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# Profiles and curvature

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# Profiles and curvature

Profiles and cross-sections

Many GIS packages provide tools for examining the profile of a surface along a selected straight line or series of straight line segments (a polyline). The form of the profile reflects: the number of observations taken along the selected line (with only two sampled points, at the start and end, the profile will be a straight line showing the slope between the two points, and as the number of observations is increased the surface will be represented in greater detail); the direction(s) of the selected line segments; and the nature of the surface itself, as represented in the surface model being utilized.

Figure 6‑14 shows a profile from south to north of the test surface TQ81NE along its westerly edge (the edge shaded gray in Figure 6‑1), in this case produced using Landserf — similar functionality is provided in many packages (e.g. TAS/Whitebox, ArcGIS).

Figure 6‑14 Profile of NS transect, TQ81NE The vertical extent is clearly exaggerated, but the profile shown does provide an indication of the slope and curvature of the surface along the selected line. The important point to note here is that the selected line does not represent the direction of maximum gradient, so values for slope and curvature apply solely to the direction chosen and will differ from calculations of slope and curvature for the surface as a whole.

In addition to single profile computation multiple-slice profiling can be performed using some packages, such as TNTMips. Figure 6‑15 illustrates this procedure using the TQ81NE dataset once more, with a selection of slices taken at 50 grid cell intervals (500m intervals as this is a 10m DEM). The profile averages for the entire input raster object are shown as the baselines against which the curve-filling is generated. In this example we have rotated the output (profiles plus relief map backdrop) through 90 degrees for clarity of viewing.

Figure 6‑15 Multiple profile computation Curvature and morphometric analysis

As with slope, curvature values depend upon the line or plane along which such calculations are made. There are several alternative measures of surface curvature. The three most frequently provided within GIS software are profile curvature, plan curvature and tangential curvature. Additional terms and measures include longitudinal curvature, cross-sectional curvature, maximum and minimum curvature, and mean curvature.

In order to clarify these various terms we have drawn some illustrative details on part of test tile TQ81NE (Figure 6‑16).

Figure 6‑16 Surface morphology At the sample location (x,y)=(600,5400) a straight line has been drawn to indicate the slope and arrows indicates the direction in which the slope has been computed (the aspect). At this point a vertical plane (i.e. a plane drawn in the Z-direction) could be constructed that is oriented in the direction of the aspect and passes through (x,y,z). It is the plane in which gravitational processes are maximized, so is of great interest in problems of hydrology, soil movement and geomorphological analyses. A plane constructed at right angles (orthogonal) to this vertical plane, i.e. a plane passing through the target point and slicing the surface horizontally (parallel to the XY-plane) maps out a line on the surface that is effectively a contour at that point. This second plane is the one for which gravitational effects are minimized. Profile and plan convexity (or curvature) is defined in terms of these two planes (see further, subsections below). A slice through the surface could also be drawn in the YZ-plane, also passing through (x,y,z). This would again generate a line where it intersected with the surface, and the somewhat confusingly named tangential curvature is defined in terms of this arrangement.

To provide an additional level of complexity and confusion to the above terms, there are a further set of curvature values that correspond to those used in the mathematical analysis of surfaces (see for example, do Carmo (1976) and/or the Mathworld web site for more details). The slope and aspect of a surface at any given point are computed by imagining that a plane can be drawn that just touches the surface. The direction this tangential plane faces (not to be confused with the YZ plane described above) defines the aspect, and the slope is the magnitude of the gradient in this direction. But the tangential plane is not aligned with the XY-plane, the YZ-plane or the ZX-plane. It sits at an angle to all three. If a pole were constructed attached to this tangential plane at the point (x,y,z) at right angles to the surface it would point in a direction known as the Normal to the surface (as shown in Figure 6‑16, white arrow). As with the three coordinate planes XY, XZ, and YZ we can now construct three planes that are orthogonal to each other based on the tangential plane and the surface Normal. One of these three is the tangential plane itself, the second is a plane through the surface Normal and in the direction of the aspect, and the last is a plane at right angles to both of these. These last two planes will cut the surface at an angle that is neither horizontal nor vertical. Where they cut the surface they will generate a line, whose curvature can be calculated. The Normal plane produce a line whose curvature at (x,y,z) is known as the longitudinal curvature, and the plane at right angles to the Normal generates a line with curvature at (x,y,z) known as the cross-sectional curvature (see further, subsection 6.2.2.6). Perhaps the simplest way of understanding the difference between the mathematical expressions of curvature and those commonly used in landscape analysis is to compare lines of latitude on a globe, which are formed by the intersection of a horizontal plane with the surface of a sphere, with great circles, which are formed by the intersection of a plane Normal to the surface, passing through the center of the sphere. The latter (corresponding to the mathematical view of a tangent plane and Normal) will always be larger circles, so have a greater radius of curvature, r, and thus a lower curvature value than the former, which correspond to the contour or landscape perspective.

