 another one hour. So the meeting is recorded from now on? Yeah, well now I will share my screen. So please use the chat later on for questions and mute yourself and your videos. Do you see my screen now? Yes, we see it. If you put it in full screen mode? Yes, now I am doing this. Do you see now? Yes, but not full screen mode. Not full screen mode? Yes. Oh, okay. Maybe it's because the second screen has to stop. Okay. Take out second screen, just a second, one more. Do we see now? Yes, perfect. Yeah, it's a full screen now. Yes. Yes, sir. Okay, wonderful. Yeah, but I don't see this full screen. I see just a few. Okay, good afternoon, everybody. Good afternoon in terms of Italian time. We are sorry that happened, but sometimes we have weather. Let me, actually, I just imposed a very in a very wrapped way. The presentation was not yet prepared, but our first part as well is done somehow. Yeah. Okay, anyway, we start the workshop, which is related to inverse problems and data assimilation in geophysical sciences. This is not a fully introductory lecture, but I would like to speak about this problem of the inversions and also specifically highlight the area which I am quite familiar. It is a computational geodynamics and particularly the inverse problems and data assimilation in geodynamics. Next, okay. When we are speaking about inversions, I think we have to determine what means the mathematical model of geophysical problems. And there are actually there is a many geophysical problems which can be described by the set of partial differential equations and the boundary and initial conditions which define in a specific domain which we choose. And the mathematical model links two things. It is a causal characteristics of geophysical process and its effects or observations which we are referring quite often. They say observations of some geophysical processes. When we're constructing the model, indeed we use the understanding of the processes. That's why we construct the model based on our physical understanding of the processes. The causal characteristics of the process include, as mentioned here, parameters of the initial boundary conditions. For example, some of them even unknown. There is a coefficient of differential equations, geometrical parameters of model domains. What are direct problems? Well, to determine, actually the aim is to determine a relationship between the causes and effects of geophysical problems. When we have a stated problem mathematical, we can solve it and we can find a solution to this mathematical problem for a given set of the coefficients, parameters, etc. In this case, solution to inverse problem entails determinations of unknowns, causes of geophysical processes or phenomena based on the observations, based on the effects of the processes. Inverse problem is opposed to a direct problem, opposite, sorry. Inverse problem is considered when the lack of information exists on the causal characteristics of information on the effects of geophysical problems. Meaning, again, observations. We have the local observations, for example, of some geophysical processes. It's not necessarily related to geodynamics, which I will be speaking today, but it could be related to any other processes in the atmosphere, in the ocean, hydrology, in the water, etc. I would like to mention here that it's one of the first applications in geophysical sciences was in fluid geophysics, I may tell. Meaning that it's in atmospheric sciences, in meteorology, in hydrology, etc. Later on, it's just several, let's say, decades ago, it moved to more solid geophysics in the area of geodynamics, geomagnetism, and other fields of geoscientific knowledge. I will be during the lecture, this one or the next lecture, I will speak more about many other different applications in solid earth in geodynamics, particularly in geophysics. Anyways, when we are speaking about inverse problems, actually, it is nothing new. For example, we speak about the machine learning or the artificial intelligence, but all these techniques are based, actually, deeply rooted in mathematics. And if we will trace and look when the first formalization came, at least to me, I mean, in a more modern way, came to us in the middle of the 18th century. And it was because of the geodesic information. We wanted to use the geodesic data to estimate shape of the earth, and they use astronomical data to infer the orbits of planets and comets. Here, I put the two names, French Laplace and the German Gauss. Actually, Gauss, I would like to tell, contributes significantly to geophysical sciences in terms of promotion geophysics. He's a student who became later a general, general buyer and served to the king of Prussia. He was approached by Gauss in terms of collecting the geodesic data, because it was very important to cross the Europe, for example, to really collect a lot of data to understand the shape of the earth, but it was not so simple. And the Gauss asked his student, and he at the same time presented this report to the king, asking king to invite other European, the kingdoms, other European states to join, to start to produce these geodesic data for the inversions in understanding the science. I mean, the basic science that you actually, at some level, were quite, quite I mean, it's influenced the development of the modern science, which is, for example, today, the geodesic. Again, telling about the basic science, where we're speaking about the theoretical theoretical physics, mathematics, and the first science particularly, because it's related to our world workshop. Finally, by the way, just speaking about Gauss, Gauss is the founder of the International Association of Geodesy in the middle of the 19th