The irregular picked seismic horizons of Figures
1 and 2 are not
defined at many *x* and *y* locations.
So to create , we must fill all the interior ``holes'' in the irregular picked seismic
horizons and also extrapolate them to the edges of the grid.

A human attempting to fill and extrapolate the data by hand would first discern, then manually extend, its dominant trends into the empty regions. Unfortunately, the human approach is a ``non-linear'' one; tough to reproduce and even tougher to encode into a computer algorithm. However, by computing a 2-D Prediction Error Filter (PEF) from the irregular picked seismic horizon, we encapsulate the spatial spectrum of the known data, and thus systematically extrapolate by imposing this spectrum on the output model. Crawley (1995) discusses a closely related example using sparse side-scan sonar bathymetry data.

First we must contend with a detail:
the data used to estimate a PEF must obey the stationarity assumption.
In other words, the spectrum of the data must
be spatially invariant in order to encapsulate the inverse spectrum
with a single PEF. Though the spatial spectra of the irregular picked
seismic horizons in this example are roughly constant, the stationarity
assumption is commonly violated for real-world problems.
I make use of data ``patching'' to subdivide
the data into smaller regions where the assumption *is*
assumed to hold, and then estimate a PEF from the data contained in
each patch Claerbout (1992). The dashed lines on Figures
1 and 2
delimit the four equal-sized patches I use in this example.

The problem of finding and
is underdetermined,
since we have only 578 known seismic data values, but 1600 model
points. However, the classical least squares solution to the problem
is valid only for *over*determined Strang (1986) systems.
To convert this underdetermined
problem to an overdetermined one, we must constrain the
output model with additional regression equations, a process known as
*regularization*. Normally the regularization operator imposes
a ``minimum-energy,'' or other similarly ``safe'' constraint on the free
model variables,
but adds little or no meaningful statistical information to the problem.
However, by using a PEF as the regularization operator, we impose
a fundamental statistical property of the known data on the
model. In symbols, the problem can be stated
through the following least squares ``fitting goals.''

(1) | ||

(2) |

In Equations (1) and (2), the output
is . The ``'' means that we minimize
the squared *L _{2}* norm of the residual.
Equation (1) forces to
match the irregular picked seismic horizon , where it is known.
is known-data ``selector'' operator, which effectively ignores the
difference wherever does not exist.
is a
so-called ``damping factor,'' which weights the effective strength
of the regularization equations (2) relative to the
``data-matching'' equations (1). The operator
is convolution with the patch-variant PEF. The problem is solved iteratively,
using a conjugate direction-type (CD) algorithm.

The result is shown in Figure 3. Now that we have the surfaces and .

Figure 3

7/5/1998