Digital Terrain Modeling: Principles and Methodology - Chapter 2 pot

DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 13 — #1 CHAPTER 2 Terrain Descriptors and Sampling Strategies To model a piece of terrain surface, first a set of data points needs to be acquired from the surface. Indeed, data acquisition is the primary (and perhaps the single most important) stagein digitalterrain modeling. For this, twostages aredistinguished, that is, sampling and measurement. Sampling refers to the selection of the location while measurement determines the coordinates of the location. Sampling will be discussed in this chapter while measurement methods will be discussed in the next chapter. Three important issues related to acquired DTM source (or raw) data are density, accuracy, and distribution. The accuracy is related to measurements. The optimum density and distribution are closely related to the characteristics of the terrain surface. For example, if a terrain is a plane, then three points on any location will be sufficient. This is not a realistic assumption and, therefore, an analysis of the terrain surface precedes the discussion of sampling strategies in this chapter. 2.1 GENERAL (QUALITATIVE) TERRAIN DESCRIPTORS In general, two basic types of descriptors may be distinguished: 1. qualitative descriptors, which are expressed in general terms, so that they are referred to as general descriptors 2. quantitative descriptors, which are those specified by numeric descriptors. In this section, a brief discussion of general descriptors is given and numeric descriptors are described in the next section. As discussed in Chapter 1, different groups of people are concerned with different attributes of the terrain surface. There- fore, a variety of general descriptors can be found based on these different interests. However, some of them are irrelevant to the concern of digital terrain modeling. Indeed, those that indicate the roughness and the coverage of terrain surface are more 13 © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 14 — #2 14 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY important in the context of terrain surface modeling. The following are some of these descriptors: 1. Descriptors based on terrain surface cover: Vegetation, water, desert, dry soil, snow, artificial or man-made features (e.g., roads, buildings, airports, etc.), and so on. 2. Descriptorsbased on genesis of landforms: Twosuch forms havebeendistinguished (Demek 1972), each of which has its own special characteristics — • endogenetic forms: formed by internal forces, including neotectonic forms, volcanic forms, and those forms resulting from deposition of hot springs • exogenetic forms: formed by external forces, including denudation forms, fluvial forms, karst forms, glacial forms, marine forms, and so on. 3. Descriptors based on physiography: Generalized regions according to the structure and characteristics of its landforms, each of which is kept as homogenous as possible and has dominant characteristics, for example, high mountains, high plateau, mountains, low mountains, hills, plateau, etc. 4. Descriptors based on other classifications. Those descriptors are so broad that they can only provide the user with some very general information about a particular landscape and thus they can only be used for general planning but not for project design. To design a particular project, more precise numeric descriptors are essential. 2.2 NUMERIC TERRAIN DESCRIPTORS The complexity of a terrain surface may be described by the concepts of roughness and irregularity and characterized by different numerical parameters. 2.2.1 Frequency Spectrum A surface can be transformed from the space domain to the frequency domain by means of a Fourier transformation. The terrain surface in its frequency domain is characterized by the frequency spectrum. The estimation of such a spectrum from equally spaced discrete (profile) data has been discussed by Frederiksen et al. (1978). The spectrum can be approximated by the following expression: S(F) = E ×F a (2.1) where F denotes the frequency at which the spectrum magnitude is S(F) and E and a are constants (i.e., characteristic parameters), which are two statistics expressing the complexity of the terrain surface (or profiles) over all of the area. Thus, they can be considered as parameters to provide more detailed information about the terrain surface, although still general in some sense. Different values for E and a can be obtained from different types of terrain surfaces. According to the study carried out by Frederiksen (1981), if the parameter a is greater than 2, the landscape is hilly with a smooth surface, and if the value of © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 15 — #3 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 15 a is smaller than 2, it indicates a flat landscape with a rough surface since the surface contains large variations with high frequency (short wavelength). The value of a provides us with general topographic information. 