£ • t A COMPARISON OF COMPUTER-BASED TERRAIN STORAGE METHODS WITH RESPECT T O THE E V A L U A T I O N O F CERTAIN GEOMORPHOMETRIC by DAVID MICHAEL MARK B . A . , Simon Fraser University, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH February, 1974 COLUMBIA OF MEASURES In p r e s e n t i n g t h i s thesis in p a r t i a l fulfilment o f the r e q u i r e m e n r s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, the L i b r a r y s h a l l make i t freely available for I agree that, reference and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f this thesis f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department: o r by h i s of representatives. this thesis for written It financial gain shall permission. Department o f Geography The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada D a t e i s understood t h a t March 11, 1974 Columbia copying o r publication not be allowed w i t h o u t my Abstract Topographic information can be d i g i t i z e d in several ways. Sampling may be surface-random (points selected according to partially or completely arbitrary criteria) or surface-specific (points selected according to their topographic significance). Surface-random sampling includes grids, contours, and randomly- located points. compared. In this study, grid sampling and surface-specific sampling are Surface behavior between sampled points is assumed to be linear. A l l aspects of surface form can be considered to reflect surface roughness. Horizontal variation includes the concepts of texture and g r a i n , w h i l e vertical variation is discussed under r e l i e f . The relationships between these are embodied in slope and the dispersion of slope magnitude and orientation. The distribution of mass w i t h i n the elevation range of a topographic surface is described under hypsometry. Parameters for investigation are selected from these categories. The variation of some selected geomorphometric parameters in southern British Columbia is examined via a stratified random sample consisting of f o r t y two 7 x 7 km areas. The values of some of these parameters are used to group the samples, and six are chosen for more detailed analysis. The relationships among the variables are examined using correlation analysis. For four geomorphometric measures (local r e l i e f , mean slope, roughness factor, and hypsometric i n t e g r a l ) , the theoretical errors involved in estimating the measures from the two selected terrain storage methods are discussed. The surface-specific point samples should produce better results than grids of reasonable densities. The latter, however, should require less d i g i t i z a t i o n time and computer storage space per point. For at least local relief and hypsometric integral, grid error should be a linear function of grid spacing. Results of empirical comparison of the methods over the six selected areas are presented. The average surface-specific point data set is found to require some 2 . 6 times as much digitization time and 3.1 times as much computer storage space as the 15 by 15 grids used in the comparison. Computed estimates obtained from both of these data bases are presented for each of the four selected parameters, together with other estimates (obtained manually) in some cases . The average errors for the methods are found to differ significantly for local relief and mean slope but not for the hypsometric integral; for a l l three measures, the grids produce larger mean errors. The assumption of a linear relationship between grid spacing and grid error is used to estimate the grid spacing which would be required fo produce the same average error as the surface-specific points. For the three parameters used, these hypothetical grids are calculated to require more computer storage space and d i g i t i z a t i o n time than the surfacespecific point data sets. The influence of the density of surface-specific points and of base map scale appear to be related to the topographic texture. For a reasonably experienced terrain analyst, the reproduceabi lity of these data sets appears to be good, although there remains a subjective element in point selection not present for grids. It is concluded that for a given amount of digitization time or computer storage space, better estimates of geomorphometric parameters can be obtained using sets of surface-specific points than using regular grids. - iii - Table of Contents Page Abstract i List of Tables vii List of Figures ' Preface x X l Chapter 1: Introduction 1 1.1: Precision of topographic map data 2 1.2: Notation 5 Computer Terrain Storage Systems 6 2.1: Digitization 6 2.2: Surface-random sampling: grids 7 2.3: Surface-random sampling: digitized contours 12 2.4: Surface-specific sampling: points and lines 13 2.5: Surface behavior 15 2.6: Computer storage of terrain information Chapter 2: 18 2.7: Comparisons of approaches 2.8: Conclusions 20 20 Chapter 3: Geomorphometric Parameters 22 3.1: The concept of "roughness" 23 3.2: Texture and grain 25 3.2.1: Grain 25 3.2.2: Texture 26 3.2.3: Drainage density (D^) 26 3.2.4: Other texture measures 28 3.3: Relief measures 29 3.3.1: Local relief (H) 29 3.3.2: A v a i l a b l e relief ( H ) a 32 Page 3.4: 3.3.3: Drainage relief (H^) 34 3.3.4: Applications of relief measures 35 Slope 36 3.4.1: Average slope: line-sampling method 36 3.4.2: Average slope: other methods 37 3.4.3: Other slope parameters 38 3.4.4: Applications of slope measures 39 3.5: Dispersion of slope magnitude and orientation 41 3.6: Hypsometry 45 3.6.1: The hypsometric curve and its variations 45 3.6.2: The hypsometric integral (HI) 47 3.6.3: Other parameters related to the hypsometric curve 49 3.6.4: Other parameters related to hypsometry 50 3.6.5: Applications of hypsometric measures 50 3.7: Chapter 4: Review and parameters to be investigated 51 Terrain V a r i a b i l i t y in Southern British Columbia, and Relationships Among Variables 52 4.1: Selection of sample areas 52 4.2: Data collection 58 4.3: Data analysis 58 4.3.1: Drainage density (D-|) 58 4.3.2: Source density (D ) and peak density (Dp) 60 4.3.3: Local relief 60 4.3.4: Mean slope ( t a n « ) 4.3.5: Hypsometric integral 4.3.6: Relationships among variables s (H) 61 (HI) °1 62 -vPage 4.4: Classification of samples and selection of areas for 66 further analysis 4.5: Description of areas selected for further analysis 66 4.5.1: Sample 8: 66 4.5.2: Sample 1 1 : Ptarmigan Creek map-area l l l e c i l l e w a e t map-area (82N/4E) 70 (83D/10W) 4.5.3: Sample 18: Manning Park map-area 70 (92H/2W) 4.5.4: Sample 24: Tatla Lake map-area (92N/15E) 4.5.5: Sample 3 1 : G h i t e z l i Lake map-area 71 71 (93E/9W) 4.5.6: Sample 4 1 : Oona River map-area 72 (103G/16W) Chapter 5: Procedures for Analysis and Theoretical Comparisons 73 of Computer Storage Systems 5.1: 5.2: Local relief (H) 73 5.1.1: Local relief: surface-specific points 74 5.1.2: Local relief: 75 5.1.3: Review regular grid 78 Mean slope (tanoc) 78 5.2.1: 79 Computational procedures 5.3: Roughness factor (R) 5.4: Hypsometric integral 80 (HI) 80 5.4.1: Hypsometric integral: surface-specific points 85 5.4.2: Hypsometric integral: regular grid 85 5.4.3: Summary 88 -vi - Page 5.5: Possibility of estimating other parameters 88 5.6: Theoretical numbers of points and triangles for triangular 89 data sets, and theoretical computer storage requirements Chapter 6: Empirical Comparisons and Computational Results 91 6.1: Digitization time and computer storage 93 6.2: Local relief (H) 95 6.3: Mean slope (tan«) 95 6.4: Roughness factor 6.5: Hypsometric integral (HI) 6.6: Comparison of errors for triangular data-sets and grids 101 6.7: Reproduceability and the influence of scale 103 6.8: Summary 106 Summary and Conclusions 108 Chapter 7: (R) References 98 98 111 Appendix I: Notation Appendix I I : Topographic and related variables for 42 areas in southern 121 124 British Columbia Appendix I l i a : Computer program 126 lllb: Triangular data-sets 135 Illc: Computer results 144 —vi I — List of Tables Table 3 . 1 : Page Sizes of arbitrarily-bounded areas over which local 31 relief was determined by various authors. 4.1: Physiographic subdivisions of southern British Columbia, 56 after Holland (1964), with sample numbers and map-areas for terrain samples analyzed in Chapter 4 . 4.2: Comparison of distribution of 42 terrain samples among 57 ten physiographic divisions with expected distribution based on division areas. 4.3: Variables included in correlation analysis. 63 4.4: Statistically significant (95 per cent level) correlation 64 coefficients among the variables listed in Table 4 . 3 . 4.5: Classification of 42 terrain samples using local relief 67 ( H ) , hypsometric integral ( H ) , and peak density (Dp). 4.6: Values of some selected geomorphometric parameters 68 for six areas selected for detailed analysis. 6.1: Numbers of points ( N y ) , boundary points (Ng) and 94 triangles ( N ^ ) for data-sets analyzed. 6.2: Estimates of local relief ( H ) , and analysis of errors in 96 these estimates. 6.3: Estimates of mean slope ( t a n o f ) , and analysis of errors 97 in these estimates. 6.4: Estimates of roughness factor (R). 6.5: Estimates of hypsometric integral ( H ) , and analysis 99 100 of errors in these estimates. 6.6: Empirical comparison of errors for triangular data-sets and 15 by 15 grids. 102 -viii- Page Table 6 . 7 : Similarity among four triangular data-sets based on 104 sample 11 for the four selected measures. 6.8: Similarity among three triangular data-sets based on sample 18 for the four selected measures. 105 -ix- List of Figures Figure 2 . 1 : Page Coefficients of determination as functions of sample size for 11 three sampling designs applied to three surfaces of varying complexity. 2.2: Map to illustrate types of surface-specific points and 14 lines. 2.3: Form of the interpolated surface between two data points for. various exponents 17 in the general i n t e r - polation formula. 3.1: Forms of surface roughness. 24 3.2: Hypothetical topographic profile illustrating various 33 relief measures. 3.3: Diagrammatic topographic profile illustrating the 40 relationships among r e l i e f , slope, and roughness. 3.4: Relationships between local relief and roughness 44 factor. 4.1: Physiographic subdivisions of southern British Columbia 54 w i t h locations of stratified random sample of terrain analyzed in Chapter 4 . 4.2: Distribution of terrain samples among the 32 55 1:50,000 scale sheets which make up a 1:250,000 scale map sheet. 4.3: Histograms for six geomorphometric parameters. 59 4.4: Correlation structure among twelve terrain and 65 related parameters. 4.5: Hypsometric curves for the six terrain samples selected for detailed analysis. 69 -xPage Figure 5 . 1 : Illustration of the distance from any point to the nearest 76 grid point. 5.2: Hypsometric curves for the portions of an inclined plane 82 w i t h i n 3 sample areas. 5.3: As in Figure 5 . 2 , but for a square-based pyramid. 84 6.1: Sample 11a, an example of a triangular data-set from °2 the Ptarmigan Creek map-area. Preface When the current project was begun, i t was the writer's intention to develop a computer system for the analysis and classification of terrain from topographic map d a t a , with the specific aim of eventually producing a quantitative physiographic map of British Columbia. Some theoretical and empirical analyses (reported herein in Chapter 5) revealed that estimating geomorphometric parameters from a regular grid could introduce considerable error. The thesis objective was therefore redefined to become an investigation of the relative merits of grids and of alternative computer terrain storage systems The results may be considered to represent a pilot study for the eventual realization of the original o b j e c t i v e . Throughout this study, the writer has benefitted greatly from discussions with his thesis supervisor, Michael Church. He and J . Ross Mackay read and commented upon drafts of the entire thesis, w h i l e Thomas K. Peucker has reviewed certain sections. Michael C . Roberts and H. O l a v Slaymaker have also provided helpful a d v i c e . Financial support was primarily provided by the Department of Geography, University of British Columbia, in the form of Teaching Assistantships. Some support was also obtained from the "Geographica Data Structure" project, Geographical Branch, O f f i c e of Naval Research, Project N O N R 710-100, principal investigator Thomas K. Peucker, Department of Geography, Simon Fraser University. Computer time was provided through the Department of Geography, University of British Columbia. -1- Chapter 1: Introduction Geomorphometry, which has been defined by Chorley et a l . (1957, p. 138) as the science "which treats the geometry of the landscape," attempts to describe quantitatively the form of the land surface; i t is a sub-discipline of geomorphology. Evans (1972, p. 18) distinguished specific geomorphometry, which measures the geometry of specific types of landforms (see for example the work of Troeh, 1964, 1965, on "landform equations"), from general geomorphometry, "the measurement and analysis of those characteristics of landforms which are applicable to any continuous rough surface." In much of the geomorphometric literature, i t has been claimed that the drainage basin represents "the fundamental geomorphic unit" (notably Chorley, 1969; see also Leopold et a l . , 1964; Williams, 1966). This view was taken to an extreme by Connelly (1968), who in a discussion of terrain statistics stated that "although i t is an oversimplification i t is certainly a valid approximation to attribute all land forms to the f l u v i a l erosion of uplifted rock masses" (p. 7 8 ) . He stated that this assumption was necessary in order to develop "a unified framework for landscape geometry. " Since about one third of the earth's land surface was glaciated during the Pleistocene (cf. F l i n t , 1971, p. 19), and as other processes such as f l u v i a l deposition, or a e o l i a n , v o l c a n i c , or periglacial action have also influenced large areas, it is the writer's opinion that a "unified framework" could only be produced i f no single process is assumed. Furthermore, the specific approach can only be applied once an area of the earth's surface has been identified as a drainage basin, an a l l u v i a l f a n , a drumlin, et cetera. The object of this study is to investigate the use of computer-stored topographic information in the evaluation of geomorphometric parameters. Computers have been w i d e l y employed in both geography and the earth sciences, and geomorphology has not been an exception. A recent book edited by Chorley -2- (1972) attests to the fact that spatial aspects of land surface form have received much a t t e n t i o n . While computers have been used i n geomorphometry, there have been few attempts to store topographic surfaces in computers and then to perform detailed quantitative analyses of land surface form. Exceptions are the works of Hormann (1969, 1971), who approximated land surfaces with sets of contiguous triangles, and of Evans (1972), whose work was based on regular square grids ("altitude matrices"). Neither of these works studied the comparative accuracy, precision, and efficiency of computer terrain storage methods, the differences between computer estimates and "standard" methods for estimating geomorphometric measures, or the relative d i g i t i z a t i o n (data gathering) times and computer storage requirements. It is the purpose of this study to review various computer terrain storage systems, and to compare the triangle and grid methods noted above. The comparison w i l l be based on the estimation of a group of landform parameters selected after a review of many such measures. For the reasons cited above, emphasis w i l l be placed upon general geomorphometric parameters, although some attention w i l l be directed toward measures based specifically upon landforms of f l u v i a l a c t i v i t y , probably the most important single class of processes which has shaped the earth's surface. All examples used in the comparisons w i l l be drawn from topographic maps of that part of British Columbia which lies south of 54 degrees latitude, mostly from 1:50,000 scale maps. Since a l l topographic data used in this study w i l l be derived from contour maps, i t seems in order to discuss briefly the precision of topographic map information. 1.1: Precision of Topographic Map Data Boesch and Kishomoto (1966) expressed elevation errors in terms of roof-mean-square errors, herein designated s_. survey s e For example, they stated that values for triangulation points are generally less than 0 . 5 m horizontally -3- and 0.1 m vertically (pp. 9 - 1 0 ) , while root-mean-square errors in map plotting range from 0.01 to 0 . 3 mm on the map (p. 12). For a 1:50,000 scale map, this would represent 0 . 5 to 15 metres on the ground. Boesch and Kishomoto stated that contour precision has two aspects: "(1) the positional error of a point on a contour, and (2) the height error of a point whose elevation is determined (or read) from the nearest contours by interpolation" (p. 14). They presented a graph of allowable standard deviations in metres as functions of ground slope for various countries and map scales (their Figure 2 ) . Thompson and Davey (1953, p. 40) cited the accuracy specifications of United States Geological Survey topographic maps as: Vertical accuracy, applied to contour maps on a l l publication scales, shall be such that not more than 10 per cent of the elevations tested shall be in error more than one-half the contour i n t e r v a l . In checking elevations taken from the map, the apparent vertical error may be decreased by assuming a horizontal displacement w i t h i n the permissible horizontal error for a map of that scale. They used the 90 per cent criterion in conjunction w i t h a table of ordinates of the normal curve to estimate the allowable s value as about 0 . 3 times the e contour i n t e r v a l . The conversion of horizontal errors into vertical ones involves the tangent of ground slope. Standards for Canadian topographic maps do not appear to be as well defined. W . A . Williamson (pers. written c o m m . , 1972) stated that the Canadian Surveys and Mapping Branch designs its maps so that "on Class A maps the contours are accurate to one-half a contour i n t e r v a l . " If it is assumed that this represents a 95 per cent confidence l e v e l , the ordinates of the normal curve can be used fo estimate the allowable root-mean-square height error as 0.255 times the contour i n t e r v a l . Williamson also stated that for Canadian Class A maps, points are to appear w i t h i n 0.5 mm of their true positions as map scale — this would represent 25 m on the ground for 1:50,000 scale maps. -4- Following Thompson and Davey's (1954, p. 43) approach, bur using this scale and a 100-foot (30.5 m) contour i n t e r v a l , the root-mean-square error for the maps used in this study should be given by: s = t ( 7 . 8 + 2 5 . 0 tan 3 ) metres (1.1) e where o is the ground slope. Another possible source of contour error is the generalization required when smaller scale maps are compiled from larger scale ones. Pannekoek (1962) discussed this, and stated that in some cases, contours should be "moved aside" in some valleys or along coasts in order to "make room" for cultural features such as roads and railways. This should not be a factor in the present study, as the map series used herein is now compiled "at publication scale" (Williamson, pers. comm.), and was formerly compiled for only a 20 per cent reduction. Errors or inconsistencies in the portrayal of the drainage net on maps may present problems in estimating drainage parameters. This problem has received more attention than has the precision of relief estimates (cf. Morisawa, 1957; Giusti and Schneider, 1965; Eyles, 1966; Gregory, 1966a, 1966b; M c C o y , 1971). Most of these writers found that the "extended drainage network" that is, the network formed by extending streams along contour crenulations, was more closely related to the drainage net determined in the f i e l d or from aerial photographs than was the "blue line network" printed on the maps (see Morisawa, 1957; Eyles, 1966). Other authors (notably Gregory) argued that the use of the extended network might lead to the inclusion of former channels not now part of the drainage system, such as "dry valleys" in karst areas or former glacial melrwater channels. Because drainage net parameters do not form an important part of the present study, analysis w i l l be simplified through the use of the "blue line" stream network shown on the topographic maps. -5- 1 .2: Notation Throughout this paper, terms and symbols are defined where they are first introduced. In addition, a complete listing of a l l symbols used w i l l be given Appendix I. Where there are "standard" symbols for variables, these w i l l be in used unless ambiguity would result. Furthermore, x and y are reserved to indicate geographical location, z elevation above sea l e v e l , N a number of objects or occurrences, D a density value, r a correlation c o e f f i c i e n t , and s a root-meansquare value. The metric system of units is employed throughout, with British units being given in some instances. Elevations obtained from the maps were in feet, but were converted to metres before analysis. -6- C h a p t e r 2 : Computer Terrain Storage Systems Topography can be considered to be a continuous surface, and thus even a small area contains an infinite number of points; the number of points which can be measured is limited by the resolving power of one's instruments and not by the surface itself. Since i t is generally not possible to specify the land surface completely, the usual objective of computer terrain storage systems is to obtain a "satisfactory" representation of the surface which w i l l minimize both the effort required to obtain the data and the computer storage requirements, while at the same time maximizing the efficiency with which some particular type of processing may be performed. In the present study, the "processing" involves the estimation of some geomorphometric parameters. The problem is really t w o f o l d : one aspect involves the collection of topographic information from maps or other sources, w h i l e the second relates to the storage, r e t r i e v a l , and processing methods employed. 2.1: Digitization Digitization can be defined as the process by which "analog measures", such as length or location on a map, are converted into " d i g i t a l , computer-usable form" (Peucker, 1972, p . 7 2 ) . Two distinct d i g i t i z a t i o n strategies are a v a i l a b l e : one involves sampling at surface-random points or lines, while the other uses surface-specific points or lines. In the surface-random approach, the points sampled are not selected on the basis of surface form but according to some partially or completely arbitrary set of c r i t e r i a . Randomly-located points are obviously surface-random, but points selected using equal increments in the x and y-directions (grid sampling) or equal increments in elevation (contours) are also generally random with respect to the surface. When surface-specific points or lines are used, knowledge of the form of the surface being sampled (usually obtained by a visual inspection of a contour map or of the land surface itself) -7is used to select points or lines which contain a maximum amount of information. These include peaks and pits, passes, ridges and course lines, and breaks of slope. 2 . 2 : Surface-random Sampling: Grids The most w i d e l y used method for storing and processing three-dimensional surfaces is probably the square g r i d , also known as the "altitude matrix" (Evans, 1972, p. 2 4 ) , or as "both a digital terrain model and a numerical map" (Connelly, 1972, p. 92). Sample points are located at the intersections of two orthogonal sets of regularly-spaced parallel lines. O n l y the altitude of the surface at each sample point must be measured and stored w i t h i n the computer — the geographical locations are determined by the grid spacing, and are implicit in the sequential position of the altitude value w i t h i n the computer storage array. A wide variety of computer programs for the processing of gridded data is a v a i l a b l e . Another advantage lies in the fact that the neighbours of a given data point, which are often required in the calculation of geomorphometric parameters, can be readily obtained, once again from the positions of points w i t h i n the computer array. The principal disadvantage of the regular grid is its tendency toward redundancy — the grid must be made sufficiently dense throughout to portray the smallest objects which must be shown anywhere w i t h i n the area covered by the g r i d . According to Tobler (1969, p. 243), the sampling theorem states that " i f a function has no spectral components of frequency higher than W , then the value of the function is completely determined by a knowledge of its values at points spaced 1/2 W a p a r t . " Thus a regular grid with a grid spacing d can only be expected to depict those variations of the surface having wavelengths of 2d or more. If the smallest significant wavelength of object one wishes to detect or portray anywhere w i t h i n a study area is of size ("wavelength") S, then the grid spacing everywhere must be 1/2 S or less. "Smoother" sub-areas of the -8- study area w i l l then contain far more points than are needed to portray their form. To improve the "resolution" of a grid by a factor f , the grid spacing must be decreased by this factor — the total number of data points is increased by a factor of f^. Tobler and Davis (1968) described a number of regular grid data sets of various types of terrain which together form a "digital terrain l i b r a r y " . Because of the wide application of this terrain storage method, the larger number of gridded terrain samples already c o l l e c t e d , and the number of computer programs a v a i l a b l e , this method w i l l be examined intensively in later chapters. At least two other grid approaches have been used: one is a "regular triangular g r i d " , while the other was termed the "variable g r i d " method by Boehm (1967, p. 404). square grid approach. The regular triangular grid has some advantages over the Each point has six neighbours which form a regular hexagon, and Mackay (1953) discussed how this form of data collection avoids the "saddle point problem" which sometimes arises in attempting to draw isopleths based on a square g r i d . The advantage in this regard is probably out- weighed by the increased complexity involved in indicating geographical location i m p l i c i t l y in the computer storage a l l o c a t i o n . Most of the drawbacks of the regular square grid would also apply to a regular triangular one. In the variable grid method, a "master" regular grid is used, but in rougher areas, denser regular grids are applied; the redundancy of the denser grid in smoother parts of the surface is thus avoided. Some preliminary analysis would be required to determine the areas in which a denser grid should be used, and how dense it should b e . If the smallest significant terrain wavelength w i t h i n each sub-area can be estimated, the sampling theorem discussed above can be used to determine the required grid spacing. This implies some knowledge of the surface form before the data are c o l l e c t e d , and thus the variable grid -9- method is not completely "surface-random", although the exact locations of the data points remain so. There is some disagreement as to the relative merits of completely random sampling of a surface, in which the locations of the sample points are random, and of the type of "surface-random" sampling represented by regular grids. Strahler (1956, p. 589-592) considered the "random co-ordinate method" and the regular grid for sampling surface slope. He stated that: " I t might be supposed that a regularly distributed sample would give coverage more uniformly representative of the entire area and would be superior to the random co-ordinate method. According to statistical principles, however, this grid sample is unsatisfactory because variance cannot be computed s i m p l y . " (p. 591) Since even the regular grid points are random w i t h respect to such surface characteristics as elevation and slope, the writer cannot understand why the variance of slopes for 100 gridded points cannot be determined in exactly the same way as for 100 randomly-located ones. Strahler also noted that the grid might produce a systematically-biased sample i f the grid lines happen to be aligned parallel to linear features in the topography, such as ridges or valleys. This latter argument was also put forward by Haan and Johnson (1966, p. 124) with reference to the sampling of elevations to be used in the construction of hypsometric curves. Because of their inherently uneven distribution, however, randomly-located points might also produce biased sampling, although the bias w i l l not be systematic — there w i l l simply be more data points in some parts of the study area than in others. W . D . Rase (personal oral c o m m . , 1970; pers. written c o m m . , 1973) investigated the relative "information contents" of randomly-located and gridded elevation samples; this unpublished study represents the only quantitative comparison known to the w r i t e r . Three surfaces were first represented by 150 x 150 grids; the surfaces were a plane, a fourth-order polynomial, and a 2 3 . 