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GEOL7333-Seismic Wave and Ray Theory - Day3 09-02-2022
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00:03 Okay, so, hello folks, is friday the second of september,
00:10 beginning the second week of instruction in seismic waves and race. And the
00:18 thing we're gonna do is to talk the questions submitted by the student,
00:26 Miss Stephanie del rio. And she , can you please explain the difference
00:33 the grass reciprocity thermal elasticity and the reciprocity of uh and the scalar reciprocity
00:40 again. So, uh let's do first thing and I'm going to find
00:47 right slides for that in this very which we looked at last saturday.
01:12 , So this is the fear expressed a formula. And let's just,
01:17 , and let me put it into mode. Okay, so there's the
01:25 and I'll explain right now what the means. We've got to places,
01:29 got a place a and a place be. Here's the force, the
01:35 vector today, and it's here's the vector at the same place sourced from
01:41 other place in between the dot product these are both factors. And that
01:48 vector dot product is equal to the one on the other end. So
01:54 is the vector reciprocity theorem of That's the general, not, not
01:59 most general statement, but most general as applied to wait. Right?
02:06 um so now uh this doesn't show what is the scalar? That's the
02:14 there. So suppose this forest vector this displacement vector are pointed in the
02:23 direction. Or maybe opposite forces down coming up in the same direction.
02:30 this uh dot part is just a product. Right? It's just the
02:34 of a time for length of Are you following that? Okay.
02:42 I need to get your nodding if following. Okay. So and
02:47 Oh, and then we're gonna arrange since we are the operator, we're
02:52 arrange for the strength of the of of the force at me to be
02:57 same as the strength of the force a. We could make it
03:00 But there's no point in that if just have the same physical source at
03:04 places, that's what we want to . And then in that case this
03:08 force magnitude cancels out on both sides the of the scalar product,
03:15 And then what's left, it says that you at a source and B
03:20 equal to um be sourced from Which means that these two day directors
03:25 the same and you can interchange source receiver, and it will still be
03:28 same. And that is the scale here and years and years in this
03:36 , we didn't call it that. call it the reciprocity. They're not
03:41 that it was really a special Uh when the force vector and the
03:46 vector pointed in the same direction as p ways, Right? So,
03:50 you're uh if you have a vertical , for example. And uh it's
03:58 energy down. That's the force And here comes the energy up from
04:02 other source. And it's coming up exactly a vertically, but it's coming
04:06 almost vertical. So the dot product the exact vertical component of that.
04:13 And uh this formula doesn't say anything the other component. The transverse
04:19 So the P wave as it comes , it's gonna have a little transverse
04:23 , cause it's not gonna be coming up, it's gonna be like
04:26 And so it's gonna have a small component. But we're not going to
04:30 that anyway, because we've got a geo fall. And and so we're
04:35 recording uh vertical component of the So, never mind that it's got
04:40 small horizontal component. Uh that's not in the Um in the statement of
04:47 theorem. So, uh what it is our vertical data at a source
04:52 B is equal to our vertical data the source. Uh so we didn't
04:58 , this was actually spelled out decades . We should have known better,
05:02 for decades we called this special case case. We called it the theory
05:08 aggressive policy and only um in 1997 we realize that that was a special
05:16 . And and uh here's how we in this context when I was processing
05:22 at Valen working for Amoco and exploration at Valhol, is that here's the
05:30 down here and over the crest of is a cloud of gas which is
05:35 up out of uh out of the over millions of years and it's lying
05:43 in the overburden. And so when p wave comes down here through that
05:49 cloud, uh it gets attenuated. when the P wave comes down here
05:54 the gas cloud is not a genuine not only is it attenuated but slow
05:59 , slow down. And so uh , the shear wave coming down outside
06:05 doesn't know about the gas cloud because outside the gas cloud. This shear
06:09 coming up from the conversion point This shear wave coming up also doesn't
06:14 about the gas cloud because it's a wave. And so, uh
06:21 uh yeah, so we have this here where uh because of that gas
06:29 in the subsurface, and because we're at converted ways instead of p
06:35 you get this uh non symmetry of uh of the data. So let's
06:41 back and look look at the cartoon . Okay, so here's this data
06:47 up. This is the data from shear wave source from be coming up
06:52 and you see it's it's perpendicular It is, it's perpendicular to the
06:57 the force today, of course, is going in this direction. And
07:02 data at a source of B is perpendicular. So this formula doesn't affect
07:12 data. The formula only concerns that in the data, which is parallel
07:20 the source because of this doctor. um the general form is called the
07:25 reciprocity is there. And uh that's to remind us that when we say
07:31 here about saying we might confuse So the best thing to do is
07:37 this vector reciprocity or the general And if these two vectors are parallel
07:47 each other degenerates to uh staler product times you and uh f cancels out
07:58 same as this effort. You have source is the same as your
08:05 And I'm seeing right now that the that I'm showing here is different than
08:10 formula here because here I used a symbol and here I use so thanks
08:17 nagging me on this, I'll go . Okay, so um and I
08:25 tell you that most geophysicists still do understand. Uh I would say most
08:34 , they say uh reciprocity there and mean scalar reciprocity don't realize it doesn't
08:43 the general day. Mhm. So of our game is in this uh
08:55 perpendicular, direct. And uh that not even mentioned up here mentioned.
09:03 uh uh not constrain our data in kind of situation, which is good
09:12 the data are very clearly non cement and either we get uh so when
09:21 confronted with this, you think aha I get a Nobel prize for for
09:25 disproving the reciprocity theorem or maybe I fired because I screwed up the data
09:32 and in our case it wasn't Didn't get any Nobel prize didn't get
09:36 , but we actually did go back read the rest of process carefully uh
09:43 not affect our day. Okay, now the next thing I wanna do
09:49 to talk about the quiz and uh I what I'd like to do,
09:55 Del Rio is to go over your so that you understand those questions here
10:02 now. And the best way to that is if you'll permit me to
10:06 your quiz. Okay, so let's that. So I'm going to stop
10:11 here and I'm gonna bring up your . Yeah, so you can turn
10:19 the recording while I'm fumbling around. .10. Yeah, No.
10:30 so we will now resume on September will resume lecture four or we
10:40 it off last Saturday. Actually this friday, september. Sure. And
10:48 left off right at this point last . So it is the convolutional model
10:54 waved congregation. We know that a that is composed of many mono frequency
11:02 and those signs and more signs. go on forever. Every one of
11:06 goes on forever. But they can combined using the mathematical process invented by
11:15 in the 19th century to make away which is localized in time. And
11:20 way that works as uh they reinforced short times and they cancel it long
11:26 . And so the resulting wave is in time in a uniform personal elastic
11:34 . These waves all traveled with the wage question. So it preserves the
11:39 shape only decreasing in aptitude during the spreading. Now when it encounters an
11:47 interface which we'll talk about tomorrow. it reflects without shape. And the
11:54 size program looks like this. This the signal which we're recording. And
12:00 is um uh uh the drag delta . And it's yes, here we
12:09 . So here's the incident amplitude. is uh the incident uh answered as
12:14 uh um it's the interface, it reflected. This is a scalar operation
12:22 and the source wave that looks like . And I mean it doesn't,
12:28 can't, I should say that you know, the source wave like
12:32 . This is some sort of localized in time. With a few wiggles
12:36 it maybe, but it's uh localized time, but it's arriving at uh
12:44 uh it's arriving at a time. actually it's arriving at a time uppercase
12:54 . Uh which we determined by looking our at our workstation screens. And
13:00 see this uh this wiggle arriving. a wiggling time. And so so
13:08 this is a single number, upper t lowercase T um uh who's on
13:17 out. And so uh how is related to uh the signal?
13:23 there's a convolutional operation here. And um oh miss del rio. I'm
13:37 going to assume that you are not um confident in your understanding of
13:45 Yeah, so uh that's very And so uh you think intuitively uh
13:58 convolution takes a function like this T. And places that function given
14:07 you know, slides that function along time axis to a certain time,
14:12 is given by uh this arrival Uh because uh delta involved with W
14:21 um uh the same, the same W. But it's uh it's uh
14:30 time uppercase T. So that is simplest case and in the real case
14:41 have um much more uh applications. let me let's step our way through
14:52 um step by step. And what have here, this formula is the
15:00 model of uh seismic wave propagation. this is the way almost all of
15:09 think about uh in twitter, we that we have a certain sort strength
15:16 is a scalar and but it might be the same in all directions,
15:21 might be different in different directions. that's why it has a theater
15:26 And that's multiplied oops uh to be , that includes some other effects that
15:35 that the operator didn't put in such as uh nonlinear behavior of the
15:42 surface materials. And also the interaction the free surface because uh this source
15:49 gonna be at or near the free . For example, if it's a
15:54 , it's going to be at the , if it's in a marine
15:58 it's uh sources told a few meters the surface. And so some of
16:03 energy goes straight down, someone goes and then down. And so that's
16:08 included in this um source strength as function of angle and it's a
16:17 but that gets involved with an initial . So this wavelength is determined by
16:25 operator and that converts this scaler into wiggle. So this part here is
16:31 outgoing wave. Excuse me, I it wrong. You see, there's
16:39 convolutional operation here. So this is a multiplication here, we have the
16:45 wave um as a function of And then what happens to that is
16:54 propagates down. And so uh here have the first convolutional operator right
17:01 And the way you should think about is that this unknown operator, Which
17:09 be complicated or it could be simple could be realistic, could be non
17:14 , whatever uh changes this wavelet, is uh launched into a wiggle displaced
17:25 time, the same wiggle displaced in and same wiggle displaced in time going
17:33 . And, you know, if subsurface is complicated, this is going
17:37 be complicated, but let's leave All the complications are all inside
17:43 And what physical effects are included in all geometric spreading transmission coefficient. So
17:49 time it passes a reflecting horizon, of it goes back up and some
17:53 it goes back down and the downward has this transmission coefficient in it and
17:59 on the distribution of velocity in the serves if it's not uniform, maybe
18:05 faster over here and slower over That makes the wave curve and focus
18:10 and all kinds of wonderful things, are less unspecified here. Also,
18:17 attenuate that's a real world issue and all included in here implicitly. And
18:23 when it gets down to the it's gonna reflect that here is a
18:29 operator. So uh that is oh converts this down going way into an
18:39 wave. Uh and then uh here the upcoming operator. And again,
18:44 a complicated, a whole bunch of stuff in there, we're gonna leave
18:48 explicitly and uh in um kind of that we're looking at. Uh there
18:54 many such reflectors and so we just some of them up. And so
18:59 example, uh reflector number one as we got here, uh certain reflection
19:09 for uh for Reflector # one, also uh there's transmission at that reflector
19:19 the the effect of the transmission that's over here, included in the download
19:25 and that happens with lots of reflection in uh subsurface environment, there's hundreds
19:33 thousands of reflections and some of them big reflections, some of our small
19:39 , some of them are separated in from other reflections. And you can
19:44 them clearly with your eyeball and some them are not. And uh if
19:50 there are two reflections um uh you know, separated by a thin
19:59 in between and the wave light reflecting one interferes with the waves reflecting off
20:04 other. Making a confused situation. going to figure all that out.
