Topographic prominence

In topography, prominence, also known as autonomous height, relative height, shoulder drop (in North America), or prime factor (in Europe), is a concept used in the categorization of hills and mountains, also known as peaks. It is a measure of the independent stature of a summit; compare topographic profile.

Definitions
There are several equivalent definitions, which are satisfactory for all but Mount Everest:


 * The prominence of a peak is the height of the peak’s summit above the lowest contour line encircling it and no higher summit.
 * If the peak's prominence is P metres, to get from the summit to any higher terrain one must descend at least P metres. Note that this implies that the prominence of any island or continental highpoint is equal to its elevation above sea level. In this definition, Mount Everest is a special case: its prominence is considered to be equal to its elevation, in order to agree with the previous definition.
 * For every ridge (or path of any kind) connecting the peak to higher terrain, find the lowest point on the ridge. This will be at a col (also called a saddle point or pass). The key col (or key saddle, or linking col, or link) is defined as the highest of these cols, along all connecting ridges. (If the peak is the highest point on a landmass, the key col will be the ocean, and the prominence of the peak is equal to its elevation.) The prominence is the difference between the elevation of the peak and the elevation of the key col. See Figure 1 below.
 * Suppose that the sea level rises to the lowest level at which the peak becomes the highest point on an island. The prominence of that peak is the height of that island. The key col represents the last isthmus connecting the island to a higher island, just before they become disconnected.

key col

In mountaineering
Prominence is interesting to some mountaineers because it is an objective measurement that is strongly correlated with the subjective significance of a summit. Peaks with low prominences are either subsidiary tops of some higher summit or relatively insignificant independent summits. Peaks with high prominences tend to be the highest points around and are likely to have extraordinary views.

For example, the world's second highest mountain is K2 (height 8,611 m, prominence 4,017 m). While Mount Everest's South Summit (height 8,749 m, prominence about 10 m) is taller, it is a subsummit of the main summit. Only summits with a sufficient degree of prominence are regarded as independent mountains.

Many lists of mountains take topographic prominence as a criterion for inclusion, or cutoff. John and Anne Nuttall's The Mountains of England and Wales uses a cutoff of 15 m (about 50 ft), and Alan Dawson's list of Marilyns uses 150 m (about 500 ft). (Dawson's list and the term "Marilyn" are limited to the British Isles.) In the contiguous United States, the famous list of "fourteeners" (14,000 foot / 4268 m peaks) uses a cutoff of 300 ft / 91 m (with some exceptions). Also in the U.S., 2000 feet (610 m) of prominence has become an informal threshold that signifies that a peak has major stature. Lists with a high topographic prominence cutoff tend to favour isolated peaks or those that are the highest point of their massif; a low value, such as the Nuttalls', results in a list with many summits that may be viewed by some as insignificant.

While the use of prominence as a cutoff to form a list of peaks ranked by elevation is standard, and is the most common use of the concept, it is also possible to use prominence as a mountain measure in itself. This generates lists of peaks ranked by prominence, which are qualitatively different from lists ranked by elevation. Such lists tend to emphasize isolated high peaks, such as range or island high points and stratovolcanoes. One advantage of a prominence-ranked list is that it needs no cutoff, since a peak with high prominence is automatically an independent peak.

Parent peak
It is common to define a peak's parent as a particular peak in the higher terrain connected to the peak by the key col. If there are several higher peaks there are various ways of defining which one is the parent. These concepts give ways of putting all peaks on a landmass into a hierarchy, showing which peaks are subpeaks of which others. For example, in Figure 1, the middle peak is a subpeak of the right peak, which is in turn a subpeak of the left peak, which is the highest point on its landmass. In that example, there is no controversy over the hierarchy; in practice, there are different definitions of parent. These different definitions follow.

A special case occurs for the highest point on an oceanic island or continent. Some sources define no parent in this case; others treat Mount Everest as the parent of every such peak with the ocean as the "key col".

Encirclement or island parentage
Also called prominence island parentage, this is the most mathematically natural definition, and is defined as follows. The key col of peak A is at the meeting place of two closed contours, one encircling A and the other containing at least one higher peak. The encirclement parent of A is the highest peak that is inside this other contour. In terms of the rising-sea model, the two contours together bound an island, with two pieces connected by an isthmus at the key col. The encirclement parent is the highest point on this entire island.

