The statistical calibration is done using dedicated scripts outside the IFS code. First, the 24/48-hour forecast error differences for a set of dates are constructed in terms of spectral

and

. This involves running a set of MARS requests and building the required GRIB files. Then, the forecast-error differences are read and processed by a Fortran statistics program that finally writes two files in GSA format: one with the coefficients of the balance operator, one with the error covariances of

and

. These files take up a couple of megabytes. They are computed for a given triangular truncation and number of levels (currently T106L31). In the covariance file there are 4 sets of vertical covariance matrices. The balance files contain one set of coefficients for the

operator, and three sets of vertical balance matrices for

and

4.3.1 Input files

The IFS needs these two GSA files to use

in e.g. the incremental analysis jobs. The

configuration described here corresponds to namelist switch LSTABAL=.true. (NAMJG), and it is identified in the

code by the string CDJBTYPE=`STABAL96'. LSTABAL=.false. would give the old

formulation. The input files must be named stabal96.cv and stabal96.bal. They are read in by sujbdat and sujbbal, respectively.

4.3.2 Namelist parameters of

Some other important namelist options in NAMJG are LCFCE (to enforce uniform background errors), L3DBGERR (to have a 3D distribution vorticity background errors), and LCORCOSU (to enforce compactly supported horizontal correlations). The switch LGUESS in NAMCT0 can be used to switch off altogether. The default is LGUESS=.true., i.e. switched on.

Figure 4.1 Calling tree for subroutine sujbcov.

4.3.3 IFS routines

Inside the IFS code,

is localized in the setups below subroutine sujbcov and in the inverse change of variable cvar2in (and its adjoint and their inverses, CVAR2INAD, CVAR2 and CVAR2AD). Calling trees are shown in Fig. 4.1 and Fig. 4.2 . The computation of the cost function and its adjoint is done in sim4d(Section 2.3 in Chapter 2 `3D variational assimilation' )---it is planned to move it to a dedicated subroutine. The sequence within the set-up routine is the following:

(i) SUJBDAT: Reads covariances from file stabal96.cv,

Interpolates in the vertical to the model's vertical levels (if necessary)

Sets humidity correlations to 0, for pressures less than 100 hPa.

(ii) SUJBCOR: Sets up spectral correlation operator

Covariance matrices (one per

) are converted to vertical correlation matrices and horizontal autocorrelation spectra. The eigenvectors and the eigenvalues of vertical correlations matrices are computed using EIGENMD and stored in FGEDVNS and FGEGVNS-arrays (yomjg), respectively, for later use in JGVCOR, JGVCORI, JGVCORAD and JGVCORIAD. The horizontal autocorrelation spectra are stored in the FGSCNM-array (yomjg), for later use in JGHCOR and JGHCORI.

(iii) SUJBSTD: Set up background error standard deviations, see Subsection 4.3.4.

(iv) SUJBBAL: Set up balance constraint. Read the file stabal96.bal and store in yomjg, for later use in balstat, balstatad, balvert, balvertad, balverti and balvertiad as part of the change of variable operator.

(v) SUJBTEST: Test of the adjoint of the change of variable, if LJBTEST=.true.

The distributed memory affects the setups below sujbdat and sujbbal when the data files are read in (by the master processor only). First, the resolution of the files is read, then the relevant arrays are allocated and the actual data is read, truncated if necessary, and broadcast. The code is designed to work at any resolution.

Figure 4.2 Calling tree for subroutine cvar2in.

In the change of variable, there is a transposition of the fields between the horizontal and vertical balance operators, balstat and balvert, repsectively. Note that the operator

is performed by calling cvar2in, so in IFS parlance

corresponds to the inverse change of variable.

4.3.4 Background error

The background standard errors are set up below sujbstd (in SUINFCE, called from SUECGES) and used in jgnr or jgnrs (and their adjoint and inverses, jgnrad and jgnrsi). In addition to the covariance files, they use a gridpoint GRIB file called errgrib in order to specify the three-dimensional error patterns. The data from the file is converted to the appropriate parameters and resolution if needed. The background error fields for some parameters (wind, height, temperature and surface pressure) are built for the screening job although they are not needed in the analysis itself. For more information, refer to the chapter on the cycling of background errors, Chapter 7 `Background, analysis and forecast errors' .

4.3.4 (a) Humidity

The humidity background errors are currently not cycled - they are computed (in SUSHFCE under JGNR) by a simple empirical formulation as a function of the temperature

and relative humidity

of the background:

(4.13)

(4.14)

The standard deviation in terms of relative humidity is then converted to specific humidity, taking the variation of

of the equation

(4.15)

where

is the relative humidity,

is the saturation water-vapour pressure at the temperature in question (Tetens' formula, Eq. (5.11) in Chapter 5 `Conventional observational constraints' ) and

is pressure.

Humidity increments are forced to be negligibly small above the tropopause to avoid a systematic drift of stratospheric humidity over extended periods of data assimilation. This is achieved by setting a very low value of

for

everywhere the pressure is lower than 70hPa, and at anyother point where the pressure is lower than 500hPa and the background temperature and pressure fields are such that the square of the buoyancy frequency exceeds