Next speaker is going to be Matt Newman. Matt is a senior research scientist and he's part of the atmosphere ocean processes team and he will be talking about was the February 2021 cold air outbreak over the central united states a sub-seasonal forecast of opportunity. Okay is that uh is that showing? Yeah it's perfect. It's not yet full screen I think. I see a bar on the right but it's big and readable. That's weird yeah I don't know why it seems to be full screen here. Oh yeah now it is. All right uh so this is work that I've been doing uh with John Albers excuse me and Sam Willow here uh and uh this is related to a couple of questions that we've been particularly interested in uh people remember my lecture the there were things I was focused on there too uh first can we identify high school uh S2S forecasts ahead of time given that on average the scale is very low it's really probably too low on average to be used in many circumstances but clearly the case to case events can be skillful. If we can do that can we in particular identify the dynamical processes responsible for these high school forecasts so that we have some confidence in where that uh prediction of high school is coming from and the example I'll use here is for NAO and finally to to uh focus on a single event was the recent cold air outbreak in central us one of these potentially high-skill forecasts so the way we do this is with something called the linear inverse model or limb where we are basically representing the system in a fairly simple way a linearly predictable portion and a stochastic unpredictable part and we're able to separate variability into these two components under our coarse grain assumption where we're basically observing and assuming that the non-linearities are fast in the system so that they're unpredictable compared to the linear predictable dynamics which are typically on a slower timescale and and that's particularly a good uh approximation for S2S timescales because what we're doing is we're not predicting individual weather events which are unpredictable beyond about two weeks but rather the aggregates of those events in a system like this anomalies can still grow and evolve through non-orthogonal eigenmode destructive to constructive interference even though all the individual eigenmodes of this linear operator here are stable and we can get this empirically that's kind of the main idea here although one could in principle do this uh in a forward sense from first principle so in this system uh forecasts are pretty simple you just solve this equation the ensemble mean uh forecast is written down simply uh then your probabilistic forecast or categorical forecast come about because the ensemble mean is undergoing some mean shift so basically the skill is entirely coming about from this predictable component and the pdf forecast are likewise entirely uh from that just shifted left or right depending on the mean signal so that immediately implies that we can use a forecast signal and noise ratio in a system like this to identify the forecast that we expect ahead of time to have high uh scale we in other words were calculating at the time of forecast it's still uh state dependent but it's not a non-linear state dependence it's an entirely linear process and uh it's obviously going to be dependent upon the forecast lead time so these are the two hind casts and forecast data sets that I'm going to be going through uh we have a limb that was trained over this period of time and we're looking at hind casts for winter time 97 2016 we're using seven-day running mean jra data listed here on the right uh and to make the forecast skill uh quasi independent we use that tenfold cross validation and we're comparing it to the eci ifs operational in 2017 which we obtained from the s2s data set and that was mean bias corrected that skill comparison for the globe has been published earlier but today we're just going to look at the nao and then we'll also look at real time forecast for the cold air outbreak event in this case we're going to be using a real time limb that's currently being transitioned to cpc it's it's running there currently which is basically the same as this limb or this limb here except we're also including north american two meter air temperature because that's what we're trying to predict and that's been running since about december of last year and then we're comparing that to a more recent version of the ifs uh so that we can get the real time uh forecasts for this period from that as well okay so first just looking at the nao uh here is uh a way of of identifying high versus low skill forecasts ahead of time we use the limb signal noise ratio and we stratify both the limb scale the limb forecasts and the nao forecasts by whether we expect them to be high skill or low skill so we're taking the top 15 percent of expected high skill forecasts and that shown is the solid line we're showing the skill of those forecasts and the dotted line shows the skill the remaining forecasts and we can see that in both these cases the limb is picking out the weeks three four five and six a high skill of the ifs and of itself it's able to identify that pretty cleanly and separate it from the lower skill of forecasts which we don't expect to be as skillful so again this is kind of immediately suggesting that at least in this case the extent to which the limb is not just predicting its own skill but also predicting the ifs skill uh suggests that that the skill is coming uh even in the ifs is coming primarily from shifts of the mean uh the ensemble mean not so much from changes in spread and skill in fact we've checked out a little bit uh for the pacific in particular and there's no obvious spread skill relationship in the ifs or obviously in the limb now we want to do is we want to diagnose where this high skill is coming from so we dig a little deeper into the dynamics of the limb we can do an eigen analysis of that limb forecast operator and look at the different eigen modes that essentially represent how different aspects of dynamics evolve uh over time and this uh plot here is just showing all the eigen values of all these modes so there's basically as one can see kind of two separate fairly nicely separate categories and we use that to do some diagnosis here we have a group of modes in blue which have typically long e-folding times so these eigen modes tend to decay on time scales longer than about a month and they typically are low frequency or even stationary modes all these modes it turns out have a large contribution from sst and stratospheric component of the limb state vector they have the other aspects as well but but they're notable because sst and stratosphere is more important in these modes whereas the remaining modes do not enlarge they have short e-folding times they have no sst component uh interestingly the mjo is in here as well which has been a common common thread in a lot of work in the past a lot of limb analysis in the past and i should notice there's also and i'll bring this up there's one there is one eigen mode which interestingly is purely stratospheric it has no sst component so what do these look like what is this stratosphere sst subspace the slower uh subspace look like this is a bit of an illustration we take a composite uh and uh of the NAM index at 10 millibars and then we look at what that composite looks like here's the typical cross section there's kind of Baldwin and Dunkerton sort of kind of quasi-dripping paint plot and one sees in fact something which is is almost entirely that downward propagating stratospheric eigen mode so that that process the that descent from the stratosphere the troposphere can be is in fact represented by just a single eigen mode at the surface it has a predominantly atlantic sea level pressure signature and like i said before there's no sst associated with that on the other hand if we do the same thing but do a composite lower down at 300 millibars we find a structure which is much more of a blend although there's some of this downward propagating component it's it's not so important as other modes which represent an sst signal more of a a projection on kind of the nonical canonical non canonical and so more of a central pacific anomaly and the nao structure is much more uh annular like and in fact what we find is that the the week three six high scale forecasts are almost entirely within this joint stratosphere sst subspace again remind you the black line is the ifs high expected scale and the orange line is the limb high expected scale so we repeated the analysis by only using the initial conditions in the stratosphere sst subspace as opposed to the remainder and almost all this scale is coming about because of that subspace so that's essentially the predictable part of the system and that means that the more that an initial condition tends to project in this stratosphere sst joint subspace a set of eigen modes the more the larger a predictable signal it'll have and therefore potentially higher scale so let's look at a particular case then we'll look at the at this february 2021 cold air outbreak this is the two week temperature anomaly it was really impressive obviously a lot of records were set particularly in northern texas had a very large societal impact and they're still fighting over it in texas i understand because the power outages and so the question we have basically is is how well did the the limb expect this could be predictable and what might have contributed to that potential forecast scale so this is what the cpc official weeks three four forecast was they do a forecast just two category they're looking for probability above or below the median temperature they weren't particularly confident for their forecast in general 55 is is pretty low although their typical probabilities are not a lot higher but clearly they were going for warmth in a broad part of the us and that was in a unreasonable forecast that's basically what all the the numerical models were showing here's the the ifs for that period very strong warming throughout most of the eastern two-thirds of the country for this period of time and in fact the cfs and the jma which are two other models cpc looks at a lot they basically were pretty similar the limb had something pretty different it was going for pretty pronounced cooling in this region and in fact had very high probabilities particularly for week four so we were getting up to 75 probability and a week later the week three probabilities we were we were in the 80s which was very high very confident for the limb and I should point out that the limb is is a reliable forecast system so that 80% basically means 80% all right so where is the limb uh getting its forecast from primarily it's getting it from from