# Presentation: 218839

## 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

## 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

## The basic recursive filter is a repetition of smoothing in one direction. The smoothing operation consists of an advancing sweep		F i = (1- ) D i +  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- ) F i +  Bi+1
- Input          F1 ................. Fi-1 Fi  Fi+1........................Fn
- Output        B1 ..................... Bi Bi+1  .......................Bn
- For decreasing i. The smoothing parameter  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

## 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 of 1st order filter (b) 1 application of 4th order filter and (c) the analytical Gaussian.

## Apply recursive filters to the global domain in Gaussian grid

- 		polar patches				zonal band
-  note: 3 requirements on the filters

## Apply recursive filters to the global domain in Gaussian grid

- 	    merge without blending		    merge with blending
- homogenous / inhomogeneous

## Multivariate relation

- Balanced part of the temperature is defined by
- 			Tb = G 
- 	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
- 			b = c 
- 	where coefficient c is function of latitude and height.
- Balanced part of the surface pressure increment is defined as
- 			Pb = W 
- 	where matrix W integrates the appropriate contribution of the stream function from each level.

## 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.

## Multivariate relation

- Vertical cross section of u and temp 		global mean fraction of balanced 					temperature and velocity potential

## Background error covariance

- The error statistics are estimated in grid space with the ‘NMC’ method. Stats of  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.

## Background error covariance

- Horizontal scales in units of 100km	       vertical scale in units of vertical grid

## Fat-tailed power spectrum

- Psfc increments with homogeneous
- Scales with single recursive filter: 			Cross validation
- Scale c

## Fat-tailed power spectrum

- Psfc increments with inhomogeneous scales with single recursive filter: scale v (left) and multiple recursive filter: fat-tail (right)

## 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).

## GDAS comparison with SSIu-component wind at 850 mb in the tropics for control(above) and experiment

## 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

## GDAS comparison with SSI

- Cross section of T increments at 100w for SSI (left) and experiment (right)

## 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

##  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