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GEOL7333-Seismic Wave and Ray Theory - Day6 09-10-2022-Part1
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00:00 So so remember that we um remember that we came to the point
00:32 we wanted to understand the effects of and we relied on this paper by
00:40 mont. Can you see mike uh cursor here let me change. Can
00:48 see my cursor moving. Okay, this important paper by gas monkey,
00:55 in 1961, has governed our understanding the effects of fluids and wave propagation
01:02 all this time over 2/3 of a . And of course we don't read
01:07 . Um I don't read German very . I can see this here,
01:13 know, elasticity of porous media, that's about my limit of Germany.
01:21 , uh this thing was uh translated 2005 and you can get this from
01:30 from the scG bookstore. I think not so expensive. And furthermore,
01:35 are collections of many other papers which be of interesting. Um yeah,
01:45 think there's papers by Lord really. so everybody believes this theory of uh
01:58 Gaz MMA. And here is the right here we say that the sheer
02:04 does not depend on flute content. remember that this is is not true
02:09 say that the sheer velocity don't depend because the sheer velocity depends not only
02:17 the modular but on the density. so the density obviously depends upon.
02:24 let's take this exactly as it's And then for the for the in
02:31 have this expression from gas mark. like it says here, it says
02:40 uh this is cited many times a in our industry and everybody believes.
02:48 um as we said, the huh support of that is weak. And
02:59 here it shows a lot of data lots of different samples showing a systematic
03:03 between the observation and the prediction. if we look at one sample um
03:11 , uh we see that uh scrapings with pressure. And so what it
03:21 is that the discrepancy here here depends um uh micro geometry in the pore
03:30 and which was not uh not included gas mine. He doesn't say anything
03:35 his uh formula about the micro geometry here the micro geometry is important.
03:44 even though we um have this consistent between the theory and the data even
03:55 we believe the theory. Almost everybody this theory anyway, because of
04:00 these kinds of data violate the low assumption that gas might make. So
04:10 sort of ignore these. And since don't have much of any data at
04:15 to um uh have speaking of low , since we don't have much
04:25 we believe gas mine anyway, precise applications at low frequency for a
04:30 long time. But in recent years have come to realize that gas mon
04:41 has a mistake in there. Um we did we still don't know how
04:50 that mistake is. Show you this . Um Right. Yes sir.
05:07 We understand. Oh and in fact have a better example than that which
05:19 here. So this is the expression everybody uh knows from jasmine. And
05:29 is a more recent result from Brown Karenga. And you see they have
05:35 different expressions in here. And uh it argues down in here that if
05:44 were the case that these three parameters savannah, campus supply and campus s
05:51 all the same then obviously in that ground and karina is the same as
05:57 . And in fact that's what they . Don't argue that if the solid
06:02 micro homogeneous that is uniform silent ice topic by the way, ice a
06:09 solid, just one mineral. And got to be an ice and tropic
06:13 , then this is true. And it's equal. Not a problem.
06:18 we know that rocks are really not homogeneous, but most people have said
06:23 these are small effects uh blocks have . These normally people say, well
06:37 just gonna ignore the fact that rocks several minerals and all of them are
06:42 a tropic and all of them are icy tropic. And so we're gonna
06:48 these um uh restrictions. And we're use gas man anyway is going to
06:56 that these three and these things are a shame but when brown and uh
07:04 that argument, they made the same mistake as gasman did when he drive
07:09 in the first place. And so so this conclusion is wrong and we
07:15 use brown fingers result Uh in four . seismic analysis. But before we
07:22 that we gotta understand what are these parameters. And so um that brings
07:33 that's a quick summary of what we about yesterday afternoon. And so um
07:40 brings us to this point here, is the next slide. And uh
07:46 the question how can we determine these in compressibility with a subscript in and
07:55 compressibility to the subject. So the news is that we don't need to
08:00 this at all because we have a result from Brown and Karina. That
08:06 this expression for uh campus of five campus M. And capital of the
08:14 . Uh huh. That's the first we've seen Kampe of the solid since
08:20 stopped talking directly about gas and these parameters in the theory. And now
08:28 see where the solid compression. So you use that result in the previous
08:35 from brian springer, then here it . And you see now no
08:39 If I hear you got a cap , a solid here and a cap
08:44 here. One second. Running I know. So the next question
09:09 is how can we determine these things ? Okay sure. No. Um
09:22 how to determine the compressibility of the . And we just do it in
09:27 same way that we we we did . We were taught by Mr Love
09:31 in 1927. We just extend his to uh heterogeneous solid which have uh
09:41 minerals and all of them are ice tropic. All of them are anti
09:45 topic. But in this we are to assume that they're randomly oriented.
09:49 the rock itself turns out to be cancer tropic. And then straightforward extension
09:55 Love himself that uh response the response the un jacketed sample is given by
10:05 compressibility of the solid. The not the rock was solid because in
10:12 unjust experiment that fluid pressure is the as the confining pressure. Yeah.
10:23 that means we're gonna do an experiment the rock uh in the laboratory.
10:29 of course we're going to there's no in doing an experiment on a rock
10:34 which is not representative of the larger . That is. We are not
10:40 sample every piece of the of the . We're going to take one or
10:45 or three pieces. And we're gonna that those are representative of the larger
10:52 . Yeah. Um Loves him is . However, this is important to
11:01 uh kappa a solid compressibility we determine here. I see that there's uh
11:11 that there's a Mhm. Another I imported some what are some
11:24 I'm another from another lecture. See I was sloppy and doing that.
11:37 this average um uh compressibility is the as we see up here.
11:47 Now after we do this, we to recognize that the solid, the
11:55 selling compressive compressive depends upon the Micra as well as the composition. What
12:02 I mean by that? Uh Let's a a solid with two minerals in
12:10 . Um And think of to ice topic minerals and one is softer than
12:17 other. And uh suppose that the geometry is such that the softer component
12:26 always located within this aggregate. Um the load bearing points within the aggregate
12:36 the stress is concentrated. If those are preferentially occupied by the softer
12:44 then when you squeeze the rock in un jacket way that says here uh
12:49 the compressibility of that solid will be than if the soft component were randomly
13:00 with asylum. And that means that we measure this sample, uh it's
13:14 be subject to that micro micro geometrical that I just said whatever it is
13:20 typically we don't know what it Typically we can't look inside the rock
13:26 figure out what is microeconomics. But already assumed that whatever it is,
13:32 going to be representative of the larger . So whatever it is that that's
13:37 we're stuck with what we've got to that it depends that the compressibility of
13:43 heterogeneous mixture of minerals, depends upon micro geometry as well as the
13:49 Yeah. Uh Here's another note which here is uh experiment on the rock
14:01 fluid is infiltrating the rock from an reservoir in order to to establish the
14:08 pressure same as pressure. So that's hydraulically open test. And what it
14:14 down here, it's okay to use hydraulically open data for a hydraulically closed
14:23 we call untrained context of wave And why is that? It's because
14:28 um the average compressibility is the same both conditions. Now, I see
14:34 that I've uh missed out another little of uh patients. So as long
14:55 we're talking only about the solid, can combine open system data with closed
15:02 data. As long as we're talking about evaluating this part. So this
15:06 how we do it in the laboratory an un jacket test. Now we
15:11 determine uh the mean compressible. Remember talked about yesterday how the M here
15:17 for mean not for mineral. And a formula which we can get from
15:25 the previous page in this um in lecture. And so if you solve
15:32 equation, uh you uh you find mean compressibility is given by quantities which
15:39 can measure. And this thing this is called skimped Inns B coefficient
15:47 after British uh physicist um from the century. No, excuse me from
15:56 20th century. Uh it stands for ratio of the fluid pressure, the
16:04 pressure in an untrained sample. So something you can measure uh put a
16:13 around the sample, you squeeze You measure the deformation of Iraq,
16:17 of uh uh since there's a jacket the sample, the sample is um
16:27 and you can measure the deformation of . And also at the same time
16:31 can measure the fluid pressure and you , what is the confining pressure?
16:36 so this is uh experiment. now we are ready to uh put
16:45 of this expression uh into the wave . And so the formula that I
16:52 back here. So this is a for compressibility capital, That's not what
17:01 want for wave complication. Uh want the inverse iveness. And it looks
17:07 if you try to make the inverse this. The left side is easy
17:10 the right side. It's complicated. if you manipulate that algebra, you
17:15 work out, it's not so And here's the answer right here.
17:24 you can see that it kind of like um gas manz equation except that
17:30 has in addition to the uh the of solid. It's got the the
17:38 compressibility in here. Also, this a kappa and all the others.
17:44 . And so uh this equation of uh reduces exactly to Gaston's equation in
17:54 case that these two are the So are you should uh this one
18:00 yeah, these two are the Then this thing reduces exactly the gas
18:07 and all this discussion turns out to useless, pointless. But now the
18:14 is, is are they the same not? Uh So we got to
18:18 experiments for that and it says that have not yet performed, been
18:26 Uh That's almost true. I have example, but I'm gonna show
18:33 But this is a great uh PhD for anybody who's interested in uh rock
18:41 . And in particular, we have capability to do these experiments at uh
18:48 the University of Houston. Um And open that somebody will take up this
18:59 . Mr wu I think it's too for Mr won't he's too far advanced
19:03 his thesis topic. But for an uh PhD candidate, this would be
19:10 great topic to measure uh this difference a bunch of rocks and see how
19:16 it is now mean compressible. It in principle on pressure. And you
19:24 look it up in any handbook why a property of the rock not on
19:29 solid. If you want to look the properties of solids, you can
19:34 those in the handbook, for you can find the compressibility of
19:39 And since porches anti psychotropic, it's give you um maybe all the anti
19:46 module on it of course. Or it will give you an average.
19:52 Anyway, you can look that up a hand for courts and for tragic
19:57 and for uh all the minerals of to us. And then you can
20:03 maybe you can um, combine those make an average compressibility for the silent
20:12 use in gas mo but you can't that for uh campus of them because
20:17 a property of the rock. uh, you got to go to
20:24 laboratory and good experiments for many mythologies many uh pressure conditions In order to
20:33 a rational application before the interpretation. , imagine yourself working for an oil
20:40 got some 40 data. And so seeing some 40 differences in the seismic
20:46 . And you want to interpret those terms of changes of poor fluids down
20:51 in the reservoir and changes of fluid pressure down there in the
20:58 And maybe it changes a ferocity. , and here you're being challenged
21:09 to use the theory which depends upon rocks down there, which you don't
21:13 a sample of all you've got is seismic data from those in that
21:19 So what you needed to do you need to have a database.
