Prediction of Indian summer monsoon rainfall: a weighted multi-model ensemble to enhance probabilistic forecast skills

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1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 1: (014) Published online 5 July 013 in Wiley Online Library (wileyonlinelibrary.com) DOI: /met.1400 Prediction of Indian summer monsoon rainfall: a weighted multi-model ensemble to enhance probabilistic forecast skills Nachiketa Acharya, a Surajit Chattopadhyay, b, * U. C. Mohanty a and Kripan Ghosh c a Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, New Delhi, India b Department of Computer Application, Pailan College of Management and Technology, Kolkata, India c Meteorological Department, Agricultural Meteorology Division India, Pune, India ABSTRACT: India gets the maximum amount of rainfall during the months of June to September (JJAS) which is known as the summer monsoon season. The erratic nature of Indian summer monsoon rainfall (ISMR), in terms of both rainfall amount and distribution, is highly responsible for the interannual variability in agricultural production as well as occurrence of floods and droughts. Accurate seasonal predictions of ISMR are required for appropriate hydrological planning and disaster management systems. Studies have revealed that probabilistic prediction, based on the products of General Circulation Models (GCMs), can be generated in a parametric as well as non-parametric manner. The present paper discusses the enhancement of probabilistic prediction by improving the potential predictable signal obtained from these GCMs. A Singular-Value-Decomposition based multiple linear regression method (SVD-MLR) has been applied to improve the signal and a simple average of all GCMs (EM) has been used as the benchmark to examine the skill of the SVD-MLR method. The potential of the proposed method has been assessed through Brier Skill Score (BSS) and Rank Probability Skill Score (RPSS). A rigorous analysis has finally revealed that SVD-MLR method has better skill than EM in predicting the typical nature of observed monsoon rainfall in extreme years. KEY WORDS Indian summer monsoon rainfall; general circulation models; multi-model ensemble; probabilistic forecast Received 11 June 01; Revised January 013; Accepted 6 March Introduction Over the years the skills of General Circulation Models (GCMs), in particular coupled systems in predicting the Indian Summer Monsoon rainfall (ISMR), have improved and in recent times they have outperformed those of existing empirical/statistical models. Krishnamurti et al. (00), Palmer et al. (004), Kang et al. (004), Kumar et al. (005), Acharya et al. (011a) and Kar et al. (01) have discussed the evolution of GCMs and their skills. The availability of several GCMs has attracted scientists over the globe to explore multi-model ensembles (MMEs) to enhance the quality of prediction. A plethora of scientific papers describes diverse approaches to generate a single reliable MME forecast that is more skillful than individual GCMs. The simplest MME approach is averaging all the individual GCMs (Peng et al., 00; Palmer et al., 004). Hagedorn et al. (005) describes the rationale behind the success of such MME techniques. Seasonal prediction MME schemes based on weighted ensemble methods are also found to exhibit higher skill for short-range and seasonal forecasting than the individual GCMs. Such methods include: the multiple regression method (Krishnamurti et al., 000; Kharin and Zwiers, 00), the singular value decomposition based regression (Yun et al., 003), the empirical orthogonal function-based regression (Yun et al., 005; Kug et al., 008), the canonical correlation analysis method (Singh et al., 01) * Correspondence to: S. Chattopadhyay, Department of Computer Application, Pailan College of Management and Technology, Kolkata , India. surajit_008@yahoo.co.in and artificial neural networks (Kumar et al., 01). Skills of the weighted MMEs to predict ISMR has been discussed in a number of papers (e.g. Chakraborty and Krishnamurti, 009; Krishnamurti et al., 009; Kar et al., 01) which show improvements in the prediction of ISMR on the seasonal and monthly scale when weighted MME schemes are used. ISMR is characterized by a large variability which results in an inherent uncertainty that can be better predicted using probabilistic forecasts as suggested by Kumar et al. (001) and Palmer et al. (004). A measure of the uncertainty is of great help for users in the agricultural/hydrological sectors, as it helps them to plan better for climate risk management. Most of the MME techniques discussed in the previous paragraph are used to produce a deterministic prediction with no measure of the uncertainty. Even though a literature survey reveals a plethora of works in the area of probabilistic prediction using GCM for seasonal scales (Doblas-Reyes et al., 