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GEOL7333-Seismic Wave and Ray Theory - Day6 09-10-2022-Part2
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00:00 this is a topic resuming after the . Um I would say that this
00:07 a topic attenuation and its connection with , which is not very intuitive.
00:15 so I would even say that probably lot of practicing show physicists don't understand
00:20 , but you will. So let go into presentation mode here and consider
00:29 following. Consider the following simple You have an impulsive source at the
00:35 in an ungrounded medium, no surface without attenuation and dispersion. So you
00:42 decompose this uh fourier source like So it's an impulsive source. So
00:50 is equal to one at T. zero and zero everywhere else. And
00:57 the uh that time series can be into this um uh spectrum and the
01:10 spectrum delta omega. This is delta . That fourier spectrum is flat,
01:17 equal to one everywhere, positive one , no imaginary part. And so
01:24 that means that to make a spike time, you just add up all
01:29 frequencies from minus infinity to plus add them up with equal weights and
01:35 all reinforce at T equals zero. they all canceled at all other
01:41 So that's uh that's what the fourier of a spike use. So let's
01:52 this source into the in homogeneous wave that we talked about several days
01:58 And uh so here here's the solution we get uh writing down the
02:09 you see it's expanding geometrical spreading, got a solitary factor uh that you
02:16 here and this is exactly the expression we uh wrote before. So there's
02:23 for the amplitude of the source and is gonna be uh constant. Um
02:29 gonna be one or you know, a stronger source, it would be
02:32 stronger constant but still be independent of for an impulsive source. So uh
02:40 can make the inverse for you, that. It's the inverse fourier
02:46 Um is uh is gonna give you through the the fourier calculus here And
02:56 these are zero was here. It's same as evil as this. And
03:03 when you uh you can look up you, you can look up these
03:08 um expressions and might be a little difficult for you to follow this fourier
03:17 inverse transform to get from um uh the spectrum back to uh time
03:27 But when you do that, the is this and you can see that
03:31 a delta function, not A. . Equals zero, but it's
03:35 Equals uh This is equal to um it's equal to one, not at
03:43 equals zero, but at time equals R or V. So, and
03:49 a decrease in amplitude as it goes . So it's just an expanding
03:54 It's a spherical shell of expanding pressure which is zero everywhere in the model
04:06 for uh at the expanding wavefront, is an impulse expanding at the rate
04:12 this philosophy. Yeah, so that makes sense that the same.
04:20 we consider that the medium is is a tentative. And let's assume it's
04:24 a constant to and no dispersion. dispersion at all. So the fourier
04:30 solution is this one I want to back and compare with this. All
04:35 did was we made this uh k here complex. So here uh put
04:42 right in here, um uh in two and remember uh we've got uh
04:52 had an I squared in here lead a minus one and we uh um
05:01 uh converted the real part of the into a real part of V.
05:09 uh and the freedoms. So making inverse transform of this gives them more
05:18 result. And you see it's got integral from minus over frequency from minus
05:23 to zero. And then another uh handle separately the uh the other the
05:30 frequencies. And if you work out intervals, you find uh that this
05:36 the answer and it's not an When you, when you graph this
05:44 , it's not an impulse. And uh furthermore, it's not a it's
05:55 zero phase wave, it at You can see it's it's uh it's
06:00 symmetrical, It starts off uh becomes zero here at negative times and the
06:09 time is given right here, the time has given uh x over
06:18 Um but it stretches for infinity in directions. And you can uh get
06:25 poor discussion of this in uh in technical by AKI and Richards, what
06:33 shows is the energy begins to arrive before the nominal arrival time, thus
06:39 the principle of causality, see the begins to arrive way back here,
06:45 you don't expect anything to arrive before before this time. And uh um
06:57 know, I misspoke, it is I misspoke, I said it was
07:03 . I said it was not symmetrical it is symmetrical. Uh we call
07:08 zero phase, whereas real data uh always uh similar shape. It's uh
07:15 loaded with a long tail extending for times. So we call that minimum
07:20 . And so these are unfeasible So one or more of our assumptions
07:26 be invalid. So let's go back look at the assumptions. We only
07:32 the linear wave equation. So uh in homogeneous equation, but the uniform
07:39 but the medium is uniform. We this an in homogeneous equation because on
07:44 right side is the source term, does not have the unknown in
07:49 That's the source term, That's that's we assume. And we assume constant
07:55 throughout the entire band. And we constant velocity throughout. And so uh
08:02 conclusion is that both of these are implausible and what we proved theoretically.
08:12 and coupled with laboratory data proves that are mathematically impossible. It is when
08:18 um when we do laboratory experiments on walks, we always find uh frequency
08:26 Q. And frequency dependent velocity at same time. And so the conclusion
08:32 because of the second law, Q be finite and positive. Um And
08:41 that both of these must also be dependent. So despite that it's very
08:50 because we have limited bandwidth in any . Uh it's uh it's very common
08:58 to approximate that you and velocity are constant within the seismic band and also
09:08 any other band you might think Like in the ultrasonic band, we
09:12 assume the velocities are constant in the band. We assume that the
09:18 but when we look, when we velocities uh compare velocities across these bands
09:26 these bands, we expect to find . So uh we expect that the
09:34 . Is going to be different in seismic van than the ultrasonic band.
09:39 velocity is gonna be different between Yeah, or whenever you talk about
09:50 in a serious way, you always involved in ideas of the mechanism
09:56 Uh And so it's always a messy , you don't ever talk about the
10:04 of elasticity. You just say, that's the that's the stiffness constant.
10:09 we always have to talk about mechanisms of insinuation. And so lots of
10:17 mechanisms have been discussed which are um are inferred to be one or another
10:25 them is inferred to be dominant in bands. So for example, um
10:32 the ultrasonic ban, the dominant mechanism attenuation is scattering off of the grain
10:42 the uh and it's different in other . So um when you do this
10:50 of analysis you're frequently almost always you that the uh there is a relationship
10:57 attenuation and dispersion which can be written this way. Take the velocity at
11:02 different frequencies. Whether their angular frequencies cyclical and you uh reduced, that's
11:13 linearly proportional to uh log log random law over the ratio of the two
11:21 . These two frequencies are the same the With the proportionality concept, which
11:27 one over Pi Q. Now, to see what this implies for seismic
11:33 take the upper frequency to be 100 , the lower frequency to be 10
11:39 . Take your vehicles 50 then the of velocity is here um at the
11:46 frequencies, putting in these numbers, in 50 right here for the
11:51 110 for the two different frequencies. turns out uh The only different from
11:59 by 1.5%. So that means to this velocity difference across this ban with
12:06 kinds of Q factors you have to doing. You have to be determined
12:12 velocities with with accuracy and precision better 1.5%. So that's a tall
12:23 Now let's do the same situation for partially saturated sediments where we've got a
12:30 smaller if you factor so now across band it's differing by 15%. So
12:38 could be measurable. Um Although it true that uh if it's a thin
12:46 , you might be not be able measure velocities in that thin layer.
12:53 so the accuracy of 15%. But can see how the in the thin
13:01 with partially saturated sediments, it's gonna high attenuation, meaning lo que.
13:06 it's also gonna have high dispersion in case 15% this. Uh And you
13:16 , because you have this high the the usable bandwidth might be
13:23 It might be that you can only up to 50 yards. So that
13:28 decrease this even small. However, you're comparing a Masonic band with the
13:37 band, then we have a big . Here's a sonic band, uh
13:44 seismic band frequencies And uh normal Brian's for the Q. And so now
13:50 see 22% uh writers. So uh the difference between uh not visible within
14:01 seismic band, not visible within the band. But when you compare velocities
14:06 two different uh bands. So what means is when you're comparing sonic data
14:15 um um seismic signing velocities to seismic you've got to be mindful of
14:23 This difference here, which could be and it's due to intrinsic attenuation In
14:34 rock in the rocks. And uh assuming it's the same cube right?
14:41 both cases. And cute. But velocities are gonna be changing by 22%
14:47 this example. Now take this same here and turn it into uh consider
14:57 frequencies which are close together and uh frequencies which are close together and consequently
15:06 velocities are close together. And turn into a differential relationship. And you
15:11 uh just rearranging this, the inverse Q is proportional to the partial derivative
15:17 velocity with respect. So here's a since attenuation comes along with dispersion,
15:34 follows that a higher value of Q larger dispersion. Miss Del rio.
15:41 that current false? Yeah, that's . Because the display the attenuation goes
15:47 1/2. Very good. So so let's talk about some mechanisms of
15:56 Now when we talk about hook, law, we assumed perfect elasticity.
16:02 that's not the only relationship between stress strain that you can imagine. And
16:08 here is a very general relationship between and strain. A generalization of
16:16 And so here we have the And here we have the modular just
16:20 we had before in the strain. now we have additional parameters to characteristic
16:26 time. And so what this says that this this is general that it
16:31 so long. Still linear. C linear between the stress and the
16:37 But it says that the this combination stress and the rate of change of
16:44 is proportional to this. Um uh of strain and the rate of change
16:52 strain. And you can bet this is gonna be complex. So naturally
17:00 can see here that Hook's law is special case of this corresponding to assuming
17:05 two characteristic times R zero. So is called the standard linear salad standard
17:14 solid. And uh you know, not surprising that we have more complicated
17:21 between stress and strain that we thought after hook, hook, hook was
17:27 first one. That was the 17th than a lot of times here for
17:32 to think about this and managed to about this. And so this is
17:38 um straightforward generalization of Hook's law to for the possibility of more complex material
17:49 . And so what you can do you can you can aspire to measure
17:53 things, um uh real materials with experiments. So as a consequence of
18:03 , we can say uh we can say the following, look what we
18:08 here. We have frequency expressed in non dimensional way because it's multiplied
18:15 you know, frequency has the physical in inverse time. So we multiplied
18:22 the square of the product of these characteristic times. Then we get a
18:27 curve here, it looks like so the and the velocity, it looks
18:32 this actually this square of the velocity low frequency is low, then uh
18:40 the interval where frequency is comparable to uh to the inverse of these,
18:48 fine, then it changes and then reaches uh asymptotic limit and goes flat
18:56 . And so the Q. Is uh proportional to the time derivative of
19:07 . So uh we showed a couple ago. The inverse of Q.
19:12 proportional to the time derivative of this here. And so here it
19:18 It has a peak just where this is changing. And the maximum value
19:25 the peak is defined in terms of characteristic times. Like so and so
19:31 either one of those characteristic times is Then the maximum value of few is
19:40 . See you right here. Uh Let's put some um put this
19:51 context for sedimentary rocks, the size band is down here, The sonic
19:57 is right here in the middle and ultrasonic band is here. So normally
20:02 think that seismic um seismic data occupies mostly flat region. And um ultrasonic
20:11 in the laboratory occupies this mostly flat . And sonic data is where it's
20:19 . And so uh of course this just um hand waving a bit because
20:27 exact uh the exact demarcations between these upon the values of these characters at
20:36 . Which are going to be different Iraq of course. Uh But that
20:41 is a general expectation. Um I there are several dispersion mechanisms which which
20:55 their peaks and they have different time here. So they have their peaks
21:00 different values of absolute frequency. So you would expect to have this sort
21:06 thing as a function of frequency. can see there's a superposition of those
21:13 of of a few functions. So sort of constant here in in the
21:19 the interim. And uh across that here, that's where the phase velocity
21:25 changing. And for lower frequencies and frequencies it doesn't change. Now.
21:31 course, the different dispersion mechanisms could um uh different absolute values of the
21:44 . And so maybe this is not , but you can see we are
21:47 limited to think of only peaks like . You can superimpose those sorts of
21:57 , Pete's with several different different mechanisms in the rock at various frequencies.
22:03 Each one of these is dominant, frequencies. So in the example from
22:08 and Richards, it ended up with phase velocity curve which was linear.
22:14 course, that's just a cartoon. you see the idea uh you can
22:21 in a real rock. The superposition many um many types of linear continuation
22:35 like I showed on the previous. for each one of them it's gonna
22:42 upon which mechanism of the generation depends the frequency. So, so you
22:49 to sonic frequencies take my word for . The dominant mode of attenuation is
22:54 fluid squirt within the pore space as p wave goes through it. Um
23:03 is the uh different parts of the pore space unequally because the pore space
23:09 a complicated shape and so parts of are thin and flat. So those
23:14 are calling cracks and they more They respond to the pressure in the
23:22 of of longitudinal stress that gets converted um in the fluid to a
23:31 And that's um um That stress affects cracks more than the round ports or
23:41 roundish ports. Of course there are spherical pores anywhere, but there are
23:45 which are more or less um equal in all dimensions. Whereas the crack
23:51 ports are defined to be those which much smaller dimension in one direction than
23:57 the other one or two directions. since the pore space is gonna be
24:07 to the imposed stress in this complicated , that means that fluid is going
24:12 be squirting from different parts of the pore space to other parts, from
24:18 cracks to the pores during the compression and the reverse in the decompression
24:25 And the word squirt is a little um overdramatic because uh the fluid doesn't
24:36 to move very far to equalize the . Okay, Just um especially from
24:44 crack to the nearest four. By , ultrasonic frequencies which are greater than
24:54 hertz. Um The dominant mode of generation is scattered from the grains.
