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The example presented in this paper does not exemplify the interesting subject of *scale-dependant*
phenomena; seismic wave propogation being one of them. In this case,
the scaling is generally in terms of temporal frequency. Specifically, estimates
of an earth property obtained by inversion of seismic data collected at different
scales will generally be different Mavko et al. (1998). Small-scale measurements (with limited
spatial coverage) of various earth
properties are obtainable from well logs, while larger-scale measurements
with good spatial coverage are provided by surface seismic data. In theory, the physics
governing the scale dependance of the problem is known; I believe that a stronger form
of multiscale regularization can effectively constrain a joint inversion of surface seismic
and well log data.
The quadtree pyramid interpolation method I presented is common sensical, but has some
strong theoretical justifications as well. The operator of equation 1
which conducts the experiment has a very simple form, and a similarly simple nullspace.
The nullspace is simply all linear combinations of the unknown model points. Least squares
regularization constrains the nullspace with a priori assumptions. The quadtree pyramid
is slightly different. Recall that there always exists a bin size such that of
equation 1 is invertible; the model at this scale is unique. Going to the
next finest scale, we can simply fill the holes with the model estimate at the coarsest
scale. In this sense, we fill the nullspace optimally in the sense that the filling is
done with ``known'' data. Analogously, in spline theory, it can be shown Ahlberg et al. (1967) that the best
estimator of the spline coefficients on a fine mesh are the coefficients on a coarser mesh.

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Stanford Exploration Project

9/5/2000