 Okay. Okay. Welcome back. Can you see the screen again? No? Not yet. I have to go to share. Now you should. Now it's okay. Okay. So again, a lot of information coming from the ground station, from the river gold station, but it's very local. It can be assimilated only locally. So to improve the estimates of the floodplain on the actual flooded area, we can use at least one, in this case, Lancet imagery. Lancet imagery has the characteristic of providing water detection at quite high spatial resolution, about 30 meters spatial resolution. Here, of course, we don't have a ground truth to verify this. Okay. Because the only information we have about the floodless tent is the one that we see from the satellite. But one advantage, as you know, the Kalman filter is reducing the uncertainty. Okay. So here you have, from a number of ensembles of model perturbed prediction of the flood wave, you have the range, okay, in the classical box plot of the area or in that area and how this reduce and also increase as an average by assimilating the Lancet imagery. There is a problem of Lancet of the assimilation of the Lancet imagery. It's the only one in time. Okay. And as I mentioned, introduction is a typical convergent system. Okay. Here you have in this pink color, the spread or the uncertainty in the estimation of the water at different locations. Okay. The ground truth and the open loop estimation, then you have the estimation of the floodway of the flood extent from the satellite. You had a sudden improvement in estimation and reducing of the uncertainty of the uncertainty. But only after only a few hours, you see the system go back to the initial uncertainty around states because of little persistency in time of the update. So of course, the strategy, the better strategy is in, I will show the final result in this final slot is to assimilate both. Okay. So to use the point of observation to, as a continuous monitoring system, okay, that compensate the little persistency of the updates. Okay. And the Lancet imagery to compensate for the sparsity of the ground observation. So here you have, for example, always in terms of the uncertainty in the spatial extent of the flood in blue, the open loop in green, if you assimilate the Lancet, the satellite observation only, so it will go back in a short time to the original open uncertainty and the improvement that you get assimilating the ground as a reverse stage observation, only one little improvement, a lot of improvement, assimilating all the information from the ground stations, and of course, the total improvement that you get assimilating both. Okay. Let me quick, okay, move to the other part of typical, I will say, family of data simulations problem, which is called geophysical invention, where again, the main goal of the simulation is not only to improve the state estimation, but to resolve uncertainties in the model parameters. Okay. Here, the problem that I will face is the problem of actual identification of quantification, or what are the causes of last subsidence that by hypothesis may be induced by excessive groundwater abstraction. So it's a really what we may call a true backward reasoning. Okay. It is really finding the clue to go back to the cause of the given problem that you don't know by that you may not even sure for that being the cause. So it's really, here, I will not spend too much time, waste too much time on this, is really a backward reasoning in an investigation sense. Here, you can read the quote from a novel from Sir Arthur Conan Doyle, was the main character, you know, it was Sherlock Holmes, that probably was the first piece of literature with the concept of backward reasoning, of invert problem was introduced. Finding the cause of the guilt of a crime from the clue that you can get on the ground after the crime has occurred. So the problem here, the study area is an aquifer, again, in central Italy, an area where there is a strong water abstraction to irrigate plant nursery, which is a very important economic activity. There have been a lot of clues about land subsidence, mainly damages to buildings and so forth. And of course, we have even in this case, satellite measurements that can detect this kind of problems. Here, we are using in the study, okay, interferometric estimates of surface deformation from two different satellites that cover different time periods, and we start from 2003 to 2010 and sent in one from 2015 to 2017. These are the velocity of surface displacement inferred by these two satellites in these two periods. The line of sight, usually the satellites have a slanted field of view with respect to the ground, and which may be, I would say, quantitative interesting is the vertical displacement. Okay, here you have a legend, and in red and yellow, you have the areas where actually subsidence is measured. That is negative velocity in the displacement of the order of several millimeters per year. How do we address the hypothesis that this subsidence may be due to excessive groundwater extraction? Okay, we set up a couple surface hydrology groundwater model fed by various forcing data, precipitation, temperature, everything that may influence the groundwater dynamics. Okay, and as states predated by this model are both the groundwater budget, but also the aquifer deformation. Aquifer is an elastic and an elastic, for certain extent, media that react to the changes in water content. Okay, and then we use the satellite surface deformation that I showed you before, and