| 00:04 | So let me uh take a look the questions which you sent me |
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| 00:12 | Thank you for that. So, I will just open my mail |
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| 00:25 | OK. So, uh the f first question is from uh Bria. |
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| 00:31 | she sent that in just this morning now. It says uh good |
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| 00:35 | My question is for marine data, we only having the stoneley surface |
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| 00:42 | marine data, only stoney surface So, uh remember that the uh |
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| 00:48 | , the stoney waves um are uh along the interfaces between uh uh rock |
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| 00:56 | and below. Uh And so, uh they are there or whether or |
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| 01:03 | um uh uh there's water up And so uh uh this uh stolen |
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| 01:11 | travels along uh the uh the surface those two uh uh uh layers in |
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| 01:19 | subsurface. But, you know, don't have any receivers there. All |
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| 01:23 | receivers are up at the top of ocean or maybe on the ocean |
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| 01:28 | Uh And so we normally don't receive uh stony waves. They are propagating |
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| 01:35 | uh but uh they uh the decay uh the amplitude decays uh away from |
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| 01:42 | boundary. So normally we don't see on our surface uh receivers, she |
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| 01:49 | on and for rail waves, what it mean physically that has a curl |
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| 01:55 | and a divergent free component? Is like having part of the P and |
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| 02:00 | wave movement? Yes, that's what means. It means that as the |
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| 02:05 | is traveling along it uh does some and it so does some diverging. |
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| 02:12 | a complicated motion much more complicated than a P wave or a sheer |
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| 02:19 | Uh And uh uh that's exactly what means. Uh It's neither a P |
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| 02:24 | nor a sheer wave, neither curl nor divergence free, but some of |
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| 02:29 | . And uh it's traveling along the boundary. And because we have |
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| 02:35 | receivers on that boundary, we receive lot of uh of railways. Um |
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| 02:41 | kind of too bad because mostly we're interested in the railway wires. |
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| 02:45 | Utah is interested in them, but of us are not, mostly we |
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| 02:49 | to get rid of them. And uh too bad, they are the |
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| 02:53 | amplitudes on our uh receivers, land . And uh so that's why uh |
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| 03:00 | need to do special processing, special , uh many sorts of techniques to |
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| 03:05 | rid of those. And we went those, those are uh uh uh |
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| 03:10 | techniques involving the sources and techniques involving receivers and techniques involving processing of all |
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| 03:17 | things. And I didn't get I didn't actually do all of them. |
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| 03:21 | , I did not do all of . Uh uh And so, |
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| 03:25 | I mean, I did not do of the ways we suppress these uh |
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| 03:33 | waves. Um That's a topic for Professor Zhou to explain. And so |
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| 03:41 | I'm, all I was doing is you how these things propagate. |
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| 03:47 | I actually didn't show you how uh uh started, remember as we did |
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| 03:51 | , as we did those uh derivation the railing wave velocity, we didn't |
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| 03:57 | a source. And in those we just said, suppose there's a |
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| 04:01 | somewhere. We don't know about it from the source. This is what's |
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| 04:05 | happen and away from the source, had uh uh these waves propagating in |
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| 04:11 | complicated way that we spent uh uh sometime yesterday figuring it out. |
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| 04:19 | um uh question that uh uh here you asked, is it like having |
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| 04:25 | of the P and sw the The short answer is yes. |
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| 04:30 | Next question is from uh uh li she says uh slide 45 slide 44 |
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| 04:42 | slide 44 and the light waves The, so let's, let's go |
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| 04:52 | to that. And, and so I wanna do is bring up that |
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| 05:13 | 44. Yeah. Shop 544. I don't think you can see this |
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| 05:37 | . I understand. Ma let let me share this. OK. |
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| 05:59 | I think I'm uh sharing that screen , you see it now. That's |
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| 06:17 | . Oh Yeah, that's right. , um so can people see this |
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| 06:48 | ? Uh um Yeah. Uh So, yeah, I have not |
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| 06:56 | . So let me present. I'm OK, Michelle uh got some. |
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| 07:19 | . So um uh Carlos can you this? Can you see that? |
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| 07:31 | . So, uh this is a where we first introduced uh uh the |
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| 07:36 | of uh ray theory. So uh uh just a second, let me |
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| 07:42 | a, a pointer, here's my . And so uh uh we are |
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| 07:53 | to uh to, I imagine uh we're gonna imagine in our minds, |
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| 07:58 | ? The, the wave is not this, the, the wave doesn't |
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| 08:02 | know anything about um the ray the, the wave is following the |
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| 08:07 | equation. But we can um think it uh using ray theory. It's |
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| 08:14 | theory and uh uh it's a good for us to uh understand what the |
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| 08:20 | is doing. And so, uh is the essential approximation here. Uh |
|
| 08:25 | , the pressure in the wave uh , in, in the wave is |
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| 08:29 | be approximated as uh a wavelet uh which is arriving at um uh an |
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| 08:36 | time, t think, think of uniform medium, uniform medium. And |
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| 08:41 | wave wavelet is arriving and it has shape, you know what the shape |
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| 08:46 | a wave looks like. And that is arriving at a time, uh |
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| 08:52 | T and it has an amplitude A so this a is a function of |
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| 08:58 | far it's traveled. And the travel is a function of how far it's |
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| 09:03 | . And we want to apply this in the high, frequently high frequency |
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| 09:10 | . That is, we wanna consider case where the wavel varies rapidly. |
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| 09:15 | can see that with variation on your uh on your workstation. And that's |
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| 09:20 | we mean by rapid and that wave moves slowly through the medium. |
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| 09:27 | and the, the time very slowly as it moves the amplitude very |
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| 09:32 | just for example, uh a uniform uh as the wave is going through |
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| 09:37 | medium, it it's doing geometrical spreading so it's getting smaller amplitude, but |
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| 09:45 | happening slowly and steadily. But inside the wave, the wavel is oscillating |
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| 09:51 | and down rapidly. So, we this um uh these ideas in the |
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| 09:58 | few slides to um uh derive the the ray theory equivalent of the uh |
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| 10:09 | uh the wave equation. And we that the icon equation. And it's |
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| 10:15 | AAA simpler uh uh uh it's AAA of the assumptions that you see |
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| 10:23 | arrival time is uh the wave is slowly through the medium as it |
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| 10:29 | it uh rapidly wiggles up and down it slowly uh uh loses amplitude. |
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| 10:38 | um we uh we derived uh the equation. It's the ray theory equivalent |
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| 10:48 | the wave equation. And uh then use that to um to understand that |
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| 10:59 | derive actually some um uh some important like we, we derive Snell's |
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| 11:06 | And so we understood that uh to Snell's law, we had to restrict |
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| 11:12 | to uh one dimensional media, flat layers. So, uh you can |
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| 11:17 | all that now, um uh Lee , does that answer your question? |
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| 11:26 | , your voice is very soft. , I, why is it called |
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| 11:32 | high frequency limit? Well, because gonna assume that this uh uh uh |
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| 11:38 | , well, let's go, let's forward in a couple of slides. |
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| 11:41 | then you'll see uh uh uh right . For example, uh we are |
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| 11:49 | to assume that at high frequency this is zero. Why is it because |
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| 11:55 | gradient uh is a small number uh the amplitude is varying slowly. So |
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| 12:01 | gradient is a small number. So gonna neglect this term. That's what |
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| 12:05 | mean by the high frequency limit. Laplace in of the amplitude is even |
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| 12:11 | . So we're gonna neglect this at two terms at a high frequency. |
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| 12:17 | that leaves only with this term. uh uh uh so here is then |
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| 12:23 | wave equation for, for W and looks like a wave ation, doesn't |
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| 12:29 | ? So there's more uh uh development , of ray theory and then there's |
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| 12:34 | complications. So we skipped over because think you don't, you don't wanna |
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| 12:39 | all those complications although you may be to, uh, uh, look |
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| 12:45 | them after class and I can, , I, uh, the reason |
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| 12:49 | put them in this course is because is probably the only time you'll ever |
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| 12:54 | them. And if you ever uh, uh, where does ray |
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| 12:58 | come from? Well, you can go back to these notes, |
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| 13:02 | and also you can go back to uh on the textbook by Sheriff and |
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| 13:06 | Dark. And uh there you'll see more discussion of great theory and you |
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| 13:13 | , uh uh right there is, actually is a good a accounting |
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| 13:20 | of how we think about race or we think about W I I think |
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| 13:27 | for most of us understanding the wave has a differential equation, I think |
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| 13:35 | um it's hard for most of us me and I think including you, |
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| 13:40 | I think that most of us can of understand uh great theory. So |
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| 13:44 | we, we understand is that uh E equation is gonna govern the uh |
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| 13:51 | the variation of the wavelength, not the data, that's the |
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| 13:57 | So this doesn't describe the data gonna up a couple of slides. So |
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| 14:04 | is the data here. And uh what, what we just uh I |
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| 14:10 | derived uh a couple slides for it the wave equation for the wavelength and |
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| 14:18 | arriving and you know, it's wiggling through the medium. And uh uh |
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| 14:23 | think we can all understand that and arriving with the travel time uppercase T |
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| 14:31 | what is uh what's the equation for ? T, what's the equation for |
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| 14:37 | uh for the arrival time? that's the icon equation which uh derived |
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| 14:43 | five slides after this. So, I think that's all I wanna say |
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| 14:49 | uh uh about uh great theory, frequency approximation. Yeah. Second. |
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| 15:09 | let me go back to uh the question. So, and here the |
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| 15:17 | . So uh I'm looking at the email, I don't think you all |
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| 15:23 | see this. Next question is what the difference between uniform homogeneous and isotopic |
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| 15:31 | or mathematically? Hm. OK. I'm not sure how to answer that |
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| 15:42 | . I know you understand what it to have. Uh uh oh |
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| 15:50 | I know you know what it means have isotropic rocks physically. You know |
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| 15:56 | you, you, you have a idea in your mind. What? |
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| 15:59 | all it means for isotropic rock, velocity of sound is the same for |
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| 16:04 | direct. Yeah. And the way think about that is just think about |
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| 16:08 | sample you can hold in your hand sandstone. And so when you look |
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| 16:15 | the sandstone, it looks like uh don't see any small scale structure in |
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| 16:21 | sandstone except the granite. And if look closely say it again, so |
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| 16:32 | it, it's isotropic but not OK. So, so now think |
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| 16:38 | isotropic but not uniform. So think think of a of a of a |
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| 16:42 | sample uh which uh has uh uh a layer of boundary and on the |
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| 16:48 | side is another sandstone or think uh uh so think of on the very |
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| 16:55 | scale, think uh it, it's uniform on a small scale because on |
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| 16:59 | small scale, you have grains and , right? So uh all uh |
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| 17:05 | the the issue of the a layer and the science and the other sandstone |
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| 17:10 | the other side, that is something uh cook would have uh uh would |
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| 17:15 | understood. OK. For him, would be like a a piece of |
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| 17:19 | of copper welded to a piece of iron, right? So uh |
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| 17:25 | would understand that kind of in homogeneity it's peace wise homogeneous. So on |
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| 17:32 | side of the boundary, it's On the other side, it's |
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| 17:35 | but it's different. Yeah, Hook have understood that. Uh but he |
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| 17:40 | not have understood uh uh our applying theory to a rock with on the |
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| 17:46 | scale, it's got both grains and right. On the small scale, |
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| 17:51 | heterogeneous everywhere you look. And furthermore , inside the uh uh the |
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| 17:57 | it's got atoms right. Uh uh inside the atoms, it's got uh |
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| 18:02 | quarks. Uh it's got uh uh and electrons, right. So all |
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| 18:08 | small scale variation is something that book not have understood or even thought |
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| 18:14 | So we are gonna talk about variation the brain scale later in this |
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| 18:21 | But for right now, we're assuming is homogeneous. So you can think |
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| 18:25 | like a glass or you can think copper. That's, that's the kind |
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| 18:30 | stuff we're talking about right now. rocks. OK. Now, uh |
|
| 18:36 | how about uh uh uh how about uniform but an isotropic? Well, |
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| 18:45 | think of a crystal. So uh uh I don't have a crystal with |
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| 18:50 | , but you have seen a, you came to class this morning in |
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| 18:54 | , you walk past the display case look at that display case when you |
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| 19:00 | uh uh at the break and you see crystals which have crystal shapes to |
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| 19:05 | . So some are cubic, some uh you know, all sorts of |
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| 19:08 | shapes and colors and everything. And uh so they look uniform, don't |
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| 19:14 | ? Uh crystal think of a crystal quart, a uniform crystal quart. |
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| 19:19 | that crystal quart is an isotropic waves in different directions uh through that quartz |
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| 19:29 | , high frequency waves, think of high frequency waves in the laboratory. |
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| 19:35 | um uh why is that? It's the crystal has small scale structure which |
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| 19:42 | can't see it has a AAA crystal structure of the atoms inside the |
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| 19:49 | And that's what's responsible for the external of the quartz, that's what's responsible |
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| 19:55 | the shiny faces on the crystal. so you can be sure that those |
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| 20:00 | atomic arrangements which make the shiny they also make for anisotropic wave propagation |
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| 20:08 | the crystal port. OK. that's not an important uh form of |
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| 20:15 | isotopy for us because normally we are at uh uh low frequency wave seismically |
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| 20:23 | sonically, we're looking at waves which are much longer wavelength than a quartz |
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| 20:31 | . So we're gonna be thinking about kinds of anisotropy. Uh right |
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| 20:35 | you can see immediately there's an anisotropy though it's homogeneous, I mean it |
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| 20:40 | homogeneous. But you know that on tiny tiny scale, it's in |
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| 20:46 | And moreover with the preferred orientation, atoms inside the crystal inside the quartz |
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| 20:52 | , they have uh uh they uh up in a atomic cells uh uh |
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| 20:59 | have AAA preferred orientation oo of the atoms oo of the atomic cells inside |
