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00:01 So hello, this is Sergeant waves race uh with a particular emphasis in

00:11 exploration. So this slide that you're at now is the introductory slide for

00:19 first lesson, but since time is , we have asked the class to

00:25 this material in advance. And so been done since we're not going to

00:31 that in this course. Uh we're not gonna cover it in

00:37 I'm gonna stop this um uh stop sharing this. And I'm gonna um

00:49 up the the next lecture. Okay. And now I'm gonna uh

01:14 share the on my screen. So this emergency notification uh slide is

01:34 in here at the request of Of H. Uh and of course

01:39 not, there's no alerts going on now we've known about it. But

01:44 they do is they ask you students the ta to update your emergency contact

01:51 . So make a note to yourself do that. So then let's begin

01:58 the actual course. So here is program for the next few weeks.

02:07 can see, we're starting off very , and then we're uh we're getting

02:13 uh standard rate theory and wave And then there's a lot of complications

02:20 . And of course in in hydrocarbon reflections are really important. So we

02:25 a lot about those here. And at the end of this part,

02:28 here, everything above this arrow is standard waves and race although that are

02:38 topics which are normally ignored in an course like this, but I think

02:45 important because we're gonna make a lot approximations and assumptions up here, which

02:50 not true. And so we're gonna to go back and do a better

02:56 to have more advanced thinking to to to address these issues. So,

03:07 the way, can you all see uh my mouse on a zoom or

03:12 I need to? So they always for lesson objectives. And uh

03:21 uh, here's the list at this here, you might you probably

03:30 you know about vectors and matrices. might not know about tensors, but

03:33 will you probably think, you know stress and strain and you know about

03:38 law. You might not know about concept of compliance is, and you

03:44 not know exactly what uh stiffness is . Of course, these are english

03:50 , you know, the english But of course, there's a technical

03:54 , meaning for that, which we're use here in this post. So

03:59 , and we're gonna mention uh an octopi briefly, and then take it

04:04 again at the end of the So the last distance. So the

04:09 question is, what is that And you probably have a good

04:13 I'm just gonna state that it's a property of a material that makes it

04:19 when you apply a stress. And it makes it recover when that stress

04:25 removed. If you push on. at my uh my hat here,

04:30 gonna push on before me, but doesn't recover. So uh that's not

04:35 elastic behavior. That's a more complicated behavior, which we're not going to

04:43 about in this course. And so what you can say is that electricity

04:49 what makes spring springing Now. This one of the major topics of classical

04:56 in the 19th century uh over 100 ago. However, the last important

05:03 was a guy named E. Love. I'm not trying all those

05:08 uh stand for, but the name might be familiar to you because we

05:15 waves in geophysics which he invented called waves. But you see, he

05:20 this book in 1928 and uh it's the best and most complete um exposition

05:32 elasticity in the literature almost 100 years now, isn't it? You can

05:38 uh it's been reprinted of course and can buy it on amazon for a

05:43 bucks. And so that that might fun. Might be worth your while

05:47 . So the next question is, are seismic waves? So there are

05:51 of stress and strain inside the rocks they're approximately elastic. And so you

05:56 see these pictures here, and you here uh the waves are going up

06:02 you can see if you look closely that near this black zone, the

06:08 squares are compressed together a little And so this is a compression wave

06:13 up and here is a shear wave up but as it moves up,

06:17 sideways and I'm sure you know these facts but you're gonna find out in

06:22 course that these ideas are seriously who simplified in the material that you've talked

06:30 before. Now in these pictures there's surface waves anywhere. Um if there's

06:37 nearby surface then that affects everything. so they're all of course in our

06:44 , there always is nearby surface. put our instruments on the surface,

06:49 ? And even if the waves are from the other side of the earth

06:52 spend most of the time in the deep interior of the earth when they

06:58 our instruments, they know about the . And when we record them,

07:03 don't record the incoming waves. No don't. We record the incoming waves

07:09 the interaction with the free service. all together in in the data.

07:13 so that's why we have to understand . Okay, so um let's first

07:19 sure that you understand the mathematical vectors and matrices. So here's a

07:25 quiz. This is uh not for but uh Stephanie tell us the answer

07:32 the question. The length of this is what uh It's b Yes.

