Probabilistic constraints for nonlinear inverse problems

The probabilistic continuous constraint framework complements the representation of uncertainty by means of intervals with a probabilistic distribution of values within such intervals. This paper describes how nonlinear inverse problems can be cast into this framework, highlighting its ability to deal with all the uncertainty aspects of such problems. In previous work we have formalized the framework, relying on simplified integration methods to characterize the uncertainty distributions. In this paper we (1) provide validated constraint-based algorithms to compute these distributions, (2) discuss approximations obtained by their hybridization with Monte-Carlo methods, and (3) obtain a better uncertainty characterization, by including methods to compute expected values and standard deviations. The paper illustrates this new methodology in Ocean Color (OC), a research area which is widely used in climate change studies and has potential applications in water quality monitoring. OC semi-analytical approaches rely on forward models that relate optically active seawater compounds (OC products) to remote sensing measurements of the sea-surface reflectance. OC products are derived by inverting the forward model on a spectral-reflectance basis. Based on a set of preliminary experiments we show that the probabilistic constraint framework is able to provide a valuable characterization of the uncertainty of all scenarios consistent with the model and the measurements. Moreover, the framework can be used to derive how measurements accuracy affects the uncertainty distribution of the retrieved OC products, which may constitute an important contribution to the OC community.