The connection between surface process rates and curvature was recognized as early as the late 19th century when work by G.K. Gilbert and W.M. Davis suggested connections between hillslope convexity and rates of denudation in mountain terrains . Efforts to define topographic structure predate these observations, however. As has been pointed out in , topographic curvature has been studied since at least the middle nineteenth century. Arthur used topographic contours to show that watershed bounding ridges are composed of “summits” (we will term these structures “domes”) connected by “knots” (we will call these “saddles”) such that each ridge line contains one more “summit” than “knot”. He argued that “immits” (we will call these “basins”) would be similarly connected by bridging saddle structures such that there is one more “immit” than connecting saddle. James Clerk similarly argued that the Earth's surface could be sorted into four shape classes; “hills” (domes), “dales” (basins), “passes” connecting hills (antiformal saddles), and “bars” connecting dales (synformal saddles), such that there will always be one more dale than bar, and one more hill than pass, thus reaching the same topological conclusion as Cayley regarding the connectivity of surface shape classes.

Today, several curvature-based metrics are used for surface classification and as an ingredient in mechanistic transport laws. As examples of classification, derived 12 curvature metrics which were used in a landscape partitioning scheme, and used geodesic curvature of topographic contours in combination with drainage area thresholding to extract channel networks from DEMs. showed that curvature is intimately connected to accumulation of overland flow, presented an extensive list of land surface curvature metrics and proposed links to topographic equilibrium, and derived curvature metrics using 2-D polynomial fits of topography for GIS applications. Such classification schemes have proven useful in surface process studies and for mapping topographic characteristics of hazard susceptibility and land use .

In mechanistic erosion models, curvature arises from continuity requirements as the divergence of a gradient driven sediment flux law . Curvature, often approximated as ∇2z where z is surface elevation, is thus used as a quantitative proxy for spatial variation in erosion rates . At the scale of orogenic provinces, erosion rates are proportional to long-wavelength surface curvature, scaled by an empirical diffusivity constant . At finer spatial scales, curvature-driven diffusion of ridges is overtaken by advective transport as drainage area increases, and sediment is transported by concentrated overland flow within the fluvial network .

We present a formal differential geometry approach that extracts geometric information from the curvature tensor directly, providing a self-consistent means of evaluating topographic form across process domains. Comparing invariants of the curvature tensor to upstream drainage area A, a quantity that underlies empirical scaling relations and process regimes provides an intuitive description of river basin development. This approach also makes a quantitative connection between the early landscape organization theories of Maxwell and Cayley and drainage area analysis methods common in fluvial geomorphology today.

Curvature formally refers to a class of mathematical operations that quantify deviations of a surface (or, more generally, a manifold) from flatness . Differential geometry and tensor calculus were in part developed to describe these operations . Intrinsic curvatures are independent of coordinate system and can be calculated using only local surface information . Extrinsic curvatures are defined with respect to the ambient space in which the surface is embedded, and thus depend on the choice of external reference frame .

On DEMs, the accurate calculation of either intrinsic or extrinsic curvatures requires careful consideration of coordinates to avoid distortions that come from projection of topography onto a map grid. The effects of projection can be seen in Fig. , which compares the distances and angles of a map projection (Fig. a) to those of the same grid lines overlaying the 3-D surface (Fig. b). In the map-view representation, the E-W and N-S grid lines are perpendicular and evenly spaced. If one were to define displacement vectors dx and dy emanating from point p along these grid lines, their combination would create a resultant displacement ds′ ending at point q′. In Fig. b, however, displacement vectors du and dv, which connect p to the same points on the surface as dx and dy respectively, are not perpendicular and their combination results in a displacement (ds) that maps to a different point (q). Neighboring grid cells therefore have non-uniform dimensions and form non-orthogonal angles.

Thus, accurate geometric calculations on topography require viewing a DEM not as a regular grid, but as a set of irregularly spaced data points sampling a surface, an approach that is similar in spirit to how elevation data are treated in landscape evolution models . To accurately define a surface, distances and angles between grid cells are not treated as uniform quantities – they are computed locally within a reference frame defined at each point, reflecting the variation in surface geometry across the domain. This specific problem of topographic projection was recognized by Euler in 1775 and motivated the work of Gauss, who, roughly fifty years after Euler's observation, established the mathematical framework for accurate geometric classification of surfaces . Using an approach similar to Gauss, we derive both intrinsic and extrinsic topographic curvature metrics built on invariant surface quantities.