07:40 good. Uh The length of this is what? Okay. Now why

07:45 it that we uh we look at take a look at this. Now

07:51 gonna go backwards and you see we we write it differently. So what's

07:56 going on there? Yeah, well actually there is a real technical

08:04 but it's not going to uh it out that for some applications uh it's

08:11 smarter to write, I write it a column victor rather than as

08:16 Um and for some applications for some of science, that's really crucial.

08:23 for us, we're gonna be doing you said, we're gonna be switching

08:26 and forth. Okay, Now, about this, what's the length of

08:30 sector as well? Yeah. Uh uh how about this? The length

08:37 a vector is given by this little , Are you familiar with that?

08:43 the dot product? It's good for . So uh Stephanie graduated from a

08:51 institution with a bachelor's degree in geophysics two years ago and she was exposed

08:58 this kind of stuff when she was undergraduate and she still remembers good for

09:03 . And so um uh that's the product. Now, I want to

09:08 at this point and uh tell you the screen, you're looking at,

09:18 you see this surrounding part here which uh surrounding the image that looks like

09:25 uh that this um image is part a a digital course. And what

09:35 says here is we're looking at page of 15 and there's some controls here

09:40 I can tell you that that is , you can buy this course from

09:46 scg uh if you do that, you shouldn't do that because the version

09:53 had was several years old and of I'm continually improving and revising it and

09:58 have not kept up, so don't that, but they wanted this set

10:03 to be um so that a student take this without any instructor uh in

10:10 room or online completely by himself, in his um uh in his office

10:18 Beijing maybe and doing it. And there's a lot of support for that

10:25 of learning, which you can see of it here, for example,

10:30 down here, uh there's a little for a narration. So since this

10:35 designed for people who english for whom is not the first language, uh

10:41 have uh feature in the ScG version they click on that, then they

10:50 scrolling down here, they see a uh written words which tell exactly what

10:58 instructor is saying. And the generation recorded pre pre recorded by an

11:05 not an expert, by a non , but he has a good voice

11:11 that's by an actor. And so the actor says is repeated in a

11:17 uh window here. So the student see the words as well as hear

11:24 and that helps a lot of people english is for whom english is a

11:28 language and then there's this other controls you see and you'll see more of

11:35 ? Uh a lot in this course particular right here. So this looks

11:43 it's a link, right? It like uh since it's underlined it looks

11:48 it's an active link and it is the scG. Uh It is in

11:54 S E. G. Version. you click that and it takes you

11:59 a definition of the mathematical dot So if you don't remember uh there

12:05 is. So uh we'll talk more these um um support capabilities in the

12:15 program because they all have uh analog the materials that we're using today.

12:26 . Mathematics so much Rebecca now we the deal with mathematics and mathematics.

12:33 matrix is a rectangular array of symbols or expressions. So individual elements

12:40 called the individual items of Carlin's So here's an element here. Here's

12:44 element here. And this is an of a three by two Matrix which

12:48 three columns and two roads. So uh let's see if you know

12:57 to do matrix algebra. So what's sum of these two matrices? Didn't

13:06 you DD as in David. And did you do that? I know

13:10 you did that. You added this to this one and got 13.

13:15 added this one to this one and six adding up the elements easy as

13:21 . But it's deceptively easy because now the product of these? Uh

13:32 Okay. Uh so so this is complicated and this is where uh people

13:42 fail. So that's why uh we're about that now and so I'm gonna

13:49 this opportunity to uh to quit this lecture and I'm going to um um

14:00 sharing this screen will come back to and I'm going to share another

14:32 I'm gonna have to browse for Sorry about that. She had this

14:38 to hand kids. Mhm. So is what I call math 101.

15:09 , so thanks for that, let I'll share my screen. Okay,

15:31 everybody can see that. So this what I call math 101. And

15:36 the pdf equivalent of this file is the blackboard for your identification. So

15:44 let's uh let's go through this. and you see we're not going to

15:52 all of mathematics. Uh we're but gonna do that, I'm gonna remind

15:58 now of the kinds of mathematics is will be using over and over again

16:05 this course because you will understand that stress is a matrix actually, it's

16:13 special kind of matrix called a And strain is also a tensor.

