GeoSci.xyz GWB Lectures, 13 Feb 2020

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From YouTube: 2020 02 13 Day7 Part1 InversionFundamentals

Description

In this lecture we present the fundamentals of geophsyical inversion.

A

Because data those are the raw data and we want to make sure that you could repeat all of the results that max and brand and Devon have been doing to get a final answer. So.

B

The max okay lovely no get this up today, but better be able seconds at all correlative GN m00n, I okay. Well, no I, don't see no sense in the popular reception ominous rock written yeah, so.

A

Today we want to talk about inversion, we're going to work in two parts. One is a simpler problem: it's called the linear, inverse problem, inversion.

B

Dilemma in watching the pivot row, the muscle, not leaning versions, are lying about lying down in Atlanta, but that the depression game.

A

The reason for this is that it sets a complete understanding of the fundamentals of any earth any inverse problem.

B

Etc, the Mazel Tov she can turn it into Josh and kidding at that time. Baby banana salute, let's see it or not it. We are a non living area accepted. It was on the world by.

A

This afternoon you will be able to apply that to the 1d DC resistivity inversion that we did in the park. The.

B

Universe, are you know the one thing that 1d 101 the resisted? That was, you know now about me. So.

A

Recognizing that maybe the mathematical level, students is not very high. The.

B

Montello, who the intended by anyone anything on a felony and that it.

A

Is difficult to proceed quickly because of language barriers and translations, the.

B

Anunnaki God, the things are: never a meter and IA, not water. Under.

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B

Panna cotta, okay, John John, I.

A

Will try to be as simple as possible and give only those things that are very important. Go.

B

Out then, there's all you, kids are looking at me about I need today, look something move on.

A

The process excuse me of inversion is going from our dataset that we have back to a model. We would call the earth an earth model that would reproduce those data right.

B

David I'm not gonna, do that ever imagine travel all the detail. You would own a computer process. Look up it's a concept developed by Romo through the lithium model organism or the cuckoo yawn. Okay, the motor love it. The insulative was a pyramid of the canonical.

A

For inversion, there are different types. We talked about parametric inversion, biometric.

B

A

This morning, I'm going to do the linear inverse problem and we're going to use the word Tekin off because he's the person who set up the framework for.

B

This see she got a move on. Oh here's, another tip, announcer, a new nightmare, the hot device keeper Tippin off. Imagine of my life in a man will.

A

Work with what is called a linear problem.

B

A

Will have an opportunity to do hands-on, an app for the linear.

B

Inverse, probably thank you, don't you do metals and economic and then.

A

We will do the DC inversion for 1d.

B

4Dz one thing version above it.

A

So how do we attack this couldn't miss this problem.

A

We have some data and we might process these data and get them ready for inversion. So.

B

Of mazo measurement without a TWA did not possible. Oh don't be silly girl. Imagine that I would do a PSA our process, Leviathan of.

A

When you collected data in the field, there were many things that needed to be done to the data to see which data were bad, maybe smooth the data to give you processed data, so process.

B

That we named I got. It did.

C

B

Not the way yeah la-di-da-di-da found epidemic Gombu, they did not home where they did their needs. How the general fancier body so.

A

We want to take those data and we want to invert them to find a model.

B

But not so you know nied data in Batmobile, inversion of aroma. So now so the model of a cloud or lower body. The.

A

Way we do this mathematically is to take our earth model region and divided up into cells. Each of each cell has a constant physical property.

B

Throughout the model ago, they telegenic enough Amahl, don't know who lives up by my way veto service TN don't need to tell them a bit of my who are like open our body. She did not move, she does without a lieutenant in the.

A

General inverse problem for DC resistivity, each of these little cubes would have its own resistivity value, and we would want to adjust what those resistivity values are.

C

A

That we reproduce the data throughout.

B

The Digimon dear Tony I, don't need a baby. The Aquarius until Adomian see male the rarities resistivity time holy. A Hodari, the ROI, resistivity and internet want to be a member to or not so. I definitely have no take sequel of sanity. Problema. Okay, oh, did that more than accountability, buddy we're.

A

Going to set this problem up as an optimization problem, where we find a solution by minimizing something, but.

B

The house Rockies, are they give them a location, a gamma short, something that I've shot a beautiful yeah a short-sighted Enoch Oh Casey, not buddy,.

A

The difficulty is that the number of cells that we have in our mathematical description can be very large, so they.

B

Cells in my body means having any up today: giac vien either, but I think you know yeah. So.

A

Maybe we've got a hundred thousand cells or a million or ten million cells that these cells that sends a quickly.

B

Put the I'm not kidding about it, yeah.

C

I thought oh come on, so they.

B

Give you the beta I like what the unite their support on today. So.

A

We will always have more cells than we have data that.

B

Sounds of the year: oh, you mean the block right, yeah.

A

So we'll have more cells, did.

B

I, put the I mean I can I need that took a mountain and yawning anybody.

A

Remember when we talked a few days ago, we talked about non-uniqueness and I said well. What would you do if you had to model parameters x and y, and the only thing that you knew was that their son was equal to one.

B

The neon even a big of X plus y equal to one so brilliant and I'm loving hearts. You don't know yeah I, never took that they know so. Okay, muted, ser me non unique. Look your your! Ever now. You need to be the one. The de marte at the example.

A

So we have more unknowns, let's so I'm. In this case the number of unknowns is equal to 2 and and the number of data is equal to 1 ohm.

B

Anima Molina being a nun, would eat knitted day and if that would be diffident what I'm your neighbor, not so,.

A

We're solving a problem in which there's more unknowns than there are data.

B

About his young, listen if an IV be seen it yeah, but how can see under the nicer my leave any boundaries. Remember that oh yeah.

B

A

Means that the inverse problem to solve it and to get a solution actually requires some other information which is called here prior information. Prior information.

B

About a child's have a tenant than I do on it. Imagine of their comma.

