i will start the seismic part of this classwith a some examples of my work in earthquake hazard reduction, where i'vebeen generating earthquake waves in a computer through the structures, andprincipally the basins that underlie most in nevada cities. and i think whatthis will do is tell you a little bit about what the earthquake waves arelikely to do: how they propagate and when
3d printed ar 15 lower receiver, they're larger and when they're smaller.so then we'll start to see how we can use those waves, not necessarily fromearthquakes but from artificial sources to be able to probe the earth and figureout actually what some of these basic structures are that put us atsuch a risk for earthquake damage here
in nevada, as well as in california, injapan, and other places. now the grayscale image in front of you is ashaded relief map of the basin that underlies reno, and i'm pointing up here- unr is about here in the northern partof the middle of the basin; this is the sparks basin over here, which is about600 meters deep; and then under west mccarran and mayberry the reno basin gets to be about to akilometer deep. there's also some pieces of the basin that are down here undersouth meadows and those areas. and four and a half years ago [2008] wehad a series of earthquakes that shook
the western neighborhoods of reno, andthis sort of colored blob here is some of the waves that were coming out of that earthquake. so this is a model thatwas made in computer clusters; and then we propagate earthquake waves throughthat model. let's observe what the what the waves do, and then we'll be ableto come back and think about how we can use those wave propagation effects andtry to learn something about the earth structure- not just from earthquakewaves but from waves that we can put into the ground ourselves. so i'm going toplay this movie and you're going to see the waves expand rapidly through thearea around reno and there's this
curious effect that a lot of the waveenergy is left sitting within those basins that underlie the urban areas. andthat's why places such as mexico city, tokyo, and kobe have shaken so terriblyin the earthquakes that have hit them. now as i go through this movie more slowlylet me rewind it back and collapse the waves back into it the mogul earthquakesource of april 25th, 2008. here the waves are just starting to hit thesurface of the earth, and if we roll this forward now i you'll see it's a verycomplex wave. we try to use simpler waves when we do seismicexploration. but these more complicated earthquake waves, they still obeythe many of the same features of
wave propagation that we will use inexploration. the wave expands across the map. yeah, i forgot to say this is a map view; andyou can see there's some sort of dim waves out here that are traveling fasterthan the others and there's brighter waves with higher amplitude that arebehind. in this class we're going to start out by looking at these first-arriving waves that travel faster. those are called p-waves. and later onwe'll take some advantage of the slower-traveling waves, called shearwaves or surface waves or s-waves. we
can watch how even by the fourth orfifth frame the p-waves are pretty much propagating out of the frame of calculationhere, and what we're looking at are principally the s-waves. it isprincipally the s-waves that are getting caught in the basins. this is aninteresting frame right here because you can see that the basinis somehow bending and actually delaying the waves. even the p-wave hereand the s-wave over here. those are being caught at theedge of the basin, and it's clear that the waves don't propagate as fast withinthe basin as they do outside the basin. you can see that here- thenice circular s-wave path outside the
basin; the s-wave "wavefront" actually,looking down at the map view of it. here in sparks where there's a basin the nice circular pathis distorted, and the wave arrivals delayed a little bit- maybe justfive percent or so of its overall time. whereas the p-waves may be evendelayed somewhat more. little scraps of p-wave left there in the basin.same thing over here too- that's in pleasant valley, the southeast side of reno.so in addition to the waves being stronger within the basin they're alsodelayed. this speaks to a rock property that we call "velocity," which hasa lot to do with the strength of the rock.
the stronger and less bendable, lesscompressible the rock, the more force you have to use to compressit, the higher the moduli of deformation; then the faster the waveswill travel. outside the basins where you have hard volcanicsand sierran granite, the the waves are travelling fairly fast.these wave speeds are on the order of 3 to 6 kilometers per second. but in thebasins the velocity might even be half of what it is outside the basin.the basins of course are filled with volcanic or in this area mostly clasticsediments. those sands, gravels and maybe tuffs and volcanic mudflowsare looser; they're not as compact,
they're not as dense, and they have lowerseismic velocities. it's a property of the rock, much like density.sediments have the lower densities and lower seismic velocities. as it turns outthey're just looser and more easily compressible, more easily sheared, and allthat. the moduli are lower. and these are the delays in the waves that thoseproduce. let me play this through again a couple more times just so youcan see the delayed wave fronts and then the energy that gets trapped at theedges of the basins, because the wave fronts propagate through pretty quickly.you can see some some energy rattling around a bitinside the basins as well.
that's a relatively simple map ofthe progress of shaking of earthquake waves. i''m going to showyou a couple more examples. here we have a very complexexample. we're looking again at a shaded relief map showing the basinstructure. elko is just off the map on the lower left side. this boundaryhere above which we don't have any basin information, that's the nevada-idahoborder. nevada on the south and idaho where we just haven't put thebasins in to the north. the map doesn't quite go to the utah border onthe right-hand side. the map is about 200 kilometers wide. also almost fiveyears ago now there was in february
2008 a magnitude 6 earthquakein the little hamlet of wells, nevada along interstate 80, about 5 hours driveeast of here. you can see that this earthquake happened in this very geologically complicated area where there are all these basins thatare part of the basin and range and provide a very complicated structurethat allows the wave motion to be channeled and and refracted around.again you see a colored area in the middle where the earthquake is going tobegin to propagate. let's watch the the waves as they refract around and getcaught in all these dozens and dozens of different basins. you can see thewaves are propagating quite slowly
through the low-velocitysediments of the basins. they're propagatingmore than two times faster in between the basins. but you'll notice thatwithin the basins where its soft there is also higher wave amplitudes and moreearthquake shaking. that's one of the reasons that 20 buildings were damaged in wells, nevada in 2008 from this earthquake. you can seethat earthquake shaking is very strongly channeled and affected bythe velocity property structure- the rock velocities that are in this area- and aregiving us lots of lots of refraction and delays in certain waves, and catchingcertain waves. others are just
accelerated right out the side of the model.again, it's the interaction here between the wave propagation and thevelocity property of the rocks that is really giving us a lot ofthe effects that we can look for. this is of course important in trying topredict where earthquakes are going to shake, and where they'll shake harder, and wherethey'll shake not as hard. which critical facilities to have to build to a higherstandard? where do we have to raise insurancerates? where can we lower insurance rates? it's also possible to lookin detail at the waves, and decide what they're telling
us about the velocity structure.when we see these high amplitude waves that are propagating slowly, then we knowthat we're in a basin. when we see lower amplitude waves that are propagatingfaster, then we know we're in between basis. we can actually explore and getsome idea about the structure by looking at these waves and how they arrive at different places on the map here. here is a another map, which is down insouthern nevada. those of you from las vegas, you might recognize the las vegasbasin. the las vegas strip is here; nellis air force base is up here. thisis the outline of the western front of
frenchman mountain, which those you fromdown south will recognize. the strip is is over here, and then bouldercity is right out down here. we got a very detailed look at the las vegas basin, and not-very-detailed looks at thebasins that are outside the las vegas basin. this is called eldoradovalley down here. there's a fairly substantial earthquake fault, which is onthe northwest side of eldorado valley and could produce easily a magnitude 6.5earthquake. there's a good fault scarp there that shows that it probablyhas ruptured relatively recently, just in the last couple thousand years.
