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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

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

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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

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Think Covariance! u1u1 u2u2 v1v1 v2v2 A

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

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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

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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 )

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

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Flavors of SVD ?

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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

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

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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

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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

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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).

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

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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

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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) ?

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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.

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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…

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

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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

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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

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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);

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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

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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

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

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

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Rugby Ball SV 1 SV n Cigar SV 1 SV n

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

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

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The California Current

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30km, 10 km, 3 km & 1km grids, 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

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

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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

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

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

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Spring SV 1 SV n Autumn SV 1 SV n

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CCS: Hessian SVs log 10 Time July 2002 July 2003 July km CCS ROMS

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

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

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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 )

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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

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