1 4-dimensional variational assimilation of satellite temperature and sea level data in the coastal...

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4-Dimensional Variational Assimilation of Satellite Temperature and Sea Level Data in the

Coastal Ocean and Adjacent Deep Sea

John WilkinJavier Zavala-Garay

Julia Levinand Weifeng Gordon Zhang

Institute of Marine and Coastal Sciences Rutgers, The State University of New Jersey

New Brunswick, NJ, USA

ROMS User MeetingGrenoble, 6-8 October 2008

http://marine.rutgers.edu

jwilkin@rutgers.edu

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ROMS* models of two Western Boundary Current regimes: East Australian Current (EAC) and Middle Atlantic Bight (MAB)

EAC:EAC:• deep ocean adjacent to deep ocean adjacent to

narrow continental shelf narrow continental shelf influenced by proximity of influenced by proximity of boundary currentboundary current

• large (250 km) mesoscale large (250 km) mesoscale eddies generated locally eddies generated locally by boundary current by boundary current separation process separation process

• 2000 x 1600 km model 2000 x 1600 km model domain; 25 km resolution domain; 25 km resolution

MAB:MAB:• wide shallow shelf wide shallow shelf

separated from Gulf separated from Gulf Stream by the Slope SeaStream by the Slope Sea

• Shelf/Slope Front (~0.3 Shelf/Slope Front (~0.3 m/s) at shelf edgem/s) at shelf edge

• Gulf Stream rings Gulf Stream rings frequently enter Slope frequently enter Slope Sea and impact shelfSea and impact shelf

• 800 x 300 km model 800 x 300 km model domain; 5 km resolutiondomain; 5 km resolution

* http://myroms.org

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44

NERACOOS

Altimetry: Jason-n, ERS

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NERACOOS

Altimetry: Jason-n, ERS

MARCOOSCODAR, gliders …

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IS4DVAR*

• Given a first guess (the forward trajectory)…

• and given the available data…and given the available data…

( )oR x

ox

*Incremental Strong Constraint 4-Dimensional Variational data assimilation

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IS4DVAR

• Given a first guess (the forward trajectory)…

• and given the available data…

• what change (or increment) to the initial what change (or increment) to the initial conditions (conditions (ICIC) produces a new forward trajectory ) produces a new forward trajectory that better fits the observations?that better fits the observations?

ox

( )oR x

( )o oR x x

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The “best fit” becomes the analysis

assimilation window

ttii = = analysis analysis initial time initial time

ttff = analysis = analysis final time final time

The strong constraint requires the trajectory satisfies the physics in ROMS. The Adjoint enforces the consistency among state variables.

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The final analysis state becomes the IC for the forecast window

assimilation window forecast

ttff = analysis = analysis final time final time

ttff + + = forecast = forecast horizon horizon

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Forecast verification is with respect to data Forecast verification is with respect to data not yet assimilatednot yet assimilated

assimilation window forecast

verification

ttff + + = forecast = forecast horizon horizon

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Basic IS4DVAR procedure:

Lagrange function

Lagrange multiplier

1

( ) ( )N

T ii i i

i

dL J

dt

xx λ N x F

( )i i t F F

( )i i t x x

( ) ( )i it i t λ λ λ

The “best fit” simulation will minimize L: model-data misfit is small, and model physics are satisfied

1 1

1

1 12 2( )

obsNT T

b b i i i i i ii

bo

JJ

J x

x x B x x H x y O H x y

J = model-data misfit

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Basic IS4DVAR procedure:

Lagrange function

Lagrange multiplier

1

1

0 ( ) 0

0

0 (0) (0) &(0)

0 ( ) 0 . .( )

ii i

i

TTi

i im m mi

b

dLNLROMS

dt

dLADROMS

dt

Lcoupling of NL AD

Li c of ADROMS

xN x F

λ

λ Nλ H O Hx y

x x

B x x λx

λx

1

( ) ( )N

T ii i i

i

dL J

dt

xx λ N x F

( )i i t F F

( )i i t x x

( ) ( )i it i t λ λ λ

At extrema of L

we require:

The “best” simulation minimizes L:

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112

1

( )

obs

b

o

T

b b

NT

i i i i i ii

J

J

J x

x x B x x

H x y O H x y

J = model-data misfit

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Basic IS4DVAR procedure:

