A logistic regression approach for monthly rainfall forecasts in meteorological subdivisions of India based on DEMETER retrospective forecasts

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 30: (2010) Published online 29 September 2009 in Wiley InterScience ( DOI: /joc.2019 A logistic regression approach for monthly rainfall forecasts in meteorological subdivisions of India based on DEMETER retrospective forecasts K. Prasad,* S. K. Dash and U. C. Mohanty Centre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi , India ABSTRACT: A multi-predictor logistic regression model has been developed for probabilistic forecasts of domain average rainfall on monthly timescale for three study regions namely, India as a whole, and two homogeneous meteorological subdivisions of India, i.e. Orissa on the east coast and Gujarat on the west coast. The time series of the monthly total observed rainfall as the predictand variable was constructed from the gridded (1 1 ) daily rainfall produced by India Meteorological Department, and those of the predictor data sets from 1-month lead forecasts of several atmospheric and oceanic variables produced by the Development of a European Multi-model Ensemble system for seasonal to interannual prediction (DEMETER) project of European Centre for Medium-Range Weather Forecasts (ECMWF). Multi-model ensembles of nine-member retrospective forecasts for the month of August generated by three constituent models of the DEMETER system, viz., ECMWF, United Kingdom Meteorological Office (UKMO) and Meteo France are used. The predictor variables (totally 36 in number) include direct model-predicted total precipitation and its inter-member standard deviation. A twostage procedure has been designed, where logistic regression is first computed for each individual variable and then for the variables ranked on the basis of Brier scores. The top-ranked variables (up to four) are used for fitting the multiple logistic regression model in a stepwise manner. The fitted model provides estimates of probability of the value of an observation exceeding a specified quantile (such as median) of the statistical distribution of the predictand variable. The model shows good performance in capturing the extreme rainfall years and appears to perform better than the direct model forecasts of total precipitation in respect of such years. Copyright 2009 Royal Meteorological Society KEY WORDS statistical downscaling; logistic regression; probabilistic forecasts; quantiles; all India average; meteorological subdivisions Received 9 July 2008; Revised 14 July 2009; Accepted 15 August Introduction The summer monsoon rainfall in India is characterized by a large inter-annual variability (IAV) and active and break cycles on sub-seasonal timescales that impact the agricultural production to a great extent. A major constraint in the implementation of the contingency crop planning strategies, which have been developed in India to minimize crop losses due to aberrant weather conditions, is believed to be the lack of advance weather information with sufficient lead time and accuracy. The agricultural community in India needs spatially and temporally differentiated weather information with a lead time of about 1 month which could be of immense value to the policy planners and agriculture related service organisations to provide critical input to farmers. Monthly forecasts of precipitation and temperature, and their timely dissemination to the farmers have, therefore, a potential benefit in agricultural planning and operations. * Correspondence to: K. Prasad, Room No. 410, Block VI, Centre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi , India. kantiprasad@hotmail.com The usual practice to meet the above goals is to provide forecasts of area average characteristics on seasonal timescales and sub-continental spatial scales obtained through general circulation models (GCMs) and/or empirical models. The direct forecasts of surface weather elements produced by the GCMs are, however, not suitable for end use as they suffer from systematic biases and have a poor skill in terms of simulating the wide fluctuations in the time series of weather elements in question, particularly on small spatial and sub-seasonal timescales. The most common approach to deal with the problem of model biases and usability of surface weather elements forecasts is to obtain the end-user products through statistical downscaling. The problem of downscaling outputs from GCMs to obtain information for weather elements like precipitation and temperature in localized areas has received increasing attention in recent years (Wilby and Wigley, 1997). The potential of GCM outputs in developing statistical models for applications like predictions of climate change scenarios and seasonal forecasting has been amply demonstrated in several studies (e.g. Murphy, 2000; Wilby and Wigley, 2000; Widmann et al., 2003; Schmidli et al., 2006). These techniques have their origin in the model output statistics (MOS)/perfect prognosis Copyright 2009 Royal Meteorological Society

2 1578 K. PRASAD ET AL. method (PPM) approaches to forecasting of surface weather elements from numerical weather prediction model outputs of flow variables for short-range weather forecasting. The fundamental basis for these downscaling techniques is the statistical relationships linking observations of local variables to the observed atmospheric circulation parameters. These relationships are then applied to the circulation simulated by a GCM in order to generate predictions of local climate (Karl et al., 1990; von Storch et al., 1993) on the assumption that these relationships remain valid under future climate conditions. To date, there is only limited experience with downscaling of seasonal predictions, whereas in climate change modelling, statistical downscaling has been applied extensively by using perfect prog approach (e.g. von Storch et al., 1993). Although the multiple linear regression has been the principal tool in most statistical downscaling studies, its use is limited by the fact that it is not capable of handling the extreme events for which very few realisations would exist. In such