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GEOL7333-Seismic Wave and Ray Theory - Day2 08-27-2022-Part3
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00:03 Okay, so we're gonna look at term of the previous solution and I
00:07 you that the phase right here is in this way. Now when you
00:13 a surface of constant phase that's called from. And when you look at
00:19 great end of the phase that points along the rate, so the greatest
00:24 the phases given by minus K So we should call K to be
00:29 gray vector instead of the wave Um wow, that's not the
00:36 It's always called the wave vector. it in points in the direction of
00:42 ray. That's sort of the definition the right. It's the gradient of
00:52 . Most of our intuition about our is based on race theory. We
00:58 some of the pictures like this from rays don't really exist. I don't
01:06 you've ever seen them. A source generates a wave of um finite
01:21 Even dynamite generates a wave look, duration. And the synthesis of many
01:28 wave look terms, it looks like . This is exactly what we show
01:33 . For radiating, radiating. Yeah, great theory is the high
01:42 approximation to the to wave theory. this is how we define it,
01:53 a term like this and separate it a wave look and an amplitude and
01:59 amplitude very slowly it goes uh very as for example, as geometrical spreading
02:08 . Then the aptitude slowly decreases It varies rapidly up and down up
02:16 down and the arrival time varies So those we're going to consider then
02:24 limiting case of wave of wave equation for high frequency and what we're gonna
02:38 is an equation for the behavior of wavelength, ignoring these other terms.
02:46 , so let's take the wave equation and accepted the source point. The
02:53 equation looks like this away from the point. We got zero for the
02:58 the uh right hand side here. let's put in there the assumption that
03:05 pressure is equal to an amplitude times . But so uh Number two times
03:12 right in here. Same thing over using changeable calculus, what we find
03:19 this this term separates into this uh times laplace. Um operating on the
03:26 black gradient. Eight times gradient uh lit. And then uh wave of
03:34 a plus, you know of We're going to assume that the amplitude
03:40 slowly in the waiver. So that that this term and this term are
03:45 zero compared to this term. So is a wave equation for the
03:56 Now remembering that the wave uh function remembering the wave, it depends upon
04:04 minus the arrival time time within the minus the arrival time, which is
04:11 call that the arrival time of of peak of the wave. So um
04:18 our wave equation involving w only and use chain will calculus to uh to
04:27 that coming from From this term See this term is the same as
04:35 one Uh Coming from the laplace in we get after applying general calculus we
04:44 out laplace in uh the arrival And then in the other term we
04:51 gradient arrival arrival time dotted with gradient arrival time. We're going to assume
04:58 this is zero because uh wave look rapidly since its very rapidly this derivative
05:08 smaller than this derivative. So we're neglect this term. And so uh
05:15 leaves these two terms here. You how we canceled out uh the second
05:21 of w with respect to time cancel it out here, cancel it
05:25 here. And so this is uh left we call this the O'connell
05:33 I connell that's um um a german . That being shot I think.
05:42 what I mean. Sure. But but uh the name of this equation
05:49 the icon of equation. And the to think about it is it's the
05:53 theory approximation that is the high frequency to the wave equation. And it
05:59 you how the arrival time changes in medium with velocity V. P.
06:13 time, the Iraq time is a of uh of uh position. And
06:22 this uh this argument I think I've position only as a function of one
06:28 act. So we want to to that time as a function of acts
06:35 replace that in the uh we want modify perturb common equation to get an
06:46 for the ray in space. So some um mathematical slides that followed here
06:55 which in my view are a little um hard to follow and also maybe
07:04 important to know. So I'm gonna over those if you're interested in that
07:09 can follow this along. And I'm to uh from those manipulations. We
07:33 skipped over. We derived from the equation. This equation for the re
07:42 which is given by this uppercase X as a function of position of.
07:50 This S. Is uh measured along rates distance measured along the rain.
07:57 is an equation for the raid So to understand what this means,
08:03 consider a case where the velocity is . And so uh it's our
08:10 So the velocity is constant. This hand side is zero. So that's
08:15 it says here, right? So says this this thing is zero which
08:20 that oh and also this is So we can take this out of
08:27 the distributive. And it says that uh the curvature of this line is
08:35 . So as you as you move the ray uh increasing the uh quantity
08:48 . Which measures your position along the , that curvature is zero. So
08:53 says that uh solution X. Factor a straight line. So let's consider
09:03 special case where velocity is only a of uh one of her that so
09:12 the uh that this specialization. Um , the rape a thick equation looks
09:20 this. So we're differentiating only with to Z. One over V.
09:27 you know, I think that the owner velocity is called the slowness.
09:31 this video of the slowness with respect Z. And it's pointing. It's
09:36 vector pointing in the Z. Here's a little trick which uh I
09:45 from a textbook, you take the product with Z vector. And so
09:51 the right hand side, uh it Z cross Z. And this is
09:55 zero. So whenever you cross the with itself, that's a zero.
10:01 uh over on the left hand side we're getting the cross product on the
10:08 here of the derivative. And so the result is this derivative with respect
10:13 uh s is equal to zero. this here is a constant. Let's
10:25 that counselor name. And uh I call that the slowness vector.
10:32 Among the slides we've passed over is definition of a slowness vector. And
10:40 I'll show you in a minute how uh slowness vector comes into the equations
10:47 are familiar with. And so the result says that uh the cross product
10:54 Z and slowness vector zero uh And uh sorry, this Z cross
11:18 is gives you the X component of p vector here. And so we
11:30 that the best name for that. called the horizontal Sloan inspector. The
11:35 Sloan is component but it's often called ray parameter. And you probably would
11:42 seen that note that those words So then uh the very path equation
11:48 this one D. Case says that horizontal gray parameter is constant along the
11:55 in terms with with which you're more . This is a statement of smells
12:02 which is valid for one day So uh let me just uh show
12:08 here that here we have a curving and as it curves along its angle
12:14 with the vertical is always data. so the horizontal component of the slowness
12:21 is one over V. P. we call the slowness at the length
12:25 the slowness vector times the sine of angle. And if velocity is a
12:31 only of depth, this thing is . So this is what I think
12:40 probably are already familiar with this Um Let's have a layered um sequence
12:51 this layered cross and let's assume that velocity increases with depth that's normal.
12:57 so in order for this thing to concept as this increases, this has
13:02 got to increase which means that data getting bigger and bigger like this.
13:07 that this is what causes raised to curve uh suckers toward the surface.
13:17 then as um they come back the so that it's a symmetrical ray path
13:25 this case and notice that the great remains unchanged despite the reflection.
13:33 as we went through that derivation, never assumed that there were no
13:38 So the horizontal component of the vector is always the same fire path going
13:50 . Now. It's uh sometimes one of those layers is not faster
14:01 that uh than the hirelings. It happen that we have a slow layer
14:09 be due to a soft pathology, be due to overpressure, maybe due
14:14 gas in the pore space. Who why it's small, but in that
14:19 the bends downwards and then uh comes out of this soft layer again,
14:28 bends towards back towards the surface. of course it's Yeah. Um in
14:40 one, which we didn't talk you might have seen this picture,
14:47 might remember this picture. This is of the few places in this course
14:52 is focused on exploration. Seismic, we talk about the whole earth.
14:57 let's look at this and see how have an earthquake epicenter and raise going
15:03 in all directions. And uh so have some color coded things here.
15:08 so you don't see the earth's crust this figure, it's too thin.
15:15 uh So you see there's mantle and and inner core. And so these
15:22 are all many upwards, like I before. Um because the deeper rocks
15:32 faster velocities raised, bend upwards. here's the Earth's outer core and the
15:40 core who me um divert uh for second. So you another personal
15:52 the inner core of the earth was in the 19 thirties, not so
15:58 ago, only 80 years ago by danish geophysicist who was a woman and
16:06 name was anna Laymon. And so of course she was underappreciated by her
16:12 colleagues being a woman and all, she was extremely smart and she looked
16:17 the data carefully and she was the who discovered the inner court from primitive
16:23 data, with only a few seismometers around the earth. And uh you
16:30 sources uh whenever they wanted to happen they wanted to have a crummy data
16:37 she discovered the earth's inner core under conditions when she was a young
16:43 Well, I actually met her for one generation removed from these giants of
16:51 . And it was the same conference I met Sir Harold Jeffreys. It
16:56 a conference held in Copenhagen and by time she was uh long since retired
17:03 in a retirement home comfortable. And she came to the conference and I
17:11 a, I met her and she invited us my wife and I and
17:19 invited us to her home for wow ! So that was quite an
17:24 . Uh and uh so when I back on um uh we do
17:32 what we do today is almost trivial to what people like her did back
17:36 the day. And of course she fighting against not only bad data but
17:42 um um prejudice from all her male . So it was amazing what she
17:48 . And so that was one of highlights of my youth to meet the
17:52 in. Now in this figure the razor p waves and the red waves
17:59 S waves. And you see that all go down and curve back
18:03 Except that right here in the uh the core there are there are no
18:12 waves. Why is that? Because core is liquid And uh uh so
18:21 doesn't transmit uh share waves. So the share waves are vanishing for takeoff
18:29 bigger than this. Now, as as it says here, these
18:37 curve up since deeper rocks have higher and they all of course follow Snell's
18:42 since the outer core is liquid. The shear waves don't propagate there.
18:49 Now the velocity in here is uh than the velocity here, because right
18:56 we have uh used view is positive here you are zero. And so
19:05 in the core k. Is but it's not enough to make up
19:08 the zero value of view. So velocity here is less. So the
19:13 following Snell's Law, the rays following law, bend down and they don't
19:22 out over here. So there's there's an area here that doesn't receive any
19:27 at all. Neither P nor s this earthquake in this place. So
19:33 is called The Shadows. So uh a consequence of Snell's Law. Now
19:47 look right here. Um That's the car and you can see that in
19:55 cartoon the rays bend down just like did here. So that implies that
20:01 the velocities should be less in the core than in the outer core.
