A prospect for super resolution

FITTING FRAMEWORK

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

$\mathbf z$

is a white seismic trace (model) at zero-offset

$\mathbf d_j$

is a red seismic trace (data) at nonzero-offset $x_j$

$\mathbf L$

is a seismic band pass filter

$\mathbf H_j$

sprays along hyperbola using a known, rough $v(z)$

$\overline{\mathbf H}_j$

sprays using a known, smooth $\overline{v}(z)$

The operator of interest is the one that transforms $\mathbf z$ to all the data $\mathbf d_j$ at all of the offsets $x_j$.

$$\mathbf d_j \quad =\quad \mathbf L \mathbf H_j \mathbf z$$ (1)

Here is a trivial idea: Estimates $\hat{\mathbf z}_j$ of $\mathbf z$ from data $\mathbf d_j$ 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 $\mathbf L \mathbf H_j$ has a large null space, about equal in size to the trace length times (the number of offsets minus one).

A prospect for super resolution