Quantifying Uncertainty Bounds in Spectral Shallow Water Bathymetry: The Profile Likelihood Framework for Semi-Analytical Models

Abstract

Accurate bathymetric mapping in shallow waters is critical for coastal monitoring, navigation safety, and environmental conservation. While semi-analytical (SA) models have proven effective for estimating water column depth (WCD) from optical remote sensing data, quantifying the associated uncertainties remains a significant challenge. This study introduces a unified framework based on profile likelihood to evaluate WCD uncertainties in Semi-Analytical (SA) models, bridging traditional variance-covariance methods with likelihood-based inference. Through controlled simulations, we systematically assess the performance of first-order statistical tests-likelihood ratio, Wald (observed and expected), and score tests-across varying environmental conditions and sensor configurations.<p>Our results demonstrate that the expected Fisher information Wald test, equivalent to classical variance-covariance propagation, provides reliable uncertainty bounds in hyperspectral settings with clear waters. However, its performance deteriorates in multispectral applications and complex water conditions, where likelihood ratio tests prove more robust. This divergence highlights fundamental limitations in the small-sample regime typical of per-pixel spectral bathymetry. Notably, even under idealized conditions with true parameter initialization, local optimization challenges in SA models can compromise both parameter estimation and uncertainty quantification, suggesting even greater challenges in practical applications with less favorable initialization. The findings underscore the importance of robust statistical inference in improving the reliability of bathymetric products for operational applications.</p>