An empirical model for the seasonal prediction of southwest monsoon rainfall over Kerala, a meteorological subdivision of India

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 28: (2008) Published online 7 August 2007 in Wiley InterScience ( An empirical model for the seasonal prediction of southwest monsoon rainfall over Kerala, a meteorological subdivision of India Lorna R. Nayagam,* Rajesh Janardanan and H. S. Ram Mohan Department of Atmospheric Sciences, Lakeside Campus, Cochin University of Science and Technology, Fine Arts Avenue, Cochin, , India ABSTRACT: There are several studies showing a skillful empirical prediction of the All India Summer Monsoon Rainfall (AISMR) based on various combination of parameters as the predictors. However, the southwest monsoon rainfall over Kerala, a meteorological subdivision of India, bears a considerably low correlation coefficient with the AISMR. This implies that the existing predictors in the long-range forecast models of the AISMR do not have much influence on the Kerala Summer Monsoon Rainfall (KSMR). This study attempts to examine the relationship of some ocean and atmospheric parameters with the rainfall and to formulate a linear multiple regression model for the long-range forecast over a small area like the Kerala. Parameters having significant correlation (significant at 1% level) with the KSMR were identified for the period The consistency of the relationship between these parameters and the KSMR was checked by doing a 21-year sliding window correlation (significant at 5% level). Using a stepwise regression method, seven predictors, explaining a significant amount of variance in the KSMR were selected and a linear multiple regression model was developed. The parameters that explain the high inter-annual variability of the KSMR are specific humidity, sensible heat net flux, relative humidity, zonal wind at 70 and 10 hpa, meridional wind and geopotential height. The characteristics of the forecast and its reliability were studied by various statistical techniques such as, Durbin Watson statistics and variance inflation factor. The model has a multiple correlation of and coefficient of determination of 88.8%. The root mean square error (RMSE) was 6.60% (15.80%), bias (BIAS) was 0.26% (6.20%), absolute error (ABSE) was 5.33% (13.15%) of mean rainfall for the training (test) period respectively. Climatological predictions were also made and the RMSE was (17.90%), BIAS ( 5.40%) and ABSE (15.16%) of mean rainfall. The selected parameters were at least 2 months prior to the monsoon season and hence have predictive value. Copyright 2007 Royal Meteorological Society KEY WORDS long-range forecast; multiple linear regression; stepwise regression; variance inflation factor; Durbin Watson statistic; KSMR Received 10 June 2006; Revised 5 May 2007; Accepted 20 May Introduction Agricultural practices and power generation in India are primarily dependent on the summer monsoon rainfall and its variability. Rainfall forecasts help farmers and water resources management personnel in planning agricultural operations for the maximum utilization of precipitation. Farmers would be more alert for rainfall extremes and thus, less vulnerable if efficient forecasts were disseminated in time. Moreover, power generation in Kerala is mainly hydel. Irrigation and power generation thus, heavily depend on the main rainy season (June September). Thus, a forecast of the monsoon rainfall over this region during the season would be of utmost benefit. Many studies have identified about 30 parameters, which bear significant correlation with all India summer monsoon rainfall (AISMR). They generally fall under the following heads viz. pressure, temperature, atmospheric * Correspondence to: Lorna R. Nayagam, Department of Atmospheric Sciences, Lakeside Campus, Cochin University of Science and Technology, Fine Arts Avenue, Cochin, , India. lorna arn@yahoo.com, lorna@cusat.ac.in circulation, ocean circulation, snow cover, etc. (Pant and Rupa Kumar, 1997). Different combinations of these parameters have been used for the long-range prediction of AISMR and many regression models have been proposed from among these variables by several investigators (Gowariker et al., 1989, 1991; Hastenrath, 1991; Parthasarathy et al., 1991; Thapliyal, 1997). Many of them were selected subjectively according to the meteorological aspects of the parameters of the rainfall and have limitations in explaining the variance in rainfall. Also, the objective selection of the variables by statistical methods is dependent on the position and size of the sample (Thapliyal, 1987; Parthasarathy et al., 1988). Based on these facts, many parameters have been used to derive forecast models for AISMR (Parthasarathy et al., 1988; Vernekar et al., 1995). However, some of these parameters lose significance when the position of the sample shifts from the training period to a much later time. There are also studies indicating some features regarding the spatial coherence of the correlation coefficients between these parameters and rainfall over the country. The high correlation coefficients are over Copyright 2007 Royal Meteorological Society

