Numerical computation of individual far‐field arrivals excited by an acoustic source in a borehole

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Abstract
In this paper, I model the acoustic logging problem and numerically compute individual arrivals at far‐field receivers. The ability to compute individual arrivals is useful for examining the sensitivities of each arrival to various factors of interest, as opposed to examining the full waveform as a whole. While the numerical computation of the mode arrivals (Peterson, 1974) and the numerical computation of the first head waves (Tsang and Rader, 1979) have been previously reported, the numerical computation of the entire set of head‐wave arrivals is new and is the major contribution of this paper. Following Roever et al. (1974) and others, the full wave field is represented as a sum of contributions from both poles and branchcuts in the complex wavenumber plane. The pole contributions correspond to mode arrivals while the branch cuts are associated with the body waves (i.e., head waves). Both the pole and branch cut contributions are computed numerically and results are presented for the cases of a slow and a fast formation. The shear event in the slow formation is found to be relatively small, consistent with observations in measured data. Contrary to existing knowledge, the shear event in the fast formation is also relatively small. The apparent strong shear arrival in the full waveforms is due primarily to the trapped mode pole in the vicinity of cutoff.

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