An illustrated introduction to general geomorphometry

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 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.

Keywords

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

I Introduction

Topography is one of the main factors controlling processes taking place in the near-surface layer of the planet. In particular, it is one of the soil forming factors (Gerrard, 1981) influencing: (1) climatic and meteorological characteristics, which control hydrological and thermal regimes of soils (Geiger, 1966); (2) prerequisites for gravity-driven overland and intrasoil lateral transport of water and other substances (Kirkby and Chorley, 1967); (3) spatial distribution of vegetation cover (Franklin, 1995); and (4) ecological patterns and processes (Ettema and Wardle, 2002). At the same time, being a result of the interaction of endogenous and exogenous processes of different scales, topography reflects the geological structure of a terrain (Brocklehurst, 2010; Burbank and Anderson, 2012).

Given this connection, qualitative and quantitative topographic information is widely used in the geosciences. Before the 1990s, topographic maps were the main source of quantitative information on topography. They were analysed using conventional geomorphometric techniques to calculate morphometric variables (e.g. slope and drainage density) and produce morphometric maps (Clarke, 1966; Gardiner and Park, 1978; Horton, 1945; Lastochkin, 1987; Mark, 1975; Strahler, 1957; Volkov, 1950). In the mid-1950s, a new research field – digital terrain modelling – emerged in photogrammetry (Miller and Leflamme, 1958; Rosenberg, 1955). DEMs (digital elevation models), two-dimensional discrete functions of elevation, became the main source of information on topography. Subsequent advances in computer, space and geophysical technologies were responsible for the transition from conventional to digital geomorphometry (Burrough, 1986; Dikau, 1988; Evans, 1972). This was substantially supported by the development of the physical and mathematical theory of the topographic surface (Evans, 1972, 1980; Gallant and Hutchinson, 2011; Jenčo, 1992; Jenčo et al., 2009; Krcho, 1973, 2001; Shary, 1991, 1995; Shary et al., 2002, 2005). As a result, geomorphometry evolved into the science of quantitative modelling and analysis of the topographic surface and relationships between topography and other natural and artificial components of geosystems.

Currently, geomorphometric methods are widely used to solve various multiscale problems of geomorphology, hydrology, remote sensing, soil science, geology, geophysics, geobotany, glaciology, oceanology, climatology, planetology and other disciplines; see reviews (Clarke and Romero, 2017; Deng, 2007; Florinsky, 1998b; Jordan, 2007; Lecours et al., 2016; Minár et al., 2016; Moore et al., 1991; Pike, 2000; Wasklewicz et al., 2013; Wilson, 2012) and books (Florinsky, 2016; Hengl and Reuter, 2009; Li et al., 2005; Wilson and Gallant, 2000).

For the successful application of geomorphometric modelling, a researcher should know the basic mathematical concepts of general geomorphometry, be aware of the system of morphometric variables and understand their physical, mathematical and geographical meanings. The aim of this paper is to review these key issues of general geomorphometry.

II Topographic surface

Real surfaces of land or submarine topography are not smooth and regular. A rigorous mathematical treatment of such surfaces can be problematic. However, in practice, it is sufficient to approximate a real surface by the topographic surface. The topographic surface is a closed, oriented, continuously differentiable, two-dimensional manifold S in the three-dimensional Euclidean space E ³.

There are five limitations valid for the topographic surface (Evans, 1980; Shary, 1995):

The topographic surface is uniquely defined by a continuous, single-valued bivariate function

z = f (x, y)

where z is elevation and x and y are the Cartesian coordinates. This means that caves, grottos and similar landforms are excluded.

The elevation function in equation (1) is smooth. This means that the topographic surface has derivatives of all orders. In practice, the first (p and q), second (r, t and s) and third (g, h, k and m) partial derivatives of elevation are used

\begin{matrix} p = \frac{\partial z}{\partial x}, & q = \frac{\partial z}{\partial y}, \\ r = \frac{\partial^{2} z}{\partial x^{2}}, & t = \frac{\partial^{2} z}{\partial y^{2}}, & s = \frac{\partial^{2} z}{\partial x \partial y}, \\ g = \frac{\partial^{3} z}{\partial x^{3}}, & h = \frac{\partial^{3} z}{\partial y^{3}}, & k = \frac{\partial^{3} z}{\partial x^{2} \partial y}, & m = \frac{\partial^{3} z}{\partial x \partial y^{2}} \end{matrix}

The topographic surface is located in a uniform gravitational field. This limitation is realistic for sufficiently small portions of the geoid, when the equipotential surface can be considered as a plane.

The planimetric sizes of the topographic surface are essentially less than the radius of the planet. It is usually assumed that the curvature of the planet may be ignored if the size of the surface portion is less than at least 0.1 of the average planetary radius. In computations, either sizes of moving windows in local computational methods or sizes of the entire area in global computational methods must comply with this condition (see section 3.2).

The topographic surface is a scale-dependent surface (Clarke, 1988). This means that a fractal component of topography is considered as a high-frequency noise.

III Morphometric variables

3.1 General

A morphometric (or topographic) variable (or attribute) is a single-valued bivariate function $ω = u (x, y)$ describing properties of the topographic surface.

In this paper, we review fundamental topographic variables associated with the theory of the topographic surface and the concept of general geomorphometry, which is defined as ‘the measurement and analysis of those characteristics of landform which are applicable to any continuous rough surface.…General geomorphometry as a whole provides a basis for the quantitative comparison…of qualitatively different landscapes….’ (Evans, 1972: 18).

There are several classifications of morphometric variables based on their intrinsic (mathematical) properties (Evans and Minár, 2011; Florinsky, 1998b, 2016: ch 2; Minár et al., 2016; Shary, 1995; Shary et al., 2002). Here, we use a modified classification of Florinsky (2016: ch 2) adopting some ideas of Evans and Minár (2011).

