Snow depth patterns over glaciers are controlled by precipitation, snow redistribution due to wind and avalanches, and the exchange of energy with the atmosphere that determines snow ablation. While many studies have advanced the understanding of ablation processes, less is known about winter snow patterns and their variability over glaciers. We analyze snow depth on Haut Glacier d'Arolla, Switzerland, in the two winter seasons 2006–2007 and 2010–2011 to (1) understand whether snow depth over an alpine glacier at the end of the accumulation season exhibits a behavior similar to the one observed on single slopes and vegetated areas; and (2) investigate the snow pattern consistency over the two accumulation seasons. We perform this analysis on a data set of high‐resolution lidar‐derived snow depth using variograms and fractal parameters. Our first main result is that snow depth patterns on the glacier exhibit a multiscale behavior, with a scale break around 20 m after which the fractal dimension increases, indicating more autocorrelated structure before the scale break than after. Second, this behavior is consistent over the two years, with fractal parameters and their spatial variability almost constant in the two seasons. We also show that snow depth patterns exhibit a distinct behavior in the glacier tongue and the upper catchment, with longer correlation distances on the tongue in the direction of the main winds, suggesting spatial distinctions that are likely induced by different processes and that should be taken into account when extrapolating snow depth from limited samples.