Article

Progress in Physical Geography 2017, Vol. 41(6) 723–752 ª The Author(s) 2017 An illustrated introduction Reprints and permission: sagepub.co.uk/journalsPermissions.nav to general geomorphometry DOI: 10.1177/0309133317733667 journals.sagepub.com/home/ppg

Igor V Florinsky Institute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Russia

Abstract 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 funda- mental 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.

Keywords Topography, geomorphometry, morphometric variable, surface, curvature, digital terrain modelling, spectral analytical method

I Introduction Topography is one of the main factors control- ling processes taking place in the near-surface Corresponding author: layer of the planet. In particular, it is one of the Igor V Florinsky, Institute of Mathematical Problems of Biology, The Keldysh Institute of Applied Mathematics, soil forming factors (Gerrard, 1981) influen- Russian Academy of Sciences, Pushchino, Moscow cing: (1) climatic and meteorological character- Region, 142290, Russia. istics, which control hydrological and thermal Email: [email protected] 724 Progress in Physical Geography 41(6) regimes of soils (Geiger, 1966); (2) prerequi- Currently, geomorphometric methods are sites for gravity-driven overland and intrasoil widely used to solve various multiscale prob- lateral transport of water and other substances lems of geomorphology, hydrology, remote sen- (Kirkby and Chorley, 1967); (3) spatial distri- sing, soil science, geology, geophysics, bution of vegetation cover (Franklin, 1995); and geobotany, glaciology, oceanology, climatol- (4) ecological patterns and processes (Ettema ogy, planetology and other disciplines; see and Wardle, 2002). At the same time, being a reviews (Clarke and Romero, 2017; Deng, result of the interaction of endogenous and 2007; Florinsky, 1998b; Jordan, 2007; Lecours exogenous processes of different scales, topo- et al., 2016; Mina´r et al., 2016; Moore et al., graphy reflects the geological structure of a 1991; Pike, 2000; Wasklewicz et al., 2013; terrain (Brocklehurst, 2010; Burbank and Wilson, 2012) and books (Florinsky, 2016; Anderson, 2012). Hengl and Reuter, 2009; Li et al., 2005; Wilson Given this connection, qualitative and quan- and Gallant, 2000). titative topographic information is widely used For the successful application of geomorpho- in the geosciences. Before the 1990s, topo- metric modelling, a researcher should know the graphic maps were the main source of quantita- basic mathematical concepts of general geo- tive information on topography. They were morphometry, be aware of the system of mor- analysed using conventional geomorphometric phometric variables and understand their techniques to calculate morphometric variables physical, mathematical and geographical mean- (e.g. slope and drainage density) and produce ings. The aim of this paper is to review these key morphometric maps (Clarke, 1966; Gardiner issues of general geomorphometry. and Park, 1978; Horton, 1945; Lastochkin, 1987; Mark, 1975; Strahler, 1957; Volkov, 1950). In the mid-1950s, a new research field II Topographic surface – digital terrain modelling – emerged in photo- grammetry (Miller and Leflamme, 1958; Real surfaces of land or submarine topography Rosenberg, 1955). DEMs (digital elevation are not smooth and regular. A rigorous mathe- models), two-dimensional discrete functions of matical treatment of such surfaces can be pro- elevation, became the main source of informa- blematic. However, in practice, it is sufficient tion on topography. Subsequent advances in to approximate a real surface by the topo- computer, space and geophysical technologies graphic surface. The topographic surface is a were responsible for the transition from conven- closed, oriented, continuously differentiable, two-dimensional manifold S in the three- tional to digital geomorphometry (Burrough, 3 1986; Dikau, 1988; Evans, 1972). This was sub- dimensional Euclidean space E . stantially supported by the development of the There are five limitations valid for the topo- physical and mathematical theory of the topo- graphic surface (Evans, 1980; Shary, 1995): graphic surface (Evans, 1972, 1980; Gallant and 1. The topographic surface is uniquely Hutchinson, 2011; Jencˇo, 1992; Jencˇo et al., defined by a continuous, single-valued 2009; Krcho, 1973, 2001; Shary, 1991, 1995; bivariate function Shary et al., 2002, 2005). As a result, geomor- phometry evolved into the science of quantita- z ¼ f ðx; yÞð1Þ tive modelling and analysis of the topographic where z is elevation and x and y are the surface and relationships between topogra- Cartesian coordinates. This means that phy and other natural and artificial compo- caves, grottos and similar landforms are nents of geosystems. excluded. Florinsky 725

2. The elevation function in equation (1) is In this paper, we review fundamental topo- smooth. This means that the topographic graphic variables associated with the theory of surface has derivatives of all orders. In the topographic surface and the concept of gen- practice, the first (p and q), second (r, t eral geomorphometry, which is defined as ‘the and s) and third (g, h, k and m) partial measurement and analysis of those characteris- derivatives of elevation are used tics of landform which are applicable to any qz qz continuous rough surface. ...General geomor- p ¼ ; q ¼ ; qx qy phometry as a whole provides a basis for the ... q2z q2z q2z quantitative comparison of qualitatively dif- r ¼ ; t ¼ ; s ¼ ; qx2 qy2 qxqy ferent landscapes ....’ (Evans, 1972: 18). There are several classifications of morpho- q3z q3z q3z q3z g ¼ ; h ¼ ; k ¼ ; m ¼ metric variables based on their intrinsic (math- qx3 qy3 qx2qy qxqy2 ematical) properties (Evans and Mina´r, 2011; ð2Þ Florinsky, 1998b, 2016: ch 2; Mina´r et al., 3. The topographic surface is located in a 2016; Shary, 1995; Shary et al., 2002). Here, uniform gravitational field. This limita- we use a modified classification of Florinsky tion is realistic for sufficiently small (2016: ch 2) adopting some ideas of Evans and portions of the geoid, when the equipoten- Mina´r (2011). tial surface can be considered as a plane. Morphometric variables can be divided into 4. The planimetric sizes of the topographic four main classes (Table 1): (1) local variables; surface are essentially less than the radius (2) non-local variables; (3) two-field specific of the planet. It is usually assumed that variables; and (4) combined variables. The the curvature of the planet may be ignored terms ‘local’ and ‘non-local’ are used regardless if the size of the surface portion is less of the study scale or model resolution. They are than at least 0.1 of the average planetary associated with the mathematical sense of a radius. In computations, either sizes of variable (c.f. the definitions of a local and a moving windows in local computational non-local variable – see sections 3.2 and 3.3). methods or sizes of the entire area in glo- Being a morphometric variable, elevation does bal computational methods must comply not belong to any class listed, but all topo- with this condition (see section 3.2). graphic attributes are derived from DEMs. 5. The topographic surface is a scale- dependent surface (Clarke, 1988). This 3.2 Local morphometric variables means that a fractal component of topography is considered as a high- A local morphometric variable is a single- frequency noise. valued bivariate function describing the geome- try of the topographic surface in the vicinity of a given point of the surface (Speight, 1974) along directions determined by one of the two pairs III Morphometric variables of mutually perpendicular normal sections (Figure 1(a) and (b)). 3.1 General A normal section is a curve formed by the A morphometric (or topographic) variable (or intersection of a surface with a plane containing attribute) is a single-valued bivariate function the normal to the surface at a given point o ¼ uðx; yÞ describing properties of the topo- (Pogorelov, 1957). At each point of the topo- graphic surface. graphic surface, an infinite number of normal 726 Progress in Physical Geography 41(6)

