Forecasting Maximum and Minimum Temperatures by Statistical Interpretation of Numerical Weather Prediction Model Output

938 WEATHER AND FORECASTING Forecasting Maximum and Minimum Temperatures by Statistical Interpretation of Numerical Weather Prediction Model Output PARVINDER MAINI, ASHOK KUMAR, L.S.RATHORE, AND S. V. SINGH National Centre for Medium Range Weather Forecasting, Department of Science and Technology, New Delhi, India 23 April 2002 and 3 February 2003 ABSTRACT The inability of a general circulation model (GCM) to predict the surface weather parameters accurately necessitates statistical interpretation of numerical weather prediction (NWP) model output. Here a system for forecasting maximum and minimum temperatures has been developed and implemented for 12 locations in India based on the perfect prog method (PPM) approach. The analyzed data from the ECMWF for a period of 6 yr (1985 90) are used to develop PPM model equations. Daily forecasts for maximum and minimum temperatures are then obtained from these equations by using T-80 model output. In order to assess the skill and quality of the temperature forecasts, an attempt has been made to verify them by employing the conditional and marginal distribution of forecasts and observations using the data of four monsoon seasons from 1997 through 2000. 1. Introduction India has an agro-based economy. Despite considerable advancements and improvements to irrigation facilities, Indian farmers are still dependent on seasonal rains, which are highly variable both in time and space. The rains adverse effects may be partially reduced if the occurrence of the events is predicted in advance and farmers are suitably advised to take ameliorative measures. Thus, the weather forecast assumes considerable importance for agricultural activities. Conventional weather forecasts based on subjective methods are normally available 1 day in advance. Although useful for other applications, these are inadequate for planning weather-based agricultural practices, as the desired lead time needed for taking precautionary measures in agriculture is longer. Considering the need for mediumrange weather forecasts for the Indian agriculture, the government of India established the National Centre for Medium Range Weather Forecasting (NCMRWF) in 1988, at New Delhi, with the major objective of developing operational medium-range weather forecasting capabilities and agrometeorological advisory services (AAS) for the farming community for each of the 127 agroclimatic zones of the country. The weather forecasts up to 3 days in advance began to be produced and distributed to various user agencies by running a global numerical weather prediction model, initially the R-40 model from the Center for Ocean Corresponding author address: Dr. Parvinder Maini, National Centre for Medium Range Weather Forecasting, Mausam Bhavan, Lodi Road, New Delhi 110 003, India. E-mail: pmaini@ncmrwf.gov.in Land Atmosphere (COLA) and later the T-80 model from the National Centers for Environmental Prediction (NCEP). The latter has been running on an operational basis from 1 June 1994 (updated to T170L28 since January 2002) for providing medium-range forecasts. NCMRWF started providing weather-based AAS on an experimental basis in 1991, through its network of agromet field units (AMFU) collocated at the State Agricultural Universities (SAU). The service was established in a phased manner, and currently NCMRWF is issuing biweekly forecasts in the medium range valid for 4 days and providing weather-based agroadvisories to the farming community through its network of AAS units spread across 82 agroclimatic zones of the country. In addition, a forecast for weekly cumulative rainfall is also given. To issue these forecasts, a forecasting system for giving objective, location-specific forecasts was developed at NCMRWF (Kumar et al. 2000) based on a man machine mix approach. The basic information used in this system is the direct model output (DMO) from the NWP (T-80) model, statistical interpretation (SI) of NWP model output, graphical output in the form of analysis and prognostic charts, synoptic interpretation, and subjective judgment of the forecaster. Surface weather parameters like maximum and minimum temperatures play an important role in agriculture. Hence, their accurate prediction is necessary for the farmer to plan his day-to-day agricultural operation so that any adverse impact due to extreme temperature events can be minimized. The variations in the daily temperatures are greatly affected by the local topography. The GCMs, though able to provide reasonably good medium-range weather forecasts, have comparatively 2003 American Meteorological Society

