## 1. Introduction

[2] The use of inverse modeling methods as a tool for estimating surface fluxes of atmospheric trace gases has become increasingly common as the need to constrain their global and regional budgets has been recognized [*Houghton et al.*, 2001; *Committee on the Science of Climate Change, Division on Earth and Life Studies, National Research Council*, 2001; *Wofsy and Harriss*, 2002]. Inverse methods attempt to deconvolute the effects of atmospheric transport and recover source fluxes (typically surface fluxes) on the basis of atmospheric measurements. Information about regions that are not being directly sampled can potentially be inferred from downwind atmospheric measurements. Inverse modeling methods have been used to estimate regional contributions to global budgets of trace gases such as CFCs, CH_{4}, and CO_{2}, and a review of recent applications is presented by *Enting* [2002, chap. 14–17].

[3] The ill-conditioned nature of the inverse problem constitutes a principal difficulty in constraining trace gas emissions. Substantially differing source/sink configurations do not necessarily lead to substantial differences in modeled mixing ratios at observational network sites. Therefore small uncertainties in the observational data correspond to much higher uncertainties in the estimated emission magnitudes [*Enting and Newsam*, 1990; *Brown*, 1993; *Hein et al.*, 1997]. In order to extract a meaningful solution, either the number of unknowns has to be decreased by substantially limiting the number of flux regions that are estimated [e.g., *Brown*, 1993; *Tans et al.*, 1990], or additional information on the sources and sinks has to be introduced into the calculation.

[4] In atmospheric science, this additional information has often been introduced by requiring that the source estimates resulting from the inversion be close to a first guess, or a priori information, on the sources. This can be done in a consistent way by adopting a classical Bayesian approach, in which all parameters are expressed as statistical probability distributions. This paper investigates the applicability of an alternate geostatistical approach, where the prior information is defined solely on the basis of a spatial and/or temporal correlation between the fluxes.

[5] In the classical Bayesian approach, the solution to the inverse problem of flux estimation is defined as the set of parameter values that represent an optimal balance between two requirements. First, the optimized, or a posteriori, fluxes should be as close as possible to the first-guess, or a priori, fluxes. Second, the measurement values that would result from the inversion-derived (a posteriori) fluxes should agree as closely as possible with the actual measured concentrations. Mathematically, this solution corresponds to the minimum of a cost function *L*_{s}, defined as

where **z** is an *n* × 1 vector of observations, **H** is a known *n* × *m* matrix, the Jacobian representing the sensitivity of the observations **z** to the function **s** (i.e., *H*_{i,j} = ∂*z*_{i}/∂*s*_{j}), **s** is an *m* × 1 vector of the discretized unknown surface flux distribution, **R** is the *n* × *n* model-data mismatch covariance, **s**_{p} is the *m* × 1 prior estimate of the flux distribution **s**, **Q** is the covariance of flux deviations from the prior estimate **s**_{p}, and the superscript *T* denotes the matrix transpose operation. Typically, both **R** and **Q** have been modeled as diagonal matrices. A solution in the form of a superposition of all statistical distributions involved can be computed, from which a posteriori means and covariances can be derived [e.g., *Enting et al.*, 1995]. The solution is [*Tarantola*, 1987; *Enting*, 2002]

where is the posterior best estimate of **s** and **V** is its posterior covariance.

[6] As will be presented in more detail in section 2.3, the geostatistical approach entails modifying the Bayesian objective function to

where **X** is a known *m* × *p* matrix, **β** are *p* × 1 unknown drift coefficients, and **Xβ** is the model of the mean of the surface flux distribution. The covariance matrix **Q** is based on a spatial and/or temporal correlation structure for the flux distribution **s** and will therefore have nonzero off-diagonal components. The inverse problem involves solving for both **β** and **s**. In addition, the parameters (e.g., variance and correlation length) of **R** and **Q** can also be estimated using the data themselves.

[7] One of the limitations of the classical Bayesian approach is that it is often difficult to estimate the prior uncertainty and model-data mismatch, making it difficult to estimate the reduction in uncertainty and the absolute a posteriori uncertainties of source magnitudes resulting from the integration of atmospheric data [*Hein et al.*, 1997; *Houweling et al.*, 1999; *Bousquet et al.*, 1999; *Rayner et al.*, 1999]. Also, similar data are sometimes used in defining and updating prior flux estimates [*Hein et al.*, 1997; *Houweling et al.*, 1999; *Bousquet et al.*, 1999], which is not strictly correct given the assumptions of the Bayesian approach. In addition, erroneous prior flux estimates can lead to estimated fluxes that are inconsistent with the atmospheric data and/or do not correspond to actual flux patterns [*Brown*, 1993]. This can be due to narrow uncertainty bounds being assigned to unrealistic prior flux estimates or to incorrect spatial flux patterns being assigned within regions, which can lead to aggregation errors [e.g., *Kaminski et al.*, 2001]. Finally, if all available data are used in defining and/or updating the prior flux estimates, no additional data are available for independently validating the obtained final flux estimates.

