GeoSci.xyz EOSC 350 (2017), 22 Nov 2016

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From YouTube: EOSC 350 EM Lecture 2

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Second lecture on electromangnetic methods in geophysics by Doug Oldenburg. Electromagnetic induction and coupling.

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Well, irrespective of which president there is in the United States they're, not going to change the fundamentals of electromagnetic, so you're not you're, not out anything on that.

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Okay, what what I want.

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Reduction so there's two there's two lectures the one that we had on Monday and one today that really probably consolidate you know almost as much material as you do in some of the rest of the course they're. Just quite a few things happening, but you know just like magnetics.

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If you look at it as a whole thing, things become kind of complicated. But if you just pick it apart piece by piece, you start to see.

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Like oh I know, I've got that kind understanding, so I want to go through again I'm, going to summarize a little bit what we did on one day and for those people who laugh I check my so but for the people who went to the lab, were you able to kind of consolidate the you know, material from the lecture into the lab is but time you've got other laughs sort of feel like okay, you know what neither so mystified anybody who look at the lab on Monday okay. So what was it like?

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At the end of the lab stuck you're lucky and knowing? What's going on there.

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Yeah, it's I think it's like anything else. That's got a number of pieces to it. The first time you see it. It's like whoa, wait, it's that and then you see it second time and then forget it, and then, by the time you see it third time.

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So here's the here's, the basic elements we need to have a time varying magnetic field that generates for us an electric field everywhere.

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If that electric field happens to be in a place where there's a good conductor, then there's going to be currents that are set up. That's called Ohm's law, so current density is equal to conductivity times, electric yeah, and once we have these currents that are set up, then those currents give rise to magnetic field. So every time you have a current, you always have a magnetic field, that's associated with it, and you can always get a sense about the direction of that field. By that right hand, rule.

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So what we were looking at is the following: we've got a transmitter which is just a loop of wire with the generator attached to it. So there is a harmonic wave that goes in there. There's it's a sinusoid, so that of course was going to give rise to a time varying magnetic field and that magnetic field comes down and impinges upon some target here and the thing that was really important about what happened here was something called the coupling.

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And the other thing was the flux, so the flux.

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Was the normal component of B? So if we have a have a loop here and we've got a magnetic field coming out like this.

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Then the flux that comes out is just the normal component of that times. What whatever the the area is, so that if the loop is like this and the magnetic field is coming up like this, then it's just the strength of D times the area.

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But if the loop is like well, maybe that's better.

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If the loop is like this and magnetic field is like this, so it's parallel to it then there's no magnetic field, that's cutting through. So therefore you know B dot and hat so n hats this you know yeah, it's just you know equal to equal to zero and in that particular case, when this is not equal to 0, then the electromagnetic force or the voltage that is produced to that is just the time rate of change of that magnetic flux.

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So an important component here is going to be this Mitter and then how that flux impinges upon the target. So this was a picture that we showed last time. So if we imagine that we've got some kind of coil of wire here, so that gives rise to magnetic fields like this. If the target move is sitting here, there's a lot of flux- that's going through so you're going to get a great big current, that's going to be generated.

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On the other hand, if your target move is like this and the magnetic field is like this, then there's no flux, that's going through there, so you're not going to get any currents so depending upon the orientation of your target. Compared to that incident thing and field you're going to get different amounts, currents that are induced.

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If we put everything together, we've got a transmitter loop, we've got our target and we got our receiver.

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What we're altum Utley interested in- and this is going to be the data- is this secondary magnetic field at the receiver compared to what the primary field is at the receiver? So it's going to be this kind of a ratio, and that depends upon quite a few things.

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The important thing is how everything is is coupled and when you do the lab you're going to see a turn that looks like this. These quantities here L are just they're called a mutual coupling coefficient that route that connects Luke 1 with Luke 2. So in this particular case here with Luke one like this there's lots of flux- that's that's coming through here when this is oriented like this.

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This coefficient is actually going to be pretty large if the this loop was oriented differently. So, let's suppose it it's almost horizontal, then there'd be no flux, that's going through! In that case, one two is very small, so these coupling coefficients depend upon sort of the relative orientations of the transmitter and receiver, and you have the option in the lab of kind of adjusting these guys to you'll have greater or lesser amounts of flux.

