GeoSci.xyz SimPEG, 23 Jun 2022

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From YouTube: Wouter Deleersnyder - Flexible, wavelet-based inversion within the SimPEG framework

Description

The salinization map of the region of Flanders, Belgium shows the depth of the interface between fresh and salt groundwater in the coastal and polder area. It serves as an exploratory tool to examine the potential of groundwater projects that improve freshwater availability in the shallow subsurface. Flanders environment agency published an updated map in 2019, based on airborne time-domain electromagnetic induction data. The result of the inversion is, however, overly smooth, which potentially conceals interesting features.

Via an inverse problem, the electromagnetic induction data can be mapped onto a conductivity profile, which serves as a proxy for salinity via petrophysical laws. The inverse problem is ill-posed and regularization improves the stability of the inversion. Based on Occam’s razor principle, a smoothing constraint is typically used with a very large number of thin layers. However, the salinity profiles in the Belgian coastal plains are sometimes sharp, impeding the correct estimation of the fresh-saltwater interface. In practice, the real underground might be either blocky or smooth, or somewhere in between. Standard constraints are thus not always appropriate.

With our novel wavelet-based inversion scheme, the original time-domain AEM data can be re-interpreted in a flexible fashion, meaning that we can easily generate an ensemble of inversion models with different types of Occam's razor minimum-structure. The flexibility is due to the wavelet basis. In simple terms, a wavelet function can be seen as a building block and a simple model is one that can be built with few building blocks of various sizes. Our proposed inversion scheme adds a regularization term that limits the number of building blocks in the wavelet-domain to make sure only the necessary complexity is retrieved. The scale-dependency makes the use of small blocks more expensive, as it corresponds to adding detail. The scheme is tuned by only one additional parameter (which determines the wavelet basis function) and can recover blocky, intermediate and smooth structures. It is also capable to recover high amplitude anomalies in combination with globally smooth profiles, a common problem for smooth inversion, and an essential feature to accurately predict the salinity.

We first demonstrated this alternative inversion scheme on 1D FDEM data and now extend it to 2D AEM data a in a saltwater intrusion context . The flexibility of the method allows for choosing the appropriate sharpness for each orientation. In Figure 1, an inversion model is shown with a relatively sharp wavelet basis (db3) in the vertical orientation, while a smoother wavelet basis (db8) is used along with the lateral orientation. Imposing the appropriate sharpness on the inversion model is crucial to obtaining a reliable estimation of the fresh-saltwater interface.

The code behind the 1D regularization has been adjusted to the SimPEG framework and is open to everyone . This allows to use this flexible regularization term for many types of inversion problems and promotes geoscience reproducibility!

Bio
Wouter Deleersnyder is a passionate Ph.D. candidate at the physics department of KU Leuven and Geology department of Ghent University. With his technical skills from his physics education (Master in Physics, KU Leuven, Belgium in 2019), he wants to tackle the problems of today. Stimulated by the importance of water for man and nature, he came into contact with groundwater and geophysics. He is working on new methods to get a better picture of what is under the ground. He is specializing in electromagnetic induction surveys, forward modelling, inversion, and uncertainty quantification.

A

There we have it and so I'll, uh give a brief intro to walter. uh So walter is a phd candidate at the department of uh at the physics department at key uh sorry, ku uh libman and the geology department at ghent university, he's motivated by groundwater problems and using geophysics to image the subsurface and he specializes in em induction surveys, uh forward modeling inversion and uncertainty, quantification, so walter. I will let you take it away.

B

Okay, okay, thank you.

B

So, let's first check: if you can all see my screen.

A

Yes looks good.

B

Okay, perfect, so thank you for having me uh at this simpex seminar, so I modded layers later uh part of k11 and kent university located in belgium and uh my talk today is going to be about a flexible wave based inversion within the simpek framework. But first let me provide some context um so yeah.

