The University of Potsdam has a new Master’s degree program “Remote Sensing, geoInformation and Visualization”. This English-language program focuses on the gathering, processing, analysing, and presentation of geoscientific spatial data by using remote-sensing technologies and data-processing methods. The program uses models and theories to assess geoinformation, and then to prepare and communicate our findings with modern means of visualization.
The new program provides a great opportunity to improve your skills in a rapidly evolving research field that also has broad applicability outside academia. If you are interested, please see this site here for further details on the program and the application for enrollment. See you in Potsdam!
Only few days left until the EGU begins, the largest European annual geoscience meeting in Vienna. In case you attend you should consider to participate the short course in geomorphometry: Getting the most out of DEMs of Difference. The course is organized by Tobias Heckmann, Paolo Tarolli and me and will be on Wednesday, 26 April, 13:30-15:00 in Room N1.
Please see here for further details on the course’s aims and scope.
Two short courses are scheduled for mid June at Potsdam University. The short courses are independent of each other; however, the topics are related and probably address a similar audience.
Geoscience investigations of point clouds, June 7-9, 2017. Instructors B. Bookhagen, R. Arrowsmith, M. Isenburg, C. Crosby.
This course will explore the acquisition, post-processing, and classification of point clouds derived from airborne and terrestrial lidar scanners and structure from motion (SfM) photogrammetry from drones. The course will take place at campus Golm (UP) and includes one day of field-data collection and two days of data post-processing and analysis.
The application is here: https://goo.gl/forms/NrRAcaASXPuseRs62. The course is sponsored by Geo-X.
Here is the flyer: PDF for more details.
Advancing understanding of geomorphology with topographic analysis emphasizing high resolution topography, June 12-15, 2017. Instructors R. Arrowsmith, W. Schwanghart, C. Crosby, B. Bookhagen.
This course will focus on advanced understanding of geomorphology with topographic analysis emphasizing high-resolution topography. The course will take place at campus Golm (UP) and includes theoretical background and analysis of digital topography using TopoToolbox in a Matlab environment. The course is sponsored by StRATEGy.
Here is the flyer: PDF for more details.
I’d be glad to see you in Potsdam!
Having prepared a stream network and equipped with a reasonable value of the m/n ratio, we are now ready to plot a chimap that visualizes the planform patterns of chi. The main interest in these maps lies in chi values near catchment divides as large differences between adjacent catchment would indicate a transient behavior of drainage basin reorganization.
Some of you might have already experimented with TopoToolbox to create chimaps. Perhaps you became exasperated with the function chiplot that is restricted to calculations with only one drainage basin and has a bewildering structure array as output. The reason for the confusing output of chiplot is that it is fairly old. At this time, I hadn’t implemented node-attribute lists that are now more common with STREAMobj methods.
Realizing this shortcoming of chiplot, I wrote the function chitransform. chitransform is what I’d refer to as a low-level function that solves the chi-equation using upstream cumulative trapezoidal integration (see the function cumtrapz). chitransform requires a STREAMobj and a flow accumulation grid as input and optionally a mn-ratio (default is 0.45) and a reference area (default is 1 sqkm). It returns a node-attribute list, i.e., a vector with chi values for each node in the STREAMobj. Node-attribute lists are intrinsically tied to the STREAMobj from which they were derived. Yet, they can be used together with several other TopoToolbox functions to produce output.
Ok, let’s derive chi values for our stream network:
A = flowacc(FD); % calculate flow accumulation c = chitransform(S,A,'mn',0.4776);
In the next step, we will plot a color stream network on a grayscale hillshade:
imageschs(DEM,,'colormap',[1 1 1],'colorbar',false,'ticklabel','nice'); hold on plotc(S,c) colormap(jet) colorbar hold off
Interestingly, there seem to be some locations with quite some differences in chi values on either side of the divide. “Victims” seem to be rather elongated catchments draining northwest. Let’s zoom into one of these locations.
Are these significant differences? Well, it seems by just looking at the range of values. However, to my knowledge no approach exists that provides a more objective way of evaluating the significance of contrasting chi values and their implications about rates of divide migration. Still, we now have a nice map that can aid our geomorphic assessment together with the tectonic and geodynamic interpretation of the Mendocino Triple Junction.
Unfortunately, I must leave the discussion to you since I am quite unfamiliar with the region. If anyone wants to share his or her interpretation, I’d be more than happy to provide space here. So far, I hope that these few posts on chimaps were useful to you and sufficiently informative to enable you to compute chimaps by yourself. In my next post, I will give a short summary and show with another example that eventually chimaps can be derived really in a few lines of code.
My last post described the math behind chi analysis. By using the integral approach to the stream power model of incision, we derived an equation that allows us to model the longitudinal river profile as a function of chi. At the end of the last post I stated that this model is a straight line if we choose the right m/n ratio, that is the concavity index. Today, I’ll show how we can obtain this m/n ratio by means of nonlinear optimization using the function chiplot.
Ideally, we find the m/n ratio using a perfectly graded stream profile that is in steady state. I scanned through a number of streams using the GUI flowpathapp and found a nice one that might correspond to Cooskie Creek that Perron and Royden (2013) used in their analysis.
