1. Estimating parameters in inversions for regional carbon fluxes Nir Y Krakauer 1*, Tapio Schneider 1, James T Randerson 2 1. California Institute of Technology 2. Earth Systems Science, University of California, Irvine * * niryk@caltech.edu niryk@caltech.edu

2. Motivation & outline Inferring carbon fluxes from patterns in atmospheric CO 2 concentrations is an inverse problem Inferring carbon fluxes from patterns in atmospheric CO 2 concentrations is an inverse problem Parameters in the inversion set-up may not be well constrained by prior information, yet the values chosen significantly affect the inferred flux patterns Parameters in the inversion set-up may not be well constrained by prior information, yet the values chosen significantly affect the inferred flux patterns Here, we explore generalized cross- validation as a method for choosing values for parameters Here, we explore generalized cross- validation as a method for choosing values for parameters

3. The linear inverse problem Ax ≈ b A transport operator that relates concentration patterns to flux magnitudes the (unknown) flux magnitudes Measurements of CO 2 concentrations, with error variance matrix C b x ≈ x 0 A prior guess for the flux distribution, with prior uncertainty variance matrix C x

4. Ambiguities in parameter choice Solving the inverse problem requires specifying Solving the inverse problem requires specifying C b, C x, x 0 Adjustable parameters include: Weight CO 2 measurements equally or differentially? How much weight to give the measurements vs. the prior guesses? Different parameter values lead to varying results for, e.g., the land-ocean and America-Eurasia distribution of the missing carbon sink

5. Generalized cross-validation (GCV) Craven and Wahba (1979): a good value of a regularization parameter in an inverse problem is the one that provides the best invariant predictions of left-out data points Craven and Wahba (1979): a good value of a regularization parameter in an inverse problem is the one that provides the best invariant predictions of left-out data points Choose the parameter values that minimize the “GCV function”: Choose the parameter values that minimize the “GCV function”: GCV = T = effective degrees of freedom

6. The TransCom 3 inversion Estimates mean-annual fluxes from 11 land and 11 ocean regions Estimates mean-annual fluxes from 11 land and 11 ocean regions Data: 1992-1996 mean CO 2 concentrations at 75 stations, and the global mean rate of increase Data: 1992-1996 mean CO 2 concentrations at 75 stations, and the global mean rate of increase Gurney et al 2002

7. Parameters we varied λ: How closely the solution would fit the prior guess x 0 λ: How closely the solution would fit the prior guess x 0 –controls size of the prior-flux variance C x  higher λ: solution will be closer to x 0 (more regularization) –TransCom value: 1 τ: How much preference to give data from low-variance (oceanic) stations τ: How much preference to give data from low-variance (oceanic) stations –controls structure of the data variance C b  0: all stations weighted equally –TransCom value: 1

8. Results: the GCV function

9. Results: inferred CO 2 flux (Pg C/ yr)

10. Results: Ocean

11. Results: equatorial land

12. overall flux distribution TransCom parameter values GCV parameter values

13. Conclusion Parameter choice accounts for part of the variability in CO 2 flux estimates derived from inverse methods Parameter choice accounts for part of the variability in CO 2 flux estimates derived from inverse methods GCV looks promising for empirically choosing parameter values in global-scale CO 2 inversions GCV looks promising for empirically choosing parameter values in global-scale CO 2 inversions GCV-based parameter choice methods should also be of use for smaller-scale (regional and local) studies GCV-based parameter choice methods should also be of use for smaller-scale (regional and local) studies