Estimation of the mesospheric wind field correlation and structure functions using multistatic specular meteor radars

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Title
Estimation of the mesospheric wind field correlation and structure functions using multistatic specular meteor radars
Title of Series
Author
Vierinen, Juha
License
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI
Publisher
Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Leibniz-Institut für Atmosphärenphysik (IAP)
Release Date
2019
Language
English

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Abstract
A method for estimating the three-dimensional mesospheric wind field covariance structure from a sparse set of specular meteor trail echo measurements is described. The measurements consist of one dimensional projections of the vector wind field measured at random points in space and time. The method for estimating the correlation function relies on cross-correlating pairs of Doppler velocity measurements, which are separated in time and space. These measurements can be used to estimate the correlation function of the mesospheric wind field at different spatial and temporal scales. The method is demonstrated using a multistatic specular meteor radar measurement data set that includes ≈105 meteor detections during a 24 hour time period. To validate the method, the results are compared with an established method for estimating horizontal mean wind. The method is also applied for estimation of high resolution temporal, horizontal, and vertical structure of the mesospheric wind. The results are compared with the expected behavior of atmospheric fluctuations at mesospheric altitudes. The temporal correlation function is used to estimate the power spectrum of mesospheric wind, which includes the semidiurnal mode and a strong peak with a 3 hour period. From the horizontal correlation function of the fluctuating wind, we estimate the structure function for the <4 hour fluctuating component and find that the spectral index is close to -5/3, which is consistent with the Kolmogorov distribution of energy within turbulence. The vertical correlation function for the fluctuating component shows that the vertical scale size of <1 hour waves is approximately 10 km for the horizontal wind components. The method described in this study expands the range of spatial and temporal scales of the mesospheric wind covariance structure that can be studied with specular meteor radars, and it may also be applied to other similar sparse and incomplete measurements of random vector fields.
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this is a on the talk about work the perhaps start about 7 5 to 7 years ago when I I came up with the break designed for something for a completely different application for for studying ionosphere of brightness sphere and that's what I do I mean I I at the time my
background in is an is an inverse problems and and and space physics and tonight through and display work between us both of those 2 2 2 topics do statistics and and and not as linear algebra and in this work is is His
work that started when cocaine invited me to come to to Cos born to
develop a new type of material and this is a major which which which hopefully off of people to convince you that this is a useful idea and also be above be commuted to you that the jurors are actually not used for primary observed reaches they're used to observe misses the point these my collaborators hopefully have not forgotten any 1 of if so I'm sorry this is a simulation of a wind fields that just I had used something else than just frequency to minus 5 thirds which is comical recall the comic of spectrum of of of energy for for turbulence but a con looks like kaffir but this is this is what we're trying to to measure and characterize we're trying to characterize the Winfield and the message here so what is the Mesosphere I looked at the year at
the titles for the speakers and I I I figured that I I problem like the good idea to put 1 slide here that explain what is the mesosphere FIL so the at atmosphere consists of different layers of troposphere which is probably the most important 1 for us because that's where we are and that's where most of the atmosphere is this the stratosphere and then there's the Mesosphere between about 50 and another kilometers and most of you might not even be aware of of of something called the Mesosphere but it happens to be what this institute actually focuses on mostly studies of the message the Eric Becker will probably tell you why it's important to study this region in this text talk and and and will be able to leverage a bit more about this but as as a more of a a little bit more of an outsider to do and this is very science community of of a dye know that this happens to be the boundary between them but the space so it at about 100 kilometers Our dynamically airplanes can fly anymore but that's that's the definition of the but it's also place where were atmospheric waves are amplified mean tidal waves are our our have 1 centimeter for 2nd velocity at ground level but they can be hundreds of other meter per 2nd in the message and then in these with these waves also break the few things that we can observe and it's also it's it's it's it's and it's in this interesting region because it's very very hard to observe its we can't fly in airplane for this region which implies satellite because it will be because there's not enough left but we can find the spacecraft because there's too much track we get to really do any in-situ measurements we can do rockets sounding rocket measurements but those of the 1 off measurement and they cost about a million dollars each so primarily this this is we use remote different remote sensing technique we use light hours we use radars user spectral measurements of the mesosphere of from from space and and from the ground and I'm going to be talking about 1 of these vitamins which is his a picture about the