what this model here investigates is-where is it going to shake in las vegas when we have that earthquakeon what's called the black hills fault? this fault is actually outside las vegas andand is much closer to boulder city than it is tothe strip, over here. i'm going to let this propagate. look down here forthe initial shaking due to the earthquake. there it comes, and it's getting funneled into the lasvegas basin. you can see the circular waves going through the springmountains, where there's no basins and not a lot ofthis velocity-property
heterogeneity to sort of disturb thewaves. so we've seen these circular waves here on the map. thewaves are getting obviously delayed and distorted within las vegas basin.they're getting caught within the eldorado valley basin andespecially in the deeper las vegas basin. they're getting caught pretty severely.look at the delays there. it just takes longer to propagate through theslow sediments with that slow velocity property. maybe half the velocity of theothers. that's that's enough exampleof the earthquake hazard calculations. how thendo we use these waves to investigate
structure? for instance, find out how deepbasins are, and find out more about that velocity property?is that tied to anything that might be interesting? can you for instance find gold based on thevelocity property? a quick answer is no, but it still can behelpful. i have some students investigating those issues right now. here we're looking at an extremelysimple and very low resolution model of a cross-section. we're going tolook at seismic wave propagation- not a map as we have- but in a cross-section.across here we have distance
across the ground surface, and thenon the vertical axis in this view we've got depth below thesurface. if i let thewaves propagate, you see that this model is supposedto be only 50 meters wide. this is the kind of scale of theexperiments that we will do in this class. we're looking not ata map but at a cross-section. to start out, we hit the ground at the topcenter, up here, and then we're watching the waves propagatein that cross section. you've probably noticed by now that there'ssomething here in the lower right, which
is reflecting a lot of those waves back.that would be since it's a model of a cavity like a tunnel or mine addit,or a utility tunnel- buried services. the point ofthis wave propagation movie is to see, what we would be looking at if we wantedto find those buried utilities? so let me bring it back here, at where we've hit the groundand we've got a wave that is traveling radially out from wherewe hit the ground, and here it is 1/50th of a second later probably.a little bit later you can see that it's basically starting to react toother discontinuities and
heterogeneities here. thevelocity property's not constant here. but the wavefront, let's sayif i track the between the black and the white, theinitial white and the black. as i track that, it's basicallysemi-circular here in cross-section. the waves are propagating in a fairlypredictable way, from the point where we hit the hammer on the ground surface, sayright here in the center at the top. down here where thattunnel is, we start to see that impinging on the waves. right here you can see thatthe waves are not propagating;
you can see they're highly delayed by thetunnel. the tunnel's full of air and air, although it's fast by our standardsat 330 meters per second, the velocity of air isone-tenth the velocity of rock. that's a big big difference. this whitepart of the wave is delayed all the way back here by the outline of thetunnel. the black part of the wave is delayed back here as well. but there'sanother thing happening too. what you'll see there as we keep going is that the wave is reflectingoff that top left side of the tunnel.
that change in velocitytakes some of the energy, in this case a lot of theenergy, out of the the initial wave that came straight from the hammerblow, and puts it into a reflected wave, which now instead ofpropagating down is propagating up! up, actually back toward the hammer. that gets to thesurface and then reflects off the surface. the surface of the earth is also a prettygood reflector. we'll collapse that back- and so that'sthe progress of these waves. if we can if we can detect that reflectedwave, as you can see here would
be hitting back off the surface where wecould measure it? if we detect that, maybe we could use the timethat the waves arrive and their curvature and all that, we could use thatto figure out where that tunnel is. essentially kinda "back calculate" and collapse that. see, here the reflection isat the surface, and if i back- calculate and collapse it backonto the top of the tunnel then i can figure out where thetunnel is. i can see the waves first, basically, reach the surface righton top of the tunnel. there's an indication there- where doesthat reflection arrive early- it
arrives earliest right over thepoint that would be the center of the reflecting structure, thecenter of the tunnel. all those are are ways that we're going to use,actually, to use these same seismic waves, really exactly the sameas earthquake waves. we're going to use them to explore the earth and look for buried structures and buried stratigraphy. now here's anotherexample. what we have here is a little bit more labeling. again, avery low resolution view. i made these a long time ago when when it wasn't soeasy to use computers to show these kinds of animations.you'll notice that this is more of a
crustal or maybe deep oil and gastype of exploration problem, perhaps even deep geothermal. there's some unknownstructure in here within this cross section again. we're not looking ata map we're looking at a cross-section. across the top we've got distanceout to ten kilometers. here's where we've hit theground with something or let off a big explosion or used "activesource" machines that will vibrate the ground. then this vertical axis isdepth. we're looking at a one-to-onesection here. the grayscale gives us the amplitude ofthe wave and up here on
the upper left you can see the timeafter the initiation, after we start hitting the ground. or when we let off theseismic charge. so i'll go ahead and let this propagate, and youcan see that there's one main wave that goes down. but even after thatpropagates out of the field of calculation here, you can see there'sthere's other things going on; there's some reflections coming up; it's alittle hard to see because the wavelengths are kind oflong compared to model. but really you can see some waves moving up.it's those reflections and other
things that are happening near thesurface here. there are some waves that are traveling along some interface here, andwe'll be able to use those to figure out what the velocities are and where thereflectors are. we can actually reconstruct the structure, notin tremendous detail, but in enough detail to actually make this techniqueuseful. so i collapse it again. here you can see there's there's one reflectioncoming back up. i think i can see another reflection coming back up, clearest righthere- that's another reflection that's coming back up- a whole series of different reflections. then there are some waves that are propagating back downfrom the surface of the earth. all of
that are things that we're going totry to figure out how to use. we'll principally do that in our in ourlabs. we've got a lab first on refraction and figuring out velocity. then we'llhave a lab on surface waves and figuring out shear-wave velocities from thosewaves that get so easily trapped in basins where they're solarge in amplitude. then we'll have a third lab that isspecifically on the reflections; and that's half the labs inthis class. we're going to examine these different pieces of the wavefield.you've got one hole [cavity] here with all these different kinds of waves. we'regoing to approach it piece by piece,
break it down, show exactlywhat's useful and what you can do with each little observed piece of wavehere. all of those examples fromthe earthquake hazard work that my students and i have been doing; theseexamples of the waves propagating on the map and propagating in cross-sections,i hope has motivated us to start looking at these waves andfigure out what we can do with them. first we have to decide how todescribe these waves. now we're in the the seismicoverheads, the "seismic overheads 1". this illustration here shows a viewof waves propagating through a medium.
