chapter 2...chapter 2 seismology 2.1 introduction . seismology is a science based on seismograms –...

CHAPTER 2

Seismology

2.1 Introduction . Seismology is a science based on seismograms – records of the vibrations of the Earth.An earthquake (or man-made explosion) sets off a series of vibrations and, in seismological parlance, istermed the source. The vibrations are modified as they are transmitted through the Earth so that the Earthcan be thought of as a kind of filter with a transmission response. Finally, the vibration is recorded bya seismograph. No instrument is capable of perfectly recording ground motion so the seismogram that isactually recorded by the seismograph is a filtered version of the true ground motion. This filtering effectof the instrument (or receiver response) should be well-determined through careful calibration and it isnow quite rare for instruments to be poorly calibrated. The transmission of seismic waves depends onthe Earth’s internal structure and the excitation of these waves depends upon the nature of the seismicsource. Because the seismogram is a mixture of the source and transmission effects it is often possible tomisinterpret features in seismograms. For example, complicated Earth structure will distort wave frontsgiving complicated seismograms, these complexities can then be misinterpreted as being due to a complicatedrupture during the earthquake. The interpretation of seismograms has advanced in a see-saw manner, attimes our knowledge of the source has been best and this has been used to refine our knowledge of Earthstructure. With a better knowledge of structure we can improve our knowledge of the source and so on.

In this chapter we shall be looking at global seismology, that is, we shall be considering the structureof the whole Earth. It should be realized, though, that seismology can be used on very small scales too.Seismograph networks may have dimensions of only a few tens of meters for engineering studies. For globalstudies we use global arrays of seismographs with dimensions of 10,000 km. Seismic sources also have atremendous range of scale. Man-made sources such as explosions can vary in size by a factor of 1012, thelargest, of course, being nuclear explosions. Natural sources such as earthquakes are observed to vary by afactor of 1018.

In the vicinity of the source the motions can be very strong. (Houses have been observed to jump offtheir foundations indicating that accelerations in excess of the acceleration due to gravity occur.) Thesevery strong motions attenuate rapidly with distance away from the source and the seismic waves we observeglobally have very small deformations associated with them (typically one part in 105 or smaller). Most ofseismology is therefore concerned with the analysis of small oscillations in the Earth.

2.2 Approximations used in modeling seismic waves . In global seismology, seismic waves with periodsof one hour to one second are typically used. These are “fast” disturbances compared with a geological timescale and the Earth responds nearly perfectly elastically to seismic disturbances. (On long time scales theEarth behaves more as a fluid.) Perfect elasticity implies that if we apply a stress to a body it will deform(strain) but when the stress is removed the body returns to its original state. This is a good approximationfor the Earth and, because we are dealing with small oscillations, we can use the theory of linear elasticity,i.e., the applied stress is linearly related to resulting strain (“Hooke’s Law”). Perfect elasticity is only anapproximation. If the Earth were perfectly elastic then seismic waves would not be attenuated. As a wavepasses through the Earth, some of its energy is converted into heat (the mechanisms of attenuation areonly poorly understood – one possibility is friction at grain boundaries in the material). This transfer ofenergy from the wave to the medium means that the medium cannot return to its original state so it is notperfectly elastic. Seismic waves are only very weakly attenuated so perfect elasticity is often an acceptableapproximation.

Another approximation that we make in modeling seismic waves is that the material in the Earth iselastically isotropic. This means that seismic disturbances pass through a material with a velocity which isindependent of the direction of propagation. This is not true of crystals but rocks are made up of aggregatesof crystals which we hope are randomly orientated giving a net isotropy to the rock. Velocity anisotropy(direction dependent velocities) has been observed in the oceanic crust and mantle. It is probably due tonon-randomly oriented fractures in the crust and to flow alignment of olivine crystals in the mantle. However,

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globally we think that isotropy is a reasonable approximation (and, incidentally, a great simplification).Another approximation that is used is that the Earth is spherically symmetric. It is surprising how well

this approximation models the data even though there are obvious aspherical structures present (ellipticityof figure, continents and oceans, etc. ). At present we have a very good idea of the spherically averagedstructure of the Earth and it is only recently that progress has been made on determining aspherical structureas a function of depth. Except near the surface, departures from sphericity appear to be small. At very longperiods (≈ 1 hr – 10 min) seismic waves are affected by the rotation of the Earth, however in most studiesthe rotation can be neglected.

An Earth model which satisfies all the above approximations is given the acronym SNREI (spherical,non-rotating, elastic isotropic). Departures of the Earth from this model appear to be small but are of greatinterest – particularly lateral variations in structure which can be used to infer the nature of large-scale flowin the mantle. We initially concentrate on 1D (SNREI) models but will look at 3D models later in the class.

2.3 Representations of the seismogram . The way we look at a seismogram depends upon the frequencyrange being used and which part of the seismogram we want to model (Remember frequency = 2π/T = ωwhere T is the period and ω is measured in radians/sec. A commonly used unit of frequency is the Hertzwhere frequency in Hz, f = 1/T ).

Fig. 2.1 Illustrations of various modes of oscillation: a) is sometimes called the "football" mode of theearth; b) is sometimes called the "breating" mode of the earth; c) illustrates a special class of modes whichconsist of shearing on concentric shells – toroidal modes

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At low frequencies, i.e., .3mHz→ 10mHz (disturbances with periods of 100 sec to about 1 hour) we usea standing wave representation of the seismogram. We think of the Earth as a bell undergoing oscillationsafter being hit. Most of these oscillations have complicated geographical shapes but a particularly simpleexample is the “breathing mode” of the Earth (see fig 2.1 for examples of mode shapes). The whole Earthexpands and contracts in this mode of oscillation. One cycle takes about 20 minutes. Another example isthe "american football" mode which is the lowest frequency mode observed (it has a period of about 57minutes) – the Earth oscillates with the shape of a football. There is also a class of modes which have nomotion in a radial direction and can be thought of as twisting on concentric shells – these modes are called"toroidal" modes whereas the breathing and football modes are examples of "spheroidal" modes.

Standing waves are used to model the whole seismogram which can actually be thought of as a sum ofdecaying cosinusoids:

u(t) =∑k

Ak cos (ωkt+ φk)e−αkt

Here, ωk and αk are the frequency and decay rate of the k’th mode (these are functions of earth structure).Ak and φk are the initial amplitude and phase of the k’th mode (these are functions of the earthquake source).

An example of some low-frequency seismograms is shown in Figure 2.2. The data have been filteredso that only disturbances with periods longer than 150 seconds are present. The dashed lines (hard to see)are predicted seismograms computed by using a model of the source and mode frequencies and decay ratescomputed for a SNREI model. Clearly, our ability to model long period seismic disturbances is very good.The small differences are due to the slight departures of the Earth from spherical symmetry.

The fact that the above seismograms are really made up of a sum of standing waves only becomes apparentif we "fourier transform" the seismogram and plot its "spectrum". The spectrum measures how much energyis at each frequency. An example is given in figure 2.3.

Fig. 2.3 The spectrum of a recording of a large deep earthquake (shown between the frequencies of 2 and6 mHz). Note that the energy is confined to "spectral peaks", each one of which is at a frequency of freeoscillation of the Earth.

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Fig. 2.2 Some low frequency seismograms of a "great" earthquake which occured 630 km under Brazilin 1994. The character codes are station name, LVZ and ABKT are stations run by UCSD in the formerSoviet Union, PFO is our nearest station near Palm Springs

When we analyze seismic disturbances in the period band 200 – 10 sec, it becomes more natural to thinkin terms of traveling waves. If you look again at the seismograms in figure 2.2, you will see large wavepackets. Figure 2.4 shows a longer time serieswhere the packets are clearer and have been labelledR1, R2, ...etc. These can be thought of as wave packets traveling around the Earth away from the source – rather likeripples traveling away from a disturbance in a lake. There are two classes of traveling waves on a sphericallysymmetric Earth, Love waves and Rayleigh waves (see fig. 2.8). Love waves are horizontally polarized(i.e., they have no vertical component of motion) and the particle motion can be thought of as a shearingmotion transverse to the direction of propagation of the wave. (Love waves are not propagated througha fluid). Rayleigh waves have particle motions which are elliptical and do have a vertical component ofmotion. The seismogram in the figure is from a sensor which measures only vertical motion and thereforecontains only Rayleigh waves. For periods shorter than 200 sec the modes which make up the Rayleigh andLove wave packets have their energy concentrated near the surface of the Earth so these wave packets arecalled surface waves.

