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GEOL7333-Seismic Wave and Ray Theory - Day4 09-03-2022-Part1
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00:00 is everybody seeing this now. So since the recording is in progress
00:07 , let me back up just a . And uh let's say first,
00:11 get rid of this. So this where we started. And uh this
00:15 where we left off yesterday, starting the discussion of two waves. Uh
00:24 I have it'll be okay, but lost my mouse. So uh two
00:31 are different because the surface in question in a bar hall is cylindrical,
00:37 flat. And of course it's not cylindrical, but it's uh it's close
00:42 to cylindrical that it makes sense to it in cylindrical coordinates, uh rather
00:50 Cartesian coordinates. And so we see um uh surface waves traveling up and
00:59 the bar hall, localized near the of the bar hall. We see
01:04 in all DSPs. And I'll just you of ES. P as a
01:07 seismic profile wherein the sources are at surface and the receivers are down
01:14 And of course we also see them sonic logs. And so uh there
01:19 many types of course, uh and they're called tube waves for obvious
01:25 And uh so now these are not we want to analyze in sonic tools
01:36 when you have a sonic tool in bar hall, you know, maybe
01:41 or break the Hughes or somebody has uh slowly the service and they're doing
01:48 of our companies don't have the capability do these kinds of logs ourselves.
01:54 we just pay one of the service do it. So, uh well
02:06 tool measures is not a tube wave it measures a body wave critically
02:13 And so we will learn more about that means in less than seven.
02:21 , so they all propagate in one . A polar down hall were both
02:26 they are analyzed using a cylindrical coordinate for obvious reasons. They decay exponentially
02:33 from the borehole wall into the formation also back into the mud. Because
02:42 have in any two boys problem, have uh the diameter of the bore
02:50 , the diameter of the cylindrical Uh that that's always going to get
02:57 our equations, we know that. so that is gonna lead to frequency
03:02 behavior because that borehole diameter provides um standard by which any particular frequency knows
03:12 it's high frequency or low frequency dependent whether its wavelength is greater than the
03:20 time. And as they're going up down, they reflect from every formation
03:28 . So let's just assume the mud continuous up and down the hall.
03:33 uh wherever there's a formation boundary and , which which was penetrated by the
03:42 that's the boundary and it reflects. if it's going up, those reflects
03:47 down and somehow transmitted and so And then here's something else you might
03:51 have thought of if we're doing this casing wherever there's a casing joint at
03:58 point, there's a double thickness of . And so it reflects off of
04:03 . And also um you know the itself is occupying the middle part of
04:10 borehole. And it's got a wire coming out the top but then it's
04:15 some sort of tool down in the and uh uh there's gonna be uh
04:23 changes in the two way behavior because changes in the tool diamond, maybe
04:32 the tool itself is maybe six inches and then the wire line is maybe
04:39 inch across. So right there there's diameter change in the tool which is
04:45 to affect the two waves going out and down. You can think that
04:50 it's propagating the partner which is in in the mud, it's going to
04:55 affected by whether or not there's a um right there so you can see
05:01 being pretty complicated. Yeah. Um most important to tube wave is a
05:08 called Stoneleigh wave because the only uh it but it should be called the
05:16 wave since it's like the Sheltie wave we talked about yesterday, remember the
05:22 wave is like, it's the only except that the upper medium is um
05:28 instead of air. And so that's situation in the borehole. So now
05:33 got a picture of half of the hall here. This is I'm going
05:36 assume the borehole is symmetrical and you see half the tool here and half
05:42 wire line here, pulling it up tube and it's being pulled up by
05:47 . Um Normally what they do is let it settle down to the
05:51 under some weight and then they pull up slowly and steadily. And uh
05:56 here is uh Uh two lens and wire line begins. So obviously the
06:04 wave is going to propagate differently up than it is here. And so
06:08 the cartoon you can see a bunch layering that's quite now an important
06:18 Mathematical complication comes from cylindrical geometry. let's just turn the geometry around so
06:25 looks like a layered halfway, a rarely way problem on a cylinder.
06:30 this looks pretty much like what the cartoon we started off with yesterday
06:36 a formation here. Never mind It's anti psychotropic and layered down
06:40 That in our simple analysis, we're going to consider that at all.
06:45 remember this is not flat. Like talked about, this is a cross
06:50 of roof. So I said that's our corner system is gonna look
06:57 It's gonna have uh actual coordinate and gonna have a radio coordinate and an
07:05 azimuth to the to the more whole . So we're gonna ignore the presence
07:12 the tool. And also the like so we're gonna look uh in
07:17 solutions like this. Remember when we the rarely way uh different really wave
07:27 love waves is really wave has displacement the plain of the figure. And
07:32 what we're gonna be looking at here way will be polarized in the
07:42 Z. Plane. Okay, so is a big difference in the
07:48 Uh In fact, we're gonna use we're gonna do very much like we
07:54 yesterday in the wave equation appears the . Um And so the application looks
07:59 this in a cylindrical coordinates. See part looks like a Cartesian part and
08:07 part looks like a shark apart. uh that's the whole thing looks
08:13 you know, driving that dishes outside scope of this course. You can
08:22 up in any in any book on physics. This definition awesome operator in
08:34 . And uh remind you here that this is uh in the two
08:39 Case there's no as mental dependence here we're going to look at waves which
08:47 independent as. So here's our it's wave equation on the left. Uh
08:58 this is gonna be further in the . And so here we're gonna have
09:03 p waves of parts in the fluid here's the the time derivative. And
09:08 is the space derivative. And uh gonna give uh rotation uh k overall
09:20 the fluid. We're gonna call decent of course uh had a body wave
09:27 through the fluid. It would be with that philosophy, but rarely waves
09:32 me the Stoneleigh wave in this uh tube wave is not gonna travel with
09:38 own philosophy, not this velocity. , so uh we're gonna we're gonna
09:44 the solution. So uh suppose that all are not uh a little bit
09:53 by this notation. So let me tell you what it is that by
09:59 zero, in the case of that standard notation, for what they
10:04 mathematician called a modified vessel function of zero. Okay, now, um
10:12 correspond to uh and uh we're doing flat problem with Cartesian coordinates. These
10:23 correspond to signs and Cozzens, but and Cozzens don't really work in the
10:30 geometry. So we have to use basis functions. Um They call modified
10:38 function. So, vessel was one these mathematicians in the 19th century and
10:44 are modified in a way to be . I don't know what, how
10:49 modified, but you can look them in a handbook, just like you
10:52 look up science and frozen. So your mind, you should think of
10:57 like science and cosign and uh tabulated . You can look them up and
11:05 functions of a non dimensional variable. the non dimensional variable uh or it
11:12 that uh that here we have a um coefficients and we're going to determine
11:21 from the boundary condition. Uh just much like we did yesterday. And
11:28 so uh you can see here uh have the radius and we have the
11:36 along along the bar. Oh by way, you can see from the
11:39 of this exponential functions that it's wiggling way along the borrow. It does
11:45 travel in any direction but in the of the world. And it's got
11:52 a wave number, component Gsfc. here's the radio component of that same
11:58 number. So uh same that same vector, so the length of the
12:05 square of the length of the victors by the pythagorean there, just like
12:10 um yesterday and that's related to the velocity, just like we did
12:19 And so uh so you can see G. Is a function of of
12:25 . So here uh with every omega gonna be a different uh G.
12:32 um here and here and so together are gonna make up, we're gonna
12:38 able to add together uh spectrum a like this to find the answer for
12:47 problem. Even the most complicated. , so now we're looking in the
12:56 as with the railing wives, the is neither curl free nor divergence free
13:02 of the boundary conditions, which we go over again. I don't think
13:07 we just mentioned curl free part will the wave equation for the scale of
13:12 . And I see this has this the scale of potential for the rock
13:18 up here. This is the scale potential. Uh And then it's gonna
13:23 a divergence free part which will obey wave equation for the effective potential in
13:29 rock. And so this has got rock p wave velocity here and a
13:36 shear wave velocity here. So you to see how this is all very
13:43 again. I want to think for to think of these as elastic
13:47 Wave is going to travel with the wind velocity. Now. Uh we're
13:57 uh you gonna uh say that uh we did before, we can uh
14:06 when we were doing the stone, flat Stoneleigh problem yesterday we decided that
14:11 only needed the transverse component of this . We called it side to its
14:17 to uh the other two vectors. In this case that transverse vector is
14:24 call that transverse component is gonna be side signify 20 in the so um
14:38 So we're gonna do like we did , we're gonna guess the solution and
14:42 the guests. So uh careful for potential for the scalar potential in the
14:51 . We got the same two best vessel function with different arguments. So
14:56 is the uh radio component of the way vector. And here is the
15:03 component. This is traveling in like before in the Z. Direction
15:09 It's got different constants and uh And otherwise it's just like what we
15:17 at before, in the foot for uh Thank you. Like component vector
15:27 . It's similar same to a vessel , modified vessel function with non dimensional
15:36 given by the radius and rate of part of the h. Each wave
15:46 here is the two more constants. see this is getting complicated. Um
15:58 are not gonna follow through this analysis the detail we did yesterday. Um
16:09 these uh these these these guesses will the wave equation. If the wave
16:16 have the length, the length of K. Wave vector is going to
16:19 related to the the velocity, wave velocity and rock. And the
16:26 of each wave vector is very similar for the sure loss. All these
16:35 together we uh all three of these wave vectors have to have the same
16:42 axel component. And it's gonna be pleasure minus omega over the Stoneleigh way
16:48 Alaska, christian wonder what we did . Boundary conditions are continuity of stress
16:55 displacement of the borehole, daughter, stress and displacement on the actors.
17:05 along the actors. We cannot have infinite stress uh at the action.
17:13 gotta go smoothly to uh some finite . And then uh as uh as
17:22 the wave continues to the other half the bar hall past the actress,
17:28 will change again and again reach the condition at the other side. Now
17:39 way out into the uh into the Uh the stress and displacement has to
17:47 to zero. We didn't say that for the flat problems that we did
18:03 . I don't think that's required. flat problems certainly didn't use it.
18:11 that will be important here because of 600%. Now suppose you had a
18:20 . Uh then that complicates things. there's yet another set of solutions for
18:26 let's enter equations for inside the tool . And so normally when people do
18:32 , they do not attempt to find solution for a real tool is too
18:37 . So they assume uh at this in the analysis, they assume the
18:44 is a solid uh solid uniform solid any machinery inside the tool. And
18:53 has a radius, they know the of the tool. And uh so
19:00 makes a complication. As you can maybe a significant country complications as we're
19:06 going to deal with that. So you uh when you do this um
19:13 this is the kind of answer you . So here is the Stoneleigh wave
19:17 a as a fraction as a ratio washing in the floor as a function
19:26 frequency for a given um for a instance of course. And um you
19:37 see here we have the phase velocity calculated for that model like this and
19:44 group velocity is calculated like this. this is uh citing paper by Cheng
19:50 Tacos 1981. So um pretty old 40 years old. So let me
19:56 tell you uh such uh interesting fact these uh Tacos was a professor at
20:05 . I. T. Which is in massachusetts. And Uh it was
20:11 very prominent your physicist now retired still by the way. Um and the
20:17 he's alive is because he married a wife about 10 years And he was
20:25 famous professor at M. IT for years. I think he retired about
20:30 years is a bit older than me so he may be retired 15.
20:39 um so um Chin was I think was a student and he was such
20:47 good student that he became his associate M. I. T. And
20:52 ran a consortium at M. Eight . I. T. For many
20:56 now Cheng was just elected president of S. E. G. Uh
21:02 month. So that's interesting. Uh assistant surpassed the master. So in
21:10 days checking and was the boss and was the subordinate. But when Cocteau's
21:18 he had never been president of the . But um but Cheng just was
21:25 president of the S. And Uh last a month ago. And
21:33 by the way he is the editor the journal Geophysics. I don't remember
21:40 another time when we had one guy was the president elect of the society
21:50 the editor at the same time. the way the scG works is we
21:54 a president elect and he's sort of the vice president except there's also a
21:59 president, but he's gonna inherit the automatically next year unless he works to
22:06 something unethical or so something like I don't think that's gonna happen.
22:12 he's gonna uh 99.9% probability he will the president of the ScG next year
22:22 he will be the first chinese president the ScG. Very interesting. And
22:29 have lots of members of course in . And of course we have um
22:34 of chinese geophysicists in this country and in europe. And uh and we
22:41 lots of chinese students in this So um let me just interrupt
22:48 Mr wu are you a member of S. E. G. Yeah
22:54 you've got to be both of, have to be members of the professional
22:59 . And I encourage you to also members of the local affiliate society,
23:05 Geophysical Society of Houston. And uh encourage you both to participate actively in
23:14 local society and to what extent is in the international society as. So
23:21 will be good for you. Professional it. Okay so uh Cheng uh
23:29 when they did this 40 years ago they were both living in Boston.
23:34 and I should I should tell you taxes retired, uh they needed somebody
23:40 to run the consortium. The consortium called Earth Resources Laboratory. And had
23:49 know about About six or 10 students it had maybe a couple of
23:56 And the major professor was Texas. after Tacos retired, they appointed some
24:05 professor to one of the uh to the consortium. And um then I
24:12 he retired and uh made an unconventional uh to be an interim director of
24:23 consortium while they were searching for a director. And the the interim director
24:29 the former executive director that is they paid um She was a high level
24:38 person and then she took care of day to day running of the consortium
24:43 the director is uh being a And so they asked her then to
24:50 the acting director of the consortium, she did very capably for about a
24:56 or two while they did a search a new director. And then once
25:01 new director came on board, she resigned of course as director of the
25:10 . But she did such an outstanding in that role that she was elected
25:15 years ago to be the president elected . And she is now the current
25:21 of ScG. Actually, she was until thursday last thursday, her term
25:27 . And um so she is now president Sug and it doesn't mean that
25:37 is free of all ties to E. G. There is a
25:43 sequence of important roles that she has perform for the S. E.
