Solving inverse problems is central to geosciences and remote sensing. Radiative transfer models (RTMs) represent mathematically the physical laws which govern the phenomena in remote sensing applications (forward models). The numerical inversion of the RTM equations is a challenging and computationally demanding problem, and for this reason, often the application of a nonlinear statistical regression is preferred. In general, regression models predict the biophysical parameter of interest from the corresponding received radiance. However, this approach does not employ the physical information encoded in the RTMs. An alternative strategy, which attempts to include the physical knowledge, consists in learning a regression model trained using data simulated by an RTM code. In this work, we introduce a nonlinear nonparametric regression model which combines the benefits of the two aforementioned approaches. The inversion is performed taking into account jointly both real observations and RTM-simulated data. The proposed Joint Gaussian Process (JGP) provides a solid framework for exploiting the regularities between the two types of data. The JGP automatically detects the relative quality of the simulated and real data, and combines them accordingly. This occurs by learning an additional hyper-parameter w.r.t. a standard GP model, and fitting parameters through maximizing the pseudo-likelihood of the real observations. The resulting scheme is both simple and robust, i.e., capable of adapting to different scenarios. The advantages of the JGP method compared to benchmark strategies are shown considering RTM-simulated and real observations in different experiments. Specifically, we consider leaf area index (LAI) retrieval from Landsat data combined with simulated data generated by the PROSAIL model.
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