you can see it's just one kind ofwave; this might be the the fast p-wave, and we might be in air or waterwhere we don't have the slower s-waves. or maybe these are justthe s-waves and the fast p-waves have already propagated out of the picture.either way, we can describe those different kinds of waveswith very similar parameters. so in this view of compressional waves we have asource- that's where the waves were let off- and we could be lookingat either a map or a cross-section. if this was across-section that would be a buried source; if this was a map
then what we'd be looking at here isa source in the middle of the map. the density of dots kind of represents,highly exaggerated of course, what happens whenthe compressional wave passes through the medium. you can see here behind thewaves, or in front of the waves there's kind of a medium density of dots.then, when the first compressional wave reaches aparticular place, the dot density is higher. then behind that, is what'scalled a "rarefaction;" the dot density goes way down.
how much compressionand rarefaction are we really talking about here?the actual strains, if you studied stress and strain, the stresses are low and the strengthsare really quite high. the moduli of rock are really very high. stresses arenot that great. they can be great for earthquake waves. but forexploration waves the stresses are not that great. so the strains are also verysmall and what we're probably looking at here is a compressional strain of10^-6. if it's positive 10^-6 it would be a compression,and negative it would be a rarefaction.
now for sure we have instruments thatcan detect these very small compressions and rarefactions very easily. asi'm talking to you there are compressions and rarefactions comingthrough the air from your computer speakers or to your eardrumfrom your headphones. those compressions and rarefactionsare are very small, but your ear is sensitive. they're probably onthe order of 10^-5 or 10^-6, and your ear is perfectlycapable of picking them up, characterizing them, timing them, anddoing the frequency analysis your ear does to figure out what i'm saying.[this applies to normal hearing.]
our instrumentation iseven more sensitive; 10^-8 strain is no problem forus to detect. we have robust relatively inexpensive,field hardened instrumentation that we can measure these compressionsand rarefactions at 10^-6, 10^-8 strain levels with no problem. that's the kind of equipment we have.you'll get to use this equipment and measure these strains.later on we'll do that in the field, we'll practice for that; and you'llanalyze the data. now what are we going to measure? i'm not sointerested yet in measuring
the exact strain. that's why i'm guessing when i'mtalking about a level of strain. really what i want tolook at is, this velocity property. i want to decidewhere the reflectors are, that are generating the reflected waves. iwant to decide: how fast did that zone of compression ofthat wave travel from the source? so i'm really not going tobe too concerned about the exact level of strain- which is soexaggerated here in this plot. really what i want to do is timethe wave. if i'm
observing out here, and this wave iscoming toward me, so i'm observing on the right hand side here, i'm going to make a recording that lookslike this- the first compression will will come at me, and then it's going torarefy- there's the rarefaction- that's going to propagate past me next; and then the secondarycompression is going to come. these waves tend to organizethemselves like this: there will be a whole series of up and down motions,compressions and rarefactions. all of those will takethese kind of sinusoidal forms.
you'll explore, ifyou're a geophysics major and you take geophysics and geodynamics 455 or 655,you'll explore why the sine wave seems to be a way that wavesorganize themselves. so there's an upswing on theseismogram- that's a compression in this case; the downswing is a rarefaction.when the wave is past it all goes back to zero- normal density.there is the zero line right through there, the gray average distance.now look at the the lower left. i can't point down there or it getscovered by the acrobat tools. look down at the lower left where i'vewritten one
equation there. v equals f lambda. this is a very basic wayof describing wave motion. you might have seen it in a physics class. i wantyou to memorize this because you're going to use this over and over again.v = f lambda waves travel at velocity v,so v is for velocity; and waves have a frequency f, in otherwords the number of cycles per second. the si unit for cycles per second is, for one cycle persecond, is one hertz. the waves also have a wavelength, lambda.
