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.