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

Summary. A modified version of the Kirchhoff–Helmholtz integral can be used to synthesize elastic wavefields in media for which velocity is a function of range, x, as well as depth, z. The essence of the method is that rays are traced from both source and receiver to some intermediate surface, σ. The field at the receiver is then given by an integral over σ, whose integrand is a particular product of the values of the source and receiver wavefields. The surface σ 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 σ. When either the source or receiver wavefield has a caustic in σ 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 σ so that caustics of the source rayfield and caustics of the receiver rayfield do not intersect on σ. We refer to 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 of using many surfaces: e.g. for two surfaces, tracing rays from the source to the first surface σ1, then from every point on σ1, to every point on the second surface σ2, then from the receiver to σ2, then integrating over the product manifold σ1σσ2. 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 σ 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 σ. 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 of the EKH procedure is that its underlying theory is very simple.

[1]  A. W. Trorey.  Diffractions for arbitrary source-receiver locations , 1977 .

[2]  Fred Hilterman,et al.  AMPLITUDES OF SEISMIC WAVES—A QUICK LOOK , 1975 .

[3]  Vlastislav Červený,et al.  Ray method in seismology , 1977 .

[4]  Fred Hilterman,et al.  THREE-DIMENSIONAL SEISMIC MODELING+ , 1970 .

[5]  Donald V. Helmberger,et al.  Applications of the Kirchhoff‐Helmholtz integral to problems in seismology , 1983 .

[6]  John R. Berryhill,et al.  DIFFRACTION RESPONSE FOR NONZERO SEPARATION OF SOURCE AND RECEIVER , 1977 .

[7]  K. Fuchs,et al.  Computation of Synthetic Seismograms with the Reflectivity Method and Comparison with Observations , 1971 .

[8]  L. Neil Frazer,et al.  On a generalization of Filon's method and the computation of the oscillatory integrals of seismology , 1984 .

[9]  William S. French,et al.  Computer migration of oblique seismic reflection profiles , 1975 .

[10]  Philip M. Morse,et al.  Introduction to the Theory of Sound Transmission , 1959 .

[11]  L. Frazer.  Two problems in WKBJ theory: the interpolation of sampled velocity profiles and the use of frequency-dependent, complex velocities , 1983 .

[12]  J. Hagedoorn,et al.  A process of seismic reflection interpretation , 1954 .

[13]  Leon Knopoff,et al.  Body Force Equivalents for Seismic Dislocations , 1964 .

[14]  V. P. Maslov,et al.  Theory of perturbations and asymptotic methods , 1972 .

[15]  G. H. F. Gardner,et al.  Elements of migration and velocity analysis , 1974 .

[16]  Mrinal K. Sen,et al.  Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium - II. Computations , 1985 .

[17]  F. Hilterman.  Interpretative lessons from three-dimensional modeling , 1982 .

[18]  C. H. Chapman,et al.  A new method for computing synthetic seismograms , 1978 .

[19]  L. Frazer,et al.  The theory of finite frequency body wave synthetic seismograms in inhomogeneous elastic media , 1980 .

[20]  P. Buchen,et al.  Use of Kirchhoff's formula for body wave calculations in the Earth , 1981 .

[21]  John R. Berryhill,et al.  Wave-equation datuming , 1979 .

[22]  William S. French,et al.  TWO‐DIMENSIONAL AND THREE‐DIMENSIONAL MIGRATION OF MODEL‐EXPERIMENT REFLECTION PROFILES , 1974 .

[23]  P. Richards,et al.  A comparison of synthetic seismograms of core phases generated by the full wave theory and by the reflectivity method , 1980 .

[24]  G. Kirchhoff.  Zur Theorie der Lichtstrahlen , 1883 .

[25]  G. McMechan,et al.  Asymptotic ray theory and synthetic seismograms for laterally varying structures: theory and application to the imperial valley, California , 1980 .

[26]  P. Richards.  Elastic Wave Solutions In Stratified Media , 1971 .

[27]  P. Giese.  Explosion Seismology in Central Europe Data and Results , 1976 .

[28]  J. H. Ansell,et al.  On the decoupling of P and S waves in inhomogeneous elastic media , 1979 .

[29]  V. Červený.  Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method , 1983 .

[30]  L. Frazer,et al.  A method for the computation of finite frequency body wave synthetic seismograms in laterally varying media , 1982 .

[31]  William A. Schneider,et al.  INTEGRAL FORMULATION FOR MIGRATION IN TWO AND THREE DIMENSIONS , 1978 .

[32]  A. W. Trorey.  A SIMPLE THEORY FOR SEISMIC DIFFRACTIONS , 1970 .