REFERENCES

Claerbout, J. F., 1976, Fundamentals of geophysical data processing: Blackwell.

Claerbout, J. F., 1992, Information from smiles: Mono-plane-annihilator weighted regression: SEP-73, 409-420.

Claerbout, J. F., 1993, 3-D local-monoplane annihilator: SEP-77, 19-26.

Claerbout, J. F., 1994, Applications of Three-Dimensional Filtering: Stanford Exploration Project.

Haskell, N. L., Nissen, S. E., Lopez, J. A., and Bahorich, M. S., 1995, 3-D seismic coherency and the imaging of sedimentological features: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1532-1534.

Marfurt, K. J., Scheet, R. M., Sharp, J. A., Cain, G. J., and Harper, M. G., 1995, Suppression of the acquisition footprint for seismic sequence attribute mapping: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 949-952.

Nissen, S. E., Haskell, N. L., Lopez, J. A., Donlon, T. J., and Bahorich, M. S., 1995, 3-D seismic coherency techniques applied to the identification and delineation of slump features: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1535-1536.

We want to prove that for a given ${\bf p} = (p_x, p_y, p_z)$ the expression  

$$(p_y \partial_z - p_z \partial_y,\ -p_x \partial_z + p_z \partial_x,\ p_x \partial_y - p_y \partial_x)\ \tilde{g}(x,y,z)$$ (2)

vanishes for all ${\bf x} = (x, y, z)$, if and only if $\tilde{g}$ is a volume of parallel planes, $\tilde{g} = g(p_x x + p_y y + p_z z)$.(${\bf p}$ represents the strike of the parallel planes; g represents a one-dimensional amplitude function).

First, we prove that expression (A-1) vanishes for a volume of parallel, planes. Applying the chain rule to the first component $(p_y \partial_z - p_z \partial_y) g(p_x x + p_y y + p_z z)$yields (p_y g' p_z - p_z g' p_y) = 0. An analogue result for the two additional components of expression (A-1) demonstrates that the entire expression vanishes for a volume $\tilde{g}$ that consists of parallel planes.

Next, we prove that if all three components of expression (A-1) vanish, than the function $\tilde{g}$ is a volume of parallel planes. We set the expression (A-1) to zero and express it as a vector product of ${\bf p}$ and the gradient of the field, $\nabla g$:

$${\bf 0} = \left(\begin{array} {c} p_y \partial_z - p_z \pa... ...partial_x\end{array} \right) \ g = ({\bf p} \times {\nabla g})$$ (3)

The vector product of ${\bf p}$ and $\nabla g$ is zero if and only if one of the vectors is the null vector, or if the vectors are parallel. Ignoring the trivial cases, we conclude that when the outputs of the three finite difference filters vanish everywhere, than the gradient of g is parallel to the constant vector ${\bf p}$. The constant ${\bf p}$ is normal to g at all locations and therefore g has to be a volume of parallel planes.