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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

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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


Roms 4d var the complete story

What is data assimilation?


Roms 4d var the complete story

Best Linear Unbiased Estimate (BLUE)

Prior hypothesis: random, unbiased, uncorrelated errors

Error std:

Find: A linear, minimum variance, unbiased estimate

is minimised

so that


Roms 4d var the complete story

Best Linear Unbiased Estimate (BLUE)

OR


Roms 4d var the complete story

Best Linear Unbiased Estimate (BLUE)

Let

OR

Posterior error:


Roms 4d var the complete story

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)


Roms 4d var the complete story

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


Roms 4d var the complete story

OR

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

At the minimum of J we have :

Obs, y

xb(t)

x(t)

xa(t)

time


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Adjoint ROMS

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Adjoint ROMS

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Covariance

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Covariance

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Tangent Linear

ROMS

Zonal shear flow


Roms 4d var the complete story

Matrix-less Operations

There are no matrix multiplications!

Tangent Linear

ROMS

Zonal shear flow


Roms 4d var the complete story

Representers

= A representer

Green’s Function

A covariance

Zonal shear flow


Roms 4d var the complete story

Solve linear system of equations!

A Tale of Two Spaces

K = Kalman Gain Matrix


Roms 4d var the complete story

Solve linear system of equations!

A Tale of Two Spaces


Roms 4d var the complete story

A Tale of Two Spaces

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

Observation space searches: Physical-space Statistical

Analysis System (4D-PSAS)


Roms 4d var the complete story

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


Roms 4d var the complete story

Representers

= A representer

Green’s Function

A covariance

Zonal shear flow


Roms 4d var the complete story

4D-Var: Two Flavours

Strong constraint:

Model is error free

Weak constraint:

Model has errors

Only practical in observation space


Roms 4d var the complete story

4D-Var Summary

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

Observation space: 4D-PSAS, R4D-Var

strong or weak


Roms 4d var the complete story

Former Secretary of Defense

Donald Rumsfeld


Roms 4d var the complete story

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 ccs

The California Current (CCS)


Roms 4d var the complete story

The California Current System (CCS)

10km grid

30km grid

Veneziani et al (2009)

Broquet et al (2009)


Roms 4d var the complete story

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)


Roms 4d var the complete story

3km grid

Chris

Edwards


Roms 4d var the complete story

Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO


Strong constraint 4d var

Strong Constraint 4D-Var


Roms 4d var the complete story

Solve linear system of equations!

A Tale of Two Spaces


Roms 4d var the complete story

CCS 4D-Var

From previous

cycle

ECCO

COAMPS


Roms 4d var the complete story

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


Roms 4d var the complete story

SST Incrementsdx(0)

Inner-loop 50

I4D-Var

4D-PSAS

R4D-Var

Model

Space

Observation

Space

Observation

Space


Roms 4d var the complete story

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


Roms 4d var the complete story

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)


Roms 4d var the complete story

Surface Flux Corrections, (I4D-Var)

Wind stress increments

(Spring, 2000-2004)

Heat flux increments

(Spring, 2000-2004)

Broquet


Weak constraint 4d var

Weak Constraint 4D-Var


Roms 4d var the complete story

Model Error h(t)

Model error prior

std in SST


Roms 4d var the complete story

Solve linear system of equations!

A Tale of Two Spaces


Roms 4d var the complete story

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


Roms 4d var the complete story

4D-Var Post-Processing

  • Observation sensitivity

  • Representer functions

  • Posterior errors


Roms 4d var the complete story

Assimilation impacts on CC

No assim

Time mean

alongshore

flow across 37N,

2000-2004

(30km)

IS4D-Var

(Broquet et al,

2009)


Roms 4d var the complete story

Observation Sensitivity

What is the sensitivity of the transport I to

variations in the observations?

What is ?


Roms 4d var the complete story

Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO


Roms 4d var the complete story

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


Roms 4d var the complete story

CalCOFI

GLOBEC

depth

Sv/deg C

Sv/psu

Sv/deg C

Sv/psu

Applications: predictability,

quality control,

array design


Roms 4d var the complete story

Observations (y)

CalCOFI &

GLOBEC

SST &

SSH

Ingleby and

Huddleston (2007)

TOPP Elephant Seals

ARGO


Roms 4d var the complete story

The Method of Representers

vector of

representer

coeffiecients

matrix of

representers

: solution of finite-amplitude

linearization of ROMS (RPROMS)


Roms 4d var the complete story

Representers

There are no matrix multiplications!

= A representer

Green’s Function

A covariance


Roms 4d var the complete story

Representer Functions

70

80

90


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


Roms 4d var the complete story

ROMS


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