Assessing the Information Content and
Impact of Observations on Ocean
Circulation Estimates using 4D-Var
Andy Moore
Dept. of Ocean Sciences
UC Santa Cruz
Acknowledgements
• Office of Naval Research
• Hernan Arango
• National Science Foundation
• Chris Edwards
• National Ocean Partnership Program
• Gregoire Broquet
• Brian Powell
• Milena Veneziani
• James Doyle
• Dave Foley
• Anthony Weaver
• Mike Fisher
• Dan Costa
• Patrick Robinson
• Javier Zavala-Garay
Outline
• ROMS 4D-Var
• Information content of observations
• Observation impacts
• Predictability
ROMS 4D-Var
(Moore et al, 2011a)
http://myroms.org
• 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)
• Adjoint of dual 4D-Var (obs impact, obs sensitivity, OSEs)
(Langland and Baker, 2004; Gelaro et al, 2007)
• Physical and ecosystem components
• Parallel (MPI)
ROMS 4D-Var
(Moore et al, 2011a)
http://myroms.org
• 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)
• Adjoint of dual 4D-Var (obs impact, obs sensitivity, OSEs)
(Langland and Baker, 2004; Gelaro et al, 2007)
• Physical and ecosystem components
• Parallel (MPI)
Lanczos Factorization
Cornelius
Lanczos
(1893-1974)
A  Vm Tm Vm
A is the Hessian or stabilized
representer matrix.
m is the number of inner-loops
The California Current
Mesoscale
eddies
Conclusions
Information content:
• Much of the current observational data
is redundant (>90%).
Observation impact:
• In situ observations represent ~10% of
total obs, but often exert considerable impact
on the coastal circulation.
Observation sensitivity & predictability:
• Satellite obs exert the largest control on
predictability of coastal circulation.
ROMS: California Current System (CCS)
4D-Var applied sequentially every 7 days: Jul 2002-Dec 2004.
COAMPS
forcing
fb(t), Bf
ECCO open
boundary
conditions
bb(t), Bb
xb(0), B
Previous
assimilation
cycle
3km, 10 km, 30km, 30 or 42 levels
Veneziani et al (2009)
Broquet et al (2009ab, 2011)
Observations (y)
CalCOFI &
GLOBEC
EN3
TOPP Elephant Seals (APB)
SST &
SSH
Ingleby and
Huddleston (2007)
ARGO
Data from Dan Costa
Observations (y)
CalCOFI &
GLOBEC
~90%
EN3
SST &
SSH
~10%
TOPP Elephant Seals (APB)
Ingleby and
Huddleston (2007)
ARGO
Data from Dan Costa
Sequential 4D-Var (2002-2004)
Observations
prior
4D-Var
Analysis
Posterior
Observations
prior
4D-Var
Analysis
Observations
prior
Posterior
4D-Var
Analysis
Posterior
7 days
Forecast
Forecast
Forecast
Maximum Likelihood Estimate
Circulation estimate from 4D-Var:
xa  xb  K  y  H (xb (t )) 
Posterior
Gain
Prior
Obs
operator
Obs
K  DG (GDG  R)
T
Prior
Cov
T
TLM at
obs points
Prior hypotheses
Obs
Cov
-1
Information Content of Obs
Degrees of freedom of obs (dof):
dof = Tr KG
Cardinali et al, 2004;
Desroziers et al., 2009
Bennett & McIntosh, 1982
Tr{KG}/Nobs vs assimilation cycle (courtesy of Lanczos vectors)
30km
upper & lower bounds
Only ~10% of obs contain independent info
At 10km resolution, dof~1-2% of Nobs
(Moore et al, 2011b)
Observation Impacts
I
7day average
transport
10km ROMS
I  Transport
increment
= (Posterior-Prior)
(Langland & Baker, 2004;
Gelaro et al., 2007)
I
KT
platform

p 1
Nobs
I p   Ii
i 1
Prior alongshore transport (CC+CUC+CJ)
Poleward
I 37 N
Equatorward
Prior cross-shore transport
Offshore
Onshore
I 500m
Analysis Cycle – Transport Increments
Poleward
Alongshore
transport
I 37 N
Equatorward
Cross-shore
transport
Offshore
I 500m
Onshore
Analysis Cycle – Observation Impacts
10km ROMS
Alongshore
transport
Poleward
I 37 N
Equatorward
Cross-shore
transport
I 500m
Offshore
Onshore
(Moore et al, 2011c)
rms
Alongshore Transport Impacts
IGW
IGW
Adjoint
CTW
(GT)
IGW
IGW
IGW
Gyre
Circulation
Adjoint
CTW
(GT)
CTW
(G)
IGW
Sv (10-5)
SSH
SST
CTW
(G)
Alongshore Transport Impacts
Sv (10-3)
Sv
Argo Salinity
CTD Salinity
Predictability
14 day forecast
ensemble
 
f 2
14
 
f
7
4D-Var
7 day forecast
ensemble
t0
t0+7
t0+14
time
Predictability due to assimilating observations during [t0,t0+7]:
    
f 2
14
f
7
2
Obs Sensitivity using (4D-Var)T
2
Example: 37N Transport Predictability
(Moore et al, 2011d)
30km ROMS

r  100 
r 0
      
f 2
14
f
7
2
f 2
14
implies 4D-Var increases predictability
Conclusions
• Much of the current observational data
is redundant (>90%).
• Caveat: Dependent on prior hypotheses.
• In situ observations represent ~10% of
total obs, but often exert considerable impact
on the coastal circulation (transport).
• Caveat: Dependent on prior hypotheses.
• Recommendation: Decimate satellite data.
• Satellite obs exert the largest control on
predictability coastal circulation (transport).
• Caveat: Dependent on prior hypotheses.