Search

A new method for generating ensemble predictions based on an ensemble of data assimilations has been developed. Using an ensemble of four-dimensional ensemble-variational minimizations provides an approach which is close to the Met Office's operational data assimilation system and less computationally expensive than other alternatives. In developing this system, several inflation schemes have been compared. One form of additive inflation, based on analysis increments, was developed and found to be very effective at increasing the overall ensemble spread and correcting systematic biases in the model. However, the analysis increments are not flow-dependent since they are randomly drawn from a long archive. It was decided to scale back their amplitude to avoid them dominating the overall performance. Of the other inflation schemes considered, it was found that relaxation-to-prior-perturbations was the most effective at maintaining the ensemble spread. However, this scheme also produced perturbations which are too large-scale and too balanced. The relaxation-to-prior-spread scheme performed well in many respects, but required a relaxation factor greater than one to produce an acceptable spread. Therefore these two schemes were combined in order to mitigate the drawbacks of each. This combination proved successful and was used in final testing of the ensemble against the currently operational ensemble transform Kalman filter (ETKF). The ETKF has its perturbations centred around a high-resolution ‘deterministic’ analysis. This was seen to be an important benefit, and the new ensemble system also benefited from being recentred around the high-resolution analysis. This recentred system has slightly lower forecast skill than the ETKF over a variety of variables, due to the fact that the spread of this ensemble is less than the spread of the ETKF ensemble. The deficiency of the spread of the new ensemble system will be addressed in ongoing work.

Outputs from a data assimilation system may be used to diagnose observation and background error statistics, as has been demonstrated by previous researchers. In this study, that technique is extended to diagnose model-error statistics using a weak-constraint data assimilation. It deals with a set of observations over a time window and uses the temporal distribution to separate model errors from errors in the background forecast. In a set of idealised tests this method is shown to be able to successfully distinguish between model and background errors. The success of this method depends on the prior assumptions included in the weak-constraint data assimilation and how well these describe the true nature of the system being modelled. Other authors have demonstrated that it can be challenging to separate observation and background errors in diagnosis schemes. Some separation of these errors is possible within the context of four-dimensional assimilation, provided that observation errors are known to be uncorrelated in time.

We show how to replace an ensemble of variational data assimilation minimizations by minimizations for the ensemble mean and the perturbations from it. Tests using the Met Office system show that this Mean–Pert method can give practically identical results at much reduced cost. This makes it a good candidate component when designing a cost-effective system for hybrid ensemble-variational data assimilation for numerical weather prediction.

Hybrid four-dimensional ensemble-variational (4DEnVar) data assimilation is a method which avoids using a linear and adjoint model by relying on an input ensemble to propagate analysis increments in time. Previous studies have shown that hybrid 4DEnVar performs worse than hybrid four-dimensional variational (4D-Var) assimilation. Given hybrid 4DEnVar's heavy reliance on the ensemble, this comparison may be affected by the quality of the input ensemble. Here we investigate how improvements to the ensemble system affect hybrid 4D-Var and how they affect the comparison with hybrid 4DEnVar. Using the Met Office's operational ensemble generation scheme (the ensemble transform Kalman filter, ETKF) it is found that hybrid 4D-Var gains little benefit from using an enlarged ensemble as input (176 as opposed to 23 members). By contrast, hybrid 4DEnVar benefits more from the increased ensemble size, and it benefits further when the weighting given to the ensemble covariance is increased. Both data assimilation methods benefit when the input ensemble is changed from using the ETKF to using an ensemble of 4DEnVars. Both schemes also show further benefit when a large ensemble (200 members) of 4DEnVars is used, and when a large weight is given to the covariance information from this ensemble. Thus, improving the ensemble covariance used in assimilation (ensemble generation method and ensemble size) and increasing its weight can have substantial benefits. Given that both hybrid 4D-Var and hybrid 4DEnVar benefit from improvements to the input ensemble, the relative performance is largely unaffected by the ensemble changes and hybrid 4D-Var performs better than hybrid 4DEnVar for all input ensembles.

