 Perfect. Thank you very much, Alik, for the kind introduction and, of course, for the opportunity to give this lecture. So I will try to share my screen, hoping that you see this. Can you see the title slide? It works. Yes, perfect. Very good. So then there we go. So as Alik already mentioned, I will give a presentation, a lecture about seismology. And in particular, I would like to talk about solving large or larger seismic English problems, not just by increasing the power of our computers, but by actually using smarter methods. And the work that I will present, of course, hasn't been done all by myself, but it's a close collaboration with my colleague Lars Gebrat, Dirk Philipp van Herveld and Andrea Zonino, so we trust us on Christian Böhm and Martin van Reen. The central theme of this lecture is really the effort to constrain the structure of the Earth using earthquake data. And since I think not all of you are seismologists, I would like to start with just a small introduction into this field. So those earthquake data are generated by earthquakes and what you see here is a numerical simulation of such an earthquake. It's in fact a 2011 Tohoku earthquake. It had a magnitude of 9.1. And literally within seconds this earthquake radiated or liberated an energy of about four to the power of 22 joule. And so this is an amount of energy that we can't imagine. And it corresponds to roughly 80 times the world energy consumption. So within a few seconds, this earthquake liberated energy that would have covered the world energy consumption for about 80 years. Now this earthquake, of course, was big enough to be recorded globally all around the Earth. For example, at the Black Forest Observatory, which is located in Southern Germany. So it's here at the position of the red star and it has an epicentral distance from the earthquake of 83.3 degrees. So roughly a quarter around the Earth. And there at the Black Forest Observatory, one could observe these seismograms. So these seismograms are recordings of ground motion at that site. And you're looking here at the vertical component at the top, the east-west component in the middle and the north-south component. And what you see is that those seismograms, those recordings of ground motion roughly fall into two parts. We have that first part that contains waves of relatively low amplitude. These are the so-called body waves that literally travel through the volume of the Earth. We see here, for example, the first arriving P wave. And all sorts of other body waves arriving at different times, different P waves, different S waves and so on and so forth. And then the later part of this seismogram is dominated by those high amplitude waves. These are surface waves that are bound to the surface of the Earth and as a consequence have larger amplitudes. This is the kind of data that we are looking at when we are trying to constrain the structure of the Earth, the deep structure of the Earth. Now a few words about the actual sources, which are those earthquakes. What you see to the left is a collection or visualization of the epicenters of about 22,000 earthquakes that have been located by the US Geological Service in 1998. So you see that most of those earthquakes, as is well known, cluster around the active boundaries of tectonic plants. There are lots of them, they like to repeat in nearly the same places and obviously they are poorly distributed. The bandwidth of those earthquakes, so the frequency content of those earthquakes, at least at a global scale, is very broad and ranges from about 0.5 millihertz to around 5 hertz. So these are 13 octaves, so it's a very broad frequency range. And of course, a priori, the position, the timing and the mechanism of those earthquakes are not known. And this has an important consequence for the inverse problem that we are trying to solve because of this in fact a coupled source and structure problem, both the source and the structure need to be constrained on the basis of those data. Now what about the receivers that we are using? Those receivers are typically called seismometers, they also come under different names, and they are poorly distributed as well. What you see here is actually the array coverage or the station coverage of North America and North Atlantic. So all the beach faults that you see are earthquakes, and all the black triangles are seismic stations, seismometers that have been installed. And you see that most of those seismometers are in rich industrialized countries, but there are very few in, for example, Africa, Eastern Europe, the Arctic, and of course there are very few in the oceans. The few that you apparently see in the oceans are actually sitting on islands. So they are poorly distributed as well. They are very variable characteristics, so they have different recording bandwidths and they suffer from different kinds of side effects. And the recording bandwidth actually quite often covers the observable bandwidth. So this is a quite nice feature. And currently I would say that the number of seismometers that are openly freely accessible at a global scale is on the order of about 10,000. This sounds a lot, but it actually isn't because of course the surface of the earth is very large, and the distribution is very heterogeneous. Now what about the medium that we are interested in? We are interested in the structure of the earths. And the earth is not a simple medium. In fact it is viscoelastic and anisotropic, so it suffers from attenuation from the conversion of kinetic energy into heat. And the earth rotates, obviously, which complicates wave propagation and it is also self-gravitating. And obviously also the earth has irregular topography, it has oceans and a fluid outer core, and those fluid parts need to be accurately coupled to the solid parts of the earth. So this suggests its evidence for the fact that the earth is a very complex medium, but when we solve an inverse problem not all of this complexity may actually be relevant. So the necessary complexity really depends on