2 824 L. R. NAYAGAM ET AL Figure 1. Correlation coefficients of AISMR with the Subdivisional rainfall of India. Kerala subdivision is shaded. northwestern and central India and the lowest over northeast and western peninsular India (Parthasarathy et al., 1991; Krishna Kumar et al., 1995). India, as a whole is too large a region to be treated as a single unit, implying that it is not spatially homogenous to rainfall (Normand, 1953; Rasmusson and Carpenter, 1983). Rainfall in some areas has negligible correlation with that in some other parts of the country. Our analysis shows that AISMR is significantly correlated with the subdivisional rainfall of northwest and central India, and a very weak relation exists over the southern part of the peninsula and northeast (windward side of the Western Ghats and The Himalayas) (Figure 1). For Kerala, the value is as low as 0.09 for an analysis period of about 54 years. This implies that the existing predictors in the long-range forecast models of the AISMR do not have much influence on the Kerala summer monsoon rainfall (KSMR). The State receives a mean rainfall of 1871 mm in the southwest monsoon season. Sixty-eight percent of the total rainfall of the state is received in this season. The variability of the southwest monsoon rainfall is very high (SD mm) compared to that of AISMR (SD 82 mm). So, there is a need for better forecasting techniques based on relevant parameters for the areas that are not well represented in the existing forecast models. During the summer monsoon months Kerala is to the windward side of the Western Ghats mountain ranges. These facts together justify the need for a long-range forecast model for Kerala, one of the southern-most meteorological subdivisions in India. Forecast of monsoon rainfall over small areas or smaller periods, such as a week, become difficult. One way to tackle this is to do different ensemble runs of numerical models. However till today, these models do not have the capability to accurately reproduce the salient features of monsoon and its variability (Rajeevan et al., 2004, Gadgil et al., 2005). So, the forecast for these smaller regions, whenever necessary, needs to be based on statistical methods.

3 SEASONAL PREDICTION OF RAINFALL OVER KERALA, INDIA 825 The approach used for long-range forecasts in India relies heavily on teleconnections that have been established through the use of statistical analysis and that have a dynamical and thermodynamical basis (Gowariker, 2002) though statistical models have many inherent limitations (Thapliyal and Kulshrestha, 1992; Krishna Kumar et al., 1995; Rajeevan, 2001; Thapliyal and Rajeevan, 2003). Rajeevan et al., 2004 have commented that on a spatial scale, we have to explore possibilities of demarcating more than the present three homogeneous regions of the country using better criteria. In this study, an attempt is made to identify parameters both global as well as regional, and to mould them into a simple and skillful multiple regression model to predict summer monsoon rainfall over the Kerala subdivision. This study is a first attempt to formulate a linear regression model for the prediction of summer monsoon rainfall over Kerala. 2. Data The subdivisional and all India rainfall (in millimeters) data (IITM-IMR) used were obtained from Parthasarathy et al. (1994), recently updated and maintained by Indian Institute of Tropical Meteorology, Pune, for the period of All India rainfall is an area-weighted average of rainfall from 29 subdivisions. Subdivisonal rainfall is computed from the area-weighted average of district rainfall, which in turn is computed from averaging data from all stations in the district. Rainfall amounts are totals for the months June, July, August and September. The data is consistent from 1871 to 1990 and is constructed from a 306-station network, with no missing data. The data for the recent period are preliminary estimates based on the subdivisional means supplied by the India Meteorological Department (IMD), which are based on a variable network. However, the IMD data have been rescaled to conform to the long-term means of the respective subdivisions in the IITM-IMR dataset. All other datasets used for identifying predictors are monthly NCEP/NCAR reanalysis at the NOAA- CIRES Climate Diagnostic Centre website for the period with spatial resolution of 2.5 longitude and latitude except sea surface temperature, which has a spatial resolution of in longitude and 94 levels in latitude. The parameters include surface parameters (sea surface temperature and surface sensible heat net flux) and multilevel parameters (specific humidity, relative humidity, zonal and Meridional components of horizontal wind and geopotential height). Specific humidity and relative humidity data are taken for eight vertical levels. Horizontal wind components and geopotential heights are used for 17 levels. All the parameters in the study are established to be related to rainfall of different regions of the globe, but all are not necessarily shown to be related to rainfall and its variability in India. 