25:02 in the seismic band. They man, we're talking about now,
25:11 strength of the squirt mechanism. That the amount of the dispersion it
25:16 and the amount of attenuation it causes on the fluid. If the pore
25:22 partially saturated with the oil, then squirt mechanism is enhanced because of the
25:29 viscosity of the oil. As the squirts from the crack to the forest
25:35 more energy to heat because of the viscosity of the oil. Furthermore,
25:44 the pore spaces partially saturated with free , then the squirt mechanism is greatly
25:50 basically because the gas gives the fluid uh space to squirt into. Does
26:01 make sense? If there's gas partial in the park space, Then when
26:06 fluid squirts around, um It's It can do so easier because it's
26:12 being resisted by fluid, which is there, there's some uh gas in
26:18 other part of the portion. So in these two context um the uh
26:27 . Factor for p waves is but two factor for share waves is
26:33 because when share wave goes to it does not compress the rock at
26:39 . It just tears a lot so uh there's no squirting of the fluid
26:46 um uh escorting. It's uh it's lot less than if there is a
26:53 being applied. So um this problem a particular interest to me because of
27:04 experience at uh as an Amoco long ago, maybe before you all were
27:12 . Um I was working for Amoco um Tulsa at the research center for
27:21 of my career with american. I in Tulsa, but at a certain
27:26 I moved to Houston Uh to the offices in Houston, which are out
27:33 of downtown near I 10 and Eldridge . And there we have, that's
27:44 um worldwide headquarters of Amoco's Exploration. now it's uh since BP bought
27:56 that's now worldwide headquarters from BP Exploration in London. And so the Houston
28:04 is I think the largest, I it's even larger than the our people
28:10 project. This data was acquired in american base and this data was
28:19 This is conventional p wave data um by Amoco uh in the early nineties
28:30 processed with the state of the art imaging from those days. And it
28:36 called a D. M. Stack. And so I'll leave it
28:39 your other professors to explain what uh that is. And so here is
28:47 reservoir down here and you can see fairly well imaged here and right in
28:53 middle it disappears and then off to side. So uh this reservoir,
29:06 prospect was discovered by Amoco In the 90s or maybe the late 80's and
29:14 saw this huge hole in the image they had no idea what it
29:25 they could uh depending on what well for for several years, they
29:34 everything they knew to get a better here in the middle. And and
29:41 um let me just give you a context. This is a limestone
29:47 Jurassic age and sediments above. And can see that it's a time section
29:54 . So this is almost three seconds this is halfway down. So above
29:59 , it looks quite normal. But at and just above the reservoir,
30:07 has this huge hole in it. eventually they got their nerve to drill
30:12 this. Can you imagine being um the drilling crew that went to drill
30:20 hole here. You know, you're into a monster, but you don't
30:25 what it is. Uh when you when you leave home to sail out
30:32 to drill this, you wanna kiss spouse tenderly goodbye and make sure your
30:39 is paid up because you might not come back from this drilling okay.
30:47 . So they did drill it. they found a very nice discovery.
30:53 found a billion barrel oil field So, that was years before I
30:58 involved. But I got involved in mid-90s because they could not make an
31:04 of this. So, our modern of this is that this lack of
31:17 quality in here comes from attenuation. understanding is that over time, over
31:26 time, the reservoir, which this a time section. So these velocities
31:31 been pushed down. Actually, these have been pushed down by uh slow
31:39 and you can see up here that doesn't happen. So, evidently the
31:45 down is happening, happening, starting with this layer here. And the
31:53 understanding is an over geologic time gas leaked up out of the reservoir,
32:00 accumulating in the overburden here. We know that um uh the reservoir looks
32:07 a broad shallow dome here, like , and that the imaging quality here
32:12 so poor because over geologic time gasses up out of the reservoir collected in
32:20 overburden here. And the effect of gas is to slow down the p
32:27 , you can see here, this in time, not a depression in
32:33 . Uh we got a uh huh section as opposed to a time
32:41 this would be just flat across So, the effect of this gas
32:49 is to slow the waves down and attenuate them. So you cannot make
32:55 a good image with this kind of . And uh so for a number
33:03 years, uh we thought, we just need better imaging uh need
33:07 do better than the and so, now the imaging technology has increased dramatically
33:14 then, and now we would try tm the first time migration on
33:20 but we would still find it would better than this, but it's still
33:24 bad because the arrivals simply aren't The energy got um disappearing out of
33:36 P wave by this uh gas saturation the overburden over the crest of the
33:47 . Yeah. Um the uh guess exist in this region in non economic
34:01 , just a few percent. So don't think even today that we can
34:05 any money by producing this gas, we can make a lot of money
34:12 figuring out how to make an image the reservoir despite this problem. And
34:19 where I came in and I was in solving that problem for ethical,
34:24 I'll show you later to know, a diagram of what's happening. We've
34:30 a P wave going down through uh on the outside of the cloud of
34:37 . And as it comes back it disappears because it's attenuated away by
34:44 high by the low value of Inside the gas. Now, about
34:54 the mid nineties, we had the that if you use converted wives,
35:00 could send a P wave down through and and let it um convert to
35:11 at this point, and this shear is gonna go up without being affected
35:16 the gas lamp. Now, of , there's uh, there's a share
35:23 convert here too. But remember the wave comes up at a steeper angle
35:29 the P wave goes down because the is long. So the shear wave
35:35 at this point is gonna come up an angle like this is gonna be
35:39 over here. It's not going to through the gas club. Uh,
35:42 conversion point in this case is over . And so uh do a survey
35:49 used converted wave energy to uh instead the p wave energy, which didn't
35:57 it. Maybe we can make a image. So what do we have
36:01 do to do a converted wave in two marine environments? We have to
36:05 a P wave source. But to uh receive this sheer way. What
36:11 have to have is ocean bottom seismic with horizontal components. And this year
36:18 coming up is going to have gonna polarized trans version. So it's gonna
36:24 polarized almost horizontally in this cartoon. course it would be like, so
36:28 exactly horizontally, but almost. And that's what we need ocean bottom seismic
36:36 , receivers with horizontal components. So we did that, um, that
36:44 uh from my team that that imaging a couple of years later in
36:49 we got uh this image here and was the first image that had ever
37:00 the first usable in it had ever made Valhol. So when I,
37:07 I went over to uh give honor the chemical offices in the chemical in
37:15 Norwegian oil capital Post of Honor on northwest coast of Norway. That's where
37:22 Amoco offices are. That's where all oil companies all the oil companies
37:29 So I gave uh showed them this , uh and these were guys who
37:34 spent the previous decade trying to produce from without ever seeing it adequately.
37:42 they stood and applauded when when they this, it was a fantastic
37:48 And my colleague over there was Olaf and uh he's still a good friend
37:56 I saw him this summer, past in Madrid. He didn't come to
38:00 SPG meeting uh two weeks ago here Houston, but I saw him in
38:05 , he's doing well. That would P anymore. So um let's see
38:13 . Uh you can see it's not perfect uh figure, but everything that
38:19 see in this figure is confirmed by bits. And so uh this is
38:26 figure that we showed to uh and sent it back to us with these
38:32 oil wells put in there and he the oil wells confirmed all the details
38:38 are imaging showed. Well, this in 1996. And uh this is
38:45 normal murat processing basically basically dicks um move out removal and stacking is all
38:53 did. And we got this this pretty good image, we can
38:57 much better images today with better um processing algorithms and done by people who
39:06 smarter at image making than I Um but the essential idea we had
39:13 to use the converted way of arrival of the P waver. So let
39:23 uh pause here and tell you a of stories associated with this image on
39:31 bottom. Seismic imaging was invented not Amoco, but by statoil, the
39:38 state oil company, which is now Ecuador. And they invented it about
39:44 in 1990 for something like that. they gave up um a talk
39:51 and the reason they did this was had a Uh reservoir of their
39:57 about 50 miles from Valhol in the Southern North Sea with a similar
40:06 a gas cloud in the older And they had the idea that if
40:09 were just Hughes could hurt a wave , they could see through the gas
40:16 . And so they, um, they invented ocean bottom seismic recording with
40:29 components. I should tell you that , there had been a previous version
40:34 ocean bottom seismometers with only a vertical and a hydrophone. And like we
40:41 you before, um, uh, need two of those, uh,
40:46 combine the two components together to eliminate water bottom multiple. What Seattle did
40:56 to uh, generalize that horizontal components back up to this line here.
41:05 presented their um, mhm results of there reservoir At the European Association
41:20 I think in 1994. And it a big hit. And I think
41:26 won an award for the best paper there and I was not there,
41:32 was in Geneva, the european Association gives their conventions in europe. And
41:42 I was not there, but my was there and he came back and
41:46 me all about it and he said you know, we can do something
41:54 that. And within a week or we had a phone call from our
41:59 in the Norway office and they said know, we have a field just
42:04 that state all field and we want uh um we want to image him
42:14 the same way they did. Would help us? So we're operations here
42:20 this, we need the help from experts in Houston. Well, I
42:25 in Houston at that time in the department. And so we uh talked
42:33 ourselves and and we said, you , this is really an interesting
42:36 We've never seen any data like this . Um but that's a research
42:43 And our our task in Amoco is exploration support, not research. So
42:53 sent this problem to our friends up Tulsa and they said we said you
42:59 should help out the Norway office to that. Well, we had a
43:04 vice president of research in Tulsa and was one of the reasons why I
43:10 Tulsa and moved to Houston and he imposed a system on them which made
43:17 uh promised to make uh to work specific problems and to yield specific results
43:26 a specific time frame. No no excuses, admitted. And so
43:34 wrote back to us or they told on the phone. Sorry. We
43:37 , we promised to work on this project. We don't have any time
43:41 this converted wave project. Sorry. we went to my boss in Houston
43:49 and he said, well and told what happened. And he said,
43:53 , you guys should do it And we said, you know,
43:57 we do that, that's research. going to be criticized by the Vice
44:01 for research up in Tulsa for stepping his toes. And so my supervisor
44:07 exploration in Houston said, never mind vice president for research. You guys
44:13 go ahead and do whatever you think best for the company. Just keep
44:17 informed so I can get you the need. Don't worry about the Vice
44:23 for research. So we did that produced this uh, reach your
44:31 But I'm skipping ahead of the Um Take yourself back to Geneva in
44:43 when Stotler was presenting their results which , you know, comparable to this
44:49 their field. And the uh, I said, I was not
44:55 but in the fall that same year came to the Sug convention and showed
45:03 same analysis and I was there for one. In fact, I was
45:08 the session in which they presented and was actually not uh not the folks
45:13 Seattle it was from the processing company was Western Chico now part of
45:21 So the western Zico uh speaker gave spiel and I'm sitting there at the
45:26 table thinking wow this is really My buddy was correct. This is
45:31 stuff. And then at the end applause and after the applause died down
45:37 buddy stood up at the back of hall and he said, wait a
45:42 you are presenting this uh analysis in of these converted waves that we see
45:53 . But isn't it true that just months ago in Geneva you presented an
45:58 of the very same data set where said that the conversion was from Peter
46:04 at the ocean bottom surface right here down as a share way and converting
46:10 not converting here coming up as a wave shear wave all the way.
46:16 you describe the same analysis analysis of same data set in terms of completely
46:24 , they bad. And I've never a speaker as embarrassed as that and
46:30 said yes, that's true. I hoping nobody would notice. And so
46:36 buddies back in the back of the he said uh well um uh so
46:44 did you change your mind? And said, well uh we designed the
46:51 knowing that the velocity ratio in these here was about two p waves twice
46:57 fast as share waves. And so we did our survey and received the
47:02 on the horizontal components over here, uh found a strong arrival about twice
47:09 p wave arrival times. The shear are arriving about twice. So we
47:15 we had it nailed and that's what presented in Geneva. She waved down
47:23 shear wave up. But then when got home to the home office,
47:26 happened to talk about some talk of of our wire line buddies and they
47:32 to us that in these kinds of , the velocities are about velocity ratio
47:36 about three times here. So that that the shear wave down, shear
47:41 up is gonna be coming in way . So just put some numbers,
47:45 about the p wave arrival comes in three seconds. And so the sheer
47:52 rival should have been coming in or , nine seconds. What they found
47:55 something coming in at six seconds halfway between. So obviously it's P down
48:00 sl so he said, once we that the velocity ratio here is so
48:05 higher. Then we were driven to explanation that the conversion happens here,
48:13 at the surface, not at the floor. So my buddies at the
48:17 of the wall, he back of hall, he doesn't let it
48:20 He says, so show us the at nine seconds for the pure share
48:28 . And they said um, oh we we we cut off our recording
48:37 at seven seconds because we knew you looking for something at six seconds.
