The resolution function in linearized born and Kirchhoff inversion

Abstract

We consider the problem of constructing the depth image of a heterogeneous model (f.ex. the subsurface) based on surface measurements. Due to a limited-aperture and band limited signals, the output from a general inversion scheme will be a distorted or blurred image. If the transmitter/receiver layout, the signal bandwidth and the (initial) background model are all known, this distortion can be computed employing the concept of resolution functions. For a general heterogeneous model this function will be space-variant. Hence, knowing the resolution function such image distortions can in principle be corrected for.

In this paper we discuss two different classes of inversion methods: the Kirchhoff technique which is in favour of reflecting boundaries, and the Born-diffraction technique which relates equally well to a discontinous model (weak contrast). We show that for both classes of methods the resolution function can be expressed by the same integral over the model's Fourier space. The key parameter is the socalled resolution vector (or scattering wavenumber), which at a particular image point is defined by the incident and scattered ray directions. For a model in favour of reflecting boundaries only scattering ray directions corresponding to Snell's law contribute asymptotically. For such a model we introduce a modified resolution function which we denote the reflector spread function. This function gives information about the resolving power of the inversion algorithm with respect to two nearby reflectors.