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ROMS 4D-Var: The Complete Story. Andy Moore Ocean Sciences Department University of California Santa Cruz & Hernan Arango IMCS, Rutgers University. Acknowledgements. ONR NSF. Chris Edwards, UCSC Jerome Fiechter, UCSC Gregoire Broquet, UCSC Milena Veneziani, UCSC

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Roms 4d var the complete story

ROMS 4D-Var: The Complete Story

Andy Moore

Ocean Sciences Department

University of California Santa Cruz

&

Hernan Arango

IMCS, Rutgers University


Acknowledgements
Acknowledgements

  • ONR

  • NSF

  • Chris Edwards, UCSC

  • Jerome Fiechter, UCSC

  • Gregoire Broquet, UCSC

  • Milena Veneziani, UCSC

  • Javier Zavala, Rutgers

  • Gordon Zhang, Rutgers

  • Julia Levin, Rutgers

  • John Wilkin, Rutgers

  • Brian Powell, U Hawaii

  • Bruce Cornuelle, Scripps

  • Art Miller, Scripps

  • Emanuele Di Lorenzo, Georgia Tech

  • Anthony Weaver, CERFACS

  • Mike Fisher, ECMWF


Outline
Outline

  • What is data assimilation?

  • Review 4-dimensional variational methods

  • Illustrative examples for California Current



Best Linear Unbiased Estimate (BLUE)

Prior hypothesis: random, unbiased, uncorrelated errors

Error std:

Find: A linear, minimum variance, unbiased estimate

is minimised

so that



Best Linear Unbiased Estimate (BLUE)

Let

OR

Posterior error:


Data Assimilation

fb(t), Bf

ROMS

bb(t), Bb

xb(0), B

Obs, y

xb(t)

x(t)

time

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


Data Assimilation

Find

initial

condition

increment

corrections

for model

error

boundary

condition

increment

forcing

increment

that minimizes the variance given by:

Tangent

Linear

Model

Obs

Error

Cov.

Innovation

Background error covariance


OR

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

At the minimum of J we have :

Obs, y

xb(t)

x(t)

xa(t)

time


Matrix-less Operations

There are no matrix multiplications!

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Adjoint ROMS

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Adjoint ROMS

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Covariance

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Covariance

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Tangent Linear

ROMS

Zonal shear flow


Matrix-less Operations

There are no matrix multiplications!

Tangent Linear

ROMS

Zonal shear flow


Representers

= A representer

Green’s Function

A covariance

Zonal shear flow


Solve linear system of equations!

A Tale of Two Spaces

K = Kalman Gain Matrix


Solve linear system of equations!

A Tale of Two Spaces


A Tale of Two Spaces

Model space searches: Incremental 4D-Var (I4D-Var)

Observation space searches: Physical-space Statistical

Analysis System (4D-PSAS)


An alternative approach in observation space:

The Method of Representers

vector of

representer

coefficients

matrix of

representers

(Bennett, 2002)

: solution of finite-amplitude

linearization of ROMS (RPROMS)

R4D-Var


Representers

= A representer

Green’s Function

A covariance

Zonal shear flow


4D-Var: Two Flavours

Strong constraint:

Model is error free

Weak constraint:

Model has errors

Only practical in observation space


4D-Var Summary

Model space: I4D-Var, strong only (IS4D-Var)

Observation space: 4D-PSAS, R4D-Var

strong or weak



Why 3 4D-Var Systems?

  • I4D-Var: traditional NWP,

  • lots of experience,

  • strong only (will phase out).

  • R4D-Var: formally most correct,

  • mathematically rigorous,

  • problems with high Ro.

  • 4D-PSAS: an excellent compromise,

  • more robust for high Ro,

  • formally suboptimal.



The California Current System (CCS)

10km grid

30km grid

Veneziani et al (2009)

Broquet et al (2009)


The California Current System (CCS)

June mean

SST (2000-2004)

10km grid

30km grid

COAMPS 10km winds; ECCO open boundary conditions

fb(t)

bb(t)

Veneziani et al (2009); Broquet et al (2009)


3km grid

Chris

Edwards


Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO



Solve linear system of equations!

A Tale of Two Spaces


CCS 4D-Var

From previous

cycle

ECCO

COAMPS


Model Space vs Observation Space

(I4D-Var vs 4D-PSAS vs R4D-Var)

Model space (~105):

Observation space (~104):

J

J

Both matrices are

conditioned the same

with respect to inversion

(Courtier, 1997)

Jmin

# iterations

# iterations

(1 outer, 50 inner,

Lh=50 km, Lv=30m)

July 2000: 4 day assimilation window

STRONG CONSTRAINT


SST Incrementsdx(0)

Inner-loop 50

I4D-Var

4D-PSAS

R4D-Var

Model

Space

Observation

Space

Observation

Space


Initial conditions vs surface forcing

vs boundary conditions

J

No assimilation

i.c.

only

i.c. + f

i.c.+ f

+ b.c.

IS4D-Var, 1 outer, 50 inner

4 day window, July 2000


Model Skill

RMS error in temperature

No assim.

Assim.

14d frcst

I4D-Var

(1 outer, 20 inner, 14d cycles

Lh=50 km, Lv=30m)

Broquet et al (2009)


Surface Flux Corrections, (I4D-Var)

Wind stress increments

(Spring, 2000-2004)

Heat flux increments

(Spring, 2000-2004)

Broquet



Model Error h(t)

Model error prior

std in SST


Solve linear system of equations!

A Tale of Two Spaces


Model Space vs Observation Space

(I4D-Var vs 4D-PSAS vs R4D-Var)

Model space (~108):

Observation space (~104):

J

J

Jmin

# iterations

# iterations

(1 outer, 50 inner,

Lh=50 km, Lv=30m)

July 2000: 4 day assimilation window

STRONG vs WEAK CONSTRAINT


4D-Var Post-Processing

  • Observation sensitivity

  • Representer functions

  • Posterior errors


Assimilation impacts on CC

No assim

Time mean

alongshore

flow across 37N,

2000-2004

(30km)

IS4D-Var

(Broquet et al,

2009)


Observation Sensitivity

What is the sensitivity of the transport I to

variations in the observations?

What is ?


Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO


Observation Sensitivity

SSH day 4

SST day 4

Sverdrups per metre

Sverdrups per degree C

Sensitivity of upper-ocean alongshore

transport across 37N, 0-500m, on day 7

to SST & SSH observations on day 4(July 2000)

Applications: predictability,

quality control,

array design


CalCOFI

GLOBEC

depth

Sv/deg C

Sv/psu

Sv/deg C

Sv/psu

Applications: predictability,

quality control,

array design


Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO


The Method of Representers

vector of

representer

coeffiecients

matrix of

representers

: solution of finite-amplitude

linearization of ROMS (RPROMS)


Representers

There are no matrix multiplications!

= A representer

Green’s Function

A covariance



Summary
Summary

  • ROMS 4D-Var system is unique

  • Powerful post-processing tools

  • All parallel

  • 4D-Var rounds out the adjoint sensitivity and generalized stability tool kits in ROMS

  • CCS, CGOA, IAS, EAC, PhilEX

  • Biological assimilation

  • Outstanding issues:

    - multivariate refinements for coastal regions

    - non-isotropic, non-homogeneous cov.

    - multiple grids

    - posterior errors



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