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ROMS 4D-Var: The Complete Story

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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

- 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

- What is data assimilation?
- Review 4-dimensional variational methods
- Illustrative examples for California Current

What is data assimilation?

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)

OR

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

Former Secretary of Defense

Donald Rumsfeld

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

Representer Functions

70

80

90

- 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

ROMS