A

So that information is sometimes geologic information.

B

I mean the teller the other day, maybe that's a absolutely no name so for.

A

Instance suppose the geology one in one dimension. So if this is depth zette and here's our resistivity, suppose that we're out.

A

The hard way- and we expect that the reasons Tiffany might maybe go down. You know something like that, so here's Jeff and here's miss Tiffany. So there's a low resistivity zone high reasons Thibodeau we're expecting something that looks like that.

B

Don't comment on it yeah one time she meant I've got an X that she made, though resistive.

C

B

Tacubaya angry, so they've got really.

A

So we can try to find a solution that has got this character. The.

B

People's Nvidia beautifully the variable tau yummy. We.

A

Would not want to app a solution. Let's say that looks.

B

Like this I love, you, okay, music,.

B

Smoochie did keep me for not so.

A

We both of these might fit the data, but one is just is not geologically acceptable, so.

B

He thought would people don't notice I? Am you grabbed I love? You cable, that came here, I mean any bugs and ideological yeah, but then I take grow pockets. They are fit to the data. I, don't like legend below so amazed enough shocked when I, probably crafty triangle, or not as a young team, but the Sanwa meaning the entire business and yeah. So.

A

The kinds of information that come that you might have are that if we go into so, let's suppose this is well. Let's do it again suppose this is the this. Is the field area, monks on carp right, maybe you know this is a hundred meters and maybe you think Oh. The the model should generally lie between. You know: 5000 meters and 300 gold meters.

B

Now these are resistivity you're at the body, Benzema thank them any holiday. Okay, okay, well, no house there any Damayanti for Allah. Only that I have achieve it.

B

A

Know overall, I think that the resistivity should be maybe someplace close to a hundred of meters, so.

B

Oh, listen, Steve, div, say: I mean I'm a cement policy, human.

A

So we could call this a reference model reference.

B

A

Alkali, so we would like to end up we'd like to find a solution that was unity close to a reference model. Reference.

B

Model- and he says all the chemo Samaj it buddy, so that's one thing that we could put in Cubano.

B

A

Other is, if you have two options: if you had two possible solutions, one of which you looked like this, so that's one and the other solution which.

A

Looks like this, which.

A

One would you take.

B

He took that my impression of their hands are up, but the magically guna, both on you, even my family, now put the doctor, know like fit.

C

Terracotta, it's.

B

Also sitting at the best solution now.

A

Because it's simpler.

B

It's producing about one whole body whenever.

A

We do inversion. We try to find a solution that has got the minimum amount of structure that still fits the data. They.

B

Don't let fit man, Thibodeaux structure, I, believe a nasal bone soup doing it I'm opening choking more nasal. What was out with your body.

A

When we do the inversions there's going to be opportunities to try to get this particular result here. Compared to this one.

B

Make a mistake officer so.

A

We're going to set the problem up so that we can attempt to get these smooth solutions, in contrast to things that are just have too much wiggliness too much structure, yeah.

B

That definite uhm, you know sitting how JEWS hydrogen or AP. We are right.

A

So all of that is going to come into this part here, the a priori information.

A

So in the notes, I'm going to talk about the statement of the inverse problem, we're going to talk about data misfit we're going to talk about something, that's called norms. These are just mathematical entities that tell us the size of something the.

B

Other of these statements are really see mentality in first problem, which in Java me you know, did a misfit for the deputy metal exactly never be Obama enormous. These are the norms around no.

A

Enormous way, I'll talk more about it and it's a way of measuring the size of something said: we have norms for measuring height right, so we have a ruler for measuring height right, so I could come along and measure everybody's height. That's that's kind of like a norm know.

B

Who thank that Mosley? Oh yes, I'm is a wannabe who they might be Julia develop over 10,000 the cool norms around.

A

These are seen Jeremy.

B

A

Go through a tea, Knopf inversion and then I'm also going to give you a flow chart which is useful for every inversion that you do.

B

So we upload Chuck payment blue time. My dog look at them: yeah I, look at voluntary, I approach, every other technical techno proach with a dame on there.

A

So here's our statement of the inverse problem that.

B

A bus problem, it was didn't I, a statement.

A

So we're given observations, D and maybe there's n of them. So this is like d1 d2. So we have a string of n observations so.

B

These are about as rough o observation, though, which is we collected one. The collected data yeah.

A

B

No come on Harley, Quinn Sam, you know you turn it I got. It did I what you're a deep observation well observed. Detail of your body and.

A

We're given some estimate of their uncertainties, you know.

B

Oh, but they don't move. Oh come on. I look mommy I,.

A

Want to stress that a number, a geophysical field- number that doesn't have an uncertainty attached to it- is incomplete because we need to know not only the number, but we need to know the error bar. So.

B

I uncertainity generally with a demography, errababu rava, a new music, my tank and a puppy. You have.

C

B

Of aluminum, you know I I'm certain. It is ready. Error, go tomita, kousei, not having my mom the.

A

Model that we are looking for we're just going to write as a generic symbol, M generic.

B

Symbol and with Obama it and.

A

Then our forward modeling is we've got some operator, some computer algorithm that often operates on that and generates data. Anything.

B

That were generated indeed, I'll be a bit up cross a slipper Tom I have enough. So what modeling in other word for Markov model is.

A

Simulation so forward, modeling simulation recitative, no, no predicted data addictive.

B

So I Ford model in general thumb. You know your holiday with the nearly who are modeling that off oh, come.

C

B

Table or observation data so young it actually or not,.

A

So here is a linear problem. Do not be too afraid of the integral I'm just going to make the falling. So the M is the thing that we're interested in and.

B

Then you'll needs another day and the D is the data. The other day that box your body and.

A

These things in here they are called kernel, functions.

B

A

They tell us about what the experiment.

B

Is like Deena, that's a couple opposed a few days occupy.

A

We're going to refer to the Diaz data. Do.