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| 21:08 | , the quartz. And that's responsible the shiny phases on the outside. |
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| 21:13 | . Um Yeah. So is, that answer your question? OK. |
|
| 21:18 | go on to the next question. should we use the scalar or |
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| 21:23 | Uh uh For example, we have scalar reciprocity theorem and the vector occupy |
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| 21:31 | uh does the Hem Holtz equation apply both situations? OK. So, |
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| 21:37 | um uh so number one, the are traveling as vector waves. So |
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| 21:45 | should. So the waves are obeying vector wave equation. However, as |
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| 21:51 | approximation, we can regard T waves um uh obeying the uh the scalar |
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| 22:00 | equation. Uh uh And that works uh uh for, for, for |
|
| 22:07 | , for imagery, for imaging our , we often as uh assume the |
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| 22:13 | wave equation or why is that? because it's easier to deal inside the |
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| 22:18 | with the scalers than with vectors. . So the, the uh the |
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| 22:26 | waves obey the scale of reciprocity which is a special case of the |
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| 22:33 | reciprocity. Of course, they also the vector reciprocity. The uh but |
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| 22:38 | sheer waves and converted waves in more ways do not uh obey the scale |
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| 22:44 | reciprocity theorem, but they do obey vector presupposing the. Now next question |
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| 22:51 | , does he Maltz equation apply? like in both situations? The answer |
|
| 22:58 | yes, the Healt equation applies. uh in all cases, you can |
|
| 23:05 | , you can always um divide any , uh feel uh any displacement which |
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| 23:14 | with uh uh time and space. can always uh uh um divide that |
|
| 23:23 | a Pearl Free Park and a divergence Park. However, however, we |
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| 23:28 | uh what we learned and you we, we didn't learn, but |
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| 23:36 | uh we will learn when we get anisotropy. If the medium is |
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| 23:41 | those two parts don't contain just P and sheer waves. We have P |
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| 23:49 | and sheer waves in both parts if an isotropic, but you don't know |
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| 23:54 | yet because so far we're just dealing isotropy. But uh um the equation |
|
| 23:59 | a very good equation and you can do that separation. But if the |
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| 24:05 | is an isotopic, then it, doesn't lead to the separation of PNS |
|
| 24:12 | and you will find that out uh to me. And uh so let |
|
| 24:27 | uh so the next question is, explain further about finding the interval velocity |
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| 24:34 | dick's differentiation. OK. Um So cost um uh pull up that slide |
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| 24:48 | 89 in lecture four. Am I ? Ok. Well, so we |
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| 24:53 | lecture four here. So uh It back four slide 89. I |
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| 25:17 | my mouth. Cheers. Well, slide 89 according to me, but |
|
| 25:40 | doesn't look like it's the slide that want. What? 90? It |
|
| 25:51 | look like the one you want. . Is this the one? Oh |
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| 26:20 | can't see. I, I think is the one. OK. |
|
| 26:25 | So let me, let me Um um No, no, |
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| 26:34 | And you sir? No, Uh Let's see which one? Uh |
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| 27:17 | 51. So let me um present one uh three. Mm Make sure |
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| 27:34 | least to OK. Let me just over night. OK. Uh uh |
|
| 27:46 | So uh uh Carlos can you see slide uh uh about this differentiation? |
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| 27:56 | , perfect. OK. So let um animate it a couple times. |
|
| 28:05 | . So I think this is a uh Lily you were asking about. |
|
| 28:10 | . So uh uh what we have at the bottom is we have uh |
|
| 28:15 | expression for the, the, the velocity of the nth layer. |
|
| 28:20 | this is the velocity of the layer above the uh uh the reflection |
|
| 28:27 | OK. And what, what do have in here? We have uh |
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| 28:31 | the arm is velocity uh coming from uh uh reflection just below. Sorry |
|
| 28:38 | that. Let me go back. is the arm o and let me |
|
| 28:42 | my uh order. OK. So is the R MS velocity from the |
|
| 28:48 | uh uh reflection just below that And here is the arm rest velocity |
|
| 28:53 | the uh uh uh reflection just above layer, top of the layer bottom |
|
| 28:58 | the layer. This is the travel to the bottom of the layer. |
|
| 29:02 | is the travel time to uh the of the layer. And uh then |
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| 29:08 | dividing by uh by, yeah, dividing by the travel time in, |
|
| 29:18 | the light. So these are so, so, so uh this |
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| 29:24 | time in the layer is the difference the travel time to the bottom and |
|
| 29:28 | travel time to the top. So on, on the uh on the |
|
| 29:33 | are uh things that you can observe you uh make the assumption that the |
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| 29:41 | that the move out velocity which you determine on your workstation is the same |
|
| 29:50 | the R MS velocity. So you can determine that move out velocity |
|
| 29:54 | all the major reflections using your workstation . And so here's uh and if |
|
| 30:01 | take that move out velocity and regard as an R MS velocity, then |
|
| 30:06 | know this one and you know this , you know everything here. And |
|
| 30:09 | you thereby deduce the interval velocity in uh the nth layer. And you |
|
| 30:17 | do that, of course, for , any of the layers. And |
|
| 30:20 | so uh that's what we mean by differentiation. Why is it called |
|
| 30:27 | Well, because we have uh the of two quantities here on the |
|
| 30:31 | Uh It's a small difference and then a, a small uh time delay |
|
| 30:36 | uh down here. So it's, like a derivative. It's the ratio |
|
| 30:41 | two small numbers. And you can that uh there might be some uncertainty |
|
| 30:49 | with determining the velocity in that but that's the way we do it |
|
| 30:54 | uh all the time either using um um the methods which I've shown here |
|
| 31:02 | involving uh uh um the move out hyperbolic move out equation that uh derived |
|
| 31:09 | dix in 1955 or uh uh co these days. Uh We uh usually |
|
| 31:18 | more elaborate methods to determine um uh velocities and we call them migration |
|
| 31:29 | And we uh we have very elaborate algorithms to do that. Professor Zhou |
|
| 31:37 | be talking about that uh for you . But it all comes down to |
|
| 31:42 | same idea that, that those are more elaborate implementations of this basic |
|
| 31:48 | So uh this is what we call differentiation. Now, the next part |
|
| 31:53 | Lily's question is uh what happens when move out velocity is not equal to |
|
| 32:00 | R MS velocity. Here, here we're uh we're uh we're gonna |
|
| 32:06 | this velocity assuming that we have the MS velocity at the bottom of the |
|
| 32:12 | and at the top of the Uh But what if that uh uh |
|
| 32:17 | that, that move out velocity, two move out velocities, which we |
|
| 32:22 | in a workstation. What happens happens it's not really the R MS |
|
| 32:26 | Well, in that case, we an error here and that's the wrong |
|
| 32:33 | . OK. So let me see . The next slide I think is |
|
| 32:38 | , so if we have the wrong , then we're gonna get the wrong |
|
| 32:43 | . You know, because we find depth by adding up all these uh |
|
| 32:47 | uh uh layer depths which you can right here. And if these |
|
| 32:52 | if these interval velocities are wrong, we're gonna get the wrong depth. |
|
| 32:58 | that's a very common thing. It's common that you find the inner velocity |
|
| 33:05 | your best techniques on your workstation and calculate the depths and it turns out |
|
| 33:11 | be wrong when you drill the sometimes it's only a little bit |
|
| 33:18 | Sometimes it's a lot wrong. And uh uh you should be prepared for |
|
| 33:25 | . And uh and so there, two sources of that kind of uh |
|
| 33:31 | to death miss time one is possibly screwed up somewhere. Uh So, |
|
| 33:38 | in that case, um with a um uh uh training on the |
|
| 33:45 | with the help of your colleagues there the company, you can avoid making |
|
| 33:50 | kinds of blenders and you're still gonna uh the incorrect depth. And so |
|
| 34:01 | a case like that, and go to this life, uh in a |
|
| 34:08 | like that, the reason for the wrong depth is you have the wrong |
|
| 34:14 | . The reason you have the wrong is that not that you had these |
|
| 34:19 | wrong, you probably have the this uh pretty accurately and this time pretty |
|
| 34:25 | . And so you can get the of them. That's this interval time |
|
| 34:28 | here pretty accurately. The uh uh major source of error is really beyond |
|
| 34:36 | control because you were forced to assume the move out velocity that you measure |
|
| 34:42 | your workstation is in fact, the MS velocity. OK. So I |
|
| 34:47 | going to show you um uh oh gonna say that the, the, |
|
| 34:55 | , the main, the main uh why the move out velocity is not |
|
| 35:03 | to the R MS velocity is And so we're gonna learn about that |
|
| 35:10 | in the course and you will uh will see very early in lecture |
|
| 35:15 | uh how it can happen that an leads to wrong. Uh uh uh |
|
| 35:22 | leads to the fact that the R velocity or excuse me, the animal |
|
| 35:26 | that you measure is not equal to R MS velocity. And so that's |
|
| 35:31 | the interval velocity that you measured are . And also you will find uh |
|
| 35:36 | recipe for dealing with that. In 10, you will not be happy |
|
| 35:42 | that recipe, but I'm gonna, telling you that's the best that we |
|
| 35:46 | do. I it, it's gonna drilling holes and drilling holes are |
|
| 35:53 | And so you're gonna make these mistakes you drill the hole. Uh And |
|
| 36:01 | you might as well be uh resigned it. And furthermore, you need |
|
| 36:07 | uh prepare your boss, you, need to prepare the expectation of the |
|
| 36:13 | that when we drill the hole, we're not gonna find that reservoir at |
|
| 36:18 | depth that we thought it was gonna . It's either either gonna be a |
|
| 36:23 | a shallower or deeper and it's you can say it's not our |
|
| 36:29 | Uh uh In order to get the depth, we have to know the |
|
| 36:36 | uh in advance and not easy to uh in fact, we, we |
|
| 36:43 | know it until after we drill the . So the boss has to be |
|
| 36:47 | for making mistakes. Now, uh uh uh uh after you've uh |
|
| 36:52 | with this course and after you've uh uh gotten some more experience, you |
|
| 36:57 | , you will be able to minimize error. Um But you never will |
|
| 37:03 | able to avoid the error completely. . So, uh, let me |
|
| 37:10 | stop sharing this and go back to the email. And next question is |
|
| 37:22 | uh this is a really good Uh It, it's about converted |
|
| 37:27 | Remember that? Remember that? Um remember that cartoon? I think I |
|
| 37:35 | wanna ask you to remember. I I wanna bring up that cartoon |
|
| 37:39 | Hold on. So let's see OK. Stop in this slide |
|
| 38:27 | Yeah, this is what I wanna and then I wanna start this presentation |
|
| 38:35 | I want to uh I think uh not sharing with you yet, |
|
| 38:46 | So, um uh I don't uh don't have here uh Utah. I |
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| 38:53 | have the, oh, what did do since, since this one |
|
| 39:02 | OK. Sure. Um On this was still here. OK. |
|
| 39:19 | . Mm OK. Uh uh Can see this picture that we, that |
|
| 39:23 | uh showed yesterday? So, uh uh what Lily is asking is uh |
|
| 39:29 | uh in this picture here. let me get a pointer. I |
|
| 39:37 | here. OK. Here we have incoming uh P wave and an outgoing |
|
| 39:42 | wave reflected and transmitted. And then also have SV waves converted and uh |
|
| 39:48 | reflection and transmission but no sh So she's asking uh uh uh can |
|
| 39:55 | have sh wave? Well, look here closely at the boundary as |
|
| 40:00 | P wave was coming in. it's jiggling the boundary in this direction |
|
| 40:06 | the direction of the P wave. that means it has a vertical component |
|
| 40:10 | a horizontal component to the displacement as P wave hits the boundary. So |
|
| 40:17 | means the boundary is moving back and this way in the plane of the |
|
| 40:22 | , right? And so that's why SV wave is, is created because |
|
| 40:27 | , the motion is in the plane the figure. Now, for an |
|
| 40:32 | wave that displacement is perpendicular to the to the um to the figure. |
|
| 40:39 | you see where my cursor is imagine uh uh the uh it uh um |
|
| 40:46 | uh there, there's no displacement in the uh uh into the screen and |
|
| 40:53 | of the screen, that's what we need to create a stage, |
|
| 40:57 | OK. You can see it's not happen here except suppose that uh uh |
|
| 41:07 | one or both of these media, or lower media. I suppose it's |
|
| 41:11 | isotropic. Yeah, if it's an , then there's P wave coming in |
|
| 41:19 | , in the plane still. But it hits this boundary it is gonna |
|
| 41:24 | some displacement out of the screen because the anisotropy. So, in that |
|
| 41:31 | , you're gonna get uh six ways out one P wave up and two |
|
| 41:36 | waves up an SV wave and an wave and three and uh uh uh |
|
| 41:45 | uh three waves going down, uh P wave and two shear wave. |
|
| 41:51 | furthermore, I can tell you, didn't tell you this before but uh |
|
| 41:55 | this is new information, those two waves going up. Let's think of |
|
| 42:01 | two shes going up and SV wave you see here and sh wave, |
|
| 42:05 | you don't see because this diagram is isotropic bodies but and anisotropic fighters is |
|
| 42:12 | gonna be an upcoming sh wave and what it's traveling at a different |
|
| 42:18 | So it's gonna have a different So you can see how things get |
|
| 42:24 | lot more complicated because in an isotopy media, there are two different sheer |
|
| 42:34 | traveling with different velocities. Wow, different polarization. That that's what we |
|
| 42:40 | when we say SV and SH uh we're describing the polarization. And |
|
| 42:46 | I'm here to tell you that neither those two sheer wave polarization are gonna |
|
| 42:53 | be lying in this planet. This uh uh here, this one lies |
|
| 42:57 | the plant, there's the polarization vector there. But if it's, if |
|
| 43:02 | uh if the medium is anisotropic, polarization factor might be sticking out of |
|
| 43:08 | plane a bit and the sh wave be sticking out of the plane a |
|
| 43:13 | . Lots of complications. So that's most courses like uh this of, |
|
| 43:23 | uh of introductory uh waves and rays talk about anisotropy at all because the |
|
| 43:32 | is a lot more complicated. That's the reason why we are concentrating on |
|
| 43:39 | only for the first nine lectures. . So in the 10th lecture, |
|
| 43:45 | gonna take up just a little bit the complications that come from an anisotropic |
|
| 43:52 | . And of course, why are doing this? Of course, it's |
|
| 43:55 | the rocks in the subs services are fact anisotropic, almost all of |
|
| 44:00 | So what we're learning now is uh an approximation uh where uh oh uh |
|
| 44:12 | Real Earth is gonna be different from picture here. And I will actually |
|
| 44:16 | you, I will tell you an story from my own experience. Uh |
|
| 44:22 | was the leader inside Amao of developing anisotropic ideas starting in 1980 maybe before |
|
| 44:31 | of you were born. And we that a secret inside Amao for six |
|
| 44:37 | . It was remarkable that we kept a secret because there's so many ways |
|
| 44:42 | secrets to leak out of companies. You know, people leave the |
|
| 44:47 | things like that and when they they take the secrets with them. |
|
| 44:51 | we've managed to keep that secret for long years. During that time, |
|
| 44:57 | was sitting in an audience at the listening to um o other people's |
|
| 45:05 | And there was a smart guy from giving a presentation about uh convert race |
|
| 45:13 | he had some field uh data from uh the office in uh uh Cono |
|
| 45:21 | uh uh in those days, uh was before the merger with Phillips and |
|
| 45:26 | international headquarters was in a small town uh Oklahoma. That's about, I |
|
| 45:32 | know, 300 miles north of where sitting here in Houston. He was |
|
| 45:38 | smart guy and he was uh giving talk about converted waves on land, |
|
| 45:44 | was uh uh uh an unusual topic those days. And he had uh |
|
| 45:50 | had flatline geometry in the subsurface and laid out a two D experiment pretty |
|
| 45:56 | like uh and he was analyzing it much in these terms and it was |
|
| 46:01 | pretty conventional, pretty boring. And just when I was about to doze |
|
| 46:07 | , he said, however, I to show you something here which I |
|
| 46:13 | understand. And um and I'm hoping can help me understand. So then |
|
| 46:20 | showed the data, uh his data three component geophones because that's what he |
|
| 46:26 | available times, three component geophones and had a vertical vibrator making a source |
|
| 46:32 | he had uh three component geo So of course, he had one |
|
| 46:36 | them, one of the horizontal components lined up with the figure just like |
|
| 46:41 | showed here. And that's he, said, of course, we were |
|
| 46:46 | to get zero data on the cross phone. He said, but look |
|
| 46:51 | the data and he showed data was strong, strong uh arrivals from the |
|
| 46:58 | line GEO phone. In this case , exactly what le le was asking |
|
| 47:06 | . He said uh uh uh we expecting zero. But look, |
|
| 47:10 | we got strong arrivals. Can you me understand that? And, and |
|
| 47:15 | was sitting in the audience and I exactly the explanation um for that it |
|
| 47:21 | uh anisotropy, but in those anisotropy was a big amical secret. |
|
| 47:27 | I happened to be sitting next to experts in anisotropy from Shell. And |
|
| 47:37 | that time, maybe still today, was regarded as the uh as the |
|
| 47:43 | advanced technically advanced company in the And these guys were high quality experts |
|
| 47:51 | an isotropy from uh uh from And they were uh uh they were |
|
| 47:58 | uh giggling in the dark. We sitting there uh uh several back in |
|
| 48:05 | dark and they were nudging each other uh in the ribs with their elbows |
|
| 48:09 | giggling and pointing to the speaker from . And they were saying that idiot |
|
| 48:14 | Arco does not know that that isn't possible that P waves do not couple |
|
| 48:20 | SS H waves. And they were exactly in these terms. Now, |
|
| 48:25 | were experts in anisotropy but in those , the only anisotropy that guys like |
|
| 48:31 | knew about was anisotropy uh uh from un fractured shales. So imagine that |
|
| 48:39 | is a sandstone up here. Incoming P wave is a sandstone down here |
|
| 48:44 | a shale un fractured. But it's uh uh uh on the small |
|
| 48:50 | it's got clay, clay platelets which aligned to be flat, preferentially aligned |
|
| 48:56 | be flat. So the shale is with different velocities for uh for all |
|
| 49:02 | uh different angles of uh propagation down . But they're all the same uh |
|
| 49:08 | uh for all hazas, it's the uh whether this figure is oriented east |
|
| 49:15 | or north south or uh no matter this uh figure is oriented in the |
|
| 49:21 | directions, it's all the same for fractured sales. That's what these guys |
|
| 49:27 | about. And it turns out that that special case of uh of uh |
|
| 49:33 | this picture is a good one. is no coupling from P to sh |
|
| 49:40 | un fractured shells block. It turns that almost all scales are fractured. |
|
| 49:46 | have asom methyl anisotropy as well as simple anisotropy that these guys knew |
|
| 49:53 | In those days, we call that simple anisotropy. We call it now |
|
| 49:58 | anisotropy because it has a polar All uh all directions about the vertical |
|
| 50:05 | axis are the same from that uh that kind of anisotropy. The old |
|
| 50:12 | name for that kind of anisotropy is , transverse isotropy. That's a bad |
|
| 50:18 | because in the name of a type anisotropy is the word isotropy. But |
|
| 50:24 | the old fashioned name and the modern is vertical polar anisotropy because it has |
|
| 50:30 | pole of symmetry. So those guys um uh from shell were amused and |
|
| 50:37 | , that idiot doesn't know that that possible. But you know, he |
|
| 50:41 | showing them the data right there in of their eyes and there was a |
|
| 50:46 | discussion in the audience. No, could understand how, what, what |
|
| 50:50 | causing that. And these guys were fun of the speaker. That idiot |
|
| 50:55 | know that's not possible. But he showing them the data. What they |
|
| 51:00 | have done was they should have said oh look at that data, that |
|
| 51:05 | is not possible according to our maybe we should think about making |
|
| 51:12 | making less restrictive assumption. Maybe we learn from listening to this smart guy |
|
| 51:18 | Arcot showing us his data, but didn't do that. They missed, |
|
| 51:23 | missed their chance. They uh they fun of the speaker giggling in the |
|
| 51:29 | and I was sitting next to them to all this and I understood all |
|
| 51:33 | that was going on, but I my mouth shut because it was an |
|
| 51:37 | micro secret. So it was a of years later when we explained to |
|
| 51:42 | community, what happens when you have rocks with uh aimy anisotropy, for |
|
| 51:49 | , fractured chairs and lecture 10. will, we will be able to |
|
| 51:54 | you some of that. Ok. let me stop. May I ask |
|
| 52:01 | a question about the what you just share? So what was the reason |
|
| 52:07 | you ended up sharing that secret? , well, the secret leaked |
|
| 52:15 | Yeah. So that, uh that's good question. OK. So here |
|
| 52:20 | are Brita working for slumber. So , I'm gonna tell you uh uh |
|
| 52:25 | some more adventures from my life. it was a year or two later |
|
| 52:31 | Amao sent me to a course just this continuing education course for people who |
|
| 52:37 | , who had jobs in the And this course was being taught by |
|
| 52:41 | professor from Stanford and it was about . And so Amao said, |
|
| 52:47 | you, you know about anti me , but just go and listen and |
|