[8] A second issue to be considered is the resolution at which fluxes are estimated. The vast majority of studies conducted up to this point have attempted to identify fluxes at continental or ocean basin scales, which can be referred to as a “big regions” perspective. As such, fluxes are aggregated into a few large regions, and emission distributions over predefined regions are assumed to be perfectly well known. The result of such a setup is that the number of unknowns, i.e., the total number of fluxes to be estimated, is greatly reduced relative to the number of surface grid cells used in the transport model. The advantage of such an approach is that it typically renders the overall problem overdetermined, in the sense that the total number of available observations is greater than the number of fluxes to be estimated. Therefore, even if certain regions are less well sampled than others, they can usually be constrained to some extent. The disadvantage is that variations in fluxes at scales smaller than the selected regions cannot be estimated. In addition, aggregation errors can occur when incorrect flux patterns are assigned within regions. If measurements are sensitive to these prescribed flux patterns, the inferred total fluxes for given regions will not be representative of the actual overall fluxes for these regions [*Kaminski et al.*, 2001; *Peylin et al.*, 2002; *Law et al.*, 2002; *Rödenbeck et al.*, 2003].

[9] As a result of these issues, certain researchers have moved toward grid-scale inversions, where the fluxes are estimated at a resolution close to that of the atmospheric transport model used [*Kaminski et al.*, 1999b; *Houweling et al.*, 1999; *Rödenbeck et al.*, 2003]. These studies have used resolutions as fine as 8° latitude by 10° longitude. In such a setup, the problem is strongly underdetermined, with the number of fluxes to be estimated being significantly greater than the number of available observations, and results in infinite variances on the recovered fluxes if no other information is used to constrain the problem. For this reason, grid-scale inversions have all relied on a Bayesian framework to introduce additional prior information into the solution and help constrain the estimates.

[10] Because of the underdetermined nature of the problem, most studies that have coupled prior flux information about grid-scale fluxes with atmospheric measurements have found that the reduction in uncertainty relative to the specified prior flux estimate uncertainty was small and the inversion yielded flux estimates that were similar to the prior flux estimates used to constrain the solution [*Kaminski et al.*, 1999b; *Houweling et al.*, 1999]. These studies have minimized dependence on prior information by modeling the uncertainties in the fluxes as fully uncorrelated between grid cells [*Kaminski et al.*, 1999b; *Houweling et al.*, 1999]. This is opposite to the big regions approach, where either fluxes over a region are assumed to be fully correlated (i.e., uniform), or their variation is assumed to be perfectly known, with a prescribed flux pattern within regions.

[11] It is reasonable to assume, however, that reality lies somewhere in between the two extremes of either perfectly correlated or completely uncorrelated fluxes at the grid scale and that small-scale spatial patterns exist in surface fluxes that the data themselves can help in defining. In fact, *Rödenbeck et al.* [2003] recently made a first attempt at introducing spatial correlations within a traditional Bayesian framework, presenting a method that required the specification of flux patterns and correlations among source strengths in addition to prior flux estimates.

[12] Spatial correlation can offer useful additional information that can be used to reduce the uncertainty of source estimates. That is precisely the goal of the geostatistical approach to inverse modeling, which uses inferred information about spatial and/or temporal correlations in the unknown function (in this case, surface fluxes of atmospheric trace gases) in addition to available measurements to constrain the estimate of the function, without specifying a prior estimate. Because prior flux estimates are not used, the inversion is strongly data-driven and sheds light on whether useful flux information can be derived from the data themselves. The feasibility of implementing such an approach for atmospheric inverse modeling, specifically for the estimation of surface fluxes of atmospheric trace gases, is the subject of this paper.

[13] The objective of this paper is twofold. First, it develops the implementation of a geostatistically based inversion method for estimating surface fluxes of atmospheric trace gases. The presented application is for the recovery of a yearly averaged global CO_{2} surface flux distribution using monthly averaged concentration measurements. This sample application uses pseudodata in order to isolate the behavior of the inversion algorithm from other factors, such as the accuracy of the transport model and the measurement error associated with observations [*Hartley and Prinn*, 1993; *Plumb and Zheng*, 1996; *Mulquiney and Norton*, 1998; *Law et al.*, 2002]. In addition, the use of pseudodata allows for a direct comparison between the “actual” fluxes (which would be unknown in a real-data case) and the fluxes inferred from the limited available measurements. Second, this paper is also intended to describe the presented methodology in enough detail to make it possible for interested parties to implement it and use it for their specific applications. To this end, several additional references, partial derivations, and examples are provided wherever practical.

[14] The remainder of this paper is organized as follows. Section 2 discusses the geostatistical approach to inverse modeling and provides a detailed description of the methodology as applied to atmospheric problems. Section 3 presents the sample pseudodata application involving the estimation of yearly averaged CO_{2} surface fluxes. Section 4 presents a discussion of the results, and section 5 draws conclusions and discusses future avenues for the application of the geostatistical approach to inverse modeling.