20:08 that's not true. The inner core very similar composition to the outer
20:13 And the only thing is it's frozen so it has a share models.
20:18 these philosophies here should be bigger and so that the refraction should be
20:26 not downwards to set of reflecting it should be reflecting hours. That's
20:35 simply an error by the guy who the cartoon. But it is true
20:42 once they get deeper into the they uh refract outwards because the
20:48 it continues to get the deeper Yeah, you will respect the time
21:07 in exploration seismic. Uh frequently we to calculate gray pants, we're going
21:17 use the equation that I just The re path equation I skipped over
21:21 derivation, but you saw the result it. Um So uh in order
21:29 calculate array of course you've got to or estimate what is the velocity.
21:34 there are three methods that they use called shooting method, exploding reflector method
21:41 source to receive a method. So differ in their boundary conditions and their
21:47 condition. So shooting is uh easy think of that intuitively. Uh simply
21:55 that equation with a set of initial . Initial conditions specified the initial
22:02 that is the shot point and the off angles. And so normally when
22:06 do this in the computer we do lots of them one after the other
22:13 the take off angles are usually like zero degrees one degree two degrees et
22:18 . And uh we want to do in polar angles for for two
22:23 And and also an asthma throw angles three days by the way, do
22:28 know what we mean by 2.5 Uh So uh you should know
22:36 So I suppose you're looking at um service model in two dimensions and somebody's
22:45 do some calculations. Uh recap calculation that model. And suppose it's like
22:56 it's got a bunch of sedimentary layers maybe it's got a salt body I
23:02 there's a solver. So the salt is intrinsically three dimensions. Well it's
23:11 common to calculate these and then in day approximation. So what that means
23:19 that the that the model is assumed be a two D. Model and
23:26 means it's the same in in and of the screen so that salt body
23:31 to infinity. Um In behind the and in front of the screen to
23:38 cross section through an infinite three Media. Well even so we're gonna
23:46 we're doing a 2.5 D calculations as rays go out, they spread out
23:51 three dimensions and the ones that are of the plane, uh we don't
23:57 them anymore, but the ones that in the plane are spreading according to
24:02 dimensional geometrical spreading. So 2.5 day uh three D. Model or the
24:09 D wave part. Now when you this, you never know where these
24:25 raids are gonna end up. If a simple model, uh there will
24:30 a simple distribution of arrival points. if it's a complicated model with assault
24:39 , for example, in the over , you never know whether those where
24:42 rays are gonna end up because of on the way down and on the
24:48 up through the irregularities of the Sure. Um when you do this
24:58 , you're gonna find places uh on surface which don't get uh illuminated by
25:06 race. And that reminds me of of an incident in my experience at
25:15 after what chemical we were actively exploring gulf of Mexico and the salt and
25:23 were getting crummy image. And uh we uh thought well the reason we're
25:32 creamy imaging below the salt bodies is we're using imaging algorithms which are which
25:41 approximations built into them. You'll learn both that such algorithms from Professor
25:51 So we improve the algorithms one x . We got rid of those
25:57 And of course that meant more computing . So we bought bigger computers,
26:02 computers and we were still getting from . So uh let's trace race.
26:13 make a model with the salt body the overburden and the reservoir underneath the
26:18 trash raised and see where they So uh we did that and we
26:23 out that the irregularities in the salt were scattered rays. So we had
26:30 poor illumination of of uh the target . So of course if you're not
26:38 uh any useful data back from uh parts of the acquisition, that's why
26:49 , that's why you're getting from So they said uh set ourselves in
26:56 to get good images. We need have better acquisition. We need to
27:01 wide azimuth acquisition. Huh. So uh lots of ways to have wide
27:13 in that position. So we defined few rays and said, okay,
27:17 we illuminate the target through this lens the overburden, which is the salt
27:25 lens, we have better options, opportunities to get an image if we're
27:32 from all directions. Seven, just as well. And so we figured
27:37 that's true, reasonably are getting poorer was because we have our azimuth acquisition
27:45 wide azimuth actresses get better images. the problem is the wide azimuth acquisition
27:52 more expensive than narrow as a You their azimuth acquisition At sea while
28:00 started off in the 60s with a streamer behind the boat and gradually over
28:06 next decades were able to run. , and an array of streamers maybe
28:14 kilometer why, But 10 km long towed behind the source book. And
28:22 way they do that is with spicy the water, which able the streamer
28:33 not directly behind the boat but off the side as well. So maybe
28:38 array of receivers maybe one kilometer 10 10 streamers in that array,
28:45 one is 10 kilometers long. So makes uh, so called narrow asthma
28:52 uh, mentions of the receiver are approximately, you know, one
28:57 11 by 10. So we need than that. So we need to
29:02 a second boat water. So we a procedure for having a second boat
29:12 off to the side, a second boat standing off to the side shooting
29:15 that same array from the side. then we tested this out the computer
29:21 Tracy Rice and figure out that that illumination of the target. And then
29:30 various perturbations of this. Um, through the ocean bottom. Seismic is
29:36 perturbation tested five different variations of wide acquisition geometry in the computer. They
29:45 work uh better than Nebraska and somewhere suitable for large fields, Some are
29:53 simple for small field etcetera. We 100 and $50 million on testing these
30:00 the field how they all work. so at that point we revealed it
30:05 the rest of the industry, what were doing and it caused a revolution
30:11 ministry very shortly all the service companies those services, all the companies find
30:18 because while the service companies were secret because more intensive acquisition, meaning more
30:27 coming into the service companies, all were happy to pay that because they
30:33 better images out of it. And they were able to see reservoirs underneath
30:43 , they could be part of So that's why that was the big
30:51 in the last two years. One the big events in the last few
30:54 , which led to my previous statement we have solved the problem of imaging
31:03 because of intensive acquisition like that, also more accurate imaging algorithms. So
31:15 started essentially a solved problem. underneath and that Now suppose you have
31:41 takeoff angle at uh say 15° and 16°. and both of those uh res
31:49 well separated after they go down and back up. Maybe you want to
31:54 , okay, let's go back and another uh shot uh shoot another rate
32:00 15.5 degrees. Maybe that'll come in to what we want. So that's
32:04 kind of argument that you that you to make. Uh So it's a
32:11 bit clumsy, I would say. another idea is called the exploding reflector
32:19 . And then here we have the wave equation. Same ray path
32:24 But the initial conditions don't specify takeoff at the sword at the surface,
32:31 instead it specifies the reflecting points and take off angles of the reflecting
32:36 usually at normal incidence, so that source and the receivers are in the
32:41 position. And so I think I some pictures of this. Yeah,
32:46 here we have a curve uh in D. We have a curved reflector
32:51 these are ray pads going off perpendicular the to the um to the surface
32:59 . And you see at a place this where it's curving down where the
33:05 where the surface is concave downwards. the uh the rays get spread out
33:13 where it's concave upwards. The rays together, they might even cost each
33:18 . Um So we'll have a lot energy here and a little energy
33:27 notice here as these waves pass through point here, they don't bother each
33:32 . They don't ricochet off each they just pass right. Sure,
33:43 the best way to do it is say I have my sources here and
33:46 have my receivers here and I wonder the reflection point was uh depending on
33:53 takeoff angle. And so uh so that case you solve the same very
34:00 equation with boundary conditions will specify the positions and the final position. This
34:06 a more difficult problem. And so why it's not done so frequently,
34:11 that's the the ideal solution right So I have some questions for
34:22 Stephanie, uh what's, what's the here? It's a limiting case of
34:28 theory. In which limit you're I don't hear you. Mhm.
34:51 That's what I'm worried. I'm worried you. I think I'm pitching this
34:56 your head because we spelled this out . It's the high frequency ler so
35:05 let's think about what we finish these and then we'll think about how to
35:10 the lecture. Uh The icon equation derived from the wave equation, assuming
35:19 same limiting case. True. And the scalar icon equation is you can
35:25 it up in your um uh in we uh just passed over and the
35:33 is C. So this uh true . And now you might remember this
35:48 is it true that the general solution the ray path equation is called Snell's
35:52 . No that's not true. That's for the case where the velocity varies
35:57 depth only. Do you remember how covered this? Is this one tour
36:08 France. This is another one of trick questions because uh if you have
36:25 um consider where the velocity varies with only do the race always bend
36:32 No, because some of those layers have slow velocities in the slow velocities
36:37 bend down. And that's true. or not it reflects a long
36:40 So that one's false. That's a question. Okay, so um I
36:46 maybe this is a good place for little break. Let's break and come
36:51 here at 3:30 and continue This. so the uh this next section is
37:00 about the equations and more about Let's take a quick break here.
37:06 back seven minutes. So let's talk move out. But you know what
37:15 out is uh um let's just look this cartoon with the uniform i psychotropic
37:26 service and a single layer here right right up and saw that with the
37:33 theorem. And you find that uh squared because T zero squared plus expert
37:40 B squared, of course T zero the vertical arrival time. And you
37:47 the depth doesn't occur any any anywhere here. That is because uh derived
37:56 way because we simply don't know the . Uh We don't know the velocity
38:01 , but we know that uh you the arrival times, you know,
38:05 offsets. And we're going to deduce velocity from this equation. Now,
38:11 an oversimplified this situation. Of This is uh more realistic where we
38:18 a layered Earth. And as I've it here? the layers are um
38:28 faster and faster as you go How do I know that? Because
38:32 ray is bending upwards meaning following Snell's that uh velocity is getting faster and
38:40 . And so the arrival time is of the one way of twice the
38:45 way arrival times uh twice the sum one way travel times. So four
38:53 . S. T. With the i is giving the travel time in
38:59 ice layer at the oblique angle. if it's got a subscript with zero
39:05 also that means it's the vertical Now what is the offset? The
39:11 is the sum of all these horizontal . So it's got in there an
39:17 signed data to uh make account for geometry. No back up here.
