МОЛОДЁЖНЫЙ ПРОЕКТ ДЛЯ ТЕХ, КТО ДЕЛАЕТ ПЕРВЫЕ ШАГИ В НАУКЕ
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2016. Том 11. Выпуск 1. Система планета Земля / 'The Earth Planet System / System Planet Erde
МЕТОДЫ — АНАЛИЗЫ — СТАНДАРТЫ / METHODS — ANALYSIS — STANDARDS / METHODEN — ANALYSEN — NORMEN
Флоринский И.В.
Иллюстрированное введение в геоморфометрию
Флоринский Игорь Васильевич, доктор технических наук, ведущий научный сотрудник, Институт математических проблем биологии РАН — Филиал Федерального исследовательского центра «Институт прикладной математики им. М.В. Келдыша РАН», Пущино, Россия 
ORCID 0000-0003-1273-5912
E-mail: iflor@psn.ru; igor-v-florinsky@j-spacetime.com
В статье рассматриваются базовые математические понятия геоморфометрии. Приводятся определения, формулы и краткие интерпретации 26-ти локальных, нелокальных, солярных и комбинированных морфометрических величин. Для иллюстрации математических понятий геоморфометрии использована цифровая модель высот (ЦМВ) горного массива Арарат, выделенная из глобальной ЦМВ SRTM1. Обработка ЦМВ проведена с помощью универсального спектрально-аналитический метода. Для всех рассмотренных морфометрических величин представлены трехмерные цветные карты Арарата.
Ключевые слова: рельеф, поверхность, геоморфометрия, морфометрическая величина, кривизна, цифровое моделирование рельефа, спектрально-аналитический метод.
Цитирование по ГОСТ Р 7.0.11—2011:
Флоринский, И. В. Иллюстрированное введение в геоморфометрию [Электронный ресурс] / И.В. Флоринский // Электронное научное издание Альманах Пространство и Время. — 2016. — Т. 11. — Вып. 1: Система планета Земля. — Стационарный сетевой адрес: 2227-9490e-aprovr_e-ast11-1.2016.71.
Florinsky I.V.
An Illustrated Introduction to Geomorphometry
Igor V. Florinsky, D.Sc. (Cartography), Principal Research Scientist, Institute of Mathematical Problems of Biology, the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Pushchino, Moscow Region, Russia ORCID 0000-0003-1273-5912
E-mail: iflor@psn.ru; igor-v-florinsky@j-spacetime.com
This paper discusses the basic mathematical concepts of geomorphometry. First, we present a brief historical overview of the progress of digital terrain modelling. Second, we introduce the notion of the topographic surface and its limitations. Then, we present definitions, formulas, and physical meanings for four main groups of morphometric variables, such as local, nonlocal, solar, and combined attributes. We discuss the following 26 morphometric variables: slope gradient, slope aspect, northerness, easterness, plan curvature, horizontal curvature, vertical curvature, mean curvature, Gaussian curvature, minimal curvature, maximal curvature, unsphericity curvature, difference curvature, vertical excess curvature, horizontal excess curvature, ring curvature, accumulation curvature, the generating function, rotor, Laplacian, catchment area, dispersive area, reflectance, insolation, topographic index, and stream power index. To illustrate mathematical concepts of geomorphometry, we used a digital elevation model (DEM) of Mount Ararat extracted from the SRTM1 DEM. The DEM was treated by the universal spectral analytical method. Three-dimensional color maps of Mount Ararat for all discussed morphometric variables are presented.
Keywords: topography, surface, geomorphometry, morphometric variable, curvature, digital terrain modelling, spectral analytical method.
Cite MLA 7:
Florinsky, I. V. "An Illustrated Introduction to Geomorphometry." Electronic Scientific Edition Almanac Space and Time 11.1(‘The Earth Planet System’) (2016). Web. <2227-9490e-aprovr_e-ast11-1.2016.71>. (In Russian).
Список литературы / References
Литература
Погорелов А.В. Дифференциальная геометрия. М.: Наука, 1974. 176 с.
Шарый П.А. Топографический метод вторых производных // Геометрия структур земной поверхности. Пущино: ПНЦ АН СССР, 1991. С. 30—60.
Шарый П.А., Степанов И.Н. О методе вторых производных в геологии // Доклады АН СССР. 1991. Т. 319. № 2. С. 456—460.