Table 1. Classification of morphometric variables. Non-local Two-field specific Combined Local variables variables variables variables Flow attributes Form attributes First-order variables GCARTI ADAISI AN AE Second-order variables kp kmin kh kmax kv H EK khe M 2 kve r Ka IS Kr C rot Third-order variables Dkh Dkv

sections can be constructed, but only two pairs of invariants. This means that they do not depend them are important for geomorphometry. on the direction of the gravitational acceleration The first pair of mutually perpendicular nor- vector. Among these are minimal curvature mal sections includes two principal sections (kmin), maximal curvature (kmax), mean curva- (Figure 1(a)) well known from differential ture (H), the Gaussian curvature (K), unspheri- geometry (Pogorelov, 1957). These are normal city curvature (M), Laplacian (r2), shape index sections with extreme – maximal and minimal – (IS), curvedness (C) and some others. bending at a given point of the surface. Flow attributes are associated with two sec- The second pair of mutually perpendicular tions dictated by gravity. These attributes are normal sections includes two ones (Figure gravity-field specific variables. Among these 1(b)) dictated by gravity (Shary, 1991). One of are slope (G), aspect (A), northwardness (AN), these two sections includes the gravitational eastwardness (AE), plan curvature (kp), horizon- acceleration vector and has a common tangent tal curvature (kh), vertical curvature (kv), differ- line with a slope line1 at a given point of the ence curvature (E), horizontal excess curvature topographic surface. The other section is per- (khe), vertical excess curvature (kve), accumula- pendicular to the first one and tangential to a tion curvature (Ka), ring curvature (Kr), rotor contour line at a given point of the topographic (rot), horizontal curvature deflection (Dkh), ver- surface. tical curvature deflection (Dkv) and some others. Local variables are divided into two types K, Ka and Kr are total curvatures; kmin, kmax, (see Table 1) – form and flow attributes – which kh, kv, khe and kve are simple curvatures; and H, are related to the two pairs of normal sections M and E are independent curvatures. Total and (Shary, 1995; Shary et al., 2002). simple curvatures can be expressed by elemen- Form attributes are associated with two prin- tary formulae using the independent curvatures cipal sections. These attributes are gravity field (Shary, 1995) (see equations (9)–(15) and Florinsky 727

Figure 1. Schemes for the definitions of local, non-local and two-field specific variables. (a) and (b) display two pairs of mutually perpendicular normal sections at a point P of the topographic surface: (a) Principal sections APA0 and BPB0; (b) Sections CPC0 and DPD0 allocated by gravity, n is the external normal, g is the gravitational acceleration vector, cl is the contour line, sl is the slope line. (c) Catchment and dispersive areas, CA and DA, are areas of figures P0AB (light grey) and P00AB (dark grey), correspondingly; b is the length of a 0 0 00 00 contour line segment AB; l1, l2, l3 and l4 are the lengths of slope lines P A, P B, AP and BP , correspondingly. (d) The position of the Sun in the sky: y is solar azimuth angle, c is solar elevation angle, and N is north direction. (19)–(23)). These 12 attributes constitute a first and second derivatives; and (3) third-order complete system of curvatures (Shary, 1995). variables – Dkh and Dkv – are functions of the Local topographic variables are functions of first, second and third derivatives. the partial derivatives of elevation (see equa- The partial derivatives of elevation (and thus tions (4)–(26)). In this regard, local variables local morphometric variables) can be estimated can be divided into three groups (Evans and from DEMs by: (1) several finite-difference Mina´r, 2011) (see Table 1): (1) first-order vari- methods using 3 3or5 5 moving windows ables – G, A, AN and AE – are functions of only (Evans, 1979, 1980; Florinsky, 1998c, 2009; the first derivatives; (2) second-order variables Mina´r et al., 2013; Shary, 1995; Shary et al., 2 – kmin, kmax, H, K, M, r , IS, C, kp, kh, kv, E, khe, 2002; Zevenbergen and Thorne, 1987); and kve, Ka, Kr and rot – are functions of both the (2) analytical computations based on DEM 728 Progress in Physical Geography 41(6) interpolation by local splines (Mita´sˇova´and the gravity field) and other fields, in particular solar Mita´sˇ, 1993) or global approximation of a irradiation and wind flow (Evans and Mina´r, 2011). DEM by high-order orthogonal polynomials Among two-field specific morphometric vari- (Florinsky and Pankratov, 2016). Comparison ables are reflectance (R) and insolation (I). These of the methods can be found elsewhere (Flor- variables are functions of the first partial deriva- insky, 1998a; Pacina, 2010; Schmidt et al., 2003). tives of elevation (see equation (2)) and angles describing the position of the Sun in the sky (Figure 1(d)). Reflectance and insolation can be 3.3 Non-local morphometric variables derived from DEMs using methods for the cal- A non-local (or regional) morphometric vari- culation of local variables (see section 3.2). able is a single-valued bivariate function Openness (Yokoyama et al., 2002), incorpor- describing a relative position of a given point ating the viewshed concept (Fisher, 1996), is on the topographic surface (Speight, 1974). also a two-field specific variable. In this case, Among non-local topographic variables are the second field is a set of unobstructed sigh- catchment area (CA) and dispersive area (DA). tlines between a given point of the topographic To determine non-local morphometric attri- surface and surrounding points. butes, one should analyse a relatively large ter- ritory with boundaries located far away from a 3.5 Combined morphometric variables given point (e.g. an entire upslope portion of a Morphometric variables can be composed from watershed) (Figure 1(c)). local and non-local variables. Such attributes Flow routing algorithms are usually applied consider both the local geometry of the topo- to estimate non-local variables. These algo- graphic surface and a relative position of a point rithms determine a route along which a flow is on the surface. distributed from a given point of the topo- Among combined morphometric variables graphic surface to downslope points. There are are topographic index (TI), stream power index several flow routing algorithms grouped into (SI) and some others. Combined variables are two types: (1) eight-node single-flow direction derived from DEMs by the sequential applica- (D8) algorithms using one of the eight possible tion of methods for non-local and local vari- directions separated by 45 to model a flow ables, followed by a combination of the results. from a given point (Jenson and Domingue, 1988; Martz and de Jong, 1988); and (2) multiple-flow direction (MFD) algorithms IV Brief description and illustration using the flow partitioning (Freeman, 1991; of morphometric variables Quinn et al., 1991). There are some methods combining D8 and MFD principles (Tarboton, 4.1 Data and methods 1997). Comparison of the algorithms can be To illustrate mathematical concepts of geomor- found elsewhere (Huang and Lee, 2016; Orlan- phometry, we used a DEM of Mount Ararat dini et al., 2012; Wilson et al., 2008). (Figure 2). The area is located between 44.2 and 44.5 E, and 39.6 and 39.8 N (the area size is 180 120, that is, 25.725 km 22.204 3.4 Two-field specific morphometric km). A spheroidal equal angular DEM was variables extracted from the quasi-global Shuttle Radar A two-field specific morphometric variable is a Topography Mission (SRTM) 1-arc-second single-valued bivariate function describing rela- DEM (Farr et al., 2007; USGS, 2015). The tions between the topographic surface (located in DEM includes 779,401 points (the matrix Florinsky 729

Figure 2. Mount Ararat, elevation.