16:18 before we get into those things let's sure you understand what the basic concepts

16:24 . So now uh the screen asked question is uh is that a

16:32 And the answer is no, it's an arrow. And uh it's an

16:39 feature of Microsoft. So sometime when in word or PowerPoint or something like

16:47 , try typing dash dash, right bracket and it will magically turn into

16:57 ash dash, right bracket, no in the train. And so who

17:02 what other tricks are like that in point inward. But this one I

17:07 uh stumbled. So now the question , is this event And uh

17:13 it has the form of a But it's notation. It's it's it's

17:21 . Now, if it's a notation uh, three apples, seven oranges

17:24 four plums, then it's not a . It's just a shopping list,

17:29 ? Or maybe and you can make laundry list. Uh that looks just

17:33 the important thing is to don't assume the notation means unless you're confident.

17:39 . But if it's notation for three in direction one and then also seven

17:45 in direction two and four units in three, then its effect. Now

17:51 have to uh agree what we mean direction one and direction to it.

17:56 that's called a court system. And most often that is three orthogonal

18:01 Sometimes we just have to and we the orientation and we agree where the

18:07 the origin is. Okay, so we have we can have a unit

18:13 be this long in the one this long in the two directions,

18:17 long in the three directions. And we label the directions of the numbers

18:24 sometimes the letters and so on. the order of directions corresponds to the

18:31 hand rule. Okay, so let see. And the right hand

18:35 So if if if this were alive would take you to uh description of

18:40 right hand and by the way, those descriptions words uh things like this

18:48 in another file which you have on blackboard called the glossary. Okay,

18:55 uh the right hand rule is like uh if you take your right hand

19:00 , index finger 12 and three. , so left hand you get a

19:07 result, 12 and three. C is point of difference. So uh

19:13 really easy to make mistakes uh in systems if you don't observe the right

19:21 rule. So uh even if you're handed use the right hand rule,

19:27 you won't you won't this is the time to be creative. You all

19:33 , stick to the standards. Now lots of coordinate systems and sometimes uh

19:40 directions are different at different points in system, for example, a spherical

19:46 system as radius, uh a polar and has with or a cylindrical system

19:57 uh linked along the cylindrical axis and and as much length. And so

20:05 you can imagine it's easiest to think it in a quarter system. Uh

20:14 here here's a cylindrical cross section of cylindrical system. Here's the here's the

20:21 axis along the axis of the price . And you can see that the

20:25 uh looking at this point, the angle and the radius sectors point in

20:31 direction is that you are over the radius vector will be pointing in

20:34 very different direction. So that's all potentially um potentially confusing, but we're

20:45 set up a mathematical machinery so it takes care of itself. You don't

20:50 to worry about. There are sometimes directions are not even orthogonal. Why

21:00 you want to have uh important system looks like that, you know,

21:05 it on this? Well, it out that there are important reasons for

21:12 instances where that is really useful. example. Uh Do you know what

21:17 you know the mineral calcite? And you seen uh usually at this point

21:23 bring with me crystal of calcite, has natural angles like this. Have

21:31 seen large crystals of calcite? And a rahm bic solid with uh directions

21:43 this. Uh You can imagine appointed of uh of calcite pointing in here

21:49 all my fingers are along the edges the calcite. So what that means

21:55 if you go 33 cell units in direction and seven cell units in this

22:02 and 17 cell directions in this movement be again at uh the corner of

22:11 uh cell inside that calcite crystal. that's a useful thing in that

22:18 Although in this case in this course won't be using things like that.

22:23 gonna be using Cartesian coordinate system. . Okay. We're gonna use these

22:30 notations and Stephanie has already encountered some these. And so uh huh What

22:39 thrown when we use all these different for these records? Okay,

22:48 intuitively, you can say um uh you're so Stephanie, when you're explaining

22:58 geophysics to your baby, Not but in a couple of years,

23:03 is gonna say, what do you ? Mommy. And uh you're gonna

23:06 explaining your physics to your kid. kids can understand words like this vector

23:14 a quantity that has both magnitude and . You know, like position or

23:19 or forced by contrast. A scalar magnitude, no uh direction, like

23:25 and temperature. And those are things kids think about. And so in

23:30 five years she'll be asking you these . So you already told us about

23:38 category in therapy and here's in here's a picture of pythagoras himself.

