ROMS 4D-Var: Past, Present &
Future
Andy Moore
UC Santa Cruz
Overview
• Past: A review of the current system.
• Present: New features coming soon.
• Future: Planned new features and
developments.
The Past….
Acknowledgements
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Hernan Arango – Rutgers University
Art Miller – Scripps
Bruce Cornuelle – Scripps
Emanuelle Di Lorenzo – GA Tech
Brian Powell – University of Hawaii
Javier Zavala-Garay - Rutgers University
Julia Levin - Rutgers University
John Wilkin - Rutgers University
Chris Edwards – UC Santa Cruz
Hajoon Song – MIT
Anthony Weaver – CERFACS
Selime Gürol – CERFACS/ECMWF
Polly Smith – University of Reading
Emilie Neveu – Savoie University
Acknowledgements
“In the beginning…”
Genesis 1.1
•
•
•
•
•
Hernan Arango – Rutgers University
Art Miller – Scripps
Bruce Cornuelle – Scripps
Emanuelle Di Lorenzo – GA Tech
Doug Nielson - Scripps
“In the beginning…”
Genesis 1.1
No grey hair!!!
Regions where ROMS 4D-Var has been used
Data Assimilation
Observations
Model
fb(t), Bf
bb(t), Bb
+
Prior
ROMS
xb(0), Bx
A complete
but
Bayes’picture
Theorem
subject to errors and
uncertainties
Incomplete picture of
Data Assimilation
the real ocean
Posterior
Data Assimilation
Observations
Model
fb(t), Bf
bb(t), Bb
Prior
+
ROMS
xb(0), Bx
The control vector:
 x (0 ) 


z 
f


 b 


Prior error covariance:
Bx

B 



Bf



B b 
Maximum Likelihood Estimate & 4D-Var
Probability
P  z y   exp   J
P z y
The cost function:
J  z  zb  B
T
Prior
Prior
error
cov.
1
z
za
 z  z b    y  H (z )  R
T
Obs
Obs
operator

Maximize P(z|y) by
minimizing J using
variational calculus
1
Obs
error
cov.
 y  H (z ) 
4D-Var Cost Function
J NL
-1
Control
  z k  z bObservation
zb  
 B  z k vector
vector
T
 y  H (z k )  R
T
J
-1
 y  H ( z k ) 
 y1 
 initial conditions


Cost function
minimum identified
using truncated


surface
forcing
Gauss-Newton
method
via inner- and outer-loops:



z  
y 
T


T open
-1
-1
boundary
conditions


  z k B  z k   G k  z k  d k-1  R  G k  z k  d k-1 




for
 zk corrections
 z k-1  z b
d k-1 m
 yodel
 H ( zerror
)
y
k-1

 N
G k  Tangent linear ROMS sampled at obs points
(generalized observation operator)
Solution
Optimal estimate:
z k  z b  K kd k
Gain matrix – primal form:
K k  B  G R G
-1
T
k
-1
k

-1
T
k
G R
Okay for strong constraint, prohibitive for weak constraint.
Gain matrix – dual form:
K k  BG
T
k
G
T
k
BG  R 
T
k
Okay for strong constraint and weak constraint.
-1
-1
Solution
Traditionally, primal form used by solving:
B
-1
 G R G k  xk  G R dk
T
k
-1
T
k
-1
Okay for strong constraint, prohibitive for weak constraint.
The dual form is appropriate for strong and weak
constraint:
G
T
k
BG  R  λ k  dk
T
k
 xk  BG λ k
T
k
The Lanczos Formulation of CG
ROMS offers both primal and dual options
In both J is minimized using Lanczos formulation of CG
General
form:
Tridiagonal
matrix:
Au  b
Primal:
Approx
solution:
-1
u
T
VT V b
V V I
T  V A-1V
A  B  G R G 
V   vi 
v i Lanczos vectors: one per
Dual:
T

T
Orthonormal
matrix:
T
-1
Primal
KA V pG
T BV
G G R
R
Dual
K  B G Vd T V
-1
p
T
TT
p
-1
d
T
T
d

-1
inner-loop
ROMS 4D-Var
•
•
•
•
Incremental (linearized 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)
• 2nd-level preconditioning for multiple outer-loops
• Diffusion operator model for prior covariances
(Derber & Bouttier, 1999; Weaver & Courtier, 2001)
•
•
•
•
Multivariate balance for prior covariance (Weaver et al, 2005)
Physical and ecosystem components
Parallel (MPI)
Moore et al (2011a,b,c, PiO); www.myroms.org
ROMS 4D-Var Diagnostic Tools
• Observation impact (Langland and Baker, 2004)
• Observation sensitivity – adjoint of 4D-Var (OSSE)
(Gelaro et al, 2004)
• Singular value decomposition (Barkmeijer et al, 1998)
• Expected errors (Moore et al., 2012)
Observation Impacts
The impact of individual obs on the analysis or
forecast can be quantified using:
Primal
Dual
K
T
 R G Vp T V
K
T
 Vd T V G B
-1
-1
p
-1
d
T
d
Conveniently computed from 4D-Var output
T
p
Observation Sensitivity
Treat 4D-Var as a function:
xk  xb  K dk 
 K
d k 
T
Quantifies sensitivity of
analysis to changes in obs
Adjoint of 4D-Var
Adjoint of 4D-Var also yields estimates of expected
errors in functions of state.
Impact of the Observations on Alongshore
Transport
Total number of obs
March 2012
Dec 2012
Observation Impact
March 2012
Ann Kristen Sperrevik (NMO)
Dec 2012
Impact of HF radar on 37N transport
Impact of HF radar on 37N transport
Impact of MODIS SST on 37N transport
The Present….
New stuff not in the svn yet
New stuff not in the svn yet
• Augmented B-Lanczos formulation
4D-Var Convergence Issues
Primal preconditioned by B has good convergence
properties:
T
-1
Preconditioned Hessian
I  G R GB