if i measure from themost compressed peak over here, to the most compressed peak on the nextpositive wave that comes by, then that is 1 lambda. lambda ismeasured in meters. it's a distance; it's a wavelength;it's a length. i could just as well- there's always morepositive and negative swings. i could go from the center of the negativeswing to the center of the next negative swing. it oughtto be pretty much the same lambda. wavelength:greek letter lambda for length l. another way to do it is to takethis first departure from
normal particle separation-zero change on the seismogram and over the top and then across thetrough, and then here we are going up again. we go we you know there's athere's an instant there where we have normal particle separation. looking at the seismogram it is easierto see this. we have normal particle separation; itgets bumped up to the peak; we swing back and briefly we getnormal particle separation again; and we go through the negative; and then we comeback to the normal particle separation- on the way up- but verybriefly were we're at normal particle
separation. so this is between here, andhere. that's also one lambda. so we can measurepeak to peak; we can measure trough to trough; or we can measure from zerocrossing the zero crossing. we'll get slightly different values-different estimates of lambda- but like with any dataset there areuncertainties and distributions of values. but it's going to be about thesame. so there's thiscardinal rule here that i want you to memorize: the velocity is equal to thefrequency times the wavelength. that's a very basic wave formula-the velocity of the wave
propagation, the frequency of the waves,the wavelength of the waves- those are basic parameters that we use to measurewaves. you can see that they're related. if i knew two of them-if i knew the frequency and the wavelength, i could calculate thevelocity. if knew the velocity, and i knew the frequency, i could take the velocitydivided by the frequency and i would get lambda. that's the usefulthing about having a relationship between these parameters. if youonly got two, you can get the third. by calculating it. now here's arelationship from optics that i'm sure
you've all heard of before. this is the reflection of waves, andwhat we're looking at here is a cross section. let me justget it to center- i can't blow it up that much. we're looking at across-section and there's this horizontal line through the middle of it. then this dashed line here isperpendicular to that horizontal line. so it's like we're down in a trench andwhere we're looking at a layer boundary. maybe we've got sand above andgravel below. or you
might find in reno diatomiteabove and clay below, for instance. so imagine you're looking at thiscross section and now here's the the boundary between the twodifferent units. we have a wave that hits fromsomewhere up above. up at the top of the trench wehit the ground, and the waves propagate along- we're tracking onelittle piece of wave energy as it gets towards the boundary. then it hits theboundary right there. then it's going to reflect. youprobably already know from optics and physics class that it reflects at thesame angle, alpha here, that it came in at.
the incident wave's path,called the "ray path," is at the same angle, alpha, from the normal-that dashed line is the normal, perpendicular to the interface- asthe reflected wave. the reflected wave is going to propagate out this way. it's got the same alpha fromthe normal. that's a p-wave coming in,reflecting to a p-wave at the same angle. when do you get a reflection? well,you get a reflection when you have some sort of interface. that means that theproperties below are not the same as the properties above. above we've got density1,rho1- that's a greek letter rho, if
you can see it- and p-velocity1.below we've got a different density, rho2, and p-velocity, vp2. it happensalso that we're going to say there's also a difference in s-velocity,and we have vs1 above and vs2 below. so clearly the densities rho1 and rho2are different, the p-velocities are different, thes-velocities are different; and that's very normal. when you've got two different kinds of rocks, it's so easy to have all theseproperties be different. you might ask, "does allof the energy of the incident
p-wave completely reflectfrom the interface? i already told you the surface of the earth isalso a reflector. you saw that there was nearly complete reflectionfrom that tunnel that had air in it, burieddown inside the rock. you can calculate theamount of the amplitude of reflection. the amplitude of reflection is ar.you have the amplitude of the incident wave ai, and you multiply it bythis factor here for ar. this factor is the difference between the media. it's rho2times vp2 minus rho1 times vp1... you might have heard the the term "seismicimpedance." the seismic impedance, or
acoustic impedance here, is thedensity times the velocity. so medium one here we're taking we'retaking rho1 and multiplying it by vp1, and that means that productis the the seismic impedance of the medium 1. and medium 2 also has aseismic impedance, which is rho2 times vp2. that's theacoustic impedance. so we're taking the difference of the acoustic impedances.what's on the denominator here? well that's the sum of the acoustic impedances.so if rho2 was equal to rho1, and vp1 wasequal to vp2- you had the same rock on both sides; same rock above as wehave
below, no difference- then thesetwo would subtract perfectly and there'd be zero left on top, which means theamplitude of the reflected wave would be zero. that kind of makes sense.there's no contrast in density or velocity; there's no contrastin impedance; there's no reflection. here this says that amplitude is zero.this simple equation here is strictly true only when the angle iszero- the incident wave is coming straight down along the normaland bounces straight back up. then you get this simpleequation here that just involves the acoustic impedances.you can see the more difference
there is between the acoustic impedances,then the stronger that reflected wave will be. it's goingto be easy for us to record reflections from strong impedance contrasts,maybe where the velocity and the density are both, say, half in thesediment as they are in the granite below. that's going to give ussediment against granite- a pretty good impedance contrast andnice strong reflections. if you just go from fine sand to coarse sand,there will be some impedance contrast, but only3% to 5%. in that case, we get a reflection coefficientof maybe 2% to 4%.
that is still detectable in manysituations, and we can go after it. but it's going to bemore difficult to find that reflection than to find a very strong reflectionfrom a strong impedance contrast, like the bottom of a sedimentary basin.there's also an equation here that tells you how the s-wave reflects.the other thing you should know is that when you have a p-wave hitting a reflector like this then yeah the reflected p-wave is atexactly the same angle alpha- alpha equals alpha. but the reflected s-wave-and the first thing you need to
realize is that an incident p-wave-will partly reflect, and some of it reflects as a p-wave- but alot more of it in most situations will reflect as an s-wave.that's another thing to watch out for. i said that the reflectioncoefficient is only good for the angle of incidence being perfectlyvertical, zero. it's a zero angle from the normalto the reflector. here's an exploration of what happens forsome typical reflector when the angle is not zero. so we begin at zero with a lowhere that looks like maybe a 7%
reflection coefficient,and go along this horizontal scale here that's theangle of reflection, or the alpha angle. you started at somelow amount of reflection, under ten percent, and it can waveraround a little bit. then suddenly you get to a certain angle, which here isabout 37â°, and the reflection amplitude jumps way up. sometimes it's very useful to tryto look at larger-angle reflections because they can be a lot stronger, a loteasier to record than these normal,
zero-angle reflections. then it will drop if you go out tolarger angles, and then gradually rise to 1.0 [100%]. at areflection coefficient of 1.0, that means that the entire incidentenergy, all of it, is reflected. none of it passes into themedium below. that's what these lowerplots are: they tell you the relative amplitude of the transmissioncoefficient, or the refraction coefficient. it begins, if this isseven percent reflection, then it might start at ninety-three percentrefracted transmission.
