ROMS 4-Deimensional Variational (4D-Var) Data Assimilation Algorithms

ROMS 4-Deimensional Variational (4D-Var)Data Assimilation Algorithms

COAWST Modeling System Training WHOI, Woods Hole, MA

August 26, 2014

Hernan G. ArangoIMCS, Rutgers University

Andrew M. MooreUniversity California Santa Cruz

• Andy Moore – UC Santa Cruz• 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

ROMS 4D-Var Team

ROMS

,y R

4D-Var Data Assimilation

bb(t), Bb

fb(t), Bf

xb(0), B

Model solutions depends on xb(0), fb(t), bb(t), h(t)

time

x(t)

Obs, y

xb(t)

xa(t)

that minimizes the variance given by:

Find ( (0), ( ), ( ), ( ))T T T Tt t t b fz x ε ε η

initialconditionincrement

boundaryconditionincrement

surfaceforcing

increment

corrections for model

error

1 11 1

2 2TTJ z D z Gz d R Gz d

diag( , , , ) b fD B B B Q

Background error covariance

TangentLinearModel

ObsErrorCov.

Innovation

bd y Hx

Data Assimilation

K = Kalman Gain Matrix

At the minimum of J we have : J z 0

Model space (control vector) search: (Nmodel x Nt) x (Nmodel x Nt)

( ) ( )T Ta

1 1 1 1b bz z D G R G G R y Hx

K

( ) ( )T Ta

1b bz z DG GDG R y Hx

K

Observation space search: (Nobs x Nobs)

OR

4D-Variational Data Assimilation (4D-Var)

ROMS 4D-Var System• Incremental (linearized about a prior) (Courtier et al., 1994)• Primal (model grid space search) and dual (observation space search) formulations

(Courtier 1997)• Primal: Incremental 4D-Var (I4D-Var)• Dual: Physical-space Statistical Analysis System, PSAS (4D-PSAS) (Da Silva et al,

1995); (4D-PSAS)T

• Dual: Indirect Representer (R4D-Var) (Egbert et al, 1994); (R4D-Var)T

• Strong and weak (dual only) constraint• Preconditioned, Lanczos formulation of conjugate gradient (Lorenc, 2003; Tshimanga

et al., 2008; Fisher, 1997)• Second-level preconditioning for multiple outer-loops• Diffusion operator mode for prior covariances (Derber and Bouttier, 1999; Weaver

and Courtier, 2001)• Multivariate balance operator for prior covariance (Weaver et al., 2001)• Background quality control (Andersson and Järvinen, 1999)• Physical and ecosystem components• Parallel (distributed-memory, MPI)• Publications: Moore et al., 2011a, b, c (Progress in Oceanography)• WikiROMS Tutorials:

https://www.myroms.org/wiki/index.php/4DVar_Tutorial_Introduction

ROMS 4D-Var Data Assimilation Systems

• I4D-Var primal formulationmodel grid space search

traditional NWP lots of experience strong constraint only (phasing out)

• R4D-Var dual formulation observations space search formally most correct

mathematically rigorous problems with high Rossby numbers strong/weak constraint and

• 4D-PSAS dual formulation observation space search an excellent compromise more robust for high Rossby numbers formally suboptimal strong/weak constraint and (4D-Var)T

I4D-Var Algorithm(Moore et al., 2011a)

R4D-Var Algorithm(Moore et al., 2011a)

4D-PSAS Algorithm(Moore et al., 2011a)

SST Increments dx(0): California Current

I4D-Var 4D-PSAS R4D-VarModel Space

Inner-loop 50

Observation Space

Observation Space

ROMS Obsy, R

fb, Bf

bb, Bb

xb, B

h, Q

Posterior

4D-Var

Priors &Hypotheses

ClippedAnalyses

Ensemble(SV, SO)

HypothesisTests

Forecast

dof Adjoint4D-Varimpact

Term balance,eigenmodes

UncertaintyAnalysis

error

Ensemble4D-Var

ROMS 4D-VAR

• Primal preconditioned by B has good convergence properties:

• Dual preconditioned by R-1 has poor convergence properties:

• Can be partly alleviated using the Minimum Residual Method (El Akkraoui et al., 2008; El Akkraoui and Gauthier, 2010)

• Restricted Preconditioned Conjugate Gradient (RPCG) ensures that dual 4D-Var converges at same rate as B-preconditioned Primal 4D-Var (Gratton and Tschimanga, 2009; Gürol et al, 2014)