century. And today, this is a part of the International Association of Geodesy. Well, but if we will move further to the 2019, 20th century, and you will see the three names which I highlighted, and which theories is significantly contributed to the inverse problems, and to the so-called the data simulation and the other field of the inversions. First of all, I would like to mention Adamar, who made the formalization of the impulse problems. The Andreti kind of contributed in terms of the solution of the impulse problems together with Jacques-Louis Lyon. And I will mention during this my talk about their contribution in some way, but particularly if we are speaking about the solution of the problems, these two or Russian and the French scientists contributed dramatically in different ways. Well, this one probably I will move to the next one, just speaking about the well and impulse problems. By the way, everything is okay with the presentation, and I speak, it's everybody hear me? Or at least Karim, can you speak? Sorry, not sorry. I have to not to connect, but it's just something to share. Okay, well, I hope everything is okay, fine, do you see my screen? No? Okay, I hope that you see my screen. I hear no one, but in the chat, yes, everything is okay. Okay, fine. I mentioned the name of Adamar, and there Adamar actually formulated properly what means well-possed problem. And in this case, he mentioned that it's well-possed problems are those which have the property of existence, uniqueness, and stability of solution of the problem. And actually, when we are speaking about impulse problems, which we were in terms of the inverse problems, and most of the inverse problems are ill-possed. And the ill-possedness is actually if the one of their properties, which I just mentioned, existence, uniqueness, and stability doesn't hold. And in this case, we call these problems the ill-possed. For example, if it is a problem is not unique, then it is impossible to solve it. We need a specific methodology to how to search for the proper solution. Stability is one of the really important issues because it's, and I will show you probably today, I will show you how stability can generate a problem. And because it's some small change in the initial conditions can develop at the quite different solutions. And that's why the stability is a very important issue. That's why there is a contribution of the Marvoi grade in the formalization of the inverse problems and to understand what our borders, what are the boundaries within which we can work. And when we cannot do something at the moment. Well, next. Okay. Data assimilation in geodynamics. If we are speaking about the data assimilation, there is a first we are speaking about data. Before an assimilation, we needed data. How to assimilate data in geodynamic models? Okay. First of all, collect geophysical, geological geodesic data and therefore assimilation in geodynamics. Well, probably, probably I have to define what means the geodynamics because it's I mean, not every participants probably know in detail what is geodynamics. Geodynamics is a reference to dynamics of the earth's interior. And there's many, many processes which are related to these dynamics among the many processes are the mental convection processes. There is a subduction. And I mean, speaking about the major processes. And then it's also the the creation and the generation of magma, the magma flow, as is the more, let's say, local scales and other processes. I will be speaking today about this, mostly about the mental plumes because they are thermal, so-called thermal convection in the mental processes. And that's why they are the models of the mental plume. The needs equations which when we try to invert the system, the set of equations as Bandria and the initial conditions becomes impossible and we need a specific methodology of solving them. That's why I will be referring today to geodynamic models keeping in mind mental convection problems. But next time, I will also show you something from the reconstruction of the subduction history and some examples of sedimentary basins and also a lecture about the volcanology. Actually, not a volcanology, but more about the lava flow and lava don't evolution and this application of inverse problems to this area. Well, first, we, as I mentioned, say it's how to assimilate data in geodynamic models. Let's start first with the collection of geophysical geodata, I would tell like that. But when we are speaking about collection, indeed, we should keep in mind to collect as much as possible high resolution and the high quality data. Because if the data are polluted by errors, these errors will propagate through your solution of inverse problems and would generate the real big problems in the solution. And sometimes just the solution explodes and there is no way to solve numerically such kind of problems. The second way to learn mathematical and computational approach to data assimilation because it's sometimes we are using some, you know, so-called the black box. It's some software which is using inversions and you invert some data. You take the data from one and then you insert in the black box and you get on the other side, there's some solution. Well, it is indeed some important issues, but for me and always I try to convince my colleagues and students that it's a very important to at least understand the mathematics and computational approaches behind all this. Not necessary to go in very deep, but at least to understand what you are doing. It's not just using the black box. That's why it was one of the reasons of this workshop to get some, let's say, simple way of understanding how we are doing when we are using some specific