For example, the encirclement parent of Mont Blanc, the highest peak in the Alps, is Mount Everest. Mont Blanc's key col is a piece of low ground near Lake Onega in northwestern Russia (at 113 m elevation), on the divide between lands draining into the Baltic and Caspian Seas. This is the meeting place of two 113 m contours, one of them encircling Mont Blanc; the other contour encircles Mount Everest. This example demonstrates that the encirclement parent can be very far away from the peak in question when the key col is low.

This means that, while simple to define, the encirclement parent often does not satisfy the intuitive requirement that the parent peak should be close to the child peak. For example, one common use of the concept of parent is to make clear the location of a peak. If we say that Peak A has Mont Blanc for a parent, we would expect to find Peak A somewhere close to Mont Blanc. This is not always the case for the various concepts of parent, and is least likely to be the case for encirclement parentage.

The encirclement parent is the highest possible parent for a peak; all other definitions pick out a (possibly different) peak on the combined island, a "closer" peak than the encirclement parent (if there is one), which is still "better" than the peak in question. The differences lie in what criteria are used to define "closer" and "better."

Prominence parentage
The (prominence) parent peak of peak A can be found by dividing the island or region in question into territories, by tracing the runoff from the key col of every peak that is more prominent than peak A. The parent is the peak whose territory peak A is in.

For hills with low prominence in Britain, a definition of 'parent Marilyn' is sometimes used to classify low hills. This is found by dividing the region of Britain in question into territories, one for each Marilyn. Once again, the parent Marilyn is the Marilyn whose territory the peak is in. Obviously, if a peak is the highest point of its island, it has no parent. Likewise, if a hill is on an island (in Britain) whose highest point is less than 150m, it has no parent Marilyn.

Prominence parentage is the only definition used in the British Isles because 'encirclement' parents break down when the key col approaches sea level. Using this definition, the parent of any low-lying bump next to the sea would be Ben Nevis - which could be said to be irrelevant and confusing. Similarly 'height' parentage is not used because there is no obvious standard for what the cutoff used should be.

This might seem arbitrary, but it gives a clear and unambiguous definition for the 'parent' of a mountain that is more significant than, connected to and reasonably close to it. It also enables one to make a 'hierarchy' of peaks going back to the highest point on the island. One such chain in the British Isles would read;

Billinge Hill → Winter Hill → Hail Storm Hill → Boulsworth Hill → Kinder Scout → Cross Fell → Helvellyn → Scafell Pike → Snowdon → Ben Nevis.

At each stage in the chain both the height and prominence are increasing.

Height parentage
Height parentage is a less widely used term. It is similar to prominence parentage, but it requires some sort of prominence cutoff criterion. The height parent is the closest peak to peak A (along all ridges connected to A) that has a greater height than A, and is above the prominence cutoff. For example, Mont Blanc's height-parent is either a minor peak in the north-west Caucasus (if the prominence cutoff is low), or Mount Elbrus (if the cutoff is high).

The disadvantage of this concept is that it goes against the intuition that a parent peak should always be more significant than its child. However it can be used to build an entire lineage for a peak which contains a great deal of information about the peak's position.

Other criteria
To choose among possible parents, instead of choosing the closest possible parent, it is possible to choose the one which requires the least descent along the ridge.

In general, the analysis of parents and lineages is intimately linked to studying the topology of watersheds. Further discussion of parents can be found in the Orometry article at peaklist.org.

Interesting prominence situations
The key col and parent peak are often close to the subpeak but this is not always the case, especially when the key col is relatively low. It is only with the advent of computer programs and geographical databases that thorough analysis has become possible.


 * The key col of Mount McKinley (also called Denali) in Alaska (6,194 m) is a 56 m col near Lake Nicaragua (unless one accepts the Panama Canal as a key col; this is a matter of contention). McKinley’s encirclement parent is Aconcagua (6,960 m), in Argentina, and its prominence is 6138 m. Put another way, to further illustrate the rising-sea model of prominence – if sea level rose 56 m North and South America would be separate continents and McKinley would be 6138 m above sea level. At a slightly lower level, the continents would still be connected, and the high point of the combined landmass would be Aconcagua, the encirclement parent. Note that, for the purposes of this article, man made structures such as the Panama Canal are not taken into account. If they were, the key col would be along the 26 m Gaillard Cut and McKinley would have a prominence of 6,168 m.

