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.

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.

† The contributions of these authors were written in the course of their employment at the Met Office, UK and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.Three data assimilation methods are compared for their ability to produce the best analysis: (i) 4DVar, four-dimensional variational data assimilation using linear and adjoint models with either a (perfect) 3D climatological background-error covariance or a 3D ensemble background-error covariance; (ii) EDA, an ensemble of 4DEnVars, which is a variational method using a 4D ensemble covariance; and (iii) the deterministic ensemble Kalman filter (DEnKF, also using a 4D ensemble covariance).The accuracy of the deterministic analysis from each method was measured for both perfect and imperfect toy model experiments. With a perfect model, 4DVar with the climatological covariance is easily beaten by the ensemble methods, due to the importance of flow-dependent background-error covariances. When model error is present, 4DVar is more competitive and its relative performance is improved by increasing the observation density. This is related to the model error representation in the background-error covariance.The dynamical time-consistency of the 4D ensemble background-error covariance is degraded by the localization, since the localization function and the nonlinear model do not commute. As a result, 4DVar with the ensemble covariance performs significantly better than the other ensemble methods when severe localization is required, i.e. for a small ensemble.

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 squareroot 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-EnVar) system for data assimilation and ensemble generation.

A mathematical model was constructed to simulate the bovine oestrous cycle by using nonlinear differential equations to describe the biological mechanisms which regulate the cycle. The model predicts circulating concentrations of gonadotrophin-releasing hormone, follicle-stimulating hormone, luteinizing hormone, oestradiol, inhibin and progesterone. These hormones collectively provide control and feedback mechanisms between the hypothalamus, pituitary gland and ovaries, which regulate ovarian follicular dynamics, corpus luteum function and ovulation. When follicular growth parameters are altered, the model predicts that cows will exhibit either two or three follicular waves per cycle, as seen in practice. Changes in other parameters allow the model to simulate: effects of nutrition on follicle recruitment and size of the ovulatory follicle; effects of negative energy balance on postpartum anoestrus; and effects of pharmacological intervention on hormone profiles and timing of ovulation. It is concluded that this model provides a sound basis for exploring factors that influence the bovine oestrous cycle in order to test hypotheses about nutritional and hormonal influences which, with further validation, should help to design dietary or pharmacological strategies for improving reproductive performance in cattle.