On the theory of drainage area for regular and non-regular points

Abstract

The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res. 47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology 219, 68–86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss’s theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.

1. Introduction

More than a century ago, Maxwell [1] stated the importance of ‘an exact knowledge of the first elements of physical geography’ and observed the prevalence of ‘loose notions on the subject’. Since then, the geomorphological and ecohydrological literature has been replete with references to topographic entities such as drainage divides, valley lines, channel heads and drainage area, but most of these basic elements of physical geography still lack a sound mathematical description, such that Maxwell’s statement remains relevant even today. Among these morphometric variables, the drainage (or catchment) area A—defined as the horizontally projected integral of all areas draining to a point [2]—is an extremely important non-local variable playing a role in several geomorphological and ecohydrological processes related to surface and subsurface water redistribution. For example, it is often used as an approximation of channel discharge in landscape evolution models [3–6] and is related to soil moisture in combination with other topographic variables [7–11]. Furthermore, water-content indices based on drainage area have been adopted in the definition of landslide susceptibility [12,13], for the determination of vegetation patterns and for biodiversity mapping (e.g. [14–19]).

Despite its importance and the numerous algorithms developed for its numerical computation from regular grid digital elevation models (DEMs) [20–29], a first theoretical description of the drainage area was provided only in 2011, with the derivation of a differential equation for the evaluation of the specific catchment area a [30]. The differential equation proposed therein has been used as a benchmark to test some numerical algorithms (namely, D8 [20,21], DEMON [25] and D$\mathrm{\infty }$ [26]), showing that all the tested methods tend to overestimate a in divergent terrain (ridges and hilltops), with the D$\mathrm{\infty }$ method being overall the most accurate. More recently, a has been obtained by mapping it into the steady-state water level that would be obtained assuming a uniform (in space and time) and unitary rainfall rate transported downslope at a constant speed in the direction opposite to the gradient [5].

The theoretical expression proposed by Gallant & Hutchinson [30] can be employed only at regular points of the watershed, thus not providing any insight on those areas characterized by either zero slope or singularities of the topographic field where different slope lines coalesce (i.e. flat areas, local maxima and minima, and ridge and valley lines when treated as singularities of the topographic surface). The first methods to deal with these non-regular points of the topographic surface were theorized in the nineteenth century, with the development of both local and non-local methods for the delineation of drainage lines [1,31–34]. These early works mainly focused on methods for the characterization of ridge and valley lines, while not directly dealing with the computation of the drainage area. Local ridge and valley definitions are based on differential geometry principles [31,33–40] but provide loci of ridge and valley lines that are interrupted at many points due to the presence of critical points and minor flat horizontal areas [41], thus not providing any insight into the treatment of non-regular points of the topographic surface. Non-local methods, on the other hand, are based on the pioneering work by Cayley [32], who explicitly focused on special points of the surface, such as local elevation maxima, minima and saddle points. He observed that, in general, there are only two special slope lines passing by a saddle, and they are a ridge and a valley line. From a dynamical systems perspective, these two special slope lines are the stable and unstable manifolds of the saddle point [42–44]. The work by Cayley [32] was continued by Maxwell [1], who stated that through each point of the surface passes a slope line going from a maximum to a minimum of the surface. Maxwell also defined basins and hills as districts whose slope lines come from the same minimum or run to the same maximum, respectively. This procedure allows the definition of ridges as ‘slope district boundaries’ [45–48]. Techniques based on the delineation of slope district boundaries have recently gained attention within the computer vision community for the analysis of image intensity functions (e.g. [46,47,49]). However, they can be broadly applied to digital terrain analysis [41,48,50–53] for the detection of ridge and valley lines (representing watershed divides and channels, respectively) and, as will be investigated in this work, for developing and testing models and algorithms for the numerical calculation of the drainage area at non-regular points. In addition, skeleton construction techniques have been shown to allow a fully automated recognition in contour-based DEMs of the complex topographic structures that can be found in real landscapes [54].

The goal of this paper is to provide a theoretical framework to define the drainage area at both regular and non-regular points of the topographic surface. The differential equation proposed by Gallant & Hutchinson [30] is here derived in a more intuitive way from the water continuity equation defined by Chen et al. [5]. The theory is then extended to non-regular points of the surface by means of a dynamical systems approach that builds on the work of Cayley [32] and Maxwell [1]. We hope that a theoretical definition of drainage area at regular and non-regular points of a given landscape may help to test and improve currently used numerical algorithms, as well as be used for the analysis of landscape evolution and stability.

The paper is organized as follows. The mathematical formulation for the definition of the drainage area at both regular and non-regular points is presented in §§2 and 3, respectively. In §4, some applications to special cases are provided, from simple analytical examples to more complex surfaces including critical and singular points. The main focus here is on continuous surfaces, while an extension to discrete surfaces (e.g. DEMs) is briefly discussed in §5 with an application to a real topography.

2. Drainage area at regular points

(a). Terminology and basic definitions

Consider a topographic surface defined by a single-valued function z=f(x,y), z being the elevation and x,y the horizontal Cartesian coordinates (with x=xi+yj). While both continuous and discrete representations are used to describe a surface topography (either of which, however, is an approximation of real land surface complexities), here we focus on a continuous description. On this topographic surface two families of lines can be defined, namely contour and slope lines [32,41].

A contour line $\mathcal{W}$ is a set of points obtained from the intersection of the topographic surface with a horizontal plane, thus being a plane curve with equation z(x,y)=const. (blue lines in figure 1). A slope (or gradient) line ($\mathcal{S}$ in figure 1) is a curve on the topographic surface for which, at every point, the direction of the tangent vector coincides with the direction of the tangential component of the gravitational force [41]. The projections of slope lines on a horizontal plane ($\mathcal{L}$ in figure 1) are the lines normally considered in terrain analysis [1,30,54]. Any contour line is orthogonal to both slope lines and slope line projections on a horizontal plane [41]. Slope lines are topographic attributes indicating flow lines in overland flows that are purely driven by gravity, as often assumed [25,26]. If overland flow is also driven by diffusional and inertial effects, however, slope and flow lines do not coincide and a clear distinction between the two concepts is needed [30,55].

Figure 1.

