## 1. Introduction

[2] To complement direct measurements of, for example, land or ocean carbon uptake, inverse modeling is widely used to estimate CO_{2} fluxes from the observed spatial and temporal variations of atmospheric CO_{2} concentrations. Different inverse modeling studies, however, have reached apparently contradictory conclusions on the longitudinal and land/ocean partitioning of CO_{2} fluxes. For example, studies using similar data sets have variously placed a sink of order 1 Pg C yr^{−1} in temperate North America [*Fan et al.*, 1998], in north Asia [*Bousquet et al.*, 1999a], or in the north Atlantic and Pacific [*Rayner et al.*, 1999].

[3] Estimating CO_{2} fluxes involves the solution of a linear system

where **b** is a vector of observed variations in CO_{2} concentrations, **ɛ** is a vector of random errors with zero mean and with covariance matrix cov(**ɛ**) = **C**_{b}, **x** is an unknown vector of CO_{2} fluxes, and **A** is a transport operator that relates CO_{2} fluxes to CO_{2} concentrations.

[4] The solution of this inverse problem is generally not well constrained by the CO_{2} concentrations (i.e., the problem is ill-posed) and must be stabilized through the use of prior information or regularity constraints [e.g., *Hansen*, 1998]. Estimates of CO_{2} fluxes are usually constrained to be close to CO_{2} fluxes specified a priori [e.g., *Enting*, 2002] by minimizing an object function of the form

consisting of the sum of the least squares object function (first term) and a penalty term (second term) that penalizes deviations of the solution **x** from a given prior estimate **x**_{0}. The covariance matrix **C**_{x} represents uncertainty about the prior estimate **x**_{0}, and the regularization parameter λ sets the weight of the term involving the prior information relative to the least squares term. (See also the online supplement.)

[5] In CO_{2} inversions, the covariance matrices **C**_{b} and **C**_{x} are usually taken to be diagonal, with diagonal entries **c**_{b} and **c**_{x} equal to assumed variances of the local CO_{2} concentration errors and of the regional prior flux distributions. The prior fluxes **x**_{0} and their assumed variances **c**_{x} are typically chosen *ad hoc* from a range of reasonable values, as are the error variances **c**_{b} of the CO_{2} concentrations. It is known that inversion results can sensitively depend on the choice of such inversion parameters [*Bousquet et al.*, 1999b; *Rayner et al.*, 1999; *Law et al.*, 2003], but a unified approach to quantifying this source of uncertainty and to choosing inversion parameters systematically has not been pursued.

[6] Several methods are available to choose inversion parameters systematically [e.g., *Hansen*, 1998, chapter 7]. One method applied in inverse problems in such fields as meteorological data assimilation [*Wahba et al.*, 1995] and geodesy [*Ditmar et al.*, 2003] is generalized cross-validation (GCV) [*Golub et al.*, 1979]. A form of leave-one-out cross-validation, GCV chooses parameters by minimizing an estimated mean-squared error of predictions with the model specified by the parameters. For the CO_{2} problem, this means that GCV chooses inversion parameters by minimizing an estimated mean-squared error of predictions of CO_{2} concentrations given estimated CO_{2} fluxes and the model of atmospheric transport. Although such predictions are generally not the objective of inverse modeling of CO_{2} fluxes, GCV provides a useful heuristic for choosing inversion parameters.

[7] We applied GCV to estimate two parameters in the TransCom 3 framework for inverse modeling of CO_{2} fluxes [*Gurney et al.*, 2002]. The two inversion parameters considered control the weighting of the prior flux estimates and the relative magnitudes of error variances of CO_{2} concentrations at different locations. We examine how the CO_{2} flux estimates depend on the values assigned to these parameters, choose parameter values by GCV, and discuss how the flux estimates thus obtained differ from those obtained with the original TransCom parameter values.