Morphometric variables can be divided into four main classes (Table 1): (1) local variables; (2) non-local variables; (3) two-field specific variables; and (4) combined variables. The terms ‘local’ and ‘non-local’ are used regardless of the study scale or model resolution. They are associated with the mathematical sense of a variable (c.f. the definitions of a local and a non-local variable – see sections 3.2 and 3.3). Being a morphometric variable, elevation does not belong to any class listed, but all topographic attributes are derived from DEMs.

Table 1.

Classification of morphometric variables.

Local variables			Non-local variables	Two-field specific variables	Combined variables
	Flow attributes	Form attributes
First-order variables	G		CA	R	TI
	A		DA	I	SI
	A_N
	A_E
Second-order variables	k_p	k_min
	k_h	k_max
	k_v	H
	E	K
	k_he	M
	k_ve	∇²
	K_a	IS
	K_r	C
	rot
Third-order variables	D_kh
Third-order variables	D_kv

3.2 Local morphometric variables

A local morphometric variable is a single-valued bivariate function describing the geometry 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 of mutually perpendicular normal sections (Figure 1(a) and (b)).

A normal section is a curve formed by the intersection of a surface with a plane containing the normal to the surface at a given point (Pogorelov, 1957). At each point of the topographic surface, an infinite number of normal sections can be constructed, but only two pairs of them are important for geomorphometry.

The first pair of mutually perpendicular normal sections includes two principal sections (Figure 1(a)) well known from differential geometry (Pogorelov, 1957). These are normal sections with extreme – maximal and minimal – bending at a given point of the surface.

The second pair of mutually perpendicular normal sections includes two ones (Figure 1(b)) dictated by gravity (Shary, 1991). One of these two sections includes the gravitational acceleration vector and has a common tangent line with a slope line¹ at a given point of the topographic surface. The other section is perpendicular to the first one and tangential to a contour line at a given point of the topographic surface.

Local variables are divided into two types (see Table 1) – form and flow attributes – which are related to the two pairs of normal sections (Shary, 1995; Shary et al., 2002).

Form attributes are associated with two principal sections. These attributes are gravity field invariants. This means that they do not depend on the direction of the gravitational acceleration vector. Among these are minimal curvature (k_min ), maximal curvature (k_max ), mean curvature (H), the Gaussian curvature (K), unsphericity curvature (M), Laplacian (∇²), shape index (IS), curvedness (C) and some others.

Flow attributes are associated with two sections dictated by gravity. These attributes are gravity-field specific variables. Among these are slope (G), aspect (A), northwardness (A_N ), eastwardness (A_E ), plan curvature (k_p ), horizontal curvature (k_h ), vertical curvature (k_v ), difference curvature (E), horizontal excess curvature (k_he ), vertical excess curvature (k_ve ), accumulation curvature (K_a ), ring curvature (K_r ), rotor (rot), horizontal curvature deflection (D_kh ), vertical curvature deflection (D_kv ) and some others.

K, K_a and K_r are total curvatures; k_min , k_max , k_h , k_v , k_he and k_ve are simple curvatures; and H, M and E are independent curvatures. Total and simple curvatures can be expressed by elementary formulae using the independent curvatures (Shary, 1995) (see equations (9)–(15) and (19)–(23)). These 12 attributes constitute a complete system of curvatures (Shary, 1995).

Local topographic variables are functions of the partial derivatives of elevation (see equations (4) –(26)). In this regard, local variables can be divided into three groups (Evans and Minár, 2011) (see Table 1): (1) first-order variables – G, A, A_N and A_E – are functions of only the first derivatives; (2) second-order variables – k_min , k_max , H, K, M, ∇², IS, C, k_p , k_h , k_v , E, k_he , k_ve , K_a , K_r and rot – are functions of both the first and second derivatives; and (3) third-order variables – D_kh and D_kv – are functions of the first, second and third derivatives.

The partial derivatives of elevation (and thus local morphometric variables) can be estimated from DEMs by: (1) several finite-difference methods using 3 × 3 or 5 × 5 moving windows (Evans, 1979, 1980; Florinsky, 1998c, 2009; Minár et al., 2013; Shary, 1995; Shary et al., 2002; Zevenbergen and Thorne, 1987); and (2) analytical computations based on DEM interpolation by local splines (Mitášová and Mitáš, 1993) or global approximation of a DEM by high-order orthogonal polynomials (Florinsky and Pankratov, 2016). Comparison of the methods can be found elsewhere (Florinsky, 1998a; Pacina, 2010; Schmidt et al., 2003).

3.3 Non-local morphometric variables

A non-local (or regional) morphometric variable is a single-valued bivariate function describing a relative position of a given point on the topographic surface (Speight, 1974).

Among non-local topographic variables are catchment area (CA) and dispersive area (DA). To determine non-local morphometric attributes, one should analyse a relatively large territory with boundaries located far away from a given point (e.g. an entire upslope portion of a watershed) (Figure 1(c)).

Flow routing algorithms are usually applied to estimate non-local variables. These algorithms determine a route along which a flow is distributed from a given point of the topographic surface to downslope points. There are several flow routing algorithms grouped into two types: (1) eight-node single-flow direction (D8) algorithms using one of the eight possible directions separated by 45º to model a flow from a given point (Jenson and Domingue, 1988; Martz and de Jong, 1988); and (2) multiple-flow direction (MFD) algorithms using the flow partitioning (Freeman, 1991; Quinn et al., 1991). There are some methods combining D8 and MFD principles (Tarboton, 1997). Comparison of the algorithms can be found elsewhere (Huang and Lee, 2016; Orlandini et al., 2012; Wilson et al., 2008).