B

A

A Miss the model and come on OVA and G is the kernel function. G sub J is the kernel function for the J data, so.

B

The other color function, look don't go out in it.

A

In the app that you'll use, this is distance along this way, and this is amplitude amplitude of the model. So this is the model, and that times to be the not yeah.

C

B

A

Can go from zero to.

B

One so I was the only one: oh I'm at water, quality of WI ye and.

A

Here is the the model, it's got, you know a box car and then maybe a Gaussian type damage.

B

I've got in tackles I really make but Bobo's I'm illegitimate, a minor yeah and the.

A

First datum or the first kernel function. The suppose looks like that. You can control that. But, let's suppose it looks like that kind.

B

Of functioning articles- and you see a general comments during the.

A

Day time associated with this model, whose fault is simply the product of these two things and then summing them up, so we take this. This goes from zero to one. So if we overlaid this on, this multiply them and sum the numbers up that gives us the value of the data so.

B

A nickel found below okay tell a mess anybody, I can imagine. Od did I a huge and everybody.

A

Shoot you're supposed to okay, so I'm going to drop the linear inverse app. It's going to look like this I'm going to give you a demo. First.

B

The linear motion Apple the most recent bone. Oh yeah, the only light and.

A

C

A

So this is the linear in inversion, app and I'm, going to start it by.

A

Restart completely because I've been running yet and I will.

A

I'd like to do a run, sell all.

B

A

Can watch first.

A

A

A

Material at the beginning of the app that talks about the purpose it talks about the forward, modeling that we that we do look come more about it later, but these these kernel functions are kind of oscillatory. They go up and down like that, but they can still they still decay.

B

C

B

Not gonna work then.

C

I tell it that I tell you like was I.

B

Bobby a little.

B

A

For for our kernel functions, they tend to look like this, so this is 0 to 1 and then the blue one starts at 0 at 1, a value of 1 that just gradually deep.

B

Decreases but yeah and without repeating it those show show us inaudible. Then you've got everybody.

A

So we'll come back to that that this is not so important, but we need to add an example. So this is.

B

What I'm, choosing you don't even it out the moon, I'm energy was already done, can't be magnetic element.

A

So we're going to set up the experiment which is and the number of kernels and.

B

A

That will be the number of data and M its yes, so n is the number of data which is the number of pearls, and you.

B

Know they got it, nobody does it now.

A

We're going to take this region 0 to 1 and we're going to divide it up into a hundred cells each of 100.

A

B

Samson Yahoo I will not get ourselves.

A

And now that's that's step. One look we'll come back to that, but that basically sets up the problem.

B

A

The next thing is to generate the model and simulate the data, so.

B

That Moldova's entertain the rhythms of jam. You know process limit right now, Wow an alias, oh, come on how to get on it.

A

So we have this set of slider bars here. You'll see.

B

We added bodies, you men, so.

A

B

A

Background kind of tells you, the level that you want to have as a background we're just going to set that equal to zero.

A

It's automatically set so.

A

Now we want to make an earth model, so this is the function here that we want to me, and the widgets up here will allow us to make this. This function block.

B

Body with shading you came here, there are multiple.

A

So in this particular case we have.

A

An M we have a boxcar. So that's what this! This is a blocks.

C

A

And then this is a Gaussian, so.

B

The money you know if you have to be in the quad, the rule of thumb, bow down low like an athlete Roxy and look alive if.

A

We wanted to make the boxcar bigger, then we could adjust the value here of M, 1 and.

B

The 1 goes re later love it. Hey penny, Bosco that it would see you as it's in a ceremony, so.

A

These three first parameters here are connected with the boxcar boy.

B

Ain't ID anybody know the abalone. He lost that bull so.

A

Let's say your buddy yeah, so we can make it bigger, so we could make it right.

B

Do you, ladies a lot of boots, you are a we.

A

Could make it 1.35.

B

You tell you got it.

A

Yeah, so we could, we could make it anything and we could also move it. So if we wanted to shift it to the right.

B

Eddie Eddie both possible a shouldn't. He have a shooter got it: okay,.

A

So that these first three runners here allow you to make something with a boxcar and move it around yeah, oh yeah,.

B

The boss, coq au vin, the hole should be straight zenith it up a little got it Bob, the body so.

A

The same is true: with this Gaussian we can make it bigger, we can shift it, we could make it skinnier I, don't.

B

See enough collective of it, the young Sheila I'll, move it off Oh. So.

A

If I wanted to make it bigger well, it's already I could make it I could make it negative to make it go up this god, negatives.

B

Are the puppet that way.

A

Or I could put it in the middle and then aptitude I could make it shift it over. I could put it over here. I'll show you that day, I could put it right in the boxcar bow.

B

Stopped at nndb.

A

I I could put it back over here. I could increase the altitude and I could make it skinnier.

B

But we have another line.

A

Could make it like that? Okay, so what this? What this panel does is, allows you to generate many different kinds of models.

B

How much get body optimized yet multi-million wolf tell up with that! Please hi everybody and this.

A

Is what we want to.

B

Find D'amato so.

A

Our goal is to find this and we're going to do that by first of all generating an experiment in which we collect some data. So.

B

They're an experiment: it was in regular that bubble of their path- arrow, you don't eat that, and.

A

Then we're going to invert those data.

B

Hey Dude: are you in vastavaiya.

A

So there's a little button here, I'm a.

B

Big hug both add noise. What is she did? The north and north of Hawaii yeah.

A

So we could, if we wanted to add noise to the data. Here's five percent plus a 440 go to the.

B

Matter, norske, 5% and I do get it. Man Flula got it up, see. What's there to right.

A

So, let's, let's go back to what our defaults were so here to here's our computers, our default, that's unclick! This now remember we had these are physical problem, which were these kernel functions. So each stick for each data, we're going to take the kernel function and multiply it into this boggle and and get they cannot.

B

Function, ability more inator video market update or modify so.