(1) Choose an

(2) Integrate NLROMS and save

(a) Choose a

(b) Integrate TLROMS and compute J

(c) Integrate ADROMS to yield

(d) Compute

(e) Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J

(f) Back to (b) until converged

(3) Compute new and back to (2) until converged

(0) (0)bx x

[0, ]t

(0)x

[0, ]t

[ ,0]t (0)(0)oJ

λx

1 (0) (0)(0)

J

B x λ

x

(0)x

(0) (0) (0) x x x

( )tx

Out

er-l

oop

(1

0)

Inne

r-lo

op

(3)

NLROMS = Non-linear forward model; TLROMS = Tangent linear; ADROMS = Adjoint

1

1

1

12

12

( )

b

o

T

b b

J

NT

i i i i i ii

J

J x

x x B x x

H x y O H x y

J = model-data misfit

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Adjoint model integration is forced by

the model-data error

xb = model state (background) at end of previous cycle, and 1st guess for the next forecast

In 4DVAR assimilation the adjoint gives the sensitivity of the initial conditions to mis-match between model and data

A descent algorithm uses this sensitivity to iteratively update the initial conditions, xa, (analysis) to minimize Jb+ Jo

Observations minus previous forecast

x

0 1 2 3 4 time

previous forecast

xb

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Three ways: Three ways:

(1) (1) The Adjoint ModelThe Adjoint Model

(2) (2) Empirical statistical correlations to generate Empirical statistical correlations to generate “synthetic XBT/CTD”“synthetic XBT/CTD”

In EAC assimilation get T(z),S(z) from In EAC assimilation get T(z),S(z) from vertical EOFs of historical CTD vertical EOFs of historical CTD observations regressed on SSH and SSTobservations regressed on SSH and SST

(3) (3) Modeling of the background covariance matrixModeling of the background covariance matrix e.g. via the hydrostatic/geostrophic relation e.g. via the hydrostatic/geostrophic relation

How is observed information (SLA, SST) transferred tounobserved state variables (velocity) andprojected from surface to subsurface?

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Mid-Atlantic Bight ROMS Model for IS4DVAR

5 km resolution for IS4DVAR5 km resolution for IS4DVAR1 km downscale forecast1 km downscale forecast

• 3-hour forecast meteorology 3-hour forecast meteorology NCEP/NAMNCEP/NAM

• daily river flow (USGS)daily river flow (USGS)• boundary tides (TPX0.7)boundary tides (TPX0.7)

• nested in ROMS MAB-GoM nested in ROMS MAB-GoM (which is nested in Global-(which is nested in Global-HyCOM*)HyCOM*)– nudging in a 30 km boundary nudging in a 30 km boundary

zonezone– radiation barotropic mode radiation barotropic mode

(*which assimilates altimetry)(*which assimilates altimetry)

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Mid-Atlantic Bight ROMS Model for IS4DVAR

~20 km outer model:ROMS MAB-GoM, or…NCOM or global HyCOM+NCODA

5 km resolution IS4DVAR model embedded in …

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Mid-Atlantic Bight ROMS Model for IS4DVAR

5 km resolution for IS4DVAR 1 km downscale for forecast

EAC eddies are resolved by AVISO EAC eddies are resolved by AVISO multi-satellite multi-satellite SLASLA

MAB MAB SLASLA is more anisotropic with is more anisotropic with shorter length scales due to flow-shorter length scales due to flow-topography interactiontopography interaction

Use along-track altimetry:Use along-track altimetry:• 4DVar uses the data at time of 4DVar uses the data at time of

satellite passsatellite pass• model “grids” along-track data by model “grids” along-track data by

simultaneously matching simultaneously matching observations and dynamical and observations and dynamical and kinematic constraints kinematic constraints

• need coastal altimetry correctionsneed coastal altimetry corrections

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Mid-Atlantic Bight ROMS Model for IS4DVAR

Model variance (without Model variance (without assimilation) is comparable assimilation) is comparable to along-track in Slope Sea, to along-track in Slope Sea, but not shelf-breakbut not shelf-break

AVISO gridded SLA differs AVISO gridded SLA differs from along-track from along-track SLASLA in in Slope Sea (4 cm) and Gulf Slope Sea (4 cm) and Gulf Stream (10 cm)Stream (10 cm)

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The altimeter anomalies are with respect to the long-term mean The altimeter anomalies are with respect to the long-term mean and therefore contain seasonal variability, but this is small and therefore contain seasonal variability, but this is small compared to the mesoscale in the MAB. compared to the mesoscale in the MAB.

