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

The eikonal equation, governing the traveltimes from a fixed source in
an isotropic heterogeneous medium, has the form

For a point source, can be chosen as the initial ray angle at
the source. Zhang (1993) shows that as a
function of spatial coordinates satisfies the simple partial
differential equation

It is important to note that for complex velocity fields, both and as functions of and become multi-valued. In this case, the multi-valued character of the ray parameter corresponds to the situation, where more than one ray from the source passes through a particular point in the subsurface. This situation presents a very difficult problem when equations (2) and (3) are solved numerically. Typically, only the first-arrival branch of the traveltime is picked in the numerical calculation. The ray tracing method is free from that limitation because it operates in the ray coordinate system. Ray tracing computes the traveltime and the corresponding ray positions and for a fixed ray parameter .

Since
and
are uniquely defined for
arbitrarily complex velocity fields, we can now make an important
mathematical transformation. Considering equations (2) and
(3) as a system and applying the general rules of calculus,
we can transform this system by substituting the inverse functions
and
for the original fields
and . The resultant expressions take the form

Comparing equations (5) and (6) with the original system (2-3) shows that equations (5) and (6) again represent the dependence of ray coordinates and Cartesian coordinates in the form of partial differential equations. However, the solutions of system (5-6) are better behaved and have a unique value for every and . These values can be computed with the conventional ray tracing. However, the ray-tracing approach is based on a system of ordinary differential equations, which represents a different mathematical model.

We use equations (5) and (6) as the basis of our wavefront tracing algorithm. The next section discusses the discretization of the differential equations and the physical interpretation we have given to the scheme.

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

2014-03-11