Carbon Cycle Data Assimilation
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Transcript of Carbon Cycle Data Assimilation
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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
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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
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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)
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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)
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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]
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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
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Errors
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4-D Var Iterative Optimization Procedure
Minimum of cost function J
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Truth Prior
Estimate (30 descent steps)
OSSE fluxes, snapshot for Jan 1st
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Prior - Truth Estimate - Truth
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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
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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