This work focuses on automating the task of estimating Polar ice thickness from airborne radar data acquired over Greenland and Antarctica. This process involves the identification and accurate selection of the ice sheet's surface location and interface between the ice sheet and the underlying bedrock for each measurement. Identifying the surface and bedrock locations in the radar imagery enables the computation of ice sheet thickness, which is important for the study of ice sheets, their volume, and how they may contribute to global climate change. The time-consuming manual approach requires sparse hand-selection of surface and bedrock interfaces by several human experts, and interpolating between the selections to save time. Two primary methods have been studied in this paper, namely, edge-based and active contour. Results are evaluated and presented using the metrics of time requirements and accuracy. Automated ice thickness estimation results from 2006 and 2007 Greenland field campaigns illustrate that the edge-based approach offers faster processing (seconds compared to minutes), but suffers from a lack of continuity and smoothness aspects that active contours provide. The active contour approach is more accurate when compared to ground truth selections provided by human experts, and has proven to be more robust to image artifacts. It is shown that both techniques offer advantages which could be integrated to yield a more effective system.