Determination of Bayesian Cramér–Rao Bounds for Estimating Uncertainties in the Bio-Optical Properties of the Water Column, the Seabed Depth and Composition in a Coastal Environment

Abstract

The monitoring of coastal areas using remote sensing techniques is an important issue to determine the bio-optical properties of the water column and the seabed composition. New hyperspectral satellite sensors (e.g., PRISMA, DESIS or EnMap) are developed to periodically observe ecosystems. The uncertainties in the retrieved geophysical products remain a key issue to release reliable data useful for the end-users. In this study, an analytical approach based on Information theory is proposed to investigate the Cramér–Rao lower Bounds (CRB) for the uncertainties in the ocean color parameters. Practically, during the inversion process, an a priori knowledge on the estimated parameters is used since their range of variation is supposed to be known. Here, a Bayesian approach is attempted to handle such a priori knowledge. A Bayesian CRB (BCRB) is derived using the Lee et al. semianalytical radiative transfer model dedicated to shallow waters. Both environmental noise and bio-optical parameters are supposed to be random vectors that follow a Gaussian distibution. The calculation of CRB and BCRB is carried out for two hyperspectral images acquired above the French mediterranean coast. The images were obtained from the recently launched hyperspectral sensors, namely the DESIS sensor (DLR Earth Sensing Imaging Spectrometer, German Aerospace Center), and PRISMA (Precursore IpperSpettrale della Mission Applicativa—ASI, Italian Space Adjency) sensor. The comparison between the usual CRB approach, the proposed BCRB approach and experimental errors obtained for the retrieved bathymetry shows the better ability of the BCRB to determine minimum error bounds.