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###### Vertical curvature

Vertical (or profile) curvature (kv) is the curvature of a normal section DPD’ having a common tangent line with a slope line sl at a given point P of the topographic surface*. The unit of measurement is m-1.

Once elevations are given by , where x and y are plane Cartesian co-ordinates, vertical curvature is a function of the partial derivatives of z:

,

where , , , , .

Vertical curvature is a measure of relative deceleration and acceleration of gravity-driven flows. Overland and intrasoil lateral flows are decelerated when kv < 0, and they are accelerated when kv > 0. Geomorphologically, kv mapping allows revealing terraces and scarps.

Like other local morphometric variables, vertical curvature can be derived from a digital elevation model (DEM) by a universal spectral analytical method as well as finite-difference methods (e.g., method 1, method 2, and method 3).

Example**. A model of vertical curvature was derived from a DEM of Mount Ararat by the universal spectral analytical method. The model includes 779,401 points (the matrix 1081 x 721); the grid spacing is 1". To deal with the large dynamic range of this variable, its values were logarithmically transformed. The vertical exaggeration of the 3D model is 2x. The data processing and modelling were carried out using the software Matlab R2008b.

#### ** Florinsky, I.V., 2016. An illustrated introduction to geomorphometry. Almamac Space and Time, 11 (1): 20 p. (in Russian, with English abstract).  Article at the journal website

For details and other examples, see:

 DIGITAL TERRAIN ANALYSIS IN SOIL SCIENCE AND GEOLOGY   2nd revised edition     I.V. Florinsky   Elsevier / Academic Press, 2016 Amsterdam, 486 p.   ISBN 978-0-12-804632-6