Amortized Probabilistic Retrieval of Atmospheric CO2 from OCO-2 Spectra Using Deep Learning with Laplace Approximations and Normalizing Flows

There is a growing urgency to track greenhouse gasses with the resolution, precision and accuracy needed to support independent verification of $CO_2$ fluxes at local to global scales. The current generation of space-based sensors, however, only provides sparse observations in space and time. This challenge has fueled interest in the potential use of data from existing missions originally developed for other applications to infer global greenhouse gas variability. The Advanced Baseline Imager (ABI) onboard the Geostationary Operational Environmental Satellite (GOES-East), operational since 2017, provides full coverage of much of the western hemisphere at 10-minute intervals from geostationary orbit across 16 spectral channels at an approximately 2 km$^2$ spatial resolution. Here, we leverage this high spatial coverage and temporal revisit to develop Deep$XCO_2$, a single-pixel, physics-guided neural network to estimate dry-air column $CO_2$ mole fraction ($XCO_2$). Deep$XCO_2$ employs a time series of GOES-East's 16 spectral bands, ECMWF ERA5 lower tropospheric meteorology, MODIS surface reflectance, solar and satellite viewing geometry, and day of year. The network was trained on collocated GOES-East and OCO-2/OCO-3 observations. Deep$XCO_2$ is able to capture realistic $XCO_2$ variability when compared against a held-out year of OCO-2 and OCO-3 observations, and against observations from the TCCON network. We also present case studies illustrating the use of Deep$XCO_2$ to observe $XCO_2$ enhancements over urban areas and drawdown over agricultural regions. Overall, while the precision of GOES-East derived $XCO_2$ can never rival that of dedicated instruments, the unprecedented combination of contiguous geographic coverage, 10-minute temporal frequency, and multi-year record offers the potential to observe aspects of atmospheric $CO_2$ variability currently unseen from space.

Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.

Forward uncertainty propagation in complex physical systems can induce structured covariance across field-valued outputs. For a probabilistic surrogate, the total predictive covariance comprises the covariance of conditional means across input realizations and the average conditional predictive covariance. Probabilistic DeepONet (Prob-DeepONet) provides lightweight uncertainty quantification by predicting pointwise Gaussian means and variances in a single forward pass, but its conditional predictive covariance is restricted to a diagonal form. To represent cross-location conditional dependence without explicitly parameterizing a full high-dimensional covariance matrix, we develop a two-step mean-variance DeepONet (two-step MV-DeepONet) through two principal modifications. First, two-step training is used to decouple output-basis learning from the input-to-coefficient mapping, together with basis orthogonalization and subspace rotation. Second, Gaussian probabilistic modeling is transferred from the high-dimensional physical output space to the low-dimensional rotated coefficient space. Mapping these probabilistic coefficients through the shared basis induces a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. A Frobenius-norm error decomposition and corresponding upper bound identify low-rank covariance compressibility, trunk-subspace approximation, finite-sample statistical error, and coefficient-space covariance estimation as the principal factors governing covariance recovery. Numerical experiments on three representative problems governed by partial differential equations (PDEs) and a hypersonic blunt-body aerothermal problem show improved generalization, more structured uncertainty bands, and accurate recovery of off-diagonal correlation patterns compared with Prob-DeepONet.