Seismic ray fields and ray field maps : theory and algorithms (2003)

The research described in this thesis covers various aspects of\nthe forward calculation of seismic ray fields and ray field maps.\nThe central theme is the solution of problems encountered in\nsmooth but complex media, i.e., media that give rise to wave front\nfolding and associated multi-pathing of rays. The ultimate aim of\nthe presented material is to enhance the efficiency of seismic\ninverse methods, by enhancing the efficiency of the forward\ncalculations. Particular emphasis is placed on the applicability\nof the ray tracing results to seismic inverse methods.\n\nAfter an overview of seismic ray theory in Chapter 2, a novel\napproach to the calculation and representation of ray field maps\nis introduced in Chapter 3. The approach is particularly useful in\ncases where ray field maps are needed for a dense distribution of\nsources at an acquisition surface, as in reflection seismics and\nborehole tomography. For such source distributions it is suggested\nto construct a single ray field map in an extended space of\nspatial coordinates and angles, rather than a number of maps in\nthe spatial domain for a range of acquisition coordinates.\n\nThe ray field map in the position/angle domain is single-valued,\nregardless of the complexity of the medium and the ray field\ninformation is organised by angles at depth rather than by points\nof emergence at the surface, which makes the maps particularly\nsuitable for use in modern seismic imaging methods. An important\nresult is that, in contrast to what is commonly assumed, obtaining\nthis information does not require the tracing of rays up towards\nthe acquisition surface. Instead, existing algorithms that trace\ndownwards can be adapted to work in the position/angle domain,\nleading to a considerable gain in efficiency.\n\nInterpolation is an important tool in both the construction and\nthe application of ray field maps. A new technique for accurate\ninterpolation using derivative information is presented in Chapter\n4. It is a hybrid of extrapolation to arbitrary order and linear\ninterpolation, and combines the advantages of both methods.\nThrough a modification of the coefficients of the Taylor\nexpansion, extrapolations from a number of locations can be\ncombined to obtain a polynomial order of accuracy that is one\nhigher than that of a single conventional Taylor expansion.\n\nIn Chapter 5 a ray field construction and mapping algorithm is\ndeveloped that extends and refines existing wave front\nconstruction methods. For ray field mapping in the spatial domain\ntwo refinements are proposed that enhance the accuracy and the\ncompleteness of the maps. The applicability in the position/angle\ndomain is investigated as well, with the unfortunate conclusion is\nthat ray field construction in its current form is not suitable\nfor that domain, due to the type of deformation in the geometrical\nstructure of the ray field.\n\nA successful algorithm for the calculation of ray field maps in\nthe position/angle domain is developed in Chapter 6. It is based\non the observation that in the position/angle domain the ray field\nmaps are single-valued and that the geometrical spreading is very\nlimited. This implies that the two most important reasons for\ndeveloping wave front construction methods in the spatial domain\nare absent in the position/angle domain. Instead, it is possible\nto use the more primitive - but more efficient - paraxial ray\nmethods.\n\nThe one-to-one mapping between position/angle coordinates and ray\nfield coordinates can be exploited in practical applications.\nCalculations that are typically performed in terms of ray field\ncoordinates can now be performed in terms of position/angle\ncoordinates and the other way around. Appendices A and B show that\nthis may be advantageous in tomography and the and the forward\ncalculation of ray fields maps directly on a grid in the\nposition/angle domain.\n\nFinally, Appendix C presents an algorithm for the calculation of\nray fields in smooth 2-D media, using a pseudo-spectral expansion\nof the wave front. This line of research was abandoned in favour\nof the ray field map methods described above. Nevertheless, it is\npresented in the thesis because its development provided useful\ninsights for the ray field map approach and some of its features\nmay be useful in other applications.