also some waste data to infer the model parameters that are unknown. These are mainly hydraulic conductivity of the aquifer and the elastic and elastic skeleton storage coefficient that are the ones that will control how much deformation you can get at the ground for changing water content. This is especially because of the sparsity of waste data and undetermined inverse problem. Okay, so the technique that we use to solve this inverse problem is a variational data simulation, the problem is nonlinear, with Tikhonov regularization. Now, I think it may be a good idea to give you also some more technical information on some of these, I would say, technical terms. Tikhonov regularization is quite a general concept that is used in solving inverse under-determined problems. Under-determined problems, by taking a long story simple, is where you have too little information with respect to the unknowns that you want to estimate with the inversion of the problem. So formally, as soon as you have, I will start with the linear, simple linear part, that you have a simple linear inverse problem that is finding the parameter theta, so that the linear problem g, the matrix g of the state x, multiplied by the state parameter theta give you some measurement z. And as you know, if the amount of measurement in z you have are too little with respect to the model parameter theta, the problem is ill-conditioned. So you want to solve this problem anyway. A technique is to add to the quadratic function that the irregular linear problem you would solve in the least square sense. An additional term, which is called Tikhonov matrix, that multiplies the unknowns and will make such that this minimization of this functional is now a well-posed problem. Now, if the problem is linear, of course, you have a closed form solution of the least square problem, which with the insertion of the Tikhonov regularization matrix will be this one. So as you can see with respect to the general solution of the least square, that will give the products of this matrix here applied to the measurements, but this needs to be inverted. You have this additional term. The additional term is the term that makes such that this product here, this matrix here, that in an ill-posed problem, it cannot be inverted. We can be now inverted with the addition of this taking more simply is like adding diagonal terms in a more simplest form to a matrix that will be, they will have a too low condition number to be inverted. Of course, we want to use this type of regularization in data simulation by ASM framework. So the problem is posed a little differently. Now you will still solve the problem of estimating the parameter z that given observation, but starting from a prior estimate theta zero. And also with the ability of characterizing the covariance of the unknown theta and the covariance of the data z, the errors of both. These of course are all terms that need to be somehow prescribed in solving this problem. Now the quadratic function to be solved is formulated a little differently. Now you want to minimize of course in the least square sense the misfit between the prediction and the measurements, but the ticker of term now is the term that has the quadratic estimation of the difference between the posterior estimate and the prior estimate that you also want to keep minimal. And of course, you have been the model linear, you get a closed form solution for this as well to give you a sort of update term with respect to the prior estimate written in this form is more clear. Okay. Now this type of regularization can be also used in nonlinear systems. Okay, I will skip this part here for gravity. Okay. In nonlinear impulse problem, now the problem is that the model is not linear. Okay. So we can write in generic form like this. And again, you want to estimate the parameter theta to minimize the misfit between the model prediction and the measurement in Z. Okay. Now the problem is that you can write in a concise form of the function to be minimized again, the misfit between model prediction and measurements and the ticker of part that is the minimization of the distance between the estimate and the prior estimates. But you cannot have a closed form solutions to this minimization problem. So this is where what we call variational approach comes from, finding the minimum of a function through a variational approach, starting the variations of this function here to be minimized. Okay. This is the result of the application of this variational with ticker of regularization in the problem I mentioned. Okay. Here are the initial parameters. I'm sorry, the output of the model prediction in terms of modulated hydraulic head level of water in the groundwater system with respect to the measured in the wells. Okay. Pretty good even in the open loop solution. Much more dispersed in the open loop solution is the amount of surface movement induced by the groundwater dynamic with respect to the one measured by the satellite. Okay. Once the satellite measurement as assimilated into the system, you get not only a better estimate of the groundwater level, but also of the movements. And in doing this, you infer, you invert for some of the unknown aquifer properties like here, for example, the storage coefficients that control the elastic response of the system. And you can see is quite a variate in space field that varies by order of magnitude that could have been hardly infer only from interpolating point measurements. Okay. So