20:10 then uh when all that babble of comes back up, it's going to
20:17 the receiver and uh receiver is going not record yeah, coming wave alone
20:30 it's probably sitting on the surface or just below the surface and in in
20:34 there's a free surface nearby. And what the instrument records is the upcoming
20:41 plus the interaction with the down with surface. And what it records is
20:46 combination and it sends the combination of wire to the computer. And then
20:52 the computer uh gets it, somebody might be doing something in the computer
20:59 the data before you ever see There might be uh filtering for example
21:07 they might be uh scaling, they be uh putting uh offset dependent uh
21:16 on there. And they also might putting on their time dependent scaling.
21:20 that the reflections from deep deep Uh they're a lot weaker than the
21:26 from shallow. And so to see that somebody might have done some gain
21:32 so that you can see it with eyeball. And so you shouldn't be
21:37 at all that when you look at seismograph uh, over the side
21:43 that's not what I came up from , from the earth. That's what
21:47 up from the earth. And then oops and interacted with a free surface
21:54 that's what we received. And then of your colleagues or somebody in the
22:00 company did something with that before you saw it. So don't make the
22:07 that what you're seeing is the actual . It's actual information, but it's
22:13 not necessarily actual data because somebody's been around with. And depending on what
22:21 plan to go with it, you to know what was done or maybe
22:24 don't care. So intuitively, that's we think of. We don't solve
22:31 wave equation in our, in our . And so because the wave equation
22:37 linear, we have all these convolutions there. This is not a bad
22:44 actually. Um thinking about wave But also I want to point out
22:51 something we often forget about noise and of that is the source generated noise
22:58 some of it is generated from nature some of it is generated by uh
23:06 another um, another company during a survey nearby. That's an interesting
23:17 So when I was, yeah, long time ago, um, I
23:25 uh, I was in the North on a seismic acquisition vessel and we
23:31 going along and we were shooting a survey and it was not a towed
23:38 survey. It was an ocean bottom survey. So we had our receiver
23:42 there on the ground and then we sailing over there shooting in a
23:50 And then there came a time when shut down and I talked to the
23:56 , I said, what's going You're still charging us? It was
23:59 by a service company. I was oil company rep, you're still charging
24:03 for your time, but you're not . Let's get off the dime
24:06 And he said, we have an with the other, um,
24:12 um, um, services company, acquisitions company. They're operating five miles
24:18 here and they're shooting their own And so, uh, we're time
24:24 . And so we just ran out time and it's their turn and so
24:28 shooting and we can see their shots our receivers and it would just confuse
24:34 . So, so we can't shoot the same time. So I saw
24:38 logic in that. So for now our turn again and we shot and
24:44 were waiting. Well that's obviously extremely , but none of us were smart
24:50 at that time to figure out a solution and to just make peace and
24:57 share and things like that. sometimes we did something clever like that
25:02 the time when they were shooting, would be moving the notes. So
25:09 okay. The other, the other didn't mind are moving notes. As
25:14 as we weren't making all right. me that. Well, that was
25:19 long time ago now. We have ways to uh, shoot at the
25:25 time. So now we can shoot clever processing technology. We can shoot
25:32 the same time as that other company there and we know how to separate
25:36 , uh, their shots from And furthermore, we can shoot our
25:44 . We can put another boat in water and shoot um, our survey
25:49 two boats instead of one boat, worrying about interfering with each other because
25:54 know how to separate those in the . And that way we complete our
25:59 in half the time we got two working. Well, maybe it's not
26:02 big deal, but sometimes it is signal. For example, if we're
26:06 in the arctic, we only have months of summer time to uh,
26:11 operate. And so it's good if can get in there with, with
26:16 big operation, Your survey never minding the energy of the other's toes.
26:22 can separate it all out. And that is a process called oppressive seismic
26:30 . You know about this. uh, you should know and
26:37 ask, uh, ask, ask Stewart about very clever process.
26:46 you know, taking account of the that our sources here and that other
26:51 over there and so that the waves there and from this stretch, these
26:55 are coming from this dress. And we can separate it all out Uh
27:02 part of the, that's part of answer. But there's more to it
27:05 they. And so um guys who that out about 10 years ago,
27:12 won an award from the sc chief that out. And it's extremely useful
27:19 oil companies should be able to acquire data in that way. They save
27:25 in the acquisition. And also, here's the thing uh whenever they decide
27:35 um are we gonna do this survey not? Survey is gonna cost so
27:39 money. Um and are we gonna value out of that? Uh you
27:46 be oil down there or maybe not the chances are 5050 down there.
27:51 we have a certain budget for spending side of me to go after that
27:57 . But if our seismic budget uh uh cost half as much because of
28:02 clever process, then maybe we have opportunity to shoot over here and and
28:10 that. Whereas if the if the operations too expensive, we have to
28:15 that opportunity by. So there's, maybe we'll pass up an opportunity to
28:20 a big oilfield over here because we uh we're unsure about whether it's even
28:29 or not. So we're not even spend the money to have a look
28:31 it, so we just pass it and give that opportunity to some other
28:36 who comes in more efficient acquisition. spend less on the acquisition, they
28:44 it's worthwhile for them and maybe they their acquisition and find out there's nothing
28:49 that could happen, but maybe they the field. So uh in our
28:54 part of the fascination of it, amounts of money depend upon um um
29:04 like us who are during the technology we don't really understand the business.
29:11 what we really understand is the but our technology enables business decision,
29:18 can be worth billions of dollars. had two ideas in my career that
29:29 um, of the order of a dollars. And uh that's kind of
29:36 to think that you're sitting there looking equations and the implications of the equations
29:42 be really big bucks. Okay, , uh, this convolutional market is
29:52 intuitive implementation. Seismic ray theory, a seismic wave theory. You don't
29:59 explicitly here any. Um Oh, that is complicated. It's just uh
30:14 just pretty simple and straightforward and uh what we have in mind when we
30:24 at the gate. Almost everybody, don't know very many people, only
30:28 few people that I know are smart to understand the wave equation to
30:32 but they can understand the convolutional Great, almost everybody. Yeah,
30:49 of the mathematics of this, uh convolutional operator here, we can slide
30:58 operators around so that um group all convolutions together in here. And actually
31:08 I have uh uh right here, includes uh the summation is now implicitly
31:19 here and we we uh group all convolutions together, slide them around,
31:25 them right and left, and in formula, and it still works and
31:29 one convolution left outside, give this bunch name and call that uh
31:35 T. And that's this is different the this is different from W zero
31:41 . C. Here's the initial waiver is the final wave wave. This
31:46 what you see on your on your street, It's gonna be different than
31:53 here, here's that W0 right this is what we started off,
32:00 as a result of property getting down proper getting back up, we've lost
32:04 frequencies and lots of things that And so what you see on your
32:09 , uh workstation screen is a combination all that stuff and we'll just give
32:16 a name. And so that looks similar to what we started off
32:20 Uh we're gonna back up here back the way up. Yeah, so
32:27 is what we started off with uh looking at the uh context more carefully
32:35 more realistic, we end up with , which is basically the same.
32:40 the main thing is different, is main thing, you can see that's
32:44 is gotta somewhere, this is now function of time. The reflectivity is
32:49 function of time, function of of times. So you have in spaceship
32:56 here and here. Another one down . But those are yielding arrivals back
33:02 the instruments so that we can describe reflectivity as a function of time and
33:08 we recognize explicitly there's noise in Yeah, so, so let's um
33:18 quick quiz here says the controversial model implicitly various reflections are well separated so
33:27 the corresponding waivers do not relax. this true or false? Yeah,
33:36 right. Uh If it were true lives would be so much simpler.
33:40 in fact we got in the Earth have uh reflectors space close together someplace
33:48 apart and all of those are shedding wavelengths going back up. And if
33:54 separation in time between these two reflectors less than the duration of the
34:00 Look at those wavelengths are gonna we have to deal with that.
34:04 nowhere in that model did we ever ? But that didn't happen. So
34:10 does happen in the real world, only valid for p waves.
34:19 that's that's also great. No word there that we say it's for P
34:23 on it could be only for shear that could be for converted waves.
34:28 the difference lies right here. Is a reflection coefficient for p waves or
34:35 coefficient here waves and converted waves or ? And we left it unspecified,
34:41 quite correct. So now you are to take a course in imaging uh
34:50 this course, maybe the next maybe they went after that. And
34:54 um um that will be a uh business. So I only want to
35:08 a few things about it here because be frank, I am not an
35:13 in seismic imaging. And um uh would say the seismic imaging is the
35:21 important job for most geophysicist make an of the subsurface on the size of
35:26 data. And when I came into business, we had primitive ideas about
35:31 to do that. And now our are much more sophisticated. So I
35:36 to just give you a glimpse of . Um company like shell or Axon
35:45 at their disposal. Doesn't literally dozens algorithms for making images. Every one
35:56 us is different. And so they're to choose the right ones for uh
36:01 each application because some of them are more expensive to run and some of
36:06 cheaper. Uh so the oldest and is what we call an emo
36:13 So remember this uh picture from uniform and from elementary geometry, you know
36:20 this is exactly the equation of hyperbole rearranging we get we can get a
36:27 factor. So the ratio of the time to the uh uh oh,
36:37 was here. And by this doesn't , how was your break reconstruction?
