Seimic Waves and Earth’s Interior

1/21/2015 Seimic Waves and Earth’s Interior

http://eqseis.geosc.psu.edu/~cammon/HTML/Classes/IntroQuakes/Notes/waves_and_interior.html 1/13

SLUEASA193 Class

Notes

Seismic Waves and Earth's Interior

Introduction

When you look at a seismogram the wiggles you see are an indication that the ground is being, or was,vibrated by seismic waves. Seismic waves are propagating vibrations that carry energy from thesource of the shaking outward in all directions. You can picture this concept by recalling the circularwaves that spread over the surface of a pond when a stone is thrown into the water. An earthquake is amore complicated process than a stone splashing into water, and the seismic waves that are set upduring an earthquake are more varied than those on the pond.

The are many different seismic waves, but all of basically of four types:

Compressional or P (for primary)Transverse or S (for secondary)LoveRayleigh

An earthquake radiates P and S waves in all directions and the interaction of the P and S waves withEarth's surface and shallow structure produces surface waves.

Near an earthquake the shaking is large and dominated by shearwaves and shortperiod surfacewaves. These are the waves that do the most damage to our buildings, highways, etc. Even in largeearthquakes the intense shaking generally lasts only a few tens of seconds, but it can last for minutesin the greatest earthquakes. At farther distances the amplitude of the seismic waves decreases as theenergy released by the earthquake spreads throughout a larger volume of Earth. Also with increasingdistance from the earthquake, the waves are separated apart in time and dispersed because P, S, andsurface waves travel at different speeds.

Seismic waves can be distinguished by a number of properties includingthe speed the waves travel, the direction that the waves move particlesas they pass by, where and where they don't propagate. We'll go througheach wave type individually to expound upon the differences.

The first two wave types, P and S , are called body waves because they



travel or propagate through the body of Earth. The latter two are calledsurface waves they the travel along Earth's surface and their amplitudedecreases with depth into Earth.

Wave Travel Times

Travel times are best conceptualized of with an analogy of an auto trip. If you have to travel 120 milesand you drive 60 mph, you'll get to your destination in two hours, if you are forced to drive at a speedof 30 mph, it will take you twice as long to arrive at your destination. The mathematical formula weuse in this problem is

driving time = (distance of trip) / (driving speed)

To apply those ideas to earthquake studies, think of the earthquake location as the starting point forthe trip and the seismometer as the place where the trip concludes. Faster waves will travel thedistance quicker and show up on the seismogram first.

travel time = (distance from earthquake to seismometer) / (seismic wave speed)

Travel time is a relative time, it is the number of minutes, seconds, etc. that the wave took to completeits journey. The arrival time is the time when we record the arrival of a wave it is an absolute time,usually referenced to Universal Coordinated Time (a 24hour time system used in many sciences).Here's an example to illustrate the difference: if two earthquakes occurred at the same place butexactly 24 hours apart, the wave travel times would be the same but the arrival times would differ byone day.

Seismic Wave Speed

Seismic waves travel fast, on the order of kilometers per second (km/s). The precise speed that aseismic wave travels depends on several factors, most important is the composition of the rock. Weare fortunate that the speed depends on the rock type because it allows us to use observations recordedon seismograms to infer the composition or range of compositions of the planet. But the process isn'talways simple, because sometimes different rock types have the same seismicwave velocity, andother factors also affect the speed, particularly temperature and pressure. Temperature tends to lowerthe speed of seismic waves and pressure tends to increase the speed. Pressure increases with depth inEarth because the weight of the rocks above gets larger with increasing depth. Usually, the effect ofpressure is the larger and in regions of uniform composition, the velocity generally increases withdepth, despite the fact that the increase of temperature with depth works to lower the wave velocity.

When I describe the different seismic wave types below I'll quote ranges of speed to indicate the rangeof values we observe in common terrestrial rocks. But you should keep in mind that the specific speedthroughout Earth will depend on composition, temperature, and pressure.

Compressional or PWaves

Pwaves are the first waves to arrive on a complete record of ground shaking because they travel thefastest (their name derives from this fact P is an abbreviation for primary, first wave to arrive). Theytypically travel at speeds between ~1 and ~14 km/sec. The slower values corresponds to a Pwavetraveling in water, the higher number represents the Pwave speed near the base of Earth's mantle.



