How Jut Works

What makes a mountain impressive? While impressiveness is inherently subjective, two factors immediately come to mind when we picture an impressive mountain: height above surroundings and steepness.

Elevation itself does not reveal much about a mountain’s rises above its surroundings. For instance, even though Mount Elbert in Colorado (elevation: 4399 m) has a higher elevation than Grand Teton in Wyoming (elevation: 4198 m), the latter is a far more dramatic peak, as it not only rises a greater height above its surroundings, but is also much steeper. For an even more extreme example, a tiny mound on the Tibetan plateau could have a higher elevation than a massive mountain that rises from close to sea level such as Mount Rainier.

John Sullivan, PD Photo

Grand Teton, Wyoming

Elevation: 4198 m  Jut: 1055 m
Rises 1825 m at angle of 35.3° above base.

Hogs555, Wikimedia Commons

Mount Elbert, Colorado

Elevation: 4401 m  Jut: 430 m
Rises 1284 m at angle of 19.6° above base.

This is where jut comes into play. Jut is an indicator of how sharply or impressively a mountain rises above its surroundings, factoring in both height and steepness. We find that the jut of Grand Teton reaches a staggering 1055 meters, surpassing Mount Elbert’s jut of 430 meters by over twofold.

Jut also allows us to define the base of a mountain in an objective way, providing objective definitions of base-to-peak height and base-to-peak steepness. For instance, the jut of Grand Teton corresponds to a rise of 1825 m at an angle of 35.3° above its base, whereas the jut of Mount Elbert corresponds to a lesser rise of 1284 m at a more gradual angle of 19.6° above its base.

So how does jut work?

Let P be the summit (highest point) of a mountain. Imagine you’re standing at some point Q on the planetary surface, looking towards P. What determines how impressively P rises above your location at Q? Obviously, the height of P above Q is an important factor. But it’s not the only factor. For two points that rise a similar height above you, the point that is closer to you—the one that you have to crane your neck more to see—will appear more impressive due to a greater angle of elevation.

Let h denote the height of P above Q, and let θ denote the angle of elevation of P above Q. Our goal is to create a function that takes in both h and θ and outputs a single number describing how impressively P rises above Q.

Let’s construct our function as such: When θ = 90°, akin to looking up a vertical cliff, the output of this function should simply be equal to h. However, the lower the value of θ, the lower the output should be as a proportion of h, as a penalty for lesser impressiveness due to a lower angle of elevation. In other words, when θ = 90°, the output should be h multiplied by 1. However, the lower the value of θ, the lower the number that h should be multiplied by. When θ = 0°, h should be multiplied by 0. What gives you 1 when you put in 90°, gives you 0 when you put in 0°, and gives you something in-between 0 and 1 when you put in something in-between 0° and 90°? That is none other than the sine of θ, or sin(θ).

Therefore, the function that describes how impressively P rises above Q is h × |sin θ|. (The absolute value merely ensures that if P is below Q, impressiveness is negative.)

Let P be the summit of Grand Teton. How impressively does P rise above Q? That depends on where Q is. If you’re standing at the summit of nearby Mt. Owen, the summit of Grand Teton has a low impressiveness of only 113 m, as Grand Teton doesn’t rise significantly above you. If you’re standing at Jenny Lake at the bottom of the Teton Range, Grand Teton has a much greater impressiveness of 670 m. If you’re standing further away at the city of Jackson, Grand Teton only has an impressiveness of 167 m due to a low angle of elevation.

That raises a question: where should you stand to maximize the impressiveness of summit of Grand Teton? If you’re standing too close, h would be low, making impressiveness low. But if you’re standing too far, θ would be low, making impressiveness low as well.

It turns out that this magic location that maximizes the impressiveness of the summit of Grand Teton is located at the bottom of Cascade Canyon, a neighboring valley just north of Grand Teton. This location is known as the base of Grand Teton. The impressiveness of the summit of Grand Teton as measured from its base, in this case equal to 1055 m, is called the jut of Grand Teton. (For those who like data visualizations, here’s a graph of the impressiveness of Grand Teton as measured from different places in its surroundings.)

Google Earth

More formally, the jut of point P is the maximum possible impressiveness of P as measured from any point on the planetary surface. A mountain with a jut of X rises as sharply/impressively as a vertical cliff of height X. So for instance, a vertical cliff of height 100 m, a 45° slope of height 141 m, and a 30° slope of height 200 m would all measure a jut of 100 m and be considered equally impressive. For math-inclined folks, here’s a geometric depiction of jut as a length.

The point on the planetary surface that maximizes the impressiveness of P is called the base of P. A mountain’s base is usually located at the bottom of a major mountain face or in a neighboring valley. The base can be thought of as a mountain’s most impressive viewpoint.

