Diffraction of axisymmetric waves in a borehole by bed boundary discontinuities

This paper presents the calculation of the diffraction of axisymmetric borehole waves by bed boundary discontinuities. The bed boundary is assumed to be horizontal and the inhomogeneities to be axially symmetric. In such a geometry, an axially symmetric source will produce only axially symmetric waves. Since the borehole is an open structure, the mode spectrum consists of a discrete part as well as a continuum. The scattering of a continuum of waves by bed boundaries is difficult to treat. The approach used in the past in treating this class of problem has been approximate in nature, or highly numerical, such as the finite-element method. We present here a systematic way to approximate the continuum of modes by discrete modes. After discretization, the scattering problem can be treated simply. Since the approach is systematic, it allows derivation of the solution to any desired degree of accuracy in theory; but in practice, it is limited by the computational resources available. We also show that our approach is variational and satisfies both the reciprocity theorem and energy conservation.