The velocity of a wave depends on the elastic properties and density of a material. If we let represent the bulk modulus of a material, the shearmodulus, and the density, then the Pwavevelocity, which we represent by , is defined by:

A modulus is a measure of how easy or difficulty it is to deforms a material. For example, the bulkmodulus is a measure of how a material changes volume when pressure is applied and is acharacteristic of a material. For example, foam rubber has a lower bulk modulus than steel.

Pwaves are sound waves, it's just that in seismology we are interested in frequencies that are lowerthan humans' range of hearing (the speed of sound in air is about 0.3 km/sec). The vibration caused byP waves is a volume change, alternating from compression to expansion in the direction that the waveis traveling. Pwaves travel through all types of media solid, liquid, or gas.

As a P-wave passes the ground is vibrated inthe direction that the wave is propagating.

SWaves

Secondary , or S waves, travel slower than P waves and are also called "shear" waves because theydon't change the volume of the material through which they propagate, they shear it. Swaves aretransverse waves because they vibrate the ground in a the direction "transverse", or perpendicular, tothe direction that the wave is traveling.

As a transverse wave passes the groundperpendicular to the direction that the waveis propagating. S-waves are transversewaves.

The Swave speed, call it , depends on the shear modulus and the density

Even though they are slower than Pwaves, the Swaves move quickly. Typical Swave propagationspeeds are on the order of 1 to 8 km/sec. The lower value corresponds to the wave speed in loose,unconsolidated sediment, the higher value is near the base of Earth's mantle.



An important distinguishing characteristic of an Swave is its inability to propagate through a fluid ora gas because a fluids and gasses cannot transmit a shear stress and Swaves are waves that shear thematerial.

In general, earthquakes generate larger shear waves than compressional waves and much of thedamage close to an earthquake is the result of strong shaking caused by shear waves.

Using P and Swaves To Locate Earthquakes

We can use the fact that P and S waves travel at different speeds to locate earthquakes. Assume aseismometer are is far enough from the earthquake that the waves travel roughly horizontally, whichis about 50 to 500 km for shallow earthquakes. When an earthquake occurs the P and S waves traveloutward from the region of the fault that ruptured and the P waves arrive at the seismometer first,followed by the Swave. Once the Swave arrives we can measure the time interval between the onsetof Pwave and the onset of Swave shaking.

The travel time of the P wave is

distance from earthquake / (Pwave speed)

The travel time of the S wave is

distance from earthquake / (Swave speed)

The difference in the arrival times of the waves is

distance from earthquake / (Swave speed) distance from earthquake / (Pwave speed)

which equals

distance from earthquake * ( 1/ (Swave speed) 1 / (Pwave speed) )

We can measure that difference from a seismogram and if we also know the speed that the wavestravel, we could calculate the distance by equating the measured time difference and the expression.For the distance range 50 to 500 km, the Swaves travel about 3.45 km/s and the Pwaves around 8km/s. The value in parentheses is then equal to about (1/3.45 1/8) or about 1/8. Thus the simple ruleof thumb for earthquakes in this distance range is the distance is about eight times the arrival time ofSwave less the arrival time of the Pwave.

That means that we can estimate the distance an earthquake is from a seismometer. The earthquakecan be in any direction, but must be the estimated distance away. Geometrically that means that theearthquake must be located on a circle surrounding the seismometer, and the radius of the circle isabout eight times the observed wave traveltime difference (in kilometers).

If we have two other seismometers which recorded the same earthquake, we could make a similarmeasurement and construct a circle of possible locations for each seismometer. Since the earthquakelocation since it must lie on each circle centered on a seismometer, if we plot three or more circles ona map we could find that the three circles will intersect at a single location the earthquake'sepicenter.



Using the "S minus P arrival time" to locate an earthquake.You need at least three stations and some idea of the Pand S velocities between the earthquake and theseismometers.

In practice we use better estimates of the speed than our simple rule of thumb and solve the problemusing algebra instead of geometry. We also can include the earthquake depth and the time thatearthquake rupture initiated (called the "origin time") into the problem.