25:50 . For the next five years like of important committees. So she she's
25:56 I think she's sitting now today on nominations committee for the S.
26:01 G. And in a few years will be the chair of nominations.
26:06 also I think she's sitting on the and awards committee in a few
26:10 she will be the chair of that uh because she is a former president
26:19 several years of declining important responsibilities for president. So that means that um
26:31 have two recent um graduates, two uh insertion. Uh My g who
26:43 now leading this society, her name ANna Shaughnessy and his name is Arthur
26:50 both excellent cheer physicist. And uh step down from that role on thursday
26:59 do is on the last day of convention, they meet at noontime and
27:06 officially hand over the responsibilities from the president. Uh and the old executive
27:14 . And the old board gets handed to the new people who have been
27:19 elected throughout the summer. And so I think we have we always have
27:26 , well it's not complete turnout, people who continue in their previous roles
27:33 then some uh turn over their roles really elected people. So that all
27:39 on thursday at lunchtime while we were lunch, they were doing this
27:46 And then uh the old uh administration and has a drink in the uh
27:55 the bar and congratulate themselves for having a good job. And then the
28:01 board has a short business meeting and they uh german and then they have
28:08 serious business meeting, I think this , this coming week we're going to
28:13 it and then monthly thereafter. So the way the sug as your work
28:19 the way. But when I was scG president, we didn't have a
28:24 of directors. All we had is meeting five people, you know,
28:30 , vice president. And so we that that was not good, That's
28:37 an appropriate governance structure. So we uh that was 19, no,
28:45 was the year 2006 I became And during that year we launched a
28:55 for reforming the uh sug governance. and it took several years for that
29:01 mature because you have to decide what want to do is to educate the
29:08 and then you have to have a for the membership. And then,
29:12 know, it takes a while, what we did. And the new
29:17 started happening, I think about 1000 mine. That's so we've been running
29:24 that pretty well. Uh there's some exceptions pretty well ever since. And
29:34 the new board, you set of has just taken office and uh in
29:43 business very shortly. Okay, so to science. Uh here we have
29:49 phase and group velocities. And you we talked yesterday about how those different
29:58 each other. And I think it's common that those two velocities are similar
30:04 not identical. And you see the way velocity is a little bit less
30:10 the mud velocity. And uh look here at zero frequency. Uh the
30:19 and group velocities are the same And also at various type treatments, they're
30:25 same and in between they're not the . So look at the band we're
30:29 about this is Kilohertz. So these are too high for seismic waves if
30:36 launch a wave like this, uh 10 kHz from uh surface source.
30:46 wouldn't go very far because of Um um in the earth. Remember
30:53 told you, I think I told that insinuation you can think of is
30:58 constant in terms of of energy loss site. And so high frequency waves
31:08 more cycles per meter than the low waves. So these high frequency waves
31:13 many cycles before they get down into uh more than a meter or
31:18 So they just die up. of course, um A source like
31:24 think about a source like dynamite impulsive that generates all all frequencies including these
31:32 these high frequencies. Most of them out. And so the only ones
31:36 propagate down thousands of feet and come thousands of feet to our reporters.
31:42 are lower frequencies in the seismic say between five hertz and 100.
31:50 it's uh 100 times 1000. it's 100 times less than wave theory
32:03 wave theory. And uh everything uh analysis comes through regardless. Now there's
32:13 um kind of mhm Really wait, another kind of tube ways which is
32:22 of like love ways and these are pseudo railing ways. Um and so
32:28 have real vectors, just just like the love problem that we had you
32:32 here uh wavefront echoing back and forth the fluid. Uh And uh in
32:40 solid you can see exponential decay. these have um or stir they're uh
32:50 not propagating in in here with as here is a ray as array and
33:01 can be certain that in the mud uh it's polarized in this direction,
33:10 not polarized, transverse. Like uh it's uh in this direction, but
33:16 it's bouncing back and forth at an in the in the mud. And
33:28 uh similar in lots of ways to love waves which we so this is
33:37 same diagram as we saw before except you see right here there's just covered
33:48 covered up stuff in the this figure a white panel. So you wouldn't
33:53 distracted. And now you see there is the whole thing. And so
33:59 is what we looked at before. then the pseudo really waves are um
34:05 have different branches here. And you see for frequencies, her frequency is
34:13 about changing the hearts in this The pseudo really wave doesn't propagate at
34:21 . And for uh uh huh good uh as a velocity which decreases with
34:33 rapidly and then look, it turns and increases again. So whenever you
34:41 uh you know, dispersion curve like , whenever you see a case where
34:52 locally independent or frequency turns around and back up, uh that's gonna mean
35:01 a lot of energy is going to concentrated in this frequency band in the
35:06 . And this was first discovered by guy named Harry. So here is
35:13 synthetic size program that shows all these . So, let's look and see
35:17 we have here is the p wave we didn't talk about but we will
35:24 about that um seven very low, low amplitude here is the shear wave
35:33 by the way, there's very stuff here which uh these are the pseudo
35:40 railways, which we just mentioned, you can see uh growing in um
35:45 amplitude. The uh the later ones growing inactive and you can see and
35:54 see that this is a little bit frequency and this is a little bit
35:59 frequency here, and the stony waves then following along behind. Just like
36:12 just like I said here, the waves are gonna be slower and most
36:16 the time most. And the the where the maximum and uh that's a
36:29 that's called the area maximum. And is happening um with these frequencies and
36:36 exactly the same frequencies as you saw the previous slime because it's the same
36:44 . Yeah, that's a very idealized . I would say that I call
36:49 difficult, but not impossible if you motivated, you could work your way
36:59 that um yourself, I think this Arthur Cheng's PhD thesis Watson. Uh
37:07 in those days it was pretty advanced . Um but you could probably work
37:12 way through it uh if you were , but then we got to recognize
37:21 in the real case uh Warhol is gonna be cylindrical. Uh And
37:29 it might not be vertical, it be d created towards the horizontal direction
37:34 maybe even uh in the horizontal and it uh So that's assuming the layers
37:42 all horizontal. For now it will a encounter. The borehole wall is
37:48 be uh nice entropic because of that . And that's gonna make it
37:56 Furthermore. Think about this. Think the problem that we had in the
38:04 , which we talked about his first with a simple sedimentary, simple sedimentary
38:16 , which is uh sitting in a field science saluted by gravity and stress
38:23 the uh scratching all the way down the vertical stress is going to be
38:31 than the horizontal stress as we talked in the real world, uh we're
38:36 have in that uh sedimentary sequence, gonna have a borehole. So it's
38:43 gets into her job analyzing stress She's gonna have she's gonna be dealing
38:51 uh data taken down home. And as soon as you introduce a more
38:59 into a stressed solid that affects the , you know, that introduces a
39:05 surface inside this stress solid where the shear stresses vanish at that cylindrical uh
39:18 cylindrical surface. So what that does it concentrates, it changes the distribution
39:25 simply because of the fact that you in there, a stress free
39:30 And suppose it's not um exactly Suppose it's uh as opposed to as
39:39 bit shoot its way through the through the rock, perhaps it didn't
39:50 a perfectly cylindrical. Okay. And suppose that the uh initially not only
40:02 manmade complications, but suppose that the stresses are not equal in this
40:09 So all of those factors mean that this redistribution of stress caused by the
40:17 hall can be complicated. And in it concentrates, concentrates some of the
40:24 of stress near the bar hall In some cases those complicated stresses change
40:34 shape of the bar hall. Uh called borehole breakouts and pieces of
40:43 of the borehole break off and fall the borehole because of this stress concentration
40:51 you know, depending on the rock ? So some rocks are strong and
40:55 doesn't happen, other rocks are weak it is come have pieces of the
41:03 the formation right there, four whole break off and fall into the
41:10 And normally uh they get extracted as are happening, you know, uh
41:16 are being uh pulled out and removed and the whole bit is being replaced
41:25 so on. So normally what happens those things get disposed of? So
41:30 don't really have pieces of rock in borehole. But um over there and
41:39 so so Stephanie's doing her borehole She should be alert to maybe the
41:49 that maybe the two horizontal stresses in borehole are not the same.
41:55 we uh we detect these bar hall . There's a tool which is called
42:06 caliper tool, that's what it's it goes down uh measures the shape
42:12 the bar hall as it goes And as it passes one of these
42:16 , it notices that for a whole inner is different in one direction than
42:22 another direction. So that's part of caliper law. And so Yeah,
42:30 been in business a long time since 1930s and has dozens and dozens of
42:36 measurements that they can sell to Uh So the service of course.
42:43 um one of those is caliber tool it's run fairly frequently so that you
42:49 blindly assume you know what question. . Um Is this also because we
42:59 this one test at work it's called miniature vein. Um I'm not sure
43:05 you're familiar with it. Would this kind of related to that? Cause
43:09 it's called like a soil saber? we um we only do it with
43:14 tube samples. Um and we have put the saber in and turn it
43:19 it measures the the sheer failure um the in the moment of failure for
43:26 only do it for clay and we got like this huge job in for
43:31 a pipeline. And yeah we just this it's a soil saber that measures
43:38 shear stress of when it fails. have to do remolded and then like
43:43 received. So I don't know if am not familiar with that. No
43:48 think it's really interesting. Uh So should familiarize yourself in a serious way
43:54 it and uh and also familiarize yourself the assumptions that you make when you
44:03 the data that you're gonna come back . Some kind of a data.
44:07 I'm not sure what it's gonna look but uh it's not what the customer
44:11 . You want that data to be in some way and you should make
44:16 that you understand what assumptions your company making as it does that interpretation.
44:24 so I'm not familiar you know uh testing companies have are been around for
44:34 long time but they're constantly thinking about they can do differently to uh help
44:42 customers and the customer standing at the of the bar hall looking down and
44:47 what's down there. And so uh like yours help them understand better what's
44:54 there. And it can be important a lot of reasons and I am
45:00 familiar with that particular. Okay, here's another real world issue. I
45:11 told you about one type of drilling , borehole breakouts. Another type is
45:16 we call mud cake. So as uh the driller is drilling, he's
45:23 in the bar whole um uh borehole . And he can adjust the density
45:30 that mud by adding or extracting solid at the top and circulating it
45:38 And so he likes to maintain. inside the bar hall it's easy to
45:44 the pressure inside the Warhol, it's that uh integration of all the pressure
45:53 to the density of the mud all way up to the surface. And
45:59 so he likes to maintain the pressure the bar hall at all depths to
46:07 a little bit higher than the natural fluid pressure in the formation as he's
46:15 because he doesn't want the formation starts come into the bar. He wants
46:20 while he's drilling. And so he he adds enough solids to the mud
46:27 make sure the pressure at all depths the hole a little bit higher and
46:33 formation pressure. What that means is he's going to be losing more whole
46:40 into the formation because the pressure is in the mud in the borehole.
46:46 the formation fluids as the formation mud into the surrounding formations. The ferocity
46:57 a tendency to filter out the particles the mud. So clogs up the
47:03 in the formation, gets clogged up blood particles. And the uh the
47:10 is happy with that because what that is the club, uh blood particles
47:16 in the uh in the formation in the outer inch or so of
47:21 formation wall are gonna reduce the So he loses less mud in the
47:27 as it progressively chokes up the ferocity silent particles from the mud. And
47:35 that uh that um cylindrical um interval formation just outside the uh Warhol which
47:51 clogged with mud particles, we call mud cake but jake. And so
48:00 in the two great problems that we talked about, we didn't have any
48:06 cake, we didn't have any anomalous physical properties close to the uh Warhol
48:13 . We just had the formation. of course for realistic problem you want
48:18 have in your model in your analysis some representation of the mud cane.
48:24 might be like an inch or so uh walk with reduced porosity and so
48:36 velocity because of the particles. Yeah there might be tortoiseshell modes now.
48:50 I have to say something about Oh normally when we um have a
49:01 tool in the more home. Normally it does is it puts a pulse
49:07 sound from the tour um out from tool towards the borehole wall and it's
49:14 same in all directions. We call a monopoly source. It's going to
49:18 a P wave wave. Of course propagates through the mud towards the borehole
49:25 . And it'll have a suitably high tool designer will arrange so that emitting
49:33 waves through the but with appropriate velocities we'll talk more about what that means
49:45 yourself. But then starting about 30 ago they um developed another kind of
49:56 of Leipold source which does not radiate equally in all directions. What it
50:05 is it emits a P wave in direction with a positive pulse and at
50:11 very same time it sucks in mud the other side. So what happens
50:16 the p wave propagates out the other or the negative polarity. Okay,
50:22 when those waves hit the uh borehole uh one of them kicks the borehole
50:29 to the left and the other one the borehole right wall to the right
50:36 that makes a traditional way of the itself is uh It's been in a
50:45 way and that deformation travels up and the more whole uh with its own
50:52 , uh which international mode um as says here and then what uh the
51:02 they do that is because further up bar hall, they have receivers and
51:08 receivers receive their especially designed to receive partial modes. And we'll talk in
51:17 seven about why they did about 30 ago. It was a big advance
51:23 a reason. That's obviously gonna make sorts of complications for the tube waves
51:29 well. And then uh we've ignored ice hockey. Of course, a
51:35 complication would have to include an eye . So that's all I want to
51:42 about two boys. So, uh do a quiz here. Let me
51:48 to you. Uh Miss Del Uh we got a B C and
51:54 says all of the bus. So soon as you see that you're gonna
51:57 alert, that d might be the , but maybe not. So,
52:01 question says, the quasi cylindrical shape a four hole naturally introduces uh a
52:09 a one day propagation of surface, that correct? Yes. And being
52:16 , is that correct? Yes. so the applause in operator in cylindrical
52:22 has solutions which are not plane but our vessel functions. And so
52:27 C. So we got a B C are all true. So the
52:32 . Now, long question, true false. Read it for yourself.
