| 00:00 | Recording. OK. So backing up backing up, we uh uh we |
|
| 00:23 | with this conclusion because of the previous , we know that uh uh uh |
|
| 00:33 | pre velocity uh for a rock containing gas is the left oops I need |
|
| 00:42 | pointer. OK. The P velocity rock K and KG is less than |
|
| 00:51 | P velocity for rock, same rock containing Bryant. And uh uh of |
|
| 00:57 | , uh the, the text down , it reminds us that uh uh |
|
| 01:02 | , to calculate, to calculate this gas dependence here, uh You have |
|
| 01:07 | know about properties of hydrocarbons under high and temperature and that's all well known |
|
| 01:13 | uh from uh many uh years of work. And for example, uh |
|
| 01:19 | rock properties uh grew up here at University of Houston has been active in |
|
| 01:25 | for many years. They know how tell you uh what are the, |
|
| 01:31 | in compressibility of any mixture of And uh um uh uh also when |
|
| 01:40 | uh partially saturated with Bryant. So that understanding, uh what we want |
|
| 01:45 | do is uh uh point out that um understanding for velocities leads to anomalously |
|
| 01:53 | reflections and to anomalous A O. , uh the anomalous black reflections um |
|
| 02:03 | uh uh uh a, a predecessor a vo when I first came into |
|
| 02:09 | business. Uh The, the only direct detection that we had was what |
|
| 02:14 | call bright spots, which is what now call a VO as expressed in |
|
| 02:20 | stack. And o we were only to trace that down to the angular |
|
| 02:26 | with offset uh shortly after I arrived the business. And because of |
|
| 02:31 | we can, we can have uh 40 seismic surveillance. And uh of |
|
| 02:36 | , that this makes a big difference the economies of our exploiting these |
|
| 02:43 | So uh I want to uh uh uh briefly what we talked about before |
|
| 02:52 | that was before we had this knowledge the effects of fluids. So remember |
|
| 02:57 | talked about for normal set of inter , we have this situation for the |
|
| 03:03 | vo intercept and the gradient. And on laboratory data, we know that |
|
| 03:08 | term uh uh dominates over this And because of that, that uh |
|
| 03:15 | gradient term has an algebraic sign opposite that of uh the instant media. |
|
| 03:21 | that's positive, this is negative and versa. Also for an interface between |
|
| 03:28 | brine and a gas on uh this like that uh uh no changes |
|
| 03:34 | in mythology, just a change from to gas. Maybe we're talking about |
|
| 03:39 | uh the top of uh uh of partially filled reservoir, same mythology above |
|
| 03:47 | below the reflection in this case, you know, that's a special |
|
| 03:51 | an extreme case and any real case be a combination of this case and |
|
| 03:56 | previous case. So uh uh uh at this case only we have the |
|
| 04:02 | expression, very same expressions for intercepts . But now we know that this |
|
| 04:08 | is zero from poor elasticity. That's we learned. Uh we saw this |
|
| 04:13 | back in electron six. Now you're the middle of uh lesson eight. |
|
| 04:17 | now you understand that this comes from essentially in 1941. So in that |
|
| 04:23 | , the algebraic sign of uh this a zero. So this always has |
|
| 04:27 | same algebraic sign as this one. uh that is what we call anomous |
|
| 04:33 | vo. So uh as we said , the real world may be more |
|
| 04:38 | , but there's lots of um um now with Avi L and it all |
|
| 04:44 | depends uh this is the essential idea there's many complications um um that those |
|
| 04:52 | deal with. That's that course not course. So here is a, |
|
| 04:56 | couple of uh of um quiz It says uh let's see. Uh |
|
| 05:06 | to Gasman, the fluid dependence of launch shoulder mous is given by |
|
| 05:11 | Is this true or false? Let turn to um uh Meade for this |
|
| 05:16 | this one true or false. I it is true. Yes, it |
|
| 05:28 | true. But that's not what I you. Remember, I told |
|
| 05:31 | I, I gave you a funnel this with Ks on the left. |
|
| 05:35 | uh yeah, but even, but answer is correct, I taught you |
|
| 05:40 | difference with cases. But you you know that this m is equal |
|
| 05:44 | K plus four thirds mu. So just add in your mind uh plus |
|
| 05:49 | thirds mu and plus four thirds new . And that difference between the frame |
|
| 05:56 | and the uh uh moduli for sheer zero. So that putting in that |
|
| 06:03 | modification makes no difference on the right , it's still true. So, |
|
| 06:12 | good. Uh Let us uh look uh first question. Uh uh Number |
|
| 06:20 | , it says uh the assertion that classic gas monetary needs to be |
|
| 06:25 | that's the assertion that I told you that I published just last year requires |
|
| 06:30 | experimental confirmation. Um uh Lee would you say that's true or |
|
| 06:37 | Yeah, I think it's true. So I'm, I'm the one who |
|
| 06:41 | out the error. Uh But even say maybe that error is important and |
|
| 06:45 | not. Uh Yeah, uh we'll out was it good for experiments |
|
| 06:51 | So, question number three, uh goes to you Carlos using either classic |
|
| 06:59 | line theory or its refinement as discussed morning, the presence of gas in |
|
| 07:04 | pores can, can lead to significant in P wave velocity and impedance. |
|
| 07:10 | that true or false? Well, think it's true. Yeah, I |
|
| 07:17 | call that true. Yeah, that , and furthermore, uh that has |
|
| 07:23 | uh responsible for a lot of uh enormous amount of discoveries of hydrocarbons. |
|
| 07:31 | tell you this story here. Uh uh uh two parts of the |
|
| 07:35 | Uh my father was the one who uh bright spots back in the early |
|
| 07:45 | forties. Imagine that about the time was uh uh born, he was |
|
| 07:52 | bright spots and he was, he a field geophysicist. He was uh |
|
| 07:58 | the crew and the crew would move and uh uh uh the office crew |
|
| 08:03 | follow the field crew and uh he look at these primitive records coming to |
|
| 08:11 | , you know, just wiggles. didn't have any computers, he didn't |
|
| 08:15 | any workstations. Uh They would um the data on photographic film during the |
|
| 08:22 | . They would develop it overnight in morning, he would look at it |
|
| 08:26 | he would say, oh well, we need to uh uh modify the |
|
| 08:33 | plan for uh today's uh um you , acquisition will uh go this way |
|
| 08:40 | of that way. So uh that's kind of conditions he was uh uh |
|
| 08:46 | under uh by current san, you , hopelessly primitive. But he, |
|
| 08:52 | , he noticed that uh whenever he make a recommendation for drilling. If |
|
| 08:58 | recommendation came out of bright reflections instead dim reflections, uh, it was |
|
| 09:04 | likely to be successful. So he fabulously successful as an oil finder. |
|
| 09:10 | found, uh, he, uh, his record for, |
|
| 09:14 | recommendations was that, uh, when recommended a, a well, there |
|
| 09:21 | a 25% chance that, that well gonna be good. So today, |
|
| 09:26 | think 25%? That's terrible. Uh If, if we had that kind |
|
| 09:31 | work to today, we would all fired. And that's true for |
|
| 09:34 | But in his day, that was uh uh you know, very rare |
|
| 09:39 | find a, a geophysicist who could with 25% accuracy where to drill. |
|
| 09:47 | he, he became a famous guy Amal and that's why they hired me |
|
| 09:52 | years later. So he, uh that was his story of developing |
|
| 09:58 | of, of inventing bright spots. The, the next stage of, |
|
| 10:03 | , of that. Uh I, gotta tell you more about that. |
|
| 10:07 | , he uh he made these recommendations of course, he was ignored for |
|
| 10:12 | years by Amaco Management. Uh What said was uh uh uh you |
|
| 10:19 | Mr Thompson, uh uh your job just to uh examine the wiggles. |
|
| 10:24 | there's any important new ideas to come , that'll happen in our research center |
|
| 10:29 | Tulsa. If you'll just please follow recipe, follow the recommended procedures, |
|
| 10:34 | would appreciate that. So he was with that kind of a management brush |
|
| 10:40 | for years and years. Uh later uh discovered uh the same thing. |
|
| 10:47 | uh here's how we learned about Uh It was the early days of |
|
| 10:53 | exploration in the Gulf of Mexico and uh the way it works in |
|
| 10:58 | is uh those uh mineral rights for uh um parcels are owned by the |
|
| 11:06 | government. And uh the, the government uh puts up for auction |
|
| 11:12 | year. A few parcels and oil bid on the rights to explore and |
|
| 11:18 | in those parcels. And people began notice that on certain parcels uh a |
|
| 11:24 | the, the they're put up for and uh there's period for uh for |
|
| 11:29 | uh exploring and you can e either with proprietary equipment or explore with the |
|
| 11:36 | companies. In those days, there a lot more done by all companies |
|
| 11:41 | . Uh And it was two D uh short streamers o over the uh |
|
| 11:47 | in the Gulf of Mexico. And began to notice uncertain um prospects. |
|
| 11:55 | was laying down large bit bets, bits and so they would bid maybe |
|
| 12:00 | million dollars. It was a big when everybody else was bidding uh uh |
|
| 12:05 | 30,000. So uh uh it was clear that the Mo Mobile knew exactly |
|
| 12:13 | they wanted to drill, they knew sort of secret that we didn't |
|
| 12:17 | So eventually that secret leaked out and was bright spots. The predecessor of |
|
| 12:22 | BO in those days, we didn't enough uh spread links to do a |
|
| 12:29 | uh uh measurement of the, of amplitude variation with offset all we we |
|
| 12:35 | at was the, the stack So, uh that's where bright spots |
|
| 12:40 | from. And uh so, uh was due to uh this um my |
|
| 12:49 | here. uh The, the gas theory is uh uh tells us that |
|
| 12:58 | you have gas in the forest that lead to significant reduction in P wave |
|
| 13:02 | and impedes hence brighter reflections. uh so uh I talked over you |
|
| 13:11 | Carl. So let me give this to you Carlos, it says uh |
|
| 13:15 | or false using either plastic gas mount , quartz, modern refinement discussed |
|
| 13:21 | the presence of gas in the phosphates lead to significant reduction in P wave |
|
| 13:26 | S wave velocity who are false. here, it says it specifically is |
|
| 13:35 | the, that's correct. That's the . Yeah, very good. So |
|
| 13:40 | gonna call this one false and, uh uh uh that, that would |
|
| 13:44 | correct. Uh It affects the P , not the S waves. |
|
| 13:49 | remember the S waves have in the density, they have the sheer |
|
| 13:54 | uh divided by uh the density and density will be affected by uh the |
|
| 14:00 | . But um uh this isn't about den density, this isn't about sheer |
|
| 14:06 | , it's about sheer velocity. And that's true. So, uh that's |
|
| 14:13 | good news here. And I'm thinking all thinking, well, what's the |
|
| 14:17 | , what's the big deal? What found out was that uh when we |
|
| 14:20 | from elasticity on homogeneous solids to poor on, in porous rocks, it's |
|
| 14:28 | making a, a minor difference. we learned is that um uh moduli |
|
| 14:35 | the density depend upon uh yeah, over the uh uh the heterogeneity that's |
|
| 14:47 | in Iraq. And in particular, sure dependence is given uh in a |
|
| 14:52 | way by vo and it's uh in and that's refined in 2023 72 years |
|
| 15:02 | . Um uh But they both lead the same conclusions here. With regard |
|
| 15:08 | exploration, we will learn about the . Uh whether or not that's an |
|
| 15:13 | refinement in the next few years. now I want to uh uh return |
|
| 15:20 | the idea of bo slow waves. I told you that Bo taught us |
|
| 15:26 | for porous rocks, we can expect kind of wave. So that's called |
|
| 15:32 | bo slow wave. So we got only P waves and sheer waves, |
|
| 15:36 | we have a mo bo slow And so uh uh that slow wave |
|
| 15:43 | because of the heterogeneity. And the he heterogeneity is we got truths versus |
|
| 15:50 | . Yeah. Right. So, way back in time, uh uh |
|
| 15:54 | uh Max Barn was a famous physicist in, in those days. And |
|
| 15:59 | asked himself the question of what happens you have uh uh a one d |
|
| 16:05 | uh object like this with beads, heavy beads uh separated by springs. |
|
| 16:11 | so I don't know why he would about that. This was uh a |
|
| 16:16 | in, in elasticity. And uh is uh mostly famous for his works |
|
| 16:22 | quantum mechanics. But here's a big and oh a classical physics which he |
|
| 16:30 | uh was interested in. And when found out that when uh as the |
|
| 16:36 | types of call it, he's got types of breeds. And you |
|
| 16:39 | he's got big ones, the heavy ones and lighter ones. And |
|
| 16:44 | the two types of beads move in that makes an ordinary move of |
|
| 16:48 | you just push on this from from the side. And uh all |
|
| 16:53 | beads are gonna be moving uh uh or less in phase. But when |
|
| 16:57 | move out of phase that makes another and it's slower. And he called |
|
| 17:03 | the optical mode because if these uh two beads are moving in opposite |
|
| 17:09 | that's uh and they're both elect electrically , of course, in his |
|
| 17:14 | And so that's gonna radiate uh um uh hello oh ra el electromagnetic |
|
| 17:24 | And so, uh and that's why called it the optical mode of |
|
| 17:29 | It's the same thing in rocks. , uh what B found in 1941 |
|
| 17:36 | when the fluid and solid move in , that's just ordinary, sound like |
|
| 17:40 | been talking about. But when they out of phase, that's the new |
|
| 17:44 | of wave which B invented and realize the same basic physics that Warren found |
|
| 17:51 | 13 years earlier. Now, uh We wanna ask where, where does |
|
| 17:58 | show up in our business? not too much because it's mostly uh |
|
| 18:03 | uh at high frequency. Uh If want to excite these things, you |
|
| 18:08 | to have high frequency, this means sonic band or ultrasonic band, probably |
|
| 18:14 | seismic band. Now, these were first detected uh in, in 1980 |
|
| 18:22 | a, a Schlumberger guy uh uh . Do you know the legend of |
|
| 18:27 | Clona? No, no, I a very good experimentalist working up in |
|
| 18:36 | . Sure. He's now retired and uh he, after all, uh |
|
| 18:41 | imagine um 39 years between the prediction these waves and the uh detection by |
|
| 18:50 | uh it was a long time for Beau to wait frankly, I don't |
|
| 18:55 | if Beau lived long enough to see happen. Now, they travel very |
|
| 19:01 | 100 m per second and with very attenuation uh uh QP not much uh |
|
| 19:08 | uh the, the uh the quality Q is less than one, we'll |
|
| 19:14 | more about the quality factors um in next lecture. So because of this |
|
| 19:20 | uh attenuation, if we, we we were to excite these ways uh |
|
| 19:27 | , uh it would be um oh would be attenuated very quickly because of |
|
| 19:35 | very low quality factor too. Uh we, we maybe need to think |
|
| 19:46 | it anyway. And seismic in the band because you can be sure that |
|
| 19:50 | every sedimentary interface, some ordinary energy converted into this kind of wave happens |
|
| 19:57 | in high frequency than low frequency, maybe some would be converted. And |
|
| 20:01 | constitutes an effective mode of attenuation. , it's gonna affect amplitudes in an |
|
| 20:10 | dependent way, which is not included Standard Avio theory. So we didn't |
|
| 20:16 | this uh uh uh in the core theory that we discussed earlier this |
|
| 20:22 | Uh uh because we assume poor uniform pressure. So the bo wave comes |
|
| 20:28 | the frequencies are high enough, the slow wave comes when you have frequencies |
|
| 20:33 | enough. So that uh the fluid is not uniform, but it's uh |
|
| 20:39 | in uh this part of the pore compared with that kind of a pore |
|
| 20:44 | . Uh We assumed earlier that that happen, we were operating in a |
|
| 20:48 | frequency and, but we cannot, the uh blood pressure does work, |
|
| 20:57 | then you're gonna get real slow waves this velocity, the velocity of those |
|
| 21:03 | is determined in part by the permeability the rock, you gotta be |
|
| 21:08 | How slow depends on the peril of rock. Now, when flow, |
|
| 21:14 | fluid flows locally, that means on grand scale during the passage of the |
|
| 21:20 | , this flow is called fluid It's a bit of a misnomer because |
|
| 21:25 | you think about squirting, you're thinking macroscopic movement of uh fluids uh from |
|
| 21:33 | part to the to the next. uh you know, the amplitudes are |
|
| 21:37 | low. So uh uh it might that we should use a different word |
|
| 21:42 | squirt because the fluid hardly moved at . But uh um when, when |
|
| 21:47 | does move, it has a big . It's responsible for the discrepancy that |
|
| 21:54 | showed earlier between ultrasonic data and gas theory. Remember this slide that I |
|
| 22:00 | earlier. So here is ultrasonic data uh in blue and the theory in |
|
| 22:07 | red showing clear discrepancy. And the for this discrepancy we now understand is |
|
| 22:14 | of fluid squirt at high frequency ultrasonic in the rock. And we uh |
|
| 22:22 | have to worry about these effects and seismic data because our frequencies in seismic |
|
| 22:29 | are so much lower than ultrasonic We uh we still have to worry |
|
| 22:36 | the other thing we talked about this . Now, when fluid flows |
|
| 22:44 | we have not only this stuff which just talked about, but it results |
|
| 22:49 | the attenuation ordinary ways which is lecture , which just happens to be the |
|
| 22:56 | topic. Ok. So let's, , let's move directly into that. |
|
| 23:02 | I will remind you that, after we leave tonight, uh, |
|
| 23:06 | afternoon, late this afternoon, you're go home and you're gonna be thinking |
|
| 23:11 | what we talked about and you're gonna down some questions. And before Friday |
|
| 23:16 | gonna send those questions to me on topic of poor elasticity and also on |
|
| 23:21 | next topic of attenuation. So I going to stop sharing here and I'm |
|
| 23:31 | to um the screen and I'm gonna up the next trial. They gonna |
|
| 24:16 | that in presentation or, and then going to um share that with you |
|
| 24:31 | . OK. Lesson nine A OK. So this is another one |
|
| 24:38 | those topics which is not uh included a standard course and ways and ray |
|
| 24:44 | because uh well, that's classical thinking we need to talk about attenuation because |
|
| 24:51 | see attenuation in our data all the . Have you look at your uh |
|
| 24:58 | workstation? You'll see that the uh coming in at long times have a |
|
| 25:05 | frequency content in the higher uh than than the data coming in at short |
|
| 25:12 | . The reason for that is that those long times uh means long way |
|
| 25:18 | , the uh uh uh the waves lost their high frequency. So now |
|
| 25:25 | gonna talk about that effect and uh what it means for us. So |
|
| 25:33 | see. So by the end of lesson, you'll understand how Books Law |
|
| 25:41 | to be modified to include a generation how this results in a wave equation |
|
| 25:47 | in includes a generation. Of if we're gonna change Hook's Law, |
|
| 25:51 | gonna change the wave equation. We're find plane wave solutions with the |
|
| 25:57 | We're going to uh uh find out that affects reflection. Remember uh uh |
|
| 26:04 | when everything we did with reflectivity um uh I know it generation assumed the |
|
| 26:11 | version of books law. Now here's , an A AAA topic which you |
|
| 26:17 | have um hm not seen coming uh turns out to be related to |
|
| 26:27 | So, dispersion is the fact that depend upon frequency. And now we |
|
| 26:32 | out according to this, that um that is connected to attenuation about |
|
| 26:40 | that's a bit of a surprise. . And then we're gonna talk about |
|
| 26:47 | of a generation. We didn't talk mechanisms of, of elasticity, did |
|
| 26:54 | ? But we're gonna be talking about of a situation. And then we're |
|
| 27:02 | uh uh point out that there's another another issue that we can call a |
|
| 27:08 | attenuation different from real generation, but looks the same if you look at |
|
| 27:16 | uh casually. So that's the program the rest of the, this |
|
| 27:23 | So most of what we are doing uh for the uh first eight lectures |
|
| 27:29 | been classic seismology equally suitable for exploration for investigations of the deep earth. |
|
| 27:38 | now we know that none of it truly suitable for exploration since it ignores |
|
| 27:43 | effects of attenuation. Now we see iteration every day in our data, |
|
| 27:51 | we normally we uh ignore it. we shouldn't ignore it. Maybe they |
|
| 27:56 | teach us something. So that's what says here and we see it every |
|
| 28:01 | . Uh the main way we see is uh because of the loss of |
|
| 28:07 | and also the loss of frequency at recording time. That is all the |
|
| 28:13 | are attenuated, but the high frequencies attenuated more than the low frequency. |
|
| 28:19 | um that's why you progressively uh don't the higher frequencies at longer reflection |
|
| 28:28 | Now, I'm, I wanna point to you that it's a good thing |
|
| 28:31 | uh sounds do die away. Otherwise the sounds ever made on earth would |
|
| 28:36 | be echoing on. Just imagine if didn't have a continuation, then all |
|
| 28:42 | sounds ever made on the surface of earth would be echoing around inside the |
|
| 28:46 | bouncing around. Uh you know, uh uh the stomping on the ground |
|
| 28:51 | the dinosaurs. Uh uh All the, the volcano of everything, |
|
| 28:56 | those noises would still be with us we didn't have attenuation. So uh |
|
| 29:02 | a real good thing that we have . It's also a good thing that |
|
| 29:06 | usually mild attenuation so that um uh can uh observe propagation of waves through |
|
| 29:16 | distances of several kilometers if the a were very high. So the things |
|
| 29:22 | out in 5 m, then we'd out of luck for exploring at least |
|
| 29:27 | the techniques that we typically use So that's what it says. We're |
|
| 29:33 | that the generation is usually weak. , um we're gonna see in |
|
| 29:41 | I think it was the sixth topic the previous list that we always get |
|
| 29:47 | when we have attenuation. But the is also usually weak. And so |
|
| 29:53 | uh we will talk more about that we get to the uh that part |
|
| 29:58 | the election. So first thing we're have to do is we're gonna have |