21:23 look up in the database and okay, for walks like these with
21:29 ages and similar porosity and similar pressure . Uh, we can expect to
21:38 um, these sorts of uh, compress abilities. That's right. So
21:49 that database is gonna be uh, um, um, ultrasonic data,
21:55 there's going to be quasi static compression . So that's a real challenge.
22:01 are lots of laboratories around the world ultrasonic. Very few that you are
22:07 . So that's a challenge from the physics community. Well, here is
22:14 only data. Almost the only data I have to uh test whether no
22:22 compressibility is same or equal to the compressible. So there's your data from
22:30 and wang. Uh These guys are the University of Wisconsin and I think
22:38 wang is now retired and he's my . So I think he's retired and
22:44 don't know anything about heart. I've met heart but I've known wong for
22:48 long, long time. Very good . And Mhm. I'm pretty sure
22:56 retired. Uh Not in the frequent with him anymore. He's my
23:01 So he's probably retired but you he might be uh still professionally accurate
23:08 there. So You can see that presenting their data as a function of
23:15 pressure. This is defining pressure and pressure. And all these different data
23:21 have different combinations of defining pressure and pressure. And you can see that
23:27 both these with them the same differential . There's always um same pressure for
23:35 there. And so you can see the solid compression also is almost constant
23:42 respect to pressure. And you expect think that it should be constant with
23:49 pressure. Almost constant but not And so that's a bit of a
23:54 because we're expecting that this should be huh Independent of pressure. And it's
24:04 but not quite. So there's two for that. One is uh errors
24:12 the experiment. And you can see scatter about the line gives you an
24:17 of experimental uncertainties. But who Maybe there is a systematic uncertainty uh
24:26 error in the experiments that will be by future experiments. Uh And so
24:33 line really should be flat. We know that for sure. Here's another
24:39 explanation for why this should be Which doesn't uh cast any uh blame
24:48 partner. Wong maybe in this experiment this rock there was what we call
24:56 process any tiny um bubbles inside the of the sample uh Cut off from
25:07 from the rest of the porosity surrounded mineral on all sides. So we
25:13 that included porosity. And you can that there might be that kind of
25:19 uh in the grain inside the grains the rock. And so as we're
25:24 pressure, that uh included porosity is get smaller of course. And so
25:30 rock is gonna get stiffer. So compressibility goes down so that's possible.
25:39 We don't know. And so we to uh consider uh we need to
25:46 a lot more terms with regard to mean compressibility determining the way that I
25:52 this one is uh definitely pressure dependent we predicted. And the two are
25:59 within the scatter down here, low . But then the mean compressibility gets
26:05 be uh Higher and higher. And the difference reaches 20% here at the
26:13 pressure. Uh maybe that's significant or not. Um We don't really care
26:21 this. What we really care about effect of this on the velocity of
26:27 . And and even in that case don't really care unless we had some
26:33 that this is typical of many And so that means we've got to
26:38 a lot more experiments. Now let point out something more about this.
26:44 shown on this graph are the compressibility for uh race captured with the voice
26:51 of the minerals and the void average the minerals. And of course they
26:55 the world war is hell. Mean going to be halfway in between
26:59 But the theory says and he'll prove a doubt that this should be the
27:05 limit. This should be the lower . But look, the data violates
27:10 . So that's the problem. Um . What could be causing that
27:19 We're measuring these are actual data on lock violate the theater theoretical limits.
27:29 um we've got to understand that. uh again the possibility is maybe that
27:37 experimenters Hartline screwed up badly. But don't think so. Uh You know
27:44 all these data points, every single point violates the limit. So um
27:55 two possibilities. These limits assume that rocks are, I mean the minerals
28:03 that's not true. We know that the minerals are vanishing and of the
28:09 that when we say this is the limit, absolute upper limit. We
28:13 include the possibility that there might be here some occluded process. Just like
28:19 said. So, uh if there included process that could account for this
28:24 difference or the fact that the minerals actually tropic, I'm not the minerals
28:30 an easy topic. We don't have theory corresponding to the theory given by
28:38 that you see here for an isotonic . So, that's another good uh
28:44 . Talking for a student at the of Houston, who wants to make
28:49 name for himself by uh by uh uh these uh so called upper moral
29:00 for the real case where the constituent are a massive topic. And uh
29:07 that's probably a lot easier problem than experimental pieces that I just outlined.
29:12 huh. But it would be if turns out that the anisotropy of the
29:20 makes a big difference. That could that could shake up a lot of
29:28 . So, yeah, I forgot remind you that in most seismic applications
29:42 interpreting for the seismic data, they have this kind of rock physics data
29:47 the laboratory. They rely on estimates this. And so that could be
29:52 big mistake. So here the difference what we should be using and what
29:58 do use in the common application is , 50% bigger. And so,
30:08 , that could be uh, that make a big difference for 40 seismic
30:16 . Never mind the difference between these . The fact that both of them
30:20 outside the work wise limits. That be a serious cause for serious worry
30:28 anybody who's doing 40 precisely. so now, after all this
30:38 we're ready to discuss uh, the to wave propagation and the good news
30:46 that everything that we previously did, can still use all we have to
30:50 is put in there for the p velocity. We put in there,
30:54 undrained um, uh, both models a function of the, what kind
31:03 pressure is what kind of fluid is there. So this is gonna be
31:08 very small number. This is the compressibility fluid. It's a small number
31:14 it's gas, it's a bigger number it's uh, oil and still bigger
31:19 if it's uh brian. And in the same way the density is
31:25 be the undrained density, which we about yesterday. And again, it's
31:30 depend upon the density of the uh, any other, whether it's
31:36 or gas or what. And we decided already, everybody agrees that
31:44 sheer modular should be independent of fluid . So this one does doesn't show
31:50 dependence. And then of corresponding thing the sheer velocity. And so we
31:55 still use everything we did in the seven lectures, all we have to
32:00 is recognize that because we're dealing with , not solid copper. We have
32:06 recognize this dependency and this dependency. of course this thing is going to
32:11 upon uh the amount of the ferocity the pressure and so on. So
32:17 of those implicit the dependency coming from hydrogenated. Uh huh. I'm gonna
32:25 implicit in the front and the module that we used module I and the
32:36 . Oh, that's really good We did not waste our time.
32:41 seven weeks. Um let me say a few more words in the case
32:51 partial saturation, which is uh pretty . Um when we um We have
33:02 in the pore space of a it's usually not um 100% gas in
33:08 pore space. It's usually uh certain with gas. And then usually there's
33:17 brian also in the pore space. often there's also oil in the pore
33:23 . And so it might be that these cases, um it makes a
33:32 difficulty. Uh it might make a difference, but uh normally what most
33:39 do and what I know about so is that this simple discussion of the
33:47 of partial saturation is good enough and don't have to worry about other um
33:57 complications than is shown here. Let just show you that because because we've
34:04 assumed that at low frequency the food is equal. And so we're gonna
34:10 that same assumption here. We're gonna that if there is a mixture of
34:14 three fluids in here, the pressure all those three fluid is gonna be
34:19 same pressure is gonna be the But of course, they have their
34:23 uh in compressed pulses. And this for brian Boyle in for gas and
34:30 that's all different. Um but for , until I've thought about this problem
34:38 more, I'm going to assume that discussion of the effects of partial saturation
34:44 good enough. And here's an implication that If you have only a small
34:52 of gas, suppose the partial set saturation of gas is only 1%.
34:58 that means this number here is only . Then even if it's so
35:10 this term is gonna dominate because these gonna be negligible, because this wanted
35:17 here is so small compared to Uh you can imagine that if the
35:23 in the pore space had the income of air breathing right now, this
35:28 be six orders of magnitude smaller than . But of course it's not,
35:33 not like that in the reservoir because there's a high ambient pressure in the
35:42 on the fluids uh coming, you , from the, from the overburden
35:48 . And so under those conditions, a high pressure in the reservoir on
35:54 fluids. This thing is uh the pressing of the brine is pretty much
36:00 same as the incan principality right at surface, you know, water that
36:06 drink, It's got in there some , uh we call it brian,
36:10 it's pretty much the the income possibility seawater this one. And this one
36:18 a lot with with the pressure. so that's uh consideration which is outside
36:27 scope of this course. And there's a high temperature inside the reservoir
36:32 in that case, uh hydrocarbons down have physical properties which uh can be
36:42 different from what you're familiar with here the service, you're familiar with motor
36:49 that you put in your car. oil down in the reservoir is not
36:53 that. And you're familiar with with gas that you breathe or you smell
37:00 surface. The gas in the reservoir definitely not like that. But there
37:04 people who make that specialty. And after dealing with all of that
37:10 we can still say uh this oxen relationship is still approximate. Now,
37:20 affects the velocity strongly because of this here. This is the expression that
37:25 showed you with the brown Karenga term there. But we didn't change this
37:29 here is the inverse of the compressibility the fluid. That's right here,
37:37 . It's here and approximately equal to . So that's right there. And
37:44 a previous page, you know, velocity has this in compressibility right in
37:49 . So all of this means that a strong fluid dependence on the
37:56 In other words, uh in a little bit of gas makes a
38:00 difference in the p wave velocity. of that, seismic data is not
38:05 good quantitative predictor of gas saturation. , little bit of gas makes a
38:13 difference and you add more gas, doesn't change that much. And
38:18 uh, measure VP maybe measure um changes is not a good quantitative
38:27 Gas saturation. And so there's a economic uh application of this Because we
38:37 want to drill into a reservoir and 1% gas saturation. We want to
38:43 some large gas saturation, 20% or , something like that. Now it
38:50 true that small amounts of gas make small difference in the density as we
38:59 about yesterday, it's very difficult to the density from seismic data. In
39:08 , you can do it using a analysis but in practice, it's proven
39:13 be very difficult for reasons which we about yesterday. A little bit.