7 km -10- square topographic sample from the Lake Louise 1:50,000 scale map sheet (grid spacing about 160 m). Samples of between about 100 and 500 points were taken from each of these populations of 22,500 points in three ways — random, "systematic stratified aligned" (regular grid) , and "systematic stratified unaligned" in which the rows and columns of the grid were not aligned with the co-ordinate axes. 50 x 50 grids (2,500 points) were then interpolated from these samples using the SYMAP program (see section 2 . 5 ) , and these were compared with the corresponding points from the original data sets using various simple statistical measures. Figure 2 . 1 plots the coefficient of determination against sample size for each of the nine cases examined by Rase. For each surface, the two systematic approaches (grids) produced considerably better results than the random co-ordinate method; the aligned samples tended to give somewhat better results than the unaligned systematic samples. This evidence strongly suggests that grid samples provide a "better" representation of a surface than do random co-ordinate samples, and appears to refute the unsubstantiated claims of Strahler (1956) and Haan and Johnson (1966). O f course, the actual values of the coefficients of determination shown in Figure 2 . 1 are at least in part dependent upon the particular interpolation model chosen to generate the 50 x 50 grids (see section 2.5). Furthermore, the problem investigated by Rase was not the same as those investigated by the other authors. From the point of view of computer storage, the random co-ordinate approach would have the added disadvantage that a l l three co-ordinates of each point must be specified and stored — the advantages of i m p l i c i t geographical location and implicit neighbours which hold true for grids are lost. -n- Figure 2 . 1 : Coefficients of determination as functions of sample size for three sampling designs (random, systematic stratified unaligned, systematic stratified aligned) applied to three surfaces of varying complexity (after W . D . Rase, unpublished study). -12- 2.3: Surface-random Sampling: Digitized Contours Contours represent another way of sampling and storing a terrain surface. It must be noted that the elevations of the contours are fixed by sea level (or other datum) and the contour i n t e r v a l , and are thus random with respect to surface features. On some maps, supplementary contours or spot elevations are used to provide the map user with additional information. The storage of topography through the use of digitized contours is of particular interest in light of recent developments in automated compiling and drafting of topographic maps. As the contours are determined using stereoplotters and plotted automatically, the succession of points along the contours could readily be stored on tape and made available for geomorphometric processing. Evans (1972, p. 23-27) discussed the relative merits of digitized contours and of altitude matrices. He noted that w h i l e the former method is superior if one wishes to know the locations of a l l pints of a certain height, it is inferior i f one wishes to know the elevation at a given location. Since the latter sort of question arises much more often in geomorphometric snalysis than the former, i t would seem that d i g i t i z e d contours are less suitable for geomorphometric analysis than are regular grids. More storage space is required per point, since only the elevation can be indicated i m p l i c i t l y , and two values per point must be e x p l i c i t l y stored. Boehm (1967) described a "contour tree ordering method" for storing surface information; this method is said to be more efficient in problems where "successive specified points are correlated, such as in line-of-sight calculations" (p. 405) than would be a storage of contour points sorted by x co-ordinates. Boehm's work w i l l be discussed further in section 2 . 7 . Computer programs are available for determining slope steepness and aspect directly from contour data (see section 3 . 4 . 2 ) ; routines for producing contours from grids are widely a v a i l a b l e , and the inverse process, that is, -13- producing grids from digitized contour data, is also possible. These processes would both involve interpolation, and the choice of the interpolation model (section 2.5) would influence the results. 2.4: Surface-specific Sampling: Points and Lines Surface-specific points can be defined as "points which furnish more information about the surface than only their co-ordinates" (Peucker, 1972, p. 23). These were termed "significant topographic points" by Hardy (1971, p. 1907). Surface-specific points include peaks and pits (maxima and minima, respectively, on the surface), passes or saddle points, stream and ridge junctions, and points where there are significant changes in the directional trends of surfacespecific lines. breaks of slope. (See Figure 2 . 2 . ) These lines include ridges, course lines, and There has been some work on the relationships among and links between various types of surface-specific points and lines on continuous, contihuously-differentiable surfaces. This was begun by Cayley (1859) and Maxwell (1870), and revived by Warntz (1966, 1968). (1972, p. 24) and Woldenberg (1972, Since both Peucker p. 327-330) have recently reviewed this work and as i t is not directly relevant to the current research, no summary w i l l be included herein. The writer knows of no work on the relative "information contents" of surface-specific and surface-random points; Peucker (1972, p. 7 2 ) , however, claimed that "surface-specific points have a higher information content than surface-random p o i n t s . " Fewer of these should be required to define a surface to a given level of precision, but there is no evidence to suggest how many fewer. Surface specific points require more storage space and d i g i t i z i n g time than do an equivalent number of grid points, since a l l three co-ordinates must be e x p l i c i t l y determined and stored. There is an element of subjectivity in the selection of surface-specific points a n d , furthermore, neighbours cannot easily be determined from the points alone. STREAM " P • STREAM JUNCTION —500 — Figure 2 . 2 : I T CONTOUR RIDGE • PEAK o RIDGE JUNCTION P A S S Map to illustrate types of surface-specific points and lines. -152.5: Surface Behavior However the sample points are chosen, one must make some assumptions about the behavior of the surface between the data points. Sometimes these assumptions are based on a theoretical or empirical knowledge of the actual surface behavior, but more often they are arbitrary. In this study only interpolating surfaces, i . e . , surfaces which pass through a l l the data points, w i l l be considered; approximating surfaces (known as trend surfaces), which do not necessarily pass through a l l the points and which are thus "smoother" than the original data, have also been applied to topography. These works have mainly been involved with attempts to determine the forms of former "erosion surfaces" now represented only by hilltops (ct\ • K i n g , 1969; Monmonier, 1969; Rodda, 1970; Tarrant, 1970). Bassett and Chorley (1971) computed trend surfaces based on 15 x 15 grids of terrain elevations in an attempt to determine different scales of variation of the topography. Such work, while interesting, is beyond the scope of the present study. As mentioned above, interpolation usually involves an arbitrary assumption about the behavior of the surface between data points. (1960, Robinson p. 186-7) stated the "standard" cartographic assumption that, in determining the*positions of isarithms from control points, linear interpolation should be used "when no evidence exists to indicate a nonlinear gradient between control p o i n t s . " Peucker (1972, p. 25) noted that linear interpolation, in particular the representation of a surface by a contiguous non-overlapping set of triangular planes, "represents the simplest, fastest, and often the least misleading interpolation m e t h o d . " Peucker goes on to point o u t , however, that such surfaces have discontinuities in the first derivative (i . e . , h a v e "breaks of slope") which may produce an "unpleasant" appearance in block diagrams or contour maps. Perhaps for this reason, most computer algorithms for producing -16dense regular grids from a less dense sample of points, (e.g. UBC X P A N D ; SYMAP) use an inverse-distance-squared weighted average of the heights of a number of surrounding data points. Since distance is determined using Pythagoras' Theorem, use of the squared distance in weighting elevations eliminates the need for a square-root determination, reducing computer processing time. A surface thus produced is continuous in the first derivative and therefore appears "smooth". A general interpolation formula may be expressed as: (2.1) 1 I where z. is the height to be determined, the z. the elevations of neighbouring points, and c. . the distance between points i and j . For linear interpolation, jJ = 1 , while in the more common interpolation algorithms discussed above,0 - 2. There has been l i t t l e i f any research into the effect of 0 -values on surface behavior; Figure 2 . 3 illustrates the influence of these values on the form of a surface between data points. This diagram suggests that different P -values may be appropriate for different types of terrain. In the absence of any work on optimal 0-values, the linear assumption, i . e . , a -value of one, w i l l be used in this study. If data are in a square g r i d , triangular planar facets for determining slope or other parameters can be produced in two different ways. In one approach, one set of diagonals is arbitrarily inserted. Turner and Miles (1967, p. 260) determined a roughness parameter for the two orientations of diagonals and found very l i t t l e difference in the results. A l t e r n a t i v e l y , additional points in the centres of the grid squares may be interpolated by averaging the four surrounding points and used to form triangles. This is done in some contouring programs in order to avoid the "saddle point problem" discussed in section 2 . 2 . -17- Figure 2 . 3 : Form of the interpolated surface between two data points (circles) for various exponents in the general interpolation formula (see equation 2 . 1 ) . -182.6: Computer Storage of Terrain Information There are a number of possible approaches to the storage of numerical terrain information. Most simply, the data may be stored d i r e c t l y , as a matrix of elevations for gridded d a t a , as lists of x and y co-ordinates for digitized contours, or as a l l three co-ordinates for surface-specific points. The surface between the points would then be determined during the processing stage after retrieval. In the case of irregularly-distributed points such as surface-specific points, however, processing w i l l be much more efficient if the neighbours of each point are indicated in some way — as already noted above, this is not required for gridded data. This can be achieved in at least two ways. Hormann (1969, 1971) stored the identification number and co-ordinates for each point. He then listed a l l the neighbours (by identification number) for some a r b i t r a r i l y chosen starting point. N e x t , for each of these neighbours, a l l adjacent points excluding the starting point are g i v e n , and the procedure is continued until every link between neighbours has been included exactly once. During processing, the computer forms triangles, beginning at the arbitrary starting point. If all neighbours of any particular point other than the first one are required, a l l previous pointer lists would have to be searched. In the basic storage system of the Geographical Data Structure^ (GDS), a l l neighbours of every point are included in that point's pointer list, making it easier to find any point's neighbours. This makes searching through the data structure easier than in Hormann's version but requires more storage space as each link appears in two pointer lists. This storage method is herein termed the "pointer mode" of the GDS. 1 Geography Branch, O f f i c e of Naval Research, Task N o . 710-100, Department of Geography, Simon Fraser University, Burnaby, British Columbia, T . K . Peucker, principal investigator -19Another approach is to store the point numbers and co-ordinates, and then to store a list of triangles, each record containing the triangle number and the identification numbers of the three points making up its vertices — this is termed the "triangle mode" of storage. could also be indicated. Other characteristics of the triangle This form of data organization is somewhat easier to prepare, and is also more efficient for " t r i a n g l e - b y - t r i a n g l e " processing required for most geomorphometric analysis. The triangle mode was used by Akin (1971), w i t h elevations replaced by precipitation values, in the calculation of the mean areal depth of precipitation. The triangle mode should be far less efficient for searching through the data structure than would be the pointer mode; computer routines for producing one data structure mode from the other are currently being developed under the GDS project. The project is also developing methods for determining the neighbours of a set of surface-specific points given only the points' co-ordinates. It is possible to produce a regular grid from a set of surface-specific points by interpolation (equation 2 . 1 ) . The results w i l l be influenced by the choice of the P-value; the appropriateness of the 0 -value of 2 used in most interpolation algorithms is suspect. Yet another approach to numerical terrain storage is to find an e x p l i c i t mathematical function or set of functions which either interpolate or approximate the surface. The coefficients of the equations, rather than the points themselves,! would be stored, and could be based on gridded or non-gridded data. Such equations can usually be differentiated, the results being equations of surface slope over the area. If a constant elevation is subtracted from the equation, the root of the resulting equation w i l l give the contour of that elevation. Junkins and Jancaitis (1971) found that this approach was an order of magnitude more efficient than the method of evaluating the surface equation at a large number of -20- grid points and then using "standard" grid contouring methods. The latter approach was used in the same context by Hardy (1971, 1972). The functions can also be integrated over the study area to determine the volume under the surface, which is of geomorphometric interest. Once a g a i n , however, there are often arbitrary assumptions about surface behavior; also, Hardy's method requires that the data points and one coefficient per point be stored, resulting in little saving of storage space, although processing may be speeded up. 2.7: Comparisons of Approaches Boehm (1967) compared five methods of surface storage: contour points sorted in the x - d i r e c t i o n (CS), "contour tree ordering" (CT), uniform grid ( U G ) , uniform grid-differential altitude ( U G D A ) , and variable grid-differential altitude ( V G ) . "Differential a l t i t u d e " means that altitude differences between neighbours rather than absolute altitudes are stored. Boehm presented an extensive table (his Table I I , p. 410) of "performance estimates" for the various methods. He then applied them to a problem in intervisibility between points on the surface, determining both storage requirements and processing speed. The grids were most efficient in terms of processing speed, w i t h the uniform grid the best, w h i l e the variable grid required the least storage. Some other c o m - parisons of methods have already been cited above. 2.8: Conclusions As Boehm (1967, p. 414) stated, "one cannot discuss the relative efficiencies of tabular representation methods without reference to the problem being s o l v e d . " Thus the results of studies by Rase (see Figure 2.1) and Boehm (see above) are not directly applicable to the problem considered here, that is, the estimation of some selected geomorphometric parameters. These parameters w i l l be selected after a review of many such measures in the next chapter. The terrain sampling and storage methods compared w i l l be the regular grid (altitude matrix) and an approach based on surface-specific points. The regular grid is -21- representative of various methods of surface-random sampling, and these two approaches are the only computer terrain storage systems which have been applied to problems of general geomorphometry (c f. Evans, 1972; Hormann, 1969, 1971). As noted above, surface behavior between data points w i l l in both cases be assumed to be linear. -22- Chapter 3: Geomorphometric Parameter In this chapter, an attempt w i l l be made to review a considerable number of geomorphometric parameters in such a way as to produce a rational classification of these measures. Attention w i l l be focussed upon two points: the amenability of the parameters to measurement based upon the computer terrain storage systems discussed above, and the probable geomorphic significance of the measures. N o attempt w i l l be made to review papers approaching landscape analysis through a set of landform "elements", " u n i t s " , or "facets" (examples of this approach include: Van Lopik and Kolb, 1959; Lebedev, 1961; Conacher, 1968; Speight, 1968; Thomas, 1969; Wong, 1969; Gerenchuk et a l . , 1970) . In cases where the units were based upon quantitative landform parameters (e.g. Speight, 1968), only the parameters w i l l be discussed. Similarly, graphical analysis methods w i l l be reviewed only where they are related to important geomorphometric parameters. Chorley (1969, p. 78) proposed that characteristics of drainage basins and drainage nets could be divided into geometrical properties, which involve the relationships among dimensional properties such as elevations, lengths, areas, and volumes, and topological properties which relate numbers of objects in the drainage net (for example, the bifurcation ratio). The latter properties w i l l not be considered herein. A l l measures of land surface form can be considered to be in some way representative of the "roughness" of the surface. This discussion w i l l thus begin with a discussion of the general concept of roughness before proceeding to actual geomorphometric parameters. -23- 3 . 1 : The Concept of "Roughness" In a general sense, roughness refers to the irregularity of a topographic surface. Stone and Dugundji (1965) and Hobson (1967) observed that roughness cannot be completely defined by any single measure, but must be represented by a "roughness vector" or set of parameters. One area may be rougher than another because i t has a shorter characteristic wavelength (finer grain or texture), a higher amplitude ( r e l i e f ) , an irregularity of ridge spacing, or sharper ridges (see Figure 3 . 1 ) . Stone and Dugundji, in a study of microrelief profiles, used five measures, while Hcbson computed 9 other measures based on three different "roughness concepts". It is convenient to discuss terrain roughness by analogy with combinations of periodic functions or spectra of the terrain. Evans (1972, p. 33-36) reviewed some of the attempts to analyze topography using spectral analysis e x p l i c i t l y . He observed (p. 36) that in practice this has not been very successful, because valleys often curve, and they converge downstream, while valley spacing w i t h i n an area is seldom regular. The general ideas of wavelength and amplitude are useful, however, and geomorphometric measures w i l l be discussed in this context. The significant wavelengths of the topography are termed grain or texture, while the amplitudes associated w i t h these wavelengths correspond to the concept of relief. The relationship between the horizontal and vertical dimensions of the topography is embodied in the land slope and the dispersion of slope magnitude and orientation, while the vertical distribution of mass under the topographic surface is contained in the concept of hypsometry. -24- Figure 3 . 1 : Forms of surface roughness. than A in some respect. B, C , D, and E are "rougher" B has a shorter wavelength, C a higher amplitude, D an irregularity in spacing, and E a "sharper"form. -25- 3 . 2 : Texture and Grain Texture and grain are terms which have been used to indicate in some way the scale of horizontal variations in the topography. These terms have been used in different contexts, and this difference is preserved i f texture is used to refer to the shortest significant wavelength in the topography and grain used for the longest significant wavelength. Texture is related to the smallest landform elements one wishes to detect, and grain to the size of area over which one measures other parameters. 3.2.1: Grain Wood and Snail (1960, p. 1) defined grain as "the size of area over which the other factors are to be measured. It is dependent on the spacing of major ridges and valleys and thus indicates texture of topography." Grain was calculated by determining the local relief w i t h i n concentric circles around a randomly-located point. Relief was plotted against diameter and, according to the authors, there w i l l generally be a "knick point" in this curve - - the diameter at this knick point w i l l be the grain ( G ) . Wood and Snell used diameter increments of one m i l e , and suggested that i f there is no knick point, relief values for the diemeters of circles centred at a number of points should be determined and averaged; "this technique w i l l produce a definite knick point so that no doubt remains as to the grain size " ( p . 5 ) . They (p.6) noted that the method is not very precise, but believed that it was better than measuring parameters such as relief for a standard arbitrary area. In the present study, " g r a i n " is also used less formally to refer to the longest significant topographic wavelength. Other parameters should be sampled over areas larger than or equal to the grain size in order to obtain representative values. -263.2.2: Texture As noted above, the term texture is herein applied in a general sense to refer to the shortest significant topographic wavelength. This should determine the grid spacing for grid sampling or the size of the triangles for surface-specific point sampling. The word "texture" has been used for a specific geomorphometric parameter. Smith (1950) proposed a texture ratio: T=N/P (3.1) where"N is the number of crenulations on the selected contour, and P is the length of the perimeter of the basin given in miles or fractions thereof" (p. 657). He "selected" the contour having the most crenulations. Smith found that the texture ratio was closely related to drainage density (see below, section 3 . 2 . 3 ) by the following empirical relationship: D = 1.658 T ' 1 1 1 5 d (3.2) Smith did not give confidence limits for the regression coefficients, but the closeness of the exponent to one suggests to the writer that the relationship may in fact be linear. The nearly linear relationship between T and the drainage density is not surprising, since the inverse of T is closely related to the average distance between contour crenulations along the selected contour. As each crenulation represents a stream in the "extended drainage n e t w o r k " , the inverse of T is closely related to the mean distance between channels, which is in turn the inverse of drainage density. 3.2.3: Drainage Density (Dd) As already noted, drainage density is closely related to texture. Drainage density, defined by Horton (1945, p. 283) as the total length of stream channels per unit area, represents a very important geomorphometric parameter. It has been found to be closely related to mean stream discharge ( c f . Carlston, 1963), mean annual precipitation ( c f . Chorley and Morgan, 1962), and -27sediment y i e l d (Abrahams, 1972). It has also been shown to increase with time on t i l l plains exposed by deglaciation (Ruhe, 1952). Roberts and Klingeman (1972) found that the total length of flowing channels at a particular time is closely related to instantaneous stream discharge. Thus drainage density for flowing channels only w i l l vary over short periods of time. Evans (1972, p. 33) suggested that i f only high order streams are considered, the inverse of valley density should provide a useful expression of overall topographic g r a i n , since the inverse of drainage density is the mean orthogonal distance between channels. In a method analogous to Wentworth's method for slope estimation (see section 3 . 4 . 1 below), Carlston and Langbein (unpub. 1960; c f . M c C o y , 1971) and McCoy (1971) used traverse sampling to obtain a rapid estimate of drainage density (see section 4 . 3 . 4 ) . Other writers have used the numbers of intersections between the drainage net and traverse lines directly without attempting to convert them to drainage density. Peltier (1962) plotted the number of drainageways per mile against mean slope and showed curves for a number of climatic or geomorphic regions; a l l traverse minima were counted, including closed depressions. Donahue (1972) determined "mean channel spacing" by counting intersections between the drainage net and a set of randomlyoriented traverse lines and dividing this into the total length of traverse. He did not, however, make a correction for the angle of intersection between traverse line and drainageway (see section 3 . 4 . 1 ) . Wood and Snell (1957, 1959, 1960) determined a parameter called "slope direction changes", the total number of minima and maxima encountered along traverse lines of constant total length. Since the profile is continuous, maxima and minima must alternate, and the number of slope direction changes is twice the number of drainageways, plus or minus one. It would be possible to convert the data of Peltier, Donahue, and Wood and Snell to drainage densities for comparison with other studies. -28- Anorher parameter very closely related to drainage density is the source density ( D ) , the number of stream sources per unit area (see Mather, s 1972, p. 311). Both this and the preceding parameter are very sensitive to the portrayal of the drainage net. As already noted in section 1 . 1 , there may be map-to-map inconsistencies in the portrayal of the drainage net, and for this reason some writers have used the "extended drainage network" formed by extending stress as indicated by the contour crenulations. introduces an element of subjectivity. This, however, The q u a l i t y of the b l u e - l i n e drainage net shown on some topographic map from southern British Columbia w i l l be investigated in the next chapter. 3.2.4: Other Texture Measures A different measure of surface texture is the number of closed h i l l t o p contours per unit area, here termed the peak density (Dp). Wood and Snell (1959) used this as one of their parameters for classifying terrain. any closed contour (other than a pit) to be a " h i l l t o p " . They considered King (1966), in her application of factor analysis to geomorphometric measures, used two peak densities: "summit dissection", which was "the number of closed summit or spur contours" (p. 4 1 ) , and " v a l l e y character", the number of closed valley contours, most of which represented drumlins. Swan (1967) mapped " h i l l frequency" as the density of hills per square m i l e . A h i l l was defined as any summit w i t h two or more closed contours, or with a difference between top and base elevations of more than 50 feet (15.2 m). Using a related measure, Ronca and Green (1970) studied the density and distribution of craters on the lunar surface. Yet another way of characterizing surface roughness is through an examination of ridges. Speight (1968) determined ridginess, the total length of ridge per unit area (analogous to drainage density) and r e t i c u l a t i o n , which was a measure of the size of "the largest connected network of crests that -29- projected into a sample area" (p. 248). He also used modified two-dimensional vector an lysis on ridge segments to measure the degree to which the ridges tended to be p a r a l l e l . 