3. Methods Linear regression model is a popular statistical method for meteorological prediction on various timescales like inter-annual, intra-seasonal, monthly, weekly etc. The main task of the development of the regression model includes the selection of predictors based on empirical relations between various parameters and rainfall, their careful and optimum selection in a stepwise regression analysis, formulation of the regression equation and verification on independent samples. Here, good predictive skill of linear regression model is achieved by careful selection of predictors and model building steps, as described below: 3.1. Selecting predictors The KSMR was taken for the period , which we refer to as training period. Spatial correlation coefficient (CC) were calculated between KSMR and the set of parameters under consideration for the period The areas that bear CC s at 1% level of significance were identified and selected for calculating indices by taking area averages of the parameter over the respective significant area (Figure 2(a) and (b)). The CC s of these indices with KSMR were checked for consistency for the entire period of analysis by doing 21-year sliding window correlation (Bell, 1977). Indices that have a significant CC at 5% level in all the 21- year sliding windows were retained. Thus, a total of 18 predictors were selected as the input to the stepwise regression analysis and are listed in Table I. Stepwise regression analysis was employed to reduce the dimensionality of these indices (Draper and Smith, 1981). Thus, the extensive basic predictor set is reduced to a candidate set by eliminating those with less influence in the variance of rainfall. The variables thus selected were, (1) specific humidity at 300 hpa during November over the south central Indian Ocean southwest of Australia, (2) surface sensible heat net flux during August over the west Siberian Plain, (3) relative humidity at 600 hpa during August over the Central Russian Upland, (4) zonal wind at 70 hpa in November over the South Pacific Ocean, (5) meridional wind at 925 hpa during October over Europe, (6) zonal wind at 10 hpa over Eurasia during March and (7) geopotential height of 250 hpa during June over the south-central Pacific Ocean. The details of the selected predictors are tabulated in Table II. All the predictors selected were for the previous year of the monsoon season except zonal wind at 10 hpa. The correlation maps of the selected variables with the KSMR are shown (Figure 2(a) and (b)). The shaded areas have CC s significant at 1%, and the selected areas are marked (rectangles in Figure 2(a) and (b)). The other shaded areas in the figures were not considered, as they do not satisfy the criteria for selection (covering large areas to take an index and persistant correlation in a 21-year CC). Figure 3. gives the 21-year sliding correlation of the seven selected parameters. The selected parameters

4 826 L. R. NAYAGAM ET AL Figure 2. (a) Spatial correlation of the Kerala Summer Monsoon Rainfall with predictor parameters (A) Specific humidity of previous November at 300 hpa, (B) Surface net heat flux, (C) Relative humidity of previous August at 600 hpa. Shaded areas have CC s significant at 1% level. Box indicates the areas selected for making indices. Others do not satisfy the criteria. (b) Same as Figure 2(a), but for (D) Zonal wind of previous November at 70 hpa, (E) Meridional wind of previous October at 925 hpa, (F) Zonal wind of March at 10 hpa, (G) Geopotential height of previous June at 250 hpa. were at least 2 months prior to the monsoon season and hence have predictive value. Multiple linear regression requires a few necessary assumptions. They are the model form, independence of residuals, homoscedasticity and normality of residuals. The first one is related to the form of the relationship of the forecast variable with the explanatory variable. If the form is incorrect the forecast and the F-test and confidence interval will not be valid. The second is if the Figure 2. (Continued). residuals are not independent the statistical viability tests are not valid. The Durbin Watson statistic can be used to see if the assumption is violated. It is defined as DW = (e t e t 1 ) 2 t=2 t=1 e t 2 (1)