Graphical representation and definitions: contour lines ($\mathcal{W}$, blue), slope lines ($\mathcal{S}$, black) and projected slope lines ($\mathcal{L}$, red) for a two-dimensional topographic surface z=f(x,y). A two-dimensional surface is here defined as a single-valued function z having a two-dimensional domain (x,y being the horizontal Cartesian coordinates). The drainage area A pertaining to a contour segment of length w at a certain arc-distance l from the hilltop is the area between the two projected slope lines (green shaded area). (Online version in colour.)

The points of the topographic surface where the local slope $|\mathrm{\nabla }z|=\sqrt{{(\mathrm{\partial }z/\mathrm{\partial }x)}^2+{(\mathrm{\partial }z/\mathrm{\partial }y)}^2}$ is equal to zero are special points (also called critical points), and they represent either peaks (local maxima), sinks (local minima) or saddles (mountain passes). Minima, saddles and maxima are also referred to as Morse critical points of index 0, 1 and 2, respectively [43]. At critical points neither contour nor slope lines can be defined. Other important points of the watershed in relation to the definition of channels are those points where the local slope is not defined. These are singular points of the surface and are characterized, for example, by the coalescence of slope lines forming a channel.

The unit tangent vector to a slope line in every point determines the vector field

$$\mathbf{\text{v}}(x,y)=-\frac{\mathrm{\nabla }z}{|\mathrm{\nabla }z|},$$ 2.1

which is defined at all points where the topographic surface is differentiable and has non-zero slope. The direction of v gives the direction of maximal decrease in elevation among all possible directions. In fact, it can be shown that the direction of maximal decrease of elevation (i.e. direction of steepest descent) is a slope line direction (i.e. −∇z) for each non-special point of the topographic surface [41].

The drainage area A is a non-local variable defined as the horizontal projection of the land surface enclosed between two slope lines that originate at a hilltop (either a common hilltop, as exemplified in figure 1, or distinct ones, as shown in figure 2 and discussed in the following sections) and are bounded at the lower end by a contour segment of length w [30]. The specific catchment area a is defined [30] as the ratio between the drainage area A and the length of the contour segment w in the limit of $w\to 0$,

$$a=\underset{w\to 0}{lim}\frac{A}{w}.$$ 2.2

The specific drainage area a (also referred to as the specific catchment area) has units of length and diverges at all those points of the topographic surface where the contour lines coalesce (namely, critical and singular points of the surface). However, at these non-regular points, while a cannot be determined, it is still possible to define the drainage area A as the integral of all areas draining to the point of interest. This will be discussed in §3, where theoretical definitions of A at non-regular points are provided.

Figure 2.

Elevation contour plot (white=high, dark grey=low) of a gradient system obtained by translating and scaling Gaussian distributions; black dashed lines are contour lines. Its dynamical systems representation is also shown: symbols represent surface maxima, minima and saddles. Red and blue lines are the stable and unstable manifolds, respectively, and were computed from integration of the vector field ±∇z (+ sign for the stable manifold, − sign for the unstable manifold), starting from small perturbations around each saddle point in the principal directions. Yellow lines in (a) are slope lines. In (a), Gauss’s theorem is applied to the surface A enclosed between the ridge line (i.e. the stable manifold) and the contour line W (black solid line), with n being the unit normal vector (positive outward) to the surface boundaries: the area of the surface A is the drainage area pertaining to W. In (b), the green area represents the drainage area A for the minimum enclosed by the red ridge lines (i.e. the basin of attraction of the minimum). (Online version in colour.)

(b). Governing equation for the specific catchment area

An intuitive way to derive an equation for a at regular points is now outlined, in part following [5]. Considering the hypothetical collection of water due to a uniform and steady rainfall r over a topographic surface, the continuity equation for this water flow reads

$$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (h\mathbf{\text{u}})=r,$$ 2.3

where t is the time, h is the water depth and u is the flow velocity. Note that the use of the water depth h is here introduced as a conceptual analogy to guide intuition, but the actual flow of water is not being modelled.

At steady-state conditions (∂h/∂t=0) and assuming that the flow is fed by a constant unitary rainfall rate falling vertically on the topographic surface, the continuity equation reduces to ∇⋅(hu)=1. If water is further assumed to flow at constant unitary speed in the opposite direction to the landscape gradient, the velocity u is defined by equation (2.1) (i.e. u=v), which can be shown to be the expression of the tangent vector to a flow line [41]. In these conditions, it is rather intuitive that the water height is simply the specific catchment area, h=a, and the continuity equation becomes an equation for the specific drainage area,

$$\mathrm{\nabla }\cdot (a\mathbf{\text{v}})=1.$$ 2.4

This equation corresponds to eqn (2) in [5] (although in that paper the expression mistakenly uses A instead of a; during the review process we also became aware that an equation equivalent to (2.4) was stated without derivation by Hutchinson et al. [56]).

It is interesting to transform the above equation into a local system of coordinates, defined by the slope lines and contour lines, allowing us to make contact with the work of Gallant & Hutchinson [30]. Accordingly, we begin by writing equation (2.4) as

$$(\mathrm{\nabla }a)\cdot \mathbf{\text{v}}+a(\mathrm{\nabla }\cdot \mathbf{\text{v}})=1,$$ 2.5

where the term ∇⋅v is the divergence of the unit normal vector to a contour line. As such, it can be shown to represent the curvature of a contour line, which is called plan (or contour) curvature k_c (this can be shown by simply computing the divergence of the vector v defined by equation (2.1) [41,57]). The curvature of the contour line is a quantitative measure of flow convergence and divergence over the topographic surface. Upon inserting ∇⋅v=k_c, equation (2.5) can be written as

$$(\mathrm{\nabla }a)\cdot \mathbf{\text{v}}+a{k}_{\mathrm{c}}=1.$$ 2.6

The term (∇a)⋅v describes the variation of a along the projected slope line l (whose direction is given by the vector v), so that (∇a)⋅v=∂a/∂l. Thus, equation (2.6) reduces to

$$\frac{\mathrm{\partial }a}{\mathrm{\partial }l}=1-k_{\mathrm{c}}a,$$ 2.7

which is the same expression derived by Gallant & Hutchinson [30]. Equation (2.7) allows us to compute the specific catchment area a at any regular point of the topographic surface by construction of slope lines and integration of equation (2.7) along the flow line from the hilltop (i.e. at l=0) to the point of interest, imposing a=0 at l=0.