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In the end, it's it's kind of a multiple sequences, that's going on, because we've got the flux from the transmitter coming in here good. So that's this l12, but now we also have to connect whatever currents are going on in here with this loop back in here. So if I've got magnetic fields that are produced by here and there's a lot of flux lines, that's going through this loop, then I'm going to be good to go. I'm gonna have good say no.

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If that move had been at a different geometry, then I wouldn't get so much signal. So the strength of what we're going to get here depends upon the coupling between the transmitter and the target, and also between the target and the receiver, and that's what gives this guy and then we're dividing it by the primary field, and that depends upon the coupling between your transmitter and the receiver. So there's a direct there's, a direct signal. It's just going a primary signals going through the transmitter to the receiver as well as this one. Oh.

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Okay, so that that was the the first part and we're going to come back at that and they're gonna sketch out what happens as we drag this system over top of a pipe. There was also another term and I don't want to get into the details of this guy here, but this this is really dependent upon what we have for a target.

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It depends upon what frequency we have kind of the resistance and the inductance of target, so for what we care about is just sort of a picture of this guy looks like, and that's what we show here.

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We have on this axis here, a response on this axis: we've got that parameter, Omega L upon R alpha and as we go up here, this quantity is getting bigger, so it could have higher frequencies or we could have lower resistances whatever, and we go down the other way we get dollars that are smaller. The important thing here is that there is a blue line that constitutes what we call the in phase or the real part. Did you question.

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No, that's just the inductance of the loop remember. We said that we're gonna have it as a circuit so that that circuit, you know it's got a resistance, and so the only things that are coupling are these guys here and that's just me so.

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The blue line represents what we're going to call the real component, which is the component. That's in phase so remember, the secondary field has got it's still harmonic and then I was trying to get one round with my hands. Last time. Pigs are in phase. You know we kind of go like this, so that's the real part. So anything that comes back it's our money. Some of it is going to be looking just. You can break in two parts of a certain amount that is in phase with this and then there's a certain amount.

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That's other things. The in phase part is governed by this blue curve, and the out of phase part looks like this. So at very low frequencies, the out of phase part, quadrature, part or imaginary part all same words or different words with the same thing are much larger than the real at very high frequencies. The real part dominates much larger than the imaginary body.

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Okay. So now we can put all that together with one more thing, and that is we're going to plot this quantity H s upon H P and we need to have a sign convention and our site Convention is going to be the following. If we're always going to have at the receiver a primary and the secondary feel if the primary and the secondary are pointing in the same direction, we're going to say it's positive, so they could use both going up or they could both be going down.

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That's fine that we're just going to plot that numbers. A positive number, if they're in opposite directions, we're going to say it's negative. So the practice the primary is going out there down or primary down secondary up there in opposite directions. So it's a negative value.

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Okay, here's our situation, we've got a pipe and we've got our instrument. That's we're going to move over this fight and we're going to sketch out this curve and the important parts is there's a positive part. There's a negative part. Another positive and there's two zero crosses.

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So here's your an thirty one transmitter receiver and.

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At the very first part and I want to break this down into two things, let's just think about the transmitter being on this side of the target, so we've got a target, think of it being a plate or something like that. We've got a transmitter like this, and so now the magnetic field is going like this right. So that's the primary field, and for that part that comes through this way magnetic field.

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It's coming in right now we're going to have currents in here that are going to counteract that so magnetic for the primaries coming in like this.

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That means we need to have a magnetic field, for instance at the top. That's coming out this way. So then, now my right hand rule says up: it could oppose this, so my magnetic field.

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From here is coming out down in no and that's giving rise to this, let's look at that magnetic field. That's coming in like that. If we're sitting up at the surface of the magnetic field here, the secondary field goes like this. So if I'm sitting here with a loop and I say, okay I've got a primary field. What direction is that that's coming down like this I've got a secondary field this, so that's coming down both in the same direction.

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Positive number that's what's happening over here and clearly that decays away, you know you simply go farther away, then everything just gets smaller. So at some women out here everything gets you know zero, and then it rises up. It's positive, it's positive, but then what happens as the receiver starts to get towards here?

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Well, as the receiver starts to get heat towards here, in fact, if at the moment that the receiver is sitting right above the plate now the magnetic fields are coming in like this, so my magnetic fields are parallel to the receiver, so there's no flux coming through so I'm going to get a zero in my data because I'm just not measuring anything, there's no flux coming through that coil okay. So that gives us this guy.

