 Okay, so our first talk of the morning session will be given by Zoltan Sylvester from the Bureau of Economic Geology down in Texas, and on meander pi, a simple model of meandering. Thank you Greg, and thank you so much for having me here. So I'm a research scientist at the Bureau of Economic Geology, which is part of the Jackson School of Geosciences at UT Austin, and I'm a member of a research group called the Quantitative Classics Laboratory, headed by Jay Kovalt, and we are funded by an industrial consortium, and I'm thankful for the sponsors of QCL. I'm also thankful to my collaborators and co-authors, Jay Kovalt, Wolderkin, Steve Hubbard, and David Morig, and what I want to do today is talk about a simple model of meandering, which I will meander pi, but it would be pretty boring and potentially short if I only talked about the model itself. So I also want to talk about how we used this model to better understand data, especially data from satellite imagery, and the other way around how looking at satellite imagery helped better understand how good the model is and what else can it help us with. I don't think I need to, in this room, I don't think I need to elaborate a lot on why meandering systems are important. If you live closer to the coast, as I do, then you probably don't live very far from a meandering river. Every one of these valleys, incised valleys along the Gulf Coast, is occupied today by beautiful meandering river. We live near meandering rivers, we live on deposits, deposited by meandering rivers. We get our water from sediments of meandering rivers, we get our oil and gas, and we store CO2 in the deposits of meandering rivers. So what are some of the points I want to touch on here? First, just talk a little bit about the model, which is, to stay the obvious, the Python model. Then this model is strongly related to how you link migration to curvature. So I want to revisit that problem a little bit. Then talk about how satellite imagery confirms or doesn't, how well the model works. And finally, talk a little bit about counterpoint bars or, in more simple terms, just downstream termination of point bars, what's going on there, and how that also links to this model. And if I think about it fundamentally, this is, as reviewer number two said correctly, this is not new. Actually there are two papers that, all I'm saying, that these two papers are amazing. And one of them is this paper by Alan Howard and Thomas Knutson, which is probably one of the first computer simulation papers in the earth sciences, sufficient conditions for river meandering of simulation approach. And this paper essentially says and shows how to do it on a computer. It says that migration rate is a function of the weighted sum of upstream curvatures. And you get back to that. The second paper is by David John Furbish. And it adds to this that because of the fact that migration rate is weighted sum of upstream curvatures, we shouldn't try to compare local migration rate with local curvature. And I think Furbish was so polite in this paper that it didn't get the attention that I think it deserves. So what I did, I kind of combined the ideas in these two papers and Pythonized this model. And probably no need to advertise Python here, but since I started using Python, it has changed my life. I'm not exaggerating. Anyway, let's get to the model. It's I think it's super simple. There are it's object based and the objects are the obvious objects that you would have in a model like this. We have a channel object, which you see several of those in this plot. We have cut-offs and then those channels and cut-offs combine to form a channel belt. And then there are some key methods that you can use. You can use on channel belts. You can migrate further an existing channel belt. You can plot it. And finally, you can build a 3D model. And when you build a 3D model, you generate another object, a 3D channel belt. And the key characteristics of a 3D channel belt is topography, stratigraphy. Every time step you have these and you can assign faces to the layers. And I just made this public on GitHub. All I'm hoping is that this is good enough that it is worth finding errors in it. So please do that. So one way to display the results is this plan new plot, which I call a stratigraphic display. You basically see everything, all the cut-offs and all the channels that move through here and the latest channel is highlighted in darker blue. Now we know that on the surface in Google Earth, you don't really see this because it's covered by vegetation and oxbows are filled up. So we created a version, a display version, where you can assign an age to basically not an age, but a rate of cover-up by vegetation over the point bars and oxbows. And you get something like this. So that versus this. And this really shows how quickly you can cover up a lot what has happened. And often in Google Earth, you don't really see the extent of meander belts. Have a quick look at an animation. It's probably a bit too fast. You can get busy if you want to watch it for too long. And it is pretty well known that these simple models generate these, sometimes we are looking very elongated meanders. And I think that's a fair criticism. The ways to tune the parameters to try to minimize that, but it's still there. But to me, it's one of the key questions. It's not so much the shape of the exact shape of every meander, but does the movement, does the change in the model, is that representative of what we see in nature? A little bit about building a 3D model. And