the amount and type of data that we're interested in, but also on the aspect of earth structure that we would like to restrain. Most frequently in global seismic tomography, in global seismic imaging, we are interested in the P velocity, so the velocity of compressional waves, but also in the velocities of SH and SB waves, which are shear waves with horizontal and vertical polarization. Now traditionally this seismic tomography has been based on the measurement of travel times and on the measurement of right here. So what one would typically do, one would take such a recording here, and then pick the arrival times of a bunch of waves. So for example of those compressional waves or of those shear waves that are arriving here, and then one would try to build an earth model that explains those travel time observations as accurately as possible. Now this is nice and good, it's a relatively simple and well behaved inverse problem. But also, you evidently throw away a lot of the information that is actually available. And as a consequence, during the past, I would say 10, 20 years, people have tried to improve this travel time tomography to what we loosely call full waveform inversion. And the goal of this full waveform inversion really is, so the ambition is to literally use every single wiggle of the seismogram that we have here, and not just the travel times of a few of the waves. So here really the goal, the ambition, the dream is to exploit as much information as possible in order to improve resolution of deep earth structure. And also to correctly account for the wave propagation physics in a complex three dimensional earth, in order to avoid forward modeling artifacts, and in the other direction to correctly account for the final frequency sensitivity and the non linearity of the problem, in order to avoid inverse modeling artifacts. So where's the broader relevance of all of this. As you are not all seismologists you may wonder, why is that, should that be interesting for you. The relevance of this of this way for my version of this full way for me is that at much larger scales at scales of about a million kilometers, you can use exactly the same methods in order to study the internal structure of the sun. At global scales, so about 10,000 kilometers we would study the dynamics evolution and composition of the earth. And at slightly smaller scales hundreds to 10,000 kilometers. Those full way from a vision models that we produce are relevant or important for earthquake source inversion for reliable tsunami warnings, but also for the monitoring of the comprehensive nuclear test and treaty. And if we go to even smaller scales, obviously the same methods can be used in exploration and reservoir geophysics for monitoring. And even smaller scales of 10 to 100 meters and engineering applications. And then at a centimeter scale, there's technology transfer from seismic full weight form inversion to medical imaging, but also the non destructive testing. And in fact, the figure that you see here. The left part has 10 centimeters scale. This is a full way from inversion image of a human breast. And to the right, you see using the same color scale a full way from inversion image of tears. And of course there are many others. So the interesting thing here is that that the methods that I will present technologies that I will present. They have relevance over about 11 orders of magnitude in spatial scale. And really the outstanding challenge and what we are doing is that the computational cost of this full way from inversion or FWI has a very poor scaling with frequency. So the scaling is somewhere between frequency to the power of four and frequency to the power of five. So at the moment we are operating at frequencies that are relatively low because this is what we can afford computationally. We want to increase the frequency in order to actually cover the frequency range that we can observe it rapidly becomes excessively expensive and totally out of scale. Now of course, one may argue that there's more slow. And so we just have to wait for me now. But you can make a little back of the envelope calculation. And even if you assume that most law continues to hold which is a little bit questionable. It will actually take many decades before we can actually solve full way for inversion problems at least that global scale within the bandwidth that we can actually observe that is down to about one hertz. So in other words, we will all be very old before we can actually solve this problem if we continue to do it the way we do now. Also as a consequence of this very poor frequency scaling. There is uncertainty quantification so an honest comprehensive uncertainty quantification using using the Bayesian or probabilistic approach is currently totally out of reach in full way. And that is unless we use frequencies that are so low that the exercise becomes useless or uninteresting. There will be a need to develop more clever algorithms and to not just wait until most law helps us to solve the problem which as I said will take a very, very long time. So what are the goals of this lecture. They are, in essence, to present you with a collection of methods that enable faster 3D for waveform inversion. But also a more complete and honest uncertainty quantification. And I will present you two classes of methods. The first one is more intelligent Monte Carlo samplers that allow us to do uncertainty quantification for interestingly large problems. And the second one is methods for accelerated forward and edge on simulations. So simulations of weight propagation and forward and edge on. So the outline of my talk is as follows. I will start with so called Hamiltonian null space charts. And what these null space shuttles are, there are methods that use artificial Hamiltonian systems in order to produce alternative models of the Earth alternative in the sense that they explain our observations, as well as the model that we may already have. And those Hamiltonian null space shuttles. They are the foundation or very