Results
The resulting slope-area plots are below. I added information deduced from the data onto the plots wherever possible. This is all done by judgment because I did not have the time to do this objectively using mathematical techniques. For this analysis I compare the slope-area plots created from my data with reference to a theoretical plot from Tucker and Bras (1998) and two plots from Brardinoni and Hassan (2006) (Figure 8 and Figures 9 and 10, respectively).
Figure 8. The typical shape of a slope-area plot for an environment in which landsliding is governed mainly by pore pressure, which would apply to my study areas (from Tucker and Bras, 1998).
Figure 9. A slope-area plot of Hesketh Creek (a creek that flows down the longitudinal axis of a glacial valley). The different process domains are denoted by C = Colluvial, F = Fluvial, HF = Hanging Fluvial (from Brardinoni and Hassan, 2006).
Figure 10. A slope-area plot of a debris flow channel following the transverse axis of a glacial valley. There is only one process domain. The dashed line indicates the boundary between two values for index of concavity (i.e. where θ must be changed in equation (1) to fit the curve) (from Brardinoni and Hassan, 2006).
The thick black line shows the slope-area relationship. It is the upper slope boundary of the data points. Any slope falling above this line would fail through the dominant process of that domain. Each kink in the line is the boundary between two areas with different dominating geomorphic processes. The fluvial part of this line follows the ‘RS’ line in Tucker and Bras’ plot above (Figure 8). I tried to create a rough line of best fit (drawn by hand and not statistically calculated) for the fluvial areas if there seemed to be a trend. With better data, the points would generally approach this line.
The thin black diagonal line in the top left of the plot is the diffusion equilibrium line (‘DS’ in Figure 8). This line represents the relationship between slope and area at which diffusional erosion is equal to uplift. Due to the logarithmic scale, the upward slope of the line indicates this area of the hillslope is convex, so slope increases as area increases (Tucker and Bras, 1998).
The thin black diagonal line nearer the middle is the saturation threshold line (‘SAT’ in Figure 8), which divides the plot in two. All points to the left of the line are from areas where soil is unsaturated. The colluvial domain (C) is where pore pressure induced landsliding occurs. It has an upper boundary that curves downwards. This curve is due to an increase in soil saturation (as contributing area increases), causing an increase in pore pressure, and a decrease in slope stability, thus lowering the maximum stable slope. All points to the right of the line are from areas where the soil is saturated. This is because a stream channel has been formed as there is enough water to create overland flow. The channel head therefore occurs where the slope-area line is intercepted by the saturation line. The slope-area line then becomes horizontal, which signifies a straight slope (i.e. no change in slope angle). This is where debris flows occur. Where the line turns downwards is the edge of the debris flow fan and the beginning of the fluvial domain (Tucker and Bras, 1998; Eaton, 406 lectures).
I have drawn in these lines wherever there is a relatively obvious trend. Many of the plots do not show an obvious debris flow or fluvial line. In these cases I grouped the two domains together as ‘fluvial.’
The vertical dashed lines are boundaries between geomorphic process domains. As well as at kinks in the slope-area line, in glaciated environments this can occur at the edge of a hanging valley. The edges of hanging valleys can be identified by large horizontal gaps in the plotted data. These gaps indicate a sudden increase in contributing drainage area, which occurs when a stream channel drops from a hanging valley into another valley.
The different process domains are denoted H = Hillslope, C = Colluvial, DF = Debris Flow, and F = Fluvial. An H with a subscript number denotes a hanging valley and is followed by a domain descriptor (for example H2C/F = colluvial and fluvial domains in the second hanging valley). MV denotes where the channel has entered a main valley (i.e. a valley much larger than previous flow).
Hillslope/Stream Channel Plots:
Figure 11. Slope-area plots of the two stream channels and their channel head areas in the black basins in Figures 4 (BC) and 5 (WA), respectively. The red points are data from the hillslope area above the stream channel heads and the blue points are data from the stream channels.
There is a visual difference between the two hillslope areas; the Washington hillslope area has a fairly obvious diffusion equilibrium line and therefore a diffusive process domain. In contrast, the BC hillslope area does not. This is probably because the BC hillslope is in actual fact a rocky cirque, not a convex hillslope typical of the Washington area. The steep drop in the BC hillslope data probably indicates the steepest cliffy part of the cirque.