3.3: Relief Measures The term relief is used to describe the vertical dimension or amplitude of topography. Evans (1972, p. 31-32) noted that the majority of relief measures depend upon the extreme values of the distribution of elevations, and would thus be sensitive to rather minor variations in estimations of these heights. He therefore proposed that the standard deviation of altitudes would provide a more stable measure of the vertical v a r i a b i l i t y of the terrain. He did observe that "the autocorrelation of altitude admittedly makes range less unreliable than i t is for random variables, since on a continuous surface a l l intermediate values between the extremes must be represented " (p. 31), but nevertheless recommended use of the standard d e v i a t i o n . A l l of the other papers reviewed herein have, however, used extreme values to characterize the vertical dimension. 3.3.1: Local Relief (H) For any f i n i t e area of a surface, the local relief is defined as the difference between the highest and lowest elevations occurring w i t h i n that area. It is important to note that local relief is always defined with respect to some particular area, and perhaps for this reason has sometimes been termed the "relative r e l i e f " (cf. Smith, 1935). This measure was apparently introduced by Partsch (1911), who termed it the reliefenergie, and was first used in the English language in 1935 in independent papers by Smith and by Huggins (1935). 1 The former author is generally credited with introducing the concept of local relief into the English language literature, but Huggins apparently presented his paper at a professional meeting some months earlier. -30These works, as well as many others (see Table 3 . 1 ) , determined local relief for arbitrarily-bounded terrain samples such as squares, circles, or latitude-longitude quadrangles. In most cases, the size of the sample area was arbitrary, although Trewartha and Smith (1941, p. 31) stated that "the size of the rectangle for which relief readings are made appears to need adjustment for the degree of coarseness or fineness of the relief p a t t e r n . " They did not indicate how the appropriate size could be determined. Wood and Snell (1960) used a variable sample area size — they first determined the "grain" of the topography (see above, section 3 . 2 . 1 ) , and then measured the relief for a circle with a radius equal to the g r a i n . Wood and Snell (1957, 1959), Peltier (1962), and Evans (1972) compared the values of local relief determined over more than one size of area. Evans (1972, p. 30) pointed out that i f the sample area "is so small (in relation to topographic wavelengths) that it is unlikely to contain a whole slope, ' r e l i e f becomes simply a measure of gradient;" in order to make relief "as distinct and non-redundant a variable as possible" (p. 3 1 ) , he recommended the use of " f a i r l y large" sample areas. The areas should definitely be larger than the texture of the topography, and preferably larger than its g r a i n . Data from Wood and Snell (1959, p. 9) support Evens' contention — they found that the correlation between relief and slope declined as the size of the area over which they were measured increased. Salisbury (1962) studied the relationship between relief and slope for g l a c i a l deposits, and found the two to be closely related for older drift sheets, t i l l plains, lake plains, and outwash, but poorly related on end moraines and sand dunes. This probably reflects the interaction of sample area size and texture. In a l l of the above examples, local relief was determined for a r b i t r a r i l y bounded sample areas; local relief has also frequently been determined for drainage basins. The minimum elevation w i l l be the basin mouth, w h i l e the -31- TABLE 3 . 1 : SIZES OF ARBITRARILY-BOUNDED AREAS OVER WHICH LOCAL RELIEF WAS DETERMINED authors date BY VARIOUS AUTHORS area (km^) type of area 1.00 1.00 2.59 3.34 7.52 10.4 10.4 15.0 23.3 25.0 25.0 25.0 32.0 34.0 65.5 93.2 203. 400. square square square square square square square quad.* square square square square * of sizes Studies using one sample size: Chen Harris Hesler & Johnson Swan Abrahams Huggins Donahue Batchelder Zakrzewska King Kaitanen Hutchinson Partsch Trewartha & Smith Smith Hammond Spreen Ahnert 1947 1969 1972 1967 1972 1935 1972 1950 1963 1966 1969 1970 1911 1941 1935 1964 1947 1970 - quad. quad. square circle square More than one size used: Gassmann & Gutersohn Evans Wood & Snell Peltier Wood & Snell 1947 1972 1957, 1959 1962 1960 * quad. = latitude-longitude quadrangle 0.25 0.63 0.81 2.59 18.3 to to to & to 28.0 62.7 414. 259. 399. square square circle square circle 13 18 8 2 7 -32- maximum is usually, but not always, located on the basin perimeter. (1960, Maxwell p. 10-11) determined the "basin r e l i e f " as the "elevation difference between the basin mouth and upper end of the diameter", where basin diameter was determined trhough a complicated set of c r i t e r i a , but was essentially "the longest dimension of the basin parallel to the principal drainage l i n e " . Since the size of drainage basins w i l l vary, many workers have found i t desirable to determine a dimensionless " r e l i e f r a t i o " or "relative relief number" by dividing the relief by some other linear dimension of the basin. The latter have included the basin diameter (defined above), basin perimeter ( M e l t o n , 1957) and square root of basin area ( M e l t o n , 1965). 3 . 3 . 2 : A v a i l a b l e Relief ( H ) n The concept of available relief was introduced by Glock (1932), and his definition was rephrased by Johnson (1933, p. 295) to read: " A v a i l a b l e relief is the vertical distance from the former position of an upland surface down to the position of adjacent graded streams. " Johnson pointed out that this could only be determined where the original upland surface could be identified from remnants and where there were "graded" streams. The latter involves the definition of the concept of " g r a d e " , which w i l l not be discussed here. Local, available and drainage relief are illustrated diagrammatically in Figure 3 . 2 . Glock stressed the importance of available relief in determining the land profile but, as Johnson noted, other factors such as drainage density and slope must also be considered. In order to determine the average available r e l i e f , one would have to construct both the " o r i g i n a l " and "streamline" surfaces (see Pannekoek, 1967, for a review of methods for constructing such surfaces), and to then divide the difference in the volumes under these surfaces by the area. -33- • Figure 3 . 2 : STREAM 'AT G R A D E ' MAXIMUM ELEVATION O STREAM OTHER Hypothetical topographic profile illustrating various relief measures. H q H is the local relief for the entire p r o f i l e , Glock's available r e l i e f , and the drainage r e l i e f . Dury's "available relief" would be the mean height of the shaded portion. -34- A different relief measure was discussed by Dury (1951), who unfortunately also used the term "available r e l i e f " ; this was defined as "that part of the landscape w h i c h , standing higher than the floors of the main valleys , may be looked on as available for destruction by the agents of erosion with reference to existing base-levels" (p. 339). working He then defined the "mean available r e l i e f " as the average height of the land above the streamline surface, computed as the difference in volumes under the actual and streamline surfaces, divided by the area. This is clearly not the same as the available relief defined by Glock (1932) and Johnson (1933). Dury (1951, p. 342-3) also discussed the depth of dissection", which is identical to the G l o c k / j o h n s o n concept of available r e l i e f . 3.3.3: Drainage Relief (H-j) Glock (1932, p. 75) also defined a measure called the drainage relief as "the vertical distance through which rain water moves over the ground from the time the water first strikes the surface until it joins a definite stream." Johnson (1933, p. 301), however, pointed out that Glock later used the term to refer to the vertical distance between adjacent divides and streams, and proposed that this latter definition be adopted (see Figure 3 . 2 ) . If in an area a l l the divides are at the elevation of the original upland surface and a l l the streams are "at g r a d e " , drainage relief w i l l equal available relief; to the latter, however, drainage relief can always be determined. in contrast Strahler (1958, p. 295) stated that "local r e l i e f , H, is a measure of vertical distance from stream to adjacent d i v i d e " ( i . e . , local relief is equivalent to drainage r e l i e f ) , but this w i l l only be true i f the sample areas upon which local relief is based are large enough to include adjacent streams and divides and yet not so large that the slopes of the streams and divides themselves significantly increase the relief w i t h i n the sample area. In Figure 3 . 2 , the area over which H is determined is "large" and H exceeds H j . Hormann (1971, p. 145) determined -35the mittlere Taltiefe ("mean valley depth") for drainage basins. First, a "roof" was constructed over the basin by linking points along the basin divide which were equidistant from the basin mouth by straight lines. The volume between this surface and the land surface wad divided by the basin area. This measure is "complementary" to Dury's "mean available r e l i e f " . 3 . 3 . 4 : Applications of Relief Measures Relief has commonly been used in a descriptive way ( e . g . Smith, 1935) or to delimit physiographic regions ( e . g . Huggins, 1935), both alone and in conjunction with other variables. Some studies have, however, related relief to landscape processes, or to other aspects of physical geography. Schumm (1954, 1963) found that sediment yield was closely related to the ratio of basin relief to basin diameter for some small drainage basins in the southwestern United States. Schumm (1956) also related sedimentyield to relief and slope for some smaller basins in the Perth Amboy badlands. Maner (1958) investigated the relationships between sediment yield and a number of basin characteristics, and found that the above relief ratio was the one most highly correlated with the dependent variable. Ahnert (1970) determined average basin relief as the mean of local relief values for 20 by 20 km squares spread over a number of drainage basins for which he had information concerning denudation rates. In the absence of stream incision, denudation w i l l reduce relief; by using the empirical relationship between denudation rates and r e l i e f , Ahnert presented theoretical curves for relief reduction as a function of time, both w i t h and without the effects of isostatic compensation. He later (1972) related these results to theoretical models for slope processes. -36- 3.4: Slope Evans (1972, p. 36) stated that "slope is perhaps the most important aspect of surface form, since surfaces are formed completely of slopes, and slope angles control the gravitational force available for geomorphic w o r k . " M a t h e m a t i c a l l y , the tangent of the slope angle (tano<) is the first derivative of a l t i t u d e , and it is as a tangent or per cent slope that this surface parameter is generally reported. Strahler (1956) also mapped slope sine, which is proportional to the downslope component of the acceleration of g r a v i t y . Strahler's (1950, 1956) work suggested that slope tangents had a normal distribution; Speight (1971), however, found that for a number of areas investigated, a log-normal distribution provided a better f i t . Unlike relief and most other geomorphometric parameters, which are only defined for f i n i t e subareas of a surface, slope is defined at every point as the slope of a plane tangent to the surface at that point. In practice, however, slope is generally measured over a f i n i t e distance, especially when data are obtained from a contour map. The size of area over which slope is measured w i l l influence the values obtained, and the effect of recording intervals on slope values was discussed by Gerrard and Robinson (1971). Mean slope was generally much less sensitive to the recording interval than was maximum slope. 3 . 4 . 1 : Average Slope: Line-Sampling Method A method for estimating average slope proposed by Wentworth (1930) has been w i d e l y a p p l i e d . The number (N) of intersections between a set of traverse lines and the contours in the sample area is counted, and the total length of the traverse lines (L) is measured. L divided by N is the mean horizontal distance between the contours, as measured along the traverse lines. This w i l l tend to be larger than the mean orthogonal distance, and therefore a "correction factor" -37must be a p p l i e d . If the traverse line intersects the contours at an angle 0 , the true inter-contour distance w i l l equal sin 0 times the traverse distance. If one assumes that a l l values of 0 between 0 and 90 degrees are equally l i k e l y , then the true mean inter-contour distance should equal L / N times the mean value of sin 0 , which is 2/TT , or 0.6366. The mean slope tangent estimate is then given by: ,an« - i _ _ (3.3) 0.6366 where I is the contour interval in the same units as L. Wentworth presented the formula for use w i t h L in miles and I in feet as: tan oc = I (N/L) 3 3 6 1 (3.4) The method gives the mean slope for an area, but has been used to construct slope isopleth maps by assigning the slope for an area to a point at the area's centre (c_f_. Smith, 1939; Calef and Newcomb, 1953; G r i f f i t h s , 1964). Other authors have used the number of contour intersections per length of traverse d i r e c t l y , without converting to an actual slope value. Wood and Snell (1957, 1960) used the "contour count" as a "measure of slope" (1957, p. 1), but in their 1959 paper converted this to slope using Wentworth's formula. Zakrzewska (1963) determined the "roughness" at a sample point as the number of contour intersections with the circumference of a circle centred at that point. 3 . 4 . 2 : Average Slope: Other Methods Raisz and Henry (1937) mapped average slope by determining areas of similar contour density (slope) subjectively. The mean slope (in feet per mile) was determined for each such area, and a choropleth map was produced. This approach has also been applied by some other writers ( c f . G r i f f i t h s , 1964). Another method which has been widely used depends upon determining the slope -38- at a sample of points distributed over the study area; these values may be averaged ( c f . (cf. Strahler, 1956; Coulson and Gross, 1967) or "contoured" Strahler, 1956; Speight, 1968). Ruhe (1950) and Rowan et_aL (1971) determined slope for traverse segments between maxima and minima along the traverses. No attempt was made in either of these studies to correct for the angle between the traverse line and the contours. In direct computer applications, a number of writers (Monmonier et a l . , 1966; Piper and Evans, 1967, c f . Evans, 1972; P a r k e t a L , 1970, 1971) have described methods for determining surface slope from d i g i t i z e d contour data. Sharpnack and A k i n (1969), as w e l l as Rase (1970, pers. oral c o m m . ) , computed both slope and aspect from an altitude matrix. Griffiths (1964) compared the "subjective method" (essentially the Raisz and Henry approach), Wentworth method and "point sampling" method. He concluded that Wentworth's method was most accurate, and that the point sampling method produced "comparable" results with less e f f o r t . 3.4.3: Other Slope Parameters Another slope parameter is the rate of change of slope, termed the "local convexity" by Evans (1972, p. 41). Mathematically, this is the second derivative of a l t i t u d e , or the first derivative of slope. Convexity can be separated into downslope convexity and cross-slope convexity (contour curvature). Evans suggested that the problem of convexity could be "solved" by f i t t i n g quadratic surfaces to 3 by 3 sections of an altitude matrix. Convexity could then be determined by differentiating the resulting quadratic equation t w i c e . Speight (1968, p. 243) examined both rate of change of slope (which he termed "slope gradient") and contour curvature. It is also possible to determine higher derivatives of a l t i t u d e , but the possible physical meaning of such higher derivatives is obscure. -39Closely related to mean slope is Strahler's ruggedness number, which was defined as HD^ by Strahler (1958, p. 289)as a result of dimensional analysis. In the case of a two-dimensional p r o f i l e , the relationship among r e l i e f , drainage density, and slope can be easily shown. In Figure 3 . 3 , H is the relief and b half the distance between channels, which equals half the inverse of D^. One thus has the mean slope given by: tan* = H/b = 2 HD or twice the ruggedness number. D (3.5) Strahler (p. 295) also introduced average slope into the ruggedness number, producing the geometry number: HDJ — 2 tan o( (3.6) If H is a reasonable estimate of the drainage relief and if the two-dimensional case can be extended to three dimensions, this geometry number should equal 0 . 5 (see equation 3 . 5 ) . The theoretical relationship is supported by the fact that Strahler found that while drainage density for his test basins ranged over two orders of magnitude, values of the geometry number remained between 0 . 4 and 1.0. 3 . 4 . 4 : A p p l i c a t i o n of Slope Measures As in the case of relief (section 3 . 3 . 4 ) , slope has been w i d e l y used in descriptive w o r k , in physiographic classification, and in m i l i t a r y work related to vehicle t r a f f i c a b i l i t y . Slope angle is a result of past or present geomorphic processes, and w i l l also influence these processes (c f. Ahnert, 1972). Indeed, the analysis of slope profile form represents an important "sub-discipline of geomorphology (for example, Institute of British Geographers, 1971; Carson and K i r k b y , 1972; Young, 1972). -40- Figure 3 . 3 : Diagrammatic topographic profile illustrating the relationships among r e l i e f , slope, and roughness (see equations 3 . 5 , 3 . 1 0 , and 3 . 1 1 ) . 3.5: Dispersion of Slope Magnitude and Orientation In addition to slope steepness, slope aspect or direction may be considered, either separately or together w i t h slope angle. Evans (1972, p. 41) proposed that the combined analysis of slope magnitude and orientation would produce "undesirable hybrid results; i t is better to separate v a r i a b i l i t y gradient from v a r i a b i l i t y in aspect." in If this is done, the aspect data should be analyzed using two-dimensional vector analysis ( c f . Curray, 1956). While such separation may be desirable in some cases, the distribution of orthogonals to the land surface (which summarize both types of information) is essentially three-dimensional, and its analysis as such would seem to be appropriate. Chapman (1952) presented a potentially useful method for examining slope steepness and aspect. Both the aspect (orientation) and slope (dip) of the land surface were determined for a sample of points on a regular g r i d . The points were then plotted on a Schmidt net and contoured in the same way as other orientation data in the earth sciences are often presented. Chapman suggested that these diagrams would probably be useful in relating slopes to structure or the effects of glacier movement, and Newell (1970) successfully used the technique in this context. One of the computer programs presented by Hobson (1967, 1972) represents a logical extension of this work, treating the perpendiculars to slope units as vectors and applying well-established mathematical approaches to the analysis of three-dimensional orientation data (cf. Fisher, 1953; Steinmetz, 1962). Unit vectors orthogonal to triangular facets formed by inserting diagonals into a regular grid were summed and the length of the vector sum (R) was determined. Hobson then calculated k (which is the estimate of the precision parameter K for the spherical normal distribution of Fisher, 1954) as: -42- k = (3.7) (N-l) /(N-R) As a surface approaches planarity, the vectors w i l l become p a r a l l e l , R w i l l approach N (the number of vectors), and k w i l l become very large. Intuitively , a plane should have a roughness of z e r o , and thus the inverse of k would represent a more "reasonable" roughness measure. Since Hobson's method was based on a regular g r i d , all triangles have the same horizontal area and similar true areas, and hence the use of unit vectors is not unreasonable. If based upon irregularly-distributed surface specific points, however, there may be a considerable variation in triangle size. It would seem appropriate to weight the vector orthogonal to each triangle by the triangle's true area. If this is done, however, k and its inverse cannot be determined through equation 3.7. Some manipulation of that equation gives: _1_ N k N-l _R_ N 100 - L(%)| (for large N) (3.8) 100 where L(%) is 100 ( R / N ) , the vector strength in per cent. For weighted vectorial analysis, L is defined as 100 times the weighted vector sum divided by the sum of the weights. It is herein proposed that the best measure of "vector dispersion" roughness is the roughness factor R, defined by: •R=100-L(%) (3.9) In the case of unit vectors and large N , R w i l l approximately equal 100 times the inverse of k. As in the case of slope, the roughness factor can be related to relief and ridge spacing through reference to Figure 3 . 3 . For R, the vertical component of each orthogonal vector w i l l equal coso<, w h i l e the horizontal components w i l l cancel out, leaving: R = 100 (1 - coso<) (3.10) -43Substituting the value for cos * gives: R = 100 (1 - ) b VH +b 2 (3.11) 2 Turner and Miles (1967) applied Hobson's (1967) vector program to t w e n t y - f i v e sample areas; twelve of these were derived from Stone and Dugundji's (1965) microterrain maps and provide a basis for examining the relationships among the parameters used by these authors, including R and H. The other thirteen samples were based on macroterrain from 1:24,000 scale maps. In addition to k, Turner and Miles determined the local relief ( H ) , and a v a r i a b i l i t y factor v , the local relief divided by the logarithm of k; writer estimated R as the inverse of k. the Linear correlation coefficients were determined among ten roughness measures for the twelve common terrain samples. In a d d i t i o n , correlations were determined separately among Stone and Dugundji's six variables(16 cases) and among the four derived from Turner and Miles (25 cases). The only statistically significant correlations based on the common terrain samples which did not reflect functional relationships were those between mean and maximum amplitude, and between H and R. The latter pair of variables were not significantly correlated over the t w e n t y - f i v e Turner and Miles samples. This is almost certainly due to the difference in scale between the 1:24,000 and microterrain maps (1 to 2 orders of magnitude). In Figure 3 . 4 , R (as estimated as the inverse of k) is plotted against H for these 25 samples and for six others analyzed in Chapter 6 . Curves of the form given in equation 3.11 for various values of b have been plotted in Figure 3 . 4 . These have been fitted "by eye" to the groups of points for each of the six scales represented. It appears that each scale has a reasonably consistent "characteristic wavelength" which influences the relationship between H and R. -44- H (metres) Figure 3 . 4 : Relationship between local relief and roughness factor. Open symbols represent micro-terrain from Turner and Miles (1967). Solid symbols are macro-terrain (circles from Turner and Miles; triangles from this study). based on equation 3 . 1 1 . Curves are -45ln a related line of research, Hayre and Moore (1961) determined theoretical scattering coefficients for t e r r a i n , based on autocorrelation functions determined from contour map data. Hayre (1962) then used observed radar return rates to estimate the roughness of the lunar surface. 3.6: Hypsometry Clarke (1966, p. 237) defined hypsometry as "the measurement of the interrelationships of area and a l t i t u d e . " Evans (1972, p. 42-48) reviewed this concept under the heading: "Regional convexity (dissection, a e r a t i o n ) . " Most of these measures, which describe aspects of the distribution of landmass with e l e v a t i o n , are based upon the hypsometric curve. 3.6.1: The Hypsometric Curve and its Variations Monkhouse and Wilkinson (1952, p. 112-115) noted that there are three common sorts of graphs used to report hypsometric data. These are: (a) the area-height curve; (b) the hypsometric (or hypsographic) curve, sometimes called the absolute hypsometric curve; (c) . the percentage hypsometric curve. The first of these methods, the area-height curve, plots the area in a band at a particular elevation against e l e v a t i o n , and by convention, elevation is plotted on the y - a x i s . If relative area is used, the diagram is a plot of the probability density function for the heights in the area. The relative frequencies of elevations are generally more easily seen on this type of curve than on the hypsometric curves. The absolute hypsometric curve is a graph of the absolute or relative area above a certain elevation plotted against that e l e v a t i o n , and is essentially a cumulative frequency for the elevations. Once a g a i n , elevation is conventional plotted on the y-axis and area (representing frequency) on the x - a x i s . -46- Clarke (1966, p. 241) pointed out that this curve does not represent an "average p r o f i l e " , since it does not record the slope between contours. Never- theless, a section of the curve w i t h a low slope indicates a larger amount of the surface at or near a particular elevation; this would generally indicate gentler slopes near that elevation. Absolute hypsometric curves have been determined for the earth's surface as a w h o l e , countries, natural regions, islands, and drainage basins. While usually plotted on simple arithmetic graph paper, various special sacles have also been employed. Tanner (1962), for example, plotted the percentage of the earth's surface area lying above certain elevations on log-probability paper, and was able to separate the curve into four Gaussian components. Chorley (1958) found that the hypso- metric curve for a drainage basin he examined plotted as a straight line on arithmetic-probability paper. The third and most w i d e l y used form of curve is the relative or percentage hypsometric curve, often termed simply the hypsometric curve. ^ It plots relative area above a height against relative height, and is the graph of the hypsometric f u n c t i o n , here termed a (h) , where h (the relative height) is defined by: z max -z . mm where z is the actual e l e v a t i o n , and z max and z . are the highest and lowest mm ° elevations, respectively, w i t h i n the study area. As in the previous cases, h is conventionally plotted on the y - a x i s . It is this form of the hypsometric curve and function upon which some important terrain parameters are based. 1 This form of the hypsometric curve is often attributed to Strahler (1952) (for example, see Chorley and M o r l e y , 1959, p. 566); relative hypsometric curves were presented earlier by Imamura (1937, c f . Evans, 1972, p . 4 2 ) , Gassmann and Gutersohn (1947), and Langbein and others (1947), only the latter being cited by Strahler. _47- 3 . 6 . 2 : The Hypsometric Integral (H) The most w i d e l y used parameter based on the hypsometric curve is the hypsometric i n t e g r a l , here designated HI. This parameter, as defined by Strahler (1952, p. 1121), is given by: i HI = / a ( h ) dh (3.13) Strahler pointed out that geometrically, this value is equal to the ratio of the volume between the land surface and a plane passing through the minimum elevation to the volume of a "reference solid" bounded by the perimeter of the area and planes passing through the minimum and maximum points. G r a p h i c a l l y , HI can be determined by measuring the area under the relative hypsometric curve. Strahler (p. 1130) proposed that the value of the hypso- metric integral reflects the "stage" of landscape development. Those areas having HI values above 0.6 were considered to be in a " y o u t h f u l " or equilibrium phase, while drainage basins in equilibrium should have hypsometric integrals between 0.6 and 0 . 3 5 . Values below 0.35 were thought to characterize a transitory "monadnock phase" in landscape development. Pike and Wilson (1971) proved that the elevation-relief ratio (E) of Wood and Snell (1960) is mathematically equal to the hypsometric integral. The former is defined by: E = Z " Z z m ? (3.14) n -z max . mm where z is the mean elevation. From equations 3.12 and 3 . 