Forecasting Forecast Error

Andy Moore & Kevin SmithUC Santa Cruz

Hernan ArangoRutgers University

Ensemble methods

Issues with ensemble methods: - how to choose the perturbations? - how large should the ensemble be? - localization and covariance inflation?

0 t

x aE

FHow can we predict forecast error ?

0 t-t

4D-Var

x

rx

x

Analysis Forecast

aE

F

y

pEOFs=left singular vectors of Q

y=right singular vectors of Q

Forecast EOFS and SVD

Equivalent to SVD of: T 1 1 r aM Ey yU

1 2 rπ UM yForecast EOFs:

T1 2 1 2 T r a r aF UM E UM E QQForecast error cov:

(t)= (0) rx M xTangent linear approx: (M=Tangent linear model)

Ehrendorfer and Tribbia (1997)Barkmeijer et al (1998)

The Inverse Analysis Error Covariance, (Ea)-1

1 T 11 a H RE B H

1 T 1 Tm m m

B H R H V T V

InverseAnalysis

errorcovariance Hessian matrix

Hessian matrix Factorization ofHessian based on

4D-Var search directions

m = the number search directions = number of 4D-Var inner-loops

PriorErrorCov.

Adjointof ROMSat obs pts

Tangentof ROMSat obs pts

ObsErrorCov.

A reduced rank approx

0 t-t

4D-Var

x

rx

x

Analysis Forecast

aE

F

y

pEOFs=left singular vectors of Qy=Hessian

Singularvectors

Forecast EOFS and SVD

Equivalent to SVD of: T 1 1 r aM Ey yU

1 2 rπ UM yForecast EOFs:

T1 2 1 2 T r a r aF UM E UM E QQForecast error cov:

(t)= (0) rx M xTangent linear approx: (M=Tangent linear model)

Barkmeijer et al (1998)

SH Baroclinically Unstable Jet

1000km

2000

km

Dx=10km, f=-10-4, b=1.6×10-11

ROMS:10km resolution20 s-levels4000 m deep

0 t-t

4D-Var

x

rx

x

Analysis Forecast

aE

F

Sequential2 day 4D-Var cyclesObs of T, 0-1000mObs spacing ds=1, 3 or 6Sampled once per dayObs error=0.3CStrong constraint15 inner-loops1-3 outer-loops

2-30 day duration

ROMS Identical Twin Set-up

ds=6, 2 day analysis cycle, 6 day forecast

Time

0 t-t

4D-Var

x

rx

x

Analysis Forecast

aE

F

Geometric Interpretation

Ti

i

x xMean squared distancebetween green and red

Referenceforecast

Another likelyforecast

(li,pi)

EOFs

0 t-t

4D-Var

x

rx

x

Analysis Forecast

aE

F

Is F Consistent with the Truth?

T x xCompare

Referenceforecast

True stateof the system

ii

with

(li,pi)

6 dayforecast

(True error)1/2

(Total EOF var)1/2

Initial timeunit norm

hyper-sphere

Final timehyper-ellipse

defined by EOFs

(0)x

t=0 t=6 d

( )tx

Reliability of F

6 day forecast

Fraction of TRUE Error Explained by EOFs

Projection of analysis error on HSVs

Projection of 2 day forecast error on EOFs

Projection of 6 day forecast error on EOFs

Projection of 12 day forecast error on EOFs

Known HSVSubspace 1

UnknownSubspace 2

a b ax x x 1T T

ax BH HBH R d Bh

h

Space NOT spanned by B

Space spanned by B

Analysis error

4D-Var Analysis:rx

Known EOF

Subspace 1

Unknown

Subspace 2

rM

rx

Forecast errorFraction of initial forecasterror explained by HSVs

Fraction of final forecast error explained by EOFs

30 day forecasts, ds=62 outer-loops, 15 inner-loops

% explained by HSVs

% explained by EOFs

(Total EOF var)1/2

(True error)1/2

Good TL approx

Can we predict the forecast error?

log10 l

Eige

nval

ue #

i

j

k

l

m

n

1 iij

j

e

1 kkl

l

e

1 mmn

n

e

1 1

1 inner innerN N

iji jinner

e eN

Eccentricity of the Marginal

Covariance Ellipses

e

e

Mean Eccentricity as a Predictor

6 day forecasts, ds=6k=2, m=15

Mean Eccentricity as a Predictor

30 day forecasts, ds=6k=2, m=15

?

Limit of Gaussian Assumption?

Linear orNon-linear?

Gaussian

Linear

SVD vs Traditional Ensemble Methods

• The cost of SVD ~ cost of a 50 member ensemble• SVD more elegant and less ad hoc than ensemble• SVD intimately linked to 4D-Var analysis• SVD does not require localization due to limited ensemble size.• Extensions needed for SVD to account for errors in: - surface forcing - open boundary conditions - ocean dynamics (weak constraint)