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 overdetermined 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.''
In Equations (1) and (2), the output is . The ``'' means that we minimize the squared L2 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 .