 It's my pleasure to introduce the next speaker who doesn't need any introduction to this group. She's a co-conspirator with me to organize this summer colloquium and it's been an absolute pleasure to work with you, Judith, to organize this. Judith is a scientist at NCAR with a joint appointment between the weather and climate division, truly an S2S bridge at NCAR. She had both worked in academia before and at an operational center, ECMWF. Her interests lie in diagnosing and representing both systematic and random model errors. She has worked across the scales from developing stochastic microphysics schemes to studying the response to an external forcing in the presence of subgroup scale variability. She's a co-chair of the working group on probability dynamics and ensemble forecasting, PDF under the World Weather Research Program. She's most known for her work in developing stochastic parameterization schemes and most proud of truly understanding the concept of noise induced drift in complex earth system prediction systems. Thanks Judith, and Judith will talk about S2S prediction and uncertainty. Thank you very much for the introduction and I have to give this back to you anytime you want the summer school organized, I will do this if Anisha is part of the team. I will turn my screen. So, I will talk about S2S predictability and uncertainty, and rather sort of giving a history of the field I thought since I have sort of the last lecture slot of the week. I'm going to review a little bit what we heard this week, and I give you sort of my perspective on what's being discussed. So, this is sort of this schematic about full cast skill as function of lead time. And we've heard about predictability from initial conditions, teleconnection patterns. And what I wanted to point out is that sort of, and at this point, sort of that that y axis is not well defined, it says, excellent, and here it says fair, and here it says poor. But this is not a quantitative measure of skill. I think it is sort of weighted by what you're interested in. So what you're interested in on sub seasonal timescales is maybe poor and seasonal fair. And so I wanted to talk a bit more about the quantification of this. So I thought I first chair some thoughts on model era, some thoughts on predictability, and then some thoughts on the physical sources of predictability. And then each I would ask you give me a strict five minute warning, please. So on short timescales, and I'm thinking here no matter where the prediction system, where the skill comes from the initialization. The system is mostly dominated by random models. And so when we run then assemble prediction systems, we, we know that there is sensitivity to the initial conditions and so we don't run a single one but we need to have an ensemble. This is the talk that Joe gave. And what we really want to do in numerical weather prediction is we want that the threat that is the variance between the different assemble members at any given forecast lead time is the same as the RMS error of the assemble mean. And we can use this spread as the uncertainty forecast of the assemble mean. So if the assemble mean error is really large, and the spread is very large then we can look at the spread and say okay this is a very unpredictable forecast. But our sample said it would be unpredictable. Because if the spread is really small, we would expect that the error is small. If this is the case we call it a perfectly reliable as all insist ensemble systems, and assemble systems tend to be under dispersive. It has gotten a lot better but they're still under dispersive near the surface. And this is why we use stochastic commodization schemes in numerical weather prediction, because what they do is, is they widen this assemble spread, you have a reliable assemble system, if, and if you then compute the skill you have a more skillful assemble prediction system. And so this really addresses the random component of model error. So what you're really doing is that in this, in this context is you're increasing the variability by adding stochastic commoditization so this is sort of the first thing what we would expect if you include random numbers. So this slide is a comment. Sort of in response to the debate last yesterday afternoon between and here and and Magdalena. This slide also pertains to a new introduction of the Lorenz system and sensitivity to initial perturbations. So, while there are many different stochastic commoditization schemes, and some of the discussion yesterday was about, should we perturb the parameters in the bulk commoditization, because we often know that those are not well constrained. We add some sometimes called ad hoc commoditization schemes like SPPT the stochastic parameter perturbations that Anish talked about, or there's skeptics which is the schematic energy backscatter scheme which I sort of added on top to represent all the processes and all the lack of upscale error growth from the past, or should we go with something more physically representative. And, and a point I'm trying to make here, I think that these ad hoc schemes work so well, and that stochastic commoditization schemes work at all, is that where the error grows is really determined by the system, and not by the particular stochastic commoditization scheme. So, I'm showing here forecast over conus I think this is brightness temperature and I'm showing this is done with work, whether model developed here. These are different lead times it's 36 to 48 hours so this is short time weather prediction not as to us. And on the top is the impact of