OCTOBER 2003 NOTES AND CORRESPONDENCE 939 less skill in forecasting surface parameters. There are several reasons why statistical intepretation of dynamical NWP output is necessary for forecasting surface weather (Wilks 1995). There are important differences between the real world and its representation in the NWP models. The NWP models necessarily simplify and homogenize surface conditions, by representing the world as an array of grid points. Due to this, small-scale effects (e.g., topography or small bodies of water) important to local weather may not be included in the NWP model. Tropical circulations/weather systems are governed by physical forcings. The physical processes, which are subgrid scale, are represented in the NWP models in the parameterized form. Thus the NWP models may not represent locations and variables for which forecasts are desired explicitly. However, statistical relationships can be developed between the information provided by the NWP models and desired forecast quantities to help alleviate these problems. The NWP models are not perfect, and their forecasts are subject to errors, some of which are systematic. This may be attributed to deficiencies in model physics. Statistical forecasts based on the NWP information can compensate for and correct some of the forecast biases. The NWP models are deterministic and hence cannot fully account for the stochastic nature of weather. But NWP information used in conjunction with statistical methods allows for quantification and expression of the uncertainty associated with different forecast situations by means of probabilistic forecasts. Due to these drawbacks of the NWP models in predicting surface parameters, work on statistical interpretation of model output was undertaken at NCMRWF. The basic principle of SI is to develop a concurrent relationship between the surface weather at a location and the contiguous upper-air circulation, thereby accounting for the deficiencies of a NWP model. There are basically three methods of doing statistical weather forecasting (Murphy and Katz 1985; Wilks 1995). One is the classical method. This approach does not use numerical model forecasts and relies purely on observed data. Before dynamical models were available, operational statistical systems were limited to the classical approach. The accuracy of any given forecast based on this approach is strongly dependent on whether significant changes occur in the atmosphere between the times of the predictor observations and the time of validity of the resultant forecast. The greater the forecast projection, the greater will be the chance of significant changes. Hence, this approach is good for short-range forecasts, but its skill falls sharply beyond a few hours. The classical approach is also used for very long-range seasonal forecasts where there is little skill in numerical model predictions. This approach has practically no skill in the medium range. The other two methods, which improve over the NWP forecasts, are being commonly used in most of the operational centers the world over for giving mediumrange forecasts. One is the perfect prog method (PPM; Klien et al. 1959) and the other is model output statistics (MOS; Glahn and Lowry 1972). Both of these methods utilize a two-step procedure. At the first stage, equations are developed between the upper-air fields and the surface parameter to be predicted. In the case of PPM, the upper-air fields are the past observations or analysis and in the MOS it is the past forecast. In the second stage these equations are used to prepare the actual forecast. It has been well established that MOS gives better forecasts than PPM because of its ability to account for some of the systematic errors in the NWP model. It has been used operationally for the past couple of decades in the United States and elsewhere. However, today s rapidly changing modeling environment prevents the wide usage of the MOS technique. This is because every time a significant change in the numerical model is made, the MOS equations have to be redeveloped. On the other hand the PPM forecasts do not deteriorate when a model is changed, and the same equation will hold good. Studies indicate that at least 2 yr of archived data from model runs are needed to derive a useful MOS equation (Jacks et al. 1990), although previous studies by Carter (1986) have indicated that stability is achieved with a larger sample of archived data. However, it is required that during the archival period, the model configuration should be kept frozen. The T80L18 spectral model, which has been operational at the center since 1994 and is run every day to produce forecasts up to 7 days, has undergone many changes in the model physics in recent years (planetary boundary layer, convection, and radiation). Under such an unstable model environment, the MOS technique is apparently not a suitable method for reinterpreting forecasts. Thus at NCMRWF, development of equations based on the PPM technique was taken up. Considering the fact that location-specific forecasts have to be given twice a week to the farming community for the subsequent 4 days, SI models have been developed for forecasting rainfall (Kumar et al. 1999) and temperature during the monsoon and winter seasons at stations located in different agroclimatic zones. Results obtained for 12 stations during the monsoon season are presented in this paper. Forecasts are also obtained from the DMO by bilinear interpolation. Comparative studies of both DMO and SI (Maini et al. 2002) over the past few years have established that SI forecasts are a definite improvement over DMO. In order to assess the quality and skill of the forecasts and to identify the relative strengths and weaknesses of the temperature forecasts, an attempt has been made to do a diagnostic verification by employing the conditional and marginal distribution of forecasts and observations based on the data of four monsoon seasons from 1997 to 2000. In section 2, the procedure used for developing the equations based on the PPM method is explained. Section 3 discusses the methodology used to

