Global Analysis of Linearized Inversion for the Acoustic Wave Equation Abstract Global Analysis of Linearized Inversion for the Acoustic Wave Equation

Global Analysis Of Linearized Inversion For The Acoustic Wave Equation by Cli ord J. Nolan 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 re ect when they impinge on a location where the acoustic velocity eld oscillates rapidly (on the scale of a wavelength). When the waves re ect back to the surface, they carry kinematical information about the location of the oscillatory velocity eld. 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 which describes how the operator maps oscillatory velocity elds to oscillatory wave elds 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 that is equivalent to the existence of a linearized inversion operator. Since the L2{adjoint of the scattering operator has inverse canonical relation, geophysicists often apply it to the scattered eld 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 eld can produce artifacts in the resulting map of the subsurface. I demonstrate this e ect numerically. I also prove that the scattering operator is continuous between a certain domain and range space i the geometrical condition on holds. Furthermore, I have shown that it is possible to map an experiment where the geometrical condition fails into another experiment where it holds. Acknowledgments There are many people that I would like to acknowledge for support and help throughout my stay at Rice. My adviser Bill Symes has helped me tremendously by sharing his knowledge and enthusiasm for mathematics. I am very grateful to him for this. My committee members have also helped me over the past years and I thank them for their interest in my work. I dedicate this thesis to my wife Elaine for her love, patience and understanding. To my parents and teachers (especially Joe Tuohy and Diarmuid Cashman), I am grateful for the opportunities they have given me. I also thank Deborah Ausman who helped me edit this thesis. I have been fortunate to meet and make friends with many people at Rice over the years. They too have played no small part in providing support during life as a graduate student. I especially want to mention Michael Pearlman, Paul Uhlig, Fredrik Saaf and Marielba Rojas. I have also bene ted from interaction with the various members of The Rice Inversion Project; M. Kern, R. Versteeg, J. Blanch, A. Sei, K. Araya, S. Minko , H. Song, M. Gockenbach, G. Bao, S. Kim, P. Ecoublet, C. Zhang, J. Qian, L. Santos, S. Liu, H. Tran, and M. Abd El-Mageed. This work was partially supported by the National Science Foundation, the O ce of Naval Research, the Airforce O ce of Scienti c Research, the Schlumberger Foundation, The Rice Inversion Project, TRIP sponsors for 1997 are: Advance Geophysical, Amoco Production Co., Conoco Inc., Cray Research Inc., Discovery Bay, Exxon Production Research Co., Interactive Network Technologies, Mobil Research and Development Corp., Shell International Research.