| 52:51 | what other people are saying. So went and uh I forgot where the |
|
| 52:55 | was. It was somewhere in uh in California I think, forgot where |
|
| 53:01 | where it was. And so uh it was a course being taught a |
|
| 53:06 | long course in an isotropy, being by this professor from Stanford. And |
|
| 53:13 | was talking all about polar anti. the unraced, shall he? He |
|
| 53:18 | know anything about realistic anti I kept mouth shut. I was just there |
|
| 53:23 | a list. So during the coffee . Um uh Another course participant from |
|
| 53:32 | came up to me and he have you heard about a guy named |
|
| 53:36 | ? And so cramping was um smart working in Scotland. And he knew |
|
| 53:45 | azimuthal anisotropy, but he knew about only in the context of earthquakes. |
|
| 53:53 | . So he had looked at a of earthquake data and he had seen |
|
| 53:58 | for propagation uh from the earth earthquake the receivers um uh indicating as in |
|
| 54:07 | an isotropy. And he had, know, he was pretty active. |
|
| 54:11 | , he gave a lot of talks uh to academic audiences, you |
|
| 54:17 | uh uh uh the, the major for academic geophysicists is called the American |
|
| 54:26 | Union. And it's a worldwide uh , it's a big society. Uh |
|
| 54:32 | and they're interested in things like earthquakes they're interested in the interior constitution of |
|
| 54:37 | earth and things like that. And why he was giving his lectures and |
|
| 54:43 | in our business was paying attention. this guy at Exxon uh from Exxon |
|
| 54:49 | up to me in the class and says, have you heard about this |
|
| 54:51 | cramping? And so I said, , yeah, I've heard about |
|
| 54:56 | you know, uh uh uh Uh uh and now I've heard about |
|
| 55:01 | . And so the Exxon guy well, we have hired him as |
|
| 55:05 | consultant and we've done some experiments um using his advice and uh you |
|
| 55:14 | what everything he says about azimuthal anisotropy true. Wow. So immediately I |
|
| 55:23 | that it had here a, a oil company listening to cramp, the |
|
| 55:30 | had spilled out. So, uh I pretended it was no big |
|
| 55:36 | I just ref refilled my coffee cup uh said, well, that's |
|
| 55:40 | And then we, we went back the classroom, but shortly after |
|
| 55:45 | while this, while the professor was , I snuck out the back and |
|
| 55:50 | went to a uh a telephone, know what a telephone is. Uh |
|
| 55:54 | you put in a quarter and it's a, a AAA wire coming out |
|
| 55:58 | back of the telephone and you put your quarter and you can make a |
|
| 56:02 | call anywhere in the, in the . Uh uh uh And uh uh |
|
| 56:09 | I, I called back to Tulsa I said our secret is out Exxon |
|
| 56:14 | listening to tramping. So uh uh immediately then Amico authorizes all of us |
|
| 56:24 | uh uh uh go public with our . And it turned out that the |
|
| 56:28 | next week was the deadline for the of abstracts for the SEG convention in |
|
| 56:36 | fall. It was uh like three later. And so by the time |
|
| 56:40 | got back to uh uh Tulsa, , I guess I came back on |
|
| 56:46 | and on Monday, uh we had uh pulled out from our doors, |
|
| 56:51 | uh uh manuscripts already prepared and waiting this moment and I hand carried them |
|
| 56:57 | the SCG uh office uh there in and I uh uh I hand them |
|
| 57:04 | the receptionist and uh with a little from Amaco management said, we think |
|
| 57:09 | is a major new development, uh deserves a special technical session at the |
|
| 57:14 | and chief. So the receptionist at SCG, of course, she doesn't |
|
| 57:19 | anything about Jewish. She accepted these she passed them on to the technical |
|
| 57:26 | for the upcoming convention. He was for uh Western Chico uh before the |
|
| 57:32 | with Slumber. And so he looked this and he said, wow, |
|
| 57:36 | looks really interesting. We had five and my, I had, I |
|
| 57:40 | the lead author and then there were other paper. So this guy, |
|
| 57:44 | technical chairman says, wow, that's . Um I'm gonna send these out |
|
| 57:49 | review to my friend at Exxon, Levitt. Now you might know Levin's |
|
| 57:55 | , famous juice of the pre preceding um one of the best. And |
|
| 58:02 | he was, he got these uh abstract for review and he says, |
|
| 58:06 | , Amao is way ahead of us time. He calls up the technical |
|
| 58:11 | and he said, he says, you uh you gotta have a, |
|
| 58:15 | special technical session with a high profile uh for the uh to present |
|
| 58:22 | However, do you mind? is however, at Exxon, we've |
|
| 58:27 | working on similar things And would you uh uh if we put in some |
|
| 58:33 | papers in the special session, even the deadline has passed? So I |
|
| 58:39 | these to the FDG at 5 p.m. the deadline day to in order to |
|
| 58:45 | sharing the credit with anybody else. that plan was defeated by the fact |
|
| 58:52 | the technical chairman sent our papers for to Levin. And he said, |
|
| 58:57 | you mind if we put in a of Exxon papers? Um even though |
|
| 59:01 | deadline has passed and the technical chairman , sure, why not? So |
|
| 59:06 | had then seven papers in that actually eight papers, eight papers. |
|
| 59:15 | the uh there were the five technical five papers from Axon 25 from |
|
| 59:21 | two from Exxon and one from a at the School of Mines in |
|
| 59:28 | And that student was absolutely brilliant. had discovered on his own at the |
|
| 59:34 | of mines without having the resources of major oil company like Amaar. And |
|
| 59:40 | discovered most of the important ideas come , and on his own. And |
|
| 59:47 | there was the, these eight So uh uh uh comes the convention |
|
| 59:53 | three months later, you know, the, in the fall, September |
|
| 59:57 | . And it was in a very session. And afterwards, Levin came |
|
| 60:01 | to me and he said, uh know, I've been coming to uh |
|
| 60:05 | uh SGG meetings for 50 years. is the most exciting technical session I've |
|
| 60:11 | um uh attended at the seg it dynamite. I'll show you more about |
|
| 60:17 | on uh uh in lecture tent. me not to waste any more time |
|
| 60:23 | that time. Retelling these same story , that that was an exciting time |
|
| 60:28 | me. So we gave these papers uh um it turns out the Exxon |
|
| 60:34 | were wrong. They had been pushed their management into uh uh publishing before |
|
| 60:40 | quite understood their data. So those Exxon papers are wrong but the five |
|
| 60:44 | papers have withstood the test of time can look them up at now. |
|
| 60:48 | , it was the uh the seg of 1986 maybe before some of you |
|
| 60:55 | born. Yeah, you can, , you're all members refugee, you |
|
| 61:00 | , you can look those up. here comes Schlumberger into the uh into |
|
| 61:08 | uh story. This is for Brianna. So uh uh we, |
|
| 61:14 | gave these uh uh dynamite talks explaining about what happens uh when the rocks |
|
| 61:21 | as little anisotropy, which most of do because of the presence of uh |
|
| 61:29 | uh unequal subsurface stresses and because of presence of frac and fractures in the |
|
| 61:37 | , most rocks have aimy anti and explained that to everybody in 1986. |
|
| 61:46 | and, and uh uh well, uh we had a large audience, |
|
| 61:53 | got a lot of publicity in the were people from slumber shade. And |
|
| 61:59 | over the, the weekend, the weekend, there was an emergency meeting |
|
| 62:04 | uh uh uh uh one of the experts of Schlumberger and in Rich, |
|
| 62:11 | uh uh in, in those the uh uh uh a major research |
|
| 62:17 | for lumber who was in Ridgefield, just outside New York City. |
|
| 62:23 | they moved it some years ago to be close to Boston. But |
|
| 62:27 | uh in those days, it was to New York City and they assembled |
|
| 62:31 | best experts in borehole sonics and they a question to these experts. They |
|
| 62:37 | , can we creep into, can measure the effects of a methyl anisotropy |
|
| 62:46 | the world? Modifying what AMAO showed surfaces, beer. And so uh |
|
| 62:57 | the experts there symbol said, maybe. And so they spent the |
|
| 63:02 | year in a crash program of uh development and clever engineering. And uh |
|
| 63:13 | uh uh they required test data at uh uh uh test sites uh wherever |
|
| 63:20 | are and in, in a year so, they came back. Uh |
|
| 63:24 | reconvene and they said, yeah, answer to that question is yes, |
|
| 63:29 | , we can see this, these in the ball. And furthermore, |
|
| 63:35 | is a major new line of business us. It's Limer J and what |
|
| 63:40 | developed at that time was what was uh what's uh uh called then. |
|
| 63:44 | now uh Rost Diol Sonic tool. so uh uh uh they thought, |
|
| 63:52 | , here's a major new line of for us, let's corner the market |
|
| 63:58 | getting a patent on all this. they put in a patent application to |
|
| 64:03 | all that. So about, about month prior to that point in |
|
| 64:11 | I am a car in Tulsa. were sitting around uh at the lunch |
|
| 64:17 | in the company cafeteria and somebody could we do this in the |
|
| 64:22 | And somebody else said, sure, not? And somebody else said, |
|
| 64:26 | , is that patentable? And we , sure, why not? And |
|
| 64:29 | in an amazing short period of the amical lawyers drew up a patent |
|
| 64:36 | and we all signed it. You , the way this works is that |
|
| 64:39 | have an invention while you're working for company, you sign over your rights |
|
| 64:43 | the convention to the company and the pays for all of the costs associated |
|
| 64:49 | getting the patent and the inventors get 100 bucks. And they, |
|
| 64:55 | uh, they get a pat on back from their boss. Uh, |
|
| 64:59 | the company is looking to make major from the uh invention. Of |
|
| 65:05 | the employee gets a regular paycheck, normally the employee does not get rich |
|
| 65:10 | of uh the invention, but the might get rich, they pay the |
|
| 65:16 | to um drop to uh um uh the patent and uh you know, |
|
| 65:21 | acquire the patent. So, so went all. Uh our luncheon meeting |
|
| 65:27 | about a month before, um Schlumberger in their patent application to the American |
|
| 65:38 | office. Our amical lawyers operated with efficiency and we put in a, |
|
| 65:46 | uh an application also within that month it takes a long time for the |
|
| 65:53 | patent office to make any decisions. about a year later, we got |
|
| 65:58 | excited call from our friends at Schlumberger Sugarland and they said, we didn't |
|
| 66:05 | you guys had cross dipole sonic tool we're sitting up there in Tulsa, |
|
| 66:12 | ? And uh uh we said uh , what tool? We don't have |
|
| 66:15 | tool. We're an oil company, not an oilfield services company. And |
|
| 66:20 | said, well, you might not a tool but you do have a |
|
| 66:23 | . And here's what happened. Our reached the patent office about two days |
|
| 66:31 | the slumbers ain't application debt. And thinking about it a year, the |
|
| 66:38 | uh uh the examiner at the depart the patent office proved uh all the |
|
| 66:45 | in the uh Schlumberger uh application and , the the other examiner who was |
|
| 66:54 | at the Amaco uh application approved those . And then after everything was all |
|
| 67:01 | about ready to forget final approval, they got in touch with each |
|
| 67:06 | then they looked for what they call art and you know, uh you |
|
| 67:13 | invent something which has already been So, uh uh when these two |
|
| 67:18 | finally got together, after thinking about for a year, then they realized |
|
| 67:23 | the Amaco invention was basically the same the Slumber invention. And it had |
|
| 67:29 | by about two or three days. they rejected, in the end, |
|
| 67:34 | rejected all of the claims. They 90 out of 91 claims for uh |
|
| 67:40 | the Schlumberger application in favor of the invention, simply because we had beaten |
|
| 67:49 | to the patent novels by a, couple of days. So, uh |
|
| 67:55 | the patent office then uh notified Slumber there application had basically been denied. |
|
| 68:03 | so on the basis of the Amaco Art. So that's why they called |
|
| 68:08 | up. So, uh uh uh a day or so later, we |
|
| 68:12 | that our uh patent had all been . So they were sitting there with |
|
| 68:17 | opportunity they had, uh you it, it seemed like it was |
|
| 68:21 | unfair because they had done a lot clever theory and a lot of clever |
|
| 68:25 | and develop the tool and acquired data so on. Uh Whereas all we |
|
| 68:29 | was we had a, a casual around the lunch table. So |
|
| 68:35 | it, it seemed like it was , however, that's why patents exist |
|
| 68:40 | that uh uh they can protect the of the inventor while still making the |
|
| 68:47 | available to society at large. So way that works is that when you |
|
| 68:53 | a company that wants to implement an . And you have another company that |
|
| 68:59 | the invention has the rights. Then two companies negotiate and the company that |
|
| 69:05 | in second will pay royalties, pay fee to the company that was first |
|
| 69:12 | using that invention and it's only a percent. It's not a big deal |
|
| 69:16 | anybody. So, uh uh that's happened after some negotiation. Amako license |
|
| 69:24 | um I invention to uh Schlumberger and to Halliburton and uh uh all the |
|
| 69:31 | firms. Uh uh So they would the Amaco technology to make money for |
|
| 69:37 | , you know, on behalf of customers. And they would pay a |
|
| 69:40 | percent of their profits to Amaco every for 17 years that's now completed. |
|
| 69:48 | that that's all uh uh uh public now. But for 17 years, |
|
| 69:53 | got um uh payments uh of the of a million dollars a year from |
|
| 70:00 | um service companies. And of the, the service company made a |
|
| 70:05 | more money than that for themselves. But uh Arcot was of the order |
|
| 70:10 | a million dollars a year, of which, you know, II |
|
| 70:13 | received a zero. Ok. Uh I always had the um attitude that |
|
| 70:20 | of those uh payment to uh Amao , uh that uh uh was because |
|
| 70:27 | me and essentially, uh Amao uh um had my services for free, |
|
| 70:37 | because of all of the payments they getting from this, these patents were |
|
| 70:44 | more than my amical seller. uh that's kind of a liberating idea |
|
| 70:49 | you think that your efforts are uh money for the company much more than |
|
| 70:54 | paying you. You don't have to that they're doing you any favors by |
|
| 70:59 | uh paying you the salary. Uh making money for the company and I |
|
| 71:04 | that uh you folks also are, making money for your companies. Um |
|
| 71:10 | Even today. Well, so uh will talk more about cross dipole sonic |
|
| 71:22 | in lecture 10. Yeah, I all, all of that discussion came |
|
| 71:28 | just this simple question by Lee asking don't we have sh waves coming out |
|
| 71:35 | this kind of situation? And I tell you that was revolutionary in, |
|
| 71:40 | , in those days, everybody uh that you would not get any data |
|
| 71:49 | on a cross dipole achiever whether it's borehole or surface seismic or anything in |
|
| 71:57 | situation. Because everybody had in their pictures like this, which they have |
|
| 72:04 | in the textbooks with an isotropic body and an isotropic body below. And |
|
| 72:11 | this, the Real Earth is not that. The Real Earth is more |
|
| 72:16 | . So we're gonna talk about that later before we do that, we |
|
| 72:21 | to uh finish up with uh uh a number of other um um uh |
|
| 72:28 | . So, oh II, I actually I I need to do before |
|
| 72:34 | get to the lecture. I need um I need to look at the |
|
| 72:41 | from Carlos. OK. All, right. Oh, that's another great |
|
| 72:48 | . So Carlos shows a picture of converted wave uh data that I showed |
|
| 72:53 | before. Remember when um uh the offsets had uh a different move out |
|
| 73:00 | the negative offsets for a, for marine um ocean bottom seismic survey. |
|
| 73:06 | that was due to uh uh uh localized velocity anomalies in the subsurface, |
|
| 73:14 | were different for P than for S in particular, it was caused by |
|
| 73:19 | , I think, I think what call a gas cloud above the |
|
| 73:24 | Um And that was all in the North Sea. So what he asks |
|
| 73:29 | , is it common in s processing process negative offsets separately from positive |
|
| 73:36 | And uh so uh uh the answer le let's go into the, so |
|
| 73:43 | of our seismic data uh does not negative offsets. For example, if |
|
| 73:49 | do a marine uh acquisition, you've the source right behind the boat and |
|
| 73:53 | got this long streamer of uh of uh of receivers behind the |
|
| 74:00 | So we can call those all positive going from the front of the boat |
|
| 74:05 | the back. Uh However, on . Um Oh by the way, |
|
| 74:10 | if you have marine acquisition with different uh uh survey design, then I |
|
| 74:17 | said suppose there's two both. no, I suppose it's another shooting |
|
| 74:22 | off to the side. Uh, , uh, uh, uh, |
|
| 74:26 | could have a whole, uh, , uh, a variety of different |
|
| 74:32 | . Some of them are positive. are negative. But, uh, |
|
| 74:36 | , in the simplest case, marine have a single set of, |
|
| 74:42 | of, of, of receivers, , uh, a, an array |
|
| 74:47 | receivers maybe 10 kilometers long and maybe kilometer wide, several streamers in, |
|
| 74:54 | there. They're all behind the boat we'll call that positive offsets. |
|
| 74:59 | on land, you can do uh uh have anything, suppose you have |
|
| 75:03 | land uh uh or maybe uh uh , suppose you have on land um |
|
| 75:12 | receiver set up and it's easy to uh having um positive and negative offset |
|
| 75:22 | between the source and the receiver uh land. And so uh uh some |
|
| 75:31 | survey designs have that. But now know from the reciprocity theorem for P |
|
| 75:39 | , we're just doing P waves. know that the reciprocity theorem guarantees us |
|
| 75:45 | a negative offset is giving you gonna you the same data as a positive |
|
| 75:50 | . That's the scalar where the prostate . So normally we uh well, |
|
| 75:57 | don't uh oh no, normally we uh process positive and negative offsets separately |
|
| 76:06 | B waves. Well, here is here's another story from my personal |
|
| 76:18 | Oh, ok. But I did, did I tell you uh |
|
| 76:34 | earlier about making images from the data I showed, I showed you the |
|
| 76:42 | . Uh And I showed you the had different move out for positive offset |
|
| 76:47 | negative offset that I uh uh did show you any images from that? |
|
| 76:57 | Let's see here. Um Well, I got started. Um mhm I'll |
|
| 77:04 | you another story from my personal So we took that data and we |
|
| 77:11 | uh we figured out the reason why positive offsets had different move out than |
|
| 77:16 | negative offsets because they obeyed the vector there, not the scalar reciprocity |
|
| 77:23 | And then uh we figured out how image uh uh the reservoir, tried |
|
| 77:30 | the positive offsets differently than the negative . And we got good images for |
|
| 77:36 | first time of that um uh reservoir Amako. It's in the nor it's |
|
| 77:45 | the Norwegian North Sea. And uh it was uh when it was first |
|
| 77:53 | discovered, it was discovered, even the data was very poor. He |
|
| 77:59 | marine P wave data was very They drilled it anyway and they discovered |
|
| 78:04 | million barrel field. Good thing. then, and they worked at this |
|
| 78:09 | 10 years or so before we finally converted wave data over that same |
|
| 78:16 | That's the data that I showed you , by the way that, that |
|
| 78:21 | field is called Valvo. It's a , famous field in our business now |
|
| 78:28 | uh we've learned so much from that . And one of the things we |
|
| 78:32 | is about uh uh a converted way acquisition and processing and imaging. So |
|
| 78:42 | uh about, after we had produced field for about 10 years, even |
|
| 78:49 | the images were very poor, we finally developed good images using converted |
|
| 78:56 | And everybody in uh uh Amao Norway very happy with that because they |
|
| 79:02 | they could see the seismic images of reservoir and they could I, they |
|
| 79:07 | develop it better. So these uh uh val field is a 5 |
|
| 79:15 | barrel field, five, a 5 barrel field instead of a 1 billion |
|
| 79:20 | . Now, now that we can it better, we can see it's |
|
| 79:23 | 5 billion barrel field of which we produced so far about 3 billion. |
|
| 79:30 | there's still 2 billion to go. of course, when BP bought |
|
| 79:34 | they bought that field as well. now BP has sold that field to |
|
| 79:40 | uh another company where BP is partners that company and uh they have |
|
| 79:47 | So yeah, but now, so uh Val Hall is officially owned by |
|
| 79:52 | other company. And uh so of , since I left BP, I |
|
| 79:58 | had no inside information, I got feeling but it was, it was |
|
| 80:03 | big moneymaker for Aero and then for and we made the first good images |
|
| 80:09 | it. So um in Norway, have a very interesting attitude towards company |
|
| 80:18 | . They uh oh and furthermore, , I gotta preface this story with |
|
| 80:22 | following uh in, in the United offshore in the United States, the |
|
| 80:31 | uh belongs to whoever uh uh finds and, and, and drills it |
|
| 80:36 | produces it. Uh So companies bid the right to uh uh explore and |
|
| 80:44 | in selected tracks offshore of the United . And so uh there might be |
|
| 80:50 | track uh uh uh uh several miles uh uh in uh the dimension |
|
| 80:57 | west, north, south and so . So, uh uh and the |
|
| 81:01 | government sells to the companies, they an auction and they sell for the |
|
| 81:05 | , the rights to any oil that be found there. And companies like |
|
| 81:10 | VP and Texaco, the companies bid those uh in a public auction and |
|
| 81:19 | uh makes the highest bid, um , they then own whatever oil is |
|
| 81:27 | . So then they have to go and uh uh drill oil and produce |
|
| 81:30 | and so on, but it's their , not in Norway, in |
|
| 81:35 | it's always, always Norwegian oil owned the Norwegian people. And so, |
|
| 81:42 | the Norwegian government uh has options where , they license the companies like a |
|
| 81:50 | , the right to explore on selected offshore Norway. But whatever oil they |
|
| 81:57 | still belongs to the people of Norway they're gonna pay the oil company to |
|
| 82:04 | it. But the oil belongs to people in Norway. So the Norwegian |
|
| 82:12 | has an interest that companies operating in waters should do that using the best |
|
| 82:21 | available. So they require the companies have uh uh bought, who ha |
|
| 82:29 | been granted licenses to produce Norwegian oil to share their technology with other companies |
|
| 82:37 | that everybody can be using the same technology. So there are no secrets |
|
| 82:44 | and uh if you operate in so that meant was that when, |
|
| 82:50 | we made these uh uh converted wave at Fo Hall, then there are |
|
| 82:59 | dot com. The Amao office in said, OK, so uh you're |
|
| 83:04 | present this in public at the next of the European Society and because it's |
|
| 83:13 | good technology, the Norwegian government will there listening and they'll say, |
|
| 83:18 | we need to give more licenses for for Norwegian oil and it need to |
|
| 83:26 | more licenses to Amaco because those guys so smart. So you see there's |
|
| 83:31 | competitive advantage for a company to be their secrets because of the Norwegian government |
