  The modelling of Earth observation data is a challenging problem, typically
approached by either purely mechanistic or purely data-driven methods.
Mechanistic models encode the domain knowledge and physical rules governing the
system. Such models, however, need the correct specification of all
interactions between variables in the problem and the appropriate
parameterization is a challenge in itself. On the other hand, machine learning
approaches are flexible data-driven tools, able to approximate arbitrarily
complex functions, but lack interpretability and struggle when data is scarce
or in extrapolation regimes. In this paper, we argue that hybrid learning
schemes that combine both approaches can address all these issues efficiently.
We introduce Gaussian process (GP) convolution models for hybrid modelling in
Earth observation (EO) problems. We specifically propose the use of a class of
GP convolution models called latent force models (LFMs) for EO time series
modelling, analysis and understanding. LFMs are hybrid models that incorporate
physical knowledge encoded in differential equations into a multioutput GP
model. LFMs can transfer information across time-series, cope with missing
observations, infer explicit latent functions forcing the system, and learn
parameterizations which are very helpful for system analysis and
interpretability. We consider time series of soil moisture from active (ASCAT)
and passive (SMOS, AMSR2) microwave satellites. We show how assuming a first
order differential equation as governing equation, the model automatically
estimates the e-folding time or decay rate related to soil moisture persistence
and discovers latent forces related to precipitation. The proposed hybrid
methodology reconciles the two main approaches in remote sensing parameter
estimation by blending statistical learning and mechanistic modeling.