Equation (2.7) is a first-order inhomogeneous differential equation, which has a solution of the form [58]

$$a(l)=\frac{C_1}{I(l)}+\frac{1}{I(l)}{\int }^{l}I(l^{\prime })\hspace{0.17em}\mathrm{d}{l}^{\prime },$$ 2.8

where $I(l)=\exp {\int }^l{k}_{\mathrm{c}}(l^{\prime })\hspace{0.17em}\mathrm{d}{l}^{\prime }$ is an integrating factor and C₁ is an integration constant, whose value is obtained imposing a=0 at l=0 (i.e. at the hilltop).

In Cartesian coordinates, the plan (or contour) curvature k_c is given by [41]

$$k_{\mathrm{c}}=-\frac{{\mathrm{\partial }}_{xx}z{({\mathrm{\partial }}_{y}z)}^2+{\mathrm{\partial }}_{yy}z{({\mathrm{\partial }}_{x}z)}^2-2{\mathrm{\partial }}_{xy}z{\mathrm{\partial }}_{x}z{\mathrm{\partial }}_{y}z}{|\mathrm{\nabla }z{|}^3},$$ 2.9

where ∂_xz=∂z/∂x, ∂_yz=∂z/∂y, ∂_xxz=∂²z/∂x², ∂_yyz=∂²z/∂y² and ∂_xyz=∂²z/∂x∂y. The plan curvature is negative for valleys, where flow converges, and positive for ridges, where flow diverges. Note that the plan curvature tends to infinity near the top of a hill or bottom of a pit (i.e. at critical points), so that at these points equation (2.7) is not defined. The plan curvature must be distinguished from the profile (or vertical) curvature, which is the curvature of the surface as we move in the gradient direction (i.e. along a slope line) [41,57]. The profile curvature of the surface does not play any role in the definition of the specific catchment area a (see equation (2.7)). However, when the ‘real’ drainage area (i.e. the actual area of the three-dimensional surface, not projected on the horizontal) is needed, the profile curvature would actually matter as it would be necessary to know how the surface curves in the three-dimensional space (as encoded in the second fundamental form [59,60]).

3. Drainage area at non-regular points

Equation (2.7) cannot be used to define the specific drainage area a at critical and singular points, where the plan curvature is not defined resulting in a being a singular function. However, as already noticed, at non-regular points it is still possible to define the drainage area A as the integral of all areas draining to the point of interest. Theoretical definitions of A at both critical and singular points are now introduced, based on the application of Gauss’s theorem, as well as building on the work of Cayley [32] and Maxwell [1] for the delineation of ridge and valley lines in the watershed.

(a). Critical points

At critical points two methods can be outlined to define the drainage area A. In the first case, Gauss’s theorem is employed to define A as the flux of a across a closed contour line. Alternatively, ridge and valley lines are constructed [1,32] and used to define A as the basin of attraction of each surface minimum.

Following the first method, a definition of A pertaining to any closed curve $\mathcal{C}$ can be obtained by applying Gauss’s theorem to the surface $A_{\mathcal{C}}$ enclosed by $\mathcal{C}$. Introducing a vector field a=av, the application of Gauss’s theorem results in

$${\iint }_{A_{\mathcal{C}}}(\mathrm{\nabla }\cdot \mathbf{\text{a}})\hspace{0.17em}\mathrm{d}{A}_{\mathcal{C}}={\oint }_{\mathcal{C}}(\mathbf{\text{a}}\cdot \mathbf{\text{n}})\hspace{0.17em}\mathrm{d}\mathcal{C},$$ 3.1

n being the outward-pointing unit normal to the curve $\mathcal{C}$ (figure 2a). As ∇⋅a=1 (equation (2.5)), the drainage area $A_{\mathcal{C}}$ pertinent to a closed curve $\mathcal{C}$ is given by the integral of all a along $\mathcal{C}$,

$$A_{\mathcal{C}}={\oint }_{\mathcal{C}}(\mathbf{\text{a}}\cdot \mathbf{\text{n}})\hspace{0.17em}\mathrm{d}\mathcal{C}.$$ 3.2

In particular, equation (3.2) can be applied to the specific case of a closed contour line $\mathcal{W}$ enclosing a minimum located at ${\mathbf{\text{x}}}_{min}$ (figure 2a), resulting in $A_{\mathcal{W}}={\oint }_{\mathcal{W}}a({\mathcal{W}}^{\prime })\hspace{0.17em}\mathrm{d}{\mathcal{W}}^{\prime }$ (note that here a⋅n=a, due to the orthogonality between contour and projected slope lines, along which a is defined). The limit of this integral where the area enclosed by $\mathcal{W}$ shrinks to the point ${\mathbf{\text{x}}}_{min}$ provides the drainage area A at the minimum, assuming that the surface is simply connected in the neighbourhood of the minimum. This definition provides a formal way to compute A at critical points through an integral of the specific drainage areas a associated with each slope line that runs into the minimum considered. The computation of a requires the knowledge of the location of surface maxima and minima and the integration of equation (2.7) along the projected slope lines from each maximum to the minimum of interest.

The specific drainage area a can be integrated along w to compute the drainage area A pertaining to any contour line bounded between w₁ and w₂ (equivalently to eqn (2) in [30]), resulting in

$$A(w_1,w_2)={\int }_{w_1}^{w_2}a(w)\hspace{0.17em}\mathrm{d}w.$$ 3.3

An alternative and more intuitive definition of A at surface minima can be delineated by partitioning the surface in basins of attraction of each minimum. In fact, the vector field described by the opposite of the elevation gradient (i.e. −∇z) defines a gradient system and, as such, can be analysed from a dynamical systems perspective. To this purpose, one should first note that, for a gradient system, critical points have real eigenvalues (no spirals or centres) and closed orbits are ruled out [42], so that critical points can only be saddles, stable or unstable fixed points of the surface. On a topographic surface, where the water flow is defined by the opposite of the gradient field (i.e. −∇z), stable fixed points represent local elevation minima, unstable fixed points are local maxima and saddle points represent, for example, mountain passes. Once the critical points are identified, their nature is determined by the Jacobian matrix of the vector field −∇z [42].