|
| 83:40 | , very different from the American So we did that, we presented |
|
| 83:45 | stuff uh uh dynamite images of val and we won an award, you |
|
| 83:55 | , we, we buckle, we the best paper award, things like |
|
| 83:59 | . And uh so meanwhile, on exhibition floor at the convention that the |
|
| 84:06 | that year was um uh held in Switzerland for the European Society. It's |
|
| 84:14 | the Eage European Association of Geoscientists and . Eage. So the convention was |
|
| 84:24 | in Geneva, that's where we presented work. And of course, they |
|
| 84:28 | a big exhibition floor and uh the oil services, uh an oilfield |
|
| 84:36 | company which had done the acquisition for . They had booth on the |
|
| 84:42 | And so uh uh they had asked , uh do you mind? Uh |
|
| 84:46 | we acquired this data for you. your data. But do we have |
|
| 84:50 | permission to uh process it and make ourselves? So, you know, |
|
| 84:58 | , it was on their computer but would not touch, it would not |
|
| 85:01 | at it without our permission. So gave them that permission. And of |
|
| 85:06 | , we expected that uh because we friendly with them, they would uh |
|
| 85:11 | good work for us in the So they processed it and they had |
|
| 85:15 | young employee who was an expert and away imaging, a recent uh graduate |
|
| 85:24 | from the University of Tulsa by the . And uh uh uh he, |
|
| 85:28 | had been hired by this uh uh to uh bring his expertise and converted |
|
| 85:37 | way of imaging into that company. so he processed the data and he |
|
| 85:41 | results which were very different from And he presented those results not in |
|
| 85:48 | technical sessions in Geneva, but he those at the booth on the exhibition |
|
| 85:55 | to whoever was passing by. And uh was hap happy to say |
|
| 86:02 | in the, in, in the that th those guys at Amao don't |
|
| 86:07 | what they're doing here are the Those guys at a do not know |
|
| 86:12 | they're doing. So at a certain , my boss was standing at, |
|
| 86:17 | the edge of the crowd at the listening to our own contractor insult us |
|
| 86:23 | public is not, my boss was happy. Uh So immediately following the |
|
| 86:29 | in Geneva, that young man was to come to Amaco Research in Tulsa |
|
| 86:36 | explain himself. And I was summoned Houston. By that time, I |
|
| 86:42 | moved from the Tulsa Research Amical research Tulsa to amical exploration in Houston. |
|
| 86:51 | I was summoned to come up to and explain myself. And so there |
|
| 86:55 | front of the bosses, we realized this young man from the service company |
|
| 87:02 | not recognized the differences between negative offsets positive offsets. He, he, |
|
| 87:08 | surely saw it but he didn't think was important. And so he processed |
|
| 87:16 | um together and um uh it, know, when he did his phd |
|
| 87:25 | , he uh uh was probably doing with synthetic data which uh didn't have |
|
| 87:31 | peculiar subsurface features that uh we had , at Val Hall. So uh |
|
| 87:36 | and his um uh uh thesis at of Tulsa, he uh uh didn't |
|
| 87:43 | this kind of difference between positive and offsets. And so when he saw |
|
| 87:47 | at Val Hall, he didn't really . Take it seriously, we took |
|
| 87:51 | seriously process the positive offsets different from negative offsets. And uh he didn't |
|
| 87:58 | and so he, his uh uh were confused because his uh gathers were |
|
| 88:04 | flat. So that's why he got different images. So uh um long |
|
| 88:11 | short, uh uh uh uh it too long before this young man was |
|
| 88:16 | longer employed by this uh service So uh that's a good lesson for |
|
| 88:22 | young people that uh when you find pu puzzling data, uh you should |
|
| 88:28 | it out, figure out the reasons it uh with your colleagues uh uh |
|
| 88:32 | the bosses watching you get in a room, roll up your sleeves and |
|
| 88:37 | out why the data looks like And uh if there are differences with |
|
| 88:43 | company, you find a way to together, collegially behind closed doors and |
|
| 88:50 | out uh why uh uh uh we these differences. I've had several instances |
|
| 88:56 | my career where uh differences in images caused by the failure of assumptions that |
|
| 89:08 | had made successfully for many years. then we were applying those to new |
|
| 89:13 | of data and getting confusing results. um I think we'll have a chance |
|
| 89:20 | talk more about that um in uh 10. So that's uh that adventure |
|
| 89:28 | my own history was a direct uh uh answer to the question that uh |
|
| 89:37 | said, I'll read it again. what he said is it common in |
|
| 89:41 | processing to handle the negative offset separately the positives? And so uh uh |
|
| 89:48 | repeating the answer for, for P , uh uh it's not common to |
|
| 89:52 | , do that separately because we have scale of reciprocity theorem. But for |
|
| 89:58 | waves, you gotta do it. I think that for you uh students |
|
| 90:04 | for including UT I, uh uh have not seen much of any converted |
|
| 90:10 | data. Um So most of what day that you're looking at is P |
|
| 90:17 | . Uh But something you can safely the scale of reciprocity theorem for E |
|
| 90:25 | and not for converted waves and incidentally for Jewish air waves either. |
|
| 90:30 | uh we will uh uh uh talk about that um in lecture 10. |
|
| 90:39 | . So uh here we uh have , we spent an hour and a |
|
| 90:44 | talking about your questions and listening to re relive uh uh adventures from my |
|
| 90:52 | . Let's see. Uh I joined when I was 40. So I |
|
| 90:57 | older than uh than anybody else Uh So I had, I had |
|
| 91:01 | the previous part of my career uh the university in New York. And |
|
| 91:07 | , but I came into Amao ignorant a lot of things that you students |
|
| 91:12 | know students already know a lot more exploration, geophysics than I knew when |
|
| 91:19 | joined Amic. They hired me anyway two reasons because my father had worked |
|
| 91:24 | AM and he had a very good and they figured that, uh, |
|
| 91:28 | hire me to get another one like . And I'm sure they were |
|
| 91:32 | But, uh, well, that, that was their motivation, |
|
| 91:37 | sure. And then the other reason in those days when I joined em |
|
| 91:41 | 1980 the industry was booming, And uh uh they were hiring anybody |
|
| 91:49 | knew how to spell geophysics and I how to spell it because I uh |
|
| 91:54 | uh academic career. But I didn't uh you know, the issues involved |
|
| 91:59 | academic geophysics are so different from the and as geo that I came into |
|
| 92:06 | um uh not knowing a lot of that you guys already know, but |
|
| 92:12 | did know some things which were that else knew. For example, I |
|
| 92:17 | about an isotropy and then I built career out of that. And uh |
|
| 92:24 | uh uh you, uh you students also have an opportunity to build a |
|
| 92:32 | if you know something that nobody else . I, I knew about anisotropy |
|
| 92:37 | I was able to see it in data. Some of the very first |
|
| 92:42 | that I ever saw inside Amer had indications of anisotropy in it. |
|
| 92:51 | and nobody else saw them. I it and cared about it. Thought |
|
| 92:56 | was interesting. They said uh they just uh sort of brushing it under |
|
| 93:00 | rock. Uh But uh so the lesson here is that when you |
|
| 93:05 | in the data, something which is peculiar and doesn't fit uh your |
|
| 93:14 | talk with your colleagues about that. does the data look like that? |
|
| 93:20 | then think about it and study and uh read the literature and read the |
|
| 93:25 | and think about it on. You find the reason the data looks like |
|
| 93:30 | is because the Real Earth doesn't obey simplifying assumptions that we've been doing. |
|
| 93:41 | we've talked a lot about how we're all this theory about waves and rays |
|
| 93:46 | the first seven lectures of this course making assumptions that uh are really |
|
| 93:54 | And so uh I'll, we will uh we will talk in the last |
|
| 94:00 | lectures, we'll talk, we'll talk more realistic, how, how to |
|
| 94:04 | more realistic assumptions and what uh effects have on the data. But you |
|
| 94:11 | be alert who features in the which are not explained by your uh |
|
| 94:23 | current understanding. And so, of , it might be that uh uh |
|
| 94:28 | a classical reason for that you find by talking to your colleagues who are |
|
| 94:32 | experienced, but then you might be like I was and stumble onto something |
|
| 94:38 | was magic. So I, I always regarded luck to be a big |
|
| 94:44 | of my success. And if you're , it will be a, a |
|
| 94:48 | of your success, but you gotta prepared for it. Uh When, |
|
| 94:52 | something, um magic comes along in of your eyes, you gotta be |
|
| 94:58 | for it. And what you're doing now in this course is you're learning |
|
| 95:02 | basics so that you will be prepared , for that. But don't make |
|
| 95:07 | mistake that those uh uh anti soy from Shell dead when they saw |
|
| 95:13 | which did not obey their understanding, made fun of the speaker instead of |
|
| 95:21 | that as motivation to improve their own . That's what they should have |
|
| 95:27 | They should have said, wow, data is impossible according to what we |
|
| 95:31 | . So, but it's data, uh uh uh so what we |
|
| 95:36 | we, we know that we made of uh of um uh assumptions in |
|
| 95:43 | our understanding. Uh uh Maybe some those assumptions were wrong. Let's think |
|
| 95:49 | that. That's what they should have . But they didn't, they, |
|
| 95:52 | wasted the opportunity. If that, an opportunity like that comes your |
|
| 95:56 | don't waste it. Try to understand the data looks like that. It |
|
| 96:02 | be a boring answer, you Uh It might be just uh so |
|
| 96:06 | , for example, uh here's an of a boring answer in the case |
|
| 96:10 | uh the Arco guy was showing the wave data, strong data on the |
|
| 96:16 | line component. Maybe the reason was the uh uh uh the instruments were |
|
| 96:22 | installed properly, maybe they weren't lined properly, right? If, if |
|
| 96:27 | line, if you think you have three components phone lined up properly and |
|
| 96:32 | not lined up properly, then uh uh you could have that kind of |
|
| 96:36 | or think about this, it could lined up properly with the arrow pointed |
|
| 96:41 | the proper direction, but maybe the is not the same in all |
|
| 96:47 | Maybe the coupling is not the My first job uh in this business |
|
| 96:52 | when I was an undergraduate student and was working on a field group in |
|
| 96:57 | Texas about 100 miles north of And my job was to install the |
|
| 97:04 | . So I would go out with bunch of earphones over my shoulder on |
|
| 97:10 | and I would come to the place a flag had been set by the |
|
| 97:14 | stamp. And I would drop one the um of the geophones on the |
|
| 97:21 | . And then I would take take it, pick it up, |
|
| 97:24 | the spike into the ground and then on it. Then I would look |
|
| 97:29 | at the um at the geophones now in the ground and it had a |
|
| 97:36 | um device on the top of it showed whether or not it was |
|
| 97:43 | it's called a leveling bubble. And would always see it wasn't quite |
|
| 97:50 | So why I would do that was kick it from the side to straighten |
|
| 97:55 | out, but it was vertical. . But you can see immediately as |
|
| 98:01 | was doing that, I was changing coupling uh because I was kicking it |
|
| 98:06 | the side, I was changing the . So the coupling wasn't the same |
|
| 98:10 | line as cross line. And so was if, if in those days |
|
| 98:14 | was vertical geophones only, you can the same principle is if you install |
|
| 98:20 | the month GEO phone with different coupling line and cross line, you could |
|
| 98:25 | the kind of data that um um Arco guy was showing and there's a |
|
| 98:35 | , but that's a B I would that is a boring solution. And |
|
| 98:39 | talked about that in the, in discussion there at uh in that technical |
|
| 98:43 | , people proposed that and the air guy said uh uh well, of |
|
| 98:48 | , uh I thought of that and went and checked and it was all |
|
| 98:52 | . Uh uh uh So uh that was uh an example of |
|
| 98:58 | of a classical explanation which could conceivably this peculiar data. And a guy |
|
| 99:05 | a smart guy. Uh And he checked out all those things and, |
|
| 99:09 | so when you find data, if doesn't fit your um understanding, then |
|
| 99:16 | , you should uh really uh work to understand what caused it and maybe |
|
| 99:21 | find it's a boring answer or maybe a, a magical answer that leads |
|
| 99:26 | a lot of uh of uh advances you uh once you answer it, |
|
| 99:32 | that that's, that's good advice. you find data that doesn't match your |
|
| 99:42 | , find out why. And probably answer is in your own mind rather |
|
| 99:47 | in the data itself. And so you uh learn in your mind how |
|
| 99:53 | eliminate the simplifying assum that you previously made. And maybe your colleague made |
|
| 100:01 | same simplifying assumption that um uh dig it, take it seriously and figure |
|
| 100:09 | why the data looks like that. for uh AO we had the uh |
|
| 100:19 | freedom, our bosses gave us the to sue our noses. And when |
|
| 100:25 | followed our noses and tried to figure why the day didn't look like |
|
| 100:29 | We came to an understanding that um eventually uh um uh uh had a |
|
| 100:42 | . Uh uh I understand it was a trillion dollars with a T, |
|
| 100:47 | trillion dollar secret we had discovered. um it revolves around this business of |
|
| 101:03 | anisotropine. You can imagine that if um subsurface has fractures in it, |
|
| 101:11 | uh the uh wave propagation is gonna a smoothly, an isotropic. And |
|
| 101:17 | if you observe that you can detect fractures, and so if you know |
|
| 101:21 | the fractures are, uh that can important uh as you're developing this |
|
| 101:26 | uh you want to uh develop the um you wanna develop the reservoir, |
|
| 101:36 | the uh the flow of fluids in subsurface caused by fractures, right. |
|
| 101:43 | The fluids can flow along the fractures lot more easily than across the |
|
| 101:48 | So we did uh we discovered the of ath an isotropy. And at |
|
| 101:55 | certain point, we convinced ourselves that was due to fracture. No, |
|
| 101:59 | else but fracturing. So we went uh Amma management with the proposition that |
|
| 102:10 | , that we should buy. Uh uh let me back up the place |
|
| 102:16 | we discovered this, that it was due the uh to fracture was here |
|
| 102:23 | Texas and, and uh uh not from here, actually, it was |
|
| 102:28 | 50 miles from here in the, the north uh in the west northwest |
|
| 102:34 | . That's where we did the which convinced us that the Azlan surgery |
|
| 102:40 | we were seeing was due to So it was in a formation which |
|
| 102:50 | call the Austin chalk. So that's limestone formation which extends broadly thousands of |
|
| 102:58 | miles across central Texas and Louisiana, known as an oil producer for years |
|
| 103:04 | years only where there's fractured. And this uh understanding, we figured |
|
| 103:11 | we know how to um uh we how to find these fractures from surface |
|
| 103:19 | data. And furthermore, the same frame as uh uh the previous |
|
| 103:25 | I told. Furthermore, we understood Exxon was gonna be, we would |
|
| 103:29 | finding out very shortly how to find fractures in the Austin trunk in thousands |
|
| 103:35 | square miles and exits. So we to Amico Management that we should go |
|
| 103:42 | buy up the mineral rights to all uh those acres, thousands and thousands |
|
| 103:48 | acres in Texas. In those the uh we could, we could |
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| 103:54 | the mineral rights that it would buy buy from the farmers. Uh I |
|
| 104:02 | in um uh in onshore us, oil beneath the farmer's land belongs to |
|
| 104:08 | farm. Normally the farmer doesn't know to uh uh buy it in and |
|
| 104:15 | it. So what he does is , he, he sells the |
|
| 104:19 | the mineral rights to oil companies number to explore it. And they, |
|
| 104:23 | , they pay uh a work, pay for the right to explore and |
|
| 104:27 | if they find something, the farmer a fraction. And uh uh so |
|
| 104:34 | mineral lights uh for exploration were selling Texas in those days for $25 an |
|
| 104:42 | . So we proposed to Aqua, should buy the mineral lights for the |
|
| 104:47 | Austin shop. Tens of thousands, of thousands of acres in Texas. |
|
| 104:53 | would have cost us about $100 So that's a lot of money. |
|
| 104:58 | um uh uh uh uh Amao could done it. So that proposal was |
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| 105:08 | all the way up through many changes management all the way up to, |
|
| 105:13 | uh head office in Chicago where it um ok. What is it? |
|
| 105:21 | rejected and it was rejected for two . One of it, at the |
|
| 105:25 | , the price of oil was very . It was about $10 barrel and |
|
| 105:31 | management had no confidence that it would be any higher than $10 a |
|
| 105:36 | And furthermore, they didn't trust our . It was brand new technology that |
|
| 105:40 | else knew about. It should have considered an advantage, but it was |
|
| 105:45 | by amicable management, a risk So they rejected. So, um |
|
| 105:52 | if they had accepted it, they have gotten the rights to all of |
|
| 105:57 | Austin chalk and get this, they have gotten all the rights to the |
|
| 106:01 | Ford shale, which lies directly below Austin chalk. So these days, |
|
| 106:07 | Eagle Ford shale is the the most uh shale producing formation in the United |
|
| 106:14 | . And uh uh if you wanna mineral rights in the Eagle Ford shale |
|
| 106:20 | days, it costs about $20,000 per . So instead of $25 an |
|
| 106:26 | it's $20,000 an acre. We would gotten that for free if we had |
|
| 106:31 | gotten the a the mineral rights for Austin chalk that uh uh right E |
|
| 106:36 | extends all the way to the center the earth. So we would have |
|
| 106:40 | the eagle Ford shale for free. see that lost opportunity, I estimate |
|
| 106:49 | amao of the order of a trillion . If they had approved that the |
|
| 106:58 | and acquired those rights, then uh would have found a lot of oil |
|
| 107:04 | maybe Aero would have later bought BP of vice versa. But they, |
|
| 107:10 | missed that opportunity. Uh It was opportunity presented to AO management by the |
|
| 107:16 | technical staff, including me and enabled the management at Aero gave us the |
|
| 107:24 | to follow our noses and, and out what made the data look like |
|
| 107:29 | . Like you, if you're you can stumble. Uh you can |
|
| 107:34 | uh features in your data that nobody . You learn how to understand those |
|
| 107:41 | and maybe it'll be um uh the to a big business opportunity for your |
|
| 107:50 | . OK. Let's see what time is. Well, so I have |
|
| 107:54 | on here for um two hours answering questions and uh uh and telling you |
|
| 108:02 | . So uh now it's time to uh resume where we left off with |
|
| 108:08 | waves, but it's also a good for a break. So uh let |
|
| 108:13 | uh stop at this moment. We'll back in 10 minutes at uh uh |
|
| 108:18 | the top of the hour and we uh uh resume talking about love |
|
| 108:24 | OK? See you see you in 10 minutes. So folks, |
|
| 108:33 | hope, I hope everybody is back the break. Uh And I hope |
|
| 108:37 | can see this slide about uh uh love waves. This is where we |
|
| 108:42 | up um uh yesterday. No. I have a confirmation that you're seeing |
|
| 108:51 | slide now about Love Wink from uh ? Do you see it? Grace |
|
| 109:06 | , do you see it? I'm sure uh where I am, they |
|
| 109:14 | see it. Um but they're not yet. Ok. So I, |
|
| 109:21 | think maybe I'll wait for a few . Um But not confident for Sega |
|
| 110:19 | Carlos is gonna be here shortly. sure let's just wait for him. |
|
| 110:33 | you for sharing your stories. Those very interesting and motivating. Uh |
|
| 110:40 | uh so, uh uh uh I telling those stories. Uh Number one |
|
| 110:46 | I'm the hero of those stories. I also, they uh they're based |
|
| 110:52 | the technology. So uh they're inspired by your questions here and uh |
|
| 111:00 | uh they bear exactly on the science we're talking about here. So let's |
|
| 111:08 | here. I'm still waiting for. , Carlos. So when, when |
|
| 111:26 | think back on my career, I'm how lucky I was uh so many |
|
| 111:31 | in my career, but I was to uh seize the luck, sees |
|
| 111:38 | opportunity. And so that's what you are doing right now. You're uh |
|
| 111:43 | yourselves so that you can uh uh luck comes your way, you'll be |
|
| 111:48 | for it. Mm. Well, think I'm gonna give uh Carlos one |
|
| 112:11 | minute I got here. OK. And Carlos made it back. |
|
| 112:29 | uh so let's begin. Remember this where we left off. Um uh |
|
| 112:37 | , uh we, we went through analysis uh looking for love waves in |
|
| 112:42 | similar way that we look for uh waves. Well, we found railway |
|
| 112:47 | and we did a bunch of analysis we came up with a formula for |
|
| 112:51 | whaling rail rail wave velocity. So , we had uh expectations, the |
|
| 112:59 | program would work for love wave. we found at this point that uh |
|
| 113:04 | uh uh the, the government equation this one and it has no love |
|
| 113:10 | solution whatsoever. So uh then what did was uh said, OK, |
|
| 113:16 | try to make it more complicated subsurface with some in homogeneity there. So |
|
| 113:22 | got uh an upper layer and a layer, upper medium has a vs |
|
| 113:27 | , lower layer has the vs And we're still gonna look for a |
|
| 113:31 | wave which is traveling in the uh this direction and uh polarized out of |
|
| 113:38 | screen like this and the thickness of layer is is D you see it |
|
| 113:43 | uh here's uh uh X is uh at this boundary, not up |
|
| 113:49 | but there's a zero at this boundary downwards so that the earth surface is |
|
| 113:56 | , at the position X three at D. So now let's do the |
|
| 114:02 | sort of uh of analysis uh with more complicated model um and take um |
|
| 114:12 | from love. I'm sure that love the same thing that we just |
|
| 114:17 | did it about 100 years ago. to an answer. There are no |
|
| 114:21 | waves, but he persisted and he , OK, let me drop the |
|
| 114:26 | that the subsurface is uniform and see I find. So what he found |
|
| 114:31 | love wave. And that's one of reasons he's famous. So, |
|
| 114:36 | so in the upper medium, we this uh wave equation in the lower |
|
| 114:40 | , we have this wave equation. what we found previously for uniform |
|
| 114:45 | And the difference is that here we vs one and here we have vs |
|
| 114:50 | . And this is the displacement, love wave displacement in the lower |
|
| 114:55 | And here we have low wave displacement the upper media. And um so |
|
| 115:03 | then we, we're gonna match so we're gonna find, we know |
|
| 115:07 | we get plane wave solutions here. get plane wave solutions here and we're |
|
| 115:11 | match them at the boundary. So what are the boundary conditions? |
|
| 115:16 | on the surface it's like uh uh had before uh and at the lower |
|
| 115:25 | we're gonna have also uh uh not of stress and displacement. So |
|
| 115:32 | , we're gonna need a, uh gonna uh uh suma um solution and |
|