000; Rajagopalan et al., 00; Kharin and Zwiers, 003; Robertson et al., 004; Palmer et al., 005; Tippett et al., 005, 006; Min et al., 009), only a few studies (Chakraborty and Krishnamurti, 009; Kar et al., 01; Acharya et al., 011a; Kulkarni et al., 01) have been carried out on probabilistic forecasting of ISMR using GCMs. Probabilistic prediction based on GCM products can be done in parametric and non-parametric ways for user defined categories. In seasonal forecasting it is common to define three equally probable mutually exclusive and collectively exhaustive categories: below-normal (BN), near-normal (NN), and above-normal (AN) (Kharin and Zwiers, 003). This categorization is based on the climatological probability density function (pdf). In the nonparametric method, these tercile 013 Royal Meteorological Society

2 Prediction of Indian summer monsoon rainfall 75 probabilities are the fractions of the ensemble members in each category. The parametric approach depends on the pdfs of the observed and the model predicted rainfalls belong. It is a common practice to assume Gaussian distribution for implementing the parametric approach in the context of seasonal prediction. Tippett et al. (006) stated that the parametric estimates of seasonal precipitation tercile category probabilities are generally more accurate than the counting estimate (non-parametric approach). In the case of parametric approaches, the mean and variance of the distribution of the available GCM outputs need to be determined. The mean of the distribution is often referred to as the potential predictable signal and the variance of the distribution referred to as the variance of stochastic noise (Kharin and Zwiers, 003). Different methods have been proposed to estimate such variance and they are well documented in the literature (Kharin and Zwiers, 003; Tippett et al., 005; Johnson and Bowler, 009). Kulkarni et al. (01) describes and compares three methodologies, Ensemble Spread (ES), Error Residual (ER) and Correlation Method (CR), to calculate the variance. The stochastic noise variance in the ES method is treated as the ensemble spread of a GCM (Kharin and Zwiers, 003), whereas the ER method uses the error between the mean of all GCMs and observation to estimate the variance (Johnson and Bowler, 009). The CR method uses the correlation between observation and signal from GCMs to calculate the variance (Tippett et al., 005). This approach is similar to that of estimating the forecast uncertainty using the standard error of a regression fit, neglecting the effect of sample size. When the correlation between observation and signal from GCMs is positive, correlation can be used to define a positive linear regression co-efficient. Kulkarni et al. (01) have used Rank Probability Skill Score (RPSS) to compare these three methods and found that the CR method is better than other schemes. In the CR method the correlation is used to compute the variance so that the spread of the forecast distribution becomes large for small correlation values and the prediction is close to the climatology. This reduces the forecast probabilities and the risk of incorrect predictions. However, the skill of the CR method is higher if the correlation between observations and signal from the GCM is further enhanced. Therefore, it can be concluded that, to improve the quality of probabilistic prediction, a high correlation between potential signal from GCM and observations is required. The present work discusses the enhancement of the probabilistic prediction skills of ISMR by improving the correlation between potential predictable signal from GCM and observations. This is basically an extension of the work of Kulkarni et al. (01), where the correlation based method has been identified as the best over other methods in calculating the forecast uncertainty. Similarly, the present study is aimed at the improvement of probabilistic prediction of ISMR applying weighted MME technique by means of improvement of the potential signal or mean of the distribution from GCM. The outline of the paper is as follows: Section describes briefly the GCMs and the observed data. Methodology for probabilistic prediction and weighted MME is described in Section 3. Section 4 elaborates the results and provides a discussion, while concluding remarks are presented in Section 5.. Data The present study is based on the eight GCMs rainfall forecasts. Three are atmospheric models only and the remaining five are coupled atmospheric-oceanic models. Model forecasts were obtained from the following organizations: International Research Institute (IRI), USA; National Center for Environmental Prediction (NCEP), USA and Japan Agency for Marine- Earth Science and Technology (JAMSTEC), Japan. For this study, lead-1 (initial conditions of May start) hindcast runs ( ) of the models for the summer monsoon seasonal rainfall (mean