This pinch-out model was created by taking planar reflection coefficients and gradually increasing dip with depth. The reflection coefficients were then convolved with the same wavelet. This creates the numerous pinch-outs seen in Figure a. This model should cause inaccuracies in the flattening method because I assume dip is not a function of time. Surprisingly, both methods in equations (5) and (7) are able to remove most of the deformation in one iteration. The results of one iteration of equation (7) is shown in figure b. To implement this iteratively, it is necessary to use equation (9) to apply weights where gaps were created by the previous flattening iteration.
Since the structure is changing with depth, one iteration will not necessarily provide sufficient information for flattening the data. This is mainly because I am assuming that dip is not a function of time by integrating on only time-slices. In this case, the output of the flattening method will need to be flattened again, possibly several times. Repeated application will also improve deficiencies in the flattening caused by inaccuracies in the dip estimation.
Figure shows the results of five iterations of the flattening method. The image shrinks with each subsequent iteration because the weight is growing to account for the smoothing of dips. This can be rectified by adding a regularization term to the fitting goal in equation (9) to fill empty spaces. Notice that after five iterations, the horizontal view is almost all white indicating that it is flat.