A

Let's see, then what the.

A

So I could I could plot the kernel functions, I can flop the model and I can also plot. The data did.

B

I tell you let them yet it Mulder Latino attic, so.

A

Those are the data without any additional noise paint see the dick. You know these are the data that you collect in the field and they're actually fairly smooth. So.

B

You're not gonna. Do daddy daddy nicely borrow your local dialect out another nipple on it. Why not Iran, but.

A

Generally, we will have noise on the data, so, let's now add some noise to these data need.

B

I'm originally yeah, we have, it did not be nice ba vitadetox, they tend to yeah. So we click this button and my daddy I don't know absolument.

A

Yeah, so that adds noise. That looks like this and maybe for this example. We want to even add a larger, noisy.

B

Does nicely takes out the stick.

A

Yeah, that's better so.

C

A

Here's our data and here's our noise, my cetera.

B

A

Know: okay, so if we think we're going to fit the data, so you know maybe any line that's kind of coming through here might.

B

Be okay: ain't, like nah I era, body yeah tomada been the album, so kept our missing, not if they don't fit good, yeah. haha Okay, so.

A

Now now we're all set to go. We've got our data, we understand our forward modelling and now, let's try to in.

B

Early Daniela be mornings, are gonna, rot the commanded, a legion of the lady there's a passage or santiana a.

B

B

How do you put floor floor floor? Oh I,.

A

Just chose something in when you go up with the instrument and 4dc resistivity. If your data are going to be voltages, there's always some both random noise value. That is happening, and so you've considered that to be the poor.

B

Floor is like is nothing written noise, but.

A

Yes, noise you're trying to estimate what the noise so.

C

The forest, of course, your.

A

Estimation of what sort of the background noises background noise.

B

How they relate to Florida masturbation? Oh.

A

Maybe not so the floor if you can imagine a dipole-dipole survey and then if the separation between electrodes gets to be large enough- and you know you're, just measuring the background noise. So that's that's your poor, lower.

B

Than that all the time machine to machine wanna go I, don't know a nearby bar, so Ducati from channel 8 I got one. Would you mind steppin on.

A

The percentage takes into account other things, for instance, if I take my potential electrode and it's my potential electrode is supposed to be here, but actually I've put it over here then there's a can be a big difference in my voltages, just because I'm well the big difference in there because I have my location specified incorrectly, but.

B

uh Generally, the person HT gyro Boone original natural Tom in the IT job or no detail. Oh dude MIT that column II haven't seen Evil movie, you know, but it's a legitimate in the end, really give a poor Aloha.

B

A

A

So here was here was our voltage: okay, that depended upon thee. So here was the current going in here and then here is the voltage member how fast it decays. It falls off as 1 upon R.

B

T, they will say nautical time you I mean tonight today.

A

Suppose that your electrode was supposed to be at one point: zero meters, one.

B

Point it'll be done by natural Tom. You know that was higher, but.

A

So it should, you should have got that point here, but actually you were out at one point, two meters which is here so.

B

One point give me down my target: I: did it so each other so.

A

This difference here, Delta V, could be. Maybe that's like five percent of what the true value was so.

B

Time in the EMF boot a snake who now time and ever look young, that's our booth, incoming water quality by Nick water. That's our hey! Who did squad unable why my father got the eighth Guantanamo will withdraw all of us in Guam.

A

So when you, when you specify errors, then it's the error that you're trying to capture has often two different things. What could be like effects? A positional error and also you have very large dynamic range in in something rinsing voltages. This might be a thousand volts that might be a hundred, and that might be one volt. So a small change in the location of where the electrode is can make a big difference.

B

I know yeah well, I'm, not a bunny and.

A

Why you need a for sometimes the data that you have not apparent resistivities, but sometimes the data that you have for geophysical surveys. You might go like.

C

A

B

You know yet it down a little and in.

A

Particular if you have a datum that goes through zero and you have so the value of five percent, for instance, for a datum, that's at zero. So remember, members are noise. Standard deviation was equal to a percent times. The absolute values of data plus support, senator.

B

V issue during attack by this one slope on me: hit'em, we have flow boundary to a buyer, who's a tabular to him. Yes, eternity night that move opposite leg up, but I'll take a floor glass of beer. So, for this point here was.

A

Here, but for this point a percentage of five percent you know gives you something that looks like this about.

B

That. But for this point.

C

A

Percentage of 5% as no AirPort zero.

B

Yama's rio5 percent lung capacity, zero, five percent plus a residual mozzarella era, an era Bathsheba Virginia and up, oh god. That would see a better one in Singapore five percent of the time, Pattaya part personally, I agree: yeah I promise is a name or a koala behind you, quite cosenza, Andy Cheney.

C

Yeah she did, but oh yes, okay, yeah.

B

Bamboo yet my purse energy flow Plaza, so.

A

Always think about floor plus a percentage. This is perhaps one of the most important things to think about in doing inversion, because I come to it in a moment.

A

A

We had our misfit okay, which was the sum over. All of the data are predicted minus the observed.

A

Divided by epsilon hi, our standard deviation extend.

B

The division that yes, I, did dog got.

A

C

Donnie, hello.

B

Come on, have you got a cool time? I feel my standard deviation in inside okay.

A

So suppose, in a misfit retirement suppose your estimated standard deviation is too long. You.

B

Know it's not in the standard deviation throat, harmonica am Nathan. Okay, my name is Louie.

A

In other words, suppose that it was supposed Absalon ID for one value, was equal to zero.

B

No standard deviation was a little bit low yeah there you have it then.

A

This quantity here would actually be infinite, and your algorithm will spend all of its time just trying to work on one data as I.

B

Noted that the couch I gotta go what.

A

Was wrong? 0 Canada.

C

Standard deviation, yes,.

B

Okay, yeah yeah, that standard deviation my bed. Well, he don't mean you come on, but I'm not doing it see the IPA my body's on it.