having stochastic parameter perturbations to the micro physics scheme. And on the bottom, we're perturbing the soil state over one particular grid box in watching Washington skate state so this is the Northwest here in this map. And what we see is that the amplitude and what we see is the difference between two assemble members with those perturbations. And so what we're seeing is that the pattern where those two trajectories spread is really very similar, there might be some differences in the amplitude. But we see for this particular case it's a particular summer initialization after 48 hours, you have you, your difference in these two trajectories is really that whether or not you have convection over the southeast of the United States. And it really didn't matter how exactly they were perturbing the system, because the error would grow in the conductively unstable regions. And so I think this is the true reason why sometimes the details of the stochastic parameterizations don't matter, because it's the flow who organizes the error growth. However, if you look more carefully we do see there's some differences especially in amplitude but we did not objectively identify the initial perturbations to be within a certain realm, it just was reasonable. So to the extent that those are different stochastic parameterization schemes or different model error schemes introduce different error growth. I also wanted to say there is skill from the mean and so here on the left side I'm showing a hypothetical forecast distribution for a particularly time doesn't matter and then the climatological distribution. So the difference, and so these have the same spread or the same variance and be thinking of this as an ensemble. And so, on the left side on the right right side is the climatological distribution and so if we look at this difference. That gives us the forecast skills so the predictive skill comes from the difference of the ensemble mean forecast to the climatological mean. There's a different source of forecast skill and it comes from the spread. So, even if the mean and the climatological mean are the same and they have different spread. There is skill in this forecast. So if you think that you have a forecast and it's five degrees C over Germany. If the climatological distribution is plus minus 10 C, then you're not sure if the presentation on the roads will freeze and you have ice and accidents or not. However, if you you can trust your forecast and you forecast says yeah it's going to be five degrees C but the forecast spread is only two degrees. You can exclude that the roads that the water on the roads will freeze and you have ice, and that is a very, very society relevant forecast. So the skill can come from the ensemble mean, but there can also be spilled skill in the spread. So, I was now talking a little bit about the shorter time scales here these months. And now I want to move sort of to systematic model errors on the long time scale. I think that if you have stochastic publicizations and here stochastic publicizations are really a way of representing unrepresented small fluctuation processes. So these S to S model are typically run at horizontal resolutions of one degree. I'm talking about things that are smaller than one degree method scale systems, high impact weather, small scale, true atmospheric variability that is not represented. And so if you have this you can really change the distribution of a system. So on the left side we have a potential well. And so, if you don't have noise there it might sort of get stuck in that left minimum. And then as you add noise, the system will have a higher variance and it can reach face space states that it couldn't reach before. So if you then look at the bottom, you see that the meat that not only the variance of these distributions are changed but also the mean. And so stochastic publicizations or neglecting raring subject scale scales can lead to an increase in your sample spread, but they also have the potential to reduce systematic model errors. The second one is very important on climatic timescales, but I also want to make the case that also on the S to S timescale. And this is really something that has not been fully explored. So Joe made this comment that fluctuation small scales can lead to a stabilization of the system and I thought I give you an example to this. This is an example for El Nino, but it's a way to understand noise induced drift. So here we are, I'm showing the new 3.4 index the spectrum of it. So we have frequencies against the power and on the left side for control control climate simulation here with CSM. It's an older version. On the right side, it's exactly the same simulation but we added a stochastic publicization. And what you can see is that it was damping these erroneously is erroneous peak in the end new band. And then to a very simple model, this is the model of a two dimensional damped harmonic oscillator forced by additive what noise, which is given here on the right side. So it only has two parameter it has your damping term, and then it has a frequency. And this is just sort of a realization to picture what's going on. And there's two things you can do you can perturb the frequency of this oscillator. So this is the equivalent of having a pendulum and then randomly changing the string. Or you can change the damping of this, and it needs to be state dependent so we could imagine that the viscosity of the environment is different in the left and the right area where the pendulum swings through maybe it swings into water or honey or it could