methodology in some specific field of geosciences. And finally they assimilate present data to restore dynamics of the Earth's mantle in geological class. That is just something which I will try to speak today in the remaining part of the lecture. Data collection, again, as I mentioned, it should be it's a quite reliable data. Well, I would like at the beginning to mention that it's not so many data like, for example, in atmospheric sciences or in oceanography, in hydrology, we have geodynamics, but still with time we have more and more, more accurate data. For example, take the seismic images, seismic tomography. With years we receive more and more exciting results or in theoretical tomography and also results related to a specific to global tomography of the Earth as well as a regional tomography and even local tomography, reflection data and so on, especially 3D seismic in exploration area becomes today the one of the exciting data and observations which we can collect and use for in Russia. Depending on the field to which you are interested, for example, the borehole data plays an important role also, for example, in the reconstruction of sedimentary bases and these data will help to do it. The heat flow data, definitely in many applications in geodynamics are quite important, especially when you are generating the model to look how model your model is reliable and to compare the model heat flow with data which are available. Composition of the crust and upper mantle is important issue and it comes from the some knowledge about your chemistry and other fields. Data from mineral physics and that's another important issue which we can use to understand, to read with the composition of the crust at least and also the upper mantle. Geodetic measurements tell us about the movements of plate and also provide the essential information about these uplifts or subsidence if it's specifically the observations and measurements now for some significant period of time, let's say several decades. Data of paleogeographic reconstruction is quite important also to check the how properly we are doing inversions of our data in terms of the data assimilation and it's how properly we construct the evolution of the deep mantle. Data on subsidence uplift, I just mentioned a few but still quite important data which normally we collect when we would like to assimilate this data into mantle convection of thermal convection processes in the mantle. Why do we actually assimilate data? You can ask it, why you assimilate data in geodynamics? We have some information from geology about the development of earth and so on. For example, the spot as the geologist could reconstruct based on the paleogeographic data and some paleomagnetic data it's possible to reconstruct the position of the earth, position of the earth and all of the plates at some specific period of time. I don't know these red lines here but well. Another issue is related to the okay related subduction of the plates and particularly here it's shown this subduction of Pacific plate under the Japanese islands and we will be speaking about this during the next lectures. Also the sedimentary basin I told that it's the importance of sedimentary basin still exists because of the hydrocarbone reservoirs and the geothermal reservoirs within the sedimentary basin and it's important to understand is how the area evolves this time. Particularly here you see there is one of the south dome where the and below the edge of the south dome. I don't know do you see my cursor and this here is a white you see some white area which is below the so-called this overhand of the south dome and this is a area white area is the area of the accumulation of oil and it's found based on the drillings that you see here is a 77 waterfalls but the idea whether it's how accumulated there what is the origin of oil what is the window of oil and gas to be generated etc and to answer to all these problems we need to reconstruct the history of the thermal and burial evolution of the sedimentary basin. Well I am very sorry I see the red line across I don't know Karim if you are online. Yes well let me check with Walter. I don't know do you see there is a red line. I see a red line. Someone made an annotation please delete it who made it should delete it. Can we avoid that this happens? For the next time I think this time it's not possible to change the settings of the meeting but if I will now stop and then come back it will be try to stop it. Do you see my screen? Yes okay fine thank you. Okay now let's very briefly state the problems and I would like to mention that it's from here you will see it's more equations but I will try to explain it just don't upright if you're not very much familiar with all equations I will explain them. First of all when as I mentioned it's when we state the problem we need to define the some model domain because we are in geophysics we are not working in the just the full space 3d space or some other that's or 2d but we are working in some specific model domain. Here I determined some parallel people as a model domain but it could be a sphere it could be tube and others but anyway anyway we determine the model domain and then the government equations which construct our model and this is a stocks equation or the momentum equation we can tell here is a p is the pressure mu is the viscosity u are the velocity r is the relay number you see there's a definition of relay number on the right hand side t is the temperature and the e it's a density vector and here alpha is a thermal expansivity coefficient g is a gravity acceleration rho is density delta t is a difference between the temperature for example temperature of the mantle at temperature