MPO 674 Lecture 20 3/26/15

3d-Var vs 4d-Var

Ensemble Kalman Filters Want flow-dependent, dynamical covariances Several different types of Kalman filter exist, all of which have a linear inference. Non-linear filters are too hard. Seek simple approximations... Ensemble Kalman filters use P f = Z f Z fT Z f is an n x k matrix containing k ensemble perturbations (about a mean state) of length n. Perturbation

P f = Z f Z fT (u’1)1 (u’1)2 (u’1)3 (v’1)1 (v’1)2 (v’1)3 (T’1)1 (T’1)2 (T’1)3 (p’1)1 (p’1)2 (p’1)3 (u’2)1 (u’2)2 (u’2)3 (v’2)1 (v’2)2 (v’2)3 (T’2)1 (T’2)2 (T’2)3 (p’2)1 (p’2)2 (p’2)3 (u’3)1 (u’3)2 (u’3)3 … … … (u’1)1 (v’1)1 (T’1)1 (p’1)1 (u’2)1 (v’2)1 (T’2)1 (p’2)1 (u’3)1 (u’1)2 (v’1)2 (T’1)2 (p’1)2 (u’2)2 (v’2)2 (T’2)2 (p’2)2 (u’3)2 (u’1)3 (v’1)3 (T’1)3 (p’1)3 (u’2)3 (v’2)3 (T’2)3 (p’2)3 (u’3) Ensemble members

P f = Z f Z fT (u’1)1 (v’1)1 (T’1)1 (p’1)1 (u’2)1 (v’2)1 (T’2)1 (p’2)1 (u’3)1 (u’1)2 (v’1)2 (T’1)2 (p’1)2 (u’2)2 (v’2)2 (T’2)2 (p’2)2 (u’3)2 (u’1)3 (v’1)3 (T’1)3 (p’1)3 (u’2)3 (v’2)3 (T’2)3 (p’2)3 (u’3)3 (u’1)1 (u’1)2 (u’1)3 (v’1)1 (v’1)2 (v’1)3 (T’1)1 (T’1)2 (T’1)3 (p’1)1 (p’1)2 (p’1)3 (u’2)1 (u’2)2 (u’2)3 (v’2)1 (v’2)2 (v’2)3 (T’2)1 (T’2)2 (T’2)3 (p’2)1 (p’2)2 (p’2)3 (u’3)1 (u’3)2 (u’3)3 … … … Ensemble members

P f = Z f Z fT (u’1)1 (v’1)1 (T’1)1 (p’1)1 (u’2)1 (v’2)1 (T’2)1 (p’2)1 (u’3)1 (u’1)2 (v’1)2 (T’1)2 (p’1)2 (u’2)2 (v’2)2 (T’2)2 (p’2)2 (u’3)2 (u’1)3 (v’1)3 (T’1)3 (p’1)3 (u’2)3 (v’2)3 (T’2)3 (p’2)3 (u’3) Ensemble members (u’1)1 (u’1)2 (u’1)3 (v’1)1 (v’1)2 (v’1)3 (T’1)1 (T’1)2 (T’1)3 (p’1)1 (p’1)2 (p’1)3 (u’2)1 (u’2)2 (u’2)3 (v’2)1 (v’2)2 (v’2)3 (T’2)1 (T’2)2 (T’2)3 (p’2)1 (p’2)2 (p’2)3 (u’3)1 (u’3)2 (u’3)3 … … …

DA in Tropical Cyclones Artificial operational methods: –Bogus Vortex (US Navy, UKMet) –Relocation (NCEP GFS) –Vortex Spin-Up (GFDL) Can one develop a physically consistent method to do DA in hurricanes?

(Brian Etherton)

WRF: PRELIMINARY RESULTS (Xuguang Wang, U. Oklahoma) (1) Assimilation of one v ob: 5 m/s higher than first guess v

(2) EnKF-based covariance of decrease in central sea level pressure with T and v

Single 850 hPa u observation (3 m/s O-F, 1m/s error) in Hurricane Ike EnKF 3d-Var hybrid Whitaker, Hamill, Kleist, Parrish, Derber, Wang

Innovation 10 m/s Ob error 0.1 m/s 4 (u,v) obs 1 degree from center in GSI 1 v ob 150km east of center in WRF/EnKF. CAVEAT: Different Storms! 3d-Var vs EnKF

GSI Analysis Increment of v (m/s) Operational L z WRF/EnKF Analysis Increment of v (m/s)

GSI Analysis Increment of v (m/s) 2 x L z WRF/EnKF Analysis Increment of v (m/s)

GSI Analysis Increment of v (m/s) 4 x L z WRF/EnKF Analysis Increment of v (m/s)

3d-Var –Geostrophic & hydrostatic correction (needed 4 “obs”). –Local horizontal correction. –Significant increment only through boundary layer. –Increasing L z improved storm structure and track forecast. EnKF –Gradient wind correction (used only 1 “obs”). –Stronger winds associated with shifting the storm further east and increasing intensity of winds. –Troposphere-deep increment: vertical error correlation lengthscale equivalent to 4x of GSI. Summary: 3d-Var vs EnKF

From Tom Hamill

Sampling Error and Covariance Localization

Petrie (1998, MS Thesis)

Example of covariance localization Background-error correlations estimated from 25 members of a 200-member ensemble exhibit a large amount of structure that does not appear to have any physical meaning. Without correction, an observation at the dotted location would produce increments across the globe. Proposed solution is element-wise multiplication of the ensemble estimates (a) with a smooth correlation function (c) to produce (d), which now resembles the large-ensemble estimate (b). This has been dubbed “covariance localization.” from Hamill, Chapter 6 of “Predictability of Weather and Climate” obs location

Why localization? Relative sampling errors depend on ensemble size, correlation. True background-error covariance of model with two grid points: Suppose there is inaccurate estimate of background-error covariance matrix and obs at grid point 1 with error covariance How do errors change with correlation and ensemble size? Relative error is large when correlation and/or ensemble size is small. Ref: Hamill et al, 2001, MWR, 129, pp From Tom Hamill

Spreads and errors as function of localization length scale Ref: Houtekamer and Mitchell, 2001, MWR, 129, pp (1) Small ensembles minimize error with short length scale, large ensemble with longer length scale. That is, less localization needed for large ensemble. (2) The longer the localization length scale, the less spread in the posterior ensemble. From Tom Hamill

Filter Divergence

Hamill et al. (2001, MWR)