situations the probabilistic type predictions, rather than quantitative values, become a preferable option. A logistic regression technique is a powerful and useful tool for deriving probabilistic forecasts. So far, the logistic regression approach has been widely used in medium-range weather forecast problems. Hamill et al. (2004) and Wilks and Hamill (2007) have applied this technique in medium-range forecast applications for precipitation and temperature using retrospective forecasts and demonstrated that the MOS-based probabilistic forecasts were skillful and highly reliable. The potential of producing useful probabilistic predictions of seasonal climate fluctuations and of applying them to crop yield forecasting has been recently illustrated by Doblas-Reyes et al. (2006). The usefulness of logistic regression analysis has also been demonstrated by Wilks (1995) and Applequist et al. (2002). In our present study, we examine the logistic regression approach for probabilistic predictions in respect of the all India average and two meteorological subdivisions of India, Orissa and Gujarat that lie at the eastern and western extremities, respectively, of north peninsular India. The choice of these two particular subdivisions, as a pilot study, was dictated by the fact that they lie in two distinct rainfall regimes and their proximity to widely separated moisture sources, namely, Bay of Bengal in the east (Orissa) and Arabian Sea in the west (Gujarat). The study makes use of domain averaged rainfall processed from daily gridded data sets obtained from India Meteorological Department (IMD), and DEME- TER (Development of a European Multi-model Ensemble for seasonal to interannual prediction) set of forecasts (Palmer et al., 2004) downloaded from the ECMWF s website. The potential of DEMETER forecast system in simulating the Indian summer monsoon rainfall (ISMR) has been investigated earlier by Xavier and Goswami (2007). The authors have reported a rather realistic representation of the features of observed rainfall climatology by the above system based on the simulations carried out in respect of ISMR from initial conditions starting 1 May. However, the authors find hardly any correspondence between the model forecasts and observed rainfall for most of the years. They further report that despite the model s ability to depict the amplitude of the IAV of Indian summer monsoon (ISM), it fails to predict the phases of observed IAV of ISM. Xavier et al. (2008) have used the DEMETER multi-model seasonal hindcasts to study the intra-seasonal variability of Asian summer monsoon. Wang et al. (2005) have conducted a study of the sea surface temperature (SST) rainfall relationships with the DEMETER data sets and concluded that despite the different physical schemes used by coupled models of the system, they are able to produce these relationships in a qualitative realistic manner. Motivated by some positive outcomes of DEMETERbased studies in summer monsoon simulation and prediction, the present work has been taken up to examine the utility of DEMETER forecasts in formulating area averaged rainfall in extended-range timescales through empirical methods. So far, the empirical models and the dynamical models have attempted to predict the area averaged rainfall on seasonal timescales and sub-continental space scale. The utility of such forecasts to the user community is limited. Our focus in the present study is to examine the potential of the GCM outputs of atmospheric and related variables in producing forecasts on monthly timescales, and for smaller areas through statistical downscaling using MOS approach. We have chosen to carry out the investigation for the month of August (forecasts generated from 1 August initial conditions) for which the data are available in the DEMETER system during the ISM (June September) period. The data and methodology including the procedure for preparing various data sets for analysis, and an outline of logistic regression approach are described in Section 2. Results are presented and discussed in Section 3. Section 4 gives the summary and concluding remarks. 2. Data and methodology 2.1. Data sources The target predictand (also known as response or dependent) variable for regression analysis in our study is the domain average monthly rainfall. The basic data used for constructing the monthly rainfall data series are the high resolution (1 1 ) daily gridded data sets prepared by IMD in the domain 6.5N 38.5N and 66.5E 100.5E. These data sets have been prepared by IMD through objective analysis utilizing measurements of 1803 rain gauge stations with a minimum 90% data availability for the analysis period based on an interpolation method proposed by Shepard (Rajeevan et al., 2006). The data sets have recently been updated to include the period Monthly gridded rainfall is obtained from the daily observed data series. The source of training data sets for computing predictor (also known as explanatory or independent) variables for the regression analysis was the DEMETER

3 MONTHLY RAINFALL FORECASTS IN INDIA BASED ON DEMETER RE-FORECASTS 1579 retrospective forecasts downloaded from the ECMWF s website. The DEMETER system comprises seven global coupled ocean atmosphere models participating in the project, which have produced the data sets, the commencement year varying from 1958 to 1980 for different models. For each year, four 9-member 6-month long ensemble forecasts are available starting on the first day of February, May, August and November at 00 : 00 h UTC, at a grid resolution of lat./long. in global domain. We have used data sets in respect of three models, namely, ECMWF, UKMO and Meteo France, for which the longest common series from 1959 to 2001 were present in the archives. The upper air data include the basic flow variables geopotential (Z), temperature (T ), wind components (u,v) and specific humidity (q) at 850, 500 and 200 hpa levels. The surface data are for the following elements: precipitation rate predicted by the model converted to equivalent monthly total precipitation (TP), mean sea level pressure (MSLP), 2-m temperature (2T ), 10-m wind components (10U, 10V ) and soil temperature