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ROMS assimilates total SSH. Therefore we need to add a mean dynamic topography (MDT) to the anomaly data prior to assimilation. This MDT is computed by 4DVAR analysis using a regional 3-D T-S climatology computed from historical hydrographic data

The altimeter anomalies are with respect to the long-term mean The altimeter anomalies are with respect to the long-term mean and therefore contain seasonal variability, but this is small and therefore contain seasonal variability, but this is small compared to the mesoscale in the MAB. compared to the mesoscale in the MAB.

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Mean Dynamic Topography (MDT) is computed by 4DVAR analysis of a regional 3-D T-S climatology computed from historical hydrographic data.

4DVAR analysis is forced with annual mean meteorology and open boundary conditions.

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High frequency variability: model and data issues

ROMS includes high frequency variability typically ROMS includes high frequency variability typically removed in altimeter processing (tides, storm surge)removed in altimeter processing (tides, storm surge)

The IS4DVAR cost function, The IS4DVAR cost function, JJ, samples this high , samples this high frequency variability, so it must be either (a) removed frequency variability, so it must be either (a) removed from the model or (b) included in the datafrom the model or (b) included in the data

Our approach:Our approach:• Run 1-year ROMS (no assimilation) forced by boundary Run 1-year ROMS (no assimilation) forced by boundary TPX0.7 tides; compute ROMS tidal harmonics TPX0.7 tides; compute ROMS tidal harmonics • de-tide along-track altimetry (developmental in MAB) de-tide along-track altimetry (developmental in MAB) • add ROMS tides to de-tided altimeter dataadd ROMS tides to de-tided altimeter data• thus the thus the observationsobservations are are adjustedadjusted to include model tide to include model tide

• assimilate – high frequency mismatch of model and assimilate – high frequency mismatch of model and altimeter is minimized and cost function is, presumably, altimeter is minimized and cost function is, presumably, dominated by sub-inertial frequency dynamics dominated by sub-inertial frequency dynamics

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High frequency variability: model and data issues

The IS4DVAR increment is to the initial conditions of The IS4DVAR increment is to the initial conditions of the analysis window, and this itself generates HF the analysis window, and this itself generates HF variability (inertial oscillations)variability (inertial oscillations)

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High frequency variability: model and data issues

The IS4DVAR increment is to the initial conditions of The IS4DVAR increment is to the initial conditions of the analysis window, and this itself generates HF the analysis window, and this itself generates HF variability (inertial oscillations)variability (inertial oscillations)

Our approach:Our approach:

• Apply a short time-domain filter to IS4DVAR initial Apply a short time-domain filter to IS4DVAR initial conditions conditions • Reduces inertial oscillations in the Slope Sea Reduces inertial oscillations in the Slope Sea butbut removes tides removes tides • Tides recover quickly Tides recover quickly

– – approach needs refinement approach needs refinement – possibly using 3-D velocity harmonic analysis of– possibly using 3-D velocity harmonic analysis of free running model free running model

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High frequency variability: model and data issues

Without a subsurface Without a subsurface synthetic-CTD synthetic-CTD relationship, the adjoint model can relationship, the adjoint model can erroneously accommodate too much of the erroneously accommodate too much of the SLA model-data misfit in the barotropic modeSLA model-data misfit in the barotropic mode

This sends gravity wave at along the This sends gravity wave at along the model perimeter model perimeter

Our approach:Our approach:

• Repeat (duplicate) the altimeter SLA observations at Repeat (duplicate) the altimeter SLA observations at t = -6 hour, t=0 and t = +6 hourt = -6 hour, t=0 and t = +6 hour but with appropriate time lags in the added tide signal but with appropriate time lags in the added tide signal • These data cannot easily be matched by a wave These data cannot easily be matched by a wave • We are effectively acknowledging the temporal correlation We are effectively acknowledging the temporal correlation of the sub-tidal altimeter SLA data of the sub-tidal altimeter SLA data

gh

gh

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High frequency variability: model and data issues