then it stays pretty steady for thistypical reflector, whatever it is, and then when it gets to that critical angle it drops to zero. notice thatbeyond the critical angle there is no p-wave energy, there's no amplitude,zero exactly, that gets into the lower medium, that is refracteddown. that's why there's all this energy for thecritical reflection. now, why doesn't the the p-wave reflection coefficient jumpup to one? well, there's that division of theenergy. some of it gets reflected as shear-wave energy, and some of it isrefracted as shear-wave energy.
all of them have the cusp right atthe critical angle. at least that stays the same in all four plots. so that's a bit on reflection.where we have a sudden change in the properties, that's where we'llhave a non-zero reflection coefficient. now, how do those waves propagate and howdo we figure out what the velocity is? how do the waves propagate, and can they get disturbed byother mechanisms than reflection? that's encapsulated inthese statements called "fermat's principle." fermat's principle is: "thewave path between any two points is the
one along which the time of travel is theleast of all possible paths." it doesn't say it's the shortest path; it saysit's the least time path. the time of travel is the least. fermat's principle is also called:"the principle of least time." let's look at a cross-sectionhere, much like that cross-section with the tunnel in it. we'llset off a source of seismic waves at the surface, up here and in the center.everywhere in this cross-section it has the sameseismic velocity property.
we're going to think about itin a very simple way. everywhere is the same velocity, so thewave isn't delayed by any basins or reflections or anything. there are no changes in thevelocity property, so the wave at 0.1 second is there,2 seconds it's there; 3 seconds. we're kind of looking here at aview of the wave fronts. and then it gets to our receiver that we've put downa well or something down here at a distance and some depth. now, if we track back the energy thatgot to that receiver, we would track it
back perpendicular to the wavefronts,back to the source. so we would climb up the hillas steeply as we could and track it back to the source. the ray pathis perpendicular to the wave front. now, where velocity is constant, thewavefronts are circular and the ray paths are straight. again, the wavefronts are circular where velocity is constant-the velocity property of rocks- the wavefronts are circular; the raypaths are radial and perfectly straight. now here's a here's a situationwhere we have high velocity on the
right-hand side of the cross-section, andlow velocity on the left-hand side of the cross section. or if you like, youcould make this a map, as i was showing you at the beginning, andmaybe we have a basin on the left and bedrock on the right. in the high-velocity part,the ray paths are circular. those circles run intothe low-velocity part. then what happens? well, you can seethe low-velocity part has the same circles. but they're closer together.it's like you know one wavelength here in the high velocity is twice as longat it is in the low-velocity part.
the velociy's like half; it takes twice asmuch time to get anywhere on the left side as it does on the right.these contours are now kind of complicated. fermat's principlesays the least time path- and this is actually quite hard tocalculate- and why waves just follow it; they just do it. it willpropagate along the interface, just inside the high-velocity area,and then it will light out here where it will find that least-time path. you cansee the least time path is still perpendicular to thewavefronts. it turns right there, where it refracts. we're seeing thatthis part right here- these are
circular up here at the top, but at thebottom we're looking at these straight refractions. this isnot the shortest path from the source to the receiver. it's the least time path. it'slike the it's like going out of your way to spend asmuch time on the freeway as possible, instead of the city streets. so here we're on thehigh-velocity freeway, and then we take an exit, that's essentially closest, anddrops right down the perpendicular
to the wavefronts, and drops down along theleast time path, to our receiver. if i had a receiver over here, i'dclimb the wavefronts to get back to the source. if i had a receiver over heresomewhere else, i'd climb their wavefronts to get back to the source,keeping the path perpendicular to the wavefront.that's fermat's principle. what we will be able to do ispredict how fermat's principle will work, from this relationship thatyou're familiar with from optics called snell's law. the part i want todiscuss first is just where i have an incident p-wave that's traveling at vp1.here again, we're looking
at the trench wall. we've got two media; medium numberone above; medium number two below. above, we have rho1, density 1, vp1, vs1. below we've gotrho2, vp2, vs2. we have an incident wave at some angle "i" or"alpha." and there's all these other waves- the reflected waves- thereflected p-wave comes out at "i" as we found out. but where does therefraction come out? snell's law allows us to calculate that. the p-wave incident angle is alpha,and snell's law says sin(alpha) over
vp1 is equal to- ok, alpha is in themedium vp1- and that ratio is equal to sin(beta) over vp2. you may haveseen snell's law in the context of optics before, where instead ofvelocity you used the "index of refraction." here i think it will be clear that thevelocity is proportional to the inverse of the index of refraction. that'swhy for angle alpha being in medium one, we divided by the velocity;and angle beta here, for the p-wave being in medium 2, we divided by thevelocity in medium 2. depending on what wave we're looking at,at what different angle, it depends on what the velocityis in that medium for that kind of wave.
