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Seismology and Seismic Imaging4. Ray theory
N. Rawlinson
Research School of Earth Sciences, ANU
Seismology lecture course – p.1/23
The ray approximation
Here, we consider the problem of how body waves (Pand S) propagate through a medium in which theelastic parameters vary with spatial location.
The elastic wave equation in a medium with spatiallyvariable properties is
ρu =∇λ(∇ · u) + ∇µ · [∇u + (∇u)T] + (λ + 2µ)∇(∇ · u)
− µ∇×∇× u
The two terms containing ∇λ and ∇µ mean that P andS motions do not decouple in heterogeneous media.
Seismology lecture course – p.2/23
However, if the scale length of variations in λ and µ arelarge compared to the seismic wavelength, then P andS can be treated separately and the elastic waveequation is simplified.
Even so, solving the elastic wave equation requiresexhaustive computational effort.
Ray theory is an alternative approach in which a pointon the wavefront is tracked rather than the completewavefield.
Ray theory is extensively used due to its simplicity,speed and applicability to a wide range of problems.
Seismology lecture course – p.3/23
Ray theory is strictly valid for media whose lengthscale variation of λ and µ is much larger than theseismic wavelength (the high frequency assumption).
At low frequencies, diffraction can be significant, andray theory is not generally valid.
Seismology lecture course – p.4/23
The eikonal equation
Consider the propagation of compressional waves inheterogeneous media.
From before, we have:
∇2Φ − 1
α2
∂2Φ
∂t2= 0
where Φ represents the scalar potential of a P-wave.
Now assume a harmonic solution of the form:
Φ = A(x)exp[−iω(T (x) + t)]
where T (x) is a phase function which describes thearbitrary distribution in space of a surface of constantphase.
Seismology lecture course – p.5/23
Substitution of the above expression into the waveequation yields:
|∇T |2 − 1
α2=
∇2A
Aω2
A similar expression can be obtained for S-waves (βinstead of α).
If we now make the high frequency approximation i.e.that ω is large enough that 1/ω2 ≈ 0, then
|∇T |2 = U2
which is known as the eikonal equation.U = slowness = 1/velocity.
Seismology lecture course – p.6/23
T (x) = constant defines surfaces called wavefronts.
∇T (x) defines raypaths.
The function T (x) has units of time and simplyrepresents the time required by the wavefront to reachx from some reference location x0.
Seismology lecture course – p.7/23
The ray equation
If we denote s as the arc length parameter along a rayand r as the position vector of the ray, then
dr
ds=
∇T
U
s
r0
1
0
rd
0s
1r
Seismology lecture course – p.8/23
Now let’s examine how the surface of constanttraveltime T varies along the ray:
dT
ds= ∇T · dr
ds=
∇T · ∇T
U= U
The above two equations may be combined to give:
d
ds
[
Udr
ds
]
= ∇U
which is referred to as the ray equation and may beused to integrate the ray trajectory through anarbitrarily varying medium in 3-D space.
Seismology lecture course – p.9/23
Fermat’s principle
Fermat’s principle states that the ray path between twopoints P and Q is a path of stationary time
tPQ =
∫ Q
P
Uds = extremum
truepath
truepath
L
t
Seismology lecture course – p.10/23
To prove Fermat’s principle, we need to show thatwhen a ray path is perturbed, the effect on traveltime issecond order
δtPQ = δ
∫ Q
P
Uds =
∫ Q
P
δUds + Uδ(ds)
r+
rδ+ +d( )rδ
+d( )rδdr+d(
)rδ
r
dr
δr
δr
r+dr
r+dr
δr
reference ray path segmentperturbed ray path segment
Seismology lecture course – p.11/23
We can show that, ignoring higher order terms, thisintegral reduces to:
δtPQ =
∫ Q
P
[
∇U − d
ds
(
Udr
ds
)]
· δrds
and the integrand equals zero by the ray equation.
Hence, the first-order perturbation of traveltime due toa perturbation in the ray path is zero, and we haveproven Fermat’s principle.