Dual preconditioned by R-1 has poor convergence
-1
T
properties:
Preconditioned stabilized
R GBG  I


representer matrix
Can be partly alleviated using the Minimum Residual
Method (El Akkraoui et al, 2008; El Akkraoui and Gauthier, 2010)
Restricted preconditioned CG ensures that dual
4D-Var converges at same rate as B-preconditioned
Primal 4D-Var (Gratton and Tschimanga, 2009)
Restricted Preconditioned Conjugate Gradient
(Gürol et al, 2013, QJRMS)
Strong Constraint
Weak Constraint
Augmented Restricted B-Lanczos
For multiple outer-loops:
New stuff not in the svn yet
• Augmented B-Lanczos formulation
• Background quality control
Background Quality Control
(Andersson and Järvinen, 1999)
fˆ 
f
PDF of in situ T innovations
y
o
i
 y
b
i

 2 ln [ f / m ax ( f )]
Transformed PDF of in situ T innovations
2

2
b
  1  
  16
2
o

2
b

  16
New stuff not in the svn yet
• Augmented B-Lanczos formulation
• Background quality control
• Biogeochemical modules:
- TL and AD of NEMURO
Hajoon Song
- log-normal 4D-Var
Ocean Tracers: Log-normal or otherwise?
Campbell (1995) – in situ ocean Chlorophyll, northern hemisphere
Assimilation of biological variables
NPZ model
• Differs from physical
variables in statistics.
– Gaussian vs skewed
non-Gaussian
• We use lognormal
transformation
• Maintains positive
definite variables and
reduces rms errors
over Gaussian
approach
Song et al. (2013)
Lognormal 4DVAR (L4DVAR) Example
•
•
PDF of biological variables is often closer to lognormal than Gaussian.
Positive-definite property is preserved in L4DVAR.
Model twin experiment. Initial surface phytoplankton concentration (log scale).
Negative values in black.
Truth
Prior
L4DVAR
Posterior
G4DVAR
Posterior
Biological Assimilation, an example
• 1 year (2000) SeaWiFS ocean color assimilation
• NPZD model
Gray color indicates cloud cover
• Being implemented in near-realtime system
1-Day SeaWiFS
Model –No Assimilation
8-Day SeaWiFS
Model –With Assimilation
Song et al. (in prep)
New stuff not in the svn yet
• Augmented B-Lanczos formulation
• Background quality control
• Biogeochemical modules:
- TL and AD of NEMURO
- log-normal 4D-Var
• Correlations on z-levels
• Improved mixed layer formulation in balance operator
• Time correlations in Q
Recent Bug Fixes
• Normalization coefficients for B
B  K bΣΛLΛ Σ K
T
T
• Open boundary adjustments in 4D-Var
T
b
The Future….
Planned Developments
Planned Developments
• Digital filter – Jc to suppress initialization shock
Thépaut, 2001)
(Gauthier &
Planned Developments
• Digital filter – Jc to suppress initialization shock
Thépaut, 2001)
• Non-diagonal R
(Gauthier &
Planned Developments
• Digital filter – Jc to suppress initialization shock
Thépaut, 2001)
• Non-diagonal R
• Bias-corrected 4D-Var (Dee, 2005)
(Gauthier &
Planned Developments
• Digital filter – Jc to suppress initialization shock
Thépaut, 2001)
• Non-diagonal R
• Bias-corrected 4D-Var (Dee, 2005)
• Time correlations in B
(Gauthier &
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
NECC
SEC
NEC
0m
100m
EUC
Equatorial Pacific
Temperature
NEC=N. Eq. Curr.
SEC=S. Eq. Curr
NECC=N. Eq. Counter
Curr.
EUC=Eq. Under Curr.
200m
0m
Observation
100m
200m
15S
Diffusion eqn with a
diffusion tensor.
EQ
15N
Weaver and Courtier (2001)
(3D-Var & 4D-Var)
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
• Combine 4D-Var and EnKF (hybrid B)
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
• Combine 4D-Var and EnKF (hybrid B)
• TL and AD of parameters
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
• Combine 4D-Var and EnKF (hybrid B)
• TL and AD of parameters
• Nested 4D-Var
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
•
•
•
•
Combine 4D-Var and EnKF (hybrid B)
TL and AD of parameters
Nested 4D-Var
POD for biogeochemistry
Biogeochemical Tracer Equation
P
t
 P
2
 u   P  A H  P  AV
2
z
2
Sources of P
 QP  SP
Sinks of P
Replace Q P  S P with an EOF decomposition
(Following Pelc, 2013)
Planned Developments
• Digital filter – Jc to suppress initialization shock
(Gauthier &
Thépaut, 2001)
•
•
•
•
Non-diagonal R
Bias-corrected 4D-Var (Dee, 2005)
Time correlations in B
Correlations rotated along isopycnals using diffusion tensor
(Weaver & Courtier, 2001)
•
•
•
•
•
Combine 4D-Var and EnKF (hybrid B)
TL and AD of parameters
Nested 4D-Var
POD for biogeochemistry
TL and AD of sea-ice model