|
| 30:03 | modify Hook's Law. So this was Law as we uh uh saw it |
|
| 30:11 | we said that uh um with law says that stress is linearly related |
|
| 30:18 | strain. Here's the, the straight and the slope of that line is |
|
| 30:22 | the compliance. And if we did the other way, uh uh the |
|
| 30:27 | of the line would be called the . And let's look at it this |
|
| 30:30 | in terms of compliance. And uh according to hook, the stress |
|
| 30:38 | the strain without delay, you apply stress and immediately without any delay, |
|
| 30:44 | got a strain or vice versa, be the strain causes the stress. |
|
| 30:51 | it be that when you squeeze on rock or on a sponge or on |
|
| 30:55 | balloon. What you're applying is a and what you're feeling, pushing back |
|
| 31:02 | the resultant stress. So, cook not know or care about the answer |
|
| 31:08 | this question, but we should, we should care. We're gonna find |
|
| 31:15 | . Um Well, I, I , you know, already let me |
|
| 31:18 | you the question when you have a of uh seismic energy in the, |
|
| 31:24 | the earth. Uh what happens to energy? Does the energy disappear or |
|
| 31:29 | it get changed to some other So I'm gonna pose that question for |
|
| 31:35 | that. When we attenuate sound in earth, what happens to the energy |
|
| 31:43 | is its uh does the energy disappear does it get changed into some other |
|
| 31:49 | ? It, it, it changes energy doesn't disappear. Yeah, energy |
|
| 31:54 | not disappear. Energy is always And if you think that you've lost |
|
| 31:58 | energy, that means you haven't looked enough and you have to uh um |
|
| 32:05 | out what happened to the energy that think you're lost. And so now |
|
| 32:09 | let me uh um turn to uh le uh if the energy is changing |
|
| 32:16 | some other form, what form is changing into? Say it again? |
|
| 32:42 | . Yes, it's changing into Yeah. So uh uh and so |
|
| 32:49 | soon as she says, thermal, know we're talking about the second law |
|
| 32:53 | thermodynamics and the second law of the says, the, that the entropy |
|
| 32:59 | a closed system always increases. And that means that we uh as |
|
| 33:04 | as the uh when it goes through rock, uh uh uh the energy |
|
| 33:11 | not disappearing, some, most of is propagating, but some of it |
|
| 33:16 | uh increases the entropy in that In other words, it increases the |
|
| 33:24 | . So, um that showed and uh the second law of thermic |
|
| 33:33 | always with us, we never can the second law. It's uh uh |
|
| 33:39 | one of the fundamental laws of And so when Cook is pretending that |
|
| 33:48 | no attenuation here, he's ignoring the law. Uh And he uh he's |
|
| 33:56 | uh so, so that's a dangerous . You don't want to be in |
|
| 34:00 | position of ignoring one of the fundamental of the universe. So this picture |
|
| 34:08 | oversimplified. Of course, when you at real rocks that they behave more |
|
| 34:13 | less like this, when you cyclically uh squeeze a rock, uh think |
|
| 34:21 | uh um any frequency you want in laboratory, imagine squeezing it, un |
|
| 34:26 | it and so on. And when un squeezes, it doesn't come exactly |
|
| 34:31 | the uh uh the same path through strain space, it has uh |
|
| 34:37 | it, there's a little bit of here. This is called a, |
|
| 34:41 | hysteresis loop, a hysteresis loop. so time is going in the direction |
|
| 34:48 | the arrow. So it's going this uh in a diagram like this, |
|
| 34:53 | can't uh you gotta have arrows showing direction of the cycling because um um |
|
| 35:02 | she comes to the wrong conclusion. now here is the point of maximum |
|
| 35:08 | right here. So this is the active here. So this is the |
|
| 35:12 | of maximum stress. But the point maximum strain comes after you go around |
|
| 35:18 | corner and come and coming back. that's later. So what we can |
|
| 35:27 | is the stress always leads the So stress causes strain, not strain |
|
| 35:34 | stress. So I think I posed you the question on the first lecture |
|
| 35:40 | we were talking about hook, the caused strain or strain causes stress and |
|
| 35:45 | forget your answers. But I think was some confusion and I don't blame |
|
| 35:51 | for that because um uh we were uh uh adopting the assumptions of h |
|
| 36:00 | pretty obvious at the time, uh and proportional restraint when you think about |
|
| 36:05 | , that's a violation of the second of thermodynamics. And so uh real |
|
| 36:11 | are more like this. Now, I've shown you here is Annie Lifts |
|
| 36:14 | of course, real walks are not be exactly elliptical like that. But |
|
| 36:19 | was easy for me to draw using um facilities that are given to me |
|
| 36:26 | Mr Bill Gates. OK. we wanna incorporate this into the previous |
|
| 36:35 | that we had developed for hooking So this is what we're gonna |
|
| 36:40 | We're gonna press hook's law just like . This is just like we did |
|
| 36:45 | . But now the stiffness is It's all we have to do and |
|
| 36:52 | . Well, so, uh, , it could be the same with |
|
| 36:55 | . And also, uh, uh, it's amazing to me that |
|
| 37:01 | can recover from the enormous mistake that made earlier. We earlier, we |
|
| 37:09 | one of the basic laws of the . And now turns out that we |
|
| 37:15 | uh uh recover from that with a simple state that when we look at |
|
| 37:22 | Law like this, we got to that um the stiffness coefficients might be |
|
| 37:34 | . So you're gonna see in a how this leads to attenuation. |
|
| 37:39 | So for isotropic rocks, we for example, the in compressibility has |
|
| 37:43 | real part and an imaginary part same the sheer motos, same with the |
|
| 37:49 | modulus real part and an imaginary So now, normally we don't write |
|
| 37:59 | uh the uh you know, the modules in this way. Instead, |
|
| 38:04 | we do is we uh uh factor the real part and that left with |
|
| 38:11 | plus I times this ratio and this , we give the name one over |
|
| 38:17 | and for the P um uh for P wave, it will be a |
|
| 38:22 | of P. And so obviously, , it says uh uh QP is |
|
| 38:30 | one divided by this one, this divided by this. OK. So |
|
| 38:35 | could have defined the inverse of But uh conventionally, we define Q |
|
| 38:41 | the way for you to remember that um uh Q stands for quality. |
|
| 38:48 | the second law says an implication of second law is that Q should |
|
| 38:53 | always be positive. Now, the Q is the less attenuation we have |
|
| 39:01 | the bigger two is means the smaller number is and the uh uh the |
|
| 39:06 | the imaginary part is compared to the part. So um now, in |
|
| 39:13 | of velocity, what we, what we have? So that the P |
|
| 39:17 | velocity squared times the density is exactly to uh the lunch models which we |
|
| 39:24 | learned how to express it in, this way here. And since the |
|
| 39:29 | is real, we can uh uh sort of factor out the density and |
|
| 39:34 | that uh the square of the velocity equal to the real part of the |
|
| 39:41 | of the velocity plus one over Um Because all we did was divide |
|
| 39:48 | density. Now, since the attenuation weak, that is Q is |
|
| 39:54 | then uh we can just take the root of that and that brings in |
|
| 39:58 | factor of, of a half right . And in um the uh |
|
| 40:05 | the solutions for the plane wave, didn't have the velocity to the uh |
|
| 40:10 | the first power we had the, inverse of velocity that's the quantity which |
|
| 40:16 | in uh the phase factor of the wave oscillator factor uh that we showed |
|
| 40:25 | . So we need the inverse of VP. So again, because |
|
| 40:30 | is large, uh we can easily this one from this one simply by |
|
| 40:36 | the sign that's Taylor series arithmetic. we've been doing all this course and |
|
| 40:42 | similar for she. So now uh the difference of course is that this |
|
| 40:51 | is different from the other Q. 22 factors are independent properties of the |
|
| 40:58 | . And normally we have to determine uh from the data and normally they |
|
| 41:05 | uh well, uh OK. I'll it that normally we have to uh |
|
| 41:11 | to uh determine them from the data normally they're in the range of say |
|
| 41:18 | to a couple of 100 four rind , well consolidated, normally pressured |
|
| 41:25 | So if it got, if it's artful gas saturation, it's a different |
|
| 41:29 | . If it's uh a poorly it's a different story. If it's |
|
| 41:34 | over pressure, it's a different But for most of our rocks in |
|
| 41:38 | seismic band, we're gonna be expecting which are a lot bigger than one |
|
| 41:47 | probably bigger than 10 and maybe bigger 30 but less than 500. |
|
| 41:54 | we're gonna be expecting values in this . We will talk later this afternoon |
|
| 42:03 | about this case here and not bri it partially saturated. With gas. |
|
| 42:09 | that's gonna make you substantially lower. that might be interesting point for us |
|
| 42:15 | think about. So. No, consider a sandstone, pure sandstone with |
|
| 42:24 | quartz and um uh Ryan. yeah, for the mineral courts, |
|
| 42:33 | queue is very high. I think in, in intuitively. Uh uh |
|
| 42:40 | think you, you have in mind you have a, um and a |
|
| 42:45 | crystal quartz, uh and you uh it from the side, it's gonna |
|
| 42:52 | for a long time because it's such perfect uh uh material. Uh uh |
|
| 43:00 | uh it's also true that for the , uh if you, if you |
|
| 43:04 | the attenuation of sound um in in a uh in a container of |
|
| 43:13 | this, your brine, it's gonna a large number also 200. But |
|
| 43:17 | , you put these two together and the brine saturated S stuff, you |
|
| 43:22 | get something in between, you get which is lower. So, |
|
| 43:27 | uh the queue for the rock is an average of the queues for its |
|
| 43:32 | . Isn't that interesting? We, uh uh w when we were talking |
|
| 43:38 | um the density, for example, a mineral assemblance, we just found |
|
| 43:43 | was an average of the densities of the inconsistent mines. So uh that |
|
| 43:53 | fails completely when we talk about The reason for that is that the |
|
| 44:03 | factor Q uh uh depends upon a interaction, an interaction between the |
|
| 44:09 | the constituents. Interesting. So, here's AAA question for uh uh for |
|
| 44:20 | and I believe uh Carlos, it's turn. So, uh let's look |
|
| 44:24 | the question and notice down here we none of the above. So, |
|
| 44:31 | oh yeah, I'll, I'll ask uh uh Carlos about part A it |
|
| 44:38 | the theory of elasticity is easily extended a 10 to media by applying it |
|
| 44:44 | rocks. Um Is that true? see. I see. I think |
|
| 44:55 | not, it's not true professor because sort of a non state uh uh |
|
| 45:03 | to rocks. What does that Uh So uh uh uh I like |
|
| 45:07 | answer and I'm turning to Brisa, how about part B the theory of |
|
| 45:14 | is easily extended to a tentative media considering that the elastic businesses are |
|
| 45:21 | Is that tur false? You that is true. Yeah. Uh |
|
| 45:38 | gonna count that as true even though did not yet show you, I |
|
| 45:42 | yet show you how those complex uh is, make it for a |
|
| 45:48 | So that was a tough question because was asking you to uh uh understand |
|
| 45:54 | yourself the implications of what I taught . See, I, I taught |
|
| 45:59 | about the complex distances but I didn't you how that leads to attenuation, |
|
| 46:06 | you figured it out on your own you're understanding, at least you're beginning |
|
| 46:11 | understand you'll understand a lot better um in a few minutes. But that's |
|
| 46:18 | kind of question I like uh a that doesn't uh rely on your memory |
|
| 46:26 | relies on your understanding, your physical and you pass the test good for |
|
| 46:31 | . So turning to Lily, this better be uh uh uh better be |
|
| 46:36 | false because we already found one. true. Uh uh So, uh |
|
| 46:42 | I want you to tell me why false. Says the theory of elasticity |
|
| 46:47 | easily extended to a tenant to medium inside the earth, considering the effects |
|
| 46:52 | high pressure. Yeah. So that's we're expecting that to be not |
|
| 46:58 | But um why? Yeah, nothing said about pressure is implying that you |
|
| 47:10 | uh that makes a situation. So , that, that this is uh |
|
| 47:15 | off the point. OK. So we are going to turn, did |
|
| 47:20 | hear something? Now we're gonna uh uh look at this business here. |
|
| 47:32 | uh Answer B and we're gonna show why Merce it is right there. |
|
| 47:37 | we're gonna do the quasi easic wave . OK. So here's the wave |
|
| 47:44 | as we defined it before. This the vector wave equation uh uh for |
|
| 47:49 | , the uh he wave displacement and got the P wave velocity in |
|
| 47:55 | And if you think about how we this, we never assumed that, |
|
| 48:00 | the P wave velocity is real. can, you can go back to |
|
| 48:05 | uh the second lecture and uh where drive this wave ation at no |
|
| 48:11 | did we ever assume that that's, a real number? So, uh |
|
| 48:17 | uh and now we're just gonna allow to be complex, why not? |
|
| 48:22 | furthermore, look at this, we assumed that it was constant. So |
|
| 48:27 | we're also gonna allow to depend upon . How about that? And you |
|
| 48:34 | verify both of these statements, go to our derivation of the wave |
|
| 48:39 | Uh We never did say whether it's or not, we did never did |
|
| 48:43 | that it's real or not. So now, we're gonna assume that uh |
|
| 48:46 | might be complex with a real part an imaginary part because we didn't assume |
|
| 48:56 | was uh uh real. We can accept that as uh uh a |
|
| 49:02 | still work uh Because of what we before and we don't have to modify |
|
| 49:08 | . So now let's look at some to that. OK. So let's |
|
| 49:13 | a solution as before. But now have a complex velocity which is also |
|
| 49:19 | dependent. So here's our N wave . Yeah, it's got, it's |
|
| 49:24 | uh it's, it's a vector displacement a function of time space and |
|
| 49:30 | And we, it's got an amplitude which is a function of frequency and |
|
| 49:36 | it's got this oscillator factor. And uh you don't see the velocity anywhere |
|
| 49:42 | . What you see is omega T or minus a dot X and a |
|
| 49:48 | uh uh has, can be written this term with three different components of |
|
| 49:54 | uh the uh the wave vector The length of it is given by |
|
| 49:59 | sum of the squares square root of sum of the squares. And that's |
|
| 50:05 | by uh Omega over VP. And we've got VP complex. So uh |
|
| 50:12 | simplicity, let's consider one D propagation positive Z direction. That means that |
|
| 50:19 | can, uh we can drop off , the plus that we only have |
|
| 50:22 | minus here. And uh we got uh uh A minus Z over |
|
| 50:29 | And uh previous slide, we um spelled that out and spell that out |
|
| 50:36 | terms of the real part of Uh And uh uh uh uh and |
|
| 50:46 | over QP, also time is a part of VP. And then um |
|
| 51:03 | separate out uh uh the, the part. So here is um uh |
|
| 51:13 | there's only real numbers in this part the uh of the phase. And |
|
| 51:19 | we still have our imaginary uh uh here. So this part here is |
|
| 51:24 | make for oscillations and this part here have any, I, you see |
|
| 51:31 | we have an I here and I where that and makes minus one. |
|
| 51:36 | there's no I in this part. that's why we separated it out. |
|
| 51:42 | this is frankly only the uh uh well, uh the whole wave is |
|
| 51:48 | in the Z direction. So of , this depends only on Z. |
|
| 51:54 | so this part is gonna lead to , but this part is gonna lead |
|
| 51:59 | attenuation because as Z gets bigger, whole thing gets smaller because of that |
|
| 52:06 | . So, and you can do similar thing for sure. And the |
|
| 52:11 | difference is that you have uh uh some subscripts s here. Now for |
|
| 52:18 | mode, since the velocity is equal the wavelength times the frequency divided by |
|
| 52:25 | pi we can rewrite that in this . So that the wavelength um is |
|
| 52:31 | explicit and I kind of like this now uh this is uh it's um |
|
| 52:39 | I look at something like this, always make sure that we have an |
|
| 52:43 | which has no dimensions to it. here we have uh dimensions of |
|
| 52:48 | And here we uh we also have of length in the denominator Q has |
|
| 52:54 | dimensions I as a no dimension. the whole thing has no dimensions. |
|
| 52:59 | is the attenuation factor only. And what you can see is that uh |
|
| 53:05 | whenever each cycle for each cycle of , uh um it goes one |
|
| 53:16 | So when we, when, when go from uh uh Z to Z |
|
| 53:20 | delta Z sit on, when we from Z to Z plus lambda, |
|
| 53:27 | uh uh that means an, an factor of uh uh of one of |
|
| 53:37 | I'm saying it wrong, sing it . When we propagate over an interval |
|
| 53:43 | Z, the interval is equal to wavelength clamor. Then this uh canceled |
|
| 53:53 | schedule and left with either the IP Q. And so for each such |
|
| 53:59 | , we're gonna reduce the amplitude by factor of uh e to the minus |
|
| 54:05 | over 50 that's about 94%. So a good thing to, to uh |
|
| 54:13 | that as waves are propagating along, are reducing in amplitude by the same |
|
| 54:22 | independent of frequency. See uh when we introduce the wavelength instead of |
|
| 54:27 | frequency, uh uh the frequency is longer explicit. So for every wavelength |
|
| 54:34 | propagation that a wave uh uh it's losing a few percent of its |
|
| 54:42 | um of the tude per cycle. , of course, we have, |
|
| 54:52 | you have higher frequencies that means shorter and you get more cycles per |
|
| 54:57 | So the higher frequencies attenuate more uh a different distance down, for |
|
| 55:03 | from surface down to the target. when we see that the, the |
|
| 55:09 | frequencies are dying out, it's because executing more cycle, not necessarily because |
|
| 55:18 | is less than QP that might be . But the, the main thing |
|
| 55:23 | that um uh shear waves execute Now, uh we specifically said we |
|
| 55:31 | propagating in the distance in the direction positive Z. So if we just |
|
| 55:37 | things out this way we can uh uh propagation in the direction of negative |
|
| 55:42 | . And also we can uh uh in our four year decompositions. We |
|
| 55:47 | include negative frequencies. Uh We put uh uh uh absolute value signs so |
|
| 55:56 | we were getting a AAA decrease in factor because we know that the uh |
|
| 56:05 | two factor are, and the real of velocity are both positive. But |
|
| 56:10 | , it often happens that we're sloppy uh we don't um uh I put |
|
| 56:17 | those um uh we, we don't in those absolute values. So I |
|
| 56:30 | this is uh uh you know, course, how we go about uh |
|
| 56:33 | uh determining velocity and when you determine in those ways that uh you're all |
|
| 56:40 | with, um you're familiar with the that you have been taught to use |
|
| 56:47 | calculating the real part of the So how would we calculate the imaginary |
|
| 56:52 | of the velocity? In other how would we calculate two? |
|
| 56:57 | let's just measure the cube by measuring loss of high frequencies. So, |
|
| 57:03 | we're going to, you know, the same expression that we had before |
|
| 57:08 | this continual fact continuity factor up in , put it inside of a bracket |
|
| 57:15 | call the whole thing uh uh uh amplitude as a function of frequency and |
|
| 57:23 | length of propagation. So this one have any Z in there. This |
|
| 57:28 | does. And then the oscillator factor here. Now we form the ratio |
|
| 57:34 | spectral component we before after and before propagated distant Zs le let me back |
|
| 57:42 | with this uh you take this thing uh so you look at your, |
|
| 57:46 | your data, look at all the uh your work or take a single |
|
| 57:49 | from your workstation and, and look a um um a reflection event and |
|
| 57:57 | a, a window that off and , take a spectrum of that wavelet |
|
| 58:02 | the window surrounding that reflection event. you take another one, you, |
|
| 58:07 | , you look deeper and look at time and find another a a prominent |
|
| 58:13 | further down longer time, make a uh spectrum for that. And then |
|
| 58:22 | uh for a given frequency, then going to uh excuse me, uh |
|
| 58:28 | , the first spectrum that you find this one, the, the, |
|
| 58:33 | second spectrum from a lower reflector is one you form this ratio of these |
|
| 58:39 | spectral components? Excuse me. Professor , I have a question regarding those |
|
| 58:45 | that you said you take the spectral . So does it matter or how |
|
| 58:52 | the, the way that we choose windows affect the result? Like it's |
|
| 58:58 | too a a and so it, never as simple as I just |
|
| 59:04 | you're always gonna say, well, I gonna look at this uh um |
|
| 59:11 | uh is a small uh uh peak between or am I just gonna look |
|
| 59:15 | the big peaks. Uh uh that's matter of judgment. And so you |
|
| 59:18 | it this way and you do it way and see how the answer changes |
|
| 59:23 | respect to your choices of the right? So, uh uh uh |
|
| 59:30 | might be sitting there at your workstation you say to your buddy, I |
|
| 59:33 | , how long should this window And he'll say, oh, it |
|
| 59:36 | be 50 milliseconds. So you try milliseconds. And then because you're a |
|
| 59:42 | person, you also try 60 milliseconds 40 milliseconds and see how things |
|
| 59:50 | Um When you change the arbitrary decision the window L right? You wanna |
|
| 59:59 | an answer that doesn't depend upon the , right? You wanna have an |
|
| 60:04 | that tells you about the medium, about uh uh your uh the way |
|
| 60:10 | handling. So you, you uh it several ways and decide for yourself |
|
| 60:15 | uh for that particular situation. Um uh are the best parameters to |
|
| 60:23 | You folks are always uh adjusting parameters in all of the processing steps. |
|
| 60:30 | furthermore, well, uh let me delay that uh comment. I'll, |
|
| 60:36 | go furthermore, uh in, in couple of seconds, we're gonna |
|
| 60:40 | form this ratio here as a function several different um frequencies. And because |
|
| 60:48 | uh uh this ratio is gonna depend uh uh when you cancel out the |
|
| 60:55 | solitary fact, when you take when look at the spectrum, you're no |