level 1 (STL1). The raw data were required to be degribed, reformatted and composited to arrange them in the form of time series in respect of each individual variable. Our aim in the present study is to provide forecasts with 1 month lead time (extendedrange timescale); therefore we presently confine ourselves to data sets pertaining to 1-month lead only. The forecasts ofthree-modelnine-member (totally 27 members) ensembles were pooled together to work out the multi-model ensemble (MME) means in respect of each element. The inter-member standard deviation of total precipitation (TPSD), as a measure of ensemble spread, was also worked out to form another predictor variable. As to the choice of MMEs rather than individual models supplying the data sets for analysis, a supporting clue was found in some of the earlier studies, where a comparison has been made between forecasts of MME systems and individual models. For example, Doblas-Reyes et al. (2000), in a study of the skill of PROVOST (prediction of climate variations on seasonal to inter-annual timescales) long-range MME integrations, found that when using the full ensemble in a probabilistic formulation, the multimodel approach offers a systematic improvement. The improvement arises both from the use of different models in the ensemble and from the higher ensemble size obtained by combining all the models for building the MME. Especially in the Tropics, a part of the skill improvement is due to the MME approach. Hagedorn et al. (2005), in a study of DEMETER MME system, demonstrated that in both deterministic and probabilistic forecast scenarios, improvements are achieved by using MMEs instead of single-model ensembles. They further demonstrated that multi-model superiority is caused not only by error compensation but in particular by its greater consistency and reliability Logistic regression technique The logistic regression sets the probability Q that a value of predictand variable V shall exceed a threshold value corresponding to a given quantile q of the statistical distribution of the predictand. This is termed as the exceedance probability. An estimate of this probability can be obtained from the logistic regression equation, following Wilks (2006), as: ˆQ t (V t q) = exp(b 0 + b 1 X t ) 1 + exp(b 0 + b 1 X t ) (1) where t = 1,...,n is the index of variables in the time series, X is an explanatory variable; exp is the exponential function and b 0 and b 1 are the fitted regression constants to be estimated empirically from a data set of predictand and explanatory variables. In the case of two explanatory variables denoted by X 1 and X 2, the equation would take the form: ˆQ t (V t q) = exp(b 0 + b 1 X 1t + b 2 X 2t ) 1 + exp(b 0 + b 1 X 1t + b 2 X 2t ) (2) and so on. The regression parameters are estimated from a data set of predictors and the associated binary data series of predictand. The binary data series of predictand is constructed by assigning to its individual series items, the binary digit 1 if the predictand value exceeds or equals q, and 0 if it is less than q. The threshold value corresponding to a pre-specified quantile q, e.g. median (the 50 th percentile, denoted as p50 in this paper), is computed from the statistical distribution of the predictand data series. The model fitting can be carried out for different quantiles, such as deciles and terciles, to work out the exceedance probabilities for various threshold limits. Equations (1) and (2) produce an S- shaped prediction surface that is bounded by 0 <Q(V q) < 1 (Wilks, 2006). A computed value of ˆQ in a future event falling nearer to zero would mean a minimal chance of the predictand exceeding the given quantile threshold q and a value nearer one would indicate the opposite. The computer program for logistic regression analysis in the present study was prepared by an extensive customisation of a Fortran 90 software package authored by Allan Miller, which was downloaded from his website ( Verification methodology Brier score and Brier skill score Verification of estimates against observations is the most important aspect of any empirical modelling scheme. The most common method used for verification of probabilistic type of forecasts is the Brier score (after Brier, 1950). Amongst several studies on the use of Brier score in the verification of probabilistic forecasts, mention may be made of Ferro Christopher (2007), which compared ensemble-based probabilistic seasonal precipitation forecasts from the DEMETER project. Based on the above works, we define the Brier score in the following manner: if ˆQ t denotes the probabilistic forecast issued in respect

4 1580 K. PRASAD ET AL. of time element t as above, the estimate of Brier score ˆB is defined as: ˆB = 1 n n ( ˆQ t I t ) 2 (3) t=1 where I t = I(V t >q); I(A) = 1ifA is true, and I(A) = 0ifA is false. Summations are carried out over t = 1,...,n. The value of Brier score varies from 0 to 1; a relatively low value near to zero indicates a good fit of the computed model to the observations. To answer a question about the efficiency of a probabilistic forecast, it is customary to test the relative skill of the forecast to a baseline forecast such as climatology or persistence, or a combination of the two. For this purpose, a skill score can be computed, which expresses a relative improvement of the computed forecast against the reference forecast. In the present study, Brier skill score (BSS) with climatology as the baseline forecast is computed as: BSS = 1 (BS reg /BS clim ) (4) where BS reg is the Brier score obtained as a result of logistic regression analysis and BS clim is the Brier score obtained by taking climatology as the baseline forecast. The climatological exceedance probabilities are simply the theoretical probabilities of exceeding a given quantile; for example, the probability of a value exceeding the lower tercile (or 33rd percentile denoted as p33) would be taken as: ˆQ t (V t p 33 ) = 0.67 (5) for all time elements t = 1,...,n, and so on. The BS clim is then computed by Equation (3) in the same manner as BS reg and used in Equation (4) to compute the BSS. BSS would range from to 1; 0 or negative value indicates no