on the surface as an example it's a difference between the two temperatures and h is a sickness of their layer here you see the sickness is h sickness of the layer where we consider where kappa is a thermal diffusivity now it's a cloud we have to supplement this thing by the continuity equation because it's we consider the incompressible fluid flow and this is a continuity equation then we indeed for the thermal convection problems we have to consider the heat equation you see here is a heat equation and the rheology rheology i use here is a very simple rheology you will see in some of my first representations the rheology can be very complicated it could be here it's the Newtonian rheology depending on the temperature and pressure actually it could be non-neutonia it would be depending on the many things like the crystals like the water like many other ingredients which are geological equations accommodate and physically well the next as you understand should be what if we construct our model when we have equations indeed we have to have also the boundary and initial conditions okay well now is there is a we can consider different equations different different the conditions here i just mentioned a few of them but it's just to understand it's what i'm speaking about that is for example the all boundaries we can have a impenetrability conditions with a perfect and then in this case this will be determined by these equations where the end is a normal vector to the boundary specific which you determine the zero heat flux for example at vertical boundary we can introduce or there is a thermal upper and low boundaries when you have the some specific specific temperature at this boundary well at initial time we normally place some temperature we don't know which that's by the way why all the problem there is the solving of the inverse problems also because when we run the models in let's say geodynamical models we run numerical models we should prescribe some initial conditions for the temperature in the mantle how we do it just to prescribe from the you know some some information and so on but we don't know exactly what is the temperature in the earth but when we have some observations for example heat flux on the top of the earth or the temperature even at the upper layers we can assimilate this information and to understand the temperatures in the earth that's that's for example why why we are interested to do such kind of things like inversions or the data simulation and well this is some temperature also can that final time well now there is a if I would like to solve this problem backward in time in terms of the reconstruction of the history of the earth dynamics of the mantle for example thermal convection or the thermal convection in the mantle we use indeed the first equation that is fox equation in compressibility equation and the energy equation and in this case it's what do you think it's what should be done to solve this problem backward in time first of all we were searching where is time here where is time here it's time and was actually only the sheet equation because it's even non-yes stocks equation contained the time but because of the very high Prandtl number I didn't hear specifically it mentioned it's because of the very high Prandtl number entering the left hand side then of the equation where the temperature where the gradian sorry derivative of velocity with respect to time is located and this Prandtl inverse Prandtl enters as a multiplier to this that in this case it's a Prandtl it's something over the 10 to the 20 and in this case the term which is related to the time disappears it means the inertia term is just that we can broke and we can get the equation the first equation which you see here which is called stock situation not non-yes stocks but stock situation okay well you will tell oh well yeah that's the equation suggested if it's the backward of time it's a change signs yeah and a changing sign it means that they are put here minus and minus and that's all it's because it's minus here it's here and you solve it but unfortunately it doesn't work it doesn't work because of the equation or heat equation is imposed when you will start to solve it backward in time and and this is a this is something which just comes from Adamar as I mentioned at he first time mentioned that this is an equation all time but if no diffusion could we do it actually yes we can do it and I will show you in some simple cases that we can do it without inversion and not only do it and all or even in the geodynamic area that people did it but actually the principal difficulties as mentioned here in the square the background is neither an auxon vector equation but heat equation as I mentioned and this is a heat equation generate power but again if no diffusion it is possible to solve for example a few decades ago this it was a published a quite interesting paper related to the backward advection or a technique which actually is a very simple you take out from the heat equation there is a diffusion and you propagate backward in time your temperature and in this case you can do something quite useful for example now you see here it is Africa it is a three-dimensional spherical shell model and you see here the Africa and how Africa moved to this present position from something like a 75 million years ago it was a position of Africa 75 million years ago and then that's a reconstruction doing what they did they solve the problem backward in time they solve this problem to understand how plate moved and particularly this African plate how it moved and how they're internally within their layer spherical layer their mantle plume actually they contributed to this