LAUSR

Retrieval of physical parameters is of paramount relevance for Earth monitoring. Statistical (machine) learning approaches have been successfully introduced in the community, because they can learn nonlinear functional relations from observational data with no strong a priori assumptions and parametric forms. However, these methods still have two relevant problems: they only consider only one nonlinear feature map of the data, which can be limiting in complex problems where inputs (e.g., radiances) and outputs (e.g., state vectors) have strong nonlinear relations, and in most of the cases, models do not incorporate the structure of the output (dependent) variables. This article proposes a kernel method that solves the two aforementioned problems for physical parameter retrieval: first, it performs multioutput regression with the desired number of connected mappings, and second, it incorporates the output variables structure via a dedicated kernel. The proposed method has a closed-form solution, and thus, neither kernel dimensionality reduction nor preimaging is necessary unlike in previous structured kernel methods. Through the definition of appropriate kernel feature mappings, we also derive a pragmatic deep structured kernel ridge regression (KRR). The method is characterized statistically using a Gaussian process (GP) treatment and providing guarantees based on the concepts of leverage scores (LSs) and effective dimension: both explain that including an output structure acts as a powerful regularizer. We illustrate the method’s performance in toy examples and remote sensing parameter estimation problems involving vegetation parameters [chlorophyll, leaf area index (LAI), and fractional vegetation cover (FVC)] from compact high-resolution imaging spectrometer (CHRIS) images and the atmospheric temperature, moisture, and ozone profiles from infrared atmospheric sounding interferometer (IASI) data.