Directional derivatives

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Directional derivatives

As noted earlier, once the components for gradient and curvature calculations have been estimated (first and second differentials) the computation of similar functions for other directions, α, is relatively straightforward. In these computations a step in the chosen direction is defined as having length ds, so the first directional derivative is obtained as follows:

If α=0 or a multiple of π this reduces to the x-component of the vector, whilst if α=π/2 or a similar multiple it reduces to the y-component (although the sign may change). In a similar manner the second derivative is defined as:

Surfer provides computation of both of these derivatives based on a user-defined angle input. This angle applies to every grid cell, so will almost always differ from the values obtained by the kind of terrain modeling computations described in Section 6.2.2, Profiles and curvature. Curvature computations based on the mathematical view of tangential curvature are also provided. Collectively these computations show how the surface changes in terms of gradient and curvature in a particular constant direction.

If the selected direction is a major compass point (NSEW) one of the sin or cos terms will be zero and the other =+1 or ‑1. The first directional derivative for the directions due north or due east would then be:

These simple expressions are those described in Section 6.1.3, Raster models and also, in the context of filtering, in Section 4.6.2, Linear spatial filtering.