The analytical solution has unique advantages in understanding the influence of terrain conditions on ground motion, but the previous research is basically limited to a single frequency steady-state response and frequency domain solution. In this paper, the probability of exceeding the surface magnification coefficient of the semicircular terrain in the hard field is studied by using the analytical method and the time domain analysis of a large number of measured strong earthquake records, and the distribution law of the concave boundary radius and the surface amplification coefficient is proposed. The wave function expansion method is used to derive the analytical solution of the peak acceleration amplification coefficient of the surface of the plane wave scattering field in the semicircular terrain. 100 strong earthquake records were extracted from the important strong earthquake network at home and abroad to form the input ground motion. According to the specification, a typical hard field is taken, and the excess probability of the surface amplification coefficient of the semicircular terrain is calculated by time-frequency transformation. The results show that the boundary radius of the concave body exists, that is, for any semi-circular sag terrain that exceeds the boundary radius, the surface acceleration amplification coefficient will achieve a stable convergence and consistent result in the engineering sense. The boundary radius of the semicircular terrain on the hard field is proportional to the shear wave velocity of the site. For depressions exceeding the boundary radius, the surface acceleration amplification factor varies from 0.3 to 2.0 depending on the angle of the incident wave and different observation points. In this paper, the distribution law of the concave boundary radius and its surface amplification coefficient is given for the first time, which can make up for the lack of understanding of the influence of local topography on ground motion.