Observations and Ocean State Estimation:

Impact, Sensitivity and Predictability

Andy Moore University of California Santa Cruz

Hernan ArangoRutgers University

Outline

• State estimation• Observation impact• Information content• Observation sensitivity• Forecast error and predictability• Examples from the California Current

Ocean State Estimation and Data Assimilation

From Bayes’ Theorem:

Ocean state vector: T, , , ,T S u v x

a bx x Kd

Posterior Prior Gain Innovation

H bd y x

Obs Obsoperator

1T T K BH HBH RKalman gain:

Priorerrorcov

Obserrorcov

TLobs

operator

may includeocean dynamicsas in 4D-VarH

Practicalities of 4D-Var

a bx x Kdidentified iteratively using 4D-Varax

Typically done using conjugate gradients (CG)

K K PracticalGain Matrix

Alternatively, view CG and 4D-Var as a function:

a bx x dK

4D-Var

Two Views of 4D-Var

a bx x Kd

a bx x dKOR

Posterior Prior Innovation

The Role of Observations

Q: What is the influence of the observations on the analysis?

Observation impact:TK

Observation sensitivity: T dK

Predictability: T d dK K

Observation Impact

Consider a function I ax

T T

T T

T T

I I

I I

I I I I

I H I

b

b

b

a b

b x

a b x

b x

x x Kd

x d K x

x x d K x

y x K x

Q: How does each obs contribute to the analysis?

of the circulation:

Observation Impact

1

1 i N i

N

g

I d d d g

g

i i id y H bx

Ti

ig I

bxK x

Observation Sensitivity

Q: How does the analysis change if the observations change?

Consider again the function I ax

I I a bx x dK

Let y y y d d d

I I I a ax x

TT

TT

=

I I I

I

I

I I

I I

a

a

a a

b

b

a x

x

x x

x d d

x d d d

x d d x

d d x

K

K K

K

K

Observation Sensitivity

Observation Sensitivity

1

1 i N i

N

h

I d d d h

h

i id y

T

ii

h I ax

d xK

Observations and Predictability

Q: How does each obs contribute to the forecast predictability?

Consider now an ensemble of forecastfunction values for obtained by perturbing priors and obs.

I I I f fx x x

I fx

Expected forecast error variance:

T2 2I I I I

f f

f

x xx E x

Forecasterror

covariance

Observations and Predictability

TT

T

aa aE I H B I H

d dR

d d

KKM M

KK

Posteriorerror

covarianceTangentLinear4D-Var

AdjointLinear4D-Var

Tf fffE DMM

Forecasterror

covariance

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

Controlpriors

Tangentlinearmodel

where:

The California Current

500m

37N

Mesoscaleeddies

The California Current

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

Veneziani et al (2009)Broquet et al (2009)Moore et al (2010)

COAMPSforcing

ECCO openboundaryconditions

fb(t), Bf

bb(t), Bb

xb(0), Bx

Previous assimilationcycle

The Regional Ocean Modeling System (ROMS)

Observations (y)

CalCOFI &GLOBEC

SST &SSH

ARGO

Ingleby andHuddleston (2007)

Data from Dan Costa

~90%

~10%TOPP Elephant Seals (APB)

EN3

Observations

DataAssimilation

Posterior

Observations

DataAssimilation

Posterior

Observations

DataAssimilation

Posterior

prior prior prior

Sequential Data Assimilation: July 2002-Dec2004

7 days

Forecast Forecast Forecast

CTD XBT

ARGO TOPP

All in situ data:July 2002 –Dec 2004

Observation Impact

Q: How does each obs contribute to the analysis?

Observation Impacts on Transport

7day average transport

I Transport increment

I

= (Posterior-Prior)

10km ROMS

Example: 37N Transport

No assim

WithData

Assimilation

JAS time meanalongshore

Flow(10km, 42 lev)

CC

CC

CUC

CUC

CC = California CurrentCUC = California Under Current

37NI

500mI

Poleward

Equatorward

Offshore

Onshore

Prior alongshore transport (CC+CUC+CJ)

Prior cross-shore transport

Analysis Cycle – Observation Impacts

37NI

500mI

Poleward

Equatorward

Offshore

Onshore

Alongshoretransport

Cross-shoretransport

10km ROMS

rms

(Moore et al, 2011c)

Sv (10-5)

Alongshore Transport Impacts

IGW

IGW

IGW

CTW(G)

AdjointCTW(GT)

SSH

GyreCirculation

yObservation

vector

x

Degrees of Freedom (dof)

“Perfect World”dof ~ Nobs

yObservation

vector

xRedundancy

dof < Nobs

Information Content of Obs

Degrees of freedom of obs (dof):

dof = Tr KH

30km

Only ~10% of obs contain independent info

Tr{KH}/Nobs vs assimilation cycle

Cardinali et al, 2004;Desroziers et al., 2009Bennett & McIntosh, 1982

upper & lower bounds

(Moore et al, 2011b)

Observation Sensitivity

Q: How does the analysis change if the observations change?

Q: How does the analysis change if the observation array changes?

Impact vs Sensitivity

Single 4D-Var cycle obs

Impact on 37N transport

Sensitivity of 37N transport to removing observations

Observation Sensitivity: An OSSE

4D-Var change in 37N transport when all SSH removed

Change in 37N transport predicted by obs sensitivity

Change in 37N transport predicted by obs impact

Observations

DataAssimilation

Posterior

Observations

DataAssimilation

Posterior

Observations

DataAssimilation

Posterior

prior prior prior

Sequential Data Assimilation: July 2002-Dec2004

7 days

Forecast Forecast Forecast

Observations and Predictability

Q: How does each obs contribute to the forecast predictability?

Forecast Ensembles

Small spread

Predictable

Large spread

Unpredictable

Expected uncertaintyof analysis

Predictability

Analysis cycleending t0+7d

Analysis cycleending t0

t0 t0+7d t0+14dt0-7d

Forecast cycleending t0+14

Forecast cycleending t0+14

Forecastensemble

Predictability

Alongshore transport

Cross-shore transport

2 2 2

14 7 14100 f f fr

0r positive impact of obs on predictability

Summary

• In situ observations have a large impact on circulation estimates, despite small number.• Adjoint operators provide considerable utility for quantifying the impact and value of ocean observations.• Routine monitoring of adjoint-based diagnostics → real-time monitoring of observing array.• Quantification of the true value of observations.