Chapter 21.3 - Assimilation for Lorenz

0:07 >> Before we conclude on data simulation, I want to just give one simple example of a simulation data simulation on the Lorenz equations. And so really what we want to do here is just walk through a little code of how we might use the common filter to get us bed best predictions and to sort of start thinking about making our forecast where we're using real measurements to improve our forecasting capabilities. So remember that this is the generic architecture. I have a model.

0:37 And that model has an error. I have an initial condition. And I can only measure it to within a certain amount. So I have an error in measuring my initial condition or estimating my initial condition. And when I take a measurement, I have noise. So look at all the problems we incur here. We have the fact that our model is not correct where it's it could be mostly correct, but it's maybe missing some physics or missing some terms.

1:02 We have the inability to completely estimate with precision the initial conditions. So we have some guess at what it is or some approximation of the initial condition it is, but we have errors. And every time we take a sensor measurement just the fact that sensors themselves are not perfect, we have an error there. So how do we handle this? Well, let's talk about it. So we've talked about the common filter. That's what the last two lectures are were about. We in fact we did it in vector form in the second lecture. And so now we're going to do this and take this as the model here.

1:35 It's the Lorenz equations as a 3 by 3 non-linear ODE system. And what we're going to assume is only one thing. Instead of dealing with we're going to have measurement noise on the sensors. And what we'll also have is initial conditions uncertainty in the initial conditions. We'll assume the perfect model. In other words, Q1 >> >> is going to be zero. We'll focus just on Q2 and Q3. In other words, suppose I don't know the initial conditions perfectly or if I have noisy measurements, what will I do? And that's what we're going to do here. In fact, if you've taken a course in dynamical systems, one of the things that you looked at right away for this equation is its sensitivity to initial conditions. So, here is for instance the dynamics of this model. This is the butterfly attractor that very famous in dynamical systems.

2:27 But what we also know is if you start with two initial conditions very near each other, so here is for instance the trajectory that you would have on this for an initial condition and if I slightly perturb those initial conditions, I get very different results. So, I'm showing you here some perturbations of the initial conditions for one of the for the variable X over 10 look eight simulations. And notice what happens, right? So, if I just perturb the initial condition slightly, I get very different results.

2:57 So, you can look at the mismatch between the light lines and the dark lines and that's the mismatch. So, you go up to about four units, five units in time, okay, but then all of a sudden they're very different from each other. And so, the sensitivity to initial data becomes very important. So, if you don't approximate the initial conditions well, this is going to be a problem for you in trying to make predictions. Already there's this sensitivity to initial data, but the fact is that if you can't be more precise with your initial conditions, it's only going to make the problem worse here.

3:32 Okay, so this is what we're going to address and what we're going to do though is as we think about dealing with the ground truth, which is the light line, suppose I have measurements once in a while to try to pin my solution down to the actual true trajectory into the system. So, for instance, here it is. Here is So, the light line that is the ground truth. This is the model with a very specific initial condition that you don't know.

4:03 You have something close to it. You can approximate it, but I just showed you that if you don't have it exactly, you'll fall right off of those that light line. The circles are measurements. And if you notice closely here, these measurements are off the line. In other words, they're noisy measurements. So, in other words, my sensors have access to the true trajectory, but they're taking a measurement and they have noise. And that's what you can see here. These measurements, some of them are off the So, for instance, this one here, it's off where it's supposed to be. In other words, they're imperfect measurements, but in general, these dots are trying to track, in fact, the the the true location of the true trajectory in the system.

4:46 And what we want to do is make those use those measurements so that instead of getting just some random trajectory, chaotic trajectory, we can pin it down to this light line, which is exactly the ground truth. So, here's how we're going to do it. We have measurement locations, right? And we're going to start thinking about our common filtering situation, which is going to use our model prediction, which is simulation of the Lorenz equations under some initial condition that's not quite right. We have our innovation. We have our measurement, which is noisy.

5:19 And again, minus the model prediction. So, here again is our common filter. So, we have to compute our noise variance. We also have to compute the P's here. And if you remember the P's, there I have to compute a Jacobian of the Lorenz equation, which you can actually do explicitly, and here it is. So, I can compute everything I need to put into this actually pretty simple framework for this model, right? This is not that hard. I have to compute these Jacobians for the model, but once I have that, I have this ability to do all of this, and this is essentially giving me the variance of the model, and then the only thing I would need to compute is the covariance the noise of the measurements itself.

6:02 Okay. And once you've done that, and you can find code for this on the GitHub, this is what you find. So, here is no data simulation. So, the the bolded line is your prediction of the future, and the light line is the actual future. And what you see here is the two fall off, and here's your error in the dynamics. So, you do pretty well up until about time 4 or 5, and then you can see the error is just all over the place. Pretty large, actually.

6:32 Versus, if you remember I put those circles here, so here's the circles. In other words, as I'm going forward in the future, I have an occasional measurement, which helps pin it down to the trajectory. Notice how well now it follows the bold line follows the light line, and here's the error now. So, in other words, what assimilation is doing is as I start to go off, it pins me back to reality. That's exactly the way you want to use data simulation, is you use your model and your data jointly, so that this will happen. So, you stick to the true trajectory all just simply by this innovation of the common filter, which tells you here's what your model's going to predict, but here's your measurement, pin it down to here. This is what you should actually be doing in your model.

7:17 Gives you a new initial state which you simulate your model from. So, that's the idea behind common filtering. Powerful idea, there's quite a bit more to dig into. There's quite an extensive theoretical background around this. There's a very nice book by Andrew Stuart Cody Law on data simulation that I If you want to get into this, that's a great place to start. but really a revolutionary tool for modern science and engineering is the ability to do a simulation. And so, you want to have this in mind as you go to real problems where you can collect real data.

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