52:37 , I'll read it realistic really waves the Earth's surface like we discussed yesterday
52:45 out in two dimensions, meaning that amplitude decreases rapidly with austin uh losses
52:54 the two horizontal directions. By Sudo rarely waves on a borehole sort
53:00 spread out in one dimension only along borehole as we talked about so that
53:05 amplitude decreases more slowly with lawsuits. these statements true or false?
53:15 Yeah. I call that trip. are good now. So that brings
53:22 to the end of the of the concerning surfaces. So we've learned uh
53:34 we thought after studying body waves. that's all good. Uh P waves
53:38 S waves were done but now we out that there are more solutions complicated
53:45 uh because of the surface of the . And what makes it complicated is
53:52 the boundary conditions which we did not and discuss before uh work so that
53:59 waves travel with surface wave velocities not not VP not Bs but the surface
54:05 velocities more complicated and more over And then we talked about the particular
54:14 waves. Hello waves and we talked worlds and uh how uh similarities coming
54:26 there's a surface there but there's also um by the cylindrical insurance. And
54:38 all of this has prepared us now this topic, reflections and refraction which
54:46 uh the most important part of our in this business. Um Sure,
54:58 we're gonna do at this point is this is a good time for a
55:04 yes, Excellent planning. So let's for uh break for 15 minutes and
55:12 come back at 9:45 and I'll uh my audio video in the interim and
55:24 be getting and I'll see you in minutes topics floor. Look at these
55:32 that we've already covered here. It's of like the basics and this is
55:37 very important, reflection and refraction is of the dominant role that reflections play
55:43 exploration, geophysics. I should remind that reflections are not so important for
55:50 seismology and global seismology, they're much interested in re fractions And but because
55:57 that, when you get done with , uh you'll think, okay,
56:02 all there is. Uh that's uh the important stuff. But look what's
56:07 next. A bunch of complications, of different complications. And then discussion
56:14 topics which we've ignored so far, have we've ignored so far the boards
56:21 the rocks, we've ignored the fact Attenuation and we've ignored the fact that
56:26 ice age. So that reminds but we are going to deal with
56:31 because of Mr Einstein. So, has on her wall in that
56:39 she has a quote from, I this is my favorite quote from
56:42 everything should be made as simple as , but not simpler. And of
56:47 she was talking there about the formulation the most basic laws of the
56:53 but we can apply this, lose to our business as well. Uh
56:58 we have looked at so far as applications of simple ideas and now we
57:06 uh and that was good according to , but if we stop there,
57:11 violating the wisdom of Einstein, because would be uh too simple an analysis
57:20 sufficient to understand our date. so that I want to um start
57:29 next topic, reflections and refraction. , so can I get some feedback
57:40 ? Uh people see the introductory slide less than six? No, we
57:48 see the outline from the last Okay, so what I have to
57:54 is um I have to get Stop sharing. Start sharing.
58:04 and I think maybe you didn't even the quote from Einstein. Uh am
58:10 correct? No, I just thought were reading it. Yeah, so
58:14 a quote from Einstein, Can you that? Okay, everything should be
58:19 as simple as possible, but not . Okay, so now I'm gonna
58:24 stop sharing and start sharing. Didn't that right before, so now you
58:35 see the interest lied for less than , is that right? Yes.
58:43 , so uh so here's what we're learn in this lesson, you'll um
58:54 be able to explain to other what are the boundary conditions at reflecting
59:00 , how these results in its simple for normal instance, planer us much
59:07 complicated formula for obliquely. So this what we normally have. This is
59:11 kind of data that we normally look , where the source receiver offset.
59:16 means the waves are traveling obliquely through medium, oblique to the to the
59:22 the reflector. And so that's an thing. And uh very complicated
59:29 But we do not follow Einstein's We do not uh think in terms
59:35 these complicated formula, but we simplify equations for the common case of weak
59:42 on either side of the reflecting And when we but we also need
59:49 consider the special case of the free . And we have to uh we
59:57 to understand the special case when the is very large. Uh See back
60:03 here, we started with normally instant and when my father was a
60:09 performing back in the thirties and the um uh that's all they looked
60:16 They didn't have the idea early on they should have a separation between the
60:22 and offset and everything was more primitive and and so as they learned over
60:27 years, they said, hey, have source receiver offsets. Uh and
60:35 impact those reflect those reflectors were at angles. And will that be
60:41 And uh I think you probably already why that's useful. Uh but ever
60:49 then we've sort of operated on the that the, the angles of incidents
61:00 not zero, but they're not very . And so we can often get
61:04 with assuming hyperbolic move out and then we wanted to do further offsets.
61:11 see the progression here, We wanted do further offsets so that we needed
61:16 have non hyperbolic move out corrections. now here we're gonna uh think about
61:24 happens when the angle of incidence is large. And then we also need
61:30 think about what happens uh when the wave is not a plane wave.
61:36 I right up here, it says waves, but you know that actually
61:41 waves are all, all of them . So what impact does that have
61:48 our analysis? And then truth to is not true that all reflecting interfaces
61:55 perfectly uh straight. Uh not only they not flat, they're they're also
62:02 . For example, if you have dipping flat interface, that's a complication
62:08 we talked about a little bit But what happens if the interface is
62:13 ? So, uh, you know um reservoirs are frequently contained within anti
62:21 structures means its structure in the surface the dome. And the structure gets
62:27 off to the sides and the oil at the top of the dome because
62:31 a lighter density than the bride. it kind of works its way to
62:36 top of the antique line. So there, you should be thinking
62:44 if it's a curved surface, can get away with ignoring that? And
62:49 only about the way uh only about on flat interface. And the answer
62:58 , well, you'll see the Okay, so that's the program in
63:04 . So the first thing is boundary . So, Uganda conditions at a
63:10 elastic innocence. So let's think about uh this scalar wave equation for the
63:18 potential looks like this. We've seen so many times now and you know
63:23 this velocity uh the way it's gonna a p wave velocity here, since
63:29 dealing with a scalar wave equation. this velocity turns out to be the
63:34 of the waves, not the velocity the particles within the wave velocity of
63:39 wave itself. And that's notation as derived it, that's a conclusion.
63:45 we first introduced this symbol as notation this ratio where M is K plus
63:52 thirds new. And we talked about how he never appears by itself in
63:58 wave equation in in a wave Always in the combination with K plus
64:05 new. So that's why we give a name. Mhm. We also
64:11 from previous work and solutions are going be a sum of plane wave terms
64:15 this. So they can be going any direction that's determined by the wave
64:21 K. You can be going up down. And because of this,
64:26 hear it oscillates. It does not increase with time uh to grow infinitely
64:38 small. It oscillates because of I and also we talked about the
64:43 that uh this is a complex number the data of course are gonna be
64:48 . So we talked about how in end when we some of these things
64:52 , we're always going to end up a real observable, but it's very
64:58 for us to talk using this complex , not have to we don't have
65:06 um consider that the sign and the sign solutions uh This way because separately
65:17 all gonna sum them up later. is the relationship between the vector and
65:27 frequency? Uh Mind you, it affects the magnitude of the way affected
65:36 of these things of these components Only the magnitude and um is a
65:45 wave velocity and the angular frequency. , suppose such a way that incident
65:54 elastic, this company, here's the wave vector. It's gonna be solving
66:02 this equation in the upper medium and uh anticipating a later result, some
66:09 it's gonna transmit down here. So so we uh say that uh that
66:15 of a way propagating in the lower , it's gonna have a very similar
66:20 and the main and the difference is here uh the physical parameter is different
66:26 uh it's gonna turn out to we have to specify for this
66:33 both the velocity and the density in . So we can expect to find
66:43 wave solution just like we uh showed slide and they're gonna have coefficients in
66:50 spectral coefficients family of solutions on both . And uh we are going to
66:57 those coefficients, determine those professions, that Bill Bay boundary conditions right
67:04 Okay, obviously we're gonna call this Z people zero at three equals now
67:12 we have an interface here. Uh interface has a normal uh normal vector
67:19 in the X direction in this case of that surface. Uh the problem
67:25 intrinsically a vector problem. So let's out the vector components which are the
67:32 of that Scaler. And so when make this gradient operation on the
67:37 it uh takes the derivative of this function and it uh primitive uh brings
67:46 the exponential function because of the magical of the oil number E. And
67:52 in front of it here we have . And vector K which comes from
67:57 differentiation of this part of the expo of course this part is independent of
68:03 . So that was so we're gonna this rector here with Scaler uh platoon
68:13 an imaginary uh oh two minus one here and vector K. Here gonna
68:20 all that together vector U. Upper you And um it's got uh um
68:32 components like that. So the three of the vector locus you are given
68:36 hair and notice here that uh you written without a vector markets uh order
68:45 then for the uh the amplitude part it out this part here uh uppercase
68:51 . V. And W. And this one shows the exponential factor separately
68:56 this one doesn't. Yeah. So boundary conditions are we going to apply
69:02 ? So these boundary conditions, I remind you are not additional ideas,
69:08 can prove that these are the right conditions by analyzing the wave equation
69:14 not something new that will spring on . Uh And so the result of
69:20 kind of analysis says that the displacement be continuous across the interface at all
69:26 . Otherwise the material would be torn the way, that's not the kind
69:29 solution we want. And uh so it is, all these components must
69:36 continuous secondly, the stress components of towel I three stress venture, that
69:45 all the components of the stress which aligned with the interface must be
69:53 If that weren't true, the vertical of stress which appears in the wave
69:58 would be interested. And so the at the surface would be infinite.
70:03 don't mind if there's a finite uh a finite acceleration of that service.
70:12 don't mind if the service wiggles up down in response to the way,
70:17 just don't want to be instantly um Um as you understand that uh grading
70:26 stress is the acceleration and so um the gradient has a jump in,
70:33 there's a discontinuity then right there at interface, the gradient of stress would
70:38 infant acceleration would be infant and have . And so here we have the
70:46 are all uh these five components of stress factor of the stress tensor must
70:54 continuous across the boundary and putting in law. Uh 13 components of
71:04 That's uh cookie and stiffness tensor or 13 for the first two industries,
71:12 as we have here and all the and all the else. Uh multiplying
71:17 all the epsilon chaos uh Nine different on the right here in general.
71:23 gonna simplify of course. Uh That's is putting hooks law in and the
71:30 thing down here for the other. , so there's still real. Is
71:38 statement true or false? I feel it's fall. You are correct.
71:49 continuity of stress and displacement. Very for students to forget that and think
71:57 and strain uh trips off the tongue easily, but that's not what we
72:04 that we just showed that stress and . So that statement is false about
72:10 . Is that true or false, . And why is it false?
72:25 it, Read it carefully. Uh it a discontinuity? Say it
72:37 I didn't hear you. Oh no, it's true. Yeah,
72:43 one is true for the display shin but now here's the core, here
72:49 the corresponding question about stress and that's I think you were thinking about when
72:53 first answered false. So this one false because only certain components of stress
72:58 to be continuous. Uh Those components are aligned with the surface itself in
73:06 to have a 31 of the at least one of the indices of
73:11 stress uh is included in that So so those components, the five
73:17 of stress which are have one of indices or three, those have to
73:26 continuous. And we didn't say anything the other components. Okay, so
73:33 that we understand the boundary conditions, think about the simplest case. Um
73:39 the simplest case because we we are to have rocks on either side of
73:43 boundaries. So the simplest case you know, have air up
73:47 but for an interface inside the earth we're going to consider um normal.
73:58 so uh we have only the displacement also gonna be vertical. So we're
74:04 call that lower case w and it's be a function of position time and
74:09 gonna have its own frequency. We change that later, but uh frequency
74:17 zero. And uh we know it's plane wave solution. So the solution
74:23 like this with uh an amplitude um the uppercase w which might also be
74:35 function of omega. And how are two items related together? Well,
74:44 we are gonna say that Kcm zero in mega zero divided by the
74:51 That's the same velocity we have and we have a plus sign.
74:55 what that means is that um uh with this minus sign, with this
75:05 sign. Uh This means that the is going down as time increases.
75:13 got to have X3 increasing in order keep the phase constant. And so
75:20 that means that this wave is going like it showed here. So this
75:26 sign and this one here uh gotta , we have to have the right
75:33 are right here. Now let's assume the boundary conditions that we just said
75:41 a reflected and transmitted wave and that a guess which we are gonna validate
75:49 . So the upcoming wave is gonna like this. Now, let's uh
75:54 look carefully. We got subscript one everywhere. We distinguish between the incident
76:00 and the reflected wave. And we a minus sign here, where we
76:05 corresponding a plus sign here. So uh is a standard convention. Let's
76:11 so that the um um displacement is well, we're gonna use that as
76:23 convention. We could choose another convention I got to tell you that over
76:27 years, many people have tried to creative at a point like this.
76:32 and they've always gotten into trouble. we're gonna follow the standard conventions have
76:37 minus sign out here and a minus here, but Uh the K one
76:45 gonna be negative. So together with minus sign, and this minus sign
76:51 the wave is traveling couples. And now here is the transmitter wave
76:57 we're gonna have subscriptions to everywhere. again, this is going to have
77:03 positive. Um not gonna have an minus sign here, just like we
77:10 here, because these waves are traveling , these two are traveling down.
77:14 that that means we're gonna want to a plus sign here, just like
77:19 had here. And again, we the minus sign here in the
77:24 But again, we have plus sign , just like we did here,
77:28 pay attention to uh screw up any these sign conventions get into trouble.