Curvature is conventionally described using the Greek letter kappa, κ, so we use this in conjunction with subscripts to define each form of curvature. The main expressions provided are based on those provided by Surfer, which in turn are based on Moore et al. (1991, 1993). For quadratic surface modeling as used in Landserf, expressions are based on Evans (1979) cited in Wood (1996). GIS packages, such as ArcGIS, may multiply curvature values by ‑100 for convenience, as recommended by Zeverbergen and Thorne (1987), to give values in the approximate range [‑1,1] with a sign that ensures that positive curvature equates to convex forms and negative curvature equates to concave forms. The value r=1/κ is the radius of curvature, hence κ can be viewed as directly related to the size of a circle or a sphere that just touches the surface at sampled points.

Wood (1996) uses the signs of the measured slope, and the curvature measures: cross-sectional, maximum and minimum curvature, to characterize surface features, i.e. to help identify peaks, ridges, passes, planar regions, channels and pits. The simplified relationships he defines are shown in Table 6‑1, where blanks indicate the value or sign of the measure has no relevance, and if a second line of conditions is included it relates to the immediate 3x3 neighborhood of the target point. Thus these various curvature measures have applications in both process modeling and feature extraction, and as such may be applied not only to physical surfaces but in some cases, other types of surface data or fields. Readers are referred to Rana and Wood (2000) and Rana (2004a) for a fuller discussion of this area and the problems associated with trying to carry out such analyses on real-world surfaces.

Table 6‑1 Morphometric features — a simplified classification

 Feature Slope Curvature Cross-sectional Maximum Minimum Peak 0 + + Ridge 0 + 0 + + Pass 0 + - Plane 0 0 0 + 0 Channel 0 0 - + - Pit 0 - -

after Wood (1996, Table 5.3). + and - indicate positive or negative curvature, 0 indicates no curvature, blank indicates not relevant

Profile curvature

The labeled (red) line in Figure 6‑16 indicates the profile curvature which is the shape of the surface in the immediate neighborhood of the sample point contained within the vertical plane. It represents the rate of change of the slope at that point in the vertical plane, and is negative if the shape is concave, positive if the shape is convex and zero if there is no slope. Profile curvature is defined as: Surfer implements the above expression using the approximations provided in Section 6.1.3, Raster models. Both ArcGIS and Landserf utilize mathematical models fitted to the surface using the 3x3 window surrounding the target point z*, in order to obtain estimates of profile curvature. These result in expressions that are defined in terms of the coefficients of the surface functions provided earlier.

For Landserf (quadratic surface model) we have: For ArcGIS (fourth order surface model) we have: where the coefficients dh are computed in the manner described in Section 6.1.3, Raster models.

Plan curvature

The plan curvature is the shape of the surface viewed as if a horizontal plane has sliced through the surface at the target point (the labeled green line in Figure 6‑16). It is essentially the curvature of a contour line at height z and location (x,y). The differential expression for plan curvature is similar to that for profile curvature: Surfer implements the above expression using the approximations provided in Section 6.1.3, Raster models. Both ArcGIS and Landserf utilize mathematical models fitted to the surface using the 3x3 window surrounding the target point z* in order to obtain estimates of profile curvature. These result in expressions that are defined in terms of the coefficients of the surface functions provided earlier.

For Landserf (quadratic surface model) we have: For ArcGIS (fourth order surface model) we have a similar looking formula, although the coefficient values will differ from those of the quadratic model: The coefficients dh, which are different for the quadratic and quartic models, are computed in the manner described in Section 6.1.3, Raster models.

Note that ArcGIS also defines a general term which it describes as curvature, defined as the difference between the plan and profile curvatures, which simplifies to κ=‑200(d+e)

Tangential curvature

Tangential curvature is again closely related to plan and profile curvature, and also to the slope, St. The differential formula is: Surfer implements the above expression using the approximations provided in Section 6.1.3, Raster models. ArcGIS and Landserf do not provide this function.

Longitudinal and cross-sectional curvature

Landserf provides these additional curvature measures based upon approximation of the surface using a local quadratic. The formulas used for each are as follows, where the letters ae refer to the coefficients of the quadratic form described in Section 6.1.3, Raster models:  Mean, maximum and minimum curvature

Mean curvature is a term used in a variety of ways. In some instances it is defined as the difference between the plan and profile curvature, or as a measure of the average curvature for specific cells (the average of the maximum and minimum curvature for that cell), or even the mean value over all cells in a grid. Likewise maximum and minimum (profile) curvatures can be defined in both local and global terms. Within Landserf local formulas for the mean, maximum and minimum values are provided based on the quadratic model used throughout the package. These again are drawn from Evans (1979) as cited by Wood (1996) and are as follows:  Hence 