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What is then going to happen if I actually measured something up over here? So I can still imagine that's transmitters on this side. If now my receiver is sitting over here now, what's going to happen, I've got a primary field. Is coming down.

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Look look my secondary field. It's kind of just looking like this, so my secondary field is pointing up. That means. I'm gonna have a negative value, so when my transmitter and receiver are straddling this guy here, I'm going to have this negative value here. Okay, so now we've got this part. I've got this part.

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The next thing is, as we move over just.

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One more if I keep moving this now so that my transmitter gets to be right. Over top of my target, I'm now also going to get a zero value but I'm going to get a zero value, because my primary field is parallel to this guy. So there's there's no time varying flux. That's going through here! So there's no currents! Nothing! So here, there's there's! There's nothing happen. So that means I get no magnetic field.

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That's from here, which means now I'm trying to have a zero and then, when I move things a little bit farther apart, so that I've got transmitter and receiver on one side, I'm a back exactly to what it was over. There.

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So that's yet, and that's why I told you on Monday that you know if you could, if you really really could sketch this curve. Oh then you really understand a lot because, even though you know you could just simply kind of draw like okay, here's here's a sketch right, there's two zero crossings, but the two zero crossings are happening for very, very different reasons. One there's just you're not capturing anything with the data, because the orientation of your receiver is in the wrong position.

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This one here you're simply not igniting the target so same thing, but very different physical causes.

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Okay, good, so that allows us to sketch out what the.

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That loves to sketch out what the sort of the structure of that curve. Let's write it, so we got something. It looked like this with plates to be on here, but now we have to come back to say, although we got this in-phase and quadrature phase part and that's where this other diagram comes in.

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What this does is tell us what the relative proportions are in the in phase and out of phase part. So if I'm sitting, if I, have a good conductor, I'm standing up over here so remember this quantity here was Omega times Sigma. So if I have a good conductor I'm sitting up over here, that means that the real part is much larger than the imaginary part, and so, when I went out and I look at my instrument, my instruments going to give me two numbers.

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The real part is actually going to look like this, and the imaginary part will look exactly the same, except it's just got a smaller amplitude, so a character of the curve will be the same, but the relative proportions of the in-phase and the flexure are different and they are governed by this particular image this year.

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Okay, so that means that you've got a way of kind of understanding what the character is of the signal, as well as getting some information about the relative strengths of the two numbers. I.

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Want to maybe just make one comment back here: one of the things that's really useful and I might ask you a bearing of that on the your. The final exam is that we've done some sketching out here, where we've got a a moving, a moving system, but I could equally well. Ask you a whole bunch of other questions. I could ask it okay, so suppose that you had just a transmitter here: okay and then now you're, going to just leave the transmitter fix but move your receiver, so I could I could do that right.

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I could just keep the transmitter and I could I could block a receiver across here, in which case I would have over here. I would have a you know. A positive value that was obtained, I still have a zero crossing, but over here I would have your just a negative value.

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Everything is going to be and the transmitter is always going to be on one side and receivers. So I'd have a very different signature, even though nothing is not going here just that same transmitter at the same currents in here, but it's just what I measured the other variant that could it could be asked is instead of having a transfer a receiver. That's just horizontal like this. I could say: okay, so what would happen if you had a vertical coil that looked like this? Maybe it was doing.

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Okay, so supposedly a disc, so we have a transmitter here and so that's going to give us this primary feel. That's coming in like that right. So we've got the current that's coming out here and in here. So now my magnetic field just secondary field. The magnetic field is looking like this.

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So that's my secondary magnetic field kind of sneaky. So now, if I measure so now what would happen if I measured the vertical component earlier, if I measured the horizontal about having my receiver in this vertical orientation. So now, instead of having a loop like this I'm gonna have a loop like this.

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So first of all, what what is? What would that sketch of the of the curve beam.

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So again we think about fake. That's our secondary feel. Maybe I'll just go out another here.

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So here is our primary. So.

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Yeah, so here's our primary feel that looks like this.

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So you find, if I'm sitting with a vertical coil, okay, then my.

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What would what would my primary field look like in my vertical coil.

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Yeah, it's good, it's you're all right!

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So here's a case where, if I actually tried to do you know HS upon H, P I, just don't have a primary a primary field through the coil I only have it only have a horizontal one if I just look to see what the what the secondary signal is from here right, so, instead of doing a chance upon H P if I just said okay well, what's my secondary with my secondary heal?