Aandahl A.R. "The Characterization of Slope Positions and Their Influence on the Total Nitrogen Content of a Few Virgin Soils of Western Iowa." Soil Science Society of America Proceedings 13 (1948): 449—454.
Band L.E. "Extraction of Channel Networks and Topographic Parameters from Digital Elevation Data." Channel Network Hydrology. Chichester: Wiley, 1993, pp. 13—42.
Beven K.J., Kirkby M.J. "A Physically-Based Variable Contributing Area Model of Basin Hydrology." Hydrological Science Bulletin 24.1 (1979): 43—69.
Burbank D.W., Anderson R.S. Tectonic Geomorphology. Chichester: Wiley–Blackwell, 2012. 454 p.
Cayley A. "On Contour and Slope Lines." The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science. Series 4 18.120 (1859): 264—268.
Clarke J.I. "Morphometry from Maps." Essays in Geomorphology. London: Heinemann, 1966, pp. 235—274.
Clarke K.C. "Scale-Based Simulation of Topographic Relief." American Cartographer 15.2 (1988): 173—181.
Deng Y. "New Trends in Digital Terrain Analysis: Landform Definition, Representation, and Classification." Progress in Physical Geography 31.4 (2007): 405—419.
Evans I.S. "General Geomorphometry, Derivations of Altitude, and Descriptive Statistics." Spatial Analysis in Geomorphology. London: Methuen, 1972, pp. 17—90.
Evans I.S. Statistical Characterization of Altitude Matrices by Computer. An Integrated System of Terrain Analysis and Slope Mapping. The Final Report on Grant DA-ERO-591-73-G0040. Durham: University of Durham, 1979. 192 p.
Farr T.G., Rosen P.A., Caro E., Crippen R., Duren R., Hensley S., Kobrick M., Paller M., Rodriguez E., Roth L., Seal D., Shaffer S., Shimada J., Umland J., Werner M., Oskin M., Burbank D., Alsdorf D. "The Shuttle Radar Topography Mission." Reviews of Geophysics 45 (2007): RG2004.
Florinsky I.V. "Combined Analysis of Digital Terrain Models and Remotely Sensed Data in Landscape Investigations." Progress in Physical Geography 22.1 (1998.a): 33—60.
Florinsky I.V. "Derivation of Topographic Variables from a Digital Elevation Model Given by a Spheroidal Trapezoidal Grid." International Journal of Geographical Information Science 12.8 (1998.b): 829—852.
Florinsky I.V. "Computation of the Third-Order Partial Derivatives from a Digital Elevation Model." International Journal of Geographical Information Science 23.2 (2009): 213—231.
Florinsky I.V. Digital Terrain Analysis in Soil Science and Geology. Amsterdam: Academic Press, 2012. 379 p.
Florinsky I.V., Pankratov A.N. "Digital Terrain Modeling with Orthogonal Polynomials." Machine Learning and Data Analysis 1.12 (2015): 1647—1659.
Franklin J. "Predictive Vegetation Mapping: Geographic Modelling of Biospatial Patterns in Relation to Environmental Gradients." Progress in Physical Geography 19.4 (1995): 474—499.
Freeman T.G. "Calculating Catchment Area with Divergent Flow Based on a Regular Grid." Computers and Geosciences 17.3 (1991): 413—422.
Gauss C.F. "Disquisitiones Generales Circa Superficies Curvas." Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 6 (1828): 99—146.
Geiger R. The Climate Near the Ground. Cambridge: Harvard University Press, 1966. 611 p.
Gerrard A.J. Soils and Landforms. An Integration of Geomorphology and Pedology. London: George Allen and Unwin, 1981. 219 p.
Hengl T., Reuter H.I., eds. Geomorphometry: Concepts, Software, Applications. Amsterdam: Elsevier, 2009. 796 p.
Horn B.K.P. "Hill Shading and the Reflectance Map." Proceedings of the IEEE 69.1 (1981): 14—47.
Horton R.E. "Erosional Development of Streams and Their Drainage Basins, Hydrophysical Approach to Quantitative Morphology." Geological Society of America Bulletin 56.3 (1945): 275—370.
Jenny B. "An Interactive Approach to Analytical Relief Shading." Cartographica 38.1–2 (2001): 67—75.
King D., Bourennane H., Isambert M., Macaire J.J. "Relationship of the Presence of a Non-Calcareous Clay-Loam Horizon to DEM Attributes in a Gently Sloping Area." Geoderma 89.1–2 (1999): 95—111.
