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Inverse models arise in various environmental applications, ranging from atmospheric modeling to geosciences. Inverse models can often incorporate predictor variables, similar to regression, to help estimate natural processes or parameters of interest from observed data. Although a large set of possible predictor variables may be included in these inverse or regression models, a core challenge is to identify a small number of predictor variables that are most informative of the model, given limited observations. This problem is typically referred to as model selection. A variety of criterion-based approaches are commonly used for model selection, but most follow a two-step process: first, select predictors using some statistical criteria, and second, solve the inverse or regression problem with these predictor variables. The first step typically requires comparing all possible combinations of candidate predictors, which quickly becomes computationally prohibitive, especially for large-scale problems. In this work, we develop a one-step approach for linear inverse modeling, where model selection and the inverse model are performed in tandem. We reformulate the problem so that the selection of a small number of relevant predictor variables is achieved via a sparsity-promoting prior. Then, we describe hybrid iterative projection methods based on flexible Krylov subspace methods for efficient optimization. These approaches are well-suited for large-scale problems with many candidate predictor variables. We evaluate our results against traditional, criteria-based approaches. We also demonstrate the applicability and potential benefits of our approach using examples from atmospheric inverse modeling based on NASA's Orbiting Carbon Observatory-2 (OCO-2) satellite. 

La Niña climate anomalies have historically been associated with substantial reductions in the atmospheric CO₂growth rate. However, the 2021 La Niña exhibited a unique near-neutral impact on the CO₂growth rate. In this study, we investigate the underlying mechanisms by using an ensemble of net CO₂fluxes constrained by CO₂observations from the Orbiting Carbon Observatory-2 in conjunction with estimates of gross primary production and fire carbon emissions. Our analysis reveals that the close-to-normal atmospheric CO₂growth rate in 2021 was the result of the compensation between increased net carbon uptake over the tropics and reduced net carbon uptake over the Northern Hemisphere mid-latitudes. Specifically, we identify that the extreme drought and warm anomalies in Europe and Asia reduced the net carbon uptake and offset 72% of the increased net carbon uptake over the tropics in 2021. This study contributes to our broader understanding of how regional processes can shape the trajectory of atmospheric CO₂concentration under climate change. 

Abstract Sulfuryl fluoride (SO₂F₂) is a synthetic pesticide and a potent greenhouse gas that is accumulating in the global atmosphere. Rising emissions are a concern since SO₂F₂has a relatively long atmospheric lifetime and a high global warming potential. The U.S. is thought to contribute substantially to global SO₂F₂emissions, but there is a paucity of information on how emissions of SO₂F₂are distributed across the U.S., and there is currently no inventory of SO₂F₂emissions for the U.S. or individual states. Here we provide an atmospheric measurement-based estimate of U.S. SO₂F₂emissions using high-precision SO₂F₂measurements from the NOAA Global Greenhouse Gas Reference Network (GGGRN) and a geostatistical inverse model. We find that California has the largest SO₂F₂emissions among all U.S. states, with the highest emissions from southern coastal California (Los Angeles, Orange, and San Diego counties). Outside of California, only very small and infrequent SO₂F₂emissions are detected by our analysis of GGGRN data. We find that California emits 60-85% of U.S. SO₂F₂emissions, at a rate of 0.26 ( ± 0.10) Gg yr⁻¹. We estimate that emissions of SO₂F₂from California are equal to 5.5–12% of global SO₂F₂emissions. 

Abstract. Atmospheric inverse modeling describes the process of estimating greenhouse gas fluxes or air pollution emissions at the Earth's surface using observations of these gases collected in the atmosphere. The launch of new satellites, the expansion of surface observation networks, and a desire for more detailed maps of surface fluxes have yielded numerous computational and statistical challenges for standard inverse modeling frameworks that were often originally designed with much smaller data sets in mind. In this article, we discuss computationally efficient methods for large-scale atmospheric inverse modeling and focus on addressing some of the main computational and practical challenges. We develop generalized hybrid projection methods, which are iterative methods for solving large-scale inverse problems, and specifically we focus on the case of estimating surface fluxes. These algorithms confer several advantages. They are efficient, in part because they converge quickly, they exploit efficient matrix–vector multiplications, and they do not require inversion of any matrices. These methods are also robust because they can accurately reconstruct surface fluxes, they are automatic since regularization or covariance matrix parameters and stopping criteria can be determined as part of the iterative algorithm, and they are flexible because they can be paired with many different types of atmospheric models. We demonstrate the benefits of generalized hybrid methods with a case study from NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite. We then address the more challenging problem of solving the inverse model when the mean of the surface fluxes is not known a priori; we do so by reformulating the problem, thereby extending the applicability of hybrid projection methods to include hierarchical priors. We further show that by exploiting mathematical relations provided by the generalized hybrid method, we can efficiently calculate an approximate posterior variance, thereby providing uncertainty information. 

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. 

Abstract Solar‐induced chlorophyll fluorescence (SIF) shows enormous promise as a proxy for photosynthesis and as a tool for modeling variability in gross primary productivity and net biosphere exchange (NBE). In this study, we explore the skill of SIF and other vegetation indicators in predicting variability in global atmospheric CO₂observations, and thus global variability in NBE. We do so using a 4‐year record of CO₂observations from NASA's Orbiting Carbon Observatory 2 satellite and using a geostatistical inverse model. We find that existing SIF products closely correlate with space‐time variability in atmospheric CO₂observations, particularly in the extratropics. In the extratropics, all SIF products exhibit greater skill in explaining variability in atmospheric CO₂observations compared to an ensemble of process‐based CO₂flux models and other vegetation indicators. With that said, other vegetation indicators, when multiplied by photosynthetically active radiation, yield similar results as SIF and may therefore be an effective structural SIF proxy at regional to global spatial scales. Furthermore, we find that using SIF as a predictor variable in the geostatistical inverse model shifts the seasonal cycle of estimated NBE and yields an earlier end to the growing season relative to other vegetation indicators. These results highlight how SIF can help constrain global‐scale variability in NBE.