I hesitated to put this in, but because it's so simple. But the idea is that, and this goes back to some old work that we did with Carlos Pirmes and Alessandro Cantelli. And this was for submarine channels. So you start with an initial channel form. Then we have a depositional surface, which is a Gaussian in this case. And wherever the Gaussian lies above this pre-existing surface, you have deposition. Then you have an erosional surface, wherever that erosional surface cuts the pre-existing surface. This is a Gaussian, this is a parabola, and then let's call this levy deposition. That is our thick here, because this is supposed to be submarine. And then after five steps, you get something like this. This all happens in a vectorized format in NumPy, so it's really quick. And you can switch out. If you don't like these parameterizations of these surfaces, you can just switch them out if you want to. So if you do that, you can generate, for example, an incised valley like this, slightly incisional river, all kinds of autogenic terraces. Yellow is roughly speaking sand. If you zoom in, you can see some of the oxbow fields and the geometries of the point bars. One of my favorite things to go on and on about is how submarine channels are not that different from fluvial channels, certainly not in plan view. And this is an example from the Bengal fan, horizon slice from a seismic, beautiful seismic data volume, compared to 27 years of the Uqayali river in Peru. And if I only gave you the center lines from the river and the center lines, reconstructed center lines from the seismic image, it would be impossible to tell which one is which. We haven't done the quantitative comparison in these patterns yet, but my intuition is that there is no fundamental difference. Therefore, we can use the Howard and Knotsen model to build models of submarine channels. Here is an agredational one. This is vertically saturated, I think like five times, big levies in submarine channel levy systems. And you can also see, because of the higher agredation rates than what you can see in rivers, the seniority combined with that high agredation rates results in quite complicated structures in 3D. And then you can start thinking about, for example, fluid flow in these kinds of lithographic successions. Here is an incising submarine channel. One of the things I didn't mention is that the model allows for variable Z. You can introduce a slope. And what happens, if you do that, if you have cutoffs, you generate nick points. It's not very obvious from this plot, but whenever you have a cutoff, you have a steep segment, just like in rivers. Difference here is that these things can be very steep on the continental slope. So these nick points can be really important in submarine channels. But let's come to this point about the relationship between curvature and migration rate. And this is a fundamental problem in fluvial geomorphology and probably the first paper that really tried to address this in a careful and quantitative way was this paper by Heppin and Nensen in 1975. They went to a number of meanders, a number of point bars in Canada, and they used dendrochronology to estimate the ages of the scrolls along these point bars. And they made a plot, which is a highly influential plot where they plotted the radius of curvature, normalized by channel width, and they plotted migration rate against it. And they said that there seems to be this relationship there at low curvatures, which is actually over here. Curvature is one over radius of curvature, of course. So low curvatures, low migration rates, then you reach a maximum migration somewhere in the middle at around three for this parameter. And then migration rate declines again if you go to higher curvatures, which are over here. And that is a little bit not intuitive because, in theory, the sharper the band, the more the stronger the centrifugal force, and you would expect just very basic physics, you would expect more migration over here. And I think one of the questions here is that why should we plot migration rate following that logic? Why should we plot it against radius of curvature instead of curvature itself? And second, as Furbish has pointed out, migration rate is not a function of only the local curvature. So plotting the local migration rate, in this case, this was averaged over all point bars, so it's fair enough. Local migration rates were not that easy to estimate at that time. But we should really think about this part. And I did a lot of research on YouTube to find ways to illustrate this problem. But here is an icy road somewhere in Russia. And soon there you will see some cars coming along here. And the question is, here is the maximum curvature point. The question is, will the cars tend to go off the road right where you have the maximum curvature? Is migration rate in phase with curvature? Or is this going to happen somewhere else? So think about that for a second. And then I will let these. I don't know if this is a setup or this happens all the time in Russia. But this is what happens. So you can see that if this was a river, the highest migration rate would be somewhere there. So it's downstream from maximum curvature. And I think that's a fundamental property of meandering. Often people come to me after I give this talk and tell me that I excellent talk. Those cars were amazing. So let's do this in a bit more scientific way. Although pretty basic manner. These are just some plots. Here is curvature, the curvature vectors plotted in black for two bends. Needless