This paper investigates the application of Robust Principal Component Analysis (RPCA) to ground penetrating radar as a means to improve GPR anomaly detection. The method consists of a preprocessing routine to smoothly align the ground and remove the ground response (haircut), followed by mapping to the frequency domain, applying RPCA, and then mapping the sparse component of the RPCA decomposition back to the time domain. A prescreener is then applied to the time-domain sparse component to perform anomaly detection. The emphasis of the RPCA algorithm on sparsity has the effect of significantly increasing the apparent signal-to-clutter ratio (SCR) as compared to the original data, thereby enabling improved anomaly detection. This method is compared to detrending (spatial-mean removal) and classical principal component analysis (PCA), and the RPCA-based processing is seen to provide substantial improvements in the apparent SCR over both of these alternative processing schemes. In particular, the algorithm has been applied to both field collected impulse GPR data and has shown significant improvement in terms of the ROC curve relative to detrending and PCA.