Variational Data Assimilation via Sparse Regularization

Transcription

1 Vritionl Dt Aimiltion vi Spre Regulriztion,, Efi Foufoul-Georgiou, Gild Lermn School of Mthemtic Deprtment of Civil Engineering June, 3

2 Evidence on Sprity of Geophyicl Signl Rinfll Prior in Wvelet Domin Hevy tiled hitogrm with lrge m round zero dbz A-A dbz/km 4 - Dim:[4, 56] Rnge: [, 5] Dim:[4, 56] Rnge: [-36, 4] R [dbz] Section A-A x [km]

3 Evidence on Sprity of Geophyicl Signl Rinfll Prior in Wvelet Domin dient domin, explined by the generlized Guin Pr. Hevy tiled hitogrm with lrge m round zero, where α, λ >. Guin (α = ) nd Lplce (α = ). BZ Zeroe c h log(pr.) log(pr.) - -5 = Ob. Fitted Guin c - - h c h [dbz] ure for mny high-reolution rinfll imge ( torm over TX nd FL). x km Extreme Jump log(pr.) log(p) b) HSTN HSTN MELB =.7 =.7 =.7 Generlized Guin p(x) = exp ( λ x α ) (α=) Guin (α=) Lplce p(x) exp ( λ x α ) α where x α α = Σ x i α Dim: [45 5] d H m= c h [dbz] [dbz]

4 MELB Evidence on Sprity of Geophyicl Signl Rinfll Prior in Wvelet Domin Rinfll Sttitic Hevy tiled hitogrm with lrge m round zero dient domin, explined by the generlized Guin, where α, λ >. Guin (α = ) nd Lplce (α = ). BZ Pr. Non-Guinity in grdient domin, explined by HSTNthe generlized Guin.5 p c exp λ c α, where α, λ >. Guin (α = ) nd Ob. Lplce (α = ). HSTN.4 - Fitted Zeroe.5 ( km) =.7 Guin Ob =.85 - Fitted Zeroe =.7 Guin.3 dbz -5 (α=) Guin = (α=) Lplce 3. Extreme -. Extreme - Jump Jump.. where x - -5 α α = Σ x i α Dim: [4, 56] Rnge: [, 5] c c c c h - - h h - - h c We tudied thi ignture for mny high-reolution c h [dbz] rinfll imge ( torm over h [dbz] TX nd FL). ure for mny high-reolution rinfll imge ( torm over TX nd FL). x km Dim: [45 5] log(pr.) Pr. log(pr.) torm over TX nd FL x km Dim: [45 6] log(pr.) log(p) b) x km Dim: [45 5] HSTN MELB =.7 =.7 log(pr.) log(pr.) log(pr.) log(p) b) HSTN MELB - - d H m= [dbz] c h [dbz] Generlized Guin p(x) = exp ( λ x α ) p(x) exp ( λ x α ) α =.7 = d H m= c h [dbz] [dbz] MELB

5 Evidence on Sprity of Geophyicl Signl Rinfll Sprity ) b) c).6 H V D. Pr.. Dim :[45, 6]; Rnge:[, 45] 4 c H 4 Dim :[45, 6]; Rnge: Sprity in the derivtive or wvelet domin

6 Evidence on Sprity of Geophyicl Signl Rinfll Sprity ) b) c).6 H V D. Pr.. Dim :[45, 6]; Rnge:[, 45] 4 c H 4 Dim :[45, 6]; Rnge:[, 45] Sprity in the derivtive or wvelet domin % of the dt contin 98% of the totl energy Sprity i trong prior knowledge. How to incorporte prity?