39:18 requires the measurement of the Avio curvature that is a parameter which is determine
39:29 only with great uncertainty in most seismic because um uh the amplitudes as a
39:38 of offset show a lot of scatter the uh systematic trends in that Avio
39:47 behavior as we describe the reflectivity. a simple um there's a simple expression
40:01 the amplitude as a function of offset this function of angle. But what
40:09 measure is not reflectivity. We measure applications and they have a lot in
40:13 are a lot of effects which are due to reflectivity. And we have
40:19 it not practical in many cases to analyze the current return and analyze the
40:30 term, especially a measured relative to interceptor. And we empirically find that
40:39 we analyze receive seismic data with a and it's guided by the reflectivity
40:49 Even though we know we're making a of mistakes, we do find that
40:54 we find uh it reduces our risk drilling dry holes. So that's been
41:03 fantastic advance in um expression geophysics accomplished my years and as I said,
41:12 was the inventor inside an echo of leo. Uh but we were preceded
41:19 experts from mobile even with all the that have been made since then we
41:29 can't measure the uh density itself very . All we can measure p wave
41:38 and how it changes uh measure. you can't measure, but we can
41:53 how the radial gradient change how the uh emphasis change with offset in the
42:01 upturn advantage if the p wave velocity Iraq decreases significantly when the saturated with
42:13 . That's what this says here. is the uh, the velocity of
42:19 gas saturated rock compared to that of brian saturated rock. And because this
42:25 is so much less than this because of the effects that we just
42:31 about. We have this big inequality . Maybe this is taking too extreme
42:41 . Maybe we shouldn't have reduced the undrained compression in untrained in compressibility all
42:52 way down to the frame compressibility. the gas is still supporting some of
42:57 load. Why would that happen? because uh progressive of the gas may
43:09 be negligible under reservoir conditions, high and high temperature. That's a topic
43:13 chemical engineering. Not in sure. of this argument here, this leads
43:22 an ominously bright reflections, anomalous Avio . And it makes 40 seismic surveillance
43:32 because of that block physics argument, can do time lapse seismic surveillance of
43:40 effects of our production on the reservoir therefore it means more efficient discovery and
43:49 of private coverage. I can tell that billions of barrels of oil have
43:53 found and billions of barrels of oil been effectively produced over the last 30
43:59 because of this argument. Lots of . I think this is a good
44:08 to stop for now. Work. what I want to do for
44:13 let's see what time it is. 9 45. Let's um stop here
44:20 reconvene at 10 o'clock sharp and we resume this argument which is coming up
44:26 right here, sure that I'm gonna sharing my screen and now I'm gonna
44:37 sharing my video. Okay, so I want to remind us about how
45:03 facts and rock physics get folded into wave propagation. You have to stop
45:16 . I'm gonna have to what? ? What? Mhm. Okay.
45:49 huh share my screen again. All this time. Yes. Okay.
46:05 in presentation road. So remember when talked about reflectivity, We uh we're
46:15 in lecture six. It seems like long time ago, uh we analyzed
46:20 reflectivity intercept and gradient and uh the uh simply depends upon the fractional jump
46:30 the p wave impedance because this p reflectivity we're talking about and the gradient
46:37 depends on on the fractional difference in wave velocity, not impedance but velocity
46:44 this term here. So, what have here here is the jump in
46:48 modules across the reflecting horizon. And is uh uh factors, scale
46:56 And you see that this number is going to be so much different from
47:01 supposed that the V. S two ratio is one half, so multiplied
47:07 two. That has to be one he squared and stillman. Well,
47:13 uh uh that's uh an approximation of BP is one half. Uh That's
47:22 approximation, which is not so bad the rocks of the deep deep
47:28 But for crystal rocks a better assumption this velocity ratio is about one
47:35 Then we have two thirds uh squared four nights, which is less than
47:41 but still you know, close to . And uh so with that
47:47 what we do is we go into laboratory and measure a bunch of
47:50 measure V. P. And S and density for a bunch of
47:54 and then sort of take them uh by two and say, ok,
47:59 assume this is the this rock is incident rock and that rock is the
48:04 rock. What do we get for jump in sheer modules and the jump
48:09 uh BP and the jump in impedance what we find from all that is
48:17 this term dominates. Usually this term bigger than any of these other terms
48:22 with a minus sign, that means the gradient here has the sign opposite
48:30 uh the impedance opposite to the to to the innocent because of this minus
48:38 . And because this term dominates. that's for brian, that's for ordinary
48:45 , most interfaces are going to be that. So that if it's if
48:49 if the reflectivity is positive at normal , it gets to be less
48:54 Maybe even change of sign goes negative offset. However, for for interfaces
49:03 has no little logic contrast at only brian uh only gasp sitting on
49:11 of brian within the pore space of reservoir. And you can imagine that
49:15 imagine a reservoir with an antique Lionel and there's an oil water contact halfway
49:23 through the reservoir. And so that up uh that kind of thing shows
49:27 on the seismic data and a flat and also anomalous li bright uh usually
49:37 um uh then now let's think about Avio behavior of that anonymously bright,
49:45 flat reflector in that case. What just learned is that the sheer modular
49:52 across that gas brian interface zero. video taught us that uh the sheer
50:06 of Iraq does not depend upon the of fluid in this interface that we're
50:13 about. The only difference across the is the food content. So this
50:17 a zero. So now what that is that the gradient is uh determined
50:25 the leading term and it has the uh sign as the intercept. Now
50:34 the real world we're gonna have lots interfaces where we have both fluid content
50:39 mythological little with a logic composition changes the reflecting rise. So that's what
50:47 says here. And this this is slide which um used to introduce the
50:56 example about uh showing a software um from some BP software that was active
51:07 uh back before I retired from I would presume that they have more
51:13 analyses at BP and at all companies than then, but uh that uh
51:21 analysis is driving all of A. . L. Analysis today. This
51:31 and uh elaborations of it more um collaborations that is what drives all of
51:41 analysis today. And companies have been it successfully to reduce risk uh in
51:53 for decades now. See I came the business and uh mid nineties so
52:03 been in this business now for 25 and and all that time. Uh
52:09 have music a pr analysis to use making billions of dollars, maybe even
52:19 trillion dollars by now of money for companies. And uh not only in
52:32 but in development as you as you to understand what's happening to the reservoir
52:39 production and during development of the Uh These rock physics ideas are critically
52:46 for understanding. Um well fine on . So Miss Del Rio questions
52:58 According to gas mont, the fluid of the longitudinal Michael's M. Is
53:03 by this family. Is that true false? This was true. How
53:12 it be true. Gasman was talking K. Not him. So I
53:18 that I literally wrote on my Why is this an M. And
53:21 A. K. I don't know I said true, it's false.
53:26 you need more self confidence here. should have said uh no sir.
53:33 answer was right for the first And here's the reason because for this
53:38 here that's K. plus four thirds and drain and this is K plus
53:43 thirds mu frame and the mute part out because according to gas mont.
53:51 those new parts are the same. here uh are you following? So
53:57 this is actually true as given by . He he told us that the
54:04 new parts of em are the same and frame. And we still believe
54:11 I believe that although uh you we have some data that um put
54:18 doubt on that. But uh I'm say that I expect that that part
54:24 for elasticity is going to remain And the only thing that's gonna change
54:30 uh the details of this form. your first answer was good. But
54:35 it is um it does require some as you vote on your notes uh
54:42 our analysis about gas mont um was terms of K and kappa. And
54:48 uh this is combining his results for . With his results for em from
54:54 . And this is still true and is what we need for p wave
55:00 . Sure. Next question. Um wu uh what's your answer to this
55:15 ? Mr wu are you with? , I would call that truth and
55:22 because I really criticized gas bound for thin um experimental proof. But the
55:32 that I talked about how experimental proof is even thinner. So we really
55:37 need further experimental confirmation. And I'll you that theory always has assumptions in
55:44 . And whenever a theory and experiment contradict each other, experiment always
55:53 Uh that argument now it could be can't argue. Well, the experiment
55:58 wrong for this and this reasons. but after all that um discussion about
56:04 quality of the experiment is over. experiments gotta take precedence over the
56:10 And if the uh experiments after being violate the theory, it means that
56:18 got to modify the theory. We've to uh go back to the theory
56:22 think about the approximations were made and were made in the theory and say
56:27 ones are those? Um uh could wrong. And journalist the theory
56:35 That's the way science progresses. So number three, uh This is for
56:43 , Miss Del Rio. So using what gas line said or what Brown
56:47 Karenga said, the presence of gas the forest can lead to significant reviews
56:53 lead a significant reduction in p wave and and mps, is that true
56:59 false? That was true. And that is uh that's gonna that
57:05 gonna be true no matter whether the which I discussed in the last few
57:11 is uh turned out to be important not. This uh statement is going
57:16 remain true and we're still gonna be to use a vo two uh increase
57:24 to decrease the risk in grilling and improve the efficiency of development.
57:33 so uh mr wu uh here's the question about shear wave velocity and
57:46 Yeah, that one is is It will make small differences but not
57:52 significant reduction. And the small differences from the density term, not from
57:57 share model. Okay, now, I started this discussion, I said
58:12 Bill taught us that because of the genetic, because we got porridge as
58:19 as grain, we're gonna have a new type of wave propagating in the
58:28 . But then I went up and about a bunch of implications. Uh
58:36 new wave in addition to the ordinary . So, I've been so between
58:40 and now we've been talking about the waves and we've seen that it makes
58:45 what I would call minor changes to understanding from the first seven lectures.
58:52 we recognize that these model I and density depend on lots of things like
58:59 and porosity and pore, fluid content so on. But that's what I
59:03 call that minor here. Uh We up a major new idea that video
59:09 said there's there's a completely new type wave which can propagate in these rocks
59:17 of the presence of ferocity. And type of wave is called a
59:22 O. Slow wave. Here's mr oh, again dr bot second wave
59:31 from the heterogeneity in the rock. uh these kinds of waves are actually
59:38 by uh famous german physicist named max back in the early part of the
59:45 century. And uh he did a simple example where he considered you have
59:52 spring and you have beads on a like this and you say uh compress
60:01 on one end and what happens? all the springs are gonna compress back
60:07 forth and the beads are gonna be back and forth. And so in
60:11 very simple case, uh what Born was that when all the beads move
60:19 of together in phase, that makes ordinary mode of sound. But when
60:24 move out of phase, that makes additional mode of sound Discovered by born
60:31 1928. And so for example, this one is moving this way,
60:35 one is moving maybe the other And can you see that this this
60:40 has a different size and mass than bead. So these two beads can
60:45 moving either together or opposite. And born called this an optical mode.
