1. Characterization of Forecast Error using Singular Value Decomposition Andy Moore and Kevin Smith University of California Santa Cruz Hernan Arango Rutgers University

2. Outline An overview of singular value decomposition (SVD) Flavors of SVD Duality of SVD Norms Unstable jet California Current

3. Singular Value Decomposition (SVD) Square Matrix: Right singular vectors: Left singular vectors: A generalization of eigenvectors for rectangular matrices. Rectangular Matrix: Important rank/ dimension info

4. Think Covariance! u1u1 u2u2 v1v1 v2v2 A

5. SVD and “Model” Errors Perfect model: Imperfect model: Errors: TLM: Tangent linear model Error State vector:

6. Singular Value Decomposition (SVD) Complimentary Function (CF) Particular integral (PI) SVD of CF – Singular vectors of initial conditions (i.e. ECMWF EFS) SV 1 SV 2 Initial Condition Covariance at t=0 SV 1 SV 2 Final Time Covariance at t=t

7. Singular Value Decomposition (SVD) Complimentary Function (CF) Particular integral (PI) SVD of PI – Stochastic optimals (SO) SO 1 (q 1 ) SO 2 (q 2 ) Model Error Covariance at t~0 Model Error Covariance at t=t SO 1 (q 1 ) SO 2 (q 2 )

8. Duality of SVD Fastest growing perturbations Dynamics of meander and eddy formation Fastest growing errors Most predictable patterns Fastest loss of predictability

9. Flavors of SVD ?

10. Initial condition error: Find the  0 that maximizes: Subject to the constraint: Equivalent to the generalized eigenvalue problem: and SVD of: FInal time norm Initial time norm

11. Flavors of SVD Model error: where Find the  0 that maximizes: Subject to the constraint: AR(1)

12. Flavors of SVD Suppose a(t) is constant in time, then SVD of: In the more general case, we require eigenvectors of: Forcing Singular Vectors Stochastic Optimals

13. Illustrative Example – A Zonal Jet 600km 360km 500m deep, f=10 -4,  =0,  x-15km,  z=100m Eastward Gaussian jet, 40km width, 1.6ms -1 SV time interval = 2 days. Energy norm, P=C. SVD: Initial SV Forcing SV Stoch Opt (white) Stoch Opt (red) t c = 2 days

14. Periodic Channel & Zonal Jet x y x z x y x z Initial Final Conservation of wave action (or pseudomomentum): Doppler shifting of (  ku) is accompanied by increase in E (Buizza and Palmer, 1995).

15. Baroclinically Unstable Jet 1000km 2000km  x=10km, f=-10 -4,  =1.6×10 -11 t=0 t=50 days SST SH

17. Initial Condition Singular Vectors Singular Vector #12 Singular Vector #11 SSH t=0t=2 days t=0t=2 days Energy norm at initial and final time

20. The Forecast Problem SV 1 SV 2 Analysis Error Covariance at t=0 SV 1 SV 2 Forecast Error Covariance at t=t t=0 t=T forecast Forecast initial condition error= analysis error EaEa F Perform SVD on: subject to: (Ehrendorfer & Tribbia, 1998) ?

21. The Inverse Analysis Error Covariance, (E a ) -1 Inverse Analysis error covariance Hessian matrix Primal space Lanczos vector expansion from 4D-Var The number Lanczos vectors = number of 4D-Var inner-loops Prior Error Cov. Adjoint of ROMS Tangent of ROMS Obs Error Cov.

22. The Forecast Error Covariance, F Experience in numerical weather prediction at ECMWF suggests that F=E is a good choice (Buizza and Palmer, 1995). We will assume the same here… … more on this later however…

23. Evolved Analysis Error Covariance t0t0 tata tftf (E a ) -1 MaMa MfMf Analysis cycle (4D-Var) Forecast cycle

24. Evolved Analysis Error Covariance We actually need the analysis error at the end of the analysis cycle: so we need the time evolved Lanczos vectors, V e. but Reorthonormalize using Gramm-Schmidt: where:and