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Fig. 2.4 A long time series from a vertical component of recording showing "surface wave" packetscorresponding to "Rayleigh" type. See figure 2.5 for an explanation

Consider the following situationConsider the following situation

The wave packet leaves the source, travels around the world and will be detected by the receiver as it

passes. The wave packet traveling around the shortest arc is calledR1 (if it is a Rayleigh wave) and hits

the receiver first. The packet traveling along the longer arc arrives next and is called R2, these packets

will then make complete orbits of the globe and arrive again (now called R3 and R4). One orbit of the

globe takes about three hours. For large events many orbits of the globe can be observed. As the wave

travels farther it becomes more attenuated and dispersed. Dispersion means that the wave packet is

initially concentrated in time and spreads out as time progresses. This is because the different frequency

components in the wave packet sample different parts of internal structure of the Earth and so travel at

different velocities. The dispersion and attenuation of the wave packets can be measured and so we can

learn about Earth structure. The amplitudes of the wave packets for different great circle paths about

the Earth tell us about the earthquake source.

In practice, only the large amplitude surface wave packets are analyzed in terms of traveling waves

so traveling wave analyses are usually only concerned with the structure near the surface of the Earth

(the upper mantle).

The analysis of the seismogram in terms of traveling and standing waves has only been done since the

middle 1950’s because a computer is required to do the calculations. The classical source of information

in seismology has been the body waves. These are the waves that arrive at the receiver before the surface

waves and can be thought of as waves which have traveled through the body of the Earth. Body waves

with periods of about 1 sec are usually used in global seismology and, at these periods, separate into

two types of waves – compressional (or P waves) and shear (or S waves). Compressional waves arelike sound waves and travel faster than shear waves, hence they arrive at the receiver first (P stands for

primary arrival). Shear waves are like light waves (and so can be polarized) – they are (usually) the

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The wave packet leaves the source, travels around the world and will be detected by the receiver as it passes.

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The wave packet traveling around the shortest arc is called R1 (if it is a Rayleigh wave) and hits the receiverfirst. The packet traveling along the longer arc arrives next and is called R2, these packets will then makecomplete orbits of the globe and arrive again (now calledR3 andR4). One orbit of the globe takes about threehours. For large events many orbits of the globe can be observed. As the wave travels farther it becomesmore attenuated and dispersed. Dispersion means that the wave packet is initially concentrated in time andspreads out as time progresses. This is because the different frequency components in the wave packetsample different parts of internal structure of the Earth and so travel at different velocities. The dispersionand attenuation of the wave packets can be measured and so we can learn about Earth structure. Theamplitudes of the wave packets for different great circle paths about the Earth tell us about the earthquakesource.

In practice, only the large amplitude surface wave packets are analyzed in terms of traveling waves sotraveling wave analyses are usually only concerned with the structure near the surface of the Earth (theupper mantle).

The analysis of the seismogram in terms of traveling and standing waves has only been done since themiddle 1950’s because a computer is required to do the calculations. The classical source of informationin seismology has been the body waves. These are the waves that arrive at the receiver before the surfacewaves and can be thought of as waves which have traveled through the body of the Earth. Body waves withperiods of about 1 sec are usually used in global seismology and, at these periods, separate into two typesof waves – compressional (or P waves) and shear (or S waves). Compressional waves are like sound wavesand travel faster than shear waves, hence they arrive at the receiver first (P stands for primary arrival).Shear waves are like light waves (and so can be polarized) – they are (usually) the secondary arrival on theseismogram. Figure 2.5 illustrates what these body wave arrivals look like.

Fig. 2.5. A 3 component seismogram illustrating the arrival of the P, PP and S etc. body waves. Notethat the timescale is now measured in minutes. Usually we measure motion in the vertcal, N-S, and E-Wdirections. In this example, the horizontal components have been rotated to a coordinate system in which onecomponent is in the direction from source to receiver (the "radial" direction) and the other is perpendicularto it (the transverse direction)

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2.4 Linear elasticity and the wave equation . The primary source of information about the elastic structureof the Earth comes from the study of the travel times of the body waves from the source to the receiver. Ifwe can measure the travel times of the body waves we can infer the wave velocity distribution with depthinside the Earth. Using the theory of linear elasticity we can relate the velocities of the waves to the elasticconstants of the material. To see how this is done we shall consider wave propagation in a one-dimensionalspring. The spring has uniform properties along its length so if we draw lines on the spring and stretch itwe get

secondary arrival on the seismogram. Figure 2.5 illustrates what these body wave arrivals look like.

Fig. 2.5. A 3 component seismogram illustrating the arrival of the P, PP and S etc. body waves. Note

that the timescale is now measured in minutes. Usually we measure motion in the vertcal, N-S, and

E-W directions. In this example, the horizontal components have been rotated to a coordinate system

in which one component is in the direction from source to receiver (the ”radial” direction) and the other

is perpendicular to it (the transverse direction)

2.4 Linear elasticity and the wave equation . The primary source of information about the elastic

structure of the Earth comes from the study of the travel times of the body waves from the source to the

receiver. If we canmeasure the travel times of the body waves we can infer the wave velocity distribution

with depth inside the Earth. Using the theory of linear elasticity we can relate the velocities of the waves

to the elastic constants of the material. To see how this is done we shall consider wave propagation in a

one-dimensional spring. The spring has uniform properties along its length so if we draw lines on the

spring and stretch it we get

Fig. 2.6 A uniformly stretched spring

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Fig. 2.6 A uniformly stretched springThe spring has a stretched length of L1 and an unstretched length of L0. Define the dimesionless "strain" εto be

ε =dU

dx

For this uniform stretching, ε is independent of position along the spring because dU doesn’t vary with x,and so U = εx. This means that

L1

L0= 1 + ε

Generally dU will be a function of x as will ε. Empirically, we find that the force required to stretch thespring is proportional to ε:

F = Kε

Here, K depends only on the material properties of the spring and the result, F = Kε, implies that ashorter spring requires a larger force to stretch it to the same length as a stretched longer spring. (In athree-dimensional medium we have to analyze the forces/unit area on a block of material (stresses) andrelate these to the deformations of the body. Therefore F = Kε is a one-dimensional version of a linearstress-strain relationship, i.e., Hooke’s Law).

For the uniformly stretched spring, F is independent of position in the spring because ε is not a functionof position. What happens if we now apply forces so that the spring is not uniformly deformed, i.e., ε = ε(x)and F = F (x)? Consider an element of the spring

The spring has a stretched length ofL1 and an unstretched length ofL0. Define the dimesionless ”strain”

! to be

! =dU

dx

For this uniform stretching, ! is independent of position along the spring because dU doesn’t vary with

x, and so U = !x. This means that

L1

L0= 1 + !

Generally dU will be a function of x as will !. Empirically, we find that the force required to stretch thespring is proportional to !:

F = K!

Here, K depends only on the material properties of the spring and the result, F = K!, implies that ashorter spring requires a larger force to stretch it to the same length as a stretched longer spring. (In a

three-dimensional medium we have to analyze the forces/unit area on a block of material (stresses) and

relate these to the deformations of the body. Therefore F = K! is a one-dimensional version of a linearstress-strain relationship, i.e., Hooke’s Law).

For the uniformly stretched spring, F is independent of position in the spring because ! is not afunction of position. What happens if we now apply forces so that the spring is not uniformly deformed,

i.e., ! = !(x) and F = F (x)? Consider an element of the spring

Fig. 2.7

Let " be the mass per unit length of the spring so the mass of the element of the spring is "#x. Thenet force on the element is

F2 ! F1 = mass " acceleration

The acceleration is the second time derivative of the displacement, i.e., u # u(x, t), thus accelerationis

$2u

$t2(x, t)

For a spring with uniform spring constantK we have

F2 ! F1 = K!(x + #x) ! K!(x) = K[!(x + #x) ! !(x)]

= "#x$2u

$t2

23

Fig. 2.7

20

Let σ be the mass per unit length of the spring so the mass of the element of the spring is σδx. The netforce on the element is

F2 − F1 = mass × acceleration

The acceleration is the second time derivative of the displacement, i.e., u→ u(x, t), thus acceleration is

∂2u

∂t2(x, t)

For a spring with uniform spring constant K we have

F2 − F1 = Kε(x+ δx)−Kε(x) = K[ε(x+ δx)− ε(x)]

= σδx∂2u

∂t2

For small deformation of the body we can expand ε(x) in a Taylor series, i.e.,

ε(x+ δx) = ε(x) +∂ε

∂xδx+ · · ·

so σδx∂2u

∂t2= Kδx

∂ε

∂x

but ε = ∂u/∂x (We have to use partial differentials because u is now a function of x and t) so

σ∂2u

∂t2= K

∂2u

∂x2

or

∂2u

∂t2=K

σ

∂2u

∂x2

K has units of force, i.e., kg ms−2 and σ has units of kg m−1 so K/σ has units of m2 s−2, i.e., it is a squaredvelocity. We can therefore write

∂2u

∂t2= c2

∂2u

∂x2where c =

√K

σ(2.1)

This is the one-dimensional wave equation and describes the propagation of a disturbance through thematerial. The speed of propagation is c. Consider a cosinusoidal disturbance of the form

u(x, t) = A cos (ωt− kx+ φ)

A and φ are the initial amplitude and phase of the disturbance (at t = 0, x = 0). ω is the frequency related tothe period, T , of the wave by T = 2π/ω. k is the wavenumber, related to the wavelength, λ, by k = 2π/λ. Itis easy to show that this is a solution to the wave equation. If we differentiate u(x, t) twice with respect to tkeeping x constant we have

∂2u

∂t2= −ω2u(x, t)

Similarly

∂2u

∂x2= −k2u(x, t)

so∂2u

∂t2=ω2

k2∂2u

∂x2

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Clearly, our cosinusoidal disturbance satisfies the one-dimensional wave equation and propagates with aspeed c = ω/k = λ/T . c is called the phase velocity and gives the speed at which a particular feature ofthe wave (e.g., a peak or a trough) travels through the medium. The propagation of arbitrarily complexpulse shapes can be studied by regarding them as a combination of cosinusoidal disturbances with differentfrequencies and wavelengths.

