Statistical Distances and Their Applications to Biophysical Parameter Estimation: Information Measures, M-Estimates, and Minimum Contrast Methods

Abstract: Radiative transfer models predicting the bidirectional reflectance factor (BRF) of leaf canopies are powerful tools that relate biophysical parameters such as leaf area index (LAI), fractional vegetation cover fV and the fraction of photosynthetically active radiation absorbed by the green parts of the vegetation canopy (fAPAR) to remotely sensed reflectance data. One of the most successful approaches to biophysical parameter estimation is the inversion of detailed radiative transfer models through the construction of Look-Up Tables (LUTs). The solution of the inverse problem requires additional information on canopy structure, soil background and leaf properties, and the relationships between these parameters and the measured reflectance data are often nonlinear. The commonly used approach for optimization of a solution is based on minimization of the least squares estimate between model and observations (referred to as cost function or distance; here we will also use the terms “statistical distance” or “divergence” or “metric”, which are common in the statistical literature). This paper investigates how least-squares minimization and alternative distances affect the solution to the inverse problem. The paper provides a comprehensive list of different cost functions from the statistical literature, which can be divided into three major classes: information measures, M-estimates and minimum contrast methods. We found that, for the conditions investigated, Least Square Estimation (LSE) is not an optimal statistical distance for the estimation of biophysical parameters. Our results indicate that other statistical distances, such as the two power measures, Hellinger, Pearson chi-squared measure, Arimoto and Koenker–Basset distances result in better estimates of biophysical parameters than LSE; in some cases the parameter estimation was improved by 15%.