So he couldn't make it to be here today, but he will arrive in two weeks. And this corresponds, in fact, to a recent evolution of IHES. Since this chair is dedicated, more or less, to applying mathematics, so here I have just taken a citation from the website of the IHES, where the aim of the chair is described.

And as it is written, the purpose is to conduct research and exchanges of views on theoretical topics, arising from problems encountered in technical domains, which could result in new applications in this area.

And in fact, I have built this talk to try to follow this citation. That is to say, I will first describe a practical problem, a technological problem in imaging. I will try to describe what the main problems, the main limitations are about this problem,

and then I will introduce the new ideas that we had with Georges Papagnikolaou to try to tackle this problem. So both of us, Georges and me, are working in wave propagation in random media. So what does it mean? It means that we consider wave propagation in complex media.

So what is a complex medium? It can be the human body, it can be the Earth, it can be the atmosphere, it's a natural medium that we don't know. So we will model it as a realization of a random process, hence the term random media. And in the point of view of applications, there are two main types of applications,

for the direct problem and the inverse problem. So for the direct problem, I've been working a lot in wireless communications, or communications through optical fibers. That's one possible application of this field.

And for, let's say, the optical fibers itself. The defects of the optical fibers are considered as problematic, and in fact they are problematic. So you may have to work a lot to understand how to propagate far in an optical fiber,

so you may use non-linear effects to compensate for random effects, stuff like that. But today I will speak about imaging, that is to say the inverse problem. So here I have written what is the purpose of sensor-array imaging. So now the idea is not to propagate far into a medium, it's to probe a medium,

to extract information about the medium from the waves that you can send through the medium. So you have an array of sources which emits waves, you have an array of receivers which record the transmitted or the reflected waves depending on the situation. And from this data that you record, you want to extract information about the medium.

So that's the purpose of imaging. So I will first give you more or less the state of the art. So first, a simple imaging situation. So since this is a Schumberg chair, I tried to put it in the geophysics context.

So you may imagine that at the surface of the Earth, you have an array of sources and receivers. So these are the circles and the triangles. And somewhere in the surface, you have a reflector that you want to image, that you want to detect and characterize. So the first part of imaging is to collect data.

So that's something that people know how to do. Nowadays, they collect a lot of data. So they put a big array of sources, a big array of receivers, and they collect all the signals that you can record in time at all the receiver's positions when you use all your point sources that you may have.

So a mathematical model for this data set is, for instance, this one, the simplest one, scalar wave equation. So you can see the source term here, a point source that emits, let's say, a short pulse f of t. You have the scalar wave equation in the left-hand side.

And in the left-hand side, you can see what I'm interested in. So you have more or less a homogeneous medium with velocity, speed of propagation, c0. And you have a local perturbation of this velocity that is centered around some point y

that you don't know, that has a shape b ref that you don't know, and that has a contrast, say, c ref that you don't know. So the purpose of imaging is, given the measured data, you want to recover the reflector, location, shape, and contrast. That is to say, what you want to do in practice is to build what the practitioners call an imaging function.

That is to say, a function of the search point, say, that will look like exactly what you are looking after. So here, just a reflector buried into the medium.

So this is what you want to do. Given the measured data, you want to build an image of your medium that will show what you are looking after. So that's a simple imaging problem. So what is done in practice? So I've tried to summarize more or less what is done in the industry.

So it can be in medical imaging, in ultrasound acoustics. It can be in seismic imaging for the imaging of the Earth. More or less, the main method is least squares imaging. What does it mean? It means that you will collect your data.

So you have your measured data on the one hand, and then you will try to match this measured data with synthetic data that you generate with a computer model of the medium. So you have a candidate, for instance, a candidate for the reflector location, y test, for the shape B, for the contrast C.

You solve the wave equation with this candidate. So you generate synthetic data. And now you compare your measured data with your synthetic data and you try to match them by a least square minimization procedure. So it's a minimization problem.

It's an inverse problem. That is usually very ill-posed. So at many levels, it can be very bad in the sense that it's not convex. There is not a unique minimizer. I mean, it can be a terrible mess. But sometimes you may have some positive results about the existence of a global minimizer,

and then the problem is to find this global minimizer. In fact, in practice, nobody does that because practitioners consider it's too complicated. So they simplify the least square inverse problem. And to simplify it, they do something that looks like a very strong approximation.

