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Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.

 

Abstract. Geostatistical inverse modeling (GIM) has become a common approach to estimating greenhouse gas fluxes at the Earth's surface using atmospheric observations. GIMs are unique relative to other commonly used approaches because they do not require a single emissions inventory or a bottom–up model to serve as an initial guess of the fluxes. Instead, a modeler can incorporate a wide range of environmental, economic, and/or land use data to estimate the fluxes. Traditionally, GIMs have been paired with in situ observations that number in the thousands or tens of thousands. However, the number of available atmospheric greenhouse gas observations has been increasing enormously as the number of satellites, airborne measurement campaigns, and in situ monitoring stations continues to increase. This era of prolific greenhouse gas observations presents computational and statistical challenges for inverse modeling frameworks that have traditionally been paired with a limited number of in situ monitoring sites. In this article, we discuss the challenges of estimating greenhouse gas fluxes using large atmospheric datasets with a particular focus on GIMs. We subsequently discuss several strategies for estimating the fluxes and quantifying uncertainties, strategies that are adapted from hydrology, applied math, or other academic fields and are compatible with a wide variety of atmospheric models. We further evaluate the accuracy and computational burden of each strategy using a synthetic CO2 case study based upon NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite. Specifically, we simultaneously estimate a full year of 3-hourly CO2 fluxes across North America in one case study – a total of 9.4×106 unknown fluxes using 9.9×104 synthetic observations. The strategies discussed here provide accurate estimates of CO2 fluxes that are comparable to fluxes calculated directly or analytically. We are also able to approximate posterior uncertainties in the fluxes, but these approximations are, typically, an over- or underestimate depending upon the strategy employed and the degree of approximation required to make the calculations manageable.