It has long been known that verification of a forecast against the sequence of analyses used to produce those forecasts can under-estimate the magnitude of forecast errors. Here we show that under certain conditions the verification of a short-range forecast against a perturbed analysis coming from an ensemble data assimilation scheme can give the same root-mean-square error as verification against the truth. This means that a perturbed analysis can be used as a reliable proxy for the truth. However, the conditions required for this result to hold are rather restrictive: the analysis must be optimal, the ensemble spread must be equal to the error in the mean, the ensemble size must be large and the forecast being verified must be the background forecast used in the data assimilation. Although these criteria are unlikely to be met exactly it becomes clear that for most cases verification against a perturbed analysis gives better results than verification against an unperturbed analysis. We demonstrate the application of these results in a idealised model framework and a numerical weather prediction context. In deriving this result we recall that an optimal (Kalman) analysis is one for which the analysis increments are uncorrelated with the analysis errors.

The Met Office has developed an ensemble-variational data assimilation method (hybrid-4DEnVar) as a potential replacement for the hybrid four-dimensional variational data assimilation (hybrid-4DVar), which is the current operational method for global NWP. Both are four-dimensional variational methods, using a hybrid combination of a fixed climatological model of background error covariances with localized covariances from an ensemble of current forecasts designed to describe the structure of “errors of the day.” The fundamental difference between the methods is their modeling of the time evolution of errors within each data assimilation window: 4DVar uses a linear model and its adjoint and 4DEnVar uses a localized linear combination of nonlinear forecasts. Both hybrid-4DVar and hybrid-4DEnVar beat their three-dimensional versions, which are equivalent, in NWP trials. With settings based on the current operational system, hybrid-4DVar performs better than hybrid-4DEnVar. Idealized experiments designed to compare the time evolution of covariances in the methods are described: the basic 4DEnVar represents the evolution of ensemble errors as well as 4DVar. However, 4DVar also represents the evolution of errors from the climatological covariances, whereas 4DEnVar does not. This difference is the main cause of the superiority of hybrid-4DVar. Another difference is that the authors’ 4DVar explicitly penalizes rapid variations in the analysis increment trajectory, while the authors’ 4DEnVar contains no dynamical constaints on imbalance. The authors describe a four-dimensional incremental analysis update (4DIAU) method that filters out the high-frequency oscillations introduced by the poorly balanced 4DEnVar increments. Possible methods for improving hybrid-4DEnVar are discussed.

The Met Office is currently developing an ensemble-based data-assimilation system for use in numerical weather prediction (NWP). In an effort to inform those developments, a simple model has been used to compare a number of different flavours of ensemble Kalman filter (EnKF). As in many previous studies, deterministic (square-root) filters are seen to perform better than stochastic (perturbed-observations) filters. Two previously proposed methods for mitigating the effects of perturbing the observations were tested. These modified stochastic filters improved performance, but were still worse than the deterministic filters. Very little difference was seen between the performance of the various deterministic filters tested. The typical approach to localization in the Local Ensemble Transform Kalman Filter (LETKF) is based solely on the distance between each observation and the model point being updated. It takes no account of the inter-observation distance and we argue that this is incorrect. However, the difference between covariance localization and observation localization is only relevant when the assimilation is not weak and hence it did not affect performance in these tests. If a hybrid covariance matrix is used to update the ‘deterministic’ forecast around which the ensemble is centred, then using the same covariance matrix when updating the perturbations is seen to be beneficial in a stochastic filter. This is not the case in the square-root filter, due to restrictions imposed by its formulation. Tests with a nonlinear observation operator indicate better behaviour when observations are assimilated simultaneously rather than sequentially. An ensemble of variational minimizations has some advantages in this situation. These results, along with other considerations, have led the Met Office to develop a 4-dimensional ensemble-variational (4D-En-Var) system for data assimilation and ensemble generation.

We have investigated a method to substantially reduce the analysis computations within the Local Ensemble Transform Kalman Filter (LETKF) framework. Instead of computing the LETKF analysis at every model grid point, we compute the analysis on a coarser grid and interpolate onto a high-resolution grid by interpolating the analysis weights of the ensemble forecast members derived from the LETKF. Because the weights vary on larger scales than the analysis increments, there is little degradation in the quality of the weight-interpolated analyses compared to the analyses derived with the high-resolution grid. The weight-interpolated analyses are more accurate than the ones derived by interpolating the analysis increments. Additional benefit from the weight-interpolation method includes improving the analysis accuracy in the data-void regions, where the standard LEKTF with the high-resolution grid gives no analysis corrections due to a lack of available observations. Copyright © Royal Meteorological Society and Crown Copyright, 2008