snell's law: another good one to memorize,just it's form right in the middle here, where we're onlyconcerned about p-waves. let's look at snell's law a little bitmore closely. we'll just look at it for p-waves only,and in fact we'll forget about the reflection for a second, and thinkabout the refraction only. we're looking our trench wall; we've gotthis horizontal interface- sorry the slide wasn't straight- and we've got theincident p-wave coming in at the angle alpha from the normalto the refracting
interface; and we've got a refractedp-wave- a transmitted p-wave- which has come through theinterface and is headed on down, but at a different angle beta. there's theangle beta right there. we just apply snell's law, and above we have a vp1; below we have vp2. so for the sine we want tostart with an angle alpha above. we want to get the anglebeta below. we start alpha, we take its sine, we multiply it by vp2, anddivide by vp1. we work out snell's law that way, and solving for the sine of beta.you can get that. so, if you want to solve for beta, you takethe sine of the incident angle, you
multiply it by vp2, you divide by vp1.that gives you the sine of beta. you just take the inverse sine-the arcsin- and that will give you the angle beta.but there's an effect here. let's consider, as we keep increasingalpha, we get an incident angle- noticehere we have an incident wave, it's at alpha, and beta is larger. so what does that mean?sin(alpha) was multiplied by vp2/vp1 and we got a larger sine, which led toa larger angle beta. what that meant was that vp2was greater than vp1, because this
ratio here had to give more than one,because sin(beta) is more than sin(alpha), so vp2 is greater than vp1.let's come back down here to this slide. there's rho1 and v1,and there's rho2 and v2. we'll stop talking about vp,so it's just whatever kind of wave. assume it's, in this class,it's a p-wave velocity. we start increasingalpha, and beta gets larger, and eventually beta hits 90degrees. then what happens? does it make any difference,whether we keep increasing it after
that? well, no. so, when does thathappen? that happens when beta is 90 degrees. so sin(90â°) is sin(alpha),or sin(90â°) is going to be v1 over v2. so sin(90â°) iswhat? it's one. the sine of zero degrees is zero but sin(90â°) is one. after putting inthat one, we can just cross that out and wehave sin(alpha) is equal to v1 over v2. that's the alpha at which the betafirst goes to 90 degrees- the refracted angle first goes to 90 degrees.
we'll call that particular alphathe "critical angle" or alpha sub c. so if you have v1 over v2 youcan get the critical angle because the sine of the critical angle, the sine ofalpha sub c, is equal to v1 over v2. now, let's suppose we had the othercase. so far, v1 is less than v2. v2 is greater. but what if v1 is greater? what if v2 was less thanv1? if v2 is less than v1 in this ratio here, and in calculating thesine of the critical angle, the ratio would be greater than one. can you have asine of any angle that's greater than one?
no, you can't. so that means there isno critical angle. there's no refraction, no matter whatyou make alpha, alpha the incident angle, you'renever going to get that refracted angle beta to 90 degrees. if v1 is greater, then alphawill always be greater than beta. that's the way it has towork. so here there's only a critical angle when it's higher velocitybelow. that's the only way there's acritical angle. so now, we know a littlebit more about how waves
are going to propagate when there arechanges in the velocity. i want to start talking about this velocityproperty and what makes some rocks low velocity what makes otherrocks high velocity. i haven't given you a section on elastic constants-you may be familiar with them. especially the geological engineers may be familiarwith elastic constants such as the incompressibility, which i call "k"; or thethe rigidity, which are which i call "mu". these are seismologist's terms. we'vegot the density rho here. the p-wave velocity is equal to the square rootof the quantity k, which is the the incompressibility, plus four-thirds timesthe rigidity mu, and divide all that
by rho and then take the square root,you've got the p-wave velocity. it's clear here thatthe higher the moduli, k and rho- the higher the incompressibility,the higher the rigidity- the higher the p-wave velocity is going to be.what's weird is that density is on the denominator. so what gives there? for higherdensity, the denominator is going to be larger. that means the ratio will be less, andthe velocity will be less. this relationship worksbest for a single crystal, or
for a massive massive kind of material.you could make this relationship work for massive concrete,that's very well cemented. the higher the density, the lower thevelocity. so very light concretes that aremade out of very light aggregatelike cinders, will have higher velocities, higher p-wave velocities; than denserconcretes. for another instance, a block of aluminum will have a highervelocity than a block of steel, because the for aluminum and steel,aluminum is lighter. let me talk about titanium and steel. titanium andsteel have roughly the same k and mu,
but titanium is quite a bit lighter, justlike aluminum is a lot lighter, so the lighter material- the lighter massivematerial- will have a higher velocity. but that doesn't work, that doesn't work when you go tomaterials like rocks; materials that are granular, that havepore space, that have fractures. this works on single crystals, and we'vegot to include some other factors when we're talking about real rocks. so what affects watervelocities? as water saturation goes up- as air gets replaced by water
in the pore space of a rock or soil,the velocity will go up. as a rock becomes more consolidated-as we weld the grains together with cements like calcite cement orquartz cement, opal cement; as consolidation increases, velocityincreases. you know that welding those grains together is going to vastlyand hugely increase the k and mu. you weld the grains together andthe k and mu can go up by a factor of ten. you weld the grains together, replace pore space by cement; you dotake the density of two- but how much is the density going to go up?maybe twenty percent, thirty percent at
most. if you've got a whole lot ofpore space to fill with cement. so the density does go upby ten percent, whereas k and mu you are increasing by a factor of 10? the velocity is going to go up. consolidation: velocity goes up. so a you desaturate a rock-you lower the water table, and thevelocity will go down. what's the opposite of consolidation?well, either weathering or fracturing. weathering takesvelocity down, so soils typically have
lower velocities than rock.and fracturing takes velocity down, because you're breaking those bondsbetween the grains, and drastically lowering the k and mu.here's some materials that i'm sure you're familiar with- some rocks andother materials. i'm giving you here the ranges of of vp: p-wavevelocity or acoustic velocity in units of kilometers per second [km/s]. to get kilofeet per second [kft/s],you multiply by a little more than three [3.28]. in u.s. industry andcommercial work, consulting
work, feet per second is therule. in scientific work and research, in academia, and globally,we're using meters. you just have to get used to both systems. this is all in kilometers per second. ok let's look first of the consolidatematerials you know it's a little bit simpler story alright so grant is you know not and i'mnot talking i'm talking about good solid sierra grande i'm not talking aboutfractured granted i'm not talking about faulted grant i'm not talking aboutdecomposed granite right talk about solid granite ok 526 km/s p wavevelocity