B

The context of my pg project is about methodology methodologically improving uh from mercenarization map and, first of all, flanders is located and that's a yellow region in belgium, which is a tiny country in europe and salinization map is shown here, and it shows the depth of the fresh saltwater interface like in this small map. Only part of the salinization map is shown, but it shows the dividability that is present and also the interface is typically located at the depth of five meters, but first I think I'll have to minimize some other screen.

B

Otherwise I cannot point to it. Okay, so, um and the problem with this salinization map is actually that the inversion result is overly smooth. It was a produced with spatially constrained inversion um and an overly smooth inversion result has a large impact on the estimation of the depth of that fresh saltwater interface, and I will try to illustrate it with this image.

B

So the blue line yeah, is a an example of an overly smooth transition from fresh to salt and, first of all, we know that we have the typical overestimation of the low electrical conductivities and the underestimation of the higher electrical conductivities, and also the definition on which we define that the interface between fresh and salt water is located has a large impact on the estimation of the depth. For example, when you take this definition, we get this estimation and why we have a slightly larger other definition.

B

We get yeah a quite different depth at which the interface occurs. While we actually know from expert knowledge that the transition between fresh and salt is actually more sharp, more like the black curve, but also actually the problem, but the definition doesn't have a large impact on the estimation of the depth. So that's a little bit the context of the problem that I will try to solve in the talk today, so that map was created with time domain airborne en data. So a helicopter flew over the boulders emitted, a trans magnetic transient magnetic field.

B

This induces eddy currents in the subsurface which are proportional with the electrical conductivity of the subsurface and those eddy currents produce a secondary magnetic field which is typically measured somewhere in the receiver coil, also below the helicopter.

B

To be more specific, the skytem dual moment system was used to produce that kind of data, and then the map was constructed with spatially constrained inversion.

B

The step to go from the data to an actual image of the subsurface is typically done with a geophysical inversion.

B

But as the fourth problem in this case is non-linear, we usually use optimization to minimize this objective function, the first term, that's typically the data misfit and also typically the least squares misfit, and it tells you to what extent an image is in agreement with the data.

B

But geophysical inversion in general is an imposed problem, so we can mostly think of an infinite set of inversion models that are in agreement with the data, and we solve this by adding this model misfit, which will try to filter out all the unrealistic images and that's typically done in occam's razor sense, meaning that if you have two inversion models that fit the data equally well, we should choose the image.

B

That is simplest and that's typically done by adding a smoothness constraint. And then we get a spatially constrained, inversion or ticking of regularization, but yeah, and this is also where we think we can solve this issue for our problem with the sharpness of the current salinization map.

B

So the goal was to develop a novel model misfit term, which was a little bit more flexible. So we want to add a tuning parameter which will allow us to get blocky smooth or in-between inversion models, and the idea is to use wavelet theory. So we will try to decompose the inverse model into the wavelet domain, which can be seen as viewing the universal model as a series of building blocks.

B

And then we combine this with the sparsity constraint, which is then combining this with okay, a simple model is, then a model that is built with the least number of building blocks, and while this principle has already been applied to geophysical inversion, we add it to novelty and that's the novelty of scale dependency, which is basically saying. Okay. If you want to do one, if you want to use smaller blocks, that's fine, but you have to pay a larger price, so they weigh more heavily in the objective function.

B

So let us delve a little bit more into the discrete wavelet transform and why it works in this context. So here I show you a very simple model with only four model parameters for simplicity and then I choose a building block in this case. It's the hard wavelet or dvc1, which really looks like a building block, and then we will start the wavelet transform, and then this is basically this.

B

This means that we have the first building block the mother wavelet, which is basically the same as the one I showed you there and then I make all these variations based on these mother wavelets and they have different sizes and also sometimes have different locations.

B

And from here you can actually already see that this building block you don't need this building block to build that part of the model, because that's flat, so that coefficient corresponding with that building block will be zero and that's like uh the sparsity that is inherent in the wavelet, transform for many uh inversion models, but I'll come back to that later, but maybe introduce some uh notation.