36:45 this is this fraction is a number than one because the offset is greater
36:50 Z. We multiply the times of offset traces by this factor. The
36:58 all come in at G. this is what we say is the
37:02 flat. So why would we want flatten the gathers? The reason is
37:06 we can average the faces sample by all over the same time, thereby
37:12 noise and the noise to what's gonna . Uh surface waves. That's uh
37:19 . We'll talk about that next That's noise. We'll talk about that
37:22 random noise. And this process of it's called stacking and it's the single
37:29 effective imaging technique. And we knew this from back in the 50s and
37:36 60s. So my father was a oil finder for ethical back in those
37:41 , knew all about this. And was the standard technique that they used
37:45 making an image and a seismic. this simple procedure is remarkably robust despite
37:54 got lots of assumptions. Here's, a listing of it. You can
37:58 the listing and I know that you know that none of these are
38:04 We assume it anyway and bass are on these incorrect assumptions. Um even
38:13 in many cases just this simple idea going to lead to a useful
38:20 Uh Let's assume that we consider many , keeping the other assumptions. These
38:26 assumptions here we're going to keep going have many layers. And the way
38:32 at each one of these uh interfaces down following Snell's law right here,
38:38 bends and refracts again. Same thing the way up. But the image
38:42 is still the midpoint as long as flat. Long. Previously we found
38:49 short offsets, we found uh this out formula. Uh this definition for
38:55 R. M. S. Philosophy um um so this is an interesting
39:06 . So I'm gonna post this to of you we have here the hyperbolic
39:14 equations both familiar with and it's got here the rms average velocity. So
39:21 this is defined for short spreads. We have longer spreads, We're gonna
39:27 more terms out. But for short , this is gonna be good
39:31 Now, here's the question why isn't short spread? Move out velocity equal
39:35 vertical average velocity instead of the M. S. So in terms
39:42 this picture, uh here's the down wave but really is an exaggeration
39:48 Uh for short spreads it's really going at angles like this and it's almost
39:55 to close to the particle. So is it? Yes, Philosophy equal
40:02 the average vertical velocity since since he's down almost vertical. So uh while
40:13 thinking about that, let me tell a story, I'm the hero of
40:17 story. So I love this Uh, so when I was first
40:22 Annika came into Amoco at age about , I've spent some time as a
40:31 before that, I don't know but I quickly acquired a reputation inside
40:37 anopheles for being a smartass. Don't how that happened. That was my
40:43 . So one of the senior guys to me a few months later or
40:49 later and he said, yeah, young whipper snappers, think you're so
40:54 smart, you got your fancy diplomas the wall and so on. So
40:58 think that the move out velocity is RMS velocity? I know because I
41:04 it should be the average velocity. it says here I said,
41:09 you know, the equation says it be the RMS loss. And he
41:13 , well, you just think of into italy, those equations are only
41:17 short offsets, all those rays are nearly vertical. So it's got the
41:22 intruder. So I said, you know, you could check it
41:29 yourself. Um, make uh, a theoretical model for the computer with
41:37 few layers, chase raised down their , Snell's law. And uh,
41:43 out the arrival time and fit it curve. And you will see it's
41:48 uh RMS philosophy and that was a unfair because those days and maybe still
41:56 , um uh senior guys didn't know to program computers. So people your
42:03 have to be accountable computers and you a lot better software, you have
42:11 like matt lives that we don't have . So that was a bit
42:17 But in those days uh they had out but the first handheld calculators about
42:29 size about the size of her phone it had of course not a screen
42:39 this, it had a little readout just at the top which gave numerical
42:46 . And uh the rest of it buttons. Uh so the first generation
42:52 come out a few years before that the second generation was actually program program
42:58 hand calculating and it costs about 1000 and the company wasn't going to spend
43:04 bucks. A young guy like So I didn't have it. But
43:09 guy was a senior technical guy and had one and he had spent some
43:14 familiarize himself with eating the instruction manual groups some simple programs and he was
43:24 by my challenge and he said, know I can do that on my
43:27 held attack like it was made by few attack uh Serious company back in
43:37 back in the 60's maybe before you were born. So uh, so
43:44 said okay go in and make yourself model realistic model with realistic velocities and
43:51 raised down there three or four layers and uh come back and tell me
43:58 so he did that. And it him about two weeks because he was
44:03 how to program and saw and uh understood that. And so he came
44:09 in two weeks in trump and he , look here's the result is not
44:13 the vertical velocity average, but it's closer to the vertical velocity and the
44:20 velocity. She's protecting both of So I said, how many significant
44:29 did you? I said was there point where you had to um write
44:34 intermediate results and then launch another module the program? Right, because a
44:40 calculator could only do a program of 20 steps. And after that you
44:44 to uh launch another model. And said, yeah, I had to
44:48 that twice or three times. So said, how many significant figures did
44:53 care? And he said, well carried you, you have to carry
44:59 . So that's crazy. We never . Yeah, but the rocks do
45:07 exactly. So you've got to uh the rocks. So I said
45:13 And so he went away and he it and he came back in triumph
45:17 a couple of years, a couple days later because he was getting and
45:22 same answer even close even closer answer the vertical verbal. I said,
45:31 many significant figures do you carry? said I carried before I said,
45:39 many have you got on this? know, I think gave 7s
45:45 You got to carry them all. that's crazy. But I'll do
45:49 So I'll do it and I'm gonna it right here in front of you
45:52 I'm pretty good at this. I'm gonna sit here at your desk
45:55 your office and I'm gonna do And after you do it, after
45:58 do, you're gonna announce to the lab at old dawn is uh smarter
46:06 all you young whippersnappers with your goddamn on the wall and you're such smart
46:13 , Sorry. I said, I'll do it. So he sat
46:16 my office in the corner of my , which looks pretty much like your
46:20 papers everywhere. And then the and punching away on his, he's pretty
46:28 at, you know, working on now for a couple of weeks.
46:32 rings, answer the phone. he's right here and passed the phone
46:36 to landline. Of course, You guys know what a landline California
46:42 like pass it over to Don. was sitting at my desk and I
46:47 off to the side and I this is for you. So you
46:52 . Oh yes, show her down . I'm in leon's show her down
46:58 . Uh so into the phone. said, I forgot my wife is
47:03 here to pick me up and we some personal errands that they're in
47:07 I forgot. And so the receptions . So immediately I knew what was
47:14 on. I had met this she was absolutely beautiful. She was
47:19 kind of woman that men's jaws dropped she walks into the room. She
47:24 a former model, still young, , beautiful, perfect hair, perfect
47:30 , perfect everything. And he, been married like 10 years. How
47:37 have you been married? Okay, you married? Okay, so,
47:44 , Stephanie already knows that after, a few years of marriage, um
47:51 very hard to impress your spouse Already knows all of your charm and
47:58 old and now she knows and she's heard all your jokes and that's
48:03 and it's very hard to impress a a longstanding and I'm speaking now I've
48:10 married for 56 years and I guarantee that it's hard to impress a spouse
48:16 long standing. She's gonna pick me at the end of today. I
48:20 she does. I hope she comes pretty confident. No, not exactly
48:26 , sure, sure Anyway that he been married about 10 years to this
48:33 gorgeous woman and I knew that he finding it hard to impress her anymore
48:40 he was gonna impress her by showing by humiliating me in front of
48:45 but old Don still has it. I hear her coming down the
48:50 high, high heels clacking and I my, my head out the
48:55 you know what? Welcome. And noticed behind her she's coming down the
49:00 , men's heads are popping out of looking what just passed. So she
49:09 and the receptionist leaves her there. I can tell she's pissed, she
49:14 want to be in her husband's subordinate's , she wants to get on about
49:18 business. Uh and so I take papers off of a chair and I
49:25 them aside. She sits there Don says to her, I'll just
49:31 a moment dear, I'm showing I'm something to leon and uh she's gonna
49:40 this for a few minutes only so punching away and I can see a
49:46 of her comes over his face and something down and says, oh
49:55 I must amendments here, I'll finish this to leon later. And he
50:00 her out of there so fast and my head spin. And and of
50:03 what happened was that when he did uh calculation of high accuracy, it
50:10 out to be the RMS philosophy, the vertical. And so he came
50:15 me and and so he had embarrassed in front of his wife instead of
50:20 . So a couple of days later came to me and he said,
50:24 know, I checked this 17 times since my wife was there, It's
50:29 correct, it's it really is the velocity. But what's wrong with the
50:36 comment that you can see right here if if this this if this life
50:42 this wave is coming down near vertically is what we assumed right here.
50:47 come it's not vertical average? So having heard this story, I want
50:53 you guys to tell me what's the ? That same question. Imagine this
51:02 new vertical, right? It's true that Greg going down here is traveling
51:10 near the vertical average. What do think? Mr wu Well, the
51:27 is that we don't measure this vertical . We have a well and if
51:34 have a vsp we we can measure but we don't normally have that.
51:39 we look at this reflected arrival and measure the horizontal and the horizontal move
51:45 . Is the dX DT Not dizzy . So as soon as we start
51:55 the F. UT. Then we in here the geometry of triangles and
51:59 bring in and stuff like that and why we get the RMS floss.
52:05 there's a similar um diagram. I'm show you when we get to uh
52:12 nice arch street in two weeks. And uh again uh Oh good.
52:37 . Mhm. Yeah. So the are going down almost vertical and almost
52:49 they travel vertically with the vertical average but we don't measure that unless we
52:54 a world. We don't measure is . D. T. We measured
53:00 movement come out. So in that you'll have a triangle which is uh
53:10 45° triangle like this, but it be an acute triangle, you know
53:14 this. But even so uh the there uh even so you have um
53:24 triangles. And so you come out the R. M. S.
53:29 of the vertical velocity because you're measuring horizontal move out, not vertical
53:36 Now apparently uh in practice we don't the R. M. S.
53:42 V. R. M. S expressed. I'm gonna back up
53:46 here's the RMS as a sum of local interval velocity. And we don't
53:53 that when we uh when we get . So instead we regard uh uh
54:03 out velocity is premature. Ized by variable quantity which was called move out
54:10 . And furthermore these days usually have offset. So we do actually need
54:15 terms out which we parameter size before the abnormal move out parameter a two
54:23 . But even so we can flatten non hyperbolically and stack them just like
54:27 did before. So the fact that have layers in the earth doesn't seriously
54:39 our ability to make images now, course the layers are not always
54:45 So here's an example of a layer is dipping and you can see that
54:48 you have a source point here, rate is gonna uh you'll have one
54:55 going down here uh and hitting right and coming back here to the to
55:01 same source point. So if you a zero offset receiver right here,
55:05 gonna be uh it's gonna be reflecting of here. Well, normally we
55:15 not only zero offset but we have find out offset receivers but we're gonna
55:19 an image point. We're going to all those to make an image point
55:24 is here, this we're gonna image reflector at this point using the source
55:30 and all the receivers. And you see that this is not directly beneath
55:35 source, directly beneath the sources. over here. So that the image
55:40 has moved up debt. So here some mathematics to analyze that. I
55:47 want to go through this, this just high school trigonometry. And uh
55:53 it tells us is that the answer that imaging uh condition is that again
56:03 get a hyperbole but the the minimum the hyperbole a is right here instead
56:10 zero offset. So the the images has moved update. How much is
56:20 ? Uh Well, it depends upon amount of the dipping amount of the
56:25 and the depth to the reflector. uh this is the reason why we
56:31 imaging, the old fashioned name for is migration because we say the image
56:37 has migrated from uh just below the update by this amount here um update
56:49 the midpoint. So we call that you're still here occasionally. That imaging
56:55 this time is migration but the idea uh the image point has migrated up
57:03 from the middle. So what seismic do is they make fuzzy representation of
57:11 subsurface reflectors hopefully accurately located in space economy. And so modern imaging algorithms
57:21 many of the assumptions we just talked and they offer you a better images
57:25 at the same time that's good. the disadvantage is they rely more strongly
57:31 getting accurate subsurface philosophies which you don't at the beginning of uh of a
57:39 uh project. You've got to use data to produce the velocities to improve
57:45 images. And so imaging is normally generative process and we have here at
57:53 University of Houston, some uh great in that and uh see some instructions
58:05 one of those is Professor joe is to give the that imaging you.
58:15 , so uh so imaging as part the process process, it also
58:22 you know, ideas like simple filtering like that, but but he's one
58:28 the world's experts on this matter. I'll leave the further discussion.
