Terahertz nondestructive testing (THz-NDT) technology is a new noninvasive and noncontact detection technology with strong penetration capability for nonmetallic and nonpolar composite materials, and has a great potential in the field of NDT. In practical detection, for complex terahertz echo signals, such as dispersive echoes and overlapping echoes, it is difficult to meet the requirements of localization accuracy for the time-of-flight only via traditional signal processing methods, such as the direct and deconvolution methods. In this context, the sparse representation method, as a new method for THz signal processing, has good localization accuracy and noise immunity. In this study, we propose a sparse representation method based on the least absolute shrinkage and selection operator (LASSO) to reconstruct an impulse response function from the complex THz echo signal and construct a double-over-complete dictionary to complete the dispersion compensation for the THz reflection signal. An amplitude decay coefficient is proposed to address the problem of inaccurate amplitude of the reconstructed impulse response function. This correction can effectively improve the peak-to-peak imaging quality in the time domain. The effectiveness of the proposed method is verified through numerical calculations and experimental analysis. The proposed method is expected to provide a novel solution for signal processing in THz-NDT.