Cmparisn f hybrid ensemble-4dvar with EnKF and 4DVar fr reginal-scale data assimilatin Jn Pterjy and Fuqing Zhang Department f Meterlgy The Pennsylvania State University Wednesday 18 th December, 2013
Intrductin Zhang and Zhang (MWR, 2012) Zhang, Zhang, and Pterjy (MWR, 2013) A prttype hybrid ensemble-4dvar (E4DVar) system utperfrmed EnKF and 4DVar in a mnth-lng eperiment ver the cntinental United States. Eperiments used the WRF mdel with a 90-km grid spacing.
Intrductin The current study eamines the perfrmance f a newly develped E4DVar system that is based n multi-incremental WRF-4DVar. EnKF, 4DVar, and E4DVar are applied fr a case f trpical cyclgenesis t cmpare the three methds at the messcale.
Cupled EnKF-Var data assimilatin Ensemble Frecast Separate ensemble int mean and perturbatins Run 3D/4DVar and EnKF separately 3D/4DVar analysis becmes psterir mean fr EnKF perturbatins Ensemble Analysis f 1,0 f f, 2,0,..., N,0 1,0! f,! 0 f f 2,0 f,...,! N,0 3D/4DVar f - 0 is used as first guess -! f f f 1,0,! 2,0,...,! N,0 are used in the cst functin EnKF f - 0 is used as first guess -! f f f 1,0,! 2,0,...,! N,0 are used t estimate the cvariance 0 a 1,0! a,! a 2,0,...,! a N,0 a 1,0, a a 2,0,..., N,0 Zhang et al. (AAS, 2009)
Hybrid cst functin Cmpsitin f hybrid increments: δ 0 = δ c 1 N 0 + (a n f n,0) N 1 n=1 The N weighting vectrs a n are cncatenated t frm the α cntrl variables fr the ensemble perturbatins α T = [a T 1, a T 2,..., a T N].
Hybrid cst functin J(δ 0 ) = β c J b (δ c 0) + β e J e (α) + J (δ 0 ) 1 = β c 2 δct 0 B 1 δ c 0 1 + β e 2 αt A 1 α + 1 τ [H t M t δ 0 d t ] T R 1 t [H t M t δ 0 d t ]. 2 t=0 d t is calculated with respect t a mdel trajectry frm f 0: d t = y t H t [M t ( f 0)]. Lrenc (QJRMS, 2003), Wang et al. (MWR, 2008)
Trpical cyclgenesis case 30 N 20 N 10 N 0 00 UTC 18 Sept. Track trpical wave trpical strm hurricane 00 UTC 9 Sept. 4DVar, EnKF, and E4DVar are cmpared fr 10 days during the genesis, rapid intensificatin and decay f Hurricane Karl (2010). 100 W 90 W 80 W 70 W 60 W 50 W Karl develped frm a trpical disturbance that initiated n 8 Sept., and frmed a trpical depressin n 18 UTC 14 Sept. Eperiments are initialized frm GFS/GDAS analyses n 18 UTC 07 Sept. Analyses are perfrmed every si hurs until 00 UTC 18 Sept.
Observatins Cnventinal bservatins 30 N 20 N 10 N 0 100 W + + + ++ + + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + 00 UTC 18 Sept. + + 90 W 80 W 70 W 60 W Field bservatins Track trpical wave trpical strm hurricane Observatins sat winds belw 500 mb sat winds abve 500 mb sundings + buys METARs 00 UTC 9 Sept. 50 W The assimilated data include cnventinal bservatins and drpsnde measurements cllected during the NSF Predepressin Investigatin f Clud Systems in the Trpics (PREDICT) field campaign. 30 N Track trpical wave 20 N 10 N 0 00 UTC 18 Sept. 00 UTC 9 Sept. trpical strm hurricane PREDICT drpsndes 10 Sept. flight missins (2) 11 Sept. flight missin 12 Sept. flight missin 13 Sept. flight missin 14 Sept. flight missin GRIP drpsndes 12 Sept. flight missin 13 Sept. flight missin 14 Sept. flight missin 16 Sept. flight missin 17 Sept. flight missin Drpsndes frm the NASA Genesis and Rapid Intensificatin Prcesses (GRIP) eperiment are used fr verificatin. 100 W 90 W 80 W 70 W 60 W 50 W
Eperiment cnfiguratin The data assimilatin is perfrmed n a 13.5-km grid spacing dmain with 34 vertical levels. Deterministic frecasts use a 4.5-km strm-fllwing nested dmain. Inner-lp iteratins (4DVar/E4DVar) use a 40.5-km grid spacing. Ensemble eperiments use 60 members with a 900-km ROI fr lcalizatin and 80% relaatin t prir perturbatins The hybrid cst functin uses 80% f the ensemble increment and 20% f the climatlgical increment.
