Two-ponit Ray Tracing Method For Horizontally Layered VTI Media

Ray tracing methods are used for travel time calculation and ray path determination,which are the fundamental requirements for seismic location and travel time tomography.Traditional ray tracing methods are generally developed on the basis of the isotropic assumption,but subsurface formations can often exhibit elasticity anisotropy.The path of a ray in an VTI medium can be complicated because the propagation velocity of a wave can change with the angle of propagation.In tectonically inactive areas,the subsurface formations can be modeled as VTI(Vertical transversely isotropy)media as the formation properties generally change only along the depth direction while show little lateral variations.In this thesis,a new bending ray tracing method for horizontally layered VTI medium is proposed.Because the ray parameter is an invariant of a ray path,for a given velocity model and source-receiver geometry,a ray path can be uniquely determined once the value of its ray parameter is found.The convention of the generic Thomsen anisotropy parameters is adopted to describe the strength of elasticity anisotropy.Theoretically speaking,the ray parameter can be obtained by solving the nonlinear offset equation using a Newton's iterative approach.However,the ray parameter and the group velocities of P,SV and SH waves in an VTI medium have very complex relationship,making it difficult to directly apply the Newton's approach to solve the offset equation.Based on the analysis of the characteristics of the formulations of group velocities in a VTI medium,it is found that the nonlinear offset equation can be simplified by using trigonometric polynomial functions to approximate the relations between the ray parameter and group velocities.With the approximated formulas,the Newton's approach can be applied to calculate the ray parameter.Numerical tests show that Newton's approach works well at near and middle offsets but becomes numerically instable at far offsets.The numerical instability at far offsets can be resolved by replacing the original ray parameter with a modified ray parameter,which is a transformation of the ray parameter.The proposed two-point ray tracing method is applicable to ray tracing of direct P,direct SH waves and SV waves without triplications in arbitrary VTI media.Generally,an accurate value of the ray parameter can be obtained within 4-6 iterations.Compared with the existing methods,the proposed new method has the advantages of high computational efficiency and accuracy as well as robustness,moreover,it is very easy to implement.