They will enforce a linearization of the forward problem. So they will try to linearize the forward problem so that now the least square imaging problem becomes quadratic. And then any engineer can minimize the quadratic function. So this is what is done in practice.

And in fact, this is what they do. So when you linearize your forward problem using Born approximation, there is a series of approximations that you can do to linearize a problem. You obtain a quadratic minimization problem. You can solve it, and you discover that the global minimizer

for this linearized problem is quite simple. You can obtain it by solving a one-way equation with, as a source, the time-reversed measured data. So you take your measured data, you time-reversed them, and now you use that as a source, and you solve the wave equation in time.

And what you will get is refocusing exactly at the reflector location. So this is what is called reverse time imaging. Okay, for practitioners, it's even too complicated. So what they do, in fact, is that they simplify. They don't want to solve the full wave equation because a three-dimensional elastic wave equation

is quite complex to solve for them. So they simplify even more the situation. And they do, in fact, what is called Kirchhoff migration. So more or less, they will do a kind of high-frequency asymptotics for the wave equation, a geometric optics-type approximation of the wave equation.

And they will simply back-propagate with travel times. So it's not even geometric optics. In geometric optics, you have an equation for the phase, for the travel times, and an equation for the amplitude. They just consider the equation for the travel times, for the amplitude, and they don't pay attention to the amplitude. And that's what is done in practice.

When you look at an echographic image of your kid, for instance, or when you look at what is done in seismic imaging, this is Kirchhoff. So it's more or less a state of the art. So no, I'm lying a little bit now. In seismic imaging, a lot of effort is put to implement reverse-time imaging.

So let's be fair. Now they are at the level of reverse-time imaging. So what is Kirchhoff? In fact, Kirchhoff is very simple. It's this formula. So that's why practitioners like it. It's so simple to implement it. So you take your data, u of t of xs. The signal recorded in time at the receiver xs

when xs is used at the point source. And simply for each search point ys, you will evaluate this data on the sum of travel times from the source point and from the receiver point. So it's the sum of travel times. It's the time it takes, more or less,

to go from the source to the search point and back to the receiver. So that's the origin of this formula. So that's very easy to implement. You can imagine that's why it's so attractive. And in fact, it's not so bad. I mean, you have seen pictures of your baby, I guess. And it's not so bad. It's particular.

It has good, more or less, good resolution properties. I mean, the details that you can find can be quite fine. It has a very good point is that it's robust to measurement noise. So that's not surprising because it comes from least-square imaging. And least-square imaging is precisely designed to be robust with respect to measurement noise.

But it has a problem. It's very sensitive to what the practitioners call clutter noise. So what is clutter noise? It's the fact that in fact your medium, the background medium, is not completely homogeneous. It has a lot of very small inhomogeneities that are responsible for wave scattering.

And when you look at your data, you see that even if your measurement device is perfect, even if there is no measurement noise, you can see fluctuations, wave fluctuations. And in fact, people consider that it is noise, but it's not noise. These are really waves that you record. These are waves that have propagated through the medium,

that have interacted with the medium, and that you record on your receiver. And then, if you have clutter noise, then Kirchhoff migration completely fails. As soon as you have a little bit of scattering, it's over. You can already see it in ultrasonic ecography.

You see a little bit of speckled noise on your image. That's typically what comes from clutter noise. You have just a little bit of wave scattering in the medium, and that's enough to regenerate this speckled noise in the image. In seismic imaging, this is a big problem, because the Earth is really heterogeneous,

and that's the main limitation in this business. So now, the problem is, what can we do? So first, let me describe to you a simple model for clutter noise. Now, you have to imagine that your medium is not only a constant background, say, one of C0 squared,

but it has also small scale, small amplitude fluctuations. And since you don't know them, you don't know exactly the medium that you are looking after, you can model it as a realization of, let's say, a zero-mean random process. So here, that could be a possible realization

of the medium in which the wave propagates. So now, a model for the data is this one. Again, the scalar wave equation, a point source, the big reflector, the reflector after which you are looking after, plus this background medium now that consists of a constant plus small fluctuations.

And now the problem is that when you have such a medium, the measured data are very noisy in the sense that the main reflection from the reflector is completely buried in the superposition

of all the tiny reflections from all the tiny fluctuations in the medium. So that's the problem that most practitioners face in this imaging business. So now comes the interesting part, mathematics.

So, it is possible to analyze such an equation, that is to say, a wave equation, in a random medium. So, Georg Papagnikolaou has worked a lot on that, in fact, before I was born.

