Atmospheric inverse modeling with known physical bounds: an example from trace gas emissions

Abstract

Many inverse problems in the atmospheric sciences involve parameters with known physical constraints. Examples include non-negativity (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. <br><br> This paper investigates the applicability of several approaches to bounded inverse problems and is also the first application of Markov chain Monte Carlo (MCMC) to estimation of atmospheric trace gas fluxes. The approaches discussed here are broadly applicable. A common method of data transformations is found to unrealistically skew estimates for the examined example application. The method of Lagrange multipliers and two 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 alternative 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

Nitrous Oxide Emissions Estimated With the CarbonTracker‐Lagrange North American Regional Inversion Framework

“…Previous inversion studies have used a range of strategies to prevent unrealistic negative fluxes, including lognormal data transformations, Lagrange multipliers, and Markov chain Monte Carlo methods (Ganesan et al, ; Miller et al, ). We experimented here with a nonnegativity enforcement algorithm that started with the standard posterior solution but then iteratively corrected the whole field for negative fluxes using Lagrange multipliers and the bounded, limited‐memory Broyden‐Fletcher‐Goldfarb‐Shanno approach (Byrd et al, ).…”

“…We experimented here with a nonnegativity enforcement algorithm that started with the standard posterior solution but then iteratively corrected the whole field for negative fluxes using Lagrange multipliers and the bounded, limited‐memory Broyden‐Fletcher‐Goldfarb‐Shanno approach (Byrd et al, ). In principle, the iterative solution should have conserved mass, with any removal of negative fluxes accompanied by weaker positive fluxes elsewhere (Miller et al, ). In practice, the nonnegativity algorithm was very slow to converge and led to an overall net increase in N 2 O emissions of about 0.3–0.4 Tg N/yr over the CT‐L domain.…”

Inverse Estimation of an Annual Cycle of California's Nitrous Oxide Emissions

“…Following Jeong et al (), we use the Just Another Gibbs Sampler system (Plummer, ) and the R statistical language (https://cran.r-project.org/) to build Markov chain Monte Carlo (MCMC) samplers for the posterior distribution in equation . The individual probability distributions in equation are described in supporting information Text S1 and convergence and accuracy of MCMC samples (50,000 samples for each parameter) are described in Text S3 (Gelman et al, ; Gelman & Hill, ; Gelman & Rubin, ; Kass et al, ; Korner‐Nievergelt et al, ; Kruschke, ; Michalak, ; Miller et al, ; Rasmussen & Williams, ).…”