Why are there so many surface curvatures?

When I first started in my Masters degree program several years ago now, I had only a very basic understanding of geomorphometry as a scientific field. It was at this time when John Lindsay, my Masters advisor, handed me a medium sized blue book and said ‘read this’. The book was Terrain Analysis : Principals and Applications by Dr. John Wilson and Dr. John Gallant. This invaluable resource was more than just a book to me, it was my first exposure to geomorphometry, and a handbook that I still frequently refer to to this day.

Chapter 3 in particular intrigued me, with its wonderfully meticulous categorization of the various primary land-surface parameters. Specifically, the topographic attributes Plan, Profile, and Tangential curvature stuck out to me, as I recognized seeing them within the WhiteboxTools software, which was being developed in the research group in which I was studying. I downloaded a test data set and began extracting these curvatures using WhiteboxTools. After extracting these land-surface parameters, and becoming more familiar with the geomorphometric literature, I thought I had a good understanding of how curvature can impact landform processes and how it can be used to identify topographic features. I thought I had a good understanding of curvature, how to use them and the methods to calculate them; however, that turned out to be wrong.

As my professional career in geomorphometry progressed over the following years, I learned that many additional curvatures existed. I’ll admit that when I first realized that the family of surface curvatures is far more extensive than the basic three to four curvatures commonly found in most GIS software, I was confused about how each can best be applied in my work as a GIS professional. Why are there so many different curvatures? During a conversation with John about surface curvatures, back when he was working on adding new curvature tools to WhiteboxTools, he pointed me to the works of Dr. Igor Florinsky. In his recent book titled “Digital Terrain Analysis in Soil Science and Geology“, and specifically Chapter 2 and Chapter 4, Dr. Florinsky does a remarkable job of describing the many types of surface curvatures in detail. It was through reading these works that I came to realize that the answer to my question as to why there are so many curvatures is that a curvature is defined by the normal section to the surface at a point, and there are essentially an infinite number of normal sections (rotated 360-degrees around the point). Of course, only a finite number of the curvatures are meaningful and have defined names. However, each of these named curvatures offer the ability to characterize the topographic form of the surface in slightly different ways, providing unique information.

Dr. Florinsky identifies approximately 16 curvature attributes, many of which I had not encountered before reading these book chapters. Through this work I gained an enormous insight into the theory behind these topographic attributes, the information that they can convey about the topographic character of a landscape, and an appreciation for the power that curvatures can have as environmental covariates in predictive modelling. I started to experiment with curvature extraction in my own work.

Many of the curvatures described by Dr. Florinksy have now been added to the WhiteboxTools‘s open-core and Extension products. In total 14 new curvature tools, on top of the four existing curvature tools (Plan, Profile, and Tangential and Total curvature), have been added to either the WhiteboxTools Open Core or the Whitebox Geospatial Extensions in the last few months. The following sections provides a brief explanation of each new curvature tool and I hope that it can help you more easily integrate these powerfully predictive land-surface parameters into your own work.

Accumulation Curvature

Accumulation Curvature is the product of profile (vertical) and tangential (horizontal) curvatures at a location (Shary, 1995). This variable has positive values, zero or greater. Florinsky (2017) states that accumulation curvature is a measure of the extent of local accumulation of flows at a given point in the topographic surface.

An example of Accumulation Curvature extracted from WhiteboxTools.

Curvedness

Curvedness (Koenderink and van Doorn, 1992) is the root mean square of maximal and minimal curvatures, and measures the magnitude of surface bending, regardless of shape (Florinsky, 2017). Curvedness is characteristically low-values for flat areas and higher for areas of sharp bending. It’s useful for mapping breaks-in-slope.

An example of Curvedness extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Difference Curvature

Difference Curvature is half of the difference between profile and tangential curvatures, sometimes called the vertical and horizontal curvatures (Shary, 1995). This variable has an unbounded range that can take either positive or negative values. Florinsky (2017) states that difference curvature measures the extent to which the relative deceleration of flows is higher than flow convergence at a given point of the topographic surface.

An example of Difference Curvature extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Gaussian Curvature

Gaussian Curvature is the product of maximal and minimal curvatures, and retains values in each point of the topographic surface after its bending without breaking, stretching, and compressing (Florinsky, 2017).

An example of Gaussian Curvature extracted from WhiteboxTools.

Generating Function

Shary and Stepanov (1991) describe Generating Function as a measure of the deflection of tangential curvature from loci of extreme curvature of the topographic surface. Florinsky (2016) demonstrated the application of this variable for identifying landscape structural lines, i.e. ridges and thalwegs, for which the generating function takes values near zero. Ridges coincide with divergent areas where generating function is approximately zero, while thalwegs are associated with convergent areas with generating function values near zero. 