940 WEATHER AND FORECASTING FIG. 2. Reference time for maximum and minimum temperatures. FIG. 1. Locations under study marked on a map of India. obtain the SI forecast from the daily run of the T-80 model including the bias correction method. Section 4 describes the approach to forecast verification based on quality, accuracy, and skill. The conclusions drawn from the study are discussed in section 5. This section also discusses some of the future plans. 2. Development of equations a. Data India is a tropical country; thus, by virtue of its location and complexity of the weather systems encountered, a large variation in the day-to-day weather is expected. The data-sparse regions like the seas on three sides and the great Himalayas on the fourth makes forecasting even more difficult. As stated earlier, in order to develop stable equations, data for at least three seasons of 6-month duration (Carter 1986) are required. In order to account for all types of variability in the weather in India, stable station-specific equations need to be developed using a longer period of record. The T-80 model was implemented at NCMRWF in 1993 and statistical interpretation of NWP models commenced in 1994. Hence due to the availability of only a limited amount of developmental data from the T-80 model, neither PPM nor MOS equations based on the T-80 model could be developed. The analyzed fields from the European Centre for Medium-Range Weather Forecasts (ECMWF) Tropical Ocean Global Atmosphere (TOGA) basic level III datasets, which are part of the ECMWF World Climate Research Programme (WCRP) level III- A Global Atmospheric Data Archive analysis, were used to develop PPM-based SI models at NCMRWF. PPM models are developed for generating forecasts of maximum and minimum temperatures for 12 locations (Fig. 1) during the monsoon season. The main statistical technique used is multiple linear regression. The regression equations are developed, using 6 yr of TOGA analyses (2.5 2.5 ) from 1985 to 1990 as the predictors, and the actual observed values of temperature as the predictands for the same period. The period of the monsoon season is taken to be June August for northwestern Indian stations and as June September for the rest of the country. TABLE 1. Meteorological parameters chosen as predictors. Level (hpa) Parameters chosen at the level 1000, 850, 700, 500 Relative humidity, temperature, and advection of temperature, zonal, and meridional wind component, vertical velocity, geopotential height, vorticity, and advection of vorticity 850 700, 700 500 Temperature gradient and advection of temperature gradient 850 500 Thickness 1000 500 Saturation deficit, precipitable water, horizontal water vapor flux divergence, mean relative humidity, rate of change of moist static energy Surface Mean sea level pressure FIG. 3. The grids considered around a station.

OCTOBER 2003 NOTES AND CORRESPONDENCE 941 TABLE 2. Predictors frequently selected during monsoon season for different predictands. Predictand No. of predictors Predictors Max temp Two to three 1000 500-hPa saturation deficit 850-hPa temperature Min temp Three to seven 850-hPa temperature, 500-hPa temperature, 850 500-hPa thickness b. Possible predictors The predictors used to develop the PPM equations include analyzed values of temperature, temperature advection, geopotential height, vorticity, advection of vorticity, wind speed, thickness, relative humidity and wind components at various levels in the lower and middle troposphere, precipitable water content, saturation deficit, and mean relative humidity between the surface and 500 hpa. The rate of change of moist static energy and horizontal water vapor flux divergence between the lower and middle troposphere (500 hpa) are some of the derived meteorological parameters that are also considered. The list of predictors is given in Table 1. c. Reference time The valid time of the predictors is taken near the time of occurrence of the maximum and minimum temperatures. Figure 2 shows the time at which the maximum and minimum temperatures are attained in 24 h on any calendar date. Based on this, the reference time at which the values of the predictor are to be considered for developing the model equations is chosen as 0000 UTC of the same day for minimum temperature and 1200 UTC of the previous day for the maximum temperature. d. Predictors selected The value of a predictor at a station is best represented by its value at the nearest grid point and the surrounding grid points. As shown in Fig. 3, nine grid points are considered around the station of interest (Tapp et al. 1986). To get the best linear combination of the values of a predictor at the nine grid points, canonical correlations (Rousseau 1982) are obtained between the predictand (maximum/minimum temperatures) of a station and the predictors at the nine grid points by using 6 yr (1985 90) of monsoon season data. The best linear combination obtained has the maximum correlation with the predictand. These linear combinations are obtained for each of the predictors to provide a new set of potential predictors. As the correlations are obtained between one predictand and several predictors, the canonical correlations in this case reduces to multiple linear regression. To eliminate predictors that contain redundant information, the new set of potential predictors is screened with the stepwise selection procedure. The predictors that explain most of the variance are selected. The selection procedure is terminated when the addition of a further variable to the prediction equation contributes less than a critical value to the percentage of variance explained by the predictors already selected (Tapp et al. 1986). In order to have a significant percentage of variance explained by the predictors selected, this value is taken as 1.0% for both maximum and minimum temperatures. Two to three predictors are selected for maximum temperature, and for the minimum temperature three to seven predictors are selected (Table 2). Saturation deficit at 1000 500 hpa and temperature at 850 hpa are most frequently selected in predicting maximum temperature, whereas for prediction of minimum temperature, temperatures at 850 and 500 hpa and 850 500-hPa thickness play a significant role. A unique set of predictors is obtained for all 12 stations. Table 3 shows the predictors obtained for three stations, namely, Udaipur, Delhi, and Jabalpur. These selected predictors are then used to develop location-specific PPM models. The regression coefficients (a i s) are obtained by linearly regressing the predictands (Y) with the selected predictors (X i s). The following linear equation is obtained: n 0 i i i 1 Y a ax. (1) Hence a unique set of coefficients is obtained for all the 12 stations under study. 3. Forecast a. Daily forecast An SI forecast for maximum and minimum temperature is obtained every day for the subsequent 4 days Station TABLE 3. Predictors selected for three stations during the monsoon season. Max temp Predictors selected Min temp Udaipur 850-hPa temp, 1000-hPa relative humidity 850-hPa temp, 1000- and 850-hPa relative humidity, 700-hPa temp Delhi 1000 500-hPa saturation deficit, 850-hPa temp 850- and 500-hPa temp, 850 500-hPa thickness, 1000 500-hPa mean relative humidity Jabalpur 1000 500-hPa saturation deficit, 850 500-hPa thickness, 1000-hPa relative humidity 1000 500-hPa saturation deficit; 850 500-hPa thickness, 1000 500-hPa mean relative humidity, 500-hPa zonal wind