motion and you see that's a present tomography which they to that's a seismic tomography they inverted into the density and around the model and that is a something which they produce as a result after 75 million years now how to prove there it's a fine or not such kind of the assimilation of data and is it possible to do this way you take this position as initial condition and you start to solve the same equations but it is now forward in time when you solve it forward in time you receive this assimilation of density which is presented in the right column and you see there is already there is a some change here particularly they found it's a really very heavy material or it is here actually the temperature mentioned because they calculate the temperature it's the some some anomaly temperature anomaly all right which is not visible at the present tomography or present temperature but if you will run a little bit further your model for example there is some model until this 126 million years and then you reconstruct that is exactly the position there is a fourth column it is a position or to the 126 million years ago how the Africa moved and how they is a mantle they reacted or vice versa how the bloom moved and the mantle and the uppermost layers reacted including the plate movement and now you would like to reconstruct it to the present position even if you will get a position more or less well fixed if you will compare with the first column but look what is a in the spherical shell circular it's the two plumes which are not visible in the seismic tomography and the conversion from seismic tomography to temperature it means this is a because of the ill postness and the small change in the initial condition can generate quite different solution that's exactly what we're shown by the people by Conrad and Gernes some time ago in J-tube application well a backward advection technique was started to be used in many different fields and there is for example Schoenberger and Kornel used it also in the mantle convection areas then it's a claus podlachikov and our group also did it with respect for salt histonyx and Kornelter Gernes I mentioned did this one I just mentioned something the first works you know after that well the many other works and so on but I I showed the first works in this area but later it was recognized that we needed something else at least after the work which Conrad and Gernes published there is a two group work already at this area it's a group of Peter Boonger and our group we work and more let's say advanced technique in the assimilation of data variational method or the adjoint method it's called and at this time we more or less were independent absolutely but after the publication we knew each other because it's a Peter now it's he's in Germany in Munich but at that time he was in Princeton and together with his colleagues he put a generalized thermal convection the inverse and we did more let's say mathematically robust in terms of the let's say less less noise less better solution well but anyway anyway they our technique finally techniques quite different but merge into the more or less stable in terms of the when you reduce these problems and come to the some so-called strong solution then your solution converge quite well and you think quite good solutions and Peter Boonger's group is well developing and also they say are now producing many interesting results as well as say a little bit later the group of Michael Gernes from Caltech started to also investigate the problems and they publish a paper first paper in 2008 related to the variational method and the application it to some interesting various probably I will show you later what means there is a variational method you will hear about variational method I hope a lot I will just speak a very simple way here and telling you it's a what it's it's a finds as I mentioned here find the best fit between the forecast model state and observation by minimizing that some objects that functional which I will show you in a few minutes and they are doing this minimization and working with adjoin to your problem you can solve quite well oh well it's well telling that it's a lot of problems indeed I will show you and but but you can really make your assimilation in internal connection particularly there is these two methods which I mentioned it's a peter method uh boonger peter boonger our group method it's based on the formulation it's a we uh use the okay well peter use the weak constraint where the errors in the model formulation take us into account the process of noise but we looked at the strong constraint where the model is assumed to be perfect it's not perfect it's not perfect strong constraint where the model is assumed to be perfect but actually actually peter did a really quite nicely they derived the general in words about the point what that finally to solve the problem they again restricted they are modeled from the weak constraint to the I mean it's restricted to the strong uh constraints and it's assuming the model is perfect and then they could solve their problem because because it's just the computationally uh very expensive becomes if you use all this all information but we have no actually so many information to have this you know develop the formulation which takes into account the noise of the processes but in some area indeed it's quite important well I talked about the assimilation of the objects the functional what means in the case of the thermal convection the mantle it means that it's here you see the norm uh j is a objects a functional or so called it's a sometimes a cost functional and t is a here it's a written actually is a solution of the forward heat equation