And in fact, Georg has developed a very, very deep analysis of this kind of problem using limit theorems and separation of scale techniques. In this type of problem, in fact, you have many different scales present. You have the wavelength, the propagation distance,

the correlation length of the medium, you may have also other characteristic length scales, but you have many different length scales present in the medium, and some of them are very different from one to each other. For instance, typically in a high frequency regime, the wavelength is much smaller than the propagation distance. In such a case, you can develop an asymptotic analysis

to understand better what happens and what is the structure of your wave, of your data set. And in particular, so you may have different regimes, so I just cited a high frequency regime, there are other regimes, and for each regime,

for each distinguished limit, in some sense, you can simplify the problem and go from this wave equation with random coefficients to a stochastic partial differential equation, so typically that can be an Ito-Schrodinger equation. What does it mean? It means that you face now a PDE, a partial differential equation,

in which the noise appears in the form of a Brownian, maybe not Brownian motion, but Brownian field. So why is it good? It is good because now, as soon as you have a Brownian motion or a Brownian field, you know that you have Ito-stochastic calculus, so you have tools to analyze the equations, and these tools will allow you to describe exactly, quantitatively,

the statistics of your data, and from this knowledge, you will be able to design new imaging functions. So what can you do first, once you have understood that? You can analyze, in some sense, any imaging function, so for instance, the Kirchhoff imaging function,

but any other type of imaging function that you may propose. So the first thing that you can do, the level zero, is to compute the mean and the variance. That's a basic statistical analysis. You can do something more clever, but let's say it's already good for... So this mean is not in exchange for the sum of the equations? Ah, sure, sure, sure.

Yes, the way you go from the statistics to the medium to the statistics of your data is highly nominal, so there is no way you can exchange the expectation. So you can analyze, let's say, the expectation of this imaging function. So that's good because that allows you to do what is called the resolution analysis that allows you to say

what is the finest detail that you can find in the image. Let's say you have two reflectors that are close to each other. What is the critical distance beyond which you will distinguish that there are two reflectors and not a big reflector? So that's the resolution analysis.

What is also very important is to compute the variance, so a higher order moment, because you want to characterize what we call the statistical stability, or an engineer would say the signal to noise ratio. That is to say, is the image suitable or not? The data are very noisy, but you want to build an image that is clear.

So that's the purpose of this statistical stability analysis. And again, to have these limit theorems and effective stochastic partial differential equations will allow you to address this kind of problem. So I told you, give me any imaging function,

I can do the analysis. Now, the more interesting problem is I want to design an imaging function that has good resolution and good statistical properties. But how much you gain by this kind of data? You will see. Answer very soon. So a few results of the multi-scale analysis,

first to show you that indeed Kirchhoff will not work, and second to indicate what you should do. When you analyze wave propagation in random media, you have more or less two main results. The first result is that the expectation of the wave, the mean wave, will decay to zero.

I mean, the more scattering you have, the more you average the wave, and it becomes of mean zero. So physicists will speak about loss of coherence. And in fact, that explains why Kirchhoff migration fails. Because Kirchhoff migration, in fact, it tries to exploit exactly this piece of the recorded data,

the trend, the mean wave. And since in the scattering media, the mean wave goes to zero, you try to image with zero, so you will not get a lot. So that's the idea of this first point. But in fact, if you have no absorption, let's assume you have no absorption,

wave energy has to go somewhere. So it goes into wave fluctuations. So in fact, what you record is just noise. It looks like a very, very noisy signal. And in fact, you shouldn't consider that as noise. Why? Because these wave fluctuations have mean zero.

But the correlation structures of these fluctuations contain the information. When you look at the correlations at nearby points, at nearby receiver points or nearby frequencies, you exhibit that these fluctuations are correlated,

and that these correlations contain a lot of information about the medium. And in fact, now we have a new point of view. Don't try to image with the mean, but exploit the wave fluctuations, the correlation structure of the wave fluctuations, and this is what we want to use to build the image. So it's a complete change of point of view. Forget about the mean.

Look at the fluctuations. Look, in fact, at the clutter noise. Don't get rid of the clutter noise. Keep it. Exploit it. And build an image with it. So it's completely new, because now you will image with correlations, cross correlations, that are quadratic in the data. So it's something.

So just as an application, I've taken that from a geophysics journal. That's a typical situation that is encountered with oil prospecting, say. So that's the situation. So people in the industry want to monitor very carefully

the noise reservoir under exploitation. They want to follow exactly how the fluids are migrating. It's very important to extract the most of it. So the idea is the following. Those people put receivers everywhere. They want to collect as much data as possible.