1 4 , i t can be seen that E is just the mean relative height (h"). Evans (1972, p. 42) pointed out that this same parameter was used much earlier by Peguy (1942, p. 462), and termed the "coefficient of relative massiveness" by M e r l i n (1965). While Srrahler's (1952) method for determining the hypsometric integral involves much -48laborious use of a planimeter to determine inter-contour areas, the e l e v a t i o n relief ratio can be determined much more q u i c k l y , with the mean elevation being determined from a sample of points. Pike and Wilson ( 1 9 7 1 , p. 1081) stated that "experience has shown that a sample of 40 to 50 elevations w i l l ensure accuracy of E t o , on the average, 0 . 0 1 , the value to which areaaltitude parameters customarily are read. " It is important that the maximum and minimum elevations are determined from an inspection of the entire sample area; gross errors in E can result i f the highest and lowest grid values are used (see section 5 . 4 ) . Evans (1972, p. 58), however, used only grid values to estimate the hypsometric integral for sub-matrices ranging from 3 by 3 (9 points) to 47 by 47 (2209 points). For the smaller sub-matrices at least, Evans 1 estimates of H are probably in serious error. Other methods for approximating the hypsometric integral or curve have been proposed. Haan and Johnson (1966) suggested that the elevations of a sample of randomly-located points could be used to construct hypsometric curves, w i t h a considerable saving in time. Chorley and Morley (1959) proposed that the hypsometric integral could be estimated by approximating the drainage basin by a simple geometric form, "the intersection of a lemiscate cylinder w i t h an inverted cone, centered at the lemniscate o r i g i n " (p. 556). The accuracy of this method depends upon the degree to which the geometrical form actually approximates the basin, particularly the fit of the lemniscate loop to the basin perimeter (Chorley et a l . , 1957). Chorley and Morley found that the method produced a systematic error, and proposed a correction factor. Turner and Miles (1967) used a computer program to interpolate a dense regular grid from a sample of points; numbers of grid points falling w i t h i n altitudinal bands were used in producing hypsometric curves. They found that their method produced results closer to planimetered values than did the corrected -49Chorley and Morley approach. It would seem that the e l e v a t i o n - r e l i e f ratio represents a more accurate and more easily applied approximation to the hypsometric integral than do the above. Furthermore, the elevation-relief ratio can be determined for arbitrarily-bounded areas (Wood and Snell, 1960; Pike and Wilson, 1971), while the Chorley and Morley method can only be used for drainage. 3.6.3: Other Parameters Related to the Hypsometric Curve A number of parameters besides the hypsometric integral have been derived from the hypsometric curve. Strahler (1952, p. 1130) noted that most hypsometric curves show a characteristic "s-shape", and proposed a parameter to indicate the sinuosity of the curve. very sinuous curves. Evans (1972, Low values of this parameter indicated p. 47-48) found a strong correlation between the hypsometric integral and the skewness of the distribution of elevations in cases having the same sinuosity. For any constant value of HI, higher skewness was associated w i t h lower values of Strahler's sinuosity parameter. Tanner (1959, 1960) suggested that the skewness and kurtosis of the height distribution function (essentially the hypsometric function) could be used to "characterize various geomorphic regions" (1960, p. 1525). Examination of Tanner's diagrams seems to confirm Evans' result that skewness is closely related to the hypsometric i n t e g r a l , and also suggests that Strahler's sinuosity parameter is closely related to kurtosis. Sinuosity, as measured by Strahler's parameter or the kurtosis, has not (to the writer's knowledge) been investigated in detail or related to other geomorphometric measures. Gassman and Gutersohn (1947) determined a parameter called the kotenstreuung. For computation, this has been shown to equal the standard deviation of the elevations, and was derived from the absolute hypsometric function. They also determined the relieffactor, which equals twice the -50kotenstreuung divided by the local r e l i e f . This is twice the standard deviation of the relative hypsometric f u n c t i o n . Gassman and Gutersohn also determined the mean elevation by using the hypsometric i n t e g r a l , "reversing" the use of the elevation-relief ratio proposed above; this method of determining the mean elevation was employed earlier by Martonne (1941). 3.6.4: Other Parameters Related to Hypsometry In addition to those related to the hypsometric curve, other parameters have been proposed to characterize the relationship between area and a l t i t u d e , sometimes also including slope. None of these have been as widely used as the hypsometric integral; since many of these have been reviewed by Clarke (1966, p. 243-248) and by Evans (1972, herein. Hammond (1964, p. 15) combined slope and height in an area-elevation measure. p. 4 4 - 4 5 ) , most w i l l not be reviewed His general profile character index was defined as the percentage of gentle slopes (tan « less than 0.08) lying above or below the mean e l e v a t i o n . Pike and Wilson ( 1 9 7 1 , p. 1079-80) noted that this index measures a similar aspect of terrain form to the hypsometric i n t e g r a l . This measure may be undefined in some areas if there are no slopes gentler than the c r i t i c a l value. 3 . 6 . 5 : A p p l i c a t i o n of Hypsometric Measures A l l or most of the parameters discussed above have been used in a simply descriptive sense or in physiographic classification. Only the hypsometric i n t e g r a l , however, has been related to geomorphic processes. HI has been determined for drainage basins. Strahler (1957, In most cases, p. 918-920) listed a number of works between 1952 and 1956 which used this parameter; none of these studies found any relationship between H and various hydrologic or sediment y i e l d measures. Chorley (1957, p. 630) measured hypsometric integrals for 27 drainage basins, but did not use this parameter in subsequent analyses nor comment on its omission. Eyles (1969) studied stream long profile -51- form, basin r e l i e f , and basin hypsometric integral for 410 fourth-order drainage basins in Malaysia. He graphed the hypsometric integral against relief and presented an "approximate curve of best f i t drawn 'by eye' " ( p . 2 9 ) . If one assumes that relief is continuously reduced w i t h time (cf. Ahnert, 1970) and that space can be substituted for t i m e , Eyles line suggests a period of 1 e q u i l i b r i u m , a monadnock phase, and an eventual return to equilibrium. An i n i t i a l inequilibrium phase does not appear to be represented in these d a t a . 3.7: Review and Parameters to Be Investigated In review, the most fundamental concepts of geomorphometry are the basic horizontal and vertical scales of the topography. Horizontal variations are encompassed by the concepts of grain (largest significant wavelength) and texture (shortest significant wavelength); grain w i l l not be investigated e x p l i c i t l y , but three measures of texture, namely drainage density (D^)/ source density ( D ) , and peak density (D^) w i l l be considered in the next chapter. s Vertical scale is generally termed " r e l i e f " ; this terrain concept w i l l be represented in further analyses by the local relief ( H ) , the most widely employed relief measure. The relationships between horizontal and vertical scale w i l l be examined through the mean slope (tan o<), while the threedimensional interaction of slope steepness and aspect w i l l be studied through the roughness factor (IR). Relatively independent from horizontal and vertical scales is the distribution of mass w i t h i n the vertical range of the topography. This concept w i l l be investigated through the hypsometric integral (HI). While there may be some redundancy among the parameters noted above, it is believed by the writer that w i t h the possible exception of g r a i n , a l l important terrain information is contained w i t h i n these parameters. next chapter, the relationships among the measures w i l l be studied. In the -52- Chapter 4 : Terrain V a r i a b i l i t y in Southern British Columbia, and Relationships among Variables Before beginning the comparison of the computer terrain storage systems, a " p i l o t study" was conducted. (1) The principal objectives of this were threefold: to provide information about terrain v a r i a b i l i t y in southern British Columbia, and thus guide in the selection of terrain samples for more detailed analysis; (2) to investigate.the relationships among the parameters selected in the preceding chapter; and (3) to provide empirical data for the evaluation of some of the theoretical errors in estimating parameters, which w i l l be discussed in the next chapter. Values for a number of geomorphometric parameters were determined for square terrain samples using simple techniques not including computer analysis. The roughness factor (IR) could therefore not be examined, but the other important parameters listed at the end of the preceding chapter were a l l studied. 4.1: Selection of Sample Areas In order to obtain a relatively unbiased sample of the terrain of southern British Columbia, a stratified random sampling design was employed. From each of the f o r t y - t w o 1:250,000 scale map sheets which cover British Columbia south of 54 degrees latitude, one of the t h i r t y - t w o 1:50,000 scale maps making up that sheet was selected with the aid of a table of random numbers. Because coverage of the area at the larger scale is incomplete, some of the randomlyselected maps were not a v a i l a b l e . In such cases, and in instances where the selected map f e l l entirely outside British Columbia, another map was p i c k e d . From each map, one 7 by 7 kilometre square of terrain was examined. If the Universal Transverse Mercator grid was printed on the map, the sample was -53- generally centred at the intersection of the two major ("10th kilometre") grid lines closest to the centre of the map; this would f a c i l i t a t e later location of the sample areas on the 1:250,000 s ale maps, i f desired. Where the grid did not appear, the sample square was usually placed over the centre of the map. Samples were relocated if more than one third of the area contained water surfaces. The locations of the f o r t y - t w o samples, together w i t h the major physiographic subdivisions of the study a r e a , are shown in Figure 4 . 1 ; two of the samples ( 4 , 9) f e l l in A l b e r t a , although the maps from which they were drawn were in part in British Columbia. Since it was not possible to adhere strictly fo the original random sample, tests were made of the randomness of the terrain samples actually used. As noted above, each 1:250,000 scale map contains t h i r t y - t w o 1:50,000 scale half-sheets (see Figure 4 . 2 ) . The numbers of these t h i r t y - t w o " c e l l s " containing exactly z e r o , one, t w o , et cetera, samples were determined and compared to the frequencies predicted according to the Poisson distribution. A chi-square test indicated that the two sets of frequencies were not significantly different at the 95 per cent l e v e l . O f twenty major physiographic divisions of British Columbia given by Holland (1964), ten occur at least in part south of 54 degrees latitude. These , together with the sample numbers and respective map-areas f a l l i n g w i t h i n each d i v i s i o n , are listed in Table 4 . 1 . The actual distribution of the f o r t y - t w o terrain samples among these ten regions was compared w i t h an even distribution based on the areas of the subdivisions, once again using the chi-square test (see Table 4 . 2 ) . The distributions were not significantly different at the 95 per cent l e v e l . The larger than expected number of samples in the first three subdivisions is probably at least in part due to the coastal locations of these regions (see Figure 4 . 1 ) . Since samples f a l l i n g on the ocean were not accepted, Figure 4 . 1 : Physiographic subdivisions of southern British Columbia (see Table 4.1) with locations of stratified random sample of terrain analyzed in Chapter 4 . -55 1 j ® © 1 ® | ® © 1 © 1 r—i i i 41 2 © 1 4 | ® i © 1 ® 1 ® ® © _ 1 1 ! © \ ® _ I j ® 1 @ ® i © 1 © © © 1 1 ® © | (29) ® 1 1 1 1 1 ® 1 ! (27) © | U i 1 : 250,000 SCALE 14 15 © 1 1 © 13 ® © i. | 1 i 1 MAP I 16 I 9E 1 (EAST) 9W 12 11 10 9 L_J_ \' (WEST) I 1 : 50,000 SCALE 5 6 7 8 f"l8| Figure 4 . 2 : 3 2 MAP SAMPLE SELECTED DETAILED 16) 4 ® OTHER TERRAIN SAMPLE 1 Distribution of terrain samples among the 32 1:50,000 scale sheets which make up a 1:250,000 scale map sheet. text for discussion. FOR ANALYSIS See ® -56- TABLE 4 . 1 : PHYSIOGRAPHIC COLUMBIA, MAP-AREAS SUBDIVISIONS OF AFTER H O L L A N D FOR TERRAIN SOUTHERN BRITISH (1964), WITH SAMPLE NUMBERS SAMPLES A N A L Y Z E D IN AND CHAPTER FOUR Western System Outer Mountain Area Insular Mountains (1) 14: 92C/16E 15:92E/15E 16: 92F/2W Coastal Trough Hecate Depression (2) 35: 102I/9E 36: 102P/9E Georgia Depression (3) 13:92B/14W Coastal Mountain Area Coast Mountains (4) 17: 92G/9E 19: 92I/5E Cascade Mountains (5) * 1 8 : 92H/2W 22: 9 2 L / 6 W 38: 103B/3E 39: 103C/16E 40: 103F/14E 37: 103A/8E * 4 1 : 103G/16W 2l:92K/6W 20: 92J/3W 23: 92M/5E 30: 93D/7E 42: 103H/3E Interior System Central Plateau and Mountain Area Hazelton Mountains (6) (no samples) Rocky Mountain Trench (7) 5: 82K/9E *11:83D/10W Southern Plateau and Mountain Area Interior Plateau (8) 6: 82L/12E 27: 9 3 A / 4 W *24:92N/15E 28:93B/9W 25: 9 2 0 / 1 6 E 29: 93C/8W 26:92P/1E *31:93E/9W Columbia Mountains (9) 1:82E/10W 3: 8 2 G / 1 2 W 2: 82F/8E 7: 82M/15E 32: 93F/9W 33: 9 3 G / 1 3 W 34: 93H/12E * 8 : 82N/4E Eastern System Rocky Mountain Area Rocky Mountains (10) 4 : 82J/1 IE 9: 8 2 0 / 4 E * 10:83C/5W indicates sample selected for more detailed analysis 12: 83E/5W -57- TABLE 4 . 2 : COMPARISON AMONG TEN OF DISTRIBUTION OF 42 TERRAIN SAMPLES PHYSIOGRAPHIC DIVISIONS WITH EXPECTED DISTRIBUTION physiographic division BASED O N D I V I S I O N AREAS per cent of area expected (e) cibserved (o) (e-o) e 1 . Insular Mountains 7.6 3 7 5.33 2. Hecate Depression 4.8 2 4 2.00 3. Georgia Depression 3.2 1 2 1.00 4. Coast Mountains 24.6 10 6 1.60 5. Cascade Mountains 1.5 1 1 0.00 6. Hazelton Mountains 0.5 0 0 7. Rocky Mountain Trench 1.6 1 2 1.00 8. Interior Plateau 33.8 14 11 0.64 9. Columbia Mountains 15.8 7 5 0.57 10 Rocky Mountains 6.6 3 4 0.33 100.0 42 42 Sums * not significant at the 95 per cent level X = 2 2 12.47* -58- samples would tend to be "concentrated" in the land areas of map sheets containing considerable water. 4.2: Data Collection As stated above, each terrain sample consisted of a 7 by 7 kilometre square; the selection of this sample area size was arbitrary. W i t h i n each area, a 7 by 7 grid w i t h a one kilometre spacing was used in determining some terrain measures. A t each of the forty-nine grid intersections, the elevation was determined, and the type of surface at the point ( e . g . land, ocean, lake, or glacier or snowfield) was also noted. The number of intersections between the grid lines and contours, and also between the grid lines and the "blue line" stream network were counted. The elevations of the highest and lowest points w i t h i n the area, the number of closed hilltop contours, the total length of streams, and the number of stream sources were also determined for each sample area. 4.3: Data Analysis 4.3.1: Drainage Density (DA) Drainage density was estimated for each sample area by measuring the total length of blue stream lines on the map, in kilometres, and dividing by the area. The number of intersections between the grid lines and the drainage net (N) was counted and divided by the total length of traverse (L). Carlston and Langbein (Unpub., 1960; c f . M c C o y , 1971) developed a theoretical equation which proposed that the drainage density should be approximated by: D =1.57N/L d (4.1) The empirical evidence collected here appears to support this equation. When a histogram of drainage density was prepared (Figure 4 . 3 a ) , the observations tended to cluster around 0 . 6 km \ having values above one. but with a number of "outliers" The writer had observed during the data collection Figure 4 . 3 : Histograms for six geomorphometric parameters. Triangles indicate the break points for three of the parameters. -60- that some of the older maps appeared to have higher drainage densities than newer ones. A statistically significant inverse correlation was found between and the year of map publication. Drainage density was also significantly correlated with mean annual precipitation at the sites, and i t was thought that this might explain the correlation between map age and drainage density, since most of the older maps were coastal. The correlation between map date and precipitation was not statistically significant, however, suggesting that the variation in the drainage net is at least in part cartographic (see section 1.1). Because of this problem, and because there were no well marked breaks in the distribution, drainage density was not used to divide the samples into groups having similar terrain. 4.3.2: Source Density (D,.) and Peak Density (Dp) The numbers of stream sources and of closed hilltop contours (peaks) were determined and divided by the land area of the sample areas. Source 2 density was found to be closely related to drainage density (r = 0 . 8 4 7 ) , but would also be dependent upon the drainage net depicted on the map, and so was not used in further analysis. A histogram for this parameter is shown in Figure 4 . 3 b . The histogram for peak density (Figure 4 . 3 c ) showed poorly developed _2 breaks at about 0.25 and 0.50 km ; these were used to classify the terrain samples. 4.3.3: Local Relief (H) The maximum and minimum elevations w i t h i n each sample area, as determined by a visual inspection of the contours, were used to determine the local r e l i e f . This should be w i t h i n one contour interval of the actual value a n d , i f the maps are accurate, must be w i t h i n two contour intervals. The maximum and minimum of the 49 grid heights were also determined; the -61difference between these was designated H * , the grid estimate of the local relief. Theoretical aspects of the relationship between the true and grid values of local relief w i l l be discussed in section 5 . 1 . Histograms of local relief were drawn for each of seven physiographic divisions, and for the combined samples (see Figure 4 . 3 d ) . The latter contained two "breaks" which were used to divide the data into three relief classes: " l o w " r e l i e f , less than 500 metres (10 samples); "moderate" r e l i e f , 500 to 1,500 metres (25 samples); " h i g h " r e l i e f , more than 1,500 metres (7 samples). 4 . 3 . 4 : Mean Slope (tan «K) The mean slope for each area was estimated using the Wentworth method (section 3 . 4 . 1 ) . The total length of traverse was 98 kilometres, except where lakes or ocean reduced the land area; in these cases, the length of traverse was reduced by 2 km for each grid intersection f a l l i n g on a water surface. The histogram for average slope (Figure 4 . 3 e ) shows a rather poorly defined break at about 0 . 3 . The high correlation between mean slope and 2 relief for the samples (r = 0.679) clearly indicates that these measures are not independent, and thus mean slope was not used in classifying the sample areas. 4.3.5: Hypsometric Integral (HI) The value of the hypsometric integral for each sample was estimated using Wood and Snell's (1960) e l e v a t i o n - r e l i e f r a t i o . The mean of the elevations of those grid points which did not f a l l on lakes or the sea was used as an estimate of the mean height of the t e r r a i n . The formula for the e l e v a t i o n - relief ratio (equation 3.14) involves both the minimum elevation and the local relief; here, the hypsometric integral was computed t w i c e : H was based on the "true" minimum and maximum elevations, while H * was based on the grid -62- estimates of these values. Theoretical errors in HI* w i l l be discussed in section 5 . 4 . Histograms for this parameter were prepared (see Figure 4 . 3 f ) , but in this case there were no clear breaks in the distribution. When Strahler's (1952) divisions at 0.35 and 0.60 were a p p l i e d , i t was found that only one sample had a hypsometric integral above 0.60 (sample 1:0.602). Thus essentially none of the areas examined were in the " y o u t h f u l " or "inequilibrium" stage. Nineteen of the f o r t y - t w o samples had HI values below 0.35 and would fall into Strahler's "monadnock phase", the remainder being essentially in e q u i l i b r i u m . While the hypsometric integral for an arbitrarily-bounded terrain sample is not necessarily the same as those of its constituent drainage basins (see section 5 . 4 ) , the value of 0.35 was nonetheless used to divide the samples into low or intermediate HI values. 4.3.6: Relationships among Variables In order to better understand the relationships among terrain and related parameters (see Table 4 . 3 ) , linear correlation coefficients among the twelve variables listed in Table 4 . 3 were computed. Table 4 . 4 indicates a l l correlation coefficients which were statistically-significant at the 95 per cent level. The correlations were then examined using the same approach as Melton (1958); Figure 4 . 4 illustrates the three isolated correlation sets which form the cores of three variable systems, namely " d r a i n a g e " , "hypsometry", and !"relief". Peak density (D ) was not significantly correlated with any other P variable. Factor analysis was also applied to the d a t a , and produced essentially the same result . -63- TABLE 4 . 3 : VARIABLES variable number 1 symbol D d INCLUDED IN CORRELATION ANALYSIS name Drainage density 2 N/L Drainage net intersections 3 Source density 5 D s D P H 6 H* G r i d estimate of local relief 7 ran <* Average slope tangent 8 HI Hypsometric integral 9 HI* Grid estimate of H 10 z Mean elevation 11 P Mean annual precipitation 12 t Year of map publication 4 Peak density Local relief -64- TABLE 4 . 4 : STATISTICALLY S I G N I F I C A N T CORRELATION D, N/L D (95 PER CENT COEFFICIENTS A M O N G THE VARIABLES D H* H H* ran* HI z LEVEL) IN p LINEAR TABLE 4.3 r 0.921 0.473 -.496 0.927 0.468 -.485 N/L - 0.554 -.479 0..824 0.419 D s D P H 0..797 0.364 H* _ - 0.987 - 0. 887 - tan-f 0.602 0. 323 HI 0.364 HI* - -.445 0.336 - z P - r -65- Dd ( DRAINAGE) N/L TEXTURE (PEAK DENSITY) H z p. t a n ot RELIEF H* HI HYPSOMETRY Figure 4 . 4 : Correlation structure among twelve terrain and related parameters (constructed in the manner proposed by M e l t o n , 1958). The outer boxes enclose isolated correlation sets; dotted lines indicate inverse correlations. -664.4: Classification of Samples and Selection of Areas for Further Analysis Three independent terrain variables, namely r e l i e f , hypsometry, and peak density, were used to divide the f o r t y - t w o samples into groups having similar terrain. The independence of the parameters is indicated by the fact 2 that the maximum r value among the three pairs was 0 . 0 6 3 . in the distributions of the variables were given above. The break points As there were three classes each for relief and peak density and two for the hypsometric i n t e g r a l , there are eighteen possible groups — of these, fifteen contained at least one sample (see Table 4 . 5 ) . An attempt was made to select six samples for further analysis (in Chapter 6) from among the classes in approximately the same ratios as the total numbers of samples; a table of random numbers was used to aid in the final selections. The exact values of a number of selected geomorphometric parameters for the selected areas are shown in Table 4 . 6 , while values for a l l f o r t y - t w o areas analyzed in this chapter are given in Appendix II . Using a polar planimeter to determine inter-contour areas, hypsometric curves were constructed for each of the six selected areas (Figure 4 . 5 ) ; based on the 49-point samples of elevations (cf. were similar to those shown. curves Haan and Johnston, 1966) The values used to construct the hypsometric curve were also used to calculate the hypsometric integral — these values w i l l be used as the "standard" to which estimates of H w i l l be compared in subsequent sections. 4.5: Description of Areas Selected for Further Analysis^ 4.5.1: Sample 8: l l l e c i l l e w a e t Map-area (82N/4E) The terrain sample from the l l l e c i l l e w a e t map-area is located in the northern part of the Selkirk Mountains subdivision of the Columbia Mountains. The minimum elevation of 2880 feet (878 m) occurs in the valley of the Incomappleux River, while the maximum (9050 feet; 2758 m) is an unnamed 1 Physiography after Holland, 1964 -67TABLE 4 . 5 : CLASSIFICATION RELIEF ( H ) , HYPSOMETRIC OF 42 TERRAIN SAMPLES USING LOCAL INTEGRAL (HI), A N D PEAK DENSITY (D ) . P NUMBERS OF OBSERVATIONS I N CLASSES ARE I N PARENTHESES; SAMPLES FOR FURTHER ANALYSIS ARE UNDERLINED. H HI D P 0.25(11) 0.25 D 0.50(21) P D P 0 . 5 0 (10) 500 m (10) 0.35 (7) 25,33 14,24,40 3,29 0.35 (3) 27 28 32 500 to 1500 m (25) 0.35 (10) 4,22,26,35,36 5,21,32,37,38 17,23 1500 m (7) 0.35 (15) 1,6,14 2,9,12,15,16, ]8,34,39,41,42 0.35 (2) 11 20 0.35 (5) 7,8,19,30 10 -68TABLE 4 . 6 : VALUES OF SOME GEOMORPHOMETRIC PARAMETERS FOR SIX AREAS SELECTED FOR DETAILED A N A L Y S I S . VALUES FOR R ARE FROM CHAPTER 6 , ALL OTHERS, FROM THIS CHAPTER. P 8 0.555 0.102 1880 0.609 13.8 0.432 11 0.549 0.143 1709 0.395 8.1 0.260 18 1.847 0.286 833 0.396 7.4 0.547 24 0.631 0.383 203 0.065 0.25 0.286 31 0.290 0.553 1195 0.225 3.1 0.355 41 0.882 0.490 869 0.400 7.6 0.395 -69- Figure 4 . 5 : Hypsometric curves for the six terrain samples selected for detailed analysis. -70peak just inside the western margin of the sample area. It is essentially an area of alpine glacial features — the higher portions show such features as cirques (two of which contain small glaciers), horns, and aretes, with u-shaped glacial troughs between. Sample 8 had the highest local relief of the six areas selected for detailed analysis, and the fourth highest over a l l . 4.5.2: Sample 1 1 : Ptarmigan Creek Map-area (83D/10W) Sample 11 contains portions of three major physiographic subdivisions. The Rocky Mountain Trench, here only 1 to 1.5 km in w i d t h , cuts across the study area from northwest to southeast; it is occupied by the southeast-flowing Canoe River, whose elevation ranges from about 2300 to 2280 feet (701-695 m). To the northeast lies a portion of the Selwyn Range of the Rocky Mountains; the maximum elevation w i t h i n the sample area north of the river is 7000 feet (2134 m) which occurs on an arete of an unnamed peak reaching 8048 feet (2453 m) just beyond the study area boundary. With the exception of the arete, the topography north of the Canoe River does not display the angularity characteristic of intense alpine glacial erosion. Such forms are present w i t h i n the sample area in the Malton Range of the Monashee Mountains ( a subdivision of the Columbia Mountains) which are found to the southwest of the Trench in this area. A horn with an elevation of 7888 feet (2404 m) represents the maximum elevation w i t h i n the sample area. The topography of sample 11 is not unlike that of the previous one (sample 8 ) , with its high relief and low peak density, but is distinguished by a considerably lower hypsometric integral (0.260) which is a result of a more prominent and level valley floor. 4.5.3: Sample 18: Manning Park Map-area (92H/2W) This sample lies w i t h i n the Hozameen Range of the Cascade Mountains. Once a g a i n , this sample area is dominated by forms produced by alpine-type g l a c i a l erosion. Here, however, the summits take on a more rounded appearance -71because they were overridden by ice during the last glacial maximum. Four summits w i t h i n the area have elevations of about 6350 feet (1935 m ) , and most valley floors are around 4300 feet (1311 m). O n l y in the northwest corner, in the v-shaped valley of the upper Skagit River,does the surface descend below 4000 feet (1219 m) to the minimum elevation of 3450 feet (1052 m). Even so, the total relief of the area (833 m) is only "moderate", according to the divisions established in section 4 . 3 . 1 ; the hypsometric integral (0.547) is by far the highest of the six samples, and the fifth highest of the 42 areas. 4.5.4: Sample 24: Tatla Lake Map-area (92N/15E) This sample lies near the western margin of the Fraser Plateau subdivision of the Interior Plateau. A t 203 metres, this area has the lowest local relief of the 42 areas studied in this chapter. The area is primarily a drumlinized t i l l plain produced by west-to-east moving ice (Tipper, 1971), with elevations of between 3100 and 3300 feet (945-1006 m); a major meltwater channel traverses the sample area leading into Tatla Lake itself, at 2985 feet (910 m) the minimum elevation in the area. This and another lake together cover some 4 per cent of the sample area. Five maxima, probably bedrock outcrops, rise above the t i l l plain to altitudes of about 3650 feet (1113 m). 