Having determined a 2-D filter, say
on the original data **mesh**,
we can now **interlace** both *t* and *x*
and expect to use the identical filter.
This is because slopes are preserved if we
replace by .Everyone knows how to interpolate data on the time mesh,
so that leaves the job of interpolation on the space mesh:
in (6)
the known (*a*,*b*,*c*,*d*,*e*) can multiply a known trace, and then the ``1''
can multiply the interlaced and unknown trace.
It is then easy to minimize the power out
by *defining* the missing trace to be
the negative of that predicted by the filter
(*a*,*b*,*c*,*d*,*e*) on the known trace.
(The spatial interpolation problem seems to be solved
regardless of the amount of the signal shift.
A ``spatial aliasing'' issue does not seem to arise.)
It is nice to think of the unknowns being under the ``1''
and the knowns being under the (*a*,*b*,*c*,*d*,*e*),
but the CG method has no trouble working backwards too.

After I became accustomed to using the CG method, I stopped thinking that the unknown data is that which is predicted, and instead began to think that the unknown data is that which minimizes the power out of the prediction filter. I ignored the question of which data values are known and which are unknown. This thinking enables a reformulation of the problem, so that interpolation on the time axis is an unnecessary step. This is the way all my programs work. Think of the filter that follows as applied on the original coarse-mesh data:

(7) |

Figure 17

10/21/1998