77:35 this is um a consistent set of . Yeah. What we need then
77:47 we need to solve for these two W one, W two. And
77:51 uh frequency Omega one. And we're use the boundary conditions here to match
77:59 solutions on this side. And on side, stop that. Um at
78:09 surface, which is uh X equals . The tangential components of displacement are
78:15 zero just by symmetry. This is vertically traveling p waves. So that's
78:20 , but for the normal part we to add together the displacements in the
78:28 part that's incoming and um reflected and those have to be the same as
78:36 single. Uh That that song here gonna be the same as single wave
78:45 war medium. And you notice here no Z here appearing no X three
78:50 because we were soon X 340. we can find a solution to this
79:00 at all times only if the frequencies all the same. So let's just
79:05 the subscription and all that will So this will be the incoming wave
79:11 frequency omega and absolutely necessary that reflected transmitted ways have to have the same
79:21 . Then uh we can divide out exponential terms and I just want to
79:29 back here so that that some comes some of this thing plus some of
79:37 thing and the minus sign is already explicitly. That's why this minus sign
79:43 here. Okay, now the continuity stress. So uh let's talk about
79:53 13 stress and uh oh So what says is uh 13 stress. You
80:13 mhm carry out the sum from hook's and you find the only terms in
80:19 are surviving are the ones which have three here and 13 year. These
80:26 all the same because of the symmetry . That gives you a two right
80:30 summing those two together and the common for this component of uh tensor is
80:38 not the common name in two component , it's a C 55 because 13
80:45 31, both of them up to five. When we switch to the
80:50 components Rotation and the definition of strain given like this. And so this
80:57 that it's got here you one that's in the horizontal direction, we know
81:02 zero because of this is normal so that 10 and this is a
81:09 in the one direction. So that's a zero. So I got uh
81:13 plus zero here. So never mind this stuff. It all gets multiplied
81:18 zero right here and parenthesis. And that's gonna be true for each
81:24 incident mode reflected and transmitted mode and same thing down here for 23
81:32 And for the 33 express we carry the sum and uh so we get
81:37 3311 term, that's a non zero in the tensor and all that.
81:45 uh and to index notation we're gonna that C 31 and here's a C
81:53 coming from this C 33 coming from and the epsilon terms are gonna involve
82:02 in the one direction. So that's be a zero here are changes in
82:05 two directions. That's gonna be And the only thing that survives after
82:09 this um something and so on is one in the three direction And the
82:15 name for this sensor, This matrix . So that's got to be continuous
82:23 the area. Yeah street in the wave. So here is uh super
82:30 here stands for incoming and we've got . So here's the incoming displacement in
82:37 vertical direction. And it um it applying the launch monitors em in the
82:44 medium. So all we're doing is this equation, the incident wave in
82:51 upper medium. And so uh we adopted the uh notation that this vertical
83:01 is given by more case W and zero reminds us that's the incoming
83:11 And from the previous page we're gonna that is a lowercase W This
83:17 Uh Thank you here and let me . Right when we do distributive,
83:35 down out of the exponential of minus K. Zero minus I K.
83:41 that appears here. So collecting all together. Uh Now what I wanna
83:52 is uh change uh notation, know M1 is uh You can be represented
84:01 terms of density and velocity in this we know that K-0 can be represented
84:06 terms of frequency and velocity in this . These are old ideas for you
84:14 . And uh collecting terms we get a density times velocity of the first
84:22 and here's the frequency right in here do that similar for the other two
84:28 . And it looks very similar. uh in a sign here which comes
84:34 our previous conventions. You can verify for yourself, the minus sign
84:39 that comes from our previous convention, that for yourself. And then what
84:46 have to have the stress that stress the incident way plus the stress from
84:50 reflected way that's happening in the upper and low medium. We have this
84:58 combining with the first equation we got displacement continent, we find this
85:05 So the ratio of the reflected the incident amplitude given by this difference
85:13 by this sum. And that is we call the p wave reflection coefficient
85:22 , correspondingly we have the transmitted Anderson by the incident amplitude is given by
85:30 minus this. And we call that transmission coffee abnormal answers. So these
85:38 should be familiar to you, although might have forgotten the derivation. So
85:45 derivation that I gave you um is good one. And it depends up
85:50 our making the right um conventions for the various minor signs have to show
86:01 notice here that the frequency has uh out this frequency hidden inside here and
86:12 inside here. Um No uh but frequency already canceled out at this
86:19 All of these have the same So these results are independent of
86:24 That's really important. So the reflected is gonna come up with the same
86:33 as the uh incident media. And when you add together all the plane
86:39 like this, that means the the reflected wave is gonna look just
86:45 incident wave. Except that it's gonna uh why not? Instead of going
86:52 ? And but it has the same of the wave, it's gonna have
86:56 different amplitude, it'll have its amplitude by this. And so this is
87:01 be a normally a small number because this product in the Uh upper medium
87:10 going to be pretty similar to this in the lower media and the difference
87:14 gonna be small compared to the And this is normally going to be
87:19 small number of the order of 10% less. And it could be positive
87:24 negative C. Haven't said here, are these is bigger. Now this
87:32 we call them peaks and we give the conventional notation uppercase Z. So
87:39 the previous result in this way Z . P. To the upper
87:45 zp two in the lower media minus . P. One in the upper
87:51 divided by the sun. And it's for you to remember which which of
87:56 is which the incident medium has the sign in china. So this jump
88:03 delta Z. P. Simply this you can recognize this sum is
88:10 Yeah. Normally the reflection coefficient is lot less than one. So normally
88:20 the transmission coefficient is functionally equal to because this number is small. Not
88:29 . And that's a good thing if we had most of the energy reflected
88:34 only a small part transmitted. That be bad because we couldn't see very
88:39 into the earth. And lucky for normally the reflection coefficient is small and
88:47 , so most of the energy continues past. Mhm. Perfect. So
88:55 other words like that one at a time. So here's our results.
89:03 you notice that if the lower uh impedance is greater than the upper impedance
89:10 uh jump is uh greater than And so the reflection coefficient is greater
89:18 and the transmission coefficient is less than . However, if the jump in
89:27 negative then the reflection coefficient here is be negative. That happens when Z
89:34 two is less than zp one. uh slower uh lower impedance below,
89:42 mp and survive. So in that the reflection purpose is negative. And
89:48 the transmission coefficient right here is greater one more amplitude goes down and comes
89:57 . How is this possible? So uh Miss del rio, have you
90:04 considered this straightforward consequence of these equations which you're presumably you're familiar with this
90:13 right here and both these results. did you ever think that when you
90:20 uh negative jumping and beating, you're have a transmission car efficient? Greater
90:28 one, like it says here or goes out than comes in. Have
90:33 ever thought about that? Does does it trouble you? Little
90:39 Yeah, it is weird. but it's so easy to see where
90:44 comes from. And you know that the Earth there's lots of alternating velocities
90:54 , and alternating densities as you go through the layers. So this must
90:58 a lot. Well, like you , it's weird. So let's think
91:04 uh just how weird this is. , uh, let's think about the
91:09 density. The energy density has is in this way. And uh we
91:17 about this um, several lectures I won't repeat that. Just remind
91:21 of what we have on the left a scale or by the way,
91:25 , this you is now energy, displacement. And I need to use
91:30 different location. The same notation we before for energy density. And now
91:38 forgot when I wrote it this that it's um, uh it looks
91:43 much like the displacement coefficient there just find 10 minutes ago, I'll go
91:51 and change. But this is the stiffness um, sensor that we had
91:58 and the same strains we had And this is a scalar. And
92:01 we have to do sums over I J and eminent lots of sons.
92:06 so, uh, these vertically traveling waves for each mode, it all
92:14 . So it looks like this. got only 33 strains. And we
92:18 only the C 3333, right So that's true for each moment.
92:25 so putting in right here, the common name for C. 333,
92:31 in right here the definition of Uh and then taking this derivative,
92:36 got an I. K. Um like a w. And it's squared
92:42 we have this product here. And that square makes a -1 and multiplied
92:48 this minus. So here's a Right here we put in the definition
92:54 em in terms of velocity. And here we put in the definition a
92:59 terms of they got, he seeing all that out, we get
93:08 omega rho omega squared divided by two . Um So that's the energy gen
93:21 each mode and from the square of frequency and the square of the uh
93:30 uh amateur. And um right here have uh solitary factors, also inside
93:39 invention of the first pound. And see where the two. So that's
93:45 energy density. Now we ask about energy flux for his mouth, so
93:49 uh that uh energy density is traveling this philosophy. And so uh energy
93:57 is going to be given by this part will just drive. So
94:06 we're gonna ask, is that energy across the boundary concern. We have
94:11 same amount of energy coming in as out downwards. And uh you're saying
94:21 wrong, we have to have the on the left hand side of the
94:25 flux and the and the reflective flux On the upper part of the medium
94:33 , is that equal to the transmitter in the lower media? If it's
94:38 equal then that means that energy is to be left behind at the
94:44 And if all the definition means the is gonna be heating up because of
94:49 incidence of this way. So that better find out that these um um
95:00 works out. We better find that flux on one side equal fox on
95:05 other side. So putting in the results that we just found uh same
95:14 mark here, uh huh The same that we just asked, it was
95:23 and then simplifying that equation uh by for yourself here and work out the
95:31 figures out, you will see the to this question right here is
95:35 we get the same energy flux on sides of this equation. Whether or
95:41 The transmission car fishing is lasting So energy is conserved within the
95:50 The same amount of energy is going the surface as coming out of the
95:55 , reflected and transmitted. So the does not yet hot, does not
96:01 cold. All the energy is retained the um in the wave. And
96:11 what that means is that uh the that the amplitude in the transmitted wave
96:18 be bigger than the aptitude and the incident wave, that's okay because it's
96:26 energy flux is dependent on this. uh this expression here? No it's
96:38 the amplitude. Uh it's the energy is important, you see right
96:43 this product same as we have And um so the fact that we
96:56 higher aptitude going out on down is a problem using this energy.
97:08 Wait, so MS del rio, this true or false? Careful,
97:26 is true. This is the weird . Yeah, this uh this is
97:31 so weird. This is sort of definition. Um uh The first part
97:36 it is the definition here. It the direction of propagation is given by
97:40 gradient of the phase. Yeah, uh I just say is now reading
97:49 carefully and um so it's a trick here, The gradient of the
98:01 Since with negative the way advances in direction of increasing spatial coordinates, uh
98:09 increasing X three. Um uh that is true. Mhm Go back
98:25 review previous slides and see that ingredient negative. Yeah, the wave advances
98:35 the direction of increasing That's true. might want to review previous items
98:47 Um is this true? The attitude a continuous plane wave is given by
98:52 magnitude At the time space position where phase is zero. Okay, so
99:01 think we didn't talk about that What I want to do at this
99:05 ? Go back out of this, did something bad here. So let
99:14 share the screen. Can you see screen again? Yes and select people
99:28 here And so we can use this slide uh you can see where slightly
99:53 . And so here is our it's creating the phase vector and the
99:58 this thing up here. So if zero then the uh the amplitude of
100:03 waves, simply by the amplitude of vector here and talking on the scale
100:08 components only traveling in the X3, this only and this is zero.
100:13 this is because phase is zero, d to the zero. So that's
100:19 one answer. The answer to the is that's false. Um So did
100:34 miss something, explain your reason? , phases zero. I'm sorry,
100:41 was thinking about everything. Never it's true. So when we have
100:45 zero here, that means each of zero, which is one. And
100:49 that means that and at such a , such a place of action,
100:56 the the amplitude of the wave is to the stellar amplitude that we have
101:01 here. Okay, fine. so I'm gonna go in presentation uh
101:21 excited. That one. Yeah. is measured in the same direction as
101:30 way affect. Yeah, that's If the deficit is positive, the
101:35 at Phase zero is parallel to the that no matter what direct. So
101:39 I want you to go back and at our our sign conventions where we
101:45 minus signs scattered around here and there decide for yourself whether this is
101:51 If all I think it's true. I if I recall correctly, that's
101:56 . And you verify for yourself, Stephanie whether or not this is true
102:02 by looking carefully at sign convention that showed on those. Yeah, this
102:12 important. You gotta get this one , tell me which one it is
102:27 . Yeah, a lot of I have to get this one
102:32 Um This one is it's the minus top minus has got to be on
102:50 and it's gonna be either this one this one. And you got to
102:53 the difference here. It all is has uh the incident medium here and
102:59 medium here. So you're right, lot of pressure but you did.
103:05 , okay, so um that's an problem. But it's not our
103:11 We've we've got to find almost all our data is obliquely incident. So
103:17 situation and we got an incident way looks like doesn't So now the difference
103:22 is that a zero is a vector the amplitude of it Is given back
103:29 or make our VP one that's a way upon at some angle. Theta
103:38 um You know what uh Uh it's 1030 is that now passed into I
104:02 this is a good time for us take a quick break. Um Want
104:07 go back to this right here. take a 10 minute break And come
104:13 in 10 minutes and we'll take up topic which is going to be
104:18 So it's a good time for a . So I will um stop sharing
104:27 I will stop my video here. , So I assume that both of
104:36 can see the screen, give me feedback. Ok, so now this
104:43 uh an important topic for us. I'm gonna put us into presentation mode
104:51 here's the situation where we have uh same uh medium above and below and
104:59 have this wave incident and we talked how this is uh uh has uh
105:06 vector k zero up here. And the length of it is given by
105:11 omega over V. P one. reflected wave is going to be over
105:17 somewhere and we don't know yet that theta one is not equal to theta
105:24 , We're gonna discover that. And length of this vector uh is gonna
105:30 um uh have the same minus sign uh and as we have the form
105:36 have a minus sign here and we have the minus sign here and we're
105:41 have a transmitted wave similar with the plus sign here as here. Same
105:47 sign. Had put a plus sign . Okay, so what we have
105:52 decide then is uh the magnitudes of vectors and the angles because we already
106:01 these omegas, all these omegas have be the same. So let's first
106:07 the issue of the angles. So is the incident wave coming in and
106:12 the wavefront looking like that and the is given here. So uh the
106:18 . So think of this as the of the one peak of the
106:22 And remember this is uh plane wave goes on forever. Science and
106:29 And uh so think of um this one of the peaks and this is
106:35 next peak. And so it's traveling distance in the time delta T.