3.4 Two-field specific morphometric variables

A two-field specific morphometric variable is a single-valued bivariate function describing relations between the topographic surface (located in the gravity field) and other fields, in particular solar irradiation and wind flow (Evans and Minár, 2011).

Among two-field specific morphometric variables are reflectance (R) and insolation (I). These variables are functions of the first partial derivatives of elevation (see equation (2)) and angles describing the position of the Sun in the sky (Figure 1(d)). Reflectance and insolation can be derived from DEMs using methods for the calculation of local variables (see section 3.2).

Openness (Yokoyama et al., 2002), incorporating the viewshed concept (Fisher, 1996), is also a two-field specific variable. In this case, the second field is a set of unobstructed sightlines between a given point of the topographic surface and surrounding points.

3.5 Combined morphometric variables

Morphometric variables can be composed from local and non-local variables. Such attributes consider both the local geometry of the topographic surface and a relative position of a point on the surface.

Among combined morphometric variables are topographic index (TI), stream power index (SI) and some others. Combined variables are derived from DEMs by the sequential application of methods for non-local and local variables, followed by a combination of the results.

IV Brief description and illustration of morphometric variables

4.1 Data and methods

To illustrate mathematical concepts of geomorphometry, we used a DEM of Mount Ararat (Figure 2). The area is located between 44.2º and 44.5º E, and 39.6º and 39.8º N (the area size is 18′ × 12′, that is, 25.725 km × 22.204 km). A spheroidal equal angular DEM was extracted from the quasi-global Shuttle Radar Topography Mission (SRTM) 1-arc-second DEM (Farr et al., 2007; USGS, 2015). The DEM includes 779,401 points (the matrix 1081 × 721). The grid spacing is 1′′, that is, the linear sizes of the 1′′ × 1′′ cell are 23.82 m × 30.84 m at the mean latitude, 39.7º N.

Elevation approximation and derivation of local and two-field specific variables were carried out by the recently developed spectral analytical method (Florinsky and Pankratov, 2016). The method is intended for the processing of regularly spaced DEMs within a single framework including DEM global approximation, de-noising, generalization and calculation of the partial derivatives of elevation. The method is based on high-order orthogonal expansions using the Chebyshev polynomials with the subsequent Fejér summation. In this study, we used 300 expansion coefficients of the original elevation function by the Chebyshev polynomials by both x- and y-axes. Calculation of non-local variables was performed by the Martz–de Jong flow routing algorithm (Martz and de Jong, 1988) adapted to spheroidal equal angular grids (Florinsky, 2017a). To derive combined morphometric variables, we sequentially applied the Martz–de Jong algorithm adapted to spheroidal equal angular grids and the spectral analytical method.

Wide dynamic ranges characterize the second- and third-order local variables as well as non-local attributes. To avoid loss of information on the spatial distribution of their values in mapping, a logarithmic transform should be applied (Shary et al., 2002)

Θ' = sign (Θ) ln (1 + 10^{n m} | Θ |)

where Θ' and Θ are transformed and original values of a variable; n = 0 for the non-local variables; n = 2,…, 9 for the second- and third-order local variables; m = 2 for the total curvatures and third-order local variables, m = 1 for other variables. Such a transformation considers that: (1) dynamic ranges of some attributes include both positive and negative values; and (2) correct mapping of variables from different classes and groups require different exponent values for the same territory and DEM resolution. In the spectral analytical method, selection of the n value depends on the size of a study area. In our case, n = 4 for third-order variables and n = 5 for second-order ones.

For the three-dimensional (3D) visualization of morphometric models, we used a 2^× vertical exaggeration and a viewpoint with 45° azimuth and 35° elevation.

To evaluate statistical interrelationships between 12 attributes of the complete system of curvatures, we performed multiple Spearman’s rank correlation analysis of their models (Table 2). Rank correlations allow the consideration of possible non-normality in curvature distributions. The sample size was 7000 points (the matrix 100 × 70); the grid spacing was 10′′.

Table 2.

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

	k_h	k_v	k_he	k_ve	E	K_a	K_r	k_min	k_max	K	H	M
k_h		0.36	0.58	–0.56	–0.72	–0.07	—	0.74	0.79	–0.05	0.88	0.04
k_v	0.36		–0.26	0.20	0.29	0.04	—	0.68	0.63	0.05	0.72	–0.06
k_he	0.58	–0.26		–0.31	–0.81	—	0.49	—	0.49	–0.27	0.28	0.56
k_ve	–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
K_a	–0.07	0.04	—	0.17	0.11		0.19	0.12	–0.17	0.78	–0.02	—
K_r	—	—	0.49	0.58	—	0.19		–0.33	0.28	–0.27	—	0.71
k_min	0.74	0.68	—	–0.47	–0.26	0.12	–0.33		0.55	0.30	0.86	–0.43
k_max	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.

Data processing was conducted with the software Matlab R2008b (^©The MathWorks Inc. 1984–2008) and LandLord 4.0 (Florinsky, 2016: 413–414). Statistical analysis was carried out by Statgraphics Plus 3.0 (^©Statistical Graphics Corp. 1994–1997).

4.2 Local morphometric variables: flow attributes

Slope (G) is an angle between the tangential and horizontal planes at a given point of the topographic surface (Shary, 1991)

G = arctan \sqrt{p^{2} + q^{2}}

Slope is a non-negative variable ranging from 0 to 90. The unit of G is degree. Slope (Figure 3(a)) determines the velocity of gravity-driven flows.