23:47 was a Greek who lived about 2500 ago founded a school of mathematics and

23:53 he taught things like vectors and And um I do not know if

24:00 made any other inventions besides this but this is a non trivial uh

24:07 . And the direction is given by ratios of these various components to the

24:14 or an equivalent other set. You imagine describing the direction in terms of

24:20 non dimensional um combination. So of the vector has a magnitude one with

24:30 the unit vector. And then we a special symbol for that. And

24:36 the inter vectors in the three quarter will are like this, see these

24:41 parents here. And so uh we answered this one and this one and

24:50 one. And uh now when you those two vectors together, Stephanie,

24:56 was like this where you have these vectors with these these components and these

25:01 and you just added them together. put this one up here, uh

25:07 to nose, just like no, gonna put the green one up

25:11 uh tail to nose. And uh answer was exactly what you said.

25:18 you could think of it, you like this. And of course it

25:23 matter whether you do the Greenman of one first and the green one second

25:28 vice versa. It doesn't matter. so here's the little formula for uh

25:35 the some of those. And that's she told us. Now here's where

25:42 It gets uh complicated and where we to drop out of elasticity to come

25:50 math 101 here is the definition of of the doctor. So it says

25:56 , there's two ways to multiply the product is a scalar and uh defined

26:03 that and you can see that if is equal to X. Then we

26:08 here X one X one and X , Y two X two X two

26:13 X three X three. And so is the square of the link.

26:19 and so here is a really important of notation. It says whenever indexes

26:29 , it means that we sum over liars so we might should have a

26:34 sign here but we don't need it we look here and in this combination

26:40 see an index repeated. So um guy who invented that idea that wherever

26:48 have um an expression involving matrix if you have a repeated index,

26:58 means you've got to some or it's if there was an I and

27:03 J. Then you were talking about IJ component of uh this is 22

27:11 eyes. So that guy was Albert . So here we are following in

27:17 footsteps of Albert Einstein over 100 years , it was about exactly 100 years

27:24 when Einstein became famous because he predicted famous equations that uh gravitational bodies been

27:38 . So we like those by a body I advance and of course that

27:47 completely unheard of at the time, he was well respected. So they

27:53 the next year there was gonna be uh so uh there's gonna be a

28:02 eclipse, wood or in front of sun, so that what that meant

28:09 you could see a star behind the , close to the sun because the

28:16 was blacked out, but you had be in the right place to do

28:19 and you to be there, you to be in the south atlantic

28:22 So the World Navy sent an expedition the south atlantic ocean to observe this

28:28 and to observe the position of the behind the eclipsed moon actually behind the

28:36 sun. And sure enough they found , it appeared in a different direction

28:41 of being in that direction. It a little bit like this and it

28:45 and it was exactly the amount which had predicted. So he immediately became

28:54 and before that he was just a with frizzy hair and after that he

28:59 famous, there was an interesting point he made two mistakes in his calculation

29:03 that And one mistake cancel out the estate. So if he if he

29:10 published uh mistaken uh thing that he have been so famous, but

29:17 he he he would have corrected it of course, but everybody would have

29:23 he's fudging it, but the fact he nailed it right off and it's

29:29 a small, just a small deviation that, he nailed it right after

29:34 became things and he also invented this that whatever and injections repeated it means

29:39 some of the world values. now we're not gonna show it

29:45 But another way to write this thing is in terms of the length of

29:50 . Length of Y times the coastline the angle between the vectors.

29:55 now here's the other way to make uh to multiply two vectors together or

30:02 by the way, this way is way that we talked about Stephanie was

30:09 quite sure. And I'm I'm supposing she would be even less sure about

30:14 cross buttock because that's more complicated. here's the definition, the eye,

30:21 component of the cross product between two X and Y is given by this

30:27 . Uh you can see it's got that the J component of X and

30:32 cake component of Y. And then funny symbol here, which I'll explain

30:36 a minute and you can see that were implied here that we're summing over

30:40 and J. Something over over J K. Here's J and K are

30:47 , I has left over here uncared . And here we are talking about

30:52 United. So what is uh the I J K. Well, it's

30:59 if any of these two indices are same and it's one if the if

31:05 list of I J K is 123 231 or 312 minus one otherwise.

31:12 if you look at this list this is a sort of natural order

31:16 if you take this one and running to the end so that gives you

31:22 . If you take this to and it around to the end gives

31:25 So these are called even permutations of . And if you do something else

31:32 them it's uh odd permutation. So in symbols then uh for all for

31:42 3 um components, uh you see looks like this, it's it's

31:49 I can never remember my this I always have to work it out

31:53 detail. And furthermore, it can shown that we're not going to show

31:58 here. It can be shown that is equivalent to X. Y.