1081 721). The grid spacing is 100, that is, the where Y0 and Y are transformed and original linear sizes of the 100 100 cell are 23.82 m values of a variable; n ¼ 0 for the non-local 30.84 m at the mean latitude, 39.7 N. variables; n ¼ 2, ...,9forthesecond-and Elevation approximation and derivation of third-order local variables; m ¼ 2 for the total local and two-field specific variables were car- curvatures and third-order local variables, m ¼ ried out by the recently developed spectral ana- 1 for other variables. Such a transformation con- lytical method (Florinsky and Pankratov, 2016). siders that: (1) dynamic ranges of some attri- The method is intended for the processing of butes include both positive and negative regularly spaced DEMs within a single frame- values; and (2) correct mapping of variables work including DEM global approximation, de- from different classes and groups require differ- noising, generalization and calculation of the ent exponent values for the same territory and partial derivatives of elevation. The method is DEM resolution. In the spectral analytical based on high-order orthogonal expansions method, selection of the n value depends on the using the Chebyshev polynomials with the sub- size of a study area. In our case, n ¼ 4 for third- sequent Feje´r summation. In this study, we used order variables and n ¼ 5 for second-order ones. 300 expansion coefficients of the original eleva- For the three-dimensional (3D) visualization tion function by the Chebyshev polynomials by of morphometric models, we used a 2 vertical both x-andy-axes. Calculation of non-local vari- exaggeration and a viewpoint with 45 azimuth ables was performed by the Martz–de Jong flow and 35 elevation. routing algorithm (Martz and de Jong, 1988) To evaluate statistical interrelationships adapted to spheroidal equal angular grids (Flor- between 12 attributes of the complete system insky, 2017a). To derive combined morphometric of curvatures, we performed multiple Spear- variables, we sequentially applied the Martz–de man’s rank correlation analysis of their models Jong algorithm adapted to spheroidal equal angu- (Table 2). Rank correlations allow the consid- lar grids and the spectral analytical method. eration of possible non-normality in curvature Wide dynamic ranges characterize the sec- distributions. The sample size was 7000 points ond- and third-order local variables as well as (the matrix 100 70); the grid spacing was 1000. non-local attributes. To avoid loss of informa- Data processing was conducted with the soft- tion on the spatial distribution of their values in ware Matlab R2008b (#The MathWorks Inc. mapping, a logarithmic transform should be 1984–2008) and LandLord 4.0 (Florinsky, applied (Shary et al., 2002) 2016: 413–414). Statistical analysis was carried out by Statgraphics Plus 3.0 (#Statistical Y0 ¼ signðYÞlnð1 þ 10nmjYjÞ ð3Þ Graphics Corp. 1994–1997). 730 Progress in Physical Geography 41(6)

Table 2. Spearman’s rank correlations between 12 attributes of the complete system of curvatures for the Mount Ararat models.

kh kv khe kve EKa Kr kmin kmax KHM kh 0.36 0.58 –0.56 –0.72 –0.07 — 0.74 0.79 –0.05 0.88 0.04 kv 0.36 –0.26 0.20 0.29 0.04 — 0.68 0.63 0.05 0.72 –0.06 khe 0.58 –0.26 –0.31 –0.81 — 0.49 — 0.49 –0.27 0.28 0.56 kve –0.56 0.20 –0.31 0.73 0.17 0.58 –0.47 –0.06 –0.14 –0.30 0.49 E –0.72 0.29 –0.81 0.73 0.11 — –0.26 –0.36 0.10 –0.36 –0.10 Ka –0.07 0.04 — 0.17 0.11 0.19 0.12 –0.17 0.78 –0.02 — Kr — — 0.49 0.58 — 0.19 –0.33 0.28 –0.27 — 0.71 kmin 0.74 0.68 — –0.47 –0.26 0.12 –0.33 0.55 0.30 0.86 –0.43 kmax 0.79 0.63 0.49 –0.06 –0.36 –0.17 0.28 0.55 –0.32 0.87 0.40 K –0.05 0.05 –0.27 –0.14 0.10 0.78 –0.27 0.30 –0.32 — –0.38 H 0.88 0.72 0.28 –0.30 –0.36 –0.02 — 0.86 0.87 — — M 0.04 –0.06 0.56 0.49 –0.10 — 0.71 –0.43 0.40 –0.38 —

*P 0.05 for statistically significant correlations; dashes are statistically insignificant correlations. 4.2 Local morphometric variables: flow Aspect is a non-negative variable ranging attributes from 0 to 360. The unit of A is degree. Aspect (Figure 3(b)) is a measure of the direction of gravity-driven flows. 1. Slope (G) is an angle between the tan- gential and horizontal planes at a given 3. Northwardness and eastwardness point of the topographic surface (Shary, aspect is a circular variable: its values 1991) range from 0 to 360, and both of these pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values correspond to the north direc- G ¼ arctan p2 þ q2 ð4Þ tion. Therefore, A cannot be used in Slope is a non-negative variable ranging linear statistical analysis. To avoid this from 0 to 90. The unit of G is degree. Slope problem, two auxiliary local indices (Figure 3(a)) determines the velocity of can be applied: northwardness (AN) and gravity-driven flows. eastwardness (AE) (Mardia, 1972) 2. Aspect (A) is an angle between the AN ¼ cos A ð6Þ

northern direction and the horizontal AE ¼ sin A ð7Þ projection of the two-dimensional vec- Northwardness (Figure 3(c)) and eastward- tor of gradient counted clockwise at a ness (Figure 3(d)) are dimensionless variables. given point of the topographic surface A is equal to 1, –1 and 0 on the northern, south- (Shary et al., 2002) N ern and eastern/western slopes, correspond- A ¼90½1 signðqÞð1 jsignðpÞjÞ ingly. AE isequalto1,–1and0onthe þ 180½1 þ signðpÞ eastern, western and northern/southern slopes, !correspondingly. These variables accentuate 180 q northern/southern and eastern/western trends signðpÞarccos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p2 þ q2 in the spatial distribution of slopes.

