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UBC Theses and Dissertations

Comparison of computer-based terrain storage methods with respect to the evaluation of certain geomorphometric… Mark, David Michael 1974

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£ • t A COMPARISON OF COMPUTER-BASED TERRAIN STORAGE METHODS WITH RESPECT TO THE EVALUATION OF CERTAIN GEOMORPHOMETRIC MEASURES by DAVID MICHAEL MARK B.A. , Simon Fraser University, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF 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 COLUMBIA February, 1974 In p resent ing t h i s thes is in p a r t i a l f u l f i l m e n t o f the requ i remen rs 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, I agree that, the L i b r a r y sha l l make i t f r e e l y a v a i l a b l e f o r reference and s t u d y . I f u r t h e r agree t h a t permission fo r ex tens ive copying o f t h i s thes is f o r s c h o l a r l y purposes may be granted by the Head of my Department: o r by h is rep resen ta t i ves . I t is understood that copying or p u b l i c a t i o n of t h i s t hes i s f o r f i n a n c i a l gain sha l l not be al lowed wi thout my w r i t t e n permiss ion . Department of Geography The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada D a t e March 11, 1974 Abstract Topographic information can be dig i t ized in several ways. Sampling may be surface-random (points selected according to part ial ly or completely arbitrary criteria) or surface-specific (points selected according to their topographic signif icance). 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 l inear. A l l aspects of surface form can be considered to reflect surface roughness. Horizontal variation includes the concepts of texture and gra in , whi le vert ical variation is discussed under re l ief . The relationships between these are embodied in slope and the dispersion of slope magnitude and or ientat ion. The distribution of mass wi th in 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 stratif ied random sample consisting of fo r ty -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 re l ie f , 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 digi t izat ion time and computer storage space per point. For at least local rel ief 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 digi t izat ion 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 wi th other estimates (obtained manually) in some cases . The average errors for the methods are found to differ signif icantly for local rel ief 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 digi t izat ion 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 reproduceabi l i ty 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 digi t izat ion time or computer storage space, better estimates of geomorphometric parameters can be obtained using sets of surface-specific points than using regular grids. - i i i -Table of Contents Page Abstract i List of Tables v i i List of Figures ' x Preface X l Chapter 1: Introduction 1 1 .1 : Precision of topographic map data 2 1.2: Notat ion 5 Chapter 2: Computer Terrain Storage Systems 6 2 . 1 : Digi t izat ion 6 2 . 2 : Surface-random sampling: grids 7 2 . 3 : Surface-random sampling: digi t ized contours 12 2 . 4 : Surface-specific sampling: points and lines 13 2 . 5 : Surface behavior 15 2 . 6 : Computer storage of terrain information 18 2 . 7 : Comparisons of approaches 20 2 . 8 : Conclusions 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 rel ief (H) 29 3 . 3 . 2 : Avai lable rel ief (H a ) 32 Page 3 . 3 . 3 : Drainage rel ief (H^) 34 3 . 3 . 4 : Applications of rel ief measures 35 3 .4 : 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 : Review and parameters to be investigated 51 Chapter 4: Terrain Var iab i l i ty in Southern British Columbia, and Relationships Among Variables 52 4 . 1 : Selection of sample areas 52 4 . 2 : Data col lect ion 58 4 . 3 : Data analysis 58 4 . 3 . 1 : Drainage density (D-|) 58 4 . 3 . 2 : Source density (D s) and peak density (Dp) 60 4 . 3 . 3 : Local rel ief (H) 60 4 . 3 . 4 : Mean slope ( t a n « ) 61 4 . 3 . 5 : Hypsometric integral (HI) °1 4 . 3 . 6 : Relationships among variables 62 - v -Page 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: l l lec i l lewaet map-area (82N/4E) 66 4 . 5 . 2 : Sample 11: Ptarmigan Creek map-area 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) 71 4 . 5 . 5 : Sample 3 1 : Ghi tez l i Lake map-area 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 : Local rel ief (H) 73 5 . 1 . 1 : Local rel ief: surface-specific points 74 5 . 1 . 2 : Local rel ief : regular grid 75 5 . 1 . 3 : Review 78 5 . 2 : Mean slope (tanoc) 78 5 . 2 . 1 : Computational procedures 79 5 .3 : Roughness factor (R) 80 5 .4 : Hypsometric integral (HI) 80 5 . 4 . 1 : Hypsometric integral: surface-specific points 85 5 . 4 . 2 : Hypsometric integral : regular grid 85 5 . 4 . 3 : Summary 88 - v i -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 : Digi t izat ion time and computer storage 93 6 .2 : Local rel ief (H) 95 6 .3 : Mean slope ( tan«) 95 6 .4 : Roughness factor (R) 98 6 .5 : Hypsometric integral (HI) 98 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 Chapter 7: Summary and Conclusions 108 References 111 Appendix I: Notat ion 121 Appendix I I : Topographic and related variables for 42 areas in southern 124 British Columbia Appendix I l i a : Computer program 126 l l l b : Triangular data-sets 135 I l lc : Computer results 144 —vi I — List of Tables Page Table 3 . 1 : Sizes of arbitrari ly-bounded areas over which local 31 rel ief was determined by various authors. 4 . 1 : Physiographic subdivisions of southern British Columbia, 56 after Holland (1964), wi th 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 wi th 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 rel ief 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 rel ief (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). 99 6 .5 : Estimates of hypsometric integral ( H ) , and analysis 100 of errors in these estimates. 6 .6 : Empirical comparison of errors for triangular data-sets 102 and 15 by 15 grids. - v i i i -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 105 on sample 18 for the four selected measures. - i x -List of Figures Page Figure 2 . 1 : Coefficients of determination as functions of sample size for 11 three sampling designs applied to three surfaces of varying complexity. 2 . 2 : Map to il lustrate types of surface-specific points and 14 lines. 2 . 3 : Form of the interpolated surface between two data 17 points for. various exponents in the general inter-polation formula. 3 . 1 : Forms of surface roughness. 24 3 .2 : Hypothetical topographic profi le i l lustrating various 33 rel ief measures. 3 .3 : Diagrammatic topographic profile i l lustrating the 40 relationships among re l ie f , slope, and roughness. 3 .4 : Relationships between local rel ief and roughness 44 factor. 4 . 1 : Physiographic subdivisions of southern British Columbia 54 wi th locations of stratif ied 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 69 for detailed analysis. - x -Page 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 incl ined plane wi th in 3 sample areas. 5 .3 : As in Figure 5 . 2 , but for a square-based pyramid. 6 . 1 : Sample 11a, an example of a triangular data-set from °2 the Ptarmigan Creek map-area. 82 84 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 data, wi th the specific aim of eventually producing a quanti tat ive 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 pi lot study for the eventual realization of the original object ive. Throughout this study, the writer has benefitted greatly from discussions wi th his thesis supervisor, Michael Church. He and J . Ross Mackay read and commented upon drafts of the entire thesis, whi le Thomas K. Peucker has reviewed certain sections. Michael C. Roberts and H. Olav Slaymaker have also provided helpful advice. 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, Of f ice 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 quanti tat ively 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 geomorpho-metry, "the measurement and analysis of those characteristics of landforms which are applicable to any continuous rough surface." In much of the geomorphometric l i terature, i t has been claimed that the drainage basin represents "the fundamental geomorphic uni t" (notably Chorley, 1969; see also Leopold et a l . , 1964; Wil l iams, 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 oversimplif ication i t is certainly a val id approximation to attribute al l land forms to the f luvial erosion of uplif ted rock masses" (p. 78). He stated that this assumption was necessary in order to develop "a unif ied framework for landscape geometry. " Since about one third of the earth's land surface was glaciated during the Pleistocene (cf. F l int , 1971, p. 19), and as other processes such as f luv ia l deposition, or aeol ian, vo lcanic, or periglacial action have also influenced large areas, i t is the writer's opinion that a "unif ied 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 identi f ied as a drainage basin, an a l luv ia l fan, 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 widely employed in both geography and the earth sciences, and geomorphology has not been an except ion. A recent book edited by Chorley - 2 -(1972) attests to the fact that spatial aspects of land surface form have received much at tent ion. Whi le computers have been used in 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 wi th sets of contiguous triangles, and of Evans (1972), whose work was based on regular square grids ("alt i tude matrices"). Neither of these works studied the comparative accuracy, precision, and eff iciency of computer terrain storage methods, the differences between computer estimates and "standard" methods for estimating geomorphometric measures, or the relat ive dig i t izat ion (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 geomorpho-metric parameters, although some attention w i l l be directed toward measures based specif ical ly upon landforms of f luvial ac t i v i t y , probably the most important single class of processes which has shaped the earth's surface. A l l 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 lat i tude, mostly from 1:50,000 scale maps. Since al l topographic data used in this study w i l l be derived from contour maps, i t seems in order to discuss brief ly 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_. For example, they stated that survey s e values for triangulation points are generally less than 0.5 m horizontally - 3 -and 0.1 m vert ical ly (pp. 9-10), whi le root-mean-square errors in map plott ing 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 interpolat ion" (p. 14). They presented a graph of al lowable standard deviations in metres as functions of ground slope for various countries and map scales (their Figure 2) . Thompson and Davey (1953, p. 40) ci ted the accuracy specifications of United States Geological Survey topographic maps as: Vert ical accuracy, applied to contour maps on al 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 interval . In checking elevations taken from the map, the apparent vert ical error may be decreased by assuming a horizontal displacement wi th in the permissible horizontal error for a map of that scale. They used the 90 per cent cri terion in conjunction wi th a table of ordinates of the normal curve to estimate the al lowable s e value as about 0.3 times the contour in terval . The conversion of horizontal errors into vert ical ones involves the tangent of ground slope. Standards for Canadian topographic maps do not appear to be as wel l def ined. W . A . Williamson (pers. wri t ten comm., 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 te rva l . " If i t is assumed that this represents a 95 per cent confidence leve l , the ordinates of the normal curve can be used fo estimate the al lowable root-mean-square height error as 0.255 times the contour in terval . Will iamson also stated that for Canadian Class A maps, points are to appear wi th in 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 in terva l , the root-mean-square error for the maps used in this study should be given by: s e = t ( 7 . 