The most obvious difference between the two graphs is the
stream
channel area. I specifically chose a stream channel with two drops at
the end of hanging valleys to see their effect on the plots. These two
drops are indicated by the sudden large increases in area at the end of
H1F and H2C/F.
Figure 12 shows these boundaries on a map of the drainage basin. Such
large gaps do not occur in the Washington plot, however there are two
smaller gaps (at around 30 km2 and 60 km2), which are caused by the
joining of tributaries into the stream in question. It is very
difficult to analyze the domains within either stream channel area as
there is a lot of scatter.
The BC stream channel plot shows some resemblance to Brardinoni and Hassan’s plot (Figure 9), however it is difficult to determine where the colluvial and fluvial boundaries are. There is much less slope-value scatter in Figure 9 than in Figure 11. This is probably because of the better measurement techniques used by Brardinoni and Hassan (2006), including field measurements and a higher resolution DEM.
Seemingly the only similarity between the two plots in Figure
11 is
that both channel heads start at about 0.1 km2. The fact that they are
so different emphasizes the impact of glacial processes on the
landscape.
Washington Basins and Large BC Basins:
The plots of Washington basins in Figure 13 (next page) show much more obvious process domains than those in Figure 11. All except for Basin #4 (WA) show a hillslope process domain, however there is one point above the diffusion line in Basin #3 (WA) and the slope of the data creating the diffusion line may just be due to the random sample. All four plots show obvious colluvial domains and two of them show obvious debris flow zones. The fluvial domains all have decreasing slope trends.
These are very clear slope-area plots in comparison to those of Large Basins #1-3 in BC (Figure 14, below). As explained in section 4 Error Analysis, the topographic anisotropy in glaciated landscapes is probably the cause of the indistinct domains in the BC plots. Only Large Basin #1 (BC) shows a distinct colluvial curve. The diffusion line in Large Basin #1 (BC) could be due to the random sampling, as there is a point above the curve, and as discussed earlier there may be no real hillslope, depending on how rocky and steep the cirque is. The hanging valleys are discernible, but had to be compared with the drainage basin maps to be sure. The only information discernible from Large Basin #3 (BC) was the saturation threshold line, however this may not be correct as there is no quantitative analysis to support it.
Figure 13. Slope-area plots of the four Washington drainage basins. These were created using random samples of 2,000 data points. The different process domains are quite distinct.
Figure 14. Slope-area plots of the large drainage basins numbered 1 to 3 in British Columbia, created with random samples of 2000 data points. These do not show very obvious domains. The plot for Large Basin #4 was very scattered and no analysis could be made.
Figure 15. Slope-area plots of the six transverse drainage basins in BC. Transverse Basins #1, #5, and #6 (on the left) are all somewhat similar. Transverse Basins #2, #3, and #4 (on the right) have much more scattered slope values.
Transverse BC Basins:
Figure 15 shows the slope-area plots of six sidewall stream basins. These were intended to help demonstrate the anisotropy in BC, but turned out looking more similar to other slope-area plots than to debris flow plots, such as Figure 10. They have roughly the same area (0.1 to 1 km2) as Brardinoni and Hassan’s debris flow channels, but show obvious colluvial and fluvial domains. I think the main reason these domains show up more obviously is that I am using data of the entire drainage basin so it is also taking into account the sides of these small basins. It looks like the ends of the channels enter the main valley, and when removing the MV zone of data from the analysis they look more similar to Figure 10. Transverse Basins #1, #5, and #6 have little variation in slope compared to the large scatter of Transverse Basins #2, #3, and #4. They are also steeper than the latter three and more similar to the plots from Brardinoni and Hassan (2006). Brardinoni and Hassan’s plots were of data from the Capilano River area near Vancouver, which receives more precipitation than the Hurley Road area. With steeper slopes and less precipitation in my study area basins, the erosional process over the past 14,000 years since the last glaciation has probably been slightly different than the areas looked at by Brardinoni and Hassan (2006). This could contribute to the difference in appearance of the slope-area plots.
Comparing the Transverse Basin plots to Large Basins #1, #2, and #3 (Figure 14) gives little information about anisotropy, as the former would all fall within the scatter of the Large Basin plots.
