__A. Chatterjee__ and __A. M. Michalak__

### SUMMARY

*Data assimilation (DA) approaches, which use a combination of numerical approximations and time-stepping schemes, are gaining in popularity for estimating CO _{2} fluxes. This study compares two state-of-the-art DA approaches (a variational and an ensemble filter approach), and finds that the variational approach is slightly more robust for obtaining CO_{2} flux estimates while the ensemble filter approach provides more reliable estimates of analysis error. The performance of the two DA approaches is highly sensitive to the complex interplay between the underlying numerical approximations and the observational characteristics (i.e., observational density, heterogeneity, and uncertainty), which has potential implications with the emerging focus on assimilation of denser but more inaccurate satellite CO_{2} observations.*

### ABSTRACT

Data assimilation (DA) approaches, including variational and the ensemble Kalman filter methods, provide a computationally efficient framework for solving the CO_{2} source–sink estimation problem. Unlike DA applications for weather prediction and constituent assimilation, however, the advantages and disadvantages of DA approaches for CO_{2} flux estimation have not been extensively explored. In this study, we compare and assess estimates from two advanced DA approaches (an ensemble square root filter and a variational technique) using a batch inverse modeling setup as a benchmark, within the context of a simple one-dimensional advection–diffusion prototypical inverse problem that has been designed to capture the nuances of a real CO_{2} flux estimation problem. Experiments are designed to identify the impact of the observational density, heterogeneity, and uncertainty, as well as operational constraints (i.e., ensemble size, number of descent iterations) on the DA estimates relative to the estimates from a batch inverse modeling scheme. No dynamical model is explicitly specified for the DA approaches to keep the problem setup analogous to a typical real CO_{2} flux estimation problem. Results demonstrate that the performance of the DA approaches depends on a complex interplay between the measurement network and the operational constraints. Overall, the variational approach (contingent on the availability of an adjoint transport model) more reliably captures the large-scale source–sink patterns. Conversely, the ensemble square root filter provides more realistic uncertainty estimates. Selection of one approach over the other must therefore be guided by the carbon science questions being asked and the operational constraints under which the approaches are being applied.