A generalized additive mixed-effect modeling approach for characterizing Sentinel-2 surface reflectances of visible light from Torch Lake in Antrim County, Michigan, 2019-2025

David J Holtschlag

A generalized additive model (GAM) and three generalized additive mixed-effect models (GAMMs) were developed and compared to describe variations of surface reflectances of visible light from Torch Lake in the blue, green, and red bands from Sentinel 2 satellite imagery. All models included fixed effects associated with water-depth intervals, satellite orbits, trend, and smooth effects associated with season by water depth, and a measure of cloudiness. The GAMM models also included random effects of strata or timing, and spatial or temporal correlation effects. The analysis was based on 287 European Space Agency’s Sentinel 2A, 2B, and 2C satellite images of Torch Lake in Antrim County, Michigan, obtained from 2019 to 2025. The approach is motivated by the recognition of multiple sources of randomness that affect model uncertainty, where, at least one of which, is separable. Primary sources of randomness include day-to-day variations in surface reflectances from the water surface and sensor measurement uncertainties. A separable source of randomness is the uncorrected atmospheric distortions through which light passes from the sun to the lake surface and back to the satellite sensors. Results indicated that the GAMM models with random effects were more accurate than the GAM model without them. The preferred GAMM model accounted for error correlation of median reflectances in 60 strata spanning the surface of Torch Lake and a random temporal intercept component associated with the uncertainties in corrections for daily atmospheric conditions. Seasonal changes in reflectances within the 0-5 ft water-depth interval had lower magnitudes and were out of phase with seasonal variability in the three deeper water-depth intervals. Changes in reflectances in shallow areas may be more indicative of changes in reflectance within benthic areas, whereas changes in deeper areas may be related to dissolved and suspended materials in the water column. The uncertainty of shallow-water depths degrades the interpretability of surface reflections from strata in shallow waters.