60:51 the reason you called that because if have electrical charges on them, then
60:57 can have a say plus charge here a minus charge here uh as those
61:02 um two different charges move relative to other. They uh emit optical ways
61:12 of the electromagnetic effects of changing um distribution of electric charge. So these
61:23 are say if these are moving together apart together and apart and so
61:27 That sheds optical light if they're And so that's why he called it
61:33 optical mode. So this is exactly to what B. O. Found
61:39 later for a rock. But it in the same for the same
61:43 Because of the hydrogenated. If these all the same, that wouldn't
61:47 But because these are different. It in the simple model and it happens
61:53 rocks because of the hydrogen rock. what we have been uh discussing is
62:03 sound when the fluid and the solid together in faith move out of
62:09 That makes what we call a. . O. Slow wave. It
62:14 only at high frequency. It's not show up in size request. So
62:25 was predicted in 1941 And was not for years and years later. Until
62:33 smart guy at a slumber did a experiment in 1980. And he actually
62:39 this. So Tom Polona is younger me. So I think he's probably
62:45 working for slumber Mike recently. I'm 80. I would guess that tom
62:51 65 or so. And recently Very clever guy. He was working
62:56 Slim Bridges Research lab which is now those days, was located in Ridgefield
63:04 , not too far from new york . Uh These days it's located and
63:12 , oh close to Harvard and I. T. And I think
63:20 no I made I think that move made from uh Ridgefield Connecticut to Cambridge
63:26 about in Around the year 2000. so this work was done in Richfield
63:34 uh Richfield was they they did very work there. It was sort of
63:40 the bell labs of the oil Now this wave travels very slowly.
63:49 at that a 10th of its per and it gets attenuated very rapidly.
63:55 it has uh has a quality factor than one. Now we haven't talked
64:02 attenuation in seismic waves yet we will that. But uh anticipating that
64:09 you probably know that in seismic waves we measure, we characterize it in
64:15 of quantity, we call Q. you can remember that sort of
64:19 Stands for quality. And so uh a a brass bell has a very
64:26 quality. Very um thank you. a bell made out a rock uh
64:35 ring like that. Just imagine a carved out of a piece of
64:41 And you hit it on the you're gonna get a clunk instead of
64:44 ring. And the reason for that uh rocks have much lower Q.
64:50 but for rocks the Q. The . Factor somewhere like um um 50
64:57 20. And so for these rocks Q. Factors really low less than
65:03 . And so we never detect Uh this be oh slow wave in
65:08 field. Nonetheless, it is important seismic since every sentiment or interface,
65:17 of the um ordinary energy is converted be all slow waves. Remember we
65:23 an incident P wave comes in and a reflected and converted share wave and
65:28 reflected and converted. Um P wave , uh reflected and transmitted. P
65:37 and reflected and transmitted share. But we didn't mention anything about the
65:42 . O. Slow wave. But according to be, oh some of
65:48 incoming energy is going to be converted be oh, slow wave. So
65:52 gonna be having outgoing, reflected and slow waves three ways up and three
65:58 down at any uh poor elastic Now, uh since we're uh doing
66:08 uh want to consider especially seismic band . So this effect is quite small
66:18 um for those frequencies much uh stronger too slow waves and higher frequency,
66:28 it's not zero. And so uh uh it's not zero. This uh
66:36 we're never going to measure these Right? So we have a reflected
66:41 . O. Slow wave in addition the reflected the wave, that slow
66:45 is gonna continue ate itself away as travels just a few, just a
66:51 centimeters maybe a few meters away from interface. So we're not, we're
66:56 gonna observe it. We're gonna observe reflected converted shear wave but it's gonna
67:01 in later than the reflected the But we never ever observed this because
67:07 um continuation so rapidly. And it's very uh strong amplitude to begin
67:16 Because uh frequency you mentioned the efficiency conversion the principle in frequency incoming where
67:26 was going to be low frequency. so this effect is going to be
67:30 and it's gonna dissipate very rapidly. it makes uh it's going to make
67:36 effective mode of attenuation. We'll talk modes of attenuation in the next
67:46 And this could be done in an in a way. And you see
67:51 has nothing to do with standard Avio . So um even though we never
68:01 observe it, it's happening and it's affect the data that we do
68:12 Why didn't we talk about it Because earlier uh we did cora lasting
68:19 . Uh Ordinary ways we assumed uniform pressure at low frequency. Yeah.
68:27 frequency when the horse road does vary the scale of the grains. That
68:34 that the fluid is gonna flow in to that. Uh And so it's
68:40 make um we call fluid squirt and gonna make um continuation of some of
68:50 energy is gonna go into viscous dissipation the fluid because the fluid is squirting
68:56 inside the pore space. Um But affecting the ordinary waves. Uh And
69:04 haven't talked about that yet, We'll about that kind of attenuation in the
69:08 lecture. But that same theory produces slow waves. And there is an
69:13 um um loss of energy from the wage gets converted to be Also here's
69:24 fact about these slow waves. Uh german by the permeability of the
69:31 So, you know, and furthermore microscopic permeability of the rock. If
69:38 have a rock which is highly you know, like a sandstone but
69:42 there's uh there's a layer boundary uh rock on the other side. That's
69:52 what I mean here. And if if it's the same kind of rock
69:57 a fault nearby filled with gouge, has uh which restricts the flow of
70:04 on on the reservoir scale. that's not what I'm talking about.
70:08 talking about the microscopic permeability of the that um um that's going to affect
70:22 velocity of the real slow way and going to affect them amount of energy
70:29 is converted out of ordinary ways into wave. And of course what it
70:35 is there's going to be in the for there's gonna be an additional physical
70:41 in addition to the K. And U. Of the fluid in the
70:46 fluid. Uh traditional parameter which um haven't talked about, we will not
70:55 about in this course on when the flows locally on the grain scale.
71:04 I said during the passage of the this type of flow is called results
71:12 . Which we're gonna talk about um . Okay let me see how our
71:22 . 33. Um I think this a good place to break for 10
71:32 . So even though we just came from the break, this is a
71:35 place to break. So when we back in 10 minutes we will take
71:41 mr wood. This is a good to stop your recording recording. Okay
71:49 this begins uh the ninth lecture. are a bit ahead of time.
71:54 think it's because of uh normally in class we have a lot more discussion
72:00 class questions. And so normally that it. Normally we're beginning this um
72:08 shirt um in the afternoon session So we're a couple of hours ahead
72:15 time. That's good. We will us time for questions from you
72:21 And there's this time for more discussion anti socks. A paper which of
72:26 is my favorite subject but uh I'd be addressing your questions so feel free
72:35 interrupt me at any time. So at the end of the previous
72:45 we talked about uh fluid squirting inside pore space uh of a porous rock
72:54 a wave is passing through and how causes attenuation. So we haven't mentioned
73:01 much in this forest but uh it's is important for reasons which we'll discuss
73:10 in a few minutes. And at end of this uh lesson you will
73:15 able to explain how hook's law gets to include attenuation. You will have
73:22 that hook himself didn't know or care about attenuation. Everything we've done up
73:29 this point is uh the first set lectures anyway, we're applying hook flaw
73:36 there's no generation in it, but gonna modify it in in uh in
73:43 but simple ways fundamental. Simple ways include continuation. And so of
73:50 once we modify hook's law, that's modify the wave equation and it's gonna
73:55 the solutions to the wave equation. so we're going to get waves which
74:01 as as they propagate, whereas before did not have. So those are
74:07 solutions that are verifying. And then we're going to also think about how
74:14 affects the reflection of ways. So is gonna be an interesting point we're
74:19 discover, for example, that if have an interface with a perfectly elastic
74:24 bird and a reflecting and and and and reflecting formation below the interface that
74:36 is gonna affect the reflected p wave though that reflected p wave never enters
74:43 attenuating medium. Right in the scenario said perfect elasticity above attenuation below looking
74:51 a reflected wave which never ever enters attenuating medium. But even so its
74:58 . And it's uh wavelength is gonna affected by that consideration which had never
75:07 directly interesting, isn't it? Here's an important point which was only um
75:18 briefly that there's gonna be an intimate between continuation and dispersion. Now,
75:27 we talk about elasticity, we never about a mechanism of the last
75:33 We just said get these module Now. When we talk about
75:38 we are going to talk about mechanisms I just mentioned one of them uh
75:43 the last lecture fluid squirt. So are others, but there is a
75:50 of attenuation that we never needed to about that before. This also there's
76:00 a there's um an effect called apparent which is a purely elastic effect but
76:07 results in loss of high frequency. And so we call that apparent
76:14 And we're going to discuss that in election. We already actually we already
76:21 it earlier in the section we called multiples. Remember how we talked about
76:27 the friendly multiples uh result and a of high frequency of the propagating wave
76:34 . So that looks like attenuation doesn't though there's no energy loss. So
76:41 not true attenuation, it's just uh converted from uh or frequency ac frequency
76:50 frequency uh in a propagating wave because the complexity of the medium is
77:00 We'll talk about that later. so except for a lecture eight,
77:09 of the Four gun has been classic equally suitable for exploration or for understanding
77:15 deep interior of the Earth. Now know that none of it is truly
77:21 for exploration since it ignores the effect attenuation. Now, I'm gonna uh
77:30 that you all have seen the effects attenuation in the limited amount of seismic
77:37 that you've seen so far in your . And the way it shows up
77:42 the data is at long recording You have low frequencies, lower frequency
77:50 a long time and lower amplitude. the amplitude gets lost partly because of
77:57 spreading. So the same energy gets out over the, expanding away
78:04 But also we lose amplitude because some the amplitude gets taken away from the
78:12 elastic deformation and it gets turned into locally by attenuation. And this happens
78:19 a way which is dependent on whereas geometrical spreading does not depend on
78:26 insinuation does depend on freedoms. So long reporting times you have fewer high
78:34 and you can normally see that with eyeball, look at any reflection seismograph
78:42 real data that you see and uh at the uh look at the wave
78:50 arriving at long recording time, will see that they have lower frequency content
78:56 the earlier ones and they might have attitudes because probably somebody has uh amplified
79:09 the traces at long recording time. it's uh it's usually done with an
79:18 algorithm which increases the game of the before uh gets to your workstation
79:29 And that's so you can see And if they didn't do that,
79:32 you would see is at long recording , you would see very low amplitude
79:39 . Uh you probably could make much better than with your eyeball. And
79:44 you know that if the amplitudes at recording times are comparable to the amplitudes
79:50 recording times, somebody has applied some of a gain function to increase the
79:57 of those uh long uh those traces long recording times so you can see
80:05 . So don't trust the amplitudes for reason. Now here's a thought,
80:11 this gain is applied as a function time um um independent of officer.