2.2.2 Fractal Dimension Fractal dimension is another statistical parameter which can be used to characterize the complexity of a curve or a surface. The discussion will start with the concept of effective dimension. It is well known that in Euclidean geometry, a curve has a dimension of 1 and a surface has a dimension of 2 regardless of its complexity. However, in reality, a very irregular curve is much longer than a straight line between the same points, and a complex surface has a much larger area than a plane over the same area. In the extreme, if a line is so irregular that it fills a plane fully, then it becomes a plane, thus having a dimension of 2. Similarly, a surface could have a dimension of 3. In fractal geometry, which was introduced by Mandelbrot (1981), the dimension- ality of an object is defined by necessity (i.e., practical need), leading to the so-called effective dimension. This can be explained by taking the example of the shape of the Earth’s surface when viewed from different distances. 1. If it is viewed from an infinite distance, the Earth appears as a point, thus having a dimension of 0. 2. If it isviewed from a position on the Moon, it appears to be a small ball, thus having a dimension of 3. 3. If the viewer comes nearer, for example, to a distance above the Earth’s surface of about 830 km (the altitude of the SPOT satellite’s orbit), the height information is extractable but not in detail. Thus, in general terms, the observer can see a mainly smooth surface with a dimension of nearly 2. 4. If the Earth’s surface is viewed on the ground, then the roughness of the surface can be seen clearly, thus the effective dimension of the surface should be greater than 2. In fractal geometry, the effective dimension could be a fraction, leading to the jargon fractal dimension or fractal. For example, the fractal dimension of a curve changes between 1 and 2, and that of a surface between 2 and 3. The fractal dimension is calculated as follows: L = C ×r 1−D (2.2a) where r is the scale of measurement (a principal unit), L is the length of measure- ment, C is a constant, and D is interpreted as the fractal dimension of the curve line. When measuring a fractal dimension of curve surface, r becomes the principal unit of surface used for measurement and the resultant area is A instead of L; the expression becomes A = C ×r 2−D (2.2b) Figure 2.1 shows an example of Koch line with a fractal dimension of 1.26. The process of curve generation is as follows: (a) draw a line with its length as a unit; © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 16 — #4 16 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY (a) (b) (c) Figure 2.1 A complex Koch line having a fractal dimension of 1.26. (a) A line with unit. (b) Divided into three line segments and mid-segment split into two. (c) Process repeated. Figure 2.2 Relationship between curvatures and complexity: the curvatures of the left two lines are 0 as the radius is infinite while the line on the right side has large curvatures as the radiuses are small. (b) divide the line into three segments; (c) the middle segment will be replaced by two polylines with length equal to 1 3 unit. The same procedure is repeatedly applied to all line segments. As a result, the line will become more and more complex, resulting in a fractal dimension of 1.26. From the discussion above, it can be concluded that a fractal dimension approach- ing 3 indicates a very complex and probably rough surface, while a simple (near planar) surface has a fractal dimension value near 2. 2.2.3 Curvature The terrain surface can be synthesized by combing terrain form elements, defined as relief unit of homogenous plan and profile curvatures (see Chapter 13 for more details). Supposea profile canbe expressed as y = f(x), thenthe curvature at position x can be computed as follows: c = d 2 y/dx 2 [1 + (dy/dx) 2 ] 3/2 (2.3) In this formula, curvature c is inversely proportional to the radius of the curve (R), that is, alargecurvatureis associated with a small radius (Figure 2.2). Thus, intuitively, it can be seen that the larger the curvature, the rougher is the surface. Therefore, curvatures can also be used asa measure for the roughness of theterrain. This criterion has already been used for terrain analysis (e.g., Dikau 1989). This is a comparatively useful method for planning DTM sampling strategies. However, a rather large volume of data (that of a DTM) needs to be available to allow the curvature values to be derived — which leads to a chicken-and-egg situation at the stage. © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 17 — #5 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 17 2.2.4 Covariance and Auto-Correlation The degree of similarity between pairs of surface points can be described by a cor- relation function. This may take many forms like covariance