| 115:38 | it's gonna have three parameters. In assumption. And then we're gonna fix |
|
| 115:43 | uh three parameters using these three One two. OK. So that's |
|
| 115:52 | this is what we're gonna try in upper medium. We're gonna try assume |
|
| 115:56 | wave solutions. Now look, we here down going and upcoming waves. |
|
| 116:02 | what's the difference here? Let's look the phase factor here, we have |
|
| 116:06 | t same thing here, Omega T have minus HX one minus H one |
|
| 116:13 | one. That means that it's gonna going in the X one positive X |
|
| 116:18 | direction. This wave here is gonna going in this direction, same thing |
|
| 116:24 | this and going in this direction Now, this one has also uh |
|
| 116:28 | X three component to the displacement uh uh to the wave vector I I |
|
| 116:36 | say and it's uh the, the is only going to be in the |
|
| 116:41 | directions, but the wave vector is gonna have a component in the vertical |
|
| 116:47 | with a different wave number. You this is different from this one H |
|
| 116:52 | versus H one and more of our that it's ne negative with the conventions |
|
| 116:58 | have. That means it's down and over here, you have a similar |
|
| 117:02 | but it's upcoming. Why it's because got a plus right there. So |
|
| 117:08 | means we're gonna be allowing waves that they're going uh to the right, |
|
| 117:13 | echoing up and down inside here. , what uh uh what can we |
|
| 117:21 | about this wave vector has 22 components that H one and H three. |
|
| 117:28 | so the square of the length of wave vector is gonna be the sum |
|
| 117:32 | the squares of those two components. that is related to the frequency by |
|
| 117:38 | S one velocity in this way, is familiar to, to you. |
|
| 117:44 | , in other words, this solution , in some of two ways, |
|
| 117:50 | one of these solves the wave equation . Some of these two and uh |
|
| 117:55 | also going to be a solution. noticed they have different constants on here |
|
| 117:59 | front and they potentially dependent on So this proposed solution solves the wave |
|
| 118:10 | if and only if we have this between the wave vector and the |
|
| 118:16 | Remember that the wave vector itself does appear in the wave equation, neither |
|
| 118:24 | the frequency. But uh uh this work if uh we have this relationship |
|
| 118:30 | that's for the upper medium only for lower medium, we only have um |
|
| 118:36 | waves because there's nothing down here to waves back up. The only reason |
|
| 118:42 | have an upcoming wave is because it's be reflected off of this boundary. |
|
| 118:47 | there's nothing to whatever uh goes down is never coming back. And it's |
|
| 118:54 | a different wave vector. It's got wave vector which we call K, |
|
| 118:58 | got two indices. Uh um uh two components K one and K |
|
| 119:04 | the square of the length of that vector is given by the sum of |
|
| 119:10 | of its components related to uh to with the S with the VS |
|
| 119:17 | Because the wave equation which governs wave down here has vs two in |
|
| 119:23 | I'm just gonna back up here. I back up here, here it |
|
| 119:28 | . Here's the wave equation down here a vs two here. The wave |
|
| 119:33 | has a vs one. OK. going back forward. So, so |
|
| 119:38 | is our trial solution and has seven here. Here's the list of the |
|
| 119:42 | parameters. It's got um uh three and four wave vector components. And |
|
| 119:50 | we're gonna use the previous three equation solve for these seven different crimes. |
|
| 120:00 | um in order to meet the boundary at all, uh horizontal positions, |
|
| 120:05 | like we had with the wave we have to find out just as |
|
| 120:10 | had with a really um a We have to find out that uh |
|
| 120:14 | H one is the same uh the, the horizontal component of the |
|
| 120:20 | vector in the upper medium has to to the horizontal component of the wave |
|
| 120:24 | in the lower medium according to the way voc that's the first time we |
|
| 120:30 | love way velocity. Now we go the logic similar to what we did |
|
| 120:35 | railways. But there's a lot of which uh I'm not gonna reproduce. |
|
| 120:41 | we come up here that VL is be the solution of this equation. |
|
| 120:45 | a pretty complicated equation. Here's the sign in here. We got a |
|
| 120:49 | of square roots and we got here the VL here here. And also |
|
| 120:54 | here and also here, like we a tangent function here. So I |
|
| 120:59 | that this whatever is inside the argument the tangent function better be non |
|
| 121:06 | that would be non dimensional. So this non dimensional where the, the |
|
| 121:11 | root is non dimensional? You can , and also you can see that |
|
| 121:15 | omega times D that's the layer of divided by VL, that's also a |
|
| 121:20 | dimension. So this is a properly tangent function here, we have the |
|
| 121:26 | of the two sheer moduli in the mediums. And uh uh unless we |
|
| 121:33 | the L again, so it's a equation. So I don't think you |
|
| 121:38 | solve you. Uh I don't think can solve that um in your |
|
| 121:42 | but here's what we're gonna do. gonna do the same geophysical thing that |
|
| 121:47 | did before we're gonna re oh Before get to that, let, let |
|
| 121:52 | point out here that uh here is frequency right in here when we did |
|
| 121:57 | um when we found the velocity for rail wave, there was no frequency |
|
| 122:03 | there. Here is the frequency. um um here, it says here |
|
| 122:09 | if you look at this closely in high frequency limit, the uh the |
|
| 122:14 | , the low velocity goes to be to the upper sheer velocity. And |
|
| 122:20 | low frequency limit it go goes to lower fre body way velocity. That's |
|
| 122:27 | . And you can test that for by considering that your omega make it |
|
| 122:33 | be very large and you'll uh uh find uh the uh uh everything simplifies |
|
| 122:41 | uh uh to where the VL then vs one as omega becomes infinite. |
|
| 122:55 | conversely, for a low frequency, do we know whether a given frequency |
|
| 123:00 | high or low? Well, we it's high or low when it's compared |
|
| 123:07 | V sub L divided by DVL divided D has the dimensions of frequency. |
|
| 123:15 | if this omega is large compared to ratio here, um uh then it's |
|
| 123:24 | large frequency and conversely, it'll be low frequency. So you see that |
|
| 123:30 | layer thickness defines a characteristic frequency so we know whether a given frequency is |
|
| 123:40 | or low compared with DV and divided the uh the love weight velocity. |
|
| 123:50 | let's rep parameter that assuming that uh the upper medium is slower than the |
|
| 123:56 | medium that's very common. So we're define a quantity uh zeta in this |
|
| 124:02 | . And then we're gonna uh uh that into the previous equation, make |
|
| 124:06 | tailor expansion in zeta. And as result of that we come up with |
|
| 124:11 | much simpler expression. And I think can look at this one and we |
|
| 124:16 | see uh uh what it's trying to us no, the uh the, |
|
| 124:22 | ratio of D to now D to wavelength uh of the sheer wavelength. |
|
| 124:38 | what this means is for larger sheer that are smaller, sheer smaller |
|
| 124:45 | zeta gets larger, uh larger uh depends in a positive way on, |
|
| 124:52 | that. Uh it is positively correlated the wavelength. So here's an example |
|
| 125:02 | using that format for three different uh different thicknesses. And we're assuming now |
|
| 125:09 | the upper case u upper uh uh is very slow and the uh lower |
|
| 125:16 | is not, is a bit So maybe these are plausible uh uh |
|
| 125:23 | numbers. Uh You can do your . Everything here is so simple, |
|
| 125:27 | can do your own. And so way vectors have this character in the |
|
| 125:31 | medium, sure enough, we have upper upgoing me an upgoing wave and |
|
| 125:37 | wave and they move together to the side and find within that layer that |
|
| 125:43 | love wave velocity, but they're propagating a velocity of vs one, they're |
|
| 125:49 | going, you know, straight to to the side, they're going, |
|
| 125:53 | they're going back and forth, going there. And the common point in |
|
| 125:58 | direction is uh uh the love way the we uh in, in, |
|
| 126:12 | order to match the conditions we have uh the, the love wave velocity |
|
| 126:18 | , is in a, it was to be equal to the upper um |
|
| 126:25 | upper shear wave velocity times H one H three. How about in the |
|
| 126:31 | me? It looks like this? decayed, the amplitude decayed the way |
|
| 126:35 | zero um in the lower medium just uh a railway. So uh uh |
|
| 126:44 | algebra for love wave is a lot complicated than the algebra for uh |
|
| 126:51 | right. And of course, um in the real subsurface, it's gonna |
|
| 127:00 | lots of layers rather than uh just layer. But you can see immediately |
|
| 127:05 | this that the that, that we to have nontrivial love wave solutions only |
|
| 127:12 | we had this layer boundary here. , since love waves are not so |
|
| 127:20 | in our data, Israeli ways, gonna stop the love wave discussion at |
|
| 127:23 | point. Uh But I'll give you few quiz questions. So here's the |
|
| 127:28 | one. this one goes to you , it says true or false. |
|
| 127:33 | is likely that one might observe a wave component on the vertical component of |
|
| 127:38 | Geophones on land. Is that true false? That's false, right? |
|
| 127:45 | we define the love wave is polarized . So it's not gonna register on |
|
| 127:50 | ver. So you, you are . So that's all I wanna say |
|
| 127:55 | love wave because love waves are not in our data for that very |
|
| 128:01 | That's the reason right there. But learned something here. We learn something |
|
| 128:06 | uh love ways. By the simple , we learned that love ways uh |
|
| 128:12 | which depend upon uh the frequency. we call that um I would call |
|
| 128:26 | phenomena dispersion because the uh uh high get ahead of the lower frequency or |
|
| 128:33 | versa. Let's uh uh let's think that and uh which, which ones |
|
| 128:39 | fast and which ones are slow? , uh going back to the railway |
|
| 128:44 | discussion, we, we had, , we found that the railway wave |
|
| 128:49 | did not, did not depend on , but we just showed that the |
|
| 128:53 | wave oh Look at that. This a typo. Uh I uh this |
|
| 129:02 | say V sub L it's a typo there. Uh You, you should |
|
| 129:10 | that in your notes and I will that for myself. Uh And, |
|
| 129:17 | , or thoughts, I mean uh during lunch, I'm gonna go back |
|
| 129:21 | correct that. That is on slide . OK. Now, why did |
|
| 129:27 | have this difference between rail waves and waves? Well, in the rail |
|
| 129:31 | discuss rail wave discussion, we had uniform half space because it was |
|
| 129:38 | There was no way to know whether given frequency it was a high frequency |
|
| 129:44 | low frequency. There was no, was no characteristic frequency and no characteristic |
|
| 129:52 | in the rail wave discussion. So didn't know whether any given frequency was |
|
| 129:58 | a high frequency or low frequency. that's why the frequency cancel out of |
|
| 130:04 | uh analysis. But um uh in real world, we have rail waves |
|
| 130:15 | in ne near surface layered formations. these layers provide characteristic links just like |
|
| 130:21 | found for the love wave free love solution. And because uh of those |
|
| 130:27 | links which come from the model, gonna have characteristic frequency. And so |
|
| 130:34 | know whether given the frequency is high or low frequency compared to those characteristic |
|
| 130:41 | which come from the lyric. And call that dispersion. And uh so |
|
| 130:49 | was laying that car this person in low way as you can just uh |
|
| 130:55 | for yourself by going back a few to the solution. Now think about |
|
| 131:02 | whatever, whenever high frequencies propagate with velocity at the low frequencies, what |
|
| 131:10 | means is that the wavel its change as they propagate. Therefore, the |
|
| 131:18 | of the wavelet is not a simple . We have just referred casually to |
|
| 131:23 | velocity of body weights uh uh and velocity of the wavelength following that same |
|
| 131:29 | wave velocity. But now we're realizing since the wavelength is changing shape, |
|
| 131:37 | simple concept of velocity is too Now, in a context where the |
|
| 131:48 | yeah, uh or the velocity is dependent. Let's uh let's consider for |
|
| 132:04 | given plane wave with this phase you see it, this phase factor |
|
| 132:10 | is defined by this difference here. satisfies the wave equation if and only |
|
| 132:17 | we have this relationship between velocity and and wave number K. And we |
|
| 132:24 | we conclude that whether or not the varies with frequency, go back to |
|
| 132:31 | we first derived this. And you'll that at no point here uh at |
|
| 132:37 | at no point, did we assume the velocity changes with frequency or |
|
| 132:44 | So this laneway travels with this But the, the wave, the |
|
| 132:53 | which is a superposition of plane waves this might travel with a different |
|
| 133:05 | we're gonna call this the phase What what, what what we just |
|
| 133:10 | and then a maximum of the phase uh uh occurs where the ver the |
|
| 133:17 | of the phase with respect to frequency zero. So putting here right in |
|
| 133:23 | , the definition of phase carrying through derivative, we find that this equation |
|
| 133:29 | is satisfied where the maximum value of occur. At this point here moves |
|
| 133:38 | the group velocity which is uh given the derivative of omega with respect to |
|
| 133:45 | . The phase velocity is the simple and the group velocity, that's the |
|
| 133:51 | that the wavel moves with depends upon derivative in terms of phase velocity. |
|
| 133:59 | the group velocity in terms of phase ? Well, uh uh let's work |
|
| 134:03 | this with the inverse of the group we call this the group slowness. |
|
| 134:08 | simply the inverse of the formula that just showed. I'm gonna go back |
|
| 134:12 | the, the formula for the, group velocity and the inverse of |
|
| 134:17 | that's this, that's the truth in to uh frequency of K. Let's |
|
| 134:23 | in there for K, let's put the previous expression for uh K in |
|
| 134:27 | of the shades velocity carry through the the uh der the this differentiation using |
|
| 134:38 | rule calculus. And we find that inverse of the group velocity is equal |
|
| 134:43 | the inverse of the phase velocity minus factor which shows how the phase velocity |
|
| 134:50 | with frequency. And we can uh this using tailors expansion. And when |
|
| 134:57 | do that, we convert the minus a plus. So uh uh this |
|
| 135:02 | normally AAA good approximation because normally this dependence is weak. So uh uh |
|
| 135:11 | this uh d derivative is less than , we call that normal dispersion and |
|
| 135:17 | root velocity is slower than the phase . But we can also have other |
|
| 135:22 | where the uh phase velocity varies in positive sense. With frequency, we |
|
| 135:27 | that inverse dispersion in that in that root velocity is faster than the phase |
|
| 135:35 | . So let me just uh uh uh show you an example. Um |
|
| 135:41 | we have two plane waves, one two, each of them traveling to |
|
| 135:48 | right with slightly different frequencies. Can see here at the uh at this |
|
| 135:54 | that the two are lined up. then as time goes on, uh |
|
| 135:58 | one gets a little bit ahead of other. See you can see this |
|
| 136:02 | peak is ahead of this one so uh we can call this time |
|
| 136:08 | And the loc these local peaks travel with the shades velocity. See this |
|
| 136:13 | line goes in the top of the wiggle, top of the wiggle top |
|
| 136:19 | the wiggle. But the envelope here the envelope of uh uh of this |
|
| 136:27 | wav that travels with a good So uh look here, the uh |
|
| 136:32 | green line is going through the uh midpoint between these two and uh uh |
|
| 136:38 | two major peaks. And that's the of the envelope up here. The |
|
| 136:44 | of the envelope is there at uh uh at, at the central |
|
| 136:49 | See how, how this wavel has shape. Here, it's got uh |
|
| 136:55 | uh the central peak is a negative uh uh it's got uh two positive |
|
| 137:01 | uh uh on both sides of of the OK Centroid of the |
|
| 137:09 | Uh But here it's a different shape . It's uh uh the center of |
|
| 137:14 | wavel is is positively. And it's again, it's, it's between |
|
| 137:19 | two peaks. So the envelope travels this group velocity. Here's another example |
|
| 137:27 | this wavelength here, this is uh running uh uh uh in time is |
|
| 137:33 | this, but the wavelength is progressing space like this. Uh at |
|
| 137:38 | the earliest times the wavel looks like and then at later times it progresses |
|
| 137:43 | then progresses from so and watch watch this little uh wiggle right here |
|
| 137:53 | the middle. Call that a phase here, you see it here and |
|
| 137:57 | it's here and here it's here. so those that phase break travels with |
|
| 138:07 | phase velocity according to this line this line here. So that that |
|
| 138:15 | connects all the lines where the P here. But look at the phase |
|
| 138:20 | here is at the beginning of the . Here, it's the middle of |
|
| 138:23 | wavelength here, it's the ladder and it's the end of the wavelength and |
|
| 138:28 | it's gone out the back end. see that's an example of how the |
|
| 138:33 | the uh wavelength is changing shape as goes along. So meanwhile, the |
|
| 138:49 | is traveling with a good velocity. this line goes through the peak of |
|
| 138:54 | envelope of the wave. So that's different velocity, that's a good velocity |
|
| 139:02 | the phase velocity uh uh is uh line here. So in this |
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| 139:09 | the phase velocity is less than the velocity because this slope is less than |
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| 139:15 | slope. Anybody have any questions about . This is very good figure coming |
|
| 139:25 | this. I'm not sure if I if I understanding correctly uh the great |
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| 139:32 | is also the envelope velocity for this , right? Say that again |
|
| 139:37 | It's uh no, I kind of question would be the weight velocity is |
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| 139:45 | same in comparison with the envelope velocity this. So, so in this |
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| 139:52 | , we would say that uh the as we're tracking this wavel arriving on |
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| 139:58 | uh uh on our workstations, we would be tracking this uh uh |
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| 140:04 | velocity here. So that's what we call the wave velocity is the group |
|
| 140:09 | of the wavelength. But uh uh uh because, and in our, |
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| 140:15 | much of our data, our sur of our surface data um uh the |
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| 140:24 | the dispersion is pretty small. So wavelets keep their shape pretty well in |
|
| 140:30 | uh surface seismic data because the frequent frequency dependence of the uh uh oh |
|
| 140:40 | of the wave velocity is very small uh uh uh most cases. So |
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| 140:46 | have in our uh data, we limited bands, right, right. |
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| 140:50 | band, our observer band only goes between five Hertz and uh 100 |
|
| 140:55 | And well, uh if we if we could see uh l lower |
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| 141:02 | or or higher frequency, then the dependence on velocity would show up more |
|
| 141:09 | in our data. Since we have limited data, it's very common that |
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| 141:14 | that band, the wavel keeps its pretty well. And you don't see |
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| 141:20 | much of our data. This kind feature in surfaces data. You do |
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| 141:26 | it in um of the surface waves there the surface waves um uh have |
|
| 141:35 | much stronger frequency dependence because of the of the arguments that I gave before |
|
| 141:42 | the an effect of of layering on wave propagation. So here is some |
|
| 141:50 | data. It's, it's a pretty picture but uh I took this from |
|
| 141:55 | and Gildart and you can see that the largest amplitudes in this data are |
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| 142:01 | coming th this band of amplitudes And that's uh uh railing waves. |
|
| 142:07 | You uh are not really interested in railing waves. What you're really interested |
|
| 142:12 | is the reflections that are are smaller . Now, maybe you can see |
|
| 142:19 | reflection maybe right here is a almost horizontal. So this is maybe |
|
| 142:25 | a hyperbolic move out here, maybe here, you see something but it |
|
| 142:30 | really hard to see anything useful in diagram. So the the regular waves |
|
| 142:38 | are arriving in this wedge of um , of offsets and arrival time. |
|
| 142:44 | that the maximum of the group philosophy is given here and the minimum of |
|
| 142:49 | group velocity is given here. And it looks very messy. It uh |
|
| 142:54 | uh uh uh frequency. See thi this frequency uh the arrivals here |
|
| 143:02 | to be uh one frequency only, arrivals are uh uh appear AAA different |
|
| 143:08 | . But uh both of them, see uh um what we call ringing |
|
| 143:14 | , which is a single frequency arriving that time man of time. |
|
| 143:20 | you see their linear move outs, they, they don't necessarily ex extrapolate |
|
| 143:25 | to the uh back to the But because uh these uh uh |
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| 143:33 | uh these frequencies here are higher shorter, uh shorter wavelength are appearing |
|
| 143:43 | you know, after these. So separated uh uh by frequency inside this |
|
| 143:51 | and we call that a dispersion. . Now, here's something that you |
|
| 143:58 | it is not really obvious, it out which we will discuss in lecture |
|
| 144:05 | . There's an intimate connection between dispersion we're talking about now. And attenuation |
|
| 144:11 | is the first time we've mentioned an . So we're gonna discuss this connection |
|
| 144:17 | lecture night for now for now. don't need to do ah or so |
|
| 144:25 | um Here we have a definition, definition of dispersion. Is that un |
|
| 144:32 | or false? But I'm not I think it's, it's just, |
|
| 144:42 | just a definition. Yeah. And it's a good definition. And |
|
| 144:46 | uh it's not a trick question but I'm glad that you're uh uh |
|
| 144:51 | looking for tricks, but uh didn't in any tricks this time. This |
|
| 144:57 | basically uh what we said for a of this person. So that was |
|
| 145:03 | . OK. So um next question to Grace says uh uh so uh |
|
| 145:12 | is this statement true or false. was dispersion in our discussion of love |
|
| 145:18 | because the layer thickness provided a standard comparison. And given wavelengths depend uh |
|
| 145:25 | differently according to whether a given wavelength short or long compared to the layer |
|
