Two point raytracing for reflection off a 3D plane

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# Introduction

For SEP-147, I calculated the response of various classic seismic algorithms on a reflection off of a plane in 3D. After wrestling with spatial geometry in old textbooks, I derived the following result from scratch using elegant, coordinate-free vector notation.

planerayfig
Figure 1.
Diagram of planar reflector and the points and vectors I use for calculating the reflected raypath.

Given a source location , a receiver location , and a plane , where is a unit normal, to find the reflection point , drop a perpendicular from to the line connecting to . Snell's Law says that running in the other direction connects to the line between and . So for some scalars and we have

Dotting onto the first two equations gives

and subtracting the first two equations produces

which can be solved directly for now that we have and . Given this , the first equation immediately yields

the desired reflection point. This can also be described in terms of the midpoint of the source and receiver as

.

 Two point raytracing for reflection off a 3D plane

Next: Converted wave reflection Up: Levin: 3D planar reflection Previous: Levin: 3D planar reflection

2012-10-29