B

um So these is what we call the wavelet coefficients and as we had a four model parameters to describe this model, we'll also also have four wavelet coefficients, which we will also store in a vector.

B

Then this is the moder wave basis function and then these are all the wavelet basis functions which have different locations or this wider and smaller building blocks and at different locations so at different translations.

B

So, let's look at another example where we have not a real, realistic geophysical model, but a linear. I will also call it a model, then I will choose a building block in this case. It's dbc2 it's a little bit more regularly shaped, but I will apply that same way. We transform to that linear model, and then I get this. So what I show here I forgot to put on the titles.

B

Apparently uh what I have here is that I have the wavelet coefficients and, what's actually quite remarkable, is that we normally need three coefficients which are non-zero to represent this signal completely, because all the other wavelet coefficients turn out to be zero.

B

This is really a dispersifying nature behind the wavelet transform, but also the first time when I saw this, I I think I thought it was quite remarkable that that irregularly shaped uh form um building block can actually represent such a linear trend and and a perfect with the perfect reconstruction.

B

So maybe I can try to convince you with the following image: I'm not sure, but I will try to so. What I did here is I I plotted those three um building blocks next to it next to each other. So I have one here from minus two to 1 and then another one one step forward and another one also one step forward, and then I multiplied it with the corresponding wavelet coefficient and indeed the first one was negative.

B

Indeed, this shape is actually flipped and when we sum them all together, you actually see the linear trend occurring between zero and one. What was also the case here between zero and one.

B

So that's a little bit of the basics of the wavelet transform, but it's important to note that simple models have a sparse representation. Usually while random models, like you just generate some random model, then also you will have like some random nature and the wavelet transform. So that's important because yeah, if the wavelength transform would be able to represent each model in this part representation that wouldn't really work, because we need the sparsity constraint in combination with the sparsity of simple or this realistic inversion models.

B

Some more definitions. I mentioned the dbc2. The two is actually standing for vanishing moments, uh which is your the p, and this tells you um in this case that, in fact, the linear trend could actually have the wavelet coefficients these ones being zero. That's actually a part of this expression and then dbg3, for example, could also exactly represent quadratic expressions and so on, and then another important aspect for the sparsity of a specific inversion model is also the compact support. But I will come back to this later.

B

Compact support is, do it of the building block itself.

B

So in this talk I will mainly use gpu wavelets, and these are defined in a specific way and actually the shape.

B

The resulting shape, which are shown for six different uh wave basis function are the result of imposing those properties, and the idea is that it is the wavelet, with the minimal compact support for a given number of vanishing moments, and that's actually a very good property so that it can really represent many inverse models in a sparse fashion, because when you have other wavelets, which have larger compact support, it will turn out that it's generally has a less sparse representation.

B

So it's less ideal to combine it with a sparsity measure. Okay, let's go to something more fun, I would say so: let's apply it. The scale dependent wavelength-based inversion to an example that you may recognize.

B

So I brought the true models uh here for each uh it's the same on each example, but then I apply our scale dependent wavelength base, inversion with all those different building blocks and for the first one, I've used the blocky building block and then eventually, our covered model will also exhibit that blocky nature, while in the other extreme, when we use dbg6, which is rather smooth.

B

We also end up with a rather smooth recovered model and also, what's also quite interesting, is that all the building blocks in between have features that are not that smooth and not that blocky, so they are rather intermediate.

B

um So that's already, I think, an advantage over the wavelength-based approach and yeah. I think you may have recognized that example. It's actually taken from the simplex documentation in the tutorials. There is a part on linear inversion, where those two examples are shown.

B

The first one, for example, is parse inversion with iteratively, related least squares, which generates blocky results, and indeed you could compare this recovered spice model with the blocky result from the wavelength-based inversion, while the more traditional linear, linearly squares, inversion results into smooth inversion results, which you then can compare with the recover model with the dbc 6.