8 + 25.0 tan 3 ) metres (1.1) 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" (Wil l iamson, 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 rel ief estimates (cf. Morisawa, 1957; Giusti and Schneider, 1965; Eyles, 1966; Gregory, 1966a, 1966b; McCoy, 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 ie ld 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 simplif ied through the use of the "blue l ine" stream network shown on the topographic maps. - 5 -1 .2 : Notat ion Throughout this paper, terms and symbols are defined where they are first introduced. In addit ion, a complete listing of a l l symbols used w i l l be given in Appendix I. Where there are "standard" symbols for variables, these w i l l be used unless ambiguity would result. Furthermore, x and y are reserved to indicate geographical locat ion, z elevation above sea leve l , N a number of objects or occurrences, D a density value, r a correlation coeff ic ient , and s a root-mean-square value. The metric system of units is employed throughout, wi th British units being given in some instances. Elevations obtained from the maps were in feet, but were converted to metres before analysis. - 6 -Chapter2: Computer Terrain Storage Systems Topography can be considered to be a continuous surface, and thus even a small area contains an inf ini te number of points; the number of points which can be measured is l imited 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, whi le at the same time maximizing the eff ic iency wi th 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 two-fo ld : one aspect involves the col lect ion of topographic information from maps or other sources, whi le the second relates to the storage, re t r ieval , and processing methods employed. 2 . 1 : Digi t izat ion Digi t izat ion can be defined as the process by which "analog measures", such as length or location on a map, are converted into "d ig i t a l , computer-usable form" (Peucker, 1972, p.72). Two distinct d ig i t izat ion strategies are avai lable: one involves sampling at surface-random points or lines, whi le 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 part ial ly or completely arbitrary set of c r i te r ia . 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) - 7 -is 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 widely used method for storing and processing three-dimensional surfaces is probably the square g r id , also known as the "alt i tude matrix" (Evans, 1972, p. 24) , or as "both a digi ta l 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. On ly the alt i tude of the surface at each sample point must be measured and stored wi th in the computer — the geographical locations are determined by the grid spacing, and are impl ic i t in the sequential position of the alt i tude value wi th in the computer storage array. A wide variety of computer programs for the processing of gridded data is avai lable. Another advantage lies in the fact that the neighbours of a given data point, which are often required in the calculat ion of geomorphometric parameters, can be readily obtained, once again from the positions of points wi th in the computer array. The principal disadvantage of the regular grid is its tendency toward redundancy — the grid must be made suff iciently dense throughout to portray the smallest objects which must be shown anywhere wi th in the area covered by the gr id . 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 apar t . " Thus a regular grid wi th a grid spacing d can only be expected to depict those variations of the surface having wavelengths of 2d or more. If the smallest signif icant wavelength of object one wishes to detect or portray anywhere wi th in 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 "digi ta l terrain l ibrary" . Because of the wide application of this terrain storage method, the larger number of gridded terrain samples already col lected, and the number of computer programs avai lable, 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 " , whi le the other was termed the "variable gr id" method by Boehm (1967, p. 404). The regular triangular grid has some advantages over the square grid approach. Each point has six neighbours which form a regular hexagon, and Mackay (1953) discussed how this form of data col lect ion avoids the "saddle point problem" which sometimes arises in attempting to draw isopleths based on a square gr id . The advantage in this regard is probably out-weighed by the increased complexity involved in indicating geographical location impl ic i t ly in the computer storage a l locat ion. 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 appl ied; 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 be. If the smallest significant terrain wavelength wi th in 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 co l lected, 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 relat ive 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 imply." (p. 591) Since even the regular grid points are random wi th 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) wi th reference to the sampling of elevations to be used in the construction of hypsometric curves. Because of their inherently uneven distr ibution, 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 comm., 1970; pers. wri t ten comm., 1973) investigated the relative "information contents" of randomly-located and gridded elevation samples; this unpublished study represents the only quanti tat ive comparison known to the wr i ter . Three surfaces were first represented by 150 x 150 grids; the surfaces were a plane, a fourth-order polynomial, and a 23.7 km - 1 0 -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 stratif ied al igned" (regular grid) , and "systematic stratif ied 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 coeff icient 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 impl ic i t geographical location and impl ic i t 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 stratif ied unaligned, systematic stratif ied aligned) applied to three surfaces of varying complexity (after W . D . Rase, unpublished study). - 1 2 -2 .3 : Surface-random Sampling: Digi t ized Contours Contours represent another way of sampling and storing a terrain surface. It must be noted that the elevations of the contours are f ixed by sea level (or other datum) and the contour in terva 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 digi t ized 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 automatical ly, 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 d igi t ized contours and of al t i tude matrices. He noted that whi le the former method is superior i f 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 locat ion. Since the latter sort of question arises much more often in geomorphometric snalysis than the former, i t would seem that d ig i t ized 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 imp l i c i t l y , and two values per point must be expl ic i t ly stored. Boehm (1967) described a "contour tree ordering method" for storing surface information; this method is said to be more ef f ic ient in problems where "successive specified points are correlated, such as in l ine-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 direct ly from contour data (see section 3 .4 .2 ) ; routines for producing contours from grids are widely avai lable, and the inverse process, that is, - 1 3 -producing grids from digi t ized contour data, is also possible. These processes would both involve interpolat ion, 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 "signif icant 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 surface-specific lines. (See Figure 2 . 2 . ) These lines include ridges, course lines, and breaks of slope. 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). Since both Peucker (1972, p. 24) and Woldenberg (1972, p. 327-330) have recently reviewed this work and as i t is not direct ly 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. 72) , however, claimed that "surface-specific points have a higher information content than surface-random points." 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 dig i t iz ing time than do an equivalent number of grid points, since al l three co-ordinates must be exp l ic i t l y determined and stored. There is an element of subject ivi ty in the selection of surface-specific points and, furthermore, neighbours cannot easily be determined from the points alone. S T R E A M RIDGE " P I T • PEAK • STREAM JUNCTION o R IDGE JUNCTION — 5 0 0 — C O N T O U R P A S S Figure 2 . 2 : Map to i l lustrate types of surface-specific points and lines. - 1 5 -2 . 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 al 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 hil ltops (ct\ • K ing, 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, whi le 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. Robinson (1960, 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 points." Peucker (1972, p. 25) noted that linear interpolat ion, 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 method." Peucker goes on to point out , however, that such surfaces have discontinuities in the first derivative (i .e . ,have "breaks of slope") which may produce an "unpleasant" appearance in block diagrams or contour maps. Perhaps for this reason, most computer algorithms for producing - 1 6 -dense regular grids from a less dense sample of points, (e.g. UBC XPAND; 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 t ime. A surface thus produced is continuous in the first derivative and therefore appears "smooth". A general interpolation formula may be expressed as: 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 interpolat ion, jJ = 1 , whi le in the more common interpolation algorithms discussed above,0 - 2. There has been l i t t le 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 id , triangular planar facets for determining slope or other parameters can be produced in two different ways. In one approach, one set of diagonals is arbitrar i ly inserted. Turner and Miles (1967, p. 260) determined a roughness parameter for the two orientations of diagonals and found very l i t t le difference in the results. A l ternat ive ly , addit ional 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 . 1 I (2.1) - 1 7 -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 ) . - 1 8 -2 . 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 rect ly , as a matrix of elevations for gridded data, as lists of x and y co-ordinates for digi t ized contours, or as al l three co-ordinates for surface-specific points. The surface between the points would then be determined during the processing stage after ret r ieval . In the case of irregularly-distr ibuted points such as surface-specific points, however, processing w i l l be much more eff ic ient 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 identi f icat ion number and co-ordinates for each point. He then listed a l l the neighbours (by identi f icat ion number) for some arb i t rar i ly -chosen starting point. Nex t , for each of these neighbours, a l l adjacent points excluding the starting point are g iven, and the procedure is continued unti l every link between neighbours has been included exactly once. During processing, the computer forms triangles, beginning at the arbitrary starting point. If al l 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 l ist, making it easier to f ind 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, Off ice of Naval Research, Task N o . 710-100, Department of Geography, Simon Fraser University, Burnaby, British Columbia, T . K . Peucker, principal investigator  - 1 9 -Another approach is to store the point numbers and co-ordinates, and then to store a list of triangles, each record containing the tr iangle number and the identi f icat ion numbers of the three points making up its vertices — this is termed the "tr iangle mode" of storage. Other characteristics of the triangle could also be indicated. This form of data organization is somewhat easier to prepare, and is also more eff ic ient for " t r iangle-by- t r iangle" processing required for most geomorphometric analysis. The triangle mode was used by Akin (1971), wi th elevations replaced by precipitation values, in the calculat ion of the mean areal depth of precipi tat ion. The triangle mode should be far less eff ic ient 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 f ind an expl ic 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 di f ferent iated, 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 e levat ion. Junkins and Jancaitis (1971) found that this approach was an order of magnitude more eff ic ient than the method of evaluating the surface equation at a large number of - 2 0 -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 again, however, there are often arbitrary assumptions about surface behavior; also, Hardy's method requires that the data points and one coeff icient per point be stored, resulting in l i t t le saving of storage space, although processing may be speeded up. 2 . 