First order born scattering away from a fluid-filled borehole

Abstract

In this paper we solve the forward scattering problem in the first order Born approximation as it applies to the multipole borehole acoustic measurement configuration. This configuration consists of a multipole source and an array of multipole receivers, both deployed on a mandrel (The tool), and where the latter is axially offset from the former by a certain distance. The tool is deployed in a fluid-filled borehole, penetrating the earth formations. Assuming an arbitrary elastodynamic scatterer away from the fluid-filled borehole, we derive explicit expressions for the far-field scattered wavefield in the borehole fluid. In principle, these expressions could be used as a fast forward operator in the development of a full waveform inversion technique. Subsequently, to further our understanding, we derive analytical expressions for the borehole fluid scattered pressure at ‘small’ source-receiver offsets, and where the scatterer is represented by a Heaviside step change in elastodynamic formation properties, having an arbitrary orientation relative to the borehole axis. Our expressions apply to tools having a multipole source and receiver of order n. Such a source/receiver consists of 2n sources/receivers, located on the circumference of the tool and azimuthally separated from one another by π/n radians. All sources may be fired with equal polarity or adjacent sources may be fired with alternate polarity. In the latter case we compare our analytical solution with 3-D finite difference simulations. We do this for the practically relevant dipole (n = 1) and quadrupole (n = 2) excitation, for the (specular) PP, SVSV and SHSH borehole acoustic scattered pressure contributions. We obtain excellent results as long as the elastodynamic contrast between scatterer and embedding is small (<30%).