Mesosphere from I saw this picture of years ago it was taken by national on the International Space Station true call this shows the atmospheres is about at about 100 kilometers you can see is this green airglow that's 55 77 on airglow from atomic oxygen and right below the screen layer is is the message with the very thin thin layer which which kind of marks the upper boundary of our of our atmosphere and you can see here that a major that's burning up in the mesosphere so make meters they burn up in the mesosphere because that's that's where the atmosphere starts be dense enough that that's ablation occurs and beaches destroyed so that's 1 optical phenomena that you can see that occurs in the message another optical phenomena you can you can see
that occurs in the mesosphere is is not to losing clouds has anyone seen Oculus in clouds a few people so if you if you look at the right season ports and lots of August September what's the right season is about that June July you can see them nights which north of its occurring I live in Tromsoe it's occurring right about me I can't really see them because we have 24 hour day life for these these these are clouds in the mesosphere which which which arms with which is scattering of sunlight of of of sunlight from from I see dust the the message and you can see that it it looks like it's it's turbulent alright so and you can even
look look at these clouds from from satellite and you can see that there is there's a lot of dynamics that occurs in the mesosphere OK so why why did I get
involved with this this is maybe my explanation after the fact of necessary because very difficult to measure of we don't have a lot of measurements between specialist of of the distribution of kinetic energy between which in approximately the spatial scale and and that spatial skills of between 1 kilometer and and hundred 200 kilometers we don't have a lot of information about the the the Connecticut about the diet the dynamic of the kinetic energy distribution of of of Mrs. Straker Floyd and we wanted to to improve that we want to make it come up with new a new technique for for bridging this gap and making this measurement OK so releases this they are methods for for measuring the missus frequent can use rockets experiments you can use lighters of spectrometers to measure Doppler-shift a spectral emission and absorption lines you can use radar scattering off of on a streak regularities in in the mesosphere using very low-frequency radars you can use larger radars like Moore's reader that I appeal Kuehlungsborn operates in and Norway to look at weekly honest turbulence carried from and turbulence you can even use we scattering from minus the without turbulence and you need a very large electron densities in you typically only see this when you have at high latitudes when you have oral presentation and finally and this method that was going to be talking about today mostly is radar scattering meteor trails using the VHF frequencies teaser frequencies between 30 and 50 megahertz and then all these different methods have advantages and disadvantages and and I think the advantage of this method is is that it is very useful for for measuring between these 2 different
scales of of of the energy just spectrum so that what what so so K here it this is a horizontal wave number that would be 10 to minus by which corresponds to a hundred kilometers of will and that corresponds to the minus 2 so that's 1 2 so that there would be a hundred meters 108 away right what is it what is
Europe greater look like this is what a major upgrade look like so it's so I'm used to working with large radar systems like receive Oracle mark our eyes gets and they're too young to be direct and 10 1 thing their view find cool but we iterate as is that it's just a bunch of small antennas that they're very inexpensive you can put them on your back yard and and you can build a Tschira and you can measure 100 thousand meters per day with but this 1 is in Austria it's these the reason why there are multiple antennas here is is in order for me to at work of we need interferometry we need to determine where the major echo comes from and and we we do this by measuring the time time of arrival between of this of the echo between these different tennis OK another critical
