Huygens wavefront tracing: A robust alternative to conventional ray tracing |

where the index corresponds to the ray parameter , corresponds to the traveltime , , is the increment in time, and is the velocity at the grid point. It is easy to notice that equation (7) simply describes a sphere (or a circle in two dimensions) with the center at and the radius . This sphere is, of course, the wavefront of a secondary Huygens source.

This observation suggests that we apply the Huygens' principle
directly to find an appropriate discretization for equation
(6). Let us consider a family of Huygens spheres, centered
at the points along the current wavefront. Mathematically, this family
is described by an equation analogous to (7), as follows:

which is clearly a semidiscrete analog of equation (6). To complete the discretization, we can represent the -derivatives in (9) by a centered finite-difference approximation. This representation yields the scheme

which supplements the previously found scheme (7) for a unique determination of the point on the -th ray and the -th wavefront. Formulas (7) and (10) define an update scheme, depicted in Figure 1. To fill the plane, the scheme needs to be initialized with one complete wavefront (around the wave source) and two boundary rays.

The solution of system (7-10) has the
explicit form

where

and

scheme
An updating scheme for HWT. Three
points on the current wavefront (, , and ) are used to advance
in the direction.
Figure 1. | |
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Figure 2 shows a geometric interpretation of formulas (7) and (10). Formula (10) is clearly a line equation. Thus, the new point in Figure 2 is defined as one of the two intersections of this line with the sphere, defined by formula (7). It is easy to show geometrically that the newly created ray segment is orthogonal to the common tangent of spheres and . Within the finite-difference approximation, the common tangent reflects local wavefront behavior.

huygens
A geometrical updating scheme for
HWT in the physical domain. Three points on the current
wavefront (, , and ) are used to compute the position of the
point. The bold lines represent equations (7) and
(10). The tangent to circle at point is parallel to
the common tangent of circles and .
Figure 2. | |
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Huygens wavefront tracing: A robust alternative to conventional ray tracing |

2014-03-11