7 : Comparisons of Approaches Boehm (1967) compared f ive methods of surface storage: contour points sorted in the x-d i rect ion (CS), "contour tree ordering" (CT), uniform grid (UG) , uniform gr id-di f ferent ia l al t i tude (UGDA) , and variable gr id-di f ferent ial alt i tude ( V G ) . "Dif ferential a l t i tude" means that alt i tude 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 intervisibi l i ty between points on the surface, determining both storage requirements and processing speed. The grids were most ef f ic ient in terms of processing speed, wi th the uniform grid the best, whi le the variable grid required the least storage. Some other com-parisons of methods have already been ci ted 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 solved." Thus the results of studies by Rase (see Figure 2.1) and Boehm (see above) are not direct ly 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 - 2 1 -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 l inear. - 2 2 -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. Attent ion 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. No attempt w i l l be made to review papers approaching landscape analysis through a set of landform "elements", "un i ts" , 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 quantitat ive 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 rat io). 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 wi th a discussion of the general concept of roughness before proceeding to actual geomorphometric parameters. - 2 3 -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 (re l ief ) , an irregularity of ridge spacing, or sharper ridges (see Figure 3 .1) . Stone and Dugundj i , in a study of microrel ief profi les, used five measures, whi le 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 exp l ic 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, whi le val ley spacing wi th in 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, whi le the amplitudes associated wi th these wavelengths correspond to the concept of re l ief . The relationship between the horizontal and vert ical dimensions of the topography is embodied in the land slope and the dispersion of slope magnitude and or ientat ion, whi le the vert ical distribution of mass under the topographic surface is contained in the concept of hypsometry. - 2 4 -Figure 3 . 1 : Forms of surface roughness. B, C, D, and E are "rougher" than A in some respect. B has a shorter wavelength, C a higher amplitude, D an irregularity in spacing, and E a "sharper"form. - 2 5 -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 signif icant 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 rel ief wi th in 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 mi le , and suggested that i f there is no knick point, rel ief 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 rel ief for a standard arbitrary area. In the present study, "gra in" is also used less formally to refer to the longest signif icant topographic wavelength. Other parameters should be sampled over areas larger than or equal to the grain size in order to obtain representative values. - 2 6 -3 . 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 rat io: 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 fol lowing empirical relationship: D d = 1.658 T1 ' - 1 1 5 (3.2) Smith did not give confidence limits for the regression coeff icients, 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 network", 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 (cf . Carlston, 1963), mean annual precipitation (c f . Chorley and Morgan, 1962), and - 2 7 -sediment y ie ld (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 f lowing channels at a particular time is closely related to instantaneous stream discharge. Thus drainage density for f lowing channels only w i l l vary over short periods of t ime. Evans (1972, p. 33) suggested that i f only high order streams are considered, the inverse of val ley density should provide a useful expression of overall topographic gra in , 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 . McCoy, 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 direct ly 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 cl imatic 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 randomly-oriented 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 profi le is continuous, maxima and minima must al ternate, 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 wi th other studies. - 2 8 -Anorher parameter very closely related to drainage density is the source density ( D s ) , the number of stream sources per unit area (see Mather, 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. This, however, introduces an element of subject iv i ty. The qual i ty of the b lue- l ine 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 hi l l top contours per unit area, here termed the peak density (Dp). Wood and Snell (1959) used this as one of their parameters for classifying terrain. They considered any closed contour (other than a pit) to be a " h i l l t o p " . 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. 41), and "val ley character", the number of closed val ley contours, most of which represented drumlins. Swan (1967) mapped "h i l l frequency" as the density of hil ls per square mi le . A h i l l was defined as any summit wi th two or more closed contours, or wi th 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 ret icu lat ion, which was a measure of the size of "the largest connected network of crests that - 2 9 -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 para l le l . 3 .3 : Relief Measures The term rel ief is used to describe the vert ical dimension or amplitude of topography. Evans (1972, p. 31-32) noted that the majority of rel ief 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 vert ical var iabi l i ty of the terrain. He did observe that "the autocorrelation of al t i tude 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 deviat ion. A l l of the other papers reviewed herein have, however, used extreme values to characterize the vert ical dimension. 3 . 3 . 1 : Local Relief (H) For any f in i te area of a surface, the local rel ief is defined as the difference between the highest and lowest elevations occurring wi th in that area. It is important to note that local rel ief is always defined with respect to some particular area, and perhaps for this reason has sometimes been termed the "relative re l ie f " (cf. Smith, 1935). This measure was apparently introduced by Partsch (1911), who termed it the rel iefenergie, 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 wi th introducing the concept of local rel ief into the English language l i terature, but Huggins apparently presented his paper at a professional meeting some months earl ier. - 3 0 -These works, as wel l as many others (see Table 3 .1 ) , determined local rel ief for arbitrari ly-bounded terrain samples such as squares, c irc les, or lat i tude-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 rel ief readings are made appears to need adjustment for the degree of coarseness or fineness of the rel ief pat tern. " 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 rel ief for a circle wi th a radius equal to the grain. Wood and Snell (1957, 1959), Peltier (1962), and Evans (1972) compared the values of local rel ief 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 i t is unl ikely to contain a whole slope, ' r e l i e f becomes simply a measure of gradient;" in order to make rel ief "as distinct and non-redundant a variable as possible" (p. 31), he recommended the use of " fa i r ly large" sample areas. The areas should def ini tely be larger than the texture of the topography, and preferably larger than its gra in . Data from Wood and Snell (1959, p. 9) support Evens' contention — they found that the correlation between rel ief 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 lacia l deposits, and found the two to be closely related for older dri f t 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 rel ief was determined for arb i t rar i ly -bounded sample areas; local rel ief has also frequently been determined for drainage basins. The minimum elevation w i l l be the basin mouth, whi le the - 3 1 -TABLE 3 . 1 : SIZES OF ARBITRARILY-BOUNDED AREAS OVER WHICH LOCAL RELIEF WAS DETERMINED BY VARIOUS AUTHORS area type of * of authors date (km^) area sizes Studies using one sample size: Chen 1947 1.00 square Harris 1969 1.00 square Hesler & Johnson 1972 2.59 square Swan 1967 3.34 square Abrahams 1972 7.52 square Huggins 1935 10.4 square Donahue 1972 10.4 square Batchelder 1950 15.0 quad . * Zakrzewska 1963 23.3 square King 1966 25.0 square Kaitanen 1969 25.0 square Hutchinson 1970 25.0 square Partsch 1911 32.0 -Trewartha & Smith 1941 34.0 quad. Smith 1935 65.5 quad. Hammond 1964 93.2 square Spreen 1947 203. circle Ahnert 1970 400. square More than one size used: Gassmann & Gutersohn 1947 Evans 1972 Wood & Snell 1957, 1959 Peltier 1962 Wood & Snell 1960 0.25 to 28.0 square 13 0.63 to 62 .7 square 18 0.81 to 414. circle 8 2.59 & 259. square 2 18.3 to 399. c i rc le 7 * quad. = lat i tude-longitude quadrangle - 3 2 -maximum is usually, but not always, located on the basin perimeter. Maxwel l (1960, p. 10-11) determined the "basin re l ief" 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 te r ia , but was essentially "the longest dimension of the basin parallel to the principal drainage l i ne " . Since the size of drainage basins w i l l vary, many workers have found i t desirable to determine a dimensionless "rel ief rat io" or "relat ive rel ief number" by dividing the rel ief by some other linear dimension of the basin. The latter have included the basin diameter (defined above), basin perimeter (Mel ton, 1957) and square root of basin area (Mel ton, 1965). 3 . 3 . 2 : Avai lable Relief (H n ) The concept of available rel ief was introduced by Glock (1932), and his defini t ion was rephrased by Johnson (1933, p. 295) to read: "Avai lable rel ief is the vert ical 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 identi f ied from remnants and where there were "graded" streams. The latter involves the definit ion of the concept of "grade", which w i l l not be discussed here. Local, available and drainage rel ief are il lustrated diagrammatically in Figure 3 . 2 . Glock stressed the importance of available rel ief in determining the land profi le but, as Johnson noted, other factors such as drainage density and slope must also be considered. In order to determine the average available re l ief , one would have to construct both the "or ig ina 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. - 3 3 -M A X I M U M E L E V A T I O N • S T R E A M ' A T G R A D E ' O O T H E R S T R E A M Figure 3 .2 : Hypothetical topographic profi le i l lustrating various rel ief measures. H is the local rel ief for the entire prof i le , H q Glock's available re l ie f , and the drainage re l ief . Dury's "avai lable rel ief" would be the mean height of the shaded port ion. - 3 4 -A different rel ief measure was discussed by Dury (1951), who unfortunately also used the term "avai lable re l ie f " ; this was defined as "that part of the landscape wh ich , standing higher than the floors of the main valleys , may be looked on as available for destruction by the agents of erosion working with reference to existing base-levels" (p. 339). He then defined the "mean available re l ief" 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 avai lable rel ief defined by Glock (1932) and Johnson (1933). Dury (1951, p. 342-3) also discussed the depth of dissection", which is identical to the Glock/ johnson concept of available re l ief . 3 . 3 . 