In three dimensions the situation is a bit more complicated. If we look at a disturbance propagatingthrough a medium we find that the displacement in the propagation direction is propagated with a speedthat is different from the speed of propagation of displacements at right angles to the direction of the wavepropagation. Displacements in the direction of propagation look like sound waves, i.e., there are alternatecompressions and dilatations of the material (see Figure 2.8). These are called compressional waves, anexample being the primary or P arrival on the seismogram. Displacements perpendicular to the direction ofwave propagation are called shear waves and are analogous to light waves. An example of a shear wave isthe secondary or S arrival on the seismogram.

The velocity of compressional waves (denoted Vp) is given by

Vp =

√Ks + 4

3µ

ρ(2.2)

where Ks is the adiabatic bulk modulus, ρ is the density and µ is the “rigidity” of the material. The velocityof shear waves (denoted Vs) is given by

Vs =

√µ

ρ(2.3)

µ is zero in a fluid so shear waves are not transmitted through a fluid. We define φ, the seismic parameter,by

φ =Ks

ρ= V 2

p −4

3V 2s (2.4)

The seismic parameter φ will be used in later chapters and can be directly calculated if Vp and Vs are knownas a function of radius.

For a physically realistic material Vp is always greater than Vs. We can show this by noting that the bulkmodulus is always positive so

Vp =

√Ks + 4

3µ

ρ>

√43µ

ρ>

√µ

ρ= Vs

Therefore Vp > Vs, i.e., compressional waves travel faster then shear waves.One of the principle uses of seismology has been to determine the internal structure of the Earth. The

observations are in terms of the time it takes for seismic waves to travel from a source to a receiver andcan be used to infer the seismic velocity throughout the Earth. This information can in turn be used to helpconstrain the thermal and chemical state of the Earth’s interior. In order to relate travel times of body wavesto the velocity-depth distributions, we use geometrical ray theory.

Ray theory is an approximate description of body wave propagation and is valid when the wavelengthof the disturbance λ is “sufficiently small.” By this we mean that the length scale on which the velocitychanges in the Earth should be large relative to the wavelength of the waves sampling it. One place wherethis is obviously violated is at a velocity discontinuity such as the core-mantle boundary. If the bodywave is sensitive to structure in the vicinity of a velocity discontinuity it will be incorrectly modeled byray theory. Ray theory can model refraction and reflection effects (which you will be familiar with fromoptics) but it doesn’t model diffraction effects which are sometimes observed on seismograms. Consider amonochromatic wave traveling with a velocity c, we have

λ = cT

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Fig. 2.8 Top two show particle motions of P and S waves – we include the particle motions for the twotypes of surface waves in the lower panels

If we want to work with small wavelengths then this equation tells us we have to use short period disturbances.At periods of about 1 second, ray theory does give an accurate description of the travel times of body wavesand so we shall employ it.

Consider a wave front propagating in a constant velocity medium:The wave front is spherical and we think of lines perpendicular to the wave front as rays. If the velocityvaries with depth the wave front becomes deformed and rays are bent (refraction), e.g.,The time it takes for a ray (or wave front) to go from the source to receiver depends upon the distributionof velocity within the Earth. If we can measure these travel times, with a little effort we can recover thevelocity distribution.

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Vs =!

µ

!(2.3)

µ is zero in a fluid so shear waves are not transmitted through a fluid. We define ", the seismic parameter,by

" =Ks

!= V 2

p ! 43V 2

s (2.4)

The seismic parameter " will be used in later chapters and can be directly calculated if Vp and Vs are

known as a function of radius.

For a physically realistic material Vp is always greater than Vs. We can show this by noting that the

bulk modulus is always positive so

Vp =

"Ks + 4

3µ

!>

"43µ

!>

!µ

!= Vs

Therefore Vp > Vs, i.e., compressional waves travel faster then shear waves.

One of the principle uses of seismology has been to determine the internal structure of the Earth. The

observations are in terms of the time it takes for seismic waves to travel from a source to a receiver and

can be used to infer the seismic velocity throughout the Earth. This information can in turn be used to

help constrain the thermal and chemical state of the Earth’s interior. In order to relate travel times of

body waves to the velocity-depth distributions, we use geometrical ray theory.

Ray theory is an approximate description of body wave propagation and is valid when the wavelength

of the disturbance # is “sufficiently small.” By this we mean that the length scale on which the velocitychanges in the Earth should be large relative to the wavelength of the waves sampling it. One place where

this is obviously violated is at a velocity discontinuity such as the core-mantle boundary. If the body

wave is sensitive to structure in the vicinity of a velocity discontinuity it will be incorrectly modeled

by ray theory. Ray theory can model refraction and reflection effects (which you will be familiar with

from optics) but it doesn’t model diffraction effects which are sometimes observed on seismograms.

Consider a monochromatic wave traveling with a velocity c, we have

# = cT

If we want to work with small wavelengths then this equation tells us we have to use short period

disturbances. At periods of about 1 second, ray theory does give an accurate description of the travel

times of body waves and so we shall employ it.

Consider a wave front propagating in a constant velocity medium:

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The wave front is spherical and we think of lines perpendicular to the wave front as rays. If the velocity

varies with depth the wave front becomes deformed and rays are bent (refraction), e.g.,

Fig. 2.9

The time it takes for a ray (or wave front) to go from the source to receiver depends upon the distribution

of velocity within the Earth. If we can measure these travel times, with a little effort we can recover the

velocity distribution.

2.5 Ray theory . First we will consider how rays travel in a flat Earth which is laterally homogeneous

and heterogeneous in depth. There is an exact transformation for the results from this case to those of a

spherically symmetric Earth but the algebra is much simpler in the flat case.

Seismic ray theory is analogous to geometrical optics so we shall start by considering what happens

to a ray of light as it passes from a medium of refractive index n1 to a medium of refractive index n2.

Fig. 2.10

You are probably familiar with Snell’s Law which gives

n1 sin !1 = n2 sin !2

For seismic waves Snell’s Law is

sin !1

V1=sin !2

V2

where V1 is the velocity in medium 1 and V2 is the velocity in medium 2. If we define the slowness, u,to be the reciprocal of the velocity, V we have

u1 sin !1 = u2 sin !2

27

Fig. 2.9

2.5 Ray theory . First we will consider how rays travel in a flat Earth which is laterally homogeneousand heterogeneous in depth. There is an exact transformation for the results from this case to those of aspherically symmetric Earth but the algebra is much simpler in the flat case.

Seismic ray theory is analogous to geometrical optics so we shall start by considering what happens to aray of light as it passes from a medium of refractive index n1 to a medium of refractive index n2.

The wave front is spherical and we think of lines perpendicular to the wave front as rays. If the velocity

varies with depth the wave front becomes deformed and rays are bent (refraction), e.g.,

Fig. 2.9

The time it takes for a ray (or wave front) to go from the source to receiver depends upon the distribution

of velocity within the Earth. If we can measure these travel times, with a little effort we can recover the

velocity distribution.

2.5 Ray theory . First we will consider how rays travel in a flat Earth which is laterally homogeneous

and heterogeneous in depth. There is an exact transformation for the results from this case to those of a

spherically symmetric Earth but the algebra is much simpler in the flat case.

Seismic ray theory is analogous to geometrical optics so we shall start by considering what happens

to a ray of light as it passes from a medium of refractive index n1 to a medium of refractive index n2.

