An illustrated introduction to general geomorphometry

Geomorphometry is widely used to solve various multiscale geoscientific problems. For the successful application of geomorphometric methods, a researcher should know the basic mathematical concepts of geomorphometry and be aware of the system of morphometric variables, as well as understand their physical, mathematical and geographical meanings. This paper reviews the basic mathematical concepts of general geomorphometry. First, we discuss the notion of the topographic surface and its limitations. Second, we present definitions, formulae and meanings for four main groups of morphometric variables, such as local, non-local, two-field specific and combined topographic attributes, and we review the following 29 fundamental morphometric variables: slope, aspect, northwardness, eastwardness, plan curvature, horizontal curvature, vertical curvature, difference curvature, horizontal excess curvature, vertical excess curvature, accumulation curvature, ring curvature, minimal curvature, maximal curvature, mean curvature, Gaussian curvature, unsphericity curvature, rotor, Laplacian, shape index, curvedness, horizontal curvature deflection, vertical curvature deflection, catchment area, dispersive area, reflectance, insolation, topographic index and stream power index. For illustrations, we use a digital elevation model (DEM) of Mount Ararat, extracted from the Shuttle Radar Topography Mission (SRTM) 1-arc-second DEM. The DEM was treated by a spectral analytical method. Finally, we briefly discuss the main paradox of general geomorphometry associated with the smoothness of the topographic surface and the non-smoothness of the real topography; application of morphometric variables; statistical aspects of geomorphometric modelling, including relationships between morphometric variables and roughness indices; and some pending problems of general geomorphometry (i.e. analysis of inner surfaces of caves, analytical description of non-local attributes and structural lines, as well as modelling on a triaxial ellipsoid). The paper can be used as a reference guide on general geomorphometry.

[1]  Carl Friedrich Gauss.  Disquisitiones generales circa superficies curvas , 1981 .

[2]  P. A. Shary,et al.  Land surface in gravity points classification by a complete system of curvatures , 1995 .

[3]  Ian M. Howat,et al.  A new bed elevation dataset for Greenland , 2012 .

[4]  David T. Sandwell,et al.  A new global bathymetry map at 15 arcsecond resolution for resolving seafloor fabric: SRTM15_PLUS , 2014 .

[5]  Andrea J. van Doorn,et al.  Two-plus-one-dimensional differential geometry , 1994, Pattern Recognition Letters.

[6]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[7]  L. Mueller,et al.  Novel Methods for Monitoring and Managing Land and Water Resources in Siberia , 2016 .

[8]  Igor V. Florinsky,et al.  A universal spectral analytical method for digital terrain modeling , 2016, Int. J. Geogr. Inf. Sci..

[9]  Mechanisms of the effects of solar radiation and terrain anisotropy on the vegetation of dark conifer forests in the Pechora-Ilych state biosphere reserve , 2013, Russian Journal of Ecology.

[10]  Kurt J. Marfurt,et al.  Seismic Attributes for Prospect Identification and Reservoir Characterization , 2007 .

[11]  Richard J. Pike,et al.  Geomorphometry -diversity in quantitative surface analysis , 2000 .

[12]  Michael P. Bishop,et al.  Remote sensing and giscience in geomorphology , 2013 .

[13]  Peter F. Fisher,et al.  Extending the applicability of viewsheds in landscape planning , 1996 .

[14]  A. J. Gerrard.  Soils and landforms: An integration of geomorphology and pedology , 1982 .

[15]  Jan Pacina,et al.  Third-order geomorphometric variables (derivatives): definition, computation and utilization of changes of curvatures , 2013, Int. J. Geogr. Inf. Sci..

[16]  P. A. Shary,et al.  Analytical and Cartographic Predictive Modeling of Arable Land Productivity , 2016 .

[17]  Alessandro Capra,et al.  Evaluation of flow direction methods against field observations of overland flow dispersion , 2012 .

[18]  C. Thorne,et al.  Quantitative analysis of land surface topography , 1987 .

[19]  D. Guida,et al.  Using object-based geomorphometry for hydro-geomorphological analysis in aMediterranean research catchment , 2016 .

