A Bayesian Optimisation Approach for Model Inversion of Hyperspectral-multidirectional Observations: the Balance with a Priori Information

Hyperspectral-multidirectional radiance observations of the land surface from space potentially form one of the richest sources of geobiophysical information possible. For soil-vegetation objects, the retrieval of this information can be simulated by radiative transfer modelling. In combination with a couple of atmospheric parameters, the surface reflectance model SLC (soil-leaf-canopy) has more than twenty degrees of freedom, which all have a potential impact on top-of-atmosphere radiance data in hyperspectralmultiangular feature space. With such a high dimensionality, model inversion methods like look-up table techniques and neural networks tend to become less practicable, and cost-function optimisation re-emerges as a viable alternative. However, model inversion by optimisation techniques is often plagued by numerical instability due to the so-called ill-posedness of the model inversion problem. In the present paper, this ill-posedness of the problem is investigated and diagnosed by means of a singular value decomposition (SVD) of the Jacobian matrix, which contains the partial derivatives of all observations with respect to the model variables. In addition, it is demonstrated how in a Bayesian approach the incorporation of a priori information can increase the numerical stability of the model inversion. This leads to an extremely efficient optimisation algorithm, which for randomly selected model variable data reaches an adequate solution in about 99% of the cases, in less than twenty iteration steps. The paper will introduce the model SLC, its coupling with the atmosphere, for which MODTRAN4 is used, and for some selected cases it will analyse the SVD results in order to explain the causes of ill-posedness. A few model inversion sequences will be presented in order to illustrate the numerical stability of the algorithm and its ability to reach a plausible solution under various circumstances. The speed of this method is still limited, but it might be applied selectively to representative pixels in a field, or to “calibrate” the fixed model parameters in a lowdimensional look-up table or neural network model inversion solution.