just be warmer air. So what you then can do is you can analytically show, and I'm not showing this here that perturbing the frequency results in a decreased memory, which is equivalent to a widening of the spectrum and no change in variance, where is perturbing the damping right results in an increased memory, which is equivalent to narrowing the spectrum and an increased variance. And so on the right side now we can see, we can think we can think of this as the unperturbed system the uncontrolled control simulation and menu 3.4, and that the stochastic permit by stations were acting as if they were perturbing the frequency of this oscillator. And that led to the reduction or the dampening stabilization of the system, leading to a spectrum that's wider and not as peaked. And if we think of this really in physical terms that that makes sense, we having these stochastic perturbations to to the wind, and to convection. So what they really do is they decouple the atmospheric from the oceanic system, and a randomization of the frequency space, and that acts as a damping and you can show this anonymity. So this is just an example how unresolved small scale atmospheric processes can impact systematic model errors. And it is one example on the stochastic community would say we have a noise induced drift, which acts as a damping and stabilizes the system. And it goes on if we have representation of. If they have stochastic permit by stations in climate models, and very often effect is similar as increasing resolution. And here is just an example. It's already 10 years old, but you can see that stochastic permit by station very often improve regime behavior and in particular blocking. It's often linked to a bias in the set 500 field, which tends to be so the zone, the flow tends to be to zonal. And as soon as the stochastic permit by station break up that zonality of the flow, the system is really able to model blocking better. And especially for blocking the effects are often very similar to increasing resolution. So now I talked about the longer time scales and systematic matter or more model error and now the s to s time scale here in the middle. I want to make the point it really is affected by both random and systematic model errors. The comment I wanted to make is yoga showed a skill scores of CSM, and she didn't compare directly to SMWF, but on the sub season as to as time scale the skill of CSM to is really similar to SMWF for, at least for to meet our temperature. And I personally think that the reason for this is that SMWF is absolutely leading when it comes to short foot to medium range forecasts because they have fantastic initializations. But on the s to s time scale, the value of the initialization, especially the atmospheric initialization is starting to get forgotten. And what plays more of a role is the systematic model errors. And if you come from a climate model, you making sure that your biases are small as possible, that you can tell a connection patterns are right, and that your variance is correct. And I think this is why we get we capture these modes of variability well in a climate model. And so although the initialization is not as perfect. Some of those reduced systematic errors are really contributing to the s to s skill. However, there's still lots of model error. So then that biases from past physics often develop in the first days to weeks so the many climate biases really can be analyzed on very short time scales. And one, one evidence for this is that s to s verification is still done on anomalies not full fields. If we would do s to s verification on full fields. So it would be a lot less and probably negative as in comparison to anomalies. And this is one of the projects in the verification tutorial is to look at how much do we gain by removing the lead time dependent bias in our models as individuals to just go with the fields, the full fields. And so point I want to make is that, while it's really important to issue s to s forecasts for society and for doing research, I think it also is a real opportunity to improve our models, because we can verify that this is on time scales where there's a lot of data, and then, and as much as errors in the representation of those processes lead to climate biases. We can really improve our models, even on climatic time scales by looking at this time scale. Some thoughts on predictability. Oh, 10 or 11 or six so I have 10 minutes or so. So this plot was made not in the context of s to s predictability but multi annual predictability by Branstad and 10. And it shows really sort of this predictability from the first and the second kind. So the red distribution is the climatological distribution under global warming. So you can see it just, let's think of it as temperature it just becomes warmer. And then this plume is showing the initial value of predictability your initializes system with lead time shown on the x axis the spread gets bigger, but as long as the spread is smaller than that of the climatology, you have skill from initial conditions. And this will obviously depend on your system here. They really have the ocean system in mind it's done for the ocean system, but conceptually the same as the case for s to s and the atmosphere. So this is also done in this context of multi annual predictability, but what you can see here, these blue lines are showing the predictability from initial conditions and so they get lost with the years, whereas the one shows the predictability of the second kind, the one coming from initial conditions. And this is purely through the