Adaptive radar processing has been shown to be useful in downward-looking radars that must detect moving targets in the midst of strong clutter returns. Compressed sensing has found applications in radar problems but has not been comprehensively studied with respect to clutter and other structured interference. The performance of compressed sensing radar techniques in the presence of clutter is explored herein and compared to existing adaptive radar processing methods, including Space-Time Adaptive Processing (STAP), via Monte Carlo exploration of target detection performance. Finally, we propose extensions to standard ?1 optimization techniques to account for known interference covariance matrix statistics. These extensions outperform current compressed sensing techniques, outperform the fully sampled, nonadaptive matched filter estimate, and approach the performance level of the fully sampled STAP estimate. However, similar detection performance can be achieved at lower computational cost by applying a linear filter using the same covariance information.