## 1. Introduction

[2] Top-down approaches allow inferring the spatiotemporal distribution of carbon dioxide fluxes at the Earth surface by combining diverse sources of information in a statistically optimal way, namely prior estimates of surface fluxes, CO_{2} concentration observations, and atmospheric transport models that link concentrations with surface fluxes [*Tans et al.*, 1990]. Due to the sparsity of the available concentration observations, the spatial extent of fluxes and the dispersive nature of the atmospheric transport, the inversion of carbon fluxes is an ill-posed inverse problem [*Enting*, 2002].

[3] The imbalance between the fluxes and observations can be alleviated either by making more observations or by reducing the effective degrees of freedom of fluxes. New observations have been increasingly collected from extended networks or satellites [*Lauvaux et al.*, 2011; *Chevallier et al.*, 2007], and continuous observations from towers may provide additional gains [*Law et al.*, 2003; *Peylin et al.*, 2005].

[4] Assigning correlations in errors of a priori (or background) fluxes, either implicitly or explicitly, reduces the number of degrees of freedom of the flux variables. For instance, the usual prescription of the flux variations within large regions (so-called ecoregions) [*Fan et al.*, 1998; *Bousquet et al.*, 2000] implements such correlations. However, imposing prior error correlations can generate aggregation errors that, in some cases, can be of the same order as the flux magnitude [*Kaminski et al.*, 2001]. There are too few independent estimates of flux variables to allow reliable modeling of the spatial statistics for flux variations at fine scales.

[5] Mesoscale or regional inversions, which enable simulation-observation comparisons [*Lauvaux et al.*, 2009b] and probably capture local meteorological or orographic scenarios, have been recently developed aiming at regional constraints on anthropogenic and biogenic carbon emissions and the coupling between regional and global scales [*Gerbig et al.*, 2003; *Lauvaux et al.*, 2008].

[6] The number of flux variables increases with finer spatiotemporal scales, which degrades the conditioning of the carbon inverse problem. As mentioned above, the dimension of the flux vector can be reduced through the aggregation of flux variables. However, it is often expected that the aggregation does not cause great loss of information. Sensitivity analyzes have been conducted with several different settings of regular resolutions for either temporal [*Gourdji et al.*, 2010] or spatial aggregations [*Tolk et al.*, 2008]. The aggregation errors, although qualified or even quantified, are not formulated explicitly for most carbon inversions.

[7] *Gerbig et al.* [2003, 2006] were the first to use heterogeneous spatial grids to lessen aggregation errors. Unrealistic correlations make the estimation of the a posteriori uncertainties of the fluxes less accurate.

[8] The adaptive spatial grid from *Gerbig et al.* [2006] is fixed and obtained with a polar projection, which is centered around one tower to adapt to the heterogeneous influence of observations. We will revise this heterogeneous inverse problem using general adaptive spatial grids with more towers that cover the domain. In this paper, the following questions are addressed: Can the adaptive grids be optimized so that the information from observations can be better propagated within the domain? What if the aggregation errors are explicitly formulated for carbon inversions? How about the role of correlations in background errors for inversions using optimal adaptive grids?

[9] Such questions are seldom investigated due to the lack of a multiscale framework for analysis. Based on a recent consistent Bayesian multiscale formalism to optimally design control space (in which control variables are to be estimated) [*Bocquet*, 2009; *Bocquet et al.*, 2011; *Bocquet and Wu*, 2011], we will construct the optimal adaptive representations of the fluxes for inversions using synthetic concentration data. Such representations are taken from a large dictionary of adaptive multiscale grids. The criterion for representation optimization is chosen to be the number of degrees of freedom for the signal (DFS) that measures the information gain from observations to resolve the unknown fluxes. Consequently the information from observations is expected to be better propagated within the domain through these optimal representations. We will then conduct carbon inversions on the optimal representations. Several issues, e.g., the information propagation, the correlations in background errors, the explicit formulation of aggregation errors, will be examined in the optimal multiscale settings. Hopefully, such optimal adaptive representations would be helpful to set up fixed multiscale grids for practical carbon flux inversions.

[10] The paper is organized as follows. In Section 2, we present the methodology for the representation optimization and the inversion in the multiscale setting. Inversions are performed in the context of the Ring 2 experiment in support of the North American Carbon Program Mid Continent Intensive (http://www.ring2.psu.edu). The experimental setup is detailed in Section 3. We report the resulting optimal representation and the corresponding inversion results in Section 4. Finally conclusions are given in Section 5.