What Makes a Mountain Measure?

Continuing the meta-examination of mountain measurements in AscenDB, this post will look at the definitions of the measurements themselves. We will compare them with one another, determining some benefits of certain measurements while pointing out their downsides when used as a method of ranking a peak's "worthiness".

Elevation

This is the most well-known of the mountain measures, and thus the most likely to give someone an intuition of the peak's climbing difficulty. This is especially true when considering the added difficulty and danger of high altitude.

Elevation is also what I am calling an independent measure. That is, if a peak suddenly self-destructs (like Mount St. Helens or Mount Mazama), the elevation of nearby peaks doesn't necessarily change. This is not the case with some other measures, such as prominence or isolation, in which the self-destruction of a highly prominent peak like Mount Mazama would likely promote the prominence of a nearby peak.

And while elevation is certainly intuitive, it does require some knowledge of the local terrain to get a good idea of impressiveness. The average height of the Tibetan plateau is around 14,800 feet, taller than most mountains in the lower 48 of the US, but a hill on the plateau is not likely to be as iconic or sought after as Mount Elbert.

Elevation is also dependent on sea level, or more correctly a model of the sea level across the Earth commonly called a reference geoid (AscenDB uses both the WGS84 and EGM2008 geoids for various calculations). With the advent of global warming, one wonders if the official height of the world's peaks will shrink as the sea level rises!

Prominence

The runner-up for popularity among climbers and mountaineers, prominence is simultaneously a less and more intuitive way to rank peaks. Less, because the technical definitions is tricky to explain and understand. More, because the idea that motivates it (summit height compared to nearby terrain) is easy to understand and helps give an idea of why the measure is valuable as opposed to plain elevation: it attempts to more accurately capture the idea of ‘peak impressiveness’.

Here’s my favorite current way to describe prominence: if you’re standing on a peak, then that peak’s prominence is the minimum amount of distance that you must descend before climbing to a higher peak (this definition assumes that all routes are climbable). In the case of Everest, the prominence is simply its elevation because there is no peak higher.

Mountains with high prominence are often exceptional viewpoints, while many high elevation peaks are simply knolls on the way to yet higher peaks (although this is sometimes impressive in its own right). And while the highest peak in an area is usually also the most prominent, climbers can prioritize different peaks in a mountain range with prominence and get both a more varied and potentially more impressive experience. This is because all the high elevation peaks in an area are usually clustered together, while the high prominence peaks are usually more spread out.

As mentioned above, prominence is a dependent measure, and a tricky one to calculate at that. For the most prominent peaks, finding the ‘key col’ becomes a matter of searching the globe, and there is some question as to whether man made structures (such as the Panama Canal) affect the resulting prominence calculations. The automated algorithms for calculating prominence are quite involved, and there are probably many improvements that could be made.

Despite the difficulty in calculating prominence, it is a worthy measure and has gained a lot of traction in the climbing and mountaineering communities.

Isolation

A peak’s isolation is the distance from the summit to the closest higher (or equally high) ground. This measure has the benefit of being easier to define and comprehend than prominence, while also being more straightforward to calculate. It is also a good way to judge peak impressiveness: a highly isolated peak should have quite a vista in every direction from its summit.

However, isolation is once again a dependent measure, so changes in the surrounding terrain can cause the peak rankings by isolation to shift dramatically. And while the method of determining isolation is obvious, it is also tedious and can take quite a lot of time to determine, both by hand and on the computer, when calculating for highly isolated peaks.

Geocentric Distance

One of the lesser known mountain measures is geocentric distance, here defined as the distance of a point on the Earth’s surface from the center of the planet. It’s a lot like elevation, but instead of using sea level, which can change over geologic time, it is relative to the Earth’s center, which really isn’t going anywhere. This is another independent measure, and not as subject to the whims of the sea as elevation. It is also quite straightforward to compute, provided you have a recent geoid dataset handy.

There are some downsides to geocentric distance as a measure. Firstly, the values are not intuitive. Most of the significant digits are contributed by the radius of the Earth, not by the relative height of the mountain. Furthermore, because the Earth is noticeably wider around the equator than between the poles, the measure heavily favors equatorial peaks. This is both interesting and limiting: the highest peak on Earth, according to geocentric distance, is Chimborazo, and the next highest peaks are mostly clustered in the Andes. Elevation has a similar problem, with many of the world’s highest mountains clustered in central Asia, but prominence and isolation can bring a greater variety of locales to the forefront.

Nonetheless, geocentric distance is a good ranking method to bring to the fore peaks which may often be overlooked.

Omnidirectional Relief and Steepness

The least known of the mountain measures covered here is also the most expensive to calculate. Omnidirectional Relief and Steepness (ORS) was first formulated by Edward Earl and David Metzler, and is intended to give a more accurate assessment of a peak’s overall ‘impressiveness’. It does this by integrating over the slopes from the summit to many nearby points, in every direction. There are several variants of this measure, some formulated as purely dependent measures. However, because it is based only on elevation values, ORS itself can be considered an independent measure. That is, changes in elevation outside a certain distance from the peak will not noticeably affect the peak’s ORS.

Honestly, this overview is does not do the measure full justice, and I wish I could. However, I think it will be much more motivating to read about it straight from the inventors’ work. For those interested in implementing it, I have made public my own Racket-based ORS module as an example.

While ORS is a great way to bring to the forefront peaks that no other measure would rank highly, it is definitely tricky to grasp intuitively, and the resulting numbers are harder to differentiate. It is also a tricky algorithm to implement, and works best with large amounts of high quality elevation data from areas that are stereotypically difficult to map from a satellite. The algorithm is also heavily compute intensive, though the concern about number-crunching time is certainly lower these days.

More?

I chose to cover the measures above because they all illustrate a different aspect of mountain ranking. Notably, they are all well-defined and not based on arbitrary human judgement. Even the smallest hill has a non-zero ORS value, though it may not be as impressive as others. And what’s more, they give a variety of different ways to help prioritize the peaks that are import in your mountain climbing journeys. High-elevation peaks are the most commonly pursued goals, but they need not be the only or the most worthy. That’s partially why AscenDB includes all of these different measures for many peaks.

There are other measures to consider if you’re interested in exploring further. For instance, prominence, isolation, and ORS can all be defined in terms of geocentric distance rather than elevation. Such definitions could give sea mounts some much-needed attention.

Until next time, I’ll see you at the summit!