rainfall of June to September) are used. Detailed descriptions of these models are presented in Acharya et al. (011a) and Kulkarni et al. (01). A brief summary of each model, including members, resolution and relevant citations, is presented in Table 1. The high resolution (1 1 ) gridded rainfall data (Rajeevan et al., 006) based on 140 rain gauge stations (with minimum 90% data availability) provided by the India Meteorological Department (IMD) are used as observational reference. Daily gridded observed data (357 grid points) over the Indian landmass is converted to seasonal total rainfall. The ensemble mean of all GCMs is bi-linearly interpolated to the observation grid point before applying the MME schemes and probabilistic approaches. 3. Methodology This section describes in detail the theoretical foundation behind the estimation of probability for the tercile categories, the variance of stochastic noise and the potential predictable signal improvement when using a weighted multi-model ensemble Calculation of probabilities for tercile categories It has already been stated that probabilistic prediction based on GCM outputs can be obtained with parametric or nonparametric methods. The present study describes the parametric approach (assuming Gaussian distribution). A commonly used Table 1. Description of GCM outputs used in the study. Model Ensemble member Resolution References CFS v1 15 (T6) Saha et al. (006) CCM3.6 4 (T4).7.8 Hurrell et al. (1998) ECHAM-MOM3AC1 4 (T4).7.8 Roeckner et al. (1996) and Pacanowski and Griffes (1998) ECHAM-MOM3DC 1 (T4).7.8 Roeckner et al. (1996) and Pacanowski and Griffes (1998) ECHAM-GML 1 (T4) Roeckner et al. (1996) and Lee and De Witt (009) SINTEX-F 9 (T106) Luo et al. (005) E4p5 (ca sst) 4 (T4) Roeckner et al. (1996) E4p5-CFS 4 (T4) Roeckner et al. (1996)

3 76 N. Acharya et al. parametric approach is based on the following equations (Kharin and Zwiers, 003): X = β + ε (1) E (X ) = E (β) as E (ε) = 0 () σ X = + σ ε as Cov (β, ε) = 0 (3) In this linear model, the observed variations of rainfall (X ) are being represented as the sum of potential predictable signal (β) and non-predictable stochastic noise (ε). The potential predictable signal β is the forecast from the GCMs and follows a Gaussian distribution with mean E(β) and variance σβ.it is assumed that stochastic noise follows Gaussian distribution with mean zero and variance, σε. It may be noted that E, σ and Cov denote the expectation, variance and covariance of the corresponding random variable, respectively. Since the mean of ε is zero, the mean of observed rainfall (X ) should be equal to the mean of signal β. The mean of the signal β is lower than the mean of the observed rainfall because of the inherent systematic and random bias of GCM. This point will be discussed in detail in the following sections. Hence, the variance of the observed rainfall (X ) can be represented as the sum of variance of signal and variance of stochastic noise. The probability of occurrence of X in any tercile category (below-normal (BN), near-normal (NN) and above-normal (AN)) given the signal (β) can be derived defining x a and x b as the upper and lower limits of the category, respectively (Kharin and Zwiers, 003). If X falls below x b in the distribution, it belongs to the below-normal category and if X stays above x a in the distribution, it belongs to the above-normal category. If the random variable X stays between the limits x a and x b, then it belongs to the near-normal category. Details of the probabilities used in this study are as follows. Let F(X β) be the (cumulative) distribution of X conditional on a given value of β, then: P X ( β) = Prob [X (x b, x a ) β] = F X (x a β) F X (x b β) (4) where, P X ( β) is the conditional probability that an event corresponding to the random variable X lies in an interval (x a, x b ) with β as the conditioning event. As the noise term ε is following a Gaussian distribution, this conditional probability may be written as: P X ( β; ) = F N ( xa β ) F N ( xb β where, F N denotes the cumulative distribution function of the standard normal distribution. If AN denotes the event when X > x a, then: P x (AN β, ) = P [(X > x a ) β, ] [ ] xa β = 1 F N [ ] β xa = F N If BN is an event for which X < x b, then: P x (BN β, ) = P [(X < x b ) β, ] [ ] xb β = F N ) (5) (6) (7) If NN denotes the event when x a < X < x b, then: P x (NN β, ) = 1 P x (B β, ) P x (A β, ) (8) Since a Gaussian distribution is assumed: x a = x b = x 1/3 σ X = FN 1 (1/3) σ X (9) one can obtain the probability of above-normal (AN ), belownormal (BN ) and near-normal (NN ) using Equations (6) (8), respectively. Therefore, the signal (β) and variance of the stochastic noise (σε ) are essential for probabilistic prediction. 