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 Di1Dn
• Output F1 ... Fi-1 Fi ...Fn
• For increasing index i, follow by a backing sweep
• B i (1-a ) F i a Bi1
• Input F1 .. Fi-1 Fi Fi1Fn
• Output B1 ... Bi Bi1 ..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 of 1st 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 in grid 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