58:35 um I have very summary of lecture , You learned how the wave equation
58:42 p wave solutions and those are our interest to us, how we can
58:48 the simple plane waves was easy to and some of them together making compact
58:54 , let's although each one of these goes on forever. And a lot
58:59 for example, when we talk about , we're going to talk only about
59:03 waves, although individual and uh acknowledge sum up all those solutions to make
59:14 reflection seismograph. Last week we talked how uh when you add a term
59:23 the wave equation which describes the then you get spreading waves instead of
59:28 waves and we uh learned a little about bri theory, just enough to
59:37 what it is and what it is a high frequency approximation. We've uh
59:44 frequency approximation, high frequency approximate Great question. Right about move
59:54 And uh this is a crucial thing uh waves are going around inside the
60:01 all the time from our own sources from other sources from natural sources,
60:06 earthquakes on the other side of the , everything. And it's crucial that
60:11 uh they superimposed on one another and disturbing each other, they just slide
60:19 through Then after we finished with the we talked about share waves and converted
60:25 and the convolutional model and then briefly to make a seismic image from these
60:32 . So that is um the end the lecture on last saturday. So
60:42 is a good time to take a . Um So let us come back
60:49 Here at 30 minutes after the hour we will take up the next topic
61:01 far. We were talking about body which are waves traveling inside the body
61:07 a lasting material. And now we're talk start talking about what happens when
61:12 a surface somewhere and that with uh the surface. So um of course
61:21 going to be important for reflections. even before we get to reflections,
61:26 want to type out waves which are with the surface in a way of
61:32 um like they don't reflect it and away from the surface. They stay
61:37 to the surface. So this is pretty mathematical flash. But I've made
61:45 as simple as as I can and think it's I think it's useful to
61:53 how this and the surface changes I mean I know you look at
62:00 simple equation, the wave equation which we had written over there and we
62:05 the solutions and that's that. What be easier. Well you're about to
62:10 out that uh any kinds of solutions that equation and the ones which involve
62:18 surface is gonna be um particular interest us because we have all kinds of
62:25 , not only um um layer batteries the air, but actually the most
62:31 surface is the free surface where we our instruments. That's the most important
62:37 because that's the biggest um uh I mean above it we got air
62:45 , we got rocks. So everywhere we got rock, Rocks on the
62:50 or maybe water on one side and . And so surface for the biggest
62:56 is the service. So uh let's about surface waves which are localized near
63:03 surface. So the end of this , you will be able to explain
63:09 that same wave equations have solutions which along the surface of the year.
63:16 a special type here which is invented a guy named Lord Really and we'll
63:22 those in some detail. And uh there's similar things, other services.
63:29 lots of uh of related applications. then you're gonna find out that because
63:35 this surface velocities turn out to have surface waves have velocities depend upon
63:45 So when we add together a bunch fourier components, uh different foreign components
63:54 different philosophies, any other frequency. uh that's that's a crucial aspect,
64:05 real world right? There's another class service where it's called Love ways
64:13 Already did we mention the guy named ? Yeah. So Love and really
64:20 uh contemporaries. No another type of . The cylindrical surface which corresponds to
64:31 world. And so we call those boys because for mysteries. So let's
64:39 first start talking philosophy. Oh, a differential equation, like the one
64:46 we have written down over there, tells how things change. But you
64:51 want to know the solution uh has , you wanna know how it
64:58 you want you want to know the , not how things change. So
65:02 means it starts off at a boundary then it changes away from that
65:07 And so uh that that boundary is be in the final um uh the
65:15 answer. And also we're going to boundaries in time. So we have
65:19 conditions as well as uh boundary And so these conditions we're gonna use
65:26 uh evaluate the various constants which appear these um solutions for example uh in
65:37 mhm. Uh Look at the simple equation and look at the plane wave
65:44 in there out in front, there's an amplitude counts. And so we
65:49 to evaluate that constant using idea using boundary conditions and the initial, we
65:58 have to do that before but now do. So the first example of
66:07 boundary conditions influence our data is The reason why that's so important is
66:13 always have our instruments at or near free surface which is a humongous um
66:20 boundary because there's nothing above because of surface there. Additional modes of sound
66:29 which we haven't talked about yet and source generated and they show up at
66:35 receiver and so in many cases they the most the most energetic um part
66:44 data on our records. So normally regard this as a noise and we
66:51 it but first of course we have understand so what are the boundary conditions
66:56 the free surface? So the first is that the sheer the stresses all
67:06 at the free service. Not not component of stress but all components of
67:11 that have a three or two threes among their industries. And it's true
67:16 at the surface which we're going to is Z equals zebra, but there
67:24 other components of stress. For tau 12 doesn't have a three in
67:29 . So that doesn't have to be zero because because the um because the
67:41 surface is perpendicular to three directions, why we have a three here and
67:48 here and three here among the boundary and of course uh stress tensor has
67:56 be symmetric. So this 31 is same as this one. So what
68:04 means all the forces on the horizontal At the service, R zero at
68:09 Times. And the reason for that statement there in english is if there
68:16 such forces then the surface would accelerate not only accelerate down, but it
68:23 it would blow up. We have acceleration. Yeah, first thing first
68:32 is uh really, so let's consider traveling in the one direction, like
68:40 see on the left of this three is pointing downwards and in two directions
68:46 so Miss Del Rio is the two pointing into the screen or out of
68:51 screen if you want to walk up the screen and hold your hand.
68:59 there Your yeah, it's 123-123 into screen. Alright now we're going to
69:21 look for solutions uh which are traveling and have displacements only in this
69:30 So if there's a displacement out of plane, uh that's not the kind
69:33 solutions we're looking for displacement direction is be some arrow like that in this
69:42 , the progression of the wait gonna exactly horizontal. So there's a picture
69:50 really. He looks kind of pleased himself, doesn't. I think he
69:55 a member of the british House awards it was such a such a um
70:06 respected scientists. And in his they didn't have a lot of technically
70:15 companies, they didn't have uh the Science Foundation is going to give grants
70:23 research. And I think that in era in order to do research,
70:28 had to find a wealthy sponsor. sponsored the research out of philanthropy.
70:34 I think his family did it. think he was born into a wealthy
70:41 . So here's a picture taken from and gelder and it looks pretty,
70:45 not a very good copy of the . But you can see in here
70:51 all this stuff is in here and are, this is low velocity and
70:56 . Why do I say it's low because it takes a long time to
71:00 a short distance. So these these of data here are traveling slowly because
71:06 takes a long time to go a distance and you can see right here
71:10 see a reflection. Can you see here is a reflection here and no
71:14 it extends. Maybe it extends right here, maybe that's the same
71:17 But obviously you can't see much of because of all these norms. So
71:24 you want to do is get rid it so we can see the reflections
71:32 . Okay, now the stresses are by Hook's law and we wrote Hook's
71:35 on the blackboard earlier today we use uh symbols for the indices, but
71:43 comes to the same thing. And , I'll remind you again that when
71:47 see a repeated index, like that means we're summing summing over M
71:52 N. And when you see an parenthesis, you better see a similar
71:57 of unmatched parentheses on the other side the equation that specifies which component
72:06 Mhm Yeah, previously we said that uh these stresses carl zero at the
72:17 . And so when you uh uh first one, we're looking for cow
72:24 . So we put a 13 right , it's the same uh same iJ
72:29 as it is here, same 13 as it is here and then we
72:34 have to sum over all the M M. And so uh when you
72:38 this some because we're insisting that that only looking at displacements in the plain
72:45 the figure, that means that all strains except for these are zero.
72:51 in this son, the only two terms of these same thing with 23
72:57 33. Yes. Now we're gonna one more thing. Uh This is
73:06 strength of the 23 strength. And this implies that some of the displacement
73:12 be in the two directions. So since uh since we're gonna look for
73:18 really waves, we're gonna assume that are zero for the study of
73:27 So then we're uh we're left with equations here. I choose to not
73:41 them. Yeah, so those terms going to lead to love ways.
73:47 since we're intent here upon studying really , uh that's not what we're looking
73:54 . So uh that's a later we're just gonna say, study of
74:02 are going to be easy. now look at this equation here,
74:07 as we had on the previous uh . And uh you recognize that epsilon
74:15 is same as epsilon 31. So that gives you two here. And
74:23 definition of epsilon three is epsilon Is this thing here with a one
74:30 and a symmetrical set of displacement And the english name for C.
74:38 . And also for C. 1313 the sheer marginal commute going through that
74:45 logic down here, we have these terms coming up and this is the
74:52 of thing. I'm gonna uh if need to puzzled by these logical
75:02 I'm gonna pass on for now that take that up with Mr wu
75:10 Okay now you see this boundary condition both the shear stress and the normal
75:20 . And because of that the solution be neither pearl free nor divergence.
75:26 how we took the wave equation for weights. And we so we Select
75:33 the partner has zero curl and all . A P. Wave. And
75:37 selected out the part that has zero . Call that a share wave.
75:42 can't do that here because of those conditions that I just showed you.
75:53 , Mr helm Holtz is still on job and he he says that the
75:58 will have girl free part and a free apartment. Uh These are gonna
76:06 together. How can they travel together I know you're looking at this and
76:13 that's the P. Wave and that's sure way. But we need for
76:17 things to travel together. So let's the next line we'll remind you
76:26 we call the curl free part uh the gradient of a vector of the
76:33 of a scalar because um um No uh there's uh back to identity which
76:43 talked about on the first day says if you have the gradient of a
76:48 , The curl of that is the . And so that means part of
76:53 um This is going to be the here. That's part of it is
76:59 be the greatest of a scalar potential similar part of it is going to
77:04 the curl of a vector potential. both of these are gonna travel at
77:10 same speed. How can that be you know this one leads to a
77:15 wave and this leads to a shear that's in the absence of surface.
77:22 we're gonna have this surface in our . So the surface is gonna make
77:26 to travel together and they will be api way nora share wave really.
77:35 the curl free part is gonna be this wave equation and the divergence free
77:41 will uh I will be this wave . You see, what is the
77:46 here? We have the scalar potential and the vector potential here, we
77:51 V. P squared here and the squared. And these are body wave
77:57 . Even though the wave is gonna up traveling with the really way these
78:03 still the material parameters which appearing in equations. So I want you to
78:09 of these as elastic parameters. Think this one as uh Moro and think
78:15 this one as well. Uh We're find these solutions for fine for side
78:25 in a way which travels horizontally only a really wave velocity. Okay,
78:36 we're asking here which components of of do we need here? Uh the
78:44 of sign is given by this expression this is just the definition of the
78:50 operator operating on a vector which we sign. She's got three vector components
78:59 it's got all these derivatives here. this is the X- three derivative of
79:04 two components of plant. Now, a really way we're going to uh
79:13 that everything in the two direction is . So that's zeros. That's uh
79:21 because I only want to look at web solutions. So by definition,
79:27 about how about this? Is it from this if there's nothing pointing in
79:32 two direction, that means uh there's driven to direction either. That gives
79:38 a here same reason that gives the here. So um the only component
79:48 need is side to a component. one here and this one here.
79:53 , so sai is gonna point out the screen or maybe into the screen
80:00 that's okay. Uh the way the is gonna stay in the screen in
80:06 13 component and the 13, The of the screen is going to involve
80:13 two on. Okay, so so write out this term explicitly in terms
80:24 the components we need. So this with three components. Here's the first
80:30 . It's got the gradient of five respect to X one. And then
80:35 back here, there's the one we minus side two divided by X.
