Elsevier

Geomorphology

Volume 102, Issues 3–4, 15 December 2008, Pages 567-577
Geomorphology

Terrain maps displaying hill-shading with curvature

https://doi.org/10.1016/j.geomorph.2008.05.046Get rights and content

Abstract

Many types of maps can be created by neighborhood operations on a continuous surface such as provided by a digital elevation model. These most commonly include first derivatives slope or aspect, and second derivatives planimetric or profile curvature. Such variables are often used in geomorphic analyses of terrain. First derivatives also provide subtle enhancements to hill-shaded maps. For example, some maps combine oblique and vertical illumination, with the latter reflecting variations in slope.

This study illustrates how second derivative maps, in conjunction with hill-shading, can cartographically enhance topographic detail. A simple conic model indicates that image-tone edges where slope or aspect varies by less than 0.5° are visible on curvature maps. Hill-shaded images combined with curvature enhance the continuity of naturally occurring tonal edges, especially in strongly illuminated areas. Variations in planimetric and profile curvature seem to be especially effective at highlighting convergent and divergent drainages and variations in erosion rate between or within sedimentary units, respectively. Shading curvature with consideration given to illumination models can add detail to hill-shaded terrain maps in a manner similar to cognitive models employed by map viewers.

Introduction

Geomorphologists and cartographers both employ variables derived from elevation data to quantify the shape or structure of topographic surfaces (Robinson et al., 1995, Wilson and Gallant, 2000, Slocum et al., 2004, Li et al., 2004). This study focuses on the cartographic representation of variables computed from local (neighborhood) operations directly from the topographic data, the “primary attributes” discussed in Wilson and Gallant (2000). Also addressed are operations that treat the terrain as a continuous function f (x,y,z) and compute first and second order derivatives.

First order derivatives include slope gradient and slope aspect. Slope, the rate of change in elevation (the z-value), quantifies the steepness of terrain. Aspect is the compass direction of steepest slope. Second order derivatives include profile and planimetric curvature, which are measured along and across the direction of maximum slope, respectively (Evans, 1972, El-Sheimy et al., 2005, Chang, 2006).

Geomorphology focuses on the patterns of these attributes and why they are significant to Earth surface processes (Table 1, column 2). Contour lines (Table 1, column 3) represent the intersection of topography with a series of planes parallel to a datum separated by a constant z-interval, while tonal contrasts and elevation derivatives, the focus of cartographic research, help visualize the terrain (Table 1, column 4).

Contours can be used to estimate slope, aspect, and curvature alone. Elevation can be read directly from contour labels, and the spacing or the change in spacing among contours indicates changes in slope and profile curvature, respectively. An orthogonal vector defined by two nearby contours on a map yields aspect, and the curve of the contour itself shows the change in aspect or planimetric curvature.

Contours, however, can require added information if the viewer is to translate them into a cognitive model of a topographic surface. For example, are two concentric, closed contours a depression or a hilltop? Determination would require reading contour values, looking for hachures, or viewing the contours in the context of the overall terrain. Also, having a priori knowledge of an area's geomorphology may be helpful (e.g. karst vs. fluvially carved terrain). One visual aid frequently applied in discriminating high vs. low is layer or hypsometric tinting (Imhof, 1965, Robinson et al., 1995), in which background colors change with elevation. Color schemes are designed to be intuitive, reflecting common variations associated with changes in elevation. For example, tints may progress from green to brown to white with increasing elevation in mountainous areas.

Contours create image tones that mimic maps with vertical illumination (Imhof, 1965). Viewers have a difficult time distinguishing shapes under such lighting. The Tobacco Root mountains of Montana are shown in Fig. 1 with contours (a) and “slope shading” from vertical illumination (b). Although some hand- and computer-contouring techniques overcome this visualization problem (e.g. Tanaka, 1932, Tanaka, 1950, Peucker et al., 1975, Yoeli, 1983, Kennelly and Kimerling, 2001, Kennelly, 2002), such techniques are not widespread.

Map users find that oblique illumination, a light source shining from a moderate angle between the horizon and zenith, and from the northwest, provides more intuitive images of the shape of the terrain (Imhof, 1965, Horn, 1981, Robinson et al., 1995, McCullagh, 1998, Slocum et al., 2004). Imhof (1965) cites westerners' preference for general room lighting when writing and drawing (from above and the left) as the reason for this psychological effect. Many map viewers see inversion of terrain when illuminated from the opposite direction. Properly illuminated, such hill-shaded or shaded relief maps are popular for their realistic look and detail (e.g. Thelin and Pike, 1991). Hill-shading is calculated as:BV=255cosθwhere BV is the 8-bit (0 to 255) grayscale brightness of each hill-shaded surface element (0 is black and 255 is white), and θ is the angle between a constant illumination vector and a normal vector to the terrain at each given locale. As such, hill-shading is a function of both slope and aspect, and its formula can also be written as:BV=255cosIsinScos(AD)+sinIcosSwhere I and D are the inclination from horizontal and azimuthal declination of the illumination vector, and S and A are the slope and aspect values.

