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Ice sounding radars are able to measure ice sheets by profiling their glaciological features from the surface to the bedrock. The current airborne and, in particular, future space-based systems are suffering from off-nadir surface clutter, which can mask the depth signal of interest. The most recent surface clutter suppression techniques are based on multi-phase-center systems combined with sophisticated coherent postprocessing. The performance of the techniques can be improved by accurate direction-of-arrival (DOA) estimates of the surface clutter. This paper deals with data-driven DOA estimation for surface clutter signals, which includes a formulation of the mathematical foundation of spatial aliasing. DOA estimation is applied to data acquired with the P-band POLarimetric Airborne Radar Ice Sounder at the Jutulstraumen Glacier, Antarctica. The effects of spatial aliasing related to a large phase center spacing are analyzed, and an unwrapping procedure is presented and applied to the data. Finally, DOA estimation of full-scene data is analyzed and used to show an along-track and incidence (off-nadir) angle dependent variation of the effective scattering center of the surface return, which is caused by a varying penetration depth.