80:41 uh derivative of X. So that's term similarly. Uh For remember we
80:51 two components zero since we're gonna look for reading so at the surface Sheer
81:04 is zero and uh this is uh definition of uh wrapped up shear stress
81:25 given by you times what's in the 11 and now you want not everyone
81:43 you want. What is that that given right here. one is this
81:49 in terms of the potentials? So they are Similar for you three in
81:54 of the potentials. Similarly for the normal stress and see different differences
82:05 Uh particularly we got lambdas and M's of you. Okay now um if
82:16 were mathematicians we would have uh elegant to solve these equations. Being
82:24 What we're gonna do is we're gonna the solution and you're gonna and then
82:29 gonna verify the guests and you can confidence in me. I am going
82:34 make the right guess but I still to show you that. That's
82:38 Let's let's assume here that the guest five is a plain white and the
82:46 for sign too is a plane wave got here subscript zero but I keep
82:52 two here because just to remind you this to now look inside here these
82:57 wave uh exponents, we got the omega but we have different wave
83:06 And but both of these in order this thing to solve the uh curl
83:13 wave equation. We gotta have a where A. Is equal to omega
83:17 v square. That's a condition on length of K. In terms of
83:23 . And remember V P squared. should think of that as just an
83:27 parameter. M overall. Uh It's the velocity of anything. The wave
83:33 not, the velocity of particles is the velocity of the wave. The
83:38 is gonna end up traveling VR. similarly for this wave vector which we
83:44 h, the length of that is to omega and B. S.
83:51 solve the problem, we need to decide these two spectral coefficients and the
83:57 of the this was our solution for for the shear stress we have the
84:07 stress is zero at the surface. in terms of these potentials, this
84:13 what it looks like. And so our guest uh from from the uh
84:19 page this derivative is gonna produce AK and K three. That comes from
84:27 uh differentiating side um in one direction three directions. Uh Because of that
84:35 number here, when you differentiate uh thing with respect to X one,
84:41 get uh the same potential appearance as and now multiplied by K one.
84:48 again with this one, you get by K three. Similarly. Over
84:53 we got uh derivatives of side And we get an H three and
85:00 three C. This is a three a three. This is one and
85:03 because we have 13 here and we three years and that's similarly for the
85:14 . So if we uh collect terms and uh we'll find them. This
85:24 just a single equation, But inside is really uh 50 times exponential.
85:36 so we expand and collect terms and and uh notice here that uh we
85:42 only from the dot product of K X vector. Uh we have only
85:50 these uh this is uh the only left because uh there might be a
85:56 two and the next to uh but uh K two is going to be
86:03 . Uh assume that every all the is in the plane and got X
86:10 equals zero because we're at the surface with only uh X one. So
86:18 is gonna be a way of traveling the X one direction, you
86:24 Yeah, here's uh the trivial So the trivial solution here, we
86:30 can find a solution just set near zero. So that worked terrific.
86:37 we call that the trivial solution. that has an important consequence. There
86:43 no really way at the surface of ocean. And uh that's one reason
86:48 marine data are usually easier to process landing, they look much better.
86:53 they don't have a railing reasons. it's simply because that comes directly out
86:59 this by assuming if new equal that's a solution that works. So
87:05 it happens in the oath no So then assuming that is not to
87:13 what we can uh divide by Then you find out the E.
87:20 . Makati and left this. Now going to assume this is true everywhere
87:26 the surface. So it must mean that a one is equal to
87:33 This uh everyone is different from H . This thing can't be true for
87:38 X. So we must have that equal to H one. And then
87:46 need to use the uh both of are equal to omega over quantity which
87:53 going to call the rarely velocity and or minus depends on whether the wave
87:58 going right or left. And you that we didn't see anything in here
88:01 a source uh source happened somewhere. around the surface waves are coming along
88:09 we're analyzing how they go uh moved here to there without worrying or caring
88:16 the source. And then once we K. One equals H one,
88:24 we can cancel out light out those And well after with this solution
88:31 So what are the unknowns here? unknowns of the case and the 50
88:37 the size 02. And the ratio the of the spectral components is in
88:43 , in terms of the um of wave vector components. Buy this for
88:53 Now, we're not done here. know that K3 square uh K three
89:02 plus K one squared plus K one is a square. So we should
89:06 subtract K one square for both And so this expression length okay,
89:14 given by mega Vp and the length and he one component we just decided
89:27 gives the really philosophy, that's this a similar thing for H.
89:33 S. Here said a V. . Here we have the same or
89:36 philosophy here. This previous argued so that um uh conclusions you can see
89:47 the the scale of potential which is like this in terms of K one
89:53 K three explicitly, it's like this now we want to separate out um
90:00 a three component and all the K component omega over pr vega. And
90:10 um that's just implementing everything we just . We do a similar thing for
90:18 too. And look this is this very similar to what we have
90:24 So that means that this side two is going to be traveling at the
90:29 speed as the 50 component. It's five component. And look here out
90:37 again it's not an exponential factor which different. Now let's think about
90:54 Ah this part is obviously wiggling away it wiggles away hearts. This is
91:02 be we're going away um radically. uh what happened to the omega?
91:13 Well um you can ask uh what to the omega up here.
91:22 uh rega yeah, being started out , multiplying everything, Here's multiplying
91:38 And so Omega Times T. No, I think I have a
91:51 here. Yeah, I have stayed , I didn't do this. Uh
92:03 was a great mistake. So I there would need, maybe I think
92:14 omega should be out here. As matter of fact, this is such
92:17 bad mistake, I'm gonna fix it right here. So don't turn off
92:21 the according, let's just fix this right here right now, what I'm
92:29 do is I'm going to, I think I'm gonna have to stop
92:43 . Uh we are gonna have to the, stop the recording when I
92:48 sharing. Will stop the recording. , go ahead. Okay, so
92:56 while the recording was off, we some mistakes here and now it's
93:02 It's got the proper behavior of the positioning of all the omegas in
93:09 And now we're going to uh to with making further progress that we're going
93:17 decide just exactly what this uh piece our is in terms of things uh
93:23 terms of material properties, stars, not uh advancing. So let's see
94:13 , this is okay, now we're now we're going to share this.
94:39 that works okay. So remember all of we got this far by
94:47 looking only at the shear stress boundary right here, shear stress boundary
94:53 Remember we had another boundary condition on normal stress. So from the previous
94:59 that was given by this and using logic as we did before, we
95:04 that and collect in terms of simple just like we did before. Um
95:11 And furthermore, using this relationship between minus lambda gives us a two mu
95:21 then using the ratio of the spectrum we got. Uh two pages
95:25 three pages earlier, that ratio which is the ratio of spectrum right
95:31 , here's that ratio spectrum and then more algebra simplifies like so collecting
95:38 Like so and uh now what I do is make this adjustment in
95:47 This m over immune is in terms body wave velocities, that's V.
95:54 squared over the square. And the uh the density is canceled out.
96:01 one square uh is uh omega squared B. P square. And so
96:11 three P squared here cancels this P squared, cancels out this V
96:15 squared here. And then left with squared over the S square followed by
96:19 these components. And if we collect further you can go over this later
96:25 yourself. And uh uh we're gonna in this case, one square by
96:33 over V. R squared here and and here And then we remember
96:41 We have these these expressions for the wave numbers K3 and yes. And
96:49 all that in there, we finally this equation for the railing, wave
96:56 . And what is this equation? , it's zero equals this collection of
97:03 of velocity and see the VR is here. Yes, this is a
97:08 wave. We asked velocity and um this other stuff. And uh so
97:16 what is the unknown in this Well, the unknown is VR everywhere
97:22 uh V. S which we assume know since we're doing this for a
97:28 medium with a known share velocity and the p velocity over here. So
97:33 an equation for video. Here's an thing. The omegas have all
97:40 We're gonna back up a little bit you see, you see omega's here
97:44 . But that's the key. They're . So you can divide them all
97:48 . So you're left with no dependence omega. So that's really um an
97:56 point. Uh It says it's uh way this is an equation for the
98:03 doesn't depend upon uh frequency. So says the way that uh um really
98:14 velocity in this case, does it on frequency non dispersing? Well,
98:23 probably know from looking at raw seismic that there's plenty of uh dispersing really
98:32 in that data, but we They're not dispersing. So what's wrong
98:39 ? Uh The answer is that this only applies to the simplest case.
98:45 In a realistic case with many layers gonna get this person. So let's
98:53 at this and this is awkward to all these square roots in here.
98:56 so let us uh take the square that equation. And uh wow,
99:04 what it looks like. So now become known as here, VR and
99:10 as a ratio with the S. here it is with as in a
99:14 with VP. And it's 1/4 order . See it's uh four third
99:19 V. R squared C. Uh imagine um squaring this all out reading
99:26 to the fourth power. The Uh The highest term is gonna have
99:33 be our to the 8th. That's 4th order, wow. And here
99:40 highest order is E. R. . Well, when you do
99:47 The lowest order term is a -2 the fourth power, which is
99:53 And on the right hand side you all this out. The lowest our
99:57 is a plus two actually plus I'm 16. And it's the
100:04 And so these these low order terms . So if you just cancel out
100:10 these low order terms then you have V. R squared in every single
100:17 . There are no terms independent of once this term cancels this term,
100:24 means you can divide by v our and get a third equation and
100:28 R squared. So that's that's good . But even so, cubic equations
100:34 really complicated. You probably know in mind or you can remember easily figured
100:39 what is the answer for the quadratic or the uh highest power of the
100:45 is square. But the high that probably don't know cubic equation of,
100:51 highest power of the unknown is to third power. But other people have
100:59 on this and they conclude that normally is normally a little bit less than
101:05 . S. So these these waves traveling to the right or maybe to
101:10 left or the combination uh with the V. R. Which is dependent
101:18 V. P. And D. . Both bodyweight VP and friday Ds
101:23 the wave is traveling with VR and traveling with a little bit less than
101:29 . S. So uh clever thing do is to uh take this result
101:38 and say okay I'm going to uh that VR is a little bit less
101:44 B. S. And I'm gonna the difference between VR es where is
101:50 quantity zeta, which you can tell a non dimensional quantity and put that
101:56 put zeta into the previous incident, out VR here with uh zeta and
102:06 taylor expand the quantity. Keep only first order. Uh So when you
102:11 that it leads to this equation here that's pretty simple. Any of us
102:16 really understand that. And you put here uh Stephanie's favorite favorite number for
102:23 velocity ratio of E. S. . E V. P. Is
102:25 half. So you square that and get 1/4 divided by at multiply times
102:32 gives you a number one. And that has subtracted from three gives you
102:36 two and two times four is So uh divided into one. So
102:43 and 18 to see how that works . And uh using that approximation
102:52 S. Is uh related. Pr , by this formula push it has
103:01 graph like this. So here's the of V. R. O.
103:04 . S. Starting as a function V. P. To be.
103:08 this is the body wave velocity And this was uh Stephanie's favorite
103:15 But uh you know, in the surface the velocity ratio repeatedly gets
103:25 And here's the way to think about . As you get close to the
103:30 , the rocks become less and less as the VP gets less. VP
103:37 never less than the philosophy of Now, as you get closer to
103:44 service. Uh these less and less rocks have smaller values of V.