to say, curvature is not constant. It goes from 0 at the inflection point and reaches the maximum and so on. Now, if you use the Harvard and Knudsen style model, weight system of upstream curvatures, then we get the migration rate, which looks like this. And you can see in this plot very nicely that migration rate, just as we saw, it's shifted downstream relative to curvature. And here is a so one of the outcome of this model is that there will be these segments like here, where the curvature vector is going in the opposite direction from the migration vector. And these are the location. These are the downstream ends of point bars, which a lot of people call certain in the sedimentological community called counterpoint bars. And we'll get back to these a bit later. But keep that in mind with these. If this works, then there is a characteristic length to these locations along the river. It's the length of this leg. And in the model, this is a quasi constant distance for a certain discharge and river size and so on. Also, we can think about how to compare curvature and migration rate. Those are the two curves. If we really want to predict migration rate from curvature, then we should take into account that there is a phase shift. It's not that complicated. So just to illustrate that point further, here are, again, those two curves. This comes from Alan Howard. The nominal migration rate is just a version of curvature so that it has the dimensions of migration rate. So if we compare a local curvature with local migration rate and we plot them in a scatter plot, then we start seeing a similar relationship to what Hicken and Manson observed, which is you reach a maximum migration rate for a given curvature. Again, this is the equivalent of curvature. So for this curvature, it looks like I have a maximum migration rate. And then if I go sharper to the right, then it starts declining. I think this is one of the reasons why you get those Hicken and Manson style plots. Now if I take into account the phase shift, just by shifting like six positions, those two curves, you get a simpler plot, which suggests that these curves are similar to each other. And they are. But this is all good. How well does this model work in nature? And I think this is an amazing time to have so much data, so much remote sensing data from all these rivers. And one of the few reasons I'm looking forward to getting older is that I can add more frames to this animation. And so we want to measure what is the kinematics of these rivers. And what we used for this is a Python package called RivaMap that Paola Pasalakwa's group developed not too long ago. Again, open source software works really well. And you can take an image like this, detect the center lines, and generate an image of the main river, and then detect the banks, and so forth. So you can get data like this through basically as long as Landsat has been around, which is more than 30 years now. So this is the first step in the analysis. Next, we want to think about how to estimate migration rates. And one way to do that is to measure these areas between the two center lines. This is not that different from measuring bank erosion if you want to just like averaging the two banks. But what we wanted to do is to measure the local migration rate and the local curvature. And to do that, to link every point on center line one to center line two, to get these vectors, use, again, open source software written by Ben Ellis and use an algorithm called dynamic time warping. This is a fantastic little tool that I think it should be used more widely in the earth sciences. And there's a lot more to this interesting discussion that is worth having, but not now. So we apply this to seven rivers in the Amazon basin. These rivers are relatively untouched by humans, relatively. So you can detect, you can see really well the kinematics. And they move fast enough to see it in 30 years. And if I zoom into the small rectangle here, this is what we see. This is the channel at, here is the channel in 1987. And I highlighted the different bends, going from inflection point to inflection point. And then I plot the channel in 2017. Again, inflection point to inflection point. This is extremely tempting to take, to say, bend 11 migrates to bend 11. That is incorrect. The curvatures around bend 11 are not affecting the position of the channel along the next bend 11. The impact of those curvatures is actually felt along this segment, which goes from zero migration point to zero migration point. So going back and forth, back and forth. And this is more obvious if you look at the curvature and migration rate plot and you see that how the, for example, bend 11 curvatures are affected in migration over here. And there is the lag, which is pretty constant along the same river segment. And I have cherry-picked, of course, that example. But we can look at 94 bends along the juror river. And you can see there is curvature, migration rate, that oblique shift from the top to the bottom. That's the lag. There are some bends that behave a little bit differently. Those are the places where you have erodability issues, I think. But other than that, it's a decent correlation, which means that, on average, if I remember well, curvature explains more than 50% of the variance in migration in these rivers. And in plan U, that's how this looks like, the quality of the prediction. There is a prediction in red. And then we plot the 2017 channel. And it's not perfect, of course. For example, over here, there seems to be