Blind deconvolution of sparse spikes is a well-known problem in the fields of seismic exploration and ultrasonic nondestructive testing. In measuring thin layer thickness of asphalt pavements using GPR, a similar problem arises; the sparse reflectivity series representing the layered structure of the pavement convolved with the radar wavelet results in masking closely spaced reflections. A successful deconvolution retrieves the reflectivity series and thus improves the time resolution and facilitates quantitative data interpretation. In this paper, we cast the convolutional model as a multidimensional data model which renders blind deconvolution via independent component analysis (ICA) possible. We use a nonlinearity related to the double exponential density whose heavy-tailed nature provides further insight into the sparse nature of the reflectivity series. The method is tested on synthetic and real GPR data from a thin PVC slab. The results attest to the accuracy of the time delay estimates and verify the high resolution of the proposed approach.