this sst stratosphere subspace and it turns out without getting into a lot of the other details this is basically from linemia uh that was occurring at this time and we had obviously very pronounced linemia and so it's picking out this very strong cooling there's a small weak mjo component and a a piece of in the stratospheric mode remember there was a sudden warming early in january and so we were still in february more at the at the tail end of that event notice by the way that these amplitudes here are double this so these are even weaker relative to this than just the colors make it look internal space obviously there's not much again we're not really expecting there to be much predictability on week three and four in the internal subspace but of course there can be a large amplitude just because we can't it's we can't predict it doesn't mean it's not there and in fact we saw that but importantly in terms of the verification we also picked up this very strong uh uh topically forced linemia signal linemia sst stratosphere subspace signal um there was a bit stronger mjo than we had predicted and pretty close uh in terms of the stratospheric mode again remember this is these amplitudes here are half of that okay so just to conclude then uh the limb can predict at the time of forecast which forecast will be skillful so it's able to identify about the the top 15 percent of high scale uh nao forecast in other words uh forecast skill that one might think would be actually usable on these weeks three and four time scales both of the limb and in the i f s and then we we can use the limb to decompose uh all anomalies into these two independent but importantly non-orthogonal uh eigenspaces uh subspaces which we can use to understand where the forecast of opportunity are coming from there's an internal subspace which dominates most of the variability again it's important to remember that most variability a weeks three and four time scale is not predictable uh on these uh time scales it's a it's this smaller subspace contained within this set of eigen modes which we've labeled this joint stratosphere sst subspace which basically consists of this downward propagating stratospheric eigen mode which has more of an Atlantic surface signature and then a group of eigen modes with joint stratosphere and sst components essentially a lot of this stratospheric component is what you might expect if you're getting wave propagation coming out of the tropics obviously as you as you're when you're 300 millibars in in the high latitudes you're you're in stratosphere so in the context of of the uh the definition that counts as a stratosphere component and then finally uh the north american uh cold air outbreak of february 2020 was clearly predictable well in advance on s2s time scale so predicting it four weeks ahead was uh was clearly clearly possible coming primarily uh from this la niña uh signal and i'll just conclude there thank you very much for the nice talk i'm always amazed um how skillful these limbs are if met designs them um the first question goes to uh arum hey thanks man for the talk and nice analysis uh maybe maybe this is a strange question so you showed that uh the limb skill is uh about the same as ifs one of the slides or for the for the nao yeah it's uh yeah yeah this one so given the these plots and all the models were predicting about normal anomalies what would make a forecaster discount the normal base forecast and go with the length uh are you mean in this case yeah yeah well they didn't in fact they went with no they did not but they should have but what what argument they have to follow i think i think uh you know we're we're trying to this has been a really uh interesting question for us trying to get a sense of what are the best uh guidance tools that we can give the forecasters right now it's one thing to just to just do this for working with cpc which is what we've been doing over the last year has taught us a lot about how people make the forecast and and one of the things that's become clear is precisely that we want to do more of this sort of diagnosis so that they have this in real time and we can say to them look the reason why we have uh strong cooling is coming from this la niña signal you know and we can show i mean potentially you can show the whole evolution of that that would be a bit much in real time but i think with time as they gain uh experience in using the tool and in a way use is the only way that a forecaster right right in the forecast that i have access to 10 different tools and five six of them which are marble based are predicting warm it is very hard for them to go out on the limb and take the something which is not not confirming with the model based forecast so anyway you know the problem yeah yeah no i agree completely whether so the week after this we were we were going cold even more strongly i mean i i there's very high probabilities and the models were going more along the same lines so what we were able to do was essentially to say to the forecaster at the time you can be more confident so they the forecaster was going cold for essentially a week three forecast but not going as strong as as the limb was saying and so we said well you can be more confident and he did take it into account which was nice that was kind of the high point for us so far i mean like i said it's a learning experience for us trying to see how how these tools are used in in real time as opposed to looking at hind casts and you know and and kind of working out diagnostics right thank you thanks so much i i have a comment and a question i i'm always what i really like about limbs is that you can decompose that in these different modes and i think you've shown a beautiful example for this year um you might have said this but i haven't fully understood it yet is there a way to decompose the ssd stratosphere modes into two modes i know it's there's a sad i it's you know after a while it gets overwhelming right i mean i'm going to sit here and show you every mode it just becomes a bit much and again these are non-orthogonal so a lot of times what's happening is you're having a couple modes that may all be kicked by noise in a particular configuration and they're evolving differently and it's the the sub the entire space that's evolving that matters um but yeah in principle we could have focused on you know any other other modes if we wanted to do that thank you what where was going is is it possible to decouple the ssd influence from the stratospheric influence in the coupled mode yeah uh i mean we've done that things like that in other contexts we haven't done that here yet but in principle it is yeah how much anish has a question yeah thanks matt yeah this is really interesting and there are lots to think about in terms of um predicting predictability and how users can use it and so my question is related to predicting predictability or predicting prediction scale from each of these modes like do we know if like from like climatology or historical forecast which modes picked up by limb like contribute more to skill and so the users can trust it more when that signal is coming up like when you decompose it into different components right that if the and so the lani near signal is strong in the limb then if that is a mode that is historically a mode that contributes to high skill in this framework and the users can trust it more whereas if it's if it's other modes then well i mean to some extent that's what this is here right basically we're saying is if the initial condition had a very high projection on this and almost no projection on that then we would say that uh we don't expect the forecast to be skillful right probably won't produce a large amplitude anyway but but uh you know the temptation would be to look at the model and say well maybe if a model had a higher amplitude that was because the ensemble was enlarged enough it didn't have enough to kind of average out to something closer to zero right or the model could happen if the limb doesn't have but in the context of the limb that's that's the difference i i get nervous about i mean i i think it's really interesting when you get one eigen mode that really looks physical because it says to me it says that that that really is a modal structure you know mode gets tossed around in our field you know all the time to mean a lot of different things and it's not really very rigorous there aren't very many modes of the system in a physical dynamical sense but it really to me it really does seem like this is in eigen mode of the system there's something physical about it you know you hit the system in a certain way you can just you're ringing a bell and you you get this eigen mode propagated there's the kind of a four-yearish and so eigen mode is is another one that you see like that there's an mjo eigen mode that comes out of a limb that seems to be robust but then other than that you know everything else is these other modes that are that are non-orthogonal because they're non-orthogonal they can cover up you know those kind of ringing modes those real eigen modes in different ways and you can get different evolution as a result and it's hard to just look at a particular eigen mode and say okay well that eigen mode is going to give me something because what really matters is how it combines with all these other eigen modes and again that's kind of that's the fundamental aspect of these non-normal systems is you can't you can't partition variability you can't partition predictability in terms of these kind of individual modes as well. Marty is there a signal of noise that's basically a function of the amplitude predicted amplitude right? Yeah yeah and and in the limb it's it's all signal noise now you know you can get more sophisticated you could in principle have a linear state dependence of the noise that would still give you linear predictability that's kind of another long story but that's hard to extract out of data you need a lot more data now you have to look at the at higher moments in order to pull that out and you can get some non-gasian behavior but it's still everything's linearly predictable but now there there can be some variation of the noise but we were kind of struck when we looked at least at the IFS and compared it to the limb in our 2019 paper that yeah it was very hard you know there's you could see a little bit of a signal spread scale relationship but it was really weak and it was really just at the extremes. Thanks. Thanks very much Matt so this concludes today's workshop thanks so much for staying around sorry we ran a little over and we're looking forward to tomorrow we're among some fantastic lectures we will hear from the students so thank you very much everyone and have a good afternoon. Thank you.


