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A new and effective analytical perturbation method is presented for the multipole acoustic logging in a transversely isotropic medium (TIM), whose symmetric principal axis is parallel to the borehole axis and the exact solutions could be found. TIM is regarded as a reference unperturbed isotropic state added to the perturbations, and three perturbation quantities about elastic modulus deviated from the isotropic medium are introduced. By searching for the zero, the first, and the second order of perturbation solutions, the multipole acoustic fields inside and outside the borehole are derived and obtained. The full waveforms in the borehole excited by a monopole, dipole, and quadrupole sources are numerical simulated by perturbation and exact solutions. It is found that the full waveforms are close to the exact solution when the first order perturbation is considered and more close to the exact solution when the second order perturbation is considered. The numerical simulation confirms that the perturbation method is a feasible method for complicated anisotropic medium in acoustic logging

Full waveform investigation of a perturbation method for acoustical multipole logging in transversely isotropic medium