Documentation on CV_OPTIONS=3 in WRF 3D Var

Wan-Shu Wu

This documentation is adopted from Wu et al ( 2002). For detailed discussion and other relevant issues please refer to the paper.

The functional to be minimized is

• J = 1/2 [x^T B^-1 x + (Hx-y)^T R^-1 (Hx-y )]

where

x is a vector of analysis increments,
• B is the background error covariance matrix,
y is a vector of the observational residuals, y=y_obs - Hx_guess
• R is the observational and representativeness error covariance matrix,
H is a transformation operator from the analysis variable to the form of the observation vector.

Definition of Analysis Variables

The analysis variables, defined on the grid, are: stream function (S); unbalanced part of velocity potential (Z_u); the unbalanced part of temperature (T_u); unbalanced part of surface pressure(P_u); and pseudo relative humidity (water vapor mixing ratio divided by the saturated value from the guess field, Dee and Da Silva, 2002 ).

Background Error Covariance

The background error B can be written as

• (VB_z B_xB_y) (VB_z B_xB_y)^T

where

• V is the standard deviations of the error. Subroutine: DA_VToX_Transforms/da_apply_be.inc
• B_x, B_y and B_z are applications of recursive filters ( Purser et al 2003a, b ) in the x, y and z directions. Subroutine: DA_VToX_Transforms/da_apply_rf.inc

The amplitudes and scales of the default background error statistics are defined as functions of latitude and height on the global domain. (default statistics file: wrfstat) These statistics are automatically interpolated to any user defined WRF domain and grid. The input parameters as1, as2, as3, as4, as5 in record11 are the tuning coefficients for the five analysis variables. These parameters are arrays with 3 elements each for variance, horizontal scale and vertical scale respectively.

Subroutine: DA_Setup_Structures/da_setup_background_errors3.inc

Calculation of Balanced Part Variables

For 3D Var in physical space, the multivariate coupling between the mass and wind variables is challenging. Since the variables are defined in physical space, it is not easy to apply a linear balance operator (Parrish & Derber 1992) which includes the inverse of the Laplacian operator. However, the relation between the mass field and the stream function is linear, so that statistical regression between the two is possible.

The balanced part of the temperature increment is defined as T_b =GS where matrix G projects stream function increments to a vertical profile of the balanced part of temperature increments. Linear regression is used to calculate the G matrix. Since the variables are defined on the grid this matrix can be latitude dependent. The balanced part of the surface pressure increment is defined as P_b = WS where matrix W integrates the appropriate contribution of the stream function from each level. The balance design is crucial; the assimilation degrades quickly without it. For example, without mass-wind balance the fit of the guess field to the surface pressure observations worsens with time and doubles in magnitude within two days (8 cycles) of the assimilation. However with statistical linear balance defined in the analysis variables, the quality of the first guess field is maintained.

A vertically localized correlation between the velocity potential and the stream function is also implemented to take into account the positive correlation between divergence and vorticity in the planetary boundary layer. The balanced part of the velocity potential is defined as Z_b = c S where coefficient c is a function of latitude and height.

Subroutine: DA_VToX_Transforms/da_transform_bal.inc

The balance projection is included as part of the forward operator H. For example, calculation of H^T R^-1 (Hx-y ) involves the following steps:

• calculate T_b, Z_b and P_b from the analysis variable S and add them to the unbalanced parts to obtain the total fields.
• convert S and Z to U and V wind components.
• observational forward model of the observation, e.g., interpolation to the observation location for most of the conventional data.
• calculate the residual, multiply by R^-1,
• apply the adjoint of the first three steps in reverse order.

Estimation of Default Background Error Covariance

The error variance is estimated in grid space by what has become known as the NMC method (Parrish and Derber 1992) . The statistics are estimated with the differences of 24 and 48-hour GFS forecasts with T170 resolution valid at the same time for 357 cases distributed over a period of one year. Both the amplitudes and the scales of the background error have to be tuned to represent the (6-hr? 3-hr?) forecast error in the guess fields. This is the reason for the tuning parameters mentioned above. The statistics that project multivariate relations among variables are also derived from the NMC method.

The variance of each variable and the variance of its second derivative are used to estimate its horizontal scales. For example, the horizontal scales of the stream function can be estimated from the variance of the vorticity and stream function.

The vertical scales are estimated with the vertical correlation of each variable. A table is built to cover the range of vertical scales for the variables. The table is then used to find the scales in vertical grid units. The filter profile and the vertical correlation are fitted locally. The scale of the best fit from the table is assigned as the scale of the variable at that vertical level for each latitude. Note that the vertical scales are locally defined so that the negative correlation further away in the vertical direction is not included.

The steps to produce the error statistics are

• use least-squares fit to find the multivariate projection arrays.
• subtract balanced part, calculated with the projection arrays, from the total error fields
• calculate the variance and scales of the unbalanced fields

REFERENCES

Dee, D. P., and A. M. Da Silva, 2002: On the choice of variable for atmospheric moisture analysis. Mon. Wea. Rev., 131, 155-171.

Parrish, D. F., and J. C. Derber, 1992: The National Meteorological Center's Spectral Statistical-interpolation Analysis System. Mon. Wea. Rev., 120, 1747-1763.

Purser, R. J., W.-S. Wu, D. F. Parrish, and N. M. Roberts, 2003a: Numerical aspects of the applications of recursive filters to variational statistical analysis. Part I: spatially homogeneous and isotropic Gaussian covariances. Mon. Wea. Rev., 131,

Purser, R. J., W.-S. Wu, D. F. Parrish, and N. M. Roberts, 2003b: Numerical aspects of the applications of recursive filters to variational statistical analysis. Part II: spatially inhomogeneous and anisotropic general covariances. Mon. Wea. Rev., 131,

Wu, W.-S., R. J. Purser, and D. F. Parrish, 2002: Three dimensional variational analysis with spatially inhomogeneous covariance. Mon. Wea. Rev. 130. 2905-2916.