To compute A at surface minima, it is then possible to partition the phase space given by the position vector x and the vector field −∇z in different regions, representing the basins of attraction of each minimum. Evaluating the solution of equation (2.3) with u=v, the basin of attraction of a surface minimum would be the set of points attracted by the minimum for $t\to \mathrm{\infty }$ [42]. Given a stable fixed point (i.e. a local elevation minimum), it is possible to define its basin of attraction, whose projection on the horizontal surface corresponds to the drainage area A pertaining to that minimum of the surface. The boundary of the basin of attraction can be found in terms of separatrices, which are the stable manifolds of the saddle points and represent ridge lines of the topographic surface [42,43,48]. Thus, the entire topography can be partitioned by connecting ridge lines (found as stable manifolds of the saddle points) and each of these regions will encompass a minimum of the topographic surface: the drainage area pertaining to each minimum is the horizontally projected area of the basin of attraction of each minimum.

An example of this second definition based on partitioning the topography in basins of attraction of local minima is shown in figure 2b, where a contour plot of a gradient system is depicted together with its fixed points. Local elevation maxima and minima are connected to saddle points through the stable and unstable manifolds, respectively. The basin of attraction of each minimum, projected on the horizontal surface, gives the total drainage area A pertaining to that minimum (green shaded area in figure 2b).

(b). Singular points

We now focus on surfaces with singularities (e.g. folds), which are loci of points of the topography where the gradient is not defined. While only one slope line passes by each regular point of the surface (figure 3a,c), at singular points slope lines from different hillslopes merge together forming a cusp (i.e. channel; figure 3b,d): this results in a being a singular function, compared with A (figure 3). Furthermore, singular points are not necessarily minima of the topographic surface so that neither the dynamical systems concept of the basin of attraction of surface minima nor Gauss’s theorem is applicable.

Figure 3.

Conceptual representation of (a) regular points of the surface (for each point of the surface passes only one slope line) and (b) singular points where slope lines converge without the presence of any local minimum or saddle, but due to singularities of the topographic surface (cusp). The horizontal projection of the surface is also shown (c,d) and the values of a and A along a contour line of length w₂−w₁ are plotted: note that while a is a smooth function for case (a,c), it becomes singular at the fold in case (b,d). In the case of the cusp A is a step function, as a finite area is instantaneously added at the singularity (green shaded area in (d)). (Online version in colour.)

The behaviour of A and a around a singularity is depicted in figure 3d and compared with the case of regular points (figure 3c). Here, the drainage area A associated with a segment of contour line w is intentionally assumed to be the same for the case with and without the singularity to highlight the different behaviour in the two cases. For regular points a is a smooth function along the segment of contour line, resulting in a gradual increase of A, computed as the integral of a along w (equation (3.3), figure 3c). On the other hand, because of the singularity A results in a step function, due to the finite area that is instantaneously added (i.e. the green shaded area in figure 3d). Thus, the specific drainage area a at the singularity is proportional to a Dirac delta function (figure 3d). This is evident from equation (2.2), where, in the limit of $w\to 0$, A remains finite, so that the ratio A/w goes to infinity.

4. Special cases

A number of explanatory cases are presented in this section to discuss specific features of the definition of the drainage area at regular, critical and singular points of topographic surfaces. The cases of the planar slope and the convergent/divergent cone already presented in [30] are reviewed. These elementary examples, for which analytical solutions are available, provide useful insights into the role of the plan curvature in the computation of a. In addition, results for a paraboloid underline the importance of the plan curvature as a mechanism of flow convergence/divergence in the calculation of the drainage area, while demonstrating that profile curvature does not play any role. The case of a two-dimensional sinusoidal surface, characterized by both convergent and divergent areas, is then analysed, along with the case of the superposition of Gaussian functions. These surfaces allow us to analyse the behaviour of more complex topographies, characterized by the presence of ridges, valleys and multiple critical points. We conclude this section with the case of a folded surface with singularities. A simple one-dimensional case is presented in appendix A to illustrate how the specific area is defined as projected on the horizontal plane and highlight the behaviour of the drainage area at critical points.

(a). Planar surface

A first simple, yet explanatory, example is given by the planar surface z=x+y. In this case, the plan curvature is equal to zero, thus the specific catchment area at a point of the surface is equal to the length of the streamline from the ridge to the point (imposing a=0 at l=0 as the boundary condition),

$$\mathrm{d}a=\mathrm{d}l\to a=l.$$ 4.1

Thus, when the surface has zero plan curvature the specific drainage area a at a point coincides with the length of the projected slope line l going from the point to the hilltop. For the planar surface, the catchment area A draining to a segment of contour line with length w is simply equal to a rectangle of area aw.

(b). Cone

The second example considers the divergent conic surface $z=-\sqrt{x^2+y^2}$ (figure 4a). The vector field (2.1) is

$$\mathbf{\text{v}}(x,y)=\frac{x\mathbf{\text{i}}+y\mathbf{\text{j}}}{\sqrt{x^2+y^2}},$$ 4.2

where i and j are unit vectors in the coordinate directions. The vector field (4.2) provides the following equation for the streamlines:

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v_y}{v_x}=\frac{y}{x}\to y=Cx,$$ 4.3

where the integration constant C can be found by imposing the passage through a generic point (x₀,y₀), so that the equation for the streamlines becomes

$$y=\frac{y_0}{x_0}x.$$ 4.4

The arclength l from the hilltop (0,0) of the projected streamline is then equal to the radius r

$$l={\int }_0^{x_0}\sqrt{1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2}\hspace{0.17em}\mathrm{d}x={\int }_0^{x_0}\sqrt{1+{\left(\frac{y_0}{x_0}\right)}^2}\hspace{0.17em}\mathrm{d}x=\sqrt{x_0^2+y_0^2}=r.$$ 4.5

The plan curvature (equation (2.9)) for the divergent conic surface is positive and equal to k_c=1/r=1/l, so that the specific catchment area a reads

$$\frac{\mathrm{d}a}{\mathrm{d}l}=1-k_{\mathrm{c}}a=1-\frac{a}{l},$$ 4.6

which, upon integration, gives

$$a=\frac{l}{2}+\frac{C}{l},$$ 4.7

as derived by Gallant & Hutchinson [30]. To find the integration constant C, we define a as the ratio between the area of the sector and its arclength,

$$a=\frac{\theta r^2/2}{\theta r}=\frac{r}{2}=\frac{r}{2}+\frac{C}{r}\to C=0\to a(r)=\frac{r}{2}.$$ 4.8

Analogously, the integration constant could be found by imposing a=0 at l=0.