7 Evidence on Sprity of Geophyicl Signl Lorenz (963) x =σ (y x) y =x (ρ z) y z =xy βz The ytem i chotic for σ =, β = 8/3, ρ = z 4 - x - y 4 x(t) Time tep x(k) 'DCT coef.' Wvenumer (k) Lorenz ytem how remrkble prity in the DCT domin N π(t )(k ) x(k) = ω k t= x(t) co N

8 Sprity in Turbulent Velocity Signl Evidence on Sprity of Geophyicl Signl Sprity in Turbulent Flow Sprity in Turbulent Velocity Signl SAFL wind tunnel over flt nd rough urfce Reynold number: 4 x^4 Neutrl BL condition Meurement t khz t the length of 5. SAFL wind tunnel over flt nd rough urfce Reynold number: 4 x^4 SAFL wind tunnel Neutrl BL condition Meurement t khz t the length of 5 Re. = 4 4 f = 5 Dt courtey of Leo Chmoro, SAFL Dt courtey of Leo Chmoro, SAFL 7 6 f= khz v [m/] 7 f= khz log(p) v [m/] nt t 3 t 6 t 9 t t Ob. = Fit nt t d d d m= m=3 3 t m=6 6 t d d m=9 m= 9 t t Ob. = Fit log(p) - d m= log(p) log(p) - log(p) - d m=3 log(p) - log(p) -5 5 d m=6 log(p) log(p) - 4 d m=9 log(p) - 4 d m=

9 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion: y = H x + v R n Bckground Model: x b R m Error: v~n, R, w~n, B Φ i proper trnformtion which prifie! l -R3D-VAR ˆ rgmin ( ) b x B x x y x x x R l -R4D-VAR N b xˆ rgmin i i( ) y x x x x Ri x B i y x b W B

10 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion Obervtion: model: y y = i H H (x x + i ) + v v R i n R n y t Bckground Model: x b R m Error: v~n, R, w~n, B Φ i proper trnformtion which prifie! Ob. l -R3D-VAR ˆ rgmin ( ) b x B x x y x x x R l -R4D-VAR N b xˆ rgmin i i( ) y x x x x Ri x B i y x b W B

11 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion Obervtion: model: y y = i H H (x x + i ) + v v R i n R n y t Bckground Model: Bckground tte: x b x b R m Rm Error: v~n, R, w~n, B Φ i proper trnformtion which prifie! t x b l -R3D-VAR ˆ rgmin ( ) b x B x x y x x x R Ob. Bckground l -R4D-VAR N b xˆ rgmin i i( ) y x x x x Ri x B i y x b W B

12 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion Obervtion: model: y y = i H H (x x + i ) + v v R i n R n Bckground Model: Bckground tte: x b x b R m Rm Error: v~n, R, w~n, B Error: v N (, R), w N (, B) Φ i proper trnformtion which prifie! l -R3D-VAR ˆ rgmin ( ) b x B x x y x x x R Ob. y t Bckground t x b l -R4D-VAR N b xˆ rgmin i i( ) y x x x x Ri x B i y x b W B

13 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion Obervtion: model: y y = i H H (x x + i ) + v v R i n R n Bckground Model: Bckground tte: x b x b R m Rm Error: v~n, R, w~n, B Error: v N (, R), w N (, B) Φ i proper trnformtion which prifie! l -R3D-VAR ˆ rgmin ( ) b x B x 4D-VARx y x x x R { l -R4D-VAR k ˆx N = rgmin y i H (x i ) b R xˆ rgmin x i i( ) i i= y x x + x b x i ix x R x B i.t. x i = M, t (x ) y t x b t M( ) B Ob. Anlyi Bckground } y x b W B

14 l-norm regulriztion Dt Aimiltion vi Spre Regulriztion Vritionl Dt Aimiltion vi Spre Regulriztion The The true true tte: tte: x xr m R m Obervtion Obervtion: model: y y = i H H (x x + i ) + v v R i n R n Bckground Model: Bckground tte: x b x b R m Rm Error: v~n, R, w~n, B Error: v N (, R), w N (, B) Φ i proper trnformtion which prifie! Φ: pre-elected bi l -R3D-VAR ˆ rgmin ( ) b x B x R4D-VARx y x x x R { l -R4D-VAR k } ˆx N = rgmin y i H (x i ) R b xˆ x rgmin i i( ) i= i y x + x b x i i x x x + λ Φx B R x B i.t. x i = M, t (x ) y t x b t M( ) Ob. Anlyi Bckground y x b W B

