We usually analyzed microtremors using a fundamental mode only. However, recent studies demonstrated that higher modes should be considered in SPAC method. In order to investigate this theoretically, we simulate microtremors numerically and applied the proposed method using higher modes. Also we applied this method to field data. In this method, we calculated apparent phase velocity from phase velocity and amplitude of higher modes, and compare the apparent velocity with the observed phase velocity. Since the phase velocity obtained from simulated data and field microtremor data is well consistent with the apparent phase velocity, the analysis method using the apparent velocity works well in S-wave velocity estimation.
Microtremor data have been applied to estimate subsurface S-wave velocity (Okada et al., 1990) in near-surface geophysics. In microtremor analysis, we have considered that fundamental mode of surface waves is predominant. In the field data analysis, however, the phase velocities obtained from spatial auto correlation (SPAC) method (Aki, 1957) are faster than the theoretically calculated velocity in high-frequency range. We could not explain the dispersion curve only considering fundamental mode of surface waves. Since it is known that the analysis considering higher modes can be consistent with this dispersion curve, it is important to establish microtremor analysis method by considering higher modes. Obuchi et al. (2004) proposed one idea considering higher modes by calculating apparent phase velocity from phase velocity and amplitude using higher mode. They further applied this idea to the field data, however, it is difficult to conduct quantitative evaluation of the analysis result using field data set. In order to overcome this difficulty, we made microtremor data set by numerical simulation and evaluate the proposed method. In this study, we simulated microtremors for two models. First model is a simple two-layer model and the fundamental mode is predominant in order to check the simulation validity. If the estimated phase velocity from simulated data set is consistent with theoretical dispersion curve of fundamental mode, we can justify our simulation approach is acceptable. Then, we show the result of the analysis using apparent phase velocity for four-layer model which predominates higher modes. Finally, we apply this analysis method to field data.
Since sources of microtremors are mainly on the surface and microtremors come from all directions equally, we simulated microtremors by setting 1000 sources randomly on the surface. To apply simulated data to SPAC method, the waveforms were recorded by the triangle array (Figure 1a). Because microtremors can be considered as surface waves, we assume that the wave propagates as a plane wave and sources are distributed randomly from radius 500 to 1000 m for the central receiver. Figure 1(b) shows geometry of array and sources. Discrete Wave-number Integral method (Bouchon and Aki, 1977) was used for waveform calculation. The source is a vertical force with 8Hz Ricker wavelet. The duration of simulated records for one source was 16 sec. Microtremors of about 10 minutes were made by superposing each waveform.