3.. Calculation of variance of stochastic noise (σ ε ) As previously discussed, Kulkarni et al. (01) found that the CR method is better than other two techniques (ES and ER) used for calculating the variance of stochastic noise (σε ). This method, originally developed by Tippett et al. (005), uses a standard error of regression between the potential predictable signal and observations to estimate the variance of stochastic noise (σε ). Practically, the variance can be estimated using the correlation between signal (β) from GCM and observations (X ) following the steps given below: cov (x, β) r = σx or, r = E (xβ) E (x) E (β) σ x E {(β + ε) β} E {(β + ε)} E (β) or, r = σx or, or, putting x = β + ε r = E ( β + εβ ) (E (β) + E (ε)) E (β) r = E ( β ) + E (εβ) {E (β)} E (ε) E (β) σ x σ x or, r = E ( β ) {E (β)} ( ) σβ + σε σβ or, r = or, r = σ β ( ( σ β σ β as E (ε) = 0 + σ ε + σ ε ( or, σ β + σε ) σβ = r or, 1 + σε = 1 r if σ β = 1 ( ) 1 or, = r 1 ) ) (10) In Equation (10), r denotes the correlation co-efficient between X and β. Therefore, variance of stochastic noise is a function of r. When the correlation between an observed series and a signal is positive, correlation can be used to define a positive linear regression co-efficient.

4 Prediction of Indian summer monsoon rainfall Methodology for improving the potential predictable signal As reported in some recent studies (Acharya et al., 011a; Kulkarni et al., 01), the GCMs have poor skill in predicting the ISMR, mainly because of the large inherent bias (systematic and random) of the GCMs. Even the MME techniques are not able to remove random errors associated with GCMs. Hence, prior to improve signal from combination of all GCMs, the systematic bias of each GCM needs to remove. In view of this, standardized anomaly of each GCM has been calculated to remove the bias as proposed by Acharya et al. (01). The simplest way to improve signal (β) is to take a simple arithmetic mean of all bias corrected GCMs. Acharya et al. (011a) showed that this simple MME technique has better skill to predict the ISMR than individual GCMs. This method (referred to as EM) has been used by Kulkarni et al. (01) for estimating β in their study and can be used as a benchmark to evaluate the other techniques. Yun et al. (003) and Acharya et al. (011b) discussed the improvements of the GCM prediction skills using the singular value decomposition based multiple regression method. This method is also tried in the present study. The entire method (referred to as SVD-MLR) is described below. Suppose Y it is i th the GCM s rainfall anomaly at time t where the total number of GCMs is m and X t is the observed rainfall anomaly at time t. The variance-covariance matrix is then denoted as C whose entries are C i,j = Yi,t T Y j,t and t=1 n is the length of the training dataset and covariance matrix between observation and model, which may be denoted as O, where: O i = n t=1 W j n X T t Y j,t (11) The Singular Value Decomposition (SVD) is applied to the computation of the regression co-efficients. The SVD of the covariance matrix C is its decomposition into a product of three matrices as: C = UW V T (1) where columns of V consist of the eigenvectors of the matrix C T C and columns of U consist of the eigenvectors of CC T.The singular values in W are square roots of eigenvalue from C T C or CC T. Since the covariance matrix C is a square symmetric matrix, C T = VWU T = C. This proves that the left and right singular vectors U and V are equal. This decomposition can be used to obtain the regression co-efficients or weights: [ ( )] 1 a = V. diag. ( U T.O ) (13) These weights are then passed on to the forecast phase to construct the final SVD-MLR predicted anomalies(s ): n S t = a i Xi,t T (14) i=1 S can now be used as the signal β to calculate the probabilities. Since the dataset contains only 7 years data ( ), a leave one out technique (recommended by the WMO standardized verification system, 00) has been implemented for cross validation. In this cross validation method, 1 year from the total dataset (consisting of 7 data points) is reserved for test and the residual dataset (consisting of 6 data points) is used as a training dataset. The training dataset has been used for calculation of all statistics. 4. Results and discussion 4.1. Signals from individual models Prior to examining the signal from the multi-model ensemble schemes (EM and SVD-MLR), the performance of individual GCMs will briefly be discussed in this section. Detailed diagnosis of all the models used in the present study showed that these GCMs are having large variability in predicting the observed rainfall climatology and interannual variability with large bias (Acharya et al., 011a; Kulkarni et al., 01). Even though almost all the models are able to capture to some extent the spatial pattern of the observed climatology, the magnitudes of predicted rainfall climatology for all the models is underestimated as it is the predicted rainfall interannual variability of each model. The temporal correlation co-efficient at each grid point can be used to determine the performance of signal. The magnitude of correlation should be at least 0.3 for being significant at 95% confidence interval for 7 years of study ( ). Almost all GCMs have poor skill (correlation below 0.3) for most of the regions of the country except at a few grid points in northern parts of India (figure not shown). 