closely related to a Monte Carlo method that I would like to present which is, which is called Hamiltonian Monte Carlo. And I will show you not only the theory behind Hamiltonian Monte Carlo, but also an application to probabilistic full way for inversion in two dimensions. And then we'll have a little break. I can't guarantee that this will be exactly after 45 minutes, but but maybe roughly, but it's a it's a natural breakpoint in my presentations I would suggest to to use that one. Then, I want to introduce you to a strategy to to tune those Hamiltonian Monte Carlo simulations automatically. So those Hamiltonian Monte Carlo simulations, they have a very large number of tuning parameters, and you can try to tune them by hand. But in fact, you can develop more intelligent algorithms that tune this Monte Carlo sampler. Essentially, while it's running so on the floor. And, and this is reproduced auto tune. Similarly, I want to introduce a numerical method that we call smoothie sound, which is a variant of spectral and simulations of seismic wave propagation. And, and these use numerical measures that are adapted to prior knowledge of the geometry of the way field. And that under favorable circumstances allows us to reduce the computational cost of forward and adjoint wayfield simulations by by orders of magnitude. And again, of course, I would be happy to have a discussion with all of you. Before I actually go into the, the hard content a few notes on the style of this presentation. You're connected virtually and I guess after one and a half years or even more pandemic. Most of you are pretty tired of virtual lectures. And, and so what you will see as a presentation that is colorful and casual. It's not hard core mathematics because I think this is not suited for this kind of presentations. My focus will really be on transmitting concepts and not so much on transmitting rigorous math. So I really want to, to just wet your appetite to get you interested. And maybe get you interested in, in having a more in depth discussion or digging out some of the references that I will show at the, at the end. So the first topic, which is on Hamiltonian null space shuttles. So first, a little bit about the preparation that a problem state, what, what is this all about what, what are the goals. And, and the first thing I want to talk about I need to talk about is the notion of the, of the effective mouth space. So the setting is as follows a matter we have some misfit functional kind and that misfit functional measures. The discrepancy the difference between data that we have observed and data that we have computed for some model of the earth need not be a seismological. And of course that misfit depends on M, which is a certain model of the earth can be a distribution of the P wave speed and S wave speed in the earth, or some electrical or magnetic properties and so on and so forth. And we assume that we have already found a model of the earth that we call M hat. So that explains our observations to within the observational errors. So we have an acceptable model and we don't bother at the moment where that acceptable model is actually coming from. And now the effective null space is the collection of all other models. So we have some M hat plus some delta and so our model that we already have, plus some perturbations that have that produce a misfit. So discrepancy between observations and some tactics that is smaller or equal to the misfit of the acceptable model that we already found, plus some epsilon, which is a misfit tolerance. There are alternative models that produce a misfit that is smaller than the one that you already found, plus possibly some little tolerance that we admit because we have observational uncertainties. And so all of those models those M plus delta M, those, these are the alternative models that we after, and all of those alternative models, they constitute our effective null space. So now the problem is, of course, how do we actually find those alternative models? How do we find those M hat plus delta M that explain our observations equally well? How do we do this? Why is this actually relevant? Why are we interested in that question? Well, obviously, this is important for uncertainty analysis. We want to know if there are alternative models that explain our data equally well, but that are potentially very different from the acceptable model that we have already found. So if there are models that are very different from the one that we already have, it means that our inferences that we are making are pretty uncertain. But also, we may use this in a constructive mode. We may want to construct alternative models that explain our data equally well, but maybe contain some new structural feature that we want to test. So how does this Hamiltonian null space shuttle actually work? What is the essence behind it? So there's a trick. It's a mathematical trick. And here comes part one of this. So what we do is we interpret any model M as the position of a space shuttle or some particle in a high dimension. So if we have an n dimensional model space, say a thousand dimensional model space, then we interpret any model M as the position in a 1000 dimensional space. And then what we do, and at this point this may seem a little bit weird, we assign a potential energy to that particle or to that space shuttle. We call that potential energy U. It depends on the position of the particle, which is M. And we simply say that this potential energy is equal to the misfit of that model. So the potential energy is equal to misfit. And we can do that because misfit is always a positive quantity and so we ensure that potential energy is positive. And then we also assign an artificial momentum. So this momentum P is totally invented. It's an auxiliary variable. The moment is just some vector. And based on this artificial momentum, we can come up with an artificial kinetic energy K. K, the kinetic energy depends on the momentum. And this is defined just as in classical mechanics as one half of the