The case for cycling the representer method with nonlinear models

Transcription

1 The case for cycling the representer ethod with nonlinear odels Hans Ngodock Scott Sith Gregg Jacobs Naval Research Lab Stennis Space Center 1

2 Outline The representer ethod (long story short) The EnKF/S ethod (long story short) A siple coparison with the EnKF/S Nonlinear reduced-gravity ocean odel The case for cycling Lorenz attractor Lorenz-4 Nonlinear reduced-gravity ocean odel NAVY Coastal Ocean Model: preliinary results Open issues Conclusion 2

3 The representer ethod (Bennett 1992) u + T Lu ( ) = F + f, t T f =, ff = C t T d = u + ε, 1 M, ε = εε =R u = u( x, t ) T T L L M M 1-1 f,n n =1 n=1 = J f ( x, t) C ( x, t, x, t ) f ( x, t ) dx dxdt dt + ε R ε f T M M λ L () u ˆ (( u ˆ x, t ) d ε + λ )W, nδ( x x n)( δ t t n) t u = = 1 n= 1 uˆ t + L( uˆ ) = F + C f λ M uxt ˆ(,) u(,) xt β r(,) xt = F + = 1 u t F + Lu = F F n M M λ T n ε + L λ = ( ux ˆ(, t) d)w, nδ( x xn) δ( t tn) t uˆ t n = 1 n= 1 n + Luˆ = F + C λ f n α T + L α = δ( x - x) δ( t- t) t α ( T ) = r + L r = C f α t r() = Ci α ( t = ) 3

4 The EnKF/S ethod (Evensen 1994) ( xt xt xt ) ( )( ) 1 ( ) ( ) a f f T f T f u ( xt, a) = u ( xt, a) + P H HP H +R d-hu ( xt, a) A = u (, ),u (, ),...u (, ) 1 a 2 a N a N f u = A = u i ( xt, a ) N i= 1 f P A-A A-A A = EnKS 1 1 N 1 a a a u ( xt, 1 1),u ( xt, 2 1),...u ( xt, N 1) a a a u ( xt, 1 2),u ( xt, 2 2),...u ( xt, N 2) = a a a u ( xt, ),u (, ),...u (, ) 1 a xt 2 a xt N a T 4

5 1.5 nonlinear reduced gravity odel 9K 16K 35 Sv 35 Sv t x x y 2 2 hu h hu hu x fhv + g h = AM + + τ drag 2 2 x 2 2 hv h hv hv y + fhu + g h = AM + τ drag t y x y h hu hv + + = t x y Δx=K, Δx=18.75K, Δt=in y Active upper 3 35Sv inflow/outflow ~4-onth periodic eddy shedding DATA network 1 3k 1d 2 3k 5d 3 k 1d 4 k 5d 5 1k 1d 6 1k 5d 7 6k 1d 8 6k 5d #data

6 nonlinear reduced gravity odel: experients setup 2-onth Data 4-year odel spin-up Initial enseble spin-up Assiilation Initial condition for the representer = initial enseble ean Assiilation challenge: nonlinearity and purposefully isplaced eddy. The EnKF, EnKS and the representer ethod will be copared based on the accuracy in the assiilation and the ensuing forecast. ( x x ) 2 t t -2 Q( x, t, x, t ) = V( xx, ) exp exp, L = 1 k, τ = 1 days, V(x,x)=.1N 2 2L τ /sec /sec IC TRUTH K K IC ASSIM K K

7 nonlinear reduced gravity odel assiilation accuracy coparison SSH U V s 1 s s 1 s Rep EnKS EnKF Data std s 1 s s 1 s Ngodock, Chen, Jacobs, 6, Ocean Modelling s s Data network 7

8 nonlinear reduced gravity odel forecast accuracy coparison (fro network 3) SSH U V s s Rep. 4 Rep. 3 Rep. 2.2 s 1 s 1 EnKF s 1 s s 1 s s 1 s

9 Cycling the representer algorith Using the representer ethod in a cycling assiilation-forecast setting: One global assiilation T T1 Xu and Daley : 1D linear advection proble with update of the initial error covariance Xu and Daley 2: 2D linear unstable barotropic proble, no initial error covariance update In both cases, the cycling solution was ore accurate (lower RMS isfit to the data/truth) than the non-cycling solution Apply the concept to nonlinear odels: Can we expect the sae RMS behavior? How does cycling interact with potential TLM issues? 9