Aspect (A) is an angle between the northern direction and the horizontal projection of the two-dimensional vector of gradient counted clockwise at a given point of the topographic surface (Shary et al., 2002)

A = - 90 [1 - sign (q)] (1 - | sign (p) |) + 180 [1 + sign (p)] - \frac{180}{π} sign (p) arccos (\frac{- q}{\sqrt{p^{2} + q^{2}}})

Aspect is a non-negative variable ranging from 0 to 360. The unit of A is degree. Aspect (Figure 3(b)) is a measure of the direction of gravity-driven flows.

Northwardness and eastwardness aspect is a circular variable: its values range from 0º to 360º, and both of these values correspond to the north direction. Therefore, A cannot be used in linear statistical analysis. To avoid this problem, two auxiliary local indices can be applied: northwardness (A_N ) and eastwardness (A_E ) (Mardia, 1972)

A_{N} = cos A

A_{E} = sin A

Northwardness (Figure 3(c)) and eastwardness (Figure 3(d)) are dimensionless variables. A_N is equal to 1, –1 and 0 on the northern, southern and eastern/western slopes, correspondingly. A_E is equal to 1, –1 and 0 on the eastern, western and northern/southern slopes, correspondingly. These variables accentuate northern/southern and eastern/western trends in the spatial distribution of slopes.

Plan curvature (k_p ) is the curvature of a contour line at a given point of the topographic surface (Evans, 1979; Krcho, 1973)

k_{p} = - \frac{q^{2} r - 2 p q s + p^{2} t}{\sqrt{{(p^{2} + q^{2})}^{3}}}

This variable can be positive, negative or zero. The unit of k_p is m^–1. Plan curvature (Figure 3(e)) is a measure of flow line² divergence. Flow lines converge where k_p < 0 and diverge where k_p > 0; k_p = 0 refers to parallel flow lines.

Horizontal (or tangential) curvature (k_h ) is the curvature of a normal section tangential to a contour line (Figure 1(b)) at a given point of the topographic surface (Krcho, 1983; Shary, 1991)

k_{h} = - \frac{q^{2} r - 2 p q s + p^{2} t}{(p^{2} + q^{2}) \sqrt{1 + p^{2} + q^{2}}}

This variable can be positive, negative or zero. The unit of k_h is m^–1. Horizontal curvature (Figure 3(f)) is a measure of flow convergence (one of the two mechanisms of flow accumulation): gravity-driven overland and intrasoil lateral flows converge where k_h < 0, and they diverge where k_h > 0. Geomorphologically, k_h mapping allows revealing ridge and valley spurs (divergence and convergence areas, correspondingly).

Vertical (or profile) curvature (k_v ) is the curvature of a normal section having a common tangent line with a slope line (Figure 1(b)) at a given point of the topographic surface (Aandahl, 1948; Shary, 1991; Speight, 1974)

k_{v} = - \frac{p^{2} r + 2 p q s + q^{2} t}{(p^{2} + q^{2}) \sqrt{{(1 + p^{2} + q^{2})}^{3}}}

This variable can be positive, negative or zero. The unit of k_v is m^–1. Vertical curvature (Figure 3(g)) is a measure of relative deceleration and acceleration of gravity-driven flows (one of the two mechanisms of flow accumulation). Overland and intrasoil lateral flows are decelerated where k_v < 0, and they are accelerated where k_v > 0. Geomorphologically, k_v mapping allows revealing terraces and scarps.

Difference curvature (E) is a half-difference of vertical and horizontal curvatures (Shary, 1995)

E = \frac{1}{2} (k_{v} - k_{h}) = \frac{q^{2} r - 2 p q s + p^{2} t}{(p^{2} + q^{2}) \sqrt{1 + p^{2} + q^{2}}} - \frac{(1 + q^{2}) r - 2 p q s + (1 + p^{2}) t}{2 \sqrt{{(1 + p^{2} + q^{2})}^{3}}}

This variable can be positive, negative or zero. The unit of E is m^–1. Difference curvature (Figure 3(h)) shows to what extent the relative deceleration of flows (measured by k_v ) is higher than flow convergence (measured by k_h ) at a given point of the topographic surface.

Horizontal excess curvature (k_he ) is a difference of horizontal and minimal curvatures (Shary, 1995)

k_{h e} = k_{h} - k_{min} = M - E

This is a non-negative variable. The unit of k_he is m^–1. Horizontal excess curvature (Figure 3(i)) shows to what extent the bending of a normal section tangential to a contour line is larger than the minimal bending at a given point of the topographic surface.

Vertical excess curvature (k_ve ) is a difference of vertical and minimal curvatures (Shary, 1995)

k_{v e} = k_{v} - k_{min} = M + E

This is a non-negative variable. The unit of k_ve is m^–1. Vertical excess curvature (Figure 3(j)) shows to what extent the bending of a normal section having a common tangent line with a slope line is larger than the minimal bending at a given point of the topographic surface.

Accumulation curvature (K_a ) is a product of vertical and horizontal curvatures (Shary, 1995)

K_{a} = k_{h} k_{v} = \frac{(q^{2} r - 2 p q s + p^{2} t) (p^{2} r + 2 p q s + q^{2} t)}{{[(p^{2} + q^{2}) (1 + p^{2} + q^{2})]}^{2}}

This variable can be positive, negative or zero. The unit of K_a is m^–2. Accumulation curvature (Figure 3(k)) is a measure of the extent of local accumulation of flows at a given point of the topographic surface.

Ring curvature (K_r ) is a product of horizontal excess and vertical excess curvatures (Shary, 1995)

K_{r} = k_{h e} k_{v e} = M^{2} - E^{2} = {[\frac{(p^{2} - q^{2}) s - p q (r - t)}{(p^{2} + q^{2}) (1 + p^{2} + q^{2})}]}^{2}

This is a non-negative variable. The unit of K_r is m^–2. Ring curvature (Figure 3(l)) describes flow line twisting.