skill. A positive value closer to 1 would indicate a high positive skill of the regression model as compared to climatology, with 1 being a perfect score The predictand variable The predictand in our study, as mentioned earlier, is the domain average monthly rainfall. The analysis has been carried out in respect of three target domains: the country as a whole, i.e. the entire grid domain 6.5N 38.5N, 66.5E 100.5E, and the meteorological subdivisions of Orissa (grid domain: 17.5N 22.5N, 82.5E 87.5E) on the east coast of India and Gujarat (grid domain: 20.5N 24.5N, 69.5E 73.5E) on the west coast. Time series from 1959 to 2001 of the domain averages of IMD rainfall (IMDR) for August of each year constituted the response variable data series. The elements of this series were standardized by subtraction of long period average (LPA) and division by standard deviation in order to remove the biases in the mean and standard deviation of the data series. The 30-year period was taken as the base period for computing LPA and standard deviation. Thus our predictand is the normalized rainfall rather than the absolute value Predictor variables The choice of predictor selection in any statistical downscaling scheme is a crucial step because the outcome of the downscaled scenario would largely depend on it. The selection process is one of the most challenging tasks and is complicated by the fact that the explanatory power of individual predictor variables may be low, and varies both spatially and temporally. Also, where a large number of variables are being considered simultaneously, the problem of inter-correlations amongst different variables, i.e. co-linearity, complicates the matter further. The current GCMs tend to provide outputs of a variety of surface and upper air variables and this has significantly increased the number and variety of potential predictors. It is the endeavour of the downscaling scheme to select a subset of a few predictors that would constitute the best combination, by a screening procedure. There are no hard and fast rules as to limiting the number of variables that should be considered in the process of screening. The basis of forming the main set of variables is to include those variables which prima facie are likely to have a sensible physical relationship with the predictand. Most of the statistical models make use of circulation flow variables as predictors, such as geopotential heights at selected isobaric levels, mean sea level pressure, specific or relative humidity and vorticity (e.g. Karl et al., 1990; Bardossy and Plate, 1992; Hay et al., 1992; Wilks, 1992; Conway et al., 1996; Katz and Parlange, 1996; Matyasovszky and Bogardi, 1996; Perica and Foufoula- Georgiou, 1996; Crane and Hewitson, 1998; Goodess and Palutikof, 1998; Kilsby et al., 1998; Wilby, 1998). Widmann et al. (2003) used GCM precipitation directly as the predictor for statistical precipitation downscaling, as in their view the GCM precipitation, in some sense, integrates all large-scale predictors. Salathe (2003) used this approach for simulation of streamflow in a rain shadow river basin. The data set of predictors in this study was assembled from the MME forecasts of the surface and upper air variables described in the earlier section. The entire set of predictors in our case has been divided into two subsets from apriori considerations. The primary subset was obtained by computing the domain average of each variable for the target area of analysis, namely, country as a whole, Orissa or Gujarat as the case may be (propinquitous, i.e. same location variables). Before computing the domain average, the MME data are re-gridded from the original grid so as to define them on the 1 1 IMDR grid. The variables considered as predictors in our analysis are the products of a dynamical system (GCMs), which intends to simulate the observed monsoon rainfall and therefore have an inherent physical linkage with the target variable, i.e. precipitation. Besides the basic flow variables provided by the MME hindcasts, a few derived variables such as vorticity, divergence and thickness between selected isobaric levels were computed. As we are dealing with the prediction of rainfall anomaly in a summer monsoon month, the consideration in framing the second subset of predictors was to

5 MONTHLY RAINFALL FORECASTS IN INDIA BASED ON DEMETER RE-FORECASTS 1581 include a few circulation parameters from the continental domain, spatially remote from the target area of analysis, which are known to have a bearing on the monsoon rainfall over India in extended timescales. This prompted us to include circulation variables such as low-level wind flow at 850 hpa over Arabian Sea, known as low-level jet (LLJ) or Somali jet; upper tropospheric (200 hpa) wind flow over the Indian peninsula and adjoining sea areas of Arabian Sea and Bay of Bengal, known as tropical easterly jet stream (TEJ) and high pressure cell in geopotential field in middle and upper troposphere over Tibetan area, known as Tibetan High. These are semi-permanent features of Indian monsoon circulation. In addition, SST over Arabian Sea, pressure gradient along the west coast of India and mean sea level pressure over the south Indian Ocean were also included. Indian monsoon on extended timescales is also strongly influenced by the anomalies of sea surface temperature over the stretch of Pacific Ocean the El Nino Southern Oscillation (ENSO) feature as a slowly varying boundary condition. As such, the SSTs over the four El Nino zones were also included in the set of predictors. In all, the list of predictors contained 36 items, including the MME predicted rainfall (TP), obtained as a monthly equivalent of the reported precipitation rate, and inter-member standard deviation of the 3 9 = 27 members of the ensemble (TPSD). Time series from 1959 to 2001 consisting of domain averages of each individual variable was constructed. 