39:39 yeah this is better with presentation So now what we have is two
39:46 here involving things that we don't know , which is the slowness, uh
39:52 it knows about the slowness but we know about the slowness. We want
39:55 eliminate uh these uh slowness terms to the time as a function of
40:03 So here's the time, some of the sum of one way travel
40:11 Mhm. And then the one way times are given in terms of the
40:16 travel times by uh this coast data the in the um um in the
40:27 and see for yourself from previous figure that comes about. And so using
40:33 trigonometry, the Kassian is the square of 1 -2 Sine Square. Using
40:41 law. Uh that that's putting the in their uh explicitly and here's the
40:50 component of the slowness vector explosively in . And in a similar way we
40:59 the offset in terms of uh same . And look here out in front
41:06 it, it's got an additional factor uh horizontal stoner's. I'll go backwards
41:13 you see the time doesn't have anything that. So what that means is
41:21 for positive offsets, P is positive negative offsets, P is negative.
41:27 of course it's the same this part is the same positive or negative.
41:34 going to simplify this offset expression with taylor expansion. Uh And so implementing
41:41 uh find that this driven this is we need is given by this so
41:48 the time the offset is given in way, see it's linear. And
41:55 um and here in the horizontal slogans it's got in here something we call
42:03 rms velocity, what is that? defined right here where the excuse
42:09 the rms velocity is defined as two by t zero vertical travel time with
42:17 some this is the sum of all layers and what we have in
42:21 the velocity of each layer. And one way travel time, vertical travel
42:26 in each layer do the same thing the time. Uh Look here,
42:34 gonna say that the parameter here is square. Oh yeah, uh small
42:42 is a square of P. And so we work through the algebra
42:47 again we encounter the same quantity V M. S, uh the RMS
42:55 . So then uh again we come to this expression for the time in
43:02 same um in those same parameters. we can solve this equation here,
43:12 for Pete, here's uh expression for . In terms of X and
43:19 And putting that into the equation, find that that the time is given
43:25 this expression. And after all of work, it's not a good
43:36 it's uh and furthermore, this is the expression that you were expecting,
43:41 were expecting to find the hyperbolic equation it's got t squared on the left
43:47 and on the right side it's got zero squared and x squared or b
43:53 . But you I fooled you here I gave you the wrong definition,
44:00 um wrong approximation. And why is wrong? It's because it's never
44:07 even in the one layer case. if you had a one layer
44:12 we know the exact solution there from layer cases to the and so uh
44:19 was not a good idea, it at the time we were doing it
44:23 it was straightforward, boring boy. it uh I think you had the
44:31 that I was gonna end up in right place, but instead I ended
44:34 in the wrong place. So let's back and see if we can get
44:39 to the right, we want a for times square because that's what shows
44:47 in the category in therapy. And the corresponding expression for I mean here
44:53 have expression for time and all we to do is we square that and
45:00 the small the lower order terms. then finally we come out to the
45:07 quality equation which is what we were for. And this one is exact
45:13 the one layer case. So that almost certainly it's going to be a
45:17 approximation for the layered case. So comes directly from the assumptions of uh
45:27 a trophy and layer and small And this is called the dicks
45:33 And this is Stuart dicks. So knew your dicks as an undergraduate,
45:38 knew your dicks. And uh so looks like a gentleman here and he
45:42 a gentleman uh in person and he well remembered and well loved by all
45:49 faculty. He died shortly after uh graduated. I don't think that my
45:56 had anything to do with his death um he was well remembered and still
46:08 , his name is mentioned in our , I would say hundreds of times
46:13 day and by geophysicists all around the . This equation, you did other
46:21 to uh this is what is So now, so this is the
46:29 of a hypothetical one layer problem, uniform no layers in here, but
46:35 has the velocity of V. M. S not its real
46:39 So that's kind of a curiosity that looks like uh looks like a real
46:45 , but it's an approximate solution. , so the actual, assuming very
46:52 is not this one, but it's race following snow stone. Now it's
47:01 from a one D model with course struck players, but subsequently it's been
47:07 to be more general than this approximately also for dipping layers and also for
47:12 psychotropic layers. And the way you to these other places is that we
47:19 the move out velocity not as this velocity but as a processing parameter and
47:25 call it the move out velocity, velocity instead of RMS velocity. So
47:33 you're talking with a colleague about move , you might be thinking move out
47:38 , animo velocity and that your colleague be thinking RMS velocity he's got in
47:45 this cartoon case that we just derived velocity. But uh that is uh
47:53 never happens except in textbooks in you're likely to have almost certainly have
48:00 actual topic layers often have dipping And um uh we can still uh
48:09 hyperbolic move out in those cases for offsets if we call it a move
48:15 velocity instead of RMS velocity. So make sure that when you're having these
48:21 , your colleague is hearing the same that you're saying. Normally these two
48:28 different. Okay, now of course varies with depth since the velocity is
48:40 with debt. And definition bottom of in is given by this. So
48:48 we have a summer, all the the layers. Uh Mhm Twice the
48:57 way travel time, vertical travel time the square of the velocity for each
49:03 and some those up. And then do. So these are like
49:07 These are waiting functions for this uh this some and here's the sum of
49:13 weights. So the sum of the here is actually uh actually the total
49:20 travel time down and up. This an player. Similarly at the at
49:26 bottom of the overlying layer, it's same sort of thing. Except it's
49:30 an N -1 here instead of an . So um uh let's go back
49:41 the inch layer and separate out out this some over in let's separate out
49:47 last term and leave the first n one time many here. This is
49:52 what appears in the expression for M R S at N -1 later
50:04 show up years page will make this . This put in the V
50:10 M. S. Uh Give this and solving for this velocity here,
50:18 find this velocity in the end layer given by the RMS velocity measure on
50:26 workstations. Actually we measure the MMO but sticking with this approximation that those
50:33 are equal, we uh look this the end layer and here's the same
50:39 at the layer just above. We with these two different times. And
50:43 this is called dicks differentiations. That's estimate for the local velocity in the
50:49 player. And these are the velocities we need to convert time to
50:56 It's not the real depth because we've all sorts of approximations here but that
51:01 call that the apparent death. And do we get these philosophies while we
51:09 we get them from uh get these velocities calculated from the RMS. Uh
51:20 has a very long time but it's the case the enemy velocity is not
51:29 to the RMS velocity. And the of that is what we call a
51:34 to depth miss time. So that you calculate the depth of the various
51:40 according to this, the apparent depth then you drill it, you usually
51:45 out that it's wrong. And uh reason strong is because some of the
51:55 we've made in deriving this were not now for many years. Hyperbolic move
52:03 was a sufficient approximation. In fact was true From the invention of Reflection
52:18 Mix in the 1920s Up to when joined the business in the mid
52:26 I joined the business in the early and almost all of that time,
52:33 hyperbolic approximations considered good enough about the I joined the business, we invented
52:44 V. O. And I think told you uh my role in
52:53 So with a V. O. actually looking at the variation amplitude with
53:01 offsets. So obviously it's a good to have lots of offsets. Obviously
53:07 a good idea to have long cables on land, it's uh obviously you
53:13 to have long distances between the source and your C. Reports. So
53:20 soon as we started to acquire these with longer offsets, then we realized
53:27 hyperbolic equation was not good enough. is you could not find move out
53:39 uh parameter which would flatten the gathers to those long offsets. Saying that
53:55 , most of the history of this , we designed the maximum offsets to
54:03 about equal to the target depth. reason was that if you have longer
54:08 and that you're wasting your money on much uh effort in the field to
54:14 out those long offsets. And uh you just didn't need them with the
54:22 of a bl surveys were extended to offstage. They gained a vo leverage
54:28 then it was found that the hyperbolic not sufficient to flatten these gatherers Long
54:35 gathers. So here's an example of , this is real data acquired at
54:41 30 or 40 years ago. And what we've done is we have uh
54:48 uh by uh state of the art at the time. We have discovered
54:56 move out velocity function, which flattened gathers at short offsets but at longer
55:03 , it was no longer flat. so that means that you do not
55:09 to stack this data from all these , You'll be stacking these things out
55:16 uh out of sync with these because have been overcorrected, remember that all
55:25 these uh these uh reflectors, they start off with move out like this
55:32 times. Further offsets. So what done is we've brought them all up
55:38 uh did it in a hyperbolic which approximately flattened the gathers for this
55:45 here. About flat out to And then beyond that it's not
55:50 Furthermore, it's lower frequency. Look that lower apparent frequency, it's gonna
55:58 out that that um feature that is is a result of uh over crush
56:10 movement. Well, you will recall um uh previous derivation that we had
56:21 there, taylor expansions. And uh uh that uh explicitly restricted oh,
56:33 of the derivation too short offsets. obviously the obvious thing to do now
56:38 we have longer offsets in the data retained another term in the taylor expansion
56:45 . So here's the first order taylor and here's the second auditor And I'll
56:51 out to you that the small quantity is being uh soon small is a
56:59 X square. Not small X. X squared. So the higher order
57:04 here has X. To the fourth . That's the next order term.
57:12 what is a two star eight A is given by uh by this um
57:21 here. Uh And you know what . R. M. S squared
57:27 ? You don't know what V. to the four where the subscript or
57:33 four is, it's not the square V. Rms square. If that
57:40 true, these two terms would This is another quantity His name.
57:47 . r. m. four whether Actually this this this is an exponent
57:58 , it's raised to the fourth But what is our M.
58:02 R. M. Four R. defined yeah defined right here is defined
58:10 a sum with V. To the power compression. R. M.
