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## Two-stage data infill

We used the two-stage data infill method Claerbout (1996). Our results are similar to Claerbout's; the interpolated data shows a loss of power near the edge of the rectangle (see Figure 1). Claerbout's Ratfor implementation uses two routines, pef and mis, which solve for the filter and the data, respectively. The abstraction of C++ and the HCL framework allows a single routine to be used for both the data and the filter:

```// Solve the missing data problem LMx = -Lk, that is L(Mx + k) = 0
// k contains the known data (and is zero where unknown)
// M is a mask operator which zeroes the known components
// MissSolve can be used for finding missing data or an unknown filter

void MissSolve(
const HCL_Vector       &k,  // known data with zeroes where unknown
HCL_Vector       &x,  // initial guess (should be pre-masked!)
// on exit x contains u, the unknown data
HCL_LinearSolver &S ) // solver used with the normal equations
{
HCL_Vector *b = L.Range().Member();    // Allocate space for b
L.Image( k, *b );                      // Set     b =  Lk
(*b).Neg();                            // Set     b = -Lk
HCL_CompLinearOpAdj A( &M, 0, &L, 0 ); // Set     A =  LM
// Solve LMx = -Lk
// (using the normal equations)
NormalSolve( A, *b, x, S );
delete b;
}
```

The first step of two-stage data infill is to create two RGFs which consist of zeroes and ones. The first has the shape of the data region, but with ones where the data is known. The second has the shape of the prediction-error filter, but with ones where the filter values are modifiable. Internal convolution is then performed with these two RGFs to give an RGF which we call the influence mask. Then we use MissSolve to find the prediction-error filter using a linear operator which is the composition of two operators. Specifically, it consists of icaf with the known data, followed by masking with the influence mask. (Notice that icaf is an operator which takes a filter as the argument).

After finding the filter, we fill in the missing data by again using MissSolve. This time, the operator is tcai with the newly-calculated filter. We employed HCL's conjugate gradient solver to find both the filter and the missing data.

Next: Nonlinear minimization Up: MISSING DATA AND UNKNOWN Previous: MISSING DATA AND UNKNOWN
Stanford Exploration Project
11/11/1997