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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.


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.


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.


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.


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.


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.


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


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.


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


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


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.


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 )


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


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.


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◦.


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


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.


The ray parameter can be measured from the gradientof a traveltime curve.

θ

T

θ


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


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


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