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The Wave Equation
2 2 = ∇
, for a P wave
2 2
L ( φ )= 0 where L is an operator.
- Using d’Alembert’s Solution:
( ) ( ,) ( )
ikx t xt Axe
ω φ
⋅− = , where the wave number k indicates the
direction of the wave
Ray Parameter and Slowness
A useful way to characterize a wave’s ray path is via its slowness, the reciprocal of the apparent velocity.
Figure 1. The arrow is used for a ray and the dashed line is used for a wavefront.
The wavenumber k indicates the direction of the ray. The angle i is both the take-
off angle and the angle of incidence.
We define the wave speed, c = ds/dt, with horizontal wave speed, cx = dx/dt, and vertical wave speed, cz =
dz/dt. Using Figure 1 we can relate the angle if incidence with the horizontal and vertical wave speed.
cp c
c
dx
dt c dx
ds i x
sin()= = = ≡
c η c
c
dz
dt c dz
ds i z
cos()= = = ≡
Here p is horizontal slowness, also known as the ray parameter, and η is vertical slowness.
c
i
c
p x
1 sin( ) ≡ = c
i
c (^) z
1 cos( ) η≡ =
The slowness vector, s = ( p , η), is composed of the horizontal and vertical slowness. Some properties of
the slowness vector:
c
s p
2
c
p +η =
However, the addition of squares of horizontal wave speed and vertical wave speed does not equal to squares
of wave speed,. In addition, we will examine critical phenomenon in reflection and refraction
with the relation
2 2 2 c (^) x + cz ≠ c
2 2
p c
η = −. In terms of wave number, each component of wave number can be
represented by horizontal and vertical slowness.
p c
k
x
x^ ω
z
z c
k
Thus, wave number speed is related to the slowness vector.
k = ( kx , kz )=(ω p ,ωη)=ω( p ,η)= ω s
Geometric Ray Theory
Remember from plane wave superposition:
φ ω
π
π
ω x t A e dkxdkyd
ikx t
−
∞
−∞
⋅− ≈
( ) ( , ) (...)
We will use a high frequency approximation, the limit as ω→ ∞ , which leads to geometric ray theory. We can
gain insight into the behavior of the seismic waves by considering the ray paths associated with them. This
approach, studying wave propagation using ray path, is called geometric ray theory. Although it does not fully
describe important aspects of wave propagation, it is widely used because it often greatly simplifies the
analysis and gives a good approximation.
A wave front connects points of equal phase: equal phase ~ equal travel time, T, from the origin.
Eikonal Equation
Consider the following solution to the wave equation, φ α φ:
2 2 = ∇
( ) ( ,) ( )
ikx t xt Ax e
ω φ
⋅−
We choose to work at a travel time, T ( x ).
( ) ( ,) ( )
iTx xt Axe
ω
−
Working to insert this expression back into the wave equation:
i T i T Ae i A Te
ω ω
− − ∇ =∇ − ∇
iT
A A T i A T A T e
ω φ ω ω ω
−
2 2 2 2 2
i T Ae
ω
− = −
^2
2 2
2 2 2 2 2 ( 2 ) α
ω ω Ak
A
⇒∇ A − A ∇ T − i ∇ A ⋅∇ T + A ∇ T =− =−
Real Imaginary
note:
k α (^) = , k β = and for the general case c
k
For information on propagation, consider just the real part.
Consider the mathematical formulation of the problem.
Hence,. Fermat’s Principle implies Snell’s Law.
The travel time curve, plotted as a function of offset, is typically a hyperbolic function. Near stationary points,
the curve is usually fairly ‘flat’, which implies that, near optimum or stationary points, the travel time is locally
insensitive to slight variations in offset. Consequently, close to a stationary point, small deviations in the ray
path can be treated as second order effects.
For example…
- The presence of a fast body embedded in a homogeneous medium
causes the reference (optimal) ray path to deflect from its original path between A and B.
- But, as a consequence of Fermat’s Principle, the change in ray path produces a second order effect on the arrival time.
- The effect on travel time of changes in wave speed (elastic
parameters) is first order.
Implication: We can generate reference ray paths assuming a homogeneous medium and use this reference
model to simplify subsequent analysis for heterogeneous media. → Important for travel time tomography.
Snell’s Law in a Spherical Medium
Ray Equation