We consider a complex kick-single-lander equation of this form, okay? And here, nu is, of course, the whole state, whole state number, eta is the random force, I will specify later, and k is, we just choose k to be a cube, n-dimensional cube, and here, n, we can just, as in the title, it's the un-victory dimension, space-width dimension, so n can be un-victory integers here. Yes, yes, yes. And for the boundary condition, we can just impose the Dirichlet's boundary condition, and we, for this special choice of domains, we can, for Dirichlet's boundary condition, we can just consider it as the, of the periodic dimensions, which means that for any x plus, and here, here, j is just n-number 1, 2, n. So for this, and for, for the random force, just 1, 2, 3, 4. The random force we just consider is a, is a smooth in space and y in time. I just, we have, b is just a set of real number, of course, and then beta, beta d is the standard Brownian motion in the complex space, which you can be written as, so here, set both these two, beta r and beta i, just the standard real Brownian motions on the real line. So here, another one is, this is the standard, standard orthogonal base of, of the Laplacian on this domain, and you can, you can be written as x is equal to d1, x is equal to x1. So, so this is the setup. What we are interested in is, in the, when the goal of, here is that we want to study the solution of this equation for new is, of course, and very small, of course, for time. The motivation of this work is, of course, is the, the motivation of the, is the, the turbulence, turbulence in water. And let's see, give some, some naive, naive observation of how we can relate this with turbulence. And the turbulent, turbulent in, in fluid is, the turbulent in fluid is, usually it's, we, of course, we need to consider the Amistok-Costle equation. And if we add the random noise here, maybe. So, and when, when the, for a turbulent, we need to let the tether in, in this limit, when we let the new go to zero, and, and the turbulence is characterized by this Euler equation. So, it's a nonlinear equation, but this, if we can just collect this force. So, it's, it's a, it's a turbulent part is determined by this nonlinear equation. And if we consider our equation when taking, taking this in this limit, is, get the equation of this form. So, here we have maybe say the slightly similar equation. So, it's also an equation that we only have nonlinear part without, linear, linear. So, it's the, the main similarity of this, this, this two equations. And because the, the fluid turbulent is completely out of touch at this moment. So, we maybe, as some standard, we can consider this equation as just a toy model or maybe a model for, for, for turbulence. So, maybe want to study this equation. And the another characteristic of turbulent is the, the, the small scale effect, which means that in any, any small, small space difference you can, you can always observe some ripple effects, no matter, no matter how, how small the, the, the scale is. So, this is, and, and if we formulate this effect mathematically is that the lower order sub-left norm, maybe L2 norm is of order one. But, and for higher sub-left norm, it's maybe much larger, much larger than one. So, it is, so, so the, the problem of turbulence can be in some sense expressed in, expressed in this close of sub-left norm's scenario. So, that's so much for the motivation. So, this is enough to justify our efforts here. And as I said, we consider the case for, not vanishing, but for small viscosity, for small viscosity. Here we have a, and for, for these equations is a work by, by, by, for this equation, this equation is a mixing, mixing, if I am mixing evolution here, is the work by Cauchy and Nasha, 2000. I will explain this later, okay. So, that's what we know about this equation already, a lot, because it's mixing. And, so, before, before going to, no, no, no, that's so much. Now, let's first stay the main result and then with just a little bit of proof. Our main, main theorem is that for any, bigger than one half the space mentioned and two, and for any, if OPI is, of course, it's, PM is finite. PM, we just defined it as, and for any, for any, for any, for any delta smaller than one over eight, this number is innocent, just taking there, and then for any delta prime smaller than one hundred, one over a hundred. This, this number is also innocent, it's just to be a small number, it's fine. And we choose kappa, positive and smaller than another number, kappa, smaller than, kappa, smaller than, and here d alpha is, if we, if we divide alpha as the minimum of m minus two and one, then d alpha is just equal to this number, and d alpha is this number. And if we can also assume that the, another thing is that alpha, alpha defined this way, alpha should be bigger, alpha n, where alpha n equal to zero for n equal to one, two and n equal to, and alpha three equal to 3.6, alpha four equal to 0.6, not equal, almost equal, approximate, alpha prime approximate in equal to one point. Then, then we can assume additionally that alpha, kappa is bigger than gamma, r, delta prime. This is the, the assumptions which you can be, you can, you can just ignore at this moment. Then for any, for this, then for this fine m, delta, delta prime, and gamma, kappa, kappa, gamma, we have a zero, new zero, if new is bigger than, smaller than new zero. Then for any, new zero belong to, just the initial data belong to the sub-left space h m, we have t, then on zero zero and and delta, new, so for any, any t bigger than, big t, we have the following statements. The first one is that we have gamma, then on t, new, everything's here, and the probability of this gamma is bigger than one minus delta. And for, for omega inside this, this set, we have, we have a maximum of the sub-left norm of this, is bigger than, bigger than new, negative kappa m. And for t, t belong to, this is the first statement, is that, for, for large enough time and for larger, larger probability, which is at least seven over eight. So, there is the sub-left norm of the solution, close at least to this power, this power of, of the viscosity constant. Yeah. And the second statement is that, second statement is that the average over time for, for t bigger than here, average over time, bigger than, so it's the, the average with respect to time and then take the expectation, at least of this order, where l is equal to two, new minus two, is, so here is this two. So, so we need to make, for this one, we, if it makes sense, we need to have this exponent, negative, positive, negative, negative, that's why we impose this constant here. Also, it might be imposed on this. Okay. This is the statement. Let's, let's comment on this statement. At least in the, for, so for, for here, we have the sub-left norm of this order. So, in some sense, that in a, in a, in a, in a space, scales of maybe delta, delta is of order, we can see a change of order one. So, this is kind of, of a characteristic of a turbulence. Okay. Yes, yes, yes, yes. Okay. Yes, yes. And for the upper bound, also upper bound is, is given, but there is no, upper bound, upper bound of this constant of this order. So, this, we can have lower bound for the, lower bound for the, for the sub-left norm, also have upper bound. Of course, you can see that there's, this one is at the best is one or five, and this one is two. All right. So, there's still a gap between the two exponents, but for this one, we can also see there is a, as a new go to zero, as new go to zero, of course, this will blow up. So, yes, yes, yes. No, no, of course, this has a constant here. So, this constant depends on initial data. No, no, no. It depends on initial data, the size of it. Yes, yes, yes. Founded by the negative power of this one. Okay. So, because, and then there's no blow up. Yes. No, no, no, no, no, no, no, no. There's no explosion, no blow up here. It's a global, it's soluble and it's have upper bound in there. So, it's normally with, with viscosity here, you, you, you don't expect explosion. So, here is so much about the statement and I think about half, one, oh, maybe 50 minutes as fast as, so I can do some nasty things as, no. Okay. So, for this statement is, the proof of this, this theorem consists of two parts. Oh, let's, let's explain them more than mixing. Mixing is here because we are going to use upper bound is in a much earlier favor. Yes, yes, yes, yes, yes, yes, yes. So, for, of course, this equation is a globally well posed. For any initial data, if we consider operator t for ability, gamma is just a set inside, inside the breast space, maybe. And no, no, it's C0 subset. And then this is just the probabilities, yo t down to, so it's the same notation with, before now is the, and then mixing means that for any initial data, there's a unique, there's unique measure on this, on this probability space, on the probability space of this continual, continual function space, such that you can merge it quickly for any, for any initial data. This is the mixing, a uniquely mixing system. There's a very, also have a very good upper bound for the, for the C, how does it say this? Supreme norm of the solution is that for any, for any t bigger than 0, any solution, any initial data, the expectation of one is more than some constant, big constant. So, this upper bound is also for the, this upper bound for the supreme norm of the solution is also, so with this two, with this two, with these two, prior, these two results, we need, we need another, these two, and we need another. So, the proof of this theorem is based on the, on these two estimate, and this mixing, and this upper bound, estimate. And now the first one is that the stationary, this stationary measure, the probability is that for the stationary measure, L2 norm is smaller than any delta constant. This constant is, it's, it's depend on, on, on b, small, small b, and just depend on small b. Okay, number two is if, if the initial L2 norm is bigger than delta, then for small enough, we have the sub-left norm of bigger than for some t, some s belong to, here we just assume that this initial data is also 0. So, omega is belong to, we have a, here we have a axis, and this omega, beta of this omega is also bigger than one. Okay, so it's two lemmas here. So, so with this mixing, we know that for with this, the first lemma is that for the stationary state, the L2 norm of the solution cannot stay, with a large probability, it cannot stay small, right? So, it can, can be small only of this, this probability. The second one is that if the L2 norm, initial data, the L2 norm is large than this one, then it will at least grow the sub-left norm, m sub-left norm at least goes up to this order in this tie interval, in this tie interval. And of course, here this probability is chosen in this way. So, if this tool we can see that for any initial data, it will be, it will close, close to the stationary solution, a stationary state. So, it will eventually, with a large probability, it will be just have, for most of the solution, it will have L2 norm bigger than this order. And then after that, we just use this lemma tool. Okay, so on that, it will at least grow up to, up to this order, right? So, it's a two-step statement. And for the first lemma, I will not prove here. So, I will just give some, how we prove the second, the second lemma here. So, now how, let's see how we choose this big omega set. Here, we just assume time is studied from, from, from zero. So, we, the omega set is, it's just chosen to be, no, it's a stationary state. Stationary state, it, it's not dependent time. So, is this, for this one, this is a stationary state? It, it does not depend on time. For any time, it's like this. So, we have, we choose the omega to be a, such that 30, more. So, here with A is for, it will be any, any positive or small, small number. You can make it to be as small. I'm pretty small and then just, and this is, this is just, we restrict the, the growth of the brown emotion in this time interval. To be up to, up to this order. Okay, for, since we have A is positive, so for, for, for gamma small enough, we always have, have this one is, we have fixed data here. And then, we want to show that for, for omega here, omega inside this set. And for initial data of this order, at least of this order, at least bigger than this one, it will close to the sub-left norm and will close to this order. So, so we use, so the claim is that, use the, assume, not true. Which means that for any omega inside this big omega, at t, smaller than, smaller than, smaller than kappa, kappa here. So, if it's not true, we have this, this, this relation. And we want to show that this will lead to contradiction. So, for the solution of this equation, we can just write as the following equation. We go to n plus plus, we can, we can write, we can write the solution of this CGR equation as this integration here. So, for the first one, oh yes, this omega is not okay. A is just equal to, okay, equation like this. So, for this equation, we need to, need to control, need to make some estimate here for, for the first one. We have, since we have this assumption here, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes. Yes, for, for this one is, this one is for the, this one is for the, the elastic