|
| 61:00 | looking at the wavel, you're looking the frequency domain. And you see |
|
| 61:04 | , there's no, um there's no terms here, there's no complex terms |
|
| 61:10 | . Everything here is real. You uh uh you determine the uh the |
|
| 61:17 | of uh depth difference between uh uh reflection and that reflection de determine that |
|
| 61:25 | um uh by uh uh estimating And so, uh by making this |
|
| 61:33 | thing on the right, we're going uh infer what this is. And |
|
| 61:37 | we're gonna infer what is uh um what, what is the cue? |
|
| 61:48 | now if you, if you choose , if you choose two reflectors, |
|
| 61:54 | are too close together, that is you have a small delta Z, |
|
| 61:59 | might get into trouble, uh You'll out for yourself that if you do |
|
| 62:04 | uh for a respected group, uh you do it for um reflection events |
|
| 62:12 | are too close together, you get answers. So you try it, |
|
| 62:17 | uh to say, OK, I forget this uh event, I'm gonna |
|
| 62:20 | down, pick a lower event and you do that, you're gonna lose |
|
| 62:26 | . So, uh uh uh when do this with a large value |
|
| 62:31 | of a delta Z between the upper and the lower reflector, um you're |
|
| 62:39 | uh only determine the average value of in between those two reflectors and the |
|
| 62:46 | the distance, the more heterogeneity is inside that average, the smaller the |
|
| 62:52 | z uh if it gets to be Z is too small, you find |
|
| 62:57 | uh uh nonsense answers. So this is a parameter that you have |
|
| 63:03 | select um uh with your good And um uh so like that, |
|
| 63:10 | having chosen a number of different you know, when you're looking at |
|
| 63:14 | data, you can't choose any delta that you want, you want to |
|
| 63:18 | a delta Z which uh corresponds to different prominent reflectors. And if you |
|
| 63:24 | to, to uh uh do something , you'll find um uh nonsense |
|
| 63:31 | So this is the kind of a that you're normally gonna get as a |
|
| 63:36 | of frequency by whoever drew this cartoon 12345678 frequencies and found that the generally |
|
| 63:48 | ratio is generally decreasing with some scatter the straight line. And uh we |
|
| 63:56 | the best fit straight line through And that, that's the slope of |
|
| 64:00 | uh logarithm of the ratio. And the Q value in by measuring the |
|
| 64:07 | of this line that's measuring how fast losing the high frequencies. Um That's |
|
| 64:18 | uh depend uh that's gonna give you an estimate for the average value of |
|
| 64:23 | inside here. That's an average over interval delta eight and also an average |
|
| 64:31 | the size of band because you're you're calculating the spectra only from here |
|
| 64:38 | here. That's the maximum frequency uh uh that you had available in the |
|
| 64:43 | . This is the minimum frequency. if you were able to look for |
|
| 64:48 | frequencies outside that you might find that uh uh instead of a straight |
|
| 64:56 | you're gonna get a curve. um normally that's not a problem. |
|
| 65:03 | , uh there's lots of scout, it's hard to see curvature in this |
|
| 65:09 | . So you normally, what you're do is settle for a straight line |
|
| 65:13 | and get uh thereby an average value Q over the interval and over the |
|
| 65:20 | of B. So now let me to li li for this quiz |
|
| 65:30 | it says the complex velocity leads to because high squared equals minus one. |
|
| 65:38 | that true? That's false. I the statement is true, but |
|
| 65:42 | it doesn't say anything about complex right? OK. So uh |
|
| 65:50 | it says if you true or high frequencies extenuated more than low frequencies |
|
| 65:56 | Q as the function of frequency is for high frequencies, is that |
|
| 66:04 | I think it's the other way Well, actually, we, we |
|
| 66:10 | say anything about the frequency dependence of . Let's back up here here, |
|
| 66:15 | have a Q which is frequency We have a, a straight line |
|
| 66:20 | . And uh the QE is in formula for that straight line. And |
|
| 66:25 | uh if there is a frequency dependence we can't see it because of the |
|
| 66:32 | . So all we can see is average over this band. And we |
|
| 66:37 | say anything about this uh uh frequency of Q. Maybe it's in there |
|
| 66:41 | maybe not. But I can tell that in most cases, when you're |
|
| 66:45 | at Q, at seismic, at seismic data, normally, you |
|
| 66:51 | do better than to assume um a value for Q independent of frequency. |
|
| 66:58 | band is not wide enough. If had a band that went from five |
|
| 67:03 | to 1000 Hertz, maybe we would some uh frequency dependence of QE. |
|
| 67:10 | normally, we don't have that for the S band, we don't |
|
| 67:14 | it for the sonic band, we have it because in the sonic |
|
| 67:18 | we might have 500 Hertz to 2000 . Again, that's a small uh |
|
| 67:23 | range of frequencies. And so normally can't see a difference in frequency in |
|
| 67:30 | because of the, within the sonic because uh the band is not wide |
|
| 67:37 | . However, it is possible that Sonic cube could be different than the |
|
| 67:43 | cue because those two bands are And furthermore, you can say the |
|
| 67:48 | thing about with the ultrasonic band. normally the ultrasonic band by itself is |
|
| 67:55 | wide enough for you to measure um dependence of Q inside that band. |
|
| 68:03 | when you compare the numbers for the band with the numbers for the Sonic |
|
| 68:08 | and the siding band, you might see differences because, uh, |
|
| 68:13 | uh, now you're looking at cross . Now this is, uh, |
|
| 68:25 | , the next question and, I remember, uh, what Carlos |
|
| 68:29 | is, he said, he thinks the other way around. Uh, |
|
| 68:32 | so this is, uh, the way around. Uh, but I |
|
| 68:35 | the answer is the same that normally don't, uh, well, we |
|
| 68:41 | see uh the sequence depends on And I will be showing you uh |
|
| 68:48 | this afternoon, I will be showing um what we should be expecting for |
|
| 68:56 | frequency dependence of Q. OK. this one, I think I |
|
| 69:03 | I, I just wanted to comment , I am in the slide 22 |
|
| 69:07 | have a comment that says that since frequencies execute more cycles per meter, |
|
| 69:14 | higher frequencies at any way more over given distance. Yeah, that's |
|
| 69:20 | But that's because of the uh wavelength uh when you have um uh uh |
|
| 69:29 | when you have high frequencies, they more wavelengths. So the A U |
|
| 69:34 | giving you the amount of um of energy loss per wavelength in that |
|
| 69:42 | So it's not happen because the wave executed more uh cycles, not because |
|
| 69:50 | key is different. So sorry, back to the, to that question |
|
| 69:56 | I was looking at those slides again I was confused. So here, |
|
| 69:59 | trick in the question is that it that that high frequency sin no |
|
| 70:06 | But it's not because it's depend, the Q, it's not of the |
|
| 70:11 | dependency of the frequency. That's It, it's, it, |
|
| 70:16 | it's because those high frequencies attenuate more execute more cycles. Mhm OK, |
|
| 70:27 | . OK. So uh uh um one goes to mesa, it says |
|
| 70:32 | frequencies that generate more than low frequencies higher frequencies execute more cycles than lower |
|
| 70:39 | . Yeah, that's true. That's we just said. Yeah. |
|
| 70:43 | So, uh uh so now we how complex stiffness is make for attenuation |
|
| 70:52 | as it propagates. Now, let's about the next thing what happens to |
|
| 71:00 | when we consider that the rocks are attenuating rocks. So we already said |
|
| 71:08 | when you have a, a sediment gas in it that's highly attenuated. |
|
| 71:13 | shoes are low for these gasses Uh uh uh uh uh uh uh |
|
| 71:20 | , uh you uh you know, that um gas saturated sediments are |
|
| 71:26 | they have slow velocity. That's the part. And now um uh we're |
|
| 71:31 | about the attenuator part, the, imaginary part of the I won't uh |
|
| 71:41 | stiffness. And now I'm telling you partially saturated rocks have high, |
|
| 71:48 | have low Q as well as low parts of the velocity. And the |
|
| 71:54 | is, well, can we use uh can we use that fact to |
|
| 71:59 | for gas saturated rock. Wouldn't that neat? No. Uh The next |
|
| 72:05 | , usually the gas reservoirs are so that there's not too much loss of |
|
| 72:10 | frequency due to a two wave propagation them. That is if it's only |
|
| 72:16 | m thick. Uh So the heat going down the back is only going |
|
| 72:22 | m. If it's vertical propagation, that's not enough to notice the loss |
|
| 72:27 | high frequencies due to the gas in layer. But should we give up |
|
| 72:39 | this line of inquire or should we thinking about it? The next question |
|
| 72:44 | here said, is there effect of on the reflectivity itself? And we're |
|
| 72:50 | about reflection coefficient, not the two propagation. OK. So let's look |
|
| 72:57 | this here is the normal reflection It's a, a jump in um |
|
| 73:03 | impedance divided by uh uh twice the competence, you can separate it out |
|
| 73:09 | the uh uh density part and the part, the density part is |
|
| 73:14 | So we can leave that alone. the velocity part now has uh um |
|
| 73:20 | real part and an immersion part. uh when we say delta VP, |
|
| 73:25 | means uh uh incident subtracted from uh reflecting. And so here is the |
|
| 73:34 | the VP two minus VP one, spelling it out in terms of uh |
|
| 73:41 | real parts and imaginary parts. And down in the denominator, we have |
|
| 73:46 | same thing with the temp plus instead a minus. So uh we introduce |
|
| 73:52 | notation, uh that means we're we're factoring out of each one of |
|
| 73:57 | terms like this, the real part then the imaginary part is proportional to |
|
| 74:02 | 1/2, 2. And where the comes from, comes from the fact |
|
| 74:08 | this is uh uh the, the as is the square root of the |
|
| 74:14 | . So that brings in a one factor here. And so here we |
|
| 74:18 | for the uh reflecting body here, have for the instant body here, |
|
| 74:22 | have the sum of them. And what we wanna do is collect the |
|
| 74:27 | and imaginary parts. OK. So there's no approximation here. And um |
|
| 74:34 | , let me see what else I'm do here. Um Oh yeah. |
|
| 74:45 | I'm gonna express this part here as the average and this part here as |
|
| 74:52 | the difference of the real parts. , uh before when we were analyzing |
|
| 75:01 | , we didn't have any terms like one, this one, we, |
|
| 75:04 | didn't have any imaginary terms in in the reflectivity coefficient. And so |
|
| 75:10 | delta V over vs that we saw , in that case, uh uh |
|
| 75:15 | we recognize that it was a, change across the reflecting horizon of the |
|
| 75:20 | part of the velocity. And now an imaginary part that we hadn't been |
|
| 75:25 | about before. Yeah. So I'm to make the approximation here that we |
|
| 75:39 | neglect this part here if the cues large and cues are bigger than uh |
|
| 75:44 | 10 or so, we're gonna ignore part, but we'll remember that, |
|
| 75:50 | we ignored it. So, uh we ever come to a case where |
|
| 75:54 | not true, we can go back , and include that part. And |
|
| 76:01 | I wanna do my favorite trick here to use a tailor expansion. We're |
|
| 76:06 | ignore this part. And up we're gonna use a tailor expansion to |
|
| 76:11 | uh separate out the, the linear in uh the cues. So when |
|
| 76:16 | do that, uh we deduce that this approximation that the reflection coefficient is |
|
| 76:25 | we had before with an imaginary which depends upon the jump in cue |
|
| 76:32 | the reflecting horizon. And down here have uh the product of the two |
|
| 76:39 | . OK. So in words, can say that when we look at |
|
| 76:47 | linearized an elastic plan reflection car that's . So uh uh we're gonna do |
|
| 76:55 | a vo problem with linear the plan . We don't want to deal with |
|
| 77:01 | exact not sole equations, we'll deal the linearized expressions. And here we |
|
| 77:08 | did the uh normal incidence term and norm at uh non normal incidence uh |
|
| 77:14 | there's a corresponding um generalization from real complex. I don't want to show |
|
| 77:20 | that here. I just wanna talk this real um uh huh oh The |
|
| 77:31 | this normal ancestor, say it it's got a real part which is |
|
| 77:36 | by uh uh the uh the, real part of the jump pen impedance |
|
| 77:43 | by the real part of the uh impedance. That's pretty much what we |
|
| 77:48 | before. But now we have another on here because now we realize that |
|
| 77:54 | rocks that we're reflecting off of uh are continuing. And so when we |
|
| 78:03 | this complex number in the reflectivity, means that the reflected wave has a |
|
| 78:10 | shape an infinite way. Wow, we had the reflection reflected wavelength looked |
|
| 78:18 | like the incident wavel with a different . And now we see that also |
|
| 78:23 | has a different shape because of this contribution to the normal of reflection of |
|
| 78:34 | . So now how uh how important this? So let's consider a case |
|
| 78:40 | the, the real part is very . So um and I can, |
|
| 78:46 | can, we can ignore this So we're looking at weak reflections here |
|
| 78:52 | we left uh uh uh with uh only this imaginary term. And so |
|
| 78:59 | means that the uh the phase which uh uh that's gonna mean that the |
|
| 79:06 | wavelength off of such an interface is shifted by 90 degrees because the real |
|
| 79:14 | is zero and the imaginary part is zero in this approximation. And um |
|
| 79:21 | uh so now we should ask ourselves OK. So we have a, |
|
| 79:26 | weak reflection, it is phase So if it's, if it's uh |
|
| 79:32 | in at a close to zero it's coming out at close to 90 |
|
| 79:36 | phase shifting. Uh And so how is that? Uh Well, so |
|
| 79:42 | just uh do it. Uh put numbers, let's set the, uh |
|
| 79:48 | uh the cue for the uh for incoming uh um uh I for the |
|
| 79:59 | media, I'm gonna set that set to 50. So that will be |
|
| 80:05 | , for example, a Brian saturated might have a Q number about like |
|
| 80:12 | , but it's the top reflection off a, a reservoir. And |
|
| 80:17 | the reservoir is a gas sand for gas sand. That's a much smaller |
|
| 80:22 | . Let's set it here at five five. OK. So then just |
|
| 80:27 | in those numbers, we find that reflection coefficient is 4.5% imaginary. |
|
| 80:35 | that's not such a small number. is not a small number. We've |
|
| 80:41 | , we've set, we assume that thing is really small at less than |
|
| 80:45 | and a lot less than 4.5% So, we might expect to find |
|
| 80:51 | significant reflection coefficient in this situation phase . Wow. Now, in this |
|
| 81:12 | , the incoming wave reflects off the of the gas sand reservoir. So |
|
| 81:19 | never propagated into the reservoir at The reflected P wave did not go |
|
| 81:24 | the uh uh reservoir at all. it did, it would have lost |
|
| 81:30 | frequency here because the, uh, Q factor in the sand is so |
|
| 81:35 | but it doesn't, uh, go , it reflects off the top. |
|
| 81:39 | has not traveled through the itinerary through reservoir, gas res reservoir, which |
|
| 81:46 | highly a generating. So it has lost any high frequencies. Wow. |
|
| 81:52 | this is a way to detect it's a way to detect gas, |
|
| 82:00 | gas reservoirs or even thick gas Yeah, let's, let's never mind |
|
| 82:06 | thinness. It's a way to detect reservoirs independent of anything else that we |
|
| 82:15 | talked about. We haven't done any vo here only uh normal instance, |
|
| 82:21 | . And we found that if uh uh in this limited case where the |
|
| 82:26 | part is negligible, the imaginary part lead to a significant uh phase shift |
|
| 82:33 | the wavel without the loss of any frequencies at all. So when you're |
|
| 82:42 | at data, I always challenge students remember this fact about reflectivity. The |
|
| 82:51 | is gonna be complex because the, rocks are janitors you can count |
|
| 82:58 | And next, next question is uh that gonna be a big enough effect |
|
| 83:03 | uh uh see it in my And we just showed that it might |
|
| 83:10 | , you could put in different numbers find different answers. But it, |
|
| 83:14 | possible that you could find a reflection efficient uh in this scenario in a |
|
| 83:24 | between two layers which don't have a uh we, we, we still |
|
| 83:34 | a, a AAA strong impedance a strong real impedance change. The |
|
| 83:42 | could come from the contrast in cues from the contrast and impedance. If |
|
| 83:48 | true, the reflected wavelength is gonna back up where the uh an appreciable |
|
| 83:54 | and it's not, it's not um , right as it goes back |
|
| 83:58 | it never went through the highly attenuated . So it's gonna go back up |
|
| 84:04 | the same attenuation that came down. it's gonna get back to the, |
|
| 84:09 | to the receivers. And how are gonna notice it? Well, it's |
|
| 84:12 | have a different shape than the Uh uh uh because uh the, |
|
| 84:19 | , the highly attenuating reservoir that it from the top of put in |
|
| 84:25 | um, a phase shifted reflection car . No. So is this a |
|
| 84:35 | way to find gas? Well, answer is in principle, it is |
|
| 84:39 | maybe not in practice. And in , nobody has ever published uh a |
|
| 84:45 | saying I discovered this gas reservoir because the uh uh effects on, on |
|
| 84:52 | attenuation on reflectivity. And we didn't it through a vo or anything like |
|
| 84:57 | . We discovered it through uh the uh the complex nature of the reflection |
|
| 85:06 | and the effects on way, what nobody has ever published a paper like |
|
| 85:14 | . Nonetheless, I think it might true. Why do why do uh |
|
| 85:19 | why do we have uh this ambiguous today? When what I just said |
|
| 85:32 | pretty clear, we should expect phase wavelengths, phase shifted reflections at normal |
|
| 85:39 | off of gas saturated reservoir. We expect that. So how come it's |
|
| 85:45 | a big deal? Well, here's reason why it's not a big |
|
| 85:48 | It's because other things might cause a effect. For example, there might |
|
| 85:53 | an interference from a nearby reflector if have a nearby reflector with a uh |
|
| 85:58 | uh uh normal instance, real car , a real normal instance, |
|
| 86:05 | car fission of opposite sign that's gonna uh uh uh uh give a reflected |
|
| 86:15 | superposition of those two top of those reflections, which is gonna look like |
|
| 86:20 | phase shifted wavel talk that good. uh That would uh look a lot |
|
| 86:28 | the one that we just uh attributed uh uh uh a situation. |
|
| 86:33 | the way you figure you, the you um uh sort that out is |
|
| 86:38 | look at the uh offset dependence and on like that, uh uh you |
|
| 86:43 | keep this in mind while you're looking real data and see that, that |
|
| 86:47 | you uh uh ever see a, puzzling um oh reflection event, maybe |
|
| 86:55 | due to the complex nature of reflection fish, just keep this in |
|
| 87:02 | You might be the one who finds way to use this to discover |
|
| 87:08 | you would be become, I can you that you would instantly become famous |
|
| 87:13 | you discover this, that this effect reliably u utilize to find uh oil |
|
| 87:21 | gas in the subsurface. It would just as much a revolution as the |
|
| 87:26 | of a bo So um if you to be famous, keep this in |
|
| 87:32 | when you're looking at data, maybe uh maybe you'll be the one who |
|
| 87:36 | it. So, uh I think one goes to Lily uh true or |
|
| 87:48 | . It says, of course, ref reflect the reflectivity must be complex |
|
| 87:53 | the density is complex, that's false we never said the density is |
|
| 87:58 | Only the stiffness is complex good for . OK. Carlos. Uh It |
|
| 88:04 | , of course, the reflectivity must complex since the stiffness is complex. |
|
| 88:09 | that true or false? Hm I think it's false. Now, |
|
| 88:21 | didn't we just say uh uh I you might not have been uh following |
|
| 88:27 | the previous discussion. Uh I'm gonna this is true since the stiffness is |
|
| 88:36 | , we've got to have a reflectivity . Uh uh Car uh I'm gonna |
|
| 88:41 | you to go back, go back the lecture material and you'll see where |
|
| 88:45 | uh we showed this explicitly. Yeah. OK. So Rosa, |
|
| 88:51 | one goes here says true or A large Q contrast at a reflecting |
|
| 88:57 | can produce a phase shifting reflection without loss of high frequencies. Is that |
|
| 89:06 | ? Yeah, that's true. And me that's what's remarkable about this whole |
|
| 89:09 | is we didn't lose any high frequencies we never propagated down through the Ainu |
|
| 89:15 | media. Uh, we, uh course lost some high frequencies, uh |
|
| 89:21 | propagating down through the overbred, but was normal. Uh We didn't propagate |
|
| 89:27 | this top reflection. Um Yeah. we didn't propagate through the highly Agenor |
|
| 89:35 | at all. What we got was phase shifted reflection with the same frequency |
|
| 89:43 | . We, we affected the, phase spectrum, not the power, |
|
| 89:47 | the uh not the power spectrum of the outcome upcoming movie. |
|
| 89:55 | So um they actually come to this which I would say this topic is |
|
| 90:09 | uh an obvious one. So what says here is there's a close con |
|
| 90:18 | link between attenuation and dispersion. So let's look at uh this simple |
|
| 90:26 | , we have an impulsive source at r equal zero and it's gonna radiate |
|
| 90:31 | in all directions, unbounded medium. , no attenuation or dispersion or, |
|
| 90:38 | anything. So when you take this of uh uh of the impulse response |
|
| 90:47 | can do a four year decomposition uh uh of this time function, you |
|
| 90:54 | what that is, that's uh zero negative times infinite for uh zero times |
|
| 91:01 | zero for a positive times. So infinite spike at equals zero, that's |
|
| 91:06 | time function. It's a peculiar one it can be for it, but |
|
| 91:10 | just like any time function, it be really decomposed um um into uh |
|
| 91:18 | uh all these different four year And uh an elementary result of uh |
|
| 91:25 | of that four analysis which you learn the first couple weeks of any |
|
| 91:31 | And uh complex analysis is that this here is flat. It is, |
|
| 91:38 | a one. And how did we that's one or we do the inverse |