At any point on a twice-differentiable surface, there exist two perpendicular directions along which the minimum and maximum normal curvatures occur . Between these directions, the curvature varies smoothly as 1κ(θ)=k1cos⁡2θ+k2sin⁡2θ, where the extrema k1 and k2 are called the principal curvatures and θ is an angular direction measured within the surface tangent plane. Equation (), known as Euler's Theorem, shows that the principal curvatures can be used to calculate normal curvature along any surface direction.

The principal curvatures can be used to calculate two other useful invariant quantities that will be more central to our analysis; the “mean” and “Gaussian” curvatures. The mean curvature, an extrinsic quantity, follows directly from Euler's Theorem and is the value about which the curvature oscillates as a function of angle on the surface (Eq. ). While it can be calculated as the average curvature of any two perpendicular paths, we define it in terms of the principal curvatures as 2KM=k1+k22.

The Gaussian curvature (KG) can be defined as the product of the principal curvatures 3KG=k1k2. This value is intrinsic, meaning it is unchanged under isometric transformations and does not depend on the actual shape of the surface in space. Instead KG captures a more subtle quality: the degree of stretching or bending required to deform a flat plane so that it conforms to the surface . Note that the units of KG (m⁻²) are not the same as KM (m⁻¹).

The mean and Gaussian curvatures together uniquely determine the local geometry as one of eight distinct shape classes (; Fig. ). Since the Gaussian curvature is the product of the two principal curvatures, it is positive only when k1 and k2 have the same sign. Positive KG thus correlates to either domes or basins, though we cannot discern which from KG alone. If KG is negative, then k1 and k2 have opposing signs and the surface is locally a saddle. As with positive KG, the orientation in space cannot be determined from this intrinsic quality. In cases where either k1 or k2 is equal to zero, KG is also zero. Such shapes comprise a class of “developable surfaces”, which are intrinsically flat and can be formed from a plane without altering surface area. Curvature thresholding to extract developable forms is a promising approach for classifying landforms. However, we do not explore this further here.

The orientation of a shape is an extrinsic quantity that can be determined from the mean curvature, allowing us to put geometric classifications based on KG into a landscape reference frame. This requires an arbitrary choice of positive curvature direction, which we choose to be positive for downward concavity consistent with differential geometry implementations in structural geology . KM is positive in two cases: when both k1 and k2 are positive, or when the higher-magnitude curvature (k1) is positive. This means that points in the landscape with KM>0 are concave down and are locally either domes or antiformal saddles. Similarly, if KM is negative, then the surface must be predominantly concave up and is either a basin or synformal saddle. More generally, the sign of the mean curvature allows us to differentiate between the divergence and convergence of surface gradient vectors.

Defining curvature rigorously on discretely sampled topography requires accounting for changes in surface orientation between neighboring points, and how that change is scaled by non-uniform distances on a complicated surface. Our derivation of principal curvatures largely follows , though we point to complementary references throughout. A position on the surface is defined as the endpoint of a position vector parameterized by uv-coordinates such as those shown in Fig. b, but referenced to a Cartesian basis as 4r(u,v)=r1e^1+r2e^2+r3e^3 where the e^i are unit vectors corresponding to easting, northing, and elevation, and u and v are coordinates following any two intersecting curves on the surface. In this case the uv-coordinates follow the lines on the map-view grid, but we do not assume orthogonality on the surface. The square of the infinitesimal arc-length between points is given by 5I=ds2=dr⋅dr=Edu2+2Fdudv+Gdv2 where E=∂r∂u⋅∂r∂u, F=∂r∂u⋅∂r∂v, and G=∂r∂v⋅∂r∂v (the metric coefficients) quantify the proportionality of distances measured on the surface to distances in the Cartesian reference frame. They can also be used to calculate the ratio of area on the surface to pixel area as α=EG-F2, a quantity we will use to calculate intrinsic drainage areas in our analysis (Sect. ).

Equation (), known as the first fundamental form or surface metric formula, is used to calculate distances and areas on the surface. This in turn can be used to scale topographic curvatures. Curvatures are calculated as a change in surface orientation along a path, defined as 6II=dr⋅N=edu2+2fdudv+gdv2, where e=∂2r∂u2⋅N, f=∂2r∂u∂v⋅N and g=∂2r∂v2⋅N (the curvature coefficients) are the projection of directional curvatures onto a unit normal vector 7N=∂r∂u×∂r∂v∂r∂u×∂r∂v.