force. And this one is the linear curves here. So, for this one, we need to, so for the first one, for the first one, we have this, it's just the square, the, the, for the first one, maybe, one. And this one is two. Last one is, no, it's three. So, for I, so, we have, it's smaller than, smaller than minus power. Then, omega here. And stream is the T, but then power. And then, this one, since we have this assumption here. So, the h1 norm is smaller than new to the minus kappa. So, we have, we have this one is just, and, and, and those here, we have nothing, three norm. So, this one is smaller than minus kappa. And from the, from the supreme norm, estimate here is always just of constant. It's independent of the, independent of the viscosity constant. So, so we have, this is smaller than, smaller than just 80, 50, go to gamma. So, here we have, we will have a norm of, of norm of no minus plus 2m kappa. Then, for the second term, second term, since we can just, we can bound it from below by, since this term is just, you take the h1 norm, h1 norm of the first one. You take the h1 norm of the first, this one will be, will be just of, of the, some L2, the L2 norm, largest one. So, and for the last one, you is using integration by part, will be just integration by part formula for, for the stochastic integration here. And the integration by part form here is just equal to kappa nu. So, we have this term. For normal integration by part formula for stochastic formula, you will have an additional term. It's called, called quadratic variance between this one and this one. Since this one is just a, a c1 function with respect to t, so the quadratic variance with respect to bounding motion is just equal to zero. So, we don't, we don't have that difference here. So, if our, our choice of omega here, this one will be just bounded by, bounded by this exponent here. So, we will, we have these two bounded by plus kappa. So, here we want, we want this term to dominate this far and this far. So, if we sum up the exponent here, we can find that if the kappa and gamma be chosen, satisfy the condition in the theorem, which is smaller than some, some, some smaller than that in relation here, then we will have a contradiction here, because this here, we will, we will have, this is of order, this is of order, of order negative power. So, the sum up here, we will have a negative order of, and we, we choose the kappa and gamma to satisfy some relation before. So, then we will have the h1 norm of this at time, at time gamma nu minus gamma of this order c to the minus nu, some exponent smaller than, smaller than gamma. So, for nu, small enough, this will be bigger than, and if this hope, this hope choose by the sub, subive embeddings for, we will have further, and this is of order nu minus gamma, and this one will have order nu minus kappa, and nu half a kappa, and this will have a nu gamma. Gamma is bigger than kappa, so we will have a contradiction here. So, this is more or less, more or less the, and the main, the main, the main, this part is mainly due to the non-linearity terms here, right. So, the growth of this sub left, sub left norm is mainly due to the non-linearity. It's not because of the random noise, or of course, not because of the dissipations here. So, here is more or less like this, and so, so with this, here this, now, now we already prove that it's, at least the, the left norm, at least we will go up to order kappa m, kappa m. Yeah, so, so it's, for, for, for pulling, for pulling the, is, remains, remain larger with respect to time, your average is, just go back to the equation and take. So, if we apply the Ito formula to, find the Ito formula, if Ito formula to this functional, for this, and then we will have the following equation, n e t, 1 t 2 is equal to, of course, we have minus 2 t 1 t 2 plus 1, and then we have this 2 t 1 t 2 plus the, the, the classic term. Here m is just the inner product in, in the sub left space. So, for this one, always have, it's smaller than some constant. So, if we already prove that in this time interval, there is at least one odd, one time is bigger than, bigger than minus kappa m. So, now we can choose 2, interval of this form t plus 0. So, in this 2, we can just take the, take the, take the, take the interaction, in the, in the section of this, for different gamma here. So, in this time interval, time intervals, we have a moment is bigger than this order, and in this moment, this time interval, we also have a moment bigger than this order. So, so if, if we, we same that for, if we same it, if there is a time, a moment, if all of this are bigger than 1 over, maybe same that, we always have 1 over 2. Then, of course, the integration with respect to t is bigger than, if we have a moment inside this one, such that the smaller than, smaller than 1 over 2. Then, since then, we can have another, another moment inside this interval is bigger than this, this up, up this order. So, if we choose, choose t, t1 to be, maybe t0, we say t0 here, and t2, the, the moments inside this interval is bigger than this order. So, this is, the, the difference of this one is of all the, at least of all the, you know, this one is just innocent. So, this one is positive, it's negative, this part is positive. So, it's, so we, we will have at least the integration of uc infinity dt from t1, t0 to t2 is at least of this one. And of course, since this is just smaller than, bigger than 0, smaller than 2, 2 nu minus gamma, then if we take the, take the, take the time average, we will have additional, additional exponent here. You will have to plus another, you divide it by, divide it by 1 over, 1 over nu for the gamma, and then you have this one. So, we, in both case, in both case, we will have, we will have the, in the time average for the, so it's like this. Okay, thank you.


Can you hear me? Can you hear me? Yes. Hi, everyone. It's great to see you. Look at all these people here at Slush. We are back this year, even bigger and better, to ask, to play back your data that you gave to us, both Slush and Kriandum in partnership, to answer the question, the most asked question. What should I pay myself? I'm Michelle Coventry. I'm the head of talent at Kriandum, one of Europe's top VC firms. And let's just run through what we're going to cover. So we'll begin by setting the scene. We'll have a quick run through how we got to this data. And then we'll talk through the results and insights. And then afterwards, I'm going to bring a founder up on stage from our portfolio, Dr. Lisa Smith from PreWave. So why do we care about getting this right? Well, in my role as head of talent, as I mentioned, I co-pilot with our founders in our portfolio. And anything to do with people and talent topics tends to come my way. We commit as one team and have been in operation for 20 years, investing in founders at the very earliest stage of your journey.  So 20 years on and over 100 active portfolio companies, we've endured three recessions. And the deal team appear to be pretty good at picking winners since we've rode along with around 20 unicorns. And as they just mentioned, currently one in six in the portfolio is currently a unicorn on its way to DecaCon status. Compensating your teams at every level and every function is a weekly, well, as she said, probably a daily ask from our founders. But definitely every week, multiple people will ask this question for your teams. And as an ecosystem, we're really well served by paid for compensation platforms. I can imagine three or four going through my head at the moment where you can log on there and see what does this person in this size company at this level, what should we be compensating them? But not so on the founder side. So it was this initiative that we took last year with Slush as to why we're here today to give you what is now 12 months worth of data. It's a free tool. And it's not just for you. It's also to point your investors to and also your board members when you're having those discussions. So we have 700 founders from almost 50 cities across Europe that have inputted into the survey. We combined this with last year's data. So total data collected now is around 1,200 data points. And it's going to remain open now. And it will be dynamic and always free for you. We also then pinged a lot of our friends that we co-invest in in the VC platform teams and other talent roles. And they also distributed it within their portfolio companies. The survey has remained exactly the same, except this year we did add one more question, which are who are your investors? Just to see if we could learn if some investors were advising more or not. But all in all, it's exactly the same. So what do we ask for? We ask for base. We ask for bonus or variable compensation. We also look at equity. We look at where you're based to see whether that matters, so your location. We look at how you spend your time. What's your main role? We look at your gender. We ask your gender. We also look at total funds raised and the last round. And then our analysts get to work. So they're not here today, but a big shout out to Richie and Kay, who spent two weeks crunching and slicing your data in various different ways to give us what we're just about to go through. All of this is live publicly tomorrow. I mentioned we started open sourcing 12 months ago. Just want to quickly show you how things have changed in the last year. Not much, to be honest. Just a 9% uptick in earlier stage founders. So bootstrapped and pre-seed stage. We've seen that data number volume increase. And the distribution across geographies has remained fairly consistent across Europe, except we have seen a small 4% uptick in founders in the Nordics. But I think before we go any further, one lesson here is one thing to remember. Top tip is when you're looking at any compensation, it's always going to be difficult to compare like for like. It's not always the same people that you're comparing like for like for. So first up here, we have salary by stage. And on the horizontal axis, you've got bootstrapped all the way through to series B. The bars are median salary, and it's all in euros. This whole presentation is in euros. And the orange line represents the 2022 data. What we see is founders' salaries generally top out a little below $150,000, which might indicate a more measured approach amidst a difficult economic and funding environment. Bootstrap companies are the highest increase in median, very modest. It's small this year. It's 4%. And while we can't be certain of the cause, it could be the profitability that's now required for the next stage, or it could be the potential lack of risk that some of the co-founders are taking. But on the precede side, we see another small uplift. And that was the only other group of 2%. And again, hard to pinpoint why this is, except we have seen a larger number of new pre-seed funds in Europe actually starting up, which may have contributed to more capital coming into that stage business. And therefore, they feel more comfortable. But it's hard to say. On this slide, we look at salary by total funding. We have total funding along the horizontal. And the bars are showing average salary. Again, the orange line indicates the median. On average, the biggest jump is 3 to 10 million raise up to 20 million euros raise at a 30% increase. And then again, almost the same, 3 to 50 million raise. The largest median jump of 72% is between 0.1 and 1 to 3 million raised. But there is more to this, which we'll come to. We'll just dive into bonuses first and then come back to that. OK, bonus by stage. First of all, the first thing to note on this slide is the vertical axis only goes up to 25%. So only one in four of you actually pay yourself a variable component to your total compensation. So a bonus, effectively. And it does this strange Nike tick thing, which again, no reason as to why given. But our estimation is investors join you along the journey at pre-seed. And the mechanism of paying yourself dividends potentially or taking a bonus out of a profitable bootstrap business then goes away. And then obviously when it comes down, it goes to one in 10. So I don't want people to get excited about paying a bonus. Just remember, it's one in 10. Only one in 10 of you pay yourself a bonus, otherwise it's non-existent. Now we look at location. So does it actually matter where you live was the question? And here we've got the geographic regions along the horizontal axis. The bars are average. Salary and the orange line is the median. So what we see is over a 12-month period, UK founders have continued to report the highest median. And then the next is the DAC region. And unfortunately for the Baltics, this is the lowest. The most affected regions of any salary decrease were the Nordics and Benelux with minus 15% over this last 12-month period. And the least affected were the UK and Southern Europe at minus 4%. So overall, medium salaries have dropped across every region. I'm so sorry to give you this news, but I think it is a reality of the world that we've been living in and the cost of living crisis that many geographies have been handling. And potentially it implies that founders are actually taking one for the team. They're taking a hit on their own salary to protect others. Then we asked you by industry. Which industry do you kind of category do you fall into? Just to see whether that has any impact on pay. And it turns out that it actually does. So along the bottom here, we've got industry, B2B SaaS, consumer and fintech. And then we have them by stage in the different bars. Over the last 12 months, we see both B2B and fintech both taking an increase of 6%. Whereas consumer has taken a decrease of 2%. Consumer salaries though at the bootstrap stage have rebounded in the sense that they were paying themselves around 8,000 a year. And now they're taking 25,000 a year. But the largest decline was the fintech founders of Series A companies, where the median was falling 18%. We then asked you how you spent your time and which role you either held the title for or which hat you wore. And we do recognize that in the very beginning as a very early stage founder, you're wearing all of the hats, multiple. And that changes over time. Or it could be that you have multiple co-founders and you each take a different hat. So when we ask what was your main role and how you do spend your time, you'll see that the CEO is ranked fourth. I don't know if this is surprising or not. To me, it's not because I work with compensation data all day. But our CRO founders, our commercial founders, the folks that are leading sales are the highest paid median at around 90K, which is 25% higher than the next well-paid, which is finance and then tech. Gender. Okay, female, green, male, gray. Nothing much has changed in the last 12 months, afraid, ashamed to say. And this is where I put my hands up as well. We see parity at pre-seed stage. So we've got stage along the bottom. We do see parity at pre-seed stage. So this is very positive. We should celebrate this. But the delta creeps in at seed of around 13%. 