| 145:31 | . Is that a a true statement or not? I think it's |
|
| 145:43 | Yeah, that is true. And a fundamental idea that uh in the |
|
| 145:48 | wave problem that we did, there no way to determine whether a given |
|
| 145:52 | is uh a higher like a high there or low. No way to |
|
| 145:58 | whether given frequency or was high or . So frequency was not part of |
|
| 146:02 | discussion. But when we have then we know that some wavelengths are |
|
| 146:08 | compared to that layer thickness, there's short. So that provides a standard |
|
| 146:14 | comparison answer that uses this dispersion fundamental why we have dispersion in the |
|
| 146:21 | So sorry, professor. The previous was true. Yeah, this one |
|
| 146:28 | true. That's, that's the please. Thanks. So you got |
|
| 146:33 | right answer for this one. coming to le le uh uh is |
|
| 146:39 | true or false? There, there no dispersion in our simple discussion. |
|
| 146:43 | first discussion of. Right. Well, no, uh uh take |
|
| 146:48 | back right, in the first discussion rail waves. Um Well, uh |
|
| 146:55 | me uh uh begin it. Our wave discussion which we did yesterday was |
|
| 147:02 | simple case. No, layers because was no time parameter in the problem |
|
| 147:08 | provide a frequency standard and no length to define a wavelength standard. So |
|
| 147:14 | didn't know whether given frequency was high low. Is that statement true or |
|
| 147:19 | ? Yeah, that one's true. . So uh and that's why we |
|
| 147:23 | no dispersion in our really way of . And it's also true why uh |
|
| 147:28 | the uh uh uh Y Lee is is uh when Utah is, is |
|
| 147:36 | looking at his data, he sees person because he's looking at real data |
|
| 147:41 | layers coming from the layered media. he's got this person in his uh |
|
| 147:46 | his data. So he deals with . That's, that's his thesis |
|
| 147:51 | to deal with it. Next question it comes to you Carlos, now |
|
| 148:00 | that we look at a seismic record which the wavelength does not change shape |
|
| 148:05 | . So, uh I know uh this wave traveling with uh uh uh |
|
| 148:13 | asked a question for the, the velocity is the same as the group |
|
| 148:17 | ? Now, is it traveling with velocity a lot less than the phase |
|
| 148:22 | , a lot greater than the phase but equal to the phase velocity or |
|
| 148:27 | of the above. And this uh this statement sort of applies to a |
|
| 148:32 | of our reflection seismic data, doesn't , the wavelength doesn't change a lot |
|
| 148:38 | it uh goes from one layer to other and I always trying to |
|
| 148:44 | it should be about the same. , about the same. Yeah. |
|
| 148:49 | uh since it's about the same, uh then it doesn't change shape and |
|
| 148:54 | not much dispersion. And um uh , uh uh we do see a |
|
| 149:01 | of uh shape of the wavelet when look at long reflection times. Think |
|
| 149:08 | your mind about long uh uh times your workstation and you always see that |
|
| 149:14 | lower frequencies at those long times and frequencies of the early time. Does |
|
| 149:21 | agree? So that factor comes from . Uh So uh the those |
|
| 149:29 | those uh arrivals coming in a long have a different shape than the wavel |
|
| 149:35 | in at short times because uh uh of attenuation. And that's an example |
|
| 149:45 | what I mentioned before. I Well, no, let me back |
|
| 149:50 | up because of that, of that uh change of shape, you |
|
| 149:56 | it's, it's losing frequencies a long . So there mu so that's due |
|
| 150:01 | attenuation. So there must be a that comes along with that, but |
|
| 150:08 | a weak dispersion. Uh And we see much of it usually in our |
|
| 150:14 | as reflection data, mostly we've got ignore that, but maybe not |
|
| 150:20 | So um that's something to keep in when you're looking at real data. |
|
| 150:26 | . John, from what we have . OK. So um I spent |
|
| 150:38 | lot of time today. Um, , remembering, uh, stories from |
|
| 150:44 | past. So, uh, because we don't wanna run out of |
|
| 150:49 | later in the course. So I'm , uh, talk only very briefly |
|
| 150:55 | two graves. So when you have truth, that is when you have |
|
| 150:59 | borehole, then you've got a surface there that the waves are gonna |
|
| 151:05 | um, uh, noticing. So , the free surface is not the |
|
| 151:09 | surface that, uh, it is to us also uh uh surfaces of |
|
| 151:16 | holes. We, when we do a borehole sonic, we have our |
|
| 151:21 | down there in the bore hole close the uh borehole wall. And so |
|
| 151:29 | waves which are traveling in that mud being affected by the uh the presence |
|
| 151:36 | the wall. So we're gonna have a surface waves traveling on the wall |
|
| 151:42 | the tube. He on the wall the borehole, not just uh not |
|
| 151:49 | at the free surface of the right borehole surface is another important surface |
|
| 151:55 | is gonna be given as surface not just um uh the railing waves |
|
| 152:02 | we've discussed so far, but uh these can be very complicated, but |
|
| 152:06 | going to uh uh mention only uh uh a couple of, of uh |
|
| 152:12 | aspects to them. So that cylindrical also has surface weight, they're seen |
|
| 152:20 | all vertical sizing profiles and all sonic . So these are called tube |
|
| 152:28 | OK. They are distinct from uh primary arrivals which uh we are gonna |
|
| 152:35 | about in lesson seven, which are give us P wave velocities. He |
|
| 152:41 | velocity is gonna be body wave, gonna be AAA certain type of body |
|
| 152:46 | traveling through the uh on the wall the formation outside the borehole wall. |
|
| 152:57 | what the data that we're mostly interested . With our uh with our um |
|
| 153:03 | tools are most, we're mostly gonna these tube waves to be noise. |
|
| 153:09 | they all propagate in one D, go up up hole or they go |
|
| 153:13 | hole or both, you gotta analyze using a cylindrical coordinate system. So |
|
| 153:18 | , there's a great example of where Cartesian coordinate system uh is uh really |
|
| 153:24 | very useful because of the, the shape. Moral. So they're gonna |
|
| 153:31 | uh uh have get to be exponentially smaller away from the borehole |
|
| 153:40 | both towards the inside of the bore and towards the outside. So we |
|
| 153:46 | a AAA tool on the inside. uh uh as the these waves uh |
|
| 153:53 | smaller and smaller and uh uh in away from the Moor Hall wall, |
|
| 153:59 | where they're gonna be encountering our But it's gonna turn out that |
|
| 154:04 | they don't um uh for AAA large of frequencies. Uh They're still noticeable |
|
| 154:13 | our tool because of the uh finite of the moral that provides um uh |
|
| 154:24 | provides a characteristic length and So, that means that some wavelengths behave longer |
|
| 154:30 | some are some be, some are and some are short compared to that |
|
| 154:35 | . So we're gonna have dispersion and gonna reflect from every formation boundary uh |
|
| 154:42 | casing joint or dual diameter change. as these waves are going up and |
|
| 154:47 | , they're reflecting off the formation and outside the, the bar. If |
|
| 154:53 | casing, there's a casing joint, know, uh during this uh logging |
|
| 154:58 | casing wherever we have two joints of together, uh um uh that makes |
|
| 155:06 | we call a casing joint. And two words are gonna reflect off of |
|
| 155:10 | . Uh And our, our tool uh gonna be dangling down there in |
|
| 155:15 | borehole on a cable, you thousands of feet of cable. And |
|
| 155:19 | we get to the tool which is than the cable so that these two |
|
| 155:23 | are gonna be reflecting off of that tool itself. So now the most |
|
| 155:31 | of these is called the stoning Uh uh uh uh first started by |
|
| 155:35 | same guy stone that we talked about . So here is a picture of |
|
| 155:40 | of the borehole. See here is uh the formation here and here is |
|
| 155:44 | mud here and the half of the here and the other half I haven't |
|
| 155:49 | . So this is half of a of a moral, but this is |
|
| 155:54 | cross section to a cylinder. So turn it around sideways and, and |
|
| 155:59 | look at the same thing here. now it looks more much more like |
|
| 156:03 | , a layered rail wave problem. it's gonna be traveling this way, |
|
| 156:07 | polarized in this plant. But remember this is uh out of the plane |
|
| 156:12 | cylindrical, not uh uh not uh . So that's gonna affect the analysis |
|
| 156:21 | a fundamental way. So we're gonna a recording system in this way, |
|
| 156:27 | got Z in this direction, radius this direction and a and auth around |
|
| 156:33 | moor hall in that direction. So this analysis, we're gonna ig ignore |
|
| 156:38 | tool and we're gonna also ignore the length. So we're gonna look for |
|
| 156:42 | plane solutions and that, and that looks like um uh the railing wave |
|
| 156:49 | uh uh a layered really railing wave for in this cylindrical gry. And |
|
| 156:56 | uh uh in the wave equation is laplacian operator and it's uh as this |
|
| 157:01 | for cylindrical symmetry, it's more complicated cylindrical symmetry than for partition symmetry. |
|
| 157:09 | here is the wave equation right here the wave equation, right, right |
|
| 157:13 | in the WAV ation is the laplacian uh so uh waves have a body |
|
| 157:20 | velocity inside the fluid given by V F. And so we're gonna guess |
|
| 157:26 | solutions are like this. Now, don't see you should be wondering what |
|
| 157:34 | these uh functions here? Well, are vessel functions. So that's a |
|
| 157:41 | complicated kind of formula, a complicated of function which you can look up |
|
| 157:49 | any handbook of math of mathematical It's kind of like cosines and |
|
| 157:55 | except it's modified vessel function named after guy vessel who invented them. And |
|
| 158:00 | of that is uh it's very These two points are very complicated and |
|
| 158:07 | we're now running short on time, gonna skip over, I'm just gonna |
|
| 158:12 | them right here and skip over them uh take us to the end of |
|
| 158:18 | 22 way discussion. So you can confident you might find it amusing to |
|
| 158:25 | this in the as homework. But we're now short on time, I'm |
|
| 158:29 | skip over it here and you can confident that things like this are not |
|
| 158:33 | appear on the uh yeah, on final exam, I'm just skipping ahead |
|
| 158:41 | now. Boundary condition discussion of boundary discussion of now, we have love |
|
| 158:47 | , different kinds of waves. This uh uh what uh a waveform might |
|
| 158:52 | like. And I, I'm just pause here and there's the P wave |
|
| 158:57 | right there and the sheer wave arrival there. But you see these two |
|
| 159:05 | , look at these, all these here. What, what have we |
|
| 159:08 | ? We've got um what we call railway and we've got only waves you |
|
| 159:22 | both of those as we, as in passing there is a special range |
|
| 159:28 | frequencies where uh the amplitudes uh where are simple. You see, this |
|
| 159:34 | stone waves from the rail waves that guy that was, that's a person's |
|
| 159:39 | , Harry, just like Raley and . OK. So, uh uh |
|
| 159:47 | all of those complications came from the cases. Uh uh in, in |
|
| 159:53 | real world, we have non cylindrical , we have deviated bore holes. |
|
| 159:58 | in the borehole wall is not perpendicular to the layer. We have stress |
|
| 160:06 | near the borehole. Remember this bore is being squeezed from the side, |
|
| 160:11 | and south, different from east and and surely that's gonna make a |
|
| 160:15 | We're gonna have drilling damage to the wall. We're gonna have what we |
|
| 160:19 | mud cake on the, on on edges of the wall. Now, |
|
| 160:25 | let me just uh stop and, ask uh uh bri to tell us |
|
| 160:29 | mud cake is. It's the residual mud that it's created on the edges |
|
| 160:41 | the, of the way the filtrated , that's very good. So, |
|
| 160:47 | knows about this because she works for . And she knows that in |
|
| 160:51 | in a bore hole, what we is uh uh uh not water circulating |
|
| 160:57 | mud circulating. And furthermore, she that the pressure inside the um uh |
|
| 161:04 | the, the mud of the borehole , is uh arranged to be higher |
|
| 161:10 | , than the native core pressure outside borehole. If it would, |
|
| 161:15 | if the pressure outside the bore hole greater than the pressure in the |
|
| 161:20 | then we would have mud, a fluid coming into the bore hole, |
|
| 161:25 | bore hole out the top of the hole. And that would be very |
|
| 161:29 | for the drillers out there to have uh to have that happening. So |
|
| 161:33 | arrange for the pressure in the be higher than the pressure in the |
|
| 161:40 | outside the ball. How do they that? They put more or less |
|
| 161:45 | uh particles in the mud, you , they, they can arrange for |
|
| 161:49 | mud to have any density than than they, than they want. |
|
| 161:52 | they arrange for it to have enough . So the pressure is higher than |
|
| 161:58 | formation fluid outside the borehole. And what that means is that gore hole |
|
| 162:06 | is continually moving from the borehole into formation because of that pressure difference. |
|
| 162:16 | as it, as it happens, as that uh uh mud flows into |
|
| 162:22 | bore hole, it filters out the the mud particles, the solid |
|
| 162:28 | in the porosity in the near war formation. So that clogs up the |
|
| 162:38 | in uh uh uh uh yeah in near surface uh rocks and then eventually |
|
| 162:47 | perm B goes away. And so you don't lose any more borehole |
|
| 162:52 | . So that is a typical situation the, the uh this formation just |
|
| 162:58 | the borehole has its pores clogged up mud and we call that mud |
|
| 163:05 | So, uh Crusader knows all about because she works for a company. |
|
| 163:16 | Then there's a uh different modes of deformation of the bore hole. Uh |
|
| 163:24 | , we're gonna talk more about that later. Uh uh And then there's |
|
| 163:30 | be anti such. And so all these complications are not included in the |
|
| 163:37 | of two waves, which was already complicated, so complicated that I skipped |
|
| 163:42 | it. OK. So, uh , we did talk a little bit |
|
| 163:49 | two waves. And so let's uh uh see what you can remember from |
|
| 163:55 | . Uh concerning the borehole surface wave waves is uh so we, we |
|
| 164:04 | a bore hole which is um almost . So that naturally introduces uh ABC |
|
| 164:12 | D uh all, all, all the above. So let me start |
|
| 164:16 | uh uh li la. So uh a true, yeah, because those |
|
| 164:22 | because the borehole is gone only in direction, surface waves can only go |
|
| 164:27 | one direction up or down. Uh it's one dimensional. So uh |
|
| 164:34 | uh how about the, the plain uh I in cylindrical coordinates? Uh |
|
| 164:41 | Does that naturally come into this problem of the shape? Yeah, I |
|
| 164:48 | it's so, yeah, if we to do this problem in terms of |
|
| 164:52 | geometry, it would be, you , just impossible to describe even um |
|
| 164:59 | cylindrical surface where we have the boundary . So we gotta have cylindrical coordinates |
|
| 165:06 | the lalos operator which appears in the equation has that cylindrical form. And |
|
| 165:12 | preceda uh is, is, is uh is a solution in the vessel |
|
| 165:17 | . The answer is yes. Uh we have all, all of the |
|
| 165:21 | cancer here. So uh uh now true, true or false, let's |
|
| 165:29 | um uh think about this. And gonna turn to um uh uh Mesa |
|
| 165:36 | I uh since, since she didn't get a chance to answer that last |
|
| 165:41 | , this one goes back to Meader it's a long question. So let's |
|
| 165:45 | it through. So realistic railways on earth's surface spread out in two dimensions |
|
| 165:51 | means uh that the amplitude decreases rapidly offset. So, uh uh we |
|
| 166:06 | in our analysis of railing waves, showed a cross section of the earth |
|
| 166:11 | and Z. But now it's looking asking about the earth's surface and spreading |
|
| 166:17 | out in who as the directions XX Y. And so the energy uh |
|
| 166:23 | it is spreading out in both west and north, south, never |
|
| 166:31 | uh depth, but uh it spreads from, from the source spreads out |
|
| 166:37 | two dimension rapidly. That's a statement uh uh of faction that's generally |
|
| 166:45 | Now, the next statement is applying concept to now, OK. So |
|
| 166:52 | , by contrast, the two waves a bar hall spread out in only |
|
| 166:57 | dimension which is along the bar So that the uh the amplitude decreases |
|
| 167:05 | more slowly with offset uh uh along borehole than the, than the real |
|
| 167:14 | waves. And the surface is that true or false since it's confined to |
|
| 167:21 | dimension instead of two dimensions. Is true or false? I think it's |
|
| 167:29 | complicated question and, and, you , I didn't, I didn't teach |
|
| 167:32 | this. This is a uh a appealing to your common sense. |
|
| 167:40 | Uh If the energy is confined to dimension as opposed to two dimensions, |
|
| 167:45 | do you think Pera is, is gonna decrease uh more slowly or less |
|
| 167:51 | , or, or more slowly Yeah. So yeah. So that |
|
| 167:56 | it's gonna have high AM. So it's uh created at a certain depth |
|
| 168:00 | the Moor Hall, it's gonna stay that moral. And uh so it's |
|
| 168:06 | be energetic and the ball makes lots uh of noises inside the ball because |
|
| 168:13 | a one D problem as opposed to two D problem. So that's |
|
| 168:20 | So this one comes to you, le oh no, we're, we're |
|
| 168:24 | with the quiz. So uh le uh we'll take you up on |
|
| 168:29 | on the next quiz in the next . So uh this is a summary |
|
| 168:33 | what we learned in the surface wave . Surface wave. Like we learned |
|
| 168:41 | uh uh the wave equation does have which travel along the surface of the |
|
| 168:48 | . And there are various types of . That's what we learned. Uh |
|
| 168:51 | boundary conditions mean that these surface waves these waves which are traveling along the |
|
| 168:59 | , they travel with surface wave Um uh more complicated than the body |
|
| 169:06 | velocities. And furthermore, dependent on in a way that the body waves |
|
| 169:12 | not. And we learned that we that how really waves and low waves |
|
| 169:19 | uh observed and uh where and so and why in the balls we have |
|
| 169:25 | waves traveling along the borehole surface which call two wave. So that brings |
|
| 169:35 | to the next topic of reflections and . And this is what makes seismic |
|
| 169:43 | so valuable, all this other what it was be would be amusing |
|
| 169:50 | a physicist. Uh uh but he think that it's uh uh all old |
|
| 169:56 | physics and that's all true. But we get to this kind of, |
|
| 170:01 | uh phenomenon coming out of the wave . And that is what makes are |
|
| 170:08 | interesting. That's what makes it possible use seismic data to find oil and |
|
| 170:15 | . And furthermore, since the um know, since the subsurface is so |
|
| 170:22 | , this is gonna be a complicated . And we like to say in |
|
| 170:26 | that the physicist uh uh stairs on uh uh uh sixth floor of our |
|
| 170:34 | , they um uh stopped studying elasticity it got too complicated for them. |
|
| 170:40 | then they, they turned their attention uh quantum mechanics and relativity. Uh |
|
| 170:46 | ago when uh elasticity got to be complicated for them. But in |
|
| 170:53 | that's the secret to our paychecks. that's what we're gonna do. So |
|
| 170:59 | this, we're gonna stop this lecture we're going to stop sharing that program |
|
| 171:12 | we're gonna start sharing the next lecture I'm gonna put that into presentation mode |
|
| 171:35 | then you can help me here find um find the account. So. |
|
| 172:10 | OK. Mhm mhm There there. that um so first movie OK. |
|
| 172:24 | people see now the entry slide for less than six? OK. So |
|
| 172:32 | this uh lesson, we're gonna learn reflections and refraction. And uh so |
|
| 172:44 | a list of objectives. At the of this lesson, you will be |
|
| 172:49 | to explain what are the boundary conditions the interface between two reflecting horizons. |
|
| 172:56 | furthermore, uh uh you'll uh find these boundary conditions result in simple formula |
|
| 173:05 | normally incident minar P waves. And . So uh uh um OK. |
|
| 173:16 | , I need uh I need a here. OK. So uh this |
|
| 173:20 | be uh uh uh we're gonna consider the, in the first instance, |
|
| 173:25 | waves uh intersecting the boundaries. And of the boundary conditions, we're gonna |
|
| 173:33 | reflections which are simple, have a formula. You probably already know it |
|
| 173:38 | normally it's in B WA and for more complicated formula for obliquely incident B |
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| 173:47 | . So remember that most of our comes from oblique incident. So the |
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| 173:52 | that we have simple expression here, so interesting, much more complicated |
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| 173:59 | In fact, it's too complicated for . But uh we are gonna simplify |
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| 174:05 | complicated equations by making the assumption that uh elastic contrast across this reflect and |
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| 174:13 | this interface right here, those con those elastic contrast are weak and we'll |
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| 174:20 | what we mean by weak. And what that means is we're gonna be |
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| 174:24 | that at the top of a the rocks are pretty similar to above |
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| 174:29 | below that uh the top reservoir And when we do that, uh |
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| 174:36 | gonna simplify these formula and you are already familiar with these simplified expressions, |
|
| 174:44 | you probably have uh some misconceptions, talk about those. Now, these |
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| 174:51 | these equations simplify even more uh when think about the free surface. And |
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| 174:56 | course, that's where our instruments So that's an important thing to think |
|
| 175:00 | . Um uh we, we're gonna uh think about what happens when the |
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| 175:07 | of incidence is large and is when have a large offset between source and |
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| 175:16 | . And then finally, we're gonna out what happens when the infinite wave |
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| 175:20 | curved right up right up here. assume plane waves. But uh but |
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| 175:24 | don't have plane waves do we in uh in the real earth, we |
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| 175:29 | localized, we have waves which are out with curved wavefront, spreading out |
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| 175:35 | a point source. So they're always . OK. And we take that |
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| 175:42 | late in the game. OK. then what happens when the interface is |
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| 175:47 | ? Uh So uh the interfaces are all flat are they and suck a |