B

But what I? What I also like about the wavelet based method is that you yeah have everything by only tuning one parameter because yeah as a q, physicist or geoscientist, you want to tackle some anti-physical inversion problem.

B

I think you always start with the trying out the ticking of regularization, because it's quite easy, you get smooth results, it's quite stable and then, for example, in this case, you realize that taken of legalization is maybe not the best decision and you think. Okay, maybe I should go for some blocky inversion and you use the other method.

B

Then I think it takes quite some effort to go from that type of inversion to the other one. You have to change from an l2 norm to an l1 norm. You will probably want to change your optimization algorithm that will throw out some different errors than you're used to um so yeah. You have again still have to learn a little bit uh how how the thing behaves to get acquainted with the new problem.

B

So I think it's also an advantage of this method that it's yeah. You have to learn at once with a specific, optimization problem, and then you can have all those things just by changing one parameter. So I think that makes the geoscientific problem, which yeah tend to become more and more complex over time.

B

Then I think this can make it a little bit more easy, so you can also use it by yourself. Actually, so you have the clear simpac framework and it makes sense that what I try, what I did with the wave based realization fits into this regularization part, and you can actually go and just pip install wabi standing for favorite based in version and then what you can do is you can take an example from the simplex website.

B

You have no your normal import statements, you add the line where you import the variable characterization and then, where you normally define, let's say simple or sparse regularization you can put wavelet regularization.

B

It also requires the mesh and then you supply that wavelet basis function. In this case it's just dbc6 and you can actually use the examples that are there and compare the results and when you want to have a little bit more of information on my github, that is a little bit more information on how you can use it and the ideas behind it.

B

Also, um you may wonder what wavelet basis function you want to use. I have already mentioned the duplicate wavelets. They are always the wavelengths that I prefer to use, but maybe for your context, you're looking for some kind of other kind of behavior, so I recommend you go to this site where you can click here on all the wavelet families and see how they look like, and you remember that the shape of the wavelet family itself has a large effect on the recovered model.

B

So maybe some there is something useful there um that you can use, and maybe you will use other wavelets than the dubstee waveless that I use maybe to end the 1d um with the basic regularization.

B

um Let me also maybe uh try to explain: what's the added value of the scale dependency over the normal wave based regularization and that's clear with this example, I think there is a black line, almost hidden behind the green line, which is a true model, and then the blue line is showing the normal wavelength based regularization without the scale dependency and the green line fits the true model actually better. So that's the scale dependent uh wave.

C

B

Regularization and the the intuitive reason behind it is that I made it more expensive to use these smaller building blocks, which has led to the uh overestimation and the too narrow peak, and also this um anomaly at the end of the inversion model.

B

So that's the intuitive reason, but also for more like theoretical background. Maybe you can check our publication from last year and the main conclusion there is that for waves, like dbg6, both methods tend to work equally well, but for the methods with the few finishing moments, for example, one dbc one two or three: it's really as an added value to have that scale dependency, and this just allows us to use the whole range of the dbc range from 1 to 6 and higher to get like an ensemble of inversion models.

B

So yeah. Let's, let's put this as the first conclusion, but let's go back to the salinization map itself, so we had the problem with the sharpness of the current salinization map.

B

So let us first look at the synthetic example which is realistic for the salinization map. So here is a synthetic example where a highly conductive salt water lens is lying on a clay layer and with some fresh water. On top, so I've used simpek to generate multi-dimensional, reliable, multi-dimensional data on an accurate 3d mesh to get some three reliable synthetic data.

B

And then I used the wave based method to invert that data in two dimensions and the extension to two dimensions. I had to make some choices and I decided to to make the choice for maximum flexibility and it's actually the same approach and, for example, the ticket of regularization implemented in simpek, where you have a separate misfit for the horizontal and for the lateral, the horizontal and vertical orientation, and this allows us to use a different wavelet.