Deterministic frecast results 7 6.5 a) u (m s 1 ) 7 6.5 b) v (m s 1 ) c) T (K) 2.8 2.6 d) q (g kg 1 ) RMSD (bservatins) 6 5.5 5 4.5 4 3.5 3 2.5 2 6 5.5 5 4.5 4 3.5 3 2.5 2 1.1 1 0.9 0.8 0.7 2.4 2.2 2 1.8 1.6 1.4 1.2 RMSD t rutine sundings and field bservatins within 800 km f strm center 1.5 0 h 24 h 48 h 72 h 1.5 0 h 24 h 48 h 72 h 0.6 0 h 24 h 48 h 72 h 1 0 h 24 h 48 h 72 h 3 e) u (m s 1 ) 3 f) v (m s 1 ) 0.7 g) T (K) 1.6 h) q (g kg 1 ) 2.8 2.8 0.65 1.5 RMSD (GDAS) 2.6 2.4 2.2 2 1.8 0 h 24 h 48 h 72 h Frecast lead time 2.6 2.4 2.2 2 1.8 1.6 0 h 24 h 48 h 72 h Frecast lead time 0.6 0.55 0.5 0.45 0 h 24 h 48 h 72 h Frecast lead time 1.4 1.3 1.2 1.1 1 EnKF 4DVar E4DVar 0.9 0 h 24 h 48 h 72 h Frecast lead time RMSD t GDAS data within 2500 km f strm center
Deterministic track and intensity frecasts 24 N EnKF 24 N 4DVar 24 N E4DVar 22 N 22 N 22 N 20 N 20 N 20 N 18 N 16 N 14 N 18 N 16 N 14 N 18 N 16 N 14 N 12 N a) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W 12 N b) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W 12 N c) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W Ma wind speed (m s 1 ) 50 40 30 Analysis time 00 UTC 12 Sept. 06 UTC 12 Sept. 12 UTC 12 Sept. 18 UTC 12 Sept. 00 UTC 13 Sept. 06 UTC 13 Sept. 12 UTC 13 Sept. 18 UTC 13 Sept. 00 UTC 14 Sept. 06 UTC 14 Sept. 12 UTC 14 Sept. 18 UTC 14 Sept. 00 UTC 15 Sept. 50 40 30 20 20 20 10 10 d) e) f) 10 12 Sept. 14 Sept. 16 Sept. 18 Sept. 12 Sept. 14 Sept. 16 Sept. 18 Sept. 12 Sept. 14 Sept. 16 Sept. 18 Sept. Frecasts t 00 UTC 18 Sept. using analysis times leading up t genesis. 50 40 30
Deterministic track and intensity frecasts 24 N EnKF 24 N 4DVar 24 N E4DVar 22 N 22 N 22 N 20 N 20 N 20 N 18 N 16 N 18 N 16 N 18 N 16 N 14 N 14 N 14 N 12 N a) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W 12 N b) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W 12 N c) 10 N 100 W 95 W 90 W 85 W 80 W 75 W 70 W 65 W 50 Analysis time 00 UTC 14 Sept. 06 UTC 14 Sept. 12 UTC 14 Sept. 18 UTC 14 Sept. 50 50 Ma wind speed (m s 1 ) 40 30 40 30 20 20 20 10 10 10 d) e) f) 12 Sept. 14 Sept. 16 Sept. 18 Sept. 12 Sept. 14 Sept. 16 Sept. 18 Sept. 12 Sept. 14 Sept. 16 Sept. 18 Sept. Frecasts t 00 UTC 18 Sept. using analysis times leading up t genesis. 40 30
4DVar and E4DVar structure functins
Balance Mean (hpa s 2 ) Standard deviatin (hpa s 2 ) Dmain rt mean square 2 p s / t 2 10 3 a) 10 4 10 5 10 6 10 4 10 5 10 6 10 7 0 2 4 6 8 10 12 b) EnKF 4DVar E4DVar 0 2 4 6 8 10 12 Frecast lead time (h) Dmain-averaged 2 P s is t 2 used t quantify gravity wave activity after initializatin. Value are averaged ver all deterministic frecasts. Standard deviatins in mean 2 P s shw amunt f t 2 variability between cycles.