knowledge if it gets warmer, I can actually predict this really well. And so you get predictability of the second time namely through knowledge of the boundary conditions. And they then made a quantification of the skill this is here this this blue, excuse me this black line where you sort of lose predictability you have sort of minimum and then you gain predictability from the second kind. And please correct me here. I think this plot has not been quantified in this form for the s to s timescale. And, and I think it would be different depending on on which regime the atmosphere is in, and it would be also different in what you, how you define your system. I think it would be very interesting to produce plots like this on the s to s timescale. And I should say what they use here is measure on the left side is relative entropy. And so this is a measure that combines information from the assemble mean, and the assemble variance, plus higher order moments so this is already a metric that is not only looking mean, but the whole distribution. I end up with some thoughts on physical sources of predictability. What I did here is, I tried to color code the sources of s to s predictability. And I have to be very clear for this troposphere. And I just color labeled predictability of the second kind and of the first kind. Obviously as we change our system, if we change it to the earth system for example, then predictability of answer would be one from initial conditions but for now on the s to s timescale I choose that particular system the troposphere, because then the strategy sphere already is a predictability of the second kind because you can think of the stratosphere forcing the atmosphere although as yoga and Amy said it is much more complex. And yesterday we heard about regimes, and I thought I wanted to share some thoughts about regimes, I think for the atmosphere, they are clearly a predictability of the first kind. Laura has shown us this plot on the left side, where they looked at forecasts where the initial state projected on let's say the NEO minus state. And they compared that to the average forecast and they got extended predictability by two to three days for states that predict on certain large scale states. And then yada has shown some skill with the CSM in predicting by weekly NEO so there is clearly intermittent skill for prediction on the s to s timescale and this is why we have this workshop and that's why we lost to Joe on Monday. So the knowledge so regimes have been studied for many, many years and there was a very nice overview of this. But typically they have been studied in the context of climate forecasts and dynamical systems. And so here's some work that I did for my PhD and it was analyzing a very long simulation from a GCM. And because we could run it out for so long, we could look at the face space tendency so this is now the face space span by the first and the fourth PC. And we saw that the system really very much had, I'm not trying to say butterfly because I'm not trying to say initial conditions but it has two special states. And one is corrected right by a zonal state and one by a block state. And this particular model is combining the P&A and the NAO because it's a first generation model. And so what the atmosphere was doing it was really circling around these states. If you removed the effect that if you ask where I'm going to be in 10 days it'll be always closer to the climate logical need. But if you look at instantaneous tendencies we saw that. So what Laura's work then really did, at least for me in the first time is to go from this climatological perspective to to an initialized perspective, and just to mimic this here. So I'm looking at daily data from the CSM, and I just looked at the states that project under the NAO at day zero these are the plus NAO those are shown in red and the minus NAO are shown in blue. And then I was following these cluster means over days over the over the sub seasonal timescale. And as you can see, they mix more and more. So if you look at the cluster mean and the location of the cluster mean you see this interesting pattern. And so you see that on average, if you ask where am I, one day later two days later three days later these cloud of initial states, you will collapse to the climatological mean which is here the 00 point. And so this this this climatological forecast is, it's not just persistent it not just gets closer and closer, but its worlds here around in some rather interesting fashion. And this is this regime predictability that Laura also showed in her face space space is based on the SMWF forecasts. Thank you so much. Yeah, I'm done. So, so, so if you then look at one dimensional projections you see this emerging of the signal on the S2S timescale. And this is related to regime predictability. And the fact that for these certain large scale patterns the atmosphere is evolving on a low dimensional manifold as opposed to as chaotic as in general. That's my summary. Think on the S2S timescale both systematic and model, systematic and sorry random model error play role. I talked a little bit about predictability of the first and second kind and it's not fully quantified in the S2S timescale. And that regimes which is a perfect example for predictability of the first kind can be detected in climatological and initialized data. And predictability related to regimes fall on the predictability of the first time and is captured by state of the art and WP models. Thank you. Great. Thanks to it. It's great. Yeah, a lot to digest and think about students any questions.