942 WEATHER AND FORECASTING FIG. 4a. Conditional quantile plots for max temp forecast at Udaipur. at an interval of 24 h. This is done by substituting the X i s in (1) by the corresponding forecast of the predictors (F i ) obtained from the T-80 model forecast valid for that particular hour. Thus, where n j 0 i ij i 1 Y a af, j 24, 48, 72, 96 h. (2) Hence in the development of the location-specific PPM models, the developmental data are provided by the ECMWF analyses, but in implementation of these models the T-80 forecast of the predictors is used. It is expected that the forecast thus obtained will be biased. This implies that the temperatures may be either over-

OCTOBER 2003 NOTES AND CORRESPONDENCE 943 Fig. 4b. Conditional quantile plots for min temp forecast at Udaipur. or underforecast and this bias should be eliminated in order to get a more accurate forecast. b. Bias removal The mean error (ME) for a sample of forecasts f and observations x is a measure of the unconditional bias in the forecasts (see the appendix). The unconditional or systematic bias can be removed by using the observations and SI forecast of the predictand obtained during two or three previous seasons. This is because bias corrections are useful only if the sample used to determine the bias is representative of the population of forecasts produced by the technique. This means that the sample used to determine the bias correction should be large. Thus in the 1997 and 1998 monsoon seasons, the bias is removed by using observed and forecasted values of the 1994, 1995, and 1996 monsoon seasons. In the 1999

944 WEATHER AND FORECASTING FIG. 5a. Conditional quantile plots for max temp forecast at Delhi. and 2000 monsoon seasons, the observed and forecast values of the 1996, 1997, and 1998 monsoon seasons are used. The value of ME obtained in each case is added to the corresponding SI forecast to obtain a biasfree SI forecast. Thus a bias-reduced SI forecast is obtained for both maximum and minimum temperatures for the 1997, 1998, 1999, and 2000 monsoon seasons. 4. Forecast verification Forecast verification is the process of determining the quality of forecasts. Analysis of the verification statistics and their components can also help in the assessment of specific strengths and weaknesses of the forecasting system (Wilks 1995). The SI forecasts obtained in the 1997, 1998, 1999,

OCTOBER 2003 NOTES AND CORRESPONDENCE 945 Fig. 5b. Conditional quantile plots for min temp forecast at Delhi. and 2000 monsoon seasons taken together for 12 stations are verified against the corresponding observations. In order to assess the forecast quality, a detailed verification is done by considering the conditional and marginal distributions of the forecasts and observations. Different aspects of forecast accuracy and skill are also presented. a. Forecast quality Conditional quantile plots Insight into fundamental characteristics of forecasting performance, as well as into the statistical characteristics of the forecasts and observations, can be obtained by examining the joint distribution of forecasts and observations (Murphy and Winkler 1987). If

946 WEATHER AND FORECASTING FIG. 6a. Conditional quantile plots for max temp forecast at Jabalpur. the forecasts and observations are denoted by f and x, respectively, then p( f, x) is the joint distribution of f and x. The information contained in p( f, x) is more accessible when it is factored into conditional and marginal distributions: p( f, x) p(x/f ) p( f ) p( f, x) p( f/x) p(x). (3) These expressions are referred to as the calibration refinement and likelihood-base-rate factorization of p( f, x). The conditional distribution p(x/ f ) [or p( f/x)] describes the relationships between the observations (or forecasts) given the forecasts (or observations) and provides information concerning several dimensions of forecast quality (Murphy et al. 1989). The marginal distributions p( f ) or p(x) describe the unconditional occurrence of the various possible forecasts or observations. In the first factorization, the distributions p(x/ f ) and p( f ) relate to two distinct characteristics of the fore-

OCTOBER 2003 NOTES AND CORRESPONDENCE 947 Fig. 6b. Conditional quantile plots for min temp forecast at Jabalpur. casts: calibration and refinement. Forecasts are said to be perfectly calibrated (or completely reliable) if E(x/ f ) f for all f, where E(x/ f ) is the expected (or mean) value of conditional distribution p(x/ f ). Thus, a temperature forecasting system is perfectly calibrated if, for each forecast value f, the mean observed temperature is equal to f. The marginal or predictive distribution of the forecasts, p( f ) relates to the refinement of the forecasts. A temperature forecasting system that produces the same forecast on each occasion is completely unrefined. However, for perfectly accurate forecasts, p( f ) is necessarily identical to the marginal distribution of the observation, p(x). In the second factorization, the distribution p( f/x) refers to the likelihood, since they indicate the likelihood that a particular forecast is associated with a given observation. The marginal distribution p(x) specifies the probability of occurrence of the respective observations. The primary objective of this paper is to bring out the forecasting performance of the SI models. In order to see