3 : Drainage Relief (H-j) Glock (1932, p. 75) also defined a measure called the drainage rel ief as "the vert ical distance through which rain water moves over the ground from the time the water first strikes the surface unti l i t joins a definite stream." Johnson (1933, p. 301), however, pointed out that Glock later used the term to refer to the vert ical distance between adjacent divides and streams, and proposed that this latter def ini t ion be adopted (see Figure 3 .2 ) . If in an area al l the divides are at the elevation of the original upland surface and al l the streams are "at grade", drainage rel ief w i l l equal available rel ief; in contrast to the latter, however, drainage rel ief can always be determined. Strahler (1958, p. 295) stated that " local re l ie f , H, is a measure of vert ical distance from stream to adjacent d iv ide" ( i . e . , local rel ief is equivalent to drainage re l ie f ) , but this w i l l only be true i f the sample areas upon which local rel ief 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 signif icantly increase the rel ief wi th in 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 - 3 5 -the mittlere Taltiefe ("mean val ley depth") for drainage basins. First, a "roof" was constructed over the basin by l inking points along the basin divide which were equidistant from the basin mouth by straight l ines. The volume between this surface and the land surface wad divided by the basin area. This measure is "complementary" to Dury's "mean available re l ie f " . 3 . 3 . 4 : Applications of Relief Measures Relief has commonly been used in a descriptive way ( e . g . Smith, 1935) or to del imit physiographic regions ( e . g . Huggins, 1935), both alone and in conjunction wi th other variables. Some studies have, however, related rel ief to landscape processes, or to other aspects of physical geography. Schumm (1954, 1963) found that sediment y ie ld was closely related to the ratio of basin rel ief to basin diameter for some small drainage basins in the southwestern United States. Schumm (1956) also related sedimentyield to rel ief and slope for some smaller basins in the Perth Amboy badlands. Maner (1958) investigated the relationships between sediment y ie ld and a number of basin characteristics, and found that the above rel ief ratio was the one most highly correlated wi th the dependent var iable. Ahnert (1970) determined average basin rel ief as the mean of local rel ief 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 rel ief; by using the empirical relationship between denudation rates and re l ief , Ahnert presented theoretical curves for rel ief reduction as a function of t ime, both wi th and without the effects of isostatic compensation. He later (1972) related these results to theoretical models for slope processes. - 3 6 -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 work . " Mathematical ly , the tangent of the slope angle (tano<) is the first derivative of a l t i tude, and i t 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 grav i ty . Strahler's (1950, 1956) work suggested that slope tangents had a normal distr ibution; Speight (1971), however, found that for a number of areas investigated, a log-normal distribution provided a better f i t . Unlike rel ief and most other geomorphometric parameters, which are only defined for f in i te subareas of a surface, slope is defined at every point as the slope of a plane tangent to the surface at that point. In pract ice, however, slope is generally measured over a f in i te 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 widely appl ied. 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" - 3 7 -must be appl ied. If the traverse l ine 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 ke ly , 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 wi th L in miles and I in feet as: I ( N / L ) tan oc = 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; Gr i f f i ths, 1964). Other authors have used the number of contour intersections per length of traverse d i rec t ly , 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 wi th 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) subject ively. 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 . Gr i f f i ths, 1964). Another method which has been widely used depends upon determining the slope - 3 8 -at a sample of points distributed over the study area; these values may be averaged (c f . Strahler, 1956; Coulson and Gross, 1967) or "contoured" (c f . 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 digi t ized contour data. Sharpnack and Akin (1969), as wel l as Rase (1970, pers. oral comm.), computed both slope and aspect from an alt i tude matr ix. Griff i ths (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 wi th less e f fo 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). Mathematical ly, this is the second derivative of a l t i tude, 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 ing quadratic surfaces to 3 by 3 sections of an alt i tude matr ix. Convexity could then be determined by differentiating the resulting quadratic equation tw ice . 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 tude, but the possible physical meaning of such higher derivatives is obscure. - 3 9 -Closely 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 pro f i le , the relationship among re l ie f , drainage density, and slope can be easily shown. In Figure 3 . 3 , H is the rel ief and b half the distance between channels, which equals half the inverse of D^. One thus has the mean slope given by: t a n * = H/b = 2 HD D (3.5) or twice the ruggedness number. Strahler (p. 295) also introduced average slope into the ruggedness number, producing the geometry number: HDJ — 2 - (3.6) tan o( If H is a reasonable estimate of the drainage rel ief and i f 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 whi le 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 : Appl icat ion of Slope Measures As in the case of rel ief (section 3 . 3 . 4 ) , slope has been widely used in descriptive work, in physiographic classif icat ion, and in mi l i tary work related to vehicle t ra f f icab 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 profi le form represents an important "sub-discipline of geomorphology (for example, Institute of British Geographers, 1971; Carson and Kirkby, 1972; Young, 1972). - 4 0 -Figure 3 .3 : Diagrammatic topographic profi le i l lustrating the relationships among re l ie f , slope, and roughness (see equations 3 . 5 , 3 .10 , and 3.11) . 3.5: Dispersion of Slope Magnitude and Orientation In addition to slope steepness, slope aspect or direction may be considered, either separately or together wi th 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 var iabi l i ty in gradient from var iabi l i ty in aspect." 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 potent ial ly 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 gr id . 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 Newel l (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 (c f . Fisher, 1953; Steinmetz, 1962). Unit vectors orthogonal to triangular facets formed by inserting diagonals into a regular gr id 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: - 4 2 -k = ( N - l ) / ( N - R ) (3.7) As a surface approaches planari ty, the vectors w i l l become para l le l , R w i l l approach N (the number of vectors), and k w i l l become very large. Intui t ively , a plane should have a roughness of zero, and thus the inverse of k would represent a more "reasonable" roughness measure. Since Hobson's method was based on a regular g r id , al l triangles have the same horizontal area and similar true areas, and hence the use of unit vectors is not unreasonable. If based upon irregularly-distr ibuted 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: 100 - L(%)| _1_ k N N - l _R_ N 100 (for large N) (3.8) 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 = 1 0 0 - 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 rel ief and ridge spacing through reference to Figure 3 . 3 . For R, the vert ical component of each orthogonal vector w i l l equal coso<, whi le the horizontal components w i l l cancel out, leaving: R = 100 (1 - coso<) (3.10) - 4 3 -Substituting the value for cos * gives: R = 100 (1 - b ) (3.11) V H 2 + b 2 Turner and Miles (1967) applied Hobson's (1967) vector program to twenty- f ive 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 rel ief (H) , and a var iabi l i ty factor v , the local rel ief divided by the logarithm of k; the writer estimated R as the inverse of k. Linear correlation coefficients were determined among ten roughness measures for the twelve common terrain samples. In addi t ion, 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 signif icantly correlated over the twenty- f ive 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 f i t ted "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. - 4 4 -H (metres) Figure 3 .4 : Relationship between local rel ief and roughness factor. Open symbols represent micro-terrain from Turner and Miles (1967). Solid symbols are macro-terrain (circles from Turner and Mi les; triangles from this study). Curves are based on equation 3 . 1 1 . - 4 5 -ln a related line of research, Hayre and Moore (1961) determined theoretical scattering coefficients for terrain, 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 ude . " Evans (1972, p. 42-48) reviewed this concept under the heading: "Regional convexity (dissection, aerat ion) . " Most of these measures, which describe aspects of the distribution of landmass with e levat ion, 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 cal led 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 levat ion, and by convention, elevation is plotted on the y-ax is . If relat ive area is used, the diagram is a plot of the probabil i ty 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 levat ion, and is essentially a cumulative frequency for the elevations. Once again, elevation is conventional plotted on the y-axis and area (representing frequency) on the x -ax is . - 4 6 -Clarke (1966, p. 241) pointed out that this curve does not represent an "average p ro f i l e " , since i t does not record the slope between contours. Never-theless, a section of the curve wi th a low slope indicates a larger amount of the surface at or near a particular elevat ion; this would generally indicate gentler slopes near that elevat ion. Absolute hypsometric curves have been determined for the earth's surface as a whole, 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-probabi l i ty 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 ar i thmetic-probabi l i ty paper. The third and most widely used form of curve is the relative or percentage hypsometric curve, often termed simply the hypsometric curve. ^ It plots relat ive area above a height against relative height, and is the graph of the hypsometric funct ion, here termed a (h) , where h (the relat ive height) is defined by: z - z . max mm where z is the actual e levat ion, and z and z . are the highest and lowest max mm ° elevations, respectively, wi th in the study area. As in the previous cases, h is conventionally plotted on the y-ax is . 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 Mor ley , 1959, p. 566); relative hypsometric curves were presented earlier by Imamura (1937, cf . Evans, 1972, p.42) , Gassmann and Gutersohn (1947), and Langbein and others (1947), only the latter being ci ted by Strahler.  _47-3 . 6 . 2 : The Hypsometric Integral (H) The most widely used parameter based on the hypsometric curve is the hypsometric in tegra 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 geometr ical ly, 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 sol id" bounded by the perimeter of the area and planes passing through the minimum and maximum points. Graphica l ly , 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 "youthful" or equil ibrium phase, whi le drainage basins in equil ibrium should have hypsometric integrals between 0.6 and 0 .35 . Values below 0.35 were thought to characterize a transitory "monadnock phase" in landscape development. Pike and Wilson (1971) proved that the elevat ion-rel ief ratio (E) of Wood and Snell (1960) is mathematically equal to the hypsometric integral . The former is defined by: E = Z " Z m ? n (3.14) z - z . max mm where z is the mean elevat ion. From equations 3.12 and 3 .14 , 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 "coeff ic ient of relat ive massiveness" by Mer l in (1965). While Srrahler's (1952) method for determining the hypsometric integral involves much - 4 8 -laborious use of a planimeter to determine inter-contour areas, the e levat ion-rel ief ratio can be determined much more qu ick ly , with the mean elevation being determined from a sample of points. Pike and Wilson (1971, p. 1081) stated that "experience has shown that a sample of 40 to 50 elevations w i l l ensure accuracy of E to , on the average, 0 . 