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

Abstract

In a medium consisting of elastic layers with irregular interfaces, Kirchhoff-Helmholtz (KH) theory can be extended to synthesize the motion due to various generalized rays. An exact elastic form of the KH integral is first derived, then various asymptotic approximations are used to convert this integral into one which can be rapidly evaluated to give the motion of a single generalized ray. The approximations used are those of geometrical optics, for propagation across layers, and the Kirchhoff or tangent-plane approximation for propagation across boundaries. It is shown how the KH method leads naturally to a generalization of our usual notion of elastic reflection and transmission coefficients. The new coefficients are functions of both angle of incidence and angle of reflection or transmission and they are derived so as to obtain coordinate-free formulae that show clearly their relation to the conventional Snell's law coefficients. The elastic KH method is applied first to the problem of a single interface, where its performance is compared to that of the Gaussian beam and Maslov methods. (For synthesizing reflections from irregular interfaces the KH method is superior because it includes signals diffracted from corners. However, when the interface is very smooth on the scale of a wavelength the Maslov and Gaussian beam methods are superior because they do not break down when there is a caustic on the reflector.) KH theory is then applied to a multilayered elastic medium and it is shown how the effects of frequency-dependent attenuation and dispersion can be incorporated into the theory by taking advantage of the approximately logarithmic variation of slowness with frequency in most earth materials. The limitations of the KH theory are discussed and some recent attempts to overcome these difficulties are reviewed. A new method for overcoming the problem of a caustic on the reflector becomes apparent when the KH integral is regarded as a member of a larger family of equivalent 1-fold integrals all of which are derivable from the same multifold path integral. Refracted or diving rays can be treated within the same formalism with equal benefit. For velocity models that are independent of one spatial direction (strike) a method is given for approximately converting 2-D results into 3-D results.

Approximate 3D Body-Wave Synthetics for Tomographic Models

“…Some recent applications involve 2D deep-earth models (Wen and Helmberger, 1998), and mapping abrupt jumps in crustal thickness (Zhu and Helmberger, 1998). Although there are many possible modifications (Frazer and Sen, 1985), the approach followed here is commonly referred to as the KirchhoffHelmholtz integral equation (Born and Wolf, 1964), with p the potential at position r.…”

Elastic Kirchhoff-Helmholtz Synthetic Seismograms

“…FRAZER and SEN, 1985;TYGEL et al, 1994). In most KH techniques, the kernel of the KH integral is constructed upon use of the Asymptotic Ray Theory (ART) (Č ERVENÝ and RAVINDRA, 1971;Č ERVENÝ et al, 1977).…”

“…Their extension was proposed in the Method of the Interference Integral (MII) (KRAVTSOV and ORLOV, 1993). Another approach is based upon the additional decomposition of the integrand to derive a multi-folded extension of the KH theory (BOSTRÖ M, 1980;KENNETT, 1984;FRAZER and SEN, 1985;WENZEL et al, 1990).…”

General representations for wavefield modeling and inversion in geophysics

“…which are the well-known Kirchhoff-Helmholtz integrals ͑Morse and Feshbach, 1953;Berkhout, 1985͒, with many applications in seismic modeling ͑Frazer and Sen, 1985;Hill and Wuenschel, 1985;Wenzel et al, 1990;Druzhinin et al, 1998͒. Equation 63 is the generalization of the Kirchhoff-Helmholtz integral for any of the wave phenomena considered in this paper.…”