T -1I G R GB Preconditioned Hessian

-1 T R GBG I Preconditioned stabilizedrepresenter matrix

4D-Var Convergence Issues

Conjugate Gradient Convergence

Congrad: Lanczos-based Conjugate Gradient algorithm (Fisher, 1998)MINRES: Lanczos-based Minimum Residual (El Akkraoui and Gauthier, 2010)RPCG: Lanczos-based Restricted Preconditioned Conjugate Gradient (Gürol et al, 2014)

Jmin

For multiple outer-loops:

Augmented Restricted B-Lanczos

ROMS 4D-Var Diagnostic Tools

• Observation impact (Langland and Baker, 2004)

• Observation sensitivity – adjoint of 4D-Var, (R4D-Var)T, (OSSE)

(Gelaro et al., 2004)

• Singular value decomposition (Barkmeijer et al., 1998)

• Expected errors (Moore et al., 2012)

Observation Sensitivity, 4D-PSAS

ADROMS forced by h (a vector correspondingto the velocity grid points that contribute tothe transport normal to 37N over the upper 500m)

Adjoint of the linearized4D-Var system, (4D-Var)T

WC13

Jan 3-7 Jan 2004, 4D-Var Cycle

• Based on (4D-Var)T

• Only available for 4D-PSAS and R4D-Var

• Quantifies the changes that would result in the circulation estimate, I, as result of changes in the observations or the observation array (Moore et al., 2011c)

• Observing System Experiments (OSEs): It can be used to predict the changes that will occur in the event of a platform failure/degradation or change in the observation array

• Adaptive sampling and observation array design

• Figure show the contribution of the observations from each platform to the total transport increment (red bar)

• The SSH observations increases the alongshore transport by ~0.55 Sv

Observation Impact: 4D-PSAS

Jan 3-7 Jan 2004, 4D-Var Cycle

WC13

• It quantifies the contribution of each observation during a 4D-Var analysis

• It yields the actual contribution of each observation to the circulation increment

• Figure show the contribution to the increment from each part of the control vector: initial conditions (IC), surface forcing (SF), and open boundary conditions (BC)

• Correcting for uncertainties in both IC and SF has the largest impact on the analysis increment

• The observation sensitivity and impact yield the same total transport increment ( I𝜹 37N)

• However, the contribution of each observation platform is different. This is due to nonlinearity and the approximation to the true gain matrix, K

Observations Impact on Alongshore Transport

Total number of obs

Observation Impact

March 2012 Dec 2012

March 2012 Dec 2012Ann Kristen Sperrevik (NMO)

Observations Impact on Alongshore Transport

Impact of HF Radar on 37N Transport

Impact of MODIS SST on 37N Transport

Regions where ROMS 4D-Var has been used

A

B

C

Grid A

• 10km resolution

• 380x400x30

Grid B

• 5km resolution

• 200x250x42

Grid C

• 5km resolution

• 198x156x42

ROMS Grids

• One of our major objectives is to produce the best ocean state estimate using observations and models (variational data assimilation)

