*** Note unusual DATE and TIME ***


                 SPECIAL DIFFERENTIAL GEOMETRY/PDE SEMINAR

                         *** THURSDAY, April 4 ***

                              Padelford C401
                              *** 2:30 pm ***

                                 Dirk Smit
      (Shell Production and Exploration Laboratory, The Netherlands)


       LINEARIZED INVERSION OF ACOUSTIC SEISMIC SIGNALS IN STRONGLY
                            HETEROGENOUS MEDIA


The mathematics of the propagation of seismic energy relevant in seismic
reflection experiments is assumed to be governed by the linear wave
equation, in which the coefficient is the sound speed of the subsurface.
Reflections in the subsurface may occur at sudden changes in the velocity.
To find the position at which reflections occurred one considers first the
so-called forward map that sends the coefficient of the wave equation to
the boundary value of its solution.  This map is highly non-linear.  For
inversion one usually considers the formal derivative of this forward map,
which is the linearized inverse problem.  Results in the past decade have
shown that the linearized forward map corresponds to a Fourier integral
operator (FIO).  The inverse problem requires the computation of the
composition of this FIO with its adjoint.  A further result states that if
the travel time from the source to a receiver specifies a unique reflection
point x, i.e. the travel time is a global function of x, then this
composition is an elliptic pseudo-differential operator of which the
inverse may be computed by standard techniques.

In many practical situations, however, a global travel time function does
not exist.  In this talk an analysis of this more general case will be
presented.  It will be shown that generically the composition of the
forward FIO and its adjoint is not itself an FIO, due to the focusing and
defocusing of seismic energy which in turn is caused by velocity variations
in the subsurface.  However, it is shown that the composed operator may be
written as a sum of a pseudo-differential operator and a remaining
non-local operator which in relevant cases turns out to be smoothing.
Furthermore, conditions are formulated under which the non-local part is
absent and an explicit formula for the inverse of the pseudo-differential
part is presented, which is suitable for numerical evaluation.


For more information about this seminar, view the Differential Geometry
Seminar World-Wide Web page at 

    https://www.math.washington.edu/~lee/Seminars/dg.html


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