Empirical Optimization of the Stumpf Method Parameter for Satellite-Derived Bathymetry

Satellite-Derived Bathymetry (SDB) based on the Stumpf log-ratio method routinely uses a fixed parameter n = 1000, a convention that has rarely been evaluated systematically. This study assesses empirically how SDB error varies with n across 31 bathymetric scenarios at 18 coastal sites with contrasting morphological and water quality conditions using Sentinel-2 imagery, ACOLITE atmospheric correction and in situ reference depths in the −0.5 to −6 m range. For each scenario, an optimal n was retrieved by minimizing RMSE against the reference bathymetry, and the response of the error to n was characterized. Two stable regions (a left and a right plateau) are identified in the error-versus-n curve, with most scenarios (26 of 31) reaching their optimum on the left plateau at small n values. Replacing n = 1000 with the site-specific optimum, the mean R² increases from 0.70 to 0.84 and reduces RMSE by 21.9% on average (~18 cm). A single generalized value n_median(LP) = 2.2, transferable across sites, recovers most of this gain; the unscaled, parameter-free case n = 1 alone already reduces RMSE by 15.9% on average relative to n = 1000 and outperforms it in 25 of the 31 cases. The margin of improvement obtainable by tuning n is strongly correlated (Spearman) with hue angle, chlorophyll-a and Trophic State Index, while turbidity and Secchi depth show no correlation, consistent with the loss of reliability of these last two indicators in optically shallow waters. The benefit of using a small n is greatest in greener, higher-chlorophyll waters, whereas in very clear, blue waters (hue > 160°, Chl-a < 2 µg/L) n = 1000 remains a defensible choice. These results support replacing n = 1000 by n ≈ 2–3 as the practical default for Stumpf-based SDB in the studied depth range and provide a water-quality-based criterion to anticipate the expected benefit of parameter tuning. These improvements make it possible to use imagery from less clear waters for SDB extraction, broadening the range of images suitable for beach monitoring.