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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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