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Okay, now as I'm coming over here, my secondary field is coming up like this and it's coming through here and so I'm going to have a very large positive as I get across here and it's going to be.

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Decreasing as I go from back before so the net overall signature is actually bringing something. So my point with all of this is that I'd, like you just to think about okay, I got transmitter. I've got something here: I'm going to get some currents now, I got the magnetic field and then now I could measure any particular component. I could measure horizontal component a vertical component whatever as I go across here, and that would give me some information about. What's there.

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Okay, so that's as much like what a kind of dwell on this I want to do, one more one more thing here to put into perspective when we do this survey.

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We've got the transmitter here and we've got an earth okay, that has got some electrical conductivity to it. In other words, the background isn't free space like we didn't think about with the with the loops, but we actually have a conductive material here. So I want to now take a look at what the effect of this background.

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Conductivity is because that it's actually going to attenuate the signal, as as we go down and there's going to be something called the skin depth which we'd already seen when we did the GPR and we're going to it's going to play a role here too,.

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So imagine something called a plane wave. So a plane wave is just a wave, that's kind of uniform in all spatial directions. And it's it's coming down on this table and it's you know it's just oscillating as it's coming down. So it's it's a a sinusoid, that's propagating down, but there's nothing happening. Laterally everything is uniform.

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So if we look to see what happens in the air for that G for the e/m 31 frequency temp, the fourth Hertz, the wavelength up here- is about 30,000 kilometers, so lambda.

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So lambda is C times a period or C divided by F and C is velocity of light. So that's actually pretty big number. So we've got a wave, that's coming down.

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It hits the earth, but in the earth there's you know it's a big conductor. So, let's suppose, even that you had ten millisiemens per meter, which is probably pretty close to what the conductivity of the ground is outside. If everything changes as this wave comes in. First of all, it starts to travel much more slowly.

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The wavelength changes and it also attenuates so as the wage traveling, the air nothing happens to it doesn't attenuate, but as soon as it gets in here, instead of traveling at the speed of light, it now travels according to this formula, and that means that for this particular case, it's about you know three times ten to the six meters per second instead of ten V eight, so it's much slower and the wavelength, instead of being thirty thousand kilometers or thirty thousand meters, is actually 314. So there's a couple of orders of magnitude different.

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So as the wave comes in, it's got a different wavelength. Much shorter it propagates more slowly and it attenuates with something called the skin depth. So we've seen that before so Delta.

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It's called the skin depth.

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Anybody remember what what it was so as as as a wave sort of propagates in and what was this in depth. The definition.

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Right so when, if this has an aptitude of Wanat surface- and this is 1 upon e, where e is 2.7, so it's about 35% or something like that. But this that this depth of which that said, the depth of which it hits that it's called the skin level.

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So the skin depth is actually given by this partner, there's another one that will use it to the GPG. So if Delta is equal to apply under root, Rose resistivity upon aft or even 501, upon Sigma.

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This is actually pretty handy formula to know it just kind of grounds you lets. You know how how much things are attenuating.

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So here's here's that expression for the for the skin depth, basically just redrawing what I had had already done, because aptitude of depth and what that means for that wage that's coming in is that you know it's still oscillating, but that you know the amplitude is just dip decaying, like this.

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Okay, so how does all of that play out when you're at the expo site with your am 31?

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We have a system that looks like this. We got transmitter receiver and there's an earth that has got. You know some particular conductivity to it, and the question is how how are we actually sampling that earth structure and how can we get information from our data, above that so I'm, going to talk about something called a response curve and show you kind of how that works?.

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So I write something called Phi sub B, so the B is vertical and.

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The vertical stands for a vertical magnetic dipoles, sometimes you'll, see this. The MD is a vertical magnetic dipole in every said that if I have a loop of current right, so if I have that loop, then that gives rise to a magnetic field. It's just like a magnet. So if I had a loop like this, then the magnet is like this.

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So the magnetic field lines go like that. For that reason, people have chosen to refer to a horizontal loop as a vertical magnetic dipole, so might be a bit confusing, but trying to get that's what that's what we mean by this.

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The second is: if that system is sitting here, there's going to be a response that that's coming from the earth and the question is at what depth is that response coming from? And that's characterized by this quantity here, which is the response function of coordinate system? So it goes up like this.