with appropriate boundary conditions and final time t final time time uh um uh tether 2 first of all I sorry I forgot to tell you when I spoke about this initial and final time because anyway we work with in the model with the two time uh or within the time window from time initial time q1 let's say to the final time uh t2 or uh tether 2 here is mentioned uh and which corresponds to the unknown yet the initial temperature distribution phi that's why it is here phi we don't know exactly what is the t temperature but we guessed some temperature and here is a tether 2 which is a time final time and then this is this is just integral form of the norm because it's the n2 norm and they say we use here this here it is a kappa you see it's uh or not kappa it's c and uh known temperature distribution at the final time for the uh initial temperature is zero that's why what it means this is actually c is uh observations speaking raise the geophysical it is our observations observations let's say related to temperature and this is a some model temperature and that's why we have to model uh uh the temperature or to find the temperature such a way uh to determine this is initial condition for temperature because our aim is to determine the phi what is a what is a uh first of all important here statement that the objective functional here has its unique minimum at phi equals t0 it means that it's a t0 it's the initial temperature but we don't know yeah it's temperature we have to find this temperature and we seek a minimum of the objective functional with respect to initial temperature and that is that's the present here there's uh there's uh really really uh the gradient of this j functional which we uh we would like to uh really uh know and to have a minimum of these objects of functional to get there our uh in our temperature uh and uh okay here uh is a formulation also called the uh adjoin or there is a principle of the variational method uh it can be shown it's the mathematical to show that if we consider the gradient of the cost function it would be a function which is a solution to the problem which is here uh listed and this is a there looks like there is uh our uh heat equation but it is not exactly heat equation and there always a specific boundary condition and the second line and the initial condition for this problem also but it could be called the final condition because it's the final for the uh problem but it's the initial for the adjoin problem and the boundary problem is referred to this one as I already mentioned as a problem is adjoin to heat equation and the what is it quite important that adjoin problem is well posed it means that we now replace the inverse problem by two sub problems direct problem and adjoin problem and how we solve it I am very sorry but it's I just observed that it's a inverse it is not a specific but it's a always it's a coming guy every time the change is one when I present these things but it's a gain in turn but it's a look it's like a garden they always invert this one but it was not related to inversion we solve the forward problem first of all we solve our problem which is already I showed but here it is mentioned only heat equation I don't put here there is a stocks equation as the incompressibility equations here is only there because that's a problem as I mentioned problem in the heat equation and we solve the heat equation for some guess function for example here vk it's written t and we find a solution well what we do next we next the solve adjoin problems backward in time we can do it because it is a well posed problem and they said and introduced into the the initial condition we introduce the solution which we found from the direct problem and we solve it and we found and now we find we find now phi k and then we may make a next step to refine this k and for this one we solve again this equation there is a forward equation and then again adjoin equations adjoin problem forward problem adjoin problem but how we refine it you can ask the app if you have to refine it and for refinement it can be used several techniques and they said we use a rather simple minimization technique here it is a alpha it is a we place it's alpha here and such a way we refine the fee and here once you refine fee you insert into the forward problem to solve it and then again adjoin problem and then it's iteration and you need to get a best solution in terms of the let's say epsilon which you will you will introduce and then you can look at the convergence to the solution well it is a some deep deep going these issues related to the this method which we use but there are definitely modifications of these things and for example Guernes modified these things and definitely there is a some modification by Boongi and so but I did the following that you have two sets of the equations one it is a forward problem and other it's adjoin problem and you solve one you insert the solution inside of adjoin you solve adjoin then you this you iterate your this solution through these two equations it's solving one and the next and refine your temperature at each time step this again speaking about the time step we indeed need our time slot divide or split into the some sub slots and within the each sub slot we solve the stocks and continue to equation to find velocity field and then we solve the heat problem which I just mentioned using the variational method this was the way how we did actually but again I mentioned there is another approach also to solve the adjoin together you know they're solving the stocks continuity equation and heat equation together and apply to the set of these equations the variational method there is also this method but we consider it's