A

Because we're trying to minimize this misfit and if some day that have assigned standard deviations that are too low, then the algorithm will continue to try to fit those points and that will not provide the best outcome. So.

B

I, instead a division to a named Bureau. It looked in a mess or not a con. Lo llamamos misfit, because I missed a communicator mama.

B

A

But this is a very, very important question and every time we do an inversion. That is the first question that comes up. What should I estimate for my errors now.

B

Yeah needs you I ate a conventional diploma, but don't ever a little, you come at me with my my earnest without people Romani, okay,.

A

So now, let's come back to our.

A

A

Okay, there that's pretty good.

A

Now we want to solve the inverse problem and remember: I said that there were two two components of this: that we've got our data misfit and then we also have our other knowledge, so ability.

B

Technically my Magnum, you go no did I, miss videos. Allow me, but my tossing laws actually attach a little bit a salami.

A

So we're going to formulate the inverse problem when we let touch.

B

A minute they must love them to.

A

Add both of these together, so here's our data misfit, they.

B

Don't miss anybody yeah.

A

So it looks like this: it's sort of what we've been been writing. Okay, so it's observed. Minus predicted, divided by the standard, deviation all squared sum all the.

B

Squares down some of that go on then Pat them. You know Bob now, I. Ask that the Senate Division is not a miss. Miss miss faithlyn misfit booyah much about it, do not go not do it for me. I love it and.

A

Then we have our geologic knowledge. This first term here is just our. We want to make a solution. That's close to a reference model generally.

B

Yeah I have the reference. What is our Journal of the Coppa Italia model come on I'm more than a nice today, I will narrow and I will check out your body and.

A

The second term here is, we don't want it too Wiggly.

B

A

We don't need to worry about the details of this just the concept, so there's a first term that tries to make things close to a reference model and a second term which tries to make it smoother. The.

B

Godly things that you got me, they say with people talking about that man, because if you turn a concept, they are open. People, easy, Ramona, misfit seemed a mystic and a geological knowledge. The table or table of sin are a sin. A below my eye. Are you never? The rock reference for attorney general metallic commentary.

C

Majdanek charity gonna.

B

Pick it up, you.

C

Know not one lion: I should shoot.

B

You too, even if my neck, smooth.

C

B

Charm over here are allowed and.

A

We're going to put those together with coefficients this is an alpha sub s first op Aston and here's an alpha sub X X yeah. So these are just two numbers. What we're gonna do? One of em you gotta go! So we've got something here something here and we're going to wait them.

B

Nobody's getting any cement making an engine of balances, you know without changing. Thank you. Okay,.

A

So we're getting almost there, we have a way to quantify our misfit and we have a way to quantify our model.

B

Home formula write that up missing I'm gonna, tell you nothing more than a coming home on a net I don't get to give Cindy a darn, it they're gonna change or I'm gonna.

A

Here's the parameters that you will play with so the first thing there's something called a mode which means either you can run or explore, follow.

B

The Fermi density, governmental body body parameter I'm, a fellow Mojave, remote motoyama body, random explorer now so.

A

This will become clear later on, because when we run it we're going to do iterations one two, three, four, five, six doctor rattling.

B

Egged enabler, Oh Cynthia's vehicle about, ran well money, but.

A

Then, after its run, we want to just explore like what does things look like at iteration for or iteration 7, but.

B

Phenomena explore and run up your mother run explore a missile explodes our duty.

A

So here's here's the buttons. If you want to run the inversion you press this after you've run the inversion you press explore and then then you're. Just looking at the results.

B

Obviously, not inversion of them is a random table. A bachelor party, you don't mind, not touch an object, each in a kind of explore, maybe Rama. Okay, they are the result exceed oh yeah,.

A

The one thing that I hadn't talked about here is that we have an optimization problem, we're going to minimize this, which is the sum of our data misfit and our model objective function or our model size. So that was so remember. We have two things.

B

A

So we've got our misfit and we would like to we.

B

Would like to make that small misfit with it then I.

A

B

Know everybody yeah.

A

And then, when we have our or model or model norm, then we want to make this small. We want to make something, that's close to a reference and we want to make the amount of Venus.

B

C

B

Hernando, well maybe you know about reference manager, okay,.

B

B

A

We want to make this small and we want to make this small, that.

B

Alone and in yellow so.

A

That could be hard to do how we're going to tackle the problem is take the sum of these, but we're going to weight them with the parameter beta.

B

So I ain't how you found today I'll, take your Barbie that'd, be a change with my body and then.

A

Clearly, we're going to have to adjust this value of beta until we get the right. Do we get something that's acceptable to us now.

B

So, okay, oh yeah, I'm, not your body.

A

So we'll see this a lot, we.

B

Have a late at night: okay,.

A

So that's how we're setting it up.

A

Okay, so now we're going to invert I've talked about the modal. We have the reference model, we've got the uncertainties, a percentage and a floor, a reference.

B

Model is she game button? She may know what she meant.

A

And then we've got Ganga, we've got observed, predicted and misfit.

B

I'm Antigone so noisy Damodar. Oh, look come on holiday, the Mohawk College Avenue team, a misfit apollomon.

A

A

The number of Danis here is just a parameter that we can choose to see how many sample, how many solutions we want to see. Emma.

B

Germain is Ramiro hunter. The weather up have a polymer battery that new or not as we have ability down what is a teenager. haha The.

A

Alpha s and alpha acts are those parameters that we just talked about, and butter eggs.

B

Nick to let it get out and I'm doing it, battery dead and.

A

These others are just basically plots that we will see.

B

A

End up in a.

B

Box Devon, oh yeah, so.

A

Here's here's the values so here's my is zero percentage and a floor. We've already done that. There's something called Chi fact: I'll come back to that, we'll set that by default, I.

B

Go to one another Lucian evolution in andalusia day and.

