Particle displacements on the wall of a borehole from incident plane waves

Abstract

Particle displacements from incident plane waves at the wall of a fluid‐filled borehole are formulated by applying the seismic reciprocity theorem to far‐field displacement fields. Such displacement fields are due to point forces acting on a fluid‐filled borehole under the assumption of long wavelengths. The displacement fields are analyzed to examine the effect of the borehole on seismic wave propagation, particularly for vertical seismic profiling (VSP) measurements. When the shortest wavelength of interest is approximately 25 times longer than the borehole’s diameter, the scattered displacements are proportional to the first power of incident frequency and borehole diameter. The maximum scattered energy occurs when an incident P‐wave propagates perpendicular to the borehole. Borehole effects on VSP measurements, such as waveform distortion, amplitude variation, and time delay, have been analyzed using the concept of a transfer function. When the shortest wavelength of interest is about 40 times longer than the borehole’s diameter, borehole effects on VSP measurements using a wall‐locking geophone are negligible.