PEAT8002 - SEISMOLOGY Lecture 7: Rays in …rses.anu.edu.au/~nick/teachdoc/lecture7.pdfLecture 7:...

PEAT8002 - SEISMOLOGYLecture 7: Rays in spherically symmetric

media

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Rays in spherically symmetric mediaIntroduction

On a global scale, the structure of the Earth is, to a goodapproximation, spherically symmetric. In this case,slowness is a function of radius only: U(r).

Seismic rays generally begin at a source on or near thesurface, and are recorded at a seismograph elsewhere onthe surface. The ray path connecting the source andreceiver is refracted and/or reflected as it traverses theEarth.

Rays in spherically symmetric mediaRay parameter

To examine the behaviour of rays in spherically symmetricmedia, we begin by investigating how the quantityr× (Udr/ds), where Udr/ds is the slowness vector, variesalong the raypath (i.e. with pathlength s)

dds

[r×

(U

drds

)]=

drds

×(

Udrds

)+ r× d

ds

(U

drds

)= 0

The first term on the RHS is zero because the crossproduct of any vector with itself is zero.The second term equals r×∇U from the ray equation,since r and ∇U are parallel for a spherically symmetricmedium, r×∇U = 0.


drds

U∆

r

ri

As a result, we reach the conclusion that

r×(

Udrds

)= constant

which means that it is preserved along the ray, and itsdirection is normal to the plane that contains dr/ds and r.


The raypath therefore lies in a plane that contains theorigin of the coordinate system.The magnitude of the preserved quantity p is referred to asthe ray parameter:

p =

∣∣∣∣r× Udrds

∣∣∣∣ = rU sin i

since for any vector a and b, |a× b| = |a||b| sin i .p is constant along the path for a particular ray.p = rU sin i is a general form of Snell’s law for sphericallysymmetric media.The radius at which the ray bottoms out is given byr = p/U since this occurs when i = 90◦.

Rays in spherically symmetric mediaTraveltime determination

The goal of this section is to determine the traveltime of araypath in a spherically symmetric medium characterisedby a continuous variation of slowness with radiusU = U(r).The gradient operator in spherical coordinates is given by

∇ =

[∂

∂r,1r

∂

∂θ,

1r sin θ

∂

∂φ

]The eikonal equation in spherical coordinates can then bewritten:

∇T ·∇T =

(∂T∂r

)2

+1r2

(∂T∂θ

)2

+1

r2 sin2 θ

(∂T∂φ

)2

= [U(r)]2


In spherically symmetric media, we can set the last term ofthe central equality in the above equation to zero sincevariations in φ do not effect T (ray travels in plane of greatcircle).In order to solve for T (r , θ), assume a separation ofvariables:

T (r , θ) = f (θ) + g(r)

Substitution into the eikonal equation gives:

r2[

dg(r)dr

]2

− r2[U(r)]2 = −[

df (θ)dθ

]2

= −k2 (1)

Both sides must be constant, because varying theindependent variable on one side will leave the other sideunaltered. Here, k2 is a separation constant.


Taking the RHS of Equation (1),

df (θ)dθ

= k so f (θ) =

∫ θ

0kdθ = kθ

From the LHS of Equation (1),[dg(r)

dr

]2

= U2 − k2

r2

so thatdg(r)

dr= ±

√U2 −

(kr

)2


These terms can now be combined to give:

T (r , θ) = kθ ±∫ r

0

√r2U2 − k2

rdr

From earlier,drds

=∇TU

Therefore

Udrds

= ∇T =

[∂T∂r

,1r

∂T∂θ

, 0]

=

[±√

U2r2 − k2

r,kr, 0

]


This expression can be rearranged to give a unit vector inthe direction of the ray:

drds

=

[±√

U2r2 − k2

Ur,

kUr

, 0

]

kUr

d rds

1is a unitvector

i

ray

U2 2r −kUr

2


From the diagram, it is evident that

sin i =k

Urcos i =

√U2r2 − k2

Urtan i =

k√U2r2 − k2

We see that k = Ur sin i = p so our separation constant issimply equal to the ray parameter. Therefore:

T (r , θ) = pθ ±∫ r

rS

√r2U2 − p2

rdr

Since T is traveltime, it must be positive and increase withdistance 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 ray turns.

rS

S R

θrR

If we differentiate the above expression with respect to θ,we get: (

∂T∂θ

)= p

which tells us that the ray parameter can be determinedfrom data if the source separation is known.


The ray parameter can be measured from the gradient of atraveltime curve.

