An order of magnitude reduction in the cost of four‐dimensional variational assimilation (4D‐Var) is required before operational implementation is possible. Preconditioning is considered and, although it offers a significant reduction in cost, it seems that it is unlikely to provide a reduction as large as an order of magnitude. An approximation to 4D‐Var, namely the incremental approach, is then considered and is shown to produce the same result at the end of the assimilation window as an extended Kalman filter in which no approximations are made in the assimilating model but in which instead a simplified evolution of the forecast error is introduced. This approach provides the flexibility for a cost—benefit trade‐off of 4D‐Var to be made.

SUMMARYIn the first of this set of three papers, the formulation of the European Centre for Medium-Range Weather Forecasts (ECMWF) implementation of 3D-Var is described. In the second, the specification of the structure function is presented, and the last is devoted to the results of the extensive numerical experimentation programme which was conducted. The 3D-Var formulation uses a spherical-harmonic expansion, much as the ECMWF optimal interpolation (01) scheme used an expansion of Bessel functions. This formulation is introduced using a convolution algebra over the sphere expressed directly in spectral space. It is shown that all features of the 0 1 statistical model can be implemented within 3D-Var. Furthermore, a non-separable statistical model is described. In the present formulation, geostrophy is accounted for through a Hough-modes separation of the gravity and Rossby components of the analysis increments. As in 01, the tropical analysis remains essentially non-divergent and with a weak mass-wind coupling. The observations used, as well as their specified statistics of errors, are presented, together with some implementation details. In the light of the results, 3D-Var was implemented operationally at the end of January 1996.

SUMMARYRecent veri cation statistics show a considerable improvement in the accuracy of forecasts from three global numerical weather prediction systems. The improvement amounts to about a 1-day gain in predictability of meansea-level pressure and 500 hPa height over the last decade in the northern hemisphere, with a similar gain over the last 3 years in the southern hemisphere. Differences between the initial analyses from the three systems have been substantially reduced.Detailed study of the European Centre for Medium-Range Weather Forecasts veri cations shows that identi able improvements in the data assimilation, model and observing systems have signi cantly increased the accuracy of both short-and medium-range forecasts, although interannual ( ow-dependent) variations in errorgrowth characteristics complicate the picture. The implied r.m.s. error of 500 hPa height analyses has fallen well below the 10 m level typical of radiosonde measurement error. Intrinsic error-doubling times, computed from the divergence of northern hemisphere forecasts started 1 day apart, exhibit a small overall reduction over the past 10 years at day two and beyond, and a much larger reduction at day one. Error-doubling times for the southern hemisphere have become generally shorter and are now similar to those for the northern hemisphere. One-day forecast errors have been reduced so much in the southern hemisphere that medium-range forecasts for the region have become almost as skilful as those for the northern hemisphere.The approach to saturation of forecast error beyond the 10-day range has been examined for sets of 21-day forecasts. When the systematic (sample-mean) component of the error is subtracted, forecast errors and the differences between successive forecasts both appear to level out near the end of the 21-day range at values close to the limit set by the natural level of variance of the atmosphere for the northern hemisphere. A number of features of the model 500 hPa height elds remain quite realistic at the three-week range. The most obvious discrepancy in mean climate is in the Paci c/North-American sector, and variance is too high in the southern hemisphere.

This paper analyses the statistical structure of the errors of the short-range wind forecasts used in the global data assimilation system at ECMWF, by verifying the forecasts against radiosonde data over North America. The kinematics of two-dimensional homogeneous turbulence is used to partition the perceived forecast errors into prediction errors which are horizontally correlated, and observational errors which are assumed to be horizontally uncorrelated. The theory further partitions the wind prediction errors into three components, viz. large-scale, rotational and divergent components, and provides a spectral description of the covariance and crosscovariance functions for stream function and velocity potential. The calculations also provide an estimate of the vertical error covariance matrices for prediction error and for radiosonde observational error, by which we mean the combined effects of instrumental error and errors of representativeness. The basic assumptions are that the forecast errors are horizontally homogeneous and that the observational errors are horizontally uncorrelated.Several important results are found. The wind prediction errors are comparable in magnitude with the wind observation errors. The prediction errors are dominated by the synoptic scales, but there is a substantial large scale wind error which reverses phase between the stratosphere and troposphere. The synoptic scale errors are largely non-divergent in the troposphere. There are good grounds for increasing the resolution of the analysis system, both in the horizontal and the vertical, over North America and other data-rich regions.

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