Global analysis of linearized inversion for the acoustic wave equation
Nolan, Clifford Joseph
Symes, William W.
Doctor of Philosophy
To predict the location of natural resources and reduce the cost of exploration, geophysicists rely on various techniques to map the internal structure of the earth. One common mapping method probes the earth's interior using an acoustic energy source (sound waves). The acoustic waves reflect when they impinge on a location where the acoustic velocity field oscillates rapidly (on the scale of a wavelength). When the waves reflect back to the surface, they carry kinematical information about the location of the oscillatory velocity field. A linearized wave equation models the scattering process and its solution operator is a Fourier integral operator. As such, the scattering operator has a canonical relation $\Lambda$ which describes how the operator maps oscillatory velocity fields to oscillatory wave fields at the surface. The goal of linearized inversion is to obtain an inverse operator (with inverse canonical relation) for the scattering operator. We give a geometrical condition on $\Lambda$ that is equivalent to the existence of a linearized inversion operator. Since the $L\sp2$-adjoint of the scattering operator has inverse canonical relation, geophysicists often apply it to the scattered field to obtain a map of the subsurface. I analyze the scattering operator using high-frequency asymptotics and show that if the geometrical condition fails, the scattering canonical relation is not injective. Therefore, application of the adjoint operator to the scattered wave field can produce artifacts in the resulting map of the subsurface. I demonstrate this effect numerically. I also prove that the scattering operator is continuous between a certain domain and range space iff the geometrical condition on $\Lambda$ holds. Furthermore, I have shown that it is possible to map an experiment where the geometrical condition fails into another experiment where it holds.
Geodesy; Geophysics; Mathematics; Acoustics