Research Vignette:
 
The TransCom3 Time-Dependent
Global CO2 Flux Inversion
É and More
David F. Baker
NCAR
12 July 2007

Outline
The TransCom3 CO2 flux inversion inter-comparison project
The fully time-dependent T3 flux inversion
Method (Òbatch least squaresÓ)
Results
Methods for bigger problems:
Kalman filter (traditional, full rank)
Ensemble filters
Variational data assimilation

TransCom3 CO2 Flux Inversions
CO2 fluxes for 22 regions, data from 78 sites
Annual-mean inversion, 1992-1996
Fixed seasonal cycle, no IAV
22 annual mean fluxes solved for
Gurney, et al Nature, 2002 & GBC, 2003
Seasonal inversion, 1992-1996
Seasonal cycle solved for, no IAV
22*12 monthly fluxes solved for
Gurney, et al, GBC, 2004
Inter-annual inversion, 1988-2003
Both seasonal cycle and IAV solved for
22*12*16 monthly fluxes solved for
Baker, et al, GBC, 2006

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Batch least-squares or
ÒBayesian synthesisÓ inversion
Optimal fluxes, x, found by minimizing:
where
giving

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Computational considerations
Transport model runs to generate H:
22 regions x 16 years x 12 months x 36 months = 152 K tracer-months (if using real winds)
22 x 12 x 36 = 9.5 K tracer-months              (using climatological winds)
Matrix inversion computations: O (N3)
N = 22 regions x 16 years x 12 months = 4.4 K
Matrix storage: O (N*M) --- 66 MB
M = 78 sites x 16 years x 12 months = 15 K

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Kalman Filter Equations
Measurement update step at time k:
 Dynamic propagation step from times k to k+1:
Put multiple months of flux in state vector xk, method becomes effectively a fixed-lag Kalman smoother

Inversion methods for the
data-rich, fine-scale problem
Kalman filter: some benefit, but long lifetimes for CO2 limit savings
Ensemble KF: full covariance matrix replaced by an approximation derived from an ensemble
Variational data assimilation (4-D Var): an iterative solution replaces the direct matrix inversion; the adjoint model computes gradients efficiently

Ensemble Kalman filter
Replace xk, Pk from the full KF with an ensemble of xk,i,  i=1,É,Nens
Add dynamic noise consistent with Qk to xk,i when propagating; add measurement noise consistent with Rk to measurements when updating, initial ensemble has a spread consistent with P0
When needed in KF equations, Pk replaced with
Replace matrix multiplications with sums of dot products
Good for non-Gaussian distributions

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Estimation as minimization
Solve for x with an approximate, iterative method rather than an exact matrix inversion
Start with guess x0, compute gradient                                   efficiently with an adjoint model, search for minimum along -„, compute new „ and repeat
Good for non-linear problems;  use conjugate gradient or BFGS approaches
Low-rank covariance matrix built up as iterations progress
As with Kalman filter, transport errors can be handled as dynamic noise

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Pros and cons, 4DVar vs.
ensemble Kalman filter (EnKF)
4DVar requires an adjoint model to back-propagate information -- this can be a royal pain 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 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

Adjoint transport model
If number of flux regions > number of measurement sites, then instead of running transport model forward in time forced by fluxes to fill H, run adjoint model backwards in time from measurement sites
What is an adjoint model?
If every step in the model can be represented as a matrix multiplication (= Ôtangent linear modelÕ), then the adjoint model is created by multiplying the transpose of the matrices together in reverse order