58:16 squared is a some second. It's like this. So this was invented
58:23 um Terry tanner. A very nice . I knew Terry dead now,
58:31 10 years. Uh He was one the generation one of a large number
58:38 Turkish geophysicist who were outstanding Geophysicist and Terry uh shown here at a at
58:48 party. Having fun with a Uh This is a good idea to
58:55 extend this uh taylor expect approximation in obvious way. Just uh just include
59:05 taylor um uh approximation term, but a problem here, which means that
59:12 large X. This means that this large X. This term is going
59:17 be dominated, But it means that large X the square time is increasing
59:28 to the 4th power of, of . So square time and large offsets
59:37 this term dominates this one. Um have the square of time increasing with
59:43 fourth power of offset, that's not , you want it to be the
59:49 of time to be increasing with the of offset like this, but with
59:54 right velocity parameter instead of the wrong . So, um before I leave
60:03 , I want to remind you that minus sign is put in here so
60:07 ada will have will be positive. need to put in there. If
60:14 didn't put the ada in here, have to put in here. In
60:18 case, you need to have a in there too drinking uh derivation of
60:29 means you've got to have this parameter there. Now, this defect as
60:37 here can be remedied by including a term uh physically motivated in the
60:44 So let's take that uh previous approximation general and Kohler uh published and add
60:54 this term right here. And so is a clever term that was done
60:59 ilya trunk and and I 26 years , seven years ago. And um
61:06 what we're gonna do is we're gonna this constant a in a clever way
61:10 as to make the correct behavior at offsets. So uh okay, so
61:34 offsets. Excuse me the shortest We got only the one. Then
61:39 X gets bigger we need to have hyperbolic corrections and as X gets still
61:44 uh um this term is gonna start dominate but as it starts to
61:52 we have this term is uh small to one because X is not yet
62:01 enough to make this term creature boy with respect to one. So uh
62:09 this term starts to grow, this still negligible but it's still larger
62:16 This gets to be large compared to one and so we can make one
62:21 then this X squared cancels two of . And so we're left with an
62:26 squared term, we still have this squared term. Here, they are
62:30 of them together. So uh following on that idea, if we define
62:35 in this way involving the horizontal velocity the rms velocity, then at the
62:44 offsets, we have the square of increasing according to the square time increasing
62:51 to the square of X. But the horizontal velocity instead of the wrong
62:57 . And that's what we want. , we don't know in advance what
63:05 the horizontal, we don't know this parameter island. So uh what are
63:16 gonna do about that? So before went into all this taylor expansion
63:25 we found the time and the offset terms of these sums on the
63:29 And previously we found the liberty solutions small P. Now we want to
63:35 the liberty solution large P. So can find the horizontal velocity as in
63:41 limit of large angles. That's a . And so the previous expressions it's
63:53 and uh putting in there. The for horizontal putting in this. And
64:04 if you look here at large angles close to pi over two, cosign
64:12 is zero at So we're dividing by and we're also dividing by zero
64:20 And so we have when we divide zero here, that's infinity. And
64:25 divided by infinity here. So we how to handle issues like that.
64:31 a it was a french mathematician in 19th century and his name was Low
64:40 tom. That's how you pronounce that french, pretty close. And so
64:48 which is explained in the glossary, you how to how to solve this
64:53 . And so this uh simplifies to expression here, from which we learn
65:00 the left side of this horizontal velocity simply equal to the Harris metric average
65:06 in the in the stock secrets. do we say that? Well,
65:12 see here is a sum over the . What's being the summits? The
65:19 layer velocities with weights of the vertical time. So here is the sum
65:24 the weights. So this is the average. So this is the average
65:30 vertical uh average of the earth metric . And now uh were specified
65:44 Yeah I guess uh taylor and no and colors um result modified bites,
65:54 and Thompson with everything specified. And only problem is that it doesn't work
66:00 well. The reason doesn't work is that has to be determined empirically and
66:10 one has to be determined empirically. uh um we still have this assumption
66:18 there and uh works pretty well. we can use these corrections but um
66:30 find it doesn't work so empirically we that it doesn't work as far out
66:44 backs in the offset equals twice the to make it work to longer
66:53 We have to recognize that um most those layers in the stack are an
67:00 tropic. And so the anti sanctuary to get in there somewhere. So
67:07 gonna be and I promised you that fewer equations, more pictures. I
67:18 obviously wrong about that. Um uh are gonna get to secondary. So
67:27 directly, oh first we'll take I'll the quiz. It says uh is
67:35 true or false is exact for a one. De ice tropic layer.
67:40 I'd say that's true. Um But does specify it does uh why does
67:52 the layer should be a flat The flat reflector doesn't say that
67:57 but I think you can infer So if you uh said that's
68:02 you get full credit for that. Now is this one true?
68:11 Well we know that's not true. did not assume that the grades are
68:15 , we assume that uh the rays bending according to Snell's law. And
68:23 so this one is definitely not Uh Well we talked about this one
68:30 in general, it's not true that equals N. M. O.
68:35 common mistake. How about this Um um is this statement true or
68:45 ? Is the abnormal murat equation? one we just showed. Is that
68:49 a second order taylor expansion or? And if so uh could we call
68:56 1/4 order without equation? And so as I explained, that's not true
69:04 it's got in there that fourth order . It's got a physically based correction
69:09 that sunken and I put in there 26 years ago. So it's not
69:17 a Taylor expansion? Oh this looks . Read this carefully um As a
69:26 approximation the abnormal movement equation has for vertical travel time. T zero.
69:33 it have 1, 2, What which of these answers is
69:55 So I'm gonna go with d you go back and check it but I
69:59 the key is correct. So here's pictures, interfering waves as we learned
70:09 the sum of two solutions is a . So here it is from the
70:13 wave equation? So that means uh a very important zone. Uh that
70:19 that we can make a four year of solutions and it's still a
70:24 That's very good. Furthermore, it that any solution can be separated into
70:30 components like plane waves and these can analyzed separately. Very important. And
70:35 the point that we're taking about the which intersect each other. For
70:41 they're coming from the different directions, simply super pose, then pass on
70:46 interaction. So here's here's an So here we have increasing time this
70:52 , increasing depth here. And so can have a way of going down
70:57 a wave coming up. So in next section, next instant of time
71:01 closely together. Next instant of they're beginning to merge. Next instant
71:06 time, they're fully, fully, merged. You see, this wave
71:11 looks pretty much like either one of then, but they continue to propagate
71:17 one coming up, continues to go up. So as it goes
71:22 it begins to separate here, it's separated and here it's completely separate.
71:27 things are passed through each other like . This is called constructive interference when
71:36 combined, positively like this, the add, so that this amplitude is
71:45 this amplitude. Plus this amplitude. what happens if they have opposite
71:53 So now this one is going down a positive central peak and the one
71:58 going up as a negative central peak , they're getting closer here, they're
72:04 to interfere with each other. And they're exactly on top of each
72:08 And although there is zero displacement, momentum in there. The downwards one
72:15 still going uh this one here is going up and this one is still
72:20 down, even though they're exactly canceling other out here and they emerged in
72:25 next uh millisecond. Here's the one up, that was this one,
72:31 still it's still tangled up with the one here, they're pretty well separated
72:36 they're well separated. And so the with the negative polarity is uh now
72:44 here started off down here, so called destructive interference because at the moment
72:51 overlap exactly cancels out, but there's momentum in there, particles are moving
73:03 different momentum at each at each And so the waves continue.
73:21 all this um entire analysis has been uh small deformation. Uh So all
73:32 nifty properties that we've discussed come from assumption near the source. The amplitudes
73:42 be large and uh so there can lots of complicated nonlinear effects which are
73:50 discussed here. And so because we it. And let's see here where
73:57 link goes to And that goes to lesson two, are we talking about
74:09 . So that that link is a one in the Sug version. Uh
74:15 I think it's it's you need to checked, you know for sure that
74:23 you try to click on this, gonna get in trouble. Okay,
74:32 um if you have large waves, can be a non-0 interaction when they
74:39 with each other. Uh these effects this are outside the scope of this
74:45 . I remember that there are there people who thought that um interfering ways
74:53 you interact with each other. And came around and tried to convince us
74:59 that at Amoco and to sell us product. And we gave them a
75:07 reception. I hope they were We said that the so called data
75:13 they had to prove their so called didn't prove it at all. And
75:19 went away disappointed. I hope we're . There's always a danger when you
75:29 you're so damn smart. Somebody comes with, oh, an advanced idea
75:39 you think is false. But it out to be true and you think
75:51 false because you have blinders on because been so good at what you've been
75:58 , that you think, you know all I've seen that happen myself.
76:07 . I can tell you one incident is particularly relevant on this point.
76:15 line into Amoco early eighties, we how important an ice ax PS for
76:27 she earrings And we kept it a inside amicable for five long years before
76:36 finally leaked out during that time I in an scG audience uh chemical session
76:44 the ScG and listen to a smart from marco talking about share wives.
76:54 so it was conventional, boring At the end of the talk,
77:00 said something extremely unusual. He I want to show you some
77:08 I don't understand it and I'm hoping you won't help me understand. So
77:14 showed some data about cheer wives, um um I understood because it came
77:22 an ice action and it came from fact that in an icy tropic rocks
77:32 are two different share waves propagating with different walls. That was the underlying
77:39 that caused the day that he was . Well I was sitting in the
77:44 in the dark next to a couple smart guys from Shell and they were
77:51 recognized experts in anisotropy why? What knew was a special case of
78:02 Uh the simple thing, they were in the simplest kind of anti socks
78:09 they had very large opinions of themselves so they giggling together in the
78:15 nudging each other in the ribs like pointing to the guy at Arco who
78:20 asking for help in an open hearted and they were giggling and they said
78:27 that guy doesn't know that can't They were looking at the data.
78:33 was showing us the data. Uh they were uh what they should
78:40 said is oh according to our that can't happen. There must be
78:45 wrong with our theory because he's showing the data. But they were arrogant
78:50 they denied the data in favor of oversimplified um theory. So I knew
78:57 what the answer was, but I keeping chemical secrets at the time.