5 SEASONAL PREDICTION OF RAINFALL OVER KERALA, INDIA 827 where, e t is the residual at the time t and e t 1 that at time t 1. The third is on the constant variance property of the residuals. The fourth is the normal distribution of the errors, which means that no residuals should lie very far from the mean distribution. If all these assumptions were held for the data under consideration the multiple linear regression would provide good forecasts (Makridakis et al., 1998). Here in this analysis, the identification of parameters is done by linear correlation analysis. So, the model form is kept linear though the atmospheric processes are not linear. All other assumptions are evaluated and reasonably held good (figures not included). The inter-correlation between the predictors leads to multi-co-linearity, i.e. the parameters are non-orthogonal. Thus, a multiple linear regression model lacks in its accuracy and may lead to unclear interpretation of the regression coefficients as measures of original effects (Mc Cuen, 1985). It imposes the problem of redundancy and unnecessary loss of degrees of freedom when they are used in large numbers (Krishna Kumar et al., 1995). Moreover the coefficients would be unstable in the sense that small data revisions could have a disproportionate effect on the calculated coefficients, and thus, the prediction. In order to reduce this and to increase the reliability of these parameters in the prediction, the multi-co-linearity is studied by the variance inflation factor (VIF) method (Fox, 1991) (Table III). It is defined as: VIF (a i ) = 1 1 R 2 (2) i where Ri 2 is the unadjusted R 2 when X i is regressed against all the other explanatory variables in the model. The VIF measures how much the variance of the estimated regression coefficients are inflated compared to situations when the independent variables are uncorrelated. Values in excess of 10.0 could significantly affect the stability of the regression coefficients (Neter et al., 1990). So, the candidates having a smaller VIF only are considered so that the parameters inhibiting interdependence is avoided to increase the credibility of the regression output. The Durbin Watson statistic checks the significance of the assumption that the residuals for successive observations are uncorrelated. Its value ranges from 0 to 4. Values above 2 indicate that there exists some negative autocorrelation and values below 2 a positive autocorrelation. If there exists any kind of significant lag one autocorrelation, then the assumption of independence of residuals is violated and the model can be improved further (Makridakis et al., 1998) Formulation of regression equation From among these refined candidate predictors, various regression models were formulated with the following general form using different iteration schemes. R j = a 0 + ( ) a i Xij + εj (3) i=1 where, R j is the dependent variable (rainfall in our case) for j = 1 to m equal time steps and X ij are the independent variables where i is the number of predictors. a o and a i are model constants and ε j the error value in that estimation. The optimum number of predictors in a statistical regression model has been a matter of debate. There are studies (Wilks, 1995; Delsole and Shukla, 2002) which recommend that the predictors should be restricted to as small a number as possible. Some studies advocate that 8 10 predictors are required for explaining a good amount of variation (70 75%) in the model development period and also limiting the root mean square error (RMSE) of the results over the independent period to a minimum (Rajeevan et al., 2004). Here, the stepwise regression has restricted the number of predictors to 7, as the variance explained by the rest is not significant. The performance of the model has been evaluated by using CC, RMSE, ABSE and BIAS for (1) training period, (2) test period. Equations used to calculate these measures are given below (Nicholls, 1984; Hastenrath, 1987) : RMSE = [ BIAS = ABSE = ( ˆR y R y ) 2 /n] 1/2 (4) y=1 ( ˆR y R y )/n (5) y=1 ˆR y R y /n (6) y=1 Where ˆR y is the model-fitted rainfall, R y is the observed rainfall and n, the number of years. The accuracy of the model forecast for the training period was also compared with the model based on climatology (average of rainfall for the period , the training period). 4. Results The analysis led to the identification of many parameters, which are potential predictors for the KSMR. All of the statistical tests imply that the parameters entered in the model are good predictors of the rainfall. The final form of the regression model is Y = (X 1 ) (X 2 ) (X 3 ) (X 4 ) (X 5 ) (X 6 ) + 4.3(X 7 ) (7) where Xs are the predictors (Table I) and Y the predicted rainfall. The regression coefficients in the model show disproportionate values because the predictors were not expressed in standardized form. Unless the input parameters into the regression model are standardized, the regression coefficients do not imply relative importance.