0 1 , the value to which area-alt i tude 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, Evans1 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, wi th a considerable saving in t ime. 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 cyl inder wi th an inverted cone, centered at the lemniscate or ig in" (p. 556). The accuracy of this method depends upon the degree to which the geometrical form actual ly approximates the basin, part icularly the f i t 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 fal l ing wi th in al t i tudinal bands were used in producing hypsometric curves. They found that their method produced results closer to planimetered values than did the corrected - 4 9 -Chorley and Morley approach. It would seem that the elevat ion-rel ief ratio represents a more accurate and more easily applied approximation to the hypsometric integral than do the above. Furthermore, the elevat ion-rel ief ratio can be determined for arbitrari ly-bounded areas (Wood and Snel l , 1960; Pike and Wilson, 1971), whi le 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. Low values of this parameter indicated very sinuous curves. Evans (1972, 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 wi th 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 in tegral , 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 detai l 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 funct ion. They also determined the rel ieffactor, which equals twice the - 5 0 -kotenstreuung divided by the local re l ief . This is twice the standard deviation of the relat ive hypsometric funct ion. Gassman and Gutersohn also determined the mean elevation by using the hypsometric integral , "reversing" the use of the elevat ion-rel ief 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 tude, 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, p. 44-45) , most w i l l not be reviewed herein. Hammond (1964, p. 15) combined slope and height in an area-elevation measure. His general profi le character index was defined as the percentage of gentle slopes (tan « less than 0.08) lying above or below the mean elevat ion. Pike and Wilson (1971, p. 1079-80) noted that this index measures a similar aspect of terrain form to the hypsometric integral . This measure may be undefined in some areas i f there are no slopes gentler than the cr i t ica l value. 3 . 6 . 5 : Appl icat ion of Hypsometric Measures A l l or most of the parameters discussed above have been used in a simply descriptive sense or in physiographic classif ication. Only the hypsometric in tegral , however, has been related to geomorphic processes. In most cases, HI has been determined for drainage basins. Strahler (1957, 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 ie ld 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 profi le - 5 1 -form, basin re l ie f , and basin hypsometric integral for 410 fourth-order drainage basins in Malaysia. He graphed the hypsometric integral against rel ief and presented an "approximate curve of best f i t drawn 'by eye' " (p . 29) . If one assumes that rel ief is continuously reduced wi th time (cf. Ahnert , 1970) and that space can be substituted for t ime, Eyles1 line suggests a period of equi l ibr ium, a monadnock phase, and an eventual return to equi l ibr ium. An in i t ia l inequil ibrium phase does not appear to be represented in these data. 3 .7 : Review and Parameters to Be Investigated In review, the most fundamental concepts of geomorphometry are the basic horizontal and vert ical scales of the topography. Horizontal variations are encompassed by the concepts of grain (largest significant wavelength) and texture (shortest signif icant wavelength); grain w i l l not be investigated exp l i c i t l y , but three measures of texture, namely drainage density (D^)/ source density ( D s ) , and peak density (D^) w i l l be considered in the next chapter. Vert ical scale is generally termed " re l ie f " ; this terrain concept w i l l be represented in further analyses by the local rel ief (H), the most widely employed rel ief measure. The relationships between horizontal and vert ical scale w i l l be examined through the mean slope (tan o<), whi le the three-dimensional interaction of slope steepness and aspect w i l l be studied through the roughness factor (IR). Relatively independent from horizontal and vert ical scales is the distribution of mass wi th in the vert ical 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, i t is believed by the writer that wi th the possible exception of gra in , a l l important terrain information is contained wi th in these parameters. In the next chapter, the relationships among the measures w i l l be studied. - 5 2 -Chapter 4: Terrain Var iabi l i ty in Southern British Columbia, and Relationships among Variables Before beginning the comparison of the computer terrain storage systems, a "p i lo t study" was conducted. The principal objectives of this were threefold: (1) to provide information about terrain var iabi l i ty 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 al l studied. 4 . 1 : Selection of Sample Areas In order to obtain a relat ively unbiased sample of the terrain of southern British Columbia, a stratif ied random sampling design was employed. From each of the for ty- two 1:250,000 scale map sheets which cover British Columbia south of 54 degrees lat i tude, one of the th i r ty - two 1:50,000 scale maps making up that sheet was selected wi th the aid of a table of random numbers. Because coverage of the area at the larger scale is incomplete, some of the randomly-selected maps were not avai lable. In such cases, and in instances where the selected map fe l l entirely outside British Columbia, another map was picked. 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 - 5 3 -generally centred at the intersection of the two major ("10th kilometre") grid lines closest to the centre of the map; this would fac i l i ta te 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 i f more than one third of the area contained water surfaces. The locations of the for ty- two samples, together wi th the major physiographic subdivisions of the study area, are shown in Figure 4 . 1 ; two of the samples (4, 9) fe l l in A lber ta , although the maps from which they were drawn were in part in British Columbia. Since i t was not possible to adhere str ict ly fo the original random sample, tests were made of the randomness of the terrain samples actual ly used. As noted above, each 1:250,000 scale map contains th i r ty - two 1:50,000 scale half-sheets (see Figure 4 . 2 ) . The numbers of these th i r ty- two "cel ls" containing exactly zero, one, two , et cetera, samples were determined and compared to the frequencies predicted according to the Poisson distr ibution. A chi-square test indicated that the two sets of frequencies were not signif icantly different at the 95 per cent level . Of twenty major physiographic divisions of British Columbia given by Holland (1964), ten occur at least in part south of 54 degrees lat i tude. These , together wi th the sample numbers and respective map-areas fa l l ing wi th in each div is ion, are listed in Table 4 . 1 . The actual distribution of the for ty- two terrain samples among these ten regions was compared wi th 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 signif icant ly different at the 95 per cent leve 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 fa l l ing on the ocean were not accepted, Figure 4 . 1 : Physiographic subdivisions of southern British Columbia (see Table 4.1) with locations of stratif ied random sample of terrain analyzed in Chapter 4. -55 1 ® 1 1 ® | ® j © © i 2 4 1 © r — i i 4 1 1 © | ® © 1 ® 1 ® 1 © i _ ! © 1 ® \ ® @ _ I ® j ® ® 1 © 1 © © 1 ® 1 © | © 1 i ® 1 ( 2 9 ) 1 1 ® 1 ( 2 7 ) U | © ! ® © 1 i © © i . | i 1 1 ® 1 1 1 : 2 5 0 , 0 0 0 S C A L E M A P 13 1 4 1 5 16 12 11 10 9 L_J_ 5 6 7 8 4 3 2 1 Figure 4 . 2 : Distribution of terrain samples among the 32 1:50,000 scale sheets which make up a 1:250,000 scale map sheet. See text for discussion. I 9 W I 9 E '\ ( W E S T ) 1 ( E A S T ) I 1 : 5 0 , 0 0 0 S C A L E M A P f"l8| S A M P L E S E L E C T E D F O R D E T A I L E D A N A L Y S I S 16) O T H E R T E R R A I N S A M P L E - 5 6 -TABLE 4 . 1 : PHYSIOGRAPHIC SUBDIVISIONS OF SOUTHERN BRITISH COLUMBIA, AFTER HOLLAND (1964), WITH SAMPLE NUMBERS A N D MAP-AREAS FOR TERRAIN SAMPLES ANALYZED IN CHAPTER FOUR Western System Outer Mountain Area Insular Mountains (1) 14: 92C/16E 22: 92L/6W 15:92E/15E 38: 103B/3E 16: 92F/2W Coastal Trough Hecate Depression (2) 35: 102I/9E 37: 103A/8E 36: 102P/9E Georgia Depression (3) 13:92B/14W 2 l : 9 2 K / 6 W Coastal Mountain Area Coast Mountains (4) 17: 92G/9E 20: 92J/3W 19: 92I/5E 23: 92M/5E Cascade Mountains (5) *18: 92H/2W 39: 103C/16E 40: 103F/14E * 4 1 : 103G/16W 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: 93A/4W 32: 93F/9W *24 :92N/15E 28:93B/9W 33: 93G/13W 25: 920/16E 29: 93C/8W 34: 93H/12E 26:92P/1E *31:93E/9W Columbia Mountains (9) 1:82E/10W 3: 82G/12W * 8 : 82N/4E 2: 82F/8E 7: 82M/15E Eastern System Rocky Mountain Area Rocky Mountains (10) 4: 82J/1 IE 10:83C/5W 12: 83E/5W 9: 820 /4E * indicates sample selected for more detailed analysis - 5 7 -TABLE 4 . 2 : COMPARISON OF DISTRIBUTION OF 42 TERRAIN SAMPLES A M O N G TEN PHYSIOGRAPHIC DIVISIONS WITH EXPECTED DISTRIBUTION BASED O N DIVISION AREAS physiographic division per cent of area expected c (e) ibserved (o) ( e - o ) 2 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 Sums 100.0 42 42 X 2 = 12.47* * not significant at the 95 per cent level - 5 8 -samples would tend to be "concentrated" in the land areas of map sheets containing considerable water. 4 . 2 : Data Collect ion As stated above, each terrain sample consisted of a 7 by 7 kilometre square; the selection of this sample area size was arbitrary. Wi th in each area, a 7 by 7 grid wi th a one kilometre spacing was used in determining some terrain measures. At 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 l ine" stream network were counted. The elevations of the highest and lowest points wi th in the area, the number of closed hi l l top 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 . McCoy, 1971) developed a theoretical equation which proposed that the drainage density should be approximated by: D d = 1 . 5 7 N / L (4.1) The empirical evidence col lected 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 \ but wi th a number of "outl iers" having values above one. The writer had observed during the data col lect ion Figure 4 .3 : Histograms for six geomorphometric parameters. Triangles indicate the break points for three of the parameters. - 6 0 -that some of the older maps appeared to have higher drainage densities than newer ones. A statist ically significant inverse correlation was found between and the year of map publ icat ion. Drainage density was also signif icantly correlated wi th 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 signif icant, 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 wel l marked breaks in the distr ibut ion, 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 hi l l top 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 .847) , 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 wi th in each sample area, as determined by a visual inspection of the contours, were used to determine the local re l ief . This should be wi th in one contour interval of the actual value and, i f the maps are accurate, must be wi th in two contour intervals. The maximum and minimum of the 49 grid heights were also determined; the - 6 1 -difference between these was designated H* , the grid estimate of the local rel ief . Theoretical aspects of the relationship between the true and grid values of local rel ief w i l l be discussed in section 5 . 