Carbon Cycle Data Assimilation

with a Variational Approach

(“4-D Var”)

David Baker

CGD/TSS

with

Scott Doney, Dave Schimel,

Britt Stephens, and Roger Dargaville

24 Sept 2004

Outline• The problem: estimate CO2 sources and sinks at fine

space/time scales (2° x 2.5°, hourly/daily)• Method:

– Why use 4-D Var? (Kalman) filtering, smoothing, and variational methods – pros and cons

– Mathematical background of 4-D Var applied to atmospheric trace gases

• Some 4-D Var results using simulated truth• Additional topics to ponder:

– 100 descent iterations 100 ensemble members?– Error estimates: 4-D Var vs. ensemble filters

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Transport: surface fluxes concentrations

fluxesconcentrations

Transport basis functions

Present FutureShift towards newer instruments/platforms:• More continuous analyzers, new cheap in situ analyzers• Aircraft, towers (flux & tall), ships/planes of opportunity• CO2-sondes, tethered balloons, etc.• Satellite-based column-integrated CO2, maybe CO2 profiles

Higher frequency with better spatial coverage -- will permit more detail to be estimated

More sensitive to continental air, detailed flow features -- synoptic meteorology, diurnal cycle must be resolved

Solve for the fluxes at the resolution of the transport model

2° x 2.5°, 25 levels, daily/hourly time step With current inversion techniques, computations grow as

O(N3)… more efficient techniques required(iterative vs. direct inversions, adjoint allows efficientgradient computation, minimal storage)

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)

Variational Data Assimilation vs. Ensemble (Kalman) filter

Pros:• Greater accuracy achieved with 2-sided

smoother than 1-sided filter• Initial transients reducedCons:• Adjoint model required• [Correlations are pre-specified, rather than

calculated, as with a Kalman filter]

4-D Var Data Assimilation Method

Find optimal fluxes u and initial CO2 field xo to minimize

subject to the dynamical constraint

wherex are state variables (CO2 concentrations),v are independent variables used in model but not optimized,z 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

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4-D Var Data Assimilation Method

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/xoo as:

The optimal u and xo may then be found iteratively by

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AdjointTransport

ForwardTransport

ForwardTransport

MeasurementSampling

MeasurementSampling

“True”Fluxes

EstimatedFluxes

ModeledConcentrations

“True”Concentrations

ModeledMeasurements

“True”Measurements

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Errors

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4-D Var Iterative Optimization Procedure

Minimum of cost function J

Truth Prior

Estimate (30 descent steps)

OSSE fluxes, snapshot for Jan 1st

Prior - Truth Estimate - Truth

Future Plans for CO2

• Assimilate remotely-sensed data

• Finer resolution (1º x 1º, or regional)

• Improve predictive capability of carbon cycle models (in two steps) by– Tying fluxes to remotely-sensed patterns– Estimating parameters in ocean and land

biosphere models using remotely-sensed fields directly as data

Atmospheric transport model

NASA/GSFC DAO ‘PCTM’ model:– Lin-Rood advection– Vertical diffusion– Simple cloud convection

• Driven by saved wind & mixing fields from DAO analyses• 6-hourly winds interpolated to 15 minute time step• 2º x 2.5º resolution, 25 vertical levels

Adjoint:• Coded manually; straight-forward because of

– Linearity of CO2 transport– Simplicity of vertical mixing routines

• Runs as fast as forward code