48:43 about that. They only saved two per shot of recording time. And
48:49 course there's expenses associated with two seconds additional recording. What um,
48:59 they had many, many shots so would have amounted to a significant increase
49:03 the budget. And so in order be efficient, they Cut off the
49:10 at seven seconds and so they completely dis analyze the whole situation. So
49:19 a lesson for you young people when doing something for the first time,
49:23 don't design the uh the process to efficient. You designed to make sure
49:29 you come back with the answers that want. So when we went out
49:34 a few months later to do the experiment at home, we recorded for
49:38 seconds. And sure enough we saw at nine seconds, nothing at 12
49:44 . We saw something in nine seconds it was very weak. And we
49:48 what everybody has to put it ever the most energetic convert away of arrivals
49:53 happening conversion here rather than at the . So then it took us a
49:59 time to figure out all this uh effects of uh converted way of uh
50:09 . And we talked a little bit that earlier in the forest, we'll
50:12 a little bit more about it uh because not tomorrow, but next friday
50:20 it turns out that anise actually plays important role. So uh before I
50:30 on, I want to tell you uh I learned later much later that
50:38 we found success before we found success this, the Vice president for research
50:47 complain to my boss in Houston. , I was not there. But
50:53 heard the story later. The vice complained to my boss who was three
50:59 lower than him in the R And he said you guys are out
51:03 line here. Research is my not yours, your your businesses applications
51:10 the exploration department. And so my had the courage to say to the
51:14 president, he said, well you set up a system that cripples
51:18 people so we can't so they can't to challenges like that. So we're
51:23 do it here. Huh? um you can imagine that the vice
51:30 was pissed at this response from this level manager. So no doubt he
51:37 to my boss's boss was Vice president Exploration in Houston. So that guy
51:43 not know me but he knew my and he backed him up. And
51:47 we went ahead and did the But I'm I can assure you that
51:50 we had not had this kind of , both scientifically and commercially for the
51:56 business unit, we would all been and I would not be sitting here
52:00 to you today. So I want tell you one more interesting story along
52:07 lines about a year later one of colleagues at a medical came to me
52:15 a confession and he said leon I've uh trying to image this Reservoir using
52:25 wave data for the past year by velocities and so on and adjusting the
52:31 model. He said I tried I 80 different variations velocities and imaging algorithms
52:39 everything. None of it um So he said you guys had this
52:47 with your very first try, very . Uh We were we were very
52:52 in all the stages here were much sophisticated these days you guys had success
52:58 we were willing to think of this a converted wave problem different in uh
53:04 many ways from a P wave And he was thinking that all we
53:08 to do was the justice thinking a bit. And he would find success
53:13 classical methods. And we said uh said to ourselves, we're gonna throw
53:19 the uh the recipe book and start about this problem from first principles.
53:26 in fact we did invent many new for conferred wave processing which are showing
53:32 here and I'll tell you more about . Um Next friday because a lot
53:40 it involves. And I started so look at this quiz was five to
53:56 time for sedimentary rocks. The dominant of attenuation in the sighting band is
54:03 fluid flow. Well, we already that that that one is true.
54:10 then you saw from the previous figure from the previous discussion that um the
54:21 is enhanced. This same mechanism is that gas is present in the four
54:26 . That's what you see right? here, but uh you see it
54:32 uh the previous image for p ways one data quality just disappeared here because
54:44 enhanced attenuation of the p waves due uh squirt global squirt flaw enhanced by
54:52 partial saturation. That's so that was true. Uh And we also saw
55:03 one's true. If we look at wave data, we might get images
55:07 are better in those freeways. And uh not necessarily, but maybe.
55:13 we saw in this example that was also. So now I want to
55:20 about apparent continuation and let me check time here. It's now 2.30.
55:27 so this is a good time for take a break. So let's come
55:32 at 2:45 and we'll talk about apparent . Mhm. So now after the
55:42 we're going to resume where we left with apparent attenuation. See how this
55:48 is italicized. Uh Somehow when I it like this, it doesn't respond
55:56 the controls. So I have to um I'm gonna try this and try
56:09 . And now here the screen see if it works this time.
56:23 , sharing the screen and now the work. Okay, remember from the
56:30 lecture we were deriving the scalar wave and here's some stuff that we um
56:38 did before and we we we came to this part where we're taking a
56:44 with respect to this uh combination which is basically the stress. And
56:53 that point we assumed that the the medium was uniform and we brought
56:58 thing outside of the uh derivative and it over here and now the derivative
57:05 is right here operating on this. you can see it's not yet,
57:10 wave equation, you can see there three space derivatives here and both space
57:16 time derivatives over here. So it some further work to get the wave
57:22 out of this, but I want concentrate on this step right here because
57:28 know the medium is not uniform, not even piecewise uniform. Normally in
57:35 business, we assume that it's piecewise and we solve the wave equation inside
57:44 piece separately. And then we handle intersection between the different uh pieces with
57:58 condition and we get reflections and refraction all that. But now I want
58:03 recognize that it's um that whole process uh not really the way to approach
58:14 Earth because the Earth has in homogeneity all scales. And so you can't
58:22 any place in the Earth which is uniform even on a small scale of
58:28 , right? There's the grains and , there's an in homogeneity. And
58:33 uh we're gonna think about in homogeneity on a larger scale, on the
58:39 scale next. So without making that when we moved from here to the
58:48 line and we get a term like did before plus uh this additional term
58:53 comes from the derivative of the stiffness with respect to X. J.
58:58 like it says in it, it's changeable calculus. So this is a
59:02 new term. So now we're going see how this term, because of
59:10 non uniformity of the media leads to attenuation. So uh you can see
59:18 in the general case it's gonna be complicated. So let's simplify ourselves down
59:23 vertical p wave propagation. And so vertical p wave propagation, the previous
59:29 simplifies to this. You can see this part is the wave equation that
59:33 saw before. And now we have additional term which comes from the derivative
59:40 the displacement. Uh comes the derivative the stiffness element with respect to um
59:50 distributive here of the vertical component of . That's just this one over
59:57 Yeah. So uh new things come because of this. So this makes
60:06 wave just like we talked about that this term is going to lead to
60:11 attenuation. We're going to assume that here is real. So uh ignore
60:17 we did earlier about. Uh complex module I I want to abandon uh
60:25 discussion of those complications, revert back the elastic case and think about how
60:32 gonna get apparent continuation out of perfectly media. This number is real.
60:40 as before we're going to assume a wave solution. And so here we
60:45 a set three, that's a that's the vertical component of the wave
60:52 . And we're gonna put this into equation of motion. And when we
60:56 we do that, that guess at solution then in that case there's two
61:03 with respect to depth results and and two factors of minus I k three
61:12 here minus I K three in the exponent leads this and in the same
61:21 we get one factor minus I k coming from this term. Right
61:29 there it is. Now, this a quadratic equation for K three.
61:36 uh where do we see that? rearranging terms? We uh we get
61:46 this uh a three squared comes from . K three to the first power
61:51 from here, and K three to zero power comes from here. And
61:55 notice that the coefficient here is imaginary before, when we ignored the in
62:03 of the medium, this term was and we just said V p squared
62:09 squared equals omega square. And that old old to you now. But
62:15 this is something new coming from this here. This is not zero.
62:22 . So I said before soon that meeting is perfectly elastic so that the
62:30 lunch, general element assistant settlement M real. Now the solution to that
62:36 equation is given by this. Uh knows how to solve a quadratic
62:42 And so because of that, three is complex even though M is
62:50 . So this quantity here comes Well you can see it's it's got
62:57 partial F. With respect to So that was zero before and it
63:03 before it was zero here and it zero here. So we got the
63:08 expression K three equals America over vp now it's not zero even though uh
63:17 non zero um uh part, let say it again, A three is
63:26 because of this, even though M real. Now in this expression,
63:34 that the velocity and the stiffness are of the medium, not of the
63:39 . K is the property of the , but M and V P.
63:44 . Properties of media. So the is specified by frequency and the choice
63:50 algebraic sign right here. So we think about propagation in the plus the
63:57 . So we select the plus sign here. Now what we're gonna do
64:05 what we do so often is to a taylor approximation and assume that these
64:11 are small in that case the square simplifies. So this linear term and
64:19 brings in a factor of -1 Where does that come from? The
64:23 comes from here and the one half from the square root and we still
64:27 this square. This quantity is going be a soon to be small.
64:34 just to make further progress, we understand what this is telling us.
64:38 make this approximation, then we average over a wavelength. And so uh
64:45 we average it over a wavelength uh This is average here, that's this
64:51 here and this gets average. That's one here. And you can see
64:57 I've done is I've multiplied this out that the omega over V.
65:01 Uh multiplied by one this term. then you get um um uh makeover
65:11 times this stuff. Uh Simple planks this, you still have omega in
65:19 ah in the Denominator here because there's omega squared here, multiplied by an
65:26 here still leaves one omega down The real part is given by this
65:32 the imaginary part is given by So this here is what we previously
65:39 the friendly multiple delay. So the that this is squared here. That
65:44 from the fact that um there's two to additional reflections in a friendly
65:50 And we talked about this friendly multiple um before a lot. And so
65:59 here it shows up again in this and it's all consistent with what we
66:05 before. So putting that all into , we have the plane wave solution
66:13 like this. Um This is the part of K three is here and
66:19 backup, here's the real part, what it says right here, the
66:23 part. And then plus I times imaginary part. So the imaginary part
66:28 showing here and IMUIM multiplied the two I together. So we got a
66:36 here. And so uh here we uh term uh leads to propagation is
66:43 we did before. And here is term leading to apparent continuation. You
66:47 see that it's getting smaller as Z . This factor is getting smaller compared
66:54 of this minus sign. But actually does depend upon this average of uh
67:01 variation uh element in. So let's a case where the stiffness shows a
67:13 . So uh any limited zone in subsurface, is it? Uh can
67:19 that the uh layers having increasing stiffness them. So in that case this
67:26 greater than zero. And this leads exponential decay in Canada frequency. It's
67:33 real or generation, but that's how decay of amplitude, even though in
67:42 no truer generation, I suppose the is moving the other way upwards.
67:49 uh so we're gonna obligation of the c direction. We select minus sign
67:57 the same sign for the imaginary term . And if the variation is small
68:02 gets to this simplification. And um in the same case where the stiffness
68:12 a positive trend, the apparent generation is still what we said before,
68:17 now Z is decreasing so his Z smaller and smaller. The amplitude growth
68:29 . So is this a problem? , it's not really a problem because
68:32 only happens over a limited range of in the um in the earth.
68:43 uh in this case what happens on way up, reverse is what happens
68:49 the way down. That doesn't happen true attenuation. Uh But it happens
68:56 this case with apparent continuation. So this case for two way to travel
69:01 this zone, the apparent continuation cancels . So that was for the case
69:09 a trend. Now let's case let's a case where the stiffness fluctuates up
69:14 down with no trend. Then the part of of uh okay, vanishes
69:22 uh if this thing is fluctuating up down with no training, the average
69:26 going to be zero and the real is frequency dependent. Um like we
69:35 before, so notice here, the average of the square of the fluctuation
69:42 non zero, the, the average the fluctuation because designed designed to be
69:49 to be zero. Since we're assuming this special application we've assumed no
69:56 So Average is two but the square that imaginary part does not average.
70:06 now just to be sort of more more explicit, let's assume uh periodically
70:16 a cycling value of of business. this is not um This is not
70:26 with uh finite jumps between the This is uh cyclical uh variation with
70:38 and co signs. And it's gonna an amplitude of the variation given by
70:45 M. And it's got a special of the variation given by H.
70:53 is uh after bed thickness, two travel time magazines. And in those
70:59 that case the stiffness derivatives given Um It's simply the derivative of
71:06 It's I times two pi over eight I time over eight. Same delta
71:15 gives the amount of the variation within cyclical in the wave vector. Uh
71:23 that into here. The wave vector uh this run by this. And
71:30 at high frequency this term goes Why? Because of high frequency omega
71:36 a large, is in the So at high frequency this goes away
71:42 we're left with this term only. that's the high frequency. So the
71:50 frequency of velocity is given by uh over k. High which is one
71:58 the average run. Uh huh. am velocity which is here, I
72:06 it wrong, this is the average of the slowness To the £-1.
72:14 this is what we're averaging is not . We're averaging slowness. This is
72:18 like uh in rate theory, the is given by the average of the
72:25 . Yeah, that's what it says . This is the very theory result
72:28 we found and here it is. Do this from huh. Which for
72:40 low frequency uh This is gonna be bigger because this term here is gonna
72:51 not negligent. And why is it necklace? Will uh This this value
72:59 not so high as it was before this uh this is positive and we
73:04 a plus here. And so the frequency phase velocity is lower because it's
73:10 know, dependent on omega over Low, here's que lo And so
73:15 phrase velocity is small is different from high frequency phase velocity because of this
73:24 here. And this minus sign comes the fact that uh this is the
73:30 of this. This plus turns into minus when we divide by uh
73:37 And this is the only multiple people that we talked about. Yeah.
73:45 there's more to be said. Uh . We've just established that the p
73:54 loss varies in frequency so there must apparent attenuation. So the very assuring
74:00 is given by this that just comes differentiating this. And so the parents
74:08 factor is related to the chill Oh yeah. Oh dispersion. Here's
74:16 dispersion and mm hmm. Uh the q. Factors related to that by
74:24 expression. And so it's non zero um the apparent continuation. We can
74:33 this the apparent attenuation due to the multiple effect. Because of this.
74:38 frequencies decay more rapidly than low Just as we just as in the
74:45 with real attenuation. Even though in analysis there is no attenuation at
74:51 All of these stiffness elements present because the material fluctuations leading to friendly multiple
75:01 . That's gonna be a function of . And it's going to lead to
75:05 apparent continuation factor. You invite that MS del rio. It says here
75:22 of motion leads in the case of betting to these things or maybe all
75:27 the above. So what do you here? I think it's only be
75:36 . Mhm. This here is We know it has. We know
75:51 apparent continuation. That's that's easy. how about a lower frequencies and
75:57 lower velocities and higher frequencies. it's higher velocities and higher.