B

Let's say a different type of structure along the vertical orientation and a different type of structure along the horizontal orientation.

B

And especially with other uh wave based methods which yeah tend to use, the higher dimensional wavelet transform the disadvantage. There is that you have the same type of structure along each orientation and I, I think, that's not always suitable for geophysical inversion.

B

So when we run the method, we can get like a whole ensemble of inversion models, let's say, for example, the first one. We have used the blocky inversion along each orientation, so we definitely also expect and see some blocking results in the inversion model.

B

The inverse model itself fits and fits the data up to the noise level, but it's not really the inverse model that we expect in real life, so maybe other wavelets combinations are more suitable.

B

For example, in this case I have changed, the lateral wavelet basis function to dbc2, and then we see definitely a sharp but smoother transition along that orientation. But you still recognize the distinct steps when you go along the vertical orientation from the dbc1 wavelet and then this example is the opposite, where I have used a very smooth basis, function along the vertical orientation and then the blocky along the horizontal one and yeah in this whole set of inversion models.

B

You may think that, for example, this combination is a good is in a good agreement with the true model, but still for this approach. Yeah there, you have still your regularization parameter and then that regularization parameter uh the relative one and simple. It's like alpha x and alpha y, but one of them is redundant, so I take them equal to one, and then I say this is the relative regularization parameter.

B

So in an ideal world, you would want to optimize for both regularization parameters, which is which, which isn't generally a problem, but uh with the flexibility of this wave based method and actually due to the specifics of the wavelet, transform and also the scale dependency.

B

It can happen that when you combine different wavelets, um the value of this um this term is not properly scaled with the value of the other term, and then you get some very um weird inversion models which you actually do not want.

B

So I tried to solve this with a calibration step and from a helicopter view, and I, the idea is as follows. So you look at the variability that you can see and and in the data itself. So if you see a lot of variation here, you expect also to see more later lateral variation in your inversion model.

B

What you can also do is to to take some random soundings and to do 1d inversion. Then you only have one legalization parameter and then you can fit up to the noise level, and then you get some idea of the complexity to expect along the vertical orientation.

B

The specifics of the calibration step itself are a little bit more involved and the details are actually described in the preprint available on research, gate and archive.

B

But at least this calibration step leads to a more objective choice of a good candidate for alpha, which I call alpha tilde and when I show this for a few weighted combinations, for example, bbc 1dbc1.

B

This is almost one, which is also a good result, showing that that when you use the same wavelet combinations along each orientation, you get yeah, not a very distinct result. So the scaling is right. But here we see when you use dpg5 in combination of dpg1, the estimated the optimal estimation is quite different than dbc 1 dbc1.

B

C

B

Something to take into account also. I would recommend that if you do this step, that you still go one step further and do some sweep along that alpha parameter to really uh yeah.

B

Let your expert knowledge uh to use your expert knowledge and, for example, in this case, what I did is I took the dbc1db q1 and I varied the gamma and the gamma multiplied with the estimated alpha leads to the real alpha present in this equation and, for example, in an extreme case here we see some very stretched out blocks along the vertical orientation and in the other extreme case, we see some very stretched out blocks in the horizontal orientation, and my decision was that actually this example with gamma equal to 5 is optimal or, as at least looks good compared to the synthetic model.

B

And then maybe let me note that when you go back, I have used this gamma throughout all other inversion models. So I only had to do this calibration once on dbc1dbc1, so I only had to do this sweep once and then I've used this gamma, which I determined there to use for all inversion models.

B

So the alpha tilde will be different for each inverse model, but the gamma will be the same. So you will be able to combine some kind of objective um yeah step with your expert knowledge and then you, when you finally end up with a suitable candidate for alpha.

B

Okay, um let's go to the actual field data itself and there I took a slightly different approach. So the field data is the data behind salinization map, so the airborne em dual moment stuff and there we will try to incorporate our prior knowledge um based on via that wave basis function. So, for example, we expect some sharp transition from fresh to solved. So maybe we should choose some sharper wavelet basis function so for the vertical orientation.

