Wave-Equation Reflection Tomography: Annihilators and Sensitivity Kernals
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deHoop_Hilst_Shen_2006GJI_ReflTomo.pdf
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Author(s)
van der Hilst, Robert D.
de Hoop, Maarten V.
Shen, Peng
Date Issued
2007
Publisher
Massachusetts Institute of Technology. Earth Resources Laboratory
Series/Report no.
Earth Resources Laboratory Industry Consortia Annual Report;2007-16
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.
Subjects
Tomography
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