4.5.5: Sample 3 1 : G h i t e z l i Lake Map-area (93E/9W) This sample is also from the Interior Plateau, but from the Quanchus Range of the Nechako Plateau. The area contains Michel Peak, at 7396 feet (2254 m) the highest point in the Nechako Plateau region. The eastern (lower) boundary of the latter subdivision was defined by Holland (1964, p. 68) as the 3000 foot (914 m) contour, and since about 5 per cent of the present sample area is part of G l a t h e l i Lake (elevation 3490 feet; 1064 m ) , the 7 by 7 km sample area contains almost the entire relief of the Nechako Plateau. relief for the study area (1190 m) is still only in the "moderate" class. Local -72- 4.5.6: Sample 4 1 : Oona River Map-area (103G/16W) Sample 41 is from Porcher Island, and ranges from a maximum elevation of 2950 feet (899 m) at Egeria Mountain to a minimum of 100 feet (30 m) near Ogden Channel. The division between the Hecate Depression and the Kitimat Ranges of the Coast Mountains is not marked by any prominent physical feature in this area. Holland (1964, p. 35) stated that "the eastern boundary of the lowland is arbitrarily taken as a generalized line along the 2000 foot contour. " Following this d e f i n i t i o n , the southwestern half of the sample area belongs to the Hecate Depression, the northeastern to the Coast Mountains; in f a c t , i t is probably more appropriate to assign the entire sample area to a transition zone between the aforementioned physiographic divisions. The area displays many cirques, some w i t h floors as low as about 500 feet (152 m), but nowhere are the ridges sharp as i n , for example, the l l l e c i l l e w a e t map-area (sample 8). Probably, cirques were formed during an early " a l p i n e " phase of g l a c i a t i o n , but later the entire area was overridden by i c e . Cirques may or may not have been re-occupied by local ice after the disappearance of the Cordilleran ice sheet from the area. -73- Chaprer 5: Procedures for Analysis and Theoretical Comparisons of Computer Systems In this chapter, the analysis procedures used to estimate the geomorphometric parameters selected for special attention w i l l be o u t l i n e d . O f the seven variables examined in the last chapter, drainage and source densities were excluded from consideration for the reasons cited above. Peak density was excluded because of computational problems, especially because the correspondence between grid maxima and actual surface maxima may not be great. The remaining four parameters which are studied in this chapter are local relief ( H ) , mean slope (tan o<), roughness factor ( R ) , and hypsometric integral ( H ) . Theoretical errors involved in estimating the parameters from a triangular network of surface-specific points and from a regular grid w i l l be discussed q u a l i t a t i v e l y , and in some cases q u a n t i t a t i v e l y . Consideration w i l l also be given to the theoretical relationships among these and related geomorphometric parameters, and to theoretical computer storage requirement. In the discussions which f o l l o w , it w i l l be assumed that topographic maps provide the only available source of information about the topography. 5.1: Local Relief (H) In estimating the "true" value of local relief from a contour map, errors can arise from a number of sources: (1) map errors, which w i l l be disregarded in the present discussion; (2) interpolation errors — the maximum possible interpolation error for both the highest and lowest point is one contour interval (expected error = 1/2 contour i n t e r v a l ) , and thus the maximum error in the local relief from this source is two contour intervals (expected error = one contour interval); -74- (3) errors due to misreading the contours — this may be one contour i n t e r v a l , or even five contour intervals i f an "index" contour is misread, for both the maximum and minimum point; (4) errors due to the mis-identification of either the maximum or minimum p o i n t , or both — for example, a particular summit may be taken to be the highest point w i t h i n the study area when in fact a higher point exists. O f these, (3) and (4) are "operator errors", and can be avoided by careful examination of the map and checking of the results; errors of types (1) and (2) are generally unavoidable, but are often small when compared to types (3) and (4). 5.1.1: Local Relief: Surface-specific Points In both this and the grid method, the estimate of the local relief is the difference between the elevations of the highest and lowest sample points. All four of the sources of error for the "true" value of local relief listed above may contribute to error in the estimate of H obtained from a set of surface-specific points. "Type 4 " errors should, however, be much less likely in the latter case than in a visual inspection of the contours. In d i g i t i z i n g an area using surface- specific points, an attempt is made to include a l l peaks and pits, as w e l l as a l l maxima and minima long the borders of the area. If a l l are i n c l u d e d , the true maximum and minimum elevations must be among them, and " type 4 " errors are eliminated. If one assumes that no "avoidable operator errors" (types 3 and 4) are present in either case, the accuracy of this method should be equal to the "standard" method (visual inspection). Otherwise, the estimate of local relief obtained from a sample of surface-specific points should tend to be more accurate than that obtained from a visual inspection of the contours; of course, in any particular case, the errors from the various sources may combine to make the visual estimate closer to the true v a l u e . -75- 5.1.2: Local Relief: Regular Grid Considerably larger errors in estimating the minimum and maximum elevations result when a regular grid is used. As noted in section 2 . 2 , grids are surface-random, and i t is highly unlikely that a grid point w i l l coincide w i t h the true maximum or minimum elevation of the study area. Since the grid maximum cannot exceed the true maximum (unless there are interpolation errors) and the grid minimum w i l l be greater than or equal to the true minimum, H*, If t is the the grid estimate of the r e l i e f , w i l l be less than or equal to H. average land slope near the maximum p o i n t , and c the distance from the maximum to the nearest grid point, the error in estimating the maximum should be given by: e max .= c tan 1 (5.1) A similar estimation may be made for e . . As an estimate of the expected ' mm r distance from an extreme point to the nearest grid point, one can use the rootmean-square distance (s^) of a l l pints from the nearest grid point. If d is the grid spacing, and i f the origin of the co-ordinate system is located at a selected grid point, s^ for a l l points closer to that grid point than to any other ( i . e . , w i t h i n the inner box in Figure 5.1) is given by: -, I (x + y ) dy dx One can further suppose that H may approximately equal o< = 0.408 d # the mean ground slope; estimates of the errors in the maximum and minimum elevations would then be: e . = e = 0.408 d tan <* mm max (5.2) v (5.3) ' -76- Figure 5 . 1 : Illustration of the distance (c) from any point (x, y) to the nearest grid point (open c i r c l e ) . other grid points. Solid circles indicate -77- and thus the expected error in the value of the local relief would be: e H = 0.816 d tan <x (5.4) The error i n , and accuracy of, the grid estimate of the local relief is theoretica a linear function of the grid spacing, as proposed by the "sampling theorem" (see section 2 . 2 ) . O f course, the mean slope (o< ) may not be a good estimate of the land slope near the extreme point. In many of the areas examined in section 4 . 3 , the minimum elevation was on a lake, the sea, or a f l o o d p l a i n , an the slope near this point (and thus also e m j ) w n a s near zero . Slopes near the maxima and the minima of most of the f o r t y - t w o samples from Chapter 4 (grid spacing 1 km) were estimated by dividing the elevation differences between the points and the nearest grid points by the horizontal distances; these values should approximate tan }f. The mean values of these angles for the maxima are similar to tan OC, in particular cases they may differ by a factor of two or more; average slope near the minima is only about one third of the mean slope. As an added complication, the closest grid point to the maximum may not be the highest grid p o i n t , and the same may hold for the minimum. In such cases, the error in the grid estimate may not be as great as expected. The empirical relationship between e ^ and tan for the f o r t y - t w o samples from Chapter 4 was e = 234.1 tanoc + 63.12 ( r = 0.245) 2 H where e ^ is in metres and d is 1000 m. (5.5) This relationship is statistically significant (95 per cent level); the large amount of "unexplained" variance is probably a result of the random factor of the actual distance from the extreme points to the nearest grid points, and of the difference between tan I and tan o< . The ratio of the mean values of e ^ and t a n * , divided by d (1000 m) is 0 . 4 2 6 , still only about half of the theoretical coefficient (equation 5 . 4 ) . -78This is probably because the slope near the minimum was often much less than the mean slope, and because in some cases the highest (or lowest) grid point was higher (or lower) than the grid point closest to the true maximum (or minimum). 5.1.3: Review In summary, the estimate of the local relief obtained from a set of surface-specific points should be as accurate as, or even more accurate t h a n , the estimate obtained through a visual inspection of the contours. For regular grids, errors due to the fact that i t is very unlikely that a grid point w i l l coincide exactly w i t h the minimum or maximum point larger than interpolation errors. w i l l tend to be much It would appear that the error in estimating H from a grid w i l l average about 0 . 4 d ran©< , which could be large in areas of steep slopes i f a relatively wide grid spacing is used. Surface-specific points should theoretically provide much better estimates of local relief than should regular grids of "reasonable"densities. Relief error for a given average slope should be a linear function of grid spacing. 5.2: Mean Slope (tan <x) Strahler (1956) determined the " t r u e " mean and standard deviation for slopes in drainage basins by measuring slope tangent at a large number of points, drawing lines of equal slope tangent (isotangents), and using a planimeter to determine the relative frequencies of the various slope classes. Means and other distributional parameters were then determined from these frequencies. Strahler then showed that the distribution of slope measurements at 100 randomlylocated points w i t h i n one study area was not significantly different from the "population" values. As noted earlier (section 3 . 4 . 2 ) , Griffiths (1964) compared this'point sampling" method to the "traverse sampling" method (Wentworth, 1930) and a "subjective" method similar to that described by Raisz and Henry (1937). -79He concluded that the Wentworth method produced the most accurate results of the three. The "isotangent" method would probably produce the best results, but as this is very time consuming, as Strahler concluded that the results of point sampling were not significantly different from this, and as Griffiths concluded that the Wentworth method was superior to the point sampling approach, the Wentworth method was used herein to provide an estimate of the " t r u e " mean slope to which computer values w i l l be compared in the next chapter. 5.2.1: Computational Procedures For the mean slope and for the subsequent two geomorphometric measures, the regular grids were first converted to a set of continguous triangular facets by inserting one set of diagonals into the g r i d ; the same analysis procedures were then used for both these triangles and the triangles based on the surface-specific points. For each t r i a n g l e , a vector orthogonal to it was determined by computing the cross product of vectors forming two edges of the triangle. The length of this vector is twice the true area of the t r i a n g l e , w h i l e the z-component of the orthogonal vector is twice the projected (map) area. Unit orthogonal vectors were determined by dividing the components by the total length, and the slopes of the triangles were computed from the z-components of these unit vectors. Three average slopes, namely unweighted, weighted by map area, and weighted by true area, were determined. weighting (cf. While the latter may represent the most logical Evans, 1972, p. 37), map area has been used by most methods, including the Wentworth approach discussed above. The accuracy of the mean slope estimate obtained from a set of triangles is highly dependent upon how closely the triangles approximate the surface. In the case of surface-specific points, the accuracy w i l l depend upon the selection of the points and the size of the triangles. Pillewizer (1972) noted that the -80- triangle method, as applied by Hormann (1971) , failed to indicate a slope asymmetry detected by f i e l d surveys and careful analysis of large-scale t o p o graphic maps. Pillewizer attributed this failure to the fact that Hormann's triangles were too large. For triangles derived from a g r i d , there w i l l be no control over the degree to which the triangles approximate the surface, except through the size, which is a function of grid spacing. 5.3: Roughness Factor (R) As pointed out in section 3 . 5 , the roughness factor is closely related to the inverse of k, Hobson's (1967, 1970) vector dispersion factor. The latter is defined only for unit vectors, and it was argued in section 3.5 that even for grids, i t would be better to weight the vectors by the true areas of the triangles. In the grid case, the map areas of a l l triangles are equal, and the use of unit vectors (cf. Hobson) should not produce results which differ greatly from weighted vector analysis. For the latter, steeper triangles w i l l be weighted more, increasing the roughness factor slightly. For triangles based on surface-specific points, the use of unit vectors w i l l be inappropriate, since the sizes of the triangles may vary considerably. In this study, both weighted and unweighted analyses were conducted, using the orthogonal vectors noted above. The only "standard" roughness value to which other methods might be compared would be Hobson's k (or its inverse), but as proposed in section 3 . 5 , this measure should be inferior to the value of R obtained from a weighted vector analysis based on surface-specific points. Thus no useful comparisons of the computer estimates to "true" values can be made as for the preceding parameters. 5.4: Hypsometric Integral ( H ) The " t r u e " value of the hypsometric integral was determined by using a planimeter to measure the areas above various elevations (that is, enclosed w i t h i n selected contours); the elevations are converted to relative values by -81- subtracting the minimum height and dividing by the local r e l i e f , w h i l e relative areas are computed by dividing by the total area. These points can be plotted to produce hypsometric curves (see Figure 4 . 5 ) , and the hypsometric integrals can then be determined by measuring the areas under the curves w i t h a planimeter, or by determining the integrals mathematically. In this study, the latter approach was used, employing the trapezoidal method for integrating a function whose values are known at a set of points. Most research using the hypsometric integral has involved drainage basins as basic units, although some studies have applied this measure to arbitrarily-bounded topographic samples as are used in the present work ( c f . Gassmann and Gutersohn, 1947; Wood and Snell, 1960; Pike and Wilson, 1971; Evans, 1972). None o f these works, however, recognized or commented upon the fact that the shape and orientation of the sample area may influence the form of the curve and sometimes the value of the i n t e g r a l , or that the hypsometric integral for a group of basins may not equal the mean of the basin values. The former fact can be illustrated by applying a square sample area w i t h two different orientations and a circle to two simple geometric forms: an inclined plane, and a square-based pyramid considerably larger than the sample area w i t h the latter centred at its apex. For the inclined plane, the hypsometric integral for a l l three samples is 0 . 5 , but the forms of the curves differ (see Figure 5 . 2 ) ; the c i r c l e and the "diagonal square" (the square w i t h a diagonal parallel to the dip of the plane) produce "s-shaped" curves which Strahler (1952) noted were characteristic of higher-order drainage basins at the equilibrium stage in the absence of structural control. Many such basins have outline forms similar to the circle or the diagonal square, and the "characteristic s-shape" is probably in part due to the influence of outline form. In the case of the pyramid, both the curve form and the hypsometric integral vary with the shape and -82- Figure 5 . 2 : Hypsometric curves for the portions of an inclined plane w i t h i n 3 sample areas. A: " p a r a l l e l " square (see inset, A ) ; B: "diagonal" square (see inset, B); C: circle. -83orientation of the sampling area (Figure 5 . 3 ) . Indeed, the curve and integral are identical for the plane and the pyramid in the case of the diagonal square sampling areas. This effect could produce considerable variation in results i f the size of the sampling area is less than or equal to the "texture" of the t o p o graphy in an area. In the present study, however, the sample areas (7 by 7 km) are considerably larger than the topographic texture of these areas. The second consideration in the case of arbitrarily-bounded sample areas is the relationship between the hypsometric integral for such an area and the integrals of its constituent drainage basins. As a simplified illustration, one can consider two adjacent basins of equal areas, minimum elevations of z e r o , and hypsometric integrals of 0.5 — the only difference is that one basin has a local relief of 500 m, the other 1000 m. The former basin w i l l have a mean elevation of 250 m, the latter 500 m — the mean elevation of the combined basins w i l l be 375 m. The total relief is 1000 m, and thus the hypso- metric integral of two basins w i l l be 0 . 3 7 5 , t w e n t y - f i v e per cent less than that of either of the individual basins. Other combinations of relative reliefs, minima, areas and integrals can produce hypsometric integrals for combined basins larger than those of the constituent basins. If the minima and hypsometric integrals are e q u a l , as w i l l be approximately the case in "equilibrium 11 topography w i t h a common local base level (the ocean, a lake, or a low-gradient f l o o d p l a i n ) , the aggregate integral w i l l always be less than the individual ones. This may in part explain the relatively large number of the f o r t y - t w o areas examined in Chapter 4 which had overall integrals below the lower limit of " e q u i l i b r i u m " (0.35) proposed by Strahler (1952) for individual basins. As noted above, the regular grids were converted to sets of triangles and analyzed using the same methods as employed for triangles based on surfacespecific points. It can be easily shown that the volume between a triangular -84- Figure 5 . 3 : As in Figure 5 . 2 , but for a square-based pyramid. Here, the hypsometric integral varies as w e l l as the curve form. -85- plane and the horizontal datum plane is equal to the product of the projected area of the triangle and the mean elevation of the three corners of the t r i a n g l e . These volumes can be summed and divided by the total area to give the mean elevation of the study area. This can then be used in the e l e v a t i o n - r e l i e f ratio formula (equation 3.14) to estimate the hypsometric integral. 5.4.1: Hypsometric Integral: Surface-specific Points It is d i f f i c u l t to determine quantitatively the theoretical precision of this method. The degree to which the value of HI determined as described above from a set of surface-specific points approximates the true value w i l l depend upon how closely the land surfaces w i t h i n the triangles formed from these points approximate planes. If the person selecting the points is careful to make sure that the contours w i t h i n each triangle are approximately parallel and equally spaced, the method should be reasonably accurate. 5.4.2: Hypsometric Integral: Regular G r i d The mean elevation determined from triangles based on a regular grid using the volumetric method outlined above w i l l be very close to the arithmetic mean of the sampled elevation values. Each point not on the outer boundary forms a vertex of exactly six triangles, and thus a l l such points are equally weighted (the projected areas of the triangles are, of course, equal); points along the boundaries are in three triangles, while corner points are in one or two. Since no attempt is made to ensure that the areas w i t h i n each triangle are even approximately planar, the estimate of the mean elevation derived from the grid should not be as accurate as that obtained from a set of surfacespecific points. The principal sources of error, however, are errors in the maximum and minimum elevations used in the e l e v a t i o n - r e l i e f ratio formula (equation 3 . 1 4 ) , errors which have been discussed above in section 5 . 1 . 2 . In the following discussion, e and e . are non-negative error terms, and an ' max mm ' a a -86- asterisk (*) is used to denote values determined from the grid alone. If one defines: z* = z - e max max max 2 = 2 min and H* = min ~f" 6 (5.6) (5.7) min mm •* - z* max min (5.8) It follows that: HP z - (z . + e . ) mm mm' (z - e ) - (z . + e . ) max max' mm mm' v (5.9) x Some algebraic manipulation of this equation yields: HI* e ] _ max + e min + - z . z z max min e mm - z . max mm z z - z mm - z . max mm (5.10) The right-hand-side of this equation is the true value of the hypsometric integral (disreqarding possible errors in z ) , and z - z . * " max mm v 3 r is H, the true local r e l i e f , giving: HI = HI* 1 - Ltt + H mm (5.11) H If HI* is to be accurate, H * must equal H , in which case either e ^ must equal zero or the following relationship must hold true: H = H* = e . mm H (5.12) The latter ratio may provide a rough estimate of hypsometry, since for the 42 samples examined in Chapter 4 it was significantly correlated with H , although the r value was only 0 . 3 4 5 . -87Some further re-arrangement of equation 5.11 yields the following expression for the relative error in H * : e HI - HI* E H T H ' 1 - HI I - e . L H J max mm H* (5.13) Equations 5.12 and 5.13 imply that if the hypsometric integral is low, the error in the minimum elevation must be less than that in the maximum i f HI* is to be a good estimate of H . In f a c t , a low hypsometric integral generally implies gentler slopes near the minimum elevation than near the maximum (see section 3 . 6 . 1 ) , which in turn implies that e . w i l l be less than e " mm max x r (see section 5 . 1 . 2 ) . x For high values of H , e . must exceed e to " min max minimize the error in the grid estimate of the hypsometric i n t e g r a l , and again this w i l l be the "expected" result. The dependence in part of these error terms upon HI should result in errors in H * being somewhat less than equation 5.13 suggests. An attempt was made to determine the relationship between the grid spacing (d) and the theoretical errors in the hypsometric integrals for a squarebased pyramid for grids parallel and diagonal to the pyramid base. The grids had odd numbers of rows and columns and were centred on the pyramid apex; meaninq that e was zero. * max When the grid minimum was used in the - — calculations, e ^ was found to be a linear function of d , once again supporting the "sampling theorem" noted in section 2 . 2 ; when the " t r u e " minimum was 2 used, the error was proportional to d . To halve the error in the former case would require four times as many points, but in the latter only twice as many. -885.4.3: Summary It is d i f f i c u l t to assess quantitatively the theoretical accuracy of the estimate of the hypsometric integral obtained from a set of surface-specific points. Accuracy w i l l depend upon how closely the triangles formed by these points approximate the land surface. For regular grids where only the grid points are used, the error in HI should tend to be a linear function of the grid spacing» (d), and is sensitive to the values of e w r m 5.5: a x and e . m i n \ (equation -i 5 . 1 3/ ) . Possibility of Estimating Other Parameters In addition to the four measures discussed above, many more of the geomorphometric parameters reviewed in Chapter 3 might be estimated from computer-stored terrain information. Among the most useful of these would be the measures of texture or grain outlined in section 3 . 2 . Most of these measures depend upon the density of peaks, pits, streams, or ridges, and are thus strongly related to surfac-especific points and lines. It should be possible to estimate these parameters rather readily from a set of surface-specific points; in the case of grids, the same approach might be applied, but many "false" peaks and pits w i l l appear in such d a t a , simply because a grid point which falls on a ridge may be surrounded by grid points on the sides of the ridge and thus appear to be a "peak" when in fact it is not. The definition of peaks, pits, ridges, and courses w i l l theoretically be much easier i f surface-specific points are stored in the "pointer mode", rather than the "triangle mode" used in the present study (see section 2 . 6 ) . For example, a peak is defined as any point which is higher than a l l its neighbours; the neighbours must therefore be known before elevation comparisons can be made. Once the number of peaks or pits, or the total length of ridges or courses, is established, it can be used to compute peak or pit density, ridginess (cf. Speight, 1968) or drainage density. also be possible to compute other roughness measures (cf. It would Hobson, 1967, 1972), -89- distributional parameters for the vectors orthogonal to the land surface other than R or k, or measures of slope asymmetry (cf. Hormann, 1971, section 4.2.4). 5 . 6 : Theoretical Numbers of Points and Triangles for Triangular Data Sets, and Theoretical Computer Storage Requirements "Euler's Law" for a contiguous set of N^. cells, N - edges and N y vertices states that: N v + N c - N - = 1 i f the "outside" is not considered to be a c e l l . should be 3N^~. sides. (5.14) If a l l cells are triangles, there Since a l l edges form sides of two triangles w i t h the exception of those edges forming the outer boundary of the study a r e a , the total number of edges is given by: 3 N + N —5= r N-= R (5.15) 2 b where N is the number of edges (and also the number of vertices) which form the boundary. Substituting this value in equation 5.14 and solving for N^. the boi D yields: N c = 2N -(N +2) y B (5.16) Thus the total number of triangles in a data-set w i l l be somewhat less than twice the number of points. One can determine the theoretical computer storage requirements of the regular g r i d , and of the "pointer mode" and "triangle mode" of the triangular data-set method. Each integer value requires one half-word of computer storage a l l o c a t i o n , while each " r e a l " or decimal value requires a f u l l word. To store the three co-ordinates (reals) and the identification number (integer) of a surface-specific point would thus require 7 half-words of computer -90- space, while each grid point needs only 2 half-words of computer storage. For the surface-specific points, either a set of pointers or a set of triangles must also be stored. The total number of pointers in a data-set w i l l be twice the number of edges ( N r J , since each edge forms a pointer of each of the vertices at its ends. Using equations 5.15 and 5 . 1 6 , the total average requirements for the pointers of a data-set can be shown to be ( 6 N y - 2 ( N g + half-words, and the total storage for the points and pointers is given by (13N y - 2N B - 6) half-words (5.17) For the "triangle mode", there are required 3 half-words for each t r i a n g l e , the number of triangles being given by equation 5 . 1 6 . The total storage requirements for the points and triangles should equal: (13N which is exactly N R y - 3N g - 6) half-words (5.18) less storage space than needed by the "pointer mode". -91Chapter 6: Empirical Comparisons and Computational Results In this chapter, the results of an empirical comparison of the two computer terrain storage methods discussed above w i l l be reported. To provide data for the comparison, the analysis procedures outlined in Chapter 5 were applied to the six topographic samples described in section 4 . 