Figure 4.

Convergent and divergent cones (a,b) and paraboloids (c,d). Blue lines are contour lines; red lines are slope lines. The green shaded area A is the drainage area pertaining to the contour segment of length w. The specific drainage area a increases linearly from the centre outward in the case of a divergent cone/paraboloid ((e), equation (4.8)), while for the convergent cone/paraboloid a grows hyperbolically as $r\to 0$ ((f), equation (4.12)). (Online version in colour.)

In the case of a convergent conic surface, $z=\sqrt{x^2+y^2}$ (figure 4b), the vector field

$$\mathbf{\text{v}}(x,y)=-\frac{x\mathbf{\text{i}}+y\mathbf{\text{j}}}{\sqrt{x^2+y^2}}$$ 4.9

provides the following equation for the streamlines:

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v_y}{v_x}=\frac{y}{x}\to y=Cx\to y=\frac{y_0}{x_0}x.$$ 4.10

The plan curvature for the convergent conic surface is negative and equal to k_c=−1/r=−1/(R−l), where l is again defined from uphill to downhill, so that r=R−l, R being the distance of the uphill point to the centre of the cone. The specific catchment area a computed according to equation (2.6) is given by da/dl=1−k_ca=1+a/(R−l), which, upon integration, results in the same expression as that obtained by Gallant & Hutchinson [30]

$$a=\frac{-l^2/2+lR}{R-l}+\frac{C}{R-l}.$$ 4.11

Imposing that a=0 at l=0, C=0, and the expression for a in the case of a convergent cone is finally

$$a=\frac{R^2-r^2}{2r},$$ 4.12

which is obviously what we would obtain by simply dividing the area of the sector by its arclength. Thus, while in the case of a divergent cone a increases linearly from the centre outwards (equation (4.8), figure 4e), for the convergent cone the r/2 term is modified by a hyperbolic growth term as the centre of the cone is approached, resulting in a highly nonlinear increase of a towards the surface minimum (equation (4.12), figure 4f).

At the critical point (0,0) (surface minimum) the specific drainage area a is not defined, as $k_{\mathrm{c}}\to -\mathrm{\infty }$. The singular behaviour of a at the minimum is also evident from the definition of a provided by equation (2.2), where the denominator w goes to zero (as the contour line at the minimum would reduce to a point). However, it is possible to compute the drainage area A at (0,0) by applying Gauss’s theorem (equation (3.2)). Considering a contour line W at a distance r from the minimum (here contour lines are simply circles of radius r), the specific drainage area at each point along the contour line is equal to a constant value (equation (4.12)) and the length of the contour line is 2πr, resulting in

$$A=\underset{r\to 0}{lim}\frac{R^2-r^2}{2r}2\pi r=\pi R^2,$$ 4.13

which is the projected area of the entire cone of radius R, as expected.

(c). Paraboloid

It is instructive to compare the previous example with the case of a divergent/convergent paraboloid to show that only the convergence/divergence of slope lines (described by k_c) plays a role in the definition of a, while the curvature of the slope lines (called profile curvature) does not affect the value of a (as we would expect from equation (2.7), which is defined in terms projected slope lines).

For the divergent paraboloid described by the equation z=−x²−y² (figure 4c), the vector field is the same as for the divergent cone (equation (4.2)) and again the projected slope line length l is equal to r. Furthermore, the plan curvature results in k_c=1/r, so that the final equation for the specific catchment area a is the same as the one for the divergent cone, equation (4.8). Analogously, the case of the convergent paraboloid z=x²+y² (figure 4d) results in the same equation as the convergent cone, equation (4.12). The equivalent behaviour of the cone and the paraboloid is also evident from inspection of figure 4, again showing that, as the drainage area is defined projected on the horizontal surface, the only surface curvature that plays a role in its definition is the plan curvature. The profile (or vertical) curvature of the surface (i.e. the curvature of the surface as we move in the gradient direction along the slope line [41,57]) does not play any role in the definition of the specific catchment area a. It becomes important only in those applications where the ‘actual’ drainage area (i.e. not projected on the horizontal) needs to be defined.

(d). Sinusoidal surface

The two-dimensional sinusoidal surface (figure 5),

$$z(x,y)=\sin (x-\frac{\pi }{2})+\sin (y-\frac{\pi }{2}),$$ 4.14

is characterized by both convergent and divergent areas, as well as by multiple critical points (surface minima, maxima and saddles). The vector field v is

$$\mathbf{\text{v}}(x,y)=-\frac{(\sin x)\mathbf{\text{i}}+(\sin y)\mathbf{\text{j}}}{\sqrt{{\sin }^{2}x+{\sin }^{2}y}}.$$ 4.15

The streamlines can be derived as

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v_y}{v_x}=\frac{\sin y}{\sin x}$$ 4.16

and integrated to give

$$y=2\hspace{0.17em}\mathrm{arccot}\left(e^{-C}\cot \left(\frac{x}{2}\right)\right).$$ 4.17

By imposing the passage through a generic point (x₀,y₀), $C(x_0,y_0)=-\ln (\cot (y_0/2)\tan (x_0/2))$. Combining this expression for C with equation (4.17) provides an equation for the streamline passing through each point, $y=2\mathrm{arccot}(\cot (y_0/2)\tan (x_0/2)\cot (x/2))$. The behaviour of the streamlines and the values of the integration constant for a subset of the topographic surface (namely, between 0<x<π and 0<y<π) are shown in figure 5c. The specific catchment area a computed by integration of equation (2.7) along the slope line going from the hilltop in (π,π) to each point of the subset between 0<x<π and 0<y<π is shown in figure 6d.

Figure 5.

(a) Three-dimensional representation of the sinusoidal surface given by equation (4.14): symbols represent critical points (△=maxima, $▽=\mathrm{minima}$, °=saddles), while red and blue lines are the stable and unstable manifolds, respectively. (b) Contour plot of the elavation field given by equation (4.14): black dashed lines are contour lines, while the green shaded area is the total area draining to the local minimum located in (0,0). (c) Vector field given by equation (4.15) and slope lines computed according to equation (4.17) for different values of the integration constant C. (Online version in colour.)

Figure 6.

(a) Drainage area A along the central streamline l having equation y=x for the two-dimensional sinusoidal surface. (b) Plan curvature and (c) specific catchment area along the central streamline l. (d) Specific catchment area for the two-dimensional sinusoidal surface computed by integration of equation (2.7). (Online version in colour.)

