Theoretical And Applied Studies Of The Multi-center Ensemble Mean Based On The TIGGE Dataset

Ensemble prediction has been widely used, and is now taking an important part of the whole numerical weather prediction system. Forecasting skill of ensemble mean is a crucial criterion to assess the ensemble prediction systems (EPSs). In this paper, theoretical and applied studies about the forecasting skill of ensemble mean based on the TIGGE dataset are organized. Questions about the skill of ensemble mean are answered theoretically, and applicated to the single-center and multi-center EPSs based on the TIGGE dataset.The equally weighted ensemble mean, that to calculate the arithmetic mean of the the constituent forecasts, is the most commonly used method in the area of ensemble forecasting. Forecasting errors are treated as random variables, in order to build up the mathematical model of the ensemble mean.Theoretical analysis shows that, the ensemble mean can not only avoid choosing the worst members, but also outperform the average skill of its constituent members. This is actually why the ensemble mean can often achieve a satisfactory skill in real forecasts. The ensemble mean cannot always outperform its best member; it occurs only when the members have similar skills and less correlated errors with each other. Forecasting skill of ensemble mean is not only depended on the skills of its constituent members, but even more determined by the error-relationship between each two members. If the members are unbiased, the MSE of the ensemble mean equals to the average error-covariance between each two members approximately.When the ensemble size increases, the MSE of ensemble mean decreases and approaches to a saturated level, on the condition that the average skill of the members and the average error-covariance between each two members remain stable. For a given saturation, the ensemble size needed to obtain a saturated skill is depended on the ratio between the average MSE of the members and the average error-covariance between each two members.It is inappropriate to add new members into an already existing ensemble blindly. The newly added members should be less correlated with each other and with the already existing ensemble mean in order to improve the skill of ensemble mean. Even if the new members have higher skills themselves, it can still depress the skill of ensemble mean if they are highly correlated with each other and with the already existing ensemble mean, and vice versa.Theories of the equally weighted ensemble mean are extended to the unequally weighted ensemble mean methods, especially for the optimal weighted ensemble mean. The relationship between the ensemble mean methods and bias-correction schemes is discussed, as well as the potential skill of bias-corrected unequally weighted ensemble mean.On the ideal assumption that the MSE of the members and the error-relationship between each two members remain stable during the training and forecasting periods, the optimal weighted ensemble mean always exists with proper physical sense; but if the members are correlated with each other, the optimal weighted ensemble mean exists only when some conditions are satisfied for the error-covariance matrix. Once the optimal weighted ensemble mean exists, it can always outperform the best individual member and the equally weighted mean; otherwise there exists one member outperforms any weighted ensemble mean methods, which implies that the ensemble mean is meaningless.The MSE of any forecast can be considered as the sum of the variance of the forecasting error and the square of the systematic error. Essentially the goal of ensemble mean, no matter equally or unequally weighted, is to eliminate the variance of the forecasting error, while the bias-correction schemes aim at eliminating the systematic error. Therefore the ensemble mean methods and the bias-correction schemes are distinguished but correlated with each other, and can be mixed together to build up a bias-corrected ensemble mean in order to obtain a satisfactory skill.Since the ideal assumption cannot be realized completely, the lower bound of the MSE of bias-corrected weighted ensemble mean methods can be obtained by regarding the forecasting period as the training period, and constructing the bias-corrected optimal weighted ensemble mean. This is actually the potential skill of any linear combination ensemble methods.Theories of the equally and unequally weighted ensemble mean are applied to the T213 EPS, which is operated by the National Meteorology Center of China and now an important member of the TIGGE dataset. Forecasting skill of ensemble mean is compared with the control run. Relationship between the skill of ensemble mean and the ensemble size is analyzed, as well as the requisite ensemble size for the ensemble mean to obtain a saturated skill. The potential skill of bias-corrected ensemble mean with both constant weights and time-varying weights are also obtained.The results show that, the equally weighted ensemble mean of the 15-member T213 EPS outperforms the control run in most leading days and for most meteorological variables, especially in medium-range forecasts. The average error-covariance between each two members is the key factor to determine the skill of ensemble mean. The error covariances between each two members are much smaller than the MSEs of the members in medium-range forecasts, which can explain why the improvement of ensemble mean is more distinctive in medium-range forecasts.When the ensemble size increases, the skill of ensemble mean improves in medium-range forecasts, but the improvement rate becomes smaller and smaller, and finally reaches a saturated skill. But in short-range forecasts, the skill of ensemble mean can hardly be improved by increasing the ensemble size.For the meteorological variables such as 500hPa geopotential height,850hPa temperature and humidity, and 200hPa wind speed, the 15 members of the T213 EPS can achieve the saturation of about 95%, which means that adding new members into the T213 EPS can only decrease the MSE of ensemble mean by 5%at most, on the condition that the skill of control run and the method of generating perturbations remain the same.Forecasting skill of equally weighted mean is close to the potential skill of bias-corrected ensemble mean methods with constant weights. Although the potential skill of methods with time-varying weights is still higher than the equally weighted mean in medium-range forecasts for some meteorological variables, this kind of methods are still difficult to use in real forecasts, due to the instability of the relative performance of each members.As a result, the equally weighted mean is still a proper choice for the T213 EPS in medium-range forecasts, because it can obtain a highly saturated skill which is better than the control run. But in short-range forecasts, the advantage of ensemble mean is inconspicuous, because the skill of control run is close to the equally weighted ensemble mean. Neither increasing the ensemble size nor improving the ensemble mean methods can ameliorate the skill of T213 ensemble mean, both in short-range and medium-range forecasts. The skill of control run and the method of generating initial perturbations should be improved instead.For the multi-center EPS based on the TIGGE dataset, equally weighted mean is compared with the single centers. The results show that, both ECMWF and NCEP have some advantages compared with other centers, but the skill of CMA is consistently poorer than ECMWF and NCEP, especially for 2-8 leading days. The equally weighted ensemble mean of the three centers outperforms its best member in most leading days and for most meteorological variables. Whether the ensemble mean can outperform the best single member is determined by the average error-relationship between each two members. For the short and medium range forecasts of humidity, and medium range forecasts of other variables, the error-covariances between each two centers are weak, which leads to a distinctively better ensemble mean than the best single center.For the forecast of humidity leading 1-10 days, and other variables leading 8-10 days, the equally weighted mean of the three centers is distinctively better than the simple average of ECMWF and NCEP, despite that the skill of CMA is consistently poorer than ECMWF and NCEP. This implies that CMA and other poorer centers may be still useful in improving the TIGGE multi-center ensemble forecasts, and the contribution comes from the lower error-covariances between different centers.The equally weighted mean is a proper choice for the multi-center EPS based on the TIGGE database, because the skill of equally weighted mean of the three centers is close to the potential skill of bias-corrected ensemble mean with time-varying weights. Even if the ideal assumption comes true, forecasting skill of ensemble mean can hardly be improved by time-varying weights and bias-correction schemes.