76:03 okay. Yeah. So uh he correct. Well done. So here
76:08 the conclusions from today's uh from this on attenuation No one. It's important
76:15 deal with frequently ignore it. But really is important. It causes the
76:24 loss of high frequencies as a result propagation through a genuine meeting. Always
76:31 by dispersion. And there are many physical mechanisms. So each one is
76:38 in one or another banda wave propagation in the in the band from ST
76:45 to sonic. The most important is flow. Also called fluids. For
76:53 addition to the velocities, the attenuation at reflectors cause the phase shift in
77:00 reflected wave. And sometimes that can significant. And finally non uniform media
77:07 apparent attenuation, whether or not there's reality. And in the special case
77:18 the where the non uniform media is , we get associated dispersion.
77:29 The transitional statement is the same, that leads to apparent attenuation also leads
77:35 anti sergeant, which is the last in this series here, remember what
77:42 did was we looked in the first lectures here, we did classical racing
77:46 . Classical raising ways. And then realized that some of the assumptions that
77:52 had made were not very realistic. we want to apply to Ocs.
77:58 we looked for elasticity, we wanted continue imperfect elasticity. So we just
78:05 that and now we turn our attention an icicle. So I'm gonna stop
78:12 and I'm gonna stop sharing. And think I want to just continue since
78:18 just now uh getting back together and I happen to have uh next
78:29 I'll cheat up right here. That works. It works. And
78:39 into presentation node. and so less 10 anti circuit. So here's our
78:48 objectives. By the end of this , you will know how to describe
78:53 common classes of anti psychotropic. You'll to learn how to find the wave
78:59 for the simplest case, which we polar and isotopic. Then you're gonna
79:06 exact solutions and then you're gonna find solutions in the case where the anti
79:12 is weak. Then you're gonna learn these anti psychotropic parameters appear in seismic
79:19 as we make the transition to weak . We're going to define new parameters
79:25 we have not yet seen. And gonna see how those appear in your
79:31 . And we're gonna see how anti affects P wave reflectivity, it affects
79:37 p wave real problem in a strong , even though the anisotropy is weak
79:44 how it affects share waves in a way and how it affects convertible waits
79:50 a fundamental. So that's a lot stuff. Good things. We
79:55 we have about five or six hours go through this. So the first
80:02 we're gonna do is tensor elasticity. I remind you that all of the
80:08 has been classic seismology equally suitable for or for understanding the deep interior.
80:16 now we know that none of it truly suitable for either case, since
80:21 ignored the effect of an eye socket out that most sediments are not massive
80:29 stones like this. But the equations they are right. The the equations
80:35 psychotropic media but most settlements are not this. Uh does anybody know where
80:43 rock is? You should know where rock is. A very famous
80:53 This is in the middle of So it's about very close to the
80:58 geometrical middle of Australia. And uh Australians call it Ayers rock for
81:06 But I suspect that the the aboriginal knew it was there before the Europeans
81:12 , and in fact they have a for it, it's called U
81:19 U R. U. And I'm sure how that translates. But um
81:25 the first thing you're about, so our equations assume that the rocks
81:32 like this in the subsurface, but fact they look more like this.
81:37 And uh it says here they're normally because you know, we're gonna be
81:43 these many layers with assigning wave which which is much longer than the average
81:51 there uh than the average layer So um uh it's gonna experience this
82:00 layering as an average, right? it's obviously gonna be a different average
82:05 or horizontal or whatever. Now um might say, okay right here we
82:13 uh sequence of layers which are more less the same, and we're gonna
82:20 those as an uniform and I see layer, and then we have a
82:24 boundary and another block type underneath this slope of weather debris. And then
82:33 I'm gonna call this is uh another uniform uh formation, And then there's
82:42 one up here. So, that be a typical uh way of thinking
82:47 these kinds of rock is piecewise But recognizing that within each layer it
82:56 be anti subtropical. Well, there might be other small scale features
83:04 than a wavelength, which might show in a data. You see these
83:09 fractures here and you can you see vertical fracture uh as half of the
83:16 , and the other half fell off the lake. And here you see
83:19 parallel fracture uh also fell in. see a second set of fractures here
83:32 this one. This whole thing is a vertical fracture where the other half
83:38 it fell into the lake out of screen towards you. So, these
83:43 small scale features with preferred orientations, are gonna make the rock mass with
83:51 as an average over seismic wave It's gonna make it. Um So
83:59 Nature tells us that all rock masses this kind of famine small scale structure
84:06 a preferred orientation. So, let's at what we have here. The
84:10 thing we have is uh layered And you can see that these layers
84:16 pretty much like layers we saw previously the cliff face. Except it's much
84:21 scale. In fact, these layers laid down in a glacial situation in
84:26 lake. And these these are yearly , yearly cycles that you can
84:32 And it's uh it's uh um sample of you can hold this in your
84:40 . Um here is crystal and you see this crystal looks to be um
84:48 , doesn't mean you can see right it. But um you know that
84:54 the inside of this crystal uh is Adams, you probably recognize this crystal
85:02 calcite. So there are atoms of and carbon and oxygen arranged in a
85:09 lattice. Uh in little tiny unit which are this shape. And those
85:15 are all lined up together. And in this case the external shape of
85:21 crystal is lined up with the internal of all those uh unit cells.
85:31 here, there's a rock where um . Has a different type of an
85:41 . Here. You see the layering this is not layering, um uh
85:48 the force of gravity. This is formed uh response to uh perpendicular to
85:58 direction of gravity, this is perpendicular a temperature gradient. This is a
86:03 rock. And it's uh These layers formed uh particular temperature gradient as cool
86:11 the subsurface. And then later somebody along collected that rock and and uh
86:18 it into a paperweight. So, the the crash put the external shape
86:24 it before. Oh, so here's a piece of wood and you
86:32 even that one has a small scale would preferred orientation. You look carefully
86:38 this piece of wood, This is out of a tree. And the
86:41 tree was going like this vertically. so the carpenter this vertically. If
86:47 look closely here, there's an elongated um cells of the tree uh quite
86:55 this direction. And so the reason carpenter cuts this way is because it's
87:00 better for building buildings. If you it in this direction for all the
87:05 scale structures, appoint advertisement, it the weight of the building better
87:12 this one looks like a sandstone. can't see any invisible uh huh uh
87:20 in here. Of course you can there are grades important and probably see
87:24 with your maybe arc eyeball. But can't see they all look as though
87:32 oriented randomly. So this rock looks it's analyzing this rock looks like it's
87:37 traffic but wanted to measure it. um is anti psychotic. The velocities
87:44 the vertical direction are faster than the across it. So we interpret that
87:50 um due to small scale cracks in rock. But you can't see with
87:57 your eyeball and um those are affecting seismic waves or in this case acoustic
88:05 , ultrasonic waves, uh even though can't see it with your eyeball.
88:09 so just to prove that we subjected brought to high pressure and anti
88:17 We went away because the cracks went , I external pressure. And so
88:23 proved to our satisfaction. The anti in this rock is due not to
88:31 orientation of the crystals repression for orientation a friend before I leave this
88:40 let me excuse myself for a You don't think you're seeing me
88:44 Or maybe you can see me walking from your computer. Now I'm
89:05 And what I want to do is sharing ST and change your views so
89:15 you can see me maximum. I am opening up my little magic
89:25 here and I have in here box magic stuff and included in this magic
89:42 . Yes, piece of calcite. looks very much like we saw
89:49 You can see right through it You can see the color of my
89:53 right through there. But if you closely, you can see that.
89:57 got got um uh angles, not right angles. You see that that's
90:04 sharp angle there. And for analyzing kind of crystal, you would obviously
90:09 to have a coordinate system which is up with the edges of this.
90:14 wrong. So that you can say as we move in one direction,
90:19 passing so many unit cells in the direction passing so many unit cells.
90:25 this rock naturally grows so that the reflect the internal symmetry of of the
90:33 structure. So starch sharing again back this screen. It's not obeying my
91:01 I slide directions. What I have do. Yes. Go back here
91:15 now if I share the screen this , now this is a famous
91:36 Uh have you all been to the Canyon? This is about a mile
91:41 the top here of the mesa, here to the bottom and you can
91:46 lots of zones here where the cliff vertical and then other zones where there's
91:51 sloping um pile of debris and uh the cliff is reduced to a slope
92:03 there is a shale. So these well consolidated, uh well cemented sand
92:09 and behind here is a shale behind , you can see that most of
92:13 section is shale, it's about a from top to bottom and most of
92:17 section is shale. When you look at one of those shales, you
92:22 that it's uh it's on the small is not uniform, it's got uh
92:32 different grains in it and it's got between the grains and the grains are
92:38 major sorts. You see these here are more or less round grains like
92:45 , more or less round grains, call those equity grains and then there's
92:49 of grains which are shaped like plates they're lying more or less parallel except
92:56 . They drape over answer, but very obvious that this um uh this
93:04 rock was uh it's a positive out um quiet waters such that the grains
93:18 of uh play settled down more or flat and they got to be more
93:24 more flat as they were buried. so this rock is intrinsically um uh
93:31 icy topic because of the preferred orientation these clay minerals. It's very clear
93:42 this way in the core, this is originally up and not this way
93:47 this way, this way is original to everybody. Remember when we saw
93:53 picture here and um this picture So I remind you that this kind
94:01 antisocial that we saw here that is here. So this this even though
94:09 uh, this is a shale behind weathered slope of um rock pieces and
94:17 this, this is a loose loose here. You would not want to
94:22 on here. If you walk down , your feet would slide and you'd
94:26 off. Uh it's a foot or feet deep of loose rock. And
94:33 behind here is the shale like we here. Now, in addition to
94:40 structure, we also have oriented french and this allows me to tell the
94:47 about how I got personally involved in such. It was the year in
94:55 before some of you were born And in those days we didn't have
95:00 the fancy workstations we had today, printed out our setting up sections on
95:08 . So in 1979 I joined uh and about a week or two after
95:17 joined chemical in Tulsa, my boss to me and he said leon here's
95:22 ticket, we're going to Denver in days. So I said what's it
95:29 ? And he said, well uh exploration office in Denver is considering making
95:36 bid on some offshore prospects offshore We don't have an office in
95:43 So the Denver office is handling and all have to understand that in the
95:48 States the offshore oil and gas is owned by anybody except the US federal
95:54 that the U. S. Federal does not want to drill for it
95:58 drill for oil and find it instead want to sell the opportunity to do
96:05 to oil companies who have the So they have auctions every year and
96:10 say okay this year we're gonna have auction on these tracks in the in
96:15 pacific ocean, atlantic ocean and gulf Mexico. And I guess also in
96:21 in the in the Barents sea and there's an offshore american waters will have
96:30 auctions and companies are invited to So had been invited to bid on
96:36 on a prospect offshore California. And exploration team there was very young in
96:43 days. Uh the whole business was and uh experienced people uh were being
96:52 away from the major companies like Amoco the smaller companies who offered them more
96:57 . And so uh large companies did want to match the pay scale of
97:06 smaller companies. So they said to , you know, they they said
97:09 their departing talent, they said uh with our blessing, we're gonna replace
97:15 with cheaper people from the universities. so we went out and hired all
97:20 oil companies went out and hired fresh , people just like uh Miss Del
97:26 . And uh you had been graduating 1979 you would have five job offers
97:34 your on your desk right now and and your husband would be thinking about
97:39 a bigger house. So these are times of course, but I'm telling
97:43 about those times. And so we out there, my boss and I
97:47 out there to advise the young exploration in Denver looking at their data with
97:54 um properties expertise to figure out is anything in this, in this data
98:02 we could help the exploration team in give them an advantage in the bidding
98:08 the auction that was coming up in a few months. And of course
98:12 was not an expert, I had been hired and why why was I
98:16 ? It was because we were losing many experienced people and they were willing
98:21 hire a person like me at the with no real world experience at
98:27 They were willing to hire me and uh probably wouldn't do that anymore because
98:33 didn't have any of the specific skills you people have. You two young
98:39 today have more skills relevant to exploration I did in 79. They hired
98:45 anyway because um they needed bodies and was willing. And so my boss
98:53 me out there and we were greeted the exploration team and they rolled out
98:58 on a large flat table. They out um a paper plot which showed
99:04 the image of the uh data offshore . And it looked like this except
99:11 was a flat sheet of paper. seismic data on here and is oriented
99:16 to west. And so it was duty era. And so just imagine
99:20 image here and it would be of the time image and the common midpoint
99:28 would be in this place. And they took out another image uh which
99:34 north to south. And but they super pose it like I'm doing here
99:39 the computer because it was in So what they did was they folded
99:44 over and then they laid it down so so what we had was a
99:49 D. Section of the subsurface this east to west and this part north
99:55 south. And it was joined right at the crossing point of the two
99:59 . Lines I skipped over that Of course we had some two
100:04 Lines running this way and some running north and south. And so they
100:09 us two sections. So and so they pointed out just where on this
100:16 was the mhm Reservoir possible reservoir looking , stay along in here somewhere.
