## 1. Introduction

[2] The use of inverse modeling methods as tools for estimating fluxes of atmospheric trace gases has become increasingly common as the need to constrain their global and regional budgets has been recognized [*Intergovernmental Panel on Climate Change* (*IPCC*), 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) based on 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] A vast majority of recent inverse modeling studies have relied on a classical Bayesian approach, where the solution to the inverse problem is defined as the set of flux values that represent an optimal balance between two requirements. The first criterion is that the optimized, or a posteriori, fluxes should be as close as possible to the first-guess (a priori) fluxes. The second is that the measurement values that would result from the inversion-derived (a posteriori) fluxes should agree as closely as possible with the actual measured concentrations. For the case where the advection scheme is linear, 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 an *n* × *m* Jacobian matrix 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 the errors associated with the prior estimate **s**_{p}, and the superscript *T* denotes the matrix transpose operation. 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.

[4] In most Bayesian studies, both **R** and **Q** have been modeled as diagonal matrices. In this case, the diagonal elements of **R** represent the model-data mismatch variance of each observation, which is the sum of the variances associated with all error components such as, for instance, the observation error, the transport modeling error, and the representation error. The diagonal elements of **Q**, on the other hand, represent the error variance of the prior flux estimates and specify the extent to which the real fluxes are expected to deviate from prior flux estimates. It is important to note that although the vast majority of Bayesian inversion studies have used diagonal covariance matrices, some errors are known to be both temporally and spatially correlated.

[5] One of the challenges of the Bayesian approach is the need to estimate the parameters defining the model-data mismatch covariance matrix **R** and the prior error covariance matrix **Q**. These covariances determine the relative weight of prior information versus available data in estimating individual fluxes, and are therefore key components in estimating the posterior covariance (and thereby the uncertainty) of these fluxes. As a result, identifying appropriate covariance parameters is essential to accurate flux estimation. The importance and challenge of accurately estimating these parameters [*Kaminski et al.*, 1999; *Rayner et al.*, 1999; *Law et al.*, 2002; *Peylin et al.*, 2002; *Engelen et al.*, 2002] and the lack of objective methods for doing so [*Rayner et al.*, 1999] have been increasingly recognized in the literature.

[6] Past inversion studies have relied on a variety of methods for estimating these parameters. In the original synthesis inversion work of *Enting et al.* [1995], the data uncertainty was based on a statistical characteristics of the NOAA flask sampling procedures [*Tans et al.*, 1990], but the uncertainty associated with errors in the atmospheric transport model could not be quantified. Many more recent studies have derived model-data mismatch from the residual standard deviation of flask samples around a smooth curve fit [e.g., *Hein et al.*, 1997; *Bousquet et al.*, 1999; *Gurney et al.*, 2002; *Peylin et al.*, 2002]. Others have relied on values independently obtained from the literature [e.g., *Kandlikar*, 1997]. The a priori flux errors have been even more difficult to quantify [*Kaminski et al.*, 1999], and the choice of prior errors has even been described as “mostly arbitrary” in some studies [*Bousquet et al.*, 1999], even though it is recognized that these parameters are crucial to the inversion. Often, researchers have applied what are considered to be “loose” priors [e.g., *Peylin et al.*, 2002; *Law et al.*, 2002] in order to yield conservative estimates of the flux uncertainties. On the basis of assessments of available data, oceanic fluxes have usually been considered more certain than terrestrial fluxes [e.g., *Kaminski et al.*, 1999]. Although considerable effort has been put into estimating covariance parameters, the specification of the prior uncertainties has been described as the “greatest single weakness” in some studies [*Rayner et al.*, 1999].

