Surface fitting in geomorphology - examples for regular-shaped volcanic landforms

In nature, several types of landforms have simple shapes: as they evolve they tend to take on an ideal, simple geometric form such as a cone, an ellipsoid or a paraboloid. Volcanic landforms are possibly the best examples of this ?ideal? geometry, since they develop as regular surface features due to the point-like (circular) or fissure-like (linear) manifestation of volcanic activity. In this paper, we present a geomorphometric method of fitting the ?ideal? surface onto the real surface of regular-shaped volcanoes through a number of case studies (Mt. Mayon, Mt. Somma, Mt. Semeru, and Mt. Cameroon). Volcanoes with circular, as well as elliptical, symmetry are addressed. For the best surface fit, we use the minimization library MINUIT which is made freely available by the CERN (European Organization for Nuclear Research). This library enables us to handle all the available surface data (every point of the digital elevation model) in a one-step, half-automated way regardless of the size of the dataset, and to consider simultaneously all the relevant parameters of the selected problem, such as the position of the center of the edifice, apex height, and cone slope, thanks to the highly performing adopted procedure. Fitting the geometric surface, along with calculating the related error, demonstrates the twofold advantage of the method. Firstly, we can determine quantitatively to what extent a given volcanic landform is regular, i.e. how much it follows an expected regular shape. Deviations from the ideal shape due to degradation (e.g. sector collapse and normal erosion) can be used in erosion rate calculations. Secondly, if we have a degraded volcanic landform, whose geometry is not clear, this method of surface fitting reconstructs the original shape with the maximum precision. Obviously, in addition to volcanic landforms, this method is also capable of constraining the shapes of other regular surface features such as aeolian, glacial or periglacial landforms.

Old Northwest and Great Lakes Region, 1785 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: A map of the north west parts of the United States of America, [by] John Fitch. It was published in 1785. Scale [ca. 1:3,000,000]. Covers the Old Northwest from the Great Lakes to Kentucky and the Mississippi River to the Allegany River and a portion of Lake Ontario. The image inside the map neatline is georeferenced to the surface of the earth and fit to the USA Contiguous Albers Equal Area Conic projection (Meters). All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, or other information associated with the principal map. This map shows features such as human settlements, forts, Native American lands, roads, drainage, proposed state boundaries, and more. Relief shown by landform drawing. Includes descriptive notes. This layer is part of a selection of digitally scanned and georeferenced historic maps of New England from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of regions, originators, ground condition dates, scales, and purposes.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 2

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet II [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng and southern Gansu Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 3

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet III [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng and southern Gansu Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 4

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet IV [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 9

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet IX [of 10] covering a portion of the Yellow River (Huang He) region in southern Gansu Sheng and northwestern Sichuan Sheng, China, including parts of Baishui Jiang and Pai Ho (Gar He). The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 5

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet V [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng and southern Gansu Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 6

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet VI [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng, southern Gansu Sheng, and northwestern Sichuan Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 7

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet VII [of 10] covering a portion of the Yellow River (Huang He) region in southern Gansu Sheng and northwestern Sichuan Sheng, China, including parts of Bailong Jiang and Tao He. Sheet VII includes a separate title: 'Cho-ni Territory, Upper and Lower T'ieh-Pu country and route to Sung-P'an, J. F. Rock, 1925-1927.' The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.

Rock map of the Yellow River (Huang He) expedition, China, 1925-1927 : Sheet 8

This layer is part of a set of georeferenced, raster images of the manuscript, paper map set entitled: Ch'ing-Hai upper Yellow River expedition : Rock and Simpson, 1925-27, [cartography by J.F. Rock]. Scale 1:250,000. This layer image is of Sheet VIII [of 10] covering a portion of the Yellow River (Huang He) region in eastern Qinghai Sheng, southern Gansu Sheng, and northwestern Sichuan Sheng, China. The map set details the route and surrounding environs of the Arnold Arboretum's "Western China" expedition led by Joseph Rock, 1924-1927. The set covers a portion of the Yellow River (Huang He) region in south central China (Qinghai, Gansu, and Sichuan shengs (a portion of historic Tibet)). It shows features, labeled variously in English, Chinese, Wade-Giles transliteration, and Tibetan, including: rivers, streams, lakes, mountains, gorges, valleys, plateaus, plains, cities, towns, villages, provincial capitals, county seats, passes, monasteries, ruin sites, native tribe locations, and more. Relief is shown by hachures, spot heights, and landform drawings. The original manuscript map set is part of the Harvard College Library, Harvard Map Collection. "Joseph Rock traced his travels for the [Arnold] Arboretum's [Western China] 1924-1927 expedition in a colorful, hand-drawn map entitled 'Ch'ing-Hai upper Yellow River expedition.' The pen-and-ink drawing was made on ten sheets that when joined form a single, irregularly-shaped map, approximately six by eight feet in size. The individual sheets are numbered, using roman numerals; on sheet VII is a second title, 'Choni Territory, Upper and Lower T'ieh-Pu country and route to Sung-Pan, J. F. Rock, 1925-1927.' Topographical and other features are identified using a combination of English, Chinese characters, Wade-Giles transliterations and Tibetan script. Rock's attractive cursive style and use of hachures, spot heights, and landform drawings to depict relief add character to the map." -- Text from the Arnold Arboretum Web site.