80:22 a common uh common thing. And so what that means is that uh
80:28 offset traces with long move on, a higher gain applied to them because
80:38 know, those far traces are appearing coming in a longer times. So
80:43 gonna be gained up more by this process. And so uh the amplitudes
80:51 been adjusted in the computer, not the earth but in the computer by
80:55 that you don't know who got his on the data before you did.
81:00 so uh there is uh an effect received amplitude as a function of
81:07 which has nothing to do with And that's an example of how received
81:14 uh differ from reflectivity ease. And when we um analyze uh received amplitudes
81:24 I showed you two or three lectures , Lecture six, I showed you
81:29 BP handled uh Avio handled real data for a VR effects without taking
81:38 Things like this. And it's just example of how uh received amplitudes during
81:48 offset for reasons that had nothing to with rueful activity. And we should
81:55 that in our mind as we look uh such data clues about what's happening
82:03 reflectivity. And it turns out that um make lots of lots of mistakes
82:11 that. And even so we can a B. O. To find
82:17 to this risk in finding out because developed workflows which find anomalies even though
82:29 um even though all these non even all these effects which don't depend on
82:39 are in the data, we still workflows that help us to find anomalies
82:45 that's what we're after anomalies in the of with this characteristic because we know
82:54 the currents of the world is It's not common in the subsurface places
83:00 the subsurface where we have accumulations of department are not normal. They're
83:06 There are normals. And we can these enormous places by using a workflow
83:11 is guided by oversimplified theory and it for us anyway to find oral.
83:20 kind of remarkable. And it's also attenuation. And so that's also kind
83:26 remarkable in this lecture, we're gonna talking about continuation. And when you
83:31 about it, it really is a thing that uh we have a generation
83:38 otherwise all the sounds that have ever made on earth would still be echoing
83:43 Of course they're going to be Consider an earth without any attenuation and
83:49 that um um all the sounds that ever made as they radiate away from
83:58 source, uh we're gonna spread out and so they're gonna be weak.
84:04 there's still gonna be echoing around inside earth every time a dinosaur stomps on
84:09 ground. That's gonna make a sound goes down into the earth and echoes
84:13 and would still be present with us this circumstance. Under this assumption.
84:18 insinuation. The only thing that would happen to that sound as some of
84:24 would come reflecting back to the surface get ready through the atmosphere and go
84:31 to outer space. But most of gonna still be inside echoing around.
84:36 could not do our scientific experiments today of that sound. So it's a
84:43 thing that does attenuate away. But it's a good thing that the attenuation
84:49 weak. So we're going to find that that statement waves lose their attitude
84:57 a small fraction in every wave o a large fraction then we would not
85:02 successful. And using seismology to find and gas or to explore the deep
85:08 of the earth. Either we'd have do something else, maybe electromagnetic
85:14 Um and so in fact that does . In fact, electric electromagnetic waves
85:21 propagate inside the earth. This course about size mints, but I can
85:27 tell you that using very similar you can use electro Magnetics to explore
85:35 the air. And there's uh two between electric electromagnetic exploration and seismic,
85:45 should say this way. There are differences between electromagnetic wave inside the
85:51 An electromagnetic and the seismic waves inside electromagnetic waves travel with a velocity,
86:00 is not so uh, which is to seismic waves. Despite what most
86:06 think seismic way, electromagnetic waves inside earth don't travel with anything near the
86:13 of life. They traveled with velocities are close to the velocity yourself.
86:20 the difference. They are highly disperse instead of this and they are highly
86:27 instead of weekly, genuine. Otherwise very similar. And so you can
86:32 the electromagnetic equation in very similar ways we've done so far in this
86:40 all of it and all you have do is recognize that those electromagnetic waves
86:47 highly attenuating and highly dispersed. so, um, since they're highly
86:58 genuine, they lose their high frequencies . So that means if you're gonna
87:05 uh exploration of the earth at high , you're only going to get penetration
87:14 a few meters or a few tens meters. And so that is called
87:20 penetrating radar. So we're operating at frequencies, But the waves are traveling
87:26 inside the earth, not at anywhere the speed of light, but at
87:31 comparable to the speed of sound, attenuate rapidly. So the uh you
87:38 get any reflections from those waves deeper a few tens of meters. So
87:44 good for, you know, finding uh coins lost coins in the dirt
87:50 it's good for framing unexploded bombs which and uh and much which are buried
88:00 the nearest sub service. That is good for finding oil because it doesn't
88:06 far enough. But you can use the same idea as that low frequency
88:13 those waves can travel down a couple kilometers and back. And um those
88:22 uh used to find subsurface oral reservoirs they have a big advantage over seismic
88:32 , which is that if there's a bit of gas in the reservoir,
88:37 makes a small effect on the electromagnetic . Whereas it makes a big effect
88:42 seismic waves that we're talking about. that you can maybe distinguish between uh
88:49 and non economic saturation of gas using techniques. The bad thing about electromagnetic
89:00 they have such low frequencies by the , the frequencies that they use are
89:05 11 cycle per second way lower than secretaries. You have to have those
89:13 frequencies in order to get the deep . And so when you have those
89:18 frequencies, it's gonna mean a lot large wavelengths. And so the resolution
89:24 of electromagnetic exploration is a lot And sorry. Mhm. So both
89:35 and bad things about electromagnetic exploration and bad thing is come mainly because of
89:45 generation. So for uh for electromagnetic , the attenuation is uh um it's
89:55 . It's not a property of the . It's a property of the equations
90:00 it has a q factor of one by two. It's one half it's
90:05 a physical property of the materials. a property of the equations. And
90:12 uh very low to high attenuation. uh so that's why it's not the
90:24 means for exploration. Seismic is generally , usually better most of the
90:31 But that doesn't mean to all the and electro Magnetics to make a very
90:36 contribution to the exploration problem in many . But our topic is seismic.
90:43 uh we have we continuation, which gonna mean a few factors much bigger
90:50 one. And we're gonna see that generation is always accompanied by dispersion,
90:57 I'll remind you is the dependence of on frequency. Because the attenuation is
91:04 dispersion is also usually weak and it's that you can't even see the difference
91:10 velocity between the maximum frequency and the frequency and the seismic bandwidth. So
91:20 that introduction, let's turn our attention hook's law. And turn it into
91:27 quasi elastic and perfectly elastic quasi elastic attenuation in hooks. So, here
91:40 uh description, a grand book says if you uh graf strasse versus
91:50 you're gonna get a straight line stress proportional restraint with some sort of up
91:59 this line. And as it's we call this slope the compliance derivative
92:06 with respect to stress, that's a . Oh, now here's a question
92:13 I posed to you earlier in this . Does the stress caused a
92:17 Or does the strain cause the So, who did not know?
92:23 care here is um Here's a picture . There's no causality here. Then
92:33 apply the stress. The strain happens , or vice versa. When you
92:39 the strain, the stress happens What didn't know or care.
92:45 let me pause here and ask for for uh mr wu to tell us
92:53 does he think that stress causes strain the strain causes stress at work?
93:10 mr. Let me let me interrupt because of your other course that you're
93:15 monitoring. We're having audio difficulties. so I'm gonna uh I'm going to
93:26 excuse you from answering that question. uh we'll have another chance maybe to
93:33 about your thoughts about this same But because of that interference, audio
93:40 , we can't hear you very So, I'm gonna ask the same
93:44 for Miss Del Rio. Uh Does think that stress causes strain or the
93:50 causes stress and women, I would that this stress would cause the strain
94:04 you're because why? Because like you're some type of force really that's causing
94:12 strain on the material. Okay so hear that and I would say that
94:18 answer is very typical. Most students will answer this way. But think
94:23 it this way I think you're you've a spring in your hands and what
94:29 just said was you push the spring the spring strings. But think of
94:36 the other way you impose the strain the spring and the spring pushes back
94:42 a force. So both of these viewpoints are exactly the same according to
94:49 . Hook and how are you? you know one of them has to
94:54 the other, we can't have instantaneous to anything because of a lot of
95:05 . So um how are you going resolve this dilemma in the in the
95:12 that I just said uh imagine a in your hands and you're just forming
95:21 . Are you applying stress or are applying strength? It's a bit hard
95:31 say, isn't it? The response the response is is really quick.
95:37 so you can't tell with your fingers what's happening there? Okay. So
95:44 uh let's this is a sort of . We're gonna answer it with experiments
95:51 with theory but with experiments but the has to be delicate. We're gonna
95:57 to go into the laboratory in a way and do some serious experiments.
96:01 can't just mess around with a slinky our hands. And uh so uh
96:08 diagram representative theory is not good So real materials behave more like this
96:16 , this is a cartoon, but more uh it's more like this have
96:20 same stress or strain, but instead a straight line here, we have
96:24 called a history since loop. And uh as time in So this is
96:31 cycling. Um the stress and strain Iraq. Just imagine in Iraq your
96:37 to squeezing it or your cycling. whatever you're doing and you're doing it
96:45 as time increases, that's the direction these arrows. Time increases following the
96:52 . And so what it says is uh the rock is going to respond
96:57 like this with an open loop, a straight line. And observe
97:04 The point of maximum stress is right . And the point of maximum strain
97:11 a little bit later as it comes uh point contest. And so what
97:19 means is that the stress, the leads the strain. The strain follows
97:26 stress. So that means that stress causing the strain. Thank you,
97:38 . Just like you said that to that we have to establish the strain
97:42 later. And so this time the isn't very much, but you can
97:46 what you can see is that you can see in the in the
97:55 , you can see that the history like this, it goes in this
98:01 direction never goes the opposite way. so that means that when you're uh
98:11 when you uh so the area inside loop is the difference between the energy
98:20 you put in and the energy that got out. So according to a
98:26 , there's no in it, there's area here. It is according to
98:32 . So that uh when if you this in a military way, you
98:39 out everything you put in. But you do it on real walks,
98:44 always get out less than you put , which means that something else happened
98:48 the energy it got turned into That is, um uh an application
99:00 the second law of thermodynamics. If rock operated in the other way so
99:06 you uh got out more than you in, that would mean it was
99:11 heat out of the uh, out the rock and going to get into
99:17 the defamation. And that doesn't happen to the second law of thermodynamics.
