Abstract
Traveltime calculations in 3-D velocity models have become more commonplace
during the past decade or so. Many schemes have been developed to
deal with the initial value problem, which consists of tracing rays
from a known source position and trajectory usually towards some
distant surface. Less attention has been given to the more difficult
problem of boundary value ray tracing in 3-D. In this case, source
and receiver positions are known and one, or more, minimum time paths
are sought between fixed endpoints. A new technique for boundary
value ray tracing is proposed. The scheme uses a common numerical
integration technique for solving the initial value problem and iteratively
updates the take-off angles until the ray passes through the receiver.
This type of 'shooting' technique is made efficient by using expressions
describing the geometrical spreading of the wavefront to determine
the relationship between the ray position at any time and the take-off
angles from the source. The use of numerical integration allows the
method to be compatible with a wide variety of structures. These
include models with velocity varying smoothly as a function of position
and those with arbitrarily orientated surfaces of discontinuity.
An examination of traveltime accuracy is given as well as a discussion
of efficiency for a few classes of velocity model. To improve upon
the first guess pair of take-off angles, a small-scale non-linear
inverse problem must be solved. The difference between the receiver
position and the arrival point of a ray, on a plane through the receiver,
describe a mis-match surface as a function of the two take-off angles
of the ray. The shape of this surface can possess local minima and
multiple 'global' minima even for relatively simple 1-D velocity
models. Its study provides some insight into the non-linearities
of a small-scale geophysical inverse problem.
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