5 , for both 15 by 15 grids (d = 500 m) and sets of surface-specific points. Figure 6 . 1 shows one of the surface-specific point data-sets; maps of the other data-sets are given in Appendix 1Mb. Samples 11 and 18 were arbitrarily selected to investigate the reproduceability of triangular data-sets and the influences of triangle size and map scale. Each of the regular grids was analyzed t w i c e , using first northwest-southeast and then northeast-southwest diagonals. The results of a l l the computer analyses conducted are given in Appendix l l l c . For the six sample areas, the differences between the computer estimates and the "standard" estimates for local r e l i e f , mean slope, and hypsometric integral were determined. For each method, the mean and standard deviations of the "errors" were determined, and the t-statistic was used to test the probability that the true mean error of each method was z e r o . If for any method this probability was 5 per cent or less, i t would be concluded that the method of estimating the parameter being tested was not v a l i d . The mean errors for the grids and triangular data-sets were compared, and the assumption that grid error is proportional to grid spacing was used to estimate the grid density which would be required to produce the precision achieved by the triangular data-sets. The hypothetical d i g i t i z a t i o n times and computer storage requirements of these hypothetical grids were then compared with those of the triangular data-sets using the time and storage estimates developed in the following section. Figure 6 . 1 : Sample 1 l a , an example of a triangular data-set from the Ptarmigan Creek map-area. -936.1: Digitization Time and Computer Storage Table 6 . 1 gives the numbers of points, boundary points, and triangles for the sets of surface-specific points used in this study. These a l l conform to the theoretical relationship given in equation 5 . 1 6 . Tests were made of the lengths of time needed to obtain the data from the topographic maps. For data to be punched on computer cards, the times cited are those required to first record the data values on a tape recorder and to then play back the tapes, writing the values on computer coding forms. The average time required to determine the elevation of a point was found to be 8.3 seconds — this should be the same for both surface-specific points and grid points. Drawing the triangular data-sets and numbering the points and triangles required 8.3 seconds per point, while an average of 8 . 0 seconds was needed to determine the vertices of each triangle. Measuring the x and y co-ordinates used an average of 12.6 seconds per p o i n t , but if should be possible to improve this considerably by using a d i g i t i z e r . Table 6 . 1 indicates that the average triangular data-set analyzed herein contained 114 points and 197 triangles. Digitization of such a data-set would theoretically require 946 seconds to draw the triangles, 1576 seconds to determine the vertices of these triangles, and 2382 seconds to d i g i t i z e the points, a total of 4904 seconds, or 4 3 . 0 seconds per point. For grids, only the elevations must be determined, and the 15 by 15 grids (225 points) should require an average of 1867 seconds. This means that the triangular data-sets required about 2 . 6 times as long to prepare as did the grids. The use of a digitizer in determining locational co-ordinates of surface-specific points should reduce this ratio somewhat. Equation 5.18 implies that the average triangular data-set stored in the "triangle m o d e " , would require 1389 half-words of computer storage a l l o c a t i o n ; -94- TABLE 6 . 1 : NUMBERS OF AND sample POINTS TRIANGLES N V BOUNDARY ( N ) FOR DATA-SETS c N B POINTS ( N ) B ANALYZED N C 8 90 25 153 11a 81 24 136 18a 119 29 207 24 142 32 250 31 114 28 198 41 138 34 240 means 114 29 197 lid 85 25 143 lib, c 29 15 41 18b, c 25 16 32 -95- the grids would require but 450 half-words. The surface-specific point d a t a - sets thus require about 3.1 times as much computer storage space as the grids. 6.2: Local Relief (H) Table 6.2 presents the estimates of local relief obtained from a visual inspection of the contours ("standard method"), from the 7 by 7 grids (d = 1000 m) used in Chapter 4 , and from the computer analyses of the 15 by 15 grids and the triangular data-sets. The results confirm those derived theoretically in section 5.1 — the triangular data-sets produce results very similar to the visual inspection method, while grid errors may be rather large. Theoretically, relief error should be a linear function of the grid spacing (d). For the six samples given here, the ratio of grid errors was somewhat less than one third when i t should in theory be one half. The difference may be fortuitous due to the random factor of distance from the extrema to the nearest grid points which influences the grid error, and to the small sample size. The t-tests indicated that none of the average errors were significantly different from z e r o , given the small sample size. Paired t-tests were used to determine whether the errors of the three estimates were significantly different from each other. A l l three pairs were significantly different at the 95 per cent l e v e l , meaning that while the grid estimates were not "significantly b a d " , the triangular data-sets produced errors significantly less than those of the grids. 6.3: Mean Slope (tan c< ) The results of slope estimation using four methods are given in Table 6 . 3 . Here, the value obtained using the line intersection method of Wentworth (1933) is the "standard" to which the computer estimates are compared (see section 5 . 2 ) . Once a g a i n , the t-tests indicated that none of the mean errors differed significantly from zero. Paired t-tests showed that the triangle and grid estimates were -96TABLE 6 . 2 : ESTIMATES OF LOCAL RELIEF ( H ) , A N D ANALYSIS OF ERRORS I N THESE ESTIMATES standard method 7x7 grid 1 5 x 15 grid triangular data-set 8 1880 1655 1853 1865 11 1709 1590 1606 1709 18 883 619 823 787 24 203 184 187 203 31 1195 1057 1192 1195 41 869 752 823 869 - 225 27 15 119 103 0 264 60 5 - 19 16 0 138 3 0 117 46 0 147 42.5 3.3 87 36.0 6.1 0.689 0.482 0.224 52% 66% 82% sample error (e): 8 11 18 24 31 - 41 e e t s p ( e = 0) -97TABLE 6 . 3 : ESTIMATES OF M E A N SLOPE ( t a n t * ) , A N D ANALYSIS OF ERRORS I N THESE ESTIMATES Wentworth method sample 15 x 15 grid NW-SE NE-SW triangular data-set 8 0.609 0.523 0.518 0.585 11 0.395 0.344 0.358 0.355 18 0.396 0.331 0.335 0.393 24 0.063 0.041 0.039 0.048 31 0.218 0.187 0.185 0.203 41 0.400 0.324 0.336 0.381 0.086 0.091 0.024 11 - 0.051 0.037 0.040 18 - 0.065 0.061 0.003 0.022 0.024 0.015 0.041 0.033 0.015 0.076 0.064 0.019 e 0.055 0.051 0.019 s e t 0.025 0.025 0.012 0.896 0.843 0.639 42% 44% 56% error (e): 8 - 24 31 41 P (e = 0) = -98significantly different. As expected, the two slope estimates obtained from the same grid using different diagonals were not significantly different at the 95 per cent l e v e l . 6.4: Roughness Factor (IR) As noted earlier in section 5 . 3 , there exists no useful "standard" value of the roughness factor to which computer estimates can be compared. Table 6 . 4 presents the results of unit vector analysis of grids (the method used by Hobson, 1967, 1972, and by Turner and M i l e s , 1967), of weighted vector analysis of grids, and of weighted vector analysis of triangular data-sets. The similarity of columns 1 and 2 (also of 3 and 4) in the Table supports Turner and Miles' contention that the orientations of the diagonals used to form the triangles has l i t t l e effect on the results. In the absence of a standard v a l u e , the claim made above in Chapters 3 and 5 , that the weighted analysis of triangular data-sets should yield the best results, cannot be substantiated empirically. The five sets of values given in Table 6 . 4 were not significantly different from each other. 6.5: Hypsometric Integral (HI) Table 6 . 5 presents the results of hypsometric analysis of the six study areas using six different methods. The standard values were obtained through the use of a polar planimeter, while the second and third columns report results obtai ned from grids in Chapter 4 . HI, based on the best available estimates of the minimum and maximum elevations, follows the approach recommended by Pike and Wilson (1971); H I * , as w e l l as the results for the 15 by 15 grids, used the grid estimates of these quantities. Pike and Wilson claimed (p. 1081) that 40 to 50 points w i l l generally produce results w i t h i n 0.01 of the true values. This claim is supported by the fact that the mean error produced by their method is 0 . 0 0 6 , and in none of the six cases did the error reach 0 . 0 1 . -99- TABLE 6 . 4 : sample ESTIMATES OF ROUGHNESS FACTOR (R) 15 x 15 grid weighted vectors unit vectors NW-SE NE-SW NW-SE NE-SW triangular data-set 8 11.36 11.23 11.90 11.85 13.80 11 6.76 6.93 7.23 7.29 8.12 18 5.75 5.79 5.96 5.97 7.37 24 0.13 0.13 0.14 0.13 0.24 31 2.27 2.26 2.44 2.43 2.92 41 5.80 5.82 6.10 6.11 7.61 means 5.35 5.36 5.63 5.63 6.68 -100TABLE 6 . 5 : ESTIMATES OF HYPSOMETRIC INTEGRAL (HI), ANALYSIS AND OF ERRORS I N THESE ESTIMATES 7 x 7 grid HI* 1 5 x 15 g r i d * NW-SE NE-SW triangular data-set standard method H 8 0.432 0.428 0.479 0.436 0.436 0.447 11 0.260 0.265 0.284 0.279 0.281 0.263 18 0.547 0.546 0.429 0.566 0.567 0.542 24 0.278 0.271 0.258 0.289 0.297 0.268 31 0.338 0.334 0.278 0.334 0.334 0.337 41 0.395 0.403 0.371 0.420 0.420 0.404 - 0.004 0.047 0.004 0.004 0.015 0.005 0.024 0.019 0.021 0.003 - 0.001 0.118 0.019 0.020 0.005 0,007 0.020 0.011 0.019 0.010 0.004 0.060 0.004 0.004 0.001 0.008 0.024 0.025 0.025 0.009 e 0.006 0.049 0.014 0.015 0.007 s e t 0.002 0.038 0.010 0.010 0.005 0.795 0.529 0.559 0.622 0.557 46% 62% 60% 56% 60% sampl e error (e): 8 11 18 24 - 31 41 P (e = 0) * these estimates are based on grid values only -101- The best of the computer estimates, those based upon the triangular data-sets, had a slightly larger average error, with the grid estimates considerably poorer. As in the cases of local relief and slope, the t-tests indicated that none of the mean errors differed significantly from zero. Unlike those parameters, however, only one of the 10 paired t-tests indicated a marginally significant difference in average errors — that was between Pike and Wilson's method and the estimate obtained from the 15 by 15 grids using the northeastsouthwest diagonals. 6.6: Comparison of Errors for Triangular Data-sets and Grids In Tables 6 . 2 , 6 . 3 , and 6 . 5 , the errors in estimating local r e l i e f , mean slope, and hypsometric integral using both regular grids and triangular data-sets were g i v e n . Table 6.6 repeats these error values and gives the ratios between the estimate errors for the two methods. According to the "Sampling Theorem" introduced in section 2 . 2 , error should be proportional to the grid spacing and the relationship should be linear. This was confirmed theoretically for two of the above three parameters in Chapter 5 . If this is applied to the grid errors noted above, one finds that in order to reduce the grid error in the estimation of local relief to the level of precision achieved by triangular data-sets, one would need to reduce the grid spacing from 500 m to 39 m. This and the values for the other two parameters are listed in Table 6 . 6 , as are other characteristics of these hypothetical grids. F i n a l l y , the values developed above in section 6.1 are used to estimate the relative digitization times and computer storage allocation requirements of these grids compared with those of triangular data-sets. For a l l three parameters, i t appears that a given level of precision can be attained with less digitization time and computer storage space using surface-specific points than using regular grids. The contrast is much more dramatic for local relief than it is for the other two parameters. -102TABLE 6 . 6 : EMPIRICAL COMPARISON DATA-SETS A N D OF ERRORS FOR TRIANGULAR 15 BY 15 GRIDS H tan <x HI 42.5 0.053 0.015 3.3 0.019 0.007 12.9 2.8 2.1 39 179 234 Mean errors: 1 5 x 1 5 grids triangular data-sets ratio Characteristics of grids theoretically required* to produce same precision as triangles: d (metres) grid size 181 x 181 40 x 40 31 x 31 ^ of grid points 32,761 1,600 961 d i g i t i z a t i o n time 55.4 2.7 1.6 storage space 47.2 2.3 1.4 Ratios of requirements of such grids to those of triangular data-sets: * assuming a linear relationship between grid error and grid spacing -1036.7: Reproduceability and the Influence of Scale As noted above, samples 11 and 18 were selected to investigate the influence of scale. For each area, an additional data-set was derived from a 1:250,000 scale map of the same area (samples l i b , 18b); next, approximately the same points were located on 1:50,000 scale maps (11c, 18c). Sample 11 was also used to examine the reproduceability of triangular data-sets by producing another such data-set of that area (sample 1 Id) w i t h approximately the same number of points as sample 1 l a . The number of points and triangles in a l l of these data-sets were given in Table 6 . 1 . In separate analyses of samples 11 (Table 6.7) and 18 (Table 6 . 8 ) , values of the four selected parameters were standardized, and the distances between sub-samples in the resulting fourvariable "phase space" were calculated. For sample 1 1 , the most similar pair was a and d , the two with similar numbers of points and triangles derived from the same map. N e x t were the distances between these and sub-sample c , derived from the same scale of map but using many less points and triangles. The most "different" data-set was 1 l b , derived from a smaller-scale map w i t h a larger contour i n t e r v a l . It seems that for this area, map scale differences are more important than the number of triangles used. the opposite conclusion was reached. For area 18 (Table 6 . 8 ) , In this case, the most similar pair was b and c , the two sub-samples with similar and lesser numbers of points derived from maps of different scales. The greatest difference was between a and b, which were from different maps and which also used different numbers of points and triangles. It would appear that for area 18, the number of triangles, or perhaps more correctly the mean size of the triangles, is more important than the differences between the 1:50,000 and 1:250,000 scale maps. Because the triangles in 18b and 18c were too large, the topography was smoothed and slopes reduced (see R and tan o< values in Table 6 . 8 ) . -104TABLE 6 . 7 : SIMILARITY A M O N G ON SAMPLE H FOUR TRIANGULAR DATA-SETS BASED 11 FOR THE FOUR SELECTED MEASURES HI tan o< R numbers of triangles points O r i g i n a l values: a 1709 0.263 0.355 8.12 81 136 b 1661 0.285 0.378 8.07 29 41 c 1709 0.265 0.367 7.59 29 41 d 1709 0.262 0.371 8.31 85 143 mean 1697 0.269 0.368 8.02 24 0.011 0.008 0.306 s Standardized values: a 0.500 -0.550 -1.684 0.326 b -1.500 1.467 1.295 0.163 c 0.500 -0.367 -0.130 -1.404 d 0.500 -0.642 0.389 0.947 Inter-pair differences: distance* rank a-b 2.000 2.017 2.979 0.163 4.119 (6) a-c 0.000 0.183 1.554 1.730 2.333 (2) a-d 0.000 0.092 2.073 0.621 2.166 (1) b-c 2.000 1.834 1.425 1.567 3.442 (5) b-d 2.000 2.109 0.906 0.784 3.144 (4) c-d 0.000 0.265 0.519 2.351 2.422 (3) * this is the distance between the samples in the four-dimensional space whose axes are the four variables Sub-samples: a , d — 1:50,000 scale, small triangles; b — 1:250,000 scale, large triangles; c — 1:50,000 scale, large triangles. -105- TABLE 6 . 8 : SIMILARITY A M O N G ON SAMPLE DATA-SETS BASED 18 FOR THE FOUR SELECTED MEASURES HI H THREE TRIANGULAR tan °< R numbers of triangles points Original values: a 878 0.542 0.397 7.39 119 207 b 838 0.517 0.257 3.45 25 32 c 823 0.544 0.295 4.53 25 32 mean 846 0.534 0.316 5.12 28 0.015 0.072 2.037 s Standardized values: a -1.126 -0.532 -1.119 -1.114 b 0.281 1.130 0.290 0.820 c 0.809 -0.665 0.815 0.295 distance Inter-pair differences: rank a-b 1.407 1.662 1.309 1.934 3.193 (3) a-c 1.935 0.133 1.934 1.409 3.080 (2) b-c 0.528 1.795 0.525 0.525 2.013 (1) Sub-samples: a — 1:50,000 scale , small triangles; b — 1:250,000 seal e , large triangles; c — 1:50,000 scale , large triangles. -106- Because only two areas were investigated, no strong conclusions can be made regarding the results and the difference between the areas. The writer proposes the following as a possible explanation for the results obtained. First of a l l , the reduction in the number of triangles was more drastic for sample 18 than for sample 11 — sub-samples l i b and 11c had about one third the number of triangles as did 11a, w h i l e 18b and 18c had only about one sixth the number in 18a. Secondly, the topography of area 18 was more complex than area 11 . This can be seen by a visual inspection of the maps in Appendix I I I , and is reflected in the fact that 52 per cent more triangles were used to characterize sample 18's topography in the basic triangular data-sets (see Table 6 . 1 ) . It is proposed that for area 1 1 , the topographic texture was sufficiently large that the larger triangles in sub-sample 11c were able to retain most of the "terrain information" present in sub-samples 1 l a and 1 I d . Differences between the map scales due to contour generalization and the larger contour interval thus predominate, making sub-sample l i b the one most distant from the others in its terrain parameters. For area 18, the finer topographic texture and larger triangles combined to make the influences of map scale relatively less important than that of the reduced number of triangles. These proposals should be tested by further investigations which are beyond the scope of the present study. 6.8: Summary Empirical tests were used to estimate the digitization times required for triangular data-sets and for regular grids. It was estimated that 4 3 . 0 seconds per point are required for the former and 8 . 3 seconds per point for the latter. The average triangular data-set would require about 2 . 6 times as long to prepare as the 15 by 15 grids used for comparisons. Theoretical considerations indicate that the former data-sets would need some 3.1 times as much computer storage space as would the grids. -107- Errors in the estimates of local r e l i e f , mean slope, and hypsometric integral were discussed, and various estimates of the roughness factor were given. In every case, the triangular data-sets based on surface-specific points gave better results than the 15 by 15 grids. Triangular data-set errors for local relief and mean slope were significantly less than those of the grids, as determined using the t-statistic. For the hypsometric i n t e g r a l , the grids produced a higher average error but the difference was not significant at the 95 per cent l e v e l . The hypothetical linear relationship between grid error and grid spacing was used to estimate the grid spacing required to equal the precision of the triangular data-set estimates of the three parameters. The d i g i t i z a t i o n times and computer storage requirements of these theoretical grids were determined, and for a l l three parameters the triangular data-sets required less time and space than did the grids. An investigation of the reproduceability of triangular data-sets and the influences of map scale and triangle size was conducted. The reproduceability was good; the relative importance of map scale and triangle size appears to be related to the complexity of the terrain. If the triangles are too large, the topography is smoothed and the effect of map scale becomes less important. Further work w i l l be required to test and quantify this proposed relationship. -108- Chapter 7: Summary and Conclusions General geomorphometry is ro be preferred over a specific approach because i t does not depend upon any single geomorphic process nor on the identification of specific types of landforms. It is therefore more applicable to arbitrarily-bounded terrain samples stored in an electronic computer. After a brief discussion of map precision and notation, approaches to computer terrain storage were discussed. This subject was reviewed in terms of digitization (data gathering) methods, actual computer storage and retrieval techniques, and assumptions about the behavior of the land surface between data points. In surface-specific sampling, points are selected which have particular significance in the topographic form — these include peaks, pits, and passes, and points along ridges and valleys. In the surface-random approach, the points are selected according to criteria independent of the surface; usually either the locations of the points are determined by some type of g r i d , or the elevations of the points to be recorded are defined (contour sampling). Completely random sampling does not appear to produce as good a representation of a surface as does the stratified random approach represented by a g r i d . G e n e r a l l y , grids require much less computer storage a l l o c a t i o n , since only one co-ordinate (the elevation) must be stored for each point. Digitized contour points require two co-ordinates, while for surface-specific points a l l three must be specified. In a d d i t i o n , the neighbours of a grid point are i m p l i c i t in its position w i t h i n the computer array, w h i l e these must be e x p l i c i t l y indicated for surface-specific points, requiring still more computer space. An arbitrary assumption about the behavior of the land surface between points is usually made; in the absence of evidence to the contrary, the linear assumption is generally the most reasonable. In the present study, the surface-specific point and regular grid approaches to computer terrain storage were compared w i t h reference to the problem of estimating some selected geomorphometric parameters. -109A large number of landform measures were reviewed, and were found fo belong to a number of basic groups. These were texture and g r a i n , r e l i e f , slope, dispersion of slope magnitude and orientation, and hypsometry. If was decided to select one parameter from each of these classes, but none of the grain and texture measures were readily adaptable to the computer methods used. Texture and grain were i m p l i c i t in the sample area size and the density of sample points. The four parameters examined e x p l i c i t l y were local relief ( H ) , mean slope (tan<x), roughness factor ( R ) , and hypsometric integral (HI). In order to select some areas for detailed analysis and to provide data for assessing the theoretical errors in the estimates of some parameters, f o r t y - t w o 7 by 7 km squares were selected from 1:50,000 scale maps of southern British Columbia using a stratified random sampling design. For each of these areas, local r e l i e f , hypsometric i n t e g r a l , mean slope, drainage density, stream source density, and peak density were estimated using manual methods. Relationships among these variables and their estimates were examined during correlation analysis. Relief, hypsometric i n t e g r a l , and peak density were used to divide the f o r t y - t w o samples into fifteen "terrain types". A stratified random sample of six areas was derived from these to provide a basis for the comparison of the computer methods, and the geomorphology of each of the six areas was briefly described. Theoretical errors involved in estimating the four selected parameters both from the triangular networks based upon surface-specific points and from regular grids were discussed, as were the actual analysis procedures employed. For local relief and the hypsometric integral at least, the precision of the grid estimates should be linearly related to the grid spacing; this probably holds true for the other measures also. The possibility of estimating other parameters, the r e l a t i o n - ships among the selected variables and between the numbers of points and triangles, and the theoretical computer storage requirements of the methods were also reviewed. -110- F i n a l l y , the results of the analysis of the topography of the six samples using the two approaches were reported. Sample 11 was used to investigate the reproduceability of the surface-specific sampling,and this and sample 18 used to study the effects of the numbers of points and the scale of the maps used. It was found that the relative importance of map scale and triangle size appears to depend upon the topographic texture. For coarse texture, map scale is more important, while for finer texture, the size of triangles used becomes dominant. This hypothesis should be tested by further research. The triangular data-sets were found to produce better estimates of the parameters than the regular grids, even though the latter averaged more than twice as many points. The average surface-specific point data-set required some 2.6 times as much d i g i t i z a t i o n time and 3. 1 times as much computer storage space as did the 15 by 15 grids. The theoretical linear relationship between grid error and grid spacing was used to estimate the grid density required to equal the precision of the triangular data-sets. These hypothetical grids would require much more time and storage space than would the data-sets based on surface-specific points (see Table 6 . 6 ) . In conclusion, i t appears that superior estimates of geomorphometric parameters can be obtained from triangular data-sets based on surface-specific points. Grids which would produce a comparable level of precision would theoretically require more d i g i t i z a t i o n time and computer storage space. For a reasonably experienced terrain analyst, the triangular data-sets appeared to show good reproduceability; further investigation w i l l be required to determine whether comparable results can be obtained using workers w i t h less training and background. Such workers should be able to produce good results using grid sampling, since this approach lacks the subjective element involved in the selection of the surface-specific points. -IllReferences Abrahams, A . D . , 1972, Drainage densities and sediment yields in eastern Australia. Australian Geogr. Studies, v . 10, p. 1 9 - 4 1 . 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B . , 1957, The dispersion of geomorphic data around measures of central tendency and its a p p l i c a t i o n . Quartermaster Res. and Eng. Command, U.S. Army Study Rept. E A - 8 . , and , 1959, Predictive methods in topographic analysis I. Relief, Slope, and Dissection on i n c h - t o - t h e mile maps in the United States. Quartermaster Res. and Eng. Command., U.S. A r m y , Tech. Rept. EP—112. -120, and , I 9 6 0 , A quantitative system for classifying landforms. Quartermaster Res. and Eng. Command, U.S. A r m y , Tech. Rept. EP-124. Young, A . , 1972, Slopes. Oliver and Boyd, Edinburgh, 1972, 288 pp. Zakrzewska, B., 1963, An analysis of landforms in a part of the central Great Plains. Annals Assoc. A m . G e o g . , v . 5 3 , p. 536-568. -121- Appendix I: Notation In this appendix, a l l variables and symbols used in the text are listed, together with their meanings or definitions. Exceptions are standard abbreviations (such as " M " for metres) which are not listed. For each entry, the text section where the sumbol first appeared is indicated in parentheses. a(h) the relative hypsometric function ( 3 . 6 . 1 ) b average distance between adjacent ridges and valleys ( 3 . 4 . 3 ) c a distance measure (2.5) D a density value (1.2) d D D drainage density ( 3 . 2 . 2 ) peak density ( 3 . 2 . 4 ) P D s d stream source density ( 3 . 2 . 3 ) E e l e v a t i o n - r e l i e f ratio ( 3 . 6 . 2 ) e H e HI grid spacing (2.2) error in the grid estimate of local relief ( 5 . 1 . 2 ) relative error in the grid estimate of the hypsometric integral ( 5 . 4 . 2 ) error in the grid estimate of the maximum elevation ( 5 . 1 . 2 ) e max e . mm f error in the grid estimate of the minimum elevation (5.1 .2) factor by which one wishes to improve grid accuracy (2.2) G grain of topography ( 3 . 2 . 1 ) H local relief ( 3 . 3 . 1 ) H* grid estimate of local relief ( 4 . 3 . 3 ) H available relief ( 3 . 3 . 2 ) H a d drainage relief ( 3 . 3 . 3 ) HI the hypsometric integral ( 3 . 6 . 2 ) H* grid estimate of the hypsometric integral ( 4 . 3 . 5 ) h relative height ( 3 . 6 . 1 ) h mean relative height ( 3 . 6 . 2 ) -122- I the contour interval ( 3 . 4 . 1 ) k the vector dispersion factor (3.5) L total length of traverse lines used in line sampling estimates of slope ( 3 . 4 . 1 ) or drainage density ( 4 . 3 . 1 ) L(%) vector strength in per cent (3.5) N number of objects or occurrences (1 .2) N^. number of cells or triangles in network (5.6) N- number of edges in network ( 5 . 