This paper presents results of a comparison between four‐dimensional variational assimilation (4D‐Var). using a 6‐hour assimilation window and simplified physics during the minimization, and three‐dimensional variational assimilation (3D‐Var). Results have been obtained at ‘operational’ resolution T213L31/T63L31. (T defines the spectral triangular truncation and L the number of levels in the vertical, with the first parameters defining the resolution of the model trajectory, and the second the resolution of the inner‐loop.) The sensitivity of the 4D‐Var performance to different set‐ups is investigated. In particular, the performance of 4D‐Var in the Tropics revealed some sensitivity to the way the adiabatic nonlinear normal‐mode initialization of the increments was performed. Going from four outer‐loops to only one (as in 3D‐Var), together with a change to the 1997 formulation of the background constraint and an initialization of only the small scales, helped to improve the 4D‐Var performance. Tropical scores then became only marginally worse for 4D‐Var than for 3D‐Var. Twelve weeks of experimentation with the one outer‐loop 4D‐Var and the 1997 background formulation have been studied. The averaged scores show a small but consistent improvement in both hemispheres at all ranges. In the short range, each two‐ to three‐week period has been found to be slightly positive throughout the troposphere. The better short‐range performance of the 4D‐Var system is also shown by the fits of the background fields to the data. More results are presented for the Atlantic Ocean area during FASTEX (the Fronts and Atlantic Storm‐Track Experiment), during which 4D‐Var is found to perform better. In individual synoptic cases corresponding to interesting Intensive Observing Periods, 4D‐Var has a clear advantage over 3D‐Var during rapid cyclogeneses. The very short‐range forecasts used as backgrounds are much closer to the data over the Atlantic for 4D‐Var than for 3D‐Var. The 4D‐Var analyses also display more day‐to‐day variability. Some structure functions are illustrated in the 4D‐Var case for a height observation inserted at the beginning, in the middle or at the end of the assimilation window. The dynamical processes seem to be relevant, even with a short 6‐hour assimilation period, which explains the better overall performance of the 4D‐Var system.

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.

SUMMARYThis paper presents results of a comparison between four-dimensional variational assimilation (4D-Var). using a 6-hour assimilation window and simplified physics during the minimization, and three-dimensional variational assimilation (3D-Var). Results have been obtained at 'operational' resolution T2 13L3 1/T63L3 1. (T defines the spectral triangular truncation and L the number of levels in the vertical, with the first parameters defining the resolution of the model trajectory, and the second the resolution of the inner-loop.) The sensitivity of the 4D-Var performance to different set-ups is investigated. In particular, the performance of 4D-Var in the Tropics revealed some sensitivity to the way the adiabatic nonlinear normal-mode initialization of the increments was performed. Going from four outer-loops to only one (as in 3D-Var), together with a change to the 1997 formulation of the background constraint and an initialization of only the small scales, helped to improve the 4D-Var performance. Tropical scores then became only marginally worse for 4D-Var than for 3D-Var. Twelve weeks of experimentation with the one outer-loop 4D-Var and the 1997 background formulation have been studied. The averaged scores show a small but consistent improvement in both hemispheres at all ranges. In the short range, each two-to three-week period has been found to be slightly positive throughout the troposphere. The better short-range performance of the 4D-Var system is also shown by the fits of the background fields to the data. More results are presented for the Atlantic Ocean area during FASTEX (the Fronts and Atlantic StormTrack Experiment), during which 4D-Var is found to perform better. In individual synoptic cases corresponding to interesting Intensive Observing Periods, 4D-Var has a clear advantage over 3D-Var during rapid cyclogeneses. The very short-range forecasts used as backgrounds are much closer to the data over the Atlantic for 4D-Var than for 3D-Var. The 4D-Var analyses also display more day-to-day variability. Some structure functions are illustrated in the 4D-Var case for a height observation inserted at the beginning, in the middle or at the end of the assimilation window. The dynamical processes seem to be relevant, even with a short 6-hour assimilation period, which explains the better overall performance of the 4D-Var system.

