Geology Reference
In-Depth Information
6.1.6 Summary of ray tracing results.
We have engaged in a substantial amount of algebra, and it is worthwhile
to recapitulate the main results and recipes. In summary, when c(x,y,z) is given,
the solution to Equation 6.5 requires our integrating seven nonlinearly coupled
ordinary differential equations in the unknowns x, y, z,
/ x,
/ y,
/ z, and
along a ray, in particular,
dx/ds = c(x,y,z) ( / x) (6.15a)
dy/ds = c(x,y,z) ( / y) (6.15b)
dz/ds = c(x,y,z) ( / z) (6.15c)
d( / x)/ds = - c -2 c x (x,y,z) (6.15d)
d( / y)/ds = - c -2 c y (x,y,z) (6.15e)
d( / z)/ds = - c -2 c z (x,y,z) (6.15f)
d /ds = 1/c(x,y,z) (6.15g)
subject to prescribed initial conditions on each dependent variable. A simple
computational scheme was outlined in Chapter 2 for systems of first-order
ordinary differential equations. The foregoing recipe is commonly used in
seismic ray tracing. It goes without saying that the implementation is
numerically intensive. We will not dwell upon these computational details in
this topic, but instead, concentrate on a number of basic issues which do arise,
that are amenable to theoretical analysis and improvement.
6.2 Format's Principle of Least Time
(via Calculus of Variations)
Fermat' s “Principle of Least Time” formally states that “time is stationary
compared with neighboring paths between the two points.” In order to apply
standard methods in the “calculus of variations” (Hildebrand, 1952; Gelfand and
Fomin, 1963), we conveniently normalize our independent variable so that the
integration is taken over the same fixed range for all paths. Let an arbitrary path
between two points (x a , y a , z a ) and (x b , y b , z b ) be specified parametrically by x
= x ( ), y = y( ), and z = z( ), where 0 < < 1. This parametric representation
(with fixed limits = 0 and 1) means that a single independent variable can be
used to formulate the minimization problem to follow, thus simplifying the
mathematics.
6.2.1 Travel time along a ray.
Equation 6.15g can be formally integrated along a ray. If is the time of
travel of the wavefront along the ray, we can certainly write
= ray ds/c(x,y,z) (6.16)
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