106:43 it's traveling at three P one. notice as it does, so uh
106:47 wavefront is sweeping across the boundary uh because of the angles involved, it's
106:54 equal to the uh this length here by the sine of this angle,
107:01 trigonometry. So we call this the velocity, which is the actual velocity
107:09 the wave divided by the sine of angle. And uh that is equal
107:13 one over the horizontal slowness. That's definition of horizontal appears. So in
107:24 to match these boundary conditions for all ones and at all times the apparent
107:30 must be the same for all So uh this is the same for
107:35 this is the incident mode, reflected and transmitted mode, they all have
107:42 have the same apparent velocity. And is a special case of Snell's
107:50 So then uh oh, sign of the incident angle is given by extra
108:04 the sign, this is the sign the incident angle. This is the
108:08 of the reflected angle and this is ratio of velocities coming from this
108:14 And so in the upper medium that is the incident velocity and the reflected
108:19 . These are the same in the medium. So that's a one.
108:23 so the sign of the of the mode is equal to the sine of
108:27 mode. Uh so the way it at equal angles. So um that's
108:39 to you or you know that this is why why that's true.
108:44 the sign of the uh transmitter wave going to be different because this ratio
108:51 not equal to one. So here's some examples. And so here we
108:57 a case where the velocity is continually . And uh I got this slide
109:05 professionally in our department and because Snell's as required, this is the statement
109:12 Snell's law in this situation. So the angle is continually increasing in this
109:19 because the velocity is continuing increasing. here here is a case where it's
109:25 decreasing uh as you put down is . So the wave is diving.
109:30 neither one of these situations is very Comments. So let's look at a
109:39 where we have uh some slow So V two is a slow layer
109:44 before is a slow layer following the . And so now the wave looks
109:49 of like this one when it reaches top of this slow layer that refract
109:54 . Then when it goes into another layer, reflect upwards then downwards again
110:01 we four are slower and upwards. uh raft bends down in the slow
110:08 . So that's uh the way the vectors work and the angles work,
110:15 it only affects the wave vectors, the amplitudes of the various smokes.
110:20 we've also got to have continuity of displacement and stress. Uh not all
110:27 but simply these components of stress. that makes two equations here. Two
110:34 here, add them up. Makes equations and all but we have only
110:38 for two free parameters left, which the amplitude of the reflected wave and
110:43 amplitude of the um transmitter wave. so you can't have uh just can't
110:53 these four equations with just two free . And so the proposed solution is
111:00 gonna work, simply not gonna So what we have to do is
111:05 have to assume that in addition there's outgoing share ways. So notice here
111:11 have this uh vector uh shows the um shear wave you three. And
111:18 looks like all these others except that it's got a K three here and
111:26 U. Three attitude vector here. you know that this this vector here
111:34 gonna be uh perpendicular to K3. perpendicular to to you one but it's
111:43 to its own Uh propagation vector. it lies in the 1 3 plan
111:51 . And so uh uh we have all three for all three waves in
112:00 upper media. We have uh these with the corresponding losses, incident wave
112:09 a P wave. That's this one . And so it's related to the
112:13 with VP one that reflected P wave uh involves VP one but it's got
112:23 minus sign here by convention because it's upwards. And uh here is the
112:30 way in the upper media, it's the same minus sign because it's going
112:35 . But it's got a V. . One down here. Sure.
112:38 velocity, shear wave in the upper down here, we have the same
112:45 wave as we had before. And we have a sure way also polarized
112:50 this plane with a wave vector given with plus omega three. S.
112:58 notice the way we've arranged the notation . Oh, industries. Uh even
113:07 are going down 02 and four. are all even The indices for the
113:14 going rates are off one and The wave vectors for the down going
113:21 have a plus plus wave vectors. Wave numbers for the growing waves have
113:29 and of course we have we have include V. S. One here
113:34 V. S. Too but we have that before so we're going to
113:40 able to find solutions only if Snell's extends to all the mod. So
113:45 the apparent velocity for the same for incident wave, reflected P wave,
113:52 P wave, reflected S. Wave transmitted s way so horizontal. Another
114:00 to write this is in terms of horizontal uh slowness, uh whatever the
114:08 velocity is given by this for all and so that's going to determine the
114:14 . So I got some movies for here. Uh you have these movies
114:20 the blackboard I think. Let me if you don't have that. So
114:25 I'm gonna do at this point is thank you. Stop sharing and share
114:52 screen, I wanna show. No want to do that. Um What
115:04 want to do. Yes I'm gonna my zoom. If I lose the
115:15 connection don't worry I'll be back I'm gonna minimize it for Mhm.
115:24 um I lost your knock here, this screen, this screen, this
115:40 and the screen and now I'm gonna the movie. Sure. Mhm.
115:57 can't play in the movie. Oh I'm hoping you can still hear me
116:31 now. For some reason the movie not play. Um So let me
116:37 the other movie to see if that's work and that one says it's not
116:52 . Okay folks I have a problem that I can't show you these movies
117:07 it's a formatting issue. The movies good, I will figure out uh
117:13 sure these movies are playable, I assumed they were playable there. Uh
117:19 format. Um But I have to into that so meanwhile I'm gonna go
117:27 here. So did you all see fumbling now? You probably didn't see
117:32 fumbling around but you heard me fumbling and so I will have to um
117:38 the movies to you uh later and make sure I have workable files and
117:45 sure that you have the right files the out. Uh So where are
117:53 on the blackboard? There should be the same as where you found these
118:02 and if they're not there it means I forgot to put them. But
118:05 okay. I probably would have put inoperable files so I'll sort this out
118:12 over lunch, I'll sort this out be okay. Those are interesting
118:19 Um and uh so we'll get it you. So uh what I want
118:28 do now is share my screen right . Yeah um can you see this
118:43 slide which is where I left you uh so I will also solve the
118:52 problem or lunchtime I use and so gonna put this into mm presentation
119:00 Yeah, so or solving those. so what we're gonna do is put
119:10 guests that I just showed you with five ways incoming and to reflected in
119:17 transform implement the wave equation using those equations for plane waves and solve the
119:28 . So this was done for the time a long time ago and the
119:32 time they did it they got wrong we didn't have a consistent set of
119:42 conventions. And so you might have a little bit irritated by the sign
119:47 that I used. But if you those sign conventions you end up with
119:52 right answer. And uh right answer found by these two. These two
119:59 not considerate. I have a picture of zipper. It's um and um
120:06 equations can be found in many modern . For example the textbooks by AKI
120:11 Richards. Uh So uh do you this textbook, AKI and Richards?
120:23 . So um I know these AKI is now retired from M.
120:27 and richards is now retired from They're both my friends. And uh
120:34 is sort of the standard textbook and and I think 09 was the second
120:41 And the first edition was sometime back the 80s. And so um um
120:51 , do you have this book? do not but I'm going to get
120:56 . It might be a bit advanced you. Uh So but you can
121:00 it cheap on amazon uh and get hard copy edition. Not not a
121:07 copy. And so uh everything that done in this course uh is consistent
121:13 what they show over there. And it's um it's focused on global
121:22 not on exploration technology, but it's a very useful tax standard tax
121:30 So, here is the answer as by AKI and Richards for the p
121:35 reflection coefficient. So look here, a mess this is, or mess
121:41 . And got so much notation. the notation is all um uh listed
121:51 here and it's got layers of So it's got a little uh small
121:56 , A, B, C, D. And then big letters
121:58 F, G, H and And they're scattered all over here.
122:02 this is such a mess that when look at this, your eyeballs roll
122:08 the back of your head. And so um this um was so um
122:22 that for years and years explorations, businesses would not touch it and there's
122:30 reasons not to touch it. Uh can talk about it just without understanding
122:36 really, it involves all six elastic . So three on each side of
122:40 boundary. And but it's a non number. Of course, the reflection
122:46 has to be non dimensional. So must mean that you have to you
122:49 rewrite this in terms of these non parameters. So the ratio of one
122:55 to the other. And then you imagine lots of ways to non dimensional
123:00 , the velocities. Uh here is one of them where you uh invite
123:04 one of these uh four velocities by one. So So there are four
123:14 scattered through there. You can see it's independent of frequency. So it's
123:20 we said uh frequency is going to out of a reflection and it's not
123:29 obvious, but it might be fun you to verify that at normal
123:34 it reduces to the previous result. all those complications go away at normal
123:40 . And it would be a fun for you to prove that to
123:47 Furthermore, if all the angles are , it's the reflection coefficient is
123:52 And uh um all the reflected wave will have the same phase as the
123:58 way. So, I want to back and look at that. We
124:02 all these angles in here, so look real. How could they not
124:06 real? Uh angles here and we've more angles here. And uh you
124:12 that all the angles are frozen There's no uh sign angle in here
124:18 . And uh all those areas. , of course they're real. How
124:22 they not be real? Well, gonna find out that it might be
124:26 of these are not real, not Cozzens. Uh in some instances,
124:34 the incident angle is always gonna be incident angle. So that means no
124:38 of the incident angle is gonna we're gonna find cases where these other
124:43 are not real. Isn't that So that comes later. Now the
124:53 law is given by this and remember we did not require uh we didn't
124:58 to know what the reflection coefficient is for um uh in order to define
125:06 the angles are. So for p angles here it is and solving this
125:13 the transmitted ankle sign data to equal sign of the incident angle with this
125:18 ratio. Yeah so I suppose that lower medium is faster than the
125:24 Um So this ratio is bigger than and then suppose that the sine of
125:30 angle is pretty large uh as we , so so far that incident angle
125:38 be anything. So let's choose a angle so that when you multiply it
125:44 um uh by this ratio which is than one, that means that the
125:51 of state A two is bigger than . You can imagine for large incident
125:57 with uh fast medium over a slow , you can have the case where
126:05 formula implies the sign of state into is bigger than one, wow.
126:11 so the Kassian uh co sign is in terms of the sign in this
126:18 . So if this thing is bigger one then uh we have a negative
126:23 underneath the square root sign. So co sign of data to under these
126:29 is gonna be imaginary, wow. remember we just uh just back here
126:38 said if all the angles are real the reflection coefficient. But now we
126:43 the case right here for some of angles are complex. And in particular
126:49 you have a complex co sign data that uh have a complex angle then
127:01 that case the post sine of that is pure imagination, otherwise it's pure
127:07 . But if in this circumstance you the case uh full sine theta two
127:13 imaginary. Yeah, where does that ? It happens in large large
127:20 And so what how how large is ? Well the where the incident angle
127:26 transitions from real to complex is at place where the sign data to uh
127:32 data to is one. So that to an incident sign at some critical
127:39 multiplied by this velocity ratio. So is that critical angle? It's that
127:45 where the sign is the sign of incident prayer triangle is given by this
127:51 relation. So at incident angles greater this. Many interesting phenomenon occur both
128:01 the reflected and transmitted light. So we don't look at these post critical
128:09 but maybe we should and I'll show , we'll talk more about that right
128:19 so far, we have ignored the of the incident away from and that's
128:28 um and we talked only about plain . Okay, so imagine a curving
128:34 from Mr fourier mr Fournier has guaranteed us that we can describe this curved
128:43 as the summer plain whites. And uh imagine um heard way from advancing
128:55 a certain direction, there's a ray perpendicular to the wave front. So
129:01 an oblique ray and a curving away and that at that particular angle where
129:07 raid direction uh say uh data zero the front. Ordinary front, it's
129:19 be a sum of plane waves. the dominant term in that sum is
129:26 uh the main way, which is parallels and away from that point perpendicular
129:32 the right in the sun. The term is the plane wave um perpendicular
129:41 to the way from tangent to the from at that point. And then
129:46 the other uh terms and that some needed to transform the uh the
129:53 that dominant plane wave into a serving , dominant term is for myself.
130:00 as it impacts a planer um plainer , like we just talked about,
130:07 we needed to talk about was the uh the interaction of that dominant plane
130:17 with the surface. And that's what been doing up to this point.
130:21 . And that works for us for of our data, all we have
130:25 think about these plane wave reflection. fishing's even though we never have plane
130:31 in our data because in the foray uh for the third wave front as
130:38 hits the wave at the circus that I'm gonna determine that for you.
130:44 is um anyway, cleaner way answer the service. So uh we can
130:54 can neglect the rest in most cases close to the source, uh the
131:03 re strong and we might have to include more terms uh there in the
131:10 process close to this source. But we're not worried about reflections close to
131:17 source uh near surface. Normally we're worried about that, but beyond the
131:25 angle near the critical angle, uh have to make a correction uh as
131:32 consider reflections near a critical angle, need to also consider this courage.
131:41 uh so in what follows, we uh add on the assumption that as
131:50 continue to look at a plane wave and transmissions means we're not very close
131:56 the line. Now we saw before very complicated expression or um for the
132:12 coefficient here is the expression for the coefficient in that same situation p down
132:18 incident and transmitted, you can see a lot um simpler, but it
132:24 involve uh the incident angle of And these these coefficients F and D
132:32 were um defined previously and the velocity over the lower medium here, but
132:39 right here, it's got the density the incident meeting. So we figured
132:46 now this is not equal to one reflection coefficient except the normal incidents in
132:54 , you gotta use this term instead was. Now these transmitted waves are
133:01 called refracted waves refracted because you if you're doing an optical experiment or
133:07 shining light on a um for example them um piece of glass, uh
133:18 light transmits into the glass at a angle and all of that is incorporated
133:26 the design of eyeglasses and contact lenses everything like that. And so those
133:33 who design their design those kinds of , all their waves refracted waves and
133:41 the same as this transmitter. Now this is the 21st century, we
133:48 have to consider uh converted coefficient. the version upon reflection. The proficient
134:00 noted like this. See for conversion for incident and expert converted reflected.