103:49 . But they don't have any lower . You can have yes, to
103:54 arbitrarily small. So ratio gets can to be large. So here it's
104:00 ratios as large as five. And see where those. So this is
104:04 near service unconsolidated rocks. And for like that the railing. Wave velocity
104:11 to be say within five or so the show and philosophy. And you
104:19 see it's never gonna uh never gonna to uh one but it's uh somewhere
104:27 90%, of uh huh. If wave body blocks is a good number
104:36 railing results. Okay. So having solved me off in terms of this
104:46 small parameter Rosetta, let's put that into the expressions we have for uh
104:53 vertical wave numbers A three NHC. working through the algebra you find that
105:02 three is a pure imaginary number. imaginary number. It has no real
105:09 to it. And you can say same thing about H. And it's
105:13 different functions of zeta. But they pure numbers. Now that has an
105:23 consequence. seven Remember that we had functional forms for the potentials where we
105:31 out the uh horizontal travel bit. that's the same for both of these
105:37 uh functions that they had different exponential out here. And now we know
105:44 K3 is a imaginary number. So here it is. And so um
105:55 me write this expression as uh um this part here and call that the
106:07 value of K three. The eye still out there. Yeah. And
106:12 same thing with the age. We're choose a negative sign here. Why
106:19 we do that? Because right here gonna uh we're gonna put This expression
106:29 K three right here And this are that are is going to make a
106:41 . And uh and this is gonna a plus one for that. So
106:47 if we choose the minus sign We end up with this minus times
106:55 K3 times seen. And that says exponential factor is going to be decreasing
107:04 amplitude as Z increased. Z equals . The amplitude is 50 as the
107:15 . Uh as you go down into end of the ground, this factor
107:22 gonna make the amplitude smaller and And you see it's not wiggling
107:26 The wiggling comes from the eye. I hear anymore because this I got
107:34 by this I Same thing for side . So they decay exponentially exponentially with
107:45 . And this is why we call particular wave solution surface wave because it
107:51 itself near the surface. So how does it reach? Well, uh
107:59 can um big these estimates. And and here is uh um washing the
108:10 squared and the wavelength of the railing , the the wavelength of the corresponding
108:18 . Your wave. And so you see that K three reaches down into
108:22 medium about six wavelengths. And um H three uh vertical component of the
108:34 uh is smaller because zeta is less one area of the spirit of
108:43 less than one. So this one at a different rate, the other
108:50 . And uh so uh putting that , you're saying that um wave reaches
108:59 only a fraction of a wave. right here. So this part is
109:17 part is real and it's moving in X direction huh, spanish uh which
109:50 go back here. So these um is the real part. This is
110:01 vertical component of the wave number and a pure imaginary quarter. Remember further
110:09 we said that the horizontal component who is a real note? It's given
110:15 omega over VR. And so the park is the horizontal uh component of
110:23 way back in the real norma And vertical component of the wave network is
110:27 imaginary number. So that's what this right here. The imaginary part of
110:34 wave vector. The vertical part is parallel from the real part, which
110:38 the horizontal part. So they call an in homogeneous wave. That's what
110:45 mathematicians say, because manufacture has two and the world partners not parallel to
110:59 . It's easy to confuse that with homogeneous equations or with non homogeneous
111:07 but this is awake and it's an homogeneous wave to a mathematician because of
111:14 feature of it. Now, if have the elastic wave a question which
111:27 um what we're dealing with elastic wave . It only allows in homage ingenious
111:35 where the imaginary part parts are perpendicular the real party here. The other
111:41 are possible with an elastic media. example, if you have an insinuated
111:46 , it's going along in this it's going on, it's decaying with
111:53 . That means that uh yeah uh has an imaginary part this uh in
112:03 direction also. And so uh if have a genuine nation, the imaginary
112:10 of the way back can be pointed any way it wants depending on
112:15 So for the elastic favorite patient, it's in a homogeneous, the two
112:21 have to be perpendicular like you Yeah, the exponential decay with
112:30 weaker lower frequencies, since both these components depend on homemaking. So uh
112:40 has a real practical consequence for earthquake . Uh they know that earthquakes are
112:49 to be making surface waves as well body waves and they can measure these
112:54 waves as a function of frequency on other side of the world. And
113:02 know that the high frequency waves are only reach down a little bit and
113:06 low frequency waves are gonna reach down lot further. And so by examining
113:12 frequency dependence of velocity of surface they can unravel the velocity structure of
113:21 earth, deeper velocities. Uh deeper of the only affect the low frequency
113:28 . Low frequency surface wave and shallower of Earth's subsurface affect growth. Here's
113:43 understand anything. If you consider a of very low frequencies. You can
113:50 talking about wave propagation of solid of and talk about the loads of free
114:00 , the oscillation of the earth. think of, think about a bell
114:06 a bell in your hand, Bella's here, you whack it and it
114:13 and it keeps on ringing and the belt is vibrated and ringing and sending
114:22 sound waves into the air, but ringing. There's no more waves propagating
114:26 the belt. It's just oscillating by . And so those are called the
114:33 of free vibration of the belt. each bell has its own before your
114:38 . The people who make the bell to make it so that it's made
114:43 of a good material like brass instead lead. Nobody wants a lead bell
114:47 there's too much intent and intensity continuation lead. So you get a good
114:53 like brass. And depending on the the size and the thickness of the
114:59 , it has a tone. And in a turkey car, they'll have
115:04 bells, each with their own They play tunes on those simple
115:09 And uh um what you're hearing is free, the free corporation of the
115:16 . And the tone that they're, tone that you hear is called the
115:22 mode of oscillation of, well the mode and it has higher harmonics.
115:30 maybe you can also hear those uh in the Earth? See if I
115:38 a slider corresponding the in the Um When a when a big earthquake
115:47 it sends waves through the body of earth and also send surface waves around
115:52 earth. And after they've made a circuit of the earth and they're all
115:58 whole earth is vibrating together. And I measured the uh frequencies of
116:09 free oscillation of the earth. And what are uh last velocities at
116:20 And so um take a take a . I'm going to ask mr you
116:26 is the approximate period of vibration of greatest mode of the earth? So
116:37 an earthquake happens in Taiwan uh waves around the earth and we can measure
116:44 in christian, we can measure vibrating hours. Gave me bigger thing.
116:51 the period of the greatest mode of here? Oh yeah that's not
117:09 In 19 days. I'm not sure you get that. Where did you
117:14 up in 19 days? Just a just to guess. Okay well you
117:21 do better than that because you know in 19 days um uh the earthquake
117:26 dissipate the attenuation will kill it off the bigger biggest earthquakes, they don't
117:33 for 19 days. So it turns uh I think it's not easy to
117:39 um um Well have you seen a a grant from uh seismographs? What's
117:51 lowest frequency you see in a seismograph a distant earthquake? Yeah.
118:03 Think of it in terms of Okay. Uh so 100 seconds is
118:11 to correspond to a mode of oscillation reaches down a few 100 kilometers.
118:20 mode is about an hour. So mode is for oscillation of the earth
118:27 this. And then there's other modes to different oscillation like this. Many
118:34 uh votes. And those are all by spherical harmonics with different industries.
118:42 mode is called 00 modes and for like this and it normally doesn't have
118:49 lot of energy but it's the lowest . You can never measure an earthquake
118:57 about a correspondence to a period of an hour. So uh of course
119:03 never see these modes of free oscillation uh hydrocarbon data instead of what we
119:12 uh surface waves. And so this a few um animated gift made by
119:21 Russell. Um I don't know but can see um that the amplitude decreases
119:31 depth and you can see the wave moving from left to right. And
119:37 I see that every individual point is in elliptical motion and it's called retrograde
119:44 it's going around counter clock seeing from sort of. And so I think
119:51 is sort of cool. So um would measure it at a kind of
119:57 like this And that's what you can look at this Too long your eyeballs
120:13 to cross now as we derived The really uh really way of velocity
120:23 non dispersant. No think what it . If you have an equation where
120:36 velocity is a function of frequency, does it know the difference between high
120:44 and low frequency? It's got to in the equation something that so that
120:51 knows that it is 10 is 10 high frequency or low frequency. It's
120:56 to have some characteristic frequency physical parameter gives a characteristic frequency and then all
121:05 other frequencies get compared to that or equivalently have a characteristic length. And
121:12 all the other frequencies have corresponding wavelengths those wavelengths are either long or
121:19 characteristic length. So in our problem , when we did there is no
121:25 frequency which you can define from the properties and there's no characteristic length.
121:34 however, if you had a layer somehow in the layer thickness would be
121:41 characteristic length. And so that would that when the consequences of that got
121:48 through all the equations would find that different frequencies have different wavelengths and they
121:56 know whether they're long or short compared this uh layer thickness. And so
122:02 didn't have any of that in this but in the real er uh it
122:06 . So uh we don't want to into that uh level of detail,
122:11 immediately you can see that the different dispersion, the frequency dependence comes from
122:20 Blair in there and and maybe it come from something else also, but
122:27 we'll get into that later. So the question. So we already derived
122:38 penis wave equations in less than four it was simple, I would say
122:43 they were simple, but really waves complicated, we slogged our way through
122:48 and uh I think probably right now quite confused about that derivation, but
122:58 you go through it on your own using the um materials which are in
123:05 blackboard, I think you'll see that all works out pretty straightforwardly and comes
123:10 with this equation, that's the third equation. But uh um uh made
123:20 uh simplification by assuming small zeta and know, it can all work out
123:27 uh reasonably, but it was a work to get there. So why
123:34 it so complicated we got here uh um is this true? Um MS
123:43 rio, did we new, did need a new wave equation?
123:52 what was it? I showed you wave equation that with a K vector
123:57 p waves. And uh I shouldn't wave equations for p waves. And
124:04 was curl free again, I said wrong, it was a wave equation
124:09 the wave factor, we call it . And that was curl free.
124:14 showed you another one which was divergence , but those that was all old
124:19 for us. We did not, need a new wave equation. So
124:25 . Is definitely true. And she uh true also. Um Let's see
124:36 um. Uh How about the trap travel in the X. Direction?
124:48 polarizing the xy plane. Okay so that's true but that doesn't answer the
124:53 and she is also true that it answer. Yeah. Now uh surface
125:01 important. The whole thing was that the problem was it was difficult because
125:07 had bounded english. Is this the condition? Well it's a trick question
125:17 that's true for some of the stress but not for all of them was
125:22 true for those stress components which have three among their industries. And so
125:28 example how 12 was not uh was in there. So that one is
125:38 . This one true or false. that one's true. This one tour
125:52 us girl. Free part has wave K. Dever industry part has wave
125:58 H. So did they propagate with philosophies? Well they don't they both
126:09 with the same velocity of er um Even though even though the wave vectors
126:17 different, They have the same horizontal K. one equals h. one
126:23 they differ in their vertical departments. 1 to ever falls. Oh they
126:42 uh Well I guess the equation is well formed but the wave travels in
126:51 horizontal direction and uh the real parts K. And H. Uh mike
127:00 do that uh K. One One R. Reel. And they
127:07 VR and they're pointing in the horizontal , imaginary parts. This one true
127:17 false. That's false. Because it . That was a trick question.