79:02 in the discussion that followed, I my mouth shut. But I was
79:06 remember today being struck about how these smart guys. Um it's not the
79:14 um because there's no simple theory. so whenever I hear somebody come across
79:21 always simplified with the theory, which think is completely. Well I always
79:27 of that, I'll tell you another actually just occurred to me again,
79:33 was at an sug meeting and a young girl came up to me speaking
79:39 a Russian accent. She said I you were in, I heard you're
79:44 to look at unconventional ideas. So I suspected uh that it's a KGB
79:53 and I'm about to fall into a track. So I said, well
80:00 can I do for you miss? said, well my father is a
80:05 geologist and he's working in the far part of Russia, in the middle
80:12 Russia, thousands of miles from any and thousands of miles from Moscow and
80:18 of miles from nowhere disappears. And this guy out in the middle of
80:23 with unconventional ideas. And he this girl says my father would like
80:30 talk with you. And so I'm suspecting a honey trap. So I
80:39 okay, so we'll meet um I'll your father. And she says I'll
80:44 . He doesn't speak english, I'll . She was actually living, I
80:50 in san Francisco at that time. it's not uncommon for young people from
80:55 to state Russia. So that that uh at the convention. And marco
81:02 a booth on the exhibition floor. I said, okay, I'll meet
81:06 in the exhibition but booth tomorrow. so that's what they showed. So
81:11 father was about my age and the was young and beautiful. And so
81:17 talked uh father and I talked with daughter is translated. And so the
81:23 has some extremely unconventional ideas to the and the upshot of the ideas was
81:31 of the things that I'm teaching you are long and that what we see
81:35 our data is something completely different. he thought that uh what we see
81:40 in our data is the effects of fractures deep inside the earth. Normally
81:48 assume that when we see fractures in earth always uh vertical or maybe obliquely
81:56 horizontal because uh talk about the this reversible stress is always the
82:06 And so if a crack is gonna , it's not gonna open this way
82:11 you have to lift the rock, gonna open this way in the direction
82:14 one of the minor, in the of the least impressive stress. And
82:20 oriented like this is gonna be a motion, but you'll never see a
82:26 . However, when you're driving along highway passing through road cut, you
82:34 see horizontal fractures uh layer mountains and might see water leaking out of these
82:41 uh fractures uh exposed in the near . Where my previous argument was not
82:49 right in the near surface, at very surface, the vertical stress is
82:54 and then the vertical stress increases. And so it's um don't get very
83:00 , no more than, say, tens of meters deep before the vertical
83:06 is the maximum. But in a cut, uh you might be shallow
83:11 so that the least compressive stress is the vertical stress. So in that
83:16 , you can open up like so so of course this, this Russian
83:20 had seen that in road cuts in and had not understood uh uh the
83:28 of it and had uh said that of the reflections that we see come
83:36 that kind of horizontal fracture, not layer boundaries, but horizontal fractures.
83:43 from that, you can tell where oil is. So of course,
83:47 immediately saw the shortcomings in his Uh and I was just about to
83:53 , shut him down when he well, I talked to Chevron and
83:58 challenged and they gave me, uh, seismic image About 30 miles
84:08 and about 10,000 ft deep. And course it had vertical exaggeration. So
84:14 can see it. And they Okay, somewhere in this long,
84:18 section, we found some arm, know where it is. You look
84:23 the section and you tell us where found on. He said, I
84:29 out. He said, I looked it for half an hour and I
84:32 , the oil is right here and some over here. And he was
84:37 right on both cases according to his . So that made me think,
84:45 , you know, maybe he's right the wrong reasons. Maybe he can
84:49 oil even though the, uh, explanation is nonsense. We don't do
84:58 experiments in order to look at the data. No, we we do
85:04 experiment to find or so even if guy has a theory which uses our
85:10 to find oil, even if his are completely wrong, we should pay
85:19 , You know, uh, in history of uh, plate tectonics was
85:26 uh, thought to be a continental . And it was invented by a
85:34 . His name was german meteorologist. he looked at the shape of the
85:41 atlantic ocean parallel coach long. And said, well, he's continents up
85:46 split apart. He told that continental and he advocated for continental drift 20
85:52 30 years and people laughed at it or 30 years before better data came
85:59 and helped us to define plate which of course is different than continental
86:04 in numerous ways. But one of things that he did was he put
86:09 a nonsense physical explanation or uh why would drift apart. And it was
86:18 nonsense. But the observation, the parallel coastlines. That was true.
86:25 so if people had taken the Parallel seriously, they could have figured out
86:33 tectonics a long, long time As a matter of fact in the
86:36 century, the English uh scholar named Beacon notice those parallel coastlines coming out
86:48 the first primitive maps that were being by the first uh global voyages.
86:58 he noticed and commented on it. don't think she offered an explanation.
87:02 he said, you know, this interesting. These coastlines look like,
87:08 , that's another example of uh data primary and the theory uh, in
87:16 of the data and theory and the conflict with each other. Then it's
87:20 the theory. That's wrong. So of this digression comes um,
87:27 actions of this last statement here that , we have to be, we
87:33 to remember that it made this approximation small uh deformation and with larger
87:41 We have other things can happen. we should be alert for those things
87:45 our data outside the scope of this . Okay, Stephanie, true or
87:55 . Read that one and tell me or false. That was false.
88:01 course. Next question that one's Yeah, that was false. The
88:20 act together phases. Don't. so now share rights. Yeah,
88:30 ahead, shear waves. It says the shear waves are similar to p
88:37 that is. They have short spread move out. Except that Number
88:42 they travel slower. There's the formula are polarized trans partially to the wave
88:48 that's uh share wives inside of a body. Polarization is not an
88:56 They can travel with any polarization they at a boundary. They reflect reflect
89:03 depending on the polarization. Because you , if you have a shear wave
89:09 a horizontal boundary like this, it's make a difference whether the polarization vector
89:16 parallel to the uh to the boundary perpendicular. It can it can lie
89:23 in this plane as it's approaching the . But when it gets the boundary
89:28 gonna interact with the boundary differently depending how it's bar. That's pretty
89:35 Now all those statements are true. furthermore, here's another important point when
89:48 look at p waves only. So waves are going to be propagating p
89:53 velocity is not share wave loss. , the shear wave properties of the
90:00 still uh affect p wave propagation at boundaries. And so the reason why
90:08 . V. O. Is interesting us because uh right in there in
90:13 HBO gradient there is the sheer weight margins and that happens you you don't
90:19 yet how that happens that you will in less than six. Usually we
90:26 to exclude the share waves from P. Wave analysis and we do
90:31 in lots of ways for example uh the marine environment there are no share
90:37 . Uh So that's good. That's marine data is usually much better than
90:41 waves. Then land data. One in land data they share waves are
90:52 recorded on vertical geophones so that uh I was your age um I had
91:00 job on a field group. Uh my job was uh G.
91:07 Phone placement engineer. Which means I a drug hustler. And so my
91:12 was to put a rack of geophones my back and then walk out along
91:18 line and plant the geophones. And those days were only interested in vertical
91:25 vertically near vertically arriving p waves. we had geophones which are only sensitive
91:32 vertical motion. So they had a um instrument on the top of a
91:37 . And I would put the steak the ground and the stomping in the
91:41 and do my best to make it so that it would record a vertical
91:48 emotionally. And so you understand that P waves as they emerge up towards
91:57 surface, they're constantly turning towards the turning towards the vertical like this because
92:05 they go up they're encountering slower and velocities. So that means according to
92:10 law, they're turning up. And when they hit the surface there,
92:15 vertical, not quite vertical, but vertical. In the same way,
92:20 waves are doing the same thing, they're doing it even more so.
92:26 the reason for that is share ways and on the share models from
92:41 Whereas p waves depend upon the functional , M or K plus four thirds
92:47 as uh in very shallow sediments. is very small because the sediments are
92:56 consolidated, But K is not so . In fact, if K.
93:01 uh if the sediments are saturated uh velocity of water, you mean the
93:09 of the ways never gets to be than the velocity of water about 1500
93:16 . But there's no lower limit on you how how low share waves can
93:24 . So, at depth velocity ratio DBS is something like three. And
93:32 great depth uh is more like you know, uh 10 miles
93:39 That ratio is more like two in kind of rocks. It's more like
93:43 . Uh And as you get very to the service to get, that
93:47 gets to be very big because S. Gets to be very
93:51 It's not uncommon to find a velocity of 10. And then for a
93:57 surface dirt Share waves traveled 10 times than P waves. And so what
94:04 means is the share wave is coming , it's even more close to
94:08 And the P wave is because the the velocity gradients for share waves are
94:15 and uh near service than for share . And for P waves. So
94:20 that means is the shear wave is up almost always vertically polarized, perpendicular
94:26 . So the polarization is uh almost the horizontal plane. And we don't
94:34 don't record that with our vertical geophones least uh manufacturers of the geo phone
94:42 everything they can to avoid getting any from a horizontal displacement horizontal um jolt
94:52 the vertical. So do that in . And furthermore, our sources are
95:00 to maximize p wave power and to shear wave. So for example,
95:06 the old days we had dynamite sources that's putting out an impulsivity. That's
95:13 putting out a lot more P waves share waves. You have a vertical
95:22 that's going to be maximizing the power radiated P waves in the vertical
95:29 So the p waves radiating from a uh vibrator or maximum vertical direction and
95:37 to be less powerful as you go shouldn't call vertical vibrator. We shouldn't
95:43 it a p wave because it also generate Children's But the sheer waves are
95:51 vibrator, the shear wave power is horizontal. So whatever shear waves is
95:58 going horizontally and they're not going down reflecting back up. So that's another
96:03 we do uh minimize shear waves in data. That's a good idea.
96:11 it's not only good idea. Another idea is to maximize the share wave
96:17 , minimize the P wave energy and at the share ways. So we
96:23 that at Amoco when when we were shortly after I joined the company,
96:28 learned to deal with anti psychotropic share . And we wanted to do some
96:35 . And so we uh we invented uh we did we did not invent
96:41 horizontal vibrated but we did refine I'll tell you who invented it.
