An asymptotic analysis of borehole sonic waveforms near a dipping bed

Abstract

An asymptotic approach is used to simulate borehole inhomogeneous plane compressional and shear waves. The reflections and mode conversions from the bed sonic waveforms in the vicinity of a dipping bed. In the boundary are calculated according to the plane wave simulation, the direct borehole signals and the bed formulas given by Aki and Richards [2). The received boundary reflected signals are calculated separately. The waveforms after the reflection are then obtained by using direct borehole signals, which contain the compressional arrivals, the shear arrivals, Stoneley and other borehole the same borehole coupling theory I31 that governs the receiver coupling. modes, are computed by a Fourier integral involving wavenumbers. The reflected signals are obtained in three From these derivations, a reciprocity relationship is steps: the first step is to calculate the borehole source found between the pressure wave in a fluid medium and radiation into formation P and SV waves; the second step the elastic waves in a solid. The reciprocity can be is to obtain the reflection and mode conversion at the bed expressed in the following equation: boundary; and, the third step is to couple the reflected waves back into the borehole receivers. A reciprocity relationship exists between the pressure wave in a fluid V(;l,w)~P’ (~l,w) = i&rr~)*F*($~)