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Published on October 3, 2007

Author: Kestrel

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Detail of a Prominence Map:  Detail of a Prominence Map PROMINENCE MAPS SHOW THE: Location of Summits Relative elevations (Prominence) of Summits Ridge Networks that connect all mountains on a continent or island Saddles, Lineage Areas, other morphometric data can be included NACIS Presentation, 10/10/03. © Aaron Maizlish INTRODUCTION:  INTRODUCTION Presented by: Aaron Maizlish Mountains are traditionally represented by their summits Mountains as landform objects have fuzzy boundaries Cartographic convention is customarily a flat, non-hierarchical representation of summit locations Location of summits (x,y,z) does not describe any characteristics of the summit relative to its surrounding terrain. Summits, as a specific class of terrain points, have unique topological characteristics that have not been fully explored I have been investigating ways to measure mountains relative to their surrounding terrain NACIS Presentation, 10/10/03. © Aaron Maizlish Summary of the Presentation:  Summary of the Presentation PROMINENCE: Relative Elevation of a Summit SUMMIT:SADDLE Relationship SURFACE ELEMENTS DIVIDE TREE: How summits fit together LINEAGE: How Summits relate to each other LINEAGE AREAS: Do mountains have boundaries? PROMINENCE DISTRIBUTION: Persistence of summits PROMINENCE MAPS: For the U.S. NACIS Presentation, 10/10/03. © Aaron Maizlish Definition of Prominence:  Definition of Prominence Prominence is a first-order derivative of elevation, representing the relative elevation of a summit Height above all surrounding terrain The elevation difference between the summit and the lowest contour that encircles it and no higher summit NACIS Presentation, 10/10/03. © Aaron Maizlish La Sal Mountains, Utah:  La Sal Mountains, Utah South Mtn. (P=1,697’), Mt. Tukuhnikivatz (P=729’), Mt. Peale (P=6,560’) Prominence is Terrain Independent:  Prominence is Terrain Independent Round Mountain has more prominence than Mount Shuksan Shuksan is more “alpine” but it has a high ridge that connects it to Baker Prominence is mathematical; not subjective Mount Baker (E=10781, P=8811) Mount Shuksan (E=9131, P=4411) Round Mountain (E=5340, P=4800) Summits and Saddles:  Summits and Saddles Central to our understanding of the topology of terrain is the unique relationship between a summit and a saddle We measure prominence from the KEY SADDLE Every summit has a unique Key Saddle. Every saddle has a unique summit. Exception is the highest summit. Therefore on a given surface the No. of Summits = No. of Key Saddles + 1 Every depression (pit) has a unique Basin Saddle; which is morphometrically the same form as a Key Saddle Thus Summits + Pits = Saddles + 2 (the mountaineer’s equation) NACIS Presentation, 10/10/03. © Aaron Maizlish Surface Networks and Surface Elements:  Surface Networks and Surface Elements Six Surface Elements - types of points on terrain Ridge Network is the set of summits, saddles, and ridges Channel Network is the set of pits and channels To derive measurement of summits we extract critical points from the ridge network: The set of summits, key saddles and their connecting ridges we call the Divide Tree Edward Earl developed an automated extraction process for the divide tree and for prominence from DEMs. NACIS Presentation, 10/10/03. © Aaron Maizlish The Divide Tree:  The Divide Tree A simplified ridge network with all summits, key saddles, connecting ridges Mathematical tree Non-arbitrary rules to ‘go-around’ enclosed basins Like a watershed boundary map, except that the critical points are summits and saddles, not drainages Map locates the 55 most prominent mountains in the Western U.S. NACIS Presentation, 10/10/03. © Aaron Maizlish The Divide Tree:  The Divide Tree Approximate extent of enclosed basins Imagine filling each basin with water until it reached the overflow point NACIS Presentation, 10/10/03. © Aaron Maizlish Lineage and Parents:  Lineage and Parents Prominence gives insight into a hierarchy of mountains the Key Saddle marks a non-arbitrary boundary of a mountain Several have wondered if summits have natural parents is every peak in fact the sub-peak of a discrete summit? Possible definitions of parent - A parent would always be higher and always be across the Key Saddle NHN - the next-higher-neighbor NPN - the next-more prominent-neighbor PIP - the “prominence island parent” In fact, each summit has a unique string of consecutively higher mountains This Lineage is the string of NHNs that form a critical path from the summit to the continental high point NACIS Presentation, 10/10/03. © Aaron Maizlish How Lineage Area Works:  How Lineage Area Works All points on the Divide Tree (summit, saddle, and ridge) and all slopes have a unique path, the Lineage, to the highest ground Trace the slope line to a summit, then follow NHN rules A listing of NHNs is partly scale dependent, but the principle is not Lineage Area is the set of all points that include a given summit in their lineage