And so they collect a lot of data, and after they collect the data, they wonder what they can do with this data. So here is a typical problem that they face. They can put receivers and sources at the surface. No problem. As many receivers and sources at the surface. They have wells in the reservoir,

and they can put receivers in the wells, but they cannot put sources in the well, because sources are in fact explosions. So you don't want to destroy your well, I guess. So this is the problem they face. They may have sources at the surface, receivers in the well. And the question is how to exploit,

as best as possible, this data set. So here is a little bit of geophysics jargon. In the Earth, in fact, the top layers are really the problem. This is what they call the overburden. This is where the scattering happens. In fact, most of the scattering happens

over the first few kilometers after the medium becomes... A few kilometers. Let's say this horizontal line is two kilometers, or something like that. Is? Yes, yes, vertical, one kilometer, and a few 10 kilometers in a cross-range.

So this is typically the situation. And they are looking after something even deeper. So this is for deep oil prospecting. So the question, what can they do with that? So they try to do what they know. That is to say Kirchhoff. So I repeat exactly the same arguments.

You have sources at the surface, receivers. Let's say a reflector, but I don't care. This is a Kirchhoff migration function. So let's try to do it. OK, here is a numerical simulation. These are not real data. I don't have real data for this problem.

So on the left, you have the configuration, the sources, the scattering medium, the receivers, and the reflector that you want to image. And on the right, I will apply Kirchhoff migration, and I will plot the image of this region.

I mean, a big square around the location of the reflector. So I would like to have a nice image of this square. So obviously, that doesn't work. So this is typically the kind of image that they get. So they get pure garbage.

There is no way from this image you can know where is the reflector. OK, now let's try to apply this idea that, in fact, you should look at the correlations rather than the data themselves. And here, I do the simplest level 0 application of this idea.

I will apply Kirchhoff to the cross-correlation of the data. So here are the cross-correlations of the data. So for each point source, I look at the data at two receivers, x0, x0 prime, and I compute the cross-correlation in time of this pair of data. And at the end, I sum over the sources.

So I get the cross-correlation of the signal recorded at x0 and x0 prime as the function of the time lag tau. And I will image with that. Here is the image. So you go from pure garbage to an image that is crystal clear.

Because here, in fact, we have put a lot of scattering. It's a highly scattering medium. So in that case, the mean wave is almost 0. That's why Kirchhoff completely fails. You just have fluctuations. But since you have fluctuations, and we know now how to use the fluctuations, you have this image. So again, this is not image processing.

There is no way from this image you can go here. You have to go back to the data and process the data in a completely different way. You adopt a new point of view and a way to build image from this kind of data. So that was an example just to show you the idea.

OK, so of course, you can do analysis. Here it was just an application with not so easy numerical simulations. You can do the analysis. So again, you adopt this separation of scale techniques.

You can do the analysis in different regimes. So it turns out that if you are in geophysics, usually you are in the random layer. This sample requires more data compared to Kirchhoff. No, I use exactly the same amount of data. I use exactly the same data on the left and on the right. It works probably quite well. Yes, exactly. Kirchhoff, I mean, here you can increase the amount of data Kirchhoff will never figure out.

But here, indeed, you need a lot of data because to compute this correlation, you need more data. But the good point of seismic imaging is that the medium does not move. So you have a little bit of time to record the data,

which is not the case, for instance, with medical imaging, where the patient is moving. OK, each domain has its own... Yes. OK, so... How much time computing is this? Oh, it's very... In fact, at the Kirchhoff level, it's extremely fast because you do that by FFT.

The migration is extremely fast. In fact, here, these are synthetic data. Here, the pain is to generate the data. But once you have the data to compute this imaging function, it is very fast. And even for seismic imaging,

where you know they collect terra octet of data, this is doable to do that. So you can do it. Amount of data is not a problem in seismic imaging. They have a lot of data. OK, so just... If you do the theoretical analysis, developing this asymptotic analysis in different regions,

OK, you will find general results and some detailed results. And the general results for practitioners are really surprising because they are counter-intuitive. What we find is that the resolution, let's say, let's start with the resolution,

is independent of the size of the array. And that, for people doing imaging, is not understandable because for all these people, when you build a microscope, a telescope, the bigger the mirror, the better the resolution. That's something everybody has in mind.