948 WEATHER AND FORECASTING TABLE 4. Yule Kendall index to check for sample skewness. Station Udaipur 0.4 Delhi Jabalpur Max temp 24 h 48 h 72 h 96 h 0.4 0.6 0.5 Min temp 24 h 48 h 72 h 96 h 0.5 0.4 0.5 1.0 0.4 0.5 the reliability of the forecasts generated by statistical interpretation of NWP products, conditional plots of p(x/ f ) are obtained for all the stations. Here the plots for three different stations for both maximum and minimum temperatures and are shown in Figs. 4 6. Quantiles, in particular the medians of the conditional distribution, provide information about conditional bias (or calibration). These quantiles also describe the way in which the variability in the observations changes as a function of the forecast. In order to assume that the median is a reasonable measure of central tendency, it is necessary to ascertain the symmetry or skewness of the data. A robust and resistant method to compute the sample skewness is the Yule Kendall index (Wilks 1995): ] [( f0.75 f 0.5) ( f0.5 f 5) YK. (4) IQR It is computed by comparing the distance between the median and each of the two (upper and lower) quartiles. IQR is the interquartile range. If the data are rightskewed, at least in the central 50% of the data, the Yule Kendall index will be greater than zero. Conversely, leftskewed data will be characterized by a negative Yule Kendall index. Table 4 gives the values of symmetry (24, 48, 72, and 96 h) obtained for both maximum and minimum temperatures at three stations, namely Udaipur, Delhi, and Jabalpur. It is seen that the measure is nearly zero in all the different ranges in which the forecast has been divided. This indicates that the distributions are reasonably symmetric and approximately normal. In such a situation, the difference between the mean and the median is quite small and median can be used as a measure to determine the bias in the distribution (Murphy et al. 1989). To obtain the conditional quantile plots, the forecast is first divided into five ranges. In each range, for every value of f, a corresponding value of p(x/ f ) is obtained. Quantiles are then obtained for the forecast and also for p(x/ f ). The conditional quantile plots in Figs. 4 6 display various quantiles of the conditional distributions of observed temperature (maximum/minimum) given the forecast temperature (maximum/minimum). These diagrams also show the 0th, 5th, 0.50th (median), 0.75th, and 0.90th conditional quantiles for three stations in India, namely, Udaipur, Delhi, and Jabalpur. The 45 line in the conditional quantile plots is included for the purpose of comparison with the median value. It is reasonable to assume that the deviation of the conditional medians from the 45 line is an indication that the forecasts are conditionally biased. Moreover, the difference between the upper and lower quartiles, (x 0.75/ f ) (x 5/ f ), and the difference between the 0.90th and 0th quantiles, (x 0.90/ f ) (x 0/ f ), describe the conditional accuracy of the forecasts. In Fig. 4a, the conditional quantile plot for maximum temperature at Udaipur shows that the median line is just below the 45 line. This indicates slight overforecasting (x 0.5/ f f ) by the SI models whereas in Fig. 4b, the median is partly above the 45 line for temperatures less than 25 C and partly below the 45 line for higher temperatures. This indicates that SI models overforecast higher values of minimum temperature at Udaipur and slightly underforecast temperature less than 25 C. Figures 5a and 5b show the conditional quantile plots of maximum and minimum temperature forecasts at Delhi. In the case of maximum temperature, the median line is almost touching the 45 line for temperatures above 34 C. This indicates that the forecasts are fairly unbiased for higher temperatures but are slightly underforecast for lower values of the observed maximum temperature. Figure 5b shows that the SI forecasts for minimum temperature at Delhi have a slight tendency toward underforecasting (x 0.5/ f f ) for all f. The conditional quantile plots of Jabalpur are shown in Figs. 6a and 6b. It is observed that the maximum temperatures are underforecast for higher observed temperatures as shown by the median line, which moves steeply after

