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
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