[7] Recently, several researchers have scaled the model-data mismatch and/or prior error covariance parameters to obtain a data misfit function that follows a χ^{2} distribution with a given number of degrees of freedom [e.g., *Rayner et al.*, 1999; *Gurney et al.*, 2002; *Peylin et al.*, 2002; *Rödenbeck et al.*, 2003]. The correct number of degrees of freedom to be used in such an analysis is equal to the total number of independent pieces of information introduced into the system (equal to the number of observational data plus the number of prior flux estimates in the case of diagonal **Q** and **R** matrices), minus the number of variables estimated in the inversion (typically equal to the number of estimated fluxes). If the covariance parameters are reasonable, the sum of the squared residuals, scaled by their uncertainties and normalized by the number of degrees of freedom, should be close to 1 (reduced chi-squared χ_{r}^{2} = 1). Some researchers have applied this test to the residuals between the available observations and those that would result from the a posteriori fluxes obtained from the inversion, while others have applied it both to these observation residuals and to residuals of the a posteriori fluxes from their prior estimates. Such tuning, however, does not yield a unique solution because more than one combination of covariance parameters can lead to the residuals having an acceptable χ_{r}^{2}. In addition, looking at the χ_{r}^{2} cannot guide the relative allocation of error between the model-data mismatch and prior flux estimates [*Rayner et al.*, 1999]. Therefore examining the variance of the residuals is a necessary, but not a sufficient, condition for evaluating the appropriateness of covariance parameters.

[8] A few recent studies have attempted to systematically quantify one or more components of the model-data mismatch. *Engelen et al.* [2002] divided the model-data mismatch covariance into four additive covariances (which were modeled as diagonal matrices) representing: (1) the observation error, (2) the error in mapping concentrations at a specific location into the measured quantity, (3) the model transport error, and (4) the representation error describing the effects of model resolution. Each of these error sources was then quantified based either on available additional information (e.g., known precision of analytical methods) or numerical experiments (e.g., comparing inversion results obtained using various transport models in order to estimate transport error). *Kaminski et al.* [2001] focused on the errors introduced as a result of imposing potentially erroneous fixed flux patterns within regions, and estimated the additional variance that should be added to the model-data mismatch covariance to account for this effect. *Krakauer et al.* [2004] estimated single scaling factors for both the model-data mismatch and prior error covariances using a generalized cross-validation approach, to show that the Transcom inversion results for the global land/ocean partitioning, particularly in the Southern Hemisphere and tropical regions, were influenced strongly by the Transcom choice of parameters. The fluxes resulting from this analysis and their associated residuals were not analyzed to determine whether these parameters result in fluxes that are consistent with the underlying statistical assumptions, and the uncertainty associated with the estimated covariance parameters was not quantified.

[9] The method that will be developed in this paper allows the available data to shed light on the covariance parameters to be used in the inversion, both in defining the model-data mismatch and the error associated with the prior flux distribution. This can be done in a consistent manner by identifying the Maximum Likelihood (ML) estimates of these parameters, given the prior flux estimates **s**_{p}, the available measurements **z**, and the sensitivity **H** of these measurements to the fluxes to be estimated. The method is applicable whenever estimates of the covariance parameters are to be obtained from the atmospheric data themselves. A ML approach has previously been used in estimating spatial drift and covariance parameters of hydrologic variables, based on limited measurements of these parameters [*Kitanidis and Lane*, 1985]. Also, a related restricted maximum likelihood (RML) approach has been used for estimating covariance parameters in geostatistical inverse modeling [e.g., *Kitanidis*, 1995; *Michalak and Kitanidis*, 2004]. Recently, this technique was applied to the estimation of covariance parameters in a geostatistical implementation of the atmospheric trace-gas inversion problem [*Michalak et al.*, 2004]. In this paper, we develop and demonstrate a maximum likelihood methodology for estimating the model-data mismatch and prior-error covariance matrix parameters needed in the application of the classical Bayesian inverse modeling approach. This method can be directly applied to all studies where the objective function is of the form presented in equation (1), which is the most common setup currently being used in atmospheric inversion studies. This method allows, for the first time, for an objective and data-driven estimation of both the model-data mismatch and prior error covariance parameters required for the solution of the flux estimation inverse problem. Several covariance parameters can be estimated simultaneously, the resulting fluxes yield residuals that follow the assumed distribution (e.g., χ_{r}^{2} = 1), and the uncertainty associated with the covariance parameters can also be estimated. In the sample application presented in this work, both **R** and **Q** are modeled as diagonal matrices, but the method is directly applicable to cases where these matrices include spatial and temporal correlation (i.e., off-diagonal terms). In such cases, parameters such as the correlation length of flux deviations from their prior estimates can also be estimated.