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.

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
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.

Figure 4

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.

11/17/1997