1 . Histograms of local rel ief 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 rel ief classes: " low" re l ie f , less than 500 metres (10 samples); "moderate" re l ie f , 500 to 1,500 metres (25 samples); "h igh" re l ie 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 fa l l ing on a water surface. The histogram for average slope (Figure 4 .3e ) shows a rather poorly defined break at about 0 . 3 . The high correlation between mean slope and 2 rel ief 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) elevat ion-rel ief ra t io . The mean of the elevations of those grid points which did not fa l l on lakes or the sea was used as an estimate of the mean height of the terrain. The formula for the e levat ion-rel ief ratio (equation 3.14) involves both the minimum elevation and the local rel ief ; here, the hypsometric integral was computed twice: H was based on the " true" minimum and maximum elevations, whi le H * was based on the grid - 6 2 -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 distr ibution. When Strahler's (1952) divisions at 0.35 and 0.60 were appl ied, 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 "youthful" or " inequi l ibr ium" stage. Nineteen of the for ty- two samples had HI values below 0.35 and would fal l into Strahler's "monadnock phase", the remainder being essentially in equi l ibr ium. While the hypsometric integral for an arbitrari ly-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 stat ist ical ly-signif icant at the 95 per cent leve l . 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 "drainage", "hypsometry", and !"rel ief" . Peak density (D ) was not signif icantly correlated wi th any other P variable. Factor analysis was also applied to the data, and produced essentially the same result . - 6 3 -TABLE 4 .3 : VARIABLES INCLUDED IN CORRELATION ANALYSIS variable number symbol name 1 D d Drainage density 2 N / L Drainage net intersections 3 D s Source density 4 D P Peak density 5 H Local rel ief 6 H* Gr id estimate of local rel ief 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 - 6 4 -TABLE 4 . 4 : STATISTICALLY SIGNIFICANT (95 PER CENT LEVEL) LINEAR CORRELATION COEFFICIENTS A M O N G THE VARIABLES IN TABLE 4 .3 D, N / L D D H H* ran* HI H* z p r 0.921 0.473 - .496 0.927 0.468 - .485 N / L - 0.554 - .479 D s _ D P - 0.987 0. .824 0.419 H 0. .797 0.364 H* - 0. 323 0.602 tan- f - 0. 887 0.364 HI - HI* - - . 445 0.336 z - P - r - 6 5 -( D R A I N A G E ) T E X T U R E ( P E A K D E N S I T Y ) Dd N/L z p. H t a n ot H* R E L I E F HI HYPSOMETRY Figure 4 . 4 : Correlation structure among twelve terrain and related parameters (constructed in the manner proposed by Me l ton , 1958). The outer boxes enclose isolated correlation sets; dotted lines indicate inverse correlations. -66-4 .4 : Classification of Samples and Selection of Areas for Further Analysis Three independent terrain variables, namely re l ie f , hypsometry, and peak density, were used to divide the for ty- two 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 0.063. The break points in the distributions of the variables were given above. As there were three classes each for rel ief and peak density and two for the hypsometric integral , there are eighteen possible groups — of these, f i f teen 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 f inal 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 for ty- two 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 ) ; curves based on the 49-point samples of elevations (cf. Haan and Johnston, 1966) were similar to those shown. 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 lec i l lewaet Map-area (82N/4E) The terrain sample from the l l lec i l lewaet 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 val ley of the Incomappleux River, whi le the maximum (9050 feet; 2758 m) is an unnamed 1 Physiography after Hol land, 1964 - 6 7 -TABLE 4 .5 : CLASSIFICATION OF 42 TERRAIN SAMPLES USING LOCAL RELIEF (H) , HYPSOMETRIC INTEGRAL (HI), A N D PEAK DENSITY (D ) . P NUMBERS OF OBSERVATIONS IN CLASSES ARE I N PARENTHESES; SAMPLES FOR FURTHER ANALYSIS ARE UNDERLINED. H HI D 0 .25(11) P 0.25 D 0.50(21) P D 0.50 (10) P 500 m 0.35 (7) 25,33 14,24,40 3,29 (10) 0.35 (3) 27 28 32 500 to 0.35 (10) 4 ,22 ,26 ,35 ,36 5,21 ,32,37,38 1500 m (25) 0.35 (15) 1,6,14 2 , 9 , 1 2 , 1 5 , 1 6 , ] 8 , 3 4 , 3 9 , 4 1 , 4 2 17,23 1500 m 0.35 (2) 11 20 (7) 0.35 (5) 7 ,8 ,19 ,30 10 - 6 8 -TABLE 4 .6 : VALUES OF SOME GEOMORPHOMETRIC PARAMETERS FOR SIX AREAS SELECTED FOR DETAILED ANALYSIS. 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 - 6 9 -Figure 4 .5 : Hypsometric curves for the six terrain samples selected for detailed analysis. - 7 0 -peak 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, wi th u-shaped glacial troughs between. Sample 8 had the highest local rel ief of the six areas selected for detailed analysis, and the fourth highest over a l l . 4 . 5 . 2 : Sample 11: 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 id th , cuts across the study area from northwest to southeast; i t 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 wi th in 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 wi th in 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 wi th an elevation of 7888 feet (2404 m) represents the maximum elevation wi th in the sample area. The topography of sample 11 is not unlike that of the previous one (sample 8 ) , wi th its high rel ief 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 f loor. 4 . 5 . 3 : Sample 18: Manning Park Map-area (92H/2W) This sample lies wi th in the Hozameen Range of the Cascade Mountains. Once again, this sample area is dominated by forms produced by alpine-type glacia l erosion. Here, however, the summits take on a more rounded appearance - 7 1 -because they were overridden by ice during the last glacial maximum. Four summits wi th in the area have elevations of about 6350 feet (1935 m), and most val ley floors are around 4300 feet (1311 m). On ly in the northwest corner, in the v-shaped val ley 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 rel ief 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 f i f th 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 rel ief 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), wi th elevations of between 3100 and 3300 feet (945-1006 m); a major meltwater channel traverses the sample area leading into Tatla Lake i tself , 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 : Ghi tez 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 Glathel i Lake (elevation 3490 feet; 1064 m), the 7 by 7 km sample area contains almost the entire rel ief of the Nechako Plateau. Local rel ief for the study area (1190 m) is st i l l only in the "moderate" class. - 7 2 -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 Kit imat 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 arbitrar i ly taken as a generalized line along the 2000 foot contour. " Following this def in i t ion , the southwestern half of the sample area belongs to the Hecate Depression, the northeastern to the Coast Mountains; in fac 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 wi th floors as low as about 500 feet (152 m), but nowhere are the ridges sharp as i n , for example, the l l leci l lewaet map-area (sample 8). Probably, cirques were formed during an early "a lp ine" phase of g lac ia t ion , but later the entire area was overridden by ice . Cirques may or may not have been re-occupied by local ice after the disappearance of the Cordil leran ice sheet from the area. - 7 3 -Chaprer 5: Procedures for Analysis and Theoretical Comparisons of Computer Systems In this chapter, the analysis procedures used to estimate the geomorpho-metric parameters selected for special attention w i l l be out l ined. 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 correspon-dence between grid maxima and actual surface maxima may not be great. The remaining four parameters which are studied in this chapter are local rel ief (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 qua l i ta t ive ly , and in some cases quant i tat ively. 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 fo l low, 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 rel ief 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 interval ) , and thus the maximum error in the local rel ief from this source is two contour intervals (expected error = one contour interval); - 7 4 -(3) errors due to misreading the contours — this may be one contour in terva l , or even f ive contour intervals i f an " index" contour is misread, for both the maximum and minimum point; (4) errors due to the mis- ident i f icat ion of either the maximum or minimum point , or both — for example, a particular summit may be taken to be the highest point wi th in 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 rel ief is the difference between the elevations of the highest and lowest sample points. A l l four of the sources of error for the "true" value of local rel ief 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 l ikely in the latter case than in a visual inspection of the contours. In d ig i t iz ing an area using surface-specific points, an attempt is made to include a l l peaks and pits, as wel l as a l l maxima and minima long the borders of the area. If a l l are included, the true maximum and minimum elevations must be among them, and " type 4" errors are el iminated. 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 rel ief 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 value. - 7 5 -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 unl ikely that a grid point w i l l coincide wi th 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* , the grid estimate of the re l ie f , w i l l be less than or equal to H. If t is the average land slope near the maximum point , and c the distance from the maximum to the nearest grid point , the error in estimating the maximum should be given by: e .= c tan 1 max (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 root-mean-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 . , wi th in the inner box in Figure 5.1) is given by: (x + y ) dy dx -, I = 0.408 d (5.2) One can further suppose that H may approximately equal o< # the mean ground slope; estimates of the errors in the maximum and minimum elevations would then be: e . = e = 0.408 d tan <* (5.3) mm max v ' - 7 6 -Figure 5 . 1 : Il lustration of the distance (c) from any point (x, y) to the nearest grid point (open c i rc le) . Solid circles indicate other grid points. - 7 7 -and thus the expected error in the value of the local rel ief would be: e H = 0.816 d tan <x (5.4) The error i n , and accuracy of , the grid estimate of the local rel ief is theoretica a linear function of the grid spacing, as proposed by the "sampling theorem" (see section 2 . 2 ) . Of 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 loodpla in, an the slope near this point (and thus also e m j n ) w a s near zero . Slopes near the maxima and the minima of most of the for ty- two 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 compl icat ion, the closest grid point to the maximum may not be the highest grid point , 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 for ty- two samples from Chapter 4 was e H = 234.1 tanoc + 63.12 ( r2 = 0.245) (5.5) where e ^ is in metres and d is 1000 m. This relationship is stat ist ical ly signif icant (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 .426, st i l l only about half of the theoretical coeff icient (equation 5 .4 ) . - 7 8 -This 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 rel ief obtained from a set of surface-specific points should be as accurate as, or even more accurate than, the estimate obtained through a visual inspection of the contours. For regular grids, errors due to the fact that i t is very unl ikely that a grid point w i l l coincide exactly wi th the minimum or maximum point w i l l tend to be much larger than interpolation errors. 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 relat ively wide grid spacing is used. Surface-specific points should theoretical ly provide much better estimates of local rel ief 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 rue" 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 relat ive 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 randomly-located points wi th in one study area was not signif icantly different from the "populat ion" values. As noted earlier (section 3 . 4 . 