There are many variations on Shepard’s original interpolation method, which is essentially the standard inverse distance (IDW) procedure. One variant introduced by Shepard involves the use of two separate powers: a lower value (generally 2) for nearby data points and a higher value (generally 4) for points further away. Another variant, implemented in some packages, adjusts the weights based on how far away the furthest point (in the entire set or within a given radius) is to be found. If this distance is R then a revised IDW formula is:
The value kj in this expression is:
This is the form used in the Groundwater Modeling System (GMS) software package.
Surfer uses a more complex procedure based on a local quadratic polynomial fit in the neighborhood of each data point. This is sometimes called the modified quadratic Shepard’s method — for an example implementation see Carlson and Foley (1991, 1992). The procedure then continues as an inverse distance model using surface values obtained from the fitted quadratic surface rather than the original data points. The result is an IDW-type interpolator without the strong bull’s-eye effect that simple IDW can suffer from. The method can be exact or approximate, depending on whether a smoothing factor is specified (see Figure 6‑31H).