The efficiency of ray tracing in a seismic model can be evaluated in terms of the ``average Lyapunov exponent'' for the model. The ``average Lyapunov exponent'' may be approximated by the square root of the corresponding Sobolev norm of the functions describing the model. We may thus construct models optimally smooth for ray tracing by minimizing the Sobolev norm during the inversion of seismic data.
The SEG/EAGE 3-D Salt Model is given in the form of 3-D velocity cube and several triangulated interfaces. A smooth version of the model parameterized by tricubic B--splines is obtained by velocity inversion with minimization of the Sobolev norm. A version containing B-spline--parameterized interfaces is obtained by including the data for the interfaces into the inversion.
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Bulant, P.: Sobolev scalar products in the construction of velocity models - application to model Hess and to SEG/EAGE Salt Model. Pure and appl. Geophys., 159 (2002), 1487-1506.
Bulant, P.: Sobolev scalar products in the construction of velocity models - application to model Hess, to SEG/EAGE Salt Model, and to model Pluto 1.5. In: Seismic Waves in Complex 3-D Structures, Report 11, pp. 133-159, Dep. Geophys., Charles Univ., Prague, 2001.