6 ) Ny number of vertices or points in network (5.6) P length of drainage basin perimeter ( 3 . 2 . 2 ) p mean annual precipitation ( 4 . 3 . 6 ) R length of vector sum (3.5) IR roughness factor (3.5) r correlation coefficient ( 1 . 2 ) S size or wavelength of smallest features one wishes to detect (2.2) s a root-mean-square value (1.2) Sj root-mean-square distance ( 5 . 1 . 2 ) s root-mean-square error (1.1) g T texture ratio ( 3 . 2 . 2 ) t year of map publication ( 4 . 3 . 6 ) V volume of landmass ( 3 . 2 . 4 ) v variability factor (3.5) W highest frequency present in a function ( 2 . 2 ) x, y geographic location co-ordinates (1 .2) z altitude above sea level (1 .2) z mean elevation ( 3 . 6 . 2 ) z max maximum elevation ( 3 . 6 . 1 ) -123z* max grid estimate of z z minimum elevation ( 3 . 6 . 1 ) ' . mm max (5.4.2) z* . mm grid estimate of z . mm (5.4.2) oc mean ground slope ( 3 . 4 . 1 ) p exponent in the general interpolation formula (2.5) t slope near the maximum or minimum point (5.1 .2) % land slope at a point (1.1) 9 angle of intersection between a traverse line and a contour or stream (3.4.1) K precision parameter for Fisher's spherical probability distribution (3.5) -124- Appendix I I : Topographic and Related Variables for 42 Areas in Southern British Columbia In the following Table, the values for twelve terrain and related variables from the f o r t y - t w o Chapter 4 are g i v e n . 7 by 7 km topographic samples examined in The parameters were listed in Table 4 . 3 , and are also included in Appendix I. indicated by the symbol The six areas analyzed in detail in Chapter 6 are while the highest and lowest value for each parameter are marked w i t h the symbols " + " and " - " , respectively. The mean and standard deviation for each variable, and the units of measurement, are indicated at the bottom of the Table. A l l of the values reported in this Table are based on the exclusion of water surfaces from the calculations. If these were included, mean slope would be reduced for those areas including lakes or the ocean, and the values of some of the other parameters would also be influenced. i -125- Area D d N/L D D s P H H* tan « H H* £ P t 0.540 0.552 0.481 0.259 0.343 1650 1818 951 1965 1046 40 40 15 45 16 64 61 3162 62 0.339 0.361 0.479 0.451 0.458 1113 1143 1683 2077+ 1967 20 55 80 30 70 58 60 61 59 66 1148 1786 91661 554 30 40 40 100 140 65 60 51 38 39 546 1169 1534 130 120 50 18 130 47 62 57 0.296 0.418 0.184 0.368 0.562 0.102 0.163 1247 884 0.167 0.602 0.388 0.041 0.055 0.224 0.408 0.694 0.417 0.653 963 329 1268 835 643 235 1113 0.405 0.091 0.462 728 0.169 0.568 0.343 0.2400.299 0.286 0.459 0.347 0.388 0.306 0.0000.224 0.102 0.163 0.167 0.184 0.122 0.102 0.326 0.333 939 1694 1880 1387 1740 820 1636 1655 1159 1423 0.241 0.445 0.609 0.527 0.504 0.364 0.352 0.428 0.405 0.347 0.317 0.510 0.837 0.122 0.102 0.143 0.408 0.333 0.122 1709 1405 375 1012 0.250 1326 1590 1207 287 938 1201 0.395 0.460 0.110 0.381 0.594 0.265 0.512 0.242 0.494 0.417 0.284 0.434 0.233 15 0.549 0.476 0.714 1.231 0.937 16 17 1.510 0.492 18' 1.847 0.676 0.684 1.020 0.337 1.163 0.378 0.459 0.306 0.796 0.286 0.163 1015 1408 883 1905 991 1143 619 1814 0.403 0.694+ 0.395 1945 1610 0.496 0.601 0.533 0.562 0.429 0.513 0.408 0.539 0.529 0.546 0.513 0.314 0.563 1.322 0.720 0.631 0.043- 0.357 0.250 0,-837 - O r 821 0.398 0.291 0.459 0.191 0.0310.000- 0.568 0.359 0.521 0.383 0.041- 686 1175 920 203366 567 -1128 832 174262 0.299 -0.376 0.342 0.064 0.042 0.251 0.272 172 70 0.286 0.371 0.281 0.309 0.297 0.365 0.270 0.233 426 341 967 1021 120 90 16 14 49 34 60 58 65 26 0.408 0.235 585 214 0.343 0.494 61 28 29 0.450 0.607+ 0.492 20 0.286 0.439 0.368 0.306 0.184 0.367 0.531 635 0.478 0.724 0.637 0.020 0.163 0.184 0.102 773 27 968 831 16 12- 65 55 0.227- 1201 15 30 0.819 0.480 0.653 0.224 0.601 1220 60 60 61 31' 0.290 0.657 0.326 0.735 2.051+ 0.255 0.106 0.104 0.041 0.553 0.521 1480 1141 23 68+ 0.490 1.408 0.409 0.268 0.485 0.344 799 1187 176 15 17 0.500 1.316+ 0.204 0.429 0.306 0.348 0.562 0.272 0.392 0.459 0.194 67 60 60 36 0.657 0.125 0.256 0.267 0.167 0.271 0.977+ 0.511 0.479 0.252 0.280 0.382 0.374 239 211 0.496 0.661 0.418 1.023 0.357 0.306 0.398 0.286 400 182 70 70 70 64 41' 42 0.882 0.569 0.531 0.316 mean 0.725 s 0.401 . unite km ^ 1 2 3 4 0.388 0.661 0.292 0.639 5 0.840 6 7 8# 9 10 0.482 n' 12 13 14 19 20 21 22 23 24# 25 32 33 34 35 36 37 38 39 40 0.776 0.555 0.586 0.508 1.420 0.531 0.581 0.250 1.089 0.583 1.510+ 0.265 1.429 0.163 0.061 287 274 0.262 0.056 0.042- 0.377 0.526 0.445 0.380 1129 754 58 65 296 2122 183 269 1972+ 1195 1054 0.225 338 272 854 613 329 192 707 471 0.101 0.043 0.257 0.256 643 0.334 0.322 860 464 0.157 0.393 0.373 0.582 0.286 738 716 655 1012 495 0.200 0.347 0.430 0.349 0.673 0.467 - 0.490 0.422 869 968 752 741 0.400 0.519 0.403 0.434 0.371 0.456 380 420 100 160+ 61 61 0.465 0.360 0.370 978 838 0.329 0.390 0.408 937 60 58 0.265 0.413 0.195 526 488 0.189 0.109 0.107 566 42 9 km ' km ^ m m km 550 559 0.060 0.607 0.540 0.243 0.562 0.386 0.559 0.302 186 m 47 105 90 125 61 61 64 64 inches y -126- Appendix I l i a : Computer Program In this appendix, the FORTRAN program used in this study (program IV program listing for the computer GEOTRI) is g i v e n . Requirements for the input data are contained in comments at the beginning of the program listing. -127: ORTRAN I V G COMPILER MAIM 02-06-74 0001 P A G E 1 0 : 1 9 : 3 2 rj**** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C £ C C C P R O G R A M : G E O T ° I p;[gon<;c; T I npTPP^ I N E A N A L Y S I S O P O S S I B L Y BfiS n W R I T T E N : T s c c D A V I D M „.M 0 C C. C C T N P L ' T : E A C H Di C A R T : JOB ^9 F A C E T S T T S .._CE^3^ ft C O L 3-5 A P H Y e g : n = : 1 o NEW =0: 10 T I T L E C A R D , ,113, P A T F O L L O W E D B Y D A T A . A 15 JF»n » . ! = W T ^ Q : C X S C A L E : 16-27 ' C N U " B E P . : . X - ° n Y S C & L E : ? 8 - 3 ° r. C A S O c U E T P E S C - S C A L E : 40-51 O . 56 C £ A L L T R I A N G L E S . A M DO C E A N 66 I _ H G = 0: =•-" : C P A R A M E T E R S U S E T P JA N G L E S C C 67-71 N'> : NUMBER O r_ 77-^6 NC t NU'-'BEP H F T I T L E , M A X I M U M T PI A N G L E So T O T R I A N G L E . P R O D U C E . D I A G O N A L S , ROWS C IN G R I D , P P D M U S E N E - S W =3: X S C A L E * F Q D E A C H D I A G O N A L S N W - S E . F ^ R M C T F L I S T , T P , I _ N ~ L E ..LIST I " N E U N I T L A K E ,.C N A'JLTS...T?...I , C A N A L Y Z E OMIT J C 8 T I n g T n E X C L U D E I T R T ^ Q ; C : T T Q\J . =1: 6T V I " i ! , ' S MNI FJ ? o5 c IN . 3 E = c P i _ o L A K E = 0: .. J O B S P A C I N G N. D E F A U L T S M E T R E S C M T H P . Y-D I R Z-0.H..EC.T IONo C C.PL'J D E F A U L T S C S P R NON F ^ rio M A T : A B O V E , MUMPER : A I _ R D A T A . R O WS P A C I N G , - P R E V I O U S N E W T C I A N G L E S TP. I A M GL E S C S P E C I F I E D . C A S " p y f U T F ECTI.OS!, O R 10 C P O I N T S R U S E S A M =1: C l P O I N T S P * T A U S E SA'-'E =1: C NFrf R E A P 3 C . I P S U R F A C E * n C ' p T H E = 9?..9..:._E.ND.0- . p U N , . . . ( F . D L L W S _ L A S T . . J O B ) C C A N D A ; T H E " V T F P S C _1_9.I1 i P F E - T R I A N G U L A T E O I T Y 3 F = 0 : C P A R A " A 3 0 R D X I ^ A T E , „ U » B . C». J O BC A P O | T; T P W H I C H 0*.TA O N G R I D Ap-K J O BR E Q U I R E S T H E nMppPHpMPTgTC S F L E T T E ™ AMCLiL I I N M A T R I X . C O L U M N S I N M A T ? IX C C TITLE C . C C C C C A R T : QgSIPEP, C C A PQE-TPI A N G M L A T F ^ - D A T A P O I N T S , NUMBER LS.....U., 7.E.6. - L A S T —TP I A N G L E S , D A T A C MtjMqco C W H I C H C C IS N O T I T L E C O C E A N , -LAS_T_ G R I D 7.F0 \ I F O N A ( N R ) IS S C ROWS n p c T H R E E ( F OP M o M r BY C A R D H A V I N G A P O I N T A S B L A N K H A V I N G W R I T T E N AT r. A<? n T P I A N G I . E V E R T I C E S , T=iAMGt.E c A I S O N A A S W R I T T E N , N B Y _ _ A _ B_L A N < " L " , D L A K E O 3 . T H E 515) CA F D 1: S r P A F . A T P aL 1 5= c, 01 T ^ O L L C W E D E A C H 1 • £ c MA S I . B E . _ _ Q L L O W E D ( I T Y ? E = ? =0W <*E O T H E R W I S E . T R I A B L E D A T A - E A C H MUST F E D U I ' - E O , MijM^rjc E Q U A L S C A * 0 _.__ P O I N T C E A C H A N DZ - C O - O R D I M A T E S . i F O R M A T 0)_ I ( I T V O E = 0 ) : R E Q U I R E D , Y , P p r ^ T t D A T A I R A N D X , C £ I DATA: C C F 76 C H A R A C T E R S . MUST Be J.MJ_EJ_T.ED_»_ O C A R P BL.AN< K Z* R C A P O , T S ? N C I F I E O M ( F O R M A T C L U P r P , O N T H E J O B AS 4 ' I T T r . v ; S I N C F M ' ) " p r 3x, ? p p C A R D . C £1 C A5_JlQT£0_AeDVE.., CCLU NS M 3-5 LAST. J ? B . I . _F L! r TO END THE OWED..."Y_.A__C R U N , A.RD..W I.TH_ _ « o o ? " _ _ T . M . C r. R F ^ T R T C T T O N S : A S WF. T T T O I . . T H F P F A R F T H FF O L L O W I N G R E S T R I CT\ " » y ^ ; . - : ORTRAN IV ^ 8 - : G CQM°ILER MAIN -MAXIMUM C 02-06-74 Nl.iM?E c -MAXIMUM N i . l ' ^ E ' —M ATP i x MAY N O T C C j__ (•'.): -IF C C £ 0 ""IVTC C 10:19:32 : (\<p) i< PAGE 0002 CQO; 0 - T3 IA NGl.r S ( N T ) N A V E ' O R E T H A N ?Q iQOO; T C ROWS (NR) DIMENSIONS ' ^ E INC-EASE3, KFF I N MI \j D - ( N R - 1 ) * (VC-1 ) CANNOT E X C F Q MO; _ N T _ S H O U L D f*E ? I E N $ I O N T H 2 * N P » D 3? '5 COLUMNS THAT: C U C " ~ £********* ************************ *^ ************ ™ DIMENSION OOOl. " X ( 5 0 0 ) , Y ( ? 0 0 ) , I ( 5 0 0 ) , I T ( ? 0 0 0 ) , I V < 1 0 0 0 ) , J V ( 1 CO 0) , K V ( 1 000 ) 1 , L V ( 1 0 0 0 > , T I T L E ( 1 ) , SU«(12) C T ' M O N . . X , Y , Z, I T f J V . K V . L V , T I T L E ,NP,NT, L A K E , I TR I RAT-O,01 " ^ 5 3 2 9 3 JT?S =0 CALL REAPER I F ( L AK Eo E Q « 999) GO T O 999 ALAKE=0. NL =0 : . , 3 iy.j 000 2 0003 0004 0005 0006 0007 000 S 99 0009 Z AX=-99^90„ M 0010 0 011 ZMIN=99999. T?MAX=3 0012 0013 : IZMIM=0 . DO 3 1 = 1 , 1 2 001 4 'J M ( .S 00: 5 3 0016 001 7 0018 0019 0020 I..) = 0 o C O N T IN',jc . I PI I TRT , E 0 . 0) r,0 TO . . 94 W R I T E ( 6 • 1 52 ) ( T I L E ( I ! ) t I T = ) , 1 9 ) 1 5 2 F O R M A T ! • 1. • r l ° A 4 , / / ) WRITE(6,500) 5 0 0 F Q R" AT ( ' NO, CORNERS L 1 AR A S L O P E ' ,/) 94 DO f- I = i , NT I F ( Z ( ! V ( I n . L E o Z V f t X ) GO T _ 6 0 Z MAX= Z( I V ( I ) ) I Z M » . X = I V ( I) T T.rilS. 7. ' t I P X t ' MAP M r 0021 . 007? 0023 0024 002 5 50 0026 0027 0028 0029 ______ 0035 0036 .0037 003 3 0039 Q040 0041 0042 Q043L OC-4 0045 J3Q4_6 • I.EIOJ.V.U. M • LE ...Z MAX ) S O T ] 6 1 Z A X = Z ( .)V ( I ) ) IZMAX=JV(I) T ( 7 t K V ( ! ) ) _ L E _ ZM-X ZMAX = Z ( K V ( I ) ) M Si 0030 .0011 003? • 0033 ; 62 " f>3 ' 6„ 65 c ) GO '> 62 T IZM/>x=K V( I I I F (.7 ( I V (I )j , GEo Z M I N L ^ 0 . _ . . : Z M I V •= Z ( IV (I )) IZMIN=IV<I) T T P ( 7 1 . I V ( T ) ) _ r , E . 7 'TNH G " T L 3 . 0 64 ; Z M I V = Z ( J V ( I )) IZMTN=JV(I) I (I (KV ( I ) ) . O G . Z M I.M..) ...G.0._I?...3_5. . c ZMIN = Z ( K V ( I U I Z M I N = KV( I ) C O N T IN') F VFC1 1 = X(J V ( I ) ) - X ( I V F C i 2=Y( J V ( VEC13=Z( JV( . : V(I)) ! ) )-Y( I V ( I) ) 11 J.- Z ( IV I I ) ) V £ C 2 1 = X ( K V ( I) ) - X ( I V ( ! ) ) V E C 2 2 = Y ( * V ( I ) ) - Y ( I VI I)) __FC? 3 =7 [ KV \ I ) . , ) . - L I V LI) ) _ : . . _ . -129OUTRAN 0047 IV G COMPILER C 02-06-74 MAIN VEC1 AND VEC2 ZM = ( Z ( I V ( ! ) ARE V )+Z(JV< C PNG C T P R S I ) ) +?( " V , 0049 0050 C v_=VECI3*V C I j TH~ THE 0051 J3_5_2_ O P * 3- THE TRIANGLE C ? l . - V E C 5 1"*V C_ 3 C CR c s S . . TR [ A N G L O O 2"*V EC 2 1 " T J c~. o E _ y F_C 1 ANP..XEC ?• AND I S . .T H P T GO TP 4 . )*'P V2 = < - 1 . ) * V 2 V3 = ( - 1 . ) * V 3 0054 HQ5.5. S.P N T . I . N ' J E THIS MAKES THF VECTOR VL = S Q R T ! V 1 * V ? + V 2 * V 2 + V OQ5 7 aT=VL/2THE L E N G T H OF 005 B D05 9_ 0060 0061 U1=V1/VL V IS 3*V?) TWICE 'J2 = V Z / V L _ 'IJP' POINT 0056 THE TRUE . AREA O F T H E TP I A N G L E , L AT : U3=V3/VL A=V3/2. T H F I I M H O " QP P Q Q J F C T E O A R E A I S H A L F THE 7-CQM 0ME !T OF V OIP=1.57 0 7963-AFSIN(U?) S =T A N ( 0 1 P ) _S S _T_HE.._.SLOPE._TANGF.NT _ INDIVIDUAL TRIANGLES ARE N1TE: I F O T H E R CHAR. A C E R I S T ! C S P F p 006 2 0063 C N D E S I T c 0064 006 5 0066.. 0067 006 8 0069 P = H I G ? _ N A L .J C I F ( V 3 . G E . 0 . 0 ) VI = ( - 1 . SI P E S P F THE T ° I A N G L E c V ?= V EC 11 * VE C2 2 - V E . I v 2 1 U ) ) ) / 3 . T H E M=AM F L ; V A T I C N OF Z M I- S *V C23-v?*.3-v~ 7"> _yj_=y_FCi C 0 Q 4 _ PAGE O O P 10:19:32 ST*TE'-1 E M S APpppp^iATF THF SHOULD BE T N S E R T E D H F ^ E : AND THE cr.,: ORIENT ATT PN P F T F I A M G L E . WRITE STATEMENT CHANGE D o ITRI.FQ.Q) ~,n T O 90 WRI'E'6,200) ITI I ) , I V l I ) • J V ( I ) , * V ( I ) , L V ( I ) , Z M , A , A T , S ' IF( r ..2.00 90 _ F O R M A T (5.I.5.»_4F.i 5,o,.e_.L . CONTINUE I F ( L V ( D o N E , 1 ) GO T O A L AKE=ALAK£ + A NL=\'L + ! ALAKE I S THE TOTAL ARE A p c L A K E S _TRT A N G L E S F A L L I N G O N . W ATE? 7 0070 I ( L A K E oE Go C ) G O T O 7 IF (I T R I . M E .0 ) WRITE ( 6 , 300) I 0R AT( TRIANGLE H ' , 1 3 , ' I S 0071 + OCEAN; N L I S T H E NUMBER PF c -0072 007? = v 1 1E0* ) GO T O 0074 00.7.5... PN A LAKE OR T H E .OCEAN AND IS 6 ' . _ . S ' J M ( D = S U M ( 1. ) + A 0.076 0077 0Q7P SUM(21-SUM(2)•AT 0079 S JM( _ ) = S L ' M ( 5 ) + S * A S U ( 6 ) = S U ' M 6 ) + S * AT S U M ( 3 ) = S U <j\)'A{ M ( 3 ) + Z 4) = S U ' M M * A 4)*S ! OOBO M ..SUM! .7.) = SUM ( 7) +1)1 00"2 0033 0QP4 <;IJM( q ) = S U M ( P ) + J ? 00B5 00P6 SU <11)= SU"M)1)+ V . / 2 . S U M ( 9 ) = S H ' M O ) 4.II-J = su''PQ)»vi/?o viMm) y S'JM( 12)=S'JM(12)+V3/2< i__ONT...LNUE___ r C THE FOLLOWING SECTICN C "°UTES. GEO n w r > RPHPMFTPTC PA°A«ETEPS EXCH'P -130- ORTRAN IV G 00?1 13 0099 0090 00° 1 15 03=52 0093 00°4 I F ( L AKE) 13,19,18 AT0T = S U M ( 1 ) * A L A K E NT A - N T - M L GO TO ? 0 ATOT = S U M m COMX!N_ME :A, >MI: SUX(I) AT OT I S THE T 0 A L MAY E X C L U D E L A K E S , NT I S THE' T O T A L N'_P">E O ^TANGLES, °LAKE = !0'0.*ALAKE/ATOT CONTINUE TN=f=LOAT( NTA1 ZM=SUM(3) /SUM( 1 ) SM1 = SU M ( 4 ) / T N S M 2 = S ' l » t 5 ) /SUM( 1 ) c c SM3=SU (6 M 0103 HYPS 3104 0105 C _r_ C C I S SLOPES: THE HYPSOMETRIC AR AT I O = S U M 1 2 ) / S J * ( A MT? I - S U M M ) / T M 1) AMT AREA I C IS NTA H E t NilMQER A<?EA, WHICH ALALY S I S F0? THE MEAN SM I . _ MVWFT G H T E O ; . SM2 0 0 WTD f?Y A ; S M 3 . o . n v AT... INTEGRAL OF THE TRIANGLES ANALYZED T H I S S E C T I O N I S FOR V E C T O R I A L A N A L Y S I S . V A R I A B L E S RELATED TO U N I T ( U N W E I G H T E D ) VECTOR ANALYST^ INCLUDE "V" I N NAMES, THOSE P E L AT 0 TNCLJPE " V " ' o TO W = 1GMTED AMAI.YSI C c C C R = VECT3R (-), __ LENGTH, K= REC!SIC'N D R F = RO:.'GHNESS FACTOR, K. T =J 0 0 / K , L = 7 G T O R PARAMETER, C 0= OR I E N T A T T O N , D=OIP OF MEAN RU=SORT(SUM(7)**2*SUM( 3)**2 + S U ( ° ) * * 2 ) RV = S0RT(SUM ( 1 3 ) * * 2 + S'JM ( 11 ) * * 2 + ? U M ( 1 2 1 * * 2 ) !jK>( TN-1 . ) / ( T N - R U ) V 0109 0110 J ? JJL 0112 0113 UKI=100./UK UL=300.*PU/TN _ V L =< 0 0 . * R V / S U M ( 2 ) 1 ' RFU=100.-UL RFV=100.-VL UL: 0114 0115 .01L6_ ,0117 OU 8 O'.i q VECTOR STRENGTH, U N I AU=SUM(7)/RU 9U = SUM(P.) / R U __J=FUM( ?.iyp.u_ AV=SUM(10)/RV 0120 B V = S U M ( 1 1 )/<?V C V = U M P 2 ) / R V IF(PU.NE.O.O) 0121 IF(A'j)ll,12,13 C GO TO _!Z2_ _1L_0U-? TO. 0123 0124 GO TO 2 4 OU=0.0 GO TO ?4 13 0 U = ° 0 . GO TO 2 4 ___._.._ NT I NUE. 3U=ATAN(AU/BU)/PAO IP C U ) 2 3 , 2 5 , 2 5 ?5 TF ("111 3"*, ? 4 . " 4 5 THE A N A L Y S I S )/SUM{ ? ) S M ' S A R E MEAN _H._I_ AX.r £H_N . H Y ? S = ( Z'*~ 7. M I V ) / H J_ 0 2_ 012 c C THE 0126 01 2 7 __28.. 0129 013 0 01 3 1 IS T 01 0 1 0106 3107 0108 0004 NTA=NT ?0 C 0095 0D96 0 0 9 7_ 009 8 0099 0100 PAGE 10:1°:32 MAIN COM°ILER 12 > 10 T VICTORS: VI. WEIGHTED BY A T e SIR! VECTOR -131"ORTRAN I V G COMPILER 013? 0133 01 3 4 01 3 5 013 6 02-06-74 MAIN 23 3? 33 I ( U>33,32,32 0U = ? 8 0 . + O U 0U=OU-t-l 0. ?4 CONTTMHC C 0005 Q _Qi.4_4_ 01.4 14 c 7 _0I_ 5 L 013 9 014 PAGE Q OiJ=A"» ^ TN( GUI / ' i D I ( R V . N E . O o O ) GO TO 1 4 I.~(A.Vt 1 5 , 1 6 , 17 15 0 V = 2 7 0 GO TO 4 4 16 p y = o , 0 . • r,o TO ^ 4 17 3 V = ^ 0 . _n._ 0. .44 013 10:19:32 o 0 0141 0142 0 1 "-3 T 5 CONTINUE OV=ATAN{AV/BV)/°A0 0146 45 IF(?V)53,44,_4 43 I («V)f3,52,52 _5.„.0V=^V«-180, 53 O V = . V + l » 0 . 44 CONTINUE 0143 0149 c JXL?J_ 01 5 1 01 5 2 t 01 QV-AP.SIM(CV)1°An C THE c c OLLOWING S E C T I O N PRINTS OUT RESULTS 0157 WRITE(6,152 ) ( TT~ L ( I T ) , 1 1 = 1 , 1 9 ) WRITE ( 6 , 6 0 3 ) *!°, N T , A T O T , PL A < E »NT A , S I !M ( i ),AMTR I 6 3 3 = O R M A ( « GENgoftt : ' , / / , ' M I I M R F P OF P O I N T S = ' , 1 0 X , 1 5 , / , ' \ " v a <ro 1 O TR IAN Gl-F S = , } 0 X , ! - - , - / • • • T_> T AL * E A = »,-i=.15-, / , ' t -CFVT L 2AKES SEA = ' , 3 X , F ? . ' = . ' AVALYS I S = ' , 10X, 1 5 , / , ' A R F A FOR A . 3 N A L Y S I S _ = «_, E 1 5_ P , / , • ' E A TOTANGLE 4 A = • , El 5 . 2 , / / ) _____ M R I T E ( 6 , 6 0 1 ) 7 I N , ! 7 M I M , 7.M A X , I Z 7 4 X , H , H Y P S , S M 1 , S M 2 ,"sM 2 , A R ATf'o""" 0158 601 C 0154 015 5 T 01^6 c D 3 C M FORMAT.' 9CATED 1ATE0 GE0 M 0' A T P O T NT AT OQINT 2 H Y P S OMETR. I C D HC M E T R Y : ' , / / , * M I N I M'J M NU^B " ' , I 4 , /, ' " i x i m . ' M M JMf} E~ • , ! 4 , / , ' L O C A L INTEGRAL = ' , F? „ 5 , / , ' T PELT F F M C A N = ' , =', 8 _ 2 , ' c f q ->2,' =« , FP S L O P E : ' , / , • c 0 0159 6Q 6 XLL6_1_ 016 2 0163 ._16.4_ F D 016 5 0166 TOTAL ELEVATION EL^VAT^ON S 2, 1 LO '. 0 r OX , / , JNWEIGHTFO - P . ? , ? , / , ' . >'F I GH : 0 BY M A P AREA 9 . 5 , / , ' „ . W E I ; HT E I 4Y T U F AREA , = P 5,7", •ARE A R A T I O =',F8,5,//) WRITE(6,602) OU,nj,UL,UK,JKl ,PFU,OV,OV,VL,»RV - ) 7 nc.»j)_T(» yc CTQR ANALYS I S : ' , / / , ' U N W E I GHT=0 ( U N I T VECTORS ) : ' , / , ' ne T IENTATION =',F?o2,/,» OTP =',F8o2,/,' LI?) 2 =' , F 3 . 2 , / , ' K =',F3.2,/,' J00/K 3F3.?,/. • ROUGHNESS. FA CT0.'_5' , F f l , 2 , //_, • VE T G.1TED_ BY T " J F._AP F,A : • , / 4 ORIENTATION =',F8.2,/,' ^1P =»,Fe.2,7t* L(«) 5 =',F8.2,/,' ROUGHNESS FACTOR = ' , F 8 . 2 > j QQ<; _ J flRC » i GO T 0 9 9 9 9 9 WR I T E ( 6 , ° 0 0 ) J O ^ S 033 FORMAT ( M ' , " = N " i OF... P I I N - __ T H I 5. . _UN_I.'-'.CJL_UDFO__j I 3 i l . . J 0 . 3 S . . ' )_._ ST 0 ° END p Ql C M MEMORY XflMJ-lLF. REQUIREMENTS ULME = !_,___. 0013A5 SE C fJ.N I S BYTES 0.R T R AN I V : G COM P I L E R 0001 0002 -13202-06-74 READER 10:19:35 PAGE 0001 S ' B ' O ' J T I N E RE I D E DHENS I O N X ( 5 0 D ) , Y ( 5 0 3 ) , 7( 5 0 0 > , I ( 1 Q O O ) , I V ( 1 O D D ) , J V (1 0 0 0 ) ,KV(1. 0 0 0 ) 1 , L V ( 1 D 0 0 > , T I T L E ( 1 ? ) , A L T ( 2 0 ,2 0 ) , I D f ? 0 , 2 0 ) t O M n \ j X , Y , 7., I T , T , J V , KV , L V» T I l . F , T , L A * , I T RI R =A ^ ( 5 , I C O ) I T Y = E , N F W " , N E . ^ T , XSCALE , Y S C A L F , Z S C A L F , L A K E , IT P I , I D I A G , V P I ,NC 0 R*-* A T ( _ I 5 » 3 E 1 2 , 5 , 5 1 5 ) I ( T T Y P E , N E . 9 3 0 ) G O T O I L» <" = o p o C T M 0003 0004 0005 0006 0007 000* 0009 0010 0011 0012 0013 00 T 4 00' 5 0016 v , 1 00 .004.6 0047 0048 004P 0050 0051 0052 0053 0054 0055 T C J I ( x C S O A L E o, 0, 0 ) X S C A „ F L F =' , 0 YS C . A L E= XS C AL E Z S C A L E = 1.0 C IF( I T Y P E . F O . l ) G O T =0 IF{>'E.yP F0.1) G O T O 3 NP=o R I O R E A 0 ( 5 , 7 0 1 ) I , X ( ! ) , Y ( I) , 7 ( I ) FORMAT! 15 ,?,F6o0) IF(T,50.0) G O T O 3 X ( I )=X ( I )*XSCA! F Y(I)=Y(T)*YSCALE Z< I ) = Z ( T ) * Z S C A L E NP=N'P+1 GO T O 2 I F M E W T . E Q . U G C TO 5 5 NT = 1 3 0026 0044 0045 C R E AD(5,103 ) ( I T L ( I ), 1= 7 , 1 9 ) OR' AT(i o>4) C I (YSCALE_EO-,0,0) IF(7SCALE.EC.0,0) 101 0043 M T 3 = T U R N 8 0019 0020 0021 0022 0023 002400 2 5 003 7 0038 0039 004.0 0041 0042- O C 1 103 •> 0027 0023 002<3 0030 0031 003? 0033 0034 00*5 0036 V p 0017 001 T V 4 102 5 55 200 201 2 02 20"* RE A D ( 5 , 1 0 2 ) I T ( M T ) , I v ( N T ) "F0R^ A T r 5 T 5 ) I F ( I T ( N T ) «.EOoOJ G O T O 5 NT=NT«-1 T . GO 0 4 \jT=VT_i ,JV(NT),KV(NT »,1V(NT) : WRITF( ,200)(TITLE(I),1=1,1^) FO R A T ( ' 7 ' , 1 ° A 4 , / ) W R I T F ( 6 ,201) N » , NT OR AT(• T H I S DATA S E T W A S T R I A N G U L A T E D M A N ( A L L Y ; i f "' ? : O 1,' P O I N T S . A N D ' , 1 3 , ' T ' I A M G L E S , ' , / / ' O P T I O N S : • I I F ( M E W ° « E 0,1) WRIT=(6,?0Z) FDR«AT(' -USES S A ^ E POINTS AS P R E C FDIN G DATA S E T ' ) I F ( V E W T . F O . 1) W P . I T E ( 6, 2 0 3 ) F_RMAT(«. .-USES. ..SA'-'E T P I A N . G L E S A S P R E C E D I N G D A T A S E T ') IF(XSCALE.NE.l.O) W R I E ( 6 , i ? 0 4 ) XSCALE -- - FOR«AT(» - X S C 4 L E = • , E 1 2 .5) I-E(YSCALE,NE,1,0) WRlTP(f,,?r>5) Y S C A L 6 U C V T 204 = 205 20 6. 207 50 F O R A T ( * M - YSC A L E = ' , E 1 . 2, 5) I F ( 7 S C A L E . M E . 1 , 0 ) W R r T F ( 6 , 206) FORMAT( •..-ZSCALE-' , F I ; „ 5 | I F ( L A K . E Q . l ) W3 I T _ ( A , ? _ 7 ) FO R A T ( * -LAKES ARE EXCLJPED p= T"PN I F ( N E W P . E O . 1 ) G O TO 61 DO 6 0 I = , N » W 1 .60 ._— 150 R.E AD I 5 ,15 D) ( A L T (.1 • J ) , J = l , N C ) FORM T(3X,15 5.0) NP=NP.*NC NR. 1= M R - 7 C A ZSCAIE F D 3 U THE ANALYSIS') N T A T N S ~ " » ,T3 OP TR AN IV G COMPILER 0056 00 5 7 0058 0059 0060 006 1 0Q6 ?_ 006 3 0064 006 5 0066 0067 J2Q.6 8_ 0069 0070 00 7 T 0072 0073 .0074 0075 0076 PAGE 0002 NC1=NC-1 N T = 2 *NR1*NCI 00 l 1=1,NR 00 51 J = l i N C 10= ( I-I )"NC«-J IDMf _,.)) = i o _X'.J? _! = - OA T . I . J t *v SC ALE Y{ 10 } = L 0 AT (NP-! )"Y SCALE" Z( I 3 ) = ALT ( I , J ) -*ZSC ALE CON INiIE CONTINUE INT = 1 DO 5 2 1 = 1 00 52 J=1,NC1 1 (TDIAG) 53, 53, 5^ C 51 61 T c IT( I N T ) _ T : ' ) T 54 I V U N T ) = IGM( I , j ) JV(INT)=IOM(i+i,j) _ YJ I N T ) = I DMJJ. + 1 J t l L LV(INT)=0 INT=INT+1 IT(INT)-TNT IV(I NT)=I DM( I , J ) JVUNTJ=TDM(I ,J + 1) J<V.UNI.)=IDM{I+1. J4-1.) LV(INT)=0 INT= INT+1 GO TO 52 IT(INT)=INT IV(INT)=TDM(I ,J) _J.VJ.LN_T" ».= i y ( I _l_+l > KV(INT)=IDM(H-1,J) LV<INT)=0 INT=INT»i IT(INT)=INT IV( INT) = IDM(I, J4-1) _J V (TNT ). = I D M ( I+1,J ) KV(INT)=IDM(I+1, LV(INT)=0 INT=INT ». 52 CONTINUF INT=INT-1 HPITE{6,?00.)(.TITI.E(.I), 1= I F ( I NT, NE. NT) 303 FORM AT ( • 301 W R I T E ( 6, 300) INT=•, I ' , B U T WRIT=(6.301) 1,19) INTENT NT=• , I 4,/) NR,M:.NQ,MT FORM AT( ' THIS DATA SET IS P.-VS^O ON A ',13,' BY ',13,' ALT IT'JOE M . T 1R IX; •, /, ' I T CONTAINS i , n , i POINTS AMn TRIANGlFS',//,»0PTI _2.0NS: '.).. -._•.: _ I (VEWPoEO.1) WPITE(6, 202) IF(NEWT.EQ.l) WPITEI6,20?) _-J-EJ_X_SCALE, EO. YSCAl E ) W IT F( 6 , ? 02 ) XSCALF FORMAT.' -SOUA^E GPIO. G'TO S P /* F1 2,• MFTPFS') ,_, „„ - -C I NG = • ,.5, IF (XSCALF.NE.YSCALE ) W RITE ( 6 , ?0 J XSCALF, YS~ A L rORVAL(.L_._.r_COL.UMN..SfiACIMjG=' , El 2, 5, • «=TRFS ; ~R_tw SPAC ING=•,c1?_ <=. • 1MET-ES * ) IF(IOIAG.EO.O) WPITE(6,?04) P Q ^ " > ( „. -USEO N3RTHWPST-SPUT(-FA.ST 01 AGONAL S ' ) S D 302 :! -3^-3 01 0 9 01 1 0 10:19:35 c 0078 0079 0103 0104 31Q5_ 0106 0107 02-06-74 c 0Q77 0081 0082 008 3 0034 0085 _036_ 0087 008 8 0039 0090 0091 _00.?__ 0093 0094 0095 0096 0397 ___L8_ 0099 0130 0107. 0102 -133- READER 304 > p V T -134- QRTPAN IV G COMPILER REACE° : 0111 0112 0113 305 C 0114 R END c TURN TOTAL MEMORY REQUIREMENTS 001B2C BYTES TIME = • 1,0:19:35 . •• , : •'' PAGE IF (10 I AG. E O . n WPITF(6,'05» FORMAT ( • -'JSEO NO-. TM * S T- F 3 U HWE S T ! A G 0 NA L S * ) IP(ZSCALE.NE.l.O) W ' I' E ( 6, 2 0 6 ) Z S C AL F 0115 COMPILE • 02-06-74 0.5 SECONDS T 0003 -135- Appendix 1Mb: Triangular Data-sets Maps of a l l the triangular data-sets analyzed in this study are g i v e n , w i t h the exception of sample 11a which was illustrated in Figure 6 . 1 . These maps are a l l at the scale of 1:50,000; areas are given in insets. 1:250,000 scale maps of the same -136- -137- Sample l i d , Ptarmigan Creek map-area (83 D/10W) Sample 18a, Manning Park map-area (92 H/2W) -144- Appendix I lie: Computer Results This appendix includes the computer output for the thirteen triangular data-set analyses and fourteen grid analyses upon which the comparisons reported in Chapter 6 were based. The titles of these output sheets are selfexplanatory . . _ _ -J_45- I L L E C 11 L E W A t T ( S A ^ L E 3 ) GENERAL: Ai JM'^FK CF P O I N T S _= 9_0_ NUMBER OF TR I ANGLE S= IS 3 TOTAL AREA = 0 o 4 3 9 92 9 6 ~ 03 PERCE NT_ L..KI. S.. + ..SEA=_ 0.3 NUMBER FOR ANALYSIS= if3 AREA FOR ANALYSIS = 0 , 439 V->29 6S 08 .HE__N TRIANGLE AREA = Oo3202S661E 06 C GEQMr'RfMHGMFXRY_L MINIM'JM ELEVATICN = 3 7 7 . 82 MA XI MUM E LE VATICM _ 2 74 3 , 2 0 "LOCAL RELlFF '= m65o'33 HYPSOMETRIC INTEGRAL = 0.44691 VFAN SLOPE: UNWE I GHT FO = 0. 5 3 3 C2 WEIGHTED BY MAP AREA = Co 58537 WEIGHTED BY TRUE AREA = 0 . 5 97 97 AREA RATIO = 1,1634 1 VECTOR ANALYSIS: UNWEIGHTED(UNIT ORIENTATION DIP VECTORS ) : = 102,50 = 31,70 = K. 100/K ROUGHNESS ~ ~ = = FACTOR^ 3 6 . 9 1 _ 7 . 59 13.18 13.09 WEIGHTED BY TRUE AREA: 0IR I Fj>l TAT ION 115,53 "'DIP • = 33.18" LU) = 36.20 ROUGHNESS F ACTQR= 13.30 LOCATED AT POINT DUMBER LOCATED AT POINT NUMBER 83 53_ . PTARM I G A N C P E E K GENERAL N U _ 3 J _ -146- 11 A ): A L L P 0 1 M1 S : JJE_ K i J M rj P R TOTAL ( S A' 1 P L E PO I NT S = 0 F " T R I A~\) G L E S ~ AREA PERCENT NUMBER " = L A K - S__ + G E FOR ' ~ 1 3 6 0 . 4 3 ° 9 Q 3 2 3 E \--_ 0 8 0 , 0 A N A L Y S I S = AREA FOR WE AN T R I A N G L E A N A L Y S I S 1 3 6 = 0 , 4 3 = 0 , AREA 9 9 3 2 C 3E 3 6 0 . 8 9 1 2 6 0 3 0 6 GEOMORPHOMETRY: MINIMUM M A X I M U M LOCAL ELEVATIC'N E LEV ATI r N RELIE'F" H Y P S O M E T R I C MEAN " I N T E G R A L 393.00 = 2 4 0 4 , 0 0 = l^o9.00"" = 0 , 2 6 3 4 1 " W E I G H T E D BY MAP W E I G H T E D BY TRUE R A T I O VECTOR A \E " = A = 0 , 3 5 5 = 0 = I , C 8 9 1 7 AR E A ' ' 0 , 4 1 4 5 3 V E C T O R S ) : O R I E N T A T I O N = 3 1 , 8 6 D I P = 8 5 , 7 6 H%)__ = ~ ' 100/K 9 0 , 9 7 = 1 1 . = ROUGHNESS W E I G H T E D 3 A N A L Y S I S : U N W E I G H T E O ( U N I T 'k' LOCATE 0 AT P O I N T L O C A T E D AT_ P O I N T S L O P E : U N W E I G H T E D AREA = FACTOR= BY TRUE _0 R I E N J £ n O N_ " DTP~ •' L I S ) ROUGHNESS • 0 0 3 , 0 9 9 . 0 3 A R E A : ^_ 6 0o35 = ay 0 e 7 = 9 1 , 3 8 E A C T O R - 3 , 1 2 37 3 ? 8 C 0 ~ NUMBER NUMBER 79 68 -147P f Ai>• 1 0 A N CP 1 S V H 11 8 ) L E ; : _ F R 0 M " l _ _ _ : ? 5 0 , 0 0 0 _ ••'•AP G E N E R A L : NUMBER 0 F PQ I'lT S NUMBER CP TR"lV>IGLSS= T O T A L i R E A NUMBER Q E A, FOR F O P MEAN 2_9 " = PeRCENJ._LfeKES...t A _f 4 1 0 „ 4 ) >99 7 7 6 E r _S = A = _ . 0.0 A M A LY S I S = A N A L Y S I S T R I A N G L E A R E A OS 4 1 = O ~ 0 . 1 1 9 , 3 0 4 '.<"• 9 9 7 7 6 E 1 6 0 E 0 3 0 7 G E Q M 3 R P H 0 M F T R Y : M I N I M U M E L E V A T I O N = 7 3 1 , 0 4 L O C A T E D A T P O I N T N U M B E R 2 9 M A X I M U M E L E V A T I O N -= 2 36 2 o.10 L O C A T E D A T P O I N T N U M 3 E P 1 9 = 1 6 6 L. 1 6 = 0 . 2 6 5 3 0 L O C A L R E L I E F H Y P S O M E T R I C MEAN I N T E G R A L S L O P E : U N W E I G H T E D " ~ W E I G H T E D RY MA ° W E I G H T E D BY T R U E AREA " = 0 . -AREA 3 - A R E A R A T I O U N W E I G H T E D ( UN I T O R I E N T A T I O N 0 , 3 9 3 6 4 = 1 . 0 8 3 2 2 V E C T O R S ) : = 3 4 J . 4 3 DIP L( 87,29 %) K ~ _ • ' " 100/K BY - Lit) R O U G H N E SS 7 F A C T O R = T R U E - 9 > r 0 7 « 7 6 A R E A : O R I E N T A T I O N " D I P 9 2 . 2 4 1 2 . 5 7 = R O U G H N E S S W E I G H T E D = 1 3 . 3 6 8.3 / 6 - c = F A C T O R = 4 3 = I i i 7 3 8 3 0 » 3 7 8 + 9 1 . 9 3 8 o 0 7 PTARMIGAN C.K ?f_K ( S A' 1 •' L r -148- _ A " O R )X'„ SAM" POINTS AS JlCl CUT 1 :5 0 , 0 0 GENERAL (VUM3F P. OF POINTS NUMBER OF TP I ANGLES= TOTAL A R F A = 0 4 39 99 64 0?: PERCENT LAKES + SE\=_ 0 NUMBER FOR ANAL YSI S= " 4 AREA FOR ANALYSIS = 0, 48999 B4GE MEAN TRIANGLE AREA = 0,3. 1951 13CE o o ?9 4 3. 08 0 1 03 C7 GEOMOR°H 0 E T R Y : M MINIMUM EI.EVATIC.-. MAXIMUM ELEVATION LOCAL RELIEF HYPSOMETRIC INTEGRAL E AN S L O P E : U N W E I G H T E D " WEIGHTED BY M A P A <E A WEIGHTED RY TRUE AREA AREA RATIO 6 9 4 , 94 2404,26 = 1709,32 = 0,26517 V = 0 o 3 7 6 ? 0 = 0 . 3 6694 = 0.38063 l . o a2ai VECTOR ANALYSIS UNWEIGHTED? UNIT VECTORS ) ORIENTATION = 18,33 DIP = 39 o 2 0 LJ •*)_ = 9 2,17 K " " " " = 12746" 100/K = 3.02 ROUGHNESS FACTOR^ 7,83 WEIGHTED BY TRUE AREA: ORIENTATION 30,5 9 "38,0 5" DI P 92,41 LiZ) 7,59 FACTOR: ROUGHNESS LOCATF 0 AT POINT NUMBER LOCATED AT POINT NUMBER 17 19 -149P T A R M I GAN G E N E R A L CR E E K NUMB E R OF P O I N T S B>;R OE TRI AL AREA TOT \LL P O I N T S : NU M lij>) : ( S A M P L E PERCENT N U M 3E k ' A N G L E S= " = L A K E S E0R + SEA A NA L Y S I A R E A FOR MEAN T R I A N G L E " 0 . 4 '"" 8 0 9 92 ' 1 4 3 4 3 6 _ . 0 8 0 . 1 5 S= A N A L Y S I S AREA 14 3 = 0 , 4 3 9 ' " > 9 2 4 8E 0 8 = 0 , 3 4 2 6 5 2 0 6 E 0 6 G E O M O R PHCME T RY M I N I M U M E L E V A T I O N M A X I M U M E L E V A T I O N L O C A L R E L I E F 1 7 0 <3 UNWE f 3 H T E 0 3 2 0 , 2 6 1 6 5 ' 0 , 3 BY MAP W E I G H T E D BY T R U E AREA D. A R E A 1 , 0 A N A L Y S I S UNW E I G H T E D ( U N I T V E C T O R S ) : O R I E N T A T I O N 4 1 o D I P 8 4 , 6 4 L ( %) = ~ W E I G H T E D 12 4 9 CI 8, R O U G H N E S S F A C T O R ^ TRUE BY O R I E N T A T I O N A R E A : = 4 9 , 6 5 D I P 3 7 L ( %) 9 . 1 . R O U G H N E S S 0 5 9 2 , 0 5 ' = F A C T O R : 3 1 1 6 3 70 7 0 Go 3 9 0 5 3 R A T I O V E C T O R K 100/K 0 S L O P E : W E I G H T E D A R E A AL I N T ! H Y P S O M E T R I C MEAN 6 9 4 o "54 2 4 0 4 o 2 4 . 3 3 6 9 3 , 3 1 9 1 4 6 0 A T P O I N T N U M B E R 3 5 L O C A T E D L O C A T E AT P O I N T N U M B E R 7 0 -150MANN f N G P A R K ( S A M PL 11 A- l«) ) : 1 : .C ,CC'J SCALE "HIGH R r SC L L T I E N " GENERAL : S LUMEiLJJ QE P C I i l l S . f i I i NUMBER ! TOTAL .... CF AREA PERCENT. NUM13ER 115. = TRIANGLES= 2C7 = 0.43«99C72E L A K E S .