Determining the threshold value in upper limit of wavenumber integration by reflection-transmission coefficient in theoretical seismograms calculation

In multi-layered half-space, the displacement field is expressed as the integration of wavenumber k using reflection-transmission (R/T) coefficient when the seismogram is calculated in numerical ways in the frequency domain. Therefore, some special methods were introduced to solve those kinds of integration as to accelerate computation. For example, when the depth of source is equal or close to that of receiver, the peak-trough averaging method (PTAM) is applied. To apply PTAM, however, one must determine the threshold value called k_c after which the integration oscillates regularly. In previous studies, k_c was estimated empirically without theoretical support. In this study, a scheme based on theoretical analysis to determine k_c is proposed, and k_c is related to the R/T coefficients. According to the generalized R/T coefficient method, violent variation of integrand will occur when the determinant of relevant R/T coefficient vanishes. We show by examples that for a given frequency, root of the determinant of the reflection coefficient on the free surface is a proper value for k_c. Compared with the method calculated with empirical formula, the new method to determine k_c will be more efficient for a given precision to prove the validity.