A wave-equation-based Kirchhoff operator

In this paper, I will study a Kirchhoff-type integral, which can be seen as a linear operator mapping angle–azimuth-dependent reflection coefficients along a reflector into reflection data for the acoustic wave equation. I will show that a minor adaptation of a construction of angle–azimuth-dependent images as proposed by Sava and Fomel leads to a left inverse of this operator, which maps primary reflection data to angle–azimuth-dependent reflection coefficients. The new construction naturally leads to a reformulation of the Kirchhoff operator, acting on space-shift-extended images, which can be implemented completely in terms of the fundamental solutions of the wave equation. I will study the composition of this new wave-equation-based Kirchhoff operator with an operator forming space-shift-extended images from data. I will show that these operators are partial inverses of each other, with their compositions being pseudo-differential operators that reconstruct suitably microlocalized versions of primary reflection data and extended images focused at space-shift zero. (paper)