Rotor (rot) is a curvature of a flow line (Shary, 1991)

r o t = \frac{(p^{2} - q^{2}) s - p q (r - t)}{\sqrt{{(p^{2} + q^{2})}^{3}}}

This variable can be positive, negative or zero. The unit of rot is m^–1. Rotor (Figure 3(m)) describes flow line twisting. A flow line turns clockwise if rot > 0, while it turns counter clockwise if rot < 0.

Horizontal curvature deflection (or generating function) (D_kh ) is a derivative of k_h by the contour line length (Florinsky, 2009; Shary and Stepanov, 1991)

D_{k h} = \frac{q^{3} g - p^{3} h + 3 p q (p m - q k)}{\sqrt{{(p^{2} + q^{2})}^{3} (1 + p^{2} + q^{2})}} - k_{h} r o t \frac{2 + 3 (p^{2} + q^{2})}{1 + p^{2} + q^{2}}

This variable can be positive, negative or zero. The unit of D_kh is m^–2. Horizontal curvature deflection (Figure 3(n)) measures the deviation of k_h from loci of the extreme curvature of the topographic surface.

Vertical curvature deflection (or tangent change of profile curvature) (D_kv ) is a derivative of k_v by the contour line length (Jenčo et al., 2009)

D_{k v} = \frac{q^{3} m - p^{3} k + 2 p q (q k - p m) - p q (q h - p g)}{\sqrt{{(p^{2} + q^{2})}^{3} {(1 + p^{2} + q^{2})}^{3}}} - r o t [\frac{2 (r + t)}{\sqrt{{(1 + p^{2} + q^{2})}^{3}}} + k_{v} \frac{2 + 5 (p^{2} + q^{2})}{(1 + p^{2} + q^{2})}]

This variable can be positive, negative or zero. The unit of D_kv is m^–2. Vertical curvature deflection (Figure 3(o)) measures the deviation of k_v from loci of the extreme curvature of the topographic surface.

4.3 Local morphometric variables: form attributes

Minimal curvature (k_min ) is a curvature of a principal section (Figure 1(a)) with the lowest value of curvature at a given point of the surface (Gauss, 1828; Shary, 1995)

k_{min} = H - M = H - \sqrt{H^{2} - K}

This variable can be positive, negative or zero. The unit of k_min is m^–1. Geomorphologically, positive values of minimal curvature (Figure 4(a)) correspond to local convex landforms, while its negative values relate to elongated concave landforms (e.g. troughs and valleys).

Maximal curvature (k_max ) is a curvature of a principal section (Figure 1(a)) with the highest value of curvature at a given point of the surface (Gauss, 1828; Shary, 1995)

k_{max} = H + M = H + \sqrt{H^{2} - K}

This variable can be positive, negative or zero. The unit of k_max is m^–1. Geomorphologically, positive values of maximal curvature (Figure 4(b)) correspond to elongated convex landforms (e.g. ridges), while its negative values relate to local concave landforms.

Mean curvature (H) is a half-sum of curvatures of any two orthogonal normal sections at a given point of the topographic surface (Shary, 1991; Young, 1805)

H = \frac{1}{2} (k_{min} + k_{max}) = \frac{1}{2} (k_{h} + k_{v}) = - \frac{(1 + q^{2}) r - 2 p q s + (1 + p^{2}) t}{2 \sqrt{{(1 + p^{2} + q^{2})}^{3}}}

This variable can be positive, negative or zero. The unit of H is m^–1. Mean curvature (Figure 4(c)) represents two accumulation mechanisms of gravity-driven substances – convergence and relative deceleration of flows – with equal weights.

Gaussian curvature (K) is a product of maximal and minimal curvatures (Gauss, 1828)

K = k_{min} k_{max} = \frac{r t - s^{2}}{{(1 + p^{2} + q^{2})}^{2}}

This variable can be positive, negative or zero. The unit of K is m^–2. According to Teorema egregium, Gaussian curvature (Figure 4(d)) retains values in each point of the surface after its bending without breaking, stretching and compressing (Gauss, 1828).

Unsphericity curvature (M) is a half-difference of maximal and minimal curvatures (Shary, 1995)

M = \frac{1}{2} (k_{max} - k_{min}) = \sqrt{H^{2} - K} = \sqrt{\frac{1}{4 {(1 + p^{2} + q^{2})}^{3}} {(r \sqrt{\frac{1 + q^{2}}{1 + p^{2}}} - t \sqrt{\frac{1 + p^{2}}{1 + q^{2}}})}^{2} (1 + p^{2} + q^{2}) + {\overset{}{{(p q r \sqrt{\frac{1 + q^{2}}{1 + p^{2}}} - 2 s \sqrt{(1 + q^{2})(1 + p^{2})} + p q t \sqrt{\frac{1 + p^{2}}{1 + q^{2}}})}^{2}}}}

This is a non-negative variable. The unit of M is m^–1. Unsphericity curvature (Figure 4(e)) shows the extent to which the shape of the surface is non-spherical at a given point.

Laplacian (∇²) is a second-order differential operator, which can be defined as the divergence of the gradient of a function z (Laplace, 1799)

\nabla^{2} = div grad = r + t

This variable can be positive, negative or zero. The unit of ∇² is m^–1. Laplacian (Figure 4(f)) measures the flux density of slope lines.

