The operator of interest is the one that creates many offsets of seismic data from a zero-offset model space.

- is a white seismic trace (model) at zero-offset
- is a red seismic trace (data) at nonzero-offset
*x*_{j} - is a seismic band pass filter
- sprays along hyperbola
using a known, rough
*v*(*z*) - sprays using a known, smooth

(1) |

Here is a trivial idea:
Estimates of from data
at different offsets *x*_{j}
have different spectral bands because of NMO stretch.
Wide offsets create low frequency.
Trouble is, these low frequencies add little spectral bandwidth.
We want extra high frequencies too.

We know a simple two-step process where one offset can be obtained from another: First moveout for one offset. Then inverse moveout for the other offset. Whenever such offset continuation works, extra offsets cannot bring us extra information. Extra traces give only redundancy.

Inversion theory says if the transformation has no null space we should be able to solve for everything. Since in practice we cannot seem to obtain that extra bandwidth, it seems that the operator has a large null space, about equal in size to the trace length times (the number of offsets minus one).

11/11/1997