Some Aspects of Band-pass Filtering of Surface Waves by Robert B. Herrmann

Narrow band-pass filtering of multi-mode surface-wave signals can be used to extract information concerning the group velocities and spectral amplitudes of each mode. A detailed examination of narrow band-pass filtering shows the effect of epicentral distance and the shape of the group-velocity curve upon the filtered signal. With this knowledge, the filtered signal can be used to make accurate estimates of the spectral amplitudes of each mode. A rectangular and a Ganssian filter are considered in detail. A numerical example is provided to indicate the values and limitations of the technique for determining the spectral amplitudes of each mode. INTRODUCTION Recent surface-wave investigations by Tsai and Aki (1969, 1970, 1971), Mitchell (1973), and others have demonstrated the wealth of information which is available in the surface-wave spectrum. When one works with data at short periods, problems in interpretation arise because of the presence of more than a single surface-wave mode. Because the surface-wave spectrum for these periods is a superposition of the modal contributions, which are complex numbers with phase heavily dependent upon distance, the resultant spectrum is highly oscillatory, making analysis difficult. It is desirable, therefore, to be able to separate the individual modes out of the ground spectrum. The only way to do this exactly is by using a linear array of detectors spaced along the same' azimuth from the source. Having this, wave number-frequency analysis can be performed to recover both phase velocities and spectral amplitudes of the individual modes as a function of frequency. When data from such an array are not available, another method can be used which is not as general and exact as the wave number-frequency analysis, but which still can provide useful data. This method is based on the fact that different frequencies of a dispersed wave train arrive at different times; the deficiency of the method is that more than one mode might arrive at the same time with the same frequency. Dziewonski et al. (1969) developed the multiple filter analysis technique to determine the group velocities of dispersed waves. Mitchell (1973) used the peak amplitude of the envelope of the filtered signal generated by this technique in order to determine the spectral amplitudes of the individual modes. The present paper investigates the use of narrow band-pass filtering for determining the spectral amplitudes and group velocities of the individual modes. THEORY The displacement caused by a propagating dispersed wave at time t and distance r may be represented by f(t, r) = ~ r) exp(ie)t) d~ = j~=o Aj(to, r) exp[i(~ot-kT) ] dco, (1) where o~ is the angular frequency, kj is the wave number of thej th mode, Aj is the complex amplitude of the j th mode, and M + 1 is the number of modes present in the signal.