Fig. 2.10


n1 sin !1 = n2 sin !2


sin !1

V1=sin !2

V2

where V1 is the velocity in medium 1 and V2 is the velocity in medium 2. If we define the slowness, u,to be the reciprocal of the velocity, V we have

u1 sin !1 = u2 sin !2

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Fig. 2.10


n1 sin θ1 = n2 sin θ2


sin θ1V1

=sin θ2V2

where V1 is the velocity in medium 1 and V2 is the velocity in medium 2. If we define the slowness, u, to bethe reciprocal of the velocity, V we have

u1 sin θ1 = u2 sin θ2

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(In fact the refractive index of a material is the speed of light in vacuum divided by the speed of light inthe medium so n is a normalized slowness.) If we now have a stack of homogeneous layers with velocityincreasing in each layer we find:

(In fact the refractive index of a material is the speed of light in vacuum divided by the speed of light

in the medium so n is a normalized slowness.) If we now have a stack of homogeneous layers withvelocity increasing in each layer we find:

Fig 2.11

u1 sin !1 = u2 sin !2 = u3 sin !3 = · · ·In a continuous medium we get

Fig. 2.12

If !0 is the take-off angle of the ray and u0 = 1/V0 is the surface slowness then u0 sin !0 = u sin ! andis constant along the ray. u sin ! is called the ray parameter, p, and a ray is uniquely specified by its rayparameter. At the “turning point” of the ray, when ! = !

2 we have

p = u0 sin !0 = uTP sin"

2= uTP

Therefore p is also the slowness at the turning point of the ray. Clearly if we can measure p, we have thevelocity at the turning point. How might we go about measuring p? Consider a wave front propagatingtowards the surface in a uniform velocity medium:

28

Fig. 2.11

u1 sin θ1 = u2 sin θ2 = u3 sin θ3 = · · ·

In a continuous medium we get



Fig 2.11


Fig. 2.12


2 we have


2= uTP


28

Fig. 2.12If θ0 is the take-off angle of the ray and u0 = 1/V0 is the surface slowness then u0 sin θ0 = u sin θ and isconstant along the ray. u sin θ is called the ray parameter, p, and a ray is uniquely specified by its rayparameter. At the “turning point” of the ray, when θ = π

2 we have

p = u0 sin θ0 = uTP sinπ

2= uTP




Fig 2.11


Fig. 2.12


2 we have


2= uTP


28Fig. 2.13From the diagram ∆ = δx sin θ but we also have ∆ = δt/uwhere u is the slowness of the medium. Therefore

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δt = δxu sin θ

orδt

δx= u sin θ = p

By measuring the difference in time of arrival of a wavefront at two stations of known distance apart, wecan measure p. This argument is only valid for a uniform velocity medium. If the velocity varies with depththen the argument generalizes to p = dT/dX, the slope of the travel-time curve.

Fig. 2.13

From the diagram ! = !x sin " but we also have ! = !t/u where u is the slowness of the medium.Therefore

!t = !xu sin "

or!t

!x= u sin " = p

By measuring the difference in time of arrival of a wavefront at two stations of known distance apart,

we can measure p. This argument is only valid for a uniform velocity medium. If the velocity varieswith depth then the argument generalizes to p = dT/dX , the slope of the travel-time curve.

Fig. 2.14

The travel time curve is a plot of the observed time it takes for a particular ray to travel from the source

to the receiver as a function of distance of that receiver from the source. The slope of this curve is the ray

parameter. In general p varies along the travel time curve which means that a different ray is responsiblefor the arrival at each distance.

Now consider a segment of the ray

Fig. 2.15

We have

dx

d!= sin " and

dz

d!= cos " = (1 ! sin 2")1/2

but

p = u sin " sodx

d!=

p

uand

dz

d!=

!1 ! p2

u2

"1/2

=1u

(u2 ! p2)1/2

29

Fig. 2.14The travel time curve is a plot of the observed time it takes for a particular ray to travel from the sourceto the receiver as a function of distance of that receiver from the source. The slope of this curve is the rayparameter. In general p varies along the travel time curve which means that a different ray is responsible forthe arrival at each distance.Now consider a segment of the ray

Fig. 2.13

From the diagram ! = !x sin " but we also have ! = !t/u where u is the slowness of the medium.Therefore

!t = !xu sin "

or!t

!x= u sin " = p

By measuring the difference in time of arrival of a wavefront at two stations of known distance apart,

we can measure p. This argument is only valid for a uniform velocity medium. If the velocity varieswith depth then the argument generalizes to p = dT/dX , the slope of the travel-time curve.

Fig. 2.14

The travel time curve is a plot of the observed time it takes for a particular ray to travel from the source

to the receiver as a function of distance of that receiver from the source. The slope of this curve is the ray

parameter. In general p varies along the travel time curve which means that a different ray is responsiblefor the arrival at each distance.

Now consider a segment of the ray

Fig. 2.15

We have

dx

d!= sin " and

dz

d!= cos " = (1 ! sin 2")1/2

but

p = u sin " sodx

d!=

p

uand

dz

d!=

!1 ! p2

u2

"1/2

=1u

(u2 ! p2)1/2

29

Fig. 2.15We have

dx

d∆= sin θ and

dz

d∆= cos θ = (1− sin 2θ)1/2

but

p = u sin θ sodx

d∆=p

uand

dz

d∆=

(1− p2

u2

)1/2

=1

u(u2 − p2)1/2

We can compute how far a ray will travel in the x direction by noting

dx

dz=

dx

d∆

d∆

dz=p

u

u

(u2 − p2)1/2

If we integrate this expression between two depths we will get the horizontal distance traveled:

26


dx

dz=

dx

d!d!dz

=p

u

u

(u2 ! p2)1/2


Fig. 2.16

X(z1, z2, p) = p

z2!

z1

dz

(u2 ! p2)1/2

(remember that u is a function of depth, u = u(z)). For a ray traveling from a source at the surface to areceiver at the surface we have

Fig. 2.17

X(p) = 2p

ZT P!

0

dz

(u2 ! p2)1/2(2.5)

where ZTP is the turning point depth. (Note that there is an integrable square root singularity at the

turning point in the integrand because at z = ZTP , u = p). To get the travel time for a given rayspecified by p we use the fact that dT = ud! for a segment of the ray, i.e.,

dT

d!= u and

dT

dz=

dT

d!d!dz

=u2

(u2 ! p2)1/2

For a ray traveling from source to receiver at the surface we have (following the logic for the calculation

of X(p))

T (p) = 2ZT P!

0

u2(z)(u2 ! p2)1/2

dz (2.6)

30

Fig. 2.16

X(z1, z2, p) = p

z2∫z1

dz

(u2 − p2)1/2



dx

dz=

dx

d!d!dz

=p

u

u

(u2 ! p2)1/2


Fig. 2.16

X(z1, z2, p) = p

z2!

z1

dz

(u2 ! p2)1/2


Fig. 2.17

X(p) = 2p

ZT P!

0

dz

(u2 ! p2)1/2(2.5)

where ZTP is the turning point depth. (Note that there is an integrable square root singularity at the

turning point in the integrand because at z = ZTP , u = p). To get the travel time for a given rayspecified by p we use the fact that dT = ud! for a segment of the ray, i.e.,

dT

d!= u and

dT

dz=

dT

d!d!dz

=u2

(u2 ! p2)1/2

For a ray traveling from source to receiver at the surface we have (following the logic for the calculation

of X(p))

T (p) = 2ZT P!

0

u2(z)(u2 ! p2)1/2

dz (2.6)

30

Fig. 2.17

X(p) = 2p

ZTP∫0

dz

(u2 − p2)1/2(2.5)

where ZTP is the turning point depth. (Note that there is an integrable square root singularity at the turningpoint in the integrand because at z = ZTP , u = p). To get the travel time for a given ray specified by p weuse the fact that dT = ud∆ for a segment of the ray, i.e.,

dT

d∆= u and

dT

dz=dT

d∆

d∆

dz=

u2

(u2 − p2)1/2

For a ray traveling from source to receiver at the surface we have (following the logic for the calculation ofX(p))

T (p) = 2

ZTP∫0

u2(z)

(u2 − p2)1/2dz (2.6)

This gives us the travel time of a ray. The combination τ(p) = T (p)− pX(p) is a useful number and is calledthe delay time. Combining the integrals for X(p) and T (p) gives

τ(p) = 2

ZTP∫0

(u2 − p2)1/2dz (2.7)

An interpretation of τ is given in this diagram:

27

This gives us the travel time of a ray. The combination !(p) = T (p)! pX(p) is a useful number and iscalled the delay time. Combining the integrals for X(p) and T (p) gives

!(p) = 2ZT P!

0

(u2 ! p2)1/2dz (2.7)

An interpretation of ! is given in this diagram:

Fig. 2.18

!(p) is the intercept on the T axis and measures the time the ray spends travelling vertically. (This canbe seen from equation 2.7 where (u2 ! p2)1/2 can be interpreted as the vertical slowness).

