Error, Error Everywhere!

As an approximate predictor of seismic traces, the convolutional model exhibits several types of error:

• 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(t₀,x) moves away from zero Hz.

The practical meaning of asymptotic error is that the convolutional model predicts the higher frequency components of the data more accurately, so that the prediction error can be reduced by more aggressive low-cut filtering. Of course this discarding of low-frequency data is only possible to a limited extent as actual data is bandlimited.

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 ``$O(\lambda)$'' to suggest proportionality of the asymptotic error to a dominant wavelength in the data. Thus

$$F[v]r(t,x)=a(t,x)r(T_0(t,x),x) + O(\lambda)$$

The single important lesson to learn from the explicit error estimates of geometric optics is that they are uniform over $C^{\infty}$-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 $C^{\infty}$-bounded set. A byproduct of the analysis will suggest explicit finite dimensional subspaces of smooth functions in which it is advantageous to seek v.