Error Analysis
DEM Resolution
Brardinoni and Hassan (2006) used both field and GIS analyses in their study, noting that field measurements gave much more accurate slope-area plots and that the 25-metre resolution DEM available for coastal BC is not a good substitute for field data. For the areas in my study, only 30-metre resolution DEMs were available. These are much less accurate than even 25-metre resolution DEMs and are probably the greatest cause for imprecision in the slope-area plots.
Void Areas
The high relief areas in the USGS elevation dataset have many small areas with missing elevation data. I downloaded three separate Washington DEMs in different formats but they all contained these void areas. The void areas are given no elevation value and drastically alter the flow direction and contributing area for stream channels. Each pixel without an elevation value is 900 m2 (30 x 30) of missing contribution area. Some of the void areas contain 20 or more pixels, which is an extremely large area to be omitting for any slope-area plot. Therefore, I specifically avoided drainage basins with void areas for my analysis.
Flow Direction Algorithm
The flow direction algorithm in ArcMap is a D8 single-flow algorithm. D8 means that flow is calculated to be in one of 8 directions (i.e. to one of the 8 pixels surrounding the pixel in question). A D-inf flow algorithm would be better as it would calculate flow in any direction. Single-flow algorithms calculate all flow from a pixel to be in the single dominant flow direction. A multiple-flow algorithm would create more realistic water flow estimation (such as divergent flow), especially for hillslopes and ridges, where there is no obvious stream channel. However, Brardinoni and Hassan (2006) found that the discrepancies between a D8 single-flow algorithm and a D-inf multiple-flow algorithm falls within the inherent uncertainty due to a 25-metre resolution DEM. The 30-metre resolution DEM I used would be even less accurate. The flow algorithm inaccuracies would only be a real problem for elevation data of much higher quality.
There was another problem with the potential to affect this
analysis. The flow direction algorithm does not seem to work very well
with large areas of low slope angles, as it creates unrealistic
parallel linear ‘streams’ that are visible after using the flow
accumulation algorithm to show stream channels. These can be seen in
Figure 16. Although this problem was minimal in the smaller valleys
used for the analysis, it probably contributed to some of the scatter
in the slope-area plots.
Flow Accumulation Algorithm
This algorithm calculates a value for each pixel, which is equal to the number of pixels whose flow directions end up within the pixel in question. A value of 10 means that water flow from 10 pixels reaches the pixel in question. To get the area in m2 these values must be multiplied by 900.
There are two problems with the flow accumulation algorithm. The first is that to create a stream channel layer, you have to omit all pixels with less than a accumulation, as stream channels are by definition areas that have enough flow accumulation (the value of a) to create a stream. The value for a must be rather high otherwise ArcMap will show stream channels where they do not actually exist. However, such high values then remove the upper part of the stream channels and the contributing hillslope areas above the stream channel head. This omitted upper part of the drainage basin is the most important zone for slope-area plots, as there are multiple process domains. Therefore, I tried combining a stream channel and a small drainage area above it to get data for all of the processes. This is very time consuming in ArcMap so I only created two of these.
Watershed Algorithm
The watershed algorithm has a bizarre problem where it omits sections of some larger watersheds (Figure 16). For this reason I was unable to make slope-area plots for very long stream channels, as much of the contributing area would be omitted.
Drainage Basins
It took a very long time to create the data for only two stream channels in ArcMap. For this reason I made most of the slope-area plots using data from an entire drainage basin instead of a line following a stream channel. For the BC drainage basins this is problematic because of the glacially forced anisotropy. Although some analysis could still be done, the plots were not very informative (Figure 14). For Washington this was not as problematic because there is no anisotropy and the plots still showed the expected domains (Figure 13).
Figure
16. The
Watershed Tool cut off the green and purple watersheds part way up the
stream channels. The boxes show the major areas of the watersheds that
were cut out. This occurred mainly for larger watersheds but seemed
semi-random. Also of note is the inaccuracy of water flow direction in
areas of low slope angle (the blue at the top of the picture). These
flow direction problems were less evident for smaller valleys.
Using a higher resolution DEM and a dedicated hydrology
toolset for
ArcGIS or a more hydrology– or geomorphology–oriented program would be
more efficient and give better results.
Background image from: <http://originalbooner.wordpress.com/2011/10/11/backpacking-to-tenquille-lake-2/>