The specific catchment area can be analytically derived along the central slope line passing by (π/2,π/2), and having equation y=x. On this slope line,

$$l=\sqrt{{(\pi -x)}^2+{(\pi -y)}^2}=\sqrt{2{(\pi -x)}^2}=\sqrt{2}(\pi -x),$$ 4.18

while the plan curvature is

$$k_{\mathrm{c}}(x)=-\frac{\cot x}{\sqrt{2}}\to k_{\mathrm{c}}(l)=-\frac{\cot (\pi -l/\sqrt{2})}{\sqrt{2}}.$$ 4.19

Using (4.19) in (2.7),

$$\frac{\mathrm{d}a}{\mathrm{d}l}=1+\frac{a\hspace{0.17em}\cot (\pi -l/\sqrt{2})}{\sqrt{2}},$$ 4.20

which, upon integration, gives

$$a(l)=-\sqrt{2}\cot \left(\frac{l}{\sqrt{2}}\right)+C\csc \left(\frac{l}{\sqrt{2}}\right).$$ 4.21

To find the integration constant C, we impose that $a\to 0$ as $l\to 0$, which provides $C=\sqrt{2}$. Thus, the specific drainage area along the central streamline reduces to

$$a(l)=\sqrt{2}\tan \left(\frac{l}{2\sqrt{2}}\right).$$ 4.22

The behaviour of k_c and a for the central slope line is displayed in figure 6b,c, which shows a slow increase of a in the first divergent part ($l<\sqrt{2}\pi /2$, k_c>0), while it rapidly grows as slope lines converge (k_c<0) to the local minimum.

The critical points of the surface described by equation (4.14) are given by x_c=nπ, y_c=nπ, with $n\in \mathbb{Z}$ (i.e. imposing −∇z=0), and their nature is defined by the Jacobian matrix of the vector field −∇z (see symbols in figure 5). Stable and unstable manifolds for each saddle point can be further computed (red and blue lines in figure 5), and provide ridge and valley lines, respectively. The basin of attraction for each minimum is then defined by connecting maxima and saddles through the stable manifolds of the saddle points. The projection on the horizontal plane of the basin of attraction is the drainage area pertinent to each minimum (as described in §3; see green area in figure 5b).

(e). Superposition of Gaussian surfaces

A more complex surface where analytical solutions are not available is provided by a superposition of three Gaussian functions,

$$z=A{e}^{-(x^2+y^2)/2}+B{e}^{-{(x+b)}^2/2\cdot d^2-{(y+b)}^2/2\cdot d^2}+C{e}^{-{(x+c)}^2/2-{(y+c)}^2/2},$$ 4.23

where b=1, c=3 and d=0.4. In particular, the three cases depicted in figure 7 are analysed: one surface obtained by superimposing three Gaussian functions (A=1, B=C=−1; figure 7a), and two surfaces given by the sum of only two Gaussian functions (C=0 and B=0, respectively; figure 7b–c). The vector field obtained by means of equation (2.1) is plotted in figure 7 for each case.

Figure 7.

Three-dimensional surface plots (a–c) and vector field (d–f) for the superposition of Gaussian surfaces (equation (4.23)). For the three surfaces, parameters were set equal to: (a,d) A=1, B=C=−1; (b,e) A=1, B=−1, C=0; (c,f) A=1, B=0, C=−1. In (d–f), the colourbar represents the surface elevation (blue= low, yellow=high), while symbols are critical points of the surface (△=maxima, $▽=\mathrm{minima}$, °=saddles). Red lines in (d,f) represent the stable manifolds (separatrices) connecting maximum and saddle and delineating the area draining to the minimum close to x=y=−1. (Online version in colour.)

For these Gaussian surfaces, the location and nature (i.e. minimum, maximum or saddle) of critical points cannot be determined analytically and are found according to the procedure described in appendix B, based on fitting a second-order polynomial at every surface point. Once saddle points are identified, it is possible to plot the stable manifolds connecting saddles to the surface maxima, thus partitioning the surface in basins of attraction for each minimum. In the first case analysed (figure 7a,d), the separatrices divide the entire domain in two basins of attraction, defining the area draining to the two minima. For the second case (figure 7b,e), the only critical points are a minimum and a maximum (no saddles are identified), so that the entire domain is the drainage area for the minimum. In the third case (figure 7c,f), only the minimum has a well-defined basin of attraction, while the remaining domain flattens in the limit $x,y\to \mathrm{\infty }$.

(f). Fold

This last example deals with singular points of the topography, where slope lines coalesce. A simple idealization of a topographic surface with singularities is given by the following function (depicted in figure 8):

$$z(x,y)=\{\begin{array}{ll}\sqrt{x^2}+x^2+y& \text{if }y\le 0\\ \sqrt{x^2+y^2}+x^2+y^2+y& \text{if }y>0,\end{array}$$ 4.24

the gradient of which is

$$\mathrm{\nabla }z(x,y)=\{\begin{array}{ll}(2x+\frac{x}{\sqrt{x^2}})\mathbf{\text{i}}+\mathbf{\text{j}}& \text{if }y\le 0\\ (2x+\frac{x}{\sqrt{x^2+y^2}})\mathbf{\text{i}}+(1+2y+\frac{y}{\sqrt{x^2+y^2}})\mathbf{\text{j}}& \text{if }y>0.\end{array}$$ 4.25

Figure 8.

Surface with a fold given by equation (4.24): (a) three-dimensional representation and (b) contour plot (white=high, dark grey=low). In (b), the black dashed lines are contour lines, the solid red lines are slope lines and the black solid line at x=0 highlights the singularity in the surface (i.e. the fold). The behaviours of k_c and a along the slope lines going from the ridge to a segment of contour line w (blue solid line) are shown in (c) and (d), respectively. (e) and (f) show a and A along the contour segment w going from P₁ to P₂ and passing through the singularity in P (at x=0). For the calculations shown in (c–f) ridges are assumed to be located in x=±2 and y=2. The green shaded area in (b) is the area that is instantaneously added in P, resulting in A being a step function (f). (Online version in colour.)