ingredient for for how meter works is is meter rigorous measure primarily something that calls of specular trailer cone so if you have a major that's flying through the atmosphere of bleeding leaving behind a trail of increased electron density this is a it or if the factor but but it only has a large radar cross-section if the incoming if if if if we have the incoming Reddy waving El-Khodary waves are organized in such a way that that that the Bragg scattering vector which is the incident k-vector 1 is in is in a k-vector plus scattered connector this this this k-vector here is perpendicular to this trail so it it satisfies this specular condition in that case we get a very large radar cross section we can measure them with the small radars and what we measure is is is a one-dimensional projection of the neutral wind so when this nature trails formed it's not moving at the at the at the velocity of the the Media it's moving at the loss of the wind because you leave behind an AppleTalk trail of increased as the net drifts along with a neutral wind a we with a major right up when we measure the Doppler shifts of this kind of transmit receive packed we measure a one-dimensional projections of the 3 dimensional misses the point and that's that's about given his so would be adopt which measure Doppler-shift and that's dots the Andes it be used it necessary control and which we want to measure plus plus a measurement noise will always have of measurement errors and customer to use you W as as Atiyah 3 different components of the neutral wind so OK so question that so if this is all you knew about the curious how would you go about improving the major that new ideas more power tolerates that would be a good idea figure antennas what else what would be like a clever way of improving so the critical point is that we we only she made me chair if if they satisfy this condition if if we have a meter that's like this oriented like that we would not see it's with this transmit receive the ideas sure more transmitters and receivers but you cheated you know you you know the answer to the so yes but you're gonna right so I mean if we bear with we build more transmitters more receivers more careful satisfied specular conditions and and this is what the the junta solver and of course each I realized and and a few years ago and and they said you know what what difficult if we build this kind of a network of we we have a combinatoric advantage every week for every if we have multiple transmitters and multiple receivers each transmitter and each receiver has a unique path enacted said that than that we we almost gets a completely new set of of nature trails and we can we can increase the number of major transformation if we increase the number of major jails that we measure we get more information about the neutral wind and the same same technique is also used for for astronomical purposes of the Seymour 8 in Canada you is the is based on a similar principle and so yes we we get a lot more beaches just by adding adding antennas making a network and and this is this is where I got in I night that the reader that I had is kind like GPS it's or or CDMA like go like cell phones work 8 instead of sending out pulses you know sending out Dirac deltas I send out noise and I designed radar which allows and I were designed to stay for another purpose but we applied it 4 meter where we the radar transmits with multiple different antennas operating on the same frequency but each 1 of them transmits a different to drive sequence and the advantage of this is that so 1st of all we get a lot more average power out of we can also that disambiguates different transmitters on the same frequency and this is a project that we was a joint collaboration colorful at MIT when I got this funding so it but this this left at the meeting we just concluded this project in 2018 and it was very successful project where we we we we work on his radar but we're not doing any magic this is again like I said this is this is a technique that sorry use for GPS and it's all your use for mobile phones I'm pretty sure military's use this kind of radars we use mortars in processing methods have basically nearly squares compressive sensing ideas which is just coming up with a sparse spaces and 4 meters it's easy it's just Kronecker Kronecker delta bases with few Kronecker deltas here's a here's a kind of reader thing across the basic bases of the reduction of processing for of multi-static radar it's low frequency so you have the the echo doesn't change very fast your equation essentially of convolution equation so this would be to test waveform delayed by distance or and this would be the echo power from range far Oracle scattering amplitude but because we have a evolved system we actually have multiple multiple paths and so on and the notes here path so we have a we have a measurement equation which is a consist of multiple different convolutions and task is the sole for and a signal and and and we might have we typically have about 5 or 6 of and it's 5 or 6 of with a pulse trade this epsilon is a Kronecker delta if this is the Kronecker delta there's no way you will ever be able solve this equation rights there is no unique solution to it if this is to random noise you you you and you know what that noise is so if it if it's a long enough sequence you would buy it will be able to come up with an overdetermined a cost problem and in solids it's quite quite easy and you can even get better results if you build a spots model model which only includes unknowns at the Rangers were for a major echoes the curve
