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Characterization of Forecast Error using Singular Value Decomposition Andy Moore and Kevin Smith University of California Santa Cruz Hernan Arango Rutgers University Slide 2 Outline An overview of singular value decomposition (SVD) Flavors of SVD Duality of SVD Norms Unstable jet California Current Slide 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 Slide 4 Think Covariance! u1u1 u2u2 v1v1 v2v2 A Slide 5 SVD and Model Errors Perfect model: Imperfect model: Errors: TLM: Tangent linear model Error State vector: Slide 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 Slide 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 ) Slide 8 Duality of SVD Fastest growing perturbations Dynamics of meander and eddy formation Fastest growing errors Most predictable patterns Fastest loss of predictability Slide 9 Flavors of SVD ? Slide 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 Slide 11 Flavors of SVD Model error: where Find the 0 that maximizes: Subject to the constraint: AR(1) Slide 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 Slide 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 Slide 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). Slide 15 Baroclinically Unstable Jet 1000km 2000km x=10km, f=-10 -4, =1.610 -11 t=0 t=50 days SST SH Slide 16 Slide 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 Slide 18 Slide 19 Slide 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) ? Slide 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. Slide 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 Slide 23 Evolved Analysis Error Covariance t0t0 tata tftf (E a ) -1 MaMa MfMf Analysis cycle (4D-Var) Forecast cycle Slide 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 Slide 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 Slide 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 Slide 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 Slide 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 Slide 29 Singular Values of 2 Day Jet Forecasts log 10 Cycle # SV # SV 1 SV n SV 1 SV n Slide 30 Singular Values of 2 Day Jet Forecasts log 10 Cycle # SV # SV 1 SV n SV 1 SV n Slide 31 Rugby Ball SV 1 SV n Cigar SV 1 SV n Slide 32 Hessian Singular Vectors SV #1 Initial SSH Final SSH Initial SSH Final SSH Initial SSH Final SSH CYCLE #1CYCLE #20 CYCLE #40 Slide 33 Hessian Singular Vectors SV #1 CYCLE #1 CYCLE #20 Initial SSH Final SSH t=2 days Forecast SSH Slide 34 The California Current Slide 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 Slide 36 Observations (y) CalCOFI & GLOBEC SST & SSH Argo TOPP Elephant Seals Ingleby and Huddleston (2007) Data from Dan Costa Slide 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 Slide 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) Slide 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 Slide 40 Spring SV 1 SV n Autumn SV 1 SV n Slide 41 CCS: Hessian SVs log 10 Time July 2002 July 2003 July 2004 30 km CCS ROMS Slide 42 CYCLE #1 SV SSH initial SV SSH final Forecast SSH 10 km CCS ROMS Slide 43 CYCLE #23 SV SSH initial SV SSH final Forecast SSH 10 km CCS ROMS Slide 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 ) Slide 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