Love Waves

Love waves are transverse waves that vibrate the ground in the horizontal direction perpendicular tothe direction that the waves are traveling. They are formed by the interaction of S waves with Earth'ssurface and shallow structure and are dispersive waves. The speed at which a dispersive wave travelsdepends on the wave's period. In general, earthquakes generate Love waves over a range of periodsfrom 1000 to a fraction of a second, and each period travels at a different velocity but the typicalrange of velocities is between 2 and 6 km/second.

Love waves are transverse and restricted to horizontalmovement - they are recorded only on seismometers thatmeasure the horizontal ground motion.

Another important characteristic of Love waves is that the amplitude of ground vibration caused by aLove wave decreases with depth they're surface waves. Like the velocity the rate of amplitudedecrease with depth also depends on the period.

Rayleigh Waves

Rayleigh waves are the slowest of all the seismic wave types and in some ways the most complicated.Like Love waves they are dispersive so the particular speed at which they travel depends on the waveperiod and the nearsurface geologic structure, and they also decrease in amplitude with depth.Typical speeds for Rayleigh waves are on the order of 1 to 5 km/s.



Rayleigh waves are similar to water waves in the ocean(before they "break" at the surf line). As a Rayleigh wavepasses, a particle moves in an elliptical trajectory that iscounterclockwise (if the wave is traveling to your right).The amplitude of Rayleigh-wave shaking decreases withdepth.

Seismic Wave Propagation

Waves on a Seismogram

As you might expect, the difference in wave speed has a profound influence on the nature ofseismograms. Since the travel time of a wave is equal to the distance the wave has traveled, dividedby the average speed the wave moved during the transit, we expect that the fastest waves arrive at aseismometer first. Thus, if we look at a seismogram, we expect to see the first wave to arrive to be aPwave (the fastest), then the Swave, and finally, the Love and Rayleigh (the slowest) waves.Although we have neglected differences in the travel path (which correspond to differences in traveldistance) and the abundance waves that reverberate within Earth, the overall character is as we havedescribed.

The fact that the waves travel at speeds which depend on the material properties (elastic moduli anddensity) allows us to use seismic wave observations to investigate the interior structure of the planet.We can look at the travel times, or the travel times and the amplitudes of waves to infer the existenceof features within the planet, and this is a active area of seismological research. To understand howwe "see" into Earth using vibrations, we must study how waves interact with the rocks that make upEarth.

Several types of interaction between waves and the subsurface geology (i.e. the rocks) are commonlyobservable on seismograms

RefractionReflectionDispersionDiffractionAttenuation

We'll examine the two simplest types of interaction refraction and reflection.

Refraction

As a wave travels through Earth, the path it takes depends on the velocity. Perhaps you recall fromhigh school a principle called Snell's law, which is the mathematical expression that allows us todetermine the path a wave takes as it is transmitted from one rock layer into another. The change indirection depends on the ratio of the wave velocities of the two different rocks.



When waves reach a boundary between different rock types, part of theenergy is transmitted across the boundary. The transmitted wavetravels in a different direction which depends on the ratio of velocitiesof the two rock types. Part of the energy is also reflected backwardsinto the region with Rock Type 1, but I haven't shown that on thisdiagram.

Refraction has an important affect on waves that travel through Earth. In general, the seismic velocityin Earth increases with depth (there are some important exceptions to this trend) and refraction ofwaves causes the path followed by body waves to curve upward.

The overall increase in seismic wave speed with depth into Earthproduces an upward curvature to rays that pass through the mantle. Anotable exception is caused by the decrease in velocity from the mantleto the core. This speed decrease bends waves backwards and creates a"P-wave Shadow Zone" between about 100° and 140° distance (1° = 111.19km).

Reflection

The second wave interaction with variations in rock type is reflection. I am sure that you are familiarwith reflected sound waves; we call them echoes. And your reflection in a mirror or pool of water iscomposed of reflected light waves. In seismology, reflections are used to prospect for petroleum andinvestigate Earth's internal structure. In some instances reflections from the boundary between themantle and crust may induce strong shaking that causes damage about 100 km from an earthquake(we call that boundary the "Moho" in honor of Mohorovicic, the scientist who discovered it).

A seismic reflection occurs when a wave impinges on a change in rock type (which usually isaccompanied by a change in seismic wave speed). Part of the energy carried by the incident wave istransmitted through the material (that's the refracted wave described above) and part is reflected backinto the medium that contained the incident wave.