127:22 was an easy trick question. This is more difficult here. Um uh
127:30 regular wave has a curl free part a diver industry part. And but
127:35 travels with with its own velocity R. And so uh does it
127:41 with the velocity between body wave velocity . P. And V.
127:48 Yeah that's false. Even though it intuitively that should be true. But
127:52 proved that is false. The railing travels with the velocity is less than
127:57 . S. Which of course is than V. P. So uh
128:00 would think that uh well here's a way to think about is traveling
128:04 And it's got the uh the curl park tangled up with the divergence free
128:10 . So it can't travel with Or V. S. Either
128:13 But you would think that would travel velocity which is intermediate. But no
128:18 travels with the velocity less than S. Okay now let's see how
128:25 doing for time. Um Yeah well let's talk about other surface waves traveling
128:37 the plane and one of them is with us. And uh so regular
128:44 really waves Schulte did Schulte waves. so what Sheltie waves are an extension
128:51 rarely waves as seen on ocean Seismic data for salty waves. The
128:56 half space is liquid, not So that means that's gonna modify the
129:03 condition. Uh like really waves, motion is confined to the 1 3
129:10 . And the boundary conditions are Kant as stress, not vanishing, a
129:14 continuity as stressed at the c force the sea floor. You've got rocks
129:20 water above and um uh and a wave propagating along that surface as called
129:29 sulky wave. And we can handle just like we did with really
129:38 but the analysis more is more The results are similar velocity is less
129:43 the share velocity decays exponentially away, going downwards and going up. That's
129:51 action point. Uh really there's no at all in the airport. Mr
129:57 , he's got a wave that's gonna only P wave components in the in
130:03 water of the interstate, but it's going to be traveling with p wave
130:09 is going to be traveling with Schultz wave velocity. And I'm uh I'm
130:14 one of the first year physicists who um in the modern era Who uh
130:23 about these things, Schultz. He his work probably say shows he lived
130:29 be 63. He probably did this when he was like 30. So
130:36 about wartime. Um and then people really pay much attention. I never
130:44 about multi ways until I saw them I saw them on ocean bottom seismic
130:50 . And I was one of the people to look at the ocean company
130:54 company Ocean bottom seismic data. And enough, we saw these travel these
130:59 traveling very, very slowly, like uh 200 m per second. I
131:06 really slowly. Yeah, but we see them in our data.
131:11 what is that? And uh, looked it up in the textbooks and
131:18 up these really type waves, surface traveling on the interface between the ocean
131:25 and the subsurface ocean rocks. And travel with a velocity, uh very
131:32 velocity because the body wave velocity and body wave velocity in a near surface
131:46 bottom of the sea is very low incidentally, that's the reason why you
131:53 see too much conversion of energy where wave is going down through the
131:58 It's the ocean for most of it , most of the energy gets
132:03 Very little gets is like uh very gets converted. And so uh,
132:14 because of the properties of the because blood is so muddy basically.
132:20 and so that's why most of the energy that we see on converter waves
132:25 from the conversion of the reflection, conversion on the sea floor down.
132:31 so uh, I think I told that the first people who looked at
132:36 wave later, they wouldn't statoil and didn't understand that point. And uh
132:45 thought they analyzed their data, convert way of imaging of their reservoir and
132:51 got a passable image a better image for uh the ways but they analyze
132:59 it in the wrong way, embarrassed in public um because they didn't understand
133:16 . So here's another type of uh type ways their internal surface ways where
133:25 a layer boundary with rock below. walked about typical reflecting boundary because that's
133:33 that's because that's the surface, it another type of surface wave. And
133:41 one is named after Stoneleigh. And what, how is it different?
133:46 It's got rocks above and rocks below as before the motion is confined to
133:52 1ft plant. And uh under our continent continuity of stress and also
134:02 of displacement because it's solid over And so that's another type of
134:09 We can handle it just like we before. Uh and the analysis just
134:14 before. I can have a picture . So this comes from Sheriff and
134:18 dark. And uh this shows that for for a model which they did
134:30 Sheriff in and were graphing as a of the ratio of sheer module light
134:37 the upper medium in the lower And as a ratio of the densities
134:41 the upper and lower medium. We get um um solutions in certain areas
134:47 this graph, for example, here here and so in those cases the
134:53 the two layers differ strongly and uh that's complicated, rarely measure these.
135:02 is it? We don't measure them because we rarely have instruments at that
135:08 . How would you do it? only way you could do, you
135:11 have a borehole and right uh next where the borehole intersects a boundary and
135:19 want to have a strong brand, supposed to have salt below and sandstone
135:25 . You want to put a few your PS two PSP tool right
135:31 you can see uh you can measure wise on this boundary, but nobody
135:39 does that. Uh you know uh you're doing that you're introducing into the
135:45 the cylindrical borehole. And so your is going to have a lot of
135:50 home surface waves in addition to uh . And then Stoneleigh uh made contributions
136:02 boho science as well. And so we'll hear more about that later.
136:08 this is uh an amusing type of , but not one that really ever
136:14 up in our business. Okay, uh Love Ways. We have time
136:20 to uh talk about love waves. so I said I said, love
136:26 the contemporary Really And so uh really the in plane displacement. And love
136:33 the out of plane displacement. So we have the same pictures we had
136:38 . Now we have the two accesses pointing out of the uh pointing into
136:45 screen and um you know this word uh this 1 3 plane is called
136:56 sagittal plane, you know this Uh it's not used too much,
137:05 sometimes it is and now you know is uh it's the plain of the
137:13 . So so we're gonna be looking uh solution to the wave equation with
137:21 the same boundary conditions as we had . But now we're looking for Love
137:26 solutions instead of really Texans. so uh this uh generate a wave
137:34 that you need across light source. then to observe, you gonna have
137:41 . Now usually we have only vertical and vertical view funds. So because
137:46 that, well, everyone has to very little attention in our business Up
137:53 maybe 20 years ago and then maybe years ago. And since then we've
138:00 um uh putting out three components So we see these all the
138:09 So let's do a brief analysis. stresses are given as before by Hook's
138:14 . And the boundary conditions are the as we had by Hook's law.
138:19 we're going to assume that all these terms are zero because those are terms
138:24 contributed to the grayling equations. And we want then uh left with only
138:32 term coming from the 23 stress. because these uh these terms are the
138:40 by symmetry. So there's the uh non trivial boundary condition is on the
138:50 stress. And uh so here here's equation. So The common name for
138:58 last 10, the stiffness tensor Elma mute. So here is uh rounder
139:06 right here. And this is uh gonna assume that this thing is zero
139:12 assumptions. So we're only left with term. And that's the love way
139:18 conditions at the surface. This is true at the surface. What
139:24 These are only this is only a wave boundary conditions. So solution will
139:31 fact be divergence trained. So I'm have a wave equation. It's exactly
139:37 we had for the shear wave body . And now we have the surface
139:41 contend with. Look here we have one component continuous. So we don't
139:52 that shear wave after potential. We talk about the displacement itself is the
140:00 itself. And there's the equation and sum of the two person L.
140:07 , reminded looking for low wave not body wasters. Just think of
140:16 as you over bro, we're not these love waves are not going to
140:23 with this cheer wave philosophy as We're going to just make a guess
140:30 , we're gonna guess the plane wave and the wave vector has this uh
140:35 square of uh the square of the of the wave vector related to
140:40 S. In this way. So need to determine these two components.
140:47 , in any way they can be . But because of the surface we're
140:52 find constraints on these, we found boundary condition. This has 3 3
141:02 . uh Gr Immediately musical zero. it's a solution. Because the left
141:12 of this zero, so that means no love waves in the marine
141:17 That's good. That's part of what marine data so lovely to work
141:22 We we can assume that the amplitude zero uh Again it's a solution but
141:31 improves implies no way at all. the empathy of zero or we can
141:35 that H three equals zero which implies H one uh is uh omega three
141:44 . So this is just an ordinary way if H three equals zero.
141:49 what we found is uh this part not going to be zero, this
141:53 oscillating i is not zero. So the only three solutions are these
142:00 So no way, No love way at all. So except that why
142:10 we have the uh mr Love didn't up at this point. He
142:16 okay, I got just this trivial here. Let's do a more interesting
142:25 , let's put in their elect So got an upper medium and a lower
142:30 and the layer boundary is right here the upper surface is up here,
142:38 equals minus D. And uh uh three equals minus dates for X.
142:48 is getting bigger and bigger as you down now uh in each uh layer
142:56 we're gonna have different wave equations because got different philosophies. Here's V.
143:01 . One, here's the S. . And so we're gonna have a
143:06 is an equation for this is the equation where the unknown is um uh
143:14 wave vector in the upper medium. love wave displacement in the upper medium
143:21 here the unknown is the love wave in the lower media. We're gonna
143:26 match match solution right here at the using those boundary conditions that we just
143:32 about. So under conditions are uh of stress uh across the upper boundary
143:45 in the air distressing zero. So gonna have Uh huh. Oh this
143:54 here uh we have to have specialist you were in the air also gonna
144:00 to be distressed at zero at this . And the stress is given in
144:06 way in terms of schumacher's of the layer and the gradient of the
144:19 Always displacement in the Opera lane. down in the lower layer we have
144:27 of stress and displacement. So here's corresponding uh depression. Far out Or
144:39 . So we're talking about the 3 stress. So here it is
144:43 the upper medium and here it is the lower medium and these two have
144:46 be continuous across this boundary. Same here for the displacement displacement has to
144:53 continuous. See it's it's um under are concerning stress and displacement not stress
145:02 strength. Why do we have displacement there instead of strength? Well if
145:08 we don't have continuity of displacement that that we're gonna have tearing of the
145:14 . So that's an earthquake. We we don't want to look at earthquakes
145:18 , We want to look at the waist. So that's not. So
145:22 gonna have we're going to assume that displacement is continuous and the stresses.
145:30 we're gonna guess a solution with these solution is gonna have free parameters And
145:37 gonna determine those parameters using these three . 1 2. Okay, so
145:46 the upper media, let's again we're do the geophysical thing, guess the
145:52 and prove the guess so. We're guess that uh there's gonna be uh
146:01 waves propagating in this media and we'll them L. One minus and
146:07 One plus both of them are gonna displacement uh out of the screen.
146:14 so uh uh you're gonna both have same wave vector, the same
146:21 One here and the same H. here. And the only difference is
146:27 the sign right here. That's a that means it's upcoming and that's
146:32 That's down going great. What is wave vector H. It's got a
146:39 . The square of the length is by Omega over v. square in
146:44 to solve the local wave equation. . About the lower meeting.