25. Hessian Singular Vectors Find the x that maximizes forecast error Subject to the constraint that (Barkmeijer et al, 1998) Solve the equivalent eigenvalue problem: where(Cholesky factorization of T) and where A + is the right generalized inverse, and The dimension of the problem is reduced to the # of 4D-Var inner- loops whole spectrum. But

26. ROMS 4D-Var Incremental (linearize about a prior) (Courtier et al, 1994) Primal & dual formulations (Courtier 1997) Primal – Incremental 4-Var (I4D-Var) Dual – PSAS (4D-PSAS) & indirect representer (R4D- Var) (Da Silva et al, 1995; Egbert et al, 1994) Strong and weak (dual only) constraint Preconditioned, Lanczos formulation of conjugate gradient (Lorenc, 2003; Tshimanga et al, 2008; Fisher, 1997) Diffusion operator model for prior covariances (Derber & Bouttier, 1999; Weaver & Courtier, 2001) Multivariate balance for prior covariance (Weaver et al, 2005) Physical and ecosystem components Parallel (MPI) Moore et al (2011a,b,c, PiO); www.myroms.org

27. Baroclinically Unstable Jet: Identical Twin 4D-Var Strong constraint primal 4D-Var 1 outer-loop, 15 inner-loops 2 day assimilation window Perfect T obs everywhere on day 0, day 1, day 2 Initial conditions only adjusted Balance operator applied

28. rms error in T rms error in u rms error in v rms error in SSH Cycle # 4D-Var No assim Forecast 4D-Var No assim Forecast 4D-Var No assim Forecast 4D-Var No assim Forecast

29. Singular Values of 2 Day Jet Forecasts log 10 Cycle # SV # SV 1 SV n SV 1 SV n

30. Singular Values of 2 Day Jet Forecasts log 10 Cycle # SV # SV 1 SV n SV 1 SV n

31. Rugby Ball SV 1 SV n Cigar SV 1 SV n

32. Hessian Singular Vectors SV #1 Initial SSH Final SSH Initial SSH Final SSH Initial SSH Final SSH CYCLE #1CYCLE #20 CYCLE #40

33. Hessian Singular Vectors SV #1 CYCLE #1 CYCLE #20 Initial SSH Final SSH t=2 days Forecast SSH

34. The California Current

35. 30km, 10 km, 3 km & 1km grids, 30- 42 levels Veneziani et al (2009) Broquet et al (2009) ERA40 and CCMP forcing SODA open boundary conditions f b (t), B f b b (t), B b x b (0), B x Previous assimilation cycle

36. Observations (y) CalCOFI & GLOBEC SST & SSH Argo TOPP Elephant Seals Ingleby and Huddleston (2007) Data from Dan Costa

37. Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior prior Sequential 4D-Var 8 day 4D-Var cycles overlapping every 4 days

38. 30 Year Reanalysis of California Current 1980-2010 Obs:Pathfinder, AMSR-E, MODIS, EN3, Aviso Forcing: ERA40, ERA-Interim, CCMP (25 km) Analysis every 4 days, 8 day overlapping assim cycles http://www.oceanmodeling.ucsc.edu Initial cost J Final cost J + Final NL J Moore et al. (2012)

39. CCS: Hessian SVs Jan 1999 June 1999 Cycle # Dec 1999 10 km CCS ROMS log 10 SV # Spring SV 1 SV n Autumn SV 1 SV n

40. Spring SV 1 SV n Autumn SV 1 SV n

41. CCS: Hessian SVs log 10 Time July 2002 July 2003 July 2004 30 km CCS ROMS

42. CYCLE #1 SV SSH initial SV SSH final Forecast SSH 10 km CCS ROMS

43. CYCLE #23 SV SSH initial SV SSH final Forecast SSH 10 km CCS ROMS

44. The Forecast Error Covariance Recall that we can express the forecast error cov. as: Posterior error covariance Tangent Linear 4D-Var Adjoint Linear 4D-Var Forecast error covariance Control priors Tangent linear model where: So the control SVD problem becomes: (computational cost equals (# inner-loops) 2 )

45. Summary SVD provides information about forecast error growth. Growing directions of the forecast error covariance error ellipsoid vary with time SV structures become smaller scale Flow and/or error dependent regimes Future work: - explicit forecast error covariance - model error and weak constraint - control singular vectors