100:27 my boss took one swift look at and he pointed right here at the
100:32 point for the two lines. He , what do you think about
100:36 And so we all looked at that and on the one section it was
100:40 bold right um reflection. And on other section perpendicular it was very
100:48 This was the the image. And the kids on the exploration team said
100:54 we didn't notice that uh is that ? And my boss said, well
101:01 don't know it could be. And looked at me and he said,
101:04 do you think? Well, I probably the most naive person in the
101:08 . I had only been on the for a week or two. This
101:12 basically the first seisint section that I ever seen and looked at seriously,
101:17 I was also the new hire and had to say something um uh
101:22 So I said, I don't it could be some kind of
101:26 So my boss said what could cause . And so of course she knew
101:33 , he knew about shale and I which would lead to a difference in
101:39 between the vertical and the horizontal And he knew that was caused by
101:44 preferred orientation of clay particles, just we saw on the previous line.
101:50 that kind of preferred orientation could not to a difference in azimuth. Like
101:57 saw here difference in east west direction to the north south direction. And
102:04 I said, well, maybe oriented . Remember we saw previous flight oriented
102:12 . And uh, so that could lead to different um seismic propagation velocities
102:22 as the folks in the restaurant. so he said, do you know
102:27 about that? And I said but I can figure it out.
102:31 that was a clever thing for me say. It gave me uh,
102:34 breathing room. So, uh, uh, spent the afternoon with the
102:42 team and then we went home even uh, the end of the day
102:46 the next day. We went And in the next couple of weeks
102:50 learned a lot about anti socks. I called up the kids on the
102:56 team back in Denver and I good news, I solved your
103:00 And they said, which problem was ? So of course they had paid
103:05 attention at all to us. Pointy folks from the research center. Uh
103:10 had only invited us because their boss suggested we should, they had checked
103:15 box. And as soon as we the uh room, probably even before
103:20 left the building, they had resumed um, conventional analysis, uh,
103:27 these features that my boss had but it was too bad because those
103:32 turned out to be important and led uh it led to my building an
103:40 professional career based on an just from lucky happenstance that days. So let's
103:49 at some of the things I learned those first two words. So if
103:56 rocks are anisotropy, we need to with the tensor wave equation. And
104:01 we still have the hell bolts theorem we still have a scale scale of
104:06 that uh that looks like this. going to assume for now that the
104:12 is uniform and so we have three . We respect the space here and
104:19 is on this side and the scale potential here. So previously we solve
104:25 problem, assuming that the stiffness sensor the ice. A tropic special
104:32 Now we have to be more realistic that. So what we wanna do
104:37 apply this equation to the simplest case elastic anisotropy, which we call polar
104:44 . It's valid for shales and thin sequences, lot for fractures like we
104:52 about. But this is um with looking back, I want to discuss
104:59 situation first and in that case the matrix looks like this, we saw
105:06 uh before I think the first uh we assume in this case that
105:12 west is the same as north So these two horizontal directions are a
105:17 one, but it's different from this and in the same way we have
105:23 share module I which are the same one which is different. And so
105:27 makes a total of four independent constants the diagonal here, plus 1/5 1
105:33 here, and which is also repeated . And then this one here we
105:39 from. So this is the simplest of geophysical interest. I'm sorry that
105:49 went from two constants, you and and near to five. It
105:55 have been nicer if we had had simpler case to learn from. And
106:01 fact there is a simpler case, gonna stop sharing here, look at
106:06 screen and we're gonna hold up another , see this crystal here, anybody
106:15 what that is. I think you know if I handed it around uh
106:23 though you're not geologist, you know geology to recognize this is a piece
106:29 fool's gold and its iron sulfide and happens to form into cubes like
106:38 So here's a nice cubic sample. this is um this turns out to
106:45 simpler representation uh the elasticity tensor than one I showed you. But we're
106:56 gonna we're not gonna discuss it because hope this works. No,
107:08 doesn't work. You got to uh sharing. Go back to this thing
107:27 okay, now I'm gonna share and I'm gonna have control of the presentation
107:52 this works. So here are five parameters. And we're gonna uh here's
108:01 the system that we resume. And the QB case that I showed had
108:07 independent uh three independent parameters instead of . Uh more complicated than I
108:17 which has just two but less complicated this one with five. But we
108:23 we don't want to discuss the cubic in geophysics because there are no features
108:29 the earth larger than the one I showed you where uh which have cubic
108:35 . So um this is the simplest . Now we know that this quantity
108:45 governs vertical pre way propagation and this governs horizontal. This is the
108:54 And then we have to share Marjolein uh fifth parameter which is like a
109:01 asse. And then um this one calculated so we have two different um
109:07 diagonal components and four uh different importance this time. Is that on,
109:19 that again? Okay, so thanks that question. Uh Mr Love a
109:31 love back in the 19 twenties introduced notation for this kind of anti
109:41 He called it G. T. . Which stands for vertical transverse
109:45 So it's vertical because the polar axis vertical. You call it trans firstly
109:51 should topic because these two transverse directions equivalent meaning these two are equal.
109:58 why these two are equal. That's very confusing name because it's a type
110:05 anisotropy. And right there in the of that type of anisotropy is the
110:10 ice, a trippy. So a name is polar anisotropy because that's that
110:17 what it is. It's anisotropy with poll of symmetry, which is this
110:22 . And because this is a polar , all of these transverse direction.
110:27 that's a better name. That's the name for VTR. And so you
110:32 see the term the description V. . I uh in lots of places
110:39 it's slowly changing towards solar energy even the simplest case the quasi P and
110:49 essentials are coupled. In other words you do a hell most decomposition into
110:57 curl free part uh and a divergence part of the display shin those are
111:02 solution. The girl free. Part the displacement vector field is not a
111:09 and neither is the divergence free. we need a better mathematical idea than
111:14 given to us by mr helm So we go back to this full
111:21 equation of motion and we're gonna guess the plane waves are a solution.
111:26 then we're going to verify the conditions the wave vector which makes this
111:31 So here's our guest right here, like we did before. And uh
111:35 our wave vector and the length of wave vector. We put this into
111:39 equation of motion and uh in So from uh those driven. He's
111:46 we have multiplication is with different um friend components over the ones that were
111:58 . And so this is three equations one. Uh So for each value
112:03 I so there's three equate iculs 12 three. And for each uh each
112:10 these three equations there uh different right side. And it's got 27 terms
112:17 we're summing over em and and and . So 27 terms on this.
112:24 you see it's pretty complicated. So thing we're gonna do is divide by
112:29 squared. So here is uh one the K squared and here's one of
112:33 pieces here. So we're gonna call ratio here the square and we're gonna
112:47 three different solutions for V depending on stiffness elements and depending on the directions
112:56 propagation which are given here, see K. J. Is a is
113:01 component of the wave vector. Is the length of the wave
113:05 So this gives uh what we call direction co sign, This is
113:10 which depends upon the direction that K . And here is a different direction
113:16 cake. So the the unknown in equation is the displacement. Here is
113:22 eye displacement. And for each of 27 terms here there's a different uh
113:29 of the of the solution you so mathematicians call this type of in equation
113:40 I can value equation. Why is ? It's because it's got the the
113:46 quantity here and here and then what's over is zero and here is a
113:53 and here is a scalar. So you have a matrix operating on the
113:59 of unknowns being equal to a scalar that uh same vector component, that's
114:11 Eigen value inflation. And in matrix away to write that is this component
114:18 with this patriots here and L called with a Sprinkle here and it's a
114:23 by 22 by two matrix. And , excuse me, it's it's a
114:32 by three. Matrix is what I see. Why is that? It's
114:36 we're summing here over J and So uh some of jan and we
114:46 uh left over uh the M and eye. And so that that's a
114:53 by three matrix call it L. uh this is the identity matrix,
114:59 ? So this is uh Matrix three 3 Matrix whose Diagonal elements are all
115:08 to one and off diagonal elements. that is the way a mathematician likes
115:16 look at this expression. So here the definition of the components of the
115:23 in matrix L. And uh So uh this has two kinds of
115:33 . The easiest solution is to okay, all components of you are
115:37 . So that's called the triggers and has a nontrivial solution only when the
115:42 of this matrix is zero. So up in the uh in the glossary
115:51 what we mean by as determined. the current is a complicated combination of
115:58 matrix elements. So when you write out explicitly it's a cubic equation In
116:10 elements of the matrix and it has solutions. Well I can values,
116:17 right here we only find nontrivial solutions three special values of blocks of B
116:26 . And those three are called Eigen and each one has associated with it
116:31 vector which we call it Eigen vector . And that is the solution for
116:36 , which comes with each of these values. Um harvey. So in
116:45 case the Eigen drivers are the squares the velocities and the Eigen vectors are
116:51 corresponding polarization vectors of the displacement, of the case but of the use
117:00 , give the direction of propagation of way and uh polarization vectors give the
117:10 of the displacement within that. When that's uh we're gonna solve that equation
117:22 the special case of pull around. start to me directly but before we
117:26 to that, let me ask you Miss del rio, uh we have
117:32 of the above or what I wanna . It's either B or none of
117:39 above. I'm not sure. Well is definitely true and uh he is
117:49 true but doesn't answer the question and is also true. That doesn't answer
117:55 question. Uh So um but B definitely true. I agree with
118:01 Now Mr will uh let me suppose one to you, A B or
118:07 or none of the above. So we're getting um um interference uh mr
118:28 between this course and your other So I'm gonna ask you to mute
118:34 microphone again and I'll talk our way this. We says here, we
118:39 to use different mathematics for the anti case because a only valid for tropic
118:47 field. No, that's not Mr Helm Holtz didn't know anything about
118:50 , octopi or anti socks or Mr that's a mathematical theory has nothing
118:56 do with physics. So that's It says that scale of potential is
119:03 a solution to the equation of Yeah, that uh that uh that
119:16 also true. The p wave solution we're gonna find here is not gonna
119:22 from the scale of potential. It's this was also false. And so
119:27 one is obviously false. So the answer is deep uh meant to you
119:36 del rio, it says true or . The Eigen value equation is just
119:41 special case of three simultaneous linear equations three unknowns which are homogeneous since all
119:48 the terms contain one of the So uh let me a minute go
119:56 here on my screen. You I you have this screen in front of
120:01 at the same time. So you the question in front of you,
120:05 gonna go back here. Mhm. here is uh the Eigen value,
120:17 and you see it is linear and homogeneous because the unknown is in every
120:23 , there's no terms over here. The source term no terms over here
120:30 without the unknown and it's linear. it's complicated with just a linear
120:36 So I'm gonna say This is a case of three. Again, this
120:43 true. Right? So now let's the solution in the special case uh
120:51 case corresponding to polar. And I so here's our special case. So
121:08 gonna apply this case to the previous what I said was from uh from
121:19 up against here is here's the equation here's the victory question but here is
121:30 one cent and this is three right? Three equations and one this
121:34 one component. This is one equation , but it's cubic. And so
121:39 gonna depend upon uh um combination of the terms in this uh matrix
121:49 And so even in the simplest even the simplest thing is too
121:55 So we have to be more clever that. We we can't just sort
122:01 uh style of a typical equation by for so let's just make a guess
122:14 um just like we had in H waves and icy tropic sizing.
122:18 might be a solution to this problem a shear wave which is polarized out
122:25 the plane of the figure for all of propagation of the shear wave is
122:33 to be polarized in the horizontal So that's what we're going to look
122:38 and see if we can find We're gonna look for a similar situation
122:42 we had with S. H. that is, it's gonna be a
122:45 which is going to be propagating in X. Three plane of the of
122:49 screen right here, the X. X. One X three plane is
122:53 screen. And it's gonna have only line component of polarization. So we're
122:59 assume that this thing here, which got any one of the uh three
123:08 any one of the three, I values, we're gonna choose it to
123:14 an Eigen value corresponding to shear wave . Why isn't she wave propagation?
123:19 we said right here, it's gonna only polarized, the only non zero
123:25 quantity in the solution in the Eigen , which corresponds to this Eigen
123:31 it's in the two directions. So that means is we're gonna have only
123:35 tool here and we're gonna have uh . Two, that is the component
123:44 K. In the two directions out the plane. That's gonna be
123:48 And now let's see if we can this equation and we'll put this,
123:53 the ice, a tropic expressions for stiffness tips in here, we named
124:01 particular Eigen value. V. H. By analogy to the icy
124:07 share wave which is incident upon a , a reflector and with polarization parallel
124:12 the plane. And so we choose because we're very clever geophysicist. Uh
124:18 we were ignorant we wouldn't be able do this, but because we're clever
124:23 going to make this assumption and see it works. So now we're gonna
124:31 for a solution for uh only non terms R. K. One and
124:36 three in the in the uh in way of it. So um but
124:45 uh requirement, we're looking for waves are propagating in the plane of the
124:50 into the previous equation and all the advantage except for this one for these
124:58 . Uh we got only K. and K three's appearing here. And
125:05 see if we have a one actually we have a one here,
125:08 got to have a one here and have for three here we've got to
125:11 a three here. That's the way sun works. And the same with
125:14 these terms, because some of these sensors are zero. Some of these
125:22 go away. So you're left with these two terms out of the
125:28 And uh this one has K one . This one has K three squared
125:32 they have different different elements here. I asked the question about.
