variables (Vapnik 1995, 1998; Haykin, 2003). The flexibility of the SVM is provided by the use of kernel functions that implicitly map the data to a higher, possibly infinite, dimensional space. SVMs have previously been widely used with good performance in pattern recognition, object recognition, and many other domains, as also in downscaling (Tripathi et al., 2006; Anandhi et al., 2008). However it is essentially a single-site regressionbased approach, with site-specific limitations in capturing precipitation amounts at comparatively drier locations. In independent testing it is seen that the SVM may be characterized by overfitting, due to which it shows worse fits to the observed distributions of daily rainfall as compared to CRF and KNN approaches. The fact that SVM gives a good performance in training and poor in testing may not only be a sign of overfitting but also of badly chosen hyperparameters. The CRF model is a stochastic downscaling model which shows comparable performance for downscaling precipitation. Notably, it does not need assumptions about distributions of climate variables as it directly models the conditional precipitation distribution and can use high-dimensional features derived from observed climatic data for probabilistic inference. Hence, it is able to use lag-1 precipitation observations and up to lag-3 climatic variable observations have been defined (as features) for the CRF model used in this study. The basic distribution for the CRF model used Table VI. Comparison of bias for observed versus CGCM3 downscaled mean daily precipitation for years Location Observed ( ) CRF downscaled 20C3M ( ) KNN downscaled 20C3M ( ) SVM downscaled 20C3M ( ) Table VII. Comparison of current and future projections for mean daily precipitation at six locations in Punjab raw and bias-corrected projections Location Current ( ) CRF A1B ( ) KNN A1B ( ) CRF A2 ( ) KNN A2 ( ) CRF B1 ( ) KNN B1 ( ) Raw projections 1 2Ð41 2Ð93 2Ð76 3Ð39 1Ð75 2Ð72 2Ð13 2 3Ð30 2Ð56 3Ð67 2Ð36 3Ð51 2Ð42 3Ð33 3 5Ð16 5Ð64 6Ð57 5Ð62 6Ð64 5Ð03 5Ð39 4 4Ð38 3Ð46 4Ð49 3Ð66 4Ð35 3Ð34 3Ð89 5 5Ð28 4Ð86 5Ð23 4Ð74 4Ð86 4Ð58 4Ð34 6 7Ð71 9Ð54 8Ð98 9Ð24 8Ð46 9Ð39 8Ð33 Bias-corrected projections 1 2Ð41 2Ð161 3Ð267 2Ð621 2Ð257 1Ð951 2Ð Ð3 3Ð511 3Ð582 3Ð311 3Ð422 3Ð371 3Ð Ð16 5Ð402 6Ð624 5Ð382 6Ð694 4Ð792 5Ð Ð38 4Ð144 4Ð767 4Ð344 4Ð627 4Ð024 4Ð Ð28 5Ð573 5Ð827 5Ð453 5Ð457 5Ð293 4Ð Ð71 7Ð765 8Ð751 7Ð465 8Ð231 7Ð615 8Ð101

13 COMPARISON OF THREE METHODS FOR DOWNSCALING DAILY PRECIPITATION 3587 (a) (b) Figure 12. Changes in future mean daily precipitation at six locations for years from years in Punjab (a) raw and (b) bias-corrected projections Figure 13. Kernel density-estimated PDFs of current versus projected daily rainfall amounts for two locations in Punjab in Equation (1) means that it belongs to the exponential family of distributions, which is still broad enough and encompasses a wide range of distributions including the Gaussian, Poisson, multinomial, and gamma distributions. Possible advantages of the CRF model not utilized in this study include use in uncertainty quantification (Raje and Mujumdar, 2010a,b) and extension to a multisite approach. However, it is seen that the model has several drawbacks including computational cost and subjectivity in the selection of feature functions. The KNNbased approach used here has several unique advantages as compared to the other two methods. It is simple and

14 3588 D. RAJE AND P. P. MUJUMDAR Figure 14. Projected PMFs of wet and dry spell lengths for at location 1 easy to implement, the method is data driven and it makes no assumptions of the underlying marginal and joint probability distributions of variables. It is also nonparametric, and through choice and sampling of nearest neighbour distributions show good skill in modelling a suite of precipitation statistics. However, its limitation lies in the choice of analogues, where future projections are hence limited to the range of current observed data. It can also be extended to a multi-site approach. Projections obtained from CRF and KNN approaches are seen to be significantly different, although they reduce somewhat after bias-correction. These differences imply that methodological characteristics as well as physical characteristics may be important in downscaling studies. CONCLUDING REMARKS GCM output is downscaled to precipitation in the Punjab region using three methods CRF, SVM, and KNN to compare the relative merits of each method and project future scenarios. Punjab state falls in the interior part of the Indian subcontinent, with very few studies on rainfall analysis for the region. It is seen that there are several assumptions, advantages, and drawbacks associated with each type of downscaling method. Our results show that different downscaling methods provide differing projections for downscaled daily precipitation. Haylock et al. (2006) also found that the inter-downscaling model differences between the future changes in downscaled precipitation indices were at least as large as the differences between the emission scenarios for a single model. Hence it would be prudent to advocate use of different types of downscaling models, global models, and emission scenarios when developing climate-change projections at the local scale. Frías et al. (2010), in their study using three downscaling techniques for the French Mediterranean basins, found that in some basins, the simulations did not agree in the sign of the anomalies and, in many others, the differences in amplitude of the anomaly were very important. They concluded that the uncertainty related to the downscaling and bias-correction of the climate simulation must be taken into account to better estimate the impact of climate change. This study projects increases in mean daily precipitation at most locations for most scenarios in Punjab. A kernel density estimation analysis of precipitation amounts projects an increase in low to medium precipitation amounts in the Punjab region. Pant and Hingane (1988) found an increasing trend in the mean annual and southwest monsoon season rainfall over most parts of the Northwest Indian region. The increasing trend was found to be significantly marked for the subdivisions constituting the peripheral areas of the Rajasthan desert. Results from this study are consistent with those trends. Some assumptions of this study need to be kept in mind for interpretation. The approach used in this study assumes that the downscaling relationship is stationary so that the observed relationship is applicable to a changed climate. It should be noted that the projected precipitation results presented in the study are from a single GCM and it is widely acknowledged that disagreements between different GCMs over regional climate changes represent a significant source of uncertainty. Therefore, over-reliance on a single GCM is not appropriate for planning and adaptation responses. Thus, future decision making should use multiple GCMs with scenarios and downscaling models to incorporate the underlying GCM, scenario, and downscaling uncertainty. ACKNOWLEDGEMENTS The work presented in this article was sponsored by the Space Applications Center (SAC), Ahmedabad, under research project ISRO/MCV/PPM/091. Funding support for the work is gratefully acknowledged. REFERENCES Anandhi A, Srinivas VV, Nanjundiah RS, Kumar DN Downscaling precipitation to river basin in India for IPCC SRES scenarios using