- physics error: seismic waves are not small amplitude pressure waves in a fluid;
- linearization error: neglect of multiple reflections and other nonlinear effects;
- deconvolution error: complete removal of the source signature is not possible;
- asymptotic error: the convolutional model becomes more accurate as
the frequency content of
*r*(*t*,_{0}*x*) moves away from zero Hz.

The following computations will introduce yet more sources of asymptotic error - and, with one exception, only asymptotic error. Therefore I will identify asymptotic error explicitly, and treat other types of modeling error as data noise. It is possible to estimate every asymptotic error explicitly, but experience suggests that these explicit estimates are not particularly useful. So instead I will use the symbol ``'' to suggest proportionality of the asymptotic error to a dominant wavelength in the data. Thus

The single important lesson to learn from the explicit error estimates
of geometric optics is that they are uniform over -bounded
sets of coefficients (meaning in this case the velocity *v*). Therefore
the velocities appearing in the sequel are restricted to vary over such a
-bounded set. A byproduct of the analysis will suggest
explicit finite dimensional subspaces of smooth functions in which
it is advantageous to seek *v*.

4/20/1999