In the common-ray approximation of the coupling ray theory, the S-wave traveltimes in the coupling equations are approximated by the first-order perturbation expansion from the common reference ray. The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in traveltimes due to the perturbation from the common reference ray. The errors of S-wave traveltimes due to the common-ray approximation may be approximated by the second-order terms in the perturbation expansion. The first-order and second-order terms in the perturbation expansion can be calculated by simple numerical quadratures along the common reference ray.
The common reference ray may be represented by the isotropic common ray or by the anisotropic common ray. The equations for the first-order and second-order perturbation expansions of traveltime from both the isotropic and anisotropic common ray to the anisotropic-ray-theory rays have already been derived. We numerically test the equations in three smooth 1-D velocity models of differing degrees of anisotropy.
We first compare the first-order and second-order perturbation expansions of traveltime from the isotropic common rays and from the anisotropic common rays to the anisotropic-ray-theory rays with the correct anisotropic-ray-theory traveltimes in order to illustrate the accuracy of the perturbation expansions and reliability of the error estimates.
Then, the synthetic seismograms obtained by the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory are compared with the more accurate coupling-ray-theory synthetic seismograms simulated by the second-order perturbation expansion of traveltime from the anisotropic common rays. In the numerical examples, the errors of the anisotropic-common-ray approximation of the coupling ray theory are considerably smaller than the errors of the isotropic-common-ray approximation.
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