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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal;page-break-after:avoid'><b><span style='font-size:16.0pt;font-family:"Tahoma","sans-serif"'>MS Thesis Defense<o:p></o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:14.0pt;text-align:center;line-height:normal'><b><span style='font-size:14.0pt;font-family:"Tahoma","sans-serif"'>INVERSE ACOUSTIC SCATTERING SERIES USING THE VOLTERRA RENORMALIZATION OF THE <o:p></o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:14.0pt;text-align:center;line-height:normal'><b><span style='font-size:14.0pt;font-family:"Tahoma","sans-serif"'>LIPPMANN-SCHWINGER EQUATION IN ONE DIMENSION<o:p></o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><o:p> </o:p></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:14.0pt;font-family:"Tahoma","sans-serif"'>Jie Yao<o:p></o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:13.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:12.0pt'>Date: Monday, July 15th, 2013<o:p></o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:13.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:12.0pt'>Location:</span></b><span style='font-size:12.0pt'> MECE Small Conference Room <o:p></o:p></span></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><span style='font-size:12.0pt'>Time: </span></b><span style='font-size:12.0pt'>10:30 am<o:p></o:p></span></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b><o:p> </o:p></b></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;line-height:normal'><b><o:p> </o:p></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b>Committee Chair: <o:p></o:p></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'>Dr. Donald Kouri<o:p></o:p></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;line-height:normal'><b><o:p> </o:p></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'><b>Committee Members: <o:p></o:p></b></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'>Dr. Bernhard Bodmann<o:p></o:p></p><p class=MsoNormal align=center style='margin-bottom:0in;margin-bottom:.0001pt;text-align:center;line-height:normal'>Dr. Fazle Hussain<o:p></o:p></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;line-height:normal'><o:p> </o:p></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;text-indent:.5in;line-height:normal'><span style='font-size:12.0pt'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;line-height:normal'><span style='font-size:12.0pt'>The inverse scattering problem has enormous importance both for practical and theoretical applications. Based on the early work of Jost and Kohn, Moses, Razavy and Prosser, Weglein and co-workers have pioneered inverse scattering series methods that do not require an assumed propagation velocity model. The only limitation of their approach appears to the finite radius of convergence of the Born-Neumann series of the acoustic Lippmann-Schwinger equation. Despite this limitation, Weglein and co-workers have made significant progress using this approach by introducing the idea of "subseries" which are associated with specific inversion tasks. Kouri showed that one could carry a renormalization transform of the Lippmann-Schwinger equation from a Fredholm into a Volterra Equation form. It can be further proved that the Born-Neumann series solution of the Volterra equation converges absolutely, irrespective of the strength of the interaction. Kouri and co-workers formulated the 1-D acoustic scattering series in terms of Volterra kernel with reflection and transmission data. Following this previous work of Kouri, higher orders of the Volterra Inverse Scattering Series (VISS) with reflection and transmission data ($R_k/T_k$) are analyzed. In addition, for the seismic exploration applications, we also extended the VISS approach to the case where only the reflection data is available. The cases of single square barriers or wells and Gauss barriers and wells are studied to illustrate how well the Volterra Inverse Scattering Series performs the inversion.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:0in;margin-bottom:.0001pt;text-indent:.5in;line-height:normal'><span style='font-size:12.0pt'><o:p> </o:p></span></p></div></body></html>