134:08 that formula is given by this in of the previous notation. And you
134:14 see immediately from this because it's got sign data out in front, multiplying
134:20 else that's gonna be zero at normal , a normal instance sine theta
134:26 So the conversion coefficient is zero and instance. And furthermore you can see
134:32 if scientist is positive, that's gonna one sign to the conversion coefficient depending
134:39 all this other stuff. We don't necessarily that this minus sign results in
134:46 negative conversion contribution depends on all this stuff. Well but for the same
134:54 In the opposite direction uh we have have a negative uh signed data.
135:00 . So in that case uh immediately that version coefficient is going to be
135:07 symmetric? I have one Uh one for positive process and the other sign
135:16 negative. Normally in our business we consider the transmitted car efficient conversion,
135:29 from P two s and transmission. don't see much of that away in
135:35 data. Um so that's not clear particularly not there. When we first
135:44 looking at ocean bottom, seismic converted 30 years ago. What? Um
135:51 the conclusion that we come to actually at a lot of converting data set
135:57 away of datasets over here? One the most of our converter wave energy
136:05 our horizontal components uh receiver. Most that is coming from conversion upon reflection
136:14 transmission. So we have here a . Is this true or false?
136:23 angles of the outgoing ways are determined Snell's law through the equality of the
136:30 velocity in all these different modes. Stephanie, is that true or
136:37 The statement is true but the equation wrong. So false. Okay,
136:42 uh yes, the statement is true uh equation is false because actually we
136:53 the same horizontal slowness in all So very good. That was a
136:58 question. Now um is this true false the exact plane of my reflection
137:07 can be written in terms of these non dimensional qualities. So the density
137:19 , isn't it supposed to be P over P. One and then the
137:23 for the last term has to be 12. So it's false. No
137:27 it's not true. It's different than I showed you. But we can
137:32 this normalization in lots of different And so so this is one.
137:37 we have to do is we have end up with three non dimensional velocities
137:43 one non dimensional density. So uh this will do it different from what
137:49 did before. Maybe in some cases might be better because this cheer wave
137:55 ratio here is a p wave velocity . And so then here is a
138:02 . P V. S velocity ratio the upper meeting. So this would
138:05 work okay for false. Since all terms and the expressions for the exact
138:12 wave reflection cards are real. The itself is really to a far uh
138:22 it's not true because the predicate here uh is false. It's not true
138:29 in general that all of the terms real. I just show you post
138:34 reflection um uh some of those times not real. So that means that
138:40 itself is not real in all in the general case. So uh
138:50 complexity is hidden within the notation. that's why it looks like it should
138:55 real. But if you go back our discussion of post critical angles,
139:01 see those uh, those post critical of those post critical angles are appearing
139:07 this expression that we looked at those are imaginary, then coefficient itself is
139:14 to be complex. So go back check the, check the equations
139:22 I'm gonna ask you to do a bit of mental arithmetic. So let's
139:28 a case at a sediment salt The incident wave is coming in with
139:35 , which is not uh not uh of typical for p wave velocities in
139:43 . Made a round number here, kilometers a second. And inside the
139:48 , it's a lot faster. So for easier, I think I'm gonna
139:52 it for uh kilometers. Uh remember formula for the critical angle and
140:04 yes. Which of these answers is best answer. Three degrees. I
140:13 hear it 30 30 degrees, Because the sine of 30 degrees is
140:19 half. Which is this ratio So this is wrong on its face
140:24 that's not an angle, that's the . So, uh 30 degrees.
140:27 that's pretty small angle. Um uh , uh, let's not jump to
140:37 conclusion that all of our data has be less than period degrees because this
140:42 an unusual interface. Most of our don't have uh, this big difference
140:49 and uh so uh furthermore uh sediments interface, you've got to consider that
140:59 , the incident velocity might be bigger this and the velocity in assault might
141:05 less than this. So, uh is not a hard limit here,
141:13 even in the case of a sediment interface. But it shows that it's
141:18 you need to worry about. If have, if you're looking at data
141:22 there's a salt body down there, might be that some of those uh
141:28 angles are most critical. Yeah, us now go on to um the
141:39 , which is uh one that we um think about more deeply. So
141:49 previous results were exact but they're almost for us because real walks are usually
141:54 exactly ice and tropic, as we to make that reasonable and Real walks
142:01 not exactly elastic either. As we to make that result, real interfaces
142:07 usually not planer and incident waves are curved. So that's potentially a problem
142:14 us. Not included in the So called exact analysis and real face
142:20 are not isolated from each other but close to other interfaces. So the
142:25 , let's are interfering the wave reflected the bottom interface is going to be
142:32 in just a little bit after the look reflected from the upper interface.
142:37 two letters are gonna interfere with each and make problems unless we deal with
142:44 . And here is the main reason we don't use those um equations is
142:50 can't get our minds around. They're too complicated. So what we're gonna
142:55 is find an appropriate approximation. most often the appropriate approximation is the
143:03 between the upper medium and lower medium elastic contracts. Furthermore, we're gonna
143:11 that the incident angle is not so that we have to worry about the
143:15 critical angle that we deal with that . And for now we're going to
143:21 that the media are actually profit perfectly and well separated from other interface.
143:27 these limiting assumptions, the crucial one actually this one right here. That's
143:32 make a big difference for us. notice that we never measure reflectivity as
143:41 function of angle. Never ever. we measure received amplitudes as functions of
143:49 . So we have to convert what receive into uh idea into other quantities
143:58 are suitable for analyzing using the reflectivity . So, our first challenge is
144:07 convert offsets to incident angle. What means is we're going to need to
144:13 what is the velocity field in the overburden above. The reflect is not
144:19 be uniforms. So the rays are going to be straight in the
144:23 They're gonna be curving down through the following Snell's law and they're gonna hit
144:29 uh we're gonna hit them reflector at angle that we don't know uh without
144:36 the calculation and how do we do ? Well, we trace rays through
144:40 velocity model to determine what incident angle to what sources say was. So
144:50 this convolutional convolutional description of wave Most of these items in here make
145:00 dependent uh variation of the amplitude. let me quiz you on this.
145:08 Stephanie, let's start at the start over here. And uh so
145:14 is the source of strength? It here, it's a function of
145:23 Uh Why is that? So give a couple of real world reasons why
145:31 source strength is gonna be a function angle from your comments. Mm I
145:44 know. Um Well okay so let turn to Utah Utah give us an
145:57 of something we do to make an dependent source. Yeah, that's a
146:12 one. So on land we're gonna vibrating uh the ground, we don't
146:18 dynamite much anymore on land and we by the way when I was
146:23 Standard practice. But now we use . And so normally we use vertical
146:31 . And the reason for that is want to concentrate the uh the p
146:36 coming off that vibrator are gonna have attitude near vertical. Of course you're
146:42 have some attitude at oblique angles. of course they're also gonna be generating
146:48 waves but uh we designed them vertically maximize the um the amplitude and vertical
146:58 near vertical direction. But there is variation. And as you go from
147:03 oblique to horizontal, it has to less and less amplitude in the,
147:09 the outgoing waves from, from a . So Utah give me a marine
147:25 advice the so I'm I'm having a time Utah hearing you because of audio
147:53 I think on your end. So me take up the, the idea
147:58 you have an air gun towed behind uh, the acquisition vessel. And
148:06 and we normally, we normally have than 11 air gun. We have
148:10 array of air guns, uh maybe dozen of them uh in an
148:14 And the, the uh the geometry the array is designed. So that
148:19 the timing of these different air guns designed to maximize the p wave um
148:27 amplitude in the vertical direction, near directions. And so there again the
148:33 function is going to be a function takeoff angle. Now there's another effect
148:39 , some of the energy from this gun array goes down and some of
148:45 goes up towards the surface. It off the uh, off the surface
148:51 comes back down again with a little following the direct uh waves downwards.
149:00 so the amount of the delay depends um uh take off angle from the
149:07 . So if it's, it's truly , you get one delay. And
149:12 it's let's say uh 20 degrees uh you're following a 20 degree rate upwards
149:18 then downwards that delay is going to a little bit longer. So the
149:23 between the uh red wave and uh so called source ghost which comes from
149:32 reflection off the surface. That's that interference is going to be angle
149:38 for that reason. So there's another of how we have angle dependent
149:44 So uh uh Wavell, it might angle dependent but let's ignore that.
149:53 the downward propagation operator, there's a of complicated stuff hidden inside here.
150:02 Stephanie, tell me why this uh propagation and see we haven't even gotten
150:07 the reflector yet, we're propagating downwards as the wave goes downwards uh is
150:16 amplitude changing as a function of ray or is it the same for
150:24 You know, it's going to be off with different uh amplitudes because of
150:28 source here. Um but as it down, is changing amplitude further um
150:37 a function of angle, What are effects in this um propagation? Um
150:48 which which could modify the uh And for two, as a function
150:55 angle Stephanie you can probably think of couple, I mean, wouldn't the
151:03 be like decreasing? Yeah of course will be decreasing. Uh from geometrical
151:11 and also decreasing from attenuation and that's be um that's going to be um
151:21 of those are gonna be dependent upon length. And so uh forgiven um
151:28 receiver offset, the path links are be longer for the old lee um
151:34 than for the um normal incidence. , naturally because of those effects that
151:42 amplitude over here is going to depend offset because of this. What's happening
151:51 ? No, not what's happening I mean in addition to what's happening
151:55 . So uh Utah, can you me another example of an effect happening
152:01 this propagation downwards which affects the amplitude a function of angle, I would
152:18 because of the 39. So the unusual from bird at a same
152:26 increase the image. That's right, right, that's over in a in
152:31 real case, the overburden is never . So what that means is that
152:37 going to be at every uh interface the one we're interested in, there's
152:44 be a transmission coffee. And so transition carp fishing is going to vary
152:48 a a a. Yeah, so then what what as a result
153:02 these and similar things for everyone? can you mute your things. Thank
153:21 . Okay, so um uh what just learned is that inside this propagation
153:29 , there's a lot of uh amplitude with offset which has nothing to do
153:35 the reflectivity. So that's in our . So we gotta keep that in
153:40 . So uh uh furthermore, uh some processing which is happening here,
153:49 what this is about and so um very common. So what uh if
154:00 an interpretation geophysicist, you're gonna be at data which has been uh previously
154:08 on by somebody else in the processing and uh they might have affected the
154:15 also. So uh when they make image, uh you know when you
154:22 an image all the amplitude information is , but but if you make a
154:29 image gather, so you you uh all the different offsets, uh getting
154:38 to stack them together to make a image. That comment image gather has
154:47 an offset variation uh which might have affected by whoever did the processing.
154:55 uh these days he's gonna be he's be doing his best to make two
155:00 preservation during the construction of this common gathered. But uh that impact he
155:12 on the amplitudes is gonna depend in crucial way on how well he's able
155:17 pre to preserve the amplitudes. And it's gonna be different for every imaging
155:29 . And so I'm not gonna talk further here. But uh what I
155:34 conclude is that if you're in charge an HBO project and they give you
155:40 data, uh you need to talk the imaging experts to uh understand what
155:49 think they have done in the computer uh these amplitudes. So that apparent
156:00 effect may or may not be something , it might be a artifact,
156:07 only propagation effects in the overburden but processing effect in the processing. So
156:15 learned today. Now most of us commercial software for analyzing amplitudes. Maybe
156:26 uh provided by Hampton Russell, Maybe provided by food grow, maybe you
156:32 your in house stuff. Uh So each of these uh software packages uh
156:40 a set of procedures, usually including . And these procedures are designed to
156:47 for many of the propagation effects that talked about so that the resulting normalized
156:53 give the reflection coefficient as a function . So they're trying to do is
157:00 rid of all this stuff so that yield this reflection coefficient as a
157:07 These procedures are usually oversimplified, just criticize those people uh without giving any
157:16 uh saying it's oversimplified. But even , I'm going to assume for the
157:22 they are adequate for the purpose, shortcomings they have. Uh Let's not
157:28 about that yet. So here is the exact expression without all the
157:36 So, I left room down here uh removing all the notation. Um
157:43 is the same formula, the exact which you can find in Aachen Rich
157:49 we're gonna reformulate it exactly in in terms here, we're gonna uh get
157:56 non dimensional density by defining the density provided by the averages and similarly for
158:06 and for Gs. And then we're find uh gonna have this this average
158:13 ratio. So three non dimensional velocities one non dimensional density and it's just
158:22 bunch of algebra to put these to um this expression into where the before
158:35 parameters are. Not those defined by richards, you know, VP one
158:40 VP two and so on. These non dimensional, that's just a bunch
158:44 algebra. And then we're going to the case of weak classic contrast by
158:50 that these fractional differences. Yeah, have three fractional differences, density,
158:58 P and V. S are all than one, a lot smaller.
159:04 then we uh thank you. Taylor of uh this expression and we assume
159:15 all the terms in the taylor expansion have these differences to more to the
159:21 power than one. Those are So we're gonna end up in uh
159:25 a simple expression only the first um first power of these three quantities.
159:36 that's this, I think you can that that's really made a substantial
159:43 So we have uh simple, simple and we have all the material complexity
159:52 uh inside these core functions which are by these expressions here. So let's
159:59 at the first one until these are or two in our first this is
160:03 normal incidence reflection. This is exactly we determine previously today. And then
160:11 got a gradient turn which is our hear that appears here, it multiplies
160:21 sign dependence uh science, great dependence their here. Now look how this
160:26 a range for for zero angle of . These two terms go away and
160:32 left with this one for small This is a small non zero
160:37 And see this is even smaller because assuming that the sine squared of theta
160:42 small. That means the tangent squared theta is also small. So we
160:46 neglect this at normal incidents are at angles and that's a clever thing to
160:52 it like this. And so this is given by this and you see
160:57 it's got in there, not only jump in VP, but also the
161:01 in density, jump in share jump in density or shooting inside
161:08 jumping density is appearing in the gradient jump in Sure. Model is given
161:16 ingredients jump in VP itself is given here and in the current term.