125:43 so okay uh denominator, is it when you if you want to answer
125:55 me this question, you have to the other um class that you're monitoring
126:01 that I can hear your voice. cannot silent because I or I can
126:09 to our writing. Okay, uh do it that way so you can
126:16 your question, write it down to yourself and we'll hold it and go
126:21 . Okay? So uh remember that we convert from this four index notation
126:27 the to index notation, this is 66 and this one is a
126:33 And uh so that a terms simplify that and yeah, more importantly than
126:49 , we have only tools here and two here because these terms are gonna
126:57 unless uh because of this uh stiffness uh the only two terms out of
127:05 four, which uh excuse me, is not four terms, this is
127:10 uh it looks like four terms but one of them is a sum over
127:15 and so when you recognize that most these terms are zero because of the
127:21 tensor, we come down to only and look at this, this is
127:25 important. Uh there's only you two the right side here and you two
127:31 the left side. So bang, did find a solution. Uh our
127:37 worked and so we can cancel out amplitude youtube for both sides and then
127:46 find out that the uh the Eigen which we label S. H is
127:53 to C. 66 times sine squared four times sine square. These are
127:57 directions of the propagation, you in the X three X one X
128:02 plan and the polarization is always in X. So our our our guest
128:10 inspired by our experience with ice, trippy with an S. H.
128:15 incident upon a plainer medium in our . And sure enough at work we
128:23 in in finding one of those three values. Uh All the three Eigen
128:32 are gonna be the squares of And so we found here that the
128:38 labeled S. H. Has a trigonometry, uh dependence on um angles
128:48 in the uh 13 player. So corresponding Eigen value is like, so
128:55 it's got yeah values. The Eigen is a porous in the two
129:02 So that leaves us with the other components which are coupled together like this
129:07 we have uh we have in this we've got both you ones and
129:13 but this is a quadratic quadratic And so we formed formed the determiner
129:20 this uh and that's a quadratic equation simpler to solve. And the determinant
129:26 that matrix is given by this. so in this two by two
129:32 the determinant has a fairly simple definition it is the three by three matrix
129:40 is much more complicated than this, the two by two determinant. And
129:46 like this, he's enough. And Here in the 2nd row, l
129:58 has index number three. Oh so it's a two by two
130:03 but the industries are not one and . The industries are one and
130:07 Okay, so that's what this So that's a quadratic equation which is
130:13 to solve. And here are the solutions. And so we're gonna,
130:17 these are uh Eigen values. The two Eigen values are the squares of
130:25 and they're gonna correspond to different modes propagation and one is like a
130:31 Wave and the other one is like S. V. Wave. Remember
130:35 SV wave we had before and um , a topic size mix. It's
130:42 shear wave which is impinge ng on in society, it's a shear wave
130:48 impinge ng on a horizontal interface with transverse um polarization of course. So
130:57 polarization is lying in the 13 And so it has a component in
131:02 V. Direction. So we call S. V. So we're gonna
131:06 the same notation here but this one gonna be uh independent of any
131:14 This is inside of a uniform polar tropic body. Now for the corresponding
131:21 tropic case, the velocity of the . D. Wave and the
131:26 H. Wave are the same but one is going to be different because
131:30 is uh the anisotropy case and the is pretty, it's a little bit
131:39 , but not too bad. Uh quantity for D. Is gonna go
131:48 and d is I think you would this complicated. It's got squares and
131:54 powers and square roots and involves lots different um elements of the plastic.
132:05 so this is the reason why nobody paid much attention to anticipate tropic
132:17 mix and exploration geophysics Until about This equation was discovered 80 years
132:29 This equation was discovered by the first ever called himself a professor of
132:36 He was a Polish guy and he teaching in the University of Krakow in
132:42 . And he, like I he was the first person who ever
132:46 himself professor of geophysics. Now, course they're a dime a dozen.
132:50 if he was the first and his specialty was seismic anisotropy by which he
132:57 this case only polar anisotropy. And derived many important results um A long
133:07 ago, it's 120 years ago he this equation and many other important results
133:14 from that, But nobody paid attention the next 80 years or hardly anybody
133:19 because of the complexity of D people up their hands at this and
133:24 you can't understand that, it's just complicated. And so um we've got
133:35 make some sort of simplifying approximations or and you can see that we have
133:42 be careful with that because look back here, the only difference in these
133:47 equations is the algebraic sign here and , so this quantity D. Is
133:53 difference between a P. Wave and S. T. Way. So
133:56 can't be casual about that. So first idea is to assume that the
134:06 the P wave assume for P wave as a p wave expands from a
134:13 source, it has elliptical wave And when you put that assumption into
134:20 , things do simplify too bad. Earth is not like that. We
134:27 lots of cases, most cases where not true. And so we need
134:33 better idea. And so yeah, sign is the only difference and we
134:41 have the third mount. So uh um remember Miss del Rio a couple
134:48 days ago, I told you there's different share waves propagating in um anti
134:54 media. So here's this one and one and then we have this P
134:59 . So these two share waves are propagate at different velocities. In general
135:06 here that each of these two in . Uh Anyway, that's what we're
135:13 interested in. The it's got four elements of the this distance element.
135:20 got one too three and four. notice right here is the shear wave
135:33 . Now this is one of the and this is the this model uh
135:42 vertical p wave propagation as we'll see a moment. Um uh And so
135:49 the icy tropic case, that's equivalent K plus four thirds new. But
135:54 is another share model by itself. 4/3 in addition to the C33.
136:01 this is a puzzlement that we would . This sheer modular in the people
136:06 following. And then it appears it appears everywhere. So we got
136:13 to share waves and one P wave the body of a uniform puller anisotropy
136:24 . Okay, so what we just is three different solutions A P wave
136:30 to share waves. And so each of them is gonna have a velocity
136:34 depends upon the angle of propagation. uh first look at this P wave
136:39 , here is the p wave um vector and here is the p wave
136:47 vector. I think you can see your screen that this displacement is not
136:53 in the direction of the propagation is a drafting error. That is what
136:59 equations say. So this is not true P way, it's not um
137:06 not a way of whose divergence is . Nonetheless, I have to tell
137:14 that uh nobody has ever figured out to either measure or make use of
137:21 little angle in it. So we're call this a p wave, even
137:26 we should call it quasi people. , now, for the share waste
137:32 share wave is a true share It's always it's polarized perpendicular to the
137:38 . That is perpendicular to the population . So that's a true share
137:43 But this uh SV mode is a share mode. Can you see with
137:50 eyeball that this polarization vector is not perpendicular to the direction of propagation is
137:59 from the um polarization direction. It different from the it does lie exactly
138:05 the plane, what? It's not perpendicular this and this angle is pretty
138:12 the same as this angle. And we call this a quasi share wave
138:17 this is a true share wave. so here we have velocities which are
138:25 to be dependent on this angle of . Um But we're gonna be um
138:32 huh. There's no asthma total All of these uh angles are only
138:39 from the polar symmetry axis. So uh uh uh is here Miss del
138:54 ? It says an ice tropic blocks abc uh no abc or all of
139:00 above or none of the above. B I didn't, did you
139:09 Well it's true but how about this ? Uh Don't the velocities all vary
139:15 angle but I thought it was only off of that um that original angle
139:22 then for see I thought there was four stiffness is that we consider not
139:27 . Okay, so let's let's go . So here are the here's the
139:33 . And of course these things all hurry with angle, there's the angle
139:38 there. Now let's go back a further all the way back, all
139:44 way back. All right back. Okay, and here is the uh
139:54 the matrix for uh that special case polar anisotropy count them up.
140:01 3, 5, 5. Okay, never mind. Okay,
140:10 um So you blew that one? thought that was an easy one.
140:22 , I got confused because that the second to last slide before this
140:28 you said there was four. So I guess I got Yeah, so
140:33 four included in here, but then the fifth one right here. Got
140:37 . Ok. Yeah, so maybe was a trick question. Okay,
140:44 now I have a quick question about angle I met mean those during
140:53 So what does that angle means? see the incident angle? It's the
141:00 of the wave propagation vector measured from polar symmetry axis. So here I'm
141:07 the same angle for all three but if you change this angle,
141:11 going to get a different velocity for one of these and uh what uh
141:19 is the angular dependence? Well, given ranking as a function of
141:23 This is the same angle and you it here and you see it
141:28 see it here. If you want estimate the ankle, how how do
141:36 estimate before you measure it? so to know what that angle
141:43 Uh Say you're interested in the angle incidence upon reflecting horizon. So you
141:49 trace rays through the overburden velocity model to the uh down to the reflector
141:59 decide how to convert offsets to because we never measure these angles,
142:04 ? Uh We measure offsets and you've to convert offsets to angles by retracing
142:12 a velocity model in the overburden. , uh it's very common that you're
142:19 sure what the velocity model is. you're gonna have errors. And if
142:25 overlying rocks or anisotropy, it's even complicated than you might, then you
142:32 have thought before if you have an topic uh velocity model in the
142:39 And if the rocks are really anti , then your conclusion about the angle
142:44 incidence is going to be wrong. we'll see a few more words about
142:51 . Uh and if if you use velocity model to estimate it. So
142:58 the p wave velocity p waves and waves have the same angle in your
143:06 well in the real Earth, uh could very well be different. But
143:12 yeah, in the real ethnic could well be different. But in the
143:15 that I just showed that was just cartoon and the real Earth, of
143:21 , uh you can have share waves at any angle and p waves propagating
143:27 any angle, Sure and not necessarily same. Absolutely not. Now it
143:40 true that if you had if you a cartoon situation with uniform overburden and
143:48 horizontal layer and you're interested in reflections a horizontal layer, at depth below
143:55 uniform or anti psychotropic layer. Uh balance point is gonna be um the
144:03 between the source and receiver. And can solve that problem with simple
144:08 Uh Not with great tracing, That is trivial, but the real problem
144:15 more complicated. You've got to trace and the res are gonna be changing
144:22 at every layer in the interface in the overburden following Snell's law. And
144:30 gonna be different for p waves for waves. So when it gets down
144:34 the reflector, uh it's gonna be different angle for the stairways than for
144:40 three wives in general. And then it's a one d problem, it
144:44 uh exactly symmetric as it comes back . Okay, so uh the big
144:54 that's been made in the past 120 here ever since that first policy of
145:01 is to recognize that the previous assumption we use to simplify those complicated
145:09 we assume elliptical p waves and real are not like that. So we
145:17 see, say with good confidence is anti sox tree they have, it's
145:23 be a week and that is, know that we found out so much
145:30 the Earth by ignoring the anti secretary assuming the anisotropy zero, surely the
145:37 um, step towards a full understanding to assume the anisotropy is weak,
145:44 zero bit small. So that's what going to consider next. Now,
145:51 the reason this was the first guy did uh size mechanics, name of
145:58 Rudzki and he lived a long time . And then much later Klaus,
146:05 um, made was the first guy that. He really was the first
146:09 all that time, um, um years difference in their ages and this
146:16 is dead, but Klaus is still us. So almost 100 years,
146:21 , a big difference between these two . And so, uh, Klaus
146:26 a class, was a soldier in german army and worldwide too, and
146:34 sure he was a terrible soldier because must have been questioning his superiors.
146:40 , he must have thought he was real pain in the buck, but
146:43 was a young man, he was into the army and then after
146:47 after peace came to europe, uh, physics and it was really
146:55 of the giants of our profession. alive, still intellectually alive, Still
147:00 after all these years and almost 100 old. Look at that. And
147:07 , um, he was the next who understood and he understood everything.
147:11 the problem is he was smarter than . And so he was happy to
147:19 deal with the exact equations. And the next guy who understood those was
147:26 Amoco, my DP colleague, joe who's still with us and whom I
147:32 um last week at the scG, still working for BP and he understands
147:39 things and there's the list of people understood is very short. It's only
147:45 people that I know of and not me what what I did was to
147:53 the appropriate approximation. And so we that electrical anisotropy doesn't work and why
148:03 ? It's because most rocks don't conform this approximation. So a better approximation
148:09 that the anti Satrapi is weak in sense. So let's define what we
148:12 by that and implement that. So if we look carefully at those previous
148:22 expressions, we can see in there following five combinations. And we're going
148:27 re parameter ize the problem in these . And since it starts off with
148:33 different module, I uh five different and settlements, we need five and
148:39 of them have the physical dimensions of . And so we're gonna call one
148:45 those V. P zero and the one we're going to call them.
148:49 then we're gonna if you look carefully can see these combinations, non dimensional
148:56 . No see uh these have physical of nothing. And so you can
149:02 that also that they uh measure the shocks me in the icy tropic
149:10 these two are the same. So thing goes to zero in the case
149:14 ice, octopi and the same for . And you can convince yourself it's
149:18 same for this one. So these non dimensional measurements of anisotropy. And
149:24 we're going to define we can xoxo as the special case of anisotropy where
149:32 things are all small compared to But first before we get there,
149:37 rewrite the exact equations with this primary . And then we're gonna make a
149:45 expansion, assuming that the both small when we do that magic happens suddenly
149:53 equations become so simple that anybody can . For example, look at the
149:59 wave here, it's got that VPc showing up here and it's got two
150:04 measures of anti shocks being there. one that we would expect, but
150:10 and this is why elliptical anisotropy doesn't because for an ellipse, there's only
150:17 electricity. And here there are two parameters which described the angular dependence.
150:25 then here's the 3rd 1 here. then for the sp term, there's
150:30 combination for us to get here. made the taylor talks and notice here
150:39 combination that we're gonna give that a and we're gonna call it sigma.
150:45 it's nothing new. It's just this . There's um there's re independent.
