 about the structural and functional complexity of river networks. Good morning. Thank you for, can you guys hear me? Yes, we can hear you. Great. Good morning. Thank you for giving me an opportunity to talk here at the summer seminar series. This work has been done in collaboration with the colleagues from the University of Central Florida. The question we are trying to address is, let me see, I don't think I can scrolled. Okay. The question we are trying to address is, does topology or in particular side branching influence the structural and functional complexity of river networks? So in the previous work, what we have seen is that if we consider two landscapes, one is very dry condition and one in very humid condition. But with the same drainage density, what we observed is that there is a signature of branching pattern or the side branching on the on the geomorphic features of this landfills, such as with function. And what we have also observed that this side branching increases with increasing bean annual precipitation. So when I say side branching, here we quantify side branching using the Tokunaga self similarity analysis, where Tokunaga suggested that the ratio of side branching to branching follows an exponential type of relationship with the C value. And here or with the Tokunaga parameter C, which we refer to as C value. So if we consider two different simple networks here, the schematic, one fully or purely branched, one the maximum possible side branching, the C value for the fully branched channel network is zero. Whereas for the side branched network is higher. If we consider this particular network, a schematic of the river network, computed C value is 1.73. So for this verb, we use 40 different bases across the United States pertaining to different climatic conditions and we computed their width function and area functions. So width function is essentially a one-dimensional representation of a channel network. As you walk along the main channel, you count number of channels intersected at a distance tag from outlet to the channel head. Whereas the area function is a cumulative drainage area at a distance x from the upstream as you walk along the channel head to the outlet. And in fact, it has been shown that these can be interchangeable. So here these are the two randomly selected basins and their channel networks and they're with function and area, incremental area function. So here we refer to structural complexity as the complexity in the arrangement of channel. Whereas the functional complexity as the complexity in the channel network as well as the ungeneralized part of the landscape as there is an open so on. So for complexity analysis, what we do is we use a sample entropy instead of the Shannon entropy as Shannon entropy does not account for the sequential arrangement of channel. So for example, if I consider these two basins which have the same total channel length and the same drainage area, thus the same drainage density and also the width function probability distribution is same. So they will have the same Shannon entropy. However, they have distinct signatures of the sample entropy for this width function and this sample entropy can be completed at different scales as well. So here is the entropy at different scales for both width function and incremental area function. And what we in from these plots we can see are two different observations. First is that the entropy for the incremental area function is overall higher than the width function and the rate of change of entropy for the area function as a function of c value is or the or the feathering is higher in the case of incremental area function. And which can be more clearly seen from this curve where we plot the slopes of these entropy as a function of c value with scales. So the red curve shows for the incremental area function whereas the blue curve shows for the width function. So from here we clearly see that this incremental area function has a higher slope for the rate of change of entropy as a function of c value as compared to the width function. So knowing that the width function tells you about the channel arrangement in a river network whereas incremental area function tells you about both channel arrangement as well as the un-generalized part of the lesson. If we take the difference between these two curves it should tell us some information about the hill slope contribution from the hill slopes to the to the complexity and this is the curve showing the difference between the slopes to the function of c value with the scales. And what we notice here is that there is peak at the scale of around 45 to 50 meters. From an independent analysis or a computation of hill slope length what we what we notice that the hill slope length can be measured as the inverse of a half of drainage density. And what we notice that this hill slope length is on the order of 56 meter which is similar to the scale at which the entropy difference peaks suggesting that indeed the hill slope adds significant complexity to the basin functionality. So in short or in summary what we have observed is that the functional complexity is higher than the structural complexity and this complexity peaks at a scale for the characteristic scale of hill slope length. With that I would be happy to take any question. These are some of the references it might be useful thank you. Okay please put questions in the chat while we're waiting. I'm sorry if you answered if you I missed it but when you showed that graph of change in slope as a function of distance and you showed that over 40 meters the slope remains constant why was that qualitatively? Let me see so here you're referring to this particular figure right? No it's the one that's slightly different right? Like after 40-45 meters this statuette because there is no more added complexity to the system so that's the maximum complexity you would add as it's at the hill slope length scale. Thank you okay um if anyone I don't think there are questions right now but as you think of them later please add them oh one more to the chat. John Shaw if you want to unmute yourself you can ask your question too. Hello Arvind can you hear me? Hi John. Hi you showed the contrast between two types of entropy sounds I'm learning myself that there's just many many many different ways of calculating entropy in a landscape system and do you have any sort of general thoughts about what type you want to use for what question? Yeah I mean so for instance in this particular case we are interested in how these channels are structured on a landscape so for that if you use Shannon entropy or an entropy which is probability based you will not be able to specifically pinpoint the spatial locations and so so for that you need to have some sort of information about some sort of metric which can contain the information about the sequential arrangement and sample entropy does that so in addition to Shannon entropy information you get some information about how these channels are arranged too so that's how I was showing in that example that I mean the width function the probability distribution of the width function can be same so they will have the same Shannon entropy but different sample entropy so yeah it depends on like what your question is so if you are interested in knowing the arrangements and so on so I would not use Shannon or something I would use some entropy