Direct reconstruction of glacier bedrock from known free surface data using the one-dimensional shallow ice approximation

Glaciers are an important component of the climate system that respond sensitively to climate change. Small glaciers respond more quickly to changing weather patterns compared to large polar ice sheets and ice caps that make them ideal to study climate variability. This study relates to the numerical reconstruction of the glacier bed in order to generate basal boundary data for ice flow models. Bedrock elevation is a paramount input parameter in glacier flow modeling to accurately capture its flow dynamics but is difficult to measure. We present an easy to implement direct numerical and analytical methodology to infer the bedrock geometry under glacial ice from the knowledge of the free surface elevation or the free surface velocity in one space dimension. The numerical and analytical methods are both based on the shallow ice approximation and require the time series of the surface mass balance distribution. Moreover, the analytical method requires the knowledge of the glacier thickness at one arbitrary location. Numerical benchmark test cases are used to verify the suitability and applicability of the algorithms, and a sensitivity analysis demonstrates the robustness of the method.