or an auto-correlation function. The auto-correlation function is described as follows: R(d) = Cov(d) V (2.4) where R(d) is the correlation coefficient of all the points with horizontal interval d, Cov(d) is the covariance of all the points with horizontal interval d, and V is the variancecalculated from all the(N)points. The mathematical functions are as follows: V =  N i=1 (Z i −M) 2 N −1 (2.5) Cov(d) =  N i=1 (Z i −M)(Z i+d −M) N −1 (2.6) where Z i is the height of point i, Z i+d is the elevation of the point with an interval of d from point i, M is the average height value of all the points, and N is the total number of points. When the value of d changes, Cov(d) and R(d) will also change because the height difference of two points with different d values is different. Covariance and auto-correlation values can be plotted against the distance between pairs of data points. Figure 2.3 is an example of auto-correlations varying with d. In general, if the value of d increases, the values of Cov(d) and R(d) will decrease. The curve is usually described (Kubik and Botman 1976) by the exponential function: Cov(d) = V ×e −2d/c (2.7) and the Gaussian model: Cov(d) = V ×e −2d 2 /c 2 (2.8) where c is the parameter indicating the correlation distance at which the value of covariance approaches 0. Therefore, the smaller the value of c, the less similar are the surface points. The value of similarity is also an indicator of the complexity of the terrain surface. The relationship between them is that the smaller the similarity over the same given distance, the more complex is the terrain surface. 2.2.5 Semivariogram The variogram is another parameter used to describe the similarity of a DTM surface, similar to (auto-)covariance. The expression for its computation is as follows: 2γ(d)=  N i=1 (Z i −Z i+d ) 2 N (2.9) © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 18 — #6 18 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY d R(d) 0 1 A B Figure 2.3 Two auto-correlation functions, whose values decrease with an increase in distance between points from 1 to 0. where γ(d)is called the semivariogram. Similar to covariance, the value of γ(d)will vary with distance. But the change in direction is opposite to the case of covariance. That is, γ(d)will increase with an increase in the value of d. The values of γ(d)can also be plotted against d, resulting a curved line. Such a curve can be approximated by an exponential function as follows: γ(d) = A ×d b (2.10) where A and b are two constants, i.e. the two parameters for the description of terrain roughness. A larger b indicates asmother terrain surface. When b is approachingzero, the terrain is very rough. Some examples of semivariograms are given in Figure 8.6. Indeed, Frederiksen et al. (1983, 1986) used the semivariogram to describe ter- rain roughness in digital terrain modeling. They also tried to connect this variable to the covariance used by Kubik and Botman (1976). 2.3 TERRAIN ROUGHNESS VECTOR: SLOPE, RELIEF, AND WAVELENGTH The numerical descriptors discussed in Section 2.2 are essentially statistical. They are computed from a sample of terrain points from the project area. Usually, some profiles are used as the sample and then a parameter is calculated from these profiles. However, there are some problems associated with this approach. One of these is that the parameters calculated from the selectedprofiles can be differentfrom thosederived from the whole surface. If one tries to compute these for the whole surface, then a sample from thewhole surface is necessary. Inthis case, the original purposeof having a terraindescriptor for project planningand design is lost. For these reasons, Li (1990) recommended slope and wavelength as the main descriptors for DTM purposes. 2.3.1 Slope, Relief, and Wavelength as a Roughness Vector The parameters for roughness or complexity of a terrain surface used in geomorphol- ogy have also been reviewed by Mark (1975). It was found that roughness cannot be completely defined by any single parameter, but must be a roughness vector or a set of parameters. © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 19 — #7 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 19 P H = amplitude Slope angle of P W H (a) (b) W = wavelength  Figure 2.4 The relationship between slope, wavelength, and relief: (a) their full relationship and (b) simplified diagram. In this set of parameters, relief is used to describe the vertical dimension (or amplitude of the topography), while the terms grain and texture (the longest and shortest significant wavelengths) are used to describe the horizontal variations (in terms of the frequency of change). The parameters for these two dimensions are connected by slope. Thus, relief, wavelength, and slope are the roughness parameters. The relationship between them can be illustrated in Figure 2.4. It can clearly be seen that the slope angle at a point on the wave varies from position to position. The following mathematical equation may be used as an approximate expression of their relationship (for a more rigorous definition, see Chapter 13): tan α = H W/2 = 2H W (2.11) where α denotes the average value of the slope angle, H is the local relief value (or the amplitude), and W is the so-called wavelength. It is clear that if two of them are known, then the third can be computed from Equation (2.11). For the reasons to be discussed in the next section, slope and wavelength together are recommended as the terrain roughness vector for DTM purposes. 