|
| 91:42 | a uh transform? We, we an integral instead of integral over |
|
| 91:48 | we have an integral over time, same factor here. But it's got |
|
| 91:52 | minus sign. See that minus there's no minus sign up here. |
|
| 91:57 | then what we're doing is we're uh that delta function that spike that I |
|
| 92:03 | before. And if you uh uh not hard to perform this in integral |
|
| 92:09 | that's a one independent of frequency. that's number is a one right |
|
| 92:15 | So let's put this uh uh analysis uh what we previously called the N |
|
| 92:24 | wave equation. So we found a for that previously. And if you |
|
| 92:30 | go back in your natural finance, solution looks like this. It has |
|
| 92:34 | oscillator part where uh uh uh uh but it uh it's oscillating in a |
|
| 92:42 | direction. So we don't, we need a vector product here. We |
|
| 92:46 | have a K times R here and going uh uh uh upwards. Uh |
|
| 92:51 | got a minus sign here. That uh it's going outwards as towards increasing |
|
| 92:58 | and because it's got a uh a term in there, it's got a |
|
| 93:02 | over R out there in front. uh uh then these constants here to |
|
| 93:07 | it uh uh um uh this one has to be the uh since this |
|
| 93:15 | a pressure pulse, this has got be a pressure uh have, have |
|
| 93:19 | dimensions of a pressure also. So uh make it look simpler by writing |
|
| 93:25 | way. So here's the fourier amplitude the source. And uh if the |
|
| 93:31 | is impulsive, that's gonna be AAA number. Uh OK. So now |
|
| 93:38 | gonna do the inverse transform. So is the uh the uh solution as |
|
| 93:45 | function of frequency. We're gonna find , the uh solution as a function |
|
| 93:49 | time by doing the inverse four you . And so all we do is |
|
| 93:56 | uh integrate. Yeah, uh we're integrating over uh the frequency. |
|
| 94:14 | And here's the oil factor. And you walk through this uh interval, |
|
| 94:21 | find that uh uh this, it doesn't take much work at |
|
| 94:27 | Just sort of um casual inspection of . Lets you see that this integral |
|
| 94:33 | , in fact the uh uh the, the fourier recomposes position, |
|
| 94:39 | put all these frequencies together with one here for the uh for the uh |
|
| 94:46 | the amplitude. But a one right that gives you the uh the a |
|
| 94:50 | function uh uh arriving not at T zero, but at T equals minus |
|
| 94:57 | over V. It's just an expanding . So thi this makes perfect |
|
| 95:05 | we can fire off a shot. And the origin and what you get |
|
| 95:09 | is AAA sphere um uh um um expanding sphere expanding outwards, decreasing it |
|
| 95:19 | amplitude with um uh according to one our, our, that's just the |
|
| 95:24 | spreading factor, just an expanding So that's um uh just working |
|
| 95:31 | the uh the source problem for it then backwards and finding that, that |
|
| 95:36 | get an expanding a shell of sound out from our soil. So that |
|
| 95:45 | for elastic medium. So now, assume that the medium has a ation |
|
| 95:50 | it. Let's assume it's a constant for all frequencies. And we're gonna |
|
| 95:55 | no dispersion whatsoever. So, putting assumptions into the solution, we know |
|
| 96:02 | that solution is here is um uh oscillator part. Uh And now we |
|
| 96:09 | an additional part here coming from the that two is uh is not infinite |
|
| 96:16 | is some sort of large um finite . And you see this part here |
|
| 96:22 | no um I in it got an over here, but we don't have |
|
| 96:27 | I here. And so this part gonna be leading to uh uh uh |
|
| 96:33 | decay as, as our increases. part here is gonna lead to smaller |
|
| 96:39 | smaller amplitudes. Now, we do 48 in uh in inverse transform. |
|
| 96:45 | uh uh uh we have an additional out here coming from the Q. |
|
| 96:50 | is what we did before. we have a new term out |
|
| 96:53 | You work through these intervals and you out that the answer looks like |
|
| 97:00 | And in graphical form, it looks this. We do not get an |
|
| 97:05 | uh uh an infinite impulse, we a finite impulse. Uh uh And |
|
| 97:11 | is the arrival time here of, , of this impulse, but it |
|
| 97:15 | forward in time and backwards in So this shows that the energy begins |
|
| 97:22 | arrive well before the arrival time. this is uh uh the arrival time |
|
| 97:28 | call it is the radius divided by real part of the velocity. |
|
| 97:33 | so see uh uh uh the energy arriving way here early and then most |
|
| 97:42 | it is arriving when you expect, some of it is arriving early. |
|
| 97:46 | does not make sense. Furthermore, symmetrical about the peak, it's zero |
|
| 97:53 | . Whereas real data is always close minimum phase. So these two conclusions |
|
| 98:02 | that analysis do not make physical So what we conclude is that one |
|
| 98:08 | the assumptions that we made to get this point must be invalid. |
|
| 98:14 | we only assume a linear wave equation a uniform medium but in homogeneous |
|
| 98:21 | right, it has a source turn , I mean by N homogeneous |
|
| 98:26 | Uh But it's linear and I don't anybody wants to argue with a linear |
|
| 98:32 | equation. We assume constant Q. that one is problematic. Maybe Q |
|
| 98:38 | not really constant. That's what, what we assume and we assume constant |
|
| 98:45 | . And so maybe that's not In fact, both of the, |
|
| 98:51 | these live are, are physically I think nobody is surprised if I |
|
| 98:57 | you that for real walk Q is be dependent on frequency. And you're |
|
| 99:05 | aware that for real walks the velocity depends upon frequency. So um uh |
|
| 99:14 | that these are implausible, we can back to the lab and uh do |
|
| 99:18 | lot more laboratory experiments and they, we're gonna prove then that both of |
|
| 99:22 | are mathematically impossible. So the conclusion , yeah, because of the second |
|
| 99:29 | , Q must be finite, we're have a generation, it's not gonna |
|
| 99:36 | infinite anywhere. I mean, Q not gonna be infinite anywhere. That |
|
| 99:41 | that a generation is not gonna be anywhere we're gonna have Q. And |
|
| 99:47 | , uh uh uh uh we conclude she gotta be frequency dependent and velocity |
|
| 99:54 | also be frequency dependent. So that uh uh came as a big surprise |
|
| 100:02 | geophysicist. And it was, it about the time that I was coming |
|
| 100:07 | this business and I was new And , so I was not so surprised |
|
| 100:12 | others, but people who had been it a long time said, |
|
| 100:16 | I've never thought about that before. , um, you know, that's |
|
| 100:21 | . So you can s see more this discussion and the textbook by Akan |
|
| 100:27 | , which we talked about before. even though this is, um, |
|
| 100:34 | uh these statements are generally true, also true that when we have a |
|
| 100:39 | bandwidth, it's a convenient uh a convenient approximation to assume that both |
|
| 100:48 | and velocity are constant within our sighted . And we can make that operationally |
|
| 100:57 | assumption. Even though we know that we look at other bands, we |
|
| 101:01 | find other values for Q and other guys from velocity. That's what it |
|
| 101:09 | here. When we consider other we can see it, but we're |
|
| 101:13 | not gonna be able to see any defensive Q or um velocity within the |
|
| 101:21 | B. No, we're gonna shortly up the issue of mechanisms of a |
|
| 101:34 | . It's gonna be a messy But for a wide variety of different |
|
| 101:41 | , this the relationship that we just concluded between attenuation and dispersion can be |
|
| 101:48 | in this way if you have a different frequencies. Um uh yeah, |
|
| 101:57 | uh sentence one and two take the and the ratio differs from one by |
|
| 102:05 | which depends upon one over Q and logarithm of the ratio of those two |
|
| 102:12 | . And so if you want to about um uh angular frequencies, uh |
|
| 102:17 | the same and uh uh uh uh frequencies is the same. You, |
|
| 102:22 | uh uh uh that doesn't make any difference we're talking about uh uh |
|
| 102:30 | cha this is the sort of change can expect to see um for almost |
|
| 102:40 | mechanism. So we're gonna um we're to postpone the issue of mechanism c |
|
| 102:47 | of mechanisms for some uh uh uh bit of time. And we're going |
|
| 102:53 | uh use this expression to understand what expect to find in our data. |
|
| 103:02 | just put in some numbers say, the frequency one at the top of |
|
| 103:05 | band and frequency two at the bottom the band and take two equals |
|
| 103:10 | And so you ask yourself uh uh is the, the ratio of velocities |
|
| 103:16 | the top of the band compared to bottom of the band? Because we |
|
| 103:20 | 50 here in the denominator and we the logarithm of uh 10 here. |
|
| 103:27 | the ratio of the frequencies that comes to be 1.5%. So, uh |
|
| 103:33 | you might not think that uh you uh detect differences in velocity between uh |
|
| 103:41 | top of the band and the bottom the band at the level of |
|
| 103:46 | If it were 10% you might be to see it. But uh |
|
| 103:51 | Probably not. So, so this why we normally don't um uh worry |
|
| 103:58 | , oh uh no, non cons and about uh velocity dispersion when we're |
|
| 104:09 | restricting our attention to a single band data, like the seismic band, |
|
| 104:16 | . Not much you put in your number, you'll find that their small |
|
| 104:21 | . However, if you have partially sediments, same bandwidth put in here |
|
| 104:27 | instead of two equals five instead of equals 50. And that puts a |
|
| 104:32 | here instead of a 50 suddenly we're about 15%. So you might be |
|
| 104:38 | to see that. Now, it be that um uh this partially century |
|
| 104:47 | is so thin that you can't estimate uh velocity such across such a thin |
|
| 104:53 | accurately. That's a different issue. the resolution issue that we talked about |
|
| 104:58 | . Uh But you, you see uh uh this is a powerful |
|
| 105:02 | You can find that you can figure the sort of, of, of |
|
| 105:08 | you can expect to find over a B by this simple formula. But |
|
| 105:16 | let's compare different bands. So here have top up, the F one |
|
| 105:20 | uh in the sonic band 2000 Hertz one is in the middle of the |
|
| 105:24 | band. So we formed this ratio , we're gonna be applying this for |
|
| 105:29 | saturated segments. So we got a down here and now we have 22% |
|
| 105:36 | . So that means that you should to find differences in se velocity compared |
|
| 105:44 | sonic velocity of the order of 20% of dispersion. OK. Now, |
|
| 105:54 | what we said earlier about, Um uh about the difference between uh |
|
| 106:06 | Backus average velocities and ray theory Remember we said that that difference is |
|
| 106:11 | to friendly multiples and that if uh if you have lots of thin |
|
| 106:17 | which we always have that we're expecting uh low velocities because of the friendly |
|
| 106:27 | effect, that's entirely separate from this in the front line multiple effect. |
|
| 106:34 | didn't have any energy being converted to . We only had um arrivals superposing |
|
| 106:43 | itself which were delayed by various delays the various thin beds, which you |
|
| 106:49 | see in the sonic bands data, can't see them in the uh in |
|
| 106:54 | seismic data. But that makes an attenuation which is also uh uh in |
|
| 107:02 | uh uh uh uh it also is to uh decrease the velocities. The |
|
| 107:10 | uh band velocities are gonna be lower the sonic band velocities and it can |
|
| 107:17 | a similar amount. So uh uh expect to find exact matches between um |
|
| 107:26 | sonic and seismic velocities because of these separate effects and some of the other |
|
| 107:33 | that we talked about before. Just remind you the, the sic velocities |
|
| 107:38 | measuring um velocity near to the the C velocities are measuring velocities way |
|
| 107:46 | from the ball hole, you a, a kilometer, two |
|
| 107:49 | 10 kilometers away from the borehole. so uh it might be that there's |
|
| 107:55 | and homogeneity going up. And so effect can also mess up the comparison |
|
| 108:01 | sonic and se velocities. So, and she uh uh things I can |
|
| 108:10 | really complicated uh because of uh of effects which we ignored with our first |
|
| 108:18 | through the theory. Now, this here, this is the difference between |
|
| 108:24 | velocity that at one frequency and a at. Well, another frequency let's |
|
| 108:31 | that the, the two frequencies are together. In that case, uh |
|
| 108:36 | uh these um uh uh in that , this ratio looks like a |
|
| 108:52 | So we can rearrange this formula so we see that the inverse of cube |
|
| 108:57 | proportional to the derivative of velocity with to frequency, see how we do |
|
| 109:04 | . So the um this number is be about one, I'm saying I |
|
| 109:27 | . If we assume these two frequencies similar, then this algorithm it can |
|
| 109:34 | uh we can express uh this in of uh uh uh uh v of |
|
| 109:41 | one equals V of omega two plus V. And this one omega one |
|
| 109:48 | one equals omega two equals omega one delta omega. Um And uh rearrange |
|
| 109:56 | in terms of differentials and this term into this. It's the logarithmic |
|
| 110:04 | The inverse of Q is proportional to logarithmic duties, which means uh the |
|
| 110:10 | dimensionalized derivative here, we have the omega and here we have omega over |
|
| 110:15 | taken together those uh things are called logarithmic derivative of V with respect to |
|
| 110:23 | multiplied by pi. So that's the form of this relationship here. Just |
|
| 110:31 | losing Taylor's air. So let's do little quiz here. And uh uh |
|
| 110:39 | it says here, true or I think uh I think Rocky, |
|
| 110:43 | one is uh uh Carlos. This uh coming to you. It says |
|
| 110:48 | the generation goes along with dispersion, follows that a higher value of Q |
|
| 110:55 | a larger dispersion. And now to that, let's just go back |
|
| 111:01 | Yeah, larger value of two means is smaller. And so this ratio |
|
| 111:07 | gonna be closer to one. So means less to this person. |
|
| 111:13 | OK. So this brings us two . But yeah, can we |
|
| 111:20 | can we back up three slides is I have a question about that when |
|
| 111:25 | mentioned that, you know, another , please. Mhm I think this |
|
| 111:32 | one. Yeah, when you yeah, that one. Yeah, |
|
| 111:37 | said that partially saturated segments. It that the segments are saturated with gas |
|
| 111:44 | this, this number of qeq equals five. It's, yeah, |
|
| 111:50 | that's typical for gas saturator that I I, that's not quite clear is |
|
| 111:56 | so thanks for pointing out? Um uh So here a mechanism of |
|
| 112:05 | . So this is a good place stop. Uh uh For a |
|
| 112:08 | Let's take a quick break here and back in 15 minutes to talk about |
|
| 112:14 | of attenuation. Now you're talking Yes, let's get started again. |
|
| 112:20 | up where we left off with mechanisms attenuation. Um uh You know, |
|
| 112:26 | don't know whether everybody is back but let's assume that everybody is back |
|
| 112:33 | we'll continue with mechanisms of a I think it's really interesting that we |
|
| 112:39 | talked about mechanisms of elasticity. Did uh uh talked about elasticity and, |
|
| 112:45 | how uh you apply a stress and get a strand, all that. |
|
| 112:49 | We never did mention anything like uh tiny particles, uh atomic particles with |
|
| 112:56 | between them. Uh We didn't say anything about the medium except what was |
|
| 113:04 | implicit in Oaks law uh proportional restrain since it's a ainu of we have |
|
| 113:14 | complex coefficients. But in there, no discussion of uh what's going on |
|
| 113:21 | the microscopic level. That's kind of . But now we have to say |
|
| 113:27 | it is that is causing this No Hook's law assumed perfect elasticity. |
|
| 113:38 | we want to do better. So gonna write down here below a more |
|
| 113:44 | consti more general constitutive equation, all equations for the standard linear solid. |
|
| 113:53 | you can see that it's got in derivative terms and it's got uh |
|
| 113:58 | a derivative of the stress with respect time and a derivative of the strain |
|
| 114:04 | the spec to time. And uh we ignore those terms, we get |
|
| 114:11 | Law. But this is a generalization H's law. Uh When we say |
|
| 114:16 | , a constitutive equation means we're talking equations which uh which uh describe the |
|
| 114:25 | of the mature. And so this one, this is more general than |
|
| 114:30 | uh that hook uh uh would have hook would have looked at this and |
|
| 114:36 | , I don't want those terms in , get rid of those terms. |
|
| 114:40 | in the 20th century people realized that apply the ideas of elasticity and poor |
|
| 114:49 | to more complicated systems, we have have a more general law. And |
|
| 114:55 | uh um this is uh uh uh express in which is, is sufficiently |
|
| 115:08 | to include and to include many different uh mechanisms. And if you look |
|
| 115:18 | here, we have a derivative of with respect of stress with respect of |
|
| 115:23 | and also some kind of a parameter . And so that uh is a |
|
| 115:29 | , it's gonna have the dimensions of , right? Because uh uh uh |
|
| 115:34 | term here has to have the same as this term here, gotta have |
|
| 115:39 | dimensions of stress. And so then gonna put in here uh uh the |
|
| 115:44 | sort of thing on the strain we're gonna have a characteristic relax relaxation |
|
| 115:49 | for strain at hooks law is just special case. Now, what does |
|
| 115:56 | look like in um uh in, graphical form? If we are |
|
| 116:05 | Uh Yeah, if we draw the , you'll, you'll recognize this as |
|
| 116:13 | modulus, the square of the uh velocity. And this has is uh |
|
| 116:19 | uh as a function of frequency down . And so this is uh um |
|
| 116:27 | the phase velocity in the relaxed So I'll tell you what that means |
|
| 116:33 | a second here is the un relaxed . And you see the curve shows |
|
| 116:37 | its functional frequency starts low and it goes through an increase and ends |
|
| 116:43 | high and flat. And uh before look at the other curve, let's |
|
| 116:49 | again, uh let's look more closely the uh at the frequency uh scale |
|
| 116:56 | here. This is non dimensional frequency the actual frequency is scaled by the |
|
| 117:03 | root of these two times. So see these are the same two times |
|
| 117:07 | define on the previous slide. And uh this uh square root has the |
|
| 117:13 | has the physical dimensions of time. so uh that time uh uh multiplied |
|
| 117:21 | omega means this is a non dimensional of frequency here. And so when |
|
| 117:26 | frequency is one, that's where the is increasing and not only at one |
|
| 117:33 | say between uh uh 0.1 and Uh uh that's where the uh |
|
| 117:39 | the velocity begins to uh deviate from the limiting ace here and heads up |
|
| 117:48 | ramp and then ends up flattening out 10 here. And as the same |
|
| 117:54 | the uh we call that upper upper uh condition, the un relaxed |
|
| 118:04 | . And we'll explain later what that . Uh uh Let's look at this |
|
| 118:09 | curve here. That is the attenuation a function of frequency and that peaks |
|
| 118:17 | uh non dimensional frequency equals one. it's uh it's a definitely a function |
|
| 118:24 | frequency, right? We said we, we wouldn't be surprised if |
|
| 118:29 | frequency, if the Q dependent on uh here is uh here's the case |
|
| 118:36 | that's explicit frequency dependent Q. And is this uh uh um maximum |
|
| 118:43 | of inverse QE maximum attenuation minimum of Q is given by this expression |
|
| 118:49 | where these TS are the same as uh uh T you saw. Uh |
|
| 118:58 | . And um so now let's uh say that uh uh put on here |
|
| 119:04 | geophysical numbers. We, we generally that the seismic band is down here |
|
| 119:08 | the relaxed regime. The ultrasonic band up here in the unreason band is |
|
| 119:16 | things are changing. And so this here is a good characterization I think |
|
| 119:26 | sedimentary rocks, if you were uh a geophysicist, you might choose some |
|
| 119:33 | , a way to characterize this. the uh uh the extenuation reaches the |
|
| 119:43 | , it's max in the middle of band where the, where the velocity |
|
| 119:48 | changing. So now let's talk about what we mean by relaxed and |
|
| 119:54 | So a good way to think about is um when we're talking about |
|
| 120:00 | the principal mechanism for a generation of wave is that uh the movement of |
|
| 120:09 | at the grain scale as the P is going through. So as the |
|
| 120:14 | wave is going through, consider any uh any mass element, say the |
|
| 120:24 | of your fist uh which has lots grains in it. Uh But it's |
|
| 120:30 | compared to the wavelength of the cell . So I, so we can |
|
| 120:36 | of the uh the uh the fluid in there to be uh uniform, |
|
| 120:44 | at high frequency, it's not Uh uh If we uh uh uh |
|
| 120:51 | think now not about seismic waves but ultrasonic waves going through that same |
|
| 120:58 | the wavelengths are much shorter wavelengths are smaller than your fist. Now, |
|
| 121:04 | wavelengths are of the order of a or so. Uh And but it |
|
| 121:10 | comprises many grains. And in uh that uh if you have a mass |
|
| 121:20 | uh the size of a centimeter and many grains and a complicated core |
|
| 121:27 | And as the wave goes through, configure, it's going to be squeezing |
|
| 121:34 | water from one part of the pore . To another because the uh the |
|
| 121:40 | parts of the pore space which are , don't deform as much as the |
|
| 121:47 | of the pore space which are like . So the crack like portion of |
|
| 121:51 | pore space is gonna be squeezed in the wave goes through, squeezing water |
|
| 121:57 | the equate pore space. And those idealize descriptions. But I think you |
|
| 122:02 | the idea because the pore space has complicated shape, fluid is gonna move |
|
| 122:09 | and as it does, so it's have viscous losses inside the fluid that's |
|
| 122:15 | convert the um uh elastic energy to uh like uh the was describing. |
|
| 122:23 | so there's a, there's a, attenuation loss and while it, while |
|
| 122:33 | doing that, uh uh I should that uh uh it, the fluid |