This is back to Lord Rayleigh, Rayleigh Resolution... Source arrays, not receiver arrays. Yes, yes, yes. But even the receiver array. In fact, the correlation radius is more important. It doesn't mean that the size of the source array or receiver array doesn't play no role. They play a role. They play a role for the stability.

You need large arrays to have enough averaging so that your correlations are well-stabilized. Can you explain that one at a time? Yes. I will go back to that later. But indeed, here, the data are required by the standard seismic survey procedure, one source at a time. I will go back to that just in the next slide.

So these are the main general results that are quite surprising from the imaging point of view. But again, we have a different point of view. We don't look at the mean wave. We look at the fluctuations. And the information about the medium is encoded in a completely different way. So that's not surprising that the resolution and stability are quite different.

Detailed results, that's something, when I say clarify the role of scattering, it's a little bit optimistic. We are still working on it. Because in fact, when you look at the results in detail, you find some regimes in which the image becomes better and better as the medium becomes more scattering.

Again, remember, we are imaging the clutter noise. So the clutter noise comes from the scattering with random fluctuations of the medium. So in fact, it's not so surprising that indeed the more scattering, the better it is. But again, for practitioners, to tell them that the more noise I have, the better it is,

that's a little bit counterintuitive. Unfortunately, that depends on the scattering regime. In randomly layered regimes, this is not true. At some point, you will lose a little bit of efficiency in the wave scattering.

Because in a layered regime, you lack of diversity. You have scattering by layers, so you don't change the angles. So scattering is not so efficient than in a three-dimensional, randomly isotropic medium. So we are working on it as soon as Georges...

Let's say point scatterers, random fluctuations that are isotropic, as the pictures I have shown you. When you have random layers, it's not as good. So we are working on it. So when I say clarify, we are in the process of...

I mean, you really have just fluctuations along, let's say, the depths. Yes, exactly. And in that case, scattering is not so good. Unfortunately, this is the case, more or less, in geophysics. So that's unfortunate, but we have to be careful. So we are very optimistic, but we have to be careful.

But you have the same information. It doesn't disappear. Is it a question of how to collect? No? No, it's not about how to collect. It's really that you... How do you extract it? The information is there, no? Yes, the information, more or less, is there. But for a given source aperture and receiver aperture, you will need a much, much broader, for instance, source aperture.

Here, you have three-dimensional scattering. So even with a relatively narrow source aperture, by multi-passing in the medium, the reflector will be illuminated with a very, very wide aperture. In a scattering medium, no way. With a layer medium, no way.

It comes from a code that is fixed by the source array. And to have a wide aperture, you really need a very wide source aperture. So, okay, but we are working on it. So when I say clarify, I should be careful. We are working on it. Okay, further results before I conclude.

So, of course, here I have just shown you... Is your wavelength roughly compared to the size of the object you have selected? How do you make it? Ah, here the object is larger than the wavelength. It's a reasonably big object compared to the wavelength. Here I have shown you level 0,

Kirchhoff migration for the cross-correlation. Of course, you can propose more evolved imaging functions, but again, based on cross-correlations, or sometimes on Wigner functions, which are just a Fourier transform of the correlation, but it contains the same information, and it's the same idea.

More interesting, you had this remark that, okay, you have one source that you shot one at a time. In fact, we don't need that. This is a standard procedure, but we don't care, in fact, about the sources. We need sources. We need signals. But that can be ambient noise sources.

And in seismic imaging, I mean, in geophysics, you have a background seismic noise. So you can use... You don't need these active sources. Just record the ambient noise on your array of seismometers, compute the cross-correlations, and that's just in us. You can build the same images.

You need even more time to average, but you don't care. You have time. Let's say you have one day or several days. That's in us. So you can really... That's really new, this idea that you can do imaging without sources. So, again, when you talk with practitioners, that becomes difficult for them.

But, in fact, we had a good piece of news. We had one particularly striking application of that for volcano monitoring. So that was just during my last visit. And we gave a warning for the Piton La Fournée

with three weeks in advance. And this happened just during our visit to Shanghai. That was October 14. So I do remember that Jean-Francois was taking pictures of me in Shanghai. And I was always with my smartphone to monitor the volcano because the eruption was happening just before,

just during the conference. So that was a demonstration that, in practice, it works. You can do imaging of the Earth without waiting for an earthquake. You don't need these strong sources to image this big region. You can just use the ambient noise.

