Continuous in time, nearest neighbor in space

Before considering interpolation in both space and time I will first consider two cases where only interpolation in space is required. In these methods the traces are assumed to be continuous functions of time.

$$\tilde{f}( ix, t ) = f(x_{ix},t)\ ;\ x \in \{ 1,\ldots,nx\}$$

The first method uses nearest neighbor interpolation in space. The value at every point on the curve is taken from the nearest trace.Figure  shows the values interpolated at every point on the trajectory. As the trajectory gets steeper a greater length of each trace is used in the integral. The trace is used from the time at which the trajectory crosses (x_ix + x_ix-1)/2 to the time at which it crosses (x_ix+1+x_ix)/2. The function $\hat{f}(x,t)$ is the function through which the line integral is actually calculated. If this is a good approximation to the true function the the integral will be close to the true integral.

$$\hat{f}( x, t ) = \tilde{f}( ix, t ),\,\, (x_{ix} + x_{ix-1})/2 < x < (x_{ix+1}+x_{ix})/2$$

 

ct-nns
Figure 4 Continuous in time, nearest neighbor in space. The value at every point on the path is taken from the nearest trace. The weighting function is a rectangle on each trace.

The weighting function on each trace is a rectangle of area equal to the average distance to the neighbor traces, note that this is because our integral is cast as an integral in x; if it were an integral in t, the rectangles would all be of unit height.

This method is closely related to the first method described by Claerbout, 1992 where a rectangular weights are used to perform an anti-aliased Kirchoff migration.