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| 175:55 | list of uh things we're gonna take . So, first, let's do |
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| 175:59 | easy, let's get into this uh um one step at a time. |
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| 176:04 | uh we're gonna think about a early elastic interface with an elastic isotropic elastic |
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| 176:12 | above and one below is one OK. Now, so we're gonna |
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| 176:20 | this with P waves. So we uh uh our wave equation looks like |
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| 176:26 | . We have a P wave velocity we have a scalar potential and, |
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| 176:30 | this is not the observable, the comes from uh finding the gradient of |
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| 176:36 | potential. But uh it'll be easy us to think about this in terms |
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| 176:41 | um this scalar wave equation. So the solutions are a, some |
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| 176:49 | plane wave terms each of which looks this. So you're familiar with this |
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| 176:56 | , and the length of this wave right here is given by the thermal |
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| 177:01 | of the components and that's related to frequency via the P wave velocity. |
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| 177:08 | . That's all familiar to you by . So here's a picture of the |
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| 177:13 | that we are looking at, we a isotropic medium above and below. |
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| 177:18 | one is characterized by a velocity and density velocity and a density we have |
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| 177:23 | P wave velocity coming in at this . So uh uh the direction of |
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| 177:28 | is indicated by the black arrow here our um and coordinate system. So |
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| 177:36 | have uh uh obligation in the 13 . OK. So we have uh |
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| 177:45 | the scalar wave equation above scalar wave below. And the only difference here |
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| 177:50 | well, the unknown is different The unknown is phi two here. |
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| 177:54 | unknown is phi one here. The parameter is VP one. And here |
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| 178:00 | material parameter is VPT notice here that specified the density also but the density |
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| 178:06 | appear here. So why did we the density? That's an interesting |
|
| 178:13 | Now, with any task space, expect to find plane wave solution and |
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| 178:18 | gonna have uh coefficients in there that gonna choose to obey the boundary |
|
| 178:24 | So uh because of that interface, is intrinsically a vector problem, so |
|
| 178:31 | write out the uh vector displacements in of the gradients of the um of |
|
| 178:38 | scale of potential. And when we , when you take the, the |
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| 178:42 | of a plane wave, uh what means is that uh uh uh |
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| 178:53 | As a uh after we take the , we get a vector and the |
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| 178:59 | is pointing in the, in the direction as K. It's got the |
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| 179:03 | here, this I comes from here the gradient operation, it's gonna have |
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| 179:07 | constant and we're gonna put all this here and call that uu vector uppercase |
|
| 179:15 | vector. So the lowercase U vector uh the amplitude uh as it changes |
|
| 179:23 | uh time and space, but the uh is a vector which only depends |
|
| 179:29 | frequency. So in terms of vector , this displacement vector has components UV |
|
| 179:38 | W and when I say here, without uh uh an arrow that means |
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| 179:43 | the X component. And so uh , in terms of uh the uppercase |
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| 179:52 | out track, uh we're gonna denote as a vector with three components, |
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| 179:57 | UVNW, all of them get multiplied this uh oscillator factor. Now, |
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| 180:05 | we're in a position to examine the conditions. So they are, |
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| 180:10 | they're not new ideas, they come the wave equation itself. And so |
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| 180:16 | first boundary condition is that the displacement be continuous across the interface at all |
|
| 180:23 | . We do not, we do want to look at solutions where the |
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| 180:28 | wave here's the interface, we have have the displacement of the interface to |
|
| 180:37 | the, the same above and So that's a boundary condition. We |
|
| 180:47 | want to think about ways which tear uh the interface. So, and |
|
| 180:56 | , these, these three components of have to all have to be |
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| 181:02 | Second boundary condition is the stress components have a three in the, among |
|
| 181:08 | indices must be continuous across the Otherwise, the vertebra gradient of stress |
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| 181:17 | the uh uh uh that appears in wave equation that would be infinite. |
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| 181:21 | so the acceleration of the interface would infinite. So we can't have |
|
| 181:27 | Uh And so uh we have this boundary condition. So here it is |
|
| 181:33 | symbols, we have ta 13, it equals Tau 31 in, in |
|
| 181:39 | of um the, in terms of hook, hook's law, in terms |
|
| 181:46 | strain and stiffness that's equal to And you see there's a sum over |
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| 181:52 | Ks and L's here that we're talking only the 13 component, that's the |
|
| 181:57 | 13 that you see here, here have 23 and here we have |
|
| 182:02 | So all of the stresses components which aligned with the boundary that is, |
|
| 182:06 | all have everything that has a three the subs in the, among the |
|
| 182:10 | , they have to be continuous across boundary. OK. Yeah. So |
|
| 182:18 | the first question, I think this comes to late. Uh This is |
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| 182:23 | true or false, says the boundary at the interface between tactic and media |
|
| 182:30 | continuity of stress and strain. Is true? Uh Well, you |
|
| 182:35 | that is a common mistake. So back up. We were talking about |
|
| 182:42 | . So that's that's continuous. But gonna back up one more slide |
|
| 182:50 | Mhm OK. So here we said is the, what we heard before |
|
| 182:57 | we said the displacement is continuous. did not say that the strain was |
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| 183:04 | straight, not displacement, not. displacement is continuous. Why it strain |
|
| 183:16 | the uh is the the uh derivative displacement strain is the root of the |
|
| 183:24 | . So we will see that the is not necessarily uh uh uh but |
|
| 183:29 | the displacement which has got to be . And the, and the strain |
|
| 183:34 | the derivative of this displacement. So what we decided about displacement. Here's |
|
| 183:39 | we decided about stress. And so uh the answer to this is uh |
|
| 183:47 | . OK. A common mistake. . Uh This one comes to you |
|
| 183:57 | uh uh Is this true or All the components of displacement but must |
|
| 184:02 | continuous. Is that true? It be false based on what you just |
|
| 184:08 | if I uh well, no. uh So here we're talking about |
|
| 184:16 | not about stress. Now, if had here. So, so if |
|
| 184:21 | had this continuity of any component of , that would be a tear in |
|
| 184:26 | interface, uh That's not what we . So, so, so uh |
|
| 184:32 | this one is true and I yeah, and now uh uh |
|
| 184:38 | this one is a similar uh uh but is now uh talking about |
|
| 184:44 | Is that one true or false? is false, that's false because it's |
|
| 184:50 | , only the components of stress which uh a three out among their uh |
|
| 184:56 | indices. Uh uh those are uh . OK. So now, um |
|
| 185:07 | uh let's talk about a special case the previous problem, normal incidents. |
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| 185:16 | here's our normal incident P wave and is uh uh the uh displacement here |
|
| 185:25 | only gonna be given by W zero we call this zero because it's the |
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| 185:41 | wave. And so it's gonna have frequency given by omega zero. And |
|
| 185:47 | gonna have an amplitude given by uppercase zero. And then it's gonna have |
|
| 185:52 | oscillator factor. And in this, the oscillation, we have the same |
|
| 185:57 | zero and we have now a K , which is this arrow here that |
|
| 186:02 | can see and that's gonna be related uh omega zero and VP one, |
|
| 186:09 | is the velocity in the, in upper medium by this expression here. |
|
| 186:15 | the bus uh makes it traveling downwards our conventions might wanna think about |
|
| 186:23 | uh uh you know, later. uh but we have a minus here |
|
| 186:27 | that means that the uh uh uh K zero is gonna be a plus |
|
| 186:32 | zero over VP one that's in the media. Now, let's just assume |
|
| 186:40 | that because of this boundary conditions, gonna have both a reflected and a |
|
| 186:44 | wave I think that's pretty obvious. here's the reflected wave. And so |
|
| 186:49 | uh look what we have here. We uh we're gonna call this uh |
|
| 186:53 | W one. So this is W . So this is W one and |
|
| 186:58 | also a function of, of uh position and time and its own frequency |
|
| 187:04 | one which is maybe different from omega and it's gonna have its own um |
|
| 187:12 | uh number, uppercase W one. with our conventions, let's make put |
|
| 187:18 | a minus one call that a minus for the, so we had no |
|
| 187:27 | here. We're gonna put a minus here. And then let's look at |
|
| 187:31 | uh at the oscillator factor. It's omega one t same omega one we |
|
| 187:39 | here. It's got AM minus K . So this K one is the |
|
| 187:46 | not the, not the one it, it before we use the |
|
| 187:51 | uh this the CYB K one to the horizontal component, uh a wave |
|
| 187:56 | uh wave traveling uh horizontally here. wave is traveling vertically with AAA vertical |
|
| 188:04 | only. And K one is related Omega one with this minus sine here |
|
| 188:10 | sine here and a minus sine That means that this wave is going |
|
| 188:16 | . So these conventions are sort of it's easy to uh make mistake uh |
|
| 188:24 | you're uh when you're putting together a like this, you got to uh |
|
| 188:28 | clear in your mind what a plus and what a minus means, |
|
| 188:33 | at, at every position in And, uh, if you, |
|
| 188:37 | you screw up somewhere you're gonna come with a wrong answer. And in |
|
| 188:42 | , uh, uh, for many , uh, back in the |
|
| 188:45 | 19 twenties, uh, people did the wrong answer because of the mistakes |
|
| 188:51 | , uh, by signing, signs here making the right sign by |
|
| 188:56 | . Well, this is just for reflected wave. How about the transmitted |
|
| 189:01 | , transmitted wave is gonna have uh we're gonna name the amplitude and her |
|
| 189:07 | . So we have an incoming W A reflected W no incoming W zero |
|
| 189:14 | W one and transmitting W-2. But gonna have uh its own frequency, |
|
| 189:21 | mega two and it's gonna have its constant uppercase W-2 and it's gonna have |
|
| 189:27 | own K vector going down which is in the same direction, but might |
|
| 189:32 | a different amplitude than this one. K two is different from K |
|
| 189:38 | The K two is related to a frequency omega two with the B |
|
| 189:43 | velocity of uh the lower medium. . So this is uh this is |
|
| 189:51 | we're going to assume for a solution this problem normal. It in, |
|
| 189:56 | P wave reflected and transmitted outgoing And uh uh we have here uh |
|
| 190:06 | determine is W 1 W-2 omega one Omega two. So we don't have |
|
| 190:12 | determine uh uh W zero, we uh make that to be any incoming |
|
| 190:17 | we want and any incoming frequency we . But these other things are gonna |
|
| 190:23 | on that. So let us put gas into our boundary conditions. |
|
| 190:31 | So the, the continuity of displacement that at the interface where X three |
|
| 190:37 | zero, let's back up here. three equals zero 00 oh uh XX |
|
| 190:48 | is the, is uh the depth uh that's this dimension here, X |
|
| 190:54 | . So at this, uh uh where X equals zero. So uh |
|
| 190:59 | the tangential components are all gonna be because uh this is, you |
|
| 191:04 | uh coming in vertically. So there's be no no displacement in the two |
|
| 191:11 | or in the one direction. So we get uh all these are zero |
|
| 191:21 | for the normal um uh the normal at the interface, uh uh We |
|
| 191:29 | say that the, the sum of two terms uh uh in the uh |
|
| 191:35 | medium is got to equal the, single term in the war meeting, |
|
| 191:39 | the clutch back up again. So sum of these two waves and here |
|
| 191:43 | a minus here and that uh that is gonna show up on the next |
|
| 191:47 | . So the sum of those uh terms here is that it be equal |
|
| 191:52 | this one down here at the That's what this says right here. |
|
| 191:58 | like each one of them has its frequency here. But we're only gonna |
|
| 192:04 | able to find solutions to this if these frequencies are the same. So |
|
| 192:10 | off the bat, we simplified our a lot uh by uh recognizing we |
|
| 192:16 | have the same frequency for all these and then we'll just drop the |
|
| 192:23 | So then uh uh uh we can out the exponential term and we get |
|
| 192:28 | equation here coming from this boundary OK. Now, we have uh |
|
| 192:35 | other powder condition is continuity of, stress. Uh And uh and we |
|
| 192:41 | two equations for Tau 13 and for 23. so, uh OK. |
|
| 192:50 | . In, in the upper we have uh uh we have this |
|
| 192:58 | element. Oh By the way, before we talk about that uh uh |
|
| 193:06 | , in either medium and for each , we're gonna have um for each |
|
| 193:11 | we're gonna have different strains. And and so in, and the upper |
|
| 193:17 | , we're gonna have a different stiffness we have in the lower medium. |
|
| 193:23 | uh uh in, in all C 1313 is equal to C |
|
| 193:29 | this is the sum or which uh required by uh uh book. And |
|
| 193:36 | we do that sum and express the in terms of displacements like so, |
|
| 193:41 | the same thing for uh the tw straight, but the tow 33 strain |
|
| 193:48 | gonna be a little bit different. uh after doing all the, uh |
|
| 193:52 | , uh after recognizing that many of uh terms are zero, we're left |
|
| 193:58 | only these three terms. Uh And , uh however, uh some of |
|
| 194:04 | are zero, We're gonna say that variation of the um uh of the |
|
| 194:12 | of the, of the displacement in , in the uh X direction with |
|
| 194:17 | to X that's gonna be zero and displacement in the Y direction with respect |
|
| 194:22 | Y it's also gonna be zero. we're only gonna be left with this |
|
| 194:26 | term. So these three expressions have lead to total stresses which are continuous |
|
| 194:37 | the interface. So the norm, normal stress for the incoming wave is |
|
| 194:42 | is given by this uh uh the zero means it's the incoming wave and |
|
| 194:48 | normal stress is given by Tau in terms of Hook's Law Hooks parameters |
|
| 194:54 | have for the upper medium, it upon M but nm sub one means |
|
| 195:00 | media and the um uh um the is given by the vertical derivative of |
|
| 195:09 | vertical displacement. I'm gonna back up here. A vertical uh displacement of |
|
| 195:16 | derivative, a vertical displacement. That's we have here. And that is |
|
| 195:21 | the incoming way with subscripts uh M the concept and zero for the incoming |
|
| 195:31 | . OK. So uh making this uh uh this is what we have |
|
| 195:37 | the incoming wave and making the derivative brings down uh uh a minus IK |
|
| 195:45 | . It is a minus IK zero comes down when I make this |
|
| 195:52 | And that's here. So uh at to equal zero, this term goes |
|
| 196:00 | . So we're left with this expression the vertical stress coming from the infinite |
|
| 196:08 | at the boundary. And now let's uh change notation express M one in |
|
| 196:18 | of uh density times square velocity. You, you know about that uh |
|
| 196:24 | M one is simply notations for Uh And the K zero is related |
|
| 196:30 | the um omega and the uh vertical by this. So we simplify all |
|
| 196:37 | . Uh and then we get this for Tau 33 for the infinite uh |
|
| 196:43 | wave. Now you can go through similarly, for the reflected wave is |
|
| 196:48 | one and for the transmitter wave is two. And so uh observe this |
|
| 196:57 | growing waves have even um uh indices and two in the upcoming wave has |
|
| 197:05 | index one. So that's a useful . Now, we have this for |
|
| 197:12 | of the three waves though the condition stress continuity means the, the, |
|
| 197:17 | stress coming from uh downgoing wave and reflected wave some together has to be |
|
| 197:23 | same as the uh uh stress of the, of the lower medium |
|
| 197:30 | the lower wave. And you combine two equations that we found. And |
|
| 197:36 | you then uh use a little bit algebra and you find that the ratio |
|
| 197:40 | the reflected ate to the infinite aptitude given by this, it's row two |
|
| 197:48 | two minus row one V one divided the sum. And so we call |
|
| 197:53 | the normal incident uh uh reflection car . And furthermore, and I |
|
| 198:00 | you know this expression how you know , where it comes from. At |
|
| 198:05 | same time, we find uh a for combining the the condition for stress |
|
| 198:12 | and for displacement continuity, we find second solution as the amplitude of the |
|
| 198:20 | wave as a ratio where the infinite is equal to one minus this |
|
| 198:26 | And we can call this the transmission fish. Now, I noticed that |
|
| 198:33 | independent of frequency. Why is It's because if you look at that |
|
| 198:39 | , there is no characteristic length in problem only in upper medium and lower |
|
| 198:45 | . And so uh uh uh uh incoming frequency doesn't know whether it's high |
|
| 198:52 | or low frequency because there is no length in the problem and no characteristic |
|
| 198:58 | time in the problem, no characteristic in the problem. So there's no |
|
| 199:04 | . Now, this product here density velocity. Remember we looked at velocity |
|
| 199:11 | not appear in all the wave equation me, density did not appear in |
|
| 199:17 | wave equation. Velocity appeared, but did not appear where did we get |
|
| 199:23 | ? I'm gonna back up a little density comes because it's hidden inside the |
|
| 199:30 | constant. And remember we expressed the in terms of, of uh uh |
|
| 199:37 | of uh uh D wave velocity. got to bring in the density. |
|
| 199:42 | that's when the density shows up. it showed up here and showed up |
|
| 199:48 | for the reflected wave and showed up for the transmitted wave and that all |
|
| 199:54 | up here. So that's where we the density even though it did not |
|
| 200:00 | in the wave equation itself. Now, we have a name for |
|
| 200:05 | product density times velocity and it's called impetus. So be expression is uh |
|
| 200:14 | uh given by the difference in in um uh above and below divided by |
|
| 200:21 | sum. And so we can express as the jump in impedance divided by |
|
| 200:27 | the average. So normally we have uh that this number is uh |
|
| 200:38 | a lot less than one. The car efficient is at um at most |
|
| 200:45 | boundaries is a lot less than So that the transmission coefficient is close |
|
| 200:51 | one that's AAA normal self. Whenever wave encounters an interface in the |
|
| 200:59 | most of it keeps on going and a small part gets reflected back up |
|
| 201:06 | . So I think this is um very familiar to you. The reflection |
|
| 201:12 | is a jump in ZP divided by the average notice that if the lower |
|
| 201:25 | has a higher impedance in the upper , then this jump right here is |
|
| 201:31 | than zero. So the reflection coefficient greater than zero. And so the |
|
| 201:38 | co efficient is less than one, minus a positive number is uh uh |
|
| 201:46 | than one. That's probably what you in your mind. But think about |
|
| 201:51 | . However, if the contrast in is less than zero, then re |
|
| 201:58 | coefficient is also less than zero and transmission coefficient is greater than one. |
|
| 202:07 | that means is more amplitude goes down out and comes in. And the |
|
| 202:13 | is, how is this positive? is this possible? So have have |
|
| 202:20 | people thought about this? I know familiar with this expression immediately, the |
|
| 202:27 | if you have a situation where the is uh uh the impedance and the |
|
| 202:34 | the lower medium is less than the in the upper medium, that's not |
|
| 202:38 | it says here. This is the where they, where lower is greater |
|
| 202:42 | ever. But you know, in earth, we have uh uh uh |
|
| 202:47 | of layers and it could happen that one of those layers, the lower |
|
| 202:52 | has a great, has a lesser than the upper medium. In that |
|
| 202:58 | , this deviation is uh negative, means that for such a uh an |
|
| 203:06 | , the reflection coefficient is less than and then transmitted co efficient is greater |
|
| 203:12 | one. See, that's what it right here, which means that more |
|
| 203:16 | goes down and out and am coming . How is this possible? So |
|
| 203:22 | me turn to, I think it's turn Carlos, how, how can |
|
| 203:28 | be that we have more altitude going and coming in? But I'm not |
|
| 203:42 | because I may hear you. Uh out loud. Yes. Professor. |
|
| 203:48 | , you don't have to answer. . Uh uh That's a puzzle, |
|
| 203:53 | it? It seemed like it's a . Uh uh So Meader, do |
|
| 203:57 | have an answer to this? I hear you thinking out loud. Um |
|
| 204:08 | , but I was thinking that it it amplifies. Say it again. |
|
| 204:14 | didn't quite, somehow it amplifies pl . So that's what it says. |
|
| 204:20 | amplifies somehow under these conditions. If lower media has a smaller impedance, |
|
| 204:29 | the uh uh the uh it that's what this is amplified and the |
|
| 204:34 | is very straightforward, right? It from this formula which I know you're |
|
| 204:39 | with. But I'm, I'm guessing you hadn't thought about it that when |
|