The menu Export > Export to workspace allows saving the extracted STREAMobj St to the workspace. Then, I use the function chiplot to see how different values of the m/n ratio affect the transformation of the river profile. Setting the ‘mnplot’ option makes chiplot return a figure with chi-transformed profiles for m/n ratios ranging from 0.1 to 0.9 in steps of 0.1 width.
Clearly, the chi-transformed profile varies from concave upward for low m/n values to convex for high m/n values. A value of 0.4-0.5 seems to be most appropriate but choosing a value would be somewhat hand-wavy. Thus, let’s call the function again, but this time without any additional arguments.
C = chiplot(S,DEM,A);
Calling the function this way will prompt it to find an optimal value of m/n. In addition, it will return a figure with the linearized profile. By default, the function uses the optimization function fminsearch that will vary m/n until it finds a value that maximizes the linear correlation coefficient between the chi-transformed profile and a straight line. So what is that value? Let’s look at the output variable C:
C = mn: 0.4776 mnci:  beta: 0.1565 betase: 2.2813e-04 a0: 1000000 ks: 114.8828 R2: 0.9988 x_nal: [581x1 double] y_nal: [581x1 double] chi_nal: [581x1 double] d_nal: [581x1 double] x: [582x1 double] y: [582x1 double] chi: [582x1 double] elev: [582x1 double] elevbl: [582x1 double] distance: [582x1 double] pred: [582x1 double] area: [582x1 double]
C is a structure array with a large number of fields. C.mn is the value we are interested in. It is 0.4776 and corresponds nicely to previously reported m/n ratios although it differs from the value 0.36 found by Perron and Royden (2013). Of course, this value might differ from river to river and you should repeat the analysis for other reaches to obtain an idea about the variability of m/n.
Ok, we have the m/n ratio now. In my next blog we will eventually produce the chimap that we initially set out for.
P.S.: We could have also used slope-area plots to derive the m/n ratio (see the function slopearea). However, along-river gradients are usually much more noisy than the profiles and power-law fitting a delicate matter. I’d definitely prefer chi-analysis over slope-area analysis.
Rivers are organized as networks. They start out as small rills and rivulets and grow downstream to become large streams unless water extraction, transmission losses or evaporation dominate. In terms of graph theory, you can think of stream networks as directed acyclic graphs (DAG). The stream network of one drainage basin is a tree routed at the outlet. To be more precise, this kind of tree is an anti-arborescence or in-tree, i.e. all directions in the directed graph lead to the outlet. Usually, stream networks are binary trees, i.e. there is a maximum of two branches or parents entering each node. However, nodes may sometimes have more than two branches at confluences where three and more rivers meet. All drainage basins form a forest of trees and each tree is a so-called strongly-connected component.
Why bother with the terms of graph theory? Because it will make a couple of TopoToolbox functions much more accessible. The function klargestconncomps, for example, has an awkward name that I adopted from graph theory. What does it do? It extracts the k largest trees from a stream network. The function STREAMobj2cell takes all trees and puts each into a single element of a cell array. FLOWobj2cell does the same for an entire flow network. The computational basis of STREAMobj and FLOWobj relies on topological sorting of directed graphs. Thanks to graph theory, this trick makes flow-related computations in TopoToolbox much faster than in most common GI-systems.
Graph theory has a lot to offer for geomorphometry and I think that this potential has not yet fully been explored.
In my previous posts (here and here), I introduced chimaps and how they are calculated in TopoToolbox. I showed that deriving a suitable stream network requires some work. Today, I will cover the dry part: the math behind.
Chi-analysis is predicated on the well-known stream power law (SPL). I will not detail the SPL but refer you to the paper of Dimitri Lague (Lague 2014). In its simplest form, the SPL shows that the energy expenditure required to incise into the stream bed is a function of stream gradient (S) and upslope area (A) and some parameters k, m and n. Combining the SPL with a simple vertical uplift rate U allows us to derive a very simple landscape evolution model for the fluvial domain:
Any change in elevation z with time t is caused by an imbalance of uplift and incision. If both processes are perfectly balanced, there is no change in elevation. We are in a state of a dynamic equilibrium or steady state:
We can rearrange this equation and solve for S=dz/dx where x is the stream distance measured from the outlet. At this time I denote that upslope area is a function of x. U and k are assumed to be spatially uniform.
The trick is then to take the integral of (dz/dx) with respect to x from a base level z(x0) (the elevation at the outlet). For convenience of units, we also introduce a reference area A0.
Now that looks complicated. Yet, by replacing the left-hand integral with z and the right-hand integral with
we can simply rewrite this equation as a function of a straight line:
A straight line? Yes… but only if we choose the right m/n ratio. I will demonstrate in the next post how to do this.
P.S.: This was really a short derivation of chi. I can recommend the paper of Perron and Royden (2013) for a more comprehensive account.
Lague, D.: The stream power river incision model: evidence, theory and beyond, Earth Surf. Process. Landforms, 39(1), 38–61, doi:10.1002/esp.3462, 2014.
Perron, J. T. and Royden, L.: An integral approach to bedrock river profile analysis, Earth Surf. Process. Landforms, 38(6), 570–576, doi:10.1002/esp.3302, 2013.