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SUMMARY:A wave equation based Kirchhoff operator and its inverse - ten Kro
 ode\, F (Shell International Exploration and Production)
DTSTART:20111215T153000Z
DTEND:20111215T160000Z
UID:TALK34979@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:In seismic imaging one tries to compute an image of the singul
 arities in the earth's subsurface from seismic data. Seismic data sets use
 d in the exploration for oil and gas usually consist of a collection of so
 urces and receivers\, which are both positioned at the surface of the eart
 h. Since each receiver records a time series\, the ideal seismic data set 
 is five dimensional: sources and receivers both have two spatial coordinat
 es and these four spatial coordinates are complemented by one time variabl
 e. \n\n Singularities in the earth give rise to scattering of incident wav
 es. The most common situation is that of re\nflection against an interface
  of discontinuity. Refl\nected and incoming waves are related via refl\nec
 tion coefficients\, which depend in general on two angles\, namely the ang
 le of incidence and the azimuth angle. Re\nflection coefficients are there
 fore also dependent on five variables\, namely three location variables an
 d two angles.\n\n The classical Kirchhoff integral can be seen as an opera
 tor mapping these angle-azimuth dependent refl\nection coefficients to sin
 gly scattered data generated and recorded at the surface. It essentially d
 epends on asymptotic quantities which can be computed via ray tracing. For
  a known velocity model\, seismic imaging comes down to nding a left inve
 rse of the Kirchhoff operator.\n\nIn this talk I will construct such a lef
 t inverse explicitly. The construction uses the well known concepts of sub
 surface offset and subsurface angle gathers and is completely implementabl
 e in a wave equation framework. Being able to perform such true amplitude 
 imaging in a wave equation based setting has signifficant advantages in tr
 uly complex geologies\, where an asymptotic approximation to the wave equa
 tion does not suffice. The construction also naturally leads to a reformul
 ation of the classical Kirchhoff operator into a wave equation based varia
 nt\, which can be used e.g. for wave equation based least squares migratio
 n. Finally\, I will discuss invertibility of the new Kirchhoff operator\, 
 i.e. I will construct a right inverse as well.\n
LOCATION:Seminar Room 1\, Newton Institute
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