While it is natural for Aconcagua to be the parent of Mount McKinley, since Mount McKinley is a major peak, consider the following situation: Peak A is a small hill on the coast of Alaska, with elevation 100 m and key col 50 m. Then the encirclement parent of Peak A is also Aconcagua, even though there will be many peaks closer to Peak A which are much higher and more prominent than Peak A (for example, Mount McKinley). This illustrates the disadvantage in using the encirclement parent.


 * Mount Whitney (4421 m) has its key col 1022 km (635 miles) away in New Mexico at 1347 m on the Continental Divide. Its encirclement parent is Pico de Orizaba (5,636 m), the highest mountain in Mexico. Orizaba’s key col is back along the Divide, in British Columbia.
 * The key col for Mount Mitchell, the highest peak of the Appalachians, is in Chicago—the low point on the divide between the St. Lawrence and Mississippi River watersheds.

Calculations and mathematics
When the key col for a peak is close to the peak itself, prominence is easily computed by hand using a topographic map. However, when the key col is far away, or when one wants to calculate the prominence of many peaks at once, a computer is quite useful. Edward Earl has written a program called WinProm which can be used to make such calculations, based on a Digital Elevation Model. The underlying mathematical theory is called "Surface Network Modeling," and is closely related to Morse Theory.

A note about methodology: when using a topographic map to determine prominence, one often has to estimate the height of the key saddle (and sometimes, the height of the peak as well) based on the contour lines. Assume for simplicity that only the saddle elevation is uncertain. There are three simple choices: the pessimistic, or clean prominence, assumes that the saddle is as high as it can be, i.e. its elevation is that of the higher contour line nearest the saddle. This gives a lower bound on the possible prominence of the peak. Optimistic prominence assumes that the saddle is as low as possible, yielding an upper bound value for the prominence. Midrange or mean prominence uses the mean of these two values.

Which methodology is used depends on the person doing the calculation and on the use to which the prominence is put. For example, if one is making a list of all peaks with at least 2,000 ft (610 m) of prominence, one would usually use the optimistic prominence, to include all possible candidates (knowing that some of these could be dropped off the list by further, more accurate, measurements).

Wet prominence and dry prominence
There are actually two varieties of topographic prominence: wet prominence and dry prominence. Wet prominence is the topographic prominence discussed in this article. Wet prominence assumes that the surface of the earth includes all permanent water, snow, and ice features. Thus, the wet prominence of the highest summit of an ocean island or landmass is always equal to the summit's elevation.

Dry prominence, on the other hand, ignores water, snow, and ice features and assumes that the surface of the earth is defined by the solid bottom of those features. The dry prominence of a summit is equal to the wet prominence of that summit unless the summit is the highest point of a landmass or island, or a summit surrounded by snow or ice. If a summit is completely surrounded by a water, snow, or ice feature, the dry prominence of that summit is equal to the wet prominence plus the depth of the highest col.

The dry prominence of Mount Everest is, by convention, equal to its wet prominence (8850 m) plus the depth of the deepest hydrologic feature (the Challenger Deep at 10,911 m), or 19,761 m.  The dry prominence of Mauna Kea is equal to its wet prominence (4205 m) plus the depth of its highest col (about 5125 m), or about 9330 m; this is the world's second greatest dry prominence after Mount Everest. The dry prominence of Aconcagua is equal to its wet prominence (6962 m) plus the depth of the highest col of the Bering Strait (about 50 m), or about 7012 m.

Dry prominence is also useful for measuring submerged seamounts. Submerged summits have both a dry topographic prominence and a topographic isolation.

The following mental exercise may illustrate the meaning of topographic prominence. Imagine you are standing at the top of a mountain peak. Let an imaginary sea level slowly rise to the top of your peak. As the imaginary sea level rises, the mountain will shrink into an ever smaller island until the island disappears at your feet. Now let the imaginary sea level slowly drop. The island will reappear with your peak as its highest point. As the sea level continues to drop, the island will grow and merge with adjacent islands. At some point, your island will touch another island with a higher summit. This point is called your peak’s key col, and the elevation rise from its key col to your summit is called its topographic prominence.