1. Introduction

Although, in general, numerical weather prediction (NWP) models are hard to beat in very short-term forecasting (up to 24 h), they do require a substantial amount of computation time and the model forecasts are not always stable at this timescale. In contrast, statistical schemes require little computation time to make a forecast and are adapted to the station’s climate but, in general, do not include nonlinear behavior. Another important advantage of statistical methods over NWP models is that the latter often produces biased forecasts, whereas the former are usually unbiased.

Due to some of these obvious advantages, statistical methods have a long application history (Murphy 1998). This paper builds on a series of studies conducted in the past, starting with a Markov-chain model predicting daily rainfall for a single station (Gabriel and Neumann 1962). Fraedrich and Müller (1983) extended this model type to predict sunshine periods and probability of precipitation (PoP). This and related studies led to the introduction of these techniques in weather services (Miller 1984; Miller and Leslie 1984; Hess et al. 1989). Besides probability of precipitation and daily sunshine, the temperature plays obviously an important role in practical weather forecasting. Therefore, many have applied statistical methods to 6–24-h temperature forecasting (Wilson 1985; Dallavalle 1996; Knüpffer 1996) as well as the long-term prediction (more than a month ahead) of monthly mean temperature (Nicholls 1980; Navato 1981; Madden 1981; Norton 1985). Thus, the first purpose of this paper is to set up a statistical model of short-term forecasting of the temperature anomaly and the PoP, and to verify these forecasts. Motivated by the forecast improvements achieved by combining independent short-term prediction schemes (Fraedrich and Leslie 1987a), a second purpose of this paper is to show the skill gained in linearly combining these statistical forecasts with the NWP model forecasts, the latter of which is the direct model output (DMO) of the Europe model (EM) of the German Weather Service (DWD). This offers an alternative to the widely used model output statistics (Glahn and Lowry 1972). Finally, the third aim is to provide and illustrate the performance of an operational statistical weather forecasting system that, associated with an interpolation method (Smith and Wessel 1990), leads to areal short-term predictions.

The outline of the paper is as follows: After a brief description of the database (section 2) the statistical models are introduced (section 3). The focus lies on the multiple regression technique for forecasting temperature anomalies and a brief review is given for the multiple regression Markov model to predict the probability of precipitation. These two statistical models are applied to single weather stations and results are presented for Hamburg Fuhlsbüttel as an example. In section 4, these forecasts are combined with the NWP model forecasts of the EM to improve the forecast accuracy. In section 5 the two statistical methods are applied to several weather stations of central Europe to obtain an areal forecast for this region. Finally, section 6 summarizes the results and highlights the important features.

2. Data

The observed data (time series) of 68 DWD stations are used for model building and forecast experiments. These observed data contain the standard synoptic parameters at 6-hourly intervals with sample lengths varying between 8 and 46 yr. The longest time series data is available for Hamburg Fuhlsbüttel and is used to present and verify the different statistical models for temperature anomalies and probability of precipitation, and for the model combination. The combination is based on numerical weather predictions utilizing forecasts of the EM (Majewski 1991). This time series of model data starts in 1991 and ends in 1995. It contains, among other parameters, the 2-m temperature to the nearest 1/10°C and four different types of precipitation, convective rain, convective snow, large-scale rain, and large-scale snow, to the nearest 1/10-mm. Therefore, the NWP probability of precipitation is defined as either 0 or 1, if any one of these four parameters has a value greater than 0 mm. The EM runs daily at 0000 UTC. Each run generates forecasts at 6-h intervals for projections of 6–48 h. To obtain forecasts for the missing generating times 0600, 1200, and 1800 UTC we use the forecasts of the corresponding 0000 UTC EM run. For example, the 6-h prediction at the 1200 UTC start time uses the 18-h prediction of the 0000 UTC run (see Fig. 1).

Strictly, the modified NWP time series cannot be compared with time series of forecasts of other models. This obvious disadvantage is outweighted by the advantage of updating the forecasts with new observations.

