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/ x) 2 + (
/ y) 2 + (
/ z) 2 = s 2 (x,y,z)
(
(6.109)
and introduce
u =
/ x
(6.110)
v =
/ y
(6.111)
so that Equation 6.109 can be rewritten as
/ z = {s 2 - u 2 - v 2 }
G(u,v)
(6.112)
Then, taking x derivatives of Equation 6.112 leads to
( z )/ x = G(u,v)/ x = G u u/ x + G v v/ x (6.113)
Now, in Equation 6.113, ( z )/ x = ( x )/ z = u/ z. Also, we find from
Equations 6.110 and 6.111 that v/ x = ( y )/ x = ( x )/ y = u/ y. Thus,
Equation 6.113 becomes
u/ z - G u u/ x - G v u/ y = 0 (6.114)
which is analogous to Equation 6.105. Again, we consider the total differential
for the phase function (x,y,z),
d = x dx + y dy + z dz
(6.115)
If we next use Equations 6.110 to 6.112, we obtain
d = u dx + v dy + G(u,v) dz
(6.116)
Thus, the total travel-time between any two points A and B is
| =
udx +
vdy +
G(u,v) dz
(6.117)
A
A
A
A
6.7.4.3 Analysis of the problem.
Consider the two-dimensional problem. From Equation 6.104, F(u) = {s 2 -
u 2 } 1/2 , and we obtain F'(u) = 1/2 {s 2 -u 2 } -1/2 (-2u) = -u {s 2 -u 2 } -1/2 . Since
Equation 6.105 can be rewritten as u/ z - F'(u) u/ x = 0, we actually have
u/ z + u {s 2 -u 2 } -1/2 u/ x = 0 (6.118)
Similar considerations apply to the three-dimensional problem. From Equation
6.112, G(u,v) = {s 2 - u 2 - v 2 } 1/2 , and G u = 1 / 2 { } -1/2 (-2u) = -u{ } -1/2 .
Similarly, G v = -v{ } -1/2 , so that u/ z - G u
u/ x - G v
u/ y = 0 in Equation
6.114 becomes
u/ z + u{s 2 - u 2 - v 2 } -1/2 u/ x + v{s 2 - u 2 - v 2 } -1/2 u/ y = 0 (6.119)
The derivation leading up to Equations 6.119 was motivated by the approach of
Van Trier and Symes (1991). Equations 6.118 and 6.119 are first-order,
nonlinear, partial differential equations with variable coefficients in s(x,z) and
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