When a wave encounters a change in material properties (seismicvelocities and or density) its energy is split into reflected and refractedwaves.

The amplitude of the reflection depends strongly on the angle that the incidence wave makes with theboundary and the contrast in material properties across the boundary. For some angles all the energycan be returned into the medium containing the incident wave.

The actual interaction between a seismic wave and a contrast in rock properties is more complicatedbecause an incident P wave generates transmitted and reflected P and Swaves and so five waves areinvolved. Likewise, when an Swave interacts with a boundary in rock properties, it too generatesreflected and refracted P and Swaves.

Dispersion

I mentioned above that surface waves are dispersive which means that different periods travel atdifferent velocities. The effects of dispersion become more noticeable with increasing distancebecause the longer travel distance spreads the energy out (it disperses the energy). Usually, the longperiods arrive first since they are sensitive to the speeds deeper in Earth, and the deeper regions aregenerally faster.

A dispersed Rayleigh wave generated by an earthquake in Alabama nearthe Gulf coast, and recorded in Missouri.

PWaves in Earth

The mathematics behind wave propagation is elegant and relatively simple, considering the fact thatsimilar mathematical tools are useful for studying light, sound, and seismic waves. We can solve theseequations or an appropriate approximation to them to compute the paths that seismic waves follow inEarth. The diagram below is an example of the paths Pwaves generated by an earthquake near Earth'ssurface would follow.



The paths of P-wave energy for a shallow earthquake located at the topof the diagram. The main chemical shells of Earth are shown bydifferent colors and regions with relatively abrupt velocity changes areshown by dashed lines. The curves show the paths of waves, and the linescrossing the rays show mark the wavefront at one minute intervals.

Note the curvature of the rays in the mantle, the complexities in the upper mantle, and the dramaticimpact of the core on the wavefronts. The decrease in velocity from the lower mantle to the outer corecasts a "shadow" on the Pwaves that extends from about 100° to 140° distance. Other waves such assurface waves and body waves reflecting off the surface are recorded in the "shadow" region, but thePwave "dies out" near 100°. Since the outer core is fluid, and Swaves cannot travel through a fluid,the "Swave shadow zone" is even larger, extending from about 100° to 180°.

Earth's Internal StructureWe have already discussed the main elements in Earth's interior, the core, the mantle, and the crust.By studying the propagation characteristics (travel times, reflection amplitudes, dispersioncharacteristics, etc.) of seismic waves for the last 90 years we have learned much about the detailednature of Earth's interior. Great progress was made quickly because for the most part Earth's interior isrelatively simple, divided into a sphere (the inner core) surrounded by roughly uniform shells of ironand rock. Models that assume the Earth is perfectly symmetric can be used to predict travel times ofPwaves that are accurate to a few seconds for a trip all the way across the planet.

The diagram below is a plot of the P and Swave velocities and the density as a function of depth intoEarth. The top of the Earth is located at 0 km depth, the center of the planet is at 6371 km.



Velocity and density variations within Earth based on seismicobservations. The main regions of Earth and important boundaries arelabeled. This model was developed in the early 1980's and is called PREMfor Preliminary Earth Reference Model.

Several important characteristics of Earth's structure are illustrated in the chart. First note that inseveral large regions such as in the lower mantle, the outer core, and inner core, the velocity smoothlyincreases with depth. The increase is a result of the effects of pressure on the seismic wave speed.Although temperature also increases with depth, the pressure increase resulting from the weight of therocks above has a greater impact and the speed increases smoothly in these regions of uniformcomposition.

The shallow part of the mantle is different; it contains several important wellestablished andrelatively abrupt velocity changes. In fact, we often divide the mantle into two regions, upper andlower, based on the level of velocity heterogeneity. The region from near 400 to 1000 km depth iscalled the transition zone and strongly affects body waves that "turn" at this depth and arrive about20°30° distant from a shallow earthquake. In this depth range the minerals that make up the mantlesilicate rocks are transformed by the increasing pressure. The atoms in these rocks rearrangethemselves into compact structures that are stable at the high pressures and the result of therearrangement is an increase in density and elastic moduli, producing an overall increase in wavespeed.