3. Results and discussions 3.1. DEMETER forecasts in simulating observed monthly rainfall over India performance evaluation As a first step in the course of our investigations, we look at the MME predictions of the domain averaged monthly total rainfall for the month of August against the corresponding observed rainfall series in respect of the three target areas of study. The time series of observed rainfall (IMDR), the corresponding lead 1 model-predicted total precipitation (TP) and standard deviation of TP (TPSD) for the all India rainfall for August are plotted in Figure 1. A prominent systematic bias in the predicted rainfall is evident; the predicted series lies much below the observed one. The systematic bias was corrected by normalizing both the series with reference to their respective LPA and standard deviation (calculated from 30-year base period ). The normalized TP and IMDR series for the three target areas of this study are presented in Figures 2 4. The simultaneous fluctuations in the two series for all India average (Figure 2) with a correlation of (statistically significant at 1% probability level) do point towards a potential skill of the DEMETER system in simulating the IAV of monthly rainfall on a sub-continental scale. However, in the case of Orissa (Figure 3) the correlation between TP and IMDR series is only 0.287, which was found to be statistically not significant, and for Gujarat (Figure 4), the correlation had a value of 0.327, which was significant only at 5% level. The low values of correlations found in the case of Orissa and Gujarat are indicative of the fact that the model skill declines as we go to smaller spatial scales. A closer scrutiny of the individual years, particularly in respect of events with large fluctuations (normalized rainfall closer to 1.5 or in excess on either side), reveals that in the case of all India series (Figure 2), there are a few occasions when the TP and IMDR peaks are in opposite phase, for example, in three major drought years 1968, 1972 and 1979 with normalized rainfall values of 2.0, 1.7 and 1.5, respectively, indicating a low skill of the prediction system in these years. On the other hand, during the excess rainfall years such as 1963 (1.9), 1973 (1.7), 1976 (1.8) and 1988 (1.3) the peaks are in phase, indicating that model simulations of observed rainfall have been skillful during the excess years. In the case of Orissa (Figure 3), the most conspicuous events, where the MME TP failed to capture the large anomalies in IMDR, were in 1997 (with a large positive anomaly of 1.5) and 1998 (with a large negative anomaly of 2.3). The years 1997 and 1998 happened to be an El Nino episode. Gujarat had 1979 as an outstanding year, when Figure 1. Time series showing monthly total rainfall for the month of August: top curve observed rainfall (IMDR); middle curve DEMETER model-predicted rainfall (TP); bottom curve inter-member standard deviation of multi-model ensemble members.

6 1582 K. PRASAD ET AL. Figure 2. Time series of normalized domain average total rainfall for the month of August: grid domain country as a whole (6.5N 38.5N, 66.5E 100.5E); base period for long period average ; solid curve observed rainfall (IMDR); broken curve model-predicted rainfall (TP); correlation coefficient between IMD and TP series Figure 3. The description of curves is as in Figure 2. grid domain Orissa (17.5N 22.5N; 82.5E 87.5E); correlation coefficient between IMD and TP series Figure 4. The description of curves is as in Figure 2. grid domain Gujarat (20.5N 24.5N; 69.5E 73.5E). correlation coefficient between IMD and TP series normalized rainfall of the highest positive magnitude 2.9 was recorded (Figure 4). The model-predicted rainfall in this case was in the same sense but much below expectation. It is interesting to note that 1979, which recorded the highest positive anomaly in the historical period in Gujarat, was a prominent drought year for the country as a whole. This only goes to support the importance and need for formulating monthly and seasonal forecasts on smaller spatial scales for their real utility. The above findings suggest that although the direct model predictions of weather elements of interest may have some useful skill in normal situations and on subcontinental spatial scales, theyfailto capturetheextremes in many cases. Therefore, it becomes necessary to fall

7 MONTHLY RAINFALL FORECASTS IN INDIA BASED ON DEMETER RE-FORECASTS 1583 back upon the strategy of obtaining the forecasts of weather elements in question via statistical downscaling methods, using model output variables as predictors, as in the MOS/PPM approach. It will be our endeavour to demonstrate in the subsequent sections that the logistic regression analysis with atmospheric variables as predictors gave probability estimates that were consistent with observations in the extreme anomaly cases outlined above, where direct model ensemble forecasts failed to capture the events Correlation analysis The explanatory power of each individual variable considered in the analysis was examined by computing the simple product moment correlation coefficients between the predictand and predictors for the three target areas, namely, country as a whole, Orissa and Gujarat (domain average rainfall in each case). The correlations were worked out based on the entire data series The correlations, which are significant at 5 and 1% levels in respect of the three domains, are shown in Table I. An important fact that emerges from Table I is that for the country as a whole, we find a large number of atmospheric predictors having statistically significant correlation. On the other hand, the list considerably shrinks when we consider the regional domains Orissa and Gujarat. In respect of the country as analysis domain, the modelpredicted total precipitation (TP) ranks the highest followedbythev component of wind at 500 hpa, 10-m u component of the wind and squared divergence at 500 hpa. For Orissa, the variables squared relative vorticity at 200 hpa, specific humidity at 500 hpa, squared divergence at 500 hpa and Tibetan High yield the highest correlations. Whereas for Gujarat, the correlations with the u component of LLJ over Arabian Sea, v component of wind at 200 hpa, squared vorticity at 200 hpa, model-predicted total precipitation and squared vorticity at 500 hpa were found to be significant. As mentioned in the preceding section, the modelpredicted total precipitation