Nearest neighbor in time, linear interpolation in space

If the function is approximated by linear interpolation in space then the integral has a different form. A single sample will influence the interpolated value within two regions. Each region is bounded by a neighboring trace and half the distance to the nearest time samples.

Within each regions the approximate function is,

$$\hat{f}( x, t ) = \tilde{f}( ix-1, it ) \left[ 1 - \frac{x... ...}( ix, it )\left[ \frac{ x - x_{ix-1} }{x_{ix}-x_{ix-1}}\right]$$

and the integral of the approximate function is given by,

$$\int_{\xi_0}^{\xi_1} \tilde{f}( ix-1, it ) \left[ 1 - \fra... ... it ) \left[ \frac{ x - x_{ix-1} }{x_{ix}-x_{ix-1}} \right]\,dx$$

This has two terms, one involving each sample value.

$$\frac{(\xi_1 - \xi_0 )}{( x_{ix-1}-x_{ix} )} \left[ \tilde{... ...tilde{f}( ix, it ) ( x_{ix-1} - \frac{\xi_0+\xi_1}{2} ) \right]$$

The weight for one sample point is the sum of the weights from the cell to its left and right. When calculating the integral along a path the paths fall into the same three classes as in the previous method. The entry and exit points of the path within a cell must be calculated, and the weights for the two samples that define the cell can then be applied. Figure  illustrates the weights for this method. Although this method is close to the triangle weighting method proposed by Claerbout, the weights in a sampled space are not exactly triangles.

 

nnt-lins
Figure 8 Nearest neighbor in time, linear interpolation in space. The function is approximated by linear interpolation in space within each cell.

Figure  shows the approximation to the original function surface that is implied by this method. If the time sampling is sufficiently fine this method will give a reasonable approximation of the true surface. When the time sampling is larger some interpolation in time must be considered.

 

linxnnt-func
Figure 9 Input data sampled every 4ms in time and 25m in space and then interpolated using nearest neighbor in time and linear interpolation in space.