6 828 L. R. NAYAGAM ET AL Table I. Details of predictors used for the stepwise regression analysis. Parameter Label Level Month CC ( ) Area Specific humidity SH300No 300mb Nov ( 1) E 120E, 30 40S Sensible heat flux SHFAu Surface Aug ( 1) E, 60 70N Meridional wind V850Ja 850mb Jan E, 20 50S Relative humidity RH600Au 600mb Aug ( 1) E, 45 60N Meridional wind V925Ja 925mb Jan E, 20 50S Zonal wind U70No 70mb Nov ( 1) W, 30 50S Meridional wind V700Ja 700mb Jan E, 20 50S Zonal wind U50No 50mb Nov ( 1) W, 30 50S Meridional wind V20No 20mb Nov ( 1) W, 30 60S Meridional wind V30No 30mb Nov ( 1) W, 30 60S Meridional wind V925Oc 925mb Oct ( 1) E, 30 55N Zonal wind U10Ma 10mb Mar E, 35 50N Zonal wind U20Ma 20mb Mar E, 30 55N Zonal wind U30Ma 30mb Mar E, 30 55N Sea surface temperature SSTAu Surface Aug ( 1) E 150W,5 30N Sea surface temperature SSTJu1 Surface Jun ( 1) W, 30 40S Geopotential height Ht 250Ju 250mb Jun ( 1) E 120W,10 25S Sea surface temperature SSTJu2 Surface Jun ( 1) E, 5 20N Table II. Details of predictors obtained after the stepwise regression analysis. Parameter Level Month CC ( ) Area Specific humidity 300 mb Nov ( 1) E, 30 40S Sensible heat net flux Surface Aug ( 1) E, 60 70N Relative humidity 600 mb Aug ( 1) E, 45 60N Zonal wind 70 mb Nov ( 1) W, 30 50S Meridional wind 925 mb Oct ( 1) E, 30 55N Zonal wind 10 mb Mar E, 35 50N Geopotential height 250 mb Jun ( 1) E 120W, 10 25S The VIF analysis of all the parameters that is retained in the stepwise regression have values around 1.5 indicating an insignificant level of multi-co-linearity (Table III). The relationship between each pair of variables can be visualized on a scatter matrix plot (Figure 4). Each column shows the relationship between the parameter listed in that column with the other seven parameters named in the respective rows. The variable on the vertical axis is the variable named in that row and the variable on the horizontal axis is the variable contained in that column. The correlation coefficients between the dependent variable and seven independent variables are listed atop the columns. From the figure it can be seen that the KSMR is positively related to all the parameters, while the independent parameters do not show much relation among themselves though feeble relations exist. The plot does not exhibit any noticeable pattern and this implies that the predictors are practically independent. So the interdependencies among the predictors, if any, are well below the potential to affect the forecast. When the t-test is conducted, the P values indicate that the coefficients of all the predictors in regression are highly significant (Table III). The model has a coefficient of determination (which is the proportion of variation in the dependant variable explained by the regression model) of 88.8% and a multiple correlation coefficient of The F statistic, which tests the significance of regression model, gives F = and the P value in the analysis of variance (ANOVA) test is very small (<0.0005). This strongly suggests that the included predictors collectively account for a significant part of the variance in the KSMR. Table III. The t-test results and variance inflation factor corresponding to the variables. Variables Sig. of t-test VIF Specific humidity Sensible heat flux Relative humidity Zonal wind Meridional wind Zonal wind Geopotential height

7 SEASONAL PREDICTION OF RAINFALL OVER KERALA, INDIA 829 Figure 3. Twenty-one-year sliding window correlations of the seven parameters in the model with Kerala summer monsoon rainfall for the period Central year of the 21-year period is shown. Climatological predictions are made and the RMSE, BIAS and ABSE were also computed. The RMSE, BIAS and ABSE for the climatological predictions are 17.90, 5.40 and 15.16% of mean rainfall, respectively. For the model development period the CC was 0.94, RMSE was 6.60%, BIAS was 0.26% and ABSE was 5.33% of mean rainfall. For the test period the CC is 0.60, which is also statistically significant. When the CC was recalculated for the period , it was found that the CC has dropped to 0.87 (significant at 1% level). The RMSE for the test period has increased to 15.80% of mean rainfall. BIAS and ABSE for the predicted rainfall are 6.20 and 13.15% of mean rainfall, respectively. In most cases, the model error is less than half of the standard deviation (SD mm) of observed rainfall. Of the 44 years, only in three years was the error more than the standard deviation. The forecast and the observed are closely related with a correlation, significant at 1% level. Figure 5. gives the standardized departures of the KSMR and its forecast. The model could capture the sign of the anomaly of extremes (±1 standard deviation) during these 44 years. The wet (dry) years are classified as years with rainfall greater (less) than 1 standard deviation. The model could represent the extreme events, including 2002 (dry year) in which other models failed. Thus the model is good to predict extreme cases too. The KSMR has a standard deviation of mm, which is 19.69% of the mean rainfall ( mm). This is twice that of the AISMR with a standard deviation of 10.4% (86.72 mm). This clearly indicates the need for a new model to predict the KSMR. The forecast of the model presented here, picks a standard deviation of 18.62% ( mm) of the mean rainfall. Thus, both the mean ( , mm) and the standard deviations (348.50, mm) of the forecast are comparable with the observed. This statistics shows that