B

I choose a rather sharp wave basis function while for the horizontal orientation yeah, it's maybe more expected to have some smooth transitions between the model parameters. So why not take a smoother every basis function and then this is actually the result that we got where these white bands are actually data. That was missing, which is which can happen on field data, and then this is also like an approach to get some result.

B

Timing, wise, I think I will skip those slides and end with a nice end of the story with some future work left recently I was awarded the brilliant marine research id grant and I'm really happy with it, because with that grant, I will be able to collect some high resolution, ground-based data, so ert borehole, loggings and also ground-based time domain em, which will enable me to um have a really high resolution image of a subsurface and then I'll use this to calibrate it against the ensemble of inversion models that I got from the airborne em data and then I will be able to make a more informed decision on what wavelet bases function in combination with the appropriate, appropriate parametrization that will be suitable, so yeah, I'm actually quite looking forward to get started on this project.

B

So if you have any questions, I would be very happy to answer them.

A

Excellent, thank you. Walter um yeah feel free to unmute uh jump on in. If you've got a question um and if lots of people do, we can do raise hands but but feel free to just uh jump jump on in.

D

Hey water: thanks for your talk, it's really interesting. Could you please go to the page 10.

B

D

uh Maybe next one? Oh yes! So if we look at the top right finger string here, the maximum value for the block model is much larger than the smooth autonomous recovery model, as well as a true model if we use the sparse norms.

D

However, if you look at your models, the maximum value for the blocking model and the smooth model looks highly consistent. Could you please explain the reason behind these observations, because this is my first time to see this.

B

um Yeah, that's actually also something we have observed in the paper you published last year is.

A

B

Even with the dbc6 okay, you get smooth results, but they are able to get higher amplitudes. There are some nice examples in that paper.

B

So when you have like a huge peak and you still use a smooth way of recovering that model, you are still able, with the wave based method to get that peak, um which is not almost not possible with the with smoothness uh regularization. So that's like a property typically of a wave based inversion.

D

Oh okay, okay, yeah! That's a really.

B

I'm not the only one who observed it by the way, so um it's confirmed by others as well.

D

Yeah, this is really interesting properties for the and.

B

That's actually very nice when I, when I can go a little bit further on this um like in this example, um there will be then, for example, a few of these coefficients, which should not be zero, but then really explain that specific peak because they correspond with the right location on that place. Where you have the peak. Imagine you have this large peak here. There will be some very distinct building blocks that will be able to explain that data and still be able to have a sparse representation.

B

So that's why the wavelet theory is able to have that peak. Well, recovered, okay, but yeah, maybe with some other examples at some other time. I can elaborate on this.

D

Yeah, it looks interesting another. Another question is related in the page 16.

D

Yeah, so the first problem is: how do you define the smoothness and the blocking? So what means what what's the center level of the smoothness is a smooth model, for example, in these eight models, so which one is a start from which which model is smooth.

B

Well, when you use, for example, this one you use smooth wavelet basis functions along each orientation as you get the smooth inversion result, I don't have the profiles now in the slides, but yeah. You also see a smoother transitions. When you put the profiles on top of each other but yeah, you can also combine them. For example, here um yeah it would be nice if I have a figure that shows like this cross-section. This cross-section will be rather smooth. Well, this cross-section will be uh blocky.

C

B

Both I mean, if you use a wave basis function with a large number of finishing moments, you will get smoothness along that orientation, so also this one will be rather smooth along that orientation.

D

I'm thinking is there any ways we can compute values and a quantity to evaluate the small sets for these models.

B

Yeah I haven't considered it. um Yeah could be yeah, maybe interesting to quantify it.

D

Yeah and another quick thing is the: I notice that the rms values looks uh different for these eight models. Yeah.

B

D

Most in the bottom right models is 1.7 and in the top middle one is 0.68.