13, one, three. Why? But that said, when we look at our respondents, we have six times more men represented in the respondents survey itself, which made us scratch our heads. And then we thought, okay, does that mean that we need to market this survey more to female founders? And the answer came back. This is representative of the industry right now. So hands up, we didn't collect any data for series B, female founders. And that's something that coming out of this is something that we commit to changing. So watch this space. Penultimate slide.  But penultimate slide. And we want to talk about ownership, which is obviously massive in terms of this is where founders get their ultimate payday. So along the bottom, along the horizontal, we've got equity percentage on the vertical and stage along the horizontal. And medium for the same time last year depicted by the orange line. Unsurprisingly, founders share decreases as your company matures through dilution. And I think it is important to mention now that the number of founders, the number of people you've co-founded your company with, that number exponentially increases the dilution factor along the journey. And just sort of stay mindful of that. Like really matters. And I don't know whether that gets educated in the early stage. I just want to land that point. Lastly, we have salary by full-time employees in terms of FTEs. On average, as your company grows, excuse me, so does your base salary. But look here at around 50, look here at around 50 to 200 employees, the average and the median plateau. And then it takes up again around 200.  And wondering why that happens. And it could just be that, reflecting the job in hands, your company, at this point, around 200, is relatively de-risk. You've found market fit. You're probably doing some sort of international expansion. And the salary increases reflective of the job in hand. And market rates. So in summary, founder salaries are down. Inflation is really starting to bite. Bonuses are on pause. And except for those that are bootstrapped. The UK still remains the highest salary spot in Europe with cost of living. Pay gap is real. Whether, just keep saying it, maybe something will change. And CROs are earning more than CEOs. You will find the founder's report on our blog, so creandom.com. And then look under stories. And this is gonna be released on Monday. So we'll put the full report with all of the findings on the website on Monday. We also, which is live now, you can check it out now, is the creandom tool, the compensation tool. So this was built by our analyst. It does contain over the last 12 months. We're gonna upload the 700 on Monday. So it'll be much better on Monday. But just to let you know that it exists. And we'd love your data. This is a free tool that you help others. You help others in your founder community. And by being data informed, your board members, your investors, and each other can actually come to the right solution for you and your company. So now, I would like to welcome to the stage Lisa Smith from PreWave. Hey, Lisa. Hi. It's great to have you here. So Lisa is, we invested in your company at the beginning of this year. Talk us through the story and how you got here today. What kind of stage? Yeah, put some color on PreWave. Yeah, sure. Yeah, hi, wow. It's exciting to be here. So yes, I'm the CEO and co-founder of PreWave. We are B2B SaaS company based in Vienna, Austria. We make global supply chains more transparent, resilient, and sustainable. And we do so by providing a technology to our enterprise clients that helps them to understand how the supply chains look like into the depth beyond the first year of suppliers, that helps them to understand where the most risky suppliers are in the supply chain networks, and it also helps them to understand what they need to do in order to mitigate that risk. So yes, Granum invested earlier this year in March into our company, and we're very grateful for that partnership. I founded the company in 2017 together with my co-founder. So we are two co-founders at PreWave. Then it's, so let's say the technology actually comes out of my PhD that happened in the five years before. And then founded the company in 2017, and then what followed was three to four tough years of finding product market fit. So yeah, not much happened, let's say, on the revenue side. And I think at the end of this, the team beginning of 2022 was about 20 people. And right now, yeah, so like 20 months later, we are 140 people and more than 150 enterprise clients. So the last 20 months were really an exciting journey for us. And what about back in the beginning, when you were thinking about founding a company and coming out of PhD Studentville, how do you and Harold have that conversation? So in the beginning, when we really just started and we didn't even have the first employee, it was let's just take enough so we can survive. So this was really very little, talking about 1,000 euros per month or something, like really just, yeah, that we could get around. Later on it was, yeah, we were working towards that we could just pay ourselves so much that we would not worry about it. So that was a bit our thinking. We always paid each other the same amount. So we're two co-founders and yeah, but this was really pre-revenue. It really fell to us. We're all working for the equity. We're not working for our salary. Otherwise we would go to some other company. So that is what felt right to us at the moment. That was also possible because we didn't have families yet to support and things like that. Families. So what happened this year? Yeah, so right, we closed around when I was nine months pregnant. So that was an exciting time, let's say, but yeah. So I got a, yeah, my second child already this year. So yes, generally the thinking about remuneration certainly changed once I have a family to support and particularly being a female founder.  And it certainly helps if you have the money basically to build up a support network. So it's particularly on the private side to get also some support for household things but also for childcare. So that really changed a bit the perspective on the remuneration side. So what did you do about it? And like, I know the answer, but I'd like you to reveal the conversation that was had to actually make that happen. Yeah, so we just talked back there. So I think founder remuneration, one thing is about the salary that you get, right? But there's also things like secondary, right? And I think there's something that is in the startup scene not really well discussed amongst founders. So that's why I wanted to share. It's something that is typical only for later rounds. So it basically means that in a round you're selling a small part of your shares, yeah? Because it can very easily happen that on paper you're already millionaire but in the end at home you can almost not pay your invoices, right? But what has happened to me as a coincident more or less we had, or to us, we had a round that was massively oversubscribed. So basically then investors were starting to ask, do you maybe wanna sell a small part of your own shares? And we were never really considering that. We didn't even know this was possible, to be honest. I always thought that, obviously, it's not gonna happen anytime soon. But yeah, this enabled us to take a bit of the chips off the table, let's say, a very small part and if you consider it in the round and so on. But for me personally, that was quite a lot of money. And that, yeah, I mean, we need to be serious.  Yeah. In my case, yeah, now two small kids at home, yeah? I was, I'm living in an area in Austria where there's not too many childcare facilities. So I was like, okay, how are we gonna do this, right? I'm a CEO of this company. We're growing quickly, hyper-grows, yeah? And then at the same time, my mom was considering, yeah, to, like, she was, she's working for more than 40 years, but she would still have three years to work. Yes. And, but she was really like, I would much rather take care, help you at home and with the children than to continue this job. So it just gave me the option to tell her, okay, maybe you can help us at home and I can financially support you for the next three years until you officially retire. And that has been, for me personally, a real game changer. And this is probably just one example of the freedom that it can give you if you're thinking about, yeah, like this second year. Thanks very much for sharing that. It's a really personal story and I think it takes great courage to sort of share that with other people that there are options if you have that discussion. How fans set their own salary also has, it's a beacon really, it's a philosophy whether you call it that, it's an approach for how you set your team's salary. So maybe we could just touch on in the final few minutes on how you think about that. And also the stage of growth that you're at bringing on new executive talent and their sort of compensation requirements. Can you just give us an insight as to how those two things marry? Yeah, so generally I think we as founders shouldn't be the best paid people in the company and we would, from my point of view, then we would do something wrong because we're trying to attract really talent on a level that, yeah, almost have respect when I send to my VPs. So yeah, so I think generally for us, the discussion about the founder remuneration was always a lot easier because we know we are working for that equity part in the end. But then VP level or even C level, that's I think the hard part because you're bringing in people from different companies with different standards from different regions and you need to create a bit of a system that makes sense, right? And I think, and it also comes down to your values, like things like that, like do you wanna give out company cars and all of these things? I think that can create quite tough conversation, that's I think quite challenging. But in the end for us, it's important that it has some logic in there. It's not transparent, let's say in our company for everyone, like in many companies. But to me, I always try to think about it if it would be transparent, it should make sense somehow. But it's certainly a big challenge. Just you've just touched on something that we haven't talked about before, but I just wanna inform everyone that not so long ago, a couple of months ago that EU directive has been that pay transparency is coming across the pond, started in the States last year and at the beginning of this year in certain regions and it is coming to Europe. It will probably be in place for 2025. So to your point about setting your approach as if it were transparent so that you can actually stand behind every decision that you've made along the way about your team. So I think it's a really important fundamental to start thinking about getting right because all companies will have to report on salaries moving forward. So I couldn't give great news, salaries are up, couldn't say that this year but just wanna say a huge thank you to Lisa for sharing her founder story on a topic that doesn't get spoken about too much and we haven't got too much data.  We do this for you so please keep that data coming and we really appreciate you contributing and checking in on it regularly and we'll continue to build it as long as you need it. So thanks very much to all of you and all of our friends and Lisa and yeah, we'll be doing Q&A next door shortly. Thanks very much indeed. Have a good day. Bye bye.