ð5Þ 4. Plan curvature (kp) is the curvature of a contour line at a given point of the Florinsky 731

Figure 3. Mount Ararat, local variables, flow attributes: (a) slope; (b) aspect; (c) northwardness; (d) eastwardness; (e) plan curvature; (f) horizontal curvature; (g) vertical curvature; (h) difference curvature; (i) horizontal excess curvature; (j) vertical excess curvature; (k) accumulation curvature; (l) ring curvature; (m) rotor; (n) horizontal curvature deflection; (o) vertical curvature deflection. 732 Progress in Physical Geography 41(6)

Figure 3. (Continued) Florinsky 733

Figure 3. (Continued) 734 Progress in Physical Geography 41(6)

topographic surface (Evans, 1979; > 0. Geomorphologically, kv mapping allows Krcho, 1973) revealing terraces and scarps. 2 2 q qr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pqs þ p t 7. Difference curvature (E)isahalf- kp ¼ ð8Þ difference of vertical and horizontal 2 2 3 ðp þ q Þ curvatures (Shary, 1995) This variable can be positive, negative or 2 2 –1 1 q r p2pqsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ p t zero. The unit of kp is m . Plan curvature (Fig- E ¼ ðkv khÞ¼ 2 2 2 2 2 ure 3(e)) is a measure of flow line2 divergence. ðp þ q Þ 1 þ p þ q Flow lines converge where k < 0 and diverge ð1 þ q2Þr 2pqs þð1 þ p2Þt p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k >0;k ¼ 0 refers to parallel flow lines. p p 2 ð1 þ p2 þ q2Þ3 5. Horizontal (or tangential) curvature ð11Þ (kh) is the curvature of a normal section tangential to a contour line (Figure This variable can be positive, negative or 1(b)) at a given point of the topographic zero. The unit of E is m–1. Difference curvature surface (Krcho, 1983; Shary, 1991) (Figure 3(h)) shows to what extent the relative deceleration of flows (measured by k ) is higher q2r 2pqs þ p2t v kh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ than flow convergence (measured by kh)ata ðp2 þ q2Þ 1 þ p2 þ q2 given point of the topographic surface. This variable can be positive, negative or zero. 8. Horizontal excess curvature (k )isa The unit of k is m–1. Horizontal curvature (Fig- he h difference of horizontal and minimal ure 3(f)) is a measure of flow convergence (one curvatures (Shary, 1995) of the two mechanisms of flow accumulation): gravity-driven overland and intrasoil lateral khe ¼ kh kmin ¼ M E ð12Þ flows converge where kh < 0, and they diverge This is a non-negative variable. The unit of where k > 0. Geomorphologically, k mapping –1 h h khe is m . Horizontal excess curvature (Figure allows revealing ridge and valley spurs (diver- 3(i)) shows to what extent the bending of a nor- gence and convergence areas, correspondingly). mal section tangential to a contour line is larger than the minimal bending at a given point of the 6. Vertical (or profile) curvature (kv) is the curvature of a normal section having a topographic surface. common tangent line with a slope line 9. Vertical excess curvature (kve) is a dif- (Figure 1(b)) at a given point of the ference of vertical and minimal curva- topographic surface (Aandahl, 1948; tures (Shary, 1995) Shary, 1991; Speight, 1974) kve ¼ kv kmin ¼ M þ E ð13Þ p2r þ 2pqs þ q2t kv ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ This is a non-negative variable. The unit of 2 2 2 2 3 –1 ðp þ q Þ ð1 þ p þ q Þ kve is m . Vertical excess curvature (Figure 3(j)) shows to what extent the bending of a nor- This variable can be positive, negative or zero. mal section having a common tangent line with The unit of k is m–1. Vertical curvature (Figure v a slope line is larger than the minimal bending at 3(g)) is a measure of relative deceleration and a given point of the topographic surface. acceleration of gravity-driven flows (one of the two mechanisms of flow accumulation). Over- 10. Accumulation curvature (Ka) is a prod- land and intrasoil lateral flows are decelerated uct of vertical and horizontal curvatures where kv < 0, and they are accelerated where kv (Shary, 1995) Florinsky 735

This variable can be positive, negative or Ka ¼ khkv –2 zero. The unit of Dkh is m . Horizontal curva- ðq2r 2pqs þ p2tÞðp2r þ 2pqs þ q2tÞ ture deflection (Figure 3(n)) measures the ¼ ½ðp2 þ q2Þð1 þ p2 þ q2Þ2 deviation of kh from loci of the extreme curva- ture of the topographic surface. ð14Þ 14. Vertical curvature deflection (or tan- This variable can be positive, negative or –2 gent change of profile curvature) (Dkv) zero. The unit of Ka is m . Accumulation cur- is a derivative of kv by the contour line vature (Figure 3(k)) is a measure of the extent of length (Jencˇo et al., 2009) local accumulation of flows at a given point of the topographic surface. q3m p3k þ 2pqðqk pmÞpqðqh pgÞ Dkv ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11. Ring curvature (K ) is a product of hor- ðp2 þ q2Þ3ð1 þ p2 þ q2Þ3 r 2 3 izontal excess and vertical excess cur- 6 2 r t 2 5 p2 q2 7 vatures (Shary, 1995) 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið þ Þ þ ð þ Þ5 rot þ kv 2 2 2 2 2 2 3 ð1 þ p þ q Þ Kr ¼ khekve ¼ M E ð1 þ p þ q Þ 2 ðp2 q2Þs pqðr tÞ ð18Þ ¼ ð15Þ ðp2 þ q2Þð1 þ p2 þ q2Þ This variable can be positive, negative or This is a non-negative variable. The unit of zero. The unit of D is m–2. Vertical curvature –2 kv Kr is m . Ring curvature (Figure 3(l)) describes deflection (Figure 3(o)) measures the deviation flow line twisting. of kv from loci of the extreme curvature of the 12. Rotor (rot) is a curvature of a flow line topographic surface. (Shary, 1991) 4.3 Local morphometric variables: form ðp2 q2Þs pqðr tÞ rot ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ attributes ðp2 þ q2Þ3 1. Minimal curvature (k ) is a curvature This variable can be positive, negative or min –1 of a principal section (Figure 1(a)) with zero. The unit of rot is m . Rotor (Figure the lowest value of curvature at a given 3(m)) describes flow line twisting. A flow line point of the surface (Gauss, 1828; Shary, turns clockwise if rot > 0, while it turns counter 1995) clockwise if rot <0. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ H M ¼ H H 2 K ð19Þ 13. Horizontal curvature deflection (or gen- min This variable can be positive, negative or erating function) (Dkh) is a derivative of zero. The unit of k is m–1. Geomorphologi- kh by the contour line length (Florinsky, min 2009; Shary and Stepanov, 1991) cally, positive values of minimal curvature (Figure 4(a)) correspond to local convex land- 3 3 q g p h þ 3pqðpm qkÞ forms, while its negative values relate to elon- Dkh ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp2 þ q2Þ3ð1 þ p2 þ q2Þ gated concave landforms (e.g. troughs and valleys). 2 þ 3ðp2 þ q2Þ khrot 2. Maximal curvature (k ) is a curvature 1 þ p2 þ q2 max of a principal section (Figure 1(a)) with ð17Þ the highest value of curvature at a given 736 Progress in Physical Geography 41(6)

