Determination of the Initial Value Ranges of Nonlinear Solutions for a Log Ratio Bathymetric Inversion Model and Bathymetry Retrieval

Abstract

The log-ratio model (LRM) with three model parameters ( ${\boldsymbol{n}}$ , ${{\boldsymbol{m}}_1},$ and ${{\boldsymbol{m}}_0}$ ) has been widely used in shallow water bathymetry of multispectral images. At present, obtaining LRM model parameters by optimization algorithms depends on the setting of initial values [ ${\boldsymbol{n}}(0)$ , ${{\boldsymbol{m}}_1}(0),$ and ${{\boldsymbol{m}}_0}(0)$ ] or initial value ranges. However, how to set the initial value can obtain model parameters more effectively and credibly, and the significance of each initial value in the optimization process is very important for LRM bathymetric retrieval method. In this article, remote sensing images and bathymetric data were used to determine the initial value ranges of credible model parameters, and then the sensitivity of each parameter was analyzed in the algorithm optimization process from the partial differential perspective, and the bathymetry was retrieved in the sea area near Ganquan Island. The results show that according to the partial differential analysis of model parameters, ${\boldsymbol{n}}$ , with the smallest change rate in three LRM parameters and the rate of ${{\boldsymbol{m}}_1}$ and ${{\boldsymbol{m}}_0}$ is at least five times of it. So ${\boldsymbol{n}}(0)$ needs more iterations to achieve the optimal. Therefore, the initial value ${\boldsymbol{n}}(0)$ determines whether the optimization method can obtain credible LRM model parameters and the algorithm convergence speed. And when ${\boldsymbol{n}}(0) \in ({1/{{\boldsymbol{R}}_{{{\bf min}}}}, 1000} ]$ ( ${{\boldsymbol{R}}_{{{\bf min}}}}$ is the minimum reflectance in blue and green bands), there must be a credible model parameter solution. The bathymetric retrieval results of the sea area with coral and sand substrates show the applicability of our initial value ranges in different substrate area.