2 ) , Griff i ths (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). - 7 9 -He 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 signif icantly different from this, and as Grif f i ths 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 rue" 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 gr id ; the same analysis procedures were then used for both these triangles and the triangles based on the surface-specific points. For each t r iangle, a vector orthogonal to i t was determined by computing the cross product of vectors forming two edges of the tr iangle. The length of this vector is twice the true area of the t r iangle, whi le 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. While the latter may represent the most logical weighting (cf. 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. Pil lewizer (1972) noted that the - 8 0 -triangle method, as applied by Hormann (1971) , fai led to indicate a slope asymmetry detected by f ie ld surveys and careful analysis of large-scale topo-graphic maps. Pil lewizer attributed this fai lure to the fact that Hormann's triangles were too large. For triangles derived from a g r id , 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 tr iangles. 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 s l ight ly. 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 rue" value of the hypsometric integral was determined by using a planimeter to measure the areas above various elevations (that is, enclosed wi th in selected contours); the elevations are converted to relat ive values by - 8 1 -subtracting the minimum height and dividing by the local re l ie f , whi le 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 wi th 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 arbitrari ly-bounded topographic samples as are used in the present work (cf . Gassmann and Gutersohn, 1947; Wood and Snel l , 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 in tegral , or that the hypso-metric integral for a group of basins may not equal the mean of the basin values. The former fact can be il lustrated by applying a square sample area wi th two different orientations and a circle to two simple geometric forms: an incl ined plane, and a square-based pyramid considerably larger than the sample area wi th the latter centred at its apex. For the incl ined 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 rc le and the "diagonal square" (the square wi th 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 equil ibrium stage in the absence of structural control . Many such basins have outl ine forms similar to the circle or the diagonal square, and the "characteristic s-shape" is probably in part due to the influence of outl ine form. In the case of the pyramid, both the curve form and the hypsometric integral vary wi th the shape and - 8 2 -Figure 5 . 2 : Hypsometric curves for the portions of an incl ined plane wi th in 3 sample areas. A : "para l le l " square (see inset, A ) ; B: "diagonal" square (see inset, B); C: c i rc le . - 8 3 -orientation 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 topo-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 arbitrari ly-bounded sample areas is the relationship between the hypsometric integral for such an area and the integrals of its constituent drainage basins. As a simplif ied i l lustrat ion, one can consider two adjacent basins of equal areas, minimum elevations of zero, and hypsometric integrals of 0.5 — the only difference is that one basin has a local rel ief 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 rel ief is 1000 m, and thus the hypso-metric integral of two basins w i l l be 0 .375, twenty- f ive per cent less than that of either of the individual basins. Other combinations of relat ive rel iefs, 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 equal , as w i l l be approximately the case in "equi l ibr ium 1 1 topography wi th a common local base level (the ocean, a lake, or a low-gradient f loodplain) , the aggregate integral w i l l always be less than the individual ones. This may in part explain the relat ively large number of the for ty- two areas examined in Chapter 4 which had overall integrals below the lower l imit of "equi l ibr ium" (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 surface-specific points. It can be easily shown that the volume between a triangular - 8 4 -Figure 5 .3 : As in Figure 5 . 2 , but for a square-based pyramid. Here, the hypsometric integral varies as wel l as the curve form. - 8 5 -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 iangle. 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 elevat ion-rel ief ratio formula (equation 3.14) to estimate the hypsometric integral . 5 . 4 . 1 : Hypsometric Integral: Surface-specific Points It is d i f f icu l t to determine quanti tat ively 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 wi th in the triangles formed from these points approximate planes. If the person selecting the points is careful to make sure that the contours wi th in each triangle are approximately parallel and equally spaced, the method should be reasonably accurate. 5 . 4 . 2 : Hypsometric Integral: Regular Gr id The mean elevation determined from triangles based on a regular grid using the volumetric method outl ined 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 tr iangles, and thus al l such points are equally weighted (the projected areas of the triangles are, of course, equal); points along the boundaries are in three triangles, whi le corner points are in one or two. Since no attempt is made to ensure that the areas wi th in 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 surface-specific points. The principal sources of error, however, are errors in the maximum and minimum elevations used in the elevat ion-rel ief ratio formula (equation 3 .14) , errors which have been discussed above in section 5 . 1 . 2 . In the fol lowing discussion, e and e . are non-negative error terms, and an a ' max mm a ' - 8 6 -asterisk (*) is used to denote values determined from the grid alone. If one defines: z * = z - e max max max (5.6) 2 = 2 ~f" 6 min min min (5.7) and H* = mm • * - z * max min (5.8) It follows that: HP z - (z . + e . ) mm mm' (z - e ) - (z . + e . ) v max max' x mm mm' (5.9) Some algebraic manipulation of this equation yields: e HI* e + e ] _ max min z - z . max min + mm z - z . max mm z - z mm z - 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 . is H, the true local re l ie f , v 3 * r " max mm giv ing: HI = HI* 1 - Ltt H + mm H (5.11) If HI* is to be accurate, H * must equal H , in which case either e ^ must equal zero or the fol lowing relationship must hold true: H = H* = e . mm (5.12) H The latter ratio may provide a rough estimate of hypsometry, since for the 42 samples examined in Chapter 4 i t was signif icantly correlated wi th H , although the r value was only 0 .345. - 8 7 -Some further re-arrangement of equation 5.11 yields the fol lowing expression for the relative error in H * : E H T ' 1 - HI I HI - HI* e - e . max mm L H J H H* (5.13) Equations 5.12 and 5.13 imply that i f 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 fac 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 x " r mm max (see section 5 . 1 . 2 ) . For high values of H , e . must exceed e to x " min max minimize the error in the grid estimate of the hypsometric integral , 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 square-based 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. When the grid minimum was used in the * max - — calculat ions, e ^ was found to be a linear function of d , once again supporting the "sampling theorem" noted in section 2 . 2 ; when the " t rue" 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. - 8 8 -5 . 4 . 3 : Summary It is d i f f icu l t to assess quanti tat ively 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 and e . (equation 5 .13) . r » w m a x m i n \ - i / 5.5 : 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 outl ined 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 appl ied, but many "false" peaks and pits w i l l appear in such data, 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 def ini t ion of peaks, pi ts, ridges, and courses w i l l theoret ical ly be much easier i f surface-specific points are stored in the "pointer mode", rather than the "tr iangle 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. It would also be possible to compute other roughness measures (cf. Hobson, 1967, 1972), - 8 9 -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^. cel ls, N - edges and N y vertices states that: N v + N c - N - = 1 (5.14) i f the "outside" is not considered to be a c e l l . If a l l cells are triangles, there should be 3N^~. sides. Since a l l edges form sides of two triangles w i th the exception of those edges forming the outer boundary of the study area, the total number of edges is given by: 3 N r + N R N - = — 5 = - (5.15) b 2 where N D is the number of edges (and also the number of vertices) which form the boi yields: the boundary. Substituting this value in equation 5.14 and solving for N^. N c = 2 N y - ( N B + 2 ) (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 "tr iangle mode" of the triangular data-set method. Each integer value requires one half-word of computer storage a l locat ion, whi le each " rea l " or decimal value requires a fu l l word. To store the three co-ordinates (reals) and the ident i f icat ion number (integer) of a surface-specific point would thus require 7 half-words of computer - 9 0 -space, whi le 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 .16 , 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 ( 1 3 N y - 2 N B - 6) half-words (5.17) For the "tr iangle mode", there are required 3 half-words for each t r iangle, the number of triangles being given by equation 5 .16 . The total storage requirements for the points and triangles should equal: ( 1 3 N y - 3 N g - 6) half-words (5.18) which is exactly N R less storage space than needed by the "pointer mode". - 9 1 -Chapter 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 arbitrar i ly 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 tw ice , 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 re l ie 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 probabil i ty that the true mean error of each method was zero. If for any method this probabil i ty was 5 per cent or less, i t would be concluded that the method of estimating the parameter being tested was not va l id . The mean errors for the grids and triangular data-sets were compared, and the assumption that grid error is proportional to gr id 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 ig i t izat ion 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 fol lowing section. Figure 6 . 1 : Sample 1 l a , an example of a triangular data-set from the Ptarmigan Creek map-area. - 9 3 -6 . 1 : Digi t izat ion 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 .16 . 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, wri t ing 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 , whi le an average of 8.0 seconds was needed to determine the vertices of each tr iangle. Measuring the x and y co-ordinates used an average of 12.6 seconds per point , but if should be possible to improve this considerably by using a d ig i t izer . Table 6.1 indicates that the average triangular data-set analyzed herein contained 114 points and 197 triangles. Digi t izat ion of such a data-set would theoretical ly require 946 seconds to draw the tr iangles, 1576 seconds to determine the vertices of these triangles, and 2382 seconds to d ig i t ize the points, a total of 4904 seconds, or 43 .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 digi t izer 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 "tr iangle mode", would require 1389 half-words of computer storage al locat ion; - 9 4 -TABLE 6 . 1 : NUMBERS OF POINTS BOUNDARY POINTS ( N B ) A N D TRIANGLES ( N c ) FOR DATA-SETS ANALYZED sample N V N B 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 l i d 85 25 143 l i b , c 29 15 41 18b, c 25 16 32 - 9 5 -the grids would require but 450 half-words. The surface-specific point data-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 rel ief 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 theoretical ly in section 5.1 — the triangular data-sets produce results very similar to the visual inspection method, whi le grid errors may be rather large. Theoret ical ly, rel ief 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 hal f . 