A

For our data, we can look at the observed and predicted data, or we could look at the misfit a.

B

That'll give us a little party where there's a observes. It's a good idea, how the detox we had at that moment you take this, come on dolly that's it yeah come on morning ha ha ha and.

A

Here's our alphas resetting right now, they're 1, & 0, make.

B

A triple chain about it and.

A

Then we have some options on how to plot. Also these.

B

Are you just a team of emoji game so.

A

B

Let's take a look, it sounds yeah.

A

So we've run it so, let's explore.

B

Explode anybody.

A

So let us first.

A

So we, let's think about how to.

A

Remember the quantity that we're trying to minimize is this spent here, which is the misfit plus beta times.

A

Our bottles, so this is our geologic noise or.

C

Geologic information.

A

And this is the date of misfit.

B

Majority, a read about it and not much I miss fate in my body and I'll tell my children, my son.

A

We can come back to some of these slides, okay.

C

So here's here's.

A

Another way of presenting things so we're going to minimize the sum of our data misfit, a nice.

B

Little money no show time. I don't mean my slow Maserati man. Oh man,.

A

And our model norm where this is a this is called a trade-off parameter and it's got values anywhere from zero to infinity. So I.

B

Have zero on e infinity Z number this URL and.

A

We want to find a solution such that Phi D is pretty close to what acceptable fits. The data.

B

A

This can be kind of tough. We have to balance things well,.

B

You know, I came up one of us on the team.

A

So think about maybe think about driving a car, so you're going to go from below mean to Yangon.

B

A

And on this axis here, you've got driving time, I.

B

Like me, my zero mountain daddy and on.

A

This axis here you've got fuel consumption.

B

The inanimate outcome, by investing or huh okay.

A

So now think about your your options. You could go, you could drive really fast, but if you drive really fast, you might burn up a lot of gasoline and your fuel consumption would be high. So.

B

A

On this graph here, this is like a trade-off breath if you had, if you were willing to have- or you want to have the smallest.

A

About the fuel could feel consumption. Okay, then it could take you a very long time. So the fuel consumption is is really small. It could take you a long time so.

B

Yeah I don't get that they need yeah man. That's.

C

The dynamic arisen and there's a pub daddy and then a lyrical.

B

Dance another: how would I ever go.

C

Dance or not, he.

B

And I name it yeah, okay, Pumbaa hydrogen. We keep go on it, you thought: does he own a? There? Was nothing so.

A

So this, if this parameter beta, is really large, then you don't you don't care about the about the time you're just carrying about the fuel consumption, and that might mean that you're just taking along.

C

A

That you're traveling very slowly and taking a long time like.

B

He was read the that's people, that's equates right, come on, our GPAs are not a table. Madrid was right now 58. If you didn't go missing or not. On the other hand, with Jennifer.

A

On the other hand, if beta was equal to zero, then the only thing you care about is time, and so you just push your foot to the gas pedal you're going as quickly as you can and you get there as fast as you can, but you've used a lot of fuel so.

B

Nothing for Mom, you say: nestle commenced them boots, rr0 with a melon. Is anybody saying.

C

But then I meanö may I'm sure mr. no-name.

B

You name it well, maybe I remain vulnerable. Young involvement, maybe around so.

A

That's that's the trade-off.

A

What's what's the best, what's the best thing to do where? How do you make a decision about where.

B

To go no more made by mr. monk I'm only missing a limb you'd want Israel found in evolution, excellent.

B

That's Tico watch our video and lady mr. Johnny time ago. No good tonight, let's see Kumamoto in the engine room. So you know minimum Warrnambool, that's see it's a chemical, so they're, not monitoring and how we can't.

B

B

Okay, good yeah.

A

B

Someone said like we care about time, so that's.

A

B

Makes the time thank.

A

B

A

If we minimize the time- okay, then we're out here, so this is time in this session. So then we're out here. So we you just huge amounts of fuel consumption. The.

B

Team is enough, so.

A

That that might not be very good if it, if you have to have 400 gallons of fuel, to get there expensive.

B

A

Anybody else, but I like.

C

The idea of tiny both.

A

Yes right, so maybe you need to have like a specific criterion. So let's so maybe the question is: why are you going to younger? Well, maybe you have a family dinner, a.

B

Movie I thought you know a lot of it. Okay be now soon as I wish. You so.

A

It's 8 o'clock in the morning, and you want to be there by 4 o'clock in the afternoon to be at the dinner.

B

A

Mention anything.

B

I am watching it well, no so.

A

That means you could find like eight hours on here. You could draw a line here and.

B

I'm still not good I'm, not missing, I'm, I'm, gonna and I'm gonna be off and that's your target value so.

A

Your target time is 8 hours and therefore you'd want to you work with this amount of fuel consumption so that you spent the least amount of fuel to get there in eight hours. So.

B

A

B

Thought Eddie my eight hours wasn't gonna. Let me answer no, you might be by my desk Evoque, no nickname. Lucky me: okay, buddy, let's see, let go look a little bit better Thanks.

A

So that's that's. The idea and you'll see this a lot through long versions. We've always got two objectives: we've got it. We want to fit the data, but then we've got these other constraints of geology and we want to kind of minimize the complexity of the job. So.

B

No names only they don't know. Oh thanks.

C

On 50mm, oh my me time: Georgina bannard, a.

B

B

Show me hey: this is.

A

B

A

On this case here, if our target value, Wireless I've got two hours here, but so I suppose it's two hours, then this is sort of an optimum point. You use the minimal amount of fuel consumption and you're there in two hours, so that.

B

Will not get on the night in a table.

A

Okay, so now, let's apply that same thing to here.

A

Let's actually just go through this, so the way in which we're going to tackle this is by successive by starting with a big beta.

C

B

Was thought about that? They did then nobody knows I'm on it.

