Seismic ray fields and ray field maps : theory and algorithms

Abstract

The research described in this thesis covers various aspects of the forward calculation of seismic ray fields and ray field maps. The central theme is the solution of problems encountered in smooth but complex media, i.e., media that give rise to wave front folding and associated multi-pathing of rays. The ultimate aim of the presented material is to enhance the efficiency of seismic inverse methods, by enhancing the efficiency of the forward calculations. Particular emphasis is placed on the applicability of the ray tracing results to seismic inverse methods.

After an overview of seismic ray theory in Chapter 2, a novel approach to the calculation and representation of ray field maps is introduced in Chapter 3. The approach is particularly useful in cases where ray field maps are needed for a dense distribution of sources at an acquisition surface, as in reflection seismics and borehole tomography. For such source distributions it is suggested to construct a single ray field map in an extended space of spatial coordinates and angles, rather than a number of maps in the spatial domain for a range of acquisition coordinates.

The ray field map in the position/angle domain is single-valued, regardless of the complexity of the medium and the ray field information is organised by angles at depth rather than by points of emergence at the surface, which makes the maps particularly suitable for use in modern seismic imaging methods. An important result is that, in contrast to what is commonly assumed, obtaining this information does not require the tracing of rays up towards the acquisition surface. Instead, existing algorithms that trace downwards can be adapted to work in the position/angle domain, leading to a considerable gain in efficiency.

Interpolation is an important tool in both the construction and the application of ray field maps. A new technique for accurate interpolation using derivative information is presented in Chapter 4. It is a hybrid of extrapolation to arbitrary order and linear interpolation, and combines the advantages of both methods. Through a modification of the coefficients of the Taylor expansion, extrapolations from a number of locations can be combined to obtain a polynomial order of accuracy that is one higher than that of a single conventional Taylor expansion.

In Chapter 5 a ray field construction and mapping algorithm is developed that extends and refines existing wave front construction methods. For ray field mapping in the spatial domain two refinements are proposed that enhance the accuracy and the completeness of the maps. The applicability in the position/angle domain is investigated as well, with the unfortunate conclusion is that ray field construction in its current form is not suitable for that domain, due to the type of deformation in the geometrical structure of the ray field.

A successful algorithm for the calculation of ray field maps in the position/angle domain is developed in Chapter 6. It is based on the observation that in the position/angle domain the ray field maps are single-valued and that the geometrical spreading is very limited. This implies that the two most important reasons for developing wave front construction methods in the spatial domain are absent in the position/angle domain. Instead, it is possible to use the more primitive - but more efficient - paraxial ray methods.

The one-to-one mapping between position/angle coordinates and ray field coordinates can be exploited in practical applications. Calculations that are typically performed in terms of ray field coordinates can now be performed in terms of position/angle coordinates and the other way around. Appendices A and B show that this may be advantageous in tomography and the and the forward calculation of ray fields maps directly on a grid in the position/angle domain.

Finally, Appendix C presents an algorithm for the calculation of ray fields in smooth 2-D media, using a pseudo-spectral expansion of the wave front. This line of research was abandoned in favour of the ray field map methods described above. Nevertheless, it is presented in the thesis because its development provided useful insights for the ray field map approach and some of its features may be useful in other applications.