ok so not a huge range you know forsolid granite and pretty fast right basalt faster still 5.4 26.4 you know uhand that's for you know solid basalt ok not to not fracture basalt not a not anot a basalt flow you know that's been that's got kilometer joining it solidbasalt ok metamorphic rocks you know a syrian roofpendant well big range right 3.5 27 . a factor of two easy okay sandstone and shale ok so these are youknow basin basin fill materials you know much lower the granite much lower thebasalt maybe not lower than maybe not much lower than this then grantedassault a lower but not maybe not much
at the upper end here and you know thereare some metamorphic rocks that are old and buried and solid but they're stilllower velocity that then some of the some of the younger sandstones andshales ok big range limestone even bigger range ok and what this rage is telling you isthat for that very rare no solid hard massive limestone you likethe redwall limestone you know just back behind the grand canyon ok and away fromany false this is it is it's as fast as this it asa result as basalt as fast as granite sixpoint oklahoma hours per second butdown in you know where its falta door or
whether it's got two cavities in it fromdissolution very low velocity to point o ok just hard to get that to that reallymassive and on undissolved limestone alright so now let's look at well okayone more thing i want to mention if i say the velocity is a six-point oh rightyou got a lot of choices ok can you can you say it's it'slimestone no can you say it's metamorphic snow you can see it's basaltyou can see it's granite right 6 . oh that could be any of those four okay ifi say the velocity is too . well uh yeah it's not it's not going to grant me thesalt can be metamorphic six but it's sandstone or shale could be either one
ok and and could be limestone could beyou know fractured buggy limestone too so so what does that mean that meansthat that the velocity is not unique you know none of these materials have aunique velocity the when you get a velocity value which is what we're goingto be measuring a lot you know with our seismic surveys really getting velocityvalues it doesn't tell you which material you have not enough ok gotta gotta gotta look at a wholebunch of other properties to to be sure ok so you know it's even worse with theunconsolidated materials right when we do seismic surveys we talk about thequote-unquote weathered layer
ok that's the soil whole or it could be deeper than thesoil ok but it's you know whatever is above the water table and so it's youknow . 3 2.9 kilometers per second 300 900 meters per second is something thatyou might just label as soil quote-unquote okay . to 52.6 ok alluviumwe got a lot of that around here that that has a huge range right . 5 i'veseen plenty of alluvium that slowed up to 2 . oh maybe even higher maybe 2.5some cases you know these ranges are out of the book and we've seen even largerranges and then when the alluvium gets cemented by caliche or calcite cementedyou know if you're from las vegas year
and you've ever tried to dig in youryard you've probably encountered this calcreit ok this caliche alright you know if you're lucky in your yard you got softerclean cheese with a velocity only two . oh that's bad enough but you know whenit gets really cemented really thick it can go up to six . right as hard as asthe hardest limestone so titanic you know factor of 3 and velocity rangethere whatever you might call clay no 1.1 a 2.5 big range factor tounsaturated sand okay that can be really low and velocity and we we might measurethat at certain places ok down to 2 200 meters per second . twokilometers per second up to you know
unsaturated sin sand you know reallysoftly loose it doesn't get above one kilometer per second saturated sandthough you know it's partially saturated it can be down as low as $money . mekm/s and as high as 2.2 km/s yet in some gravel you you can raise the losseslittle bit but you know maybe not maybe not distinctly so okay saturated sandgravel you know it's another huge range of velocity depending on this saturation that case glacial till whereyou take that sand and gravel and you had a bunch of clay right that can lowerthe velocity just because it just says 1.7 km/s for saturated still doesn'tmean that that's all it could be but
maybe there isn't a huge range there youknow okay well then you compact it and it cango up to 2.1 now so here here you know it's clear that saturation has a bigeffect on the velocities of unconsolidated materials especially okayand why is that all right well here's here's thevelocity of some fluids alright so one thing about two fluids they basically bydefinition have a shear velocity is 0 right so i i said shear velocity isdefined to be 0 for any fluid alright some common fluids that will encounterour surveys are water and air they're there but they're both fluids
ok water has a velocity that you knowvarious from 1.4 km/s to 1.6 kilometers per second 1.4 if it's you know freshand hot and 1.6 if its cold and and and quite briny you know most water isreally very very close to 1.5 kilometers per second in velocity that's one ofthose indicators velocities that lets us figure out all right now what does thismean if we take you know sand particles or courts right so they're going to looklike each sand particle as an internal velocity of between five and sixkilometers per second right we put it together in one consolidated eunos andyou don't like the surface of the sand mountain you know just east of here andok was totally unsaturated very very
loose like the you know like the toplayer of sand mountain de ok then it's going to be at point 2kilometers per second p wave velocity ok now if we take thatsame loose sand and we fill it with we fill the pore space with water ok so it rains on sand mountain and thewater is running off sand mountain ok then suddenly we have saturatedcompletely saturated sand and it's a and its velocity can easily be 1.5 okay whyis that because it's then the the water in the pore space that's controlling thevelocity of the sand ok it's very loose and ok then then eventhough the sand grains of cells are five
kilometers per second internally they'restill going to rub against each other and you know the p wave velocity isgoing to be the same as the velocity of water there's a lot of water in that sand ok same thing with with air saturatedmaterials you know you saturate material with air you take the the velocitycouldn't get lower than the velocity of air itself which is you know . 32.3 4.32.34 km/s and the higher velocity where the air is less dense ok such as an elevation up here in renocompared to sea-level uh most
experiments are done you know the thevelocity of the of the error is not really noticeably different from . 3 3330 meters per second so that's pretty good and what's remarkable is that thatunsaturated sand even though it's still full of air can have a lower velocityand a class a few years ago also approved that that unsaturated dry you know veryfriable clay can have a similarly low-velocity i think we got a velocityat point 2244 a bunch of really dry clay alright you know in a in a canal banklevy alright so that's about the only case where we can actually get a rockvelocity right unsaturated sand sand
mountain i mean it's it's still rockright technical definition that rock velocity is less than the velocity ofair otherwise that velocity of error is really a floor to the velocities that wecan have you know we really don't see velocities of hardly anything that's aslow as the velocity of air ok alright you take those fluids and youyou freeze them right you take water you freeze it it becomes ice and ice is amassive but fairly light of you know crystal and that so it has a pretty highvelocity it's pretty stiff and it's also light and massive and so ice itself willhave velocity like granite 526 km/s ok snow is a sedimentary rock right