99:24 these histories and swims always go in direction. And so, uh let's
99:35 uh what I said is that the compliance is given by the the
99:41 of this. Uh But let's think uh the slope of this curve,
99:48 instantaneous slope. So you can imagine if you have an instantaneous slope,
99:53 change it to here. Following my . That's on the way to increasing
100:00 . Then on the way, decreasing , it would be following this tangent
100:04 here, which is a difference. you see the compliance, the apparent
100:11 depends upon your loading or unloading the . And none of this was included
100:18 our analysis of uh books law because law assumed that this history viciously loop
100:26 completely closed, just a straight line of this loop, which is so
100:32 can see that that's gonna make um everything we did in the first seven
100:40 , because now we see that it's false. Uh the assumption that we
100:46 the whole thing on was hooked. now we can see that real rocks
100:51 behave like said. And of course is just a simplification. Real rocks
100:57 more complicated behavior than this. But this is uh hello aspect of real
101:09 that I want to custom now up uh rocks for materials like copper and
101:17 . This history since luke, uh close to a straight line like hook
101:23 . But for real rocks, it's it's obvious in the data. You
101:28 have to look too hard at your to see this kind of effect difference
101:33 loading and unloading in uh a cyclical . And of course it's easy to
101:41 . Uh It's easier to do this the laboratory at low frequencies, you
101:46 have your sample and use just music . And you can do it with
101:53 low frequency, high amateurs, low , whatever you want, that's gonna
101:59 your under your control in the And you can find all sorts of
102:07 differences depending on these variables. Good high pressure, low pressure, high
102:12 , low pressure, low temperature. these things are um interest too block
102:21 . But from our point of all I'm just gonna say is that
102:25 we learned from this cartoon. stress causes strain, not racers.
102:35 we going to implement that in our ? Well, the best way to
102:43 it is to simply allow these elastic I to be complex. So this
102:50 now books law just like we had before, but we're gonna take every
102:54 of these tense elements and allow them be complex with a real part and
102:59 imaginary. And so uh this imaginary here, that's a real number was
103:07 by spirit of -1. So all that real number here with all that
103:13 imaginary part of um stiffness tensor It's gonna be different for every
103:21 J. M. And M. then here's the real part. And
103:25 course it's gonna be the same as . That would be a real
103:28 And then the Medicare part compliance. so uh this it should it will
103:36 be obvious to you right now. . How this leads to attenuation but
103:43 will be shortly. Yeah. So that was for the general elasticity that's
103:51 specialized isotopic rocks. And so from both Marcus and the K. Models
103:57 the modular. We can separate into real part and an imaginary part.
104:04 that the Mhm is governs that A velocity through this expression. And
104:17 the the functional modular is complex it's got to mean that the density
104:25 or the velocity is also complex. it's common to uh not consider these
104:36 parts as written but the factor out real park and express this ratio here
104:44 one over Q. And since module M. Is governing the P wave
104:50 , we're gonna call this um subscript on here. And of course it
104:56 be different for Sherwood. And um this Q. Factor is defined as
105:03 ratio. And for rocks it's it's greater than well because of the second
105:10 of thermodynamics, it's greater than And for real rocks it's greater than
105:16 . Usually a lot greater than Usually a number like 20 or 50
105:20 something like that rocks. And that's because if this number were small we'd
105:26 a large complex would have a large part and would have a the implications
105:33 that would be larger attenuation. So gonna proceed with the assumption based on
105:40 That the continuation is not zero but weak. And so that means that
105:47 is going to be large compared to always know it's always always positive.
105:54 if you do some sort of experiment you think proves that Q. Is
105:59 , that means you made a mistake . Either that or you get a
106:03 prize for disproving the second law of , which is probably not gonna
106:08 Uh go back and check what you and you will conclude that Q.
106:15 awesome. And you will usually conclude most seismic instances that it's a lot
106:24 than one. So now, in of velocity, uh this is what
106:29 have for uh modular and the And we're going to factor out the
106:35 part of this is factually out there here, roe V. B squared
106:54 identically equal to. And so this exactly the same as we said on
106:59 previous slide, we just put in the recognition that N. Is equal
107:03 roe V square. Now we're going uh uh huh Got the convention that
107:12 density itself is real. Um So is that um a critical assumption or
107:29 you can imagine that you have a machine, like a geo phone.
107:36 you can recognize that you jiggle the phone and it's gonna give some sort
107:42 a complicated response coming out the wire to the recording truck. And so
107:48 can modify that complicated machine via phone with a complex modules, complex
107:58 What rock, you know, a . F. O. Has all
108:03 of stuff in it, it's got , it's got corals, it's got
108:06 and that and it's complicated compared to to a rock. Well, Iraq
108:11 also complicated but not in the same . So we're going to uh adopt
108:18 I think it's not a probably never change to uh to think of the
108:27 in Iraq as a real quantity independent frequency and independent of uh There's no
108:37 of cyclical density. Yes, I the density does change as uh the
108:51 goes through. It does compress Uh that I'm gonna speculating uh on
109:07 the accuracy of this assumption we're gonna is outside the scope of this
109:14 So we're gonna assume that S. . Is real. And all of
109:19 complexity that you see up here comes the velocity. Now we're gonna do
109:27 little bit of um of uh taylor and we're going to assume that this
109:35 , the queue is large. So over Q. Is small. So
109:39 means that when we take the uh real part of the velocity itself instead
109:46 the square of the velocity, if do a taylor expansion, all we
109:51 a one half here because we're using first hour and send of the
109:57 And um when you think about taylor's , Taylor's expansion is valid. Um
110:07 in the case of complex media, quantities. This quantity is small compared
110:14 one, still going to get the itself compared to the square of the
110:23 is going to depend on a real and imaginary part where the imaginary part
110:28 one over twice this. And then the same way uh we take the
110:35 of velocity, which is the slowness say uh it's different. It depends
110:41 the real part here and uh it's same um imaginary part, but it's
110:47 a minus here etcetera plus. Uh of the way taylor's um um oxidation
110:56 . And this is the point that need for plane wave face loss.
111:03 you can say a similar thing for waves. Here's the shear wave is
111:08 real part and a Q factor for waves. And the shear wave velocity
111:15 is given by one over this real . Uh times that's right. Plus
111:25 half, I think few for And uh these uh these few factors
111:34 properties of the rock, not independent . Uh They're independent quantities of the
111:42 . And they, you know, is a this is a property of
111:46 rock and this is a property of rock, which you can determine
111:51 Now let me divert a moment to to the back to electro mit electromagnetic
112:02 in electromagnetic waves. This old factor identically one because of the structure of
112:11 uh of the equations. So what means is the Q factor for electromagnetic
112:17 is identically one half. So that half times two makes one. And
112:22 that that comes out of the out the equations which are slightly different than
112:28 equations from uh saving wave propagation. that said that means that like the
112:39 ways, it's not true that the is a factor of media. It's
112:45 it's a property of the equation These are too just like VPN GSR
112:51 properties, easter interference. They might related. That is you might say
112:56 in Iraq with high I. P. It's also gonna have like
113:00 but uh even so it will be um independent very late because of
113:12 We've got to determine them from the . Okay, so here it gives
113:19 wider range than I said. But that's because uh we have a wider
113:25 here, but across the seismic we're going to expect to find values
113:30 in here for either two P or two S. And I can tell
113:36 that normally they are similar to each . And we're going to get more
113:43 in share waves than in p waves share waves have higher uh border
113:54 So more attenuation and fear than P per meter per cycle. It's gonna
114:01 similar because these two things are How would you go about major leaves
114:23 the data? I think you probably have in your mind an idea how
114:29 do that because we already said that you look at the data, look
114:33 any reflection um um dataset then you think of an image or you can
114:39 of raw data and you will see at low at long reflection times.
114:46 you're gonna have lower frequencies. So you can do in principle is take
114:51 short window around uh the data at times in a short window around the
114:58 . Long times. Getting a spectrum each of those in each of those
115:03 windows and then uh make a measure how the uh spectrum has lost its
115:13 frequencies. And so uh uh spectrum at long reflection time will have a
115:23 fall off uh frequency of amplitude with frequency. And so by doing that
115:32 have found a measure of the average between the two time windows.
115:42 And so uh normally want to know properties With high resolution as possible.
115:53 what you do is you move those windows closer together in the in the
115:58 say have one window between uh two and 2.10 seconds and another one between
116:08 the other one between um huh 3 , 3.1 2nd. And that that
116:17 you get average value for Q. the depth interval corresponding to um Uh
116:29 time window from two seconds, three . Uh Here's here's the problem.
116:35 well so um the first problem is still low resolution you really want to
116:44 any time your estimated physical quantity want higher resolution than that. So you
116:49 the windows closer together and as you them closer together you can see you
116:53 to be more and more uncertain about differences because you're gonna be measuring the
117:00 with certain uncertainty. And the closer get those time windows the more of
117:10 noise in the data is going to your estimate of the difference in that
117:15 . So there's gonna be a limit how much resolution you can get.
117:20 not gonna be um you can see the parliament isn't going to be inherently
117:30 resolution. Never gonna get down to level of the reservoir layer. And
117:37 here's another problem in this in the . I know you're thinking of those
117:45 in terms of primary reflections but suppose the data are multiples. So those
117:52 um multiples are gonna be spending a of their time in the shower part
117:57 the rock, not the interval um you're thinking of that death. And
118:04 the average continuation that you get out method just said is not gonna be
118:11 average cute factors for that layer for interval because it's got arrivals coming in
118:19 those times. Uh from multiples which a lot of their uh travel time
118:30 that interval. So before you do you're gonna want to remove the multiples
118:36 she can never remove them perfectly. you can. Mhm. Whatever you
118:44 is going to improve the situation that never be perfect. Another thing you
118:49 do to say it's okay. I'm make an image of this and in
118:54 image uh as I make the I'm gonna be eliminating multiples in in
119:01 normal way. That migration of certain reduces multiples, not eliminate them,
119:12 further whatever multiples are remaining. But when you do that and take your
119:19 your time windows or equivalently your depth up in this image. You're gonna
119:26 a change in the frequency. But uh you got to go back to
119:30 guy, your colleagues who did the and and ask him what does your
119:36 process due to the spectral content which in the data but which is in
119:45 algorithm used for imaging. And so need to have a good conversation with
119:50 guy who is the expert or Use the output of his data.