and here's an example of that this is a again this is this is some refers back to what I what I said about the different committees not talking to each other this is a reduction of processing technique that still used today it's this is how you read apostasy radar and here's typically will will rights maximum likelihood estimate this is how this is their idea of of linear least squares and then It's not like that so but they still use user because it 4 point like pocket it it is optimal target dreary point like and this results in in in sidelobes and this is from a poster by by that that you will be shown by work we go or could today at the poster session if we do it at least linear least squares estimates we slightly better results of we get rid of all the side lobes which occur because properly solving a least squares equation and if we do a sparse model we can also improve get better signal-to-noise ratio and and and it's even more meters so this is this is an example of of of of this new material that we developed and and 1 minute we see about 40 40 echoes other advantages this rate
is that there are no ranged up when we do use pulse straight it's it's it you have all the prompt so would have with that with the Nyquist sampling we don't have those problems with with with the random noise It's fall into our apply to radio interference of of a random signal does not correlate with everything anything else except the same instance of that random single that's and sold with radiant if random transmitted signal will not correlate with radio interference that's why this rate tolerance of interference it's well-suited for Malta study networks will power 2 components open source small it's an inexpensive and more details and in in this 2 papers and is it is a picture was taken during the last campaign by York Trotter and he set of all these antennas in in a in a pure tree on on the on and that that's the that's the receiver systems so also commercial off-the-shelf found other advantages of this type of rate is
that so it allows us to to seperate multiple different transmitters on the same frequency and in addition to building a multi-static network with transmitters and receivers located hundreds of kilometers away from each other we can also build a system where where we have multiple transmitters located near each other and then they don't radio interferometry configuration and if each 1 of these trumpeters transmitted difference different single we can't in the end we get scattering from a meter trailer go on with 1 antenna with 1 antenna we can actually measure signals from originally from multiple different transmit antennas and that allows us to with a single receiver antenna allows us to measure the light the angle of transmission general of departure for the for the radio and that gives us another way of determining the position of of we detail and this is really useful of Cook itself gel realize that this is useful because when we when he started building a network here we went to different places and said can I can I install took a lot transplatin on your backyard and everyone said no often hope it will it will the birds will not like it and you know my my heart kids might get hurt would would this cut the part of the different it that that's a traditional system but if with this type of inverse system where we have multiple transmitters transmitting different codes department of departure we can we can install 1 single passive antenna with that of radiate which only receives and we can determine the position of features using the system so I think this is this is your really useful for actually building a network in those schools we can go to amateurs we can go to different people we can easily convince them to install 1 antenna
that looks like this we can convince them to enter land is the land and in a like this if it's at the top that that's radiating 10 kilowatts of 50 kilowatts nobody will like the OK here's a here's a measurement from
from and of the 1st flight measurement from last last summer's campaign of this is kind of a proof of concept we show that the method works we we gotta roles we did a good amount of nature's 30 thousand meters per day at I think is is actually slightly better than then the normal the Long Beach area
OK I'm starting and will be 1 more prick was it here Reynolds decomposition is is because composition of the wind field into 2 components of so if this is a Winfield you this is a two-dimensional and feel it can be decomposed into a low frequency component so this is low spatial frequencies in high spatial frequencies there's no really this all official definition of what this frequency offset should be that depends on on the observer depends on what you wanna do but it's it's a very handy way of of of of grouping your signal in the you're the wind field into a mean wind and a fluctuating wind and then they can be both studied separately somebody using you over what you bar as as as a mean wind and and a new you prime as fluctuating wind OK I am again just to remind you that this is a radar measurements this is the this is a meteorite measurements a 1 dimensional projections so Doppler-shift that we measure is carried out the one-dimensional of projection of the this good to win at position p make major gives us what 1 projection a position of the wind how do we get information about the mean wind this is a standard technique of mean wind amplitude estimation here we assume that your win feel consists of 2 main components horizontal horizontal components and then it consists of fluctuating components and when you do this you get this kind of measurement equation it's it's on the x component of k-vector times so you and the mean wind and you can you can shovel all these fluctuating component enters into side and 0 mean random noise and and Europe this this becomes your measurement equation which is is very simple and this is this is what goes into your ear you know it's all of these fluctuation component you can treat them as noise 0 mean random noise you get a very simple linear relationship between lots of different Doppler-shift measurements but with different k-vector at different positions and then to mean you can do a standard maximum likelihood estimator time and then you can get a mean wind 1 of the