A

So we're going to start with a large beta and therefore only be concerned with this this term new, so beta is large. Then we're not very concerned about the misfit okay, so we're gonna we're going to this up. We're gonna minimize this, but we're going to start off with a very large value of beta. The.

B

Minimizer looked in it, I might be done that, but that Dean is not so.

A

Now watch what happens did.

B

A

So here we have the model that we're looking for.

B

Their loop sounded a more that goes anywhere.

A

And at iteration one where we've got a very large beta, here's our predicted model, but.

B

Being done about diesel biodiesel and then.

A

Here's the here's, the predicted data and here's the observations.

B

So I'm on life, yeah, oh yeah, yeah, oh yeah, come on t-bone, oh yeah, oh yeah, honey till it be a male sativa. So we do a terrible.

A

B

It or my family, you know, if they look about this axis, is the misfit. So do you need extra time, Laney mataró and they spit that? What is you man.

A

And on this axis we are plotting the model norm, the model norm, the size of the model, the PLA.

B

In Matera the norm, save of module in norms in Mali, 89.9 kwar, you, okay,.

A

And for this particular beta we're starting up up here.

B

So maintain a better co-op pose a lot okay, my mother ain't, nobody Sally, but no now.

A

We're going to decrease, we're going to change beta and then we're going to see what happens as we.

A

Progressively get better here and better results, there.

A

Maybe this is okay, so now I'm going to increase or decrease beta.

A

I've changed the plot. This one might be easier so on this axis we now have beta so.

B

Anyway, you want other.

A

Beta and we're starting off with a very high bit ego, we don't mean this on its I, get it okay and you see the green dot. That tells us what the value of beta is and.

B

They say ah nothing, they know about it, sort out of your body and on this axis we have the best fit, but the Eddy, not many new mines. Are we a misfit that was today.

A

So as I reduce beta because emotions.

B

A

Then I'm going to get slightly better.

A

Predictions between the observed and predicted and then we'll get better on the model so.

B

We haven't gone on an info that that woman they died man that down and if what we don't.

C

B

Without a limit.

A

But look at this so now I'm at iteration, ten out of ten little tiny so beta is down is thank you. Thank you, yeah iteration 19. This is the value of a beta here. They done that one young boy so.

B

We're reducing the misfit, the misfit shorts on a hobby, and you can see we've done some kind of oppose yeah I said they have already said eyes on D honey I'll, be here, maja movie may inaudible.

A

So we're better here we're trying to get to the black dots and we've got the blue, and you can see we're starting to get some information about this feature here. On that.

B

Feature so I Teague now come as a modern in it. The nipple penny the dilated and aneisa Lobby Oh. Tell me I, better change it immediately. Okay,.

A

So we're we're doing better. So, let's, but let's make beta smaller without.

B

The network, the.

A

Other, let me keep reducing.

A

They so now the value of beta is here and the misfit as considerably got better. So the observed and predicted are looking like this, and this is actually not. You know we're starting to pick things out right. We I mean. Maybe if you thought about this in the aquifer, this is death, then maybe there's near surface aquifer and then maybe something in 50.

B

Per the auto expert pilot radicalism- okay, but by the news, media, mama ya, think maybe she.

C

B

You see this with honey, beautiful.

A

And so now let's go a couple more.

A

I want to stop here.

A

So here's a result.

A

That is, it will come out as a default result, usually in in the programs, and this horizontal dashed line here is the expected value of the misfit. Given the uncertainties of June applied, you.

B

Mean that it doesn't line yep I am the I know. I said this ended if I'm unity, misfit yeah.

A

So remember the misfit 5e is the sum I equal one to n.

A

Di predicted minus di observed divided by epsilon I.

A

A

If you have the correct answer, if you have the true value, then this quantity in here the misfit divided by the standard deviation, should be a Gaussian random variable with 0 mean and unit standard deviation.

B

A

Ahead, yes, it's fine if they don't understand the universe and but say the words. So what I'm saying is, if the if you have a solution so that you have the bump that the.

C

Predicted mode.

A

Where it worries equal to the real true, true data, then that difference divided by the standard deviation, indigenous.

B

Allocated on the mule is how you would.

A

Be a Gaussian random variable with 0 mean and unit standard deviation, mean and unit standard deviation,.

B

B

A

And why is that important? Because, statistically there is an expected value for this. So if I, if I, give the whole process many many many times and then I looked at what the expected value of 5g is that I find that is approximately equal to and the number of data.

B

So I'm on for the process with ain't ain't ain't, no, damn it P, naught or not, and I might be like Hawaii for me, laughs, okay, I'm, not sure what is that II wasn't either. Oh, the.

A

Expected value so, as you could kind of think about, is the average, so the expected value of random variables I is one faux fur and Exide.

A

C

Heads tails that there's bills.

A

Here it isn't all done. Oh yes, true, the cultures are good point.

B

A

It's possibility if I took everybody in this room and I said what is the average height of everybody. Then you'd know what to do right. You measure each person find the find the average value okay and then you could also so that would be the mean or the expected value, and then the variance of that or the standard deviation could be obtained by looking at the differences between that mean value and each person. So.

B

You are got the mean value in every.

C

Druggy you can Amish it yeah.

B

Yeah yeah classy person who ran busy day in admission, this algorithm, yeah I'm, sorry I'm, not a puppy, create your a detectable. Let me young guys giving them a new job without standard deviation. Comment, never get my populist Robbie buzzing. You wouldn't be every sublime harmony. You never.

A

Never except love me on eBay.

B

Hello, yes, then, a deviation so.

A

I would say that the expected value, if I look at it. Everybody in here I'd say that the expected value of the height of people here is 5, the 7 inches thick I, say: ok, so that would be Mike's but expectations 5, feet 7, but emotionally.

B

Yeah yeah yeah, TV, okay, so.

A

The expected value of this misfit is approximately equal to the number of data.

A

Okay, look I'm summing up here. I've got this stuff in here. If I've done, everything correctly and I've got a good standard deviation, then the expected value of this number is equal to n. The number data need a.