The surface defined by equation (4.24) is singular at x=0 for y≤0. In fact, here the gradient (equation (4.25)) is not defined and it changes sign passing through x=0, where the surface forms a cusp with a behaviour analogous to the one exemplified in figure 3b. At these points, the specific catchment area a is a singular function ($k_{\mathrm{c}}\to -\mathrm{\infty }$ in equation (2.7)), while A can still be determined as discussed in §3b (see also figure 3). An example is provided in figure 8, where a and A are computed along a contour line w between the points P₁ and P₂ (blue line in figure 8b). It is here assumed that the domain is bounded between x=±2 and y=2 (i.e. the location of ridges). Along w the surface is singular in P. The specific drainage area a is computed by integration of equation (2.7) along the projected slope lines going from the ridge to each point of the contour segment (figure 8e), while A is evaluated as the integral of a along w (equation (3.3)). At the singularity, the drainage area A is a step function (similarly to the conceptualization of figure 3d), as the whole area above the point P is instantaneously added (green shaded area in figure 8b). The specific drainage area increases from P₁ to P as a result of both the increased length of the projected slope line and the more convergent terrain (more negative k_c values, as shown in figure 8c). At P, a is proportional to a Dirac delta function. From P to P₂, a mirrors the behaviour between P₁ and P as a result of geometric symmetry around x=0.

5. Application to a real topographic surface

The analysis of real topographies requires moving from a continuous to a discrete representation of the surface, imposed by the finite resolution of a DEM. Topographic data are discrete approximations of terrain surfaces (typical elevation models are regular grid DEMs, triangulated irregular networks or contour-based DEMs) and the computation of the drainage area for these discrete surfaces requires the development of appropriate numerical models. It is beyond the scope here to even attempt a discussion of a rigorous translation of the mathematical concepts above to numerical algorithms, although some peculiar differences between discrete and continuous surfaces are discussed while presenting the results of a test case. The latter considers a portion (approx. 700 m by 700 m) of the Calhoun Critical Zone Observatory (CZO) located in South Carolina, USA (figure 9a), where a 1 m resolution DEM is available [61].

Figure 9.

(a) Location of the Calhoun CZO and (b) three-dimensional representation of the smoothed topographic surface analysed. (Online version in colour.)

As a first step towards making the translation from the previous theory to the discrete case, a moving average of size 20 m by 20 m was applied to smooth the topographic surface (figure 9b) in order to limit the number of critical points but still provide an informative test case. The computation of the specific catchment area a by means of equation (2.7) is feasible on the entire domain, excluding critical points of the surface where $k_{\mathrm{c}}\to \pm \mathrm{\infty }$. This requires the construction of slope lines passing through each DEM grid cell, with computational costs much higher than the commonly used flow routing algorithms [30]. An example of this calculation is provided for a subregion of the DEM relative to a contour line of length w (see figure 10a for location of the subregion and figure 11 for results). After computing the slope line going from each point of the contour line to the local surface maximum, equation (2.7) is integrated along each projected slope line, from the local maximum to the contour segment w and imposing a=0 at the hilltop. The values of the plan curvature k_c and the specific drainage area a along these slope lines are shown in figure 11a,b. Near the hilltop k_c is positive (i.e. divergent topography) and the specific drainage area along the slope line slowly increases. Moving downhill, as the plan curvature decreases and becomes negative (i.e. convergent topography), a rapidly grows (note the higher values of a in the central slope lines characterized by highly convergent terrain). The behaviour of both a and A along the contour line w is also shown in figure 11c,d (analogous to the conceptual representation of figure 3). A is here computed integrating a along w, according to equation (3.3). The central part of the contour line is characterized by a strong increase of a due to the convergent terrain configuration, resulting also in a steep increase of A (figure 11c,d), not too dissimilar from figure 3c. The value of A computed by means of equations (3.3) and (2.7) is also in good agreement with the area of the polygon enclosed by w and the slope lines (figure 11d), confirming the validity of equation (2.7) for the computation of the specific drainage area. Gauss’s theorem was here applied to find A along a contour line of length w, but the same procedure can be applied to compute the drainage area pertaining to any local minimum. In this case, a closed contour encompassing the minimum must be considered, as discussed in §3a.

Figure 10.

(a) Dynamical systems representation of the topography shown in figure 9b: the contour plot represents elevation; black dashed lines are contour lines; symbols represent surface maxima, minima and saddles, while blue and red lines are the unstable and stable manifolds. The green contour line w is used for the analysis of figure 11. (b) Elevation profiles along sections AB, CD and EF ($z_{min}$ being the minimum elevation in each section). (Online version in colour.)

Figure 11.

Computation of a and A along a contour line w (see figure 10a for the location of w). (a) and (b) show the contour curvature k_c (equation (2.9)) and the specific drainage area a (obtained by integration of equation (2.7)) along the slope lines going from discrete points of w to the local maximum (red triangle). Black solid lines in (c) and (d) show the values of a and A along the contour line w (analogously to the conceptual representation of figure 3d). A is computed by integration of a along the contour line w and is compared with the total area of the polygon enclosed by w and the two external projected slope lines, i.e. the area inside the green lines in figure 10 (grey dashed line, computed using the Matlab polyarea function). (Online version in colour.)

Regarding the computation of A at critical points, the dynamical systems concepts introduced in §3a can also be applied to define the drainage area at each minimum as its basin of attraction as well as to find ridge and valley lines based on the stable and unstable manifolds of saddle points. Critical points of the surface are shown in figure 10. Their location and nature were identified according to the procedure delineated in appendix B. From each saddle the stable and unstable manifolds can be constructed and provide a way to define ridge and valley lines (as defined by Cayley [32] and Maxwell [1]) on the entire domain: stable manifolds represent ridge lines connecting saddle points and local maxima, while unstable manifolds are valley lines connecting saddle points to local surface minima (figure 10). It is important to note, however, that, while stable and unstable manifolds can be used to identify some ridge and valley lines, they do not account for those ridges and valleys characterized by coalescence of streamlines without any associated saddle point [51] (e.g. the example of the fold). Once the stable manifolds are detected, the entire domain can be partitioned into basins of attraction of each surface minimum, thus allowing the reconstruction of the drainage area pertaining to each local surface minimum [43].