the cool things with with these types of major Internet which is a because we have so many new so many many more majors that we had before instead of just
estimating 1 of mean went back to the connect the estimates of wind fields of the a vector field of of of mean wind and this this is a paper by interest over and and who itself this is still there at all so then the really cool part is that this has hasn't really done before being before and and you this was only enabled by by this realization that you know if we build these major great on networks we we get more meters and good theory as a as a as a post about this Winfield inversion of tonight and new users essentially different regularization worry you minimizes deal the Laplacian to to estimate these when fields OK you can also ask people look
at the fluctuating components of the Doppler-shift by subtracting 4 1st estimating the mean wind and then subtracting the mean wind from your Doppler-shift measurements and what you get is you only left with fluctuating components of the Winfield at that point and now we get to the year how much time do I have left 20 minutes of the no we get to the so the these are all existing techniques now we get to the new contribution that some of this paper that that we have submitted if you want to if you treat your this is frequent feel like as a random process as a stationary random process of the is is a is a is a random vector and in it's it's completely characterized by its 2nd moment if it's Gulshan and you have lost at time t in position P multiplied by the transpose of the velocity at time T plus tau some time lag in plan locked in position life so as is a is a spatial I. T. Tao is a is a position get you get this kind of set of of 6 unique correlation functions these the on the diagonal you have all the correlation functions of the the Ms. frequent so here you have so no meridional and and vertical of the correlation functions as a function of time delay bundles this list in between measurements and spatial displacement between measurements and then you have these cross-correlation part so these these correlation functions would completely characterize this process if we assume that it a 0 mean Gaussian random process so the Gokey had the idea that you know what point we try to estimate these and I thought that's a good idea and I decided to help with its and and left we wrote all these equations if we have 2 different radio measurements in J. R. these at 2 different beaches measured at 2 different times and 2 different positions so they have a big they're one-dimensional projections of the of the missus TriQuint u W is included there and the K vectors basically the measurement geometries included there too and we have measurement noise a site Inc's J. R. R are assumed to be Gaussian 0 mean independent random variables if we now take a product of these 2 we get these all these terms these last 2 rows are all 0 mean the only non-zero mean mean terms here but assuming that all I use not J. Phys is chip is is J then we have a problem but not quite in our case White is not J so this this last term is is not the is is also 0 mean not because what if you multiply 2 Gaussian random depend a Gaussian random variables that the product of the they're there means that the mean is is so these last 3 rows result and in 0 in this 0 mean so if we look at these these elements here of and we take the mean value where he took the mean value here so we know that the G you you always correlation functions come from from the different the different components of the 2nd moment of of this velocity we can no we can express the multiple gate to measure to radial Doppler-shift measurements will the play with each other as there the mean value plus some random variable this rather variables not Gulshan books with 0 mean of we can then make a we can express to settling the near of relationship where we have all that that the 2 K vectors of the 2 different measurements we can have we can we can stack multiple different measurements of different measurement there so this would be a measurment 1 time measurement to measurement wanted measurement 3 and so forth if all of them have a habit of all these missions have a similar time-lag in spatial lag they're all science and and we make the white and stationary assumption of of this correlation functions they're all the measurement they're all measurements of the same unknown quantity these correlation functions this is not completely new I mean something you are all along these lines have has been done by Denise for sonority in in the in the late nineties a wing Hawking actually considers this part of the equation but but he only considers a case where all i and j equals hard is is so I equals doing and the only considers that special case and and like Nichols has has also worked on this with England's we're now applying this for meteorite hours I think the jury was really getting a huge benefit that from this method again you can use a linear least years quote the least-squares