96:45 was Kanako. And so those days was separate from phillips. Both of
96:52 were headquartered. Both of them were companies headquartered in small towns in northeast
97:00 for Paulson and a good technical staff . And they invented uh horizontal
97:10 And their idea was this, he okay we can do pretty good p
97:15 imaging but we'd like to go do than imaging. We like to get
97:21 information out of the seismic data for , we'd like to know what is
97:26 pathology in each square. So we that the velocity ratio in different pathologies
97:34 different. So in order to measure , let's also go out and do
97:40 ways shear wave surveys Uh and use shear wave velocities two in combination with
97:48 P wave velocity to deduce use That was the idea back in the
97:55 in the 70s. And so they horizontal vibrant and so pretty much like
98:03 P. Wave vibrator, have a truck, truck drives up to the
98:09 point stopped, lowers the vibrating the ground raises the weight of the
98:16 . Yeah truck is bearing down on vibrating bad and then it turns on
98:22 vibrator actuator and it vibrates like this of like this. And of course
98:28 gonna do it vibrating cross line. got the wheels of the truck running
98:34 way down the road and they've got when they stop and make their uh
98:39 point, they're vibrating cross to that what they want to do Is to
98:45 S. H. one. That the plan. And so that's actually
98:50 a bad idea turned out to be not particularly satisfactory but uh it's
98:58 Good idea. Uh One more I one more piece of when they went
99:06 with their brand new fancy horizontal Of course they went first to nearby
99:14 in north west Oklahoma which had excellent wave data call and then require the
99:22 using um ideas, record taken from the data was garbage. They tried
99:33 again and again garbage. They said need we're gonna need some help to
99:42 this problem. So they put together consortium of american oil companies and there
99:48 lots of them in those days. some followers and an uncle was one
99:53 them. It was called the Kanako shoot. And so we all put
99:57 money together and aid for Kanako to around and to acquire uh share wave
100:05 just like I said. and uh sites all known to have good p
100:14 there. And so they did that him about a year or so to
100:19 that and they shared the data with the sponsors and the data was
100:25 I think 19 out of 20 sites was just garbage and maybe uh between
100:32 site, not so bad, but can't Can't live with failure rate of
100:41 . So they terminated the project and gave up on sheer weight. So
100:52 eventually learned the reason for that So I'll tell you about that later
101:00 the course. Yeah, yeah, ahead. So zooming now, uh
101:11 how we minimize share waves in So expanding on the previous uh
101:21 Uh Oh my God, the vibration the power um antenna properties of
101:30 If you have a horizontal vibrator, maximizes the power downwards as share waves
101:38 it does produce horizontal uh horizontally traveling waves. And uh conversely a vertical
101:46 is the opposite vertical vibrator. Uh the p wave power vertically. And
101:55 share wave power is maximized at oblique . Now in processing we discriminate against
102:09 waves in lots of ways. For , ground role is a surface wave
102:15 has lots of shear wave energy, of sheer energy in it. And
102:19 we filter out the ground roll using . K. Filters and other methods
102:25 then we stack and we migrate with velocities uh reflections. Now for anti
102:34 media there is uh Excuse me. that's how we uh we discriminate against
102:41 waves most of the time. And a good idea for most of the
102:46 . Another good idea is to uh freeways, maximum shear waves. But
102:53 just told you some stories about how idea did not work out well In
102:59 uh in the early 1980s. And reason was because of anti sanctuary.
103:05 because the rocks are anti psychotropic and since uh shear waves, I have
103:15 more. Um And I should topic uh than P waves. Um We're
103:25 uh we're gonna postpone the discussion of ways until later in the course and
103:34 on for most of the course on wee but here's a little quiz.
103:40 So Stephanie uh What's your answer That's true. How about this?
103:54 . How about this? This is an ice a tropic formation. No
104:07 one's true because it doesn't know which is which. Um But when it
104:13 close to a boundary then it knows normal direction to the boundary is different
104:18 parallel directions of the boundary. And it uh those two polarization act different
104:30 a boundary. So then there's convert , okay, we will learn,
104:35 will learn in lesson six. Right we are in less than four.
104:39 So in lesson six we're going to that because of the boundary conditions had
104:43 reflect on the horizon. We generally to find converted ways. For
104:48 when a P wave gets any uh Here's the boundary between ice,
104:55 tropic rocks over ice. A tropic , here's a reflected P wave and
105:01 transmitted P wave. But also there's reflected SV wave and a transmitted sc
105:07 . And that comes directly out of equation. So uh that's what we
105:13 mode conversion. So the energy of incoming wave is partitioned among these four
105:20 modes. And that's why the P Avio uh reflectivity includes the sheer modules
105:30 its reflectivity because this this reflected P has a different um aptitude because it's
105:40 to share some of its energy with S. P. Wave. And
105:44 S. P. Wave as well uh the transmitted P wave. So
105:50 why HBO is interesting. Sure, convert waves are important in exploration.
106:00 in two contacts which are ocean bottom and shear wave logging. So I
106:06 to uh talk about those. So a cartoon about ocean bottom seismic.
106:14 here's a source ship and of course source boat is going to be putting
106:19 p waves through the water and here's reservoir down here. And uh so
106:26 got a P wave going down through water, down through the rock,
106:29 and coming up to the receiver. you see the receiver is a four
106:35 receiver. It's got three verticals, vector components here plus the hydrophone.
106:42 is that? The reason is because this p wave reflects here, not
106:48 do we have this upcoming wave, we have this one which flashes past
106:54 uh the sea floor up to the reflects back down with uh high with
107:01 large reflection coefficient of course. And coming back down and it uh it
107:07 the receiver a few milliseconds later. know, depending on the water
107:15 Now that the reason we want to four components is because the hydrophone can't
107:23 which direction the wave is coming It just measures pressure. But the
107:29 geo phone can tell the difference between upcoming and a down going way.
107:34 what that means is that by combining data from the hydrophone and from the
107:42 geo phone, you can separate out primary from the water layer multiple.
107:49 there's uh that's a big uh bigger of ideas. But you already know
107:56 essential idea. The hydrophone can't tell way it's coming from. But the
108:01 vector uh can so we combine these datasets, we call it the
108:08 Z. Component, dizzy or in , they call it the PZ uh
108:16 that gives you oh reflecting arrival without interference from the ocean bottom multiple from
108:28 ocean surface. Now, in we have this converted share wave which
108:34 this one converts at the sea floor the way down, comes down as
108:38 share wave bounces pretty close to this comes up as a share wave and
108:44 gets recorded on both of these components normally you don't know how normally these
108:51 are not installed with good control over orientation of the horizontal components. You
109:02 out what the orientation is from the itself. And furthermore, you
109:09 in a wide as um survey waves be coming in from any direction.
109:14 , you know, you need to components here to measure these sheer ones
109:22 shivers. But there's another converted mode is right here which converts upon reflection
109:28 this point. So it's it's a wave all the way down to this
109:31 and it comes up to uh to uh as a share wave. And
109:37 course the shear wave doesn't propagate neither of these shear waves propagates into the
109:42 column. Now it turns out that most in most cases this arrival,
109:48 converter arrival is much more energetic than one. And the reason for that
109:54 that that efficiency of conversion here, that there's a soft model air there
110:01 top, usually the uh amplitude and transmitted converted share wave is low.
110:09 so normally this one is much more than this. So we call this
110:14 convert mode, we call it a . And so here's a picture of
110:21 sea wave geometry that you can find a in a textbook. We have
110:27 nice a tropic homogeneous layer. We a P wave shown here and this
110:34 wave is reflected at the midpoint But the the sea wave is reflecting
110:40 here and coming up as a share , an SV wave and the position
110:47 this bounce point is determined by these where these angles have to uh a
110:56 of Snell's law is that the ratio these angles here. The sign of
111:01 angles here is the same as the ratio in the overburden. So smells
111:08 determines where this balance plan is. other words, it's a physical
111:12 not a geometrical argument. This is geometrical argument, the midpoint, but
111:17 is a physical argument here. So what that means, is you have
111:20 have the physics right? So for , it would be different if there's
111:25 layered medium here than the uniform it would be different if it's an
111:31 tropic and if it's icy tropic uh this position of this bounce point depends
111:40 the physics through this argument here, gotta have the physics sure. When
111:48 first got into this business of converter uh most important um most recent most
111:59 contribution was that these guys were test and bela. Uh and what they
112:04 us a closed expression uh for where uh where that conversion point is as
112:13 function of source receiver offset and it out to be from their work.
112:19 The conversion point depends upon the So uh for reflectors, for shallow
112:28 , the balance point is very close the receiver. And so the energy
112:35 going down as a p wave all way over here and coming up it's
112:38 shear wave. But then this balance gets a further and further from the
112:45 as you go down and it goes emphatically to this point here, which
112:50 called the asymptotic conversion point. And different from the midpoint which over here
112:58 there's a closed expression for that I'm drawing some rays. Mhm.
113:07 because of this it's not easy to a common conversion point gathered. Now
113:13 course you realize that for p wave , all you need to do is
113:18 sort the traces so that you have common midpoint uh gather you have all
113:23 of uh think of a two D and you have all sorts of source
113:29 and receiver position. And so you sort them so that you find all
113:34 who have the same midpoint and that you uh common image gather because they
113:43 a common midpoint but what we want a common reflection prank gather. And
113:48 can tell from from this diagram that a more complicated thing to construct.