Lauren Silvio is in her second season as the Titans Nutrition Coordinator. She quickly built respect around the Titans with her knowledge, her background, her ability, and her professionalism. But she also occupies an important role to today's football players. These guys understand that nutrition is a major part of their preparation. I would say nutrition is a huge role. You know, one thing is, I'm here as a resource for them. I can't, you know, have them do everything. Like I can tell them what to do and I can educate them on what to do, but they have to actually take that knowledge and use it themselves. Lauren Silvio earned her undergraduate degree at Alabama and a master's degree at LSU. After three years of internships, she was hired full-time at LSU before handling nutrition at Auburn for six years. The Titans hired her prior to the 2022 season. The team quickly learned that multitasking may be her most impressive skill. A normal day sees Lauren Silvio never stop. I'll go in and work on anything I need to from like a meal plan standpoint or maybe talk with the chefs in the kitchen about what menus we're doing during the season. They'll come out for a snack prior to practice because we practice right in the middle of the day. And so it's kind of an interesting scheduling situation with food, making sure they're eating the right things before they go out to practice and then coming back in from practice and having another kind of lunch situation. So you want to make sure that you're telling them what to eat that's not going to hurt their stomach before they go out and run, making sure that there are things they're going to have enough energy to go out there because sometimes they worry that if they eat too much in the mornings they're not going to feel good at practice, but there's kind of that balance in helping each guy understand what they need to be doing from that standpoint. After practice, we'll come back in and have lunch. I'll be in there if they have any questions about stuff, what to put on their plate. During this past off season, she came up with a plan for every player. They talk goals, nutrition, staying consistent with the plan, etc. And Lauren Silvio is a constant presence. She isn't just in her office or in the cafeteria. She's at practice, in meeting rooms, on the sidelines during games, on road trips, you name it. Yeah, it's a huge relationship building to be present at all times and in different spaces.  If you're not in the meal room, if you're not at practice helping them, that's something that's really important to build those relationships and trust because nutrition can be a thing that people don't really want to talk about. It may be more fitting to call Lauren Silvio the Titans nutrition coach. Her trust and respect among Titans players seems to be growing by the day. As long as we can continue to emphasize nutrition, I think that's the most important thing. And I have great support from Coach Frable and Frank in the weight room and the athletic training staff as well. So being able to have all of those people kind of supporting nutrition and being able to help the guys is the ultimate goal. And I think that the more that I'm here, the more that we can implement new things and, you know, educate more and kind of find where we want to increase the nutrition's role. Lauren Silvio, important to this football team. Yeah, absolutely. She does a fantastic job. I love her energy. I love her spirit. I love the fact that, you know, during practice, she's out there hustling and helping Todd and his staff and coordinates the meals and plans the meals. And, you know, we get these rookies and a lot of them don't have the same type of nutrition that the SEC schools or Big Ten schools or bigger schools had. And, you know, she teaches them how to go shopping and cook and make meals at home in the off season or during the season at night when maybe they're not here at work. So just a great role and, again, no job is too small for Lauren and we appreciate what she does. Her first role is making sure that the players are healthy internally and that, you know, they're trying to eat the right things and have the right supplements and then anything else that she can do, she does.


Hello everyone, welcome to the NPTEL course on groundwater hydrology and management. This is week one, lecture five. With today's lecture, we would be completing the first week wherein we looked at the introduction of groundwater hydrology, some concepts of groundwater and why this course is important, how the course is structured and what you need to take away from this learning. Let's continue from where we discussed in the last lecture. We stopped at the system's representation of groundwater hydrology as a flowchart. So we said precipitation can come in after leaving some losses for interception and interception storage, part of the precipitation goes through through fall and hits the land surface. Once it hits the land surface, the water is available for infiltration. Infiltration is also driven by the gravity because gravity wants to pull things down. Same from the atmosphere, rainfall comes down because of gravity and then it hits the land, pulls more gravity to further move the water down. So infiltration happens, part of the water goes as overland flow, which means the water goes on the surface and goes into the oceans, rivers, whatever is nearby and then the infiltrated water goes into the soil, converts the soil from unsaturated to saturated. What is unsaturated? The soil and rock media has porous space in it and when water moves down, water will push the air and fill it with water. Not all can air can be driven out, but it can be pushed and then water takes up the space. Soil also attracts water so that plants can take up the water for evapotranspiration and trees also and other living organisms like worms under the ground. So once this soil profile converts from unsaturated to saturated, you slowly hit the groundwater table and the groundwater recharge. So below the groundwater table, all the porous space is full of water. So groundwater recharge further into this aquifer or the groundwater table will raise the groundwater table. So now you see it raising groundwater table. So this precipitation has converted through the infiltration into the unsaturated soil and then into the aquifer, groundwater storage. What happens to the water? Because of water wants to be in the most stable potential, it will try to see if it is in a high potential, it will go to a low potential until it is balanced. The oceans, lakes, etc. are sometimes at a very low potential, especially the sea is at a zero potential. So water would start to move and that contributes to base flow. The some soil moisture would contribute to interflow and then you have the overland flow all combined together in the channel storage like rivers streams and goes out as runoff. You have some of the water taken up by plants which go as evapotranspiration. Let's look at the base flow points. This is what I wanted to continue from last lecture.  So a river is like this where there's no dots, stream is like this, there's no dots. So this is surface water, this is groundwater. In groundwater you still have the solid face, the solid particles, water flows through it. It can only flow when the water table is achieved. So you could see the water table is achieved and water moves. Where does it move? It moves to the stream, the arrow marks show it because your stream is at a low potential compared to the high potential in the water table. So water flows from high to low potential, same here but in this case, let's finish this case, you have trees and agricultural area. So you have better infiltration and plants taking up the water but more important because you have infiltration high in an agricultural area or a forested area, water can recharge fast. When you come to a desert system, you still have a stream, your stream is there but your recharge is less maybe because of high temperature, low rainfall and or the infiltration is not happening as good as in your forested area because you can see cactus, cactus is seen in deserts. So that is an indication of low rainfall. So what is happening is already the water table is much, much lower than the stream level. So now the stream water potential is high and groundwater is low. So the opposite happens which means from the stream the water comes down. So when base flow occurs, it is only from groundwater into the stream but that doesn't happen always, the stream can also contribute water down which is called recharge. We'll just look at base flow for now. So understand the concept of base flow wherein we have water table and the water table is at a higher potential. So it has to give in to the stream. Moving on, we also would discuss in the lecture series, a base flow separation to understand the component of base flow in hydrographs. So what we do actually is we look at a time series of discharge in the stream. Discharge is the water flow coming in the stream. You would have a peak happening when the rainfall is done and the peak rainfall is done. So you have a low discharge then a climbing limb wherein the discharge climbs and at a peak it stops and then comes down. So this is the peak discharge, the flood top level in your stream. Then it slowly comes on. It doesn't come down fast as your peak is reached because rainfall is occurring at a high pace. So it comes, once after rainfall slows down, the process is slow. So once you come down, what you notice is that your base flow component is already inside your discharge. Let's look at how it looks. So when there is no rainfall, your stream is still flowing. How can a stream flow or a river flow when there is no input to the system? Because your groundwater is giving water. So in the previous diagram also, you didn't see rainfall on the top. Groundwater is flowing because it is at a high potential and no potential. It has nothing to do with your rainfall. So that is what is happening. So this line is your base flow. When there is no rainfall, you still have discharge. But as soon as rainfall picks up, the base flow component goes down. Why would that be? It is because when you raise your level in your stream because of rainfall. So when rainfall comes, the river level goes up. This level would go up. So once it goes up, it is at a higher potential compared to the groundwater. So initially it was like this. The groundwater is giving water into the stream, but now your stream level is high. It gives to the water. So or pushes or pushes, it gives water to the base flow component or it prevents base flow from giving water to the river. So your amount of base flow comes down because your peak is happening.  