OCTOBER 2003 NOTES AND CORRESPONDENCE 949 FIG. 7. Std dev of the forecasts and observations as a function of lead time. crossing 34 C. In the case of minimum temperature at Jabalpur, the SI forecasts exhibit a tendency toward slight overforecasting; that is, (x 0.5/ f f ) for all f. In general it is observed that for all three stations the median line is almost touching the 45 line, indicating that even though the temperature forecasts are either over- or underforecast, the amount of deviation of the forecast from the observed is considerably less. The 0th and 0.90th quartiles show that forecasts are fairly symmetric as they are equidistant from the 45 line and are conditionally accurate. The traditional summary measures such as mean and variance for both forecasts and observations are used to summarize the marginal distributions p(x) and p( f ). The means and standard deviations of the observed and forecasts for both maximum and minimum temperatures are obtained for the three stations Udaipur, Delhi, and Jabalpur. It is found that the mean forecast temperature in the case of both maximum and minimum temperature is within a range of C of the mean of the corresponding observed temperature for all lead times. This indicates that the SI forecasts are relatively unbiased. Comparison of the standard deviation of forecast and observed temperature (maximum/minimum) is shown in Fig. 7. It is seen that the standard deviation of the observed temperature is slightly more ( 1 C) than the corresponding forecasts except in the case of maximum temperature at Delhi where it is otherwise. This shows

950 WEATHER AND FORECASTING TABLE 5. Skill of max and min temp forecasts during the monsoon periods of 1997 2000. Station Akola Anand Bhuvaneshwar Coimbatore Delhi Hisar Jabalpur Ludhiana Pantnagar Raipur Ranchi Udaipur Max temp Lead time 24 h 48 h 72 h 96 h 0 0.54 0.41 9 2 7 0.40 0.52 0.57 0.47 0.58 0.44 3 0.48 5 7 7 6 8 0.44 0.46 0.46 0.51 1 7 0.43 8 7 5 0 0.40 7 6 0.46 0.44 4 1 3 0 1 7 3 4 2 4 0.41 7 1 Min temp Lead time 24 h 48 h 72 h 96 h 3 0.59 4 9 3 1 9 1 0.45 6 0.44 7 0 0.54 7 9 2 2 5 4 2 7 0.41 8 7 0.51 2 1 8 5 0 1 2 6 0.45 5 8 0.47 2 8 2 9 3 6 1 9 0.45 4 that the forecast temperatures are able to account for the variability in the observed temperatures within reasonable limits. The reduced variability of the observed maximum temperature over Delhi can be attributed to fewer rainy days in Delhi, which affects the maximum temperature. b. Forecast skill Skill is the accuracy of the forecasts of interest relative to the accuracy of forecasts produced by a standard of reference such as climatology or persistence. The skill of an SI forecast is found for all 12 stations using MSE clim as the reference forecast where MSE clim is as given in the appendix. The subscript clim indicates climatology. Hence, MSE SSclim 1. (5) MSE clim The skill scores are given in Table 5. It is seen that the skill scores for both maximum and minimum temperatures are reasonably high for 24-h forecasts but decrease with increasing lead time. In the case of maximum temperature it is around 0.4 on the average and in the case of minimum temperature it is around. It is also observed that the skill of the maximum temperature is higher for all stations as compared to the skill of the minimum temperatures. The skill scores for Bhuvaneshvar and Coimbatore are as low as. This can be attributed to certain environmental features around the station, which may not be well represented by the model. c. Forecast accuracy Accuracy refers to the average correspondence between individual pairs of forecasts and observations over the verification sample. Some of the common measures of accuracy are mean absolute error (MAE), mean square error (MSE), and root-mean-square error (rmse) (see the appendix). The correlation coefficient and rmse are obtained for maximum and minimum temperatures between the observed temperatures of the 1997 2000 monsoon seasons and the forecast temperatures for the same period. Table 6a gives the correlation coefficient for both maximum and minimum temperatures versus the corresponding observations. On average the correlation for maximum temperature is around 0.75 and for minimum temperature it is around 0.60 for all lead times except for Coimbatore where the correlation coefficient for minimum temperature is quite low. It is also observed that with increasing projection the correlation coefficient also diminishes, indicating that the degree of association between the forecasts and the observations decreases with increasing lead time. The low value of correlation in Coimbatore can be attributed to its unique geographical location in the southern part of the Indian peninsula, which is surrounded by the Western Ghats on the west and the Eastern Ghats on the east. Coimbatore lies on the leeward side of the Western Ghats. The temperatures are greatly affected by the winds. Occasionally, katabatic winds flow downward from the Ghats to the station during early morning. This may bring down the minimum temperature. The existence of a gap called the Palghat Gap in the Western Ghats southwest of Coimbatore further adds to the variability in the minimum temperature. The SI model developed for Coimbatore is not able to account for this occasional downdraft of winds and the sudden advection of cold air from the Palghat Gap to the station leading to variability in the minimum temperature. Hence due to differences in the relationship between the observed and forecast, the correlation of minimum temperature in Coimbatore is quite low. Table 6b gives the rmse values of all 12 stations. It is observed that the rmse is in the range of 1 2 C and increases with lead time. Also the rmse values of the minimum temperature are lower than the corresponding rmse values of the maximum temperatures. This shows that during the monsoon season the correspondence be-