We could write a computer program to computeX(p), T (p) and !(p) by approximating the velocitydistribution by a set of homogeneous layers:

Fig. 2.19

The integrals then become sums over layers:

X(p) = 2p"

i

hi

(u2i ! p2)1/2

for ui > p

T (p) = 2"

i

hiu2i

(u2i ! p2)1/2

for ui > p

!(p) = 2"

i

hi(u2i ! p2)1/2 for ui > p

31

Fig. 2.18τ(p) is the intercept on the T axis and measures the time the ray spends travelling vertically. (This can beseen from equation 2.7 where (u2 − p2)1/2 can be interpreted as the vertical slowness).

We could write a computer program to compute X(p), T (p) and τ(p) by approximating the velocitydistribution by a set of homogeneous layers:

This gives us the travel time of a ray. The combination !(p) = T (p)! pX(p) is a useful number and iscalled the delay time. Combining the integrals for X(p) and T (p) gives

!(p) = 2ZT P!

0

(u2 ! p2)1/2dz (2.7)

An interpretation of ! is given in this diagram:

Fig. 2.18

!(p) is the intercept on the T axis and measures the time the ray spends travelling vertically. (This canbe seen from equation 2.7 where (u2 ! p2)1/2 can be interpreted as the vertical slowness).

We could write a computer program to computeX(p), T (p) and !(p) by approximating the velocitydistribution by a set of homogeneous layers:

Fig. 2.19

The integrals then become sums over layers:

X(p) = 2p"

i

hi

(u2i ! p2)1/2

for ui > p

T (p) = 2"

i

hiu2i

(u2i ! p2)1/2

for ui > p

!(p) = 2"

i


31

Fig. 2.19The integrals then become sums over layers:

X(p) = 2p∑i

hi(u2i − p2)1/2

for ui > p

T (p) = 2∑i

hiu2i

(u2i − p2)1/2for ui > p

τ(p) = 2∑i

hi(u2i − p2)1/2 for ui > p

The condition ui > p must be imposed so that we include in our sum only those layers above the turningpoint of the ray. (Remember u = p at the turning point and u < p below the turning point).τ is useful as it has a very simple behavior when plotted against p. To see this we note that

dτ

dp= −2p

ZTP∫0

dz

(u2 − p2)1/2= −X(p) and

d2τ

dp2= −dX

dp

As X is always positive, dτ/dp is always negative so τ decreases as p increases:

28

The condition ui > pmust be imposed so that we include in our sum only those layers above the turningpoint of the ray. (Remember u = p at the turning point and u < p below the turning point).! is useful as it has a very simple behavior when plotted against p. To see this we note that

d!

dp= !2p

ZT P!

0

dz

(u2 ! p2)1/2= !X(p) and

d2!

dp2= !dX

dp

As X is always positive, d!/dp is always negative so ! decreases as p increases:

Fig. 2.20

In simple structures, where the velocity increases smoothly with depth, dX/dp is usually negative. Tosee this, consider the diagram:

Fig. 2.21

i.e., X increases as p decreases. If this is the case for a branch of the travel time curve, the branch issaid to be prograde (if dX/dp is positive the branch is said to be retrograde). For a prograde branchthe !(p) curve is concave upwards as

Fig. 2.22

2.6 Travel-time curves and complicated structures . Consider a velocity structure with a rapid velocity

32

Fig. 2.20

In simple structures, where the velocity increases smoothly with depth, dX/dp is usually negative. To seethis, consider the diagram:


d!

dp= !2p

ZT P!

0

dz

(u2 ! p2)1/2= !X(p) and

d2!

dp2= !dX

dp


Fig. 2.20


Fig. 2.21


Fig. 2.22


32

Fig. 2.21

i.e., X increases as p decreases. If this is the case for a branch of the travel time curve, the branch is said tobe prograde (if dX/dp is positive the branch is said to be retrograde). For a prograde branch the τ(p) curveis concave upwards as


d!

dp= !2p

ZT P!

0

dz

(u2 ! p2)1/2= !X(p) and

d2!

dp2= !dX

dp


Fig. 2.20


Fig. 2.21


Fig. 2.22


32

Fig. 2.22

29

2.6 Travel-time curves and complicated structures . Consider a velocity structure with a rapid velocityincrease in a depth zone:increase in a depth zone:

Fig. 2.23

The travel-time curve for this situation looks a bit more complicated and dX/dp will change sign as pvaries.

Fig. 2.24

This feature is called a triplication as there are three arrivals at a given distance. When dX/dp = 0 alarge number of rays converge at the same point giving a large amplitude arrival – this point is called a

caustic. The !(p) curve is much simpler, the points at which dX/dp = 0 correspond to inflections inthe curve, i.e.,

Fig. 2.25

A steep velocity increase as shown above is sometimes modeled using a step discontinuity in velocity

33

Fig. 2.23The travel-time curve for this situation looks a bit more complicated and dX/dp will change sign as p varies.

increase in a depth zone:

Fig. 2.23


Fig. 2.24



Fig. 2.25


33

Fig. 2.24This feature is called a triplication as there are three arrivals at a given distance. When dX/dp = 0 a largenumber of rays converge at the same point giving a large amplitude arrival – this point is called a caustic.The τ(p) curve is much simpler, the points at which dX/dp = 0 correspond to inflections in the curve, i.e.,

increase in a depth zone:

Fig. 2.23


Fig. 2.24



Fig. 2.25


33

Fig. 2.25A steep velocity increase as shown above is sometimes modeled using a step discontinuity in velocity andhomogeneous layers, i.e.,

30

and homogeneous layers, i.e.,

Fig. 2.26

For a single jump in velocity, we get several kinds of rays. The most obvious is the direct ray which

travels along the surface. As p decreases we get reflections off the discontinuity. These are calledpost-critical reflection for reasons which will become obvious. As the takeoff angle of the ray steepens

further (p decreases) we will arrive at the situation where the ray is turned along the discontinuity. Theangle at which this occurs is called the critical angle and this happens when

u1 sin !c = u2 sin"

2= u2 (u1 =

1v1

, u2 =1v2

)

This can be thought of as a disturbance traveling along the interface (this is called a “head wave”) and

energy is radiated toward the surface as the disturbance propagates. As p decreases yet further we getreflections off the interface again but these are now called pre-critical reflections. The travel time curve

for this situation looks very similar to the previous one but the direct wave and head wave branches are

straight lines and, in principle, go on forever.

Fig. 2.27

Suppose now we have a low velocity zone. In the case when a layer is underlain by a layer of lower

34

Fig. 2.26

For a single jump in velocity, we get several kinds of rays. The most obvious is the direct ray which travelsalong the surface. As p decreases we get reflections off the discontinuity. These are called post-criticalreflection for reasons which will become obvious. As the takeoff angle of the ray steepens further (pdecreases) we will arrive at the situation where the ray is turned along the discontinuity. The angle at whichthis occurs is called the critical angle and this happens when

u1 sin θc = u2 sinπ

2= u2 (u1 =

1

v1, u2 =

1

v2)

This can be thought of as a disturbance traveling along the interface (this is called a “head wave”) and energyis radiated toward the surface as the disturbance propagates. As p decreases yet further we get reflections offthe interface again but these are now called pre-critical reflections. The travel time curve for this situationlooks very similar to the previous one but the direct wave and head wave branches are straight lines and, inprinciple, go on forever.

and homogeneous layers, i.e.,

Fig. 2.26

For a single jump in velocity, we get several kinds of rays. The most obvious is the direct ray which

travels along the surface. As p decreases we get reflections off the discontinuity. These are calledpost-critical reflection for reasons which will become obvious. As the takeoff angle of the ray steepens

further (p decreases) we will arrive at the situation where the ray is turned along the discontinuity. Theangle at which this occurs is called the critical angle and this happens when

u1 sin !c = u2 sin"

2= u2 (u1 =

1v1

, u2 =1v2

)

This can be thought of as a disturbance traveling along the interface (this is called a “head wave”) and

energy is radiated toward the surface as the disturbance propagates. As p decreases yet further we getreflections off the interface again but these are now called pre-critical reflections. The travel time curve

for this situation looks very similar to the previous one but the direct wave and head wave branches are

straight lines and, in principle, go on forever.

