Global distributions of glacier properties: A stochastic scaling paradigm

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Abstract
Many problems in global climate and Earth systems science require knowledge of regional‐ or global‐scale distributions of glacier properties, which includes mass balance, ice velocity, flux, thickness, volume, and surface area, among others. With roughly 160,000 glaciers worldwide, obtaining information on the global probability distributions of most ice properties is expensive and often infeasible. Only surface area distributions are relatively easy to measure, either by direct observation or remote sensing. While other properties are difficult to observe, this work shows that scaling relationships from the continuum dynamics of ice can link the distribution of surface areas to the global and regional distributions of any other continuum property. Some data inventories already exist for constructing reasonable distributions of glacier sizes, and this analysis presents theoretical arguments based on glacier network topologies (similar in concept to river networks) to suggest a power law times an exponential distribution of surface areas (in agreement with the data). Therefore, by using existing data or the theoretical distributions of surface areas, specific distributions for other glacier properties in any region of the world can be constructed. As an example, predictions (up to a scaling constant) are made for the distribution of glacier volumes, characteristic thicknesses, characteristic velocities, and characteristic response times in the Alps.