|
| 122:40 | to do that a as the uh the fluid is uh as the wave |
|
| 122:44 | passing through. If the frequency is high, it's not enough time for |
|
| 122:49 | fluid to move from one part of rock to another one. So we |
|
| 122:53 | that the a band, the, um the mechanism. In this |
|
| 123:04 | food court hasn't had time to happen the wave has come and gone. |
|
| 123:09 | that's un relaxed at the other end the spectrum down here uh at the |
|
| 123:13 | uh the lower end of the that's when uh those frequencies are so |
|
| 123:19 | that uh uh pressure equalization has already . Uh while the wave is going |
|
| 123:27 | and at intermediate frequencies that's partially So at low, at low enough |
|
| 123:37 | , we say that mechanism of attenuation relaxed. So you think, |
|
| 123:47 | this is pretty easy. Uh I'm to uh do a bunch of experiments |
|
| 123:51 | a bunch of frequencies and phrase and determine these curves. And therefore, |
|
| 123:57 | gonna determine what are the values of sub uh tau and T sub |
|
| 124:04 | Well, it turns out to be more complicated than that because real walks |
|
| 124:09 | have a single dispersive mechanism like but they have several. And so |
|
| 124:15 | you have several dispersion mechanisms, all the same total, um all with |
|
| 124:21 | same uh maximum uh for a uh it, it's gonna look like |
|
| 124:28 | like this. You can see that is a superposition of many bell shaped |
|
| 124:32 | like that. And when you superpose this uh or uh uh several curves |
|
| 124:41 | this, and you're also gonna have superpose several curves like this. And |
|
| 124:45 | you get then an extended ramp and velocity and uh uh you only get |
|
| 124:52 | have constant velocity when you've uh got frequencies which are higher than uh the |
|
| 125:02 | the maximum frequency for the uh the which is represented by this little tique |
|
| 125:11 | . And you can say, say same thing about, you know, |
|
| 125:16 | uh about the low frequency end. for rocks, um the general conclusion |
|
| 125:23 | , let, let me say here , at so frequencies, the dominant |
|
| 125:29 | mechanism of um of uh attenuation comes the scattering of waves off of the |
|
| 125:39 | grains. The fluid has not had to move uh uh at ultrasonic |
|
| 125:47 | So you can consider the ultrasonic frequencies the limit, they don't move at |
|
| 125:52 | . And so the pore pressure is be different within the cracks and within |
|
| 125:57 | pores as the food has not had to pressure equalize. But still, |
|
| 126:03 | will be uh scattering off of the high frequency. So that's the mechanism |
|
| 126:13 | scattering uh uh uh uh in the band in the sonic and the seismic |
|
| 126:22 | . The uh the dominant mechanism of is uh fluid squirt. Let me |
|
| 126:33 | that again. I I said it in the sonic band. The dominant |
|
| 126:38 | mechanism is smooth squirt. And when get down to the sonic band, |
|
| 126:43 | we hope is that smooth squirt has happened. It's all gone away and |
|
| 126:47 | have uniform pore pressure at uh uh the grain scale uh uh for uh |
|
| 126:55 | seismic waves. Now, it's still frequencies. You can imagine that the |
|
| 127:00 | might flow over long distances, more a grain scale, it might flow |
|
| 127:06 | uh uh the, the distances uh to uh a wavelength. So that's |
|
| 127:13 | long way for the fluid to, flow. And so that means |
|
| 127:18 | very low frequencies. We don't see sorts of low frequencies in our kinds |
|
| 127:23 | data. So um oh I should uh uh at um I should say |
|
| 127:44 | when we have sonic band frequencies, frequencies have long wavelengths. So whenever |
|
| 127:55 | have a long wavelength that of propagating the earth, it's always almost always |
|
| 128:01 | to encounter heterogeneity, heterogeneity of uh rock types. Like if you have |
|
| 128:10 | wave, which is 100 m you know, there's gonna be lots |
|
| 128:14 | different Ortho in uh uh contained within cycle of such a wave. So |
|
| 128:22 | that case, it's a bit um really difficult to say. Um what |
|
| 128:33 | is the mechanism of attenuation at sonic because of the heterogeneity, which is |
|
| 128:44 | gonna be affecting those waves also. um the uh there is gonna be |
|
| 128:50 | effect of apparent attenuation also in seismic uh band, we're gonna have this |
|
| 128:59 | of friendly multiples that we talked about seriously. So that's why this statement |
|
| 129:05 | extends the fluid start mechanism down into uh into the uh seismic band. |
|
| 129:13 | It's uh it, it's a bit whether fluid skirt stops it at 10 |
|
| 129:19 | or maybe it stops at 100 Hertz somehow, but it's definitely present in |
|
| 129:24 | sonic band. So uh at ultrasonic , this is what I said um |
|
| 129:32 | scattering. Now, let's think about seismic b, the strength of the |
|
| 129:40 | squirt me mechanism depends upon what kind food we're talking about. If it's |
|
| 129:47 | with oil, then the squirt mechanism enhanced because of the greater viscosity of |
|
| 129:52 | compared to Bryan. Furthermore, if pore space is partly saturated with |
|
| 130:00 | then the squirt mechanism is greatly enhanced if you have gas in the pore |
|
| 130:06 | , basically that move, that gives the uh fluid room to move |
|
| 130:12 | And so uh that mechanism is greatly . So in these situations, uh |
|
| 130:17 | the, the, the quality factor P waves is low, but for |
|
| 130:24 | waves, it's normal. So I'm take you back to um uh to |
|
| 130:30 | hall and to show you what a state of the art image looked like |
|
| 130:36 | Val Hall in 1994. So this uh about the time that I was |
|
| 130:45 | the oil industry uh previous to I had been a professor in the |
|
| 130:49 | University of New York and I joined Amaco, I think in 1995. |
|
| 130:58 | I didn't know these guys. Uh I didn't know at that time, |
|
| 131:01 | didn't know this guy uh barred since . He's become a very good |
|
| 131:05 | And by the way, he no works for VP. Uh But at |
|
| 131:09 | time, he was working for Amao Norway and he was in charge of |
|
| 131:15 | this field here. And it's in southern part of the Norwegian North |
|
| 131:22 | It's near where the boundaries, the of Norway and Denmark and the United |
|
| 131:33 | in the middle of the North This is not too far from that |
|
| 131:38 | . And so their first images look this. You can see uh uh |
|
| 131:45 | fidelity of this image. You have confidence what's in here. Uh Because |
|
| 131:57 | is a time section, not um not a uh depth section. We |
|
| 132:03 | didn't know how to do depth migration that time. I'd say, I |
|
| 132:07 | say we were learning how to do migration. It was very common to |
|
| 132:11 | at um sections in uh as, time migrated sections. And so the |
|
| 132:19 | uh uh is this uh structural depth is this caused by a late arrival |
|
| 132:27 | because of low velocity up here? an open question when you look at |
|
| 132:34 | and you can see up here that uh uh reflection seem to be going |
|
| 132:39 | across, there's no stroke, there's depression here. And you see this |
|
| 132:44 | 1400 milliseconds down. So uh the uh reservoir is gonna turn out |
|
| 132:51 | be down here at 2800. So is halfway down. And above |
|
| 132:56 | the ex the data quality was What kind of, of um imaging |
|
| 133:01 | this? This was done with the of the art at that time, |
|
| 133:06 | was called ad mo stack dip move stack. And it's uh uh a |
|
| 133:13 | of what we showed earlier about uh stacking traces with uh uh dick move |
|
| 133:21 | . So all those early ideas were enough for simple um substructure. But |
|
| 133:28 | we explored more and more, we more and more complicated um uh structures |
|
| 133:34 | the subsurface. And this is an . And so we're gonna need better |
|
| 133:39 | algorithm than DMO to see this. you know, we're not gonna wait |
|
| 133:45 | that happens. We're gonna drill it . So when, when Amaco was |
|
| 133:51 | that, I think this was uh in the two D era. I |
|
| 133:54 | it was two D data. And Amao uh was uh uh doing that |
|
| 134:09 | expression in the early part of the nineties, got this image in |
|
| 134:16 | And you know, some manager is say we're gonna drill that sucker, |
|
| 134:21 | they had no idea what was down . Some kind of a monster down |
|
| 134:25 | that it might be, you severely overpressure. Nobody knew what they |
|
| 134:30 | gonna get. Some adventuresome interpreter had a line through there and some lines |
|
| 134:37 | there. But nobody was willing to whether that was a structural depth or |
|
| 134:43 | , a velocity push down. Nobody . So you can uh try to |
|
| 134:50 | yourself as the uh as the staff who's going out on the drill |
|
| 134:57 | And so uh uh uh uh before leave the office, uh uh |
|
| 135:02 | you look over this image and the says, uh OK, Ralph, |
|
| 135:07 | gonna be our ge expert on the ship when, when we drill into |
|
| 135:13 | . So, the first thing you do is you wanna go back home |
|
| 135:16 | make sure your medical insurance is fully up and you, you kiss your |
|
| 135:22 | and kids tenderly goodbye and you say , I hope to see you in |
|
| 135:27 | weeks time and then you uh uh then you go up out and |
|
| 135:32 | So that's what they did. And discover that all of this was velocity |
|
| 135:38 | out and um uh attenuation due to uh you know, uh uh just |
|
| 135:48 | to, to Hewa attenuation here. the structures were really structures that looked |
|
| 135:54 | this across here, mild structure, like this. So uh they came |
|
| 136:02 | uh and they said, honey, home and uh uh um uh it |
|
| 136:06 | a million barrel AAA billion barrel oil . Great discovery by Amako in the |
|
| 136:14 | , early nineties based on this kind data. So the top of the |
|
| 136:20 | here is here. And by by the way off to the |
|
| 136:23 | uh it's good imaging simple structures and only the crest of the energy is |
|
| 136:29 | well, OK, it's messed Well, we have come to realize |
|
| 136:33 | the uh well, uh that this data quality comes from what we call |
|
| 136:39 | ga a cloud of gas. this is the peak, the, |
|
| 136:43 | actual reservoir looks like this sort of , a gentle dome. This is |
|
| 136:48 | crest of the dome right in You can't see it on this image |
|
| 136:54 | at the crest gasses leaked up out the crest of the dome over G |
|
| 136:59 | time collecting here and the overburden, over here, but only here and |
|
| 137:04 | the crest and not up here. the uh uh the reflectors go through |
|
| 137:10 | fine up here. So the starting here on down the um the gas |
|
| 137:16 | collected in there and it does two . It uh makes the wave, |
|
| 137:20 | P waves slow down and, and attenuates them. So it says slows |
|
| 137:27 | waves down and it's and retain. uh this is not the only place |
|
| 137:32 | this in the North Sea, there's be 20 places like this and one |
|
| 137:39 | them is owned by um mhm Before tell you that part of the |
|
| 137:52 | I'll, I'll, I'll complete that one of them nearby about 20 miles |
|
| 137:56 | was owned by conical and it looked similar and it was also a major |
|
| 138:01 | reservoir hiding beneath a cloud of gas that. So, um this is |
|
| 138:07 | picture of what we uh uh uh was happening. We had P waves |
|
| 138:13 | down uh and outside the gas cloud up through the gas cloud, uh |
|
| 138:19 | gas cloud. So it slows the uh the P wave down and it |
|
| 138:23 | lowers the quality. So the waves not only slowed down but they're |
|
| 138:30 | So, um, over, um, uh uh uh I said |
|
| 138:47 | miles away, there's a, a field, um, owned by Cono |
|
| 138:51 | called Eco Fsk. Um Norwegian by the way, Val is a |
|
| 138:58 | name, uh, that refers to , the, the, the home |
|
| 139:03 | the gods. So that when gods die in battle, they are taken |
|
| 139:09 | , uh, what we might call . It's called Val. And uh |
|
| 139:13 | that's uh in Norse mythology. That's it's called. Uh Sometime you might |
|
| 139:23 | to go to the Val Hall bar the rice campus. It's a, |
|
| 139:28 | hidden bar. Nobody knows where it . It has uh uh no signs |
|
| 139:34 | indicate where it is. But if find the right student say, take |
|
| 139:40 | to the Val Hall bar, you see that. OK. So, |
|
| 139:46 | Ko had a well, like had field like this, but they didn't |
|
| 139:50 | what to do with it either, maybe 50 miles away. State Oil |
|
| 139:56 | a field. So sta oil is Norwegian State Oil Company. And uh |
|
| 140:02 | they had the bright idea that let's um explore for, oh, uh |
|
| 140:15 | see if we can get a better of our reservoir that looked pretty much |
|
| 140:21 | this one over and they, they uh uh this field was called uh |
|
| 140:29 | hm So on the tip of my , but I can't bring it off |
|
| 140:40 | tip of my toelle, that's what field is called. Toleen. Another |
|
| 140:45 | name, don't know what, what means, but their field looks a |
|
| 140:49 | like this. They had the same of image data quality, but they |
|
| 140:53 | the bright idea let us explore for with sheer waves. Why? It's |
|
| 141:01 | if this is a shear wave going instead of a P wave, it's |
|
| 141:04 | compressing the uh rocks as it goes , it's shearing the rocks sideways. |
|
| 141:10 | gonna come down here and reflect and back. And so when it goes |
|
| 141:13 | the cloud of gas, it's gonna shearing the cloud back and forth, |
|
| 141:18 | compressing it in any way. So sheer wave doesn't care what is the |
|
| 141:24 | in the pore space, just like talked on the previous uh lecture lecture |
|
| 141:30 | eight about uh or elasticity. We that uh uh the nature of the |
|
| 141:38 | has a very minimal effect on sheer . It affects the density, but |
|
| 141:42 | does not affect the shear modulus. let's, they said, let's do |
|
| 141:48 | wave exploration and get a good shear image of our field at Toma |
|
| 141:55 | Well, the problem is that is there is a uh a ocean uh |
|
| 142:01 | there and you can't send sheer way the ocean. So they said, |
|
| 142:07 | , let's invent ocean bottom seismic receivers put the receivers on the sea |
|
| 142:15 | then we'll have our sources generating um P waves in the ocean water, |
|
| 142:23 | like before both P waves will go through the water, they'll convert to |
|
| 142:28 | waves at the sea floor. They'll down into the sea floor and uh |
|
| 142:35 | uh uh up through, uh down the uh c far down to the |
|
| 142:41 | , reflect, come back through the cloud, unaffected and received by our |
|
| 142:46 | component receivers easy. And so they it and they had success and they |
|
| 142:58 | their results at a meeting of the Ex uh Geophysical Society. And they |
|
| 143:04 | great applause for that successful imaging at through the gas cloud. What she |
|
| 143:13 | , I was not there, but colleague was there and he came back |
|
| 143:17 | told me all about it. He that's great stuff. And so shortly |
|
| 143:22 | our uh our friends at the chemical in Norway said, hey, let's |
|
| 143:28 | that. And uh and so um they said, will you help us |
|
| 143:34 | that? Well, we, we know how to do uh ocean bottom |
|
| 143:38 | uh side. So we, we , I was working in Houston at |
|
| 143:42 | at that time, you know, AMMO Corp office is out in West |
|
| 143:47 | uh about 10 miles from where we here. So we agreed to |
|
| 143:53 | we agreed to help the Amer Norway acquire and process um uh energy you're |
|
| 144:02 | uh a converted cheer. And so the process of that, we learned |
|
| 144:08 | important we learn that when the heat goes down through the water and hits |
|
| 144:15 | sea floor, most of it doesn't to, to um shear waves at |
|
| 144:21 | point. Most of it can uh as a P wave, the shear |
|
| 144:25 | only like 5% something like like it's converted to uh shear waves. |
|
| 144:32 | so by the time those shear waves back uh from the deep reflector, |
|
| 144:38 | they're all dissipated, they, they just too low amplitude to uh see |
|
| 144:43 | , you know, they, they uh they, they um a as |
|
| 144:49 | propagate, they lose energy faster than P wave because they're making more cycles |
|
| 144:54 | meter than the sea than the P are. And so they uh attenuate |
|
| 145:01 | more in ordinary rocks like we see , ordinary rocks, they attenuate more |
|
| 145:06 | P waves. So when those sheer came back, we detected them, |
|
| 145:11 | very, very weak it, but were lucky. Uh the, the |
|
| 145:17 | uh helped us in another way because found out that if we ignore that |
|
| 145:22 | and if we consider P waves which convert of the sea floor, but |
|
| 145:27 | all the way down to the reflector convert to S waves right here coming |
|
| 145:34 | through the uh uh red to to the gas cloud as a sheer |
|
| 145:39 | converted at the reflector that one we see. So here's the, here's |
|
| 145:44 | diagram of that. So here's our wave coming down. You see, |
|
| 145:47 | conversion point is different than the We talked about that a few lectures |
|
| 145:53 | , coming up as a sheer wave transversely to the propagation direction, it |
|
| 145:59 | through the sheer wave and it doesn't care what fluid is there, uh |
|
| 146:04 | or oil or gas or what it's continue. So it comes up |
|
| 146:08 | And so that makes uh uh uh wave arrival And it, it, |
|
| 146:13 | all the possibilities for converted wave conversion . This is what we call the |
|
| 146:19 | wave. That's the one which which converts upon reflection. That's this |
|
| 146:25 | here. So we made images at hall using that idea. And here |
|
| 146:30 | the images, this one came This was our first image, I |
|
| 146:37 | say it was very naive imaging, at least it recognized the uh uh |
|
| 146:46 | of converted waves. It's coming back than the uh uh P wave |
|
| 146:52 | See, let's see, this is , oh this is converted to P |
|
| 146:56 | times. So the, this is a, a time section. Uh |
|
| 147:00 | is uh 3000 milliseconds. Uh So converted to P wave time. So |
|
| 147:06 | looks normal to us. And sure , we see a low dome across |
|
| 147:11 | . And so maybe there's some issues uh uh with better imaging techniques developed |
|
| 147:20 | then, since 1996 we see that absolutely continuous across here. So all |
|
| 147:28 | did was normal move out processing. was in charge of the team that |
|
| 147:33 | this processing and I am not a uh I am not an expert in |
|
| 147:40 | wave imaging. That's why they, don't have me to be teaching the |
|
| 147:44 | in imaging. They have Professor he is a much more expert than |
|
| 147:49 | in imaging. All I knew how do was remove the move out swimming |
|
| 147:55 | layers of these are pretty much flat um I stacked them up according to |
|
| 148:02 | . That's all I knew. But recognized that the image point was uh |
|
| 148:08 | over here instead of here at the . And of course, we recognize |
|
| 148:13 | the uh wave coming up is gonna coming up as a sheer wave. |
|
| 148:21 | um you can see here that the this overburden here, here's to where |
|
| 148:28 | gas cloud is. Nothing really unusual , not very good imaging, but |
|
| 148:33 | not a disaster like we showed Here is the uh the top of |
|
| 148:37 | gas clouds coming in here. And uh um below, this is the |
|
| 148:44 | cloud, it's bad but not I would say. And so all |
|
| 148:49 | these um um um our boreholes posted this image by uh barred after we |
|
| 148:58 | the image. And he said these four of the boreholes that we have |
|
| 149:04 | they uh they validate exactly what this is. And so when we showed |
|
| 149:09 | image, uh um uh to uh went over there to Norway, I |
|
| 149:16 | I went alone, went over there Norway and uh um uh convened every |
|
| 149:22 | together to look at our um uh novel processing of converted wave data from |
|
| 149:31 | . And to show that to the people in Norway who were trying to |
|
| 149:36 | a living by producing here, they never seen an image like this |
|
| 149:42 | What they saw was a terrible image I showed you five slides ago. |
|
| 149:48 | I can tell you they stood and when they, when they saw this |
|
| 149:52 | because now they knew they could manage field properly knowing how it was shaped |
|
| 149:58 | uh in true space in, in depth. I'll tell you another uh |
|
| 150:07 | incident that comes from this. Uh told you about um I think I |
|
| 150:13 | you about this um of the image the uh the incident where the, |
|
| 150:19 | young man who had processed uh the data for our acquisition contractor. He |
|
| 150:26 | it got different results because he didn't the intricacies of converted wave propagation. |
|
| 150:33 | I talked about that some time uh several elections ago. Uh Now |
|
| 150:39 | wanna tell you more uh one more incident there. Uh A friend of |
|
| 150:46 | in Houston took the same data set he uh worked with it and worked |
|
| 150:53 | it and worked with it. He to me a year later and he |
|
| 150:57 | Leon, I've tried to uh uh this data set oh with 80 different |
|
| 151:06 | of velocity, 80 different velocity None of them showed any uh uh |
|
| 151:13 | like this one did. And you got this one on your first |
|
| 151:18 | So that all that has made a impression on me. He was thinking |
|
| 151:23 | this data set was pretty much like data sets that he had seen. |
|
| 151:28 | just gonna tweak his procedures and he's solve the uh flattening problem. And |
|
| 151:35 | , you know, he's gonna uh low velocities, uh you know, |
|
| 151:39 | the crest of the reservoir, it work and it didn't work and it |
|
| 151:43 | work, it didn't work. And 80 tries he gave up, we |
|
| 151:49 | success on our first try because it a new type of data. It |
|
| 151:53 | converted web date, it's not web date. It was converted web |
|
| 151:58 | with many different issues associated with And we, we discovered those and |
|
| 152:04 | dealt with them and we came up this image on our first trial. |
|
| 152:09 | you should do that too. Uh you look at uh uh a new |
|
| 152:15 | , but with the old type of , uh you're gonna be able to |
|
| 152:19 | that problem by tweaking the old But if you're looking, if you |
|
| 152:24 | a new type of data that you seen before, you should look at |
|