10 Cycling the representer algorith: benefits Background fro previous cycle s nonlinear forecast: better than the corresponding portion of global background Shorter cycles: stable + accurate TLM Solution ssh loc 5 Background ssh loc 5 Truth Non-cycling Cycling More accurate analysis Cost reduction: 1) saller iniization proble: conjugate gradient non-cycling Nlog(N) N= #data in entire tie interval cycling Nlog(Ncy) Ncy= #data within a cycle 2) One outer loop!?!! Cycling Non-cycling Data May require ore outer loops in early cycles 1

11 Cycling the representer algorith: Lorenz attractor 11

12 Cycling the representer algorith: Lorenz attractor cycles 1 cycles cycles 1 cycles 4 cycles 2 cycles 4 cycles 2 cycles Weak constraint Strong constraint Ngodock, Sith, Jacobs, Month. Wea. Rev. (7) 12

13 Cycling the representer algorith: Lorenz & Eanuel 1998 X t i ( ) = X X X X + 8, i = 1,... 4 i+ 1 i 2 i 1 i Δt =.5 ~ 6hr 13

14 Cycling the representer algorith: 1.5 nonlinear reduced gravity 4-onth RMS Last 2-onth RMS SSH U SSH U s s s s s s s s Network s 1.5 Network Non-cycling.5 Network Cycling 3 outer loops each cycle Cycling 3 outer loops only in first cycle s 1.5 Network 14

15 1.5 nonlinear reduced gravity: truth solution, network 1 15 Non-cycling Cycling K K /sec K.6.2 K /sec K K K.6.2 K K K All data, network 1

16 nonlinear reduced gravity: truth solution after 3 onths K K Network 3 U, V, SSH data /sec K K Network 2 U, V, SSH data /sec K K Network 1 U, V, SSH data K K Network 1 SSH data K K 16

17 1/8 Global Navy Coastal Ocean Model (NCOM G8) NCOM based on POM and SZM configuration anaged by NRL coded for efficient scalability, portability on parallel architecture NCOM G8 vertical coordinate blends terrain following siga + fixed depth z free surface 4 aterial levels: 19 σ over 21 z, σ-z interface at , logarithic stretching Leap-frog + Asselin filter 1 axiu upper layer thickness curvilinear horizontal grid 14 k resolution at 45 N (~1/8 lat) NRL DBDB2 bathyetry Arctic overlap and cell aspect ratio ~1 other options third-order upwind advection, Sagorinsky horizontal ixing MYL2 + Large ixing 17

18 4DVAR prototype: SEED setup Global NCOM 1/8 o resolution Intra- Aerica Sea (IAS) NCOM 1/24 o Mississippi Bight odel Nested Mississippi Bight, SEED Doain ADCP array Global NCOM Intra Aerica Sea 18

19 Misfit of background and solution relative to data Misfit of background and solution relative to data 19

20 Where we are going 4DVAR or 3DVAR with high resolution 4DVAR high resolution with In-situ data 3DVAR assiilation Existing operational data streas Existing operational data streas + other available in situ data Gliders, HF Radar, Moorings

21 Cycling the representer algorith: Open issues Initial error covariance update Forecasted open boundary conditions for nested odels ay be incopatible with corrected BCs 21

22 Conclusion The representer ethod is can be applied to operational NWP and ocean 4DVAR assiilation in place of the gradient descent, eventually at the sae cost, and we do not need the perfect odel assuption The deonstration with the NAVY Coastal Ocean Model is coing shortly Stay tuned WE ARE LOOKING FOR POST-DOCS 22

23 THE END 23

24 Initial tie 4DVAR prototype: SEED Final tie Model Data No assiilation of data at initial tie. Only coparison of data and assiilated solution Assiilation No assiilation 24

25 Hurricane Ivan 25

26 Tangent Linearization of INCOM All nonlinear ters are tangent linearized using the first order approxiation of Taylor s expansion, except ML2.5. Background fields needed for tangent linearization are read in fro the nested IAS NCOM odel. Switches that deterine which open boundary condition are used (Flather, advective, Orlanski, and relaxation) are deterined using background fields. Tangent linearized odel is ore stable when horizontal ixing coefficients are coputed fro linearized Sagorinski versus read in as background fields. 26

27 Experient Details: 5 day experient that treated initial and boundary conditions of velocity and wind stress as weak constraint easureents were assiilated. The CG took 542 iterations to converge, which corresponds to 12.7% of the total nuber of easureents. This experient took 1.5 days on this laptop. 27

28 Mobile Gulfport Pensacola Destin N 29N In situ observations Hurricane IVAN (9-14-4) track 28N 9W 89W 88W 87W 86W 28

29 Exaple coparison of data, background, and solution In general, the aplitude of solutions is closer to data In general, the direction of solutions is closer to data 29

30 3