Next: Constrained solution with weights Up: Lomask: Improved flattening with Previous: Introduction

# Methodology

Constraints were first applied to the flattening problem in Lomask and Guitton (2006). In the 3D Gauss-Newton flattening approach, data is flattened by iterating over the following equations iterate {
 (1) (2) (3)
} ,
where the subscript k denotes the iteration number, is the estimated time-shift found in the least-squares sense so that its gradient matches the dip , and is an adjustable regularization controlling the amount of conformity in the time (or depth) dimension. is a 3D gradient operator with an parameter. The residual update in equation (1) can be rewritten as
 (4)

I wish to add a model mask to prevent changes to specific areas of an initial field. This initial field can be picks from any source. In general, they may come from a manually picked horizon or group of horizons. These initial constraints do not have to be continuous surfaces but instead could be isolated picks, such as well-to-seismic ties. To apply the mask I follow a similar development to Claerbout (1999) as
 (5) (6) (7) (8) (9)
The resulting equations are now iterate {
 (10) (11) (12)
} .

Because the mask in equation (11) is non-stationary, I solve it using preconditioned conjugate gradients. The preconditioner efficiently exploits Discrete Cosine Transforms(DCTs) to invert a Laplacian operator Lomask and Fomel (2006) as:
 (13)
where is the 3D discrete cosine transform and
 (14)
J is the real component of the Z-transform of the 3D finite difference approximation to the Laplacian with adjustable regularization.

Following an approach for weighted 2D phase unwrapping by Ghiglia and Romero (1994), I define and so that equation (11) becomes
 (15)
For each non-linear Gauss-Newton iteration, the least-squares solution to equation (11) is solved by iterating over the following preconditioned conjugate gradients computation template:


iterate {
solve   using equation (13)
subtract off reference trace
if first iteration {

} else{

}

}

Here, k is the iteration number starting at k=0. It is necessary to subtract off the reference trace at each iteration because the Fourier based solution using equation (13) has no non-stationary information such as the location of the reference trace. Subtracting the reference trace removes a zero frequency shift. Nonetheless, in practice equation (13) makes an adequate preconditioner because is often close to .

Convergence can be determined using the same criterion described in Lomask et al. (2006). Alternatively, an adequate stopping criterion would be to consider only the norm of the residual . This is essentially the average of the divergence of the remaining dips, i.e. ,
 (16)
where is the size of the data cube.