Integrating Domain Knowledge in Data-Driven Earth Observation With Process Convolutions

The modeling of Earth observation (EO) 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 are scarce or in extrapolation regimes. In this article, 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 modeling in EO problems. We specifically propose the use of a class of GP convolution models called
latent force models
(LFMs) for EO time series modeling, 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 illustrate the performance in two case studies. First, we consider time series of soil moisture (SM) from active Advanced Scatterometer (ASCAT) and passive [SM and ocean salinity (SMOS), advanced microwave scanning radiometer-2 (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 SM persistence and discovers latent forces related to precipitation. In the second case study, we show how the model can fill in gaps of leaf area index (LAI) and Fraction of Absorbed Photosynthetically Active Radiation (fAPAR) from moderate resolution imaging spectroradiometer (MODIS) optical time series by exploiting their relations across different spatial and temporal domains. The proposed hybrid methodology reconciles the two main approaches in remote-sensing parameter estimation by blending statistical learning and mechanistic modeling.