TP, which is often looked upon as providing the direct forecasts of actual precipitation (Widmann et al., 2003), turned out to be statistically significant and at the top rank in respect of the country as a whole. It was significant in case of Gujarat also but with much less magnitude and comparatively at a lower rank. It does not appear in the list of significant correlations for Orissa. Some of the variables like sea surface temperatures over Pacific (the El Nino factor), which are believed to have a strong influence on the Indian monsoon, also do not appear in the list of significantly correlated variables. It should be mentioned in this context that although Pacific SSTs do have a teleconnection with Indian monsoon rainfall in a qualitative sense, the quantitative relationship as expressed in correlations is not very significant, and there is no one to one relationship between the two. Table I. Significant correlations between predictand (domain average monthly rainfall) and predictor variables for the three regions of study, computed from data series Predictor variable Product moment correlation Country as whole (grid domain: 6.5N 38.5N, 66.5E 100.5) TP V U D PGRD V LLVB TPSD TEJ LLUB D TBHI Orissa (grid domain: 17.5N 22.5N, 82.5E 87.5E) Vo Q D TBHI Gujarat (grid domain: 20.5N 24.5N, 69.5E 73.5E) LLUA V Vo TP Vo Significant at 1% level; Significant at 5% level. TP, total monthly precipitation; V500, v component of wind at 500 hpa; 10U, u component of wind at 10-m level; D500 2, squared divergence at 500 hpa; PGRD, pressure gradient along west coast (difference of mean sea level pressure between monsoon heat low region: 25N 30N, 70E 75E and south Arabian Sea region: Equator 10N, 70E 75E); V850, v component of wind at 850 hpa; LLVB, v component of lowlevel wind flow (850 hpa) over Bay of Bengal; TPSD, inter-member standard deviation of TP (27 members of the DEMETER ensemble forecasts); TEJ, upper tropospheric (200 hpa) wind flow in tropical easterly jet stream region of Indian monsoon circulation; LLUB, u component of low-level wind flow (850 hpa) over Bay of Bengal; D850 2, squared divergence at 850 hpa; TBHI, thickness between 850 and 500 hpa levels over Tibetan region (30N 40N, 90E 120E), designated as Tibetan High; Vo200 2, squared vorticity at 200 hpa; Q500, specific humidity at 500 hpa; D200 2, squared divergence at 200 hpa; LLUA, u component of 850 hpa wind in the low-level jet region over Arabian Sea; V 200: v component of wind at 200 hpa; Vo500 2, squared vorticity at 500 hpa Logistic regression analysis The logistic regression analysis was applied to the normalized domain average precipitation as the target variable and the set of predictands as outlined in Section 2. Quantile limits for computing exceedance probabilities (as defined in Section 2.2) were set at seven percentiles: the 10th (lower decile), 25th (lower quartile), 33rd (lower tercile), 50th (median), 67th (upper tercile), 75th (upper quartile) and 90th (upper decile) of the statistical distribution of observed normalized rainfall (IMDR). The designated percentiles were computed from the sorted IMDR data series for the respective target area.

8 1584 K. PRASAD ET AL. Regression fitting was carried out recursively for each of the above quantile limits with the corresponding binary data series of IMDR constructed as explained in Section 2.2. We used the cross-validation approach (Wilks, 1995) to ensure the independence of training and evaluation data. In this approach, model fitting is carried out iteratively with the given data sample with one successive year excluded in each iteration. The regression estimates of excluded years at each step constitute the elements of verification sample. In our case the total length of the series was 43 years (from 1959 to 2001) and the training samples comprised 42 years (one verification year successively excluded in each iteration). This produced 43 data items in the verification sample the exceedance probabilities for the specified quantile. Brier scores (Section 2.3) were computed from the items of the verification samples. A two-stage procedure was designed in the selection of variables to constitute the final regression model. In the first stage the single-predictor equation (Equation (1)) was fitted with each individual variable separately in order to rank the variables on the basis of Brier scores (computed from cross-validated verification samples). The second stage consisted of fitting the multiple predictor model (Equation (2)) by entering the ranked variables in succession, beginning with the single topranked variable followed by the combination of two, three and up to four variables in order of their ranking. The model fitting was carried out for all seven quantile limits as described earlier. Although the ranking of variables in the screening procedure was not necessarily the same for the seven quantile limits, the top-ranked variables were found to be generally the same. For the purpose of second stage analysis, the screened variables corresponding to the median, obtained at the first stage, were chosen as the predictor variables for all seven quantiles. The above analysis was carried out in respect of the three target domains separately. The fitting of successive multiple predictor models brought out that a combination of one to three variables was an optimum number that yielded the lowest Brier scores. The screening procedure delivered the top-ranked variables as follows: (1) for the country domain: meridional component of wind at 500 hpa, 10M zonal wind component, squared divergence at 500 hpa, model-predicted total precipitation, with a Brier score value of 0.207; (2) for Orissa: squared relative vorticity at 200 hpa, zonal component of LLJ over Arabian Sea, squared divergence at 500 hpa, specific humidity at 500 hpa, with a Brier score value of and (iii) for Gujarat: squared relative vorticity at 200 hpa, meridional component of wind at 200 hpa, squared vorticity at 500 hpa, zonal component of LLJ over Arabian Sea, with a Brier score value of It is important to clarify at this point that the variables entering the regression and their ranking as per the criterion adopted for