Although hill-shaded maps are rich in visual information, and useful for solving specific geomorphological problems (Oguchi et al., 2003, Smith and Clark, 2005, Van Den Eeckhaut et al., 2005), default computer applications may limit cartographic freedom to enhance particular features of importance (Thelin and Pike, 1991, Allan, 1992, Patterson, 2004). Some cartographers such as those with the Swiss Institute of Cartography, elevated manual hill-shading to an art form, including combining hill-shading with slope shading to highlight steep terrain (Imhof, 1965). Computer-assisted cartographers combine the two in a similar manner to provide additional information about the shape of the terrain in a visually harmonious manner (Patterson and Herman, 2004). I include in Fig. 1 examples of the Tobacco Root Mountains with hill-shading (c), and hill-shading combined with slope shading (d). The latter adds some visual emphasis to ridge lines and valleys, especially those oriented parallel to the direction of illumination (i.e. northwest to southeast).

Representing aspect on a map is a challenge because aspect is a cyclical variable; north has a value of both 0° and 360°. Cartographic researchers have addressed this problem by devising new aspect color schemes, where luminosity or lightness of colors approximates or complements tonal variations associated with hill-shading. Moellering and Kimerling (1990) chose easily discriminated colors, and varied them with aspect such that the color luminance approximates the hill-shading brightness. Brewer and Marlow (1993) varied colors' lightness with aspect and chroma with slope. Kennelly and Kimerling's (2004) aspect color scheme was designed to complement hill-shading. Subtle variations in the luminosity of aspect colors highlight linear terrain features oriented parallel to the direction of illumination; these features are difficult to discern in traditional hill-shaded maps.

Such cartographic research falls under the umbrella of scientific or geographic visualization (Lo and Young, 2002, Slocum et al., 2004), which includes methods of computation, cognition and graphic design (Buttenfield and Mackaness, 1991). Although geographic visualization of hill-shading with slope and aspect is well established, similar efforts with second derivative maps are not. Curvature map images provide detailed information about certain physical aspects of the terrain, but not in the context of overall shape. Maps often represent curvature in shades of gray, in a method similar to slope shaded maps (e.g. Wilson and Gallant, 2000, Li et al., 2004, El-Sheimy et al., 2005). Others use a diverging color scheme (Brewer, 2005), taking advantage of curvature values diverging away from flat terrain towards both concave upward and convex upward. Such a color scheme may vary from white in the center (flat) to dark blue (concave upward) and dark red (convex upward) at its ends (e.g. Wood, 1999).

This paper explores methods by which planimetric and profile curvature can be effectively imaged with hill-shading. The methods are based on illumination models, and have the potential to add useful information to the map. Such an image should match the viewer's cognitive model and help him/her understand the geographic context of patterns imaged. I begin by exploring the graphic limitations of first derivative maps, especially slope maps, on hill-shaded images from synthetic data. I then demonstrate how the same subtle variations in shape show more distinct patterns on second derivative maps. Finally, I show how these second derivative maps can be used to highlight physical features of real world data on hill-shaded maps.

Section snippets

Methods and models

To demonstrate that second derivative maps enhance patterns not obvious on hill-shaded maps or first derivative maps combined with hill-shading, I chose the simple geometric model of a cone. It is created from concentric contours with an interval of 50 m and with subsequent variations in radius of 100 m. The lowest contour is at a z-value of zero and the apex has a value of 1000 m.

From these contours, the cone everywhere would slope at 26.57°. To include a slope variation, all contours below

Results

I used hill-shading with curvature enhancements to map a portion of the Mauna Kea volcano, Hawaii and the Grand Canyon, Arizona. Input data are part of the National Elevation Dataset obtained from the U.S. Geological Survey National Map Seamless Distribution System website <http://seamless.usgs.gov>. The Hawaiian digital elevation model (DEM) has a resolution of one arc second, or approximately 29.8 m. The Grand Canyon DEM has a resolution of one-third arc second, or approximately 9.2 m. The

Discussion

The images, their means of creation, and applications of the mapping techniques described here are similar to well established traditions in cartography and remote sensing. Adding curvature to hill-shaded images reprises the delineation of skeletal lines and the use of rock shading in the Swiss tradition. Highlighting local areas of rapid change in curvature is similar to edge-enhancement filtering in digital image processing. The usefulness of such images likely lies with the ability of the

Conclusions

Subtle details of the terrain are important to both geomorphologists and cartographers. Many quantitative techniques for defining topographic attributes and displaying images are available to both disciplines. This paper shows that changes in slope gradient of as little as 0.5° can be recognized in maps of profile curvature. These highlighted tonal edges do not disappear on the illuminated side of terrain if combined with traditional hill-shading methods. Combining hill-shading and curvature

Acknowledgements

I thank Dr. Takashi Oguchi of the University of Tokyo for editorial guidance and comments; Dr. Richard Pike of the U.S. Geological Survey for helpful feedback; George Billingsley and Susan Priest of the U.S. Geological Survey for providing the color palette for the map of the Grand Canyon; and Susan Vuke of the Montana Bureau of Mines and Geology for identifying prospective areas for detailed geologic and topographic comparison.

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