momentum transposed times the inverse of the mass times the momentum. And the important is that inverse of the mass is actually the mass matrix. And so it's, it's, it's a generalization of the scalar mass that we are used to. And so now we have, we have invented a potential energy, and we have invented the kinetic energy. And then we say that we have some initial momentum, so just imagine that the space shuttle is sitting in its insert position M. And then we say at that position, it has not only an initial potential energy, but it has an initial momentum. And that this initial momentum is chosen such that the kinetic energy corresponding to that initial momentum P hand is just equal to our misfit tolerance. But this is this epsilon that we had before. So we go back. So here we have our little misfit tolerance over here. So we choose the initial momentum, such that the initial kinetic energy is just equal to that tolerance. Well now we have invented potential energy and the kinetic energy, we can add both. And this gives us total energy, which in classical mechanics typically is called the atomic. This is one of the trick inventing essentially a couple of quantities. Now part two is letting this space shuttle actually fly through model space. So we have the total energy age. And we can then compute a trajectory of that space shuttle of the space shuttle by solving Hamilton's equations from classic classical mechanics. I'm sure most of you have seen those the time derivative of the position of our model is equal to the momentum derivative of the total energy. So this is the first of Hamilton's equations. And the second Hamilton equation is that the time derivative of the momentum is equal to minus the space derivative of the Hamilton. But then we can we can solve those equations, and we know that age is actually constant along a trajectory. So age to total energy does not change as the space shuttle flies through models. And that's basically what this implies. So we have here the total energy age. After some time team. So the total energy depends on position and momentum, which in turn depend on time. And the total energy is equal to the potential energy, which we define to be equal to the misfit as a function of position as a function of model. The kinetic energy, which as we have seen before is one half momentum transpose times the inverse of the mass matrix times the momentum. Let's carry this a little bit further. Then we know that age, the total energy is preserved. This means that the total energy after some time key must be equal to the total energy at the beginning of the trajectory. The initial position of that not special. And this initial total energy is equal to the initial potential energy, plus the initial kinetic energy. And then remember, we cross out a few terms, this initial potential energy, we define to be equal to the misfit tolerance. And then we have this equation here just crossing out a few terms that are not so interesting anymore. And then what we see is that since epsilon is a no since. This kinetic energy here is a positive quantity. So the kinetic energy as a function of time is always positive. And to account, we find that our misfit as a function of time is smaller or equal than the misfit at the initial position, plus our little tolerance. And this is exactly what we wanted. So this means as this space shuttle flies through model space, it produces misfits that are always smaller or equal than the initial misfit at the initial position, plus that little tolerance. So this means that all models along that trajectory are indeed within the effective null space. This is exactly what we wanted. So what does this look like in practice. It is important to note that depending on some choices that we make some subjective choices, this space shuttle probes different parts of the null space. And for example, if we say that our tolerance epsilon, so epsilon is, if this is zero, then we can actually show that this null space shuttle is equal to a gradient descent. And different types of descent methods that we have, for example, conjugate gradients or new methods or steepest descent, they depend on the kind of mass matrix. So the null space shuttle that I just presented is a generalization of gradient descent. We can also prescribe the takeoff direction of the null space shuttle. I won't show this here. And this corresponds to adding specific features to the model. After it has been constructed. And what we can also do by different choices of mass matrix is to probe either rougher or smoother parts of the non space. Here comes an example. And that example is from a non linear travel time. So at the bottom, you have receivers as sources. So these are explosions or earthquakes. And the waves from those earthquakes are recorded by some receivers that you see here on top. So these are such monitors. And what we did here is we constructed a tomographic image that you see here in color. And the tomographic image, which is the seismic wave speed in kilometer per second already explains our observations to within the uncertainties. So this is our initial model. This is an initial model that we already like. And then we add some initial momentum key hat. And this initial momentum is just this little blow that you see. And then we let that null space shuttle fly. And after four artificial seconds, which corresponds to 500 iterations, we see that the initial model here has changed to this model. By construction, this alternative model that we obtain after four seconds explains the observations equally well to within the uncertainties. So we have constructed an alternative model. And in fact, during this iteration we have constructed 500 alternative models but of course I can't show all of them. Now the, what you see here below is a diagram of the evolution of the different energies. We have the initial kinetic energy that is sitting here. The initial potential energy, which is the initial misfit, it's plotted here in green, and the total energy in blue is the Hamiltonians. That's