Three dimensional variational analysis with spatially inhomogeneous covariances

1. Three dimensional variational analysis with spatially inhomogeneous covariances Wan-Shu Wu, R. James Purser and Dave F. Parrish Introduction Apply recursive filter to a global domain Multivariate relation Background error covariance Covariance with fat-tailed power spectrum Comparison with SSI Conclusion

2. Analysis system produces an analysis through the minimization of an objective function given by J = xT B-1 x + ( H x – y ) T R-1 ( H x – 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 – H xguess R is the observational and representativeness error covariance matrix H is the transformation operator from the analysis variable to the form of the observations. Goal: make minimal adjustment of the first guess to fit the information in the data Analysis variables: stream function, velocity potential, temp, q/qs(guess), Psfc

3. The basic recursive filter is a repetition of smoothing in one direction. The smoothing operation consists of an advancing sweep F i = (1-a ) D i + a Fi-1 Input D1 …………….. Di-1 Di Di+1……………………Dn Output F1 ……………... Fi-1 Fi …..…….………………Fn For increasing index i, follow by a backing sweep B i = (1-a ) F i + a Bi+1 Input F1 …………….. Fi-1 Fi Fi+1……………………Fn Output B1 ………………... Bi Bi+1 .….………………Bn For decreasing i. The smoothing parameter a lies between 0 and 1 and is related to the correlation length of the smoothing response function. Requirements on the filter: • Accommodation of geographically adaptive horizontal scale • Amplitude control • Numerical-artifact-free boundary treatment

4. Gaussian: An isotropic response is obtained by sequentially applying filter in ‘x’ and in ‘y’. No other profile shape possesses this simplifying property. The result on the Cartesian grid of (a) 4 application of1st order filter (b) 1 application of 4th order filter and (c) the analytical Gaussian.

5. Apply recursive filters to the global domain in Gaussian grid polar patches zonal band note: 3 requirements on the filters

6. Apply recursive filters to the global domain in Gaussian grid merge without blending merge with blending homogenous / inhomogeneous

7. Multivariate relation • Balanced part of the temperature is defined by Tb = G y where G is an empirical matrix that projects increments of stream function at one level to a vertical profile of balanced part of temperature increments. G is latitude and height dependent. • Balanced part of the velocity potential is defined as cb = c y where coefficient c is function of latitude and height. • Balanced part of the surface pressure increment is defined as Pb = W y where matrix W integrates the appropriate contribution of the stream function from each level.

8. Multivariate relation U (left) and v(right) increments at sigma level 0.267, of a 1 m/s westerly wind observational residual at 50N and 330 E at 250 mb.

9. Multivariate relation Vertical cross section of u and temp global mean fraction of balanced temperature and velocity potential

10. Background error covariance The error statistics are estimated ingrid space with the ‘NMC’ method. Stats of y are shown. Stats are function of latitude and sigma level. The error variance (m4 s-2) is larger in mid-latitudes than in the tropics and larger in the southern hemisphere than in the northern. The horizontal scales are larger in the tropics, and increases with height. The vertical scales are larger in the mid-latitude, and decrease with height.

11. Background error covariance Horizontal scales in units of 100km vertical scale in units of vertical grid

12. Fat-tailed power spectrum Psfc increments with homogeneous Scales with single recursive filter: Cross validation Scale c

13. Fat-tailed power spectrum Psfc increments with inhomogeneous scales with single recursive filter: scale v (left) and multiple recursive filter: fat-tail (right)

14. GDAS comparison with SSI Two sets of T62 assimilation cycled for 19 days to produce 2 weeks verifiable 5-day forecasts. 14-case means are 0.750/0.751 and 0.728/0.716 (left); 8.04/8.50 and 3.95/4.55 m/s (right).

15. GDAS comparison with SSIu-component wind at 850 mb in the tropics for control(above) and experiment

16. GDAS comparison with SSI Differences: • localized vertical correlations, dropping negative correlations farther away in the vertical • mixture of spectral approach (stratospheric levels) and model grid space filter approach • Dropping explicit calculation of linear balance operator in statistical multivariate relations • Dropping scale dependent multivariate relations and non-separability (different vertical scale for each horizontal scale) • … Can these differences and mixture give rise to inconsistencies? -- leading to increase noise problems during the initial integration of the model

17. GDAS comparison with SSI Cross section of T increments at 100w for SSI (left) and experiment (right)

18. GDAS comparison with SSI Global RMS divergence of analysis and forecast after the first time step analysis_1 forecast_1 analysis_2 forecast_2 Exp 8.593e-6 8.361e-6 8.763e-6 8.583e-6 Cntll 8.082e-6 7.962e-6 7.644e-6 7.966e-6 Global mean convective precipitation 0-3 hr 3-6 hr 6-9 hr Exp 0.2443 0.2331 0.2459 Cntl 0.2449 0.2363 0.2449

19. Conclusion • Gain: freedom in spatial variation of covariance Price: limited freedom in specifying the shape of the error statistics in wave number space. (The limitation is partially over come by applying multiple recursive filters for structure function) • In extra-tropics 3D Var in physical space can be as effective as in spectral space. Spatial variation in error stats is beneficial to forecasts in the tropics. • Straightforward to apply to a regional domain. chance to test in parallel system