Single-valued definition of the multivalued function for borehole acoustic waves in transversely isotropic formations

Abstract

It is useful to extract all components, including compressional, shear, and guided waves, from the full waveforms when we investigate the acoustic log data. The component waves can be simulated by calculating the contributions from poles and branch points of the borehole acoustic function according to Cauchy’s theorem. For such an algorithm to be implemented, the multi-valued function for the borehole wave field in the frequency-axial-wavenumber domain has to be rendered single-valued first. Assuming that the borehole axis is parallel to the symmetry axis of transverse isotropy, this paper derives the branch points of the borehole acoustic function. We discover that the number and the locations of those branch points are determined by the relation among the formation parameters c33, c44, ɛ, and δ. Thus the single-valued definitions in the acoustic-wave computation are sorted into two different cases. After building the Riemann surface related to each radial wavenumber, we give the single-valued definition of the borehole acoustic function inside and on the integration contour based on the radiation condition. In a formation with δ > ɛ + c44/2c33, if we choose the integration contour and the single-valued definition of the acoustic function in the way used in isotropic cases, the simulation results of component waves will be wrong.