Fig. 2.27

Suppose now we have a low velocity zone. In the case when a layer is underlain by a layer of lower

34

Fig. 2.27

Suppose now we have a low velocity zone. In the case when a layer is underlain by a layer of lower velocity,the ray is refracted towards the vertical:

31

velocity, the ray is refracted towards the vertical:

Fig. 2.28

If we have a low velocity zone we get the following effect:

Fig. 2.29

Note that the rays which emerge on either side of the shadow zone have almost identical ray parameters

so the slopes of the travel times curves on either side of the shadow zone are identical. There is at least

one major global low velocity zone inside the Earth. At the core-mantle boundary the compressional

velocity drops from 13km/sec to less than 10km/sec. The result is a shadow zone for P -waves extendingfrom a distance range of 99! ! 145!. The effect on the !(p) curve is to give it a discontinuity:

Fig. 2.30

35

Fig. 2.28



Fig. 2.28


Fig. 2.29





Fig. 2.30

35

Fig. 2.29

Note that the rays which emerge on either side of the shadow zone have almost identical ray parameters sothe slopes of the travel times curves on either side of the shadow zone are identical. There is at least onemajor global low velocity zone inside the Earth. At the core-mantle boundary the compressional velocitydrops from 13km/sec to less than 10km/sec. The result is a shadow zone for P -waves extending from adistance range of 99◦ → 145◦. The effect on the τ(p) curve is to give it a discontinuity:

32


Fig. 2.28


Fig. 2.29





Fig. 2.30

35

Fig. 2.30To complete our catalog of structures we consider the effect of a discontinuity in seismic properties.This gives rise to reflections and conversions as well as refractions. The following diagram sums up thepossibilities for an incident P wave.

To complete our catalog of structures we consider the effect of a discontinuity in seismic properties.

This gives rise to reflections and conversions as well as refractions. The following diagram sums up the

possibilities for an incident P wave.

Fig. 2.31

Note that p remains constant for all ray types, i.e.,

p = uP1 sin !i = uP

1 sin !Pr

so !i = !Pr, but we also have

p = uP2 sin !PR = uS

1 sin !Sr = uS2 sin !SR

2.7 Flat-Earth transformation . Until now we have done all our analysis for a flat Earth and, as

mentioned earlier there is an exact mapping to the spherical case. Usually our programs are written for

a flat Earth as the equations are simpler so we want a method of converting a spherical model to the

equivalent flat structure. If r is the radius variable and a is the radius of the Earth we can define a newflat Earth depth variable z by

z = !a ln! r

a

"

and V fp (z) = Vp(r)

a

r

and V fS (z) = VS(r)

a

rwhere the superscript f indicates a property in the equivalent flat Earth. This transformation retains allthe geometrical properties of rays. The integrals for T (p),X(p) and "(p) can be derived for a sphericalEarth and for reference we give them here:

T (p) = 2a#

RT P

u

$1 ! p2

u2r2

%!1/2

dr

X(p) = 2a#

RT P

p

r2u

$1 ! p2

u2r2

%!1/2

dr

36

Fig. 2.31Note that p remains constant for all ray types, i.e.,

p = uP1 sin θi = uP1 sin θPrso θi = θPr, but we also have

p = uP2 sin θPR = uS1 sin θSr = uS2 sin θSR2.7 Flat-Earth transformation . Until now we have done all our analysis for a flat Earth and, as mentionedearlier there is an exact mapping to the spherical case. Usually our programs are written for a flat Earthas the equations are simpler so we want a method of converting a spherical model to the equivalent flatstructure. If r is the radius variable and a is the radius of the Earth we can define a new flat Earth depthvariable z by

z = −a ln( ra

)and V fp (z) = Vp(r)

a

r

and V fS (z) = VS(r)a

r

33

where the superscript f indicates a property in the equivalent flat Earth. This transformation retains all thegeometrical properties of rays. The integrals for T (p), X(p) and τ(p) can be derived for a spherical Earthand for reference we give them here:

T (p) = 2

a∫RTP

u

(1− p2

u2r2

)−1/2

dr

X(p) = 2

a∫RTP

p

r2u

(1− p2

u2r2

)−1/2

dr

τ(p) = 2

a∫RTP

u

(1− p2

u2r2

)1/2

dr

where p = ur sin θ and RTP is the radius at which the ray turns.For studies of shallow structure the transformation has little effect on the velocities and depth variable

and the Earth can be treated as flat.

2.8 Estimating travel-times . The discussion thus far should have illustrated the fact that travel-time curvescan be quite complicated and contain a lot of information. (Features such as triplications and shadow zonesexist in the real Earth travel-time curves.) How do we go about estimating travel time curves? To do this weclearly must know the origin time and location of the source. Nowadays an ideal kind of source for gettingtravel times is a nuclear explosion but these were not available in the past. Earthquake locations and traveltimes have improved in a see-saw fashion.

To start the process some events with known or estimated locations are required. These are obtainedby determining the locality of most intense shaking by inspecting the damage after an earthquake and bynoting the time on stopped clocks, etc. In southern California there is a network of seismographs and localtravel-times could be found in this region. Local travel times look like:

!(p) = 2a!

RT P

u

"1 ! p2

u2r2

#1/2

dr

where p = ur sin " and RTP is the radius at which the ray turns.

For studies of shallow structure the transformation has little effect on the velocities and depth variable

and the Earth can be treated as flat.

2.8 Estimating travel-times . The discussion thus far should have illustrated the fact that travel-time

curves can be quite complicated and contain a lot of information. (Features such as triplications and

shadow zones exist in the real Earth travel-time curves.) How do we go about estimating travel time

curves? To do this we clearly must know the origin time and location of the source. Nowadays an ideal

kind of source for getting travel times is a nuclear explosion but these were not available in the past.

Earthquake locations and travel times have improved in a see-saw fashion.

To start the process some events with known or estimated locations are required. These are obtained

by determining the locality of most intense shaking by inspecting the damage after an earthquake and

by noting the time on stopped clocks, etc. In southern California there is a network of seismographs and

local travel-times could be found in this region. Local travel times look like:

Fig. 2.32

When several events are available, consistency between the travel-times from the different events can

be checked and a best set of travel-times chosen. Once we have these preliminary travel-times we can

relocate all the events by moving the earthquakes around until the differences between the measured

travel times and the predicted times (travel time residuals) are minimized (a least-square procedure can

be used to do this). These are now our improved estimates of location. We then reinspect the residuals to

make sure there are no systematic differences between the predicted travel times and observed times. If

there are, we adjust the travel time curves accordingly and relocate the events. Convergence to a “best”

set of locations and travel times occurs after two or three iterations of this cycle. Global travel times

can be determined using the locations derived from local networks. The method of improving the travel

times at large distances from the event is the same as for the local travel times. A global set of travel

37

Fig. 2.32When several events are available, consistency between the travel-times from the different events can bechecked and a best set of travel-times chosen. Once we have these preliminary travel-times we can relocateall the events by moving the earthquakes around until the differences between the measured travel times andthe predicted times (travel time residuals) are minimized (a least-square procedure can be used to do this).These are now our improved estimates of location. We then reinspect the residuals to make sure there areno systematic differences between the predicted travel times and observed times. If there are, we adjust thetravel time curves accordingly and relocate the events. Convergence to a “best” set of locations and traveltimes occurs after two or three iterations of this cycle. Global travel times can be determined using thelocations derived from local networks. The method of improving the travel times at large distances from theevent is the same as for the local travel times. A global set of travel times is shown in Figure 2.33.

34

Fig. 2.33. A global set of travel times

35

A simple method for locating local events once travel times curves have been obtained is the three circlemethod. This involves measuring the difference in time of arrival between the S arrival and P arrival at atleast three stations. As can be seen from the diagram this S − P time is characteristic of the distance of thestation from the event. With three stations we can draw circles around each station with radii correspondingto the distances from the event. These circles should intersect at the location of the Earthquake, e.g.,

Earthquake, e.g.,

Fig. 2.34

Implicit in this discussion is that the events are all shallow. Travel time curves for deep events are

systematically different from the travel times of shallow events, i.e.,

Fig. 2.35

At stations near to the source, the travel time of a ray from a shallow event (A) is shorter than the travel

time from a deep event (B). At stations far away, the opposite is true so the travel time curves for the

two types look like

Fig. 2.36

To avoid biasing our estimates of travel times, we must be able to distinguish between deep and shallow

events. Shallow earthquakes generate large surface waves with periods of about 20 sec. These waves

have their energy concentrated in the lithosphere. Earthquakes occurring below the lithosphere do not

39

Fig. 2.34Implicit in this discussion is that the events are all shallow. Travel time curves for deep events are


Earthquake, e.g.,

Fig. 2.34



Fig. 2.35



two types look like

Fig. 2.36




39

Fig. 2.35At stations near to the source, the travel time of a ray from a shallow event (A) is shorter than the travel timefrom a deep event (B). At stations far away, the opposite is true so the travel time curves for the two typeslook like

Earthquake, e.g.,

Fig. 2.34



Fig. 2.35



two types look like

Fig. 2.36




39

Fig. 2.36To avoid biasing our estimates of travel times, we must be able to distinguish between deep and shallowevents. Shallow earthquakes generate large surface waves with periods of about 20 sec. These waves havetheir energy concentrated in the lithosphere. Earthquakes occurring below the lithosphere do not excite

36

these surface waves very well and the lack of 20 sec surface waves is a good clue that an event is deep.Another feature of deep earthquakes is a small arrival after the main P arrival, this arrival is called pP (“littlep big P”) and is due to a reflection from the surface:

excite these surface waves very well and the lack of 20 sec surface waves is a good clue that an event

is deep. Another feature of deep earthquakes is a small arrival after the main P arrival, this arrival is

called pP (“little p big P ”) and is due to a reflection from the surface:

Fig. 2.37

In shallow earthquakes, the surface reflection and the direct P arrival are so close together that they give

a single pulse on the seismogram. This is not true for deep events and pP can be used to estimate the

depth of the event.