113:55 in order to construct that you've got know the velocities throughout the overburden just
114:01 order to form a gathered just in to choose from from the traces you
114:09 the traces you want. But in to form this gather you need to
114:13 the velocity. So you see there's a problem. So that means that
114:18 workflow is required. Now there is simple formula for the democratic conversion
114:29 So let's develop some notation here, is uh uppercase gamble with a subscript
114:36 gamma zero is the velocity is the of vertical velocities gamma with such
114:43 M. O. Is the ratio move out losses And gamma effective is
114:49 square of the move out gamma divided the vertical gamma. And then the
114:54 conversion point as a ratio with the receiver offset is gamma effective divided by
115:01 plus gamma effect. So those are ideas are presented For the first time
115:13 a famous paper which I wrote in And before that um um standard reference
115:23 a testament Bela but ever since then reference has been uh 1995.
115:32 what else can we say about converting ? So, um for a modest
115:37 , the hybrid move out is So we still are using Taylor expansions
115:43 . Uh Mr Taylor was a mathematician he didn't know anything about seismic
115:49 And so uh we can use his to find um move out as a
116:00 function. The same sorts of approximations we had before. And the only
116:05 is that the velocity parameter here is sea wave move out of laws.
116:11 what is that? In terms of we uh no more about?
116:20 it's this expression here which is this the square of the p wave move
116:26 velocity function. Um That's what you know about. And I think you
116:30 be surprised that there's a corresponding shear move out velocity function. And then
116:37 those have to be combined together with gamma zeros like this to find the
116:44 function for the sea wave moving Uh We might not have all that
116:51 right? You might not have, probably have a p wave move out
116:56 function. Well uh and do a tomorrow and looking for converted waves.
117:02 probably gonna do it in a place we have p wave data already.
117:07 we probably already know this but we don't know the share waves move out
117:12 we probably don't know this stock uh velocity ratio. So that's what we're
117:19 to have to learn from the data for a uniform icy tropic layer,
117:25 expression simplifies a lot down to this where uh the square of the sea
117:32 about velocity is equal to the product the P wave times the shear
117:38 But that's only for a uniform I topic layer which of course is not
117:42 case for the realer. What else we say about convertible as well?
117:50 amplitude is zero. For normal if you have a P wave going
117:54 vertically and coming back as a as share wave, well it doesn't know
118:00 the upcoming share wave is it doesn't whether it should be um oscillating sideways
118:06 way or this way or this way what. And since since it doesn't
118:10 it can't make up its mind the of this converted share wave at normal
118:15 is zero and only got um you've got finite aptitude for the upcoming share
118:28 for oblique incidence. Furthermore, it's opposite polarity for positive and negative
118:37 So you have to have your receiver on the sea floor and you have
118:43 source going along here sending a P down. So as it hits this
118:47 wave it's going to be pushing the in this negative direction. When it
118:52 over here it's going to be sending wave down, hitting, pushing the
118:55 in the positive horizontal. So obviously gonna lead to opposite polarity for both
119:04 of a common receiver gap. So there you have a violation of the
119:13 of reciprocity theorem. So if you're change source and receiver position, Uh
119:19 not gonna leave the data and change , we're going to um uh multiplied
119:24 -1. And there might be other as well. So uh right,
119:36 should have known for years and years the reciprocity theorem has stated this way
119:44 only for p ways. So um I'll tell you some interesting stories about
119:56 . Uh I think not today, not sure exactly what slides come up
120:01 , but we'll see. I do a very interesting story on this very
120:06 which I'll get to in the proper because of this polarity reversal. It's
120:14 when you have split spread surveys and very easy to get split spread surveys
120:19 ocean bottom side acquisition, not with streamer acquisition. Ocean bottom uh
120:27 you got your instruments sitting there on sea floor and you're sailing over with
120:32 ship. Obviously you're gonna get split spread acquisition. So in order to
120:38 that to make the two sides of um of a split spread survey looked
120:45 much like it's common to multiply one by minus one so that you can
120:51 flatten it stacking. Oh, so here's a good example, There's an
120:57 and I will tell you my story just now. So this is real
121:04 from uh and uh, here's the that in 19, In 1986.
121:24 , No, it was, it about 1990, 1995. Somewhere in
121:33 , in there. Yeah, So here's the date in 1999.
121:38 um uh, Regional company status invented bottom seismic for exploration purposes and Seattle
121:52 now named Ecuador. You name themselves few years ago. First.
122:00 that's what it was called status on days. And they invented at a
122:06 of theirs in the Norwegian North Sea not far from one of our amicable
122:13 . And in both fields had similar difficulties. I can tell you what
122:19 are. Both fields were simple structures this, uh, limestone reservoirs and
122:27 geologic time, gas had leaked up of the reservoir into the overburden and
122:33 lodged in the pore space in the . And what that meant was that
122:39 overburden had a small concentration of gas in it. And that killed the
122:45 waves going through. So you cannot a good image of the reservoir with
122:51 ways. And so stuttle had the idea, let's do it with converted
122:57 because the converted waves when uh, down as a p wave, that's
123:01 outside the gas clouds. And when comes up as a shear wave through
123:06 gas cloud that share wave won't care it's got gas or liquid or whatever
123:11 the forest since it's not compressing the , it's sharing the rocks as it
123:16 up. So that was a good . And they, uh, they
123:20 the equipment. He had a lot smart guys on their payroll, invented
123:25 equipment, went out, did the survey and it was a smashing
123:33 And um, so, uh, had, they presented that at the
123:40 the annual meeting of the european I think it was in,
123:46 I don't know where it was. did not attend that. But we
123:51 people who attended that meeting and you've got to do this. We've
123:55 our field named Valhol, very close their field with the same problems.
124:00 do it there. So we commissioned world's second ocean bottom seismic experiment.
124:11 , so oil company purposes. here's another interesting side bet.
124:20 uh, Central did their work. did it in the most straightforward
124:26 They had autonomous underwater vehicles swim down the ocean floor being piloted by somebody
124:35 the surface. And they would have mechanical arm and they planted a geo
124:40 in the mud jack analog of what , he would do on land.
124:50 it worked. But it was very recently. Didn't. An interesting
124:58 He said. They said to the who did this, He said,
125:03 you tired of being uh, receiving salary, would you, would you
125:10 be rich? And so they said we'd rather be rich. So they
125:15 okay you can leave Seattle with our go and set up a company on
125:22 own. We will be your first . We will give you all the
125:27 property rights all the patents and everything you developed as standalone employees and we'll
125:32 your first employees go for it. make yourself rich live a great
125:38 So they did that except Wild. I was nervous about going out on
125:45 own. And he signed up with large uh services company called PTS also
125:56 and signed him on as a vice for ocean bottom size. The other
126:02 set up their their uh uh private . And eventually they went back because
126:10 ideas weren't quite right. Meanwhile this P. G. S. Got
126:14 big salary from P. G. . And they said he got to
126:19 and they said okay so what's your going to be? And he said
126:24 the way we did it at was good because it was too expensive.
126:29 got to have a better way to it. So my idea is that
126:33 not gonna put instruments on stakes in mud. We're gonna put instruments to
126:39 on the sea floor. So we're have an instrument package about the size
126:43 a suitcase. And we're gonna have gonna pull that along the sea floor
126:48 a cable from the ship. And going to have a line of these
126:52 and we'll pull them along the we have a two D. Acquisitions
126:59 uh we'll pull them into place, stop, we'll do our shots and
127:03 we'll pull them forward and repeat. that will be a lot more efficient
127:08 putting uh individual uh cheer phone on . So PGS said what you
127:18 You're just gonna have them sitting there the sea floor uncoupled. Well that
127:24 . And he said no I don't it'll work, but if it does
127:28 it'll be fabulous because it'll be So they said amazing when PTS said
127:35 for it. So that's what they . They constructed a set of uh
127:41 like that. And America was their customer. And so that's what they
127:45 . They had 50 of these on string about about 50 or 100 m
127:52 . Drug them into place uh park their ran uh seismic shots over
127:59 them for and so on. And this is the data for uh coming
128:03 that. And this is a velocity of a sort which you have never
128:12 , you can see that it's got uh it's not perfectly flat and over
128:20 is uh is what we call the spectrum. So um uh Stephanie,
128:28 you familiar with with plots like this we have for all times. And
128:33 all different velocities. We calculate we the gatherers according to the hyperbolic move
128:42 . And then we uh we calculate quantity which um uh measures the
128:53 And uh so that quantity is um shown in colors here and we do
129:00 for all the velocities. And for of the velocities is not very
129:04 So uh those things are colored blue , but wherever you have a maximum
129:10 the colors right here, that's the velocity. So that's why we call
129:14 a velocity spectrum. All different And so for every one of these
129:19 we do it for all those different . And then we say okay that's
129:22 right velocity at this step, that's right velocity here and so on.
129:28 then using those uh different velocities as function of time, we flatten the
129:35 . And so this is what the look like. Common midpoint gather.
129:40 you see it's not flat. And know you've seen non flat gathers
129:46 but what makes this one so uh is that this is a split spread
129:52 and we have negative offsets on this and positive offsets for this side.
129:56 we have successfully flattened using this set philosophies, we have successfully flattened the
130:03 offsets and those uh were both velocities too slow uh to uh awfully flattened
130:13 positive offsets. So the reciprocity the scalar reciprocity theorem says that for
130:21 split spread? common midpoint gather like , it's got to be symmetric so
130:26 when your insurance sort source and receiver data is unchanged. But this is
130:31 not one symmetric. Sure that's a . So we thought we must be
130:45 something wrong. When I first produced figure, I immediately saw there was
130:51 problem. So I went to my world, noted geophysicists working for amazon
130:57 I was not noted then or now good at seismic image. That's why
131:03 not teaching. The force is He's an expert. I'm not.
131:10 guys were also experts. They look leon the image spokesman gather has
131:17 be symmetric because of the reciprocity which is absolutely true. No way
131:22 get away from the rest of processor . You must have screwed up
131:25 You must have screwed up the geometry . Go back and check your
131:29 Finding mistake, fix your mistake. be okay. So of course I
131:33 them. And I went back and everything. No mistakes. So finally
131:38 desperation and desperation, I read the of porosity theory and I found out
131:45 this statement about symmetric split spread gathers the special case applying only for p
131:53 . They're now called the skin of reciprocity does not apply to this kind
131:59 data. So I want you to back here at the uh at the
132:06 and you can see that uh we this line of high velocity, but
132:11 could have picked this line to see interpreter could have chosen that line.