Graphite in "lead" pencils and diamonds are a more common example of atomsrearranging themselves under different conditions they are both composed of carbon.



The different arrangement and bonding of the carbon atoms in the two materials producesdramatically different properties. Diamonds are formed under enormous pressures; allnatural diamonds formed at depths of about 150200 km, and were brought to the surfaceby volcanic activity. At the high pressures the carbon atoms are squeezed into a tightarrangement that makes them one of the hardest materials. In contrast, the lowpressurearrangement of carbon in graphite creates the slippery, soft character of "lead" that we usefor pencils.

The two largest contrasts in material properties in the Earth system are located near the surface andthe coremantle boundary. Both are compositional boundaries and the coremantle boundary is thelarger contrast. Other sharp contrasts are observable, the innercore outercore boundary is relativelysharp, and velocities increase from the liquid to the solid.

Models of Earth's Heterogeneity

The PREM model is a useful reference for understanding the main features of Earth. More recentefforts have focused on estimating the lateral variations in wave speed within the shells that make upthe reference model. These approaches are often based on seismic tomography, which is a way ofmapping out the variations in structure using observations from large numbers of seismograms. Thebasic idea is to use observed delayed (or early) arrival times (delayed with respect to the referencemodel) to locate regions of relatively fast and relatively slow seismic wave speed.

The idea is illustrated in the cartoon to the left. Waves arerepresented by arrows and are traveling from left to right. Thosethat travel through the slow region are slowed down, and hence willbe recorded later on the a seismogram.

The same ideas are used in medical CAT scan imaging ofhuman bodies, but the observed quantity in a CAT scan is not atravel time, but the amount of xray absorption. Ultrasoundimaging is identical to Pwave tomography, it's just that inseismology we don't have the choice of where are wave sourcesare located we just exploit earthquakes.

In the two decades tomography has been applied to Earth studies on many scales, from looking atsmall regions of Earth's crust that may contain petroleum, to imaging the entire planet. On a globalscale, we might expect that the shallow parts of the mantle would correlate with the major structuralfeatures we can observe at the surface the plate boundaries.

In regions where material is rising from the mantle, it should be warmer, and the velocity should belower, in regions that are old and cold, such as beneath many of the old parts of continents, we wouldexpect to see faster regions (assuming that temperature is the only difference). The actual variationsare influenced by both temperature and composition variations, but they agree well with the ideas ofplate tectonics, particularly at the divergent boundaries or oceanic spreading ridges.



Map of the variations in seismic shear-wave speed with respect to the value in PREM at 100 km depth. Thewarm colors (red, orange, and yellow) show regions with slower than normal speeds, the darker regions arefaster than normal. Note the correlation with plate boundaries and surface heat flow. (Model S12 WM13,from W.-J. Su, R. L. Woodward and A. M. Dziewonski, Degree-12 Model of Shear Velocity Heterogeneity inthe Mantle, Journal of Geophysical Research, vol. 99(4) 4945-4980, 1994).

The velocities deeper in the Earth have also be imaged. The next map shows the variations at 2,880km depth , in the mantle just above the coremantle boundary. The color scale is the same but notehow the lowermantle velocity variations are more subdued than those in the more heterogeneousupper mantle. Also, note that the correlation with surface tectonics is gone, as you would expect for acomplex convective system such as Earth's mantle.



Map of the variations in seismic shear-wave speed with respect to the value in PREM at 2,880 km depth,just above the core mantle boundary. The warm colors (red, orange, and yellow) show regions with slowerthan normal speeds, the darker regions are faster than normal. Note the correlation with plate boundariesand surface heat flow. (Model S12 WM13, from W.-J. Su, R. L. Woodward and A. M. Dziewonski, Degree-12Model of Shear Velocity Heterogeneity in the Mantle, Journal of Geophysical Research, vol. 99(4) 4945-4980, 1994).

These variations are actually quite small, on the order of a few percent, so the basic idea of Earthbeing a spherically stratified planet are well founded. In the crust, the variations are larger and canreach tens of percent. The crust is the material extracted from the mantle over the last 4.5 billion yearsand it contains a great diversity of structures that are often apparent when you study the rocks exposedat the surface.

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Prepared by: Charles J. Ammon (Last updated:8/1/101 )

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Seimic Waves and Earth’s Interior