An example of Generating Function extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Horizontal Excess Curvature

Horizontal Excess Curvature is the difference of tangential (horizontal) and minimal curvatures at a location (Shary, 1995). This variable has positive values, zero or greater. Florinsky (2017) states that horizontal excess curvature measures the extent to which the bending of a normal section tangential to a contour line is larger than the minimal bending at a given point of the surface. 

An example of Horizontal Excess Curvature extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Maximal Curvature

Maximal Curvature is the curvature of a principal section with the highest value of curvature at a given point of the topographic surface (Florinsky, 2017). The values of this curvature are unbounded, and positive values correspond to ridge positions while negative values are indicative of closed depressions.

An example of Maximal Curvature extracted from WhiteboxTools.

Mean Curvature

Mean Curvature is the average of any mutually orthogonal normal sections, such as profile and tangential curvature (Wilson, 2018). This variable has an unbounded range that can take either positive or negative values. Florinsky (2017) states that mean curvature represents the two accumulation mechanisms of gravity-driven substances, convergence and relative deceleration of flows, with equal weights.

An example of Mean Curvature extracted from WhiteboxTools.

Minimal Curvature

Minimal Curvature is the curvature of a principal section with the lowest value of curvature at a given point of the topographic surface (Florinsky, 2017). The values of this curvature are unbounded, and positive values correspond to hills while negative values are indicative of valley positions (Florinsky, 2016).

An example of Minimal Curvature extracted from WhiteboxTools.

Ring Curvature

Ring Curvature is the product of horizontal excess and vertical excess curvatures (Shary, 1995). Like Rotor, ring curvature is used to measure flow line twisting.

An example of Ring Curvature extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Rotor

Rotor describes the degree to which a flow line twists (Shary, 1991). Rotor has an unbounded range, with positive values indicating that a flow line turns clockwise and negative values indicating flow lines that turn counter clockwise (Florinsky, 2017).

An example of Rotor extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Shape Index

This tool calculates the Shape Index (Koenderink and van Doorn, 1992) from a digital elevation model (DEM). This variable ranges from -1 to 1, with positive values indicative of convex landforms, negative values corresponding to concave landforms (Florinsky, 2017). Absolute values from 0.5 to 1.0 are associated with elliptic surfaces (hills and closed depressions), while absolute values from 0.0 to 0.5 are typical of hyperbolic surface form (saddles). Shape index is a dimensionless variable and has utility in landform classification applications.

An example of Shape Index extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Unsphericty

Unsphericity curvature describes the degree to which the shape of the topographic surface is nonspherical at a given point (Shary, 1995). It is calculated as half the difference between the MaximalCurvature and the MinimalCurvature.

An example of Unsphercity extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

Vertical Excess Curvature

Vertical Excess Curvature is the difference of profile (vertical) and minimal curvatures at a location (Shary, 1995). This variable has positive values, zero or greater. Florinsky (2017) states that vertical excess curvature measures the extent to which 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 surface.

An example of Vertical Excess Curvature extracted from the WhiteboxTools DEM and Spatial Hydrology Extension.

By no means is this an exhaustive list of surface curvatures that exist in literature or exist within GIS software, however this blog intends to shine light on the various surface curvatures that are now available in WhiteboxTools and to explain what they can be used to measure. While many of the available curvatures exist within the WhiteboxTools open core, some requires a software license to the DEM and Spatial Hydrology Extension or the General Toolset Extension to use. More information about licensing can be found on our Extension pricing page. The following table below, lists all surface curvatures available in WhiteboxTools and which require a license. If you are curious how to extract these tools in WhiteboxTools, check out our Youtube video! Don’t forget to like and subscribe to our channel.

Tool NameAvailability
Accumulation CurvatureLicense required: DEM and Spatial Hydrology Extension or the General Toolset Extension
CurvednessLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Difference CurvatureLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Gaussian CurvatureAvailable in the WhiteboxTools open-core
Generating FunctionLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Horizontal Excess CurvatureLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Maximal CurvatureAvailable in the WhiteboxTools open-core
Mean CurvatureAvailable in the WhiteboxTools open-core
Minimal CurvatureAvailable in the WhiteboxTools open-core
Plan CurvatureAvailable in the WhiteboxTools open-core
Profile CurvatureAvailable in the WhiteboxTools open-core
Ring CurvatureLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
RotorLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Shape IndexLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Tangential CurvatureAvailable in the WhiteboxTools open-core
Total CurvatureAvailable in the WhiteboxTools open-core
Unsphericity License requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Vertical Excess CurvatureLicense requires: DEM and Spatial Hydrology Extension or the General Toolset Extension
Share this post