3. Single-station models

Statistical single-station forecast models are developed for both the temperature and the probability of precipitation. A multiple regression technique is used for the prediction of temperature anomalies (R model). This technique, in combination with a Markov process, is used to predict PoP (M model). After a description of the two models, the performance is presented using the weather station Hamburg Fuhlsbüttel as an example. A particular emphasis is taken on the joint (probability) distributions of forecasts and observations (Murphy and Winkler 1987). The denoted marginal and conditional distributions are used “to identify the strengths and weaknesses in the forecasts to provide the modeler and forecaster with feedback and a basis for improving the quality of such predictions” (Murphy et al. 1989). Furthermore, the skill of both models is analyzed with reference to simple standard forecast models, like the climate mean and persistence.

a. Temperature forecast: Multiple regression model and its verification

A multiple regression method (backward elimination) is used to build the forecast model for temperature anomalies. The anomalies are defined as deviations from a mean averaged over all years ( j = 1, 2, . . . , J) for a given calendar day (i = 1, 2, . . . , 366) and each observation time (k = 0000, 0600, 1200, 1800 UTC) separately. Thus, for a certain day i of a year j at the main observation time k one obtains the temperature anomaly T_Aijk:

View Expanded

where T_ijk is the observed temperature. To smooth these four annual cycles (Fig. 2) associated with the four main observation time k, a 7-day running mean is applied.

1) Multiple regression model

A classic multiple regression scheme (Draper and Smith 1981; Perrone and Miller 1985; Wilks 1995; von Storch and Zwiers 1998) is applied to each of the four temperature anomalies and to the whole year (annual model). In addition, it is possible to fit the model for each individual season (winter is defined as December–February; spring as March–May; summer as June–August; and fall as September–November); this will only be used for the application of the areal forecast (section 5). Furthermore, the time series data is divided into a model building part (1949–79) and an independent verification part (say 1980 to 1995 for Hamburg Fuhlsbüttel). The standard routine synoptic observation provides 20 parameters for applying the regression scheme. For practical use the number of covariates (or predictors) is reduced as follows: First, the comprehensive version of the model is built with all 20 covariates. Second, the covariates are sorted with respect to their importance [see Eq. (2)] using only the 0000 UTC start time with the 12-h projection. Third, models are reconstructed by successively reducing the number of the less important covariates. Fourth, the final model version is obtained when its rms errors (for all start and lead time combinations in the independent dataset) begin to show changes in the second digit (compared with the comprehensive model performance). Here it should be noted, that the ranking of the predictors [with respect to their importance, Eq. (2)] is different for each combination of start and lead times. Thus, a set of 12 regression coefficients are estimated for each start time (0000, 0600, 1200, and 1800 UTC) and for the 6-, 12-, 18-, and 24-h forecast projections. There are two types of covariates: (a) direct use of the synoptic data (e.g., cloud cover from the 0–9 synoptic code number as the predictor value) and (b) derived covariates. We use the dewpoint anomaly (treated exactly like the temperature) and the pressure anomaly from long-term annual mean value. The natural logarithm is applied to the covariate visibility due to the exponential characteristics (which is also shown in the use of the exponential code table). The meridional and zonal wind is derived from the wind speed and the wind direction.

Table 1 shows the 12 input covariates taken at the observation time 0000 UTC for the 12-h prediction of the annual Hamburg Fuhlsbüttel model. The covariates (i = 1, . . . , 12) are sorted with respect to their importance I_i, which is defined as

View Expanded

where the a_i are the corresponding regression coefficients and the X_it are the values of the covariates and all cases t = 1, . . . , T in the independent forecast sample. Additionally, Table 1 shows the corresponding regression coefficients for the 0000 UTC observation time and the 12-h forecast. The temperature is the most important covariate representing the persistence of the system. If a positive anomaly is observed, the positive regression coefficient causes a positive contribution for the forecast. The other covariates represent the climatological setting of the station. For example, the meridional wind component contributes negatively to the forecast. Two artificial covariates—the sine and cosine of the date—are used to test whether there is a remaining small contribution to the annual cycle, which appears not to be the case so that the main part of the annual cycle is excluded when utilizing temperature anomalies.

2) Forecast verification

The basis of forecast verification for the independent sample 1980–95 is the joint distribution of the forecasts and the observations, from which the traditional summary performance measures can be deduced.

In Fig. 3 the bivariate histogram for the 12-h predictions versus observations is presented for the annual R model together with the corresponding scatter diagram using 1°C class intervals. The scatter about the diagonal gives a qualitative impression of the model reliability. Note that even for extreme anomaly conditions the scatter does not increase essentially and the joint distribution appears to be symmetric about and along the diagonal. We recognize that the dispersion of the errors is nearly Gaussian around the climate mean state of zero temperature anomaly indicating minimal or no forecast errors. The largest joint relative frequencies occur in the domain where the error between observation and forecast is below 1°C. Thus, all off-diagonal frequencies of the joint distribution correspond to forecasts with errors larger than 1°C. Adding all relative frequencies of the error interval ±2°C, we obtain 80% of the 6-h forecasts (not shown) and 70% of the 12-h forecasts. The 18- and 24-h predictions (not displayed) show similar behavior with the difference being, that the joint distributions become wider and plainer due to deteriorating forecast accuracy.