OCTOBER 2003 NOTES AND CORRESPONDENCE 951 TABLE 6. Verification of maximum and minimum temperature forecasts during the monsoon seasons 1997 2000 as given by SI forecasts using (a) correlation coefficient and (b) rmse. Station (a) Correlation coefficient Akola Anand Bhuvaneshwar Coimbatore Delhi Hisar Jabalpur Ludhiana Pantnagar Raipur Ranchi Udaipur (b) Rmse Akola Anand Buvaneshwar Coimbatore Delhi Hisar Jabalpur Ludhiana Pantnagar Raipur Ranchi Udaipur Max temp Lead time 24 h 48 h 72 h 96 h 0.74 0.83 0.72 0.57 0.70 0.74 0.79 0.65 0.72 0.83 0.82 0.84 2.61 1.70 1.87 1.69 2.46 2.50 2.49 2.53 2.36 2.17 1.77 1.98 0.71 0.79 0.67 0.55 0.67 0.71 0.78 0.62 0.69 0.82 0.79 0.81 2.73 1.81 1.97 1.70 2.74 2.72 2.54 2.72 2.64 2.18 1.92 2.19 0.69 0.76 0.63 0.51 0.63 0.66 0.79 0.59 0.66 0.81 0.76 0.77 2.83 1.89 2.07 1.73 2.96 2.99 2.49 2.89 2.87 2.19 2.04 2.30 0.69 0.73 0.58 0.44 0.60 0.64 0.77 0.57 0.65 0.80 0.74 0.76 2.94 2.06 2.18 1.80 3.10 3.12 2.61 3.00 2.90 2.29 2.16 2.50 Min temp Lead time 24 h 48 h 72 h 96 h 0.63 0.56 0.51 4 0.57 0.59 0.63 0.51 0.45 0.62 0.56 0.62 1.26 1.08 1.03 1.28 1.98 2.19 1.24 2.03 1.45 1.12 1.17 1.41 0.63 0.50 0.46 5 0.53 0.59 0.61 0.47 3 0.58 0.53 0.59 1.28 1.14 1.07 1.36 2.13 2.16 1.28 2.12 1.61 1.19 1.19 1.49 0.63 0.47 0.40 8 0.45 0.55 0.60 0.45 5 0.59 0.56 0.58 1.31 1.18 1.10 1.43 2.30 2.27 1.33 2.16 1.61 1.19 1.19 1.52 0.62 0.44 6 4 0.42 0.53 0.61 0.42 4 0.58 0.58 0.57 1.37 1.24 1.13 1.45 2.39 2.37 1.38 2.22 1.62 1.25 1.25 1.61 tween the observed and the forecast is better in the case of minimum temperature than of maximum temperature. 5. Conclusions and future plans In this paper certain characteristics of a PPM-based operational method for forecasting surface temperatures (maximum minimum), introduced for the first time in India at NCMRWF, have been described. In order to provide insight into the basic characteristics of the forecast, a detailed verification of the temperature forecast has been done. The examination of the conditional distribution p(x/ f ) shows that the maximum temperature forecasts are relatively unbiased whereas the minimum temperature forecasts are slightly biased and need improvement. The skill scores of the temperature forecasts further support this fact. It is observed that the maximum temperatures have good skill and the minimum temperatures have comparatively lower but usable skill. The results obtained are encouraging and hence increase the confidence in providing the surface temperature forecasts to the farming community by following this objective method on an operational basis. In order to improve the temperature forecasts, further tests with different techniques like the Kalman filter and neural networks are being undertaken, and PPM models based upon T-80 analyses are already in the process of development. In the future, attempts will be made to develop MOS equations based on the operational T-80 model output and its upgraded version. A new statistical interpretation system called the updateable MOS is also being developed at various other centers. This new interpretation technique (Wilson and Vallée 2002) addresses the drawback of the MOS technique and permits the rapid adaptation of the statistical forecast to changes in the formulation of the numerical model. Attempts will be made in the future to implement this new technique in India. Recent studies (Stensrud and Skindlov 1996) have shown that gridpoint prediction of temperature forecasts from a mesoscale model can be improved by a simple running mean bias. This procedure is more useful at certain geographic locations over which the model temperature forecasts happen to be biased systematically. This will be tested from the results of the mesoscale models [Eta and the fifth-generation Pennsylvania State University National Center for Atmospheric Research Mesoscale Model (MM5)] currently operational at NCMRWF. Acknowledgments. The authors gratefully acknowledge the help of Dr. U. C. Mohanty, Indian Institute of Technology, New Delhi, for providing the ECMWF analysis data. Thanks are also due to Dr. L. H. Prakash and Ms. M. Dasgupta of NCMRWF, for help with the graphics. Finally, the authors wish to thank the National