Shape index (IS) is a continual form of the discrete Gaussian landform classification (Koenderink and van Doorn, 1992)

I S = \frac{2}{π} arctan (\frac{k_{max}^{} + k_{min}^{}}{k_{max}^{} - k_{min}^{}})

This is a dimensionless variable ranging from –1 to 1 (Figure 4(g)). Its positive values relate to convex landforms, while its negative ones correspond to concave landforms; its absolute values from 0.5 to 1 are associated with elliptic surfaces (hills and closed depressions), whereas its absolute values from 0 to 0.5 relate to hyperbolic ones (saddles).

Curvedness (C) is the root mean square of maximal and minimal curvatures (Koenderink and van Doorn, 1992)

C = \sqrt{\frac{k_{max}^{2} + k_{min}^{2}}{2}}

This is a non-negative variable. The unit of C is m^–1. Curvedness (Figure 4(h)) measures the magnitude of surface bending regardless of its shape. Flat areas have low values of C, while areas with sharp bending are marked by high values of C.

4.4 Non-local morphometric variables

Catchment area (CA) is an area of a closed figure formed by a contour segment at a given point of the topographic surface and two flow lines coming from upslope to the contour segment ends (Figure 1(c)) (Speight, 1974). This is a non-negative variable. The unit of CA is m². The catchment area (Figure 5(a)) is a measure of the contributing area.

Dispersive area (DA) is an area of a closed figure formed by a contour segment at a given point of the topographic surface and two flow lines going down slope from the contour segment ends (Figure 1(c)) (Speight, 1974). This is a non-negative variable. The unit of DA is m². The dispersive area (Figure 5(b)) is a measure of a downslope area potentially exposed by flows passing through a given point.

4.5 Two-field specific morphometric variables

Reflectance (R) is a measure of the brightness of an illuminated surface (Horn, 1981). For the case of the ideal diffusion (the Lambertian surface)

R = \frac{1 - p sin θ ctg ψ - q cos θ ctg ψ}{\sqrt{1 + p^{2} + q^{2}} \sqrt{1 + {(sin θ ctg ψ)}^{2} + {(cos θ ctg ψ)}^{2}}}

where θ and ψ are the solar azimuth and elevation angles (Figure 1(d)), correspondingly. This is a non-negative dimensionless variable normalized for the range from 0 to 1. Reflectance maps (Figure 6(a)) clearly and plastically display the topography.

Insolation (I) is a proportion of maximal direct solar irradiation at the Sun’s position determined by solar azimuth and solar elevation (Figure 1(d)) (Shary et al., 2005)

\begin{array}{l} I = 50 {1 + sign [\sin ψ - \cos ψ (p \sin θ + q \cos θ)]} \\ \times \frac{[\sin ψ - \cos ψ (p \sin θ + q \cos θ)]}{\sqrt{1 + p^{2} + q^{2}}} \end{array}

This is a non-negative variable ranging from 0 to 100. The unit of I is per cent. Insolation (Figure 6(b)) characterizes a perpendicularity of the incidence of solar rays on the topographic surface.

4.6 Combined morphometric variables

Topographic index (TI) is a ratio of catchment area to slope at a given point of the topographic surface (Beven and Kirkby, 1979)

T I = ln [1 + C A / (10^{- 3} + tg G)]

This is a non-negative dimensionless variable. The term 10^–3 is used to avoid division by zero for the case of horizontal planes. The topographic index (Figure 7(a)) is a measure of the extent of flow accumulation in TOPMODEL, a concept of distributed hydrological modelling (Beven and Kirkby, 1979). TI reaches high values in areas with high values of CA at low values of G (e.g. a terrain with a large upslope contributing area and flat local topography).

Stream power index (SI) is a product of catchment area and slope at a given point of the topographic surface (Moore et al., 1991)

S I = ln (1 + C A \cdot tg G)

This is a non-negative dimensionless variable. Stream power index (Figure 7(b)) is a measure of potential flow erosion and related landscape processes. SI reaches high values in areas with high values of both CA and G (e.g. a highly sloped terrain with a large upslope contributing area).

V Discussion

5.1 The main paradox of general geomorphometry

It is quite obvious that the real topography is non-smooth and so non-differentiable (Shary, 2008). This means that it cannot have partial derivatives and so local morphometric variables, which are functions of the partial derivatives (sections 4.2 and 4.3). However, introducing the concept of the topographic surface and accepting the condition of its smoothness (section II), we are able not only to calculate local variables, but also to apply these ‘non-existent and abstract’ attributes to study and model relationships between topography and properties of other components of geosystems (for examples, see section 5.2). Topography influences soil and other landscape properties mainly via gravity-driven overland and intrasoil lateral migration and accumulation of water. Dependences of soil and other landscape properties on ‘non-existent’ morphometric attributes may be explained by a hypothetical assumption that real soil hydrological processes also ‘smooth’ the topography ‘ignoring’ its minor non-smooth details.

Computationally, the non-smoothness of the real topography is considered in finite-difference methods calculating local variables with 3 × 3 or 5 × 5 moving windows (Evans, 1980; Florinsky, 1998c, 2009; Minár et al., 2013; Shary, 1995; Shary et al., 2002). In these algorithms, the smoothness condition of the topographic surface should be met within a moving window only (a polynomial is approximated to 9 or 25 points of a window; there is no approximation between windows). In the same manner, the non-smoothness of the real topography is considered in calculations of its spatial statistical metrics (so-called roughness indices, see section 5.3). Such calculations are also performed with moving windows.

Concepts and methods of fractal geometry (Gao and Xia, 1996; Mandelbrot, 1967) were sometimes applied in general geomorphometry (Klinkenberg, 1992; Mark and Aronson, 1984; McClean and Evans, 2000; Xu et al., 1993). In particular, there were ideas to use fractal dimensions of a terrain as a morphometric index. However, Clarke (1988) clearly demonstrated that a fractal component of topography is important for landscape simulation only. In scientific studies, a fractal component of topography can be considered a high-frequency noise. Thus, we use such a limitation (#5) in the concept of the topographic surface (see section II).

