Characterization of Forecast Error using Singular Value Decomposition

Andy Moore and Kevin SmithUniversity of California Santa Cruz

Hernan ArangoRutgers University

Outline

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

Singular Value Decomposition (SVD)

Ae eSquare Matrix:

Right singular vectors: Au vLeft singular vectors: T A v u

T 2A Au u T 2AA v vTA UΣV

A generalization of eigenvectors for rectangularmatrices.

-1A EΛERectangular Matrix:

Σ Important rank/dimension info

Think Covariance!

u1

u2 v1

v2

A

p pd dt Mx x

SVD and “Model” Errors

Perfect model:

( )d dt M t x x εImperfect model:

( ) ( ) ( )pt t t x x x

( )d dt t x M x ε

Errors:

TLM:

Tangent linearmodel

Error

State vector: T, , , ,T S u v x

0

( ) (0, ) (0) ( , ) ( )t

t t t d x M x M ε

Singular Value Decomposition (SVD)

ComplimentaryFunction (CF)

Particular integral(PI)

SVD of CF – Singular vectors of initial conditions (i.e. ECMWF EFS)

SV 1

SV 2

Initial ConditionCovariance

at t=0

SV 1

SV 2

Final TimeCovariance

at t=t

Singular Value Decomposition (SVD)

ComplimentaryFunction (CF)

Particular integral(PI)

SVD of PI – Stochastic optimals (SO)

SO 1 (q1)

SO 2(q2)

Model ErrorCovariance

at t~0

Model ErrorCovariance

at t=t

SO 1 (q1)

SO 2(q2)

0

( ) (0, ) (0) ( , ) ( )t

t t t d x M x M ε

Duality of SVD

Fastest growingperturbations

Dynamics ofmeander and

eddy formation

Fastest growingerrors

Most predictablepatterns

Fastest lossof predictability

Flavors of SVD

( )d dt t x M x ε

?

Flavors of SVD

0 (0)d dt x M x εInitial condition error:

Find the 0 that maximizes:T ( ) ( )t t x P x

Subject to the constraint:

T T0 0 (0) (0) 1 ε Cε x C x

Equivalent to the generalized eigenvalue problem:

T (0)= (0) M PM x C xand SVD of: 1 2 1 2P MC

FInal time norm

Initial time norm

Illustrative Example – A Zonal Jet600km

360k

m500m deep, f=10-4, =0, x-15km, z=100mEastward Gaussian jet, 40km width, 1.6ms-1

SV time interval = 2 days. Energy norm, P=C.

TM PMSVD:

T

dt dt M P MSVD:

T dtM PMSVD:

SVD:| '| T 'ct t t

t te dtdt M PM

Initial SV

Forcing SV

Stoch Opt (white)

Stoch Opt (red)

tc= 2 days

Periodic Channel & Zonal Jet

x

y

xz

x

y

xz

Initial Final

Conservation of wave action(or pseudomomentum):

Doppler shifting of (ku) isaccompanied by increase in E(Buizza and Palmer, 1995).

E

ku

Baroclinically Unstable Jet

1000km

2000

km

x=10km, f=-10-4, =1.6×10-11

t=0 t=50 days

SST SST

SH

Initial Condition Singular Vectors

Singular Vector #12 Singular Vector #12

Singular Vector #11Singular Vector #11

SSH SSH

SSH SSH

t=0 t=2 days

t=0 t=2 days

Energy norm at initial and final time

The Forecast Problem

SV 1

SV 2

Analysis ErrorCovariance

at t=0

SV 1

SV 2

Forecast ErrorCovariance

at t=t

t=0 t=T

forecast

Forecast initialcondition error=

analysis error

1T 1

ax E x

Tf fM FM

Ea F

Perform SVD on:

subject to:

(Ehrendorfer & Tribbia, 1998)

?

fM

The Inverse Analysis Error Covariance, (Ea)-1

1 1 T 1 a GE RD G

1 T 1 T D G R G VTV

InverseAnalysis

errorcovariance Hessian matrix

Hessian matrix Primal spaceLanczos vectorexpansion from

4D-Var

The number Lanczos vectors= number of 4D-Var inner-loops

PriorErrorCov.

Adjointof

ROMS

Tangentof

ROMS

ObsErrorCov.

The Forecast Error Covariance, F

Experience in numerical weather prediction atECMWF suggests that F=E is a good choice(Buizza and Palmer, 1995).

