Abstract

I herein propose a new method for estimating the true Green’s function of a near-surface layered structure for SH waves at the borehole receiver depth from the apparent Green’s function that contains downgoing waves from the upper layers and the free surface. These Green’s functions are defined using transfer functions in the z-transform domain. The information required for the proposed method is a set of pole frequencies that are the frequencies at which the transfer function (the input and output of which are borehole and surface recordings, respectively) has poles. The real part of the true Green’s function is related to the gain function of this transfer function for a perfectly elastic lossless medium, and the imaginary part is connected to the real part by means of a Kramers–Kronig relation. The results obtained by applying the proposed method to actual borehole receiver array recordings show that convolutions of the impulse response of the estimated reciprocal of the Green’s function and surface recordings successfully output upcoming and downgoing waves at the borehole receiver depth. As a result, the synthesized borehole wave is in good agreement with the recorded borehole wave.

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