161:23 this is different from this one because one contains the density as well.
161:30 to get here, we also use two uh relations from general calculus,
161:38 in Z jump in Z P is to jump in and VP. And
161:44 plus the jump in density, all are normalized that you show here and
161:50 the jump in mu is given in of the jump nvs and density like
161:57 , and you of course you're going recognize that this jump in uh VP
162:04 given explicitly right here. That's that's jump in VP. And here's the
162:09 value of V. P. So now this is a foreigner we can
162:24 with. This is simple enough for to uh to understand. Uh And
162:29 is in fact the standard way of at at Avio amplitudes in terms of
162:36 simplified expression for um reflecting. But other ways to do it. For
162:45 , you might be you might uh somewhere the reflectivity grading is given in
162:54 of physical properties like this, where is now the jump in Parsons ratio
163:00 Parsons ratio is non dimensional. So don't have to divide by average Parsons
163:07 in parcels ratio followed by this complicated here involving junction Parsons ratio and jumps
163:14 density. It also involves the average ratio here here. So now this
163:23 uh quite simple if we have average ratio is 1/3. All the thing
163:30 away. So the second part of is right. This is quite
163:34 but that's commonly not true in all of rocks. For example, if
163:39 velocity ratio of the average velocity ratio a number like three. And this
163:44 typical of shallow marine sediments. Yesterday talked uh Stephanie's elected number two for
163:51 . You can uh again, uh that's again for yourself, what happens
164:01 the velocity ratio is two? What you get gonna get for the approximate
164:06 ? It's going to be different from but it's not gonna be one
164:10 This is a more typical number for marine sediments. And in that case
164:15 expression this term does not go away the results of this, which is
164:23 um find this term with this And you get this which is arguably
164:34 no more simple. And the straight uh statement from right here, we're
164:43 talk about it in these terms. I want to show you one more
164:47 that you might see in lecture. might see the great intern given in
164:52 of the jumps in the LeMay I'm going to claim that again,
164:58 is more complicated than the first phone lectures. So now using that um
165:10 Previous expressions for intercept great encouragement, can find these um jobs directly here
165:21 jump in VP jumping, rope jumping . S. In terms of the
165:27 and the intercept and the grading We use these these terms here to
165:38 this. And in some cases you , well this is what we really
165:44 . You really want these jobs in parts. So it looks easy,
165:49 it's difficult in practice because of the here are those same three relationships.
165:56 this is the one we probably want know most the jump in VP but
166:00 curvature is usually the worst determined of of the Avio activist, usually this
166:09 current return is very poorly determined. that's bad what we want. It
166:15 upon what we, what we have confidence. Best determined uh of these
166:23 parameters is innocent and it's frequently not ours, but usually impossible to
166:31 the grating. Yeah, be absolutely honest with you. We don't really
166:42 these three observables. And why is ? It's because the seismic trace has
166:47 unknown scale factor. Usually somebody before ever see the day that some traces
166:54 have been re scaled to lie somewhere plus or -1000. So if you're
167:01 on your workstation and you're looking at seismic session, you put your mouth
167:05 not a seismic section on seismic You put your mouse over any
167:10 And uh it's gonna come back with number probably between plus or minus
167:16 But reflect activities usually are much Four orders of magnitude smaller. So
167:23 we can determine our certain ratios such the ratio of the gradient to the
167:28 . In that case the unknown scale cancels out. So this is uh
167:34 is the number that we can And with great uncertainty, we can
167:42 the non dimensional uh curvature. But of the uncertainties, you rarely see
167:49 analyzing this. So we usually have one reliable seismic amplitude attribute which is
168:07 relative value of the intercept. Before we should go any further or
168:16 should recognize that that we don't really to find physical properties. What we
168:21 want to find is highly. So not get um let's not get overly
168:31 . Um Finding these property jumps what we want to find is
168:37 And so we know that hydrocarbons are common in the subsurface, there are
168:42 . So let's use a V. . To find these anomalies. So
168:51 consider only the intercept and ingredients. here's the intercept the agreement and we're
168:55 do it for these two occasions. sedimentary interfaces where there's brian on both
169:02 of the interface. That's the normal . That's going to be happening in
169:07 all reflect that you see are gonna reflections off of brian brian interface.
169:14 some of them are going to have carton on one side of the
169:18 Let's look at the uh special case there's gas on the other side because
169:23 is really a lot different from brian the physics department. And so normal
169:32 interface, here's our two expressions for ingredients and based on laboratory data art
169:43 a bunch of rocks and measure their . P and V. S and
169:48 the in their density and reduce the modules. And normally and then pick
169:53 of those at random. And normally gonna find that this uh this difference
170:00 here, it's so big that the term dominates the first. So that
170:06 that the ingredient has the same algebraic as the second term and that means
170:13 has the opposite uh algebraic sign on intercept because of this minus sign.
170:21 reading has algebra sign opposite to that's the normal text. But for
170:30 grind gas energies Looking at the same equations, this term is gonna be
170:36 and we're going to learn about that from in lesson eight. You see
170:43 again, if it's the same mythology both sides of the interface, um
170:50 this term is gonna consider dominating. term is gonna be zero. So
170:54 algebraic sign of the ingredient is gonna the same as this, which is
171:00 be the same as this. So such interfaces the algebraic sign has the
171:08 , it is the same for the and the interest and the real world
171:14 be complicated. For example, there be a little ology difference as well
171:18 a different across property. Yeah. . Well let me show you uh
171:35 software with a particular example. This software from BP. And this is
171:41 this is not Bp secrets. It's published uh and uh my friend did
171:48 analysis and what what this is a D. Analysis. And so what
171:53 see here is uh two D. section so that it's an image and
172:01 never mind how they made an But you know that behind every one
172:05 these image points there lies a gathered midpoint gather. Common image point gather
172:12 this is all pretty flat line. these common midpoint gathers, uh pretty
172:18 approximations to common image gather and um that's not showing you what you see
172:29 the result um some flattening those gatherers summing them up to make uh point
172:38 the image. Remember that every point this image has behind it again.
172:44 so when the interpreter first brings this , you don't see any colors here
172:48 this is blank. And all you is is the uh oh is the
172:58 ? And let's look over here and what we have. This is a
173:01 plot where we have the uh the and the great on this cross
173:07 And when he first brings this up all black. So then what he
173:11 does is he notices, wow, at this attitude anomaly right here in
173:15 middle. Uh is that gonna is that due to hydrocarbons or is
173:22 due to something else? Some length anomaly in the service? We're gonna
173:28 to find that out. So the thing he does is he draws uh
173:32 box around the anomaly. And as as he completes the yellow box
173:38 every point interior to the yellow box lights up over here, all these
173:44 are yellow and these points here on cross line, give that intercept.
173:51 the gradient lying behind every one of points inside the yellow box and you
173:57 see it's a diffuse cloud of yellow . So the next thing the interpreter
174:10 is he draws an interior green doesn't matter what the shape is,
174:16 according to his expert, um there are no hydrocarbons inside here.
174:23 all of these, all of these here are simply brian brian interfaces caused
174:30 um uh with a logic contrast across boundaries. And he could have gone
174:36 also, he just chose this and soon as he completes this yellow
174:40 this green box, some of these here turned green. These are
174:46 these are not still yellow and but got some green points here and notice
174:51 the green points form a tight cloud the middle of this diffuse cloud and
174:58 has this orientation, it's an elongated uh in the Yeah, second quadrant
175:09 the fourth part. Remember how, you used to name these in
175:13 So this is this is the first or both of these things are positive
175:18 and by the way, why not is zero, this is the
175:21 So all these are positive here. are positive here. Over here,
175:26 intercept is uh negative over here both negative and over here the ingredient is
175:34 . So this is order 123 and and literal. Odjick um plowed always
175:42 oriented like some and then he notices the interpreter notices that uh he's got
175:51 bunch of yellow points outside here and here. Uh and these are these
175:58 contain hydrocarbons. It guarantees that these points have no hardcovers. And hear
176:07 often this has a negative slope to tight elongated clown. Now he uh
176:16 works so now what he does is he's gonna select on the cross plan
176:23 points here. And as soon as completes, as soon as he selects
176:28 on the cross plot, the corresponding show up over here. Now,
176:34 is the important part because if these these uh these were these points were
176:40 . Now he's uh selected them. now they're purple. If those things
176:45 from all over here, it would be noise. But you see,
176:49 don't come from all over there, from the top of the structure.
176:54 there's a pattern there. And that's and he immediately suspects that these points
177:06 the top of the structure um correspond anomalous fluids, i. E.
177:17 . And how do we know they're ? We don't know that until we
177:21 at the cross plant. And so have a vo characteristics uh with negative
177:32 , intercept and negative uh negative intercept negative grading. Yeah, he was
177:43 by a simple minded theory that says these terms are all in the third
177:48 should be um uh I think it's a Class three anomaly. Not
177:54 Somebody made a classification, but look , we got uh these are probably
177:59 same anomaly here. Uh Nature doesn't like classifications. It's uh has smooth
178:06 and so these are probably the same . And this is probably the
178:11 the bottom of the anomaly. It's those, but these are more interesting
178:15 the bottom is flat and the top uh structure. So um by the
178:26 , I don't know whether this is time series or gets here, I
178:32 know that uh similar results or time . That's So what we've done is
178:44 found that the anomalous points in the plot correspond to the top of the
178:49 in real space. So we've been by and by the way, this
178:54 exactly what we uh decided back Uh we took this miniaturized expression for
179:07 reflectivity um or efficient. And then applied reasoning to those reflectivity curves and
179:17 found out that if there's a if comes from a a brine gas
179:23 yeah, yeah, reflectivity should have same physical sign. Uh So the
179:31 and the intersect and that's what we found right here. Same algebraic sign
179:38 the intercept and the gradient. And analyzing the reflectivity, that's what we
179:45 . But this is data. right. Who knows what um uh
179:54 other effects might be here. These received amplitudes. These are not
179:59 These are if you look at every of these uh it's a big
180:05 See the intercept here as dimensions that up to 5000 here and -5000
180:12 So these are big numbers. It be uh reflectivity is, this is
180:19 data but guided by the period for ease. We have uh been able
180:26 empirically locate on harmless shows in the . So this is uh not a
180:34 conclusion but it does substantially lower the . So we call this the qualitative
180:41 your program often works even though there many cases which uh don't correspond to
180:49 kind of analysis and those always puzzled . Uh so let me pause at
180:58 point and tell you the following Um um so this analysis I gave
181:09 was from D. P. Uh was uh those grassroots published and the
181:18 part of the 21st century after repeat America. Well um uh they come
181:30 from chemical analysis. Sin was Emma heritage guy. I was an and
181:38 fact I am the the inventor area . So that's an interesting story which
181:49 will now tell you. So I Amoco in um she in 1980,
182:01 when I joined aven and I joined in the rock physics department in the
182:07 physics section of the geophysics department of research center in Tulsa. So the
182:16 department was in the Houston and new and Denver and uh and regional offices
182:24 . Research center was impulse. So joined um chemical at a time when
182:35 uh well business was moving and um there are lots of small oil companies
182:45 are hiring hiring experienced people away from major oil companies by offering them larger
182:55 often including royalty. So if you're experienced guy working for Amoco you get
182:59 offer from uh from a small company and they say well we're going to
183:04 your salary by 50%. And if you make successful recommendations on where
183:10 are grill you're gonna get like a of the disk of that discovery.
183:18 that can, that can be pretty money. You could get rich by
183:24 a good geophysicist. Whereas uh continue work for a big work on being
183:30 hammock, oh you're gonna be comfortable you're not gonna be rich. So
183:36 were hiring away the experienced people and the major companies were replacing the experienced
183:43 with new hires and so they would recruiters to the universities like and all
183:49 universities in the, in the oil , which is what we call the
183:54 , Oklahoma and lucy and museum. those universities uh getting regular visits from
184:03 recruiters from the major oil companies and up the graduating seniors as fast as
184:09 could. That's right. And that enough. So they started expanding their
184:17 activities eventually they got as far north new york where I was teaching in
184:21 state University of new york and I teaching of course uh curiosity and uh
184:31 birth of deep interior of the earth earthquakes. So that's what I was
184:38 . And also the rock properties of very deep rocks. They got up
184:45 my university and they liked the quality the graduates that they were producing and
184:50 they would hire some and then next they would come back with those same
184:57 new hires in tow now with one experience and they would make friendly greetings
185:03 all their friends who are not yet . And and they would give a
185:10 showing all the exciting stuff they've been at the oil company in the previous
185:17 . And I'm sitting there in the thinking, you know, uh why
185:22 these kids have all the fun? I could do that. So
185:26 when the recruiter uh I said okay who wants to join us? Uh
185:33 year I held up my hand and they said, oh well we hadn't
185:38 about how in faculty, but let's . And so we did talk and
185:42 I got an offer to join Amoco's Center in Tulsa, roughly double my
185:50 salary. So uh my wife and had to decide whether we wanted to
185:57 this or not at this time. was also working as a staff member
186:01 the university and she was actually much famous at that time than I
186:07 She was uh she was the premed premed adviser and she was exceptional.