4.. Signal from multi-model ensemble The temporal correlation co-efficient at each of 357 grid points (Figure 1) is calculated to assess the signals obtained using the EM and SVD-MLR methods. The skills of both the methods have been evaluated based on leave one out cross validation technique applied on entire study period ( ). For EM method, the correlation is significant (0.3 at 95% confidence interval) only in some parts of northern and northeastern regions of the country. Similarly, over southern India, signals from this method show significant correlation only at a few grid points. Results reveal that the EM method is not able to show any noticeable improvement in skill compared to the individual GCMs as equal weight is given to each model. The same weight being assigned to the models having high as well as poor skill allows only little improvement in this method. On the contrary, the SVD-MLR method is based on the multiple regression technique where weights are assigned to the models on the basis of their skill in the training period (using leave one out cross validation). Therefore, the SVD-MLR method shows much better correlation compared to the EM method. The number of grid points having significant correlation is increased appreciably in this method. The skill is improved, especially over the southern and western parts of the country. However, for north and northeast India, the skill is as good as it is for the EM method. The skill of probabilistic predictions can be assessed using a number of scores, among which are the Brier Skill Score (BSS) and Rank Probability Skill Score (RPSS). These scores are introduced and discussed in relation to the EM and SVD- MLR in the following section Brier skill score (BSS) Brier Skill Score (BSS) (Murphy, 1973) assesses the skill of probabilistic prediction in each individual category (tercile category) with respect to some reference forecast. BSS is defined as: BSS = 1 BS BS REF (15)

5 78 N. Acharya et al. EM SVD MLR Figure 1. Temporal correlation between the observed rainfall and the potential signal from EM and SVD-MLR during hindcast period ( ). Areas with correlation values with > 0. have been shaded. This figure is available in colour online at wileyonlinelibrary.com/journal/joc where BS represents Brier Score defined as: BS = 1 n n (f i O i ) i=1 (16) with n being the number of forecast, f i the i th probability forecast and O i the observed frequency (O i = 1 if the event occur and 0 otherwise). In other words, Brier Score is the mean squared error in probabilistic space. BSS represents the level of improvement of the Brier Score compared to that of a reference forecast, BS REF which is generally the climatological probability (i.e for each of the tercile category). For a perfect forecast the value of BSS is 1, whereas 0 refers to no improvement over the reference forecast. A negative value of BSS indicates that the forecast strategies are worse than the reference strategy (climatological probabilities). Positive BSS represents better forecast with respect to climatology. The BSS is calculated for EM and SVD-MLR for each tercile category (Figure ). In terms of BSS, EM has a noticeable number of grids where the score is positive for the BN and AN category (14 grid points in BN and 143 grid points in AN category), especially, some grid points over northern parts of the country with BSS greater than 0.. The number of grid points with positive BSS is further increased in SVD-MLR compared to EM for both the categories (133 grid points in BN and 148 grid points in AN category). The SVD-MLR scheme has the same number of grid points over north India with BSS > 0. as obtained using EM method. However, the SVD-MLR method provides better results over other parts of India. Especially, some grid points in the north-eastern; western and southern parts of the country show BSS > 0.. Furthermore, SVD-MLR also shows a large number of grid points (14 in BN and 17 in AN) with BSS > 0. when compared to EM (7 in BN and 6 in AN). Therefore, it can be concluded that SVD-MLR has higher skills than EM under BN and AN category. On the other hand, in the NN category, even though both the MME methods have a large number of grid points with positive BSS compared to BN and AN categories, the magnitude of BSS is very small (less than 0.05). The comparatively poor skill in the NN category is due to the fact that this category is bounded on both the sides while the AN and BN categories are unbounded