B

Yeah, that's true. uh That's that's mine, that's mainly because yeah. We need to generate all these inversion models and it takes quite a while. So you have like endings, uh you have some optimization to do so. The discretization for the absolute regularization parameter was not small enough. Let's say to really hit the root me square error equal to one each time.

B

So that's the reason why it's not always equal to one.

D

Yeah, but so are they so my point is: are these do you fit the data in the same levels if they they cannot fit the data in the same levels? Probably it's unfair to comparing these models.

B

um Yeah but yeah in fact, uh that's that's the fair point. It should be more ideal if they all are equal to one um but yeah that would take um yeah. Maybe.

D

Quite some extra.

B

Effort to to generate these inversion models, yeah.

D

B

D

Yeah, maybe in the small range 0.8.

B

Yeah, you never know when it will when it will hit. So that's because yeah you don't know what the values of the model misfits will be. So it's always a little bit of a guess and then suddenly it goes really quick and then it overshoots or in the shoots and yeah you have a too low value. Yeah.

E

We just need to have a smarter, uh smarter way to come back up right. You're using a cooling schedule for beta is that the other.

B

Reason, father yeah.

E

Yeah yeah we need to be able to just at the end. You know have a little little time to bring it back to target basically yeah. Okay,.

B

Yeah that would be uh yeah more suitable in this in this case, and also save some computation time. I suppose.

E

Well and then at the end, because you'll need to reiterate to just readjust, but anyway,.

C

Yeah, it's a linear. It's a linear system right.

E

The whole thing is linear. Is that right? I'm sorry is the iteration process.

D

Linear is it the linear.

B

E

That you're solving with.

B

uh I'm not sure what you what you mean actually.

E

Well, is it like your your regularization term, is it model dependent or is it is just a linear? You know, linear operators that are inside.

B

Is it yeah? I guess there are linear operators inside the model. Misfit is not the computation issue. That's really fast. If that's answered your question.

E

Yeah, I'm guessing it's a tough narrative, yeah! That's.

B

E

um Great talk by the way uh I need to jump. I need to quit to go to another meeting soon. So I'll, just tell you two things, one. It sounds like uh you're really ready for a pr to bring your stuff to. uh uh It looks like your radarization class is like perfectly in line with what we're doing so. It would be great if you have a pull request and bring your stuff right. So then uh we can help. You uh basically maintain it over time uh and we can use it.

E

You know more uh more readily mm-hmm.

B

E

And then the second thing that I wanted to ask you oh yeah, so common to the alpha values. So usually we use the alpha values to uh yeah sure as a parameter for people to tweak, but at the same time there's a lot of like dimensionality. That happens. You know. Let's say you have a model regularization and you have a gradient term. We use the alphas to scale those dimensionally with with each other so that they can. They can play nice right. Let's say the standard alphas is going to be one over dx square.

E

You know that's kind of like the the usual the usual thing, yeah and here you're kind of combining different wavelet functions. So is there any dimensionality big offsets between let's say db1 and db5.

B

Because that's usually.

E

Where alpha is coming in yeah sorry yeah.

B

I think the problem is is actually the scale dependency here because, depending on the length of the signal, so if you have more soundings, let's say you have a wider image and a wider mesh, but this will have an effect on the amount of wavelet levels.

B

So the amount of times you you take smaller building blocks and when you take a smaller building block, the way that you will assign to it will be a power law so two to the yeah, let's say fourth and sixth and so on- and this is, I think, a little bit the reason why it doesn't scale like you. You would expect.

E

Right, but is there any like yeah, I'm just gonna, throw it out there if there's any uh obvious constants that needs to be accounted for. If you're going to combine, let's say a db1 with a db5 right, like you know, let's say one has peak amplitude of a thousand and the other one has a peak absolute of one. Then you know you know implicitly that your alphas needs to have. At least you know a thousand coefficients in front of them to combine them. Does that.

