A Tomographic Velocity Inversion for Unstacked Data
, by William S. Harlan and Robert Burridge
Tomography will resolve intermediata interval velocity changes better than
amplitude inversions or root-mean-square methods. Velocity information from
reflections off continuous beds appear over offset, h. information from
diffractions over midpoint, y. Scattering surfaces with high curvature
contain the most velocity information at depth; thus, the data reduction used
should allow arbitrarily shaped scatters.
By performing local three-dimensional slant stacks in the seismic data cube,
over midpoint, offset, and time, onw may recognize segments of wavefronts as
sufficiently coherent for signal. From the arrival angles, midpoints, offsets,
and total travel-times of these segments, one may assign corresponding simple
raypaths. These raypaths make no assumption of the shape of the scattering surface.
Simple raypaths suffice, without later perturbation, for out resolution. The data
cube reduces to a list of raypath parameters, d. Each genuine event produces a
family of raypaths, whose spatial distribution correspond to the accuracy with which
one can estimate exit angles. This family of raypaths preserves resolution infromation
through the non-linear conversion to model coordinates and allows a robust solution
to the following least-squares inversion. We may emphasize diffraction information by
first estimating and removing continuous bed reflections with local two-dimensional
slant stacks.
One defines an eart nodelm gm as a partition of interval slownessness. A transform,
L, which sums this model along raypaths found in the data should reproduce the
travel-times in d. The g best minimizes (Lg - d)^2. We easily find the adjoint of L
and a steepest descent algorithm for the best slowness model.
STANFORD