Kirkby M.J., Chorley R.J. "Throughflow, Overland Flow and Erosion." Bulletin of the International Association of Scientific Hydrology 12.3 (1967): 5—21.
Krcho J. "Morphometric Analysis of Relief on the Basis of Geometric Aspect of Field Theory." Acta Geographica Universitatis Comenianae. Geographico-Physica 1 (1973): 7—233.
Krcho J. "Teoretická koncepcia a interdisciplinárne aplikácie komplexného digitálneho modelu reliéfu pri modelovaní dvojdimenzionálnych polí." Geografický Časopis 35.3 (1983): 265—291.
Li Z., Zhu Q., Gold C. Digital Terrain Modeling: Principles and Methodology. New York: CRC Press, 2005. 323 p.
Mark D.M. "Geomorphometric Parameters: A Review and Evaluation." Geografiska Annaler. Series A 57.3–4 (1975): 165—177.
Martz L.W., de Jong E. "CATCH: A Fortran Program for Measuring Catchment Area from Digital Elevation Models." Computers and Geosciences 14.5 (1988): 627—640.
McCullagh M.J. "Terrain and Surface Modelling Systems: Theory and Practice." Photogrammetric Record 12.72 (1988): 747—779.
Moore I.D., Grayson R.B., Ladson A.R. "Digital Terrain Modelling: A Review of Hydrological, Geomorphological and Biological Applications." Hydrological Processes 5.1 (1991): 3—30.
O’Callaghan J.F., Mark D.M. "The Extraction of Drainage Networks from Digital Elevation Data." Computer Vision, Graphics, and Image Processing 28.3 (1984): 323—344.
Penck W. Morphological Analysis of Land Forms. A Contribution to Physical Geology. London: Macmillan, 1953. 429 p.
Quinn P.F., Beven K.J., Chevallier P., Planchon O. "The Prediction of Hillslope Flowpaths for Distributed Modelling Using Digital Terrain Models." Hydrological Processes 5.1 (1991): 59—80.
Rosenberg P. "Information Theory and Electronic Photogrammetry." Photogrammetric Engineering 21.4 (1955): 543—555.
Shary P.A. "Land Surface in Gravity Points Classification by a Complete System of Curvatures." Mathematical Geology 27.3 (1995): 373—390.
Shary P.A., Sharaya L.S., Mitusov A.V. "Fundamental Quantitative Methods of Land Surface Analysis." Geoderma 107.1–2 (2002): 1—32.
Shary P.A., Sharaya L.S., Mitusov A.V. "The Problem of Scale-Specific and Scale-Free Approaches in Geomorphometry." Geografia Fisica e Dinamica Quaternaria 28.1 (2005): 81—101.
Speight J.G. "A Parametric Approach to Landform Regions." Progress in Geomorphology: Papers in Honour of D.L. Linton. London: Institute of British Geographers. 1974, pp. 213—230.
Speight J.G. "The Role of Topography in Controlling Throughflow Generation: A Discussion." Earth Surface Processes 5.2 (1980): 187—191.
Strahler A.N. "Quantitative Analysis of Watershed Geomorphology. " Transactions of the American Geophysical Union 38.6 (1957): 913—920.
Tarboton D.G. "A New Method for the Determination of Flow Directions and Upslope Areas in Grid Digital Elevation Models." Water Resources Research 33.2 (1997): 309—319.
USGS. Earth Explorer. Sioux Fall: Earth Resources Observation and Science (EROS) Center, U.S. Geological Survey, 2015. Web. <http://earthexplorer.usgs.gov>.
Wilson J.P., Gallant J.C., eds. Terrain Analysis: Principles and Applications. New York: Wiley, 2000. 479 p.
Young T. "An Essay on the Cohesion of Fluids." Philosophical Transactions of Royal Society of London. Pt I 95 (1805): 65—87.
Zevenbergen L.W., Thorne C.R. "Quantitative Analysis of Land Surface Topography." Earth Surface Processes and Landforms 12.1 (1987): 47—56.
References
Aandahl A.R. "The Characterization of Slope Positions and Their Influence on the Total Nitrogen Content of a Few Virgin Soils of Western Iowa." Soil Science Society of America Proceedings 13 (1948): 449—454.
