Research Vignette: The TransCom3 Time-Dependent Global CO 2 Flux Inversion … and More David F. Baker NCAR 12 July 2007 David F. Baker NCAR 12 July 2007

Outline  The TransCom3 CO 2 flux inversion inter- comparison project  The fully time-dependent T3 flux inversion  Method (“batch least squares”)  Results  Methods for bigger problems:  Kalman filter (traditional, full rank)  Ensemble filters  Variational data assimilation  The TransCom3 CO 2 flux inversion inter- comparison project  The fully time-dependent T3 flux inversion  Method (“batch least squares”)  Results  Methods for bigger problems:  Kalman filter (traditional, full rank)  Ensemble filters  Variational data assimilation

TransCom3 CO 2 Flux Inversions CO 2 fluxes for 22 regions, data from 78 sites  Annual-mean inversion,  Fixed seasonal cycle, no IAV  22 annual mean fluxes solved for  Gurney, et al Nature, 2002 & GBC, 2003  Seasonal inversion,  Seasonal cycle solved for, no IAV  22*12 monthly fluxes solved for  Gurney, et al, GBC, 2004  Inter-annual inversion,  Both seasonal cycle and IAV solved for  22*12*16 monthly fluxes solved for  Baker, et al, GBC, 2006 CO 2 fluxes for 22 regions, data from 78 sites  Annual-mean inversion,  Fixed seasonal cycle, no IAV  22 annual mean fluxes solved for  Gurney, et al Nature, 2002 & GBC, 2003  Seasonal inversion,  Seasonal cycle solved for, no IAV  22*12 monthly fluxes solved for  Gurney, et al, GBC, 2004  Inter-annual inversion,  Both seasonal cycle and IAV solved for  22*12*16 monthly fluxes solved for  Baker, et al, GBC, 2006

TransCom3 interannual inversion setup: Solve for monthly CO 2 fluxes from 22 regions, , using monthly concentrations from 78 sites

Fossil fuel input Atmospheric storage Land + ocean uptake 0.45*FF FF = Atmos + Ocean + Land Bio

NH SummerNH Winter CO 2 Uptake & Release

0 HH Form of the Batch Measurement Equations fluxes concentrations Transport basis functions

Batch least-squares or “Bayesian synthesis” inversion Optimal fluxes, x, found by minimizing: where giving Optimal fluxes, x, found by minimizing: where giving ^

Computation of the interannual variability (IAV): Europe Monthly flux Deseasonalized flux IAV Between-model error Estimation error [PgC/yr]

Total flux IAV (land+ocean) [PgC/yr]

Ocean Flux IAV Land flux IAV [PgC/yr]

Flux IAV [PgC/yr], 11 land regions

Flux IAV [PgC/yr], 11 ocean regions

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 (N 3 )  N = 22 regions x 16 years x 12 months = 4.4 K  Matrix storage: O (N*M) MB  M = 78 sites x 16 years x 12 months = 15 K  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 (N 3 )  N = 22 regions x 16 years x 12 months = 4.4 K  Matrix storage: O (N*M) MB  M = 78 sites x 16 years x 12 months = 15 K

0 HH Kalman Filter/Smoother

0 HH

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 x k, method becomes effectively a fixed-lag Kalman smoother Measurement update step at time k: Dynamic propagation step from times k to k+1: Put multiple months of flux in state vector x k, method becomes effectively a fixed-lag Kalman smoother Error Time  = tangent linear model

Inversion methods for the data-rich, fine-scale problem  Kalman filter: some benefit, but long lifetimes for CO 2 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  Kalman filter: some benefit, but long lifetimes for CO 2 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 x k, P k from the full KF with an ensemble of x k,i, i=1,…,N ens  Add dynamic noise consistent with Q k to x k,i when propagating; add measurement noise consistent with R k to measurements when updating, initial ensemble has a spread consistent with P 0  When needed in KF equations, P k replaced with  Replace matrix multiplications with sums of dot products  Good for non-Gaussian distributions  Replace x k, P k from the full KF with an ensemble of x k,i, i=1,…,N ens  Add dynamic noise consistent with Q k to x k,i when propagating; add measurement noise consistent with R k to measurements when updating, initial ensemble has a spread consistent with P 0  When needed in KF equations, P k replaced with  Replace matrix multiplications with sums of dot products  Good for non-Gaussian distributions

For retrospective analyses, a 2-sided smoother gives more accurate estimates than a 1-sided filter. (Gelb, 1974) Kalman filter vs. Kalman smoother

Estimation as minimization  Solve for x with an approximate, iterative method rather than an exact matrix inversion  Start with guess x 0, 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  Solve for x with an approximate, iterative method rather than an exact matrix inversion  Start with guess x 0, 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

° ° ° ° 00 22 11 33 x2x2 x1x1 x3x3 x0x0 Adjoint Transport Forward Transport Forward Transport Measurement Sampling Measurement Sampling “True” Fluxes Estimated Fluxes Modeled Concentrations “True” Concentrations Modeled Measurements “True” Measurements Assumed Measurement Errors Weighted Measurement Residuals  /(Error) 2 Adjoint Fluxes =  Flux Update 4-D Var Iterative Optimization Procedure Minimum of cost function J

4-D Var Data Assimilation Method Find optimal fluxes u and initial CO 2 field x o to minimize subject to the dynamical constraint where x are state variables (CO 2 concentrations), h is a vector of measurement operators z are the observations, R is the covariance matrix for z, u o is an a priori estimate of the fluxes, P uo is the covariance matrix for u o, x o is an a priori estimate of the initial concentrations, P xo is the covariance matrix for x o,  is the transition matrix (representing the transport model), and G distributes the current fluxes into the bottom layer of the model

4-D Var Data Assimilation Method Adjoin the dynamical constraints to J using Lagrange multipliers Setting  F/  x i = 0 gives an equation for i, the adjoint of x i : The adjoints to the control variables are given by  F/  u i and  F/  x o o as: The gradient vector is given by  F/  u i and  F/  x o o, so the optimal u and x o may then be found using one’s favorite descent method. I use the BFGS method in order to obtain an estimate of the leading terms in the covariance matrix.

· [ kg CO 2 m -2 s -1 ] OSSE Results For Five CO 2 Measurement Networks

Pros and cons, 4DVar vs. ensemble Kalman filter (EnKF) royal pain  4DVar requires an adjoint model to back-propagate information -- this can be a royal pain to develop! less efficient  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 royal pain  4DVar requires an adjoint model to back-propagate information -- this can be a royal pain to develop! less efficient  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 (= ‘tangent linear model’), then the adjoint model is created by multiplying the transpose of the matrices together in reverse order  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 (= ‘tangent linear model’), then the adjoint model is created by multiplying the transpose of the matrices together in reverse order FWD * * * * * * ADJ flux grid measurement sites