Equation () is called the second fundamental form, and measures changes in the orientation of the tangent plane in the direction of ds. Combining the information in Eqs. () and () as 8κ=-III=-drds⋅dNds=-edu2+2fdudv+gdv2Edu2+2Fdudv+Gdv2 allows us to fully characterize the local shape of a surface in 3-D space. The coefficients of the second fundamental form (e, f and g; Eq. ) are the directional curvatures where e and g correspond to curvature along the E-W and N-S grid lines respectively, and f is a cross term that accounts for directional covariance. These values are scaled by the coefficients of the first fundamental form (E, F, and G; Eq. ), which maps lengths on the coordinate grid to lengths on the surface.

The directions of the principal curvatures can be found algebraically by defining a parameter λ=dv/du and rewriting Eq. () as 9κ(λ)=edu2+2fλdu2+gλ2du2Edu2+2Fλdu2+Gλ2du2=e+2fλ+gλ2E+2Fλ+Gλ2.

Since the principal curvatures correspond to extrema where dκ/dλ=0 we differentiate Eq. () with respect to λ and set the result equal to zero giving 10dκdλ=E+2Fλ+Gλ2(f+gλ)-e+2fλ+gλ2(F+Gλ)=0, a quadratic equation in λ whose roots correspond to the principal curvature directions. Recalling that λ=dv/du these values can be equated to angles in our local uv-coordinate system and can thus reference principal curvature orientations within the map-view grid.

Magnitudes of the principal curvatures are found through a similar approach. Since dκ/dλ=0 along the principal directions, Eqs. () and () are combined to give a simpler expression for the curvature 11κ=f+gλF+Gλ.

Recognizing that 12E+2Fλ+Gλ2=(E+Fλ)+λ(F+Gλ) and 13e+2fλ+gλ2=(e+fλ)+λ(f+gλ), Eq. () can be rearranged to show 14κ=f+gλF+Gλ=e+fλE+Fλ.

The two expression for curvature given by Eq. () are rearranged as 15f-κF+λ(g-κG)=0 and 16e-κE+f-κF=0, respectively. Multiplying Eqs. () and () by du (with λ=dv/du) we arrive at a system of linear equations in our original uv-coordinate system 17e-κEf-κFf-κFg-κGdudv=00.

This has a non-trivial solution only if the determinant of the coefficient matrix is zero. The corresponding quadratic equation in κ 18(EG-F2)κ2-(gE-2fF+eG)κ+(eg-f2)=0 has roots that are the principal curvatures. By convention, we take the more positive of these roots to be k1, while the less positive curvature is k2.

To calculate DEM curvatures, it is necessary to do some degree of smoothing to remove artifacts of the gridding process . We use 8.1 m resolution DEM data freely available through the National Map (https://apps.nationalmap.gov/downloader/, last access: 16 November 2024). While higher resolution LiDAR (Light Detection and Ranging) data are available in the study area, the coarser dataset is sufficient for resolving geometric trends and ridge-valley landforms at the scale of fluvial basins.

There are many established approaches to DEM smoothing, including b-spline fitting , wavelets , selective denoising , and TIN interpolation . We choose to filter the data using a Discrete Fourier Transform (DFT; also a contribution of Gauss; ), which decomposes discretely sampled signals into sums of harmonic functions. Smoothing is accomplished via low-pass filtering, where information at wavelengths smaller than a defined cutoff is removed. Fourier methods have been extensively applied in geomorphology toward the identification of characteristic process scales , landform classification , and the assessment of topographic controls on mass transport mechanics .

One challenge of Fourier methods is that harmonic functions do not naturally respect the finite nature of a DEM. Tapering of the data is thus required to obtain zero elevation at the boundaries prior to applying a DFT. It is common to accomplish this by convolving the DEM grid with a 2-D raised cosine (aka Hanning window), such that the resulting topography is equal to its actual value only at center of the grid, and approaches zero at the margins .

A downside of this approach is that it does not preserve the spectral power of landscape features. Fortunately, this effect can be minimized by first reflecting the topographic grid along each coordinate axis, then tapering the data only in reflected portions that fall outside the limits of the original DEM . While spurious signals at wavelengths greater than the DEM are not eliminated, this windowing approach minimizes smaller scale distortion within the study area. We use a Tukey window (implemented as window2 in Matlab), which consists of a boxcar function convolved with a cosine taper along the margins .