a much complicated compared to this simple method because stocks equation and the continuity equations doesn't depend on temperature oh sorry it doesn't depend on time that's why we we need really to concentrate in the solution focus solution solution of heat equation and once we know temperature we can solve the easily stocks and continue to equations well let me show you a few slides related to the mantle bloom of illusion it is a rather old work which were done uh uh some some uh 10 15 years ago uh and related to the mantle bloom first of all it's here there is a some um work by uh uh Montelli showing how deep can be plumes uh there is a plumes normally originated at the core mantle boundary it's a very uh uh uh hot material which starts to go up for and the point is that it's because the core mantle boundary it is a thermal boundary layer and a small perturbation and this thermal boundary layer can generate the uh the uh lighter material compared to surrounding to go up and then with a high temperature material becomes lighter if considering that composition is the same and there is a goal and this can go up in such a way generating the mantle plumes and this is a just a seismic tomography of a few mantle plumes which were observed at that time and definitely there this analysis is refined today and uh but but I just would like to show it's a hard work really well now it is uh uh look look they say here I put the variational technique which we I just spoke here it is a uh uh that we we reconstruct these uh using using this present day correct and then we are doing backward with diffusion and we find a some solution and then what we are doing we are going back to reconstruct these things but how we did the present day because again it was a some test of the mantle plume we wouldn't like immediately to take the plume which comes from the seismic tomography because it is a polluted with errors of seismic tomography that's why what we did initially we generated a forward model that is a forward model we started with some noise and this generated a mantle plume and then we consider okay finally stop that's that's something which is uh for us and then we will now propagate this is a upward to see there is a how we can use a variational approach but if we will look at their similar things using their uh this is again the same same present day and then it is a backward advection with no diffusion then you can see that at the some period of time still you have a after 12 million years let's say still you have but again for this relay number is depending on the relay number it's a it will be longer or shorter period of time of the rather stay more or less stable result but then this results become the worse uh and uh the the the conditional reconstruction is a significantly different than the uh what what relational approach and here it's just a show of the restoration errors which are the uh differences between the forward model and the our reconstruction and you see that it's I'm very sorry it's a very small probably you know it's but I can tell you that it's here it's the reconstruction error it's a very small compared to what we observed in when we are running the backward advection model without diffusion that's that's uh well I will this one sorry okay well so now now it's a few words about the problems because it's the problems you can see here you know even even when reconstructed it's not bad I will tell but you see is a some small noise arrived uh this uh at the area uh you see that that's it we want to try to understand what is the reason for this and the uh actually actually uh look consider that you have uh in your forward models uh you know it is a some function which looks like at like a red line you know it means that it at the point tipping point this is a you know there is a no derivatives and if you will start to uh make this uh you know you you solve this problem and then you will solve backward you will receive a smooth solution but not a sharp like that a change in the uh in the in function and you see that it's that when you go backward in time then you will get a smoother solution but you will never get this small key you will get a smaller and smaller and but for this you need a lot of acceleration I will uh I don't know uh probably later I will show sorry it was not prepared as a lecture well because it's I just wanted to replace my colleague and uh I don't remember it's here and later I will show you in some slides probably there is really some uh uh problem generated but look when we start to force our four uh to force our model to get this peak suddenly becomes such kind of things and uh we didn't understand we initially looked at it all that's probably something in the model and so on but later we found the issue in the mathematically well developed by Samarsky and his group is the one of the well-known Russian mathematician which explained the reason of this I am not interesting right now and I have no time to explain this in some detail but I would like just to tell never force such kind of things to get exact solution which you because it's never you can get this solution uh well that is the effect of the heat diffusion ah well that's something which I wanted to tell uh but very briefly again there is the effect of the heated diffusion you know it's well we we started from this for example and then it's we are going this way and the plumes initially had a very sick uh tails and the smaller smaller heads and then the tails become thinner you see uh and heads becomes large like a very very similar to uh two mushrooms and then with time if you decrease for example the relay number they start and it means that it's there is a more source