  Significant uncertainty in climate prediction and cloud physics is tied to
observational gaps relating to shallow scattered clouds. Addressing these
challenges requires remote sensing of their three-dimensional (3D)
heterogeneous volumetric scattering content. This calls for passive scattering
computed tomography (CT). We design a learning-based model (ProbCT) to achieve
CT of such clouds, based on noisy multi-view spaceborne images. ProbCT infers -
for the first time - the posterior probability distribution of the
heterogeneous extinction coefficient, per 3D location. This yields arbitrary
valuable statistics, e.g., the 3D field of the most probable extinction and its
uncertainty. ProbCT uses a neural-field representation, making essentially
real-time inference. ProbCT undergoes supervised training by a new labeled
multi-class database of physics-based volumetric fields of clouds and their
corresponding images. To improve out-of-distribution inference, we incorporate
self-supervised learning through differential rendering. We demonstrate the
approach in simulations and on real-world data, and indicate the relevance of
3D recovery and uncertainty to precipitation and renewable energy.
