Visualizing Topography by Openness:A New Application of Image Processing to Digital Elevation Models

A new parameter, here termed openness, expressing the degree of dominance or enclosure of a location on an irregular surface, is developed to visualize topographic character. Openness is an angular measure of the relation between surface relief and horizontal distance. For angles less than 90", it is equivalent to the internal angle of a cone, its apex at a DEM location, constrained by neighboring elevations within a specified radial distance. Openness incorporates the terrain line-of-sight, or viewshed, concept and is calculated from multiple zenith and nadir angles-here along eight azimuths. Openness has two viewer perspectives. Positive values, expressing openness above the surface, are high for convex forms, whereas negative values describe this attribute below the surface and are high for concave forms. Openness values are mapped by gray-scale tones. The emphasis of terrain convexity and concavity in openness maps facilitates the interpretation of landforms on the Earth's surface and its seafloor, and on the planets, as well as features on any irregular surface-such as those generated by industrial procedures. Introduction Relief maps and the parametric characterization of topography are essential to the interpretation of the land surface. Hillshaded maps initially were drafted by landscape artists, but map precision and reproducibility were limited by the manual technique and its requisite cartographic skill (Raisz, 1931; Imhof, 1965; Alpha and Winter, 1971). The advent of the electronic computer and the digital elevation model (DEM) overcame these restrictions, and machine-made shaded-relief maps now are widely used to display topography (Yoeli, 1967; Horn and Brooks, 1989; Thelin and Pike, 1991; Vigil et al., 2000). The technique is not without drawbacks of its own. Because a directional light source is required, for example, ridges and valleys intersecting that source are shown clearly while features parallel to the source can be difficult to identify. Attempts to mitigate this problem by multiple light sources have been only partially successful (Mark, 1992; Moore and Mark, 1992; Riehle et al., 1997). Other types of neighborhood operations centered on DEM grid points yield digital maps that display or extract topographic features (Tobler, 1969; Peucker and Douglas, 1975; R. Yokoyama and M. Shirasawa are with the Department of Computer and Information Science, Faculty of Engineering, Iwate University, 4-3-5, Ueda, Morioka, Iwate, 020-8551 Japan ([email protected]. jp). R.J. Pike is with the U.S. Geological Survey, MIS 975, 345 Middlefield Road, Menlo Park, CA 94025. Weibel and Heller, 1991; Blaszczynski 1997; Guth, 2001). Terrain slope, curvature in both the XY and Z domains, relative relief, and slope azimuth are among the measures most commonly mapped from DEMs (Evans, 1972; Pike, 1988). Algorithms that implement the much-used DEM-to-watershed transformation, now an essential tool for hydrologic and geomorphic modeling, incorporate several such measures Uenson and Domingue, 1988; ESRI, 1992). Like automated reliefshading, however, all of these parameters are sensitive to the vagaries inherent in gridded height data (Carter, 1992). In particular, slope maps computed from DEMs commonly have an unsatisfactory wormlike appearance that reflects the method of contour gridding and obscures the true configuration of the land surface. In this paper, we introduce a new concept of surface representation, provisionally termed openness for want of a better name. It is a novel method for digital terrain modelingactually an image-processing technique-by which values of surface openness are calculated from a DEM, displayed in map form, and used to visualize landscapes. The resulting maps of openness superficially resemble digital images of shaded relief or slope angle, but emphasize dominant surface concavities and convexities. Values of openness require no light source, thus removing one limitation of relief shading, and are less affected by the DEM noise that afflicts most other parameters. After describing the technique, we present several maps of surface openness and discuss how they may be applied to the analysis and interpretation of topography. Zenith and Nadir Angles Before defining topographic openness, we must establish some angular relations between the point locations along individual DEM-derived profiles, and do this for viewer perspectives both above and below the surface. The first quantity to be extracted from these relations is elevation angle 8. Figure 1 shows a sample of terrain heights in a DEM arrayed north-south and east-west at a constant spacing M, projected in Universal Transverse Mercator (UTM) coordinates, and centered on a point of interest (double circle). Each point in the DEM is described by (i, j, H) where i and j are column and row numbers and H is elevation. Figure 2 defines the geometric relation in profile between any two points A(iA,jA, HA) and B(iB, je, HB) in the DEM, here showing a case where HA < HB. Horizontal distance P between A and B is given by Photogrammetric Engineering & Remote Sensing Vol. 68, No. 3, March 2002, pp. 257-265. 0099-lll2/02/6803-257$3.00/0 O 2002 American Society for Photogrammetry and Remote Sensing PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING March 2002 257 Figure 2. Elevation angle 0 between two points on a DEM. Terrain height A(iA, , i, Hr) is at the origin of the calculation; height B(ie, je, Hg) l ies at some horizontal distance P on the DEvl. d is assumed to be positive 'for H^ < H', and negative for HB < HA. Figure 3. Surface openness defined in terms of zenith and nadir angles. Zeni th angle o@. or (90 oF) , and nadir angle o$, or (9O + o6r), calculated along one of eight azimuths D wi th in radia l imi t l in F igure 1; dots are height a long terrain profi le. Positive openness is the mean value of oQ, along the eight sampling directions; negative openness is the corresponding mean of eight values of pry'. ' Elevation angles DBr and pD1 can be positive or negative, depending on the character of the topography around the central