logistic regression analysis may not necessarily be the same as those based on simple correlations (Table I) Prediction of extreme events Our main objective in this study is to demonstrate the capability of a logistic regression model in providing skillful probabilistic forecasts of the anomalies of precipitation in the target zones. Of particular interest are the cases of large anomalies as identified in Section 3.1. With this view, the events with large peaks in observed rainfall series, both positive and negative, were picked up. The logistic regression fitting provides cross-validated estimates of exceedance probabilities corresponding to the chosen seven quantile levels at each point of the time series. The exceedance probabilities against quantiles were plotted for the selected extreme years. Figures 5 7 contain these plots in respect of all India, Orissa and Gujarat, respectively. The legend entries carry the normalized rainfall value of the concerned year alongside. The abscissa in these diagrams has the seven quantile limits denoted by p10, p25, etc., the left ordinate represents the exceedance probability and the right ordinate the normalized rainfall. The curves read the predicted probability (left ordinate) of the observed value of normalized rainfall exceeding a given quantile limit of its statistical distribution. Horizontal dashes in the diagrams correspond to the numerical values (right ordinate) of the respective quantiles. The data plots for several extreme years have resulted in two categories of events (positive anomalies and negative anomalies) producing two sets of curves appearing in two distinct clusters (Figures 5 7). In the entire region beyond the median (50th percentile), and in respect of most of the years in the region below the median, the cluster of curves representing positive anomalies (shown by solid lines) lies on the higher side of the probability scale relative to the one representing negative anomalies (shown by broken lines), with one or two exceptions as seen in the diagram in respect of Orissa (Figure 6). This implies that the exceedance probabilities estimated by the logistic regression analysis at any quantile limit above and including the median are higher in the positive anomalies cases than those of the negative anomalies cases, testifying to a good skill of the developed logistic regression model. Attention needs to be focused on the exceptional cases outlined in Section 3.1, namely, the three major drought years 1968, 1972 and 1979 for all India, the two contrasting years 1997 (high positive anomaly) and 1998 (high negative anomaly) for Orissa and the highest positive anomaly year 1979 for Gujarat, where the model-predicted TP failed to represent the observation. First, considering the all India case, the exceedance probabilities corresponding to median are 6.4, 27 and 40.9% for the years 1968, 1972 and 1979, respectively. Although in 1979 this probability is somewhat higher (still less than 50%), in the other two years it is quite low, consistent with the observed negative anomaly. In the case of Orissa, the exceedance probability of median is 92.4% in 1997 (excess rainfall year) and only 23.4% in 1998 (deficit year). For Gujarat, the probability of exceeding the median works out as 67.9%

9 MONTHLY RAINFALL FORECASTS IN INDIA BASED ON DEMETER RE-FORECASTS 1585 Figure 5. Exceedance probability versus quantile limits by cross-validated logistic regression analysis in respect of extreme rainfall years: grid domain country as a whole; solid curves excess years; broken curves deficit years; graph provides the probability (left ordinate) of an observation exceeding a given value of quantile corresponding to p10 (10th percentile), p25 (25th percentile), etc. on the abscissa; horizontal dashes in the diagram indicate the numerical values (right ordinate) of respective quantiles in the statistical distribution of observed normalized rainfall; figures in braces in the legend box at bottom are the values of normalized rainfall for the year; top-ranking predictor variables entering the regression meridional component of wind at 500 hpa, 10M zonal wind component, squared divergence at 500 hpa, model-predicted total precipitation; Brier score corresponding to median Figure 6. The description of curves is as in Figure 5. grid domain Orissa; top-ranking predictor variables entering the regression squared relative vorticity at 200 hpa, zonal component of low-level jet over Arabian Sea, squared divergence at 500 hpa, specific humidity at 500 hpa; Brier score corresponding to median in 1979 (excess year). Thus in all the above cases, the logistic regression model was able to capture the extreme events, which were not well represented by direct model forecasts, establishing its superiority over the latter Comparison of forecasts produced by atmospheric variables versus model-predicted precipitation as explanatory variables Some of the earlier studies on the application of logistic regression in medium-range forecasts were based on the ensemble mean forecasts of precipitation itself as the predictor (Hamill et al., 2004) and in one study, the ensemble spread as an additional predictor (Wilks and Hamill, 2007). Taking a clue from these studies, we examined the potential of the ensemble mean precipitation (TP) and its standard deviation (TPSD) for fitting a two-predictor model in a way similar to the case of one subdivision namely Orissa. The results obtained are shown in Figure 8. The performance of the model based on these two variables, vis-à-vis the ones selected as per the screening criteria set in our study, is self-revealing

