Geology Reference
In-Depth Information
Recapitulation.
Key implications may have well been buried behind the
facade of subscripts. Thus, it is convenient to introduce the simplified notation
P=
x
(1.205)
q=
y
(1.206)
r=
z
(1.207)
s=
t
(1.208)
so that Equation 1.181 becomes
H(
,p,q,r,s,x,y,z,t) = 0
(1.209)
The implied recipe for solving the
partial
differential equation in Equation
1.181 calls for the equivalent solution of a system of coupled first-order
ordinary
differential equations in the nine “independent” unknowns shown in
the argument of Equation 1.209, namely,
dx/d
=H
p
(1.210)
dy/d
=H
q
(1.211)
dz/d
=H
r
(1.212)
dt/d
=H
s
(1.213)
dp/d
=- H
x
- p H
(1.214)
dq/d
=- H
y
- q H
(1.215)
dr/d
=- H
z
- r H
(1.216)
ds/d
=- H
t
- s H
(1.217)
d
/d
=p H
p
+ q H
q
+ r H
r
+ s H
s
(1.218)
where Equation 1.218 is obtained by simplifying Equation 1.183 with the
expressions in Equations 1.210 to 1.213. Again, the p, q, r, s, x, y, z, t and
derivatives of H( ,p,q,r,s,x,y,z,t) are regarded as
independent
partial derivatives.
Since H is known, the expressions for the right sides of Equations 1.210 to 1.218
are explicitly available.
When initial values for , p, q, r, s, x, y, z, and t are prescribed, the solution
can be obtained by the simultaneous integration of
nine
first-order ordinary
differential equations in the
time-like
coordinate (simpler H functions, of
course, will reduce the number of coupled equations). In our subsequent study
of geophysical applications, this integration is carried out for the “eikonal
equation” ( / x)
2
+ ( / y)
2
+ ( / z)
2
= 1/c(x,y,z)
2
corresponding to the three-
dimensional model
2
P/ t
2
- c
2
(
2
P/ x
2
+
2
P/ y
2
+
2
P/ z
2
) = 0 for normal
stresses in the earth.
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