Figure 4. Mount Ararat, local variables, form attributes: (a) minimal curvature; (b) maximal curvature; (c) mean curvature; (d) Gaussian curvature; (e) unsphericity curvature; (f) Laplacian; (g) shape index; (h) curvedness. Florinsky 737

Figure 4. (Continued)

point of the surface (Gauss, 1828; Shary, 1 1 1995) H ¼ ðkmin þ kmaxÞ¼ ðkh þ kvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 kmax ¼ H þ M ¼ H þ H K ð20Þ ð1 þ q Þr 2pqs þð1 þ p Þt ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ This variable can be positive, negative or 2 2 3 –1 2 ð1 þ p þ q Þ zero. The unit of kmax is m . Geomorpho- logically, positive values of maximal curva- This variable can be positive, negative or zero. ture (Figure 4(b)) correspond to elongated The unit of H is m–1. Mean curvature (Figure 4(c)) convex landforms (e.g. ridges), while its represents two accumulation mechanisms of negative values relate to local concave gravity-driven substances – convergence and rela- landforms. tive deceleration of flows – with equal weights. 3. Mean curvature (H) is a half-sum of 4. Gaussian curvature (K) is a product of max- curvatures of any two orthogonal nor- imal and minimal curvatures (Gauss, 1828) mal sections at a given point of the rt s2 topographic surface (Shary, 1991; K ¼ kminkmax ¼ ð22Þ 2 2 2 Young, 1805) ð1 þ p þ q Þ 738 Progress in Physical Geography 41(6)

This variable can be positive, negative or breaking, stretching and compressing zero. The unit of K is m–2. According to (Gauss, 1828). Teorema egregium, Gaussian curvature 5. Unsphericity curvature (M)isahalf- (Figure 4(d)) retains values in each point difference of maximal and minimal cur- of the surface after its bending without vatures (Shary, 1995)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 M ¼ ðkmax kminÞ¼ H K 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 8 sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi! 9 u > 2 > u > 1 þ q2 1 þ p2 > u > r t 1 p2 q2 > u > 2 2 ð þ þ Þ > u < 1 þ p 1 þ q = u 1 ¼ u 3 > sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi!> u4ð1 þ p2 þ q2Þ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > t > 1 þ q2 1 þ p2 > :> þ pqr 2s ð1 þ q2Þð1 þ p2Þ þ pqt ;> 1 þ p2 1 þ q2 ð23Þ

This is a non-negative variable. The unit of its absolute values from 0 to 0.5 relate to hyper- Mism–1. Unsphericity curvature (Figure 4(e)) bolic ones (saddles). shows the extent to which the shape of the 8. Curvedness (C) is the root mean square surface is non-spherical at a given point. of maximal and minimal curvatures 6. Laplacian ( 2) is a second-order differ- (Koenderink and van Doorn, 1992) r rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ential operator, which can be defined as k2 þ k2 the divergence of the gradient of a func- C ¼ max min ð26Þ tion z (Laplace, 1799) 2 r2 ¼ div grad ¼ r þ t ð24Þ This is a non-negative variable. The unit of C is m–1. Curvedness (Figure 4(h)) measures the mag- This variable can be positive, negative or –1 nitude of surface bending regardless of its shape. zero. The unit of r2 is m . Laplacian (Figure Flat areas have low values of C, while areas with 4(f)) measures the flux density of slope lines. sharp bending are marked by high values of C. 7. Shape index (IS) is a continual form of the discrete Gaussian landform classification 4.4 Non-local morphometric variables (Koenderink and van Doorn, 1992)  1. Catchment area (CA)isanareaofa 2 k þ k IS ¼ arctan max min ð25Þ closed figure formed by a contour seg- p kmax kmin ment at a given point of the topographic This is a dimensionless variable ranging from surface and two flow lines coming from –1 to 1 (Figure 4(g)). Its positive values relate to upslope to the contour segment ends convex landforms, while its negative ones cor- (Figure 1(c)) (Speight, 1974). This is a respond to concave landforms; its absolute val- non-negative variable. The unit of CA ues from 0.5 to 1 are associated with elliptic is m2. The catchment area (Figure 5(a)) surfaces (hills and closed depressions), whereas is a measure of the contributing area. Florinsky 739

Figure 5. Mount Ararat, non-local variables: (a) catchment area; (b) dispersive area.

2. Dispersive area (DA)isanareaofaclosed This is a non-negative dimensionless vari- figure formed by a contour segment at a able normalized for the range from 0 to 1. given point of the topographic surface and Reflectance maps (Figure 6(a)) clearly and two flow lines going down slope from the plastically display the topography. contour segment ends (Figure 1(c)) 2. Insolation (I) is a proportion of maximal (Speight, 1974). This is a non-negative vari- 2 direct solar irradiation at the Sun’s position able. The unit of DA is m . The dispersive determined by solar azimuth and solar ele- area (Figure 5(b)) is a measure of a down- vation (Figure 1(d)) (Shary et al., 2005) slope area potentially exposed by flows passing through a given point. I ¼ 50f1 þ sign½sin c cos cðpsin y þ qcos yÞg ½sin c cos cðpsin y þ qcos yÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4.5 Two-field specific morphometric 1 þ p2 þ q2 variables ð28Þ

1. Reflectance (R)isameasureofthe This is a non-negative variable ranging from 0 brightness of an illuminated surface to 100. The unit of I is per cent. Insolation (Figure (Horn, 1981). For the case of the ideal 6(b)) characterizes a perpendicularity of the inci- diffusion (the Lambertian surface) dence of solar rays on the topographic surface. 1 psin y ctg c qcos y ctg c R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ p2 þ q2 1 þðsin y ctg cÞ2 þðcos y ctg cÞ2 4.6 Combined morphometric variables ð27Þ where y and c are the solar azimuth and eleva- 1. Topographic index (TI)isaratioof tion angles (Figure 1(d)), correspondingly. catchment area to slope at a given point 740 Progress in Physical Geography 41(6)

Figure 6. Mount Ararat, two-field specific variables: (a) reflectance (the Lambertian model; y ¼ 45; c ¼ 35); (b) insolation (y ¼ 90; c ¼ 20).

