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liubainian銀蟲 (小有名氣)
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Statistical methods can provide probabilistic forecasts without the need for ensemble forecasts. The climatological distribution has been mentioned previously; another example is a single deterministic forecast turned into a probabilistic forecast by dressing it with a distribution of past forecast errors valid for the same region, season and forecast range. However, such simple statistical methods will always yield the same spread. Some flow-dependence could be introduced in the error statistics by stratifying the sample according to the predicted atmospheric state but sample size is likely to be a major issue with this approach. In contrast, ensembles consisting of multiple integrations of numerical weather prediction models do exhibit temporal and spatial variations of the spread because the conditions of the flow itself modulate the dispersion of nearby atmospheric states. The realism of the spread will depend on how well the sources of uncertainty are represented in the ensemble prediction system. An example of the variability of the ensemble spread is given in (Fig. 3). It shows two consecutive ensemble forecasts of hurricane Katrina’s track. Katrina, made landfall near New Orleans on 29 August 2005, 12 UTC. The ensemble forecast issued on 26 August, 00 UTC exhibit a large dispersion of tracks. This dispersion is substantially reduced in the ensemble forecast initialised at 12 UTC on 26 August. The actual track of Katrina was close to the mode of the distribution predicted by the later forecast. Even for a perfect ensemble, one cannot expect a very high correlation between the ensemble mean error and the spread if the correlation is computed from pairs of error and spread for individual cases because one essentially compares the standard deviation of a distribution with the magnitude of individual realisations. The correlation of spread and ensemble mean error for a perfect ensemble increases with the case-to-case variability of the spread [25]. It is more appropriate to assess the validity of (7) by considering the sample mean RMS errors conditioned on the predicted spread. Such a diagnostic was performed previously for the ECMWF EPS [26]. It is repeated here using data from a more recent version of the operational ECMWF EPS. The data consists of 89 ensemble forecasts (Feb–April 2006) for 500 hPa geopotential height in the Northern Hemisphere mid-latitudes (35–65N). As the true state of the atmosphere is unavailable, the ensemble mean error is estimated from analyses, i.e. estimates of the initial state obtained with ECMWF’s four-dimensional variational assimilation system. For the considered variable and forecast ranges this is considered to be an acceptable approximation. The data is analysed on a regular 2.5×2.5 latitude-longitude grid—2160 grid points in total. Thus the entire sample consists of 192240 pairs of spread and ensemble mean error. The data has been stratified according to the predicted spread and divided into 20 equally populated bins, where bin boundaries are given by the 5%- ,10%,. . . ,95%-percentiles of the spread distribution. Then, the RMS error of the ensemble mean and the RMS spread are computed for each bin. Figure 4 shows the relationship between error and spread for the 2-day, 5-day and 10-day forecast. At all three forecast ranges, the ensemble spread contains useful information about variations of the width of the distribution of the ensemble mean error. The larger the spread of the ensemble the larger is the average ensemble mean RMS error. At the later forecast ranges (5–10 days), the average RMS error is fairly accurately predicted by the ensemble standard deviations. At the shorter forecast ranges (2-days), the spread is less reliable in predicting the variability of the width of the ensemble mean error distribution. For cases with large (small) ensemble spread, the average RMS error is systematically lower (higher) than the spread. The comparison shown in Fig. 4 does not distinguish between temporal variations of the spread and spatial variations. Some of the skill in predicting variations of the width of the forecast error distribution can arise from correctly predicting the seasonally averaged geographic variations. We consider now the geopotential normalised by the time-average RMS spread as new variable in order to focus on the aspect of the temporal variability only. By definition, the time-averaged spread of the normalised geopotential is geographically uniform. The relationship between spread and average RMS error for the normalised geopotential is shown in Fig. 5. The normalisation reduces the range over which error and spread vary. For instance, the ratio of RMS error in the bin with largest 5% spread to the RMS error in the bin with lowest 5% spread is about 4.5 without normalisation at day 5; this is reduced to about 2.7 with normalisation. With increasing forecast range, the distribution of the predicted spread narrows. Eventually, as the ensemble converges to the climatological distribution, the spread will be almost constant except for sampling uncertainty and moderate intra-seasonal variations. Figure 5 demonstrates that the EPS is indeed providing information about flow-dependent variations of the width of the ensemble mean error distribution. At the same time, the diagram reveals systematic errors of the flow-dependent error bars predicted by the ensemble. Both calibration as well as improvements of the initial uncertainty representation and model uncertainty representation are expected to lead to significant further improvements of the statistical consistency and thus increase the capability to predict flow-dependent variations of forecast uncertainty. |
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