2.3.2 The Adequacy of the Terrain Roughness Vector for DTM Purposes From both the theoretical and the practical points of view, slope, altitude, and wavelength are the important parameters for terrain description. In geomorphology, Evan (1981) states a useful description of the landform at any point is given by altitude and the surface derivatives, i.e. slope and convexity (curvature) Slope is defined by a plane tangent to the surface at a given point and is completely specified by the two components: gradient (vertical component) and aspect (plane component) Gradient is essentially the first vertical derivative of the altitude surface while aspect is the first horizontal derivative. Further, land surface properties are specified by convexity (positive and negative convexity — concavity). These are the changes in gradient at a point (in profile) © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 20 — #8 20 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY and the aspect (in the plane tangential to the contour passing through the point). In other words, they are second derivatives. These five attributes (altitude, gradient, aspect, profile convexity, and plane convexity) are the main elements used to describe terrain surfaces. Among them, slope, comprising of both gradient and aspect, is the fundamental attribute. Gradient should be measured at the steepest direction. However, when taking the gradient of a profile or in a specific direction, it is actually the vector of the gradient and aspect that is obtained and used. Therefore, the term slope or slope angle is used in this context to refer to the gradient in any specific direction. The importance of slope has also been realized by others. As quoted by Evans (1972), Strahler (1956) pointed out that “slope is perhaps the most important aspect of surface form, since surfaces may be formed completely from slope angles .” Slope is the first derivative of altitude on the terrain surface. It shows the rate of change in height of the terrain over distance. From the practical point of view, using slope (and relief) as the main terrain descriptor for DTM purposes can be justified for the following reasons: 1. Traditionally, slope has been recognized as very important and used in surveying and mapping. For example, map specifications for contours are given in terms of slope angle all over the world. 2. In the determination of vertical contour intervals (CIs) for topographic maps, slope and relief (height range) are the two main parameters considered. For example, Table 2.1 is a classification system adopted by the Chinese State Bureau of Sur- veying and Mapping (SBSM) in its specifications for 1:50,000 topographic maps. 3. In DTM practice, many researchers (e.g., Ackermann 1979; Ley 1986; Li 1990, 1993b) have noted the high correlation between DTM errors and the mean slope angle of the region. 2.3.3 Estimation of Slope To use slope together with wavelength or relief to describe terrain, two problems related to the estimation of its values need to be considered, that is, availability and variability. By availability we mean that slope values should be available or estimated before sampling takes place, to assist in the determination of sampling intervals. If a DTM exists in an area, then the slope values for DTM points can be computed and the average can be used as the representative (Zhu et al. 1999). Otherwise, slope may be estimated from a stereo model formed by a pair of aerial photographs with overlap (see Chapter 3) or from contour maps. The method proposed by Wentworth (1930) is still widely used to estimate the average slope of an area from the contour maps. Table 2.1 Terrain Classification by Means of Slope and Relief Terrain Type CI (m) Slope ( ◦ ) Relief (Height Range) (m) Plain 10 (5) <2 <80 Upland 10 2–680–300 Hill 20 6–25 300–600 Mountain 20 >25 >600 © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 21 — #9 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 21 The average slope value (α) of a homogeneous are can be estimated as follows: α = arctan  H × L A  (2.12) where H is the contour interval, L is the total length of contours in the area and A is the size of the area. If there is no contour map for such an area, then the slope may be estimated from an aerial photograph. Some of the methods that are available for measurement of slope from aerial