Geology Reference
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i
kkk
(k) =
i
kkkk
(k) =
i
kkkkk
(k) = ... = 0 (6.76)
All of these simplifications serve to reduce our general high-order kinematic
wave modulation to mathematically tractable form. In particular, Equations
6.62 and 6.63 simplify to
ck c
2
ka
x
/(2f
0
a) - 1/2 c
2
k
x
/(2f
0
)
(6.77)
a
2
/ t + c a
2
/ x = - c
2
k
2
a
2
/(2f
0
) + c
2
aa
xx
/(2f
0
)
(6.78)
For
dynamically steady problems
excited by a constant frequency
0
, the
propagation equations reduce to
ck
c
2
ka
x
/(2f
0
a) - 1/2 c
2
k
x
/(2f
0
) =
0
(6.79)
c da
2
/dx = - c
2
k
2
a
2
/(2f
0
) + c
2
aa
xx
/(2f
0
) (6.80)
which is a set of nonlinearly coupled ordinary differential equations in a(x) and
k(x) easily solved by conventional numerical methods. We will not pursue
computational solutions in this chapter, but rather, discuss the consequences
associated with different high-order terms in a general manner, and establish
their special relevance to ray tracing, cross-well tomography, and
first arrival
time
analysis. Before delving into these seismic issues, we set the stage for our
discussion by referring to some fundamental ideas.
6.6 Subtle High-Order Effects
The role of high-order derivatives in the theory of nonlinear equations
forms a relatively new area of study in applied mathematics just decades old.
Many of the already established ideas, understandably, have not yet found their
way into engineering practice, by virtue of their highly specialized nature. The
following discussion expands upon ideas elucidated in Chin (1980), developed
in the contexts of water waves and general continuum mechanics, and Chin
(1993a,b), developed in the context of low-order saturation discontinuities and
high-order capillary pressure effects in reservoir simulation.
6.6.1 A low-order nonlinear wave equation.
Let us consider the first-order, convective, nonlinear wave equation for a
function u(x,t) satisfying
u/ t + u u/ x = 0
(6.81)
Equation 6.81 is not unlike
k/ t +
r
k
(k) k/ x = 0 (6.82)
describing low-order wave conservation in a homogeneous medium. Equation
6.81 possesses a simple general solution assuming very arbitrary initial
conditions. To construct it, we recall that the total differential for any function
u(x,t) satisfies du = u/ t dt + u/ x dx. Thus,
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