Rivix.com: The Rivix Mass Flux Method

The RiverTools Mass Flux Method

The image at the top of this page shows a side-by-side comparison of total contributing area grids computed using the Rivix Mass Flux Method, the D-Infinity method and the D8 method, for the Keystone Ski Resort. Unlike the D8 and D-Infinity methods, there are no pixellated artifacts in the Rivix Mass Flux Method image. Note that no smoothing of any kind has been performed in any of these images. Our method just works, even in complex topography. See our Image Gallery for more images that compare these methods, especially the one for Mount Sopris.

The Problem: Divergent vs. Convergent Topography

On a hillslope flowlines may first diverge (spread out) before converging into channels at the bottom of the hill. RiverTools provides routines for analyzing both convergent and divergent flow. For convergent flow and the identification of individual channels, channel networks and basin boundaries, the well-known D8 algorithm is used. This algorithm tracks flow from each pixel to one of its eight neighbor pixels. The RiverTools implementation of the D8 algorithm is efficient and powerful, and can assign reasonable flow directions even in very complex and ambiguous situations without user intervention. However, the D8 algorithm is based on two simplifying assumptions that do not capture the geometry of divergent flow over hillslopes. These simplifications are: (1) the use of eight discrete flow angles, and (2) all flow leaving any given pixel (or the area contributing flow to the pixel) is assumed to flow into a single downstream neighbor pixel. While these simplifications are valid in regions where the flow is convergent, it is possible to do better on hillslopes where the flow is divergent by using an algorithm that allows (1) continuous flow angles, and (2) flow partitioning to two or more neighbor pixels.

The D-Infinity algorithms proposed by Tarboton (1997) provide a pragmatic and approximate solution to this problem, but they do not solve it in a rigorous manner. That is, the rule for partioning mass between two neighbor pixels is somewhat arbitrary and not based on mass balance. These algorithms strike a compromise between the limitations of D8 flow and other problems that occur when flow to more than two neighbor pixels is permitted. They are available within RiverTools via the Extract → D-Infinity Grids dialog, and can be used to create a continuous angle flow grid, a slope grid, and a contributing area grid.

In RiverTools 3.0, a new method for partioning flow between neighbor pixels was introduced, called simply the Mass Flux Method, based on ideas from computational fluid dynamics. It has some basic elements in common with the DEMON algorithm that was proposed by Costa-Cabral and Burges (1994). This method treats each pixel as a control volume and uses a rigorous mass balance that

divides each pixel into four quarter-pixels,
computes a continuous flow angle for each, and then
computes total contributing area (TCA) and specific contributing area (SCA) using the actual fraction of flow that would pass through each of a pixel's four edges under a uniform rainrate.

The accuracy of this contributing area does not depend on whether rainrate is actually uniform or not, it comes from recognizing that this problem can be reformulated as an equivalent fluid dynamics problem that must obey conservation of mass. In this case, the total contributing area (TCA) is equivalent to a volumetric discharge of water, Q, divided by a constant rainrate, R, i.e. A = (Q/R). As with the D-Infinity method, flow from a given pixel (grid cell) will typically be partitioned between two neighbor pixels, except in the case of pixels that are single-pixel peaks or that lie on drainage divides. However, with the mass flux method, flow will only be to nearest neighbors, as opposed to one nearest and one diagonal neighbor as is the case with the D-infinity method. This is consistent with the Divergence Theorem, which is a foundational concept used in computational fluid dynamics. Any multiple-flow direction algorithm that sends flow to more than two downhill neighbors (except on peaks and ridges) or to diagonal neighbors is provably incorrect and subject to a problem called overdispersion.

Note that we are speaking in terms of flows and mass balance even though our objective is to compute contributing areas on divergent surfaces as accurately as possible. The new Mass Flux algorithms are available in RiverTools via the Extract → Mass Flux Grids dialog. In order to handle peaks and linear ridges as accurately as possible, grids based on quarter-pixels are created and saved (with QP for quarter-pixel in their name, along with their own RTI files) as an intermediate step. These have twice as many columns and rows as the original DEM, but are then integrated to create flow angle and area grids that have the same dimensions as the DEM. Continuous angle grids from the D-Infinity and Mass Flux methods look quite similar, but there can be significant differences between the area grids computed by the two methods as seen in the figure at the top of this page. Both the D-Infinity and Mass Flux algorithms use a D8 flow grid to resolve ambiguous flow situations in flats and pits.

While the D8, D-Infinity, and Mass Flux methods all give significantly different results on a divergent surface, as occurs near peaks and ridgelines, then all give similar values for the contributing areas at the outlets of larger, channelized watersheds.

The Carbondale story on our Stories Page describes how our mass flux method was instrumental in protecting the water supply for the town of Carbondale, Colorado.

More information on the D8 and D-Infinity algorithms can be found here.

Here are two images that compare the interactive Vector Zoom tools for both the D8 and Rivix Mass Flux methods.

References

Gruber, S. and S.D. Peckham (2009) Land-surface parameters and objects specific to hydrology, In: Hengl, T. and Reuter, H.I. (Eds), Geomorphometry: Concepts, Software and Applications, Chapter 7, Developments in Soil Science, vol. 33, Elsevier, 171-194, http://dx.doi.org/10.1016/S0166-2481(08)00007-X.

Peckham, S.D. (2013) Mathematical surfaces for which specific and total contributing area can be computed: Testing contributing area algorithms, Proceedings of Geomorphometry 2013, Nanjing, China, pp. O-11-1 to O-11-4, http://geomorphometry.org/Peckham2013, October 7.

Peckham, S.D. (2011) Monkey, starfish and octopus saddles, Proceedings of Geomorphometry 2011, Redlands, CA, pp. 31-34, http://geomorphometry.org/Peckham2011b.

You can download some of these and other papers from our Downloads Page.