[20]  C. Jacobi,et al.  Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution. , 1839 .

[21]  The geodesic boundary value problem and its solution on a triaxial ellipsoid , 2013 .

[22]  Ian S. Evans,et al.  Relations between land surface properties: Altitude, slope and curvature , 1999 .

[23]  A. N. Strahler.  Quantitative analysis of watershed geomorphology , 1957 .

[24]  Mike Kirkby,et al.  THROUGHFLOW, OVERLAND FLOW AND EROSION , 1967 .

[25]  A. R. Aandahl,et al.  The Characterization of Slope Positions and Their Influence on the Total Nitrogen Content of a Few Virgin Soils of Western Iowa , 1949 .

[26]  MAPPING WORLDS WITH IRREGULAR SHAPES , 1998 .

[27]  I. Florinsky.  Global Lineaments: Application of Digital Terrain Modelling , 2008 .

[28]  A. Roberts.  Curvature attributes and their application to 3D interpreted horizons , 2001 .

[29]  Andrea J. van Doorn,et al.  Surface shape and curvature scales , 1992, Image Vis. Comput..

[30]  Ute Christina Herzfeld,et al.  Automated geostatistical seafloor classification—principles, parameters, feature vectors, and discrimination criteria , 1996 .

[31]  I. Florinsky.  Digital Terrain Modeling , 2012 .

[32]  T. Vincenty.  DIRECT AND INVERSE SOLUTIONS OF GEODESICS ON THE ELLIPSOID WITH APPLICATION OF NESTED EQUATIONS , 1975 .

[33]  I. Colomina,et al.  Unmanned aerial systems for photogrammetry and remote sensing: A review , 2014 .

[34]  John P. Wilson,et al.  Water in the Landscape: A Review of Contemporary Flow Routing Algorithms , 2008 .

[35]  Jochen Schmidt,et al.  Comparison of polynomial models for land surface curvature calculation , 2003, Int. J. Geogr. Inf. Sci..

[36]  G. Neumann,et al.  FIRST GLOBAL DIGITAL ELEVATION MODEL OF MERCURY , 2016 .

[37]  Charles Hansen,et al.  The Visualization Handbook , 2011 .

[38]  Thierry Toutin,et al.  ASTER DEMs for geomatic and geoscientific applications: a review , 2008 .

[39]  P. A. Shary,et al.  Statistical evaluation of the relationships between spatial variability in the organic carbon content in gray forest soils, soil density, concentrations of heavy metals, and topography , 2013, Eurasian Soil Science.

[40]  S. K. Jenson,et al.  Extracting topographic structure from digital elevation data for geographic information-system analysis , 1988 .

[41]  Vanessa Lucieer,et al.  A review of marine geomorphometry, the quantitative study of the seafloor , 2016 .

[42]  H. Bork,et al.  Correlation of colluvial deposits with the modern land surface and the problem of slope profile description , 2014 .

[43]  P. Pizor.  Principles of Geographical Information Systems for Land Resources Assessment. , 1987 .

[44]  J. Franklin.  Predictive vegetation mapping: geographic modelling of biospatial patterns in relation to environmental gradients , 1995 .

[45]  L. Martz,et al.  CATCH: a FORTRAN program for measuring catchment area from digital elevation models , 1988 .

[46]  R. Horton.  EROSIONAL DEVELOPMENT OF STREAMS AND THEIR DRAINAGE BASINS; HYDROPHYSICAL APPROACH TO QUANTITATIVE MORPHOLOGY , 1945 .

[47]  Jozef Krcho,et al.  Morphometric analysis of relief on the basis of geometric aspect of field theory , 1973 .

[48]  A. A. Kokhanov,et al.  Cartographic and geodetic methods to characterize the potential landing sites for the future Russian missions Luna-Glob and Luna-Resurs , 2015 .

[49]  D. Mark,et al.  Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping , 1984 .

[50]  Andrea Tribe,et al.  Automated recognition of valley lines and drainage networks from grid digital elevation models: a review and a new method , 1992 .