Cst Apprimating a minimum t the 4DVar/E4DVar cst functin requires many iteratins f the tangent linear and adjint mdel (can be epensive). Inner-lp iteratins are averaged ver the 40 data assimilatin cycles: 4DVar E4DVar Mean iteratins 37.5 25.6 Mean analysis time (256 cres) 2236 s 1566 s
Cnclusins A tw-way cupled hybrid methd (E4DVar) is fund t utperfrm the benchmark EnKF and 4DVar systems fr a 10-day trpical cyclgenesis and rapid intensificatin event. E4DVar des nt require a lng assimilatin windw t develp flw-dependent infrmatin. It may als use bservatins at the beginning f the time windw mre affectively than 4DVar. The ensemble infrmatin imprves initial cnditin balance ver 4DVar and EnKF and reduces the number f iteratins required t minimize the 4-D cst functin. Future research will fcus n ptimal windw length, cnditining, and cmparisn with 4D-ens-Var.
Particle filtering Apprimate mments f the psterir errr distributin using f ( t ) = f ( t )p( t y t )d t, N wt n f (t n ). n=1 Fr the simplest particle filter, the weights are given by w n t = p(y t n t ) Nn=1 (y t n t ).
Lcal likelihd particle filter (LLPF) Prblem: unless the ensemble is relatively large, it will eventually lse track f the signal, in which case, the weights becme cncentrated n a small number f particles. This prblem ccurs faster when the dimensins f and y are large (Snyder et al. 2008, MWR). Pssible slutin: use a likelihd functin that decays epnentially away frm bservatin lcatins. Filter degenerecy is then restricted t lcal regins f the dmain.
Lcal likelihd particle filter (LLPF) The Lrenze-96 mdel is used t test this idea and cmpare with EnKF: d t+1,i dt Eperiment details: 100 variables = ( t,i+1 t,i 1 ) t,i 1 t,i + 8. Observatins are taken frm a truth run at every third grid pint, with added Gaussian nise (σ = 1). These bservatins are assimilated every 24 h (dt = 0.2) fr 1000 cycles. Relaatin cefficient and lcalizatin radius fr EnKF are tuned fr each ensemble size.
Lcal likelihd particle filter (LLPF) N e = 50 30 cycle mean RMSE 2 1.8 1.6 1.4 1.2 1 0.8 0.6 EnKF LLPF 0.4 0 5 10 15 20 25 30 30 day averaging perid RMSEs are averaged every 30 cycles.
Lcal likelihd particle filter (LLPF) N e = 100 30 cycle mean RMSE 2 1.8 1.6 1.4 1.2 1 0.8 0.6 EnKF LLPF 0.4 0 5 10 15 20 25 30 30 day averaging perid RMSEs are averaged every 30 cycles.
Lcal likelihd particle filter (LLPF) N e = 200 30 cycle mean RMSE 2 1.8 1.6 1.4 1.2 1 0.8 0.6 EnKF LLPF 0.4 0 5 10 15 20 25 30 30 day averaging perid RMSEs are averaged every 30 cycles.
Lcal likelihd particle filter (LLPF) N e = 300 30 cycle mean RMSE 2 1.8 1.6 1.4 1.2 1 0.8 0.6 EnKF LLPF 0.4 0 5 10 15 20 25 30 30 day averaging perid RMSEs are averaged every 30 cycles.
Cnclusins A lcalized likelihd apprach t particle filtering is being investigated fr systems that cntain a large spatial dimensin. This methd makes use f a distance-dependent likelihd functin that can be applied t prevent filter degeneracy (much like lcalizatin in EnKF). Much mre wrk needs t be dne; e.g., resampling, inflatin, better chices fr likelihd functin, etc.
Etra A simple lcalizatin eample is t use a functin that decays epnentially away frm the lcatin f an bservatin, s that the likelihd f the the i th variable given the j th bservatin is written p(y t,j n t,i) = [p(y t,j n t ) 1 N e N y ]ep( d i,j R ) + 1 N e N y, where d i,j is the physical distance between the bservatin and mdel grid pint, and R is a tunable lcalizatin radius.
Etra EnKF (N = 60) 4DVar E4DVar (N = 60) Mean number f bservatins 26140 26140 26140 Mean inner-lp iteratins N/A 37.4 25.6 Number f NLM runs 60 2 62 Mean TLM time (s) N/A 17.3 17.3 Mean ADM time (s) N/A 39.5 39.5 Mean NLM time (s) 13.4 13.4 13.4 Mean IO (s) N/A 2.2 2.2 Mean analysis time (s) 840 2236 1566 TOTAL TIME (s): 1644 2263 2397