photographs have been reviewed by Turner (1997). By variability we mean that slope values may vary from place to place so that the slope estimate that is representative for one area may not be suitable for another. In this case, average values may be used as suggested by Ley (1986). If slope varies too greatly in an area, then the area should be divided into smaller parts for slope estimation. Different sampling strategies could be applied to each area. 2.4 THEORETICAL BASIS FOR SURFACE SAMPLING After estimating slope and relief (height range), the wavelengths of terrain variation can be computed. These parameters are used to determine the sampling strategy and intervals for data acquisition. First, some theories related to surface sampling are discussed. 2.4.1 Theoretical Background for Sampling From the theoretical point of view, a point on the terrain surface is 0-D, thus without size, while a terrain surface comprises an infinite number of points. If full information about the geometry of a terrain surface is required, it is necessary to measure an infin- ite number of points. This means that it is impossible to obtain full information about the terrain surface. However, in practice, a point measured on a surface represents the height over an area of a certain size; therefore, it is possible to use a set of finite points to represent the surface. Indeed, in most cases, full or complete information about the terrain surface is not required for a specific DTM project, so it is necessary only to measure enough data points to represent the surface to the required degree of accuracy and fidelity. The problem a DTM specialist is concerned withishow to adequately represent the terrain surface by a limited number of elevation points, that is, what sampling interval to use with a known surface (or profile). The fundamental sampling theorem that is being widely used in mathematics, statistics, engineering, and other related disciplines can be used as the theoretical basis. The sampling theorem can be stated as follows: If a function g(x) is sampled at an interval of d, then the variations at frequencies higher than 1/(2d) cannot be reconstructed from the sampled data. That is, when sampling takes two samples (i.e., points) from each period of waves with the highest frequency in the function g(x), the original g(x) can be completely reconstructed withthe sampled data. In the case of terrainmodeling, if aterrain profile is long enoughto berepresentativeof thelocal terrain, it canthen berepresented bythe © 2005 by CRC Press DITM: “tf1732_c002” — 2004/10/22 — 16:36 — page 22 — #10 22 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY Figure 2.5 The relationship between the least sampling interval and the highest functional frequency. Left: sampling interval is less than half the functional frequency so that full reconstruction is possible; right: sampling interval is larger than half the functional frequency so that information about the function is lost. sum of its sine and cosine waves. If it is assumed that the number of terms in this sum is finite, there is, therefore, a maximum frequency value, F , for this set of sinusoidal. According to the sampling theorem, the terrain profile can be completely reconstruc- ted if the sampling interval along the profile is smaller than 1/(2F) (see Figure 2.5, left). Therefore, extending this idea to surfaces, the sampling theorem can also be used to determine the sampling interval between profiles to obtain adequate inform- ation about a terrain surface. In contrast, if a terrain profile is sampled at an interval of d, then the terrain information with a wavelength less than 2d will be completely lost (Figure 2.5, right). Therefore, as Peucker (1972) has pointed out, “a given regular grid of sampling points can depict only those variations of the data with wave lengths of twice the sampling interval or more.” 2.4.2 Sampling from Different Points of View Points on a terrain surface can be viewed in various ways from the differing view- points inherent in subjects such as statistics, geometry, topographic, science, etc. Therefore, different sampling methods can be designed and evaluated according to each of these different viewpoints as follows (Li 1990): 1. statistics-based sampling 2. geometry-based sampling 3. feature-based sampling. From the statistical point of view, a terrain surface is a population (called a samplespace) andthe sampling can becarried out either randomlyor systematically. The population can then be studied by the sampled data. In random sampling, any sampled point is selected by a chance mechanism with known chance of selection. The chance of selection may differ from point to point. If the chance is equal for all sampled points, it is referred to as simple random sampling. In systematic sampling, the points are selected in a specially designed way, each with a chance of 100% probability of being selected. Other possible sampling strategies are stratified sampling and cluster sampling. However, they are not suitable for terrain modeling and thus are omitted here. From the geometric point of view, a terrain surface can be represented by different geometric patterns, either regular or irregular in nature. The regular pattern can be subdivided into 1-D or 2-D patterns. If sampling is conducted with a regular pattern © 2005 by CRC Press [...]