If a given image volume is approximated by a plane layer volume, we can find the normal vector by minimizing following expectation over the volume:
where denotes the angle between and .We will have to decide on an orientation for , since both and minimize the expectation.
Since consists of only two independent variables, small expectation volumes will yield reliable estimates of the normal vector, or the corresponding dip. In particular, the minimum is independent of amplitude and polarity changes of the gradient . For example, in Figure 3, the gradient varies within the window while the cross product is zero everywhere.