The fk-based direct exact stiffness matrix method for broadband seismogram synthesis of a multi-scale crustal structure due to finite fault kinematic sources

Broadband seismogram synthesis of the multi-layer and multi-scale has been a longstanding focus in seismic engineering area for decades, which also has many difficulties to overcome in computing range and efficiency. This paper proposes the direct exact stiffness matrix method, a form of the frequency-wavenumber method (abbreviated as fk), for synthesizing seismograms. First, the direct exact stiffness matrix of the multi-layered and multi-scale underground structure is constructed in the frequency-wavenumber domain, and the sub-fault source (the earthquake fault is divided into the sub-fault source) is equivalent to the displacement-stress discontinuity to describe the dislocation. Furthermore, the “direct stiffness method” is used to determine the receiver’s motion, which establishes Green’s function (the relationship between the source and the receiver). The divergence index term in the stiffness matrix is modified to overcome the solution inefficiency associated with the thick layer and the high-frequency. The accuracy is verified by comparing the proposed method results with three examples in the publications, involving different source types and depth scales. The kinematics hybrid source model, based on the GP14.3 model, is introduced to generate a finite fault source model. The 2023 Turkey M_w 7.8 earthquake source model is established, and the applicability of the fk method in the near-fault seismic simulation is validated by comparing it with the time history and spectrum records. The M_w 7.0 dip-slip fault source model is based on the GP14.3 hybrid source generation which is set in a multi-scale layered half-space. The study area comprises a near-fault area with 122 receivers. Conclusions are drawn based on the analysis of the normalized response spectrum, peak spectral acceleration, and difference rate by changing the shallow velocity structure. The results show that the shallow velocity structure results in an average amplification of 5.1% in the fault-parallel direction and 7.1% in the fault-perpendicular direction.