Jut of Well-Known Mountains

Maps

In the contiguous U.S., the Appalachians are relatively tame as a result of old age, with all peaks measuring below 500 m of jut. The Colorado Rockies have a lower jut than more northerly sections of the Rockies (i.e., Grand Teton National Park, Glacier National Park, Canadian Rockies), as the peaks in Colorado rise less steeply from a high plateau. Summits along the Eastern Sierra of California measure jut values between 700 m and 1000 m. Places with jut exceeding 1000 m include the North Cascades, Mt. Rainier (jut = 1332 m at Point Success; greatest in the contiguous U.S.), Glacier National Park, Grand Teton National Park, Yosemite National Park, and Mt. San Jacinto.

In the rest of North America, the Canadian Rockies measure a significantly greater jut than their American counterparts, with several summits measuring over 1500 m of jut and Mt. Robson (jut: 1915 m) measuring the greatest jut in the Rockies. Mountains in Alaska and Northwest Canada have an even greater jut, measuring the greatest values outside Asia. The greatest jut in North America belongs to the North Peak of Denali (jut: 2548 m), with Mount St. Elias and Mt. Logan following close behind. In general, jut-to-elevation ratio increases with greater proximity to the poles, likely due to glaciers carving steeper mountains and deeper valleys. This phenomena is also evident in the Andes, where peaks with over 1500 m of jut are scattered throughout the range, with the Southern Andes measuring a comparable jut to the Northern and Central Andes despite having only about half the elevation. The greatest jut in South America belongs to Padreyoc, a lesser-known summit next to the Apurimac Canyon in Peru, with jut possibly exceeding 2000 m by a slight amount. In Europe, the Alps measure a very similar jut as the Canadian Rockies, with a handful of summits exceeding 1500 m of jut. The greatest jut in the Alps belongs to the Jungfrau (jut: 1856 m).

Asia is home to numerous mountains exceeding 2000 m of jut. Even though Mt. Everest has the highest elevation, it doesn’t have the greatest jut, as its immediate base is already quite high up in the Himalaya. Meanwhile, some other Himalayan peaks rise from deep gorges that are much lower in elevation, making them more impressive. The most impressive peaks with over 2700 m of jut include Nanga Parbat, Dhaulagiri, Machapuchare, Gyala Peri, Rakaposhi, Haramosh, and a few summits on the Annapurna Massif. The greatest jut in the world belongs to Annapurna Fang (jut: 3395 m), the apex of the largest mountain face on Earth.

For a taste of what approximately 3000 m of jut feels like, check out this incredible photo sphere of Machapuchare.

Definition of Height and Angle of Elevation

Note that jut does not define the height of P above Q is as the elevation difference between P and Q. Instead, the height of P above Q refers to height above the horizon, described as follows:

Let the horizon of Q be a flat plane that passes through Q, to which the direction of gravity at Q is perpendicular to. The height of P above Q refers to how much P rises relative to this plane in the direction opposite of gravity at Q. For instance, the height of Mount Everest above the Dead Sea would be -1809 kilometers (that’s with a negative!), despite Mount Everest having a much higher elevation. That’s because if you’re standing at the Dead Sea, Mount Everest would be below your horizon due to planetary curvature.

Similarly, the angle of elevation of P above Q is defined as angle of elevation above the horizon of Q.

Why are height and angle of elevation defined as such? The reason is that on planets without a sea level, elevation is defined rather arbitrarily. Depending on how elevation is defined, elevation differences can differ. Furthermore, on irregularly shaped asteroids, the notion of distance on the planetary surface breaks down, making it futile to use a “flat earth approximation” when calculating angle of elevation. Hence, jut is based purely on gravity and the actual planetary surface, providing non-arbitrary measurements on any terrestrial planet or asteroid.

Definition of the Planetary Surface

Jut defines the planetary surface in a way that excludes cave walls and surfaces underneath overhangs. The planetary surface is defined as all parts of a planet that can be hit by falling raindrops (provided these raindrops are infinitesimally small and fell along the gravity field lines). Hence, cave walls and surfaces underneath overhangs are not included. This definition of the planetary surface aligns well with existing digital elevation models.

FAQ

Q: How does jut differ from prominence, an existing metric?

A: Prominence is actually a bit of a misnomer. A more suitable name for prominence is independence, as it determines whether a mountain rises independently enough to be considered the summit of a mountain, rather than being a subpeak or non-summit point.

For instance, consider Annapurna Fang, which measures a jut of 3395 m, the greatest in the world. In contrast, its prominence is only 267 m, as Annapurna Fang is a subpeak of the Annapurna Massif. Other examples of high-jut, low-prominence points include the nose of El Capitan or the rim of the Grand Canyon.