150:54 and I see traffic parameters And two and I should tropic philosophies. So
151:02 see how this uh these right here the three equations that we just showed
151:10 that I've got here signal in Now let's look at vertical incidents,
151:15 incidents signed data zero. So this goes away. This term goes away
151:20 only the one. So um uh left with the vertical p. Wave
151:27 is called Vp zero. So would clever to uh call this parameter
151:33 P. Zero. A few pages . Okay now let's look at the
151:38 ways signed eight equals zero. So get V. S. Zero and
151:43 same for S. H. So two waves travel traveling vertically have the
151:48 velocity. So for a vertical share way of propagation. Uh they travel
151:57 . Now let's look at horizontal horizontal got uh signed data equals one.
152:05 coast data equals zero. So this goes away, this goes to
152:09 And so we get for the horizontal wave velocity is one plus epsilon.
152:16 was the vertical falls. Now let's uh click next at V.
152:24 H. So we got signed eight one. So we got gamma plus
152:30 times V. S. S. the S. H. Velocity differs
152:35 uh huh vertically trapped. Shh. of what we call gamble, it's
152:43 of the three parameters. But now at uh SV way we got
152:50 V. We got scientific wonder coast equals zero. So this term goes
152:55 and we're left with only the S. Zero. It's the same
152:59 we had over here. But in these different. Sure. Let's draw
153:09 pictures. So here we have a at this point and we have a
153:14 front that goes out like so and gone down 2000 m after a certain
153:24 of time. And horizontally it's gone . Why is that? Because horizontal
153:31 faster than word of them there. horizontal if epsilon is greater than uh
153:41 positive, this is going to be than this. So it was only
153:48 travels further. Also shown a circle which is coming out exactly 2000.
153:54 that's a way front and here's an topic circle just to guide your
154:00 Okay, now let's this looks like lips, doesn't it? But it's
154:06 uh before I get there it's the V. P. Zero implied by
154:12 of these. Now uh the black looks like any lips but it's not
154:20 me lay on here any lips. using the magic that's provided to me
154:26 mr Bill gates, I can draw exactly lips here. I selected
154:31 So it has the same um verges has the same vertical velocity and the
154:38 horizontal velocity. But in between it's from the ellipse. So it turns
154:46 that black is not any limps. there's the horizontal velocity and the vertical
154:57 different but it's not in lips. next thing I'm gonna do is lay
155:02 here a different ellipse which I call delta ellipse and that's also elliptical and
155:09 matches here with vertical velocity, but comes out over here and what is
155:18 ellipse? This is an ellipse with electricity 5%. Which I've done to
155:24 , the green curve has electricity 15% delta has 5%. Uh you see
155:33 um uh the wavefront, the black sticks closer to the delta ellipse at
155:43 for these small angles, it's closer the delta lips than it is to
155:47 epsilon ellipse. And then eventually it to peel away from the delta lips
155:54 to end up on the epsilon. so that's the way that the wavefront
156:01 , they're all together here and it off close to the delta air lips
156:06 ends up on the lips. So have the right to ask.
156:13 so what is this velocity here, is different from the vertical velocity by
156:19 factor one plus delta. But you , no energy has arrived here,
156:24 energy has come and gone. The is way out here and this ice
156:28 tropic uh, case here, that's a guide you're on. So,
156:34 this distance out here is 15% greater this, this is 5% greater and
156:43 have the right to ask yourself and me so what, what does this
156:48 represent Now, before we answer that , I want to make two alternative
156:58 . One here, here's the equation we start off. And one thing
157:02 gonna do is use trigonometry to I that science squared equals one minus cosine
157:08 and I also know that cosign squared one minus sine squared. So implement
157:15 . And uh, that means that but in here one minus science script
157:21 puts a minus delta times sine of fourth over here. So this combination
157:29 , we call that, I'm gonna that the name ada prime epsilon minus
157:34 . And we call that the weak of any electricity because if the
157:39 if the wavefront were elliptical, This be a zero and would only have
157:45 science fair term. And so the and electricity comes from the fact that
157:50 things are not um, the same real walks. So this number is
157:56 to be a non zero number and gonna call it a to prime.
158:03 , I'm gonna reserve the term ada a related quantity, We're gonna call
158:10 a to prime this simply. And I'm gonna reformulate in another way.
158:16 I'm gonna assume that if we have angles, we're gonna neglect this
158:23 whether or not this is a zero for small angles, the sign of
158:27 sign of a small angle is gonna a small number, sine squared is
158:31 small number and sign of the fourth even a small smaller number. So
158:35 gonna neglect this turn, regardless of car fishing because the Trigon metric part
158:44 so small, that's gonna be true small angles. So since it's 4:30
158:54 , this is a good place for to uh break. And so uh
159:00 going to break here for 15 So I'll see you at 4:45 and
159:07 come back and pick up at this point. I'm gonna stop sharing here
159:14 I'm gonna stop my video here. . And then go into presentation
159:24 Okay, so so far we talked an abstract problem of uh what kinds
159:33 plane wave solutions exist in polar anti media. Now we want to think
159:39 how these parameters are gonna show up our data. Okay, So uh
159:48 have basically two kinds of data, kinds of seismic data. We measure
159:55 times as a function of offset. call that move out and then we
160:01 amplitudes. So first let's consider arrival . So here is uh the cartoon
160:10 we're gonna use to analyze this problem the first instance, we've got a
160:16 problem as we had before, homogeneous , flat layer down here. But
160:22 it's an issue trump polar anisha And uh we ask where is it
160:29 to uh what's the move out of um event as the offset increases in
160:40 first place because it's homogeneous, we that the bounce point right here is
160:46 the middle. That's a geometrical argument doesn't make much physics at all uh
160:53 it's homogeneous and because the wave front because the wave type is the same
160:58 as it is here. We know bounce, The bounce point is gonna
161:02 right here in the middle. And , we have this hyperbolic new body
161:11 that we saw before. And in case where the overburden is aisha tropic
161:17 this simple problem. This is just and it's um uh given by the
161:25 theorem and in a more complicated Aisa scenario, layers and so on.
161:32 recognize that it's the first term in taylor expansion and the parameter of this
161:38 term is called the move out velocity over the move out to listen,
161:44 the derivative of the square time with to the square of distance. This
161:49 the small parameter Yeah, evaluated at zero offset and that's one over the
161:57 out velocity square. And so that's we had for the icy tropic
162:02 And now all we do is we this same cartoon for the uh and
162:09 should drop the case and there we that the move out velocity is given
162:15 this expression vertical velocity times one plus delta anisotropy prem So you remember this
162:28 that we had before I told you story about uh how I had this
162:35 concerning layers of um uh icy tropic and asking themselves why isn't the short
162:45 velocity equal to the vertical velocity? in the short spread instance, all
162:50 rays Are traveling almost vertically. So could be. So so you would
162:59 that the move out lost it would v. zero. And we talked
163:04 this uh a lot in connection with layered antisocial problem. And it's the
163:10 answer here as as it is even though these rays traveling near vertically
163:19 traveling with almost a vertical velocity, we measure instead, you know,
163:25 is reminding of what we did about layer problem. In that case we
163:30 the issue by saying what we measure uh the horizontal movement. And it's
163:36 same here for the layer. What measuring is the horizontal move out and
163:42 differs from the vertical velocity Because of factor the anti ship traffic factor one
163:50 dealt so in this case it's uh similar to the multi layer anti psychotropic
163:58 , but in this case the the medium is uniform homogeneous. And
164:08 now let's um um look at some implications of this simple little formula.
164:23 the first place we did find hyperbolic spread move out even though the layers
164:30 psychotropic. So don't get confused about hyperbolic move out and century. If
164:41 um uh the offsets are short, gonna get hyperbolic move out. Whether
164:48 the subsurface is Aissa tropic or an tropic whether it's homogeneous or layered or
164:56 you're gonna get acrobatic move out. the in this case the anti surgery
165:03 hidden inside the move out velocity. measure this, we don't measure these
165:08 things separately. So you can do hyperbolic uh move out analysis and come
165:15 with a move out velocity. It's got inside there some anisotropy because probably
165:22 overburden is huh? Now, if ignore that and go ahead and calculate
165:31 time from the depth, you're gonna the wrong answer because the depth requires
165:37 arrival time, vertical arrival time, , you know, times the vertical
165:42 which you don't know, you don't this part. All you know is
165:45 part. So if you ignore the delta and you uh estimating depths using
165:53 move out velocity and the correct vertical time. You're going to get the
165:57 depth. We call that a time depth this time. Very common in
166:03 business. And there's only two. two to uh to possibilities. One
166:11 screwed up somewhere in your analysis and the more likely uh subsurface is an
166:18 property not mr trump. Now, an innocent point. The the anti
166:24 is magnified in the move out because the following argument. Consider the anti
166:30 philosophy that we just talked about a minutes ago. And let's restrict ourselves
166:35 short off sense. So hyperbolic move with short offsets and all the rays
166:41 traveling down and back with velocities following expression because we're gonna ignore the higher
166:47 expression here with the Science Square. Square, ignore that. And um
166:56 let's consider a case where the maximum uh maximum offset has a maximum angle
167:03 30 degrees, 30 degrees. So sine of 30 degrees is one half
167:09 the square is 1/4. So let's a case where the Parameter Delta has
167:15 value 10%. So what that means 10% times 1/4 means all the grades
167:22 traveling down and up with velocities within of the vertical velocity. But even
167:32 the move out velocity differs from the velocity by the full 10 in that
167:39 . The effect of the anisotropy is in the move out. Which is
167:45 primary observable a viable times as a of move out. That's our primary
167:51 . And you see the reason that is this formula doesn't have the Science
167:56 data in there. It's just one delta. So this is not multiplied
168:01 the small factor which is comes from fair data. It's it's just one
168:05 delta. So the and gets magnified our primary observable, which is the
168:12 out philosophy. That's a real bummer that's the primary reason for um uh
168:23 time to death miss time because neglected this sub sign interpreting without velocity in
168:34 of vertical velocities without the corresponding delta . Now the problem that we talked
168:41 so far is unrealistic uniform overburden. let's talk about um layers. So
168:48 we have 1234 layers. Each one them is laterally homogeneous, but it's
168:53 psychotropic, polar. Anti psychotropic. so then let's see what do we
168:59 to move out velocity in this Well we find is equal to the
169:04 velocity with a correction factor which I've here average value of delta averaging down
169:11 the layers uh with a subscript M. S. Because there's some
169:16 some weights inside this uh inside this it's a weighted average and the weights
169:24 indicating with the subsequent primaries. And you do what dix did and and
169:32 some interval velocities by looking at how move out velocity varies from this reflected
169:39 this one. And this one then find interval velocities that look like
169:43 every one of them has the local vertical velocity. That's what the subscript
169:50 out here means local vertical velocity times plus delta. This is the delta
169:57 for this third layer. And similarly the other layers. So you get
170:04 when you convert from um average an velocities to interval velocities. You don't
170:13 away from the and you still have anti sex in there for each of
170:18 layers. So when you if you to use these interval velocities to convert
170:25 to death, you would still make mistake time, death miss time each
170:32 of those internal velocities has an interval for that lamb in it. So
170:39 can't escape the anti sexually by computing velocities. And you can't escape it
170:44 considering on short offsets. Because we've assumed that uh we've already assumed that
170:54 velocity expression is taken from the derivative the move out times in the limit
171:05 small offsets. Now you can measure anti sunscreen. How do you do
171:14 by comparing the move out velocity with vertical velocity? How do you get
171:19 ? Well you you run a VSP the well is drilled and the difference
171:24 these two is due to the uh parameter of an ice onscreen as it
171:32 your depth. Of course it's gonna different in every life. So uh
171:41 a plan. Let's look at longer , observe the non hyperbolic move out
171:46 use that to estimate delta. So sort of a plausible plan. So
171:51 we do that, we take data this. So this is real data
171:55 medical data from a long time And it has been flattened with an
172:00 velocity function at um near offsets. that uh those uh move out velocities
172:10 been moved from your offset. But can see that for the far
172:14 We've overcorrected, overcorrected the move out move out used to look curved down
172:21 and now it's flat out to this and then it curves up so that's
172:27 . Uh And so um uh we do like what we did before.
172:32 consider 1/4 order taylor expansion. This really a second order taylor expansion because
172:39 small quantity being uh uh expanding is squared. So this is the first
172:46 here is the second order term but can uh norman called a core check
172:52 expansion because of this form we got new parameter to determine here but as
172:59 talked about before in the ascii tropic case um This is not a good
173:06 because at the largest offsets the square time is increasing with the fourth power
173:11 offset, we want the square of to increase with the second power of
173:16 . And that happens if we add physically based term here. So that
173:21 we get to the furthest offsets this here dominates the one. And so
173:26 have expert behavior in the denominator cancels the excess up here and to the
173:32 X is up here. And so what expert behavior on right um velocity
173:39 of the wrong velocity. And we're do that um uh cleverly by uh
173:48 the proper values for a four and . And when it turns out that
173:55 we do that uh the a is be proportional to the a. four
174:00 this um uh proportionality constant. And this is the horizontal velocity and this
174:07 move out washing. And so um uh this will ensure that the longest
174:18 the square time is increasing with the wall set with the horizontal philosophy in
174:25 . That's what we want. Now the case where we have a uniform
174:33 an extra traffic layer, this simplifies to the following form and you can
174:41 this, it looks the same except changed notation. So instead of a
174:46 here we call this -28. I'm go back here. So instead of
174:51 a four, it's a -280. just a change of location. The
174:58 thing is down here we have instead a new parameter a to me instead
175:05 a new parameter a here's our new a we have a to appearing
175:13 So this is really important. Um you want to implement this expression,
175:19 have two independent, you have to um find from the data,
175:25 spread, move out velocity, a parameter and a print. So I
175:31 this with a colleague years ago and was excited because that's a real challenge
175:38 a to a processing geophysicist to find each reflector. These three parameters at
175:48 vertical rival time. Basically not possible do that. But my colleague
175:55 Amoco uh published this with me a time ago left. Amoco went to
176:00 colorado school of mines and one of uh I think his very first student
176:08 up with this approximation here. And is ada in terms of things we've
176:15 mentioned, it's a to prime divided one plus 28. Remember ada prime
176:20 just this simple difference here. So processor only has to estimate two parameters
176:29 move out parameters for every reflected move out velocity and ada parameter.
