Local Topographic Position: Examining Terrain With Elevation Residuals

I can still recall the moment that I first got my copy of John Wilson and John Gallant’s now seminal book, Terrain Analysis: Principles and Applications, back when I was a PhD student nearly twenty years ago. (John Wilson has recently released the new book on terrain modelling and it’s even better than the first!) This book had such a tremendous impact on my career. It inspired me to work in the field of geomorphometry–the discipline that focuses on extracting information from topographic information, usually in the form of digital elevation models (DEMs), to better understand landscape processes.

I read every page of that book multiple times, but none more-so than Chapter 3. In this chapter Gallant and Wilson masterfully layout the various classes of what would eventually become known as land-surface parameters, that is, DEM-derived topographic attributes. There’s a dark corner of that chapter, Section 3.2 Elevation Residual Analysis, that I think doesn’t get enough attention. It comes after the much more attention grabbing sections on morphological parameters, like slope and curvature, and the flow parameters, like catchment area. But elevation residuals are so very cool! Elevation residuals are metrics for estimating local topographic position, or the height of a location relative to its local surroundings. Some examples of these metrics include difference from mean elevation (DIFF), deviation from mean elevation (DEV), and elevation percentile (EP). Many of the so-called ruggedness indices are really just variations of these elevation residuals.

What’s the deal with local topographic position and elevation residuals?

So, what is it that’s so wonderful about these simple elevation residual parameters? They’re scale dependent. And I don’t mean that they’re scale sensitive, like say, slope and curvature. Slope and curvature are well known to be sensitive to the resolution of a DEM, but this is an artifact of the way that they are calculated, using 3×3 windows. By definition, slope and curvature are expressed for a point on the surface, not an area.

The elevation residuals are different. They are defined for a point, but are expressed relative to a local area, or neighbourhood, which determines the scale of the analysis. The scale variation is actual information, and not just a limitation of the way they are measured! The scaling of these metrics provides a truly interesting way of carving out the many overlapping scales of natural variability that make up the topography of a landscape. When you, for example, calculate DEV from a DEM, you need to specify a neighbourhood size, and this will result in emphasizing topographic variability that matches that scale while de-emphasizing all other scales above or below the test scale.

Take a look at the image below. It shows the pattern of DEV derived from a shuttle-radar topography mission (SRTM) DEM of a section of southwestern Ontario and the northern US. Here, I’ve used a neighbourhood size of about 28 km to calculate this DEV map. You can see that there are certain topographic features that are emphasized as either locally low-lying (negative values in blue) or locally elevated (positive values in red) at this scale. Features, like bedrock folds, eroded river valleys, and depositional moraines, are all evident in this image.

A measure of local topographic position. certain topographic features that are emphasized as either locally low-lying (negative values in blue) or locally elevated (positive values in red) at this scale. Features, like bedrock folds, eroded river valleys, and depositional moraines, are all evident in this image.

The TopographicPositionAnimation tool for visualization a DEV scale stack

But what if you wanted to use this approach to visualize all of the scales of topographic variability? Essentially you’d be creating a ‘scale stack‘ of elevation residual images. That can be quite a tricky thing to calculate and to work with. Fortunately, Whitebox Geospatial has just the tool to handle this situation–the TopographicPositionAnimation tool, which is part of the General Toolset Extension for WhiteboxTools. This tool will efficiently calculate DEV across a defined scale range and then visualizes that scale stack as an animation. For example, take a look at the GIF below.

This is the Whitebox Geospatial, Topographic Position Animation tool. This tool calculates deviation from mean across a defined scale range and then visualizes that scale stack as an animation.

In reality, the TopographicPositionAnimation tool outputs its GIF embedded in HTML, Javascript and CSS in a way that allows viewers to zoom into the image and pan around. See here for an example.

Maximum elevation deviation

In 2015, I introduced the concept of maximum elevation deviation, DEVmax. DEVmax collapses a DEV scale stack into a single raster, where each pixel is assigned the maximum absolute DEV value for that pixel within the scale range. Therefore, DEVmax is an example of a locally adaptive, scale-optimized topographic attribute. That is, rather than depicting the pattern of local topographic position at a single homogeneous scale, DEVmax displays the pattern of local topographic position at a heterogeneous scale that is optimized for each pixel in the input DEM. By optimized, we mean that it is the DEV value at the scale for which that site is most elevated or low-lying within the landscape. This has a really interesting effect of capturing a maximum amount of topographic information within a single land surface parameter. For example, the animation below shows the pattern of DEVmax for the same SRTM DEM.

This is the Whitebox Geospatial, Topographic Position Animation tool. This tool calculates maximum deviation from mean across a defined scale range and then visualizes that scale stack as an animation. This is the multiscale equivalent to deviation from mean.

Notice how, as the pattern begins to stabilize at the largest tested scales, the image simultaneously shows fine-scale topographic features along with the larger-scale features. Every pixel is represented at the appropriate scale, that is, the scale at which it is most outstanding in the landscape. Notice, that the TopographicPositionAnimation tool has an option to output a DEVmax animation as well as the homogeneous scale DEV pattern.

The concept of the scale heterogeneous, locally scale optimized land surface parameter.
The concept of the scale heterogeneous, locally scale optimized land surface parameter.

DEVmax is also the secret behind the now famous multi-scale topographic position colour composite images (MTPCC) that came out of that 2015 Geomorphology paper of mine. You can see these images plastered all around the Whitebox Geospatial Inc. website. They are certainly among the visualization for which I have received the most user questions, because they are so stunning.

DEM DTM Multiscale Local Topographic Position

I hope that you’ve enjoyed my tangent into the realm of the often-overlook Elevation Residual Analysis. The usefulness of this set of land surface parameters for visualizing and understanding terrains is truly remarkable. Whenever I am presented with a new DEM of a landscape that I am unfamiliar with, these are the attributes that I first extract from the DEM as I build my understanding of the landscape’s geomorphology and surficial geology. If you are using these tools yourself in interesting applications, we’d love to hear about it. Either contact us here or send us a tweet at @whiteboxgeo. Questions and comments are also welcome in Google Groups forum.

In closing, I’ll leave you with this beautiful full-scale MTPCC map. I can’t tell you how many hours I’ve spent zoomed in, lost in the glorious topographic details of this map! Right-click and select ‘open in a new tab‘ and then zoom in and see for yourself. You will no-doubt curse me for the lost time that you spend gazing at the image!

Multiscale topographic Position of Eastern North America
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