An example of a high-prominence, low-jut point is Dome Argus in Antarctica. Since it’s the highest point on the Antarctic ice cap, its prominence is a high value of 1639 m. However, the Antarctic ice cap gains its elevation so gradually that a person standing at Dome Argus would find their surroundings flatter than Kansas. Not surprisingly, the jut of Dome Argus is only 0.2 m.

Derek B, The Trek Blog

Annapurna Fang, Nepal

Jut: 3395 m  Prominence: 267 m

Stephen Hudson, Wikimedia Commons

Dome Argus, Antarctica

Jut: 0.2 m  Prominence: 1639 m

Q: Does jut account for viewshed obstruction? Is the summit always visible from the base?

A: No; the reason is that while jut is designed to capture the subjective quality of visual impressiveness, it also serves as an objective measure of local relief. Every jut value corresponds to a base-to-peak height and base-to-peak steepness; whether a rise is obstructed or not does not change the fact that a mountain rises a certain height at a certain angle. Furthermore, viewshed is a bit arbitrary, as it depends on how tall you are. If you’re an ant standing at the base of a mountain, even a tiny pebble can block your view. Therefore, jut ignores viewshed blockages entirely.

While the summit may not necessarily be visible from the base, there are usually places close to the base where the summit is visible. There are usually also places where both the summit and base are visible (such as from a nearby peak or an airplane), allowing one to view the full extent of a mountain’s local relief.

Q: Why jut? Why not base-to-peak height instead?

A: Imagine we have two mountains: Mountain A and Mountain B. Mountain A rises 1 km in 1 km of horizontal distance on one side, and rises 1.5 km in 5 km of horizontal distance on the other side. Mountain B rises 0.5 km in 5 km on one side, and 1.25 km in 5 km on the other side. We’d be inclined to say that mountain A has a base-to-peak height of 1 km, as even though the 1.5 km in 5 km rise is higher, the 1 km in 1 km rise is much steeper and more immediate. Looking at Mountain B in the same manner, we’d say it has a base-to-peak height of 1.25 km, outscoring Mountain A’s base-to-peak-height. But this statement is incredibly misleading, as Mountain A rises more than Mountain B on each of its respective sides.

Rather, if we wanted to measure a mountain’s rise above its base, we should not be using height as our metric, but instead the same criteria we use to select the base in the first place—a combination of height and steepness. And that’s exactly what jut does. Mountain A having a greater jut than Mountain B simply means that Mountain A has a more immediate rise that qualifies more for base status. Said differently, jut measures how based a mountain is.

Q: Could a non-summit location on a mountain have a greater jut than the summit?

A: Yes, and in fact this happens quite often! For instance, the cliff edge of El Capitan in Yosemite measures a greater jut than the “peak,” which is further away from the precipitous drop. Point Success on Mount Rainier measures a greater jut than the summit, as it is closer to the sharply rising Southwest Face. The challenge is identifying these places with a potentially greater jut, as they are not always marked on maps.

Q: How does the jut of mountains on Earth compare with mountains on other planets and asteroids?

A: Generally speaking, smaller planets have lower gravity, allowing taller mountains to form. On other planets, elevation fails to provide meaningful mountain measurements due to the lack of a sea level to base zero-elevation off of. However, jut works consistently on any terrestrial planet or asteroid.

Due to continual asteroid bombardments and lack of erosion, average jut values on the Moon are greater than those on Earth. Much of the lunar surface is covered by mountains comparable to the Andes or Himalayas in scale. The greatest jut on the Moon belongs to an unnamed summit approximately 200 kilometers northwest of the Lippmann Crater, with a jut of 2600 m.

Mars is a land of topographic extremes. The planet is home to Valles Marineris, a massive canyon with five times the jut of the Grand Canyon, with some locations measuring over 3500 m of jut. The massive volcano Olympus Mons has a jut of only 947 m, much lower than one would expect. This is because Olympus Mons rises so gradually that due to planetary curvature, its summit is close to the horizon of a person standing at its foot.

Due to their low gravity and irregular shapes, asteroids and other small rocky objects can have an even greater jut. On the asteroid Vesta, the rim of the Rheasilvia crater has a jut of over 10,000 m. On Miranda, a moon of Uranus, the cliff Verona Rupes has a jut of over 14,000 m.

Acknowledgements

Jut measurements are made in Google Earth Engine using the MERIT digital elevation model globally except Antarctica, where GLO-30 is used. Locations are provided by GeoNames.

Jut is inspired by the omnidirectional relief and steepness (ORS) / spire measure, developed by Edward Earl and David Metzler for a similar purpose as jut. Jut is designed to be easier to understand and use.

For more information about jut, as well as several other new metrics for measuring mountains on Earth and beyond, check out this research paper. If you find jut interesting, you might also be interested in on-top-of-the-world mountains.

Thank you to individuals on Reddit for contributing great feedback on making jut more accessible!