A VariationalCarbon Data Assimilation

System (CDAS-4DVar)

David Baker&

David SchimelNCAR / Terrestrial Sciences Section,

Scott DoneyWoods Hole Oceanographic Institution

19 Sept 2006

Main Points

• A 4D-Var assimilation system to compute time-varying surface fluxes, based around the PCTM transport model driven by GEOS-DAS3 fields

• Extensively tested using simulated data -- works!!

• Used to do OSSEs for OCO satellite and other potential observing networks

• Method can solve for other control variables besides fluxes if models available

• Caveat: a true adjoint to the full (non-linear) advection scheme is still needed

Outline

• 4DVar assimilation method– PCTM transport model– Mathematical details– Perfect-model experiments

• to test it works• to assess the impact of data errors• to assess the impact of the prior

– OSSE Experiments• current and future in situ networks• OCO satellite• dense surface coverage

• 4DVar vs. EnKF: pros & cons

Atmospheric Transport Model• Parameterized Chemical Transport Model

(PCTM; Kawa, et al, 2005)• Driven by reanalyzed met fields from NASA/Goddard’s old

GEOS-DAS 3 scheme• Lin-Rood finite volume advection scheme• Vertical mixing: diffusion plus a simple cloud

convection scheme• Exact adjoint for linear advection case• Basic resolution 2°x 2.5°, 25 layers, t »30 min, with

ability to reduce resolution to– 4°x 5°, t »60 min – 6°x 10°, t »120 min <-- we’ll use this in the example– 12°x 15°, t »180 min

• Measurements binned at t resolution• Adjoint coded manually -- runs as quickly as forward

model

4-D Var Data Assimilation Method (Baker, et al, 2006)

Find optimal fluxes u and initial CO2 field xo to minimize

subject to the dynamical constraint

wherex are state variables (CO2 concentrations),h is a vector of measurement operatorsz are the observations,R is the covariance matrix for z,

uo is an a priori estimate of the fluxes,

Puo is the covariance matrix for uo,

xo is an a priori estimate of the initial concentrations,

Pxo is the covariance matrix for xo,

is the transition matrix (representing the transport model), andG distributes the current fluxes into the bottom layer of the model

€

J = (h j (x j ) − z j )TR j

−1(h j (x j ) − z j )j

∑

+ (ui − uio)TP

u io

−1(ui − uio)

i= 0

I−1

∑ + (x0 − x0o)TP

x 0o

−1(x0 − x0o)

€

x i+1 = Φ i+1i x i + G iui ≡ di(x i,ui), i = 0, L ,I −1

4-D Var Data Assimilation Method (Baker, et al, 2006)

Adjoin the dynamical constraints to the cost function using Lagrange multipliers

Setting F/xi = 0 gives an equation for i, the adjoint of xi:

The adjoints to the control variables are given by F/ui and F/xo as:

The gradient vector is given by F/ui and F/xoo, so the optimal u and x0 may

then be found using one’s favorite descent method. I have been using the BFGS method, since it conveniently gives an estimate of the leading terms in the covariance matrix.

€

F ≡ J + λ i+1T (di(x i,ui) − x i+1)

i= 0

I−1

∑

€

i = [Φ i+1i ]T λ i+1 +

∂hi∂x i

T

R j−1(hi(x i) − z j )δ ij i = I −1,K ,1

λ I =∂hI∂x I

T

R j−1(hI (x I ) − z j )δIj ≈ 0

€

∂F∂ui

T

= G iT λ i+1 + P

u io

−1(ui − uio), i = 0,K ,I −1

∂F

∂x0

T

= Φ10T λ1 + P

x 0o

−1(x0 − x0o) + R j

−T (h0 − z j )δ0 j

°

°

°

°

0

2

1

3

x2

x1

x3

x0

AdjointTransport

ForwardTransport

ForwardTransport

MeasurementSampling

MeasurementSampling

“True”Fluxes

EstimatedFluxes

ModeledConcentrations

“True”Concentrations

ModeledMeasurements

“True”Measurements

AssumedMeasurement

Errors

WeightedMeasurement

Residuals

/(Error)2

AdjointFluxes=

FluxUpdate

4-D Var Iterative Optimization Procedure

Minimum of cost function J

Perfect-model Experiments

• 2-hourly measurements in the lowest model level at 6°x 10°, 1 ppm error (1)