+ ..SEA-.. FOR AN AL Y 3 I S= AREA FOR ANALYSIS MEAN TRIANGLE = AREA 2C7 0 . 4 3 ° 9 9 C 7 2F _. MINIMUM ELEVATION = 1051.66 MAXIMUM ELEVATION - 19 3 5 . ^ 3 RELIEF = HYPSOMETRIC INTEGRAL . .NEA.N..S.LC.P.E UNWEIGHTED W E I G H T E D BY W E I G H T E D BY AREA . 7 7. MAP A R E A T R U E AREA UNWEIGHTEC (UNIT ORIENTATION D IP 0 . . <2 26 = = = 0.3 £9. 2 0 . 3 9 3 15 0.3.747 - 1.C79S4 VECTORS ): 24.16 89.27 L m 92 K 1 3 . 72 7.29 7 .25 WEIGHTED FACTOR- BY TRUE ORIENTATION AREA: 351,0 1 F AC TOR - 39.52 9 2.61 7 .39 ( ? ) ROUGHNESS .75 = DI P L LOCATED AT POINT LJJ^IJ:I.._AT_j _ 0 j N T NUMBER NUMBER S2 = ANALYSIS: 100/K ROUGHNESS _ .... . RATIO VECTOR C8 •= 0 . 23 6 71 C4_t_F_ 06 .._ GEC.M0R_PH CM£I_R_Y_:_ LOCAL C8 . ..0.0 2 __15 •151 M A N N I N G P A R K I S A M P L E 1 8 ) : l : ? 5 C , C C 0 M— SC A L E MAP G E N E R A L : N U M B E R CF P O I N T S NUMBER EE T R I A N G L E S ^ TOTAL il AREA = _ 5 . 22 0 . 4 3 9 9 9 9 C H E P E R C E N T ....L A K E S . + ...S E \ = N U M B E R FOR AREA FOR _M E A N T R I C8 0 , 0 . . ANA LY S I S - A N A L Y S I S A N G L E AREA 3 2 = 0 . 4 R 9 5 c c c 4E C8 = 0 . 1 3 3 1 2 C7 4 7 0 E -GECM.GR£HCME0L!1_ M I N I M U M E L E V A T I O N = 1 0 6 6 . £ 0 L O C A T E E AT P O I N T MJ M PER NAX E L E V - 19 C 5. CO L O C A T E 0 A T P O IN N U MB ER_ I MUM L O C A L AT I O N R E L I E F H Y P S O M E T R I C = I N T E G R A L 83 = 8 . 2 0 0 . 5 1 7 3 1 _.f'E.AN.„S.LOP.EjL._ U N W E I G H TEO W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A AREA AREA R A T I O J V E C T O R i UNW E I = 0 . 2 6 0 1 5 •= C . 2 : 5 1 9 = 0 . 2 T 7 r = A N A L Y S I S : GHTE 0 ( UN I T V E C T O R S ) : O R I E N T A T I O N = 3 9 . 8 2 D I P - 3 9 . 2 7 L.(J.) = 9 6 . 4 1 K = 1 0 0 / K •= A O U GHNESS W E I G H T E D BY TRUE A R E A : = D I P = R O U G H N E S S 7 3 . 7 1 3 . 5 9 ORIENTATION L I S ) 2 6 . 9 F A C T O R = 37.29 3 9 26 1 . 0 3 5 8 2 . 3 9 - 9 6 . 5 5 F AC TOR= 3 . 4 5 T M A N N I N G P A R K ( S A M i > l . f i 1.8): C - l . : . C , G C O S C A L E , APPRO X . S A M F P>'.NTS G E N E R A L J>JJjM.'3_E R O F P Q I v| T S NUMBER CF T R I A N G L E S ^ T O T A L A R E A 2 2 = 0. 4 3 9 9 < PERCENT . L_AK ES.__+_ S E A " N U M BE R FOR A N A [ Y S I S = A R E A FOR A N A L Y S I S MEAN T R I A N G L E AREA > 9 C 4 E C3 CO " = 0 . = 0 . 153 "' 4 39 9 9 9 0 4 E 1 2 A 7 C E 3 2 CS C 7 .-GE0.MJBPH0Mr3T.B_Y_ M I N I MUM JiAXj___UM_ L O C A L E L E V A T I O N 1 C 6 6 . 3 0 L O C A T E C EL E V A T 1 8 3 9 . 7 6 L O C A T E D ION R E L I E F H Y P S O M E T R I C = I N T E G R A L 8 2 2 . 9 6 •= 0 . 5 4 3 . 2 = 0 . 3 C E 5 2 X E A J S L S k Q P f : L M W E I G H T E D W E I G H T E D B Y MAP W E I G H T E D BY T R U E A R E A AREA 0 . 2 9 8 1 . C 4 C ( UN I T AT V E C TO R S ) I O N 35 5 P 2 0 100/K W E I G H T E D 18_ . 1 1 4 . 9 ESS F A C T O R - BY T R U E O R I E N T A T I O N A R E A = 10-6 8 9 L(*) 95 E A C T O R 7 4 . 3 2 P R O U G H N E S S 1 . 7 4 __95_. K ROUGHN .5 8 3 .LULL D I 0 . 2 9 5 4 6 = ANALYSIS I G H T E O R I ENT D I - R A T I C VECTOR UNWE A ^ F A - .3 JL 2 . 4 3 5 2 7 4 6 AT Al. P 0 I N T P O I N T NOME ER 2 NUMB FR 3 AS 18 6 -153T A T L A L A K E ( S A M P L E 2 4 ) : ' i N C L U O TNG L A K E S G E N E R A L : N U M B E R OF P O I N T S NUMBER OF T R I A N G L E S = T O T A L j 4 ? A R E A R E K C E N T . NUMBER " = L A K E S . FUR 0 o 4 ± _ . S E A=_. 399 B 9 2 n E . F O R A N A L Y S I S MEAN T R I A N G L E 8 2 5 0 = 0 . = 0 . 1 9 5 5 A R E A 0 0 3 . 7 6 A N A L Y S I S ^ A R E A 2 5 4 3 9 9 3 9 2 SE 9 56 9E C 8 0 6 . G E O M O R P H O M E T R Y : M I N I M U M E L E V A T I O N MAX I E L E V A T I O N MUM L O C A L R E L I E F H Y P S O M E T R I C MEAN U N 9 0 9c S3 1 1 1 2 . 5 2 I N T E G R A L = 2 0 2 . 6 9 = 0 . 2 6 7 3 9 S L O P E : W E I G H T E W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A D " ' " = ' A R E A = A R E A = R A T I O V E C T O R = A N A L Y S I S UN WE I G H T E D ( U N I T 0 . C 7 4 7 0 0 . 1 . 0 : V E C T O R S ) : O R I E N T A T I O N 4 5 . 5 2 D I P 8 9 . 6 4 ICS) 9 9 . 4 1 K " = J.OO/K = R O U G H N E S S W E I G H T E D BY L(35) R O U G H N E S S 0 . 6 0 TRUE 0 . 5 9 A R E A : O R I E N T A T I O N DIP " 1 6 7 . 9 9 E A C T O R = 0 4 7 7 2 0 . 0 4 3 C 3 3 4 1 . 5 9 = """ 89.T7 = 9 9 . 7 6 F A C T O R = 0 . 2 4 02 '»6 L O C A T E O AT P O I LO C AT AT P O I N T F 0 NT NUMBE R N U M B E R .. TATLA LAKE -154- ( S A M P L E 2 4 ) : E X C L U D ING L A K E S GENERAL: NUMBER OF P O I N T S • \.±Z NUMBER CF" 1'RIANGLE S = " 250 TOTAL A R E A = 0, 4 3 9 ° 9C C 8E 0 5 P E R C E N J L A K E S +_ 5 A.. . . __ 3, 7 6 N U M3 E R F 0 R A MALY S I S= 23 8 A R E A FOR A N A L Y S I S = 0, 4 7 1 5 9 C C 8 E 0 3 MEAN T R I A N G L E AREA = 0 1 9 3 1 4 7 C 6 E 0 6 C O GEOMORPHOMETRY M I N I MUM E L E V A T I O N MAXIMUM E L E V A T I O N L O C A L RELIEF H Y P S O M E T R I C INTEGRAL M E A N SLOPE: UNWE T GHTED * = W E I G H T E D B Y MAP AREA - J l L l i L L T ED BY TR iJE AREA P A T I O VECTOR AR E A 909 3 3 1112,52 = 20 2,69 = 0,27530 0 0 , C 7 3 46 = 0 , 0 4 5 58 = 0, ANALYSIS: UNWEIGHTED(UNIT OR I EN TAT ION DIP L m K 100/K R O U G H N E S S VECT0 R S) 45,52 39»63 99,33 159,92 0,63 F ACTOR: 0,62 W E I G H T E D BY T R U E ARE A: ORIENTATION = 3 41,5 9 DIP = 39,76 L.S) = 99,75 R O U G H N E S S F AC TO R= 0,2 5 ! C 49 90 1,0 02 5 5 LOCATED LOCATED AT AT 0 I N T NUMBER 7 "01 NT NUMBER 1 '6 P -155G U I T E 7. L I L A K E ( SAM ' L E E I I N C L U D I ) N'.. L A K E S G E N E R A L N U M B E R CE N U M B E R OF T O T A L 1 1 4 P U I N T S TR I A N G L E S 19 6 1 3, 4 A R E A P E R C E N T N U M B E R _ L A K E S _ _ j F O R 8 9 9 9 1 3 6 E SEA = OS 4 o 7 A N A L Y S I S - A R E A FOR A N A L Y S I S M E A N T R I A N G L E A R E A 19 = 0, 4 8 9 9 9 13 6 E = 0, 2 4 7 4 7 C 3 0 7E 8 8 3 0 6 G E O M O R P H O M E T R Y : MI MI MUM E L E V A T I O N 1 0 6 3 MAX I MUM E L E V A T I O N 2 2 ~> e. , = 1 1 9 40 3 2 = 0.3 3 7 - 0 = 0 , 2 L O C A L R E L I E F H Y P S O M E T R I C MEAN U N I N T E G R A L 5,7 23 S L O P E : W E I G H T E D W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A " " ~ ~ = 0, A R E A A R E A R A T I O V E C T O R A N A L Y S I S : V E C T O R S ) O R I E N T A T I O N 7 . 9 2 8.3, 2 9 D I P 9 5 , 6 5 %) K = •= 100/K R O U G H N E S S W E I G H T E D 22,88 4.37 F A C T O R - BY T R U E O R I E N T A T I O N 4 . 3 5 A R E A : = 5, S3 D I P 8 5 . 4 9 L ( 9 7 , 0 8 %) R O U G H N E S S 27 o l o U N W E I G H T E D . U N I T L ( 7 5 0 F A C T O R - 2 . 9 2 5 1 4 2 G 2 5 9 15 9 0 0 3 3 3 1 L O C A T E D AT P O I N T NI.J M B E R L O C A T E D AT P O I N U M B E R N T 2 7 2 -156- GH i TF 7 I. [ I.AK E ( S V -1 H. E EiX C1L ) U U I N'' LAKES GENERAL: Q F POINTS NUMRcf 1 1 4 NUMBER OF T R I A N G L E S TOTAL A REA - 1 9 3 0 4 39 9 9 I 5 2 E 03 P F R C E N T L AK E S + _ S E A = NU -3E"R " F O R " A N A L Y S I S ~ AREA FOR A N A L Y S I S = MEAN T R I A N G L E AREA = 4, 7 8 18 5 0.4<!6554CcE 08 0 . 2 5 U 9 1 3 7 E 06 V 0 GEOMORPHOMETRY: MINIMUM MAXIMUM LOCAL ELEVATION ELEVATION 0 RELIEF = HYPSOMETRIC INTEGRAL = MEAN S L O P E : UNWErGHT E D " " " ' " " ~ = W E I G H T E D 3 Y MAP A R E A W E I G H T E D BY T R U E A R E A = AREA RATIO VECTOR 106 3 75 2 2 5 Eo 57 119 4.S2 0.354 13 0.29448 0.2 1277 0.22638 lo 03493 ANALYSIS: UNWEIGHTED(UNIT OR I E N T AT I O N DIP VECTORS): L( %> 7 . 02 82.79 95.40 21.61 K = 100/K - 4.63 FACTOR- 4 . 6 0 ROUGHNESS W E I G H T E D BY T R U E 0 R_I EM T A T I O N "DID" ~ " ROUGHNESS = AREA: = 5 . 83 35,27 FACTOR- 96.9 5 3.05 LOCATED LOCATED AT AT POI NT POINT NUMBER NUMBER 2 72 -10/O C N A K I VEP. ( S A M P L E 4 1. ) G E N E R A L : N U M B E R GIF NUMBER T O T A L p E - RCE N T T A N G L E S L A K E S . FQ»? ME A N TP 2 4 1 0 o 4 F O R A R E A G TR A R E A N U M B E R .13.8. P H I N T S E n Y S I S A N A L Y S I S E c B 0 6 C E *.._S.:.-.V A N A L I A N G L 3 9 A R E A 0 O B _Q.Q r 2 4 0 -- 0 . 4 = 0 . 8 9 9 8 * 6 3 E 2 0 4 1 6 2 3 I E 0 8 0 6 E . 0 M J 3 R f H_0 M E T R Y _ M E L E V A T I O N M A X I M U M . M I N I M U E L E V A T I O N L O C A L M E A N . A T L O C A T E D AT = 8 6 8 . 5 3 I N T E G R A L = 0 . 4 C 3 3 0 S L O P E : U N W E I G H T E D * W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A L O C A T E D 1 6 3 Co R E L I E F H Y P S O M E T R I C 4 3 3 9 9 o = A R E A 0 . 3 9 1 6 4 = U 2 A 0 . 3 8 0 = 1 . C 8 2 6 0 R A T I O V E C T O R A N A L Y S I S : U N W E I G H T E D ( U N I T V E C T O R S ) : 2 2 0 . O R I E N T A T I O N 1 7 DIP 3 8 L ( S) 9 2 . 4 2 K 1 3 . 1 4 100/K W E I G H T E D . 0 9 7.61 R O U G H N E S S 7.53 F A C T O R ^ BY T R U E ' O R I E N T A T I O N A R E A : = 2 6 0 . 6 9 DI P 8 L I %) 9 2 . 3 9 R O U G H N E S S 9 0 0 . 3 9 1 2 2 F A C T O R ^ 3."72 7 . 6 1 POINT POINT N U M B E R 1 3 1 N U M B E R 7 3 ILLECILLEWAET ( SAMPLE ^3) G P.To , N W - S E 7)1 'AGON A'fs GENERAL NUMBER CF POINTS 225 NUMBER GF T R I A N G L E S = 392 TOTAL AREA = 0. 43997936E 08 P E R C E N T L A K E S _ _ SEA = _ _ 0 0 NUMBER" F O R " A~NALYS"I'S= " 3 9 2 " O AREA MEAN FOR A N A L Y S I S T R I A N G L E AREA = = 0 . 439-9 7 93 6E 0. 12499469E 08 06 GEOMORPHOMETRY MINIMUM MAXIMUM ELEVATION ELEVATION = = LOCAL RELIEF HYPSOMETRIC INTEGRAL MEAN SLOPE: UNWEIGHTED W E I G H T E D BY W E I G H T E D BY AREA RATIO VECTOR MAP A R E A TRUE AREA 3 9 0 . 0 2 2 7 4 = = 1853.18 0.43536 = 0.52344 = = 0.52345 0.5 2900 1.143C7 ANALYSIS UNWEIGHTED(UNIT ORIENTATION DIP Lit) K " 100/K ROUGHNESS WEIGHTED BY TRUE OR I E N T A T_I_0N DIP lit) ROUGHNESS VECTORS): = 114.94 = 8 3 . 1 7 _= §8.64_ " = ~ 3.78 11.39 FACTOR= " 11.36 AREA: = _1_15, 07 =" 3o 2 0 " 3 3 . 2 3 = 8 3 . 1 0 FACTOR= 11.90 L O C A T E D A T P O I N T N U M B E R 2 0 7 L O C AT f D AT P O I N T N U M B E R 9 2 . I L L E C I L L E W A E T -159- ( S A M P L E 3 ) : G R I D , N E - S W D I A G O N A L S G E N E R A L : N U MB E R OF P O I N T NUMBER OF TR T O T A L NUMBER GEO = F 0 R 0 4-399 7 93 6E o _ ...... 0 8 0 » 0 A N A L Y S I S - A N A L Y S I S J R i A N G L E _ A R _ E A 3 9 2 - 0 . 4 8 9 9 7 9 3 6 E 0 8 - 0 . 1 2 4 9 9 4 6 9 E 0 6 MORPHCMEJ.RY: M I N I M U M E L E V A T I O N - M A X I M U M E L E V A T I O N = 2 7 4 3 . = 1 8 5 L O C A L R E L I E F H Y P S O M E T R I C N E A N I N T E G R A L = " • " W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A 8 9 0 . 0 2 3 . 1 3 0 . 4 3 5 7 6 = A R E A A R E A 0 . 5 1 8 4 3 - 0 . 5 = 0 . 5 3 6 5 4 R A T I O V E C T O R A N A L Y S I S : U N I T V E C T O R S ) O R I E N T A T I O N = D I P = 8 3 . 2 2 - 83.77 Lm _ K ~ " " R O U G H N E S S R O U G H N E S S ' 8 . 38' 1 1 . 2 6 F A C T O R - 1 1 . 2 3 BY L(5g) ' - T R U E O R I E N T A T I O N ~ 1 1 3 . 0 5 """"= 1 0 0 / K W E I G H T E D 1 8 4 4 = 1 . 1 4 2 3 7 U N W E I G H T E D ^ D I P " 2 0 S L O P E : U N W E I G H T E D _ 111 3 9 2 L A K E S _ + _ . S E A - F O R MEAN - A R E A P E R C E N T A R E A S I A N G L E S - ~ _ A R E A : _= = 1 1 5 ^ 0 7 _ 8 3 . 2 3 = 8 8 . 1 5 F A C T O R - 1 1 . 8 5 L O C A T E D A T P O I N T NUMBER 2 0 7 L O C A T E D AT P O I N T N U M B E R 9 2 _ f P T A R M I G A N ! GENERAL: ! > ( i I I . - 1 6 0 - CREEK (3V-1PLE 11 T? GRT6 , . OI&^HNW-SF >^TAC('NTLS NUMBER CF POINTS ?2 5_ NUMBER OF TRI ANGLE S= 392 TOTAL AREA = 0.43997936E 03 P ERCENJ . LAKES.* SEA= ... . 0 . 0 NUMBER FOR ANALYSIS= 392 AREA FOR ANALYSIS = 0.439}7936E 08 MEAN TRIANGLE AREA = 0.12499469E 06 G E O M O J R P H _CM E LRU MINIMUM ELEVATION 694.94 — 2 3 0 l o 24 MAXIMUM ELEV ATI ON — 1 6 C 6 . 30 LOCAL RELIEF 0,27922 HYPSOMETRIC INTEGRAL MEAN SLOPE: — 'bo~3 4 3S7 UNWEIGHTED 0 . 3 43 69 WEIGHTEO BY MAP AREA 0 . 3 60 39 WEIGHTED BY T R U E AREA = AREA RATIO 1 . 0 78 58 UNWEIGHTEDIUNIT VECTORS): ORIENTATION = 53.92 DIP = 8 8 . 4 9 l(%) = 9 3 , 2 4 K = 100/K = ROUGHNESS FACTOR= W E I G H T E D BY T R U E " " LIZ) R O U G H N E S S ~ 14.75 6.78 6.76 A R E A : O R I E N T j A T _ I O N "DlP LOCATED LOCATED — VECTOR ANALYSIS: V. ~ T : ^ 0 = ' = F A C T O R = 55_._6 2 ="87.94 92.77 7.23 AT AT — POINT POINT - 5 NUMBER 22 NUMBER 1. ° o - 1 6 1 P T A R M I G A N C R E E K ( S A M PLE 3 1 ) : G R I D , 1 : 5 0 , 0 0 0 , NE-SW DIAGONALS G E N E R A L : v f N U M B E R OF NUM13ER OF T O T A L P O I N T S A R E A P E R C E N T N U M B E R 22 5 3 9 2 5E 0 8 0 • o. T R I A N GLE S 0 . L A K E S FOR +• 4 3 9 9 7 9 3 S E A - A N A L Y S I S - A R E A F O R A N A L Y S I S MEAN T R I A N G L E - A R E A = 39 2 0 . 4 3 9 9 7 9 3 6 E 0 8 0 . 0 6 1 2 4 9 9 4 6 9E G E O M O R P H O M E T R Y : M I N I M U M E L E V A T I O N = 4 . 9 4 L O C A T E D AT M A X I M U M E L E V A T I O N = 2 3 0 1 . 2 4 L O C A T E D AT = 1 6 0 6 . 3 0 = 0 . 2 8 0 5 4 L O C A L R E L I E F H Y P S O M E T R I C MEAN I N T E G R A L W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A A R E A R A T I O V E C T O R 0 , 3 5 8 3 6 0 . 3 5 8 3 7 = 0 . 3 7 1 5 2 = 1 . C 7 9 3 2 NU MPFR 2 2 1 ° 5 9 V E C T O R S ) : 5 8 . 6 4 D I P 8 3 . 3 9 L 9 3 . 0 7 m 1 4 . 4 0 100/K 6 . 9 5 R O U G H N E S S W E I G H T E D BY F A C T O R - T R U E 6 . 9 3 A R E A : O R I E N T A T I O N 5 5 , 6 2 D I P 8 7 , 9 4 9 2 , 7 1 U ) R O U G H N E S S V. - O R I E N T A T I O N K i• N U M B E R A N A L Y S I S : U N W E I G H T E D . U N I T L POINT POINT S L O P E : U N W E I G H T E O A R E A 6 9 F A C T O R - 7 , 2 9 * _ "PTARMI ~CR GAN "(SVIP'tE EEK T l V T -162OR I I : 2 50,"o 6 7 " 0 0 ~ ~~NW^S E D f >^G0>« Al'."S G E N E R A L NUMBER OF P O I N T S NUMBER OF T R I A N G L E S T O T A L A R E A P E R CE_N J N UM BER FOR +__S_E A j A N A L Y F O R MEAN T R I A N G L E ST A R E A E L E V A T I O N M A X I M U M E L E V A T I O N " H Y P S O M E T R I C 3 . 4 8 9 9 7 9 3 6 E = 0 . 1 2 4 9 9 4 6 9 E 7 0 l o 2 3 6 2 o I N T E G R A L S L O P E : ____ U N WE I G H T E D BY MAP W E I G H T E D RY T R U E 1 6 6 1 . 1 6 = 0 . 2 7 4 6 4 •= 0 . 3 5 4 6 0 - — A R E A A R E A = 0 . 3 7 2 7 5 1 . C 8 1 9 5 A N A L Y S I S : ' U N W E I G H T E D I U N I T V E C T O R S ) : O R I E N T A T I O N = O I P = _ ~ 1 0 0 / K W E I G H T E D 5 3 . 0 1 8 8 . 6 9 3 . 0 1 = 1 4 . 2 6 T R U E 6 . 9 9 A R E A : 0 R I E N T A T I O N =_ D I P - L ( ^ ) R O U G H N E S S 7 . 0 1 F A C T O R = BY 5 _= = R O U G H N E S S 2 0 = R A T I O " C4 _ """ W E I G H T E D V E C T O R 0 8 3 = R E L I E F A R E A 2 0 9 2 0 3 0 6 RY: M I N I M U M V E A N 5 39 0 • S= A N A L Y S I S GEOMOR P H G M E T L O C A L 22 0 . 4 3 9 9 7 9 3 6 E L A K E S , A R E A K 1 _ 5 1 ~~' .JU 8 3 . 0 4 = 9 2 . 4 8 F A C T O R = 7 . 5 2 L O C A T E D L O C A T E D AT AT P O I NT NUMBER 2 2 5 PU I N T N U M B E R 1 9 9 . -163P T A R M I G A N C R E E K ( S A M P L E 1 1 _._ G R I D , 1 : 2 5 0 , 0 0 0 , ... N E - S W .. D I A G O N A L S G E N E R A L : .NUMBER OF N U M B E R OF T O T A L I AN GLE 0 . 4 FOR FOR T R I A N G L E GEO MqRPHC.MET.RY_ I S = A R E A ; M I N I M U M E L E V A T I O N M A X I M U M E L E V A T I O N MEAN 0 . 7 9 3 6 E 1 2 4 9 9 4 6 9 E L O C A T E D A T P O I AT P O I N T 16 0 . 2 7 5 3 9 073 W E I G H T E D BY T R U E 0.3 A R E A V E C T O R 0. A R E A = R A T I O U N W E I G H T E D ( U N I T V E C T O R S ) : O R I E N T A T I O N = 5 3 . 3 0 DIP = 88.52 9 2 .83 ~ = 1 3 . 9 2 Li%) _____ _ K 1 0 0 / K R O U G H N E S S W E I G H T E D = 7 . 1 9 F A C T O R - 7 . 1 7 BY T R U E A R E A : O R I E N T AT I O N ~ " l i t ) R O U G H N E S S _ = " 5 1 * 0 1 _ = " 8 3 . 0 4 = 9 2 . 4 1 F A C T O R - 6 7 6 3" 6 7 6 9 3 8 2 2 9 1 . 0 8 2 7 4 A N A L Y S I S : 7 . 5 9 6 L O C A T E D SLOPE_: MAP 0 G4 1 6 6 1 o BY 2 0 8 2 . 2 0 7 0 l o 2 36 I N T E G R A L W E I G H T E D D I P 8 3 9 0 . 4 3 9 9 U N W E I G H T E D A R E A 2 0 OoO R E L I E F H Y P S O M E T R I C 39 9 3 6 E A N A L Y S I S ^ ANA L Y S _MEAN L O C A L 3 9 9 7 L A K E S S E A ; N U M B E R ' S- A R E A PERCE N T A R E A 221. P O I N T S TR NT N U M B E R N U M B E R 225 199 . M A N N I N G PARK ( S A M P L E 1 3 ) -164- ........ NW-SE in, DIAGONALS G E N E R A L : NUMBER OF P O I N T S 2 2 5 NUMBER CF T R I A N G L E S ^ 3 9 2 T O T A L A R E A P E R C E N T NUMBER 0 , 4 8 9 9 7 9 3 6 E L A K E S _ - t - _ S E A = F O R A N A L Y S I S " : A R E A F O R A N A L Y S I S MEAN T R I A N G L E A R E A 0 8 OoO 39 = 0 , 4 3 9 9 7 9 3 6 E = 0 , 1 2 4 9 9 4 6 9 E 2 0 8 0 6 G E O M O R P H C M E T R Y : M I N I M U M E L E V A T I O N M A X I M U M E L E V A T I O N L O C A L R E L I E F - H Y P S O M E T R I C M E A N _ I N T E G R A L W E I G H T E D BY M A P W E I G H T E D BY T R U E A R E A P O I N T N U M B E R 2 A T " H I N T N U M B E R 8 82 2o 9 6 6 5 8 9 — 0 , 3 3 1 0 3 DIP L(%) K 100/K A 0 , 3 = — E A " R O U G H N E S S W E I G H T E D — F A C T O R - BY I E N T A T T R U E I O N " — 5,75 A R E A : - " 3_50, = " " " " 8 9 %) R O U G H N E S S 1 , 0 6 3 4 7 343,67 = 8 9 , 3 9 = 94,25 17,33 = 5,77 = F A C T O R - " 2 1 0 9 0 , 3 3 3 4 5 V E C T O R S ) : O R I E N T A T I O N V. A T L O C A T E D 0 , 5 A R E A A N A L Y S I S : U N W E I G H T E O f U N I T L ( L O C A T E D 7 6 = = R A T I O V E C T O R D I P 3 0 1 8 8 9 o S L O P E : U N W E I G H T E D 0R 1 0 6 6 o 60 , 3 4 9 4 , 0 4 5 , 9 6 - • • — — - - f G E N E R A L : N U M 3 E R O F P0INT__S___ N U M3 E R OF T R I A N G L E T O T A L A R E A P E R C E N T 22_5 " = L A K E S NUMBER" = S= F O R M E AN T R I A N G L E A N A L Y S I S GEOMOR PHOMET 0 . 4 3 9 9 " " 3 9 2 7 9 3 6 E " . = A R E A 0 8 0 = o 0 3 9 2 0 . 4 3 9 9 7 9 3 6 E 0 8 0 0 6 0 1 24-99 4 6 9 E R Y : M I N I M U M E L E V A T I O N 1 0 6 6 . 3 0 L O C A T E D AT P O I N T N U M B E R 2 M A X I M U M E L E V A T I O N 1 8 3 9 „ 7 6 L O C A T E D AT P O I N T N U M B E R 8 L O C A L R E L I E F H Y P S O M E T R I C MEAN 3 2 2 o = I N T E G R A L 0 . 3 W E I G H T E D BY M A P W E I G H T E D B Y T R U E A R E A AREA — AREA 0 . 3 4 1 3 5 1 . 0 6 3 6 1 A N A L Y S I S : U N W E I G H T E D ( U N I T V E C T O R S ) O R I E N T A T I O N 3 5 1 . 7 2 D I P 8 9 , Li%) 9 4 . 2 1 K = " ~ 1 0 0 / K R O U G H N E S S I P - ~ Ll%) R O U G H N E S S _ 1 7 . 2 3 5 , 8 0 F A C T O R = 5 . 7 9 T R U E A R E A : = O R I E N T A T I O N D 3 6 = BY W E I G H T E O 3 4 7 3 0 , 3 3 4 7 5 R A T I O V E C T O R " 9 6 0 . 5 6 6 9 1 S L O P E : U N W E I G H T E D V. ~ + _ S E A = _ F O R ' A N A L Y S I S = A R E A ' ~ ~ "= •= F A C T O R = 3 5 0 . 6 0 " 8 9 . 3 4 9 4 . 0 3 5 . 9 7 . .._ -166- T AT LA LAKE (SAMPLE 2 4 ) : OR 1 0 NW-SE OI-AOONALS GENERAL NUMBER OF NUMBER OF TOTAL POINTS TP I A N G L E S = AREA PERCENT NUMBER 39 2 0 , 4 3 9 R 7 9 3 6 E LAXE S FOR t ANALY S I S= AREA FOR ANALYSIS MEAN TRIANGLE AREA OB 0 . 9 S E \= 3 9 2" = 0 . 4 3 9 9 7 S 3 6 E 0 8 = 0 . 1 2 4 9 9 4 6 9 E 0 6 G E O M 0R P HOME_TR11 M I N I M U M E L E V A T I O N = MAXIMUM ELEVA T 10 N = LOCAL RELIEF HYPSOMETRIC - INTEGRAL SLOPE:_ MEAN " " WEIGHTED BY MAP AREA WEIGHTED BY TRUE '""'"'"=" AREA = DIP = 0 , 0 4 C 3 4 3 . 3 8 9 9 . 8 7 K = 7 4 0 . 0 8 1 0 0 / K 0 . 1 4 FACTOR= TRUE 0 . 1 3 AREA: OR I E N T A T I O N I P ~ ROUGHNESS _34_8o 5 9 " FACTOR =' " 3 9 . 7 9 = 9 9 . 6 7 3 87 l.OCllo" 3 9 . 3 0 L ( %) BY P O I N T N U M B E R 3 0 AT P O I N T N U M B E R l_ 1 " 0 , 0 4 0 8 1 VECTORS): ORIENTATION D 0.'0 40 8 = = UNWEIGHTED.UNIT WEIGHTED A T 0 . 2 9 8 1 0 A N A L Y S I S : ROUGHNESS L O C A T E D jEjOjC A T E D 18 7 , 4 5 = RATIO VECTOR 3 28 _ U N W E I G H T E D " " " " AREA JO 5 . 3 1 0 9 7 , 0 . 1 3 ' ; T AT LA I L A K E -167- 2 4 ) : G R I D N E - S W D I A G O N A L S G E N E R A L : v NJJMBER OF P O I N T S NUMBER OF T R I A N G L E S - T O T A L i ' = A R E A P E R C E N T I ( S A M P L E N U M B E R 2 2 5 . = L A K E S FOR 0 , 4 S 9 9 7 93 6E. + _SEA_=__ F O R A N A L Y S I S M E A N T R I A N G L E 8 3 9 = A R E A 2 0 0 . 0 . A N A L Y S I S - A R E A 39 0 . = 4 3 ' - 9 7 9.3 6 F 0 „ 1 2 4 9 9 4 6 9 E 2 0 8 0 6 G E O M O R P H O M E T R Y : M I N I M U M E L E V A T I O N M A X I M U M E L E V A T I O N L O C A L zz R E L I E F H Y P S O M E T R I C MEAN = I N T E G R A L S L O P E : UNWE BY MAP W E I G H T E D BY T R U E A R E A A R E A R A T I O V E C T O R = 0.0 zz 0 . 0 3 9 4 1 — 1.001-30 V E C T O R S ) : = D I P = 8 9 . 7 9 L ( 2 ) = 9 9 . 8 7 - ~ 7 100/K W E I G H T E D 3 4 3 . 3 3 4 3 . 5 2 = R O U G H N E S S 0.13 F A C T O R - BY T R U E 0.13 A R E A : O R I E N T A T I O N - D I P = 3 9 . 7 9 LiZ) - 9 9 . 8 7 F A C T O R - 0 . 1 3 R O U G H N E S S L O C A T E D AT AT r>01 ° Q I NT NT NUMBER 3 0 N U M B E R 1 0 . 2 9 6 3 0 O R I E N T A T I O N ~ L O C A T E D 3 . 8 7 . 4 5 _ A N A L Y S I S : U N W E I G H T E D ( U N I T K 9 c 8 3 zz " O . C 3 9 3 5 I G H T E D W E I G H T E D A R E A v. 9 0 1 0 9 7 . 2 8 3 4 8 . 5 9 3 9 3 5 - -- — -•• - ( G H I TF.TL"i GENERAL ("SAMPT'E" LAKE" = NUMBER OF P O I N T S OF T R I A N G L E ' S - AREA PERCENT NUMBER AREA KAL S 22_5 " = L A K E S FOR FOR MEAN sTTTTlA G O : NUMBER" TOTAL ) ~ : ~ ~ G R i 0, " NW- 3 1 + ~ A N A L Y S I S 2 0 8 _ 0 . 0 ' = AREA 39 0 . 4 8 9 9 7 9 3 6 E SEA=_ A N A L Y S I S - T R I A N G L E " ~ ' 39 2 0 . 4 3 9 9 7 9 3 6 E - 0 8 0 6 > 4 9 9 4 6 9 E 0 . GEOMORPHOMETRY: M I N I M U M MAXIMUM LOCAL 1 0 6 zz E L E V A T I O N 52 1 1 9 1 . 7 7 zz I N T E G R A L 3 . 7 5 2 25 5, = R E L I E F H Y P S O M E T R I C MEAN _ E L E V A T I O N 0 . 3 3 3 9 5 S L O P E : U N W E I G H T E D W E I G H T E D BY MAP W E I G H T E D BY T R U E AREA AREA AREA R A T I O VECTOR — 0 . 1 8 6 8 = 0 . 1 8 6 8 6 .zz 0 . 1 9 4 3 6 zz 1 . 0 AN A L Y S I S : U N W E I G H T E D I U N I T V E C T O R S ) : ORIENTATION - 2.44 DIP = 8 5 . 8 9 ' = "43,96 L( %) K 100/K = ROUGHNESS F A C T O R - W E I G H T E D BY TRUE nRIENTjAJ)_ON_ . ^ p Lit) ROUGHNESS 97.73 2.27 2.27 A R E A : = —-- 2 . 2 6 g = 9 7 , 5 6 F A C T O R - 5 2 . 4 4 28 5 03 L O C A T E D AT P O I N T N U B E R 1 7 L O C A T E D AT POI NUMBER 1 5 2 NT M GHITEZLI 31): LAKE -169GRTOY NE-'SW" DTAGONALS" G E N E R A L : NUMBER OF NUMBER OF T O T A L I AN GLE S- A R E A Go 4 3 9 9 P E R C E N T _ L A K E S N U M B E R FOR +_ 793 SEA= 39 2 0 3 6 E _ )o A N A L Y S I S ^ A R E A F O R MEAN T R I A N G L E 5 22 P O I N T S TR A N A L Y S I S A R E A = Go 4 3 9 9 = 0 . 1 2 4 9 9 4 6 9 E 7 o r 6 E 0 39 2 0 8 0 6 G E O M O R P H O M E T R Y : MINIMUM MAXIMUM L O C A L E L E V A T I O N 1 0 6 3 o 7 5 E L E V A T I O N 2 25 5 . 5 2 R E L I E F 1 1 9 1 . 7 7 H Y P S O M E T R I C MEAN 0 . 3 3 4 3 1 I N T E G R A L S L O P E : 0 . 1 8 4 U N W E I G H T E D W E I G H T E D BY MAP W E I G H T E D BY T R U E A R E A A R E A 1 . 0 A N A L Y S I S : U N W E I G H T E O ( U N I T V E C T O R S ) O R I E N T A T I O N = 2 . 8 2 D I P = 8 5 . 8 8 L I S ) K " * _= 9 7 . 7 4 = 4 4 o " G 3 ~ 100/K = R O U G H N E S S W E I G H T E D BY 2 . 2 7 2 . 2 6 ARE T R U E _ - Lm A : 2.26 _= D I P R O U G H N E S S , F A C T O R * O R I E N T A T I O N " 0 . A R E A R A T I O V E C T O R 35Y53 9 7 . 5 7 F A C T O R ^ 7 4 0 . 1 8 4 7 5 2.43 1 9 2 4 0 2 8 0 1 LOCATED AT LOCATED AT P O I N T NUMBER P O I N T NUMBER 1 7 152 __ 0C1NA R I V E R ( S A M P L E 4 1 ) : - l / U - G R I D , N W - S E DIAGGN'ATS G E N E R A L : V, ( " ' NUMRER OF PO I N T S N U M B E R OF T R I A N G L E S * T O T A L ' j _ N A R E A P E R C E N T U M B E =_ = L A K E S R FOR F O R +•_ ?_25 0 . 4 A R E A ; MEAN_TR_I A N G L E S.E;\= A N A L Y S I S A R E A 3 0 2 0 3 = 0 . 0 6 1 2 4 9 9 4 6 9E 3 0 . M E X R Y J = E L E V A T = II N R E L I E F H Y P S O M E T R I C L O C A T E D AT P O I N T N U M B E R 2 2 5 L O C A T E D AT P O I N T N U M B E R 1 1 3 3 2 2 o 9 6 I N T E G R A L = U N W E I G H T E O 0 . 4 1 9 * " 0 7 3 W E I G H T E D BY MAP W E I G H T E D BY T R U E AREA -AREA R A T I O A N A L Y S I S 1 4 * 11%) ~ = 1 . 0 6 5 2 4 8 8 . _ K ' " " " ' " 0 . 3 2 4 4 5 0 . 3 3 4 9 5 2 6 6 . 8 0 D I P " " " ~ 9 1 7 . 2 0 F A C T O R * 5 . 8 0 5 . R O U G H N E S S W E I G H T E D BY T R U E ~~ 11%) R O U G H N E S S " ~ 8 1 A R E A : = _263_. OR I EjNTAT IQN_ ~ 7 7 ° 2 0 4 = 1 0 0 / K 44" = V E C T O R S ) : O R I E N T A T I O N 2 4 - : U N W E I G H T E C I U N I f DIP' 4 8 8 5 3 . 4 4 S L O P E : V E C T O R 3 0 . * 8 9 9 7 9 3 6 E E L E V A T I O N A R E A 2 0 = M A X I M U M MEAN 3 9 0 . 0 M I N I M U M L O C A L 3 9 9 7 92 6E A N A L Y S I S * : G E 0 M O R P_HC . 8 0 =" 83.30' = 9 3 . 9 0 F A C T O R * 6 . 1 0 -171OON A R I V E R 41): ( S A M P L E OR I!.), NE-SW D I A G O N A L S GENERAL NUMBER OF POINTS 225 392 NUMBER O F T R I A N G L E S = 0, 4 3 9 9 7 9 3 6 E OS TOTAL AREA 0 . 0 P E R C E N T L A K E S_ + _SEA_=_ _ 39 2' N U M B E R " F O R "A N A L Y S I 3 = " " AREA FOR A N A L Y S I S = 0 4 8 9 793 6E 0 3 06 MEAN T R I A N G L E A R E A = 0 1 2 4 9 9 4 6 9 E C GEOMORPHCMETRY: MINIMUM MAXIMUM LOCAL ELEVATION ELEVATION RELIEF HYPSOMETRIC 3 22.96 INTEGRAL MEAN S L O P E : UNWEIGHTED W E I G H T E D BY W E I G H T E D BY AREA RATIO VECTOR 3C< A 3 95 3, 4 4 0.42CC2 0 . 3 26 3 2 MAP A R E A TRUE AREA 0 , 326 34 0.3.3638 1 . 0 6 5 33 ANALYSIS: UNWEIGHTED.UNIT ORIENTATION DIP LIZ) K 100/K ROUGHNESS WEIGHTED 17.13 5.84 FACTOR* BY TRUE ORIENTATION DI P " 11%) ROUGHNESS VECTORS ): = 26 3.92 88.30 94.18 5.82 AREA: =_ 26_3_.3_0 • = . 3 8 . 3 0 = FACTOR* 93.89 6.11 L O C A T E D A T P O ! NT L O C A T E D AT POINT NUMBER 113
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Comparison of computer-based terrain storage methods with respect to the evaluation of certain geomorphometric… Mark, David Michael 1974
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Title | Comparison of computer-based terrain storage methods with respect to the evaluation of certain geomorphometric measures |
Creator |
Mark, David Michael |
Date Issued | 1974 |
Description | Topographic information can be digitized in several ways. Sampling may be surface-random (points selected according to partially or completely arbitrary criteria) or surface-specific (points selected according to their topographic significance). Surface-random sampling includes grids, contours, and randomly-located points. In this study, grid sampling and surface-specific sampling are compared. Surface behavior between sampled points is assumed to be linear. All aspects of surface form can be considered to reflect surface roughness. Horizontal variation includes the concepts of texture and grain, while vertical variation is discussed under relief. The relationships between these are embodied in slope and the dispersion of slope magnitude and orientation. The distribution of mass within the elevation range of a topographic surface is described under hypsometry. Parameters for investigation are selected from these categories. The variation of some selected geomorphometric parameters in southern British Columbia is examined via a stratified random sample consisting of forty-two 7 x 7 km areas. The values of some of these parameters are used to group the samples, and six are chosen for more detailed analysis. The relationships among the variables are examined using correlation analysis. For four geomorphometric measures (local relief, mean slope, roughness factor, and hypsometric integral), the theoretical errors involved in estimating the measures from the two selected terrain storage methods are discussed. The surface-specific point samples should produce better results than grids of reasonable densities. The latter, however, should require less digitization time and computer storage space per point. For at least local relief and hypsometric integral, grid error should be a linear function of grid spacing. Results of empirical comparison of the methods over the six selected areas are presented. The average surface-specific point data set is found to require some 2.6 times as much digitization time and 3.1 times as much computer storage space as the 15 by 15 grids used in the comparison. Computed estimates obtained from both of these data bases are presented for each of the four selected parameters, together with other estimates (obtained manually) in some cases. The average errors for the methods are found to differ significantly for local relief and mean slope but not for the hypsometric integral; for all three measures, the grids produce larger mean errors. The assumption of a linear relationship between grid spacing and grid error is used to estimate the grid spacing which would be required fo produce the same average error as the surface-specific points. For the three parameters used, these hypothetical grids are calculated to require more computer storage space and digitization time than the surface-specific point data sets. The influence of the density of surface-specific points and of base map scale appear to be related to the topographic texture. For a reasonably experienced terrain analyst, the reproduceability of these data sets appears to be good, although there remains a subjective element in point selection not present for grids. It is concluded that for a given amount of digitization time or computer storage space, better estimates of geomorphometric parameters can be obtained using sets of surface-specific points than using regular grids. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093157 |
URI | http://hdl.handle.net/2429/18980 |
Degree |
Master of Arts - MA |
Program |
Geography |
Affiliation |
Arts, Faculty of Geography, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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