Objective frequentist uncertainty quantification for atmospheric CO$_2$ retrievals

The steadily increasing amount of atmospheric carbon dioxide (CO$_2$) is affecting the global climate system and threatening the long-term sustainability of Earth's ecosystem. In order to better understand the sources and sinks of CO$_2$, NASA operates the Orbiting Carbon Observatory-2 & 3 satellites to monitor CO$_2$ from space. These satellites make passive radiance measurements of the sunlight reflected off the Earth's surface in different spectral bands, which are then inverted to obtain estimates of the atmospheric CO$_2$ concentration. In this work, we first analyze the current operational retrieval procedure, which uses prior knowledge in the form of probability distributions on the relevant atmospheric state variables to regularize the underlying ill-posed inverse problem, and demonstrate that the resulting uncertainties might be poorly calibrated both at individual locations and over a spatial region. To alleviate these issues, we propose a new method that uses known physical constraints on the state variables and direct inversion of the target functionals of the CO$_2$ profile to construct well-calibrated frequentist confidence intervals based on convex programming. Furthermore, we study the influence of individual nuisance state variables on the length of the confidence intervals and identify certain key variables that can greatly reduce the final uncertainty given additional deterministic or probabilistic constraints, and develop a principled framework to incorporate such information into our method.