Ensem ble- - Based Data Assim ilation Based Data Assim ilation - PDF document

Ensem ble- - Based Data Assim ilation Based Data Assim ilation Ensem ble Martin Ehrendorfer Ehrendorfer Martin Department of Meteorology and Department of Meteorology and Data Assimilation Research Centre (DARC) Data Assimilation Research Centre (DARC) University of Reading University of Reading Workshop on Workshop on Mathematical Advancement in Geophysical Data Assimilation Mathematical Advancement in Geophysical Data Assimilation Banff International Research Station for Banff International Research Station for Mathematical Innovation and Discovery (BIRS) Mathematical Innovation and Discovery (BIRS) 3 – 3 – 8 February 2008 8 February 2008 Banff, Alberta, Canada Banff, Alberta, Canada 5 February 5 February 2008 2008 http:/ / darc.nerc.ac.uk/ http:/ / w w w .m et.reading.ac.uk/ o u t l i n e • 1 m otivation – im portance of DA • 2 BT and KF • 3 ensem ble-based KF – EnKF • 4 specific issues – degrading xb ps and m odes • 5 EnKF variants and directions • 6 sum m ary and discussion Ensemble-Based Data Assimilation, Banff, 5 February 2008 2 1

10-day & 9-day ECMWF forecast for 05/02/2008/1200UTC D+10 ECMWF forecast for 05 February 2008 → 1-day forecast error different from zero → model has sensitive dependence upon I.C. 50m D+10 10m 05/02 D+1 26/01 27/01 216 h D+9 ECMWF forecast for 05 February 2008 Ensemble-Based Data Assimilation, Banff, 5 February 2008 3 ERA-4 0 Re-Analysis Dec – Feb clim atology 7 0 m / s Ensemble-Based Data Assimilation, Banff, 5 February 2008 4 2

The January 0 8 Tellus I ssue Ensemble-Based Data Assimilation, Banff, 5 February 2008 5 s o m e d e v e l o p m e n t s • LeDimet & Talagrand 1986 • Talagrand & Courtier 1 9 8 7 • Tokyo 1995 (WMO-II) • 4DVAR ECMWF 2 5 Novem ber 1 9 9 7 Ensemble-Based Data Assimilation, Banff, 5 February 2008 6 3

adjoint code for grad J atime : do jt = ntstep , 0 , -1 innovation do k=1,3 ; x(k)=xs(k);xf(k)=xfs(k);xt(k)=xts(k) ;enddo jtp1 = jt+1 ; x (iobs) = x (iobs) + d (jtp1) call adtimstp1 ( x,xf,n,xt,dt,eps,jtp1 ) do k=1,3 ; xs(k)=x(k);xfs(k)=xf(k);xts(k)=xt(k) ;enddo irefc = jt ; call adrhs1t (x,xt,n) do k=1,3 ; xs(k) = xs(k) + x(k) ; xts(k)=0.0 ; enddo enddo atime xs (iobs) = xs (iobs) + d (0) see fortran printouts grad J Ensemble-Based Data Assimilation, Banff, 5 February 2008 7 o u t l i n e • 1 m otivation – im portance of DA • 2 BT and KF • 3 ensem ble-based KF – EnKF • 4 specific issues – degrading xb ps and m odes • 5 EnKF variants and directions • 6 sum m ary and discussion Ensemble-Based Data Assimilation, Banff, 5 February 2008 8 4

DA – Bayesian Estim ation Theory PRI OR B x b B background error covariance m atrix ANALYSI S, P a x a P a analysis error covariance m atrix DATA y Ensemble-Based Data Assimilation, Banff, 5 February 2008 9 data assim ilation in 1 D observation background gain analysis innovation bg error analysis increm ent Ensemble-Based Data Assimilation, Banff, 5 February 2008 10 5

com bination of tw o estim ators recursive filtering Ensemble-Based Data Assimilation, Banff, 5 February 2008 11 Kalm an ( 1 9 6 0 ) filter – recursive updating x initialization observation gain computation prediction state update covariance update analysis Ensemble-Based Data Assimilation, Banff, 5 February 2008 12 6

o u t l i n e • 1 m otivation – im portance of DA • 2 BT and KF • 3 ensem ble-based KF – EnKF • 4 specific issues – degrading xb ps and m odes • 5 EnKF variants and directions • 6 sum m ary and discussion Ensemble-Based Data Assimilation, Banff, 5 February 2008 13 Ensem ble KF … EnKF – conceptually PRIOR B x b P a x a ANALYSIS, P a x a DATA y Ensemble-Based Data Assimilation, Banff, 5 February 2008 14 7