• Iterate 30 descent steps, 1-year-long run, starts Jan 1st

• 4 cases– Case 1 -- No measurement noise added, no prior– Case 2 -- Add measurement noise added, no prior– Case 3 -- Add noise, and apply a prior– Case 4 -- No noise, but apply a prior

• Designed to test the method and understand the impact of data errors and the usefulness of the prior

• Case 3 is the most realistic and will be used to do OSSEs for several possible future networks

Estimated - True Flux

No data errorsNo prior constraint

W/ data errorsW/ prior constraint

No data errorsW/ prior constraint

W/ data errorsNo prior constraint

Prior – True Flux |Prior – Truth|

|Est.-Truth|-|Prior-Truth|

· 10-8 [ kg CO2 m-2 s-1 ]

Assimilation resultsusing dense surface

measurements

Iteration

1.0

60

Relative |Estimate-Truth|

Current in situ

Extended in situ

OCO-column

OCO-surface

Dense-column

Dense-surface

No data error, no prior

With data error, no prior

With data error, with prior

No data error, with prior

Experiment #1: Convergence of the flux error

OSSE Experiments

• Use Case 3 from above to test more-realistic measurement networks:

• Current in situ network• Extended version of current network• OCO satellite• 2-hourly 6° x 10° column measurements• 2-hourly 6° x 10° in situ measurements (from the perfect-model experiments above)

• An “OSSE” (observing system simulation experiment) tells you how well your measurements should constrain the fluxes. Only random error part, not biases

The five networks tested in Experiment #2

1. “Current in situ” – the current in situ CO2 network, with ~100 weekly flask sites and ~30 continuous sites.

2. “Extended in situ” – an extended version of the current in situ network, with continuous analyzers placed around the globe in places with wide footprints (on tall towers or aircraft over land, or in the middle of the ocean). Variable precision, 0.5-20 ppmv (1). ~180 measurements/hour.

3. “OCO” – simulated measurements from the polar-orbiting OCO satellite. 16 north-south day-time ground tracks per day. Column-integrated concentrations with a 1 ppmv (1) precision. ~70 measurements/hour.

4. “Dense-column” – every horizontal grid location gets one column-integrated measurement per hour, at 1 ppmv (1) precision.

5. “Dense-surface” – as in 4., but with measurement taken in the bottom model layer.

Dense coverage, 6°x 10°, 2-hourly

OCO Groundtrack,Jan 1st

(Boxes at 6x 10)

Across 1 day

5 days 2 days

Extended Surface Network

· 10-8 [ kg CO2 m-2 s-1 ]

OSSE ResultsFor Five CO2

MeasurementNetworks

For retrospective analyses, a 2-sided smoother gives more accurate estimates than a 1-sided filter.

The 4-D Var method is 2-sided, like a smoother.

(Gelb, 1974)

Why use a variational approach? 4DVar vs. ensemble Kalman filter

(EnKF) • 4DVar requires an adjoint model to back-propagate information -- this can be a royal royal painpain to develop!

• The EnKF can get around needing an adjoint by using a filter-lag rather than a fixed-interval Kalman smoother. However, the need to propagate multiple time steps in the state makes it less efficientless efficient than the 4DVar method

• Both give a low-rank estimate of the a posteriori covariance matrix

• Both can account for dynamic errors• Both calculate time-evolving correlations between the state and the measurements

References

• Baker, D., Doney, S., and Schimel, D., Variational Data Assimilation for Atmospheric CO2, Tellus-B, 2006 (in press)

• Gelb, A., et al., Applied Optimal Estimation, The M.I.T. Press, 1974, 374 pp.

• Kawa, S.R., Erickson, D.J., Pawson, S., and Zhu, Z., Global CO2 transport simulations using meteorological data from the NASA data assimilation system, J. Geophys. Res., 109 (D18): D18312, Sep 29, 2004.