The Kirchhoff–helmholtz Integral Pair

Abstract

The Kirchhoff-Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff-Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff-Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff-Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff-Helmholtz integral.

Seismic velocity estimation and time to depth conversion of time‐migrated images

“…The Kirchhoff migration operator can be derived from asymptotic inversion of the Born scattering approximation (Miller et al, 1987;, from inversion of the Kirchhoff-Helmholtz integral (Tygel et al, 2001), or from geometrical considerations (Tygel et al, 1996). It order to implement operator (1), it is necessary to define the background velocity model for computing the traveltime and amplitude functions.…”

Modelamentos de Kirchhoff/Born para propagação de ondas

“…Talvez a aproximação mais usada em sísmica seja a Teoria dos Raios (veja, por exemplo, Cerven:Y, 1995), a qual fornece uma descrição de propagação de ondas em alta-frequência. Problemas mais gerais usam a representação integral do campo de onda (Frazer & Sen, 1985;Langenberg, 1986;Tygel et al, 1994). Através de certas hipóteses e simplificações (como, por exemplo, pequenos contrastes nos parâmetros, alta frequência, etc.)…”