176:36 he does that for every vertical So that guy's name was all
176:41 if you were paying attention how Khalifa on campus at the University of Houston
176:48 thursday, this is his professor, was my former chemical colleague went to
176:54 Colorado School of Mines and his first was all curriculum. Now it often
177:02 that when a new professor comes to , his first student is the best
177:08 whatever he whatever comes his way and students after that is never as outstanding
177:14 the first one. Well, Al went on to uh famous career and
177:20 he's a famous professor back in Saudi . And so he came to Houston
177:27 as part of his job as professor geophysics in Saudi Arabia. He was
177:32 the ScG convention two weeks ago then stayed for a couple of weeks and
177:38 did us the courtesy of business at University of Houston on thursday gave a
177:44 interesting talk, not on anti sanctuary on machine learning and that was very
177:50 and I don't believe that you all there along either of you there
177:59 it was Oh good. Yeah. you agree that was an interesting
178:04 Yeah, yeah. Was a very guy. And and here's an interesting
178:12 thing uh of course is uh Saudi is a Russian jew. And you
178:20 imagine that these two guys might disagree a number of political um issues,
178:28 they never talk about those issues. only talked about the geophysics together and
178:33 been good colleagues for all this time over over 25 years by now.
178:43 uh let's compare with what we did in Chapter four. Uh back in
178:49 four, we didn't have any anti , but we did have this uh
178:55 taylor expansion. And remember we had same term then as we're talking about
178:59 and we had uh this fourth order with the correction factor here and the
179:07 A was all determined and had all uh determined. But uh we had
179:14 challenge. Uh um You see this factor does not need to be determined
179:23 . It comes from uh this factor uh the horizontal velocity and the horizontal
179:32 is given by the average of the velocities. This average over the
179:37 So that's all fully specified. And only have to determine this parameter.
179:42 this one now with this new We don't have to we don't have
179:48 use any of this machinery down Uh We have only this term,
179:56 now it's uh wait a second before say that back in the lecture
180:04 This term came from ray bending and we have an additional two ray
180:09 We have anisotropy in there. So with the anti surgery included, the
180:17 move out is simpler than we had . This is what we had
180:20 And now we see, we don't to determine this thing down here.
180:24 don't have to calculate it. It directly from this one, so we
180:28 have to determine these two parameters. uh actually yeah, that's all.
180:40 here we have the move out loss . This has nothing to do with
180:44 squared anymore. This is a calculated the layers. This is actually what
180:49 measure over here. So it's all now. That's assuming homogeneous layer above
180:58 reflector. That's this is this one in the stations or whatever. But
181:13 was for only 11 light. And now we um I understand now we
181:23 that uh as an approximation, we use the same idea where this uh
181:30 have uh an effective ada which is over all the upper layers. And
181:37 great thing is it's the same same down here. So now we we
181:41 to estimate the short spread move out and the ada parameter for every major
181:49 of the horizon. When we do , there's gonna be uh contributions to
182:02 empirically determined a to promote, that's one which flattens to gather at far
182:10 . And this parameter which we determined both re bending and M.
182:16 Search. And when you apply that can see that flattening works a lot
182:22 . It's not perfect for all but now it's flat from here,
182:27 the way out to here. So is another student, our Khalifa home
182:43 among the authors which at all have is on there. So this is
182:48 very interesting um Example which they published 22 years ago. And so in
182:58 first place, uh this is uh an icy topic, wave propagation through
183:06 complicated model. And it's a two . Model you can see here and
183:10 you can see assault body here and fault here and so on. And
183:16 uh it's what we call 2.5 the so that the model is two
183:23 but the waves propagate out from the is we're gonna have sources and receivers
183:29 along here. Uh but the waves propagating out in three dimensions.
183:34 what that implies is that this model be uh in variant uh into the
183:42 and out of the screen. And course that is not realistic if it's
183:46 to be complicated like this in cross in the real Earth, you
183:50 it's going to be complicated in third as well. But leaving that
183:55 let's consider some modeling for this 2.5 uh model. And so we're gonna
184:04 we're gonna present the model in four here. So this panel shows in
184:11 of gray, the vertical p way , piecewise constant. And here it
184:17 delta piecewise constant and epsilon piecewise and piecewise cuts. Look here, the
184:26 is a lot simpler than either delta epsilon. So you can bet that
184:32 they did was they assumed this distribution a dis and probably assume this distribution
184:38 deltas. And then calculated this And so the numbers here are not
184:45 up to uh 15% in delta and to 8% in ADA. Okay,
184:54 now, with this model, we're calculate uh forward modeling of data using
185:06 reasonable guys offsets and starts location all here. And then we're gonna take
185:13 data and um um try to make image. The original model, which
185:21 course, we know what it We know what we're looking for.
185:24 gonna try to make an image of model uh from the model data with
185:31 imaging algorithms. Actually, we're gonna the same imaging algorithms but with different
185:37 . Ation and I want to concentrate this fall right in here. So
185:46 the results. So there's the fault you see. And so the first
185:52 you can see is not a very image is it's got artifacts all over
185:58 . This one was calculated using I , Kirk off migration. Uh and
186:05 an issue topic migration with the correct of the uh they knew what the
186:12 values are because they put them into model. And so this is what
186:16 get um with the best um imaging available at the time, which was
186:25 up not very good but may be . So that's panel A. In
186:34 next panel B. They didn't, pretended that they didn't know what are
186:42 correct values. And they found from data they found the best the
186:47 M. O. And they assumed that delta was zero. They could
186:53 this from the data. Short spread out. They could measure ada from
186:57 data but they didn't know epsilon and separately. So they just assumed delta
187:05 zero. So um you can see it's not a bad image. But
187:12 can you see this image? This part right here is too low.
187:16 should be here. So this red gives uh guide your eye horizontally.
187:23 this image point should be up So they got the wrong depth Because
187:31 assume dealt equal zero. And so in those days, that was a
187:37 assumption we always always assumed that um . The anti start the traffic parameters
187:53 all zero back in the in the and then going into the late eighties
187:59 the nineties. We commonly assumed that we don't know any better, let's
188:03 set Golf equals zero. We can ada from the far offset move
188:09 measure the N. M. From the Nero set move out.
188:14 with those fitted parameters, we got pretty good model except it was coming
188:20 at the wrong depth. So now here in the art. See,
188:28 go back to the bad old days assume that all the antiseptic apartment uh
188:34 and you can see that uh the is really bad. It's got artifacts
188:40 over it. Look at these artifacts . It's just not a good
188:46 But that is what we did. what we could do basically assuming I
188:52 with a fitted move out philosophy and the Anasazi zero. So now here
189:01 the 4th, let's assume that we a wellbore here and so we know
189:09 vertical velocities in this zone. So can find the true vertical velocities.
189:16 now we are making an image with two vertical velocities. We're still still
189:23 them. And I saw parameters or . So this is aisha tropic migration
189:30 the true vertical velocities and it's a image. Look at all the artifacts
189:36 . If we're gonna make uh an with only one parameter. We want
189:41 use the best parameter which we find the data by finding the best
189:45 Move out velocity. You were ignoring one and we're using instead the true
189:50 velocities that we got from the So it's a bad image. But
189:55 here it's appearing at the correct This image point. Is this tied
190:04 the same amount as we had up ? Because we assume the delta equal
190:10 . Mhm. We assume that delta zero both here and here. Uh
190:16 here we've migrated. We migrated with vertical velocities. But that's not the
190:24 to correct for my for move out this anti psychotic problem for move
190:29 You have to correct with we have correct the move out with the best
190:35 animal velocity, not the vertical So that's why this image is so
190:40 . But even though it's poor, does have the intersection at the
190:47 So this is a good um set four pictures which shows the effects of
190:54 different parameter. Ization of the same received data calculated synthetically from a known
191:05 and all with the same over the imaging algorithm in this case Kerkhoff
191:15 But you learn more about in your and image your course and image aircraft
191:23 . Uh We can do better now if we did the same analysis using
191:29 . T. M. All these would be better but we still have
191:35 same kind of death miss tie here the same time. Yeah. So
191:47 to finish up today um let's answer question. So Miss del rio,
191:53 already stumbled on this once and uh it carefully now and answer the
192:01 Just read it carefully. Would it b. 5? Oh because it
192:09 here P velocity. Remember you saw for the p velocity? We only
192:14 four and furthermore it says weekly. in for a week a p wave
192:22 we only need three. So this a big improvement that when we um
192:29 we do the elastic uh solution for waves, we do need four,
192:34 don't need five, but we do four for the elastic for the exact
192:39 , but for the week equation uh only need three and and they're not
192:46 not the C. One ones and C. Three things, there's those
192:50 that we called? Uh excellent delta gamma. Well now we don't need
192:55 for the p wave problem, but do need vp zero for the D
193:00 problem, so we need V P , delta and epsilon. So the
193:06 answer is that for three. So this is one of those questions where
193:10 have to uh read it carefully and on all the special cases which is
193:18 right here and right here. So I don't know for p waves and
193:24 waves both, we need five for exact problem for p waves only its
193:30 And several weekend three. Now mr this is for you. uh we
193:40 a B. C. Or all the above. And so uh logical
193:45 right answer for this statement. And start affects V. P V N
193:51 O. And I think the I you miss daria and I start to
194:06 affects non parabolic move out which of statements describes this anti psychotic effect.
194:12 we have a B. C. . To know none of the
194:16 So I know all of the So we've got to have one
194:21 Um see it says the non hyperbolic depends upon the near offset primer.
194:35 delta. That's not true that this this affects the hyperbolic term at the
194:40 . So this one is false. , you pass that one by.
194:46 So this is true but it's uh also depends upon not only epsilon but
194:55 delta in the combination that we call so this is the one I'm looking
195:00 . Yeah. Yeah and so this is wrong because it's involving SV rates
195:08 people. Okay so um this is good place to stop for today um
195:16 because this next topic is a big . And so let's take this one
195:24 at um 1 30 on friday. so I don't have a quiz for
195:30 this afternoon but I will be expecting each of you on friday. Uh
195:36 written questions and I know that mr had several questions coming up. So
195:43 write all those down and we'll address directly on friday afternoon and then uh
195:50 the end of the day friday I hand out to you um final exam
195:58 I think we want to meet in at the university of friend. Is
196:03 okay with everybody? Yeah we'll meet person and I will hand it out
196:09 you um uh and paper you like on paper or do you like
196:16 Um um Electronic I prefer it email I I just uploaded on my ipad
196:23 do it there. Okay. Yeah if it's paper then you got to
196:28 it and all that. So I say at the end of the day
196:31 friday I will send you both the exam and uh by by email.
196:41 it will be uh sort of like we've seen before for the quizzes,
196:47 it will cover all the topics with emphasis on these last four topics.
196:55 it will it will count for 50% your grade. So so far you've
197:01 25% and 25% so far. So final example uh account for 50% of
197:09 final grade and presuming you both you'll get whatever badges and belts come
197:15 that. I don't think I understand system, but anyway we are consistent
197:21 that because mr Professor Van has uh that way. And so we will
197:31 all that on friday afternoon at the of the day and the topic that
197:37 be uh the final exam will be before open book, unlimited time.
197:46 be uh an approximate um estimated time me to hours. So it'll be
197:55 than what you've seen before. Uh you will have you can spend as
198:00 time as you like between friday afternoon the next Wednesday. And um let's
198:10 . Then you will both hand it to uh MS. W.
198:17 You will uh hand yours in by to um Mr Wu and mr wu
198:25 forward it to me before midnight on . And I think in the interim
198:31 you see me friday, you're gonna your new course directly saturday morning.
198:36 I correct about that? No there's course on saturday. Okay that's
198:44 So you're not gonna be distracted by else, You'll have all day saturday
198:48 day sunday and then um you're working Tuesday Wednesday. So you gonna be
198:57 focused on that. But you'll have to consult together. And again let
199:03 encourage you to uh consult together um ask questions of each other and learn
199:13 each other. And then when you down to take the exam you have
199:18 your reference materials in front of you have what I've given you and
199:22 had whatever anything you want. Uh any textbook, any class notes,
199:29 just no other person helping you will that and finish it back to me
199:36 uh Wednesday evening before Wednesday evening. then I will um get it back
199:45 you. I'm gonna be pretty busy those times so there might be a
199:49 . Well we'll see how quickly I respond Anyway. That's the program.
199:55 I will see you all again next at 1:30 on campus. Okay?
200:04 you. So with that I'm gonna sharing
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