this is the final comparison on how the improved estimate explain the measurement ground displacement spatial pattern that is quite a good coherence for from the displacement, the subsidence predicted by the groundwater model and the one that is measured by the satellite. Okay. I will not go into the detail on how what is the statistics on how you measure the spatial coherence of a predicted field with respect to measurement that you have. But also we obtained quite a good coherence or agreement also in the time behavior of the displacement, both in the first years measured with the envisage satellite in red dots are the measured by the satellites in blue, the displacement at different key points in blue, the one predicted by the model. And here in the other period with the other satellite. So at the end, okay, still referring on the concept of finding the guilt of a crime, you know that especially in the US judicial system is the concept of means, motive and opportunity that are used to convince the jury of the guilt in a criminal proceeding. Here we may say that the motive relies in the aquifer deformation physics. Okay. There is a motive upon which an aquifer may induce land deformation. The opportunity stands in the fact that the spatial and temporal coherence of subsidence and groundwater dynamic, they have a similar temporal spatial behavior. And the means, which is the true result of this assimilation is the actual capability also in quantitative terms of the aquifer system to induce that amount of deformation. So we may conclude that the aquifer and the abstraction of water from aquifer was the guilt for the land subsidence. Okay. Let me go in the remaining time to the more complex problem. As I mentioned, many hydrologic applications are characterized by both insufficient data with sufficient observation and uncertain model parameters. So many hydrologic problems and data simulation hydrologic problem need to address geophysical inversion issues that is parameter estimation or initial state estimation. We call also an initial state estimation as a geophysical inversion problem and Bayesian or state estimation problems. Okay. Together. Okay. Now, to solve together state estimation and geophysical inversion, you may essentially follow two main type of approaches. In the dynamic filtering, the Kalman type of filtering, linear or nonlinear, you may use what is called filter augmentation. Okay. It's like assuming that the parameters are states, dynamic states as well. If you want the parameter estimated like this, you eventually write an evolution equation for that parameter that tells that evolution is null, that is the parameter needs to remain at the same value. But then you keep updating its values in a Kalman filtering sense. Or the other approach is the variational simulation with an adjoint. Okay. What is the advantage of using an adjoint? We'll see in a while. Okay. I will properly have time not to show both example, but at least I will try to show the first one. Okay. Where satellite measurements are used for mapping surface soil moisture control and energy balance for soil atmosphere interaction. That is a problem that needs to estimate essentially how much some information on the soil moisture and how the soil moisture control the flux in terms of heat and vapor from the earth's surface toward the atmosphere. And here, of course, there is a problem of state estimation, soil, of parameter estimation regarding the vegetation, regarding the type of turbulent flux you may have and so forth. So the typical combined inversion state estimation problem. Okay. Starting from the first one, a number of studies including this one I participated to over many years is using the lens surface temperature that can be detected from different type of satellites. I showed you before in the first part of the lecture the problem of filtering of improving this type of information using again this type of measurement thermal infrared of the lens surface to infer information on soil moisture and how this soil moisture controls the surface turbulent fluxes. What is the guiding principle? First of all, that you expect and here you have a thermal image of irrigated crops that because the presence of water, wet surface will be much colder than warm surface for similar atmospheric and radiation conditions. Why is this? Okay. The reason for this is contained what is called the surface energy balance. The energy that comes from the sun at the red surface may be in part absorbed by the ground media in terms of ground heat flux and in parts dissipated back toward the atmosphere with two main fluxes. What we call the latent heat flux, the flux of vapor and the sensible heat flux. Okay. Now for a given amount of energy, the partition between these two flux will depend on the soil moisture. If the soil moisture is high, you will have a lot of dissipation in terms of latent heat flux and little dissipation in terms of sensible heat flux. If the soil is dry, you cannot dissipate through latent heat flux. The sensible heat flux needs to increase to maintain energy balance and to increase the sensible heat flux, the ground surface needs to heats up. Okay. So this can be detected by measuring the urinal cycle of the lens-to-face temperature. The higher is the amplitude and the higher is the daytime temperature, the more likely is the soil to be dry. Of course, you experience walking in somewhere on the beach on the dry sand or walking on the wet sand near the shore. Okay. There's