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In this paper, we investigate the performance of two robust beamformers, Chen's Quadratically Constrained Beamformer and Shahbazpanahi's General Rank Beamformer, on their performance on clutter rejection in Radar Depth Sounding of ice sheets. The investigation is carried out using both simulation and measured data with our Multichannel Coherent Radar Depth Sounder/Imager (MCoRDS/I). The results show that the two robust beamformers outperform conventional sample matrix inversion (SMI) and Loaded SMI beamformers in resolving the issue of signal self-nulling. However, the effectiveness of both robust beamformers on clutter cancellation was greatly reduced in situations of small antenna array and small number of data samples. By using MUSIC cum geometry to estimate the directions of arrivals (DOAs) of the dominant clutter signals, we have shown via simulations that we can enhance the two robust beamformers by incorporating linear constraints at the clutter signal DOAs to improve SINR by more than 10 dB.I.