Sensitivity Analysis of Wave-equation Tomography: A Multi-scale Approach |
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Authors: | Valeriy Brytik Maarten V de Hoop Mikko Salo |
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Institution: | (1) Eaton–Peabody Laboratory of Auditory Physiology, Massachusetts Eye and Ear Infirmary, Boston, MA 02114, USA;(2) Department of Otology and Laryngology, Harvard Medical School, Boston, MA 02115, USA;(3) Department of Physics, Purdue University, West Lafayette, IN 47907, USA;(4) Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093, USA;(5) National Center for Physical Acoustics, University of Mississippi, University, MI 38677, USA |
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Abstract: | Earthquakes, viewed as passive sources, or controlled sources, like explosions, excite seismic body waves in the earth. One
detects these waves at seismic stations distributed over the earth’s surface. Wave-equation tomography is derived from cross
correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events.
The times corresponding with the maxima of these cross correlations replace the notion of residual travel times used as data
in traditional tomography. Using first-order perturbation, we develop an analysis of the mapping from a wavespeed contrast
(between the “true” and reference models) to these maxima. We develop a construction using curvelets, while establishing a
connection with the geodesic X-ray transform. We then introduce the adjoint mapping, which defines the imaging of wavespeed
variations from “finite-frequency travel time” residuals. The key underlying component is the construction of the Fréchet
derivative of the solution to the seismic Cauchy initial value problem in wavespeed models of limited smoothness. The construction
developed in this paper essentially clarifies how a wavespeed model is probed by the method of wave-equation tomography. |
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