As indicated above, traditional performance measures also play an important role in verification schemes. Moreover, they are summary measures of the joint distributions and can facilitate comparisons among prediction lead times and between different models. Table 2 shows the root-mean-square error (rms error) and the correlation coefficient between observations and forecasts. In this context the rms error is more important, because the correlation coefficient is more phase and less amplitude dependent. The R-model disposes of the highest accuracy for all lead times. The 6-h persistence and the first-order autoregressive or AR(1) process, which is a combination of an optimally weighted persistence and climate model plus white noise, show the same behavior. Only the climate mean forecast has an initial error due to the concept of this model. The performance of the 6-h persistence model indicates the system’s tendency to remain in its initial state. The improvement of the AR(1) process being significantly lower compared to the R-model justifies the use of the more complicated R model.

b. PoP forecasts: Multiple regression Markovmodel and its verification

The PoP forecast technique is based on Markov-chain models (Gabriel and Neumann 1962; Fraedrich and Müller 1983) utilizing time series X_t = i, which can be defined for discrete states i (h or j) = 1, . . . , s occurring at discrete time steps t = 1, . . . , n (or m). In the theory of Markov chains, see, for example, Kemeny et al. (1976), the observed event X_t = i, which can be regarded as the outcome of a trial at time t, depends only on the outcome of the directly preceding trial t − 1. Thus, the outcome is not associated with a fixed probability, but with a pair corresponding to a conditional (or transition) probability (prob):

View Expanded

The transition probabilities p_ij can be estimated by the maximum likelihood method (Anderson and Goodman 1958; Kemeny et al. 1976). Thus, the Markov-chain model consists of a transition probability matrix mapping an initial state probability vector linearly into a predicted state probability vector that defines a probability distribution: the vector components correspond to discrete classes or states i = 1, . . . , s of the probability distribution; they are nonnegative and sum up to unity.

In this study, the Markov-chain model utilizes three classes of discrete weather states (cumulus, stratus, and rain; see Table 3), which are mutually exclusive and collectively exhaustive.

Combining the two variables W1 (weather condition of the past 6 h) and WW (weather condition of the observation time) has the advantage that they cover the whole 6-h time interval compared to the amount of rain (RR) frequently measured over a 12-h interval (World Meteorological Organization 1988). The division of the“no rain” class into two classes is supported by the fact that the covariate, low clouds (C_L), has a strong correspondence with the precipitation. The initial condition is defined with respect to the classes, WW and C_L, whereas the final condition uses WW, C_L, and W1 to determine the transition probabilities at the corresponding lead time of a first-order Markov-chain model forecast. For each class (0, 1, and 2) the Markov-chain model provides a transition probability for each forecast projection. For example, the 6-h forecast of an annual model yields for Hamburg Fulhsbüttel

View Expanded

For example, the probability of class 2 (rain) given an observation of class 1 (stratus) is 33% for the next 6 h.

1) Multiple regression Markov model

This Markov-chain model is now extended by combination with a standard multiple regression technique (Miller 1984; Miller and Leslie 1985; Kirk and Fraedrich 1998). The PoP (Kirk and Fraedrich 1998) for a given state j = 1, 2, 3 at current time t and for h hours ahead is

View Expanded

where X_i( j, t, h) are the covariates and a_i( j, t, h) the corresponding regression coefficients. These regression coefficients are estimated for each class (cumulus, stratus, and rain) and the 12 single-station standard synoptic data. Like the R model, it is possible to establish seasonal models (only used in section 5). This leads to 12 regression coefficients for each class and for each forecast projection. In contrast to the temperature forecast predicting for a time point (say the temperature in 6 h) the PoP forecast predicts for a time interval (say the PoP during the next 6-h time interval). Table 4 lists the 12 covariates of the 12-h prediction of an annual model ranked in order of their importance. Here, the same modifications apply to the synoptic parameters as for the R-model, except for the temperature and the dewpoint anomalies being replaced by their actual values.