Band L.E. "Extraction of Channel Networks and Topographic Parameters from Digital Elevation Data." Channel Network Hydrology. Chichester: Wiley, 1993, pp. 13—42.
Beven K.J., Kirkby M.J. "A Physically-Based Variable Contributing Area Model of Basin Hydrology." Hydrological Science Bulletin 24.1 (1979): 43—69.
Burbank D.W., Anderson R.S. Tectonic Geomorphology. Chichester: Wiley–Blackwell, 2012. 454 p.
Cayley A. "On Contour and Slope Lines." The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science. Series 4 18.120 (1859): 264—268.
Clarke J.I. "Morphometry from Maps." Essays in Geomorphology. London: Heinemann, 1966, pp. 235—274.
Clarke K.C. "Scale-Based Simulation of Topographic Relief." American Cartographer 15.2 (1988): 173—181.
Deng Y. "New Trends in Digital Terrain Analysis: Landform Definition, Representation, and Classification." Progress in Physical Geography 31.4 (2007): 405—419.
Evans I.S. "General Geomorphometry, Derivations of Altitude, and Descriptive Statistics." Spatial Analysis in Geomorphology. London: Methuen, 1972, pp. 17—90.
Evans I.S. Statistical Characterization of Altitude Matrices by Computer. An Integrated System of Terrain Analysis and Slope Mapping. The Final Report on Grant DA-ERO-591-73-G0040. Durham: University of Durham, 1979. 192 p.
Farr T.G., Rosen P.A., Caro E., Crippen R., Duren R., Hensley S., Kobrick M., Paller M., Rodriguez E., Roth L., Seal D., Shaffer S., Shimada J., Umland J., Werner M., Oskin M., Burbank D., Alsdorf D. "The Shuttle Radar Topography Mission." Reviews of Geophysics 45 (2007): RG2004.
Florinsky I.V. "Combined Analysis of Digital Terrain Models and Remotely Sensed Data in Landscape Investigations." Progress in Physical Geography 22.1 (1998): 33—60.
Florinsky I.V. "Computation of the Third-Order Partial Derivatives from a Digital Elevation Model." International Journal of Geographical Information Science 23.2 (2009): 213—231.
Florinsky I.V. "Derivation of Topographic Variables from a Digital Elevation Model Given by a Spheroidal Trapezoidal Grid." International Journal of Geographical Information Science 12.8 (1998): 829—852.
Florinsky I.V. Digital Terrain Analysis in Soil Science and Geology. Amsterdam: Academic Press, 2012. 379 p.
Florinsky I.V., Pankratov A.N. "Digital Terrain Modeling with Orthogonal Polynomials." Machine Learning and Data Analysis 1.12 (2015): 1647—1659.
Franklin J. "Predictive Vegetation Mapping: Geographic Modelling of Biospatial Patterns in Relation to Environmental Gradients." Progress in Physical Geography 19.4 (1995): 474—499.
Freeman T.G. "Calculating Catchment Area with Divergent Flow Based on a Regular Grid." Computers and Geosciences 17.3 (1991): 413—422.
Gauss C.F. "Disquisitiones Generales Circa Superficies Curvas." Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 6 (1828): 99—146.
Geiger R. The Climate Near the Ground. Cambridge: Harvard University Press, 1966. 611 p.
Gerrard A.J. Soils and Landforms. An Integration of Geomorphology and Pedology. London: George Allen and Unwin, 1981. 219 p.
Hengl T., Reuter H.I., eds. Geomorphometry: Concepts, Software, Applications. Amsterdam: Elsevier, 2009. 796 p.
Horn B.K.P. "Hill Shading and the Reflectance Map." Proceedings of the IEEE 69.1 (1981): 14—47.
Horton R.E. "Erosional Development of Streams and Their Drainage Basins, Hydrophysical Approach to Quantitative Morphology." Geological Society of America Bulletin 56.3 (1945): 275—370.
Jenny B. "An Interactive Approach to Analytical Relief Shading." Cartographica 38.1–2 (2001): 67—75.
King D., Bourennane H., Isambert M., Macaire J.J. "Relationship of the Presence of a Non-Calcareous Clay-Loam Horizon to DEM Attributes in a Gently Sloping Area." Geoderma 89.1–2 (1999): 95—111.
Kirkby M.J., Chorley R.J. "Throughflow, Overland Flow and Erosion." Bulletin of the International Association of Scientific Hydrology 12.3 (1967): 5—21.
