Extension to N-D

Lloyd's method would not lead to an interesting result for a signal that has a uniform distribution. The result would be a set of evenly-spaced points that span the range of the signal. Lloyd's method is effective for a signal that tends to cluster around a few selected values (what turns into the codebook). The velocity selection problem is effectively handled by the Lloyd approach because the earth is composed of geologic layers. The velocity range within a geologic layer tend to be relatively constant. Lloyd's algorithm automatically identifies these clusters.

In some problems we have several parameters at every model location. If these parameters are correlated, a clustering techniques such as Lloyd's approach can be effective. Extending the 1-D approach described above to a 3-D case is relatively simple:

• The codebook entries become vectors rather than scalers.
• To find the nearest point, we do a vector distance calculation.
• Instead of averaging the points attached to a given codebook, we find the centroid of the attached points.

These changes are all making an assumption that the range of each element of our vector field is approximately the same. If this assumption isn't true, for example when our vector field consists of velocity (ranging from 1500-4500 m/s) and $\delta$ (ranging from 0-.1), we need to scale the $\delta$ value so their range is approximately equal to velocity. If we do not, the $\delta$values will have such a small effect in our calculations that the algorithm reduces to a 1-D velocity solution.