As can be seen from this example, different rays are given different names. We must be aware of all

possible ray types to make a complete catalog. Note that “sharp” discontinuities in seismic properties

give refraction, reflection and conversion of one ray type to another, i.e., P ! S. The definition of“sharp” refers to a change which occurs over a depth range which is small compared with the wavelength

of the wave sampling it. An example of a sharp discontinuity would be the core-mantle boundary. An

example of a strong change in velocity would be a phase transition in the upper mantle such as the rapid

change in properties at a depth of 400 km. Consider the following ray traveling through the Earth.

Fig. 2.38

Suppose that the ray consists of compressional wave motion in the mantle legs of the ray and, because

the outer core is fluid and does not sustain shear waves, the leg of the ray in the outer core must also be

compressional wave motion.

We use the following notation

Refracted rays

P compressional in mantle

S shear in mantle

K compressional in the outer core

I compressional in the inner core

J shear in the inner core

40

Fig. 2.37In shallow earthquakes, the surface reflection and the direct P arrival are so close together that they give asingle pulse on the seismogram. This is not true for deep events and pP can be used to estimate the depthof the event.

As can be seen from this example, different rays are given different names. We must be aware of allpossible ray types to make a complete catalog. Note that “sharp” discontinuities in seismic properties giverefraction, reflection and conversion of one ray type to another, i.e., P → S. The definition of “sharp”refers to a change which occurs over a depth range which is small compared with the wavelength of thewave sampling it. An example of a sharp discontinuity would be the core-mantle boundary. An exampleof a strong change in velocity would be a phase transition in the upper mantle such as the rapid change inproperties at a depth of 400 km. Consider the following ray traveling through the Earth.

excite these surface waves very well and the lack of 20 sec surface waves is a good clue that an event

is deep. Another feature of deep earthquakes is a small arrival after the main P arrival, this arrival is

called pP (“little p big P ”) and is due to a reflection from the surface:

Fig. 2.37

In shallow earthquakes, the surface reflection and the direct P arrival are so close together that they give

a single pulse on the seismogram. This is not true for deep events and pP can be used to estimate the

depth of the event.

As can be seen from this example, different rays are given different names. We must be aware of all

possible ray types to make a complete catalog. Note that “sharp” discontinuities in seismic properties

give refraction, reflection and conversion of one ray type to another, i.e., P ! S. The definition of“sharp” refers to a change which occurs over a depth range which is small compared with the wavelength

of the wave sampling it. An example of a sharp discontinuity would be the core-mantle boundary. An

example of a strong change in velocity would be a phase transition in the upper mantle such as the rapid

change in properties at a depth of 400 km. Consider the following ray traveling through the Earth.

Fig. 2.38

Suppose that the ray consists of compressional wave motion in the mantle legs of the ray and, because

the outer core is fluid and does not sustain shear waves, the leg of the ray in the outer core must also be

compressional wave motion.

We use the following notation

Refracted rays

P compressional in mantle

S shear in mantle

K compressional in the outer core

I compressional in the inner core

J shear in the inner core

40

Fig. 2.38Suppose that the ray consists of compressional wave motion in the mantle legs of the ray and, becausethe outer core is fluid and does not sustain shear waves, the leg of the ray in the outer core must also becompressional wave motion.We use the following notation

Refracted raysP compressional in mantleS shear in mantleK compressional in the outer coreI compressional in the inner coreJ shear in the inner core

Reflected raysc reflection from the core-mantle boundaryi reflection from the inner core boundary

37

The ray in the diagram is called PKP. Here are two other examples

Reflected rays

c reflection from the core-mantle boundary

i reflection from the inner core boundary


Fig. 2.39

An underside reflection does not have a symbol so aP -wave which bounces off the surface once betweenthe source and receiver is called PP if it does not enter the core, e.g.,

Fig. 2.40

(You may be wondering what distinguishes PP from pP . A ray leg that emerges upwards from the

source is given a lower case letter while a ray leg that emerges downwards from the source is given an

upper case letter.)

We are now in a position to discuss the global travel time curves shown in figure 2.33. Consider the

41

Fig. 2.39

An underside reflection does not have a symbol so a P -wave which bounces off the surface once betweenthe source and receiver is called PP if it does not enter the core, e.g.,

Reflected rays

c reflection from the core-mantle boundary

i reflection from the inner core boundary


Fig. 2.39

An underside reflection does not have a symbol so aP -wave which bounces off the surface once betweenthe source and receiver is called PP if it does not enter the core, e.g.,

Fig. 2.40

(You may be wondering what distinguishes PP from pP . A ray leg that emerges upwards from the

source is given a lower case letter while a ray leg that emerges downwards from the source is given an

upper case letter.)

We are now in a position to discuss the global travel time curves shown in figure 2.33. Consider the

41

Fig. 2.40

(You may be wondering what distinguishes PP from pP . A ray leg that emerges upwards from the sourceis given a lower case letter while a ray leg that emerges downwards from the source is given an upper caseletter.)

We are now in a position to discuss the global travel time curves shown in figure 2.33. Consider theP -wave travel time curve. Enlarged it looks like

38

P -wave travel time curve. Enlarged it looks like

Fig. 2.41

There are triplications near 20! (not shown in the original) due to the sharp velocity jumps associatedwith the major upper mantle discontinuities (at !400km and !670km depth). The true geometrical

P -wave arrival then extends to an epicentral distance of about 100!. The PcP branch corresponds to

waves which are reflected off the core-mantle boundary. P wave arrivals are observed beyond 100! andcorrespond to diffraction effects around the core-mantle boundary. These arrivals are not predicted by

geometrical ray theory but, just as light can be diffracted around corners, so can sound waves. The Pwave branch continues on at larger distances as PKP and the offset is due to the lower velocity of Pwaves in the outer core. You will also note from figure 2.33 that the S-wave travel time curve stops ata distance of about 100!. This tells us that, not only does the outer core have a lower P wave velocity

than the mantle, but also that the outer core does not transmit shear waves, i.e., it is a fluid.

2.9 Solving for velocity . Until now we have been considering the “forward” problem, i.e., given a

velocity structure, compute observable quantities such as the travel time of a ray or !(p). Wemust be ableto solve the forward problem before we can consider the real problem of interest – the “inverse” problem,

i.e., given observed quantities such as travel times or !(p), what is the velocity/depth distribution? Thesevelocities give us the elastic properties of the Earth which can be used in other fields of geophysics. (For

example we will use the “seismic parameter” " to compute adiabatic temperature profiles in the Earth.)Inverse problems are what geophysics is all about, i.e., inferring the properties of an inaccessible Earth

from a finite amount of inaccurate data.

In the early 1900’s it was realized by several investigators (Wiechert, Herglotz, Bateman) that the

so-called Abel transform pair could be used to recover velocity profiles from travel time data. The

method requires a complete travel time data set which is perfectly accurate and a velocity profile which

is everywhere increasing with depth. The Abel transform pair can be written:

t(y) =a!

y

f(#)d#"# # y

(2.8)

f(#) = # 1$

d

d#

"

#$a!

!

t(y)dy"y # #

%

&' (2.9)

Note that t(y)must be continuous, t(a)must be zero and t(y)must have a finite derivative with, at most,a finite number of discontinuities.

42

Fig. 2.41There are triplications near 20◦ (not shown in the original) due to the sharp velocity jumps associated withthe major upper mantle discontinuities (at ∼400km and ∼670km depth). The true geometrical P -wavearrival then extends to an epicentral distance of about 100◦. The PcP branch corresponds to waves whichare reflected off the core-mantle boundary. P wave arrivals are observed beyond 100◦ and correspond todiffraction effects around the core-mantle boundary. These arrivals are not predicted by geometrical raytheory but, just as light can be diffracted around corners, so can sound waves. The P wave branch continueson at larger distances as PKP and the offset is due to the lower velocity of P waves in the outer core. Youwill also note from figure 2.33 that the S-wave travel time curve stops at a distance of about 100◦. This tellsus that, not only does the outer core have a lower P wave velocity than the mantle, but also that the outercore does not transmit shear waves, i.e., it is a fluid.