θ

T

θ


From Fermat’s principle, ∂T/∂p = 0 - traveltime isstationary with respect to variations in the path, so

∂T∂p

= θ ±∫ r

rS

−p

r√

r2U2 − p2dr = 0

Therefore:

θ = ±p∫ r

rS

dr

r√

r2U2 − p2

Substitution back into the expression for T now gives:

T = ±∫ r

rS

[p2

r√

r2U2 − p2+

√r2U2 − p2

r

]dr

= ±∫ r

rS

p2 + r2U2 − p2

r√

r2U2 − p2dr


Thus, we end up with an expression that allows us tocompute traveltime using only p and U(r):

T = ±∫ r

rS

rU2√r2U2 − p2

dr

If the ray begins and ends at the surface, then theascending and descending contributions are equal.

re

T1

S R

rer0

T2

∆


The total traveltime is thus given by (c =wavespeed):

T = 2∫ re

r0

rU2√r2U2 − p2

dr = 2∫ re

r0

r/c√r2 − c2p2

dr

If ∆ is the total angular distance covered by the ray, then:

∆ = 2∫ re

r0

pc/r√r2 − c2p2

dr

since from before:

θ = ±p∫ r

rS

dr

r√

r2U2 − p2

Rays in spherically symmetric mediaBody waves in the Earth

Beginning in the early 1900s, travel time tables for theEarth were compiled by combining data from manyearthquakes observed at various epicentral distances.These seismological observations provide the basis for ourfirst-order understanding of Earth structure i.e. A layeredEarth composed of a thin crust, a mantle, a liquid outercore and a solid inner core.

mantle

outer core

inner core

crust


The classic Jeffreys-Bullen model of the internal seismicstructure of the Earth, derived from earthquake traveltimeobservations, was first published in 1940.The JB model divided the mantle into an upper and lowermantle, each with smooth velocity gradients. The transitionzone between these two regions features rapid velocityincrease with depth.Below the core-mantle boundary (CMB), the core wasdivided into an outer core and inner core.Subsequent studies have also derived sphericallysymmetric models of seismic structure, including IASP91and AK135. These later models have improved resolutionin the mantle transition zone and core, but otherwiseconfirm the results obtained by Jeffreys and Bullen.


COMPARISON OF THE JB AND IASP91 SPHERICALLY SYMMETRIC EARTH MODELS


Wavetrains from distant earthquakes (or teleseisms) arelikely to contain a variety of arrivals that can be attributedto features present in spherically symmetric Earth modelslike IASP91.

Rays in spherically symmetric mediaBody wave nomenclature

P

S

K

I

J

PPP

PP

SS

SP

SSS

PS

pP

sP

Compressional wave

Shear wave

P−wave through outer core

P−wave through inner core

S−wave through inner core

P−wave reflected at free surface

S−wave reflected at free surface

P−wave reflected at free surface twice

S−wave reflected at free surface twice S−wave reflected at free surface as P−wave

P−wave reflected at free surface as S−wave

P−wave upgoing from source, reflected at free surface

S−wave upgoing from source, converted to P at freesurface

DescriptionName


P−wave diffracted along core−mantle boundary

DescriptionName

c Wave reflected at core−mantle boundary (e.g. ScS)

i Wave reflected at inner core boundary (e.g. PKiKP)

P’ Abbreviation for PKP

P or Pd diff

Pg P−wave refracted in crust

Sg S−wave refracted in crust

Pn P−wave refracted at Moho (head wave)

Sn S−wave refracted at Moho (head wave)

Rays in spherically symmetric mediaIASP91 traveltime curves

Rays in spherically symmetric mediaCore shadow zone

For purely transmissive phases, the core shadow zoneoccurs between about 98◦ and 145◦.In reality, body waves are observed in the shadow zonee.g. PP, SS etc., although usually only for larger events.Their amplitudes may also vary significantly due to thepresence of lateral structure.

Rays in spherically symmetric mediaCore phases

The contrast in properties between the solid mantle andthe liquid core, which has lower velocity than the overridingmantle, makes the core well suited to seismological studiesusing refracted, transmitted, converted and diffractedarrivals.The inner core boundary also represents a strong contrastin physical properties, and along with the outer coreboundary is the subject of much research.The strongly contrasting structure of the core means that avariety of subtly different phases can be identified, andused to study structure. These phases have anothersub-class of nomenclature (e.g. PKP-AB, PKP-DF), whichare based on the various branches of the traveltime curvesthey generate.

Rays in spherically symmetric mediaCore phases

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