Bilinear interpolation in time and space

In the case of bilinear interpolation, the approximate function is given by,

Figure  illustrates the geometry associated with bilinear interpolation within a cell.

 

bilin
Figure 10 Bilinear interpolation. The function is approximated by bilinear interpolation within a cell defined by the four corner samples. The interpolation weights depend on the coordinates at which the path enters an leaves the cell.

The path within the cell, t(x), is approximated by a linear segment,

$$\hat{t}(x) = t(\xi_0) \left[1 - \frac{x - \xi_0}{\xi_1-\xi_0}\right] + t(\xi_1)\left[ \frac{x - \xi_0}{\xi_1-\xi_0} \right]$$

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 (x_ix,t_it) is:

$$( \xi_1 - \xi_0 ) \frac{ 3 ( t_{it+1} \xi_0 + \tau_1 x_{ix... ...\tau_1 \xi_0 } { 6( t_{it} - t_{it+1} ) ( x_{ix+1} - x_{ix} ) }$$

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.

 

bilin-func
Figure 11 Input data sampled every 4ms in time and 25m in space and then interpolated using bilinear interpolation in space and time.