4D-Var is a temporal extension of 3D-Var. Observations are organized in one-hour time-slots as described in Section 3.2. The cost-function now measures the distance between a model trajectory and the available information (background, observations) over an assimilation interval or window. For a 12-hour window (as currently used), it is either (03UTC-15UTC) or (15UTC-03UTC). Eq. (1.2) (see Chapter 1 `Incremental formulation of 3D/4D variational assimilation-an overview' is replaced by

(3.1)

with subscript i the time index. Each i corresponds to one-hour time slot.

is as before the increment at low resolution at initial time, and

the increment evolved according to the tangent linear model from the initial time to time index i.

and

are the covariance matrices of observation errors at time index i and of background errors respectively.

is a suitable linear approximation at time index i of the observation operator

. The innovation vector is given at each time step by

, where

is the background propagated in time using the full nonlinear model and

is the observation vector at time index i. As SYNOP and DRIBU time sequences of surface pressure and height data are now used with serial correlation of observation error, the observation costfunction computation for those data spans all time slots. Eq. (3.1) therefore needs generalising, as has been done in the paper by Järvinen et al. (1999).

The minimization is performed in the same way as in 3D-Var. However, it works fully in terms of increments, a configuration which is activated by the switches L131Tl and LOBSTL, and involves running the tangent-linear and adjoint models iteratively as explained in Section 2.3 of Chapter 2 `3D variational assimilation' , and using the tangent-linear observation operators.

A way to account in the final 4D-Var analysis for some non-linearities is to define a series of minimization problems

(3.2)

with superscript n the minimization index.

is the current estimate of the atmospheric flow. It is equal to the background for the first minimization.

is the innovation vector, computed by integrating the model at high resolution from the current estimate. The way the increment is added to the current estimate is similar to that used in 3D-Var (see Chapter 1 `Incremental formulation of 3D/4D variational assimilation-an overview' .

(3.3)

The number of times the trajectory is updated, i.e. the number of outer-loops (which corresponds to the number of minimizations performed), is typically a number between one and four. In operational 4D-Var the number of outer loops is two.

This can be controlled in the prepIFS set-up, together with the number of inner-loops (iterations of m1qn3) within each minimization. One outer-loop corresponds to what is normally done in 3D-Var. The number of inner-loops should then be 70 as in 3D-Var. The most standard 4D-Var uses two outer-loops. The first minimization runs with the simplified physics on 50 inner-loops. The second minimization runs with the more complete linear physics on 25 inner-loops. Switches for the two sets of physics will be given in Section 3.4.

The variational quality-control (Chapter 2 `3D variational assimilation' Section 2.6) is switched on at the default iteration number (40) in the first minimization. It is activated from the first iteration in the subsequent minimizations.

The final 4D-Var trajectory is post-processed every 3 hours. Fields called 4v are created with initial date and time the start of the window (03UTC or 15UTC) and steps every 3 hours. The 4v field valid at 12UTC or 00UTC, is then renamed as the final analysis (type=an) for the atmospheric fields and the waves. The cycling from one cycle to the next is performed by taking these analysis fields, together with the surface fields updated by the SST, snow and soil moisture analyses as input to a 12-hour forecast which produces the background for the next cycle.

The analysis and forecast error calculations are performed as explained in Chapter 7 `Background, analysis and forecast errors' , with the inclusion of the time dimension in the minimization. The analysis error variances are available at the beginning of each window, and the forecast error variances at the end.

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20.03.2002

DATA ASSIMILATION

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Table of contents

3.1 Introduction