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 signif icantly different from zero, given the small sample size. Paired t-tests were used to determine whether the errors of the three estimates were signif icantly different from each other. A l l three pairs were signif icantly different at the 95 per cent leve l , meaning that whi le the grid estimates were not "signif icantly b a d " , the triangular data-sets produced errors signif icantly 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 again, the t-tests indicated that none of the mean errors differed signif icant ly from zero. Paired t-tests showed that the tr iangle and grid estimates were - 9 6 -TABLE 6 .2 : ESTIMATES OF LOCAL RELIEF (H), A N D ANALYSIS OF ERRORS I N THESE ESTIMATES standard 7 x 7 15x 15 triangular sample method grid grid 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 error (e): 8 - 225 27 15 11 - 119 103 0 18 - 264 60 5 24 - 19 16 0 31 - 138 3 0 41 - 117 46 0 e 147 42.5 3.3 s e 87 36.0 6.1 t 0.689 0.482 0.224 p ( e = 0) 52% 66% 82% - 9 7 -TABLE 6 . 3 : ESTIMATES OF MEAN SLOPE ( t a n t * ) , A N D ANALYSIS OF ERRORS IN THESE ESTIMATES sample Wentworth method 15 NW-SE x 15 grid 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 error (e): 8 - 0.086 0.091 0.024 11 - 0.051 0.037 0.040 18 - 0.065 0.061 0.003 24 - 0.022 0.024 0.015 31 - 0.041 0.033 0.015 41 - 0.076 0.064 0.019 e 0.055 0.051 0.019 s e 0.025 0.025 0.012 t 0.896 0.843 0.639 P (e = = 0) 42% 44% 56% - 9 8 -signif icantly di f ferent. As expected, the two slope estimates obtained from the same grid using different diagonals were not signif icantly different at the 95 per cent level . 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 Mi les , 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 Mi les ' contention that the orientations of the diagonals used to form the triangles has l i t t le effect on the results. In the absence of a standard value, the claim made above in Chapters 3 and 5, that the weighted analysis of triangular data-sets should y ield the best results, cannot be substantiated empir ical ly. The f ive sets of values given in Table 6 .4 were not signif icantly 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, whi le 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); HI* , as wel l as the results for the 15 by 15 grids, used the grid estimates of these quantit ies. Pike and Wilson claimed (p. 1081) that 40 to 50 points w i l l generally produce results wi th in 0.01 of the true values. This claim is supported by the fact that the mean error produced by their method is 0 .006, and in none of the six cases did the error reach 0 . 0 1 . - 9 9 -TABLE 6 .4 : ESTIMATES OF ROUGHNESS FACTOR (R) sample 15 x unit vectors NW-SE NE-SW 15 grid weighted vectors 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 - 1 0 0 -TABLE 6 .5 : ESTIMATES OF HYPSOMETRIC INTEGRAL (HI), A N D ANALYSIS OF ERRORS IN THESE ESTIMATES sampl standard e method 7 x H 7 grid HI* 15x NW-SE 15 g r id* NE-SW triangular data-set 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 error (e): 8 - 0.004 0.047 0.004 0.004 0.015 11 - 0.005 0.024 0.019 0.021 0.003 18 - 0.001 0.118 0.019 0.020 0.005 24 - 0,007 0.020 0.011 0.019 0.010 31 - 0.004 0.060 0.004 0.004 0.001 41 - 0.008 0.024 0.025 0.025 0.009 e 0.006 0.049 0.014 0.015 0.007 s e 0.002 0.038 0.010 0.010 0.005 t 0.795 0.529 0.559 0.622 0.557 P (e = 0) 46% 62% 60% 56% 60% * these estimates are based on grid values only - 1 0 1 -The best of the computer estimates, those based upon the triangular data-sets, had a sl ightly larger average error, wi th the grid estimates considerably poorer. As in the cases of local rel ief and slope, the t-tests indicated that none of the mean errors differed signif icantly 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 northeast-southwest 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 re l ie f , mean slope, and hypsometric integral using both regular grids and triangular data-sets were g iven. 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 l inear. This was confirmed theoretical ly 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 rel ief 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 inal ly , the values developed above in section 6.1 are used to estimate the relative digi t izat ion times and computer storage al location 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 digi t izat ion time and computer storage space using surface-specific points than using regular grids. The contrast is much more dramatic for local rel ief than it is for the other two parameters. -102-TABLE 6 .6 : EMPIRICAL COMPARISON OF ERRORS FOR TRIANGULAR DATA-SETS A N D 15 BY 15 GRIDS H tan <x HI Mean errors: 1 5 x 1 5 grids triangular data-sets ratio Characteristics of grids theoretical ly required* to produce same precision as triangles: d (metres) grid size ^ of grid points Ratios of requirements of such grids to those of triangular data-sets: d ig i t izat ion time storage space 42.5 0.053 0.015 3.3 0.019 0.007 12.9 2 .8 2.1 39 179 234 181 x 181 40 x 40 31 x 31 32,761 1,600 961 55 .4 2 .7 1.6 47.2 2 .3 1.4 * assuming a linear relationship between grid error and grid spacing -103-6 .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) wi th 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 four-variable "phase space" were calculated. For sample 11 , the most similar pair was a and d , the two with similar numbers of points and triangles derived from the same map. Next 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 "di f ferent" data-set was 1 l b , derived from a smaller-scale map wi th a larger contour in terval . It seems that for this area, map scale differences are more important than the number of triangles used. For area 18 (Table 6 . 8 ) , the opposite conclusion was reached. In this case, the most similar pair was b and c, the two sub-samples wi th 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 ) . -104 -TABLE 6 . 7 : SIMILARITY A M O N G FOUR TRIANGULAR DATA-SETS BASED O N SAMPLE 11 FOR THE FOUR SELECTED MEASURES numbers of H HI tan o< R points triangles Or ig inal 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 s 24 0.011 0.008 0.306 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 THREE TRIANGULAR DATA-SETS BASED O N SAMPLE 18 FOR THE FOUR SELECTED MEASURES numbers of H HI tan °< R points triangles 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 s 28 0.015 0.072 2.037 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 Inter-pair differences: distance 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 fol lowing 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, whi le 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 suff iciently large that the larger triangles in sub-sample 11c were able to retain most of the "terrain information" present in sub-samples 1 la 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 relat ively 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 digi t izat ion times required for triangular data-sets and for regular grids. It was estimated that 43 .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 re l ie f , mean slope, and hypsometric integral were discussed, and various estimates of the roughness factor were g iven. 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 rel ief and mean slope were signif icantly less than those of the grids, as determined using the t-stat ist ic. For the hypsometric integral , the grids produced a higher average error but the difference was not significant at the 95 per cent level . 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 digi t izat ion 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 ident i f icat ion of specific types of landforms. It is therefore more applicable to arbitrari ly-bounded terrain samples stored in an electronic computer. After a brief discussion of map precision and notat ion, approaches to computer terrain storage were discussed. This subject was reviewed in terms of digi t izat ion (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 val leys. In the surface-random approach, the points are selected according to cr i ter ia 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 stratif ied random approach represented by a g r id . General ly , grids require much less computer storage a l locat ion, since only one co-ordinate (the elevation) must be stored for each point. Digit ized contour points require two co-ordinates, whi le for surface-specific points a l l three must be specif ied. In addi t ion, the neighbours of a grid point are impl ic i t in its position wi th in the computer array, whi le these must be exp l ic i t l y indicated for surface-specific points, requiring st i l l 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 wi th reference to the problem of estimating some selected geomorphometric parameters. -109-A large number of landform measures were reviewed, and were found fo belong to a number of basic groups. These were texture and gra in , re l ief , slope, dispersion of slope magnitude and or ientat ion, 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 impl ic i t in the sample area size and the density of sample points. The four parameters examined exp l ic i t ly were local rel ief (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, for ty- two 7 by 7 km squares were selected from 1:50,000 scale maps of southern British Columbia using a stratif ied random sampling design. For each of these areas, local re l ie f , hypsometric integral , 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 integral , and peak density were used to divide the for ty- two samples into f i f teen "terrain types". A stratif ied 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 brief ly 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 rel ief and the hypsometric integral at least, the precision of the grid estimates should be l inearly related to the grid spacing; this probably holds true for the other measures also. The possibility of estimating other parameters, the re la t ion-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-Final ly , 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, whi le 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 dig i t izat ion 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 ) . 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G e o g . , v . 53, p. 536-568. - 1 2 1 -Appendix I: Notat ion In this appendix, a l l variables and symbols used in the text are l isted, together wi th their meanings or definit ions. Exceptions are standard abbreviations (such as " M " for metres) which are not l isted. 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 drainage density (3 .2.2) D P peak density (3 .2.4) D s stream source density (3.2.3) d grid spacing (2.2) E elevat ion-rel ief ratio (3 .6.2) e H error in the grid estimate of local rel ief (5 .1.2) e HI relat ive error in the grid estimate of the hypsometric integral (5 .4 .2) e max error in the grid estimate of the maximum elevation (5.1.2) e . mm error in the grid estimate of the minimum elevation (5.1 .2) f factor by which one wishes to improve grid accuracy (2.2) G grain of topography (3.2.1) H local rel ief (3 .3 .1) H* grid estimate of local rel ief (4 .3.3) H a available rel ief (3 .3.2) H d drainage rel ief (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) N y 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 g root-mean-square error (1.1) 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 maximum elevation (3 .6 .1) max -123-z* grid estimate of z (5 .4 .2) max max z . minimum elevation (3.6.1) mm ' z* . grid estimate of z . (5 .4 .2) mm mm 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 probabil i ty distribution (3.5) -124-Appendix I I : Topographic and Related Variables for 42 Areas in Southern British Columbia In the fol lowing Table, the values for twelve terrain and related variables from the for ty- two 7 by 7 km topographic samples examined in Chapter 4 are g iven. The parameters were listed in Table 4 . 3 , and are also included in Appendix I. The six areas analyzed in detail in Chapter 6 are indicated by the symbol whi le the highest and lowest value for each parameter are marked wi th the symbols "+ " and " - " , respectively. The mean and standard deviation for each var iable, 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 inf luenced. i - 1 2 5 -Area D d N / L D s D P H H * tan « H H* £ P t 1 0 . 3 8 8 0 . 2 9 6 0 . 1 0 2 0 . 1 6 3 