| 204:45 | ne when the reflection coefficient is the transmission coefficient is greater than |
|
| 204:51 | So like, and, and uh she says, somehow it amplifies, |
|
| 204:56 | very weird. It seemed like it's . But, you know, the |
|
| 205:01 | we went, uh we went through carefully, you're familiar with the |
|
| 205:08 | but there's an implication that you probably think about So could it be that |
|
| 205:15 | this um what says that somehow this in this circumstance, the interface amplified |
|
| 205:24 | incoming signal? Wow, that seemed it's impossible. But let's, let's |
|
| 205:31 | about this more. So the energy in that incoming way is given by |
|
| 205:43 | um it is the change in energy to the presence of the wave it's |
|
| 205:51 | by this sum of, of um uh of uh stress and strain. |
|
| 205:58 | uh uh the stress is shown here Hook's Law so that the strain uh |
|
| 206:03 | is appearing here a quadratic form. here's the system on it. And |
|
| 206:07 | see all these sums over I and and M and N. So |
|
| 206:11 | the left is a scr and on right is also a scalar with sums |
|
| 206:16 | all these things. Now for a traveling E wave, uh This is |
|
| 206:21 | simplified, all these sums go away we're left with only a term like |
|
| 206:26 | for each mode as a term like . So uh uh uh put in |
|
| 206:40 | for uh uh uh the strain is in the square of the strain is |
|
| 206:45 | by the square of DWDX three. this is the vertical displacement, taking |
|
| 206:52 | AAA diff a derivative with the vertical the vertical direction squared. And then |
|
| 207:00 | C 33 is gonna uh come down an M uh uh putting in |
|
| 207:05 | the English name for C 33 and carrying out these derivatives, we have |
|
| 207:12 | square. And uh so uh uh , I uh uh when it gets |
|
| 207:19 | , it's a minus one and it out this minus one. So the |
|
| 207:23 | thing simplifies down to this expression Uh For each mode, it's, |
|
| 207:29 | AAA square of the incoming amplitude times square of the P times the local |
|
| 207:38 | with the one half. So now energy flux for each mode is given |
|
| 207:43 | this, it's the velocity of that times this change of energy. So |
|
| 207:49 | we do is multiply what we found the previous slide by the velocity. |
|
| 207:55 | . So now uh uh we're gonna is the energy flux across this boundary |
|
| 208:00 | algebraically, we have the infinite flux . The reflected flux here with a |
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| 208:06 | sign because it's uh uh uh it's up and the transmitter flux here. |
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| 208:13 | um intuitively, we think that the has to be conserved there. So |
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| 208:21 | the energy that's coming in, it's go out no more, no |
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| 208:28 | But I remember that we found just minute ago, we found that the |
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| 208:36 | increases in the downgoing wave under certain . And that what that causes the |
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| 208:43 | . How is this possible? that turned us to this discussion of |
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| 208:48 | and energy flux and we came to question of uh uh is energy |
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| 208:56 | Um It can served at the boundary terms of the plane wave parameters. |
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| 209:04 | same question is given by this. uh I just uh uh put in |
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| 209:09 | each mode but in this expression, we uh just define and then answer |
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| 209:16 | question, this simplifies to uh uh expression here, here you see ZP |
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| 209:23 | . That's right here. Here you uh uh uh a another uh another |
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| 209:32 | one that's this one here. Oh where, where does this reflection coefficient |
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| 209:36 | from? That's the ratio of this to this one. So we divide |
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| 209:40 | by W one that gives us this coefficient. And we ask ourselves the |
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| 209:45 | is that the same on the left and the right side. Well, |
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| 209:51 | can work out the algebra and you see that the answer is true. |
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| 209:56 | thing is, is this equation throw this equate this question mark. Now |
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| 210:02 | equation is true whether or not the coefficient is positive or negative. So |
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| 210:10 | we can conclude from this is that situation that we had before with amplification |
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| 210:18 | the lower me under the circumstance uh where the impedance was less. That's |
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| 210:29 | a problem because even in that we're gonna have conservation of energy at |
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| 210:35 | , at the reflecting horizon. So amplitude is bigger, but the stiffness |
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| 210:40 | smaller. So the energy is the incoming and outgoing. And so that's |
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| 210:46 | good thing. We wanna have energy at that boundary we don't want for |
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| 210:53 | incoming ways to deposit any energy in interface that would heat up the |
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| 211:01 | And we, that's, we don't that to happen. And sure |
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| 211:06 | this says uh that it does not . Professor uh uh a question regarding |
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| 211:12 | . So does that mean that the in amplitude or the, well, |
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| 211:16 | amplification doesn't have to do anything with energy? Well, it does, |
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| 211:24 | it, it all works out to proper the uh the, the high |
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| 211:29 | in that lower medium that you fly as an amplification. That means high |
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| 211:35 | . But it doesn't necessarily mean high . The energy is different than amplitude |
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| 211:42 | the energy is concerned uh uh the amount of energy uh comes in uh |
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| 211:49 | the incoming wave as goes out with two other wave. So energy is |
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| 211:54 | even though the amplitude seems to be , it is bigger in that |
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| 211:59 | But uh it's not a problem. just the way the equations tell us |
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| 212:05 | amplitude should be. OK. So me see here. I think I |
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| 212:11 | my laser. Yeah, here's my . OK. So let's uh let's |
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| 212:21 | a quiz. Now, uh this I think comes to um uh Carlos |
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| 212:28 | . Whose turn is it Lily OK. Li Lily's turn. |
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| 212:34 | So, is this true or Uh the direction of propagation of a |
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| 212:38 | waving is given by the gradient of phase? Uh uh That part is |
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| 212:47 | . Mhm If the uh if the of the phase is negative, that |
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| 212:54 | the wave advances in the direction of spatial coordinate, it is increasing |
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| 213:00 | So this is a complicated question. so I'm going uh so the answer |
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| 213:04 | this is truth. And so uh want you to go back over the |
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| 213:12 | uh the material and verify number one the direction is given by the gradient |
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| 213:17 | the fade. And number two, that gradient is negative, it means |
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| 213:23 | the wave advances in the direction of space increasing co so that is |
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| 213:29 | And that is, that principle lies the reason why we had minuses in |
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| 213:36 | places in our proposed solution and pluses others. So, uh but I |
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| 213:42 | say that it's, it's uh there's of opportunities here to get the wrong |
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| 213:47 | . So, II, I want you all to, to study this |
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| 213:52 | uh uh making your notes that this is true and then go back and |
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| 213:56 | about it and uh convince yourself that parts of that is true. So |
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| 214:02 | I'm gonna turn to Carlos for the one is of this true or |
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| 214:08 | The amplitude of a continuous plane wave given by its magnitude at any time |
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| 214:16 | space position where the phase is OK. So uh uh to, |
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| 214:22 | answer that le le let's go back , let's go back here and go |
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| 214:27 | back, way back, way way back, way back, way |
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| 214:41 | . OK. So here's a, uh uh for the incoming way, |
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| 214:47 | phase is this part here in uh uh um parentheses. And so the |
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| 214:55 | uh way back there in, in quiz, the question was uh that |
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| 214:59 | place where the phase is zero. imagine this is zero, this is |
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| 215:03 | to the I times zero. And course, that's a one ee to |
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| 215:07 | zero is one. And what that is that uh the, OK, |
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| 215:14 | amplitude of the, of the way that place where phase equals zero, |
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| 215:21 | simply given by this amplitude factor W . So that's the way we have |
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| 215:26 | this up. And so the answer that uh uh uh so we didn't |
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| 215:33 | it in that way. So that's we had to go back to look |
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| 215:36 | the way we had things set up verify that at the place where the |
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| 215:41 | is zero. The amplitude of the is given by this uh uh um |
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| 215:49 | factor in front. So now I'm go forward now all the, all |
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| 215:53 | uh uh skip over all this all this stuff, all this |
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| 216:01 | all this stuff, all this OK. Yeah. So this is |
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| 216:07 | we get. So uh uh so uh this is again a, a |
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| 216:14 | of the thinking that led behind and led to uh the way we defined |
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| 216:19 | those parts. OK. Uh By way, uh I think about |
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| 216:27 | Um Yeah. Well, we did for a day. Uh uh |
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| 216:33 | let me just give you this as assignment. I just convinced you, |
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| 216:39 | hope, convince you that the answer this is true. And I showed |
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| 216:43 | that by looking at our uh definitions the incoming phase. Now, I |
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| 216:49 | for you to apply that same logic the uh uh to the reflected phase |
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| 216:59 | uh to the re reflected way and is the answer still free that uh |
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| 217:04 | that for a homework assign. For uh uh I have a question |
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| 217:10 | , for this one. But even if the phase is not |
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| 217:15 | we will still have a value of amplitude, right? Of course. |
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| 217:20 | , of course. But it, it won't be the uh uh uh |
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| 217:24 | pl uh that will give the magnitude other spacetime positions where the faith is |
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| 217:30 | non zero. But uh uh that be different, that magnitude would be |
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| 217:36 | than the amplitude factor uh which I back there. Yeah. OK. |
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| 217:44 | now this one comes to your bea is this true or false? Uh |
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| 217:49 | polarity of A P wave is measured the same direction as its wave |
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| 217:56 | If the amplitude is positive, if amplitude is positive, the displacement at |
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| 218:05 | zero is parallel to the wave no matter what direction that wave vector |
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| 218:10 | . OK. So, uh uh , so uh I'm, I'm gonna |
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| 218:15 | take you off the hook crusader. uh the homework question that I just |
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| 218:21 | you is uh uh uh is connected the answer to this. So for |
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| 218:28 | , go back and look at the the same uh diagram, the same |
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| 218:34 | of the, of the model. you'll see that uh uh uh there's |
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| 218:41 | minus signs appearing here and there in expression for the reflected wave compared to |
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| 218:47 | incoming wave. And uh so use um um use that information in that |
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| 218:59 | to answer this question. I would it's a hard question. Uh But |
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| 219:04 | gonna need to uh uh to look uh uh how we have that set |
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| 219:10 | . And uh so um uh uh give you, I'll give you a |
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| 219:16 | . The answer to this is also . And I want you to verify |
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| 219:21 | using the uh uh the cartoon that just looked at and with comparing |
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| 219:27 | what that means in terms of um um of the reflected wave and the |
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| 219:34 | wave from incoming wave and reflected And then uh by the way, |
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| 219:39 | go on and apply it also for transmitted wave. So that's your homework |
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| 219:45 | for everybody. The next question. this is the question about your |
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| 219:54 | Mhm So uh is the answer to one A B or C? And |
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| 219:58 | this one comes to Lili. Uh says C uh and so now I'm |
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| 220:05 | , uh so why didn't you do , well, because C has the |
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| 220:10 | sign here. A has the minus here. We can't, uh uh |
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| 220:14 | two are gonna be similar to each . So we can't have a small |
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| 220:18 | in the, the denominator. The number has to be in the |
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| 220:24 | OK. Now, here's the hard . Uh I would for B we |
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| 220:30 | have a minus sign here, but gave the answer to be C instead |
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| 220:36 | B, tell me why you did . Yeah. Yeah. So uh |
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| 220:44 | uh so her memory is correct and hope your memory is correct. When |
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| 220:49 | , when you look at this uh reflection coefficient, it's gotta be uh |
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| 220:53 | reflecting medium minus the incident medium and medium plus, I mean, that |
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| 221:01 | matter here down here. We have me, we have uh an incident |
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| 221:06 | minus reflecting medium. That's wrong because the um uh becau uh this answer |
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| 221:13 | wrong. This is the one that derived and we derived it using all |
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| 221:17 | conventions that we set up in the first statement of the problem and |
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| 221:27 | Uh uh So uh this is the answer and if you don't follow the |
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| 221:33 | , you get into trouble. So why the previous two questions were about |
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| 221:37 | conventions for assigning all those minus signs . But not there in the, |
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| 221:43 | the model. OK. So I say that there's a lot of |
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| 221:51 | um, simplicity in this normal incidence . Our, our trial answer came |
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| 221:57 | to be very simple. That's uh , but there was, it was |
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| 222:02 | because we set it up with minus here and, and not there. |
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| 222:10 | if we had not done it in the right way, we would |
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| 222:13 | would have got a different answer, we did do it the right |
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| 222:17 | So we came up with the right which you know, from other, |
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| 222:21 | or some other work. This is answer. OK. So that was |
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| 222:26 | for the problem of normal incidents. , that turns out to be a |
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| 222:33 | easy problem compared to the problem for uh for oblique incidents. So oblique |
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| 222:42 | , uh uh is most of our as oblique incidents. So let's set |
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| 222:47 | that problem here. We have exactly same setup here. We have an |
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| 222:52 | , uh a wave coming in obliquely let's assume for an out for a |
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| 222:58 | that we're gonna have outgoing waves like . And so then we're gonna have |
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| 223:05 | uh uh we're gonna have uh solve for two displacements, which is uh |
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| 223:11 | , the magnitude of U two, magnitude of U one and these angles |
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| 223:18 | you see here. OK. but that is a statement of the |
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| 223:26 | and um uh um the solution to problem is gonna be a little bit |
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| 223:34 | . So let's stop here and break lunch and I'll see you all |
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| 223:39 | at two o'clock Eastern time and we'll up, uh, uh, of |
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| 223:45 | these quantities given this definition cause rock of the problem. OK. |
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| 223:56 | um, that's, that's good. so this is slide 28 out of |
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| 224:06 | and 20 slides for this lecture. see, we've just scratched the surface |
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| 224:10 | reflection. So we're gonna pause right where we've set up the problems for |
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| 224:15 | incident. You can look at uh and see if we set it |
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| 224:18 | properly. You see, we have um, minus sign here and we |
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| 224:25 | ac uh minus sign here. maybe we should have had a plus |
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| 224:30 | here. Oh, well, maybe gonna make up for it with a |
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| 224:33 | sign here. Uh So, uh think about all that and come back |
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| 224:38 | it at uh two o'clock, uh some time and then we'll work on |
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| 224:42 | till six o'clock, uh, and break for a week. So for |
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| 224:47 | that's where we're gonna stop and I'm stop sharing at this point and I'm |
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| 224:52 | gonna leave you at this point and gonna go get lunch and maybe you |
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| 224:57 | your lunch or maybe you're gonna get anyway. Um, I'll see you |
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| 225:00 | two |
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