estimate for for the unknowns this is not the likelihood estimate on 4 unfortunate because we can make the assumption anymore that that are known as Gaussian but it's close enough post often and this is really a a fast way to do it we really can have millions of of these measurements in a earlier them vision equation so we kind of have to have a fast way of solving it OK what kind Alaskan can considers the here 2 different different though with that I think about forming lines if this is 1 nature measurements position PRI in time t I it is giving us information about the that's G. I and P I and and if I if I make a set to 0 to not and indeed not a set that includes an interval in time and space I can I get a set of measurements for each measurement enacted the measurements I can I can form another set of measurements which are in some kind of a box for the spatial displacement and that that is me that gives me a set of measurements to correlate and they will give tell me something about FIL they'll they'll give me a measurement of the all the color of the correlation functions at that spatial light can also the temporal like there's so it also make make do another thing I can also just I can ignore the direction of the spatial like I could say that what I want my spatial I just to be horizontal distance and this is useful especially for the Mesosphere because it's very stratified so that again what we do this method this matrix equation we had 1 1 such matrix equation for for what given lacked with some lag resolution and topic temporal and spatial light resolution of where these useful 1st of all that the winner Kantian theorems that relates correlation functions 2 spectra so if we can make and a temporal the correlation function with 0 spatial lag that's actually a that's a temporal or spectral over fluctuations in in the wind they also do spatial spectra we do especial of we transform of this of spatial autocorrelation function at 0 temporal lack we get a spatial you can also due to the 2 dimensional or or of 240 Menschel correlation functions if we want to but the interesting thing is that these these correlation function does 0 either correlation functions is related to the kinetic energy in the mass frequent so this has units of meters squared divided by 2nd squared and if you multiply that but this year 0 either the correlation functions with the new travesti contact you get Julesburg per cubic meter in in kinetic energy you can also use these correlation functions at different lots to estimate how how the kinetic energy correlates in time and space OK so numbers show some
results from the 1st campaign the really cool thing about this this this radar that we build which is a rate of network is is that what we got about 100 thousand useful measurements and 1 day this is is about 10 times more than you get with a typical rate so I'd consider have 10 times improvement as as a as a some kind of yardstick of of success so I think it was a it was a really really useful thing that we did these are all these all these points indicates the locations of Mitch's measured over 24 hours and just to demonstrate so I have some
plots with 5 intervals so this is the 1st 5 seconds we have to define reaches of 1 meter here 1 major here these are all the stations that we had in the network this next 5 seconds so 3 more meters the next 5 seconds to be more 304 1 tears we just keep getting reaches you know almost 1 per 2nd and every time we get to meet cheers for example if if I assume that 5 seconds is simultaneous for in in terms of misses the point I can now form used these 2 majors for example to to that to estimate the spatial correlation of the Miss effect window and in this direction I can use those 2 to estimate the spatial correlation of this length scale and an an enactment scale right so if we have a 100 thousand meters we have a lot of errors that we can use of course so there are correlations within within them and and that's a really a little bit of a topic of future study but we can
do a lot here's here's some histograms of of how many how many measurement there's we conform so this is temporal distance and the spatial distance supporters' spatial horizontal distance and I means that assume that I'm forcing the images to be within a one-kilometer altitude range so obviously we have a lot of beaches which are in the same place with 0 time delay because it's the same meteorites let's say 10 2nd time difference and and what let's say
on the 2nd time difference and think kilometer spatial if we already have in a close thousand meters at this altitude range and here with the court considering you know 10 hour time delay and 100-kilometre accepts horizontal separation we have we have million million almost a million pairs so we could form so we have a lot of a a lot of measurement there's the we can use to to get statistics but to get different lines of of the correlation function with on adult with
100 does majors you can already do a better job with the mean wind and here's a mean wind done with 30 minute time resolution 1 kilometer altitude resolution no overlap between the bins is a 4 hour average mean wind so this very Dylan's wind same data but with for our if I subtract the four our mean wind from half and half an hour or no but you have to know our mean wind the residuals show me