at one side. The prediction of such a narrow category (NN) (close to the mean of the distribution) often does not match with the observation (Van den Dool and Toth, 1991; Kar et al., 006). Van den Dool and Toth (1991) summarized the possible reasons for poor skill in NN category as: (1) the likelihood of observations escaping the forecast class is considerably higher for a closed (NN) than for an open ended class (AN or BN); () the low skill near the mean of the distribution (NN) occurs as forecast methods generally turn out to have more or less uniform error while the observations tend to have frequency distributions peaked near the normal value Rank probability skill score (RPSS) The Rank Probability Skill Score (RPSS), measures the cumulative squared error between the categorical forecast probabilities and the observed categorical probabilities relative to a reference (or standard baseline) forecast (Weigel et al., 007). The RPSS is defined as: RPSS = 1 RPS (17) RPS REF where the RPS is the Rank Probability Score of a forecast while RPS REF is the Rank Probability Score for a reference forecast. As with BSS, the climatological forecast is in general used as the reference. RPS is given as: RPS = K ( ) Pj O j (18) j =1 where, K is number of categories of the probabilistic forecast, P j and O j are the cumulative probabilities of forecasts and observations falling in the category j. RPSS is calculated for all categories (above normal (AN), near normal (NN), or below normal (BN)) cumulatively; whereas, BSS is calculated for each category, separately. Like BSS, when the value of RPSS equals to 1, it implies that the observed category is always predicted with 100% confidence. RPSS 0 implies that the prediction skill is same as climatological prediction and a score < 0 means that the forecast system performs worse than climatology. RPSS values obtained from EM and SVD-MLR are presented in Figure 3. RPSS displays results similar to those observed BSS, that is an improvement in skill for SVD-MLR when compared to EM. The SVD-MLR method shows a large number of grid points with higher skills over the southern, western and

6 Prediction of Indian summer monsoon rainfall 79 Figure. Brier skill score (BSS) for below norrmal (BN), near normal (NN) and above normal (AN) category of EM and SVD-MLR during hindcast period ( ). Areas with correlation values with > 0 have been shaded. This figure is available in colour online at wileyonlinelibrary.com/journal/met EM SVD MLR Figure 3. Rank probability skill score (RPSS) of EM and SVD-MLR during hindcast period ( ). Areas with correlation values with > 0 have been shaded. This figure is available in colour online at wileyonlinelibrary.com/journal/met north-eastern parts of the country; whereas, in the EM method these regions have no skill. SVD-MLR not only improves the skill in new regions, but also has the same skills as the EM method in northern part of the country. The number of grids having 0 < RPSS < 0.3 for both the MME methods are shown in Figure 4. It is evident that there is not much improvement in SVD-MLR (148 points) compared to the EM (134 points) in terms of total number of grid points having RPSS > 0. However, the number of points having higher RPSS value is larger in SVD-MLR than EM (Figure 4). To quantify the relative improvement (in percentage) of skill in SVD-MLR with respect to EM, the difference between the number of grid points of SVD-MLR and EM divided by the grid points of EM and multiplied by 100 for each category i.e. > 0, > 0.05,... >0.3 is calculated. The improvement in first category (> 0) is very low (10.5%), but in the other categories it is quite impressive:

7 730 N. Acharya et al. Figure 4. Number of grid point having Rank probability skill score (RPSS) > of EM and SVD-MLR during hindcast period ( ). Figure 6. Probabilities of below-normal (P(BN)), near-normal (P(NN)) and above-normal (P(AN)) by EM and SVD-MLR for excess monsoon year (1988 and 1994). Dotted line represents the climatological probability (0.33). Figure 5. Probabilities of below-normal (P(BN)), near-normal (P(NN)) and above-normal (P(AN)) by EM and SVD-MLR for deficit monsoon year (1987 and 00). Dotted line represents the climatological probability (0.33). 