10 1586 K. PRASAD ET AL. Figure 7. The description of curves is as in Figure 5. grid domain-gujarat; top-ranking predictor variables entering the regression squared relative vorticity at 200 hpa, meridional component of wind at 200 hpa, squared vorticity at 500 hpa, zonal component of low-level jet over Arabian Sea; Brier score corresponding to median Figure 8. The description of curves is as in Figure 5. grid domain Orissa; predictor variables forced as model-predicted total precipitation and inter-member standard deviation of total precipitation; Brier score corresponding to median (compare Figures 6 and 8). In this case although the curves corresponding to the two contrasting years 1982 (excess) and 1987 (deficit) are in right positions in the diagram relative to each other, those for 1997 (excess) and 1998 (deficit) show a response that is opposite to observed anomalies the curve for 1998 on the higher side on probability scale and the one for 1997 on the lower side. The Brier score has a value of 0.261, which is much larger than the value of based on atmospheric variables. A model using only TP and TPSD as predictor variables was thus a failure in these two important events. This suggests that using the total precipitation as the sole predictor in this type of models, as used by some other workers, may not be the best choice and the model based on other model output variables may perform better Brier skill score In order to evaluate the relative skill of probability forecasts with reference to climatology as the baseline forecast, the BSSs, as defined in Section 2.3, were computed for the median in respect of the three target areas. Plots of BSS are shown in Figure 9. Positive values of BSS are indicative of a higher skill of the fitted model over that of climatology. 4. Summary and concluding remarks In this study we have attempted to examine the potential of a multi-predictor logistic regression model in generating probabilistic forecasts of precipitation anomaly in the month of August for all India and two of its selected meteorological subdivisions, namely, Orissa and Gujarat with a lead time of 1 month. DEMETER retrospective forecasts have been used for the purpose. Time series of domain average rainfall for August as predictand and a large number of basic flow variables and derived parameters from within the domain under consideration,

11 MONTHLY RAINFALL FORECASTS IN INDIA BASED ON DEMETER RE-FORECASTS 1587 Figure 9. Brier skill scores of logistic regression forecast probability estimates for median relative to the climatological probabilities as the base line forecasts in respect of the three target areas of study. and variables from the continental domain of large-scale global monsoon circulation were constructed. The multimodel forecasts (ensemble of ECMWF, UKMO and Meteo France) of 1 month lead from 1959 to 2001 formed the core of the data set used for training the model. Seven quantile limits of the statistical distribution of normalized rainfall in the three domains individually were set for computation of exceedance probabilities from the fitted regression model. The seven quantile limits chosen are the lower decile, lower quartile, lower tercile, median, upper tercile, upper quartile and upper decile. The ranking and selection of variables to form the predictors was done on the basis of Brier scores, which were computed from a series of independent verification sample obtained by fitting the logistic regression by a cross-validation approach. One of the key objectives of the present study was to evaluate the potential of the logistic regression model in capturing extreme events of rainfall anomalies in the inter-annual time series of monthly total rainfall and compare its performance vis-à-vis the direct model forecasts of the parameter in question produced by a MME system comprising the GCMs (DEMETER). Several years with large anomalies (both positive and negative) were picked up from the time series of normalized rainfall of all the three regions under study and plots of exceedance probabilities in each category of quantile limits were prepared for the selected years individually. The plots generated two sets of curves in two distinct clusters corresponding to positive anomaly cases (lying in the higher side of the probability scale) and negative anomaly cases (lying in comparatively lower side of the probability scale), thereby indicating consistency with the observed anomalies and a good skill of the fitted model. A revealing outcome of the analysis was that some of the extreme events that failed to show up as having been well represented by the direct model forecasts were captured by the probability forecasts in logistic regression analysis based on atmospheric variables, thus establishing its superiority over the direct MME forecasts. A comparison of the two sets of regression models in a case study, one with the atmospheric predictor variables selected as per the screening criteria designed in our work and the other with ensemble mean monthly precipitation and its spread as the predictor variables given by the DEMETER system, and suggested by some other workers, indicated that the forecasts produced by the latter were inferior in that the extreme events were not well captured. This leads us to believe that the GCM simulated precipitation may not be the most powerful predictor of observed precipitation vis-à-vis the circulation-based predictors in statistical regression techniques. It would be pertinent to mention as a concluding remark that the present study was confined to only 1 month, i.e. August, as the data sets in the DEMETER system used for our work were available only for this month of the Indian monsoon period (June September). Further, the study was confined to only 1-month lead time in keeping with our objective of producing downscaled forecasts for extended range, i.e. monthly timescales. From operational point of view, although the GCM forecasts of 1 month lead, if available right at the beginning of the month, could be utilized for producing downscaled forecasts of the target weather element a month in advance, in an ideal situation one should be able to generate the forecasts with at least a couple of months lead time for the information to be more useful. We recognize the need to extend this work for greater lead times and with various other GCMs outputs currently available. We used only three models in the DEMETER system due to uniformity of data length in respect of these models. Combining the data sets from several other GCMs in an MME approach may lead us to a better skill, which remains a future work item. A serious limitation of the empirical models like the one developed in our present study remains linked