From the Hagedoorn imaging technique to Kirchhoff migration and inversion

The seminal 1954 paper by J.G. Hagedoorn introduced a heuristic for seismic reflector imaging. That heuristic was a construction technique – a 'string construction' or 'ruler and compass' method – for finding reflectors as an envelope of equal traveltime curves defined by events on a seismic trace. Later, Kirchhoff migration was developed. This method is based on an integral representation of the solution of the wave equation. For decades Kirchhoff migration has been one of the most popular methods for imaging seismic data. Parallel with the development of Kirchhoff wave-equation migration has been that of Kirchhoff inversion, which has as its objectives both structural imaging and the recovery of angle-dependent reflection coefficients. The relationship between Kirchhoff migration/inversion and Hagedoorn's constructive technique has only recently been explored. This paper addresses this relationship, presenting the mathematical structure that the Kirchhoff approach adds to Hagedoorn's constructive method and showing the relationship between the two.