Topographic prominence?

I use this term so often, that I thought I should make a page for it.  Topographic prominence is a way of rating a peak relative to other terrain, by considering how high a peak stands above the saddles that connect it to higher terrain.  Imagine a land mass where there are only three mountain peaks as follows:
Assume the black bar at the bottom is sea level, and that in fact, this mountain range is an island in the ocean.  P1 has an elevation of 5,000 feet above sea level, P2 has an elevation of 2,500 feet, and P3 has an elevation of 7,000 feet.  As for the saddles, S13 is at an elevation of 1,000 feet and S23 is at an elevation of 1,500 feet.

Obviously P3 is the highest peak on this landmass, so its prominence is 7,000 feet, relative to sea level.  P2 connects to two higher peaks, but the key is that we use the highest saddle from which to count the prominence.  Thus, we have to use S23 to determine the prominence of P2, which is 2,500-1,500 = 1,000 feet.  Basically, every peak other than the highest peak on a landmass has a single “key saddle” that connects it a single higher peak.  Many peaks may connect to the same higher peak, but each of the lower peaks will have its own key saddle.  Because S23 is higher than S13, from P1’s perspective, P2 is part of the P2/P3 massif to which it connects via S13.  Thus, the prominence of P1 is 5,000-1,000 = 4,000 feet.  So, we can make the following table:

```Peak   Elev    Prom    Key saddle
P1    5,000   4,000     Ocean
P2    2,500   1,000      S23
P3    7,000   7,000      S13```

Thus, if you only want to climb peaks with more than, say, 5,000 feet of prominence, you only climb P3.  If you want to climb peaks with more than, say, 3,000 feet of prominence, you climb P1 and P3.  If your prominence criterion is 1,000 feet or less, you want to climb all three.

In reality, people often do choose a particular prominence threshold upon which to base their peakbagging.  The normal minimum threshold which has become widely accepted is 300 feet.  The Lists of John site contains every known peak down to this level for the lower 48 plus Hawai’i (Alaska is in the works), plus named peaks lower than that threshold.  However, many peakbaggers favor peaks with 1,000, 2,000, or even 5,000 feet of prominence.  (The latter are sometimes called ‘ultras’ and only a few people have climbed all 57 of them in the lower 48.)  A peak can have a high altitude, but small prominence if it is just a bump near a higher peak.  On the other hand, a relatively low peak can have a relatively high prominence if it is an island or rises out of very low terrain on all sides.