5.2 Application of morphometric variables

Let us briefly consider the application of the reviewed morphometric variables. Slope and aspect have been well known in geosciences for many decades, and so there is no need to specify their fields of application. Curvatures are systematically used in geomorphic studies to describe, analyse and model landforms and their evolution (Burian et al., 2015; Elmahdy and Mohamed, 2013; Evans, 1980; Guida et al., 2016; Melis et al., 2014; Mitusov et al., 2013, 2014; Prasicek et al., 2014; Temovski and Milevski, 2015). In soil science and ecology, curvatures are regularly applied to study relationships in the topography–soil–vegetation system and to perform predictive soil and vegetation mapping (Behrens et al., 2010; Florinsky et al., 2002; Moore et al., 1993; Omelko et al., 2012; Sharaya and Shary, 2011; Shary and Pinskii, 2013; Shary and Smirnov, 2013; Shary et al., 2016; Stumpf et al., 2017). In structural geology, curvatures are utilized to reveal hidden faults as well as to study fold geometry (Bergbauer, 2007; Florinsky, 1996; Lisle and Toimil, 2007; Mynatt et al., 2007; Roberts, 2001; Stewart and Podolski, 1998). Application of horizontal and vertical curvature deflections may be useful to recognize thalweg and crest lines (Florinsky, 2009; Minár et al., 2013). Reflectance maps are well known in cartography as hill-shaded maps (Jenny, 2001). Insolation is utilized to describe the thermal regime of slopes in geobotanical and agricultural research (Shary and Smirnov, 2013; Shary et al., 2016). Catchment and dispersive areas as well as topographic index are widely used in hydrological and related soil, plant and geomorphic studies (Behrens et al., 2010; Beven, 1997; Florinsky et al., 2002; Mitusov et al., 2013, 2014; Moore et al., 1993; Omelko et al., 2012; Sharaya and Shary, 2011; Shary and Smirnov, 2013). Stream power index is applied in erosion and soil research (Florinsky et al., 2002; Moore et al., 1993; Omelko et al., 2012).

Comprehensive reviews of the application of the morphometric variables can be found elsewhere (Florinsky, 1998b, 2016: pts II and III; Hengl and Reuter, 2009: pt. III; Moore et al., 1991; Wilson and Gallant, 2000).

5.3 Statistical aspects of general geomorphometry

It is clear that some topographic variables can be expressed as linear combinations of others (equations (4) –(30)). For example, mean curvature is a combination of horizontal and vertical curvatures (equation (21)). So, it is not surprising that the results of correlation analysis of models from the complete system of curvatures demonstrate rather high relations between some attributes (Table 2); see, for example, correlation coefficients between k_h and k_min , k_he and E, k_max and H, as well as M and K_r . Thus, questions arise of (1) an information redundancy of the variables discussed in section IV, and (2) the selection of topographic attributes for a particular (e.g. soil) study.

First, although statistical relationships between morphometric variables are useful in geomorphic research (Csillik et al., 2015; Evans, 1980, 1998; Evans and Cox, 1999), such statistics cannot be utilized to select particular variables. Such a selection should be based on their physical/mathematical interpretations. Second, it is a priori impossible to know which particular morphometric variables control, for example, a soil property under given natural conditions. To search topographic attributes controlling the property, it is reasonable to utilize a representative set of the attributes at the first stage of a study (e.g. soil predictive modelling with morphometric variables as predictors; Florinsky, 2016: ch 11; McBratney et al., 2003). When governing variables are found by a correlation analysis, one can reduce their number because it is incorrect to use together, for instance, horizontal, vertical and mean curvatures in a predictive (e.g. regression) model of the property (Florinsky, 2016: 309–311; Shary and Pinskii, 2013).

To delineate and quantitatively describe terrains of different morphologic structure as well as geological composition and age, digital models and maps of topographic (or surface) roughness are used in geomorphic, geological and planetary studies (Fa et al., 2016; Frankel and Dolan, 2007; Grohmann et al., 2011; Herzfeld and Higginson, 1996; Karachevtseva et al., 2015; Kreslavsky and Head, 2000; McKean and Roering, 2004; Trevisani and Rocca, 2015). To avoid confusion, we should stress that topographic roughness is not a morphometric variable. It is a generalized term, without a clear definition, which is usually applied to denote some spatial statistical metrics of a morphometric model (e.g. median absolute value of slope, eigenvalue ratios of external normals, standard deviation of elevation, slope, etc.). Models of topographic roughness are usually calculated using windows moving along a morphometric model. The principal difference between morphometric variables and topographic roughness is as follows: Morphometric variables mathematically describe ‘geometry’ of the topographic surface and its physically interpretable relationships with other components of geosystems, while topographic roughness statistically describes ‘spatial variability’ of a particular morphometric property of the topographic surface in the vicinity of a given point of the surface.

5.4 Some pending problems of general geomorphometry

Advances in the terrestrial light detection and ranging (LiDAR) technology (Hodgetts, 2013) have allowed the production of 3D high-resolution models of caves and grottos. Some researchers try to use geomorphometric modelling to analyse the geometry of inner surfaces of these objects (Brook et al., 2017; Gallay et al., 2015, 2016). However, such an application of geomorphometry violates the limitation #1 stating that the topographic surface is uniquely defined by a single-valued function (see section II). It is obvious that caves and grottos cannot be directly described by a single-valued elevation function because, at least, two values of elevation correspond to a pair of planimetric coordinates.