| 152:30 | with open eyes and figure this thing offer issues which were not even part |
|
| 152:38 | my previous thinking because it's a new data. So that's what we did |
|
| 152:43 | 1996 and had this success. by the way, when we have |
|
| 152:52 | success of the, just the fact we have a successful image here tells |
|
| 152:58 | that the problem is attenuation in the . It tells us that the reason |
|
| 153:06 | didn't get um the heat wave images because we had no P wave energy |
|
| 153:13 | after it was attenuated by the cloud gas. Before, if you have |
|
| 153:21 | poor imaging with P and good imaging convert with C waves, that essentially |
|
| 153:27 | that the problem with the P waves coming from uh the gas CL. |
|
| 153:33 | so now that says, OK, we know there's lots of gas |
|
| 153:36 | Uh Is this maybe um commercial area we attend in it, intentionally tried |
|
| 153:44 | reduce the gas here? Um Would be worth our while? We think |
|
| 153:51 | answer is no, we think the of gas here is at the level |
|
| 153:56 | 1% and probably not commercially attractive to to produce it. But who |
|
| 154:03 | maybe um maybe the they would decide the future that it's uh worthwhile. |
|
| 154:13 | I should tell you that in the couple of years BP has sold this |
|
| 154:17 | to another company where they have only uh ownership of that company and that |
|
| 154:25 | is uh trying to get the last billion barrels, few barrels out of |
|
| 154:30 | . I should say that when this was discovered, it was a 1 |
|
| 154:35 | barrel field. Now, it's recognized be a 5 billion barrel field because |
|
| 154:40 | can, we can see it better modern techniques including converted wave techniques and |
|
| 154:46 | better imaging techniques. Um Of NMO processing NMO imaging would be considered |
|
| 154:54 | extremely naive today. But in uh in 1996 it was only a bit |
|
| 155:02 | and the fact that it was successful didn't well its parts everybody but uh |
|
| 155:08 | not the uh the surprise came from solving the the sea wave problems, |
|
| 155:14 | from solving the imaging problems. um let's have a quiz. Let |
|
| 155:27 | turn to you Verda uh true or . It says for segment rocks, |
|
| 155:34 | dominant mechanism of continuation in the se in P waves is fluid squirt. |
|
| 155:41 | that true or false? It is . Yeah, that's true. Uh |
|
| 155:46 | you Lily, for S rocks, dominant mechanism of a generation in the |
|
| 155:52 | band for P waves is enhanced if is present in the pore space who |
|
| 155:57 | false. That's also true that that's abundantly true at val Hall. So |
|
| 156:04 | Carlos for a generative sedimentary rock, waves might, may give images, |
|
| 156:11 | give images which is superior to those B waves, true or false. |
|
| 156:16 | would say it's true. And I gave you an example. And so |
|
| 156:20 | are many other examples. Uh And uh uh it turns out that converted |
|
| 156:28 | are useful for a number of uh solve a number of imaging problems of |
|
| 156:34 | this one is the most prominent, uh uh the gas cloud problem. |
|
| 156:38 | are others which are uh less um common, you know. But uh |
|
| 156:46 | working with us now for 20 we can say that uh converted wave |
|
| 156:51 | are useful mainly in the context where have overburdened gas. So let us |
|
| 157:00 | turn our attention to apparent attenuation. remind you what we mean by |
|
| 157:06 | This is a situation which does not converting elastic energy to heat. This |
|
| 157:14 | a apparent a generation and remind you this slide coming back from uh lecture |
|
| 157:21 | that as we were deriving here uh the wave equation, uh We uh |
|
| 157:28 | found that uh if the, if medium is uniform, we can take |
|
| 157:35 | stiffness tensor outside of this operator, it over here and then what's left |
|
| 157:41 | gonna turn out to be um uh the wave equation I'll back up |
|
| 157:50 | Um um This is not really the equation you can see here that there |
|
| 157:54 | three derivatives with the spec to space of two, you know that uh |
|
| 158:00 | but uh uh if you go back lecture three, you'll see how we |
|
| 158:05 | uh uh found the wave equation scalar equation with only uh one not three |
|
| 158:13 | of 27 term, but one equation for the uh uh scalar potential. |
|
| 158:21 | And that's all the story back in lecture three. So uh also from |
|
| 158:28 | same lecture, we realized that if the medium is non uniform, we |
|
| 158:34 | that problem by uh including another term the uh result. Uh If this |
|
| 158:40 | depends upon space, then we're going have um AAA derivative of that thing |
|
| 158:47 | respect to space. In addition to term, this term is gonna lead |
|
| 158:52 | uh uh the wave equation. And is a new term which comes from |
|
| 158:58 | non uniformity of the media. So now we're gonna see how this leads |
|
| 159:03 | apparent attenuation. OK. For a isotropic medium with vertical P wave |
|
| 159:11 | this becomes like this. So here have only the uh uh the, |
|
| 159:17 | vertical component of displacement. So we that lowercase W. So that's here |
|
| 159:23 | that's here and it's also here. you see this is only one derivative |
|
| 159:27 | respect to depth. And over here a derivative with respect to depth of |
|
| 159:33 | longitudinal modulus. And so this looks if, if we ignore this, |
|
| 159:39 | this is a zero right here, a locally uniform medium. So we |
|
| 159:44 | the wave equation here. But uh now we're gonna consider cases where this |
|
| 159:50 | not zero. So this is the equation and this term is going to |
|
| 159:55 | to apparent attenuation. So let's assume we have plane wave solution. And |
|
| 160:02 | think you're familiar with this and we only uh a K three component of |
|
| 160:07 | wave vector here because uh uh we're propagating in the vertical direction. So |
|
| 160:15 | put that into the equation of not the equation, not the wave |
|
| 160:19 | . We put that into the equation motion, which has this additional term |
|
| 160:23 | here. And um uh so, from two derivatives with respect to |
|
| 160:30 | we get uh a minus uh we I omega squared and from two derivatives |
|
| 160:36 | to K three, we get minus three squared. That's what we saw |
|
| 160:41 | the uh uh uh from the wave . And then we have this additional |
|
| 160:48 | where um uh we have um a was back to Z and only one |
|
| 160:56 | of um um of the displaced uh . So uh simplifying that we get |
|
| 161:05 | quadratic equation for the vertical uh component the wave vector. And it has |
|
| 161:12 | term in there which we didn't see . So uh let's assume that the |
|
| 161:21 | uh the um the stiffness uh the stiffness M is real that way |
|
| 161:28 | there's gonna be no true attenuation at . OK. So whatever comes out |
|
| 161:34 | this, whatever attenuation comes out of is gonna be apparent attenuation. So |
|
| 161:40 | everybody here knows how to solve a equation. I think you remember that |
|
| 161:45 | high school days. And it looks uh this term here, close to |
|
| 161:50 | the square root of this term Now, where does this come |
|
| 161:58 | This comes, I'm gonna back This comes from right here. That's |
|
| 162:06 | ordinary solution to a quadratic equation is take the coefficient of the linear |
|
| 162:13 | Here's the, here's the second order , here's the linear term, here's |
|
| 162:18 | uh the term with no unknown at in it. And so uh uh |
|
| 162:24 | answer to the um to the solution the quad, that quadratic equation is |
|
| 162:31 | here. You see it's um Even though this thing is assumed to |
|
| 162:37 | real, we get uh complex. Now, in this expression, the |
|
| 162:44 | and the stiffness are properties of the not of the wave. Where do |
|
| 162:48 | see the properties of the wave Well, the frequency, that's part |
|
| 162:51 | the wave, that's, that's not of the uh uh uh of |
|
| 162:59 | that's not part of the media, part of the wave. And so |
|
| 163:03 | the wave is specified by the frequency by the choice of algebraic sign in |
|
| 163:09 | second term, you see, I a typo here, I need to |
|
| 163:12 | back and, and remove that type . And so we're gonna consider propagation |
|
| 163:18 | the plus Z direction. So we're um let the plus. Yeah. |
|
| 163:27 | uh let's now um use our old to tailor approximation. So, in |
|
| 163:34 | case, we're gonna repla we're gonna this um square root with this term |
|
| 163:42 | , which has the same minus sign you see right here and everything |
|
| 163:48 | that's what you see here. But you go, uh what you, |
|
| 163:50 | new is the one half that comes the fact that this is a square |
|
| 163:55 | or one half. OK. So gonna average this over a wavelength. |
|
| 164:04 | so um uh um uh thi this is the average here, that's uh |
|
| 164:13 | swap places. So this is the here. That's the real because you |
|
| 164:17 | this is the real part here, is the imaginary part. So I'm |
|
| 164:21 | move the imaginary part over here. uh we, we, this is |
|
| 164:26 | , the real part of the average of the wave factor as it goes |
|
| 164:32 | this a gen homogeneous sequence. And is the imaginary part. So this |
|
| 164:40 | the friendly multiple delay that we talked before. If you look uh uh |
|
| 164:46 | at uh look back what we did . You see that we had uh |
|
| 164:51 | uh uh a real part of the factor depending on uh uh the square |
|
| 164:56 | the uh of the reflectivity. And is the same friendly multiples delay written |
|
| 165:05 | different language, different uh terms. again, I see I made a |
|
| 165:10 | out here. I forgot to have , a closed quote. Right |
|
| 165:18 | So now the plane wave solution looks this. This is our definition of |
|
| 165:25 | plane wave solution got an uppercase W front and the lowercase W means the |
|
| 165:31 | . And this is the amplitude. so we're gonna separate out the real |
|
| 165:36 | parts and the imaginary part is over . And you see that because of |
|
| 165:44 | , I multiplied by this, we have uh high squared minus one |
|
| 165:51 | here. So this is decreasing as um uh as the wave propagates |
|
| 166:01 | So this is not like a surface , a sur surface wave oscillates as |
|
| 166:06 | goes along um horizontally and it decays the amplitude de decays away vertically with |
|
| 166:14 | term looks more or less like But here we have decrease in amplitude |
|
| 166:23 | it goes down. So that's making attenuation. So remember this is |
|
| 166:34 | that's real. Everything is real here this term, but we do get |
|
| 166:40 | situation because of this factor with the sign here. Now let's consider a |
|
| 166:45 | where the stiffness shows a trend. uh uh and that means that uh |
|
| 166:51 | is gonna be greater than zero. this means that this thing is gonna |
|
| 166:58 | greater than zero exponential decay of amplitude of frequency, no true attenuation because |
|
| 167:08 | is real. OK. Now, assume suppose the wave is moving the |
|
| 167:17 | way coming back up. So in propagation in the minus Z direction, |
|
| 167:22 | gonna select the minus sign, same uh uh uh they have the same |
|
| 167:29 | for the uh imaginary term and uh the same thing but the uh you |
|
| 167:34 | , uh variation is small and get of the square root. And uh |
|
| 167:40 | so, um now, at the case, same case where the stiffness |
|
| 167:50 | a positive tr the apparent A generation still um um the same as we |
|
| 167:58 | before, but now Z is the wave is coming back up. |
|
| 168:04 | now, uh um yeah, um the trend is positive, this part |
|
| 168:10 | positive, but as the wave goes smaller Z here, the amplitude is |
|
| 168:16 | exponentially. Wow. So if it down at lost amplitude, but as |
|
| 168:22 | comes back up to the same sequence rocks, it recovers the, the |
|
| 168:29 | isn't that strange. So this apparent cancels itself out for a reflection problem |
|
| 168:38 | A VSP which is, you one way propagation uh that would be |
|
| 168:44 | would be a different scenario. But this case, the apparent cancellation cancels |
|
| 168:53 | . That's interesting. I'll bet that as a surprise to came as a |
|
| 168:56 | to me when I was working through . So now let's consider a case |
|
| 169:01 | the, the stiffness fluctuates, but has no tr so then the imaginary |
|
| 169:06 | of P three vanishes because this, part uh uh averages to zero because |
|
| 169:12 | , there's some places where in increases some places where it decreases, no |
|
| 169:17 | at all. So that means that part is zero. Now, let's |
|
| 169:25 | at the real part, the real is frequency dependent with a nonzero part |
|
| 169:30 | because we're squaring this driven for downgoing , it looks like this and then |
|
| 169:37 | square is nonzero. OK. uh uh that means we're having the |
|
| 169:43 | multiple delay effect when we have uh fluctuating stiffness with no tramp. It's |
|
| 169:56 | friendly multiple delay because of that minus here. Now, let's assume, |
|
| 170:02 | let's be more specific, instead of saying fluctuates, let's say there's a |
|
| 170:07 | uh variation in the stiffness. And let's describe the stiffness as a constant |
|
| 170:14 | of stiffness with uh uh uh uh attitude, uh a verbal part which |
|
| 170:22 | oscillates uh uh with a bit thickness H over two. So that's a |
|
| 170:27 | to describe a, a sinusoidal variation um a nonzero average in the |
|
| 170:37 | So then the stiffest derivative looks like where this part goes away. And |
|
| 170:42 | have only the uh the uh the the, the delta M, that's |
|
| 170:47 | delta M here. And so the looks like that and then the wave |
|
| 170:52 | looks like. So, um uh so this part here, um excuse |
|
| 170:59 | , oh yeah. Uh the way it has only uh see that's uh |
|
| 171:20 | the wave factor is looking like. yeah. So I noticed that we |
|
| 171:25 | the uh the frequency here. And at high frequency, that term goes |
|
| 171:33 | zero because this uh uh this frequency large. And so we're left with |
|
| 171:40 | this term here. That's the average uh at the average of the |
|
| 171:47 | This is the ray theory average and is the average of the slowness of |
|
| 171:51 | high frequencies. That's what we found lecture three for the ray theory. |
|
| 171:59 | result. And so um no, this is the high frequency wave |
|
| 172:10 | So uh recognizing that the velocity is to the wave vector by this relationship |
|
| 172:17 | , we find that the velocity is by the average over the layers of |
|
| 172:22 | inverse of velocity. In other slowness. And then of course, |
|
| 172:28 | have to take the inverse of This is the ra ra result which |
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| 172:32 | found previously. So I'm gonna show back here, remember this slide here |
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| 172:37 | we derive the theory by uh uh derive the velocity in a coarse layer |
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| 172:43 | sequence, we call it coarse layer the layers are thick compared to the |
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| 172:49 | wavelength. So this is the high limit and we found the same |
|
| 172:56 | Now, going back to uh back this situation, I'm gonna go back |
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| 173:02 | slide back on, slide back So we're looking at this. Now |
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| 173:06 | gonna look at the low frequency situation this number is low, here's low |
|
| 173:16 | . And uh um um uh we not gonna neglect this because this is |
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| 173:22 | low number and the low frequency phase is low uh because of that. |
|
| 173:33 | , um a second here, Um This is the low frequency wave |
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| 173:43 | , see. Uh And so we're gonna neglect this. We're gonna keep |
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| 173:47 | , we're not gonna let it go uh a frequency go to zero, |
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| 173:51 | we're gonna say it's low. So still in there. And uh because |
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| 173:57 | this term, the low frequency velocity um um less than the high frequency |
|
| 174:09 | as the friend, the same friendly multiple delay. So now that's all |
|
| 174:17 | the propagation. How about the Well, you know that the uh |
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| 174:25 | that uh we have the, the for attenuation is related to dispersion by |
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| 174:32 | formula here. So let's apply that the previous situation here. Uh We're |
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| 174:39 | take the derivative of the low frequency with respect to frequency and we're gonna |
|
| 174:47 | get nothing here because the uh the frequency part does not depend on the |
|
| 174:53 | , but there is a frequency dependence in here. So, working through |
|
| 174:58 | derivative, that derivative is given by to the apparent two factor from the |
|
| 175:11 | representation on page 53. So uh have your uh your notes go back |
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| 175:16 | page 53. You'll see that the uh the apparent attenuation is related |
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| 175:23 | the velocity dispersion by this term And so we get friendly multiple attenuation |
|
| 175:31 | well as friendly multiple delay, higher decay more rapidly than low frequencies |
|
| 175:37 | And you know that this doesn't come converting plastic energy to heat. It |
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| 175:45 | from superposing many, many friendly multiples each other with different time delays. |
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| 175:53 | they uh uh those different time delays the effect of killing out the high |
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| 176:01 | . And that's what we, we here for the, uh that's what |
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| 176:05 | found here for the velocity. The is uh is slower than the high |
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| 176:12 | , high frequency velocity. And the the amplitudes are also down because of |
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| 176:18 | . So when we get our display seismic arrivals on our workstation, we |
|
| 176:30 | that the uh uh uh uh at reflection times, we see that |
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| 176:37 | the frequency content is less and that's to a combination of two attenuation and |
|
| 176:45 | attenuation. This is the apparent attenuation . Oh um La Cross um consider |
|
| 176:59 | quiz statement of uh uh begins with equation of motion, not the equation |
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| 177:06 | not the wave equation, but the of motion with that additional term leads |
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| 177:12 | in the case of cyclical bedding ABC A all of the above. |
|
| 177:18 | uh uh number one, let me to you uh Bria, how about |
|
| 177:24 | ? Uh uh is this true lower at higher frequencies? It is |
|
| 177:31 | Hm Who is that true? what the friendly multiples are gonna make |
|
| 177:38 | a delay. So the, the multiples are going to slow down |
|
| 177:45 | uh, um, uh the low waves. So it's gonna be at |
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| 177:50 | frequencies at lower velocities. So this is false, you've got fueled by |
|
| 177:59 | trickery in the question. So, , le le how about Lee? |
|
| 178:04 | this one true? That one is . So, uh, so this |
|
| 178:11 | better be false. Um, uh sure enough, uh, that one |
|
| 178:14 | false. The uh just the fact we have cyclical betting does not necessarily |
|
| 178:21 | we have attenuation unless the beds themselves a generation in them. The, |
|
| 178:29 | um the apparent uh vellos of high due to the cyclical bedding leads to |
|
| 178:41 | current degeneration without any loss of energy heat. It does lose high frequencies |
|
| 178:47 | it propagates. It doesn't lose any that to heat. It just loses |
|
| 178:51 | to the superposition of the many, friendly multiples on top of each other |
|
| 178:57 | verbal uh um verbal delays that makes attenuation even when there is no true |
|
| 179:06 | . So, as conclusions, uh have learned that a generation is an |
|
| 179:12 | part of any realistic seismology. It the relative loss of high frequencies. |
|
| 179:20 | always accompanied by dispersion and its velocity upon frequency, many different physical |
|
| 179:31 | But in the seismic toson band, most important is fluid squirt. Here's |
|
| 179:39 | you probably didn't see coming at all you have a generation differences at reflectors |
|
| 179:45 | causes a phase shift of the reflected . Now, this might be important |
|
| 179:52 | because we know that for partially saturated below the top of the reservoir, |
|
| 179:59 | gonna make the and a generation difference abo uh for P waves, so |
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| 180:06 | above the reservoir, we're gonna have attenuation inside the reservoir, we're gonna |
|
| 180:12 | high attenuation. But for that reflected , it doesn't go through from the |
|
| 180:17 | reflected wave from the top doesn't cause loss of high frequency as it says |
|
| 180:25 | , because that wave didn't go through highly reflective, highly attenuated medium, |
|
| 180:31 | goes back up to the normally attenuated . And so, uh even so |
|
| 180:37 | can detect that difference in uh uh uh attenuation at the reflecting horizon with |
|
| 180:45 | resolution, I might say because of phase shift. And so that could |
|
| 180:51 | uh a, a big economic We could use this fact, find |
|
| 180:55 | lot of uh gas which we didn't before. If we look more carefully |
|
| 181:03 | uh historical data, I think we do that. I, I think |
|
| 181:06 | is worth uh uh uh a student um thesis, not sure if it |
|
| 181:14 | be worth it for uh Schlumberger to versa to work on that or for |
|
| 181:23 | uh uh Carlo's company to pay in work for that because these two people |