B

Number, a number of number of gauges, yep.

A

So I'm for here for Phi, D I'm measuring the sum of these different students divided by standard deviation and swearing, so I've got this I've got this thing.

B

Expected solution: anybody, okay,.

A

So I so now this is this is good. Now I have at least got on this axis of misfit. Some number that okay, that's kind of what I'm expecting to have for a good result.

B

Come on girl, chained to that or not and.

A

That number is the sitting here in my biological.

B

A

That's not bad. Look at your.

B

Job, you do any send anybody, think I survived yeah.

A

Here's the truth: here's the true model, here's the recovered model and no we've got energy here. Energy here.

B

A

Then look at the fit to the data. It's he's.

B

Out yeah everything with you like Bob, it's.

A

Not too bad, it's not great, but.

B

It's not too bad, yeah, Eva, calm with it or not. So.

A

At this point, you could think oh I've solved my inverse problem.

B

But I think mom's, no ok, gonna be in faster than either of you seem to be thought. You know the killer, yeah.

A

Here's here's what I think the earth is like. Here's my fit to the data. Well,.

B

Not hope I got that up, be it optimal. Job I do both under the rope yeah, be your D yeah B.

A

Okay, so we've solved our inverse problem so.

B

That doesn't need much problem. He was deaf. Another piece in I have enough, but.

A

We could could maybe do better, because we noticed that.

A

We could reduce the misfit here. You know a little bit more and not well. We could reduce this misfit a little bit Morin and see what we got so.

B

Let's change the day that is, she name is oh man, so he lucky Ljubljana City on business. So.

A

Let's change beta.

A

So now we're actually we're reducing beta a little bit more and you can see this is actually a better solution. Right we've got a little bit closer here and the misfit is it's.

B

Better so this way down showed a miserable comic level police tell a body.

A

So what might think that? Why don't we just keep reducing beta, there's.

B

No way down with us up so time you don't look sane I know.

A

So, what's good watch what happens as I decrease beta, so the more I decrease beta, the better my fit between my observations and predictive data without.

B

So down hit it on me come on. Indeed, a do Britney said body.

A

So what do you think about that? It's still okay,.

B

Go to yeah babe I got.

A

So remember, as this green dot goes down, we're reducing the misfit, the mystic.

B

Will show time enough to buy today as a young donkey Oh it'll be a sweeper shortly. So.

A

These are piece the blue and the black are getting closer. I know.

B

Yeah I'm it man yeah.

A

A

So now keep this keep going down.

A

A

So now watch what happens to the model morning.

B

Downloading TV see.

A

How the model is changing, so the model is now changing, but we're not really making any difference in the misfit. We we're fitting the data, but as well as we can do and look what's happened to the model. Well, then, about its development, the model is garbage.

B

A

This point all we're doing is over fitting the data we're fitting the noise, so daddy.

B

B

A

You were captain at just fitting the data, you'd say ah good job. Look at this Dean.

B

Dean: okay, come on.

B

But you've overfit the data and you're, not alone jump in your top and now.

A

Here's the model that you get it's just like this, it's complete garbage so.

B

A mom or no Samuel, oh whoa, Oh My, those home yeah Bobby, so.

A

You need to guard against that.

A

This is totally unacceptable. You've done all this work to invert the data, but the choice of the portrayed off parameter means that you've overfit the data and you now have a very poor result, but nobody.

B

Need a do-over fit good yeah there is that God knows I subtract.

A

So you can play around the other curve that we will use. A lot is called the T knopf curve.

B

A

A plot of 5 D against 5m5 da.

B

By Benjamin and.

A

As when beta is very large, we are up up here and then the character of this curve is such that, if we, if we reduce by D without increasing by M the structure, then we're probably getting working with signal in the data so find.

B

D was so cute, so we may find.

A

So if 5 years changing but Phi M is not increasing that we're not putting Denver, probably still working with signal in the data.

B

A

Like good stuff in the data.

B

Remaining with identity and my mind is read that warm up yeah yeah got it I'm a flirt. Yeah Bob, you don't slow down my video, my buddy.

C

Pick up the funk and gum of yahwah's.

B

Your daddy found a house, it's a janitor body, count it down. Yeah, okay,.

A

So what we could probably take a break. That's a lot! That's a lot of stuff. Okay, so the plan is we'll take a short break. Coco.

A

We forgot to ask you to print out the got the slides, Wow, the slides too, but the question.

C

A

There, on the course can can people access the course.

A

Waiting: okay, so can everybody access the GSI science.

C

Courses to watch this side of the Odyssey okay.

A

Dad you have course material costing the DiBiase Bobby buck and then here's here's bibliography the linear inversion and the nominated version and the parametric inversion are up there for slides. So maybe we don't need to print these out.

A

It actually with you guys, then you could write.

A

Oh yeah questions, so here's the questions on linear, inverse, steering.

B

Being a university / unit but put the question.

A

So what a flight you will have a little gray and then I want you to come back and open up my binder and download the inversion app and then step through this question. So basically, what this question is is I've asked you to to generate a model, so the first part with all the kernels and stuff will leave no for now.

A

Just use that as a default, then I want to Jim I, want you to generate a model and data, so you're just going to use a boxcar and then you're going to assign noise and then you're going to then go ahead and add noise and carry through the inversion and we'll repeat the same thing: where you're going to try to recover the boxcar and get garbage over.

B

So the total break me about it break down again I miss it on my bed find out a point. You know I think I'm actually happy with Donna look about it being deposited Oh.

B

A

That you do here is exactly what you will do for when we give you the 1d, DC, resistivity and version code, as well as the 2d DC resistivity, well-known.

B

Aha table oh I, wanna do do D. Resistivity was a question about that long. Yeah.

A

Okay, so let's, let's take a short break, maybe fifteen minutes, okay, goodbye and then tried to do the exercise.