2.9 Solving for velocity . Until now we have been considering the “forward” problem, i.e., given a velocitystructure, compute observable quantities such as the travel time of a ray or τ(p). We must be able to solvethe forward problem before we can consider the real problem of interest – the “inverse” problem, i.e., givenobserved quantities such as travel times or τ(p), what is the velocity/depth distribution? These velocitiesgive us the elastic properties of the Earth which can be used in other fields of geophysics. (For example wewill use the “seismic parameter” φ to compute adiabatic temperature profiles in the Earth.) Inverse problemsare what geophysics is all about, i.e., inferring the properties of an inaccessible Earth from a finite amountof inaccurate data.

In the early 1900’s it was realized by several investigators (Wiechert, Herglotz, Bateman) that the so-called Abel transform pair could be used to recover velocity profiles from travel time data. The methodrequires a complete travel time data set which is perfectly accurate and a velocity profile which is everywhereincreasing with depth. The Abel transform pair can be written:

t(y) =

a∫y

f(ξ)dξ√ξ − y

(2.8)

f(ξ) = − 1

π

d

dξ

a∫ξ

t(y)dy√y − ξ

(2.9)

Note that t(y) must be continuous, t(a) must be zero and t(y) must have a finite derivative with, at most, afinite number of discontinuities.To apply the transform to travel time data we note that

39

X(p)

2p=

ZTP∫0

dz

(u2 − p2)1/2

If we let ξ = u2 we have

X(p)

2p=

p∫u20

dz

dξ

dξ

(ξ − p2)1/2= −

u20∫

p2

dz

dξ

dξ

(ξ − p2)1/2

where u0 is the slowness at z = 0 (the surface). If we now let y = p2, a = u20 and t(y) = X(p)/2p ourequation becomes equivalent to equation 2.8 with

f(ξ) = −dzdξ

Substituting these variable changes into equation 2.9 gives

−dzdξ

= − 1

π

d

dξ

u20∫

ξ

X(p)/2p√p2 − ξ

dp2

As dp2 = 2pdp, we can rewrite this as

dz =1

πd

u0∫ξ1/2

X(p)√p2 − ξ

dp

where z is a function of ξ. If we remember that ξ = u2 and integrate we have

z(u) =1

π

u0∫u

X(p)√p2 − u2

dp+ C

where C is a constant. C can be evaluated by putting u = u0. The integral is zero and we have z(u0) = C = 0because u0 is the surface slowness at which z = 0. Finally, we obtain

z(u) =1

π

u0∫u

X(p)√p2 − u2

dp (2.10)

Our solution is in terms of depth as a function of slowness. To calculate a velocity-depth profile fromsome continuous X(p) data, we choose many different values of u and evaluate the above integral (probablynumerically) to give the depths at which these values of u occur. Unless we are very careful in how X(p)is specified (remember it must be continuous and exact), the solution can be unstable. In fact it is possibleto get unphysical structures such as multiple-valued velocity/depth functions with this method as illustratedon the following page.

40

fact it is possible to get unphysical structures such as multiple-valued velocity/depth functions with this

method.

Fig. 2.42

We shall suggest an alternative way of retrieving the velocity structure from !(p) data. Suppose wehave a travel-time curve T (X), we can compute p = dT/dX and !(p) = T (p)!pX(p) so it is a simplematter in principle to get !(p) data from T (X) data. (In practice, differentiation of an empirical curve isnumerically unstable and better methods exist for estimating !(p) from T (X) data.) Consider equation2.7

!(p) = 2ZT P!

0

(u2 ! p2)1/2dz

We wish to estimate u(z) from !(p). This relation looks nonlinear, i.e., !(p) is not linearly related tou(z); however, looks can be deceiving. Suppose we look for z(u), i.e., depth as a function of slowness– the problem can then be regarded as a linear one and is consequently much easier to solve. This is

easy to see if we parameterize the velocity structure as a set of homogeneous layers

!(p) = 2"

i


Suppose we have N data !j(pj)j = 1, N and we specify ui in each layer. We can write the equation

for the jth datum as

!j ="

i

hiGij

(where Gij = 2(u2i ! p2

j )1/2 and can be computed). This equation represents a set of simultaneous

equations where the unknowns are the layer thicknesses, hi, and there are N equations. Clearly if

N is greater than or equal to the number of layers, the equation can be solved and a velocity structure

determined. The success of thismethod depends upon how clever you are at parameterizing the structure,

errors in the data and independence of the data. Your parameterization allows a unique model to be

constructed but it is only one of an infinite number which will fit a finite dataset. A modern velocity

44

Fig. 2.42

We shall suggest an alternative way of retrieving the velocity structure from τ(p) data. Suppose we havea travel-time curve T (X), we can compute p = dT/dX and τ(p) = T (p) − pX(p) so it is a simple matter inprinciple to get τ(p) data from T (X) data. (In practice, differentiation of an empirical curve is numericallyunstable and better methods exist for estimating τ(p) from T (X) data.) Consider equation 2.7

τ(p) = 2

ZTP∫0

(u2 − p2)1/2dz

We wish to estimate u(z) from τ(p). This relation looks nonlinear, i.e., τ(p) is not linearly related to u(z);however, looks can be deceiving. Suppose we look for z(u), i.e., depth as a function of slowness – theproblem can then be regarded as a linear one and is consequently much easier to solve. This is easy to see ifwe parameterize the velocity structure as a set of homogeneous layers, each with thickness hi and slownessui.

τ(p) = 2∑i

hi(u2i − p2)1/2 for ui > p

Suppose we have N data τj(pj)j = 1, N and we specify ui in each layer. We can write the equation for thejth datum as

τj =∑i

hiGij

(where Gij = 2(u2i − p2j )1/2 and can be computed). This equation represents a set of simultaneous equationswhere the unknowns are the layer thicknesses, hi, and there are N equations. Clearly if N is greater than orequal to the number of layers, the equation can be solved and a velocity structure determined. The successof this method depends upon how clever you are at parameterizing the structure, errors in the data andindependence of the data. Your parameterization allows a unique model to be constructed but it is only oneof an infinite number which will fit a finite dataset. A modern velocity model of the whole Earth is shownin figure 2.43.

41

model of the whole Earth is shown in figure 2.43.

Fig. 2.43. Seismic velocities and density vs. radius for PREM shown as solid lines, and SP6, which is

a modified version of the IASP91 model, shown as dashed lines. PREM and SP6 are in close agreement

throughout the lower mantle and core, with the largest differences occuring in the gradients just above

and below the core-mantle boundary. PREM contains transverse isotropy in the uppermost mantle

between 24.4 and 220 km depth; the lines plotted show the minimum and maximum velocities. PREM

has three major discontinuties in the upper mantle at depths of 220, 400 and 670 km, and an S-wave

low-velocity zone between 80 and 220 km. SP6, like IASP91, contains upper-mantle discontinuities

only at 410 and 660 km, with no low-velocity zone.

Modern spherical Earth models are built using large data sets of free-oscillation frequencies, surface-

wave phase velocities, and the travel-times of body waves. Model PREM is shown Figure 2.43. This

model has a pronounced 220 km which does not appear to be a global feature of the real Earth. Other

spherical Earth models of seismic velocity alone (IASP91 and SP6) have been generated from travel

time data compiled by the International Seismological Commission (ISC).

45

Fig. 2.43. Seismic velocities and density vs. radius for PREM shown as solid lines, and SP6, which isa modified version of the IASP91 model, shown as dashed lines. PREM and SP6 are in close agreementthroughout the lower mantle and core, with the largest differences occuring in the gradients just above andbelow the core-mantle boundary. PREM contains transverse isotropy in the uppermost mantle between 24.4and 220 km depth; the lines plotted show the minimum and maximum velocities. PREM has three majordiscontinuties in the upper mantle at depths of 220, 400 and 670 km, and an S-wave low-velocity zonebetween 80 and 220 km. SP6, like IASP91, contains upper-mantle discontinuities only at 410 and 660 km,with no low-velocity zone.

Modern spherical Earth models are built using large data sets of free-oscillation frequencies, surface-wavephase velocities, and the travel-times of body waves. Model PREM is shown Figure 2.43. This model hasa pronounced 220 km which does not appear to be a global feature of the real Earth. Other spherical Earthmodels of seismic velocity alone (IASP91 and SP6) have been generated from travel time data compiled bythe International Seismological Centre (ISC).

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chapter 2...chapter 2 seismology 2.1 introduction . seismology is a science based on seismograms –...

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