We will assume the same here…

… more on this later however…

Evolved Analysis Error Covariance

t0 ta tf

(Ea)-1 (Ea)-1

Ma Mf

Analysiscycle

(4D-Var)

Forecastcycle

Evolved Analysis Error Covariance

1 T T T Ta

a a a a e eM E M M VTV M V TV

We actually need the analysis error at the end of theanalysis cycle:

so we need the time evolved Lanczos vectors, Ve.

T 1 V D V I T 1 e eV D V Ibut

Reorthonormalize using Gramm-Schmidt:T T Te e g gV TV V PTP V

where: g eV V P T g gV V Iand

Hessian Singular VectorsT T f fx M EM xFind the x that maximizes forecast error

Subject to the constraint that 1T 1

ax E x(Barkmeijer et al, 1998)

Solve the equivalent eigenvalue problem:

1 2 -1 -T T T -1 T 1 2 e f f eS L P V M EM V P L S w w

where TT LSL (Cholesky factorization of T)

and1 2 T T ew S L PV x A x and x A w

where A+ is the right generalized inverse, and AA I

The dimension ofthe problem is reduced to the #of 4D-Var inner-loops wholespectrum.

1 1 T T a e eE V P T P VBut

Baroclinically Unstable Jet:Identical Twin 4D-Var

Strong constraint primal 4D-Var1 outer-loop, 15 inner-loops2 day assimilation windowPerfect T obs everywhere onday 0, day 1, day 2Initial conditions only adjustedBalance operator applied

rms error in T

rms error in u rms error in v

rms error in SSH

Cycle #

Cycle #

Cycle #

Cycle #

4D-Var

No assim

Forecast4D-Var

No assim

Forecast

4D-Var

No assim

Forecast4D-Var

No assim

Forecast

Singular Values of 2 Day Jet Forecasts

log10

Cycle #

SV # SV1 SV1

SVnSVn

SV1 SV1

SVnSVn

Rugby Ball

SV1 SV1

SVnSVn

Cigar

SV1 SV1

SVnSVn

Hessian Singular Vectors

SV #1 SV #1

SV #1

Initial SSH Final SSH Initial SSH Final SSH

Initial SSH Final SSH

CYCLE #1 CYCLE #20

CYCLE #40

Hessian Singular Vectors

SV #1

SV #1

CYCLE #1

CYCLE #20

Initial SSH Final SSHt=2 daysForecast SSH

The California Current

30km, 10 km, 3 km & 1km grids, 30- 42 levels

Veneziani et al (2009)Broquet et al (2009)

ERA40 and CCMP forcing

SODA openboundaryconditions

fb(t), Bf

bb(t), Bb

xb(0), Bx

Previous assimilationcycle

Observations (y)

CalCOFI &GLOBEC

SST &SSH

Argo

TOPP Elephant Seals

Ingleby andHuddleston (2007)

Data from Dan Costa

Observations

4D-VarAnalysis

Posterior

Observations

4D-VarAnalysis

Posterior

Observations

4D-VarAnalysis

Posterior

prior prior prior

Sequential 4D-Var

8 day 4D-Var cyclesoverlapping every 4 days

30 Year Reanalysis of California Current1980-2010

Obs:Pathfinder, AMSR-E, MODIS, EN3, AvisoForcing: ERA40, ERA-Interim, CCMP (25 km)Analysis every 4 days, 8 day overlapping assim cycleshttp://www.oceanmodeling.ucsc.edu

Initial cost J

Final cost J

+ Final NL J

Moore et al. (2012)

CCS: Hessian SVs

Jan1999

June1999

Cycle #

Dec1999

10 kmCCS ROMS

log10 SV #

Spring

SV1 SV1

SVnSVn

Autumn

SV1 SV1

SVnSVn

Spring

SV1 SV1

SVnSVn

Autumn

SV1 SV1

SVnSVn

CYCLE #1

SV SSH initial

SV SSH final

Forecast SSH

10 kmCCS ROMS

CYCLE #23

SV SSH initial

SV SSH final

Forecast SSH

10 kmCCS ROMS

The Forecast Error Covariance

TT

T

aa aE I G B I G

d dR

d d

KKM M

KK

Recall that we can express the forecast error cov. as:

Posteriorerror

covarianceTangentLinear4D-Var

AdjointLinear4D-Var

Tf fffE DMM

Forecasterror

covariance

diag , , ,f b f aD E B B B

Controlpriors

Tangentlinearmodel

where:

1 2 1 T T T 1 2 ff fS L V D VL S w wM M

So the control SVD problem becomes:

(computational cost equals (# inner-loops)2)

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