### S.M. Miller, __A.M. Michalak__ and P.J. Levi

### SUMMARY

Atmospheric inverse modeling is a statistical approach that can be used to trace atmospheric measurements of trace gases back to the magnitude and patterns of fluxes (i.e. emissions and uptake) of these gases at the Earth surface. Some of these quantities have known physical upper or lower bounds. This work explores the applicability of various existing approaches to enforcing such bounds within the context of atmospheric inverse modeling, using the problem of estimating methane emissions in North America as a prototypical example. The examined methods include data transformations, the method of Lagrange multipliers, and two Markov chain Monte Carlo (MCMC) approaches.

### ABSTRACT

Many inverse problems in the atmospheric sciences involve parameters with known physical constraints. Examples include nonnegativity (e.g., emissions of some urban air pollutants) or upward limits implied by reaction or solubility constants. However, probabilistic inverse modeling approaches based on Gaussian assumptions cannot incorporate such bounds and thus often produce unrealistic results. The atmospheric literature lacks consensus on the best means to overcome this problem, and existing atmospheric studies rely on a limited number of the possible methods with little examination of the relative merits of each.

This paper investigates the applicability of several approaches to bounded inverse problems. A common method of data transformations is found to unrealistically skew estimates for the examined example application. The method of Lagrange multipliers and two Markov chain Monte Carlo (MCMC) methods yield more realistic and accurate results. In general, the examined MCMC approaches produce the most realistic result but can require substantial computational time. Lagrange multipliers offer an appealing option for large, computationally intensive problems when exact uncertainty bounds are less central to the analysis. A synthetic data inversion of US anthropogenic methane emissions illustrates the strengths and weaknesses of each approach.