[51]  D. Wardle,et al.  Spatial soil ecology , 2002 .

[52]  T. Behrens,et al.  Uncertainty-guided sampling to improve digital soil maps , 2017 .

[53]  I. Evans,et al.  Transformation (normalization) of slope gradient and surface curvatures, automated for statistical analyses from DEMs , 2015 .

[54]  Jay Gao,et al.  Fractals in physical geography , 1996 .

[55]  S. V. Filippov,et al.  A desktop system of virtual morphometric globes for Mars and the Moon , 2017 .

[56]  John P. Wilson,et al.  Terrain analysis : principles and applications , 2000 .

[57]  A. C. Esq..  XL. On contour and slope lines , 1859 .

[58]  Igor V. Florinsky,et al.  Prediction of soil properties by digital terrain modelling , 2002, Environ. Model. Softw..

[59]  M. Temovski,et al.  DEM based geomorphometric analyses of karst surface in the Republic of Macedonia , 2015 .

[60]  Brian Klinkenberg,et al.  Fractals and morphometric measures: is there a relationship? , 1992 .

[61]  David R. Montgomery,et al.  Multi-scale curvature for automated identification of glaciated mountain landscapes☆ , 2014, Geomorphology.

[62]  T. Young.  III. An essay on the cohesion of fluids , 1805, Philosophical Transactions of the Royal Society of London.

[63]  P. A. Shary,et al.  Models of Topography , 2008 .

[64]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[65]  Jan J. Koenderink,et al.  Local features of smooth shapes: ridges and courses , 1993, Optics & Photonics.

[66]  Takeo Tadono,et al.  PRECISE GLOBAL DEM GENERATION BY ALOS PRISM , 2014 .

[67]  V. Gardiner,et al.  Drainage basin morphometry , 1978 .

[68]  I. Florinsky.  Global Morphometric Maps of Mars, Venus, and the Moon , 2008 .

[69]  Charles F. F. Karney,et al.  Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen , 1825, 0908.1823.

[70]  D. Tarboton.  A new method for the determination of flow directions and upslope areas in grid digital elevation models , 1997 .

[71]  Gerhard Krieger,et al.  TanDEM-X: The New Global DEM Takes Shape , 2014, IEEE Geoscience and Remote Sensing Magazine.

[72]  I. Evans.  General geomorphometry, derivatives of altitude, and descriptive statistics , 2019, Spatial Analysis in Geomorphology.

[73]  I. Evans.  Statistical Characterization of Altitude Matrices by Computer. Report 6. An Integrated System of Terrain Analysis and Slope Mapping. , 1979 .

[74]  S. A. Stewart,et al.  Curvature analysis of gridded geological surfaces , 1998, Geological Society, London, Special Publications.

[75]  Keith C. Clarke,et al.  Scale-Based Simulation of Topographic Relief , 1988 .

[76]  L. Sjöberg.  New solutions to the direct and indirect geodetic problems on the ellipsoid , 2006 .

[77]  Mark W. Smith,et al.  Structure from motion photogrammetry in physical geography , 2016 .

[78]  Igor V. Florinsky,et al.  Quantitative topographic method of fault morphology recognition , 1996 .

[79]  I. Moore,et al.  Digital terrain modelling: A review of hydrological, geomorphological, and biological applications , 1991 .

[80]  Igor V. Florinsky,et al.  Spheroidal equal angular DEMs: The specificity of morphometric treatment , 2017, Trans. GIS.

[81]  Gyozo Jordan,et al.  Digital Terrain Analysis in a GIS Environment. Concepts and Development , 2007 .

[82]  Gegam Vagramovich Bagratuni.  COURSE IN SPHEROIDAL GEODESY , 1967 .

[83]  P. A. Shary,et al.  Fundamental quantitative methods of land surface analysis , 2002 .

[84]  Igor V. Florinsky,et al.  Computation of the third‐order partial derivatives from a digital elevation model , 2009, Int. J. Geogr. Inf. Sci..

[85]  J. Poesen,et al.  Relationships of attributes of gullies with morphomoetric variables , 2015 .

[86]  R. H. Shaw.  The Climate Near the Ground , 1957 .

[87]  F. Nitsche,et al.  The International Bathymetric Chart of the Southern Ocean (IBCSO) Version 1.0 , 2013 .

[88]  Patrick Cozzi,et al.  3D Engine Design for Virtual Globes , 2011 .

[89]  K. L. Frankel,et al.  Characterizing arid region alluvial fan surface roughness with airborne laser swath mapping digital topographic data , 2007 .

[90]  H. Mitásová,et al.  Interpolation by regularized spline with tension: I. Theory and implementation , 1993 .

[91]  Igor V. Florinsky,et al.  Derivation of Topographic Variables from a Digital Elevation Model Given by a Spheriodal Trapezoidal Grid , 1998, Int. J. Geogr. Inf. Sci..

[92]  Igor V. Florinsky,et al.  Digital Terrain Analysis in Soil Science and Geology , 2011 .

[93]  Michael F. Hutchinson,et al.  A differential equation for specific catchment area , 2011 .

[94]  S. Elmahdy,et al.  Remote sensing and GIS applications of surface and near-surface hydromorphological features in Darfur region, Sudan , 2013 .

[95]  K. Mardia.  Statistics of Directional Data , 1972 .

[96]  David E. Smith,et al.  Initial observations from the Lunar Orbiter Laser Altimeter (LOLA) , 2010 .

[97]  Qing Zhu,et al.  Digital terrain modeling - principles and methodology , 2004 .

[98]  M. Summerfield.  Tectonic geomorphology , 1991 .

[99]  Thomas A. Hennig,et al.  The Shuttle Radar Topography Mission , 2001, Digital Earth Moving.

[100]  D. Fabre,et al.  Global Bathymetry and Elevation Data at 30 Arc Seconds Resolution: SRTM30_PLUS , 2009 .

[101]  S. Bergbauer.  Testing the predictive capability of curvature analyses , 2007 .

[102]  David M. Mark,et al.  Geomorphometric Parameters: A Review and Evaluation , 1975 .

[103]  Carlos Henrique Grohmann,et al.  Multiscale Analysis of Topographic Surface Roughness in the Midland Valley, Scotland , 2011, IEEE Transactions on Geoscience and Remote Sensing.

[104]  D. Pollard,et al.  Using differential geometry to describe 3-D folds , 2007 .

[105]  T. G. Freeman,et al.  Calculating catchment area with divergent flow based on a regular grid , 1991 .

[106]  Thomas Young,et al.  An Essay on the Cohesion of Fluids , 1800 .

[107]  Igor V. Florinsky,et al.  Combined analysis of digital terrain models and remotely sensed data in landscape investigations , 1998 .

[108]  N. C. Toimil,et al.  Defining folds on three-dimensional surfaces , 2007 .

[109]  Kwan Tun Lee,et al.  Distinctions of geomorphological properties caused by different flow-direction predictions from digital elevation models , 2016, Int. J. Geogr. Inf. Sci..

[110]  S. D. Iyer,et al.  Tectonics and Geomorphology , 2008 .

[111]  A-Xing Zhu,et al.  Multi-scale digital terrain analysis and feature selection for digital soil mapping , 2010 .

[112]  M. Shirasawa,et al.  Visualizing topography by openness: A new application of image processing to digital elevation models , 2002 .

[113]  Wenzhe Fa,et al.  Topographic roughness of the northern high latitudes of Mercury from MESSENGER Laser Altimeter data , 2016 .

[114]  Interpolation by Regularized Spline with Tension � , 2022 .

[115]  I. Florinsky,et al.  A web system of virtual morphometric globes for Mars and the Moon , 2017 .

[116]  Boleslo E. Romero,et al.  On the topology of topography: a review , 2017 .

[117]  David E. Smith,et al.  The global topography of Mars and implications for surface evolution. , 1999, Science.

[118]  Ian S. Evans,et al.  Apparent Fractal Dimensions from Continental Scale Digital Elevation Models Using Variogram Methods , 2000, Trans. GIS.

[119]  T. Oguchi,et al.  3.6 Digital Terrain Modeling , 2013 .

[120]  Joan Serrat,et al.  Evaluation of Methods for Ridge and Valley Detection , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[121]  P. A. Sharyi,et al.  Geomorphometric study of the spatial organization of forest ecosystems , 2011, Russian Journal of Ecology.

[122]  T. Hengl,et al.  Geomorphometry: Concepts, software, applications , 2009 .

[123]  Bo Sun,et al.  Bedmap2: improved ice bed, surface and thickness datasets for Antarctica , 2012 .

[124]  J. McKean,et al.  Objective landslide detection and surface morphology mapping using high-resolution airborne laser altimetry , 2004 .

[125]  P. Tarolli.  High-resolution topography for understanding Earth surface processes: Opportunities and challenges , 2014 .

[126]  H. Bork,et al.  Detection of land surface memory by correlations between thickness of colluvial deposits and morphometric variables , 2013 .

[127]  K. Beven,et al.  A physically based, variable contributing area model of basin hydrology , 1979 .

[128]  Berthold K. P. Horn,et al.  Hill shading and the reflectance map , 1981, Proceedings of the IEEE.

[129]  B. Minasny,et al.  On digital soil mapping , 2003 .

[130]  Angelo Camerlenghi,et al.  The International Bathymetric Chart of the Arctic Ocean (IBCAO) Version 3.0 , 2012 .

[131]  Xiaoye Liu,et al.  Airborne LiDAR for DEM generation: some critical issues , 2008 .

[132]  Y. Deng,et al.  New trends in digital terrain analysis: landform definition, representation, and classification , 2007 .

[133]  Keith Beven,et al.  TOPMODEL : a critique. , 1997 .

[134]  Igor V. Florinsky,et al.  Accuracy of Local Topographic Variables Derived from Digital Elevation Models , 1998, Int. J. Geogr. Inf. Sci..

[135]  Igor V. Florinsky,et al.  A web-system of virtual morphometric globes , 2017 .

[136]  Andrea Gavioli,et al.  Analytical basis for determining slope lines in grid digital elevation models , 2014 .

[137]  K. Beven,et al.  THE PREDICTION OF HILLSLOPE FLOW PATHS FOR DISTRIBUTED HYDROLOGICAL MODELLING USING DIGITAL TERRAIN MODELS , 1991 .

[138]  M. T. Melis,et al.  Tracing the boundaries of Cenozoic volcanic edifices from Sardinia (Italy): a geomorphometric contribution , 2014 .

[139]  M. Rocca,et al.  MAD: robust image texture analysis for applications in high resolution geomorphometry , 2015, Comput. Geosci..

[140]  David Hodgetts,et al.  Laser scanning and digital outcrop geology in the petroleum industry: A review , 2013 .

[141]  Ján Kaňuk,et al.  Geomorphometric analysis of cave ceiling channels mapped with 3-D terrestrial laser scanning , 2016 .

[142]  I. Moore,et al.  Fractals, fractal dimensions and landscapes — a review , 1993 .

[143]  Joseph L. Awange,et al.  Environmental Monitoring using GNSS: Global Navigation Satellite Systems , 2012 .

[144]  Jozef Minár,et al.  A classification of geomorphometric variables , 2011 .

[145]  J. Maxwell.  L. On hills and dales: To the editors of the Philosophical Magazine and Journal , 1870 .

[146]  R. Wheate.  An Interactive Approach to Analytical Relief Shading , 2004 .

[147]  James W. Head,et al.  Kilometer‐scale roughness of Mars: Results from MOLA data analysis , 2000 .

[148]  Gary A. Peterson,et al.  Soil Attribute Prediction Using Terrain Analysis , 1993 .

[149]  Charles F. F. Karney.  Algorithms for geodesics , 2011, Journal of Geodesy.

[150]  P. Thomas.  Shapes of small satellites , 1987 .

[151]  Jürgen Oberst,et al.  The Phobos information system , 2014 .

[152]  Martin Jakobsson,et al.  General Bathymetric Chart of the Oceans (GEBCO) – Mapping the Global Seafloor , 2014 .