Elastic Wavefield Inversion of Reflection and Transmission Data
, by Peter Mora

Elastic inversion of seismic data can be performed by finding the earth
parameters (P- and S-wave velocities and density) that minimize the
square error between the wavefield computed using this model and
the observed wavefield (Tarantola (1984) and Mora (1986a)).
The resultant model is a maximum probability
solution provided the assumptions implied using least squares,
namely Gaussian probability distributions of the data and model parameters,
are valid. This is not a bad assumption considering the elastic inversion algorithm
assumes the elastic wave equation and therefore
accounts for S-waves, mode conversions, head waves and rayleigh waves etc.
that are normally considered as coherent noise.
As the number of shot profiles used in an inversion
is increased,
the signal to noise ratio increases and there are more illumination
angles of the seismic waves on the subsurface. More illumination angles implies
a more complete picture of the subsurface can be obtained. Even using
a single shot profile, Mora (1986a) showed with some synthetic studies
that a reasonable resolution can be achieved (both spatially and between
the P- and S-wave velocities).
When many shots are used to invert surface seismic data (reflection data),
the result of the inversion is comparable
to the expected result a prestack elastic shot profile depth migration,
namely, a high frequency image of P- and S-wave velocities and density.
However, compared to migration, artifacts are smaller, the P and S-wave
velocity images are better resolved from one another,
the magnitudes of the velocity and density
perturbations have significance in an absolute
sense and there is a slight increase in the lower frequency components of
the image making this result easier to interpret (see also Mora (1986a)).
Synthetic studies show that when signal to noise ratios are high, that
an inversion of just a few shots profiles results in a \fIcomplete\fR
image of the subsurface (whereas conventional processing (stack and migration)
would have required at least an order of magnitude more data (Ronen (1985)
comes to the same conclusion)).
In comparison to the case of inversion of reflection data where only the high
frequency part of the model can be resolved, inversion of transmission data
yields a result containing \fIboth high and low frequencies\fR (and hence
blockiness in the model due to layering and other gross features).
Transmission data (such as offset VSP and/or well to well data)
contains direct P- and S-waves that are most affected by the
low frequency blocky velocity perturbations which cause significant delays
to these direct waves. If multiple shots are used and hence the direct waves
have illuminated the subsurface at many different angles, then there is enough
redundancy to resolve these gross features (this
is comparable to elastic diffraction tomography,
see Devaney (1984) for the case of linearized
acoustic diffraction tomography).
Synthetic studies show that even using offset VSP's
from only two wells located on either side of a
region of interest, it is
possible to obtain an inversion result that is almost identical to
the true solution.