The cross product expression may not be the most obvious approach for the dip estimation of a layered volume. I emphasize the cross product, because it allows a formulation of dip estimation by Prediction Error filters (see next section). However, other methods do exist. For example, Claerbout 1992 shows a method based on the volume's gradient and crosscorrelation function.
In general, estimating the gradient of a layered volume is harder than estimating its dip. The gradient information includes the dip information (the direction of the gradient vector) but also includes the gradient amplitude. As Figures 2 and 3 illustrate, the gradient amplitude varies within a plane layer volume even though the direction of the gradient vector remains constant. Consequently, we can estimate a single meaningful direction on an expectation volume encompassing the entire window, while the average gradient in that window would be close to zero. To average the gradient in the given window, we would have to restrict the window size to a fairly small area in which the amplitude of the gradient vector is nearly constant.
A normalized version of the amplitude vector, f(x) / |f(x)|, involves a division by zero at locations where .
On the other hand, the cross product formulation to measure the dip of a 3-D function is insensitive to the gradient amplitude and depends only on the gradient direction.