952 WEATHER AND FORECASTING Centers for Environmental Prediction (NCEP) for providing an initial version of the T-80 model and the Center for Ocean Land Atmosphere for providing the R- 40 model. APPENDIX Verification Measures Bias refers to the relationship between the average forecast and the average observation over a verification data sample. Bias or the mean error (ME) (Glahn et al. 1991) is defined as ME f x, (A1) in which f is the average of forecasts and x is the average of observations. When f x,me 0, and the forecasts are unconditionally unbiased. A positive bias (ME 0) indicates that the predicted temperature is warmer on average, whereas a negative bias (ME 0) indicates that it is cooler on average. It is a measure of the unconditional (or systematic; or overall) bias in the forecasts. The mean square error (MSE) for a sample of data is defined as follows: 2 MSE ( f x). (A2) MSE is a measure of the accuracy of the forecasts. The root-mean-square error of the forecasts, rmse, is the square root of MSE. MSE rmse 0 for completely accurate forecasts. If the measure of accuracy is denoted as S, then the skill score SS is given as (Sc S f) Sf SS 1, (A3) S S c where S f and S c are the accuracy of the forecasts of interest and of the climatological forecasts, respectively. In the (A3), S can be replaced by any of the measures of accuracy such as mean absolute error (MAE), MSE, or rmse. In this paper skill is found using the MSE clim as the reference forecast where MSE clim is given as n 1 2 k n k 1 MSE (o o ), (A4) clim where o is the long-term climatological value of the maximum or minimum temperature at each station. Here the subscript clim indicates climatology. Hence, c MSE SSclim 1, (A5) MSE clim where SS is a measure of skill. Note that SS 0 when the forecasts of interest and the climatological reference forecasts are equally accurate, and it is positive when the accuracy of the former exceeds that of the latter. REFERENCES Carter, G. M., 1986: Moving towards a more responsive statistical guidance system. Preprints, 11th Conf. on Weather Forecasting and Analysis, Kansas City, MO, Amer. Meteor. Soc., 39 45. Glahn, H. R., and D. A. Lowry, 1972: The use of model output statistics (MOS) in objective weather forecasting. J. Appl. Meteor., 11, 1203 1211., A. H. Murphy, L. J. Wilson, and J. S. Jensenius Jr., 1991: Interpretation and verification for nonprobabilistic forecasts. Lectures, WMO Training Workshop on Interpretation of NWP Products in Terms of Local Weather Phenomena and Their Verification, Wageningen, Netherlands, AREP/RDP/PTD, PSMP 34, WMO/TD 421, XI91 XI925. Jacks, E., J. B. Bower, V. J. Dagostro, J. P. Dallavalle, M. C. Erickson, and J. C. Su, 1990: New NGM-based MOS guidance for maximum/minimum temperature, probability of precipitation, cloud amount, and sea surface wind. Wea. Forecasting, 5, 128 138. Klien, W. H., B. M. Lewis, and I. Enger, 1959: Objective prediction of 5-day mean temperature during winter. J. Meteor., 16, 672 682. Kumar, A., P. Maini, and S. V. Singh, 1999: An operational model for forecasting probability of precipitation and yes/no forecast. Wea. Forecasting, 14, 38 48.,, L. S. Rathore, and S. V. Singh, 2000: An operational medium range local weather forecasting system developed in India. Int. J. Climatol., 20, 73 87. Maini, P., A. Kumar, S. V. Singh, and L. S. Rathore, 2002: Statistical interpretation of NWP products in India. Meteor. Appl., 9, 21 31. Murphy, A. H., and R. W. Katz, 1985: Probability, Statistics, and Decision Making in the Atmospheric Sciences. Westview Press, 545 pp., and R. L. Winkler, 1987: A general framework for forecast verification. Mon. Wea. Rev., 115, 1330 1338., G. B. Barbara, and Y.-S. Chen, 1989: Diagnostic verification of temperature forecasts. Wea. Forecasting, 4, 485 501. Rousseau, D., 1982: Work on the statistical adaptation for local forecasts in France. Proc. Workshop on Statistical Interpretation of Numerical Weather Prediction Products, Reading, United Kingdom, ECMWF, 395 415. Stensrud, D. J., and J. A. Skindlov, 1996: Gridpoint predictions of high temperature from a mesoscale model. Wea. Forecasting, 11, 103 110. Tapp, R. G., F. Woodcock, and G. A. Mills, 1986: Application of MOS to precipitation prediction in Australia. Mon. Wea. Rev., 114, 50 61. Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences. An Introduction. Academic Press, 467 pp. Wilson, L. J., and M. Vallée, 2002: The Canadian Updateable Model Output Statistics (UMOS) system: Design and development tests. Wea. Forecasting, 17, 206 222.