Our approach:Our approach:

• Repeat (duplicate) the altimeter SLA observations at Repeat (duplicate) the altimeter SLA observations at t = -6 hour, t=0 and t = +6 hourt = -6 hour, t=0 and t = +6 hour but with appropriate time lags in the added tide signal but with appropriate time lags in the added tide signal • These data cannot easily be matched by a wave These data cannot easily be matched by a wave • We are effectively acknowledging the temporal correlation We are effectively acknowledging the temporal correlation of the sub-tidal altimeter SLA data of the sub-tidal altimeter SLA data

gh

gh

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Other possible HF issues:Other possible HF issues:

• Should we include sea level air pressure in ROMS and Should we include sea level air pressure in ROMS and attempt to model the inverse barometer response? attempt to model the inverse barometer response?

• What about remote HF variability from coastal trapped What about remote HF variability from coastal trapped waves? waves? – – regional MOG2D?regional MOG2D?

High frequency variability: model and data issues

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Sequential assimilation of SLA and SST

Before attempting assimilation of COOS observatory in situ data for a full reanalysis, we are assimilating satellite data (SSH and SST) to tune for the assimilation parameters (e.g., horizontal and vertical de-correlation scales, assimilation window, etc.).

We use the unassimilated hydrographic data to evaluate how well the adjoint model is propagating information between variables, and in space and time. This experiment also serves as a prototype for a real-time forecast system.

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Sequential assimilation of SLA and SST

• Reference time is days after 01-01-2006

• 3-day assimilation

window (AW)

• Daily MW+IR blended SST (available real time)

• SSH = Dynamic topography + ROMS tides + Jason-1 SLA (repeated three times)

• For the first AW we just assimilate SST to allow the tides to ramp up.

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Sequential assimilation of SLA and SST

First AW (0<=t<=3 days)

Observed SST ROMS SST and currents at 200 m

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Sequential assimilation of SLA and SST

Second AW (3<=t<=6 days)

Observed SST ROMS SST and currents at 200 m

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Sequential assimilation of SLA and SST

Second AW (3<=t<=6 days)

Observed SST ROMS SST and currents at 200 m

Jason-1 data

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Sequential assimilation of SLA and SST

Second AW (3<=t<=6 days)

Observed SST ROMS SST and currents at 200 m

Jason-1 data

XBT transect(NOT assimilated)

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Sequential assimilation of SLA and SST

ROMS-IS4DVAR fits reasonably well the assimilated observations (SST and SSH), but how well does it represent unassimilated data?

ROMS solutions along the transect positions [lon,lat,time]

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Sequential assimilation of SLA and SST

ROMS-IS4DVAR fits reasonably well the assimilated observations (SST and SSH), but how well does it represent unassimilated data?

ROMS solutions along the transect positions [lon,lat,time]

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Future work

Evaluation of the forward model shows significant forecast skill error due to biases in the boundary and surface forcing.

We will therefore use climatology as an additional data source in the assimilation procedure.

After bias correction, we will assimilate all the available COOS data (CODAR surface currents, glider T-S, CTD, XBT, moorings) for 2006 to produce a MAB reanalysis.

In a true operational forecast system the state of the atmosphere and boundary forcing is not known in advance, so we are developing techniques to produce adjoint-based ensemble techniques that will allow us to place error bars to the forecast.

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SummaryAssimilation of gridded AVISO SLA is not appropriate in MAB

because of length/time scales of variability and anisotropy

Assimilation of along-track SSH successful but requires consideration of …– tidal signal in data-model (lest it dominate cost function)– time filtering IS4DVAR increment to reduce inertial oscillations – time correlation of SLA obs to suppress waves in adjoint solution

Subsurface projection (in addition to Adjoint) is in development using multi-variate background covariance– Less straightforward than in EAC …

– Shelf/Slope Front and shelf mean circulation reach seafloor so geostrophic balance must acknowledge the bathymetry

ROMS User MeetingGrenoble, 6-8 October 2008

jwilkin@rutgers.edu http://marine.rutgers.edu