Note, that the covariate intercept, which is the most important covariate, indicates the transition probability that a pure Markov model would have without regression. All other covariates improve the performance of the M model significantly and reflect the climatological experience.

2) Forecast verification

To evaluate the PoP forecasts for the independent sample 1980–95 the joint distribution p(δ, ϕ) of observations δ and forecasts ϕ is decomposed into the two conditional distributions p(δ|ϕ), p(ϕ|δ) and the two marginal distributions p(ϕ), p(δ). The conditional distribution p(δ|ϕ) of the observations given the forecasts is called reliability; the conditional distribution p(ϕ|δ) represents the forecasts given the observations and identifies as the discrimination ability or the conditional sharpness (Murphy and Winkler 1987). The diagonal line represents the perfect reliability, that is, p(δ = 1|ϕ) = ϕ for all ϕ. The lower panel of Fig. 4 displays the reliability for 12-h prediction with small overforecasting [ϕ > p(δ = 1|ϕ)] at lower probabilities and small underforecasting [ϕ < p(δ = 1|ϕ)] for the rest.

The discrimination diagram illustrates that the model is able to differentiate between dry and wet days but that the general sharpness is relatively poor because the climatological chance of rain for the next 12 h is almost 50% in Hamburg. The traditional performance measure of probability forecasts, the half-Brier score (see Table 5), shows a significant improvement of the M model over standard reference forecasts. Note that the PoP forecasts are made for increasing time intervals (6, 12, and 24 h).

4. Forecast combination

The numerical weather predictions and the single-station stochastic model forecasts are combined in an error minimizing fashion. This procedure consists of two steps: downscaling and combination. The NWP model forecasts are areal forecasts whereas the forecasts of the statistical models are single-station forecasts. Before combination, it is necessary that these forecasts possess the same spatial representation. That is, the models’ forecasts needs to be compatible in their space and time structure. Here, we apply the downscaling method reducing an areal forecast to a point forecast. This procedure is illustrated for the temperature anomaly at the station Hamburg Fuhlsbüttel. Next, the linear combination scheme is applied (appendix) to the temperature predictions (Thompson 1977; Fraedrich and Smith 1989) and to probability forecasts following Fraedrich and Leslie (1987a).

5. Area forecasts: Method and case study

The R model and the M model are fitted for 68 weather stations to deduce an areal forecast of temperature anomaly and of probability of precipitation for central Europe. Because of the different sample lengths, we choose two-thirds of the time series data as the model building part and one-third as the verification part.

6. Summary and conclusions

Two statistical schemes are introduced for short-term (up to 24 h) single-station forecasts: a multiple regression model predicting temperature anomalies and a probability of precipitation forecast system. The verification of independent forecasts is shown in some detail using the most accurate scheme for the weather station Hamburg Fuhlsbüttel as an example. The verification methods applied are outlined by Murphy and Winkler (1987). The joint distribution allows an interpretation of the forecast performance itself. Even if extreme temperature anomalies are observed, the scatter of the corresponding forecasts increases only slightly. For example, 80% of the 6-h forecasts lie in a range of ±2°C. The rms error of the temperature forecasts achieves a magnitude of 1.7°C for the 6-h forecast (2.2°C for the 12-h forecast) and the half-Brier score of the PoP forecasts yields 0.13 for the 6-h forecast (0.15 for the 12-h forecast). The comparison with simple standard statistical models like the climate mean shows a significant improvement of accuracy obtained by the more advanced statistical models.

As the performance of the statistical models shows good results, the linear combination of these models with numerical weather predictions achieves further improvements in forecast accuracy. The combination temperature forecasts yield a 14% gain for the 24-h prediction with respect to the R model alone, and 17% gain with respect to NWP model. The combined PoP forecast achieves 18% with respect to the R model and 33% with respect to NWP model. While statistical models based only on observations are independent of NWP models, the MOS technique, to provide the best results, requires a recomputation whenever the NWP model is changing. Therefore, the linear combination scheme offers an alternative way to improve the direct model output (DMO) instead of the widely used model output statistics or Kalman filtering methods because only the weights need to be recomputed whenever the NWP model version changes.

The results of the statistical weather forecasting system illustrate the possibility of deriving an areal forecast for Europe using a standard interpolation technique that is economical in computing time. Due to the small technical expense, such schemes may be suitably introduced to regional weather forecast centers. Extensions are possible if the observations of neighboring weather stations are included. An operational statistical system running every 6 h is shown in the Internet (http://www.dkrz.de/wetter/prognosen/).