This study is concerned with the computation of seismic first-arrival traveltimes in anisotropic media using finite difference eikonal methods. For this purpose, different numerical schemes that directly solve the eikonal equation are implemented and assessed numerically. Subsequently, they are used for pre-stack depth migration on synthetic and field data.
The thesis starts with a detailed examination of different finite difference methods that have gained popularity in scientific literature for computing seismic traveltimes in isotropic media. The most appropriate for an extension towards anisotropic media are found to be the so-called Fast Marching/Sweeping methods. Both schemes rely on different iteration strategies, but incorporate the same upwind finite difference Godunov schemes that are implemented up to the second order. As a result, the derived methods exhibit high numerical accuracy and perform robustly even in highly contrasted velocity models.
Subsequently, the methods are adapted for transversely isotropic media with vertical (VTI) and tilted (TTI) symmetry axes, respectively. Therefore, two different formulations for approximating the anisotropic phase velocities are tested, which are the weakly-anisotropic and the pseudo-acoustic approximation. As expected, the pseudo-acoustic formulation shows superior accuracy especially for strongly anisotropic media. Moreover, it turns out that the tested eikonal schemes are generally more accurate than anisotropic ray tracing approaches, since they do not require an approximation of the group velocity.
Numerical experiments are carried out on homogeneous models with varying strengths of anisotropy and the industrial BP 2007 benchmark model. They show that the computed eikonal traveltimes are in good agreement with independent results from finite difference modelling of the isotropic and anisotropic elastic wave equations, and traveltimes estimated by ray-based wavefront construction, respectively. The computational performance of the TI eikonal schemes is largely increased compared to their original isotropic implementations, which is due to the algebraic complexity of the anisotropic phase velocity formulations. At this point, the Fast Marching Method is found to be more efficient on models containing up to 50 million grid points. For larger models, the anisotropic Fast Sweeping implementation gradually becomes advantageous. Here, both techniques perform independently well of the structural complexity of the underlying velocity model.
The final step of this thesis is the application of the developed eikonal schemes in pre-stack depth migration. A synthetic experiment over a VTI/TTI layer-cake model demonstrates that the traveltime computation leads to accurate imaging results including a tilted, strongly anisotropic shale layer. The experiment shows further that the estimation of anisotropic velocity models solely from surface reflection data is highly ambiguous. In a second example, the eikonal solvers are applied for depth imaging of two-dimensional field data that were acquired for geothermal exploration in southern Tuscany, Italy. The developed methods also produce clear imaging results in this setting, which illustrates their general applicability for pre-stack depth imaging, particularly in challenging environments