|
| 181:27 | highly paid but uh uh uh students paid hardly anything. So uh uh |
|
| 181:35 | would be this is a great project a student to figure out whether uh |
|
| 181:40 | kind of uh reflection off the top a oh yeah, gas saturated |
|
| 181:48 | Does that constitute um an expiration clue not? If the answer is |
|
| 181:55 | then we can try maybe make a of money. That person will become |
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| 182:00 | if the answer is no, because confusion from other effects like uh a |
|
| 182:06 | thin bed, nearby thin bed If, if you can't reliably um |
|
| 182:13 | use this phase shift on reflection, it's still an interesting thesis. And |
|
| 182:27 | , uh when we have lots of uh layers like we always do, |
|
| 182:31 | causes a pa uh uh uh parent . And uh uh uh uh in |
|
| 182:39 | case where the uh the uh the cycle call, we can see very |
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| 182:44 | that uh that leads to apparent attenuation well as uh real dispersion. |
|
| 182:51 | So I'm sure you have lots of along these lines. But um we |
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| 183:01 | enough time now, I'm gonna ask to hold your questions, send them |
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| 183:05 | me by email for next Friday. I'll, I'll tell you what's gonna |
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| 183:12 | . Now, we're gonna uh we 40 minutes left. We can begin |
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| 183:17 | discussion of anisotropy. That's a big . I give five day courses, |
|
| 183:24 | full day courses on that. You get a half a day. So |
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| 183:28 | see that it's gonna be a full a day, next Friday afternoon. |
|
| 183:36 | the end of that day, you not be experts in seismic uh wave |
|
| 183:43 | in case of seismic uh anti but you will know some important |
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| 183:49 | And so I hope it, it you to learn more about anisotropy because |
|
| 183:56 | ? Because the rocks are an isotropic especially these days when we're looking at |
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| 184:02 | with angle, we have to consider and we have, of course, |
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| 184:08 | have variations with angle of incidence and have variations with angle of asthma of |
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| 184:15 | propagation. And all of these are which were completely um off the radar |
|
| 184:23 | I joined Amao in 1979. But was lucky and I found uh a |
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| 184:32 | the first week I was with The very first data set that I |
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| 184:39 | showed the effects of anisotropy. And of my background, I could see |
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| 184:46 | when others couldn't see it. Other who were much more expert than me |
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| 184:51 | all of the topics of exploration to , maybe they had been working for |
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| 184:58 | for 10 years. They were real class experts, but they had not |
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| 185:03 | their minds pre prepared to think Maybe those subserous rocks are Amish |
|
| 185:10 | When we look at it, we find it and um come on before |
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| 185:21 | uh before about 1979 or so, 78. Only a few people in |
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| 185:29 | uh the world thought about anisotropy and people were largely regarded as cranks, |
|
| 185:36 | was paying any attention to them. so now anisotropy is a mainstream |
|
| 185:42 | Uh Lots of people know about We have lots of experts around the |
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| 185:47 | uh uh and uh um mainstream. you will just get your toe dipped |
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| 185:56 | this in the next um lecture uh will happen on Friday. Also on |
|
| 186:07 | , you can anticipate receiving from me examination and here's the way it's gonna |
|
| 186:15 | . I'm gonna send you the exam email. I don't like to do |
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| 186:19 | because, um, I like for to have it as a piece of |
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| 186:25 | copy in your hand, which is an envelope. So you don't see |
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| 186:30 | and you don't see it until you to take the exam. But |
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| 186:38 | uh uh the exam is gonna be book unlimited time. So you can |
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| 186:45 | your notes, you can have you can have anything open while you |
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| 186:49 | it. Take as much time as want to do this exam. I'm |
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| 186:54 | try to make it where I think can do it in three hours. |
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| 187:01 | usually, well, most people take than three hours, but I do |
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| 187:04 | best to make it a three hour . And in court, you |
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| 187:08 | the students could rush through it. might make their first pass in two |
|
| 187:13 | . But then they say, I have some more time, let |
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| 187:15 | check this and let me check And so that checking process usually ends |
|
| 187:20 | to be more than three hours. when you take the exam set aside |
|
| 187:25 | time for yourself where you're gonna be and you're not gonna have other people |
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| 187:33 | around bothering you with other things and have a block of time, |
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| 187:39 | a generous block of time for you work on this. At that |
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| 187:43 | you open the exam from your email until this time. Uh uh I |
|
| 187:49 | tell you by the way, this is gonna be in um um uh |
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| 187:54 | attachment to the email. And so uh I think it will, it |
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| 187:58 | be a PDF attachment. And um the smart thing for you to do |
|
| 188:07 | uh uh as soon as you open up, soon as you open the |
|
| 188:11 | , print off the attachment and then with it. Uh And uh with |
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| 188:17 | and pencil, there will be plenty space on the paper for you to |
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| 188:21 | whatever you need to do to solve questions. And then when you're |
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| 188:27 | uh you scan it and send it to not, not send it to |
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| 188:32 | , send it to uh Utah. when he gets all three of |
|
| 188:37 | he's gonna send them to me at time. So you can choose your |
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| 188:41 | . The uh the due date I will be on the Wednesday following the |
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| 188:46 | . So a Wednesday 10 days from . That's when it will be due |
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| 188:50 | the end of Wednesday, midnight on Wednesday. But we're doing this by |
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| 188:55 | because Carlos is over there in, , um, uh, Colombia |
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| 189:01 | uh, uh, it, we've to send him the exam by |
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| 189:05 | So that means we don't want it be an unfair, uh, advantage |
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| 189:10 | disadvantage that he has. So we're do it all by email, uh |
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| 189:15 | to le le here at the And as a bonus, uh uh |
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| 189:20 | pres it does not have to drive to from Richmond. She'll get it |
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| 189:24 | her email. Everybody will handle it the same way after dawn. Right |
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| 189:29 | there. I took 3.5 hours. I took 2.5 hours, whatever. |
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| 189:35 | uh, you, uh, wanna take, you take it, |
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| 189:39 | 10 hours if you want, write off and there'll be a space on |
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| 189:42 | exam to say what, how much you took. That's for my information |
|
| 189:47 | there's no points taken or removed from , uh, the scores because of |
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| 189:55 | time. I just, I'm interested the time. Send it back to |
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| 189:59 | by email when he has all three them. He will send it to |
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| 190:06 | . That's gonna happen after class on sometime Friday evening. I'll send it |
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| 190:12 | all of you. I know your address. So I'll just send to |
|
| 190:17 | so that leaves us with half an to begin the discussion of anti |
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| 190:22 | So, what I'm gonna do is gonna stop sharing here and I'm going |
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| 190:29 | bring up the powerpoint and then I'm watch and I start to be, |
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| 190:51 | sure did. I'm gonna put myself presentation mode and then I'm going to |
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| 191:01 | this and I'm going to bring back Zoom session and I'm gonna share my |
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| 191:11 | and there is less than 10 We're almost done here and share |
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| 191:19 | So professor, we just don't have in the canvas. That's right. |
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| 191:27 | not in the compass, but it be um uh uh tonight or uh |
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| 191:33 | or maybe tomorrow. I, I put it in canvas tomorrow. And |
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| 191:38 | , um um sorry, uh uh have to just watch the screen here |
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| 191:43 | anything here. And uh again, having difficulty getting this to me full |
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| 191:53 | . Um uh Utah help me. do I can't remember what it is |
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| 191:58 | you do to make this full You go to that one. |
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| 192:04 | That's it. So everybody sees this less than 10. OK. So |
|
| 192:10 | uh we do, we have uh time left before quitting time today to |
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| 192:15 | to begin this topic. So, the uh uh the course objectives uh |
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| 192:23 | . Are you gonna uh be able explain to your friends? Um uh |
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| 192:29 | are the common classes of anisotropic rocks how to find the wave equation. |
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| 192:36 | Remember that if it's uh if it's , then it's the same in all |
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| 192:41 | . No problem. But if it's isotropic, it's gonna be different in |
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| 192:45 | directions. So that is uh that why we have different classes of anisotropy |
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| 192:53 | it says here. And then we're find a wave equation. We're gonna |
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| 192:57 | what we did before uh to find wave equation and then we're gonna find |
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| 193:09 | and then we're gonna find simpler So if we have, we're gonna |
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| 193:16 | that if the anisotropy is weak, the solution, the expression for the |
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| 193:21 | as a function of angle is gonna a lot simpler than these exact |
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| 193:27 | As a matter of fact, when see these exact solutions, you will |
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| 193:30 | quite unhappy, but you will be when you see these simple solutions. |
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| 193:35 | in fact, this is a kind geophysical approximation that we often make. |
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| 193:40 | recognize that most of the oil that's been found has been found by ignoring |
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| 193:46 | , right? So uh uh it be AAA small effect. And so |
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| 193:55 | assume that it's small but non that's what this is. And then |
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| 193:59 | find that even though it's small, gonna have uh uh a noticeable effects |
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| 194:05 | our seismic data. And we're gonna uh uh a particular particular attention to |
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| 194:15 | effects of an I start on the Avio problem. And then uh |
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| 194:22 | basically, all of this is gonna for P waves. But then we're |
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| 194:26 | gonna sh show how it affects cheer and it's gonna be a fundamental |
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| 194:34 | And the reason is because we're gonna that in anisotropic media, there are |
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| 194:39 | different, share ways, still only P wave but uh two different sho |
|
| 194:45 | . So that's a fundamental difference. then because uh C wave is a |
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| 194:50 | of PNS, uh there's gonna be effects there. And that's the program |
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| 194:56 | can see it's a full program. uh It takes me five full days |
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| 195:01 | instruction to explain all this uh You get half a day. So |
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| 195:07 | uh let's see what we can learn high points. OK. So everything |
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| 195:14 | done, the first nine lectures have classic discussion of seismology equally suitable for |
|
| 195:22 | or the deep interior. But none it is truly suitable for either one |
|
| 195:29 | previous discussion has ignored anti our friends do, who are interested in the |
|
| 195:35 | earth. They uh mostly ignore not all of them anymore. Um |
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| 195:42 | Anisotropy is getting to be a more topic in um uh academic geophysics. |
|
| 195:50 | shouldn't say it that way. I say uh geophysics which is motivated by |
|
| 195:56 | about the earth in exploration Geophysics. motivated by uh we wanna have a |
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| 196:03 | driven geophysics who motivated by finding by geophysics to solve socially important problems. |
|
| 196:16 | , you know, this of that hydrocarbons are normally found in se |
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| 196:20 | Iraq, such rocks are normally an . Now, the anti syncopy comes |
|
| 196:27 | these different features. It comes from bed layering. You can see uh |
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| 196:31 | in this outcrop, many thin And so the the seismic wavelength is |
|
| 196:38 | stretch from the top of this cliff the bottom of this cliff. And |
|
| 196:43 | uh we can say that in uh this layer in this zone, it |
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| 196:48 | uh propagates as though it's uniform. gonna be reflecting off of boundaries like |
|
| 196:57 | . Uh But uh maybe this is , but you can see the |
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| 197:01 | many layers in there. And so layers are gonna mean that this uh |
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| 197:07 | velocity is traveling with an average velocity there. But it's obviously gonna be |
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| 197:12 | different average if it's a if it's vertically compared to obliquely compared to |
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| 197:20 | Now, among these layers, there's be shales and the shales are gonna |
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| 197:24 | intrinsically an iso and then furthermore, gonna be other features in the rock |
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| 197:32 | are s have a small scale compared the seismic wavelength with the preferred |
|
| 197:39 | And so you can see this joint here. It's it uh it has |
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| 197:44 | fractured the layers here to here. looks like it didn't penetrate into here |
|
| 197:50 | maybe it did uh did, didn't here. So it's limited top and |
|
| 197:55 | by um uh pathology considerations, but can be very long as you, |
|
| 198:03 | , if we were able to dig into this rock, we would find |
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| 198:07 | this uh, uh joint goes into rock quite a long ways. How |
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| 198:12 | we know that? Well, look here, here's another one and another |
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| 198:16 | probably oriented in the same way. look here is half of one bet |
|
| 198:21 | didn't see this. This is half a fracture and the other half has |
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| 198:25 | into the lock. This happens to in Ireland. So this is a |
|
| 198:30 | on a lake and you can see here, same thing and these are |
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| 198:34 | uh are more or less parallel So these are small scale structures smaller |
|
| 198:40 | a seismic wavelength with a preferred And these are gonna lead to uh |
|
| 198:48 | musical variations in in uh and uh . Obviously, if they had just |
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| 198:55 | fractured layers, then all the asthma would be the same. But when |
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| 199:00 | have uh uh joints in here, obviously gonna make Lauth and Iar. |
|
| 199:08 | , mother Nature tells us that all masses possess a fabric. So let's |
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| 199:15 | at this set of hand scents. is a set here. You can |
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| 199:18 | the layers in here. And so can, you can see obviously that |
|
| 199:22 | gonna have uh uh velocities which are vertically than horizontally just because the fact |
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| 199:28 | the layers are all horizontal. And this is a small scale version of |
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| 199:34 | laying that we saw in the left in the previous figure. So let's |
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| 199:42 | at this. This is a piece metamorphic rock and it has layers |
|
| 199:46 | But these layers are not necessarily As a matter of fact, these |
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| 199:51 | are uh these are horizontal because they but during the time when this sedimentary |
|
| 199:58 | was being created, gravity was always downwards. And so the layers are |
|
| 200:04 | uh horizontal layers. Gravity is not big issue here. These layers which |
|
| 200:10 | see in here are um perpendicular to uh a gradient and not a gradient |
|
| 200:20 | the pressure but a gradient in So as this mor metamorphic rock |
|
| 200:25 | it made uh different layers perpendicular to gradient of temperature. Furthermore, you |
|
| 200:32 | see the uh the outside shape of . This is a paperweight by the |
|
| 200:37 | and the craftsman who made this paperweight on there, some smooth sides here |
|
| 200:42 | here and here and here. And did that with his machinery is uh |
|
| 200:50 | AAA Craftsman's machinery. And uh uh outer faces are not necessarily related to |
|
| 201:01 | inner structure of the sample. But over here at this crystal. This |
|
| 201:10 | looks like it's homogeneous, matter of , you can see right through it |
|
| 201:16 | you uh uh the it it's got crystal faces here and here and here |
|
| 201:21 | those are determined by the internal distribution the atoms, the atoms here are |
|
| 201:28 | up in uh in uh atomic all lined up with the cells are |
|
| 201:35 | sh uh shaped in this same shape as the, the external uh uh |
|
| 201:42 | . So the the external phases of thing are determined by the internal |
|
| 201:52 | And so uh when I uh come the lecture tomorrow, uh on |
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| 201:58 | I will bring with me this piece rock. And uh we will see |
|
| 202:03 | least those of us here in Houston see that uh the uh we can |
|
| 202:09 | optical an iso every it by looking through this crystal. Now, here's |
|
| 202:16 | piece of wood. And so this not a rock but uh also as |
|