146:53 only a down going way. There's be no waves coming up from uh
147:00 the center of the year. Why this one coming up? Because it
147:04 off of here? No reflector down ? And so there's only a down
147:09 way. So I've got a minus here, same as this minus sign
147:13 . I noticed we have a different factor here. K. and uh
147:20 through K is given by Omega and two. That's the sheer velocity down
147:26 . And uh this solution here has match with this solution here. Thank
147:34 . So it's got seven Parameters. count them Amber through here. That's
147:44 one, the other amplitude and then third amplitude. And then all these
147:49 vectors H H one, H K. One, and K seven
147:56 . Yeah. So we're gonna make max these boundary conditions here and
148:03 . Um all different access. We've to have uh that the horizontal components
148:10 Ace in the same. So that the love, weight loss. The
148:17 of the logic is similar to what did, but more complicated. And
148:21 at the end of that complicated analysis is given as the solution of this
148:27 . It's a pretty messy equation. here, it's got square roots and
148:31 on. It's got tangents. And got uh here's the layer length,
148:36 ? A layer thickness right in And it's got VL as a function
148:41 a ratio with DS two. And a ratio with GS one. You
148:46 see any VP in here anywhere, you? So that's a complicated
148:53 So what you can see is in high frequency limit, it's got to
148:57 a high frequency limit is the sheer in the upper medium. And the
149:01 frequency limit is uh your velocity of lower limit. And I think you
149:07 see that for yourself by assuming that omega is either very high or very
149:17 . Um We're gonna re parameters that we did before saying, the love
149:23 velocity is given by the stairway of ever media where the zeta correction.
149:31 we're going to assume that zeta is . And then we're going to,
149:36 we said before, like a taylor of zeta and come up with
149:53 And what happened to the frequency? , frequency is uh he's inside
150:03 frequency is, oh, Washington times frequency frequency is inside. And what
150:20 is the ratio of this wave So, this layer fix. So
150:28 didn't have that in the railroad We didn't have a characteristic length like
150:34 have here. And para length in problem is the thickness of the
150:41 which is characteristic of the model. if that thickness is uh it's an
150:48 thickness, then everything simple, everything away. And it turns out to
150:53 uh really way Philosophy, simply equal the body weight. Philosophy.
151:01 Here here's some uh some characteristic occurs a function. This gives the really
151:09 velocity as a ratio with the upper share boss. You see, it's
151:18 going to be a bit bigger than and as a function of frequency it
151:22 like this. And uh so this for uh a layer of thickness of
151:30 10, that's 10 m and 20 and 40 m. And these are
151:39 higher velocity. You see, you that we're assuming that V.
151:43 One is less than B. S . I think that's true if I'm
151:50 . Mr would you remember uh if PS one is greater than V.
151:55 to do we still have a Okay, good. Anyway,
152:01 we we could decide by looking at equation, but I think we won't
152:04 that. Yeah. Yeah. The vectors have this character that uh VL
152:23 , the wave is going in this , but uh the wave is going
152:30 upwards and downwards reflecting uh in in pattern as traveling with um body wave
152:38 , V. S uh but it's traveling um horizontal. The horizontal component
152:45 that is Yeah. And um Uh Oops, hello, it decays
152:57 . Exponential with depth alone. This is trying the way back to the
153:09 of the way back to K. is consequences. Okay, thanks.
153:17 . So let's uh take a good for files, it is likely that
153:24 might observe love waves on the vertical of a fearful or false. Of
153:31 , because we're love wave is gonna uh horizontally problems. Okay, so
153:39 for uh for loved ones. How dispersion In the earlier discussion, Railway
153:49 did not depend on frequency, but love wave velocity did. So,
153:54 why. Say it again. In really discussion, there was no characteristic
154:00 or any characteristic wavelength so that a frequency omega would not know whether it's
154:07 high frequency and low frequency in layered . These layers provide cashless links,
154:16 dispersion always occurs. And surface waves really waves and love waves. And
154:25 fact we saw explicitly that it was very positive dispersion. Now, what
154:36 the consequences of dispersed? If if a wave is traveling, disperse
154:46 fully? High frequencies are going to to different philosophies and the low
154:52 So that means that wave let's change shape, right? So if the
154:59 is changing shape, it's not quite what we mean by the velocity.
155:09 If the waves, if the waves all the same, then the wave
155:13 retains its shape and if it's got uh it looks like a with the
155:19 peak which is maximum, then the of that peak gives the velocity of
155:24 way. Pretty clear. If the that is changing shape as it
155:30 It's not quite clear what we mean the velocity. In fact, there's
155:36 , there's two, wow, two , two velocities which we can get
155:47 . And so to get there. uh rewrite this expression here for uh
155:55 a scalar way, but this is solitary function and we'll call that omega
156:00 minus cake. We're gonna call that phase of definition. So here it
156:06 in the I five and five is function finds a function of frequency,
156:12 and space. This expression satisfies the equation. If uh velocity and the
156:23 equation is given by this ratio omega K. You came in the definition
156:31 the face. This is true whether not the velocity of various worth
156:38 We proved that this expression satisfies the equation with material property V squared if
156:49 only if vehicles omega over K. to make that proof, we did
156:54 have to specify whether or not he with. So if we were in
157:01 live expression here, uh you see this uh link carries you back to
157:08 right to the right equation in the election, but we're not gonna go
157:12 . And since uh we're not um up inside the uh all this learning
157:24 , we're gonna call this same ratio for us meg. Okay, is
157:32 gives the phase. Lossy for any for your component with frequency omega in
157:39 west. That's the faithful action. velocity of phase occurs where this derivative
157:50 of face with respect to omega is by this expression here. Uh Here's
158:00 definition face. We take the derivative that respect to omega. And so
158:05 here, out of the first time get in and out of the second
158:09 we get minus decay. The omega X. This point it moves with
158:20 velocity X over T, which is same as D omega DK.
158:26 um because X. So this is here, uh X over T equals
158:37 omega DK because of this. And uh we identify that uh this point
158:46 maximum value of phase moves with the velocity defined in this way, which
158:52 course is different from this one except some special case. Um where if
158:58 was uh omega was given back uh , in a in a special case
159:09 no consequence were the same, but real materials there uh always different.
159:18 we don't have to say at this what causes this variation in this state
159:24 in some cases uh wave vector K upon omega and we will learn more
159:31 this in the uh eighth lecture about elasticity. Once an important ideas about
159:42 this should be non-0. So uh is the definition of, of group
159:53 or more precisely group slowness. And put right in here for K.
159:59 put the definition that we found Found previously K. In terms of
160:04 and phase velocity work out the implications this using chain rule calculus. And
160:12 find the relationship between uh group slowness face slowness. Is this difference right
160:23 ? Using taylor's approximation. We can these uh minus one. Said group
160:31 is given by phase philosophy with the depending on the frequency dependence of the
160:38 loss. So uh this derivative could either positive or negative. And so
160:46 say that if it's uh if that is negative, we call that normal
160:52 . I'm not sure I know why is uh label is there? But
161:01 seemed normal, unusual for somebody and they call that normal dispersion. If
161:08 lack of this creation is negative, means for higher frequencies it goes
161:24 Um that's not that's not usual for wave types. That's the definition.
161:31 when high frequencies go slower uh that the normal dispersion. If high frequencies
161:38 faster that leads to inverse dispersion. , now let's uh uh let's see
161:49 consequences here, I have two slacks uh sign function and you can see
161:58 your eyeball that this one is a bit shorter uh wavelength than this.
162:06 , now when you combine the two , you get this and you can
162:12 that right here the two super pose reinforce and then they reinforce negatively.
162:18 then as you get over here they of cancel each other accuracy right over
162:22 . These two cancel each other And then uh further along here they
162:28 back in sync again. So the peak travels with the phase velocity.
162:38 you see this red line lines up peak here and here and here is
162:44 same peak that travels with phase The peak of the envelope travels
162:54 So this green line is going through maximum of this waiver here. It's
163:02 you know what I mean? Uh I say the envelope just make a
163:06 curve here, connecting all these peaks then another smooth curve uh connecting all
163:12 cross that's called the envelope. And the green line is uh goes through
163:19 uh peak of the of the envelope and the peak of the envelope here
163:26 the peak of the envelope here. that's a different philosophy, that's the
163:30 philosophy. The different frequencies travel with velocities. The wave of changes
163:42 So here's uh here's a wave moving right to left as it moves along
163:49 uh oh x to higher. And these change points right in here.
164:13 you see this little uh nick right , that's the same neck here and
164:18 same neck here. And so those those phase points all travel with the
164:24 phase velocity. The envelope travels with group loss. And so as the
164:32 goes along, looking at the The little knick point right here is
164:35 the beginning of the of the of wavelength and then it's sliding backwards through
164:42 wave it and then it goes back the back end. Call that a
164:49 break slides backward to the envelope. uh what is changing shape? That's
164:59 example of of a change shape because phase philosophy is less than the group
165:12 . Now here you've seen this uh before from Sheriff and guild art and
165:19 um uh it's a messy picture but gonna uh show you these are all
165:27 the surface waves in here and they they're traveling with a maximum uh maximum
165:34 philosophy is given by this. You out here you see far offset short
165:39 you see a reflection and uh service are mhm. Within this ban of
165:55 . And you can see in here a lot of uh what we call
165:59 us um And so you don't see outside of these limits. Action in
166:06 minimum. Within each. Uh Within these limits you do see linear
166:18 out separated by frequency. So uh . I have frequencies here and the
166:27 frequencies. So see with your Um dispersion. Well this might be
166:44 to you but there's an intimate connection dispersion and attenuation And we will address
166:54 connection in less than nine. Not . So how about this? For
167:08 ? That statement. True or Yeah, that's true. That's
167:17 That's the definition about this. In false. There was dispersion in our
167:23 of blood ways because the layer thickness a standard or characteristic different wavelength behaving
167:32 according to whether it's short or long this standard. Is that statement
167:41 Lots of words there. But uh that's uh that is logically true.
167:50 get dispersion because some wavelengths have some have wavelengths long compared to the standard
167:58 others short agreements. And here's the of that. In our simple discussion
168:04 railways, there was no no time in the problem to provide a frequency
168:10 and no length parameter to divide. politics also. Yeah, supposed to
168:26 looking inside me correct or the way doesn't change this way was traveling with
168:32 velocity where the group velocity is less a lot less than this year.
168:36 velocity about equal. A lot more none of the Yeah, about
168:43 Yeah. So um now in um excitement data, you always have a
168:50 . So you always have dispersion. so that means that high frequencies are
168:56 be traveling with different velocities and low but inciting data. Normally you only
169:02 a limited bandwidth. You don't have frequencies and uh, cycling sequences in
169:11 same data set. So anything is you look at, it always happens
169:16 there is a limited man. And that limit fan, you might not
169:20 able to see with your eyeball the effects, they're probably in there,
169:25 they have to be teased out You won't see them easily with your
169:32 normally in most datasets. Because uh the man with his limited, you
169:39 see dispersion actually, you will see generation, you'll see that uh high
169:46 go away and uh low frequencies but you you probably won't see much
169:52 the way of dispersion. Okay, this makes bigger topic. So I
170:00 this is a good time to And this is just pointing out that
170:04 going to carry these ideas about surfaces the more we have the cylindrical
170:10 So that's a cylinder. That's the . There's bound to be surface waves
170:15 with it because it's too because it's , we're going to call them to
170:20 ways and they're going to have different because of the cylindrical geometry. So
170:25 just useful to remember that Not all are flat, most important, non
170:31 surfaces. Uh Okay, so that's good place to start to stop and
170:40 call it a day for today And resume in the morning at 8:30.
170:47 right. And you're gonna supply the
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