Ensem ble Kalm an Filter - Equations Analysis Step Gain is formed without ever explicitly forming covariances Forecast Step + noise Houtekamer & Mitchell 2005 QJ Ensemble-Based Data Assimilation, Banff, 5 February 2008 15 EnKF – the analysis step Stochastic update algorithm s – Houtekam er & Mitchell 1 9 9 8 , 2 0 0 5 • K can be form ed w ithout ever explicitly com puting the full background B • Also: possibility of serial processing Determ inistic update algorithm s – update in a w ay that generates the sam e analysis error covariances that w ould have been obtained from the KF assum ing the KF‘s B is m odelled from the background ensem ble → square-root filters ( Tippett et al. 2 0 0 3 ) Ensemble-Based Data Assimilation, Banff, 5 February 2008 16 8

attractiveness of the EnKF Fundam entally statistical and physical Com putationally parallel Ability to account for som e aspects of nonlinearity – but fundam entally still least-squares Extendable to treat m odel error Circum vention of current B m odel restrictions Error statistics are flow -dependent and anisotropic Prim ary EnKF issues are treatm ent of sm all ensem ble sizes and treatm ent of m odel error Ensemble-Based Data Assimilation, Banff, 5 February 2008 17 attractiveness of the EnKF Fundam entally statistical and physical Com putationally parallel Ability to account for som e aspects of nonlinearity – but fundam entally still least-squares Extendable to treat m odel error Circum vention of current B m odel restrictions Error statistics are flow -dependent and anisotropic Prim ary EnKF issues are treatm ent of sm all ensem ble sizes and treatm ent of m odel error Ensemble-Based Data Assimilation, Banff, 5 February 2008 18 9

o u t l i n e • 1 m otivation – im portance of DA • 2 BT and KF • 3 ensem ble-based KF – EnKF • 4 specific issues – degrading xb ps and m odes • 5 EnKF variants and directions • 6 sum m ary and discussion Ensemble-Based Data Assimilation, Banff, 5 February 2008 19 specific issues I m pact of finite ensem ble size * – w hich com plications w ould disappear if ensem ble size w ere infinite? ( a) Sam pling error/ noise in covariance estim ation * ( b) Covariance localization * ( c) Balance and im balance * ( d) Filter divergence and ensem ble spread * ( e) I nflation, Model error and inbreeding ( use factor slightly > 1 ) ( f) Serial processing of observations ( g) Assim ilation of surface pressure ( h) Use of Lorenz and sim plified m odels ( i) Deficiency in grow th of analysis errors ( j) Ease of im plem entation and costs Ensemble-Based Data Assimilation, Banff, 5 February 2008 20 10

( a) Sam pling noise in estim ated covariances noise random sam pling: dom inates error in ( co) variances is proportional to 1 / N this leads to spurious – long-distance correlations spurious effects of – observations Lorenc 2003 Hamill 2006 Ensemble-Based Data Assimilation, Banff, 5 February 2008 21 ( b) covariance localization I m provem ent of noisy covariances – trying to elim inate artifacts of lim ited sam ple size Schur/ Hadam ard product – A = B o C – I f B and C are covariance m atrices then so is A to act as an heuristic attem pt → – so that distant features do not appear as dynam ically interrelated exam ples: Lorenc, Ham ill, Beck/ Ehrendorfer Ensemble-Based Data Assimilation, Banff, 5 February 2008 22 11

sam pled covariances m odified by Schur prd. Lorenc 2003 A = B o C C B A C … taken from Gaspari/Cohn Ensemble-Based Data Assimilation, Banff, 5 February 2008 23 localization exam ple: Ham ill 2 0 0 6 A = B o C C correlation function noisy B correlations A filtered 2 0 0 m em bers correlations Ensemble-Based Data Assimilation, Banff, 5 February 2008 24 12

height correlations 5 0 0 hPa derived from ensem ble integrations ( D+ 4 ) operational EPS ( N= 2 5 ) sam pling, N= 2 5 , M= 5 0 sam pling, N= 5 0 , M= 1 0 0 spurious correlations removed Ensemble-Based Data Assimilation, Banff, 5 February 2008 25 ( c) Balance and im balance Localization causes im balance – Schur product → • height increm ents fall off m ore rapidly w ith distance • require increased w ind increm ents for geostrophic balance • but w ind increm ents are reduced by Schur product factor Exam ples: – Effect of localization on geostrophic balance – significantly subgeostrophic w ind increm ents ( Lorenc 2 0 0 3 ) – PE m odel: im balance less for less severe localization ( Mitchell et al. 2 0 0 2 ) [ m easure ( ?) ] Ensemble-Based Data Assimilation, Banff, 5 February 2008 26 13