The basic idea of *domain relief and steepness* (DRS, or "ruggedness") is to calculate the average
ORS value of every
point in a region. However there are some subtleties to this process.
(Click the link above for explanations, data, and
acknowledgements regarding ORS.)

First, let's assume that we have a fixed region K for which we want to find the ruggedness. We can think of any point in the region as a reference point for ORS; intuitively, think of standing at that point and evaluating the impressiveness of the view based on steepness and local relief of the surrounding terrain. The simplest thing to do would be to take a simple average of all of the resulting ORS values. Instead it turns out to be better to take a root-mean-square (RMS) average of the ORS values at each point. The result is called the ruggedness R(K) of the region.

The resulting measure rewards both relief and steepness. In particular it is different from a commonly quoted measure of ruggedness, namely average slope. Average slope would reward a very low-relief but steep washboard landscape the same as it rewards a landscape with the same slope but much more relief. DRS rewards the landscape with higher relief more highly than the one with less relief. We consider this a very desirable feature.

So far so good, if we have a given region K. However if we just start with wanting to find "the ruggedness of the Alps" or "the ruggedness of the North Cascades" we have problem. Mountain ranges do not have precisely delineated borders or a well-defined size. So what exactly should we use for the region K, say, in the case of the Alps?

There are two separate issues we can identify. First, how big a region should we take, i.e. what area A should K have? We will postpone dealing with this until below. Second, once we choose a fixed A, exactly what are the right boundaries for K? For example, should it include a lot of the foothills or not?

Here is our strategy: always let a range show its best face for comparison. So what we do is the following, in the case of the Alps, to be specific. First, find the largest possible region anyone could think of as "the Alps." Then, given a choice of area A, find the most rugged region K with the specified area A which fits inside that huge starting region. That will be our version of "the Alps" at size A.

Now back to the first question, what size A to choose? This is where we have decided not to decide. We don't see a good way to nail down exactly how big a region to take, what outlying areas to include or exclude, or how far down into the foothills to go. So we have calculated the DRS value of various "versions" of each range, as a function of the area A.

Here is how this affects the DRS calculation: as the area A gets smaller, we are selecting a more and more rugged core region of the range, so the DRS value increases. So we do not get a well-defined single value for "the ruggedness of the Alps" or "the ruggedness of the North Cascades." But we do get an interval of DRS values, and a graph of DRS as a function of area A. This allows us to still make meaningful statements and comparisons, as we do on the main DRS page.

For example, consider the Sierra Nevada (in California) and the mountains of Colorado. Even the largest version of the Sierra Nevada (48700 km^2, DRS = 95m) beats the smallest version of the Colorado mountains (36200 km^2, DRS=79m). Also, the sizes are not completely different. Hence it makes sense to say that the Sierra Nevada are more rugged than the Colorado mountains.

In the case of the North Cascades and the West Chugach, it is harder to make a clear-cut comparison. The North Cascades come out a little ahead on a head-to-head basis, but the differences are within the margin of error for the calculation.

A word about units and accuracy: the units of DRS are the same as the units of height; we use meters. To get a feel for the meaning of the number, note that the ORS value of a point on top of a conical peak of height H and slope 1 (45 degrees) is equal to H/2. So for example, when I say that the DRS of the Alps is 177m, that means roughly that the "impressiveness" of the average point in the Alps is comparable to standing atop a conical peak of height 354 meters and slope 1. That's pretty rugged!

The accuracy of most of these DRS values is plus or minus 5 percent, sometimes up to 10 percent. Since the main use of the numbers is for rough comparison, this seems sufficient.