In civil engineering, ground-penetrating radar is widely used for road pavement survey. In contrast to the existing literature, this paper takes account of the influence of interface roughness (surface and interlayer roughness) within the scope of the data processing of radar signals. The rigorous electromagnetic method PILE (propagation inside layer expansion) provides the simulated data that show the influence of the interface roughness on the backscattered primary echoes of stratified media, the interface roughness provides a continuous frequency decay of the magnitude of the echoes. The observed frequency variations of the radar magnitude introduce some shape distortion on the radar waveform. The latter variations can be modelled by exponential function, which provides satisfactory results for a narrow bandwidth (2 GHz). An adaptation of the root-MUSIC algorithm is proposed. As a result, it is allowed to jointly estimate the time delay and the interface roughness. Then, the algorithm is tested on the data simulated by the PILE method, and numerical examples are provided to assess the performance of this algorithm. The associated results show that the proposed algorithm can estimate both the time delay and roughness parameters with a small relative error.