119:56 don't want to use the output of process without having a good conversation with
120:02 . Well or her. Well what did to the data that might be
120:12 . Here's another important point. This saturated rocks. That value for
120:20 P. A substantial loan. Not us but for pete. And the
120:27 for that is that in a partial of rock, Oh we'll be
120:34 Um We will learn shorten. You what I think. I think I'm
120:41 to uh go home um justifying this um until later in the lecture Because
120:53 a number of ideas that we have get past before we uh perfectly
120:59 But it can be a substantial It could be in the range of
121:05 instead of 30-20. We'll talk about later. Yeah, let's think about
121:15 . See if we can understand Um Confortable. So for brian,
121:25 the generation is pretty high for uh you measure the insinuation in the laboratory
121:32 brian, you're gonna get a factor two. He writes like 200.
121:39 then if you do the same thing courts, it's very hot. Uh
121:44 cure for any mineral is going to very high. And so you think
121:49 for sandstone, it's gonna be like abs somewhere in between the But
121:53 it's lower. It's outside these. immediately they say, oh wow,
122:03 can it be that the sandstone made some some quartz and some brine doesn't
122:09 a Q. Factor which is in between here. So obviously the queue
122:15 Iraq is not an average of these . It's the consequence of a nonlinear
122:21 between these constituents. So that Ryan going to be interacting non non linearly
122:30 the coarse grains. So tell you in um later in the election.
122:42 here is a quiz. Let me , let me turn to uh Miss
122:47 Rio. And uh as the theory easily extended to attenuated media by
122:53 B. Or C. Or none the above me. Yeah, so
122:58 me. So um these others completely you're a cracked. And so um
123:06 what we did. We simply made module I two B. Compliance.
123:14 now let's apply that to the wave . So here is our vector wave
123:19 with uh the velocity parameter in here by M over raw. Now when
123:26 derive this, we never assume that are real. You can go back
123:31 the to the second lecture, maybe third election whenever we it was the
123:36 lecture when we developed the wave equation go through that. And we never
123:41 assume that this thing is real. didn't assume any of this stuff is
123:46 . So then we say, oh good. We never did restrict our
123:50 of this to real module. I let's just recognize that in real rocks
123:56 that we have real rocks. After poor elastic lecture. Uh Now we
124:04 that got real rocks and we're gonna um bye josh to be complex.
124:13 gonna keep density than me. And , we didn't assume that it's constant
124:20 respect to the frequency. So we allow it to be frequency dependent.
124:25 back and look at that derivation. can see that nowhere when we derived
124:30 that we ever assume. Remember this the equation. This doesn't have uh
124:35 way we have plane wave solution but didn't assume plane land solutions here and
124:41 never assume whether or not this thing independent of frequency. We're going to
124:48 a family of solutions to this plane solutions and we're gonna know because of
124:56 foyer that we can construct any solution of some of these plane wave solutions
125:03 none of the plane wave solutions uh that quantity and which appears in
125:10 that the VP that appears in there frequency independent. So our previous solution
125:17 works. That's very good. We have plane waves. So here are
125:20 plane wave solutions same as before. with a complex velocity, here's our
125:26 wave solution. It's got vector quantity a function of time and three dimensional
125:34 and frequency. And this is one wave, our family of plane
125:39 it's got an amplitude which is a here. And it's got an exponential
125:45 with oscillates because of the scrotum -1 here. And that's got a phase
125:51 function here, which has got negative or minus. Uh vector K
125:58 position vector X. Which is this X. And I'll remind you this
126:04 the displacement, this is the position the vector X has three components.
126:09 why it's uh quite a general Uh They lose in the direction of
126:19 . So lose in three dimensions. length of the Uh vector is given
126:27 Omega three p. And now we this quantity here is complex, so
126:33 means this one is going to So now for for simplicity, let's
126:40 vertical propagation in the downward direction. raising and simplified for this. And
126:48 now we're gonna put in here for put in here a complex VP got
126:56 complex real part plus I times It's the same same Z as we
127:03 here. And now we have two uh Q. P times the real
127:08 of VP that comes from the previous that we did. So we're gonna
127:14 uh separate out this part here over . Now this part here is actually
127:24 here and it's uh we got a change. Now we're emphasizing that
127:31 we have the real part of vP into Z as we factored out uh
127:39 here. But look what we have , we have uh we have plus
127:44 here and we have an eye here multiplying my eye here. So that
127:50 that makes a minus sign here with eyes left over. And we're gonna
127:57 that this is causing attenuation as Z as that wave goes down. This
128:04 is gonna get smaller and smaller depending cue and depending on vp also depending
128:11 omega. And so this is the that we did the first seven lectures
128:16 this now we have a new term that and decreasing all the amplitudes by
128:25 factor here uh is here similarly we do the same thing for sure.
128:32 the only difference is we have uh velocities here and share a queue
128:38 Same thing. And again, you , same minus. So here is
128:46 attenuation factor and let's rewrite the velocity the wavelength times the angular frequency divided
128:55 two. So this is uh this the free, this right here is
129:01 cyclical frequency. And so when we that into this expression we see that
129:09 insinuation factor depends upon Z divided by . And in there is uh personality
129:19 1/2. And there's also um Thank . So that means that the amplitude
129:28 multiplied by by the fraction. Eat minus pi over Q. For each
129:34 of propagation. So let's put in number, let's put in for two
129:39 50. And so um e to minus five or 50 is 500.94.
129:46 uh in this example we lost 6% the energy heard cycle. Uh And
129:57 you know, so you could also on frequency. Right? But you
130:01 let's assume that you is a constant to 50. And so under these
130:07 we we lose 6% of the energy each cycle. So p wave going
130:20 . Uh 10 wavelengths coming back 10 will have this quantity multiplying 20
130:30 Uh 200.94 raised to the uh to 20th power. So if Z.
130:41 equal to 20 land uh Z. equal to 10 wavelengths going down And
130:49 10 going back up, we need multiply this .94 by itself 20
130:55 And so that's going to be And that's why we have to gain
131:01 lose a lot of atmosphere. And in order to see it on our
131:06 will have to apply on um a factor so that we can expand those
131:14 construction times back to something. We see with our our ball. And
131:20 you can see immediately that that's gonna upon frequency. So that if we
131:25 higher frequencies shorter wavelengths and in that scenario it's going to be more
131:32 And so they continue eight more and typical exploration um data sense where we're
131:45 at reflection times for a few You will easily see that the high
131:51 which are present at short reflection times mostly gone by the time you get
131:57 to uh reflection there at all because this. Now. A little bit
132:07 there. And uh so the attenuation should be written like this where we're
132:14 about that the absolute value of the just doesn't matter what whether it's going
132:19 or up. It's still gonna be and it doesn't matter whether the frequencies
132:25 positive or negative, it's still gonna a continuing we're still gonna get a
132:29 sign. So I know what you're back here. You're thinking okay,
132:35 can see it when Z is when the wave is coming back
132:38 uh Z is decreasing. That's going make a change. But no um
132:45 not gonna happen that uh amplitudes and frequencies come back as it propagates back
132:53 . That's not gonna instead it continues decrease. But you're gonna see some
133:03 uh expression just like I had on previous line, ignoring that fact.
133:09 that that's not gonna happen. So just like wait uh uh said
133:17 you can hear is sort of a for measuring the loss of high
133:21 Here's the expression we had, let's this whole part out here. Got
133:28 attenuation factor. Now included in this . Usually oscillation. We're gonna call
133:34 uh the attitude as a function of and position, frequency, position.
133:42 in true. When you propagate this , uh evaluate this thing before and
133:51 . Um is propagated then. Um I said before here is uh change
133:59 sea between these two applications um function distance between the distance here of the
134:10 reflections at one seconds compared to reflections 1.5 seconds. So um reflections at
134:20 ft compared to reflections at 6000 That's the different disease and here it
134:27 . And so this is going to an average Q. And also an
134:31 philosophy over that interval. So right you can see that higher frequencies are
134:38 to attenuate faster. Factor is going be bigger. And so using uh
134:48 this and using several frequencies from calculating spectrum at this depth at this depth
134:57 the spectrum. Uh you can estimate average people over this death. It's
135:06 , you know, here's a schematic . So here's the log of these
135:13 attitudes uh at different greetings, calculate I won't take long. Yeah,
135:24 need to take the log of both here. That needs this. Um
135:31 and there's no cartoon with scattered And so we have the best fit
135:36 . It's gonna average Q. Hope averaging over the width of the psychic
135:44 and over the interval. That And so um if this interval is
135:54 short there's gonna be more scamp. so you can't uh got a good
136:04 of average to over a short. um here's a question for you,
136:18 Del Rio, is this statement true false? Is it true?
136:25 yeah, I think it is There's a lot more to it.
136:28 uh but you get uh the essential is that in the plane wave solution
136:35 had I squared equals my and put . That part of the solution.
136:41 that leads to generation. Yeah, would call that truth. Go on
136:46 del rio next. Yeah, that is really false because you don't know
136:55 yet. But I told you that uh in principle it's gonna vary with
137:02 but we have a narrow band in last in the surgery bandwidth. And
137:07 of course that ban uh Q. uh does it very very much.
137:14 this is true for P. And S. And uh so it's very
137:19 to assume that Q. Is a across the seismic ban. And so
137:24 we do lose higher frequencies more than frequency. But it's not because
137:29 Is lower. It's because it executes cycles per meter. You're correct about
137:39 . And the same answer. How this? Um This is true.
137:52 , so that brings us to the of attenuation and reflection. Yes.
138:11 think this is a good place to a break. So let's take a
138:15 minute break and come back at And then we're gonna break around 12
138:20 for lunch. So uh don't don't and try to make yourself a
138:25 Let's uh let's Stop here for 10 because this is a convenient place to
138:33 and take a bathroom break and come at noon to continue at this
138:39 So I'm gonna at this point stop video just now. Let me start
138:46 again. Uh Just before we we were talking about attenuation in
138:51 Now we're gonna talk about attenuation in . Here we are. And um
139:04 , presentation mode. And so, I'm asking the question, if you
139:11 a sedimentary layer of gassing, it's attenuated. And of course, we
139:15 know it's slow from the previous argument uh in this morning. Um and
139:20 course you knew that before, but was maybe provided you some further understanding
139:26 why that's true. But now it the statement that uh gassy sediments are
139:35 regenerative as well as slow. Does attenuation lead to an exploration clue?
