Ground-penetrating radar (GPR) is a kind of high-frequency electromagnetic detection technology. It is mainly used to locate targets and interfaces in underground structures. In addition to the effective signals reflected from the subsurface objects or interfaces, the GPR signals in field work also include noise and different clutters, such as antenna-coupled waves, ground clutters, and radio-frequency interference, which have similar wavelet spectral characteristics with the target signals. Clutter and noise seriously interfere with the target’s response signal. The singular value decomposition (SVD) filtering method can select appropriate singular values and characteristic components corresponding to the effective signals for signal reconstruction to filter the GPR data. However, the conventional time-domain SVD method introduces fake signals when eliminating direct waves, and does not have good suppression of random noise around non-horizontal phase axes. Here, an SVD method based on the Hankel matrix in the local frequency domain of GPR data is proposed. Different numerical models and real field GPR data were handled using the proposed method. Based on the power of fake signals introduced via different processes, qualitative and quantitative analyses were carried out. The comparison shows that the newly proposed method could improve efforts to suppress random noise around non-horizontal phase reflection events and weaken the horizontal fake signals introduced by eliminating clutter such as ground waves.