One may ignore this limitation working with form attributes (e.g. Gallay et al., 2015, 2016), which are gravity field invariants (see section 3.2). However, the limitation becomes important if one works with flow attributes because these are gravity-field specific variables (section 3.2). In water dynamics/runoff on cave ceilings and walls, forces of surface tension can play a dominant role, rather than gravity-driven overland lateral transport of liquids controlled by horizontal and vertical curvatures (section 4.2). This calls into question the adequacy of utilizing these morphometric variables (e.g. Brook et al., 2017) to study hydrological processes in caves, grottos and niches, at least for microtopography of ceilings. This problem requires in-depth study and development of recommendations for geomorphometric modelling in speleology and related disciplines.

There are three other interrelated pending problems.

A set of the third-order local variables was introduced by Jenčo et al. (2009). In general, these attributes describe deviations of the second-order variables from loci of the extreme curvature of the topographic surface (Florinsky, 2009; Minár et al., 2013; Shary and Stepanov, 1991). Only two of them have been discussed here, horizontal and vertical curvature deflections (section 4.2), because their interpretation and practical application are more or less clear: their zero values correlate with crest and thalweg lines. Other third-order variables are still insufficiently studied.

Comparing with the theory of local variables (Jenčo et al., 2009; Shary, 1995), a mathematical theory for non-local attributes is still little developed (Gallant and Hutchinson, 2011; Koshel and Entin, 2017; Orlandini et al., 2014; Peckham, 2013). In particular, Gallant and Hutchinson (2011) proposed a differential equation for calculating specific catchment area. However, a general analytical theory of non-local morphometric variables still does not exist. The diversity of flow routing algorithms for calculating non-local variables (section 3.3) is a result of underdevelopment of an appropriate mathematical theory.

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

VI Conclusions

In the last 25 years, great progress has been made in the field of geomorphometry.

A physical and mathematical theory of the topographic surface has substantially evolved (Evans, 1972, 1980; Gallant and Hutchinson, 2011; Jenčo, 1992; Jenčo et al., 2009; Koenderink and van Doorn, 1992, 1994; Krcho, 1973, 2001; Shary, 1991, 1995; Shary et al., 2002, 2005).

Effective algorithms for calculating attributes from DEMs have been developed (Evans, 1980; Florinsky, 1998c, 2009; Florinsky and Pankratov, 2016; Freeman, 1991; Martz and de Jong, 1988; Minár et al., 2013; Shary, 1995; Tarboton, 1997).

High- and super high-resolution DEMs have become widely available (Tarolli, 2014) owing to advances in LiDAR technology (Liu, 2008), real-time kinematic global navigation satellite system (GNSS) surveys (Awange, 2012: ch 8), areal surveys based on unmanned aerial vehicles (UAVs) (Colomina and Molina, 2014), and structure-from-motion (SfM) techniques (Smith et al., 2016). UAVs and SfM introduce a low-cost alternative to manned aerial surveys and conventional photogrammetry. UAV/SfM-derived, photogrammetrically sound DEMs can be successfully used for further geomorphometric modelling: it is possible to produce noiseless, well-readable and interpretable models of slope and curvatures (with the resolution of 5–20 cm) for grassy areas with separately standing groups of trees, shrubs and other objects (Florinsky et al., 2017b).

Quasi-global and global, medium- and high-resolution DEMs of the Earth (SRTM1 (Farr et al., 2007), Advanced Spaceborne Thermal Emission and Reflection Radiometer Global DEM (ASTER GDEM) (Toutin, 2008), SRTM30_PLUS (Becker et al., 2009), SRTM15_PLUS (Olson et al., 2014), WorldDEM (Zink et al., 2014) and Advanced Land Observing Satellite World 3D model (ALOS World 3D) (Tadono et al., 2014)) have been produced and become available.

There are new bathymetric DEMs (Arndt et al., 2013; Jakobsson et al., 2012; Weatherall et al., 2014). Submarine topography influences ocean currents, distribution of perennial ice and sediment migration. Submarine valleys participate in the gravity-driven transport of substances from land to ocean. Geomorphometric modelling of submarine topography can provide new opportunities for oceanological, marine geomorphological and marine geological studies.

There are new DEMs for the ice bed topography of Greenland (Bamber et al., 2013) and Antarctica (Fretwell et al., 2013). In this case, application of geomorphometric methods can produce new results for understanding both glaciological processes and geological structure of glacier-covered terrains.

Advances in 3D seismic surveys offer a wide field of activity in applying principles of geomorphometric modelling to subterranean surfaces (Chopra and Marfurt, 2007) that can give a new impetus to geological research.

Global, extra-terrestrial medium-resolution DEMs for Mars (Smith et al., 1999), the Moon (Smith et al., 2010), Phobos (Karachevtseva et al., 2014) and Mercury (Becker et al., 2016) are available. Geomorphometry can provide additional tools for comparative planetary studies.

Morphometric globes for the Earth, Mars and the Moon have been developed (Florinsky and Filippov, 2017; Florinsky et al., 2017a) owing to advances in scientific visualization (Hansen and Johnson, 2005) and virtual globe (Cozzi and Ring, 2011) technologies. Such virtual globes can be easily utilized by users without special training in geomorphometry.

The above achievements and future possibilities, as well as reproducibility, relative simplicity and flexibility of geomorphometric methods, determine their potential for geosciences. It should be realized, however, that the governing factor for the evolution of digital terrain analysis is advances in the theory of the topographic surface, which lays a rigorous physical and mathematical foundation for both computation algorithms and applied issues of topographic modelling.

Footnotes

Acknowledgements

The author is grateful to Ian Evans and two anonymous reviewers for constructive criticism.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the Russian Foundation for Basic Research, Grant number 15-07-02484.

Notes

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