  Due to the irregular space-time sampling of sea surface observations, the
reconstruction of sea surface dynamics is a challenging inverse problem. While
satellite altimetry provides a direct observation of the sea surface height
(SSH), which relates to the divergence-free component of sea surface currents,
the associated sampling pattern prevents from retrieving fine-scale sea surface
dynamics, typically below a 10-day time scale. By contrast, other satellite
sensors provide higher-resolution observations of sea surface tracers such as
sea surface temperature (SST). Multimodal inversion schemes then arise as an
appealing strategy. Though theoretical evidence supports the existence of an
explicit relationship between sea surface temperature and sea surface dynamics
under specific dynamical regimes, the generalization to the variety of upper
ocean dynamical regimes is complex. Here, we investigate this issue from a
physics-informed learning perspective. We introduce a trainable multimodal
inversion scheme for the reconstruction of sea surface dynamics from
multi-source satellite-derived observations. The proposed 4DVarNet schemes
combine a variational formulation involving trainable observation and a priori
terms with a trainable gradient-based solver. We report an application to the
reconstruction of the divergence-free component of sea surface dynamics from
satellite-derived SSH and SST data. An observing system simulation experiment
for a Gulf Stream region supports the relevance of our approach compared with
state-of-the-art schemes. We report relative improvement greater than 50%
compared with the operational altimetry product in terms of root mean square
error and resolved space-time scales. We discuss further the application and
extension of the proposed approach for the reconstruction and forecasting of
geophysical dynamics from irregularly-sampled satellite observations.
