Question that you asked

How much spatial autocorrelation exists in historic streambed changes in the Andrews Forest, both in the across-stream direction and the along-stream direction?

Name of the tool or approach that you used

I addressed each direction of the problems separately using. Since this resulted in one-dimensional, evenly spaced or faux-evenly spaced data, I just used the acf function in R to analyze the data.

Critique of the method – what was useful, what was not?

The methods provided an interesting look at patterns within the cross section data set. However, I think the one-dimensional approach fails to capture interactions between across- and along- channel changes as well as any temporal autocorrelation.

Description of steps you followed to complete the analysis and results you obtained.

For my project, I am looking at changes in the streambed along 4 stream reaches (black boxes) along two streams (blue lines) in the Andrews forest. The watershed of the larger stream, Lookout Creek is shown in black.

Researchers in the Andrews forest repeatedly surveyed fixed cross sections along the reaches from 1978 to 2011. In this exercise, I worked with a subset of the data from 1978 to 1998. The cartoon below does not contain real data but shows the arrangement of numbered reaches (blue lines) and their fixed endpoint stakes (blue dots) along a stream.

Part A – across-channel changes

Each individual cross section spans from bank to bank and show the topography and bathymetry across the width of the stream. Below are two consecutive years of cross-sectional profiles of the stream bed at Lower Lookout Creek, cross section # 3.

Profile from 1995.

Profile from 1996.

It looks like there was a lot of change between these two profiles. The active stream channel appears to have avulse from one side of the cross section to the other.

I compared the two profiles to see how much elevation was gained or lost.

Here grey line represents the surface of the stream bed in 1995, and the black line represents the surface of the stream bed in 1996. The area show in red represents material that was eroded away, and the green area represents material that was newly deposited.

We can simplify the plot above to show only the positive or negative elevation change across the channel. The figure below shows the same information as the figure above, but directly in terms of elevation change.

You can clearly see from this figure that this site at this year had one broad area of elevation gain or deposition, and one broad area of elevation loss or erosion. This isn’t precisely an attraction/repulsion phenomenon, but there can be positive feedbacks associated with erosion and deposition. For example, sediment deposited in a stream may locally obstruct or slow down the water passing by it and encourage deposition in adjacent areas.

Now, not every pair of years had this *much* elevation change, but also, not every pair of years shows such broad regions of positive or negative elevation change. Here is a figure of the elevation change at the same cross section between 1996 and 1997. You can see that elevation change is both smaller in magnitude and shows up in smaller patches.

I used spatial autocorrelation metrics to investigate the patch size. Note that autocorrelation at some scales is always expected when looking at topographic change. Even if you had extremely precise and accurate measurements, you would expect the stream bed change at one point to be strongly correlated with the change a few cm away both because the two points probably experience very similar forces and also because gravity tends to smooth out topographic irregularities in nature. But in most scenarios, you would probably not expect to find a correlation between the changes in a streambed with a change at a point 100 m away from the bank.

I picked two distances that seemed potentially interesting: 1 meter and 5 meters. Then I investigated autocorrelation at 1m and 5m lag distances using the acf function in R.

These lag distances are superimposed on the first change graph in the figure below.

I expected that all cross sections would have higher autocorrelation at the shorter lag distance than the longer one, but I also expected that a set of cross sections with very large patch sizes might show relatively higher autocorrelation at the larger lag distance compared to the small-patched cross sections. I need to spend more time looking into how the magnitude of the changes influences the autocorrelation function so that I can understand it better.

I repeated the process for every cross section at every year. Results are shown below.

The cross sections showed mostly positive autocorrelation at 1 m. Some years showed more autocorrelation across the board than others, and some individual cross sections seemed like they frequently had high autocorrelation values at multiple years.

Different patterns emerge at this spatial scale. At a 5 m lag, neutral or negative autocorrelation was much more common.

 

Part B – along-channel changes

I also wanted to look at autocorrelation between cross sections. Was the change at one cross section related to the change at cross sections immediately up- or down-stream?

Because I don’t have true geographic locations of the cross sections, I couldn’t do any fancy two dimensional work like interpolating the data into a river bathymetry model. Instead, I consolidated the cross-sectional change data into a single metric for each cross section.

Here is another look at the change plot I showed earlier. This is the same plot as before, but I’ve colored the positive values green to show deposition and the negative values red to show erosion.

I simplified this plot by removing the horizontal across-stream component and summing the red and green areas separately.

Then I had a choice: I could add the red and green areas together to show the total area of the cross section that changed in any way, or I could subtract the red area from the green area to show the net deposition in the cross section. Each of these metrics could be interesting for different reasons. For example, the total area of change might be relevant for ecological processes, but the net overal change could be relevant for tracking net sediment transport in these streams.

I started by adding the boxes together to calculate the total area of change.

In this way, I made a score for every cross section for every year pair. Hypothetical scores for the cartoon river are shown below.

Then I used the acf function again in R to determine how correlated adjacent cross section scores were with each other, compared to more distant cross section scores. This method is not fully spatial explicit, since I did not include any information about how far apart each cross section was from its neighbors. I just assumed that they were roughly equidistant within each reach. Indeed, the distance between cross sections can become hard to define when they are placed along curved parts of the stream.

I calculated these values with every year pair on each of the four reaches. I then repeated it with the subtraction method described above.

For the “addition” method, autocorrelation values were generally close to neutral with some reaches in some years tending towards positive values. Positive values could occur if changes in one reach directly influenced an adjacent reach (i.e. if upstream channel change influenced the hydraulics directly downstream) or if adjacent cross sections were influenced by similar other factors (i.e. two adjacent cross sections placed on a steep part of the reach might experience more similar changes to each other than two cross sections placed in a shallower part of the reach.)

The pattern is different for net change, where negative autocorrelation is more common. I think this is very interesting, because it implies that parts of the stream that had net deposition were more likely to be adjacent to parts of the stream that had net erosion. This could be an expression of a sink/source phenomenon where material scoured from one area is transported and deposited immediately downstream at the next cross section.