When moving from a continuous to a discrete surface, the detection of those valley lines related to singularities of the surface translates into finding a suitable criterion to infer the coalescence of streamlines. With reference to the three cross sections depicted in figure 10b, the increasing curvature of the topographic surface as slope lines merge together suggests that the discrete counterpart of the fold might be identified, at least in this case, based on local surface curvature. Figure 12 shows the behaviour of a and k_c along four unstable manifolds. When these slope lines merge together (points A and B in figure 12a), a decrease in k_c (indicating convergence of streamlines) is observed, which translates into a sudden increase of a (see insets in figure 12b,c). Such an increase is even more evident as the minimum is approached, where more slope lines coalesce into higher-order channels.

Figure 12.

Behaviour of (b) a and (c) k_c along the unstable manifolds (a) of saddle points 1–4 (circles). The blue triangle represents a surface minimum. Insets show the sudden variation of a and k_c where the four slope lines merge together (point B). Note that the four streamlines from the minimum to point B are very close to each other but do not overlie each other completely due to the discrete nature of the surface analysed, resulting in small differences in the progressive distance from the minimum to point B (in b and c). (Online version in colour.)

6. Discussion and conclusion

Despite the large number of numerical algorithms developed for its computation and the manifold of its geomorphological and ecohydrological applications, the specific drainage area was lacking an analytical definition until 2011 [30]. Its differential equation for regular points [30] was here rederived in a simpler and more intuitive way from a steady-state continuity equation. The definition of the drainage area was then extended to critical and singular points of the topographic surface.

The theoretical tools used here can be easily extended to discrete surfaces. For singular points the discrete nature of the surface requires an ad hoc definition of singularities. This coalescence of streamlines is related to the detection of channel lines in digital topographies and is crucial for hydrological and geomorphological applications [62]. While in the continuous case such behaviour is easily identifiable by the singularity of the surface (e.g. the example of the cusp in §4f), for a discrete surface it depends on the definition of empirical criteria. On the one hand, in geomorphology the automatic detection of channel heads and river networks from discrete DEMs (e.g. [4,20,62–65]) is typically based on imposing either a constant or a slope-dependent threshold on the drainage area. Within the computer vision literature, on the other hand, the channel delineation has been based on high convergence of streamlines (as theorized by Rothe [66] for valley lines). In this case, those grid cells that are crossed by a minimum number of streamlines are identified as channel lines (for an application see [67]). The sudden variation of a and k_c where slope lines merge together (figure 12b,c) suggests that the two methods for automatic detection of channels from DEMs are not too dissimilar. In fact, as is evident from figure 12c, the coalescence of streamlines translates into an increase of a, so that the detection of channel lines in terms of either merging streamlines or critical support areas (used as a surrogate for overland flow responsible for erosion and sediment transport) is intimately connected.

We hope this work may be useful towards a better ‘knowledge of the first elements of physical geography’ [1]. The theoretical definitions provided here can be used as a benchmark for the evaluation of current numerical methods for the definition of drainage area, as well as for advancing the theoretical analysis of landscape evolution dynamics and stability and channel formation theory.

Appendix A. One-dimensional case

A one-dimensional case is here analysed as a simple illustration of the fact that the drainage area is defined as projected on the horizontal plane and to highlight the singular character of the specific drainage area at local minima. In the one-dimensional case A coincides with the specific drainage area a. In any one-dimensional case, the drainage area is simply the distance (projected on the horizontal axis) of each point x from the closest peak, given that there is no change in gradient sign going from the point x to the peak. In fact, in this case the topographic surface is only a function of x and there is no plan curvature to be defined. Thus, equation (2.7) simply reduces to da=dl, so that the specific drainage area a is equal to the length of the projected slope line l starting from a maximum and going to a minimum. Note that the slope line is defined projected on the horizontal surface, so that a is the projected arclength going from each maximum to the minimum.

As an example, consider the elevation field given by the superposition of two sinusoidal waves (figure 13a). The corresponding specific drainage area a is shown in figure 13b, where it is evident that a is a singular function at local minima of the surface where flows from different hillslopes converge and are summed.

Figure 13.

(a) Elevation field z and (b) specific drainage area a for the one-dimensional surface given by $z=\sin (x-\pi /2)+0.5\sin (6x-\pi /2)$. Red and blue dots in (a) are the local maxima and minima, respectively. (Online version in colour.)

Appendix B. Detection of critical points

Where the locations of the critical points cannot be computed analytically, the following numerical procedure is adopted. At each point (x_g,y_g) of the discretized surface, a second-order polynomial is fitted to the nine points comprising (x_g,y_g) and the eight neighbours. The polynomial surface has the following equation:

$$Z(x,y)=p_1+p_{2}x+p_{3}y+p_4{x}^2+p_{5}xy+p_6{y}^2.$$ B 1

For each surface point the critical point (x_c,y_c) of the fitted surface can be easily found by imposing −∇Z=0, where

$$-\mathrm{\nabla }Z=-(p_2+2p_{4}x+p_{5}y)\mathbf{\text{i}}-(p_3+2p_{6}y+p_{5}x)\mathbf{\text{j}}.$$ B 2

If the critical point lies between x_g±dx/2 and y_g±dy/2, then the grid point is recognized as a critical point. The nature of the critical point is then defined by means of the Jacobian of the vector field (B.2)

$$J(x_{\mathrm{c}},y_{\mathrm{c}})=\left[\begin{array}{cc}-2p_4& -p_5\\ -p_5& -2p_6\end{array}\right],$$ B 3

which has trace τ=−2(p₄+p₆) and determinant $\mathrm{\Delta }=4p_4{p}_6-p_5^2$. When Δ<0, the critical point is a saddle; when Δ>0 and τ>0, it is a surface maximum; and when Δ>0 and τ<0, it is a surface minimum.

Data accessibility

The Calhoun DEM [61] was made available by the OpenTopography Facility with support from the National Science Foundation under NSF Award nos. 1226353 and 1225810.

Authors' contributions

All the authors contributed equally to conceiving and designing the study. S.B. performed the analyses and wrote an initial draft of the paper, to which all the authors contributed edits at all stages. All the authors helped to interpret the results.

Competing interests

We have no competing interests.

Funding

S.B. and A.P. acknowledge support from the US National Science Foundation (FESD EAR-1338694). A.P. also acknowledges support from the USDA Agricultural Research Service cooperative agreement 58-6408-3-027; National Science Foundation (NSF) grant nos. CBET-1033467, EAR-1331846 and EAR-1316258; and the Duke WISeNet grant no. DGE-1068871.