The same way your stream gets rainfall, your groundwater also gets rainfall in a delayed days, number of days, you can see. So rainfall stopped here, but the peak happened two, three days later. And after three, four days, your base flow kicks in. So now your stream is at a higher. It was preventing the base flow from coming in and then slowly your base flow comes down. So now your groundwater comes in. So this can be explained by this diagram. When the peak happens in your stream, your base flow is at the lowest. The base flow contribution into the stream is low. Then after when the recession limb comes, when your groundwater is being picked up by base flow. But most importantly, after your peak, your stream discharge hits the recession, which means it's climbing down. While it climbs down, your base flow increases and then comes back to the same line. So if you imagine drawing a straight line, it will be the base flow component.  And then again, your base flow kicks in when there's no rainfall. So this is called the hydrograph separation method. There are equations for it. What we need to drive out here is all streams, perennial streams, not seasonal, all perennial streams would have a base flow component because not everyday rainfall occurs. So rainfall doesn't directly flow into your stream, but it flows through the groundwater and comes out as base flow. Now the tricky question, how does this look? How does this discharge look in a situation where you do not have a perennial stream? So more or less, it is a seasonal stream. So for a seasonal stream, let me draw it. Your discharge will not start at non-zero. It will be zero. So when the rainfall occurs or before the rainfall, it is zero and then it will peak and then go down to zero. Okay. So there is no base flow component before. However, there could be some base flow here coming from the groundwater component. And when the peak is attained, it falls into the recession. But again, it goes back to zero. It slowly goes back to zero with your base flow component. But both the discharge and the base flow is zero. That is why you do not see a flow when there is no flow in the river. So the base flow is zero. That is why you don't see a flow in the river. Okay. So moving on, let me put back my spotlight. So moving on, you have the groundwater relocation. Okay. So what is the groundwater relocation? You have seen the base flow component. Now you could also see that groundwater doesn't always have one aquifer or one water table. You can imagine to have another saturated zone below the aquifer. So I will not get into the detail because this would be a very focused discussion. But what I wanted to get is groundwater is being relocated throughout its journey. So from the precipitation, water is called precipitation when it comes from the atmosphere into the top surface. Once it hits the top surface, it becomes infiltrated water. Then it takes the name as soil moisture, as you can see here. After soil moisture, it gets into perpulation and goes into the groundwater. This one and this one. Within the groundwater, it doesn't stay there because of the potential difference. It starts to move. And that movement gives way to groundwater flow, groundwater hydrology. And at the end, it can go to base flow and into oceans or rivers or lakes, depending on where it's situated. And within the groundwater, different layers can be created depending on the layering of the rocks. So suppose you have a layer of rock, which is impervious, which means it doesn't allow water to flow. So all the infiltration cannot pass on this layer. So it has to go this way. Same way your groundwater is coming, it cannot go up because this layer is preventing. Same this layer, aquilute impervious layer is preventing it from going down. So it moves lateral. So you have two lateral flows because of the presence of an impervious layer. So groundwater relocates. It relocates to the place where it has less resistance and where it can flow. So with this, I would like to recap the week one. So you've seen the introduction to the course as why we need to study groundwater as a resource, why we need to study groundwater hydrology. Let's do a short recap. We did the course introduction on why it is necessary and especially for India. We did some examples of a case for India why and how much we extract. We looked at a particular state to look at how much water is available, how much water is the demand and what is the stage of development. We also looked at course breakups and linkages between the weeks. So we go strategically from one week to the other week and we carry on the learnings so that at the end you take the combined learning to manage groundwater. We gave an introduction to groundwater. Water constitutes groundwater and we discussed the porous space and water gets into the porous space which initially had air. Now it has water and that water constitutes the groundwater. The groundwater hydrology which means groundwater movement occurs when these pore spaces are connected. Okay, breakup of groundwater and total water budget. We showed this example from FAO where you saw that all water is 97.5% is salt. So if you take 100% water 97.5 percentage is salt water in oceans. Only 2.5% is freshwater. Of the 2.5% only 21% is usable through groundwater and easily accessible surface water. So surface water is only 1% which is your lake soil moisture. Soil moisture can also combine into the top surface. Okay, so the easily accessible water could be 1% whereas the 20% is groundwater. Of the 20% groundwater only half of it can be easily accessible. So if you run the numbers anywhere from 0.5% to 0.7% of the total water available in the planet is groundwater that can be easily accessed. So of the freshwater groundwater still is the biggest contributor and so that is why there is tremendous stress on groundwater for agriculture, domestic use, industrial use, you name it groundwater has been exploited. So if we don't understand the relevance of groundwater in the overall budget, we are losing the groundwater. Think about if you pull the water out and water the plants or you use it for industry. It goes back as precipitation through and via evapotranspiration. So you apply it through the plants, it evapotranspires, goes to clouds and then forms precipitation. Where does the precipitation go? Is it going back to the groundwater? Not. It goes back to rivers and lakes. So you have depleted the groundwater, you are not putting back the water, but you are relocating the water as surface water at a much faster pace, which took a long, long time for groundwater to recharge. So increasing demands are occurring. There is tremendous increase for groundwater because climate change is impacting the world. There are many, many floods, often floods and droughts. So let's take an example. If you have floods, the whole city is flooded, which has bad water on it. Which water are you going to drink? You're not going to take that water and purify it and drink it. People go to wells because the floodwater takes time for it to pollute the groundwater. So if I want to drink during a flood, I would rather go to the groundwater, take it, purify it and drink it. If you're taking your flood water, it can be contaminated. Then come to the drought. As the name suggests, there's no rainfall. There's no water in the surface. So how do you irrigate? How do you drink? What do you have to go to the groundwater? So in both extremes, groundwater is kind of a mitigation and adaptation tool. It creates a buffer for you. Okay. Animals do not have that luxury. If they don't find water, they have to migrate. So you see elephants migrating along the western guards. You see animals dying, walking kilometers without water. They cannot dig and then get water. They find for a water pole. A water pole is somewhere where the groundwater comes in and discharges. So it is very, very important groundwater in the terms of climate change. The water stress has increased tremendously. Of course, the baseline notices shows that most of the countries that are under agricultural livelihood options are staring at water stress either high or extremely high. And when there is water stress, people go eventually to the groundwater, assuming that it is an unlimited supply of water. So what initially was stopping them from taking groundwater? The technology was not good enough. You had pumps that can only pull water to a couple of meters. The process was expensive. You have to put the bore well. You had to put a pump, energy to pump the water was all expensive. Right now, it is the opposite. A borehole can be drilled at any location just by a rig which is put on the back of a truck. Initially it was a big lorry which would come bigger hole or dig a rig and then put the borehole. But now I've seen very small sized trucks with high power machine that can drill anywhere in the urban city, put your over well and then go ahead. The same way, the pumps have become more efficient, more powerful and less expensive. So both sides, it is hitting the groundwater. The easy access and cheap pumps that can literally pull all the water out. So you have pumps on one side and then you have access to groundwater on the other side. One more aspect I'll throw in here. There are low cost energy options for pumps, diesel power pumps, solar power pumps. You name it, there are multiple different power sources for groundwater now. So all would eventually get in and take all the water out and do not want to put it back in through artificial means or natural means. And that is one aspect we will be covering in the lectures. Let's take this debate with data. So in this paper, what you see is groundwater irrigation map and you see the percentage of area, almost 100% in Gujarat, Rajasthan, Punjab, Haryana and your Mahashtra, the Vidarbha regions, the dry regions, some parts of West Bengal, even along the Ganges. You see a very heavy percentage of area is irrigated by groundwater. It could be the double crop, triple crop. So but still, irrigation is happening. And even if it is one time, you can put it as 100% of that area is covered by groundwater irrigation. Is this reflected in other means? Yes. Let's take the net area irrigated from different sources. You could see from 1950 to 2010. And the net area irrigated initially was with canals. A canal is a distribution system through a dam. So you would construct a big dam, a large irrigation dam along a river, and then water would be stored. Once water is stored up to a particular level before the rubby season, which is because non monsoon season, they