The Discrete Fourier Transform (DFT) is calculated as 19Z(kx,ky)=∑p=0Nx-1∑q=0Ny-1zpΔx,qΔye-2πikxpNx+kyqNy, where Nx and Ny are the number of grid cells in each direction, p and q are array indices, Δx and Δy are the grid spacings in each direction, and kx and ky are the wavenumbers in the respective x and y directions . Each value in the output array given by the above equation is associated with a frequency in x and y directions with magnitudes 20fx=kxNxΔx,fy=kyNyΔy. These frequencies can then be used to define a radial frequency as 21fr=fx2+fy2. The DFT periodogram is then given by 22Pf(kx,ky)=1Nx2Ny2|Z(kx,ky)|2

Following we apply a half-Gaussian filter based on radial frequencies 23Flow=1,f<f1,exp⁡-(f-f1)22σ2,f≥f1, where σ=13|f2-f1| is the standard deviation. The filter is convolved with the radial frequency spectrum before the filtered spectrum is reverse transformed and the original domain of the DEM is extracted from the windowed representation to yield a low-pass filtered raster of topography.

Once the landscape has been filtered, the invariant curvature metrics outlined in Sect.  and are calculated on each DEM pixel. Curvature values are binned by drainage surface-area, calculated using the D-infinity algorithm implemented in the TopoToolbox MATLAB function library , with pixel values weighted by the surface area ratio (α) defined in Sect. .

To explore the impact of low pass filter scale, we first smooth the landscape to 50 m, then increase the low-pass filter cutoff by increments of 50 up to 500 m and look for shifting patterns in the area-space evolution of topographic geometry (Fig. ). While the magnitudes of curvature and slope vary with increased filtering, general trends in these metrics are similar across this range of filter cutoffs. This suggests that the filter cutoff parameter does not strongly alter landscape geometric structure. However, while the magnitudes of mean curvature decrease systematically with increasing filter cutoff, the main extrema in Gaussian curvature have the greatest magnitudes at a cutoff of 200 m, perhaps indicating a characteristic curvature scale in the landscape.

Based on these observations, we perform all further analysis on topography low-pass filtered to 200 m. This filter scale allows us to identify landscape features that span hillslope and fluvial process regimes, but inhibits our ability to analyze topography at finer scales. Map-view distributions of mean and Gaussian curvatures, principal curvatures, tangent plane slope, and upstream drainage area for a DEM filtered to 200 m are shown in Fig. . The MATLAB code used to filter and compute curvatures on the landscape is publicly available , in addition to a Python implementation .

In this work, we have proposed a geometrically self-consistent framework for defining topography through connection to formal surface theory. A benefit of intrinsic geometric methods is that they eliminate the projection distortion of surface properties computed in map-view, which is the most common way to calculate topographic slope and curvature. The severity of such projection distortion depends on the geometry of the surface itself, as map-view approximations are quite valid in low-slope regions but are less so in steep topography. Below, we compare our method to these common approaches and quantify where on landscapes the intrinsic geometric perspective is likely informative.

Noting significant and systematic variation in shape class distributions and curvature metrics with upstream drainage area (Fig. a–c), we now explore landscape segmentation using inflection points in the mean and Gaussian curvatures. This is motivated by the physical assumption that signs of both KM and KG have implications for mass transport phenomena. The sign of KM records the divergence versus convergence of local gradients, while the sign of KG differentiates between stable and unstable “critical points” that influence how the surface responds to disturbances . As our partitioning approach is rooted in geometry, we choose a labeling scheme Σij based solely on curvature invariants. Subscripts indicate the curvature used (i=G for KG and i=M for KM), while superscripts (j) correspond to the number of previous zero crossings in area space (j=0 corresponds to zero drainage area).

In this section we show that mean curvature also provides a way of understanding connections between geometry and process in fluvial topography. We decompose the landscape into two regions (ΣM0 and ΣM1) separated by the single inflection in KM at drainage areas of 7.40×102 m². Results are shown in Fig. . Alignment between this point in area-space and the peak of the slope curve in Fig. a is consistent with the idea that curvature decreases as hillslope profiles approach an angle-of-repose (∼ 30° inferred from the peak of the slope curve in Figs. –), above which loose material is gravitationally unstable . Downhill of this point, slope decreases and unconsolidated material will tend to collect as colluvium at the head of the channel network (region ΣG2).

Partitioning the landscape this way reveals surprising symmetries in both shape class distributions and surface geometry metrics for the two regions (Fig. d–e). The landscape is equally distributed about this zero crossing. For our filtering method and scale, 50 % of points are above or below the most probable drainage area value in our study area. Figure a shows a map of the study area divided into concave and convex domains based on this area threshold. Probability distributions of slope, mean curvature, and Gaussian curvature for the two regions are shown in Fig. b–d. While the slope and Gaussian curvature are similarly distributed in the concave and convex landscape regions, we see that the mean curvature has mirrored distributions such that the integrated mean curvature in the landscape is approximately zero.