a huge difference in temperature on the sand temperature. I am very sorry. Could you please explain what means the latent heat flux and sensible flux? What is the difference? Yes. Okay. Latent heat flux is the, I would say, is the heat content of evaporation. Okay. It's essentially the process of evaporation from the surface and the vegetation that in terms of energetic content is just the amount of water which is evaporated multiplied by the latent heat of vaporization. Okay. It's essentially the reason why we start sweating when we are in too much warm conditions to dissipate heat from our body. Okay. It's the amount of heat which is dissipated through evaporation. Sensible heat flux is the convection of heat by simply temperature differences. Okay. It may be. Both processes is diffusion processes. Both. Well, they are diffusion processes but they are the turbulent diffusion processes. Okay. Thank you. Okay. There is a lot of turbulence here which is also one of the parameters that are uncertain that need to be estimated to have a correct representation of this process. Okay. But the basic principle is that the the urinal cycle of lens surface temperature bears in information or whether you are in this condition or in this condition. Okay. Now, I understand that the time is running very fast. Okay. Let me ask Alec or other old attendants if I should spend time on explaining what is a multivariational approach for data simulation within a joint. Well, I try to explain but I think it's better if you will explain this particular case and this particular case just to understand the flow. Okay. Okay. Let me go quickly. Okay. Yes. Again, here you have a non-linear prediction model with some uncertain parameter and states and an observation. Okay. Again, we split what are the unknowns that can be observed, and there are no that cannot be observed. That may be states or parameters. Okay. Now, again, we want to minimize a cost function that first of all as in the ticker of regularization has both the misfit between the model prediction and the measurements. Okay. And also the update on the priors of the model parameters. But you also want to bring in with respect to the simple case I showed before with the linear ticker of regularization, the fact that you notice the dynamics of the model. Okay. And you bring in this information to the Lagrange multipliers. Okay. So you have the Lagrange multipliers that multiply the misfit between the dynamics and the model. Okay. With the estimated parameters. Okay. So you have a functional that brings in both the observation and the model prediction. And you want to minimize this functional, of course, is a non-linear problem. So you need to use a variational approach. A variational approach means finding the derivative of all of this functional with respect to all the unknowns, the states, the unknown parameters, and the Lagrange multipliers. Okay. Now we taking some algebra that I will not do here at the end to estimate not numerically this variations, but precisely, you may end up with this system of equation. Well, you have, of course, your forward prediction model. You have a backward adjoint model. It's called the adjoint model because it has a similar structure with respect to the forward prediction model, but the unknowns are the Lagrange multiplier, and it is a linearization of the non-linear prediction model. It has the minus signs, so it needs to be integrated back in time. Okay. Now, the parameter update is proportional to the Lagrange multiplier. So you usually solve this problem by iterating through this equation until lambda vanishes. Why lambda vanishes? Because the forcing term of the adjoint model is the misfit between the model prediction and the measurements. Okay. So if the model is predicting exactly this forcing of the backward adjoint model will be zero, and this will have a null solution. Lambda equal to zero, it means that you don't need to update anymore the states that you want to estimate. So usually iterate among this equation to converge toward a final estimate of the parameters that you need to estimate or the states that you need to estimate. Okay. You can apply this very general concept to any complex model provided that you have the capability to code in terms of actual writing a model coding of this equation here. Okay. You need to explicitly code this variation, for example, or this variation here if you want to have an efficient solution system. Okay. In this case, we have a forward model that was based on the heat diffusion into the soil. Okay. With boundary condition provided by surface energy balance, I discussed it before, with distinction between the bare soil and the vegetation. There is a lot of control of the vegetation characteristic in this flux. We know well that a forest will evaporate much more than a grass. Okay. So you need to predict somehow what kind of vegetation are you dealing with. So there is also a vegetation dynamics model and the retrieved daily states parameters are parameters that controls, as I said, related to soil moisture between the latent heat and the total flux that is the distinction, the partitioning between the latent heat flux and the sensible heat flux from the vegetation and from the bare soil and the evolution of the capability of the vegetation to evaporate. Okay. This parameter here is called fraction of photosynthetically active radiation, which is the amount of energy that is used by the plants for the photosynthesis. Okay. Why