The adjoint method has been used to calculate the sensitivity of short-range forecast errors to the initial conditions. The gradient of the energy of the day 2 forecast error with respect to the initial conditions can be interpreted as a sum of rapidly growing components of the analysis error. An analysis modified by subtracting an appropriately scaled vector, proportional to the gradient, provides initial conditions for a 'sensitivity integration' that can be used to diagnose the effect of initial-data errors on forecast errors.Statistics of sensitivity calculations for the month of April 1994 characterize the sensitivity patterns as smallscale, middle or lower tropospheric structures which are tilted in the vertical. The general pattern of these structures is known to be associated with the fastest possible growth of forecast error. When used as initial perturbations, they evolve rapidly into synoptic-scale structures, propagating both downstream and to higher atmospheric levels.On average, the sensitivity integration corrects for about a tenth of the day 2 forecast error, which indicates that indeed not all of the error is in the fastest-amplifying modes. But the fraction of the error corrected at day 2 is important for an improvement in the medium-range, as this fraction continues to grow substantially in the non-linear regime. These results have proved that there is still scope for great improvement in the medium-range forecast, particularly over Europe, by a better description of the initial conditions. The sensitivity experimentation suggests that many cases of major forecast-errors may be explained by defects in the analysis. A small but well-chosen change in the analysis can frequently improve the forecast quality.

SUMMARYRecent data assimilation developments which have taken place at numerical weather-prediction centres are briefly discussed, from the perspectives of both the importance of data and algorithmic developments. The increase in quality and quantity of satellite data is seen to play a major role in the improvement of forecast performance, particularly in the southern hemisphere. Further optimization of the use of observations is possible through the proper evaluation of data impact and the optimization of the amount of data to be assimilated. The generalized advent of four-dimensional variational assimilation is presented, and trends in the specification of error statistics are described. Finally, a more interactive forecasting system including an adaptive component is a new challenge to bring additional improvement to the forecasting of high-impact weather.

SUMMARYIn this paper, the optimal con gurations of model resolution, observation resolution and observation density are investigated in a simple one-dimensional framework. In this context, the representativeness error is formalized and estimated before being used in the analysis-error formulation. Some optimal and suboptimal assimilationschemes, differing from different approximations of observation-error covariance and observation operator, are compared. The optimal observation-extent is determined as a function of model resolution. Increasing the observation density is usually bene cial, except for suboptimal schemes similar to the ones used in operational practice. The impact of thinning the observations with correlated error is also studied from a suboptimal viewpoint.

Several networks of Global Positioning System receiving stations over Europe send their data to several processing centers to generate atmospheric Zenith Total Delay (ZTD) observations. Thanks to the efforts of the Targeting Optimal Use of Global Positioning System Humidity measurements in meteorology project, these observations combining surface pressure and total precipitable water information in the atmosphere have been delivered to the operational meteorological centers in near real‐time since 2004. This paper presents forecast impact trials of such ZTD observations in a global Four‐Dimensional Variational (4DVAR) assimilation and forecasting system. The implementation of the ZTD assimilation in the 4DVAR system is described, including a preprocessing developed specifically for the ZTD data. The preprocessing involves a time averaging procedure of the observations in order to ensure consistency with the resolution of the 4DVAR, a bias correction, and a station selection based on χ2 tests of the normality of the observation minus first‐guess differences. Three forecast trials were conducted: winter, spring, and summer 2005. These trials cover various meteorological conditions and a total of about 10 weeks of assimilation. All three trials suggest a positive impact of the ZTD data in helping constrain the synoptic circulation in 1 to 4 day forecasts. In the spring and the summer trials, the impact of the ZTD data also shows positively on the prediction of precipitation patterns as indicated by improved Quantitative Precipitation Forecast scores for total precipitation forecasts over France between +12 and +36 hours. We also assess in this paper ZTD observation and background errors.

SUMMARYAdvanced infrared sounders will provide thousands of radiance data at every observation location. The number of individual pieces of information is not usable in an operational numerical weather-prediction context, and we have investigated the possibilities of choosing an 'optimal' subset of data. These issues have been addressed in the context of optimal linear estimation theory, using simulated Infrared Atmospheric Sounding Interferometer data. Several methods have been tried to select a set of the most useful channels for each individual atmospheric pro le. These are two methods based on the data resolution matrix, one method based on the Jacobian matrix, and one iterative method selecting sequentially the channels with largest information content. The Jacobian method and the iterative method were found to be the most suitable for the problem. The iterative method was demonstrated to always produce the best results, but at a larger cost than the Jacobian method. To test the robustness of the iterative method, a variant has been tried. It consists in building a mean channel selection aimed at optimizing the results over the whole database, and then applying to each pro le this 'constant' selection. Results show that this 'constant' iterative method is very promising, with results of intermediate quality between the ones obtained for the optimal iterative method and the Jacobian method. The practical advantage of this method for operational purposes is that the same set of channels can be used for various atmospheric pro les.