Calculation of Conical (mach) Wave Displacement Fields Radiated by Borehole Sources in Slow Formations and Inhomogeneous Media

Stationary phase solutions for the radiation patterns of borehole sources are commonly used to study the far-field seismic wavefields produced in crosshole or reverse VSP experiments, but they break down when the formation shear wave velocity is less than the tube wave velocity in the source borehole. This is because the tube wave, not the primary source, radiates the dominant shear wave signal in the form of large amplitude conical waves, which are also called Mach waves. I model this effect by considering the tube wave to be a moving secondary point source generated by the primary source of acoustic energy. A discretization of the source well allows a numerical solution of the integral equation which yields the displacement field by a general source distributed in space and time. The time at which each point source in the discretization emits energy is determined by the group velocity of the tube wave, while the radiation of the individual sources 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 Summation Method (RSM). Comparison of RSM results with full waveform synthetic seismograms computed with the discrete wavenumber method confirms the accuracy of this method. Unlike the discrete wavenumber method, however, the use of ray tracing in the RSM allows computation of the Mach wave arrivals for inhomogeneous layered media as well as homogeneous models, including the waves generated by reflections of the Mach waves at interfaces and from the reflections of the tube wave itself. The interactions of the conical waves with interfaces can show unusual patterns of arrivals which would not be predicted from ordinary point source behavior.