inorganic solid and made it but themineral ice and it can have a and of course when when you're looking at at asnowbank you know like this fresh powder snow that we're all hoping for it's been you know since december thatwe've had some good powder on the slopes here and that powder snow you know it'sa it is it's an extremely unconsolable dated very very loose and it has avelocity that's that's lower than the lower than that of air okay . twokilometers per second of course you compact the snow has beenhappening on the slopes lately with the freeze-thaw cycles
and the velocity can get you know closeto that of solid ice right as the snow gets closer to to be solid ice so that's that that huge range there youknow so snow is a sedimentary rock or soil if you like and and like like likea you know clastic sediments and soils has got a huge range depending on itsphysical condition just noticed your wave velocities all i express them herein terms of of the vp the vs over vp ratio ok for us massive crystal and rockthe dso vp ratio is about 1 over the square root of 34.6 ok for sedimentaryrocks is somewhat less about half so the the shear velocity will be about halfthe ppl hace from consolidated materials
it can be . for or actually i've seen itas small as 1 15 which is you know really remarkably low so the you knowthe shear velocity can drop off very quickly in a in unconsolidated materials here's a little of famous chart that canhelp you to take rocks that are sedimentary rocks that are of interestto oil exploration ok and so here we're looking at depth in in thousands of feetthat's a you know the deepest wells are at least in california are twelve toeighteen thousand feet there's some wells that are even deeper than that inin oklahoma some gas wells and there are you know mostly oil and geothermal wellsand nevada are going to be between three
and six thousand feet deep so this this chart addresses that quite well andthe end on the left hand side the velocity is given in kill ft/s thousandsof feet per second so you know 15,000 feet per second is just under fivekilometers per second right that's the upper range five six kilometers persecond is is as high as these sandstones and shales get ok and so you know as youbarry sandstones and shales right the volt feet they get more lithified andand so the velocity goes up that's what these individual curves are are areshowing ok these are from you know tens hundreds of thousands of measurement ofvelocity versus velocity and in well
logs k using sonic loggers ok the other thingthat that you might notice here is that the the youngest rocks are lower invelocity ok the oldest rocks here are higher invelocity and if you take a if you take a depth of burial at sea and you make itin feat and you take the age of the formation in years ok so that's going to be in the millionsmaybe hundreds of billions ok and then here's this constant k whichis you know when when zzz and feed and tease in in years then it's a hundred25.3 ok so you take the depth in feet
that's going to be thousands of feet xmillions of years and you take the power of 16 right so you know you get to getmillions but then you you basically take its sixth root and so i guess it down tosingle digits again and or tens anyway and you multiply it byhundred 25.3 that's k there and you get the ship the velocity in in feet persecond ok so this is a little rule of thumbhere that that you can easily use to try to predict what your what your a yourpeople hace is going to be you might ask yourself alright you know how muchreflection coefficient would i see between a post eocene sandstone videoseat sandstone rights than you and
they're the same death right so you youyou seen sandstone is what 60 million years old right so and your your postyou seen standstill maybe it's a quaternary it's only two million rightso you you figure that in you know two-million-year-old sandstone you youput in 2 million for that and and maybe you're observing it at a hundred feetdepth right so you put izzy and and you get it you're going to get value outokay and then you put in your 40 you put in 60 million for the ecg sandstone okand go through this you're going to get a higher value right and so that you cango ahead and and see whether you're gonna get a reasonable reflectioncoefficient ok one last thing in today's
lecture is the relationship of velocityto porosity ok so iãve talked about you know how in soils and an unconsolidatedrocks you know the saturation makes a difference and and you know how looseand porous it is also makes a difference and there's a formal relationship thatyou that you can use right it's called this wily time-averaged relationship andbasically you factory you assume you know you have the some kind of flu the pore space you know usually waterbride ok so that's the velocity of the fluid vsub f see let me let me focus in a bit here
yeah so a visa bath is the velocity ofthe fluid in the pore space v sub m is the velocity of the of the rock matrix ok so you know maybe we're going to tryto compose a sandstone and figure its velocity you know loose and is probablyfive kilometers per second and well i'm sorry the the sand grains themselveshave an internal velocity you know to the extent their massive day of thevelocity of five kilometers per second okay and maybe the process the thirtypercent porosity is full of water ok and notice here that this this iscalled a time-averaged relationship because the inverse of of the velocityis really time and notice that we're
adding up all these inverses of velocityso we got a proportional average right so we take one over the velocity of theof the of the fluid right and we we multiply that by the fluid fractionwhich is the porosity ok so maybe that's thirty percent okay and then we take oneover the the matrix velocity maybe that's one or five kilometers per secondand we multiply that by a point-by-point 2 by 1 minus the porosity which isseventy percent okay so . x points7 add the two together right and that thatgives us a time-averaged slowness so then we we just convert that to getthe the time average velocity so i think what this is telling you here is thatthe the higher the porosity the lower
the velocity right we're looking at1-over velocity here and the higher the porosity the the lower the velocityusually you know our fluid is slower than our that our matrix right you know1.54 water versus five kilometers per second for san grades so we can use thisnow to see you know how much how much prossie we have ok and so here's areciprocal velocity micro seconds per foot which we get straight from soniclog you know we we measure it that way and then here is a porosity ok and youknow it's really hard to get porosity above thirty percent right got to havesome kind of diatomaceous earth right which we got a lot of here at reno andit's still pretty close
ok these little triangles right there soand then here's a here's regular sandstone right which could easily be 0porosity right if all the grains are perfectly cemented together okay and so we have you know we start at$time at a hive at a high velocity right low reciprocal velocity right noticethat this scale increases to the left which is kinda weird but there it is sohigh velocity at zero porosity and then we climb up you know when we climb up inporosity we get to you know the the microseconds per foot is starting in at250 here and then by the time you get a 3d print three percent porositymicroseconds per foot is it a hundred so
that's you know half the velocity right twicethe time half the velocity so a big effect especially in the materials thatwill be surveying and measuring big effective porosity and you can see itreally counts you know whether you have air or water in the pores because youknow air has one fifth the velocity the
p wave velocity of water you know . 3 3vs 1.5 kilometers per second ok so that's plenty for uh thisintroductory seismic lecture and the next lecture we'll continue to examinethese basic facts of seismic wave propagation
