Wave-Equation Reflection Tomography: Annihilators and Sensitivity Kernals

Abstract

In seismic tomography, the finite frequency content of broad-band data leads to interference effects in the process of medium reconstruction, which are ignored in traditional ray theoretical implementations. Various ways of looking at these effects in the framework of transmission tomography can be found in the literature. Here,we consider inverse scattering of bodywaves to develop a method of wave-equation reflection tomography with broad-band waveform data— which in exploration seismics is identified as a method of wave-equation migration velocity analysis. In the transition from transmission to reflection tomography the usual cross correlation between modelled and observed waveforms of a particular phase arrival is replaced by the action of operators (annihilators) to the observed broad-bandwave fields. Using the generalized screen expansion for one-way wave propagation, we develop the Fréchet (or sensitivity) kernel, and show how it can be evaluated with an adjoint state method. We cast the reflection tomography into an optimization procedure; the kernel appears in the gradient of this procedure.We include a numerical example of evaluating the kernel in a modified Marmousi model, which illustrates the complex dependency of the kernel on frequency band and, hence, scale. In heterogeneous media the kernels reflect proper wave dynamics and do not reveal a self-similar dependence on frequency: low-frequency wave components sample preferentially the smoother parts of the model, whereas the high-frequency data are—as expected—more sensitive to the stronger heterogeneity.We develop the concept for acoustic waves but there are no inherent limitations for the extension to the fully elastic case.