what you see here. And then by construction, as this null space shuttle flies, the total energy in blue remains constant. And then the misfit or the potential energy increases a little bit, but it still remains below the threshold that we have set by construction. And then you can keep this running and you see that then along this trajectory, the potential energy or the misfit fluctuates, but it always remains below the tolerance. And then as this null space shuttle flies, we produce lots of alternative models. So this one after four seconds, this one after 24 seconds, after 48 seconds and so on and so forth. This can also be shown in the form of a movie. This is here. So every little snapshot that you see in this movie is an alternative tomographic model. It explains the observations just as well as the model that you have seen at the beginning. So you have a mechanism to produce literally thousands, hundreds of thousands of alternative models. The point here is, so some preliminary conclusions, the point is that we can explore the efficient null space of even of this nonlinear English problem and produce alternative models without Monte Carlo sample. So it's a lot more efficient than Monte Carlo sample. It manifests on the somewhat counterintuitive construction of an artificial hematomene system where the data misfit corresponds to the potential energy and the misfit tolerance that we admit corresponds to the kinetic energy. We have, I have explained that this is a hybrid between gradient descent Monte Carlo methods. In fact, if the tolerance, the misfit tolerance epsilon is zero, then this null space shuttle becomes a gradient descent method. And what you will see then in the next chapter is that if we draw those random momentum repeatedly in a random fashion, then we end up with an algorithm that is called Hamiltonian Monte Carlo. We have different user defined modes of operation of this mouth space shuttle. And like, and with the help of this, we can do quantitative hypothesis testing. We can construct alternative models with specific features that we would like to see, but we can also produce alternative models that are either smooth or rough. So there's a lot of potential to, to play with that method, and to use it in a way that is useful for specific applications. So let's go into the to the next chapter, which is Hamiltonian Monte Carlo and specifically applied to seismic tomography. For those of you who don't, who haven't seen this picture of what you see here on this slide is the Monte Carlo casino, after which Monte Carlo methods have been, have been named. So what is the motivation of Hamiltonian Monte Carlo. The efficient sampling of model space. And this can be exemplified with a toy example. So this Hamiltonian Monte Carlo method, it was introduced as hybrid Monte Carlo so under a different name and quantum mechanics 1987, but it took a surprisingly long time so really about three decades before it actually leaked into into geophysics. So it's a method that allows us to sample the posterior probability distribution of an inverse problem. And the motivation in essence is a well known deficiency of the metropolis. I want to illustrate this deficiency here. So assume we have a probability density it's just two dimensional that is that is shown here. So we want to explore this two dimensional probability density using how much using metropolis hastings. Essentially we have two choices. So we can make small steps baby steps through model space. This has the advantage that almost all of the steps will be accepted. But, on the other hand, our exploration of model space is very, very slow. But then, in order to accelerate model space exploration one may think that we can do just bigger steps, as shown in this diagram. But if we do bigger steps, then the likelihood that the next model is actually accepted in this metropolis hastings algorithm becomes very low. So as a consequence, in the end, we also explore very slow because our acceptance rate is so slow is so low. So Hamiltonian Monte Carlo tries to fix this problem by basically taking advantage of derivative information. And this allows Hamiltonian Monte Carlo to make both long distance moves and have a high acceptance rate. And this is exactly the combination of properties that we need in order to solve high dimensional inverse problems as we typically encounter in geophysics. So in conceptual introduction to Hamiltonian Monte Carlo, which in principle you have already seen in the previous chapter, it's a variant of this Hamiltonian null space shuttle. We start with an initial model M. In the null space shuttle we have chosen this deterministically. It was a model that we already liked that we already found acceptable. Now in Hamiltonian Monte Carlo, the initial model is chosen randomly. So say it's, it's sitting up here. And then we have a misfit. And we say that this misfit is the potential energy of that initial model. So the same strategy here, we have a misfit and we now simply say that we call this misfit the potential energy. For our brain, we may say that this initial model is the position of a high dimensional space shuttle and that space shuttle orbits around a planet, which is, which corresponds to the maximum likelihood model. And then we assign a random momentum to that space shuttle, which means that we kick the space shuttle in this direction as before as in the Hamiltonian null space shuttle. This is a totally auxiliary quantity. And this initial momentum then defines the kinetic energy of the space shuttle. And then we solve Hamilton's equations. So Hamilton's equations then let the space shuttle move along a trajectory through this high dimensional space, which is our model space. And, and as this null space shuttle, the space shuttle flies, it preserves this total energy. So we move towards a new test model. And then just as or similar to the Metropolis Hastings algorithm, we evaluate a Metropolis rule. If this move to this test model is rejected, we go back to where we came from. And