Stationary phase solutions for the radiation patterns of borehole sources are commonlyused to study the far-field seismic wavefields produced in crosshole or reverse VSPexperiments, but they break down when the formation shear wave velocity is less thanthe tube wave velocity in the source borehole. This is because the tube wave, not theprimary source, radiates the dominant shear wave signal in the form of large amplitudeconical waves, which are also called Mach waves. I model this effect by considering thetube wave to be a moving secondary point source generated by the primary source ofacoustic energy. A discretization of the source well allows a numerical solution of theintegral equation which yields the displacement field by a general source distributed inspace and time. The time at which each point source in the discretization emits energy isdetermined by the group velocity of the tube wave, while the radiation of the individualsources is characterized by the stress field induced by the tube wave at the borehole wall. An integration along the borehole of these point sources then yields the observed Mach wave arrivals. Since this method involves the summation of shear wave ray arrivals from the many point sources along the borehole, the method is called the Ray SummationMethod (RSM). Comparison of RSM results with full waveform synthetic seismogramscomputed with the discrete wavenumber method confirms the accuracy of this method.Unlike the discrete wavenumber method, however, the use of ray tracing in the RSMallows computation of the Mach wave arrivals for inhomogeneous layered media as wellas homogeneous models, including the waves generated by reflections of the Mach wavesat interfaces and from the reflections of the tube wave itself. The interactions of theconical waves with interfaces can show unusual patterns of arrivals which would not bepredicted from ordinary point source behavior.
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