Major Straits and Passages

① Mindoro Strait ~420m

② Panay Strait ~570m

③ Sibutu Passage ~320m

④ Dipolog Strait ~504m

⑤ Surigao Strait ~60m

⑥ San Bernadino Strait ~80m

⑦ Tablas Strait ~565m

⑧ Verde Island Passage

~70m

Cruise CTD TowedADCP

MooredADCP

Glider APEXFloater

UnderwaySurface

T,S

TowedCTD

Time

Exploratory2007

Jun 2007

Joint Cruise 2007

Dec 2007

RegionalIOP 2008

Jan 2008

RegionalIOP 2009

Feb – Mar 2009

Observations

• SST satellite data• SSH altimetry• HF Radar currents

P

WX

PP

P

PP P P

P

PPP

X

XXXX W

WWW

X

M

M

Processed for data assimilation

Not suitable for data assimilation because of tides

Not assimilated

Instrument malfunction

Satellite-derived SST Products

RMSE=0.75oC

Sparse and Incomplete Observations

Jun 6–Jul 3, 2007

• CTD• EM-APEX• Gliders

UK Met Office

EN3 dataset

Averaged Sea Surface Temperature

June 26 – July 22, 2007

Arango et al., 2011

Remarks

Averaged Sea Surface Salinity

June 26 – July 22, 2007

Arango et al., 2011

20 60 100 140

0

-100

-200

-300

De

pth

Station Numbers

Salinity Observations

20 60 100 140

rms error = 0.17573

Model minus Observations

20 60 100 140

rms error = 0.090601

Model DA minus Observations0.5

0.25

0

-0.25

-0.5

4DVar Assimilation: SalinityModel Before DA

20 60 100 140

Model After DA

34.9

34.6

34.3

34

33.7

33.420 60 100 140

49%

20 60 100 140

0

-100

-200

-300

De

pth

Station Numbers

Temperature Observations

20 60 100 140

rms error = 2.132

Model minus Observations

20 60 100 140

rms error = 1.3227

Model DA minus Observations4

2

0

-2

-4

4DVar Assimilation: TemperatureModel Before DA

20 60 100 140

Model After DA

30

25

20

15

10

5

20 60 100 140

38%

Observations used in comparison: ship , glider, and APEX

Forecast skill

Remarks

• To our knowledge ROMS is the only ocean community model offering all three 4D-Var systems, (4D-Var)T, and other adjoint-based algorithms

• ROMS 4D-Var Systems: I4D-Var, R4D-Var, 4D-PSAS• Give nearly identical solutions for the same error hypothesis

(Courtier, 1997 dual formulation)• Fully parallel (MPI)• Multivariate Balance Operator: unobserved variables

information is extracted from directly observed data using linear balance relationships (Weaver et al., 2006)

• Efficient Lanczos-based conjugate gradient algorithms• Limited-Memory Preconditioners (LMP): Spectral and Ritz

(Tshimanga et al., 2008)• (4D-Var)T is available for R4D-Var and 4D-PSAS systems used

for observation sensitivity, OSEs, adaptive sampling, and posterior error covariance analysis

• Digital filter – Jc to suppress initialization shock (Gauthier and

Thépaut, 2001)

• Non-diagonal R

• Bias-corrected 4D-Var (Dee, 2005)

• Time correlations in B

• Correlations rotated along isopycnals using diffusion tensor (Weaver and Courtier, 2001)

• Combine 4D-Var and EnKF (hybrid B)

• TL and AD of parameters

• Nested 4D-Var

• Proper Orthogonal Decomposition (POD) for biogeochemistry source and since terms (Pelc, 2013)

• TL and AD of sea-ice model

Planned Developments

PhilEX Summary

• The Philippine Archipelago is very complex and challenging for modeling and predict

• ROMS forecasts without data assimilation are usually saltier at the surface when compared with the observations. The thermocline is somewhat diffused.

• The 4D-Var data assimilation corrects these problems:

• RMSE in temperature is decreased between 35% to 42%

• RMSE in salinity is decreased between 40% to 49%

• Excessive salt flux from prescribed lateral boundary conditions for salinity

• There are large areas in need of sampling in time and space to support and evaluate an ocean prediction system for the Philippine Archipelago

Publications

Arango, H.G., J.C. Levin, E.N. Curchitser, B. Zhang, A.M. Moore, W. Han, A.L. Gordon, C.M. Lee, and J.B. Girton, 2011: Development of a Hindcast/Forecast Model for the Philippine Archipelago, oceanography, 20(1), 58-69, doi:10.5670/oceanog.2011.04. Fiechter, J., G. Broquet, A.M. Moore, and H.G. Arango, 2011: A data assimilative, coupled physical-biological model for the Coastal Gulf of Alaska, Dyn. Atmos. Ocean, 52, 95-118. Moore, A. M., H. G. Arango, and G. Broquet, 2011: Analysis and forecast error estimates derived from the adjoint of 4D-Var, Mon. Weather Rev., accepted. Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, A.T. Weaver, and J. Zavala-Garay, 2011a: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part I: System overview and formulation, Prog. Oceanogr., 91, 34-49, doi:10.1016/j.pocean.2011.05.004. Moore, A.M., H.G. Arango, G. Broquet, C. Edwards, M. Veneziani, B.S. Powell, D. Foley, J. Doyle, D. Costa, and P. Robinson, 2011b: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part II: Performance and Applications to the California Current System, Prog. Oceanogr., 91, 50-73, doi:10.1016/j.pocean.2011.05.003. Moore, A.M., H.G. Arango, G. Broquet, C. Edwards, M. Veneziani, B.S. Powell, D. Foley, J. Doyle, D. Costa, and P. Robinson, 2011c: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part III: Observation impact and observation sensitivity in the California Current System, Prog. Oceanogr., 91, 74-94, doi:10.1016/j.pocean.2011.05.005.  Zavala-Garay, J., J. L. Wilkin, and H. G. Arango, 2011: Predictability of mesoscale variability in the East Australia Current given strong-constraint data assimilation, J. Phys. Oceanog., accepted. Zhang, W.G., J.L. Wilkin, H.G. Arango, 2010: Toward an integrated observation and modeling system in the New York Bight using variational methods. Part I: 4DVAR data assimilation, Ocean Modeling, 35, 119-133.