We show that it is possible, and often desirable, to accommodate arbitrary P-wave anisotropy in practical seismic data processing, including migration and full waveform inversion. We propose a practical concept of arbitrary P-wave anisotropy, providing a sufficient degree of freedom to explain and reproduce observed anisotropic seismic signatures to a high degree of accuracy. The key to this concept is the proposed P-wave anisotropy parameterization (A-parameters) that, together with the use of the weak-anisotropy approximation, leads to significantly simplified theory. Here as an example, we use a simple and transparent formula relating P-wave traveltimes and 15 P-wave A-parameters. The formula is used in the inversion scheme, which does not require any a priori information about anisotropy symmetry and its orientation. We test applicability of the proposed scheme on a blind inversion of synthetic P-wave traveltimes generated for VSP experiments in homogeneous models. Three models of varying anisotropy are used: tilted orthorhombic and triclinic models of moderate anisotropy (~ 10%), and orthorhombic model with a horizontal plane of symmetry, of strong anisotropy (> 25%). In all cases, the inversion yields all 15 P-wave A-parameters, which make reconstruction of corresponding phase-velocity surfaces possible with a high degree of accuracy. The inversion scheme is robust with respect to noise and source distribution. Its quality depends on the angular illumination of the medium. The results of the inversion are applicable, for example, in migration or as a starting model for inversion methods, such as full waveform inversion, if a model refinement is desired.
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