Non-arbitrary and non-scale dependent Maxwell said that watercourses form the boundaries of hills Trace the two watercourse lines from the key saddle and from all secondary saddles, until they merge or reach the ocean All points within this area count the summit in their lineage NACIS Presentation, 10/10/03. © Aaron Maizlish Computing Lineage Area:  Computing Lineage Area Lineage Area is a natural region for which the summit is the highest point Note that Mt. Diablo and Loma Prieta have only the one (key) saddle. Copernicus Peak also has only one saddle. Loma Prieta and Mt. Diablo are both in the Copernicus lineage area. Both are lower elevation peaks with greater prominence. Mt. Pinos (Ventura Co., CA) has two saddles; its own and the San Gorgonio KS. Computation of the lineage area involves enclosed basins. Lineage area is the “system with fewest constraints” that describes a precise region around a summit. NACIS Presentation, 10/10/03. © Aaron Maizlish Computing the PALA:  Computing the PALA Lineage Area has few constraints: a small summit might have a large L.A. if it is on the flank of a large summit, e.g. Olancha Peak Prominence-Adjusted Lineage Area (“Domain”) excludes the L.A. of lower, more prominent summits (for example Copernicus would lose Diablo and Loma Prieta) Selection rules for PALAs same as L.A. but with additional saddles Olancha Peak (E=12123 P=3083) has key saddle + 2 saddles to lower, more prominent summits Domain is now proportional to the prominence of the mountain Olancha PALA is ±2,500 mi2, the LA is ±150,000 mi2 Many mountains have the same LA and PALA, e.g. Mt. Diablo NACIS Presentation, 10/10/03. © Aaron Maizlish Distribution of Mountains:  Distribution of Mountains Prominence gives us a non-arbitrary dataset and methodology to compare summits as discrete topological phenomena Summits occur in all terrain types, at all elevations Some questions: How is prominence distributed across the terrain manifold? How are mountains distributed by elevation? Is the distribution of summits dependent on morphogenic processes, or is it a largely independent mathematical occurrence? Can the number of mountains be predicted? NACIS Presentation, 10/10/03. © Aaron Maizlish Distribution of Prominence:  Distribution of Prominence Exponential growth of low-prominence summits Prediction is imprecise, perhaps very terrain dependent based on our limited datasets we might predict 15-20,000 hills in the contiguous US of P>500’, and 40-60,000 hills of P>300’ NACIS Presentation, 10/10/03. © Aaron Maizlish Mtn. Distribution by Elevation:  Mtn. Distribution by Elevation Combined Western U.S. Only Eastern U.S. Only NACIS Presentation, 10/10/03. © Aaron Maizlish Distribution Curves:  Distribution Curves U.S. dataset appears to show perfect bell-curve distribution Distribution might change with addition of lesser hills Data is a natural ‘branch’ of divide tree. Most data has bias due to artificial political boundaries We have larger datasets for Washington, Arizona and Great Britain NACIS Presentation, 10/10/03. © Aaron Maizlish PROMINENCE MAPS:  PROMINENCE MAPS I completed a series of maps for all of the United States Plots the 1,234 summits with 2,000’ prominence Many hikers avidly hiking based on prominence (more popular in Great Britain) I am experimenting with Large-Scale Maps Mapping Goals: Would like to see worldwide data to P=5,000 (1,000-1,500 peaks), will require clean SRTM data Working on Switzerland, select other areas to P=2,000 Thinking about large-scale atlas if there is any interest NACIS Presentation, 10/10/03. © Aaron Maizlish CALIFORNIA PROM MAP:  CALIFORNIA PROM MAP P=5,000 9 Mt. Whitney Mt. Shasta San Jacinto Peak San Gorgonio Peak White Mtn. Peak Mt. San Antonio Telescope Peak Lassen Peak Mt. Eddy P=4,000 7 P=3,000 30 P=2,000 118 (e) 5 CA PEAKS 169 OREGON PROM MAP:  P=5,000 4 Mt. Hood Sacajawea Peak Mt Jefferson South Sister P=4,000 4 P=3,000 14 P=2,000 51 (e) 3 OR PEAKS 76 OREGON PROM MAP WASHINGTON PROM MAP:  P=5,000 7 Mt. Rainier (#1) Mt. Baker Mt. Adams Mt. Olympus Glacier Peak Mt. Stuart Abercrombie Mtn. P=4,000 11 P=3,000 34 P=2,000 92 (e) 4 WA PEAKS 148 WASHINGTON PROM MAP P 5,000 Map:  P 5,000 Map (shown) 5,000’ + 57 (not shown) 4,000-4,999 84 3,000-3,999 253 2,000-2,999 799 (e) 41 U.S. PEAKS 1,234 Large Scale (all hills over 300’ prominence):  Large Scale (all hills over 300’ prominence) Uses for Prominence Theory?:  Uses for Prominence Theory? Mapping A non-arbitrary ordering system for mountains. Inclusion of high prominences on maps (unfortunately not all are named.) Thematic mapping could employ descriptive prominence concepts Earth Sciences Geomorphology, geology: correlation of prominence to other functions Biogeography - interpretive tool for species isolation and landscape level modeling Terrain modeling Visibility studies, siting studies, movable terrain objects, data compression, planetary mapping Mathematics Developments in Morse topology based on hierarchical structures Or just as “pure geography”! NACIS Presentation, 10/10/03. © Aaron Maizlish

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