Better slope-area plots

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Plotting upslope area vs. slope in logarithmic axis has been the de-facto standard approach to river-profile analysis. In theory, if a river profile is in a perfect steady state, then this plot should yield a straight line. The TopoToolbox function slopearea calculates these plots. In addition, the function fits a power law S = k*A^(-theta). However, the function has drawbacks, and I want to address one of these in this post.

Slope-area plots require stream gradient, the first derivative of the river profile. If elevation values of the profile have errors, then this uncertainty becomes even more significant in the first derivative. To alleviate this problem, slope is usually averaged in bins. The function slopearea calculates the width of these bins based on upslope area so that bins are equally spaced in log space. However, this approach entails that bin width increases for larger upslope areas. Thus, for larger upslope areas, gradients are averaged over larger stream distances which may hide some of the structures that we wish to detect.

Another approach is to calculate slope averages over river segments of predefined length. While this is not supported by the function slopearea (yet), I’ll show here how to calculate it. The new function labelreach comes in handy here.

Let’s first load a DEM, derive a stream network.

DEM = GRIDobj('srtm_bigtujunga30m_utm11.tif');
FD = FLOWobj(DEM,'preprocess','carve');
S = STREAMobj(FD,'minarea',1000);
S = removeshortstreams(S,1000);
DEM = imposemin(S,DEM);
A = flowacc(FD);

Then we calculate node-attribute lists for upslope area and gradient.

a = getnal(S,A)*(DEM.cellsize^2);
g = gradient(S,DEM);

The function labelreach calculates a label for each reach in a STREAMobj. The option ‘seglength’ allows us to subdivide the labels into reaches of ~2000 m length. Note, however, that the function separates reaches at confluences so that segments may not always be exactly 2000 m.

label = labelreach(S,'seglength',2000);

The function accumarray is one of my favorite functions. Here we use it to aggregate the values in the node-attribute lists g and a based on label. As an aggregation function, we use the average, while we express errors as the standard error.

gg = accumarray(label,g,[],@mean);
ggs = accumarray(label,g,[],@(x) std(x)/sqrt(numel(x)));

ag = accumarray(label,a,[],@mean);
ags = accumarray(label,a,[],@(x) std(x)/sqrt(numel(x)));

Finally, let’s plot the results.

plot([ag ag]',[gg+ggs max(gg-ggs,0.0001)]','color',[.7 .7 .7]);
hold on
plot([ag-ags ag+ags]',[gg gg]','color',[.7 .7 .7]);
plot(ag,gg,'o','MarkerFaceColor',[.7 .7 .7])
hold off
set(gca,'Xscale','log','Yscale','log');
xlabel('Area [m^3]')
ylabel('Gradient [m m^{-1}]')
ylim([1e-3 0.6])
slopearea.png
Distance-binned slope-area plot of rivers in the Big Tujunga area

Now that looks much better than area-binned slope-area plots. Give it a try…

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