more migration than what you would expect based on the model. Note that there is a recent cutoff here. So this is similar to what John Schwenk has found that there you have a cutoff migration rates increase. And this analysis shows that it increases further to what you would expect from curvature alone. And finally, let's talk for a couple of minutes about counterpoint bars. These are locations where you have deposition along the concave bank. Usually, classically, these are associated with erosional confinement. So in this place, there's clearly some kind of boundary over here, which is guiding these bends to migrate downstream, translate downstream. There is an example from an outcrop where mudier sediments are darker, and there's no tides involved. There are no tides involved in that case. We know that. So why do some enders translate downstream if you have no confinement? And the model basically suggests that you don't necessarily need confinement to have downstream translation. All you need to consider is that the point of maximum migration, the red, is downstream from the point of maximum curvature, which is black. In big bends like these, most of these, it still results in expansion. But in small bends like this one, which in this case is related to a cut-off, it results in translation. Because the red dot falls on the downstream limb of the meander. Here's a counterpoint bar from the Peace River. And Daryl Smith and others have studied these things extensively. There are some cores that they have taken. And you can see how it goes from sandy to muddy along this bar. We can plot up roughly where the boundaries are. And mathematically speaking, this hasn't really been quantified before. But essentially, you can quantify it by saying that whenever the vectors are pointing in the same direction, that's a point bar. And then as I go downstream, they go into opposite directions. So we can try to do that. And we just call this a bar type index. So there is curvature, migration rate. If you multiply the two, we get the red curve. Whenever the red curve goes negative, that's where the counterpoint bars are expected to be. And the more negative that is, the bigger the absolute value, the more important this effect is going to be. And that's the illustration in plan U. Whenever you see the brown colors, that's where you would expect more muddy sediment, more counterpoint bar style deposition. And here is an example of a cutoff in the model. And where it's blue, you can see that those are the places where you have deposition along the concave bank. Those are the places where you expect counterpoint bars. Cutoffs and other perturbances are important for the development of translation and this more muddy deposition. Here is an example from an actual river. There blue means counterpoint bars. And you can see how this perturbation here results in lots of translation and lots of counterpoint bars. Here's another one. Lots of downstream translation and weird deposition with lakes and stagnant water and probably separation zones. There is a whole story there that a lot of aspects of which are not fully understood. And finally, the reason we want to build 3D models and understand the distribution of the bar types is one of the reasons is that if you are interested in fluid flow in porous media, then you can play games like this where your 3D model slides through many times. If you put a pressure gradient on this, you can see how the fluid front is going to propagate through. And that's what we are watching here. And I try this many times now to predict how this is going to evolve through time and I cannot. Like you just have to do it, you have to watch it, and you have to play with it. And I forgot to put that here, but this is again a piece of open source, entirely free software, in this case in MATLAB, called MATLAB Reservations Toolbox. Super cool stuff. So to wrap it up, talk about the model a little bit. Courage or migration rate relation, relatively simple. Satellite imagery and model, the match between them is relatively good. And content point bars are relatively common. Thank you. OK, we have time for a couple of questions. So Jaya was asking if we have tied these two dynamics, if you can take into account, for example, that at a cutoff you would have a steeper slope and so on. The answer is no, for me, probably. But yeah, it's something to, what I'm definitely interested in is to maybe start with building, not developing this further, but building, just taking some models of the shelf and looking at some of these meander geometries and see what really happens in 3D, how the secondary circulation works. One of the disconnects is that test how well it works in the Amazon basin. That is one time step. And you develop those very elongated meanders through a longer time evolution. So using only a short time makes it easier. The other thing is that in the model, in this model anyway, there are no irritability variations. And I think in the Amazon basin, most of these flood plains are actually pretty homogeneous overall. Unless you touch the edge of the insides really, things are pretty erodable, but still there are gonna be minor variations. And once you have those, it is enough to have a big tree over here and it can result, I think, in the curvature flips. And once the curvature flips, then the meander becomes segmented, right? Now you have new bands. Totally throws off the model doesn't capture it, but maybe that's something that wouldn't be so hard to experiment with. Great, thanks again, Sylvester.