Higher-Order Vario Functions and Geostatistical Classification, Applied in the Study of Snow and Ice Surface Roughness

Study of the surface roughness of snow fields, glaciers, and ice sheets requires measurement and analysis of the surface's three-dimensional features, anisotropies, and complex microtopography. Observing that the notions of relief and surface roughness differ only with respect to scale, we consider surface roughness a spatial variable defined as the derivative of (micro)topography. In a project aimed at collecting subscale information for satellite data, we designed and built the Glacier Roughness Sensor, a multichannel instrument to measure snow and ice surface roughness at 0.2~m across-track, 0.1~m along-track resolution and subcentimeter vertical accuracy, with differential kinematic GPS data for positioning. Geostatistical surface classification is aimed at distinguishing objects - surface provinces or surface types - objectively and automatically. The basic idea is to calculate spatial structure functions from surface data and extract parameters from those functions that constitute a feature vector. If feature vectors can be designed to capture characteristic properties of surface types, then a classification of surface provinces is possible. Application in a moving-window operation facilitates segmentation of a given study area into surface provinces. Application to time series of surface data provides a means to study morphogenetic processes and changes in environmental conditions. The geostatistical surface classification has been applied success- fully to ice-surface roughness data collected on the Greenland Inland Ice, in the drainage basin of Jakobshavns Isbr\ae , the world's fastest moving glacier. Surface structures on the glacier are huge crevasses, visible in SAR data, and surface structures in the slow-moving ice of the drainage basin are still 1-2 meters in relief, so the mathematical problem of extracting characteristic parameters from variograms of roughness data is well- posed. In a study of snow-surface roughness data from Niwot Ridge, Colorado Front Range, the same problem is ill-posed, because the snow surface develops morphologic features that are small relative to the resolution of the survey device. Definition of vario functions of higher order provides a solution to this problem. There is a wide range of applications: (a) High-resolution surface structures influence return signals of satellite sensors, and on this data much of the monitoring of changes in the large ice sheets is based. (b) Geostatistical classification yields a characterization of typical and distinct ice surface provinces, including sastrugi, blue ice fields, ablation channels, and combined forms. (c) Seasonal comparison facilitates understanding of ice-surface morphogenetic processes, such as the interaction of wind and ablation processes, as carried out for Jakobshavn Isbr\ae , Greenland. (d) Self-organizational processes in snow can be distinguished from the influence of ground patterns on winter and summer snow surface morphogenesis in an alpine snowfield. (e) Surface roughness characteristics influence energy exchange, heat transfer, melting in the snowpack, and thus are important input variables in snow-hydrologic models.