would slowly release water in channels, in channels to areas where farmers have procured land around the channels for agriculture. It doesn't go to every farm. Wherever the channel goes, it will give the water. You can put a lift irrigation or you put a pump and then take the water that is up to you. But the channel doesn't go to every farm. So what has happened is the canals area has increased from 1950s. That was the good revolution which was happening. But then it stopped. It didn't increase much. Same way, your tanks, the tanks which would store water and then distribute it or store water when you pump it, also did not increase. In fact, it decreased. What has increased is your tube wells. The other wells are your shallow wells, which is your top part of your groundwater. The shallow wells are not taking much water, but it did increase from 1950s to 2010, a little bit stagnated. What has happened and is still continuing to happen is the access to tube wells. Tube well irrigation has boomed suddenly and it just exponentially increased and it's going at a very, very alarming and unsustainable rate. And then this is the distribution of energy sources, which state uses what percentage of energy is used by different sources. I was talking about solar. So you have solar pumps, windmills, diesel, electric pumps. Manual is when you step on it or cyclic and take water. Or you can also have animals going around and then pumping. So what is more important is to understand the area irrigated by tube wells has tremendously increased. And these two are groundwater resources, whereas these two are surface water. So on the overall, groundwater use has increased tremendously. This doesn't look good for the water stress image. So 44% of India is looking at high to extremely high water stress as per the WRI. And that areas where they've demarcated coincide exactly with your groundwater map. So this is a different data source. This is a different data source. But what matches with them is that the water, maybe the surface water has already dried out. And that is why they have been using, they have been excessively using groundwater. And you could see the same regions having more groundwater use or abuse, I would say, because it is at overexploited stages. What you call overexploited is if you have 100 rupees put in your bank, you're taking 120 rupees out. So you're taking more than the water that came in. And at a quicker rate, so your 100 rupees would have come after you worked hard for one month, you put it in. And then you take it and buy it with something within a minute. It's not going to be used for the whole month. That's what I'm trying to say. And also excess 120. So you are actually eating in your deposit, or you're eating as a credit, which you have to eventually fill the next month. Groundwater works in a similar way, except that your account is leaky, which means your 100 rupees doesn't stay there. There are some leaks, which leak out, like an interest, like a credit card interest. So it is very, very important to understand that if you have a water stress, you should not overstress your groundwater. And if you are using your groundwater over, then the recharge rate, there has to be some mechanism to put back water into it or reduce your consumption. That would be my aim to drive across this lecture when we finish. Let's do the other comparison. If you look at this area and this area, it matches. So wherever the groundwater irrigation is happening 100%, it is overexploited, which means it is going beyond the natural recharge rate. This is highly unsustainable. So this is a central groundwater report, and this is another study's report. So if you look at that, and you look at where areas are, this could be your agricultural data, agricultural census data, irrigation department data. What you could see is wherever the blocks, wherever the spaces, which is 100%, matches almost exactly with your overexploited blocks in India as per 2017, which is the latest CGWB yearbook. The other point I wanted to take away here before we conclude is that groundwater takes its time for recharge. It doesn't recharge fast. You can put a pump and take the water within a month, or your abbey season is three months, but the water that comes into your pump will take one year. So whatever water you used from the previous year, you cannot use successfully within a couple of years. So that is what has happened because of the technology and development in pumps. You have put the pumps, you are using the water, yes, but is it an annual water? No, it is a water that has been stored for years. So you could see that some aquifers or some storage units, the water takes days and years to come, whereas some units it would take centuries and some unit would take millennia, a thousand years for the water to come. And all these have been proved through groundwater dating studies, wherein radioisotopes have been used to date the water like carbon dating. And they have found that the water that in particularly very deep locations have taken a thousand years to recharge. So with that, we're going to conclude on a driving statement for the need for this course. Groundwater occurrence is limited. And also your food production is increasing because the population is increasing, tremendous stress on the farmers. So what can they do? They have to eventually go to groundwater aquifers to take the water, irrigate, and then feed you. So the growing population puts tremendous stress on increasing the food production in the country. So don't waste food. And also coming back, the farmers have no choice.  Groundwater recharge takes a long time. So it is not as quick as you pump the water against recharge. Groundwater is sustaining livelihoods because farmers, industries, etc., they take this water for their use. Even urban centers, when the public supply through pipes is failing, they take groundwater to support their livelihoods and sanitation. Groundwater is a buffer to climate change and serves as a tool for mitigation and adaptation. It is one of the best, I would say, which is available. And India is the highest groundwater extractor in the world. It extracts almost 245 kilometer cube, which is more than the combination of the next two countries, which is US and China put together. So let's think about that and see if this course is very, very important and needed. With this, I'd like to conclude the course for the first week. And the introduction part is over and we'll be jumping into the specifics on groundwater hydrology in the weeks to come. Thank you.


You had to do a lot of time. You did nine years? One thing, no, no, no. Let me tell you. So when he read about it in the paper, he had a stroke, and it paralyzed the whole left side of his body. Yeah, we on boss talk, one-on-one. One-on-one. Yeah, we going to talk.  She was four when he died. So like four years after he got out. So when he did the 15 years, how long were you locked up at that time? I had never got locked up. That's what I'm saying. He took the whole charge. But watch this. You get secret in daddy the very next year by the fades. Wow. They were after you from the get-go. My daddy had told me that.  If you don't stop what you're doing, then I lied to him. I was like, I promise I'm going to stop because I'm going to. Because even after all of that and how you felt when he went and how many years he got knowing that he was a preacher, you still didn't stop. I stopped for like 30 days because he asked me to. And I was like, I tried. So why did you go back? Because that's all I wanted to do. That's all she knew. And I say all I knew, I've always had a hustle mentality. That's right. And I always wanted to make some money. It was basically all I knew, but that's what I wanted to do. You know what I'm saying? So then now here I am, then got caught up by the feds. And I'm stuck like my name. And I'm stuck like too. Your daddy did nine years and you, for how long were you out while he was in before you was eight? Not even a year. So you wasn't even out a year, but that had to be a tough year. Just being with you. It was. I think I would still go like I leave quiet her some Saturdays. I'm driving to the prison. Straight to him. To go see my dad. To go see my dad. Because I was his, you know, I was a daddy's girl. And so. Was your mama mad at you? Yeah. Of course. She was mad at me and him. Yeah. Because she divorced him. I can imagine how she felt about you the whole time. How did that, how did that make you feel? What she did? Like even though she was mad at me, she never like, she would still keep my kids. Oh, she didn't treat you terrible. I don't want you terrible.  She don't.  She like, but this was going to happen because that's how you live in. So you just got to take it for what is work unless you change your life. And so she never got to see me change because she died while I was in prison. Wow. At the age of 42. So when you young, yeah, but let me go back. I want to, I want to go back to your dad being locked up all those years. You go to prison yourself while he's, when you get locked up and go to prison while he's locked up, who got, who gets our first, you get out first, right? No, he did. He got out first. So you had to do a lot of time. You did nine years. No, no, no. Let me tell you. So when I, when he read about it in the paper, he had a stroke and it paralyzed the whole outside of his body. Wow. And then he knew you was going to get caught up because he just was hurt. Right. He knew it. Because I mean, he knew it. He knew it. Because he said it to you. But he never, I'm still reminding him, still going to visit him, put money. I'm doing everything I'm, I'm thinking it's necessary. But then when he, when he read it, once he got in the hospital, he wrote the federal judge and he said, whatever time you're going to give her, just add it to the time that I already have because I don't want my daughter going to prison. Wow. And what did, what did that work? That judge, when I went before the judge, the judge told me, he said, you're a witness to society. You're going. He said, you're going. You are. He probably knew that you was the reason your dad was even in there. Of course he did. Wow. Yeah. It's crazy.  No, he was just semi positive stuff. He was like, yo, don't give up. No, he didn't say, yo, but he was like, listen, don't give up, nothing beats a failure, but a try. At least you didn't get the life sentence that they offered you. You know, you got an opportunity to get out and make it right. Yeah. Like you got so much talent. Do whatever it is. Do make money with your talent. If you know how to flip the drugs, you can flip other stuff.