are showing this slide here? Not to go into the details, but just to show that in that very general formulation, you may plug quite complex prediction equation, many parameters to estimate of different kinds, different type of observation and so forth. So just showing some results to give you a hint of what are these kinds of applications. This is the site south of Mali to the border with Burkina. It's called the Gurma site, which is a field facility that has been run for several years to also to provide ground measurement to a very important satellite mission. Okay. And to have a feedback of the capability of the mission to measure hydrological variables in a very sensitive region, the sub-Saharan Africa and near to the desert to monitor also the apparently increasing extent of the desert to carry us. Okay. These are the type of results that you get from this kind of assimilation. These are different times, different period of times of about 30 days each. Okay. Of the growing of the vegetation, which is detected through this method. And here, according to the growing of vegetation, the amount of evaporation. Okay. So you have the capability of detecting more temporal trends and spatial gradient from the northern more desertic part to the southern more green part. Okay. This is how, for example, these retrievals of the partitioning that we know to be related to the amount of soil moisture compared to actual soil moisture measurement taken in different days from an independent satellite measurement that directly measure soil moisture that we use for verification. That is not the perfect match, of course, but you can clearly recognize some distinct spatial patterns of the fields. And of course, there is also some ground verification with, as I mentioned, this was a site for a quite extensive field campaign with quite precise measurement also from the ground of this fluxes to, for example, two sites. And here is how this retrieval capability is how this algorithm is capable of retrieving not just the average values of a given period, but even the deurnal cycle of the values relevant fluxes. Okay. Here, for example, is the comparison of the retrieved from this algorithm. The continuous line with respect to the measurement in the site here, where all the four terms of the energy balance were measured and how these are retrieved. Notice here, coherent with these changes here in the amount of moisture and vegetation activity. Okay. How you have a satellite switch from a dry regime where most of the dissipated energy is in the form of sensible heat flux through this moist regime where most of the dissipated energy is in term of latent heat flux. Okay. Very coherently with what is measured at the ground. I have still have 10 minutes to go, right, Alec? So let me at least give a hint of what would be an even more complex variational assimilation approach. Okay. For a very, I would say, hydrologic specific application, which is flood forecasting. Okay. This type of flood forecasting problems are solved by quite detailed hydrologic model that needs to resolve the formation of runoff and propagation of runoff along the river network at quite spatial, a quite very high spatial resolution. This, for example, is a model is used operationally to provide real-time flood forecasting in a basin near Firenze. Actually, Firenze is here. Okay. And the model is run at a resolution of 500 by 500 meters. Pretty high resolution with about 8,000 different river reach lengths, which are modeled in this quite detailed model. We deal the processes that are involved in the runoff formation. Quite a bit complex model. I'll put the complexity of this model halfway between a model like the one I showed you before and an atmospheric model. Now, the problem is that we have very little information in case of flood events to use to improve the prediction. To improve the prediction of flood wave, you need actual flood wave measurements. As I mentioned in the introduction, river gouges are becoming more and more sparse in space. In this case, we have only five gauges that are operationally and routinely used with reliable river discharge information. We did some hint card experiments to see how the flood prediction can be improved through data simulation technique. I'm showing you this picture that shows for 16 different high-flow events that recall that peak flow at the downstream gauge and the basin average cumulative precipitation. I always show this picture in, for example, when there are also meteorologists present to see and to convince them that predicting flood is not just predicting rainfall. If you are predicting rainfall quite well, you are predicting the amount of rainfall, but for similar amount of rainfall, you may still have a huge variability in terms of runoff response and flood formation. For this similar amount of rainfall, about 70 millimeters average on the basin, in a given event, it has been a peak flow of 500 cubic meters per second. In another, from a precipitation point of view, similar event almost four times more. So you need a hydrologic model to resolve this type of variance. Again, here there are some techniques I will skip. The possibility of having predetermined flow paths that help in reducing the spatial problem into a multivariate one-dimensional problem. Again, problem of localizations of the ground measurements I mentioned before and so forth. Again, here we still have the problem of estimating both states and parameters. So we go again with the