186:16 was amazing at getting and graduating premed into medical school and she was famous
186:26 over the Northeast for her success. in fact she was the President elect
186:32 the Society of Premed advisers there for Northeast. And the university was so
186:38 of her work. They were advertising the high school seniors come to come
186:44 new york, we will get you medical school, you don't have to
186:49 your family's money at a private school Columbia. And that was really working
186:55 that was the only accepted in those the top 7% of the students from
187:01 high schools in New York. So you had to be a straight a
187:04 to get into the State University State University of New York at
187:11 It was all the public I've been school were good and those are the
187:17 of graduates we were producing for the companies from the geology department. Now
187:23 had this offer to join um ethically Tulsa or not. And it was
187:30 problem because she was not famous. wife was not famous in Oklahoma.
187:35 was famous in the Northeast and I know anything at all about the oil
187:41 . What I knew about was deep furthermore, I had a tenured
187:48 I had a lifetime job um at in New York. And what America
187:55 offering me was a 30 day But we decided that we would give
187:59 a try. And so we went Tulsa and um um one of the
188:06 things that happened one we had an office in New Orleans and then and
188:19 we were exploring in the gulf of . Uh we're partners with mobile on
188:30 certain play in the gulf of So you've got to understand that an
188:34 company, it has various temporary partnerships various other oil companies uh with respect
188:42 certain specific um prospects. So at time a major oil company like chemical
188:51 be involved in a dozen or two partnerships with other oil companies. Each
188:58 focused on a certain area. In case we were focused on a certain
189:03 in the gulf of Mexico. Uh uh only partner was mobile and in
189:11 partnership like this, one of the is designated as operating partner. And
189:16 others are support partners. So we the only support partner in this hall
189:23 . They were the operating and so a certain required some data and in
189:30 days oil companies did most of the in house. So at a certain
189:35 in the processing mobile guy calls up Amoco partner says, hey ralph come
189:43 over. I want you to look our our processing on this prospect.
189:52 the Amoco guy, they set a and time the american guy walks down
189:56 street in new Orleans and they reach other. You know their friends,
190:00 been colleagues for a long time, have a cup of coffee, they
190:03 some gossip. And then the mobile rolls out on the table uh two
190:09 . Cross section uh printed on paper those days and probably in color color
190:19 advance party from black and white to . And he rolls it out and
190:24 says this is where we are so in the process. And um uh
190:33 look at it and uh this is the first time they've seen an image
190:39 this. Uh This is a better they've seen before because processing department has
190:44 working and refining velocities and so So they have a pretty good
190:49 And then the the beef. Then mobile guy takes a second uh paper
190:57 it out on top of the first is this is the Monet diagram for
191:01 Monet plot showing the shear wave problem it's got a lot of different colors
191:08 it. But the Amoco guy's jaw because he knows this is a marine
191:13 , they are not receiving any share arrivals. And so he's amazed that
191:19 has this uh shear wave section. guy sees on the Amoco face surprised
191:27 he realizes that by mistake he has some proprietary mobile technology which he should
191:35 have shown to this amicable. They the operators, they have no obligation
191:41 show the results of proprietary systems. this? So he makes a
191:50 He says, oh, this must something else. He rolls it out
191:53 puts it away, must be some of mistake. And they go on
191:56 the meeting following the standard agenda. the MMO guy knows that he has
192:03 something that's dynamite. So as soon he gets out of the meeting,
192:06 runs back to the office, tells boss, those guys at mobile,
192:11 know how to uh huh they know to get sharing properties in a marine
192:19 , wow. So uh, that's in the day and so 8:00 the
192:24 morning, my boss has fallen up Tulsa brains and the project gets assigned
192:31 me because I'm new in the Everybody else has their plateful of other
192:36 . Uh, and they were supposed make progress on those projects. My
192:41 is not yet full because I'm new the house. That comes to
192:44 So very quickly I uh, figured , I think one of the things
192:49 I did was I went to high richards and uh, it's right there
192:53 , spelled out in terms which are too different than what I've showed here
192:59 without the hydrocarbon, um, which we talked about just half an
193:09 because arcane richards were deep Earth curiosity driven, not utility driven.
193:16 so they didn't know or care much the oil business but they had most
193:20 the analysis that I showed you So I very quickly I realized the
193:27 argument that I just showed you in last um half pounds. I didn't
193:35 the software example of course much months and we didn't even have the word
193:42 . B. O. In those we had a different name for this
193:47 inside. Mm but we realized that we uh analyze our amplitudes as a
193:56 offset, maybe it gives a clue to what's down there. In
194:02 so we can do better than making image. We can tell you something
194:06 the physical characteristics of what's down there particular. Maybe we can tell you
194:11 or not hydra apartments are really there you see an anti client in the
194:16 service image, you know that um hydro counts might be there. That's
194:24 detection of hydrocarbons. This was potentially detection of amplitudes by looking at
194:35 direct detection of heart departments by looking the amplitudes as they vary with officer
194:42 dynamite. So immediately what Amoco did they scrambled the team of experts and
194:47 went down to new Orleans and they an office off site from the
194:53 they're in a warehouse somewhere. They close and they got a bunch of
195:01 and medical data for they had previously required in the gulf of Mexico and
195:06 had drilling results from those episodes and were successes and some are not.
195:13 so now we look at the data lay behind those drilling operations and we
195:21 that that uh where the drilling had discovery of harder conference, the corresponding
195:30 data had what we now call positive in it. We hadn't known previously
195:36 look for it. Now we did we saw immediately that if we only
195:42 wells which have positive Avio will have better success, It will be 100%
195:48 it's much better success rate. So , this uh analysis with a secret
195:58 offsite in your on that was accomplished just a few weeks and we realized
196:05 we now have a way to do action and higher departments before drilling from
196:11 data. Not foolproof but really useful . And so immediately then the project
196:22 in Tulsa was taken away from me I was in new hire and uh
196:28 important new development like this couldn't possibly left in the hands of a new
196:34 . So they gave to one of old guys, cancel him out of
196:40 previous projects, gave him this So, um as a result,
196:46 didn't make any more progress in the for the next 30 years. There
196:54 a problem which I saw even in days, which is clearly indicated in
197:01 example that we're looking at there is problem. Um Let's look at bubble
197:09 trend here, We have a tight and we have no hydrocarbons anywhere inside
197:15 green box, but the slope is steep. Something is wrong.
197:22 remember that. This is received amplitudes . This is got in it.
197:27 of the effects of propagation and everything in there in these amplitudes. And
197:33 what we're analyzing. And we're calling uh reflectivity gradient and intercept, but
197:39 not, it's received aptitude Grady and the slope in the figure is about
197:47 -5. You look here, you see it's about the slope of -5
197:53 it goes down five units for every unit. It goes silent For laboratory
197:59 . We expect about -1. So wrong? Well, and the reason
198:10 wrong is because we've neglected all those effects on amplitude coming from other effects
198:19 um propagation effects. And we ignored that. We applied our reasoning for
198:27 , not to reflectivity data but to amplitudes. So there's a magic way
198:34 correct for all that to what I power point process. So I'm gonna
198:40 this file this image right here and gonna grab it with my mouse right
198:46 and press everything further. So that's I've done here. So now now
198:53 are compressed compressed vertically. And look the little logic trend has about my
199:00 . So how about that? But all the uh anomalous points are still
199:06 and they still collect over here at top of the structure. So we
199:12 for all of our many mistakes and pathology trend with a correct slope just
199:18 compressing damage. And the anomaly is anomalous. Amazing. So uh uh
199:27 we we found here is a way uh look at a non lowest places
199:33 the earth in terms of amplitude And it successfully defines fluid anonymous by
199:48 . Even though we've made serious We didn't correct for propagation effects very
199:56 . Maybe not at all. And we neglected an eye socket. Now
200:01 about this. We're analyzing amplitude variation a function of offset. It means
200:08 variation as a function of angle, we're assuming that the velocity does not
200:13 everything. We know that in the of ours. He does very
200:18 That is uh um local anisotropy. shouldn't we really be analyzing the reflectivity
200:29 anti psychotropic instead of the tropics. , and they took the project away
200:39 me in the same week. when as they were uh probably finding
200:49 Avio really does conventional Ariel and we have today still today. And basically
200:55 same way as we did then it does reduce risk, But at the
201:02 in the same week when we came that conclusion for certain and and Amoco
201:08 1980, or 81 maybe before you were born in, we understood
201:17 Uh and uh but I thought in same week, I ask myself these
201:23 . Can we do better if we if we don't neglect and I
201:28 we aspire to doing better if we to avoid these mistakes, the answer
201:32 yes. And we'll talk we'll talk that election. Now, let's see
201:40 , I'm probably good. So the as we talked about is the property
201:49 the interface. You know, it's those deltas ease and delta V's and
201:53 muse in there. It's not a of either medium. It depends on
201:58 different differences. So many people in profession find it easier to consider the
202:05 properties themselves rather than these interface So that the layer properties can be
202:11 from the inner properties via a process we call seismic inversion. Now,
202:16 a bit of a fancy term that does not the same as mathematical
202:21 but that's what we call it. we end up with layer properties,
202:26 inner property. So we don't say doing this, we don't say that
202:31 uh this year marvelous jumps across that . We say the share market has
202:40 value in the upper layer and then other value in the lower. So
202:45 a computation, which has its own . And that's all I'm gonna say
202:51 that, except that you will learn about that inversion. And of course
202:57 A B L. I think you that in this sequence here, You
203:01 have somebody talking about a B Um in an entire force you'll learn
203:08 things like that. But everything you here will apply to those computed layer
203:14 . So let's have a quick quiz and then uh have a break.
203:23 in a common reflection point, common midpoint gather conversion to offsets two
203:29 requires an accurate estimate of velocity, above and below reflecting interface.
203:35 is that true or false? Huh didn't hear you, What do you
203:40 ? True. Well now let me you, why do we need to
203:44 the velocities below the reflecting interface for purpose? We're converting offense to
203:51 So why do we need the uh velocities below to make this conversion?
204:05 it something to do with smells? , I know it's something to do
204:09 the fact that this conversion here all the angles uh involves the rays that
204:16 down and back up and they never below. So we don't know uh
204:22 this computation, converting our sense we do not need to know the
204:27 below the reflecting interface. In fact gonna learn opens when we do the
204:32 Avio analysis but for converting assist we don't need to know this.
204:38 the answer to this one is So this I would say this is
204:41 a trick question but it's a question does uh require you to read it
204:48 and think it through and decide whether of these points which are mentioned
204:55 this conversion, this accurate estimate above below. Are those all necessary to
205:02 this? And the answer is false you do not need to know the
205:06 below in order to convert process. next question, if your analysis is
205:16 done on migrated pre stacked gatherers to or maybe, well, no,
205:27 gonna go with maybe because when we the migration, we got a better
205:31 , but who knows what we did the amplitude. So uh, answer
205:36 question about what did we do to amplitudes in the migration? You gotta
205:43 to your local experts and he will you whether or not he has confidence
205:47 those migrated applications. And if he thought about it. And then you
205:53 , well, I'm not even gonna at this until you give me a
205:55 answer than that. And so then week or two or three later,
205:59 gonna come back and he said, , um, uh, I think
206:04 attitudes are pretty good in the after . So you can go ahead and
206:09 at pre stank migrated priest and gather you might face something bad. He
206:15 , oh my God, I didn't what I was doing to the
206:18 So the best answer for this is because it depends upon this other
206:26 doing his job well. And you want to just assume he did his
206:31 . Well, you want to talk him about it and say,
206:34 ralph, when you do these what did you do to the attitude
206:39 you're given to? Okay, number in this little rise expression, the
206:49 wave properties appear where intercept graded both term. All is it both of
207:01 ? Well, okay, so we saw it explicitly in the
207:06 We saw the jump in mu explicitly the grading. Now tell me where
207:11 saw uh in the innocent the intercept you remember is the one we actually
207:17 ourselves. That's the delta Z over term. And yeah, so you
207:25 say, well, the busy includes VP and the VP includes K plus
207:30 thirds mu but that's not fair because , that four thirds mu always appears
207:37 combination with K. So we call , give it a name and we
207:41 it M. And so I'm going say that there are no shear wave
207:45 inside the intercept, but only in gradient. And then uh in the
207:50 term, the same logic appears to , the answer, oops, the
207:55 is b okay, before The grading , the R two term, maybe
208:03 in terms of portions ratio. Although expression is either, you know,
208:09 any additional approximation. Although this expression either more complicated or contains further assumptions
208:17 the former looked at. So the to that is uh true. That's
208:22 true statement. And you can go to the previous slides too. So
208:31 easy to estimate. Uh I don't this word, that's easy to estimate
208:38 sheer property jumps at the interface by simple algebraic combination of Avio gradient
208:44 Is that true or false? I like nothing in this lecture was
208:49 So like on the. Okay. . So I gave you actually a
208:55 but the formula, the formula and the curvature. And so the curvature
209:01 never well determined. So the answer looking for here is false. Well
209:07 , that's what I was thinking. was like nothing looks good because it
209:11 easy With a simple combination. But fact the 90s because what you rely
209:17 is the curvature, which is Next question. The relationship between Avio
209:25 grade. It depends among other things whether or not the food cut that
209:30 across the interface. Is that Yeah, that one is true.
209:37 that is the basic answer. That's basic reason why a vo is so
209:42 in our business that this gradient the changes and the inter ship changes
209:49 But uh the relationship between the interception changes. If there's a fluid content
209:56 , then you're likely to have the to have the same sign as the
210:01 the intercept. That is if it's then with offset it gets more positive
210:08 , that's more negative, and that's we call positive, a video signal
210:14 a food content change and it doesn't . Uh 100% of the time that
210:21 happens so commonly. That is a good way to reduce the risk,
210:29 Oh, really? Okay, so is a good place for us to
210:38 . Yeah, it's 12 30 so stop right here and pick up here
210:42 1 30 we'll go through to 5 . And I noticed that we didn't
210:47 any rain yet, but I think rain is still coming, so I'm
210:51 everybody's safe at home, so I'm uh stop sharing here.
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