of the topographic surface (Beven and measure of potential flow erosion and related Kirkby, 1979) landscape processes. SI reaches high values in areas with high values of both CA and G (e.g. a TI ¼ ln½1 þ CA=ð103 þ tgGÞ ð29Þ highly sloped terrain with a large upslope con- This is a non-negative dimensionless vari- tributing area). able. The term 10–3 is used to avoid division by zero for the case of horizontal planes. The V Discussion topographic index (Figure 7(a)) is a measure of the extent of flow accumulation in TOPMO- 5.1 The main paradox of general DEL, a concept of distributed hydrological geomorphometry modelling (Beven and Kirkby, 1979). TI It is quite obvious that the real topography is non- reaches high values in areas with high values smooth and so non-differentiable (Shary, 2008). of CA at low values of G (e.g. a terrain with a This means that it cannot have partial derivatives large upslope contributing area and flat local and so local morphometric variables, which are topography). functions of the partial derivatives (sections 4.2 2. Stream power index (SI) is a product of and 4.3). However, introducing the concept of catchment area and slope at a given point the topographic surface and accepting the condi- of the topographic surface (Moore et al., tion of its smoothness (section II), we are able not 1991) only to calculate local variables, but also to apply these ‘non-existent and abstract’ attributes to SI ¼ lnð1 þ CA tgGÞð30Þ study and model relationships between topogra- phy and properties of other components of This is a non-negative dimensionless vari- geosystems (for examples, see section 5.2). able. Stream power index (Figure 7(b)) is a Topography influences soil and other landscape Florinsky 741

Figure 7. Mount Ararat, combined variables: (a) topographic index; (b) stream power index. properties mainly via gravity-driven overland Concepts and methods of fractal geometry (Gao and intrasoil lateral migration and accumulation and Xia, 1996; Mandelbrot, 1967) were sometimes of water. Dependences of soil and other land- applied in general geomorphometry (Klinkenberg, scape properties on ‘non-existent’ morpho- 1992; Mark and Aronson, 1984; McClean and metric attributes may be explained by a Evans, 2000; Xu et al., 1993). In particular, there hypothetical assumption that real soil hydrolo- were ideas to use fractal dimensions of a terrain as gical processes also ‘smooth’ the topography a morphometric index. However, Clarke (1988) ‘ignoring’ its minor non-smooth details. clearly demonstrated that a fractal component of Computationally, the non-smoothness of the topography is important for landscape simulation real topography is considered in finite- only. In scientific studies, a fractal component of difference methods calculating local variables topography can be considered a high-frequency with 3 3or5 5 moving windows (Evans, noise. Thus, we use such a limitation (#5) in the 1980; Florinsky, 1998c, 2009; Mina´r et al., concept of the topographic surface (see section II). 2013; Shary, 1995; Shary et al., 2002). In these algorithms, the smoothness condition of the topographic surface should be met within a 5.2 Application of morphometric variables moving window only (a polynomial is approxi- Let us briefly consider the application of the mated to 9 or 25 points of a window; there is no reviewed morphometric variables. Slope and approximation between windows). In the same aspect have been well known in geosciences for manner, the non-smoothness of the real topo- many decades, and so there is no need to specify graphy is considered in calculations of its their fields of application. Curvatures are sys- spatial statistical metrics (so-called roughness tematically used in geomorphic studies to indices, see section 5.3). Such calculations describe, analyse and model landforms and their are also performed with moving windows. evolution (Burian et al., 2015; Elmahdy and 742 Progress in Physical Geography 41(6)

Mohamed, 2013; Evans, 1980; Guida et al., (equations (4)–(30)). For example, mean curva- 2016; Melis et al., 2014; Mitusov et al., 2013, ture is a combination of horizontal and vertical 2014; Prasicek et al., 2014; Temovski and Mile- curvatures (equation (21)). So, it is not surpris- vski, 2015). In soil science and ecology, curva- ing that the results of correlation analysis of tures are regularly applied to study relationships models from the complete system of curvatures in the topography–soil–vegetation system and to demonstrate rather high relations between some perform predictive soil and vegetation mapping attributes (Table 2); see, for example, correla- (Behrens et al., 2010; Florinsky et al., 2002; tion coefficients between kh and kmin, khe and E, Moore et al., 1993; Omelko et al., 2012; Sharaya kmax and H, as well as M and Kr. Thus, questions and Shary, 2011; Shary and Pinskii, 2013; Shary arise of (1) an information redundancy of the and Smirnov, 2013; Shary et al., 2016; Stumpf variables discussed in section IV, and (2) the et al., 2017). In structural geology, curvatures are selection of topographic attributes for a partic- utilized to reveal hidden faults as well as to study ular (e.g. soil) study. fold geometry (Bergbauer, 2007; Florinsky, First, although statistical relationships 1996; Lisle and Toimil, 2007; Mynatt et al., between morphometric variables are useful in 2007; Roberts, 2001; Stewart and Podolski, geomorphic research (Csillik et al., 2015; 1998). Application of horizontal and vertical cur- Evans, 1980, 1998; Evans and Cox, 1999), such vature deflections may be useful to recognize statistics cannot be utilized to select particular thalweg and crest lines (Florinsky, 2009; Mina´r variables. Such a selection should be based on et al., 2013). Reflectance maps are well known in their physical/mathematical interpretations. cartography as hill-shaded maps (Jenny, 2001). Second, it is a priori impossible to know which Insolation is utilized to describe the thermal particular morphometric variables control, for regime of slopes in geobotanical and agricultural example, a soil property under given natural research (Shary and Smirnov, 2013; Shary et al., conditions. To search topographic attributes 2016). Catchment and dispersive areas as well as controlling the property, it is reasonable to uti- topographic index are widely used in hydrologi- lize a representative set of the attributes at the cal and related soil, plant and geomorphic studies first stage of a study (e.g. soil predictive model- (Behrens et al., 2010; Beven, 1997; Florinsky ling with morphometric variables as predictors; et al., 2002; Mitusov et al., 2013, 2014; Moore Florinsky, 2016: ch 11; McBratney et al., 2003). et al., 1993; Omelko et al., 2012; Sharaya and When governing variables are found by a corre- Shary, 2011; Shary and Smirnov, 2013). Stream lation analysis, one can reduce their number power index is applied in erosion and soil because it is incorrect to use together, for research (Florinsky et al., 2002; Moore et al., instance, horizontal, vertical and mean curva- 1993; Omelko et al., 2012). tures in a predictive (e.g. regression) model of Comprehensive reviews of the application of the property (Florinsky, 2016: 309–311; Shary the morphometric variables can be found else- and Pinskii, 2013). where (Florinsky, 1998b, 2016: pts II and III; To delineate and quantitatively describe ter- Hengl and Reuter, 2009: pt. III; Moore et al., rains of different morphologic structure as well 1991; Wilson and Gallant, 2000). as geological composition and age, digital mod- els and maps of topographic (or surface) rough- ness are used in geomorphic, geological and 5.3 Statistical aspects of general planetary studies (Fa et al., 2016; Frankel and geomorphometry Dolan, 2007; Grohmann et al., 2011; Herzfeld It is clear that some topographic variables can and Higginson, 1996; Karachevtseva et al., be expressed as linear combinations of others 2015; Kreslavsky and Head, 2000; McKean and Florinsky 743

