In the case of bilinear interpolation, the approximate function is given by,
Figure illustrates the geometry associated with bilinear interpolation within a cell.
The path within the cell, t(x), is approximated by a linear segment,
The Mathematica code in the appendix will calculate the appropriate weights for each sample point. For example, the weight to be applied to the sample point (xix,tit) is:
Figure shows the approximation to the original function surface that is implied by this method. For this set of parameters it is a reasonable approximation of the original surface. In general bilinear interpolation is a reasonable interpolation method for sampling intervals up to about half of the Nyquist frequency. If the data has high dips and high frequencies the approximation may not be valid.