otherwise, we move on. Moving on means we again kick the space shuttle in a random direction by giving it a random momentum, and then it flies off in a different direction. And then I see, I think you see where this is going. It just flies from one position to another and so on and so forth, and traverses model space. What are the important features of this algorithm. The first one is that all those trajectories always orbit around plausible models. In, in, in simple words, the space shuttle always stays near, near the earth, or if you use a different picture, the earth always stays around the side, yes. This is what enables long distance moves. And so we, we move along very long trajectories, but all of those trajectories are still close to the maximum likelihood model. So so they're still very likely to actually be useful to be meaningful. So we have, we can have long distance moves. And as a consequence, we have fast model space exploration. We can move fast. We have long moves. We can move a long way and those long distance moves are actually being accepted with a high probability. A disadvantage, maybe that we actually require derivatives of the forward problem. So derivatives of the year of the potential energy with respect to our model parameters, because this derivative is needed in the solution of this problem. But in fact, it turns out that this is a very small disadvantage, because we can compute those derivatives very efficiently using the job techniques. That's pretty doable. So, towards applications, how is this useful. And, and what I want to show you is an application or synthetic application of this Hamiltonian Monte Carlo to full wave form inversion in two dimensions. For example, here we use to demand to the elastic wave propagation behind that this is classical stack of grid finite difference method with a maximum frequency of 50 hertz. We interested in three physical model parameters key velocity as velocity density, and we discretize our model using 10,800 reports. This means that the total model space dimension is 32,400 which is actually pretty large for a Monte Carlo. It is synthetic. So we are using a synthetic model a constructed model in order to produce artificial data. And here you see aspects of this model on top. This is the distribution of as velocity. So it's in a box that is 250 meter wide and 125 meters deep. And, and you see this, this sort of mimic of a geological structure, all the sources are here at the bottom, and all the black triangles near the top, the surface are the other receivers. And below you see the distribution of density that we put it. And then using this Hamiltonian Monte Carlo sampler, we can produce many aspects of the posterior distribution, for example, the mean value of as velocity, and this mean value, which, of course, would not be the only statistics that you look that you would look at. Very closely reproduces the target model that we plugged in. And, and something similar is true for density, but of course density classically can be can be reconstructed only much more food. And, and what we also obtain are quantifiers of uncertainty. So for example here, the standard deviation of as velocity, or the standard deviation of density. And so this is very nice, especially the density part, because as I said, density classically is extremely difficult to reconstruct the reconstruction here of course it's not perfect, but at least it is honest. And we get an honest quantification of uncertainty. Also here's some preliminary conclusions before we go into the break. This Hamiltonian Monte Carlo method is a randomized version of the Hamiltonian null space shuttle, where you successfully draw random momentum, so you kick the space shuttle randomly in different directions. What we have seen is that we can treat all of magnitude 10,000 model parameters in the two dimensional full wave form inversion. And that without any super computing. So we can solve a meaningful full wave form inversion problem, probabilistically in a Bayesian way without requiring a super computer. So the example that you have seen before, ran on the laptop within a couple of days. And so it is comparatively cheap. And what we obtain are not only estimates of the parameters, but also honest estimates of the uncertainties and those parameters. And that includes estimates of density, which is very important from a from a geophysical perspective. And those uncertainty estimates that is also important here are free from any bias that would come from subtractive regularizations of the English. Now why does it work so well why is it interesting for geophysics, and that interest really comes from the computational cost needed to generate an independent sample. So an independent model of the earth. Metropolis Hastings, which has been developed many decades ago, this scales as the power of n squared, the scales is n squared. So when n is the number of model parameters, then the number of samples of Monte Carlo samples that you need to draw before you actually see a new independent model scales as n to the power of two. And this means that with Metropolis Hastings, you cannot solve high dimensional geophysical inverse problems, simply because the number of samples that you would need to produce grows extremely fast. In contrast for Hamiltonian Monte Carlo the scaling is much more favorable. The scaling is approximately n to the power of five over four. Almost not quite, but almost a linear scale. And the scaling property is what makes Hamiltonian Monte Carlo so well suited for high dimensional geophysical inverse problems. So this is the point here. Now what you see what you will see in the next chapter after the break is really more in depth. And the generation of those of those independent samples, how can we even accelerate this, and how can we optimize the generation of those of those independent samples. And as I said, this will come after the break which I think is about 10 minutes. And, and this will then be about this topic about auto tuning Hamiltonian Monte Carlo.