B

Make sense yeah it makes sense, and it's something that I tried to solve with various methods yeah that made sense and then eventually didn't work at all. So you.

E

Did it empirically yeah this.

B

Is like the the best um I could find. I.

E

Got you yeah sounds good perfect. All right, I gotta go thank good talk, congrats, okay, yeah, good luck with the finishing pg bye.

A

Maybe we wait for others to come up with questions. One thing um when you were sort of pointing out where in the right or where in simpeg these changes take place, I appointed primarily to the regularization, but if I'm understanding correctly, would it also be a mapping um that you need to apply or is.

B

It yeah and in the current implementation I don't need the mapping I used to do it. I used to optimize in the wavelet space, but when you do this, when you go to multiple dimensions, yeah, this doesn't work. When you take this approach, so it's basically um yeah it's basically the chain rule that solves the problem, which is important, which is and the derivative the derivative of the model misfit. So it really fits really nicely. I would say into simpac, um doesn't require an additional map.

A

So you would still presumably invert for like log conductivity yeah.

C

But then the regularization.

A

Is where all the wavelet um magic is happening? Oh cool.

C

D

So I'm wondering, did you use the sim pack optimization modules to solve this cost function?.

B

D

You write yeah yeah.

B

Not for this, because when I was developing it yeah things get always too messy, and then you work with your own codes and actually for this one. I worked with the bfgs in sci-fi, which also, which is all yeah, is great actually for this purpose and also, I think, a little bit faster than than the simplex for this case.

D

C

May also ask a question: hello, everybody. I know walter quite well because I'm one of his promoters, but but I'm coming from a more let's say, theoretical physics, a mathematical viewpoint. uh Actually here you have slidestop and vertical and horizontal horizontal direction.

C

What prohibits using genuine 2d wavelets? Do you win or lose something by not doing that or doing it.

B

Why I don't consider 2d wavelet, transform.

C

No, no, I know, but would that change anything? Could that help or is it computationally harder? The 2d.

B

Yes well, first of all, then you're really limited to square images and, for example, when we go to this image, it's definitely not square. We have 300 soundings and the discretization and the depth is only 64.

B

um and if you were to need to extend it to like 300, also in in that orientation, um when you use analytical models, for example yeah, it's not really. um I mean it's not really computationally efficient, so I think it's better to have a method that is suitable to not to do not do some combination, computational overkill, I don't care yeah, also some other people use shirtlets, which then also take into account some directionality, but the amount of model parameters that you have.

B

It's explode, yeah yeah and then you need to optimize for all of them yeah, but not much gain for this application. Okay, thanks.

A

Folks have any other questions. Yeah.

C

B

Right and you think it would be straightforward to extend it to 3d or you.

E

B

Any difficulties yeah, I don't expect any difficulties. um Well, I also didn't expect that difficult difficulties to go to 2d, but then the calibration was like a hard hard thing.

E

A

B

I expect that I can do the same thing, which I will actually also try um for for this step. I will try to do some to get some 3d electrical model to use this, to compare with the high resolution data.

C

A

And here just to clarify, when you said tuna or 2d, was it 1d, ford simulations and then 2d regularization.

B

uh Yeah, I think I won't use two and a half d, meaning that you have only um variations in depth and one direction and then all constant along one direction. um But the imaging itself use a 1d for model, and then I call it quasi-2d.

A

C

A

Any any last questions a minute or so.

A

A

If not uh thanks so much boulter, it's really uh exciting to see your work and so cool to see. I think you know one of the things we strive for and the ideas we had with simpek is that it'd be modular, so people can plug in new pieces, um but it's really nice to actually see that that that's possible and you've done some really cool work and so yeah to dom's point.

A

If it's something you're interested in, including in the main, um I think we should absolutely uh follow up on that and chat through that um and best of luck with the next stages. Congrats on your award, I'm really excited to see where, where you go and where you continue to go with this.

B

Okay, thank you.

A

All right thanks so much I'll stop the recording here.