| 202:21 | uh uh uh uh tree grew, grew, gravity was always pointing |
|
| 202:27 | And so as the tree grew, arranged its internal structure, the the |
|
| 202:33 | of uh the, the wood here arranged so that it's strong in this |
|
| 202:39 | , not so strong in this Uh You can see the, the |
|
| 202:44 | uh the branches came out here. here those are the hor branches which |
|
| 202:48 | gonna lead to horizontal branching, but trunk of the tree is vertical like |
|
| 202:53 | . And uh it's stronger in this than in uh the horizontal directions. |
|
| 202:59 | you've ever worked with wood, you about how you need to respect the |
|
| 203:03 | of the wood as you work as you work with it. And |
|
| 203:07 | can guarantee you that uh uh if have a house, if you live |
|
| 203:12 | a house where the wood frame, always arranged so that the wood is |
|
| 203:16 | out of the tree so that the strong direction of the tree is |
|
| 203:22 | in the strong direction of the lumber hold up the the to hold up |
|
| 203:28 | him. Uh house better. here's a piece of sandstone and this |
|
| 203:35 | , in fact, Baria sandstone, talked about that several time before you |
|
| 203:39 | at it. And you do not with your eyeball, any preferred any |
|
| 203:44 | structure or maybe you can maybe rarely the grains in here, but they're |
|
| 203:49 | randomly oriented. So this rock is except that when you measure it, |
|
| 203:55 | not really isotropic. If you measure uh uh with uh send uh he |
|
| 204:01 | down this way, they travel faster P waves across this. The reason |
|
| 204:08 | because this core was taken out of ground where the coring tool is a |
|
| 204:14 | core. And so the sides of are free and it's a tall |
|
| 204:19 | So the, the stress, whatever was in the rock mass before this |
|
| 204:24 | uh uh taken out of the rock . It has released the stress on |
|
| 204:30 | sides, but it hasn't released the on the top. So that means |
|
| 204:34 | cracks have all open inside this sandstone they are generally cracks with um um |
|
| 204:46 | are not radio cracks, they are ranks. So if you think of |
|
| 204:52 | as uh flat cracks, the the flat sides are always perpendicular or |
|
| 204:58 | perpendicular to this external radius here. uh if you look down on the |
|
| 205:06 | from the top, uh you should see the top edges of the |
|
| 205:09 | you don't see the flat sides of crack. So since the cracks have |
|
| 205:15 | small scale structure with preferred orientation that leads to anisotropy. And by the |
|
| 205:21 | , we verify everything I said here the craft by squeezing the rock. |
|
| 205:26 | when we squeeze it back to its pressure, the anisotropy goes away, |
|
| 205:31 | proves that it's due to uh You can't see them with your |
|
| 205:38 | you can't even see them under a uh because you make two dimensional slices |
|
| 205:43 | your microscope. So you don't see crafts, but you see them with |
|
| 205:48 | sound waves vertically as opposed to Now, when you look at any |
|
| 205:56 | , here's a typical outcrop and it like these rocks are homogeneous. And |
|
| 206:02 | see here, doesn't that look to a homogeneous uh the shales? Uh |
|
| 206:08 | should back up here. You see cliffs here where the rock has uh |
|
| 206:13 | is not shale, but it uh enough calcite cement to hold itself up |
|
| 206:20 | a vertical cliff. But over here see uh uh the rock has |
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| 206:25 | So if you go, if you away all this stuff here, you'll |
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| 206:30 | a shale here. Uh uh And reason I know that is because it |
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| 206:35 | in this pattern with uh uh uh we call this an alluvial sand, |
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| 206:42 | contrast, this is not shall because has a vertical cliff. So um |
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| 206:49 | that's what it says here, behind one of these slope, sloping debris |
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| 206:55 | that's a shale. So uh uh at this section, most of this |
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| 207:00 | is shale. Now, I call a typical outcrop. Of course, |
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| 207:05 | is not typical. This is one the most spectacular outcrops in the |
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| 207:09 | It's a mile from the top to . This is the Grand Canyon. |
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| 207:14 | , let's look behind uh this uh slope, find the uh the un |
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| 207:20 | shale and we find um um that looks on a small scale like |
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| 207:27 | So you can see grains of but most of it is grains of |
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| 207:33 | and the grains of clay are shaped platelets. And during the sedimentary |
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| 207:38 | these platelets fell out of the ocean onto the sea floor and mostly |
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| 207:46 | very few of them landed on their . And then the there were mixed |
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| 207:50 | a few equipped grains of quartz or Planica here and here. But you |
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| 207:56 | tell from the orientation of the clay that the original um I don't know |
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| 208:03 | of gravity is that maybe when you at this, maybe this is really |
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| 208:15 | originally up in, in in this , maybe this was the top and |
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| 208:19 | was the bottom. But I'm sure none of you thought that this direction |
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| 208:23 | up because you can see the orientation a Lady uh Grant. And so |
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| 208:28 | is gonna mean uh intrinsic anisotropy much the crystal that we saw. All |
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| 208:37 | these uh uh uh grains of course crystals. And so the the crystal |
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| 208:44 | anisotropy of this uh clay platelet also to the anti isotopy. Here's a |
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| 208:50 | of quartz, I think this is , not sure either quartz or |
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| 208:54 | either way this grain is anisotropic in . But when you look at all |
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| 209:02 | grains, they are uh all uh oriented so that they yeah, |
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| 209:09 | The anti oxy internal anisotropy of these does not of these equant grains does |
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| 209:16 | contribute to the anti oc of the because they're all ran pointed because their |
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| 209:24 | shape is more or less the same all directions. You can see that |
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| 209:29 | and here however, that, that platelets are thin and flat and the |
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| 209:34 | sides are always uh horizontal. And , each of them is an anisotropic |
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| 209:41 | itself intrinsic an itself. So this is an isotropic and like we said |
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| 209:49 | , most of this section is I think it's true that something like |
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| 209:54 | of all the sedimentary rocks in the are shale and a small percentage are |
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| 210:02 | and a small percentage are carbonates. . So now how are we going |
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| 210:10 | deal with these anti sat topic We can't use the isotropic um wave |
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| 210:20 | . I think I misspoke that we, we cannot use the |
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| 210:25 | We request, we're gonna need the wave equation. And so let's try |
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| 210:34 | separate out P waves and sheer waves assuming that this stiffness constant is |
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| 210:44 | Uh I just now called it a constant. This is really a stiffness |
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| 210:50 | three by three by three by And you know that we enjoyed representing |
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| 210:58 | on a piece of paper as a by six matrix. So all the |
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| 211:04 | in that six by six matrix what uh uh is uh represented by the |
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| 211:10 | uh to me, let me back up. All the information in this |
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| 211:16 | rank tensor is represented in the second matrix. So that second uh |
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| 211:26 | we can plot on a piece of . And I'm gonna show you some |
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| 211:30 | and think about it since it's has two indices, we can't even think |
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| 211:35 | this because it has four indices. , if you remember that, uh |
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| 211:40 | we looked at that six by six for isotropic rocks, there were only |
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| 211:46 | uh there was a uh a lot zeros in there, there were three |
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| 211:52 | and three muses and three lamin out that, we get two P, |
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| 211:59 | types of waves two waves and sheer , we never find uh a wave |
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| 212:04 | is propagating according to LAMBDA, we a wave which is propagating according to |
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| 212:10 | and one propagating according to view, are respectively PNS waves. Although um |
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| 212:17 | the LAMBDA is just another concept which in there and is useful for uh |
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| 212:24 | some uh rock mechanics problems but not anything involving wave propagation. No. |
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| 212:34 | did we get to PNS waves? , we had that simple stiffness matrix |
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| 212:40 | , and uh from that stiffness we made a simple isotropic stiffness |
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| 212:46 | And we put that into this equation . And we used Helmholtz uh uh |
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| 212:53 | uh which says that whenever you have like this quantity um side guinea um |
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| 213:05 | , what hes theorem says is that uh whenever you have a vector field |
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| 213:12 | varies with XY and Z, you always separate it into two parts. |
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| 213:17 | what hub Holt's term says. Go and check it. We have two |
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| 213:21 | , the part which is uh uh uh divergence free and a part which |
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| 213:28 | curl free, the curl free part the scalar of uh of an existing |
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| 213:34 | for the scalar potential. And this still applies even though this thing is |
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| 213:42 | isot we solved uh uh this equation it's isotropic. But now we need |
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| 213:52 | be more realistic. OK. So us consider the simplest case of |
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| 214:00 | So this is the simplest case of it's gonna apply to thin un |
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| 214:06 | fractured shales and un fractured, thin sequence. So that's gonna lead to |
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| 214:12 | what we call polar anti sore because has a pole of symmetry and all |
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| 214:18 | horizontal uh directions are equivalent. And because as these uh uh these rocks |
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| 214:26 | formed, gravity was always pointing down the downward direction to be a preferred |
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| 214:33 | under that um that assumption where the is polar anisotropic, it has a |
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| 214:43 | symmetry, you can rotate uh uh uh uh around this vertical axis and |
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| 214:51 | get the same properties independent of That's what we have assumed here as |
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| 214:56 | the simplest special case. So right , we've assumed axis one and two |
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| 215:05 | uh soon to be equivalent. And the way, that's the same reason |
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| 215:09 | we have uh these two elements, elements down here for the uh uh |
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| 215:17 | mous. And this element is gonna a controlling P wave, vertical P |
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| 215:24 | uh uh velocities. This one is be in controlling horizontal by velocities. |
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| 215:32 | gonna need these others for oblique he velocities, right? Most of most |
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| 215:38 | our data is gonna be uh traveling . So we're gonna need these three |
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| 215:44 | then we have two different share ways here. Um uh We'll talk about |
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| 215:51 | , the shear wave complications, then get to shear wave propagation on |
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| 215:59 | Now count them up how many um how, how many uh independent parameters |
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| 216:06 | we have? Well, we have true 345 parameters. So I, |
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| 216:16 | regret to tell you that the simplest of an isotopy doesn't have just three |
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| 216:25 | numbers. It has five different So um I I misspoke just |
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| 216:34 | there is a case of anisotropy which only three different numbers scattered among uh |
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| 216:42 | the uh elements here. So that's bad. It uh going from two |
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| 216:48 | two to that other case with That's not bad, but that's not |
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| 216:53 | case which we ever encounter in That's the case of cubic anisotropy. |
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| 216:59 | we do have cubic crystals. If uh as you walk down the hall |
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| 217:03 | class, look at in the uh cabinet there. I think we have |
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| 217:07 | cubic crystals in that cabinet. And they have very simple anti. But |
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| 217:14 | in geophysics, this is the simplest that we have. And it turned |
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| 217:19 | that this one governs the vertical P velocity. This one governs the horizontal |
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| 217:25 | wave velocity. We have two different moduli with two different shear waves. |
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| 217:31 | then we have this one here which repeated here, but it's like a |
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| 217:36 | parameter and it's not gonna uh clo , actually, um this one is |
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| 217:44 | show up in wave propagation, it gonna show up in wave propagation. |
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| 217:49 | gonna name it C 13. we're not gonna name it after lame |
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| 217:55 | Lama didn't know anything about that, only knew about isotopic. So now |
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| 218:01 | is the simplest case, we're gonna more complicated cases. And by the |
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| 218:07 | of the day and of the evening Friday, you will be able to |
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| 218:12 | manage this case and further more uh , more realistic cases. Why do |
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| 218:19 | say more realistic? Because these shas usually not frac uh not the these |
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| 218:26 | usually have fractures in them with preferred . That means the shales have lower |
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| 218:33 | than is indicated here. Uh But is gonna be for un fractured shas |
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| 218:38 | unraced tin bed sequence. We're gonna this matrix as a first step towards |
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| 218:47 | an isotropic wave of now. Too . Even for the simplest case, |
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| 218:52 | quad, it has gonna have a P wave and two quasi estimate |
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| 218:58 | but they are coupled together. That the curl free part of the field |
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| 219:04 | he gave us. That's not a . So we need to have a |
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| 219:09 | math math mathematical idea. But here's beginnings of that better mathematical idea. |
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| 219:16 | gonna go back to the full tensor of motion and we're gonna have it |
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| 219:23 | gonna be the homogeneous equation of So uh it's gonna be this is |
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| 219:28 | be a homogeneous with respect to no spatial dependence on that. But |
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| 219:35 | not gonna be the isotropic special case we looked at before. It's gonna |
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| 219:41 | , it's gonna have these kinds of in it. So we'll take that |
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| 219:48 | on Friday. Uh, uh, we'll be able to read ahead, |
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| 219:53 | this is a good place for us stop for. Now. Um, |
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| 219:57 | can see what's, uh, what's go ahead. We're gonna guess we're |
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| 220:01 | assume, uh, plane waves and , uh, this assumption into here |
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| 220:06 | see if that works. And, , you will see, um uh |
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| 220:11 | gets, it's not gonna be as uh simple and straightforward as we found |
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| 220:17 | rocks. That's the reason why we this discussion of anisotropy until after you've |
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| 220:25 | your minds around uh uh the uh consequences of assuming is arre I think |
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| 220:34 | were a lot of complications that we at here uh with uh um isotropic |
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| 220:41 | . And so now we're gonna see different kinds of complications arising from an |
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| 220:48 | . And of course, we're driven do that because the rocks are in |
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| 220:51 | to drink. If the rocks were , we would happily leave this |
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| 220:56 | Well, let's leave it for the uh for there for today. And |
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| 221:00 | I'm gonna stop sharing this and I'm uh say goodbye. Uh uh You |
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| 221:06 | send me some questions arising from the today. Uh during the week |
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| 221:11 | I will post onto canvas the 10th . I didn't do it before because |
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| 221:17 | didn't know we would need it But, you know, we didn't |
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| 221:21 | it very much. You will have shortly. Um, and, |
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| 221:25 | so tomorrow morning after church you'll have chance to, uh, to read |
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| 221:32 | . Ok. So I think that is a good time for, for |
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| 221:36 | to say goodbye and leave you uh, for Fri until Friday at |
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| 221:41 | . |
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