the inter-annual variability of the KSMR is well represented by this model. The standardized departures of the actual and model fitted rainfall shows 7 years (1964, 79, 80, 82, 95, 96 and 2003) with the departure of the model forecast opposite to that of actual rainfall (Figure 5). But over most of the years the observed values are very close to the mean so that a small deviation of the forecast from the observed rainfall may end up with an opposite sign of the anomaly. Moreover, none of these years were extreme and they lie within the range of one standard deviation of normal monsoon. The model showed the maximum departure for the year Scatter plots of residuals with the predictors and fitted values (figure not shown), helps us to visualize the patterns in the residuals. If some organized pattern is present it means that there exists some relation between the fitted values and the residuals so that the model can be further improved by extracting the relationship. It was noted that they do not bear a pattern with the fitted values. The relationship in the residuals is also checked using Durbin Watson statistics. This gives a value of 2.099, which suggests that the lag 1 autocorrelation is negligibly small in the residuals. 5. Discussions and Conclusion Only a few have attempted to predict the rainfall on smaller scales, as the variability of rainfall is high and the stability of predictors are less. In this work, an attempt has been made to predict the rainfall over a subdivision of India, for which only highly correlated parameters are selected. These analyses and observations led to the formulation of a new regression model which is a first attempt for Kerala, a meteorological subdivision in India, could capture the sign of the extremes in the rainfall, like those in the years 1997, 2002 and The percentage of the variation in rainfall that can be explained by the regression equation is 88.8%, useful for the prediction of the KSMR. Researchers have mentioned that the need for making long-range forecasts compel them to derive empirical relations between various phenomena of the ocean atmosphere system though a physical relationship is not well understood. However, such empirical relations

8 830 L. R. NAYAGAM ET AL Figure 4. Scatter plot matrix of the KSMR and predictor variables. The variable on the vertical axis is the variable named in that row; the variable on the horizontal axis is the variable named in that column. SH300No-specific humidity of previous November at 300 hpa, SHFAu-surface sensible heat net flux, RH600Au-relative humidity of previous August at 600 hpa, U70No-Zonal wind of previous November at 70 hpa, V925Oc-Meridional wind of previous October at 925 hpa, U10Ma-Zonal wind of March at 10 hpa and HT250Ju-geopotential height of previous June at 250 hpa. Values on top of the columns represent the respective correlation coefficients of the KSMR with each predictor. Bottom labels represent the units of the variables. Figure 5. Standardized departures of the KSMR and model fitted values ( ). The bar indicates the standardized departure of observed rainfall and the line indicates the standardized departure of the model fitted rainfall. One unit corresponds to mm of rainfall. have unveiled some important processes in the atmosphere. It, therefore, seems desirable that the existence of such empirical relations be discussed, for it may provide some insight into the physical and dynamical processes beneath. The independent variables having lags as large as 7 months have been tried based on their highly

9 SEASONAL PREDICTION OF RAINFALL OVER KERALA, INDIA 831 significant statistical persistence throughout the model development period. Though the atmospheric variables do not have such large climate memory to influence a process after a few months, they may be having an influence on some boundary parameters having a large memory, which in turn may influence the subsequent processes. According to Tang (2004) the memory mechanism of land cannot compete with the ocean, but the atmosphere acts as a bridge that connects the extratropical parameters with the summer precipitation in different regions. The impact is similar to the effect of boundary variables such as sea surface temperature and soil moisture. So, further detailed analysis is needed to clearly identify the causative relations of the predictors identified with the predictant and the physical reasons behind them. New approaches that include non-linear relationships and dynamical variables from model simulations can be included in the existing statistical models to improve the skill of the models. A better prediction is possible by a better understanding of the processes underlying air-sea interaction and identification of new predictors that are stable with the advance of time and captures the high variability of the subdivisional rainfall. In this conjuncture, we have to rely on statistical models for the forecasts of monsoon, with the continual updating of the models. Acknowledgements This research was supported under the University Grants Commission Special Assistance Programme (UGC SAP- 115) to the Department. The authors are grateful to Dr D. S. Pai, Director, LRF Division, IMD, Pune, for the benefit of several useful discussions. We thank Prof. Jon Ahlquist of Florida State University, and Mark R. Jury, Department of Geography, University of Zululand for their valuable suggestions. 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