... are convex and concave points Peak Course line Ridge lines Figure 2. 6 Terrain feature points and lines © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 23 — #11 24 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY (a) 40 30 20 (b) 40 A B C 30 E F A B C E F 20 Figure 2. 7 Points (e.g., C) on a ridge line being local maxima (a) (b) (c) (d) Figure 2. 8 Slope changes at F-S points (peaks,... been proposed later (Makarovic 1975), such as the so-called random-variation, parabolic, distance, and contour criteria Of course, other criteria may also be used as the basis of the sampling strategy for a particular type of terrain © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 25 — #13 26 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY Progressive sampling can solve part of... line VIPs and representatives Figure 2. 9 Patterns of sampled data points © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 27 — #15 28 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY The data sets that are sampled along rivers, break lines, or feature lines all belong to this pattern Actually, it is not an independent pattern, but rather a supplemental one that is F-S For example,... available for selection, such as regular or rectangular grids These patterns can be classified in different ways Figure 2. 9 shows one such classification © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 26 — #14 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 27 Regular 2- D data are produced by means of regular grid or progressive sampling The resulting pattern could be a rectangular grid,... measurement, instruments used, and technique adopted A B Frequency Figure 2. 10 Cutoff frequency: the swing approaching 0 © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 28 — #16 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 29 Technique means the field survey, photogrammetry, or map digitization Generally speaking, data acquired by field survey are usually the most accurate and data acquired by... Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 24 — # 12 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 25 organizations (e.g., military survey organizations) where speed of data acquisition is of prime importance 2. 5 .2 Sampling with One Dimension Fixed: Contouring and Profiling In analog photogrammetry, stereo models are constructed from a pair of aerial photographs and direct measurement of contours... composed of a finite number of points, and the information content of these points may vary with their positions Therefore, surface points are classified into two groups, one of which comprises feature-specific (F-S) (or surface-specific) points (and lines) while the other comprises random points An F-S point is a local extrema point on the terrain surface, such as peaks, pits, and passes These points may not... motion In dynamic mode, the data acquired are usually of much lower accuracy There will be more discussion on data measurement and the accuracy of measured data using different techniques in the next chapter © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 29 — #17 ... average points The lines connecting certain types of F-S points are referred to as feature-specific lines, such as ridge lines, course lines (rivers, valleys, ravines, etc.), break lines, and so on Figure 2. 6 shows the F-S points and lines Ridge lines are the lines connecting pairs of points such that the points on them are local maxima (see Figure 2. 7) Similarly, course lines are linking pairs or strings... former is very efficient in measurement and the latter is very effective in surface representation Such a combination is referred to as composite sampling In this way, abrupt changes — specific features on the terrain such as ridges, break lines, etc — are sampled selectively And the values and F-S points — peaks, passes and hollows — are added to the regular grid-sampled data Indeed, there are two types . chicken -and- egg situation at the stage. © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 17 — #5 TERRAIN DESCRIPTORS AND SAMPLING STRATEGIES 17 2. 2.4 Covariance and Auto-Correlation The. convex and concave points. Ridge lines Course line Pea k Figure 2. 6 Terrain feature points and lines. © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 24 — # 12 24 DIGITAL TERRAIN. 2. 9 Patterns of sampled data points. © 20 05 by CRC Press DITM: “tf17 32_ c0 02 — 20 04/10 /22 — 16:36 — page 28 — #16 28 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY The data sets that are sampled