Wave propagation in laterally heterogeneous layered media

Abstract

This paper derives a solution of the seismic response of an acoustic medium with 2-D geometry using Kirchhoff's representation of the wavefield. The medium can be composed of several layers, separated by curved interfaces.The difference between the approach presented here and other Kirchhoff-type solutions of the multilayer case is that we take care of all diffractive effects during wave propagation, including those during transmission from source to reflector and from reflector to receiver.A crucial point in this method is the replacement of the Kirchhoff-Helmholtz formula, that requires both the knowledge of the wavefield and its directional derivative at the medium's interfaces, by the Rayleigh-Sommerfeld representation where only the wavefield must be known o r approximated. The propagation of acoustic or elastic waves through a stack of layers with curved interfaces can then be described by a sequence of plane-wave compositions or plane-wave decompositions. This point of view implies a reinterpretation of Kirchhoff's formula, in terms of plane waves emanating from a reflector, excited by an incident wavefield.The outlined theory provides a convenient formulation for the transmission and reflection of waves in a multilayered laterally inhomogeneous medium with good accuracy as long as the radius of curvature of the boundaries clearly exceeds the dominant wavelength and if the distance between interfaces is larger than a few wavelengths.

Elastic Kirchhoff-Helmholtz Synthetic Seismograms

“…The IBEM results were obtained by adapting the formulation of SÁ NCHEZ-SESMA andCAMPILLO (1991) andPEDERSEN et al (1996) to an explosion in the model geometry shown in Figure 4b. In Figure 5b, we compare the zero-offset 2 Hz wave forms computed by these two methods and by the WKBJ technique used in the previous example (see also WENZEL et al, 1990). It is apparent from Figure 4, that the performance of KH is satisfying near the caustic where the ART is not valid.…”

“…Next, the components (8) are evaluated upon use of the well-known elementary source solutions whenever possible. In more complicated models, these components are expanded into the WKBJ series (SINGH and CHAPMAN, 1988;WENZEL et al, 1990). By putting the source location x% into a complex space, we may also construct an analytical extension of the diverging wave propagating away from this source (FELSEN, 1984;WU, 1985).…”

“…A plane-wave formulation of eq. (10) (WENZEL et al, 1990) leads to the WKBJ technique (SINGH and Vol. 151, 1998 CHAPMAN, 1988) which was applied to simulate the qP and qSV wave propagation due to a point source in such model.…”

“…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.…”