Plane waves in three dimensions

GRADIENT ALONG THE BEDDING PLANE

The LOMOPLAN (LOcal MOnoPLane ANnihilator) filter in three dimensions is a deconvolution filter that takes a volume in and produces two volumes out. The $x$ -output volume results from a first order prediction-error filter on the $x$ -axis, and the $y$ -output volume is likewise on the $y$ -axis.

Although I conceived of 2-D LOMOPLAN as the ``ultimate'' optimization criterion for inversion applications in reflection seismology of sedimentary sections, it turned out that it was more useful in data interpretation and in data-quality inspection. In this study, I sought to evaluate usefulness with three-dimensional data such as 3-D stacks or migrated volumes, or 2-D prestack data.

In experimenting with 3-D LOMOPLAN, I came upon a conceptual oversimplification, which although it is not precisely correct, gives a suitable feeling of the meaning of the operator. Imagine that the earth was flat horizontal layers, except for occasional faults. Then, to find the faults you might invoke the horizontal gradient of the 3-D continuum of data. The horizontal components of gradient vanish except at a fault, where their relative magnitudes tell you the orientation of the fault. Instead of using the gradient vector, you could use prediction-error filters of first order (two components) along $x$ and $y$ directions. 3-D LOMOPLAN is like this, but the flat horizontal bedding may be dipping or curved. No output is produced (ideally) except at faults. The 3-D LOMOPLAN is like the gradient along the plane of the bedding. It is nonzero where the bedding has an intrinsic change.

LOMOPLAN flags the bedding where there is an intrinsic change.

Plane waves in three dimensions