15 l-norm regulriztion Qudrtic Progrmming Auming Φx = c R m, then the bove problem cn be rewritten, minimize z { } ct Qc + b T c + λ c where, Q = Φ T ( B + H T R H ) ) Φ nd b = Φ (B T x b + HT R y. () Hving c = u v, where u = mx (c, ) R m nd v = mx ( c, ) R m nd then w = [u T, vt ]T, the more tndrd QP formultion of the problem i immeditely followed : minimize w { [ wt Q Q Q Q ] ( [ w + λ m + b b ]) T w } ubject to w. () Obtining ŵ = [û T, ˆvT ]T R m the olution of (), one cn eily recover ĉ = û ˆv nd thu the initil tte of interet ˆx = Φ ĉ.

16 l-norm regulriztion Grdient Projection Method

17 l-norm regulriztion Advection-Diffuion Eqution Flt nd Qudrtic Top-ht (prity in wvelet) x(, t) + x(, t) = ɛ x(, t) t x(, ) = x ()

18 l-norm regulriztion Advection-Diffuion Eqution Flt nd Qudrtic Top-ht (prity in wvelet) x(, t) + x(, t) = ɛ x(, t) t x(, ) = x ()

19 l-norm regulriztion Advection-Diffuion Eqution Flt nd Qudrtic Top-ht (prity in wvelet) Window inuoid nd Squred Exponentil (prity in DCT) x(, t) + x(, t) = ɛ x(, t) t x(, ) = x ()

20 l-norm regulriztion Sytem Eqution Model Eqution: x i = M,ix, where M,i = A,iD,i Obervtion Model: y i = Hx i + v, with v N (, R) H = Rn m.5 y i t

21 l-norm regulriztion White Bckground Error x (B = σb I, R = σ ri ), where σ b =. (SNR =.5 db) nd σ r =.8 (SNR = db) ).5 4D VAR x x x ).5 4D VAR b).5 c) R4D VAR 4D VAR.5 c) x x x b) R4D VAR 4D VAR d) R4D VAR d).5 x R4D VAR

22 l-norm regulriztion White Bckground Error x.5 e) (B = σ b I, R = σ ri ), where σ b =. (SNR =.5 db) nd σ r =.8 (SNR = db).5 4D VAR e) 4D VAR.5 f) R4D VAR x x x f) g) R4D VAR x x 4D VAR x g) 4D VAR h) R4D VAR h) R4D VAR.5 x

23 l-norm regulriztion White Bckground Error (B = σ b I, R = σ ri ), where σ b =. (SNR =.5 db) nd σ r =.8 (SNR = db) White Bckground Error MSE r MAE r BIAS r R4D-Vr 4D-Vr R4D-Vr 4D-Vr R4D-Vr 4D-Vr FTH QTH WS SE e 5 Tble : Expected vlue of the MSE r, MAE r, nd BIAS r, for 3 independent run.

24 l-norm regulriztion Correlted Bckground Error 8 6 AR(): B ij = e α i j AR(): B ij = e α i j ( + α i j ) AR() AR() 5 ) b) κ (B) 4 m=4 to 4 κ (B) 5 m=4 to α α

25 l-norm regulriztion Correlted Bckground Error AR().5 ) Bckground.5 d) Bckground x b.5 x b x.5.5 b) 4D VAR x.5 e) 4D VAR c) R4D VAR.5 f) R4D VAR x.5 x

26 l-norm regulriztion Correlted Bckground Error AR() Top pnel: FTH Bottom pnel: WS (B = σ b C b, R = σ r I ), where σ b =. (SNR =.5 db) nd σ r =.8 (SNR = db)..5 ) R4D Vr (R) 4D Vr (C)..5. b) c).5 MSE r. MAE r. BIAS r. C.5 C R.5 C R.5 R α α α MSE r x 3 d) e) f) C.5 C 4 R.4 R.3 MAE r BIAS r R C α α α

27 l-norm regulriztion Thnk You