We have thus far focused on documenting Oregon Coast Range landscape segmentation in drainage area from a curvature perspective. A clear corollary to this is to ask specifically about the emergent channel and ridge network structures that manifest from this drainage area segmentation. It is well established that curvature provides a powerful tool for extracting continuous concave-up structures and deriving definitions of channel networks that are self-consistent throughout the landscape . Our methods are suitable for this task as well, and for the parallel extraction of concave-down ridge network structures , but we will not pursue that objective here.

Instead, we will focus on the strikingly even partitioning of mean curvature between concave up structures (channels) and concave down structures (ridges). These structures are themselves composed entirely of alternating basins and synformal saddles (in channels) and domes and antiformal saddles (on ridges). Figure a shows a close-up of our study area around Franklin Creek to demonstrate this pattern. While the size distribution of these alternating shape classes within a channel or ridge is variably sensitive to lowpass filter threshold, the shape classes themselves are much more robust as they reflect zero crossings in KM and KG whose positions are insensitive to filter cutoff (Fig. ), particularly in the case of mean curvature. This alternating pattern of local shape classes, originally recognized qualitatively by and , manifests clearly in channel and ridge network geometry.

Figure b–c plots in blue the elevation of Franklin Creek and its south ridge as a function of distance from the most downstream point of the creek (where it intersects the Umpqua river). The drainage area (red curves) along these structures (calculated using the intrinsic area calculation method in Sect. ) reflects expectations: discontinuities in drainage area along the channel correspond to tributary junctions while ridge-top drainage area deviates from one grid cell only in saddles (up to 8 grid cells long here) between local maxima.

Figure d–e plots the signed principle curvatures for ridge and channel. An immediate comparison of note is that local minima in k1 for the channel and k2 for the ridge correspond to basins and antiformal saddles, respectively (circles). These structures align with tributary junctions in the channel, and lie directly upslope of 1st order channel heads on the ridge. This indicates that the curvature shape classes reflect structural changes in network geometry for both concave and convex topography. Neither structure – the local basins at channel junctions or saddles on ridgetops corresponding to transitions from hillslopes to channels – have been previously described to our knowledge. As these geometries describe changes in curvatures associated with branching structures in channels and ridges, their locations are minimally sensitive to the filter cutoff used for smoothing the DEM.

Figure f–g then plots the Gaussian (KG) and mean (KM) curvatures along the channel and ridge. The notable comparison in this case is that local maxima in KG and KM are anticorrelated along the channel and correlated along the ridge. This symmetry reflects the paired shape classes in either structure.

Comparing channel and ridge geometries we see that, in both cases, the two measures of curvature (principle curvatures or invariants) are near mirror-images of each other. One metric oscillates around zero (a principle curvature in Fig. d–e and KG in Fig. f–g), while the other is strictly positive (for ridge) or negative (for channel), although still oscillatory. The along-channel and along-ridge envelope of this latter metric varies non-monotonically as expected for a small dendritic drainage basin, but also exhibits coincident channel widening (decrease in the magnitude of k2) and ridge narrowing (increase in k1 towards the mouth of Franklin Creek (distances ≲ 2300 m).

Finally, while the development of curvature-driven process models is outside the scope of this work, it is informative to compare the observed curvature of channel and ridge to theoretical models. Figure c–e show best fitting power laws to ridge and channel, after for bedrock channel longitudinal profile and models such as those of for interfluvial ridge elevations. The fit constants are a=6 m, b1=2.84×10-7 m^−1.4, c1=-1.4, b2=41.88 m^0.72, c2=0.28.

For Franklin Creek, while the elevation profile is well-approximated by a stream power-law fit, the resulting curvature (obtained by differentiating the longitudinal profile twice with respect to alongstream distance x) does not capture oscillations observed in k1, the along-channel principal curvature. However, the average value of the stream power model curvature (1.1×10-5 m⁻¹) is close to the average value of k1 (8.3×10-5 m⁻¹) extracted from the DEM (we expect an even closer match if tributaries are included in the stream power model, e.g., ). Thus, the steady state model approximates the average concavity of the true channel geometry, despite much larger curvature oscillations associated with local basin structures at tributary junctions.

Similarly, a power-law fit to the Franklin Creek south ridge profile in Fig. c well represents the elevation but fails to capture the smaller scale curvature oscillations oscillations between domes and antiformal saddles. The average value of this fit (-3.3×10-4 m⁻¹) is within 7 % of the average value of k2 (-3.5×10-4 m⁻¹) extracted from the DEM, reflecting the overall concave down nature of the along-ridge curvature. These results suggest that standard fluvial process models, while missing physical ingredients at smaller scale, capture network-scale curvatures of channels and ridges.