Estimating Estimating parameters in parameters in inversions for inversions for

regional carbon regional carbon fluxes fluxes

Nir Y KrakauerNir Y Krakauer1*1*, Tapio , Tapio SchneiderSchneider11, James T , James T

RandersonRanderson22

1. California Institute of Technology1. California Institute of Technology2. Earth Systems Science, University of 2. Earth Systems Science, University of

California, IrvineCalifornia, Irvine

* * niryk@caltech.edu

Motivation & outlineMotivation & outline Inferring carbon fluxes from patterns Inferring carbon fluxes from patterns

in atmospheric COin atmospheric CO22 concentrations is concentrations is an inverse probleman inverse problem

Parameters in the inversion set-up Parameters in the inversion set-up may not be well constrained by prior may not be well constrained by prior information, yet the values chosen information, yet the values chosen significantly affect the inferred flux significantly affect the inferred flux patternspatterns

Here, we explore generalized cross-Here, we explore generalized cross-validation as a method for choosing validation as a method for choosing values for parametersvalues for parameters

The linear inverse problemThe linear inverse problem

Ax ≈ b

A transport operator thatrelates concentration patterns to flux magnitudes

the (unknown) flux magnitudes

Measurements of CO2 concentrations, with error variance matrix Cb

x ≈ x0

A prior guess for the flux distribution, with prioruncertainty variance matrix Cx

Ambiguities in parameter Ambiguities in parameter choicechoice Solving the inverse problem requires Solving the inverse problem requires

specifying specifying Cb, Cx, x0

Adjustable parameters include: Weight CO2 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

Generalized cross-validation Generalized cross-validation (GCV)(GCV)

Craven and Wahba (1979): a good Craven and Wahba (1979): a good value of a regularization parameter in value of a regularization parameter in an inverse problem is the one that an inverse problem is the one that provides the best invariant predictions provides the best invariant predictions of left-out data pointsof left-out data points

Choose the parameter values that Choose the parameter values that minimize the “GCV function”:minimize the “GCV function”:

GCV = T = effective degrees of freedom

The TransCom 3 inversionThe TransCom 3 inversion

Estimates mean-annual fluxes from 11 Estimates mean-annual fluxes from 11 land and 11 ocean regionsland and 11 ocean regions

Data: 1992-1996 mean COData: 1992-1996 mean CO22 concentrations at 75 stations, and the concentrations at 75 stations, and the global mean rate of increaseglobal mean rate of increase

Gurney et al 2002

Parameters we variedParameters we varied λλ: How closely the solution would fit : How closely the solution would fit

the prior guess the prior guess xx00

– controls size of the prior-flux variance controls size of the prior-flux variance CCxx higher higher λλ: solution will be closer to : solution will be closer to xx00 (more (more

regularization)regularization)

– TransCom value: 1TransCom value: 1

ττ: How much preference to give data : How much preference to give data from low-variance (oceanic) stationsfrom low-variance (oceanic) stations– controls structure of the data variance controls structure of the data variance CCbb

0: all stations weighted equally0: all stations weighted equally

– TransCom value: 1 TransCom value: 1

Results: the GCV functionResults: the GCV function

Results: inferred COResults: inferred CO22 flux (Pg C/ flux (Pg C/ yr)yr)

Results: OceanResults: Ocean

Results: equatorial landResults: equatorial land

overall flux distributionoverall flux distributionTransCom parametervalues

GCV parametervalues

ConclusionConclusion Parameter choice accounts for part of Parameter choice accounts for part of

the variability in COthe variability in CO22 flux estimates flux estimates derived from inverse methodsderived from inverse methods

GCV looks promising for empirically GCV looks promising for empirically choosing parameter values in global-choosing parameter values in global-scale COscale CO22 inversions inversions

GCV-based parameter choice methods GCV-based parameter choice methods should also be of use for smaller-scale should also be of use for smaller-scale (regional and local) studies(regional and local) studies