66.7% (> 0.05), 11.5% (> 0.1), 00% (> 0.15 and > 0.) and 100% (> 0.5 and > 0.3). Moreover, although SVD-MLR skills have in general improved, value of BSS or RPSS is never higher than 0.3. Since the signal improvement is based on the hindcast skill of each model, the above poor skill may be attributed to the poor skill of individual models. The overall performance of the MME methods for the entire period ( ) can be complemented with the assessment of their ability to capture the extreme rainfall values of the ISMR. During the study period ( ) there are two major deficit years, 1987 and 00 and two major excess rainfall years, 1994 and Probabilities of below-normal, near-normal and above-normal for EM and SVD-MLR for deficit and excess monsoon years are plotted in Figures 5 and 6, respectively. It is found that both the methods have higher values of probability in the appropriate category, that is below normal (BN) for deficit years and above normal (AN) in excess years, than in the other two categories. Moreover, for both extreme years, the probability (below or above normal for deficit or excess rainfall years, respectively) value of SVD-MLR is higher than EM. Although, in general, it is observed that the signals are not so strong, at least MME could explain the below and above normal rainfall signals in both the extreme cases. However, these signals are not distinguishable by climatological probability (0.33). Even though MMEs have enhanced the probabilistic prediction in extreme years, their contribution is not as strong as expected. While the above discussion presents an improvement of the potential predictable signal using weighted MME, the method has also some limitations. In the SVD-MLR scheme, the regression co-efficients depend on the estimates of covariance of a GCM with observation in the training period. This estimation could be degraded, if the training was carried out with a poor hindcast from the individual GCMs. Yun et al. (003) showed that prediction skill is improved when a higher quality training dataset is deployed for the evaluation of the multi-model bias statistics. However, it should be kept in mind that the total period of data used is only 8 years ( ), which may not be very large for estimation of regression co-efficients. 5. Concluding remarks Probabilistic predictions and the associated information on uncertainty are particularly relevant for the Indian summer monsoon rainfall forecasts. In recent years, probabilistic forecasts based on the outputs of GCMs have gained considerable attention. The present study has looked at ways to improve the probabilistic prediction by improving the potential predictable signal coming from individual GCMs. This study is motivated by the findings of Kulkarni et al. (01), which reported that the correlation based method is better than the Ensemble Spread and Error Residual methods to calculate the uncertainty associated to probabilistic approaches. The purpose of the present study was twofold: firstly, to calculate the uncertainty of prediction and secondly, to improve the potential predictable signals. The paper describes the use of the Singular-Value-Decomposition based multiple linear regression method (SVD-MLR). Outputs from eight GCMs and observations have been used in the present study. A simple average of all GCMs (EM) provided the benchmark to evaluate the improvement of the SVD-MLR method. It was found that SVD-MLR significantly improves the signal in terms of temporal correlation. Subsequently, a parametric probabilistic prediction (assuming Gaussian distribution) method was implemented on both EM and SVD-MLR and the two systems were evaluated using the Brier Skill Score (BSS)

8 Prediction of Indian summer monsoon rainfall 731 and Rank Probability Skill Score (RPSS). Both scores showed moderate improvement of the SVD-MLR method compared to EM. The skill of the SVD-MLR and EM methods in extreme rainfall years, i.e. 1987, 1988, 1994 and 00, was analysed. Years 1987 and 00 were characterized by deficit rainfall, whereas years 1988 and 1994 were characterized by excess rainfall. The study reveals that SVD-MLR predictions are more skillful to predict the typical nature of observed rainfall; even though the improvement was not as large as expected (the highest value of BSS or RPSS is 0.3). This may be due to the dependence of the regression co-efficients on the estimates of the GCMs covariance with the observation during the training period, which could be degraded if the training is executed with a poor hindcast from individual models. Moreover, only 7 years ( ) have been used for the study, which may not be sufficiently large for estimation of regression co-efficients. This study can be extended by implementing more sophisticated weighted multi-model ensemble schemes for further improvement of skills. Acknowledgements The study is conducted as part of a research project entitled Development and Application of Extended Range Weather Forecasting System for Climate Risk Management in Agriculture, sponsored by the Department of Agriculture and Cooperation, Government of India. Gridded rain data have been obtained from India Meteorological Department. We gratefully acknowledge the IRI modelling and prediction group led by D. Dewitt for making six of their GCM-based seasonal forecasting systems available to this study, as well as the IRI Data Library group led by B. Blumenthal. We acknowledge the particular contributions of D. Lee, H. Liu, and M. Bell. 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