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Acoustic and elastic Kirchhoff integrals for primary wave field extrapolation


, : Acoustic and elastic Kirchhoff integrals for primary wave field extrapolation. SEG Abstracts 1988(1): 1066-1069

The conventional Kirchhoff integral is based on the two-way wave equation. It formulates how the acoustic pressure in a point A inside a closed surface S can be calculated when the acoustic wave field is known on S. In this paper we decompose the two-way wave field on S into one-way wave fields. Next, by applying the one-way wave equations on S, the conventional Kirchhoff integral is considerably simplified. The resulting expressions describe forward extrapolation of primary waves through arbitrarily inhomogeneous fluids. The theory is also extended for the full elastic case, yielding expressions for forward extrapolation of primary P- and S-waves through arbitrarily inhomogeneous, anisotropic solids. Surprisingly, these full elastic one-way Kirchhoff integrals have the same simple form as the acoustic one-way Kirchhoff integral. For sampled wave fields, the one-way Kirchhoff integrals (acoustic and full elastic) can be represented as a matrix operator acting on a data vector. This matrix operator and its inverse play an important role in a new full elastic seismic processing scheme as proposed by Berkhout and Wapenaar (1988b).

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