A Kirchhoff method for the computation of finite-frequency body wave synthetic seismograms in laterally inhomogeneous media

Abstract

A modified version of the Kirchhoff-Helmholtz integral can be used t o synthesize elastic wavefields in niedia for which velocity is a function of range, x, aswell as depth, z. The essence of the method is that rays are traced from both source and receiver to some intermediate surface, S . The field at the receiver is then given by an integral over C, whose integrand is a particular product of the values of the source and receiver wavefields. The surface 2: is not a reflector since the medium is continuous across it. Geometrical ray theory (GRT) is used to calculate the source and receiver wavefields on 2 . When either the source or receiver wavefield has a caustic in C then the GRT amplitude is infinite and, in theory, the method breaks down. However, numerical breakdown can be avoided by parameterizing the GRT amplitudes so that their singularities are integrable and choosing C so that caustics of the source rayfield and caustics of the receiver rayfield d o not intersect on C. We refer t o this alternative as the extended Kirchhoff-Helmholtz (EKH) method. For reasons of economy EKH may be a practical alternative to the more theoretically correct procedure o f using many surfaces: e.g. for two surfaces, tracing rays fr-om the source t o the first surface C, , then from every point on C, , to every point on the second surface C,, then from the receiver to C,, then integrating over the product manifold C, x C 2 .In this paper we give examples of the errors that arise when caustics on C are treated as integrable singularities. First the EKH method is compared with the WKBJ method for a stratified medium, then the EKH method is compared with the ordinary Kirchhoff-Helmholtz method where C intersects no caustics. Errors in the EKH method take the form of small spurious phases which generally arrive later in time than correct arrivals. The arrival times of these error phases can be changed by adjusting C. For some velocity models these phases can be eliminated completely.The EKH method is not as fast as the Maslov (extended WKBJ) method because of the amount of ray tracing needed. However, one of the attractive features o f the EKH procedure is that its underlying theory is very simple.

Kirchhoff?Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium ? I. Theory

“…The nature of the breakdown is discussed in Section 4.4. Numerical examples will be given in Sen & Frazer (in preparation), and have already been given for a similar situation by Frazer & Sinton (1984).…”

“…Scott & Helmberger (1983) used the KH integral to model body wave reflections from mountclin topography and spdl from nuclear blasts. Application of the method to the synthesis of finite frequency body wave synthetic seismograms in media with laterally inhomogeneous but continuous velocity was made by Sinton & Frazer (1981), Haddon (1982), Zherniak (1983), and Frazer & Sinton (1984).…”

Causality, caustics, and the structure of transient wave fields

“…These wave-field expansions may be obtained using several techniques (Berry and Upstill, 1980). These include the Maslov ( 1965Maslov ( , 1972 phase space construction (see also Kravtsov, 1968;Gorman et al, 1981;Chapman and Drummond, 1982), the Kirchhoff integral method (Born and Wolf, 1975;Frazer and Sinton, 1984), the Gaussian beam expansion technique (Cerveny et al, 1982;Madariaga, 1984), and the use of Feynman path integrals (Feynman and Hibbs, 1965;Berry and Mount, 1972). We will return to this point in the final section.…”

Improved applicability of ray tracing in seismic acquisition, imaging, and interpretation

“…The two-way seismic response for a given source point, receiver point, and recording time can then be obtained by approximate Kirchhoff modeling, which may be formulated as a wave-type specific integral ͑Hilterman, 1975; Trorey, 1977;Frazer and Sinton, 1984;Frazer and Sen, 1985;Eaton and Clarke, 2000;Schleicher et al, 2001͒. The numerical integration can be implemented in various ways.…”