Seismology lecture course – p.12/23
Snell’s law
We can use Fermat’s principle to derive Snell’s law,which describes the refraction of a ray path at aninterface between media of different wavespeeds.
x1
B( , z2)x2
i2 t2
i1t1
v1
x=X
=0zv2
A( , z1)
O( , 0X )
Seismology lecture course – p.13/23
The total traveltime T between A and B is given by
T =
√
z2
1+ (X − x1)2
v1
+
√
z2
2+ (X − x2)2
v2
Fermat’s principle says that dT/dX = 0, so
X − x1
v1
√
z2
1+ (X − x1)2
+X − x2
v2
√
z2
2+ (X − x2)2
= 0
This gives us Snell’s law:
sin i1v1
=sin i2v2
Seismology lecture course – p.14/23
Rays in spherically symmetric media
On a global scale, the structure of the Earth is, to agood approximation, spherically symmetric. In thiscase, slowness is a function of radius only: U(r).
Let us start by looking at how the quantity
r ×(
Udr
ds
)
varies along the raypathSeismology lecture course – p.15/23
Ray parameter
In other words, how does it vary with the arc lengthparameter s?
d
ds
[
r ×(
Udr
ds
)]
=dr
ds×
(
Udr
ds
)
+ r × d
ds
(
Udr
ds
)
= 0
Thus,
r ×(
Udr
ds
)
= constant
so this quantity is preserved along the ray, and itsdirection is normal to the plane that contains dr/ds andr.
Seismology lecture course – p.16/23
The raypath therefore lies in a plane that contains theorigin of the coordinate system.
The magnitude of the preserved quantity p is referredto as the ray parameter:
∣
∣
∣
∣
r × Udr
ds
∣
∣
∣
∣
= rU sin i
p is constant along the path for a particular ray.
p = rU sin i is a general form of Snell’s law forspherically symmetric media.
The radius at which the ray bottoms out is given byr = p/U since this occurs when i = 90◦.
Seismology lecture course – p.17/23
Traveltime determination
The eikonal equation in spherical coordinates withU(r) is
(
∂T
∂r
)2
+1
r2
(
∂T
∂θ
)2
= [U(r)]2
To solve the eikonal equation, assume a separation ofvariables, i.e.
T (r, θ) = f(θ) + g(r)
Substitution into the eikonal equation yields:
T (r, θ) = kθ ±∫ r
rS
√r2U2 − k2
rdr
where k2 is a separation constant. Seismology lecture course – p.18/23
Since T is traveltime, it must be positive and increasewith distance along the ray.
Take +ve for ascending ray since dr > 0.Take −ve for descending ray since dr < 0.
Therefore, the integral needs to be split up if the rayturns.
rS
S R
θrR
Seismology lecture course – p.19/23
It can be shown that k = p, the ray parameter, so
T (r, θ) = pθ ±∫ r
rS
√
r2U2 − p2
rdr
If we differentiate the above expression with respect toθ, we get:
(
∂T
∂θ
)
= p
which tells us that the ray parameter can bedetermined from data if the source separation isknown.
Seismology lecture course – p.20/23
The ray parameter can be measured from the gradientof a traveltime curve.
θ
T
θ
Seismology lecture course – p.21/23
If we now make use of Fermat’s principle, which can bestated here as ∂T/∂p = 0, we can differentiate ourintegral expression with respect to p to give
T = ±∫ r
rS
rU2
√
r2U2 − p2dr
If the ray begins and ends at the surface, then theascending and descending contributions are equal.
∆
2T
0rer
RS
1T
er
Seismology lecture course – p.22/23
The total traveltime is thus given by (c =wavespeed):
T = 2
∫ re
r0
rU2
√
r2U2 − p2dr = 2
∫ re
r0
r/c√
r2 − c2p2dr
If ∆ is the total angular distance covered by the ray,then:
∆ = 2
∫ re
r0
pc/r√
r2 − c2p2dr
since from before:
θ = ±p
∫ r
rS
dr
r√
r2U2 − p2
Seismology lecture course – p.23/23