132:16 is unusual to find a trend like which is consistent. It's not unusual
132:25 find certain times where there's uh an velocity uh with a high uh coherence
132:36 it uh at low velocities. And normally think that's caused by multiple,
132:43 stopped together at low velocities. But is happening along a trend. So
132:49 do you suppose if we were to that low velocity trend here, by
132:54 way, I should tell you that side here is polarity reverse. It
132:58 moral. And look here. It have just like yeah, we predicted
133:05 amplitude at normal incidence, positive offsets under correct? No, there are
133:12 velocity trends. So let's pick the trend. So here is the other
133:16 pick. And now the positive offsets flat and the negative offsets are turned
133:24 . So this data is exposing a of converted waves. Sea waves,
133:34 is not shared with p waves, is that um that um spit spread
133:43 are not necessarily uh not necessarily And furthermore, look here, we've
133:51 some significant reflection energy at um normal . How about that? Yeah.
134:06 . Thank you. Okay, can just explain the and so how is
134:18 ? Probably what are you? so we've got a bunch of ocean
134:29 seismometers on the sea floor. We're at the horizontal in line component.
134:35 . And uh we've got the boat over, and so it's got positive
134:39 and negative offsets. So uh those gonna inherently have different polarity on two
134:47 . So we we reverse polarity on side here for the negative offsets.
134:54 actually don't know whether we reverse the on this one or on this
134:58 but we multiplied one set by so that they would look sort of
135:06 the way. And then we uh that gather, excuse me. Up
135:14 this point we don't have any but then we sort them just like
135:18 p waves and we uh we we together into a common midpoint, gather
135:24 the traces with both positive and negative , and then we plot those like
135:29 uh as a function of offset. this is zero offset, the large
135:36 offset, large positive houses all with same midpoint now for each one of
135:44 . Uh So now that's our And so now we try to
135:48 and so we flatten it with all different velocities from uh small velocities to
135:54 velocities. And we calculate the various which determines how flat they are.
136:01 so that quantity is, is shown in colors. So uh that quantity
136:06 a maximum here here where it's And so um an interpreter has picked
136:14 here here and here he's picked um literally interpreted in train. You can
136:20 uh chosen with his interpreters judgment, saying, okay, this uh maximum
136:27 here means that that's the right velocity this one means it's the right
136:32 So we'll just interpret it like so obvious. And then uh now we
136:38 a velocity function which varies as a of time. And we just flatten
136:45 gather at all times according to this function. And that's what you see
136:50 . Uh these here are flattened, philosophy, these down here are
136:56 this philosophy etcetera, uh straightforward hyperbolic . And when we do that,
137:04 find that sure enough, it's not . We did a pretty good job
137:11 flattening one side, but we did poor job of flattening the other
137:15 particularly long positive offsets. This is corrected. So what that means is
137:22 flattening it was bending way down like . And now that it's been partly
137:28 . It's not bending down as but it certainly is bending down.
137:34 so then on the next slide it's slightly different position, but it's the
137:38 idea that the interpreter is now picking slow train and now what what he's
137:45 is he's flattened the positive offsets. now the negative offsets are overcorrected.
137:51 went from curbing down to curbing So an obvious violation. Uh the
137:59 of of the scale and reciprocity Now, I will remind you what
138:12 you sure for uh so using these , so we recognized as soon as
138:28 saw this that we had to develop thinking well beyond conventional p way of
138:35 , we have to uh make our using sea wave ideas, not p
138:40 values. And there were many other that we had to make. I
138:45 think we have time to talk about too much in this course, Maybe
138:49 . Uh but uh out of we got the first successful images of
138:56 previous to that, it had been completely invisible for a medical seismologists and
139:08 made the first good uh image are . And this was after it had
139:15 producing for 10 years. They had basically producing blind. And so um
139:25 Norway is um a special place to business because the oil offshore Norway is
139:36 owned by anybody except the state of nation of normal. And they are
139:44 to produce it as efficiently and profitably possible and also as safely as
139:51 And so they give licenses, they licenses to companies including Seattle which is
139:58 company and american companies like and british and uh companies like that will buy
140:08 rights to explore and produce for oil specific, lots of receiving tracks in
140:16 raging waters. Norway requires that companies operate in their waters, apply the
140:27 technology available. And furthermore they require if you learn new technology, you've
140:34 to share it with your competitors who also producing Norwegian arm and if you
140:41 share then you don't get another chance uh, bid on other Norwegian
140:51 So because of that pressure from the government, he um, ah went
141:00 explained our success and the reasons for success, all the new ideas that
141:05 developed in order to get these that balance went way beyond what anybody
141:10 had ever done. And we presented at the meeting of the European
141:17 I think it was in 1996, me. And uh, we got
141:29 award for best paper at the Meanwhile, PTS at the booth on
141:39 exhibitor form and we had given them to take the data which they had
141:46 for us process it on their make their own images and to show
141:51 to whatever they wanted, gave him permission. So the young man who
141:55 that work was making his presentation of imaging of our data in the booth
142:03 the PGS booth and he had very images, very different images. And
142:10 told all the people who were listening his presentation in the booth that Amoco
142:15 know what they're doing because he thought to be an expert. He was
142:19 recent graduate of uh, american University his, his thesis was converted wave
142:29 . So he thought he was an and he thought we didn't know what
142:33 were doing. And he told all his listeners that Amoco doesn't know what
142:38 doing, well included in his audience find the time we're amicable executives.
142:43 did not appreciate hearing Amoco criticized like in public by our own contractor.
142:51 so, uh, after the this young man was summoned from Norway
142:58 by that time I had left chemical research in Tulsa and I was
143:03 for a chemical exploration in Houston. I was summoned from Houston to figure
143:11 the reasons for this difference. And it turned out the reasons were that
143:19 young man from PGS had not understood business about non symmetric split spread
143:27 He had assumed that the scale of theorem applied also to convert ways and
143:34 done a faulty velocity analysis because of . And that's why his images look
143:41 different. And so, um, , we figured that out in real
143:46 by, uh, of course the were listening to um, discussion in
143:52 room like this, not a room this room with the rows of city
143:57 speakers standing at the front and first guy and then me and so
144:02 And so we figured out these differences , uh, in view of our
144:08 . And so, uh, the man went home with his tail between
144:12 legs and uh, a few months he was no longer working for that
144:20 . And so there's a lesson here when you have um scientific difference
144:25 the best way to resolve them is closed doors, sitting around tables like
144:31 with your sleeves rolled up instead of at international conventions, saying the other
144:37 don't know what they're talking about. now let me explain to you why
144:41 is that that uh scale of reciprocity doesn't work for this kind of
144:48 Here's the statement of the reciprocity as I explained it yesterday. Remember
144:53 a vector, its its identity between relations. The force at a uh
145:01 into the data at a source from equals the corresponding product on the other
145:09 . So here's our situation that we involved. Here's our reservoir down here
145:14 here's the gas cloud up above here's A. And position B. And
145:19 for position A. The waves are down, it's p waves up through
145:27 gas cloud of share waves. And share wave doesn't care whether gas cloud
145:31 uh is there or not. So being received over here. Then in
145:38 on the other side we have sources be going down through the gas cloud
145:44 a p way, being slowed up this gas plant and coming up as
145:50 share wave. And so these share are um uh the same. The
145:58 wave velocity is the same as the wave velocities. But these p wave
146:02 are less now. Uh let's look see uh ah this formula applies to
146:14 situation. It says here that the signal you which is parallel to the
146:23 f uh is uh coming out of dark product. And it says that
146:31 signal here which is parallel to the at B. That product is also
146:39 but this data is source perpendicular. example, the sources is this source
146:44 pointing down this way, but the share wave is perpendicular to that.
146:52 this says that this uh this identity applies to the source parallel components which
147:00 essentially zero in. So all this zero equals zero. Our data is
147:07 is perpendicular to our forces so that data is not um constrained at all
147:15 this reciprocity. So that is why data is not symmetrical in the spirit
147:23 gather because it doesn't have to obey scale of reciprocity theorem. It raised
147:29 vector reciprocity theorem, but basically the reciprocity theorem does not affect, it
147:36 affects the source parallel components. Said there's been a lot of advances since
147:44 . But uh we found uh the to uh to this uh when Valhalla
147:56 first discovered, it was a one barrel field of which so far after
148:02 30 years, $3 billion have been and they still have two more,
148:08 billion more to produce. And the for this improvement over time is that
148:16 now understand the razor where better because have better imaging and this is part
148:22 the better imaging, not the complete , but it's uh part of the
148:25 energy. And so there is billions dollars of uh profits to Amoco and
148:33 BP coming from advanced your physical So I showed you uh this this
148:46 here for fairways and I give you exactly the same argument for converted
148:54 So um still quick quiz. Uh this uh is this true Stephanie?
149:09 , that was true. Every single you're gonna be converting pts if it
149:15 a lasting discount. Yeah, So I I know I'm pressing you
149:23 we're almost out of time. Um how about this? This is a
149:31 question. Read that one carefully. . Well it's a trick question because
149:42 it does, it's displaced from the point towards the receiver, not towards
149:47 so be alert to those issues in quiz. Okay, so think about
149:55 single word in the quiz and we're to hand out the quiz. So
150:00 one is obviously true. Uh this false because the vector reciprocity theorem is
150:09 for all seaways for always. now, so we've run out of
150:15 today we still have a lot of to do, but uh we can't
150:21 up the next topic in the time have left. So we're gonna stop
150:24 right here and take up right at point next friday. And so you
150:31 stop the recording now, Utah and I will
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