and a more heat source coming up more for example the plumes start to decay and they say there is a decay there's a scene like a disappearance of plumes you know and this effect actually was found also but by and the why in their work what they did experimentally and uh they they observed that's initially very strong mantle blue which went up and then they started to disappear with time in turn there's a temperature the temperature increases and then we see the disappearance of the uh head plumes and we also reconstructed such uh cases here is a shown how we did this reconstruction uh I hope that you see this small movie but anyway we took at this very diffused mantle plume and restored them to rather prominent state and this is also the important because sometimes we see in the mantle tomography as some red dots but we don't understand what it means and the means actually but at least at least when we look at such kind of the model means that probably here was a uh some mantle plumes of the set of mantle plumes which now it's the more or less diffused but we have a some pieces which is a still remains like here uh okay I will not speak about the number iteration just to tell very simple that as you mentioned as I mentioned the variational approach based on iterations and the iterations it is a truly very important because it's if you are iterating forever like you know thousand times and so on it's a very time consuming that's why the better your technique the more faster you can converge and the number of iterations something like 10 to let's say 20 is a reliable number but in some cases when you have a smooth at the beginning smooth initial condition you can converge to a very proper solution look here is a logarithmic scale within the few time steps you know oh time iterations it's a really really fantastic but look it is a theoretical things but in the normal case you have a much more iteration to need to be done to get some something like 10 to the minus 6 etc I will show you instead of my case I will show you this leo uh uh and it is a student of professor gurness from caltech what they did in 2008 and also use the same technique which I just mentioned and they tried to reconstruct the position of the slab subduction and they did it quite properly and they say they say uh look they say how they say slab uh bigniz bigniz the north uh uh america they say subducting and it was a history and it's 100 years ago 100 million years ago and this is a present position and how they also the change of the massimetry they looked at the change of the data and so and they fitted rather well with observations it was a one of the uh application of this technique to the real subduction zone I will tell you later also about the our application to the japanese islands and the subduction of the pacific slab uh I mean it's on the other side of the pacific it's also also quite interesting where the work which were done but uh not not today because it's I am already over the time but I would like to finish with this slide which I just mentioned you about the step function like function and the look here is a presented actually burgers equation it is not the equation which we saw but we decided to solve a nonlinear equation you know and to to look at the issue when we have a smooth uh at the beginning smooth the uh initial condition which is on the left part side then it is a uh the the the uh curve with a uh non-differentiable point and another the step function type the uh curve which we see and uh look here the convergence or the uh time iterations just a few a few time iterations and you can get a rather good solution here it's a difference uh between the what what the solution of your problem and the real real uh real solution oh well uh and here here you see that you need at least a thousand iterations look to get to this position but you will never get this one before you're destroying your solution and in the term of the step function you will get something again similar uh thousand thousand iterations but still you can use some a priori information if you know about your solution and you reduce the number of iterations but it is a really very very uh time consuming that's why what we have to do somehow to smooth our initial conditions before we start data simulation but unfortunately this uh uh specific i mean it's definitely you can use a uh high order uh there is a joint and so on and in this case the solution will be smooth by the already the uh technique but still still there is a it is a quite uh uh interesting point is how to avoid this noise this and how to solve the problem is it any other way that we ask ourselves definitely we knew that we can make a more complicated there is a joint and uh they say you know all the calculation of hessian and so on and then we we can resolve this problem as the many are doing it quite well but with today the maybe there is another technique which allowed to do it and uh unfortunately i have no time but it's the next time i will start for these things i will be speaking about the quasi-reversibility method and this comes the third person who started this it is a uh uh professor lions together with lathe at lions they introduce this quasi-reversibility method of the solution of inverse problems and how how to deal with this and finally for this methodology which i wanted to tell during the my talk i will speak about the also i will compare these methods for the backward convex uh backward advection method variational method uh or a joint method at the quasi-reversibility method well i will stop right now because it's uh we are now over of time uh and uh thank you for your attendance i am open now for your questions okay thank you