these kind of oscillation which for approximately 3 hours and they have they have different ways of different vertical wavelength and then the the tidal waves of so we think this is real of item and notching the before I I the corpus rough has seen something like this would would Morrissey but anyway this is this is interesting
and to validate our method I compare the mean wind so so from the mean wind you can you can get the 0 either the correlation function with the same meaning so I compared my my method which I called the Winfield correlation function fair version but would this well known how well that'd technique for mean when estimation and I estimated that the U. U. B. the Ubi the horizontal correlation functions and with the same data I get pretty much the same result even though these 2 methods are completely different is to kinetic energy by way as a function of time and altitude OK so
and if I do the scatter plots the scatter plots show that it's a one-to-one relationship so the method is is on unbiased which is good so we can then go go forward and do things that we cannot do would mean when this nation so 1st of all 1 thing that you can do is you can estimate temporal correlations so this is a temporal all the correlation function of the zonal and real wood and you can see that it has a 12 hour cycle this time on the x-axis and y-axis is correlation if I look at the sub for our fluctuating component when I can see this kind of damped oscillation which is that's pretty much has that 3 or of period and if I look at someone our watches the fluctuating component I can see this kind of a dance all the correlation functions I can already tell from this how much how much kinetic energy there is in these different grape also about up to 20 per cent of the energy is in this but sub for our components of the total of wind energy these this light light years old so this these dark lines these are large scale so I I I I did the bidding in such a way that I could have anywhere a 500-kilometer horizontal distance that the the the light lionized by by force them to be within a 100-kilometre radius so that I got like I can measure how much more energy is there when I consider a small scales compared to large scales in the something that is a little bit of a little bit more kinetic energy that the correlates spot on larger scales so this is really a you know all-singing all-dancing a measure of kinetic energy that it temporal scales and and also spatial skills the correlation functions are related something called the structure function so that's the definition of structure function on Komarov already used it in his 1941 paper to derive its two-thirds law if you if this is to the 2nd power and these are means you can relate the structure function to the 0 lag the correlation functions and and the minus 2 2 times the cross-correlation function at that at a lot at that lack of talent that OK I'm almost there OK so why is is is the structure function important will 1st of all it very we know not we now know how to estimate these correlation functions with different spatial and temporal as which we haven't had before we if we can do this now we can all we can go back to the structure and we can actually relate what we measured into the structure function we now have a structure function of the misses the wind at different temporal spatial skills structure from is useful because you know maybe a month to mathematician but I've not very good at that for you transforming a a spectrum of this form I think it's because if 2 K 0 but this I get into the problems but with the structure function you don't actually have to if if you have a a spec spectral model which is a power law you don't have to Poitras' weakened if you're if you're structure function is a power-law P minus 1 that corresponds to a spectral model of K 2 minus P so we all we need to do is we need to look at this structure function the see see what the spectral index is within Fourier transform involved the windowing no no practical nuisances that are involved with spectral analysis of come into play here and so here's here's an example of a horizontal all the correlation function of a fluctuating component of the wind this is a so and radio wind you can see that too and and this is the light so this is horizontal lie between 0 and 500 kilometers with 25-kilometer like spacing you can see that the where they give 12 and a half kilometers like this because that the windy correlates as a function of distance and if I now convert this correlation function into a structure function I see and then a plot on log log scales there words a large by so it's not statistically significant but the mean value follows the 2 thirds power loss of this indicates that in the horizontal direction but in this the wind is is in agreement with the core of theory theory for tervalent of this edition of education and turbulence for many minutes so I have is gun where were we have lunch next so that I should be actually very maybe I will skip this questions failing that we've done with the last thing I wanna say is that we've we've not by by the improved provision to able to work with the I P and and participate in and and and all of these different advancements in the technology of of of me during our technology so I think we have IP in my opinion this is the world leading leading Institute for for doing when measurements would materials and in addition to light as as well and and modeling so that's about it and
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