You can do... So passive reflector imaging is this idea that you don't need active sources. So I will show you a result. And of course, you can also use higher order correlations. So that depends. Let's imagine that you show that the wave fluctuations are Gaussian distributed.

Then it's over. You have shown that it means zero. You have extracted the sudden order correlation. It's over. You have extracted all the possible information. It turns out that this is not true. The wave fluctuations are not Gaussian. So there is still some information in higher order correlations. So, again, we are working on that,

on correlations of codas, of tails of correlations. Again, this is under progress, but there is some more information encoded in the wave fluctuations. So this is this idea that you don't need active sources. So you have to turn your... Sorry, you have to turn your head by 90 degrees.

But this is the same situation. You have the sources, the receivers, one reflector. But here, these guys are ambient noise sources. So they emit stationary random signals. So they emit pure noise. And this is typically what you record on each receiver.

You emit noise, you record noise. That's very nice. But in the correlation, you have information. So, again, you compute the correlations between these signals. And, in fact, you're right. Here, I have 300. In fact, I need to go to 10,000

to compute the correlations with quite high accuracy. And with that, I apply, again, Kirchhoff. And I get this image. So I can localize the reflector with just ambient noise sources. Okay, conclusion. So, that's the end.

So the main idea is that in scattering media, you should try to use a correlation of the data rather than the data themselves. You have to consider these correlations because otherwise you will not make it. These new ideas that you can use in ambient noise sources, sometimes scattering can help.

So this is very good for us, George and me, who are experts in wave propagation in random media. We are happy to be able to claim such a statement. And you have very counter-intuitive results. For practitioners, sometimes they have a hard time. So just in 30 seconds,

this is an application that George has in mind. So now you have to turn your head by 190 degrees, or 180. So everything is upside down. Sources are here. Receivers are in the middle. The target is at the top.

And the medium is random, just here. So this is the same picture as this one, but completely reversed. And this is for a completely different application. This is for the space surveillance program in the US. They want to get a very nice image of the satellites. So it's not to image the Earth from a satellite.

It's to image the satellite from the Earth. And they have problems because of the turbulent atmosphere. So what you can do is to put high frequency radars on the Earth. But because of the turbulent atmosphere you will get this noisy image. An idea

would be to put the radar on a flying platform at high altitude, 15 kilometers. That would be okay, but the radar is quite heavy. It requires a lot of power. So to have a per minute monitoring on a balloon or something like that, it's not very easy.

But with this idea, you don't need to put the whole radar on the flying platform. You can let the sources on the ground. Just put an antenna on the top. That's light. Just a receiver. And with that you should be able to do that kind of imaging. So that's exactly the same problem, but

with different waves, different settings, different types of fluctuations. This is a random paraxial regime here in that case. But again, this should work. So hopefully we will find a way to implement that in the next few months. And I thank you for your attention.

Thank you. Are there questions? Do you have any questions about sonar? Ah, sonar. The problem with the sonar is that the sea is moving.

The heterogeneity, the small-scale fluctuations in the sea are moving. And that's the main problem because here as Earth said and as was remarked, you need some time, you need a lot of data in order to be able to average and to extract your cross-correlation. If your medium is moving

while you record the data, that's not good. So we still have problems with that. I think that some people try to do that in Scripps Institute in San Diego. So they have measured more or less the correlation time of the medium. By how much time do you have

to do that and you don't have a lot of time. You just have a few minutes. After a few minutes the medium has changed and it's over. So that's a problem for underwater acoustics. Medium is changing in that case. Is this a problem also from the atmosphere?

Ah, for the atmosphere? Yes, that could be a problem indeed. But here in underwater acoustics in seismic imaging you use acoustic or elastic waves. So you need, let's say, a few hours. Here you use electromagnetic waves.

That's not the same time scale. Here you don't need one hour. Here it's very fast in fact. Another question. Could you use this method for astronomical observations? Oh, I don't. Ah, to compensate instead of adaptive optics?

I don't know. I have not. It's not completely obvious because it's really optics. And here you need to record the phases. In optics, you record the intensity. So you need a way to record that's not completely obvious.

Here in acoustics, electromagnetism, you record the complexity. So what's the difference in electromagnetism and optics? In optics, you record the wave intensity. In electromagnetism, for radar, you record... It's radar. You record everything. So that's different.

When you mention the higher order fluctuations, you talk about fourth order. So third order is the same phenomenon as the average. Yes, it's zero. Yes, indeed. If not, we thank the speaker again.