Krcho J. "Morphometric Analysis of Relief on the Basis of Geometric Aspect of Field Theory." Acta Geographica Universitatis Comenianae. Geographico-Physica 1 (1973): 7—233.
Krcho J. "Teoretická koncepcia a interdisciplinárne aplikácie komplexného digitálneho modelu reliéfu pri modelovaní dvojdimenzionálnych polí." Geografický Časopis 35.3 (1983): 265—291.
Li Z., Zhu Q., Gold C. Digital Terrain Modeling: Principles and Methodology. New York: CRC Press, 2005. 323 p.
Mark D.M. "Geomorphometric Parameters: A Review and Evaluation." Geografiska Annaler. Series A 57.3–4 (1975): 165—177.
Martz L.W., de Jong E. "CATCH: A Fortran Program for Measuring Catchment Area from Digital Elevation Models." Computers and Geosciences 14.5 (1988): 627—640.
McCullagh M.J. "Terrain and Surface Modelling Systems: Theory and Practice." Photogrammetric Record 12.72 (1988): 747—779.
Moore I.D., Grayson R.B., Ladson A.R. "Digital Terrain Modelling: A Review of Hydrological, Geomorphological and Biological Applications." Hydrological Processes 5.1 (1991): 3—30.
O’Callaghan J.F., Mark D.M. "The Extraction of Drainage Networks from Digital Elevation Data." Computer Vision, Graphics, and Image Processing 28.3 (1984): 323—344.
Penck W. Morphological Analysis of Land Forms. A Contribution to Physical Geology. London: Macmillan, 1953. 429 p.
Pogorelov A.V. Differential Geometry. Moscow: Nauka Publisher, 1974. 176 p. (In Russian).
Quinn P.F., Beven K.J., Chevallier P., Planchon O. "The Prediction of Hillslope Flowpaths for Distributed Modelling Using Digital Terrain Models." Hydrological Processes 5.1 (1991): 59—80.
Rosenberg P. "Information Theory and Electronic Photogrammetry." Photogrammetric Engineering 21.4 (1955): 543—555.
Shary P.A. "Land Surface in Gravity Points Classification by a Complete System of Curvatures." Mathematical Geology 27.3 (1995): 373—390.
Shary P.A. "The Second Derivative Topographic Method." The Geometry of the Earth Surface Structures. Pushchino: Pushchino Research Centre Publisher, 1991, pp. 30—60. (In Russian).
Shary P.A., Sharaya L.S., Mitusov A.V. "Fundamental Quantitative Methods of Land Surface Analysis." Geoderma 107.1–2 (2002): 1—32.
Shary P.A., Sharaya L.S., Mitusov A.V. "The Problem of Scale-Specific and Scale-Free Approaches in Geomorphometry." Geografia Fisica e Dinamica Quaternaria 28.1 (2005): 81—101.
Shary P.A., Stepanov I.N. "Application of the Method of Second Derivatives in Geology." Doklady Earth Sciences 320.7 (1991): 87—92. (In Russian).
Speight J.G. "A Parametric Approach to Landform Regions." Progress in Geomorphology: Papers in Honour of D.L. Linton. London: Institute of British Geographers. 1974, pp. 213–230.
Speight J.G. "The Role of Topography in Controlling Throughflow Generation: A Discussion." Earth Surface Processes 5.2 (1980): 187—191.
Strahler A.N. "Quantitative Analysis of Watershed Geomorphology." Transactions of the American Geophysical Union 38.6 (1957): 913—920.
Tarboton D.G. "A New Method for the Determination of Flow Directions and Upslope Areas in Grid Digital Elevation Models. Water Resources Research 33.2 (1997): 309—319.
USGS. Earth Explorer. Sioux Fall: Earth Resources Observation and Science (EROS) Center, U.S. Geological Survey, 2015. Web. <http://earthexplorer.usgs.gov>.
Wilson J.P., Gallant J.C., eds. Terrain Analysis: Principles and Applications. New York: Wiley, 2000. 479 p.
Young T. "An Essay on the Cohesion of Fluids." Philosophical Transactions of Royal Society of London. Pt I 95 (1805): 65—87.
Zevenbergen L.W., Thorne C.R. "Quantitative Analysis of Land Surface Topography." Earth Surface Processes and Landforms 12.1 (1987): 47—56.
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