139:43 , now, let's let me back on here. Uh I think it's
139:48 think now is the time. now is not the time. Uh
139:55 , I'm gonna ask you to take on uh as an unsupported statement so
140:02 that sediments with gas in it have generation lo que higher generation because of
140:10 gas. Uh Without justification. Please that for now, the justification is
140:20 come later in the lecture when we about mechanisms of attenuation. Okay,
140:29 , uh for now, take it an unsupportive statement and ask yourself what
140:35 this lead to an exploration clue. , usually gas reservoirs are so thin
140:42 you don't lose much high frequency due propagation. Right? If the if
140:47 reservoir layers only say 20 m you're not gonna lose much high frequency
140:56 propagating through that, even from a reflection where it's gone through two
141:01 There's just not enough path length to enough loss of high frequencies so you
141:06 see it in the data. But ask ourselves is there an effect of
141:13 on reflectivity itself? So let's think normal incidence reflection only. It's the
141:21 in impedance as a function of uh the relative change of impedance. We
141:27 separate it into a relative change in and velocity. And so this part
141:33 really going to change that. But this part is gonna have um uh
141:41 velocities above and below a reflecting Yeah, so I didn't see
141:48 We're gonna look at reflection at a an interface which is a continuation both
141:55 and below reflecting. Okay, so is above that instant media here is
142:03 and down below is the sum of . And all we did here is
142:07 out the real and imaginary parts inside . And now. What uh what
142:13 done is I've um factored out the factors. So here is the one
142:22 uh 1/2 um uh to be for reflecting medium and for the incident media
142:32 now I'm going to collect the real imaginary parts. So here's one factor
142:37 hear having this difference here. And let's see then I'm going to give
142:44 name to this. This is uh jump in the real part of VP
142:51 . It is again. And now can see after these manipulations. you
142:54 see that the complex parts are nicely here. Yeah, let's consider the
143:04 . The normal case where the queues large. So that means that these
143:10 cues are um uh we're gonna uh gonna neglect this because that's you as
143:18 and that Q. Is large but not gonna neglect it up here because
143:23 the minus sign because the minus Uh this might make a significant difference
143:29 in the case where the queues are . So let's do that, let's
143:36 it here, let's keep it here after collecting terms and so on.
143:42 we find is that uh the normal reflectivity has a real part which is
143:51 we looked at before but it also an imaginary part. And look
143:56 it's got a delta Q. And got in here a QP two and
143:59 QP one, where does that come ? That comes from? Um simplifying
144:04 with the assumption that these two terms large but not infinitely large. And
144:10 you're gonna get a difference depending on um the difference in Q. And
144:18 the difference in v. And uh assume that some things are negligible,
144:25 that uh we're going to assume in that this thing is uh VP one
144:33 similar to VP two, that's a velocity distinction as we did before.
144:40 the only uh part of this difference is important is the difference in the
144:46 themselves. And that shows up right and you see it shows up in
144:52 numerator. And then in the denominator have the product of the queue.
144:57 if these are large, if each these is large then the product is
145:02 be even larger. So um um of that it's normally a negligible
145:13 That's why we we you will hardly see this fact of complex reflection coefficient
145:24 um in any study. And of this is just for the normal incidence
145:29 and there's you could do a similar for the greater term if you
145:36 What we just decided was that after did all the linear ization that we
145:40 about the an elastic plainly reflection coefficient complex. And of course it's also
145:47 that like I said, no normal . Uh you can say some other
145:51 about shipping. Now, what this is that the reflected wave like is
145:57 shifted because of this complex this um when we multiply the incident wavelength by
146:13 reflection factor, it's gonna shape and gonna do it um from it's gonna
146:21 that shape of the reflected white Yeah. Just to see that,
146:27 that into focus. Let's consider a um where the real part of this
146:33 very small, right? So we're consider real part of this is
146:40 And so then what we're left here a reflection coefficient which is all
146:48 And so when you multiply this times spectrum of the incoming uh Well you
146:57 a 90° phase shift if it comes , if it were to come in
147:01 phase, if you've done something to the incoming way, done something in
147:06 computer, the incoming wave zero then the reflective way is gonna be
147:12 a 90° phase shift because this American . Mhm. So we can call
147:20 um uh this case here, but real part is very small, we
147:25 call that a cure reflection because only from the differences in to not from
147:31 because here we have assumed this part negligible. So to estimate the
147:38 let's just assume that for the upper the queue is normal number like 50
147:44 for the lower q. Uh it's lot smaller, let's just choose
147:50 Then this reflection coefficient turns out to minus 4.5% times I which is not
148:00 , you know? All of the uh coefficients have numbers like this,
148:07 the magnitude is um A lot less one. Uh And uh previously we
148:16 only cases where this real numbers are . Now we're assuming the case where
148:22 real numbers is actually negligible and still gonna be a reflection, a key
148:29 which is a comparable side to the reflections. So now in a real
148:37 we're gonna have uh non negligible um negligible real part. And uh the
148:49 part. May or may not be right here. We got some motivations
148:54 examine this. Now, this effect a possible exploration. Yeah. And
149:03 clue would be that if we look them reflecting a reflective way his spectrum
149:11 a lot different than whose whose space looks a lot different than other
149:18 Maybe that's due to this effect. quick question about the previous equation.
149:31 how do you know there's uh positiveness minus 90 degree shift from the
149:39 Uh Because yeah, so uh let's it as an example um an incoming
150:05 lit which is a record Waveland. , and it has a zero phase
150:13 zero phase um spectrum which means that phases zero. Uh and the spectrum
150:21 only real. Okay, now upon that spectrum gets multiplied in this example
150:30 an imaginary number. So now the spectrum has uh is all imaginary.
150:38 all real, but all imaginary. so that we described that circumstance as
150:46 90° phase shift that it went from real polar measurements You get like for
150:57 your example below you get a -4.5% it is not 90 positive 99-19
151:11 Oh yeah, well uh so um let's see in this case uh um
151:20 delta QP is gonna be a QP um Q. P two minus
151:29 P. One. So uh that uh Uh 5 -50. So um
151:40 I'm showing is 50 minus five. have that wrong actually, thank you
151:44 that. Uh um I made a . If you did this on a
151:53 I would uh downgrade you on So downgrade myself and this this should
152:00 should be um uh five minus So that should be plus 5.5%.
152:07 you very much for that. Mr I will uh fix that up right
152:12 ? Uh Matter of fact, that's an important thing, I want to
152:14 sure I I uh don't forget to that. So I'm gonna do this
152:20 , I'm gonna make it uh All man, it's 50 that's gonna be
152:32 now I'm going yes. Okay so it's right, thank you very much
152:39 that question and for finding my So you understand now that uh that
152:50 when you when you multiply um a , any spectrum by an imaginary
152:57 you change the uh you change the of that by 90 degrees. It
153:06 uh zero phase for example. Now uh 90 degrees or maybe 90 minus
153:12 degrees because of um multiplying the spectrum an imaginary and sir it was in
153:23 time to me if it was um wavelength, it becomes an anti symmetric
153:31 . We saw that before in discussing uh the interference between the top and
153:38 of the wedge. So the top bottom of the wedge have reflectivity is
153:44 instance reflectivity of opposite sign. Then combine to make uh as the top
153:53 bottom reflections merged together as the wedge in thickness. There comes a point
154:04 the combined wave looks um like it's and this metric instead of symmetric.
154:13 then as you decrease the thickness of rates further, the amplitude of that
154:19 away. But it still has anti time signature, tender man, it
154:28 anti symmetric even though the individual wavelengths symmetric because they have opposite polarity.
154:35 that leads to uh To a 90° shift in the reflection wave form for
154:46 thin bet. And the same thing happening here because of attenuation here.
154:52 have only one reflector. There's no between anything but it's a complex
154:59 And so um in this example it's a negligible amplitude. They did make
155:10 a significant change in that. Um because you can change yeah. Form
155:25 the way of the shape of the . Now let's think more about this
155:32 example, the same example uh gonna on top reflection. And that's what
155:39 did here. That's what I stopped . Now this reflected wave which has
155:48 , this reflected wave which has this coefficient that wave never penetrated through the
155:55 reservoir. So it didn't lose any its high frequencies at all. So
156:02 we got was a phase shift that with the same frequency content. So
156:08 amplitude spectrum was unchanged and face that was this team. So it's
156:17 time shifted wavelength. And so maybe could be a good exploration flu.
156:24 I have posed this question to many over the years and nobody has ever
156:30 to me, oh I've seen that and nobody has ever come back to
156:35 six months later and I said, know, I looked for that and
156:37 found it and the reason for uh that failure is because other things might
156:43 a similar effect. For example, I said here, interference from nearby
156:48 also makes a face shape, a shift upon reflection from a thin
156:54 So that so the the glue is an obvious clue. Uh You've got
157:02 say if you see that sort of , this sort of thing, a
157:07 shifted weight, you have to uh something to help yourself figure it out
157:15 it's due to attenuation or to these effects. Like the thin bed
157:20 And so maybe that would be, that would involve analyzing uh the offset
157:27 of this space shift. Uh That's . This is a good master stations
157:34 to understand how this phenomenon um differs uh the thin bed form. And
157:46 um Miss del Rio, you are to be doing a capstone project I
157:55 later in your studies. And so thing is for you to do a
158:03 of some data, for example from current um from your current board.
158:11 but uh if that is not bright in some for some reason, this
158:18 be an interesting topic, a theoretical analysis instead of of data based
158:27 I would say that most of our projects are based on data coming from
158:31 employers of the various students. And the advisor for those capstone projects is
158:42 and he does a lot of those probably does no more than a dozen
158:47 every term. So the final comment nevertheless, you should keep this in
158:55 because you might be the one who a way to use this to discover
159:01 . So you would become famous. uh here's a question I said.
159:09 course the reflectivity must be complex since density is complex. Is that true
159:14 false? Yeah, it's false because density is not complex. At least
159:20 assume it's not. The density is . But of course the reflectivity must
159:24 complex because the velocity is complex or stiffness is complex, which is given
159:30 the next uh quiz question. So one is true. Now, how
159:35 this one, Miss Del Rio? large queue contrast at reflecting horizon can
159:43 a face shifted reflection without the loss high frequencies at all. Sure.
159:49 , that's the example that we just and that's kind of interesting. So
159:53 is an effective Q. That does involve loss of high frequency. And
159:59 you can see it's a high high measure of uh of uh phew.
160:08 don't have to uh look for the loss of frequencies with propagation distance.
160:16 happening right there at the reflecting So that can be a very valuable
160:22 see high resolution measure a few. , so the next topic is Attenuation
160:29 dispersion. So that's a big So we're gonna do that after
160:33 So let me um stop sharing this stop my video. And uh so
160:44 will see you back here at 1 give you time for lunch and a
160:50 relaxation. And we will pick up connection between attenuation and dispersion at that
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