variational approach. The main parameter to estimate, which is actually not really a parameter, but is a state, but is treated as something that is not observed. So from this technique point of view, it's like a parameter, is the amount of runoff that is coming from the yield slope into the channels. Now, we also try to explore the possibility of having a better estimate not only of the runoff on the channel network, but also of the yield slope runoff all over the basin. This is difficult to adjoin, but at least mass conservation and rainfall distribution need to be maintained. So what we also try this sort of mixed variable variational and particle filter is a Monte Carlo type of filter approach. What we do is I will try to synthesize in the scheme here. We use hydrometric data with the variational simulation within our joints to get increment of analysis of river flows. Then in perturbing soil moisture and rainfall interpolation parameters, we get an estimate of the produced runoff. Then we can do a sort of likelihood match between the many ensemble members of the produced runoff with the one that the variational simulation of hydrometric data will tell what is the best estimate of the total runoff. In a sense, we have a variational approach that provides the likelihood for the Monte Carlo assimilation. This will provide at the same time an improvement in the estimate of the river runoff and the analysis of improvement of the state estimation in terms of yield slope runoff and soil moisture. These are a few examples. These are hidden experiments as I mentioned on real flood events, on how the prediction of the flood hydrograph for a given event at a different gaujin station will depend on how early you assimilate. Of course, here we are talking about prediction. We want to improve the capability of predicting at least towards the outlet to improve the prediction of the peak flow before that course. So assimilating rivets this chart before the arrival of the peak. Of course, you have different performance and you expect to have better prediction as you start assimilating more and more as the flood wave develops. Here, I will not have time to go into the details. You can find that in the studies I mentioned at the beginning. You have the details on how, for example, the performance of the data simulation system in terms of improved root mean square error of prediction changes in terms of lead times. Of course, you expect the longer is the lead time, the worse will be the performance and the less will be the benefit of the data simulation. But except for some peculiar events, you still we are still able to detect quite strong improvements, even for quite long lead times in the prediction capabilities of the system. Okay, this is an example of the sort of analysis of the helos slope runoff that you can get by with that mixed technique I mentioned, again still assimilating only river observation at this point. It is through this type of quite advanced variational and Monte Carlo assimilation together, we can use this point observation to improve the estimation of a quite distributed field like the helos slope runoff production. Okay, I think I am about finished, I still have one minute. So as just as concluding remarks, I will mention some trending, what I call trending topics, at least some of them that I see out there in conferences, in recent publications and so forth on the three ingredients that make a data simulation system, the models, the data and the techniques. On the models, as I mentioned at the beginning, one characteristics of many of the geologic community is to have in time gave birth to a plethora to a jungle of many different prominent specific models. And these are strong weaknesses. Now there is a strong effort in building what are called digital tweets, okay, also in the geologic community. So there is an open path still to be completed toward the holistic or application-independent geological model. It's a big effort that is ongoing that will involve a close cooperation between geologic and atmospheric science and other geophysical sciences community. There are big advancement also in the data that can be used to be assimilated in geologic models. Mainly, for example, user-generated contents, okay, the data that you can get from a sensor and that you bring every day with you and can add that continuously sense the environment starting from your smartphone, okay, they have temperature and pressure sensors. What are called opportunity data? A typical one, for example, is attenuation signals from satellites, from for radio transmissions, citizen science, involvement of citizens in collecting data, okay. In terms of data technology techniques, as you know, the big novelty is the joint use of physically based model and data simulation technique and artificial intelligence learning algorithm. I know that you will have a specific lecture on this, okay. I would just mention that most of the of this application I see out there in data simulation benefiting from learning algorithm is mostly in the sequential data simulation approach, okay. Where you have iterations, okay, and each iteration, the system would provide additional information on how the system itself reacts to each iteration, okay. A learning algorithm can learn from how the system converges on how a covariance matrix evolves in a sequential common filter to improve, for example, the convergence to reduce the computational efforts and so forth. Okay, I think I will finish here.