32:03 signed dated uh all multiplying times the vector in the Z direction where Z

32:10 perfected X. Y. So those concepts which we will use later in

32:15 point. So I already answered this . And so so much for

32:23 So mathematics, let's go back to vector mathematically in a vector, which

32:30 to different components if you make a choice support system, according to the

32:35 transformation rule. So let's think let's that this is what a mathematician,

32:40 say right here. So let's examine that means. Uh Remember we can

32:45 any coordinate system that we wish if talking about position. The position of

32:53 corner of my, of my So uh we can define that in

33:02 of any coordinate system we like and completely arbitrary and the laptop does not

33:08 , it knows where its corner right. Uh So we want to

33:13 this uh set up our mathematics so we don't uh commit ourselves to a

33:23 an arbitrary choice of things. So let's talk about this uh in three

33:31 . And here's our coordinate system you can see X one here and

33:35 three here. And can you see two is pointing out of the screen

33:40 is it pointing into the screen? uh Stephanie tell me whether X two

33:46 pointing out of the screen or into screen if we have a right handed

33:50 system or you're guessing so hold up right hand and point it in the

34:00 one direction, point your index finger the X one direction and then uh

34:07 uh your your middle finger in the two direction and it is uh is

34:14 three now pointing out or down? . Okay. So now turn now

34:19 your hand over sir. Excuse pointed ? Yeah. Okay. Yeah yeah

34:30 right pointing out right, and that's way you have to figure it out

34:34 your right hand every time you just answer that question. So now this

34:43 is sort of a logical order system we set up here. Um but

34:49 the physics doesn't care. So we to set up the physics so that

34:54 independent of our choice of of a system. So let's consider this two

35:03 vector. Uh Consider the the position shown here and the length of the

35:08 is this like we said now consider choice of foreign system rotate. So

35:15 can see that it's it's the same . I'm gonna go back and

35:20 You see it's the same vector but we've got a different quarter system and

35:25 gonna call it now in this quarter , we're gonna call it vector

35:29 Prime with elements vector elements, it's prime and X two prime. And

35:38 uh you see the new voting system the same as the old program system

35:46 . We could have changed the the origin too. But uh we didn't

35:51 that in this example. It's only further more than the second reporting system

35:56 also orthogonal, right by our So that only the only difference is

36:02 rotation by a positive rotation angle. . So if we hold up our

36:09 like this with our Z. Pointing of the screen uh uh counterclockwise angles

36:19 positive. So here are the two coordinates written out with their components.

36:30 you obviously you want to know what the uh relationship between these sets of

36:35 and so you can work it out your high school knowledge of plane geometry

36:43 this is what you find that uh prime is given by this former ex

36:48 minister platform. So there are those transformer questions. These these may be

37:00 in the following way for the ice is equal to the sum here.

37:05 showing the some explicitly of the repeated . Uh and then so we have

37:16 matrix here uh with J component It is right down here. Um

37:25 eyes count the rows and J count column. So this right here is

37:37 rule for transformation with rotation like So quantity to a mathematician and quantity

37:43 a vector if it transforms for it to another court system like this,

37:52 we talked about. So that array called the matrix and we denote it

37:59 their Children. If it has two we say that it has ranked two

38:06 we can work through opponents like If each index counts up to

38:11 we say that it is a dimension . So uh one we showed there

38:19 right two dimensions. So uh Normally gonna be um looking at matrices which

38:33 the same rank and dimensions either two 2 or three. Now it's easy

38:44 imagine matrices of rank three or more you can't write them on the

38:51 All right, so we have, we have three indices here, it's

38:55 to imagine that that's possible. But third index uh here would would count

39:02 depth, can you see that here depth one and here is depth to

39:06 all the so but that's hard to on the screen, and I can

39:12 you that we're going to see before day is over. We're going to

39:17 matrices with rank four. And so I can't write that at all.

39:25 either in a two dimensional screen or three dimensional uh picture like this,

39:31 four dimensions. So you can imagine four dimensional that uh matrices are ranked

39:40 are gonna cause special problems for Okay, so uh now this one

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