Roering, 2004; Trevisani and Rocca, 2015). To if one works with flow attributes because these avoid confusion, we should stress that topo- are gravity-field specific variables (section 3.2). graphic roughness is not a morphometric vari- In water dynamics/runoff on cave ceilings and able. It is a generalized term, without a clear walls, forces of surface tension can play a domi- definition, which is usually applied to denote nant role, rather than gravity-driven overland some spatial statistical metrics of a morpho- lateral transport of liquids controlled by hori- metric model (e.g. median absolute value of zontal and vertical curvatures (section 4.2). This slope, eigenvalue ratios of external normals, calls into question the adequacy of utilizing standard deviation of elevation, slope, etc.). these morphometric variables (e.g. Brook Models of topographic roughness are usually et al., 2017) to study hydrological processes in calculated using windows moving along a mor- caves, grottos and niches, at least for microto- phometric model. The principal difference pography of ceilings. This problem requires between morphometric variables and topo- in-depth study and development of recommen- graphic roughness is as follows: Morphometric dations for geomorphometric modelling in spe- variables mathematically describe ‘geometry’ leology and related disciplines. of the topographic surface and its physically There are three other interrelated pending interpretable relationships with other compo- problems. nents of geosystems, while topographic rough- ness statistically describes ‘spatial variability’ (1) A set of the third-order local variables of a particular morphometric property of the was introduced by Jencˇo et al. (2009). topographic surface in the vicinity of a given In general, these attributes describe point of the surface. deviations of the second-order variables from loci of the extreme curvature of the topographic surface (Florinsky, 5.4 Some pending problems of general 2009; Mina´r et al., 2013; Shary and Ste- geomorphometry panov, 1991). Only two of them have Advances in the terrestrial light detection and been discussed here, horizontal and ver- ranging (LiDAR) technology (Hodgetts, 2013) tical curvature deflections (section 4.2), have allowed the production of 3D high- because their interpretation and practi- resolution models of caves and grottos. Some cal application are more or less clear: researchers try to use geomorphometric model- their zero values correlate with crest ling to analyse the geometry of inner surfaces of and thalweg lines. Other third-order these objects (Brook et al., 2017; Gallay et al., variables are still insufficiently studied. 2015, 2016). However, such an application of (2) Comparing with the theory of local geomorphometry violates the limitation #1 stat- variables (Jencˇo et al., 2009; Shary, ing that the topographic surface is uniquely 1995), a mathematical theory for non- defined by a single-valued function (see section local attributes is still little developed II). It is obvious that caves and grottos cannot be (Gallant and Hutchinson, 2011; Koshel directly described by a single-valued elevation and Entin, 2017; Orlandini et al., 2014; function because, at least, two values of elevation Peckham, 2013). In particular, Gallant correspond to a pair of planimetric coordinates. and Hutchinson (2011) proposed a dif- One may ignore this limitation working with ferential equation for calculating spe- form attributes (e.g. Gallay et al., 2015, 2016), cific catchment area. However, a which are gravity field invariants (see section general analytical theory of non-local 3.2). However, the limitation becomes important morphometric variables still does not 744 Progress in Physical Geography 41(6)

exist. The diversity of flow routing Existing algorithms of geomorpho- algorithms for calculating non-local metry can be applied to DEMs given variables (section 3.3) is a result of on plane square grids (Evans, 1980; underdevelopment of an appropriate Florinsky, 2009; Freeman, 1991; Martz mathematical theory. and de Jong, 1988; Mina´r et al., 2013; (3) Loci of extreme curvature of the topo- Quinn et al., 1991; Shary, 1995; Tarbo- graphic surface may define four types ton, 1997; Zevenbergen and Thorne, of structural lines: crests (or ridges), 1987) as well as spheroidal equal angu- thalwegs (or courses), as well as top and lar grids located on an ellipsoid of rev- bottom edges (or breaks) of slopes. olution and a sphere (Florinsky, 1998c, Although structural lines have been 2017a). Geomorphometric computa- mathematically studied for many tions on spheroidal equal angular grids decades, an analytical solution based are trivial for the Earth, Mars, the on local differential geometric criteria Moon, Venus and Mercury (Florinsky, has not yet been found (see reviews: 2008a, 2008b; Florinsky and Filippov, Koenderink and van Doorn, 1993, 2017; Florinsky et al., 2017a). This is 1994). At the same time, crests and thal- because forms of the above mentioned wegs can be considered as two topolo- celestial bodies can be described by an gically connected tree-like hierarchical ellipsoid of revolution or a sphere. For structures (Clarke and Romero, 2017). these surfaces, there are well-developed Maxwell (1870) defined a ridge as a theory and computational algorithms to slope line connecting a sequence of solve the inverse geodetic problem local maximal and saddle points, and a (Bagratuni, 1967; Bessel, 1825; Kar- thalweg as a slope line connecting a ney, 2013; Morozov, 1979; Sjo¨berg, sequence of local minimal and saddle 2006; Vincenty, 1975). Formulae for points. Rothe (1915) argued that crests the solution of this problem are used and thalwegs are singular solutions of to determine parameters of moving the differential equation of the slope windows in morphometric calculations lines. Koenderink and van Doorn (Florinsky, 2017a). At the same time, it (1993) supposed that crests and thal- is advisable to apply a triaxial ellipsoid wegs are the special type of slope lines for describing forms of small moons where other ones converge, and they and asteroids (Bugaevsky, 1999; also argued that local differential geo- Stooke, 1998; Thomas, 1989). How- metric criteria for crests and thalwegs ever, for the case of a triaxial ellipsoid, cannot exist. Since the problem of ana- solutions of the inverse geodetic prob- lytical description of structural lines has lem are presented in general form only not been resolved, practical derivation (Bespalov, 1980; Jacobi, 1839; Karney, of crests and thalwegs is mainly carried 2012; Krasovsky, 1902; Panou, 2013; out by flow routing algorithms, similar Shebuev, 1896). This makes difficult to calculations of non-local variables geomorphometric modelling of small (see reviews: Clarke and Romero, moons and asteroids. Thus, the next 2017; Lo´pez et al., 1999; Tribe, 1992). item on the geomorphometric agenda Hopefully, these three interrelated is development of computational algo- problems will be solved in the near rithms for modelling on a surface of a future. triaxial ellipsoid (Florinsky, 2017b). Florinsky 745

VI Conclusions Spaceborne Thermal Emission and In the last 25 years, great progress has been Reflection Radiometer Global DEM made in the field of geomorphometry. (ASTER GDEM) (Toutin, 2008), SRTM30_PLUS (Becker et al., 2009), (1) A physical and mathematical theory of SRTM15_PLUS (Olson et al., 2014), the topographic surface has substan- WorldDEM (Zink et al., 2014) and tially evolved (Evans, 1972, 1980; Gal- Advanced Land Observing Satellite lant and Hutchinson, 2011; Jencˇo, World 3D model (ALOS World 3D) 1992; Jencˇo et al., 2009; Koenderink (Tadono et al., 2014)) have been pro- and van Doorn, 1992, 1994; Krcho, duced and become available. 1973, 2001; Shary, 1991, 1995; Shary (5) There are new bathymetric DEMs et al., 2002, 2005). (Arndt et al., 2013; Jakobsson et al., (2) Effective algorithms for calculating 2012; Weatherall et al., 2014). Submar- attributes from DEMs have been devel- ine topography influences ocean cur- oped (Evans, 1980; Florinsky, 1998c, rents, distribution of perennial ice and 2009; Florinsky and Pankratov, 2016; sediment migration. Submarine valleys Freeman, 1991; Martz and de Jong, participate in the gravity-driven trans- 1988; Mina´r et al., 2013; Shary, 1995; port of substances from land to ocean. Tarboton, 1997). Geomorphometric modelling of sub- (3) High- and super high-resolution DEMs marine topography can provide new have become widely available (Tarolli, opportunities for oceanological, marine 2014) owing to advances in LiDAR geomorphological and marine geologi- technology (Liu, 2008), real-time kine- cal studies. matic global navigation satellite system (6) There are new DEMs for the ice bed (GNSS) surveys (Awange, 2012: ch 8), topography of Greenland (Bamber areal surveys based on unmanned aerial et al., 2013) and Antarctica (Fretwell vehicles (UAVs) (Colomina and Molina, et al., 2013). In this case, application 2014), and structure-from-motion (SfM) of geomorphometric methods can pro- techniques (Smith et al., 2016). UAVs duce new results for understanding both and SfM introduce a low-cost alternative glaciological processes and geological to manned aerial surveys and conven- structure of glacier-covered terrains. tional photogrammetry. UAV/SfM- (7) Advances in 3D seismic surveys offer a derived, photogrammetrically sound wide field of activity in applying prin- DEMs can be successfully used for fur- ciples of geomorphometric modelling ther geomorphometric modelling: it is to subterranean surfaces (Chopra and possible to produce noiseless, well- Marfurt, 2007) that can give a new readable and interpretable models of impetus to geological research. slope and curvatures (with the resolution (8) Global, extra-terrestrial medium- of 5–20 cm) for grassy areas with sepa- resolution DEMs for Mars (Smith rately standing groups of trees, shrubs et al., 1999), the Moon (Smith et al., and other objects (Florinsky et al., 2010), Phobos (Karachevtseva et al., 2017b). 2014) and Mercury (Becker et al., (4) Quasi-global and global, medium- and 2016) are available. Geomorphometry high-resolution DEMs of the Earth can provide additional tools for com- (SRTM1 (Farr et al., 2007), Advanced parative planetary studies. 746 Progress in Physical Geography 41(6)

(9) Morphometric globes for the Earth, 2. A flow line is a plane curve, a projection of a slope line Mars and the Moon have been devel- to a horizontal plane. oped (Florinsky and Filippov, 2017; Florinsky et al., 2017a) owing to References advances in scientific visualization Aandahl AR (1948) The characterization of slope positions (Hansen and Johnson, 2005) and virtual and their influence on the total nitrogen content of a few globe (Cozzi and Ring, 2011) technol- virgin soils of western Iowa. Soil Science Society of ogies. Such virtual globes can be easily America Proceedings 13: 449–454. utilized by users without special train- Arndt JE, Schenke HW, Jakobsson M, et al. (2013) The ing in geomorphometry. International Bathymetric Chart of the Southern Ocean (IBCSO) Version 1.0: A new bathymetric compilation The above achievements and future possibi- covering circum-Antarctic waters. Geophysical Research lities, as well as reproducibility, relative simpli- Letters 40: 3111–3117. city and flexibility of geomorphometric Awange JL (2012) Environmental Monitoring Using GNSS: Global navigation satellite systems. Berlin: Springer. methods, determine their potential for geos- Bagratuni GV (1967) Course in Spheroidal Geodesy. ciences. It should be realized, however, that the Wright-Patterson, OH: Foreign Technology Division. governing factor for the evolution of digital ter- Bamber JL, Griggs JA, Hurkmans RTWL, et al. (2013) A rain analysis is advances in the theory of the new bed elevation data-set for Greenland. Cryosphere topographic surface, which lays a rigorous 7: 499–510. physical and mathematical foundation for both Becker JJ, Sandwell DT, Smith WHF, et al. (2009) Global computation algorithms and applied issues of bathymetry and elevation data at 30 arc seconds topographic modelling. resolution: SRTM30_PLUS. Marine Geodesy 32: 355–371. Acknowledgements Becker KJ, Robinson MS, Becker TL, et al. (2016) First global digital elevation model of Mercury. In: 47th The author is grateful to Ian Evans and two anon- Lunar and Planetary Science Conference, 21–25 ymous reviewers for constructive criticism. March 2016, Woodlands, Texas. Houston, TX: Lunar and Planetary Institute: # 2959. Declaration of Conflicting Interests Behrens T, Zhu AX, Schmidt K, et al. (2010) Multi-scale The author declared no potential conflicts of interest digital terrain analysis and feature selection for digital with respect to the research, authorship, and/or pub- soil mapping. Geoderma 155: 175–185. lication of this article. Bergbauer S (2007) Testing the predictive capability of curvature analyses. Geological Society of London Funding Special Publications 292: 185–202. Bespalov NA (1980) Methods for Solving Problems of The author disclosed receipt of the following finan- Spheroidal Geodesy. Moscow: Nedra (in Russian). cial support for the research, authorship, and/or Bessel FW (1825) U¨ ber die Berechnung der geographischen publication of this article: This study was supported La¨ngen und Breiten aus geoda¨tischen Vermessungen. by the Russian Foundation for Basic Research, Grant Astronomische Nachrichten 4: 241–254. number 15-07-02484. Beven K (1997) TOPMODEL: A critique. Hydrological Processes 11: 1069–1085. Notes Beven KJ and Kirkby MJ (1979) A physically-based 1. A slope line is a space curve on the topographic surface. variable contributing area model of basin hydrology. At each point of the curve, the direction of the tangent to Hydrological Science Bulletin 24: 43–69. the curve coincides with the direction of the tangential Brocklehurst SH (2010) Tectonics and geomorphology. component of the gravitational force (Cayley, 1859). Progress in Physical Geography 34: 357–383. Florinsky 747

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