Kirchhoff scattering series: Insight into the multiple attenuation method

There are two basic integral equations used to represent wavefields in theoretical seismology: the Lippmann-Schwinger integral equation and the representation theorem. The Born scattering series currently used for attenuating free-surface multiples has been derived from the Lippmann-Schwinger integral equation. Similarly, we have used the representation theorem here to derive a Kirchhoff scattering series for attenuating free-surface multiples in towed-streamer data. The Kirchhoff series for attenuating free-surface multiples is, in theory, equivalent to the Born series; most important, like the Born series, it does not require any knowledge of the subsurface. However, it still provides useful insight into the multiple-attenuation methods because the form of some quantities involved in the Kirchhoff series is different from the form in the Born series. For example, in towed-streamer seismic data, the Kirchhoff series requires measurements of the vertical derivative of the pressure field in addition to the pressure field itself, whereas the Born series requires only the pressure field and tries to obtain its vertical derivative by multiplying the pressure field in the f-k domain by the vertical wavenumber (generally known, in the context of multiple attenuation, as the obliquity factor). The existing numerical implementations of the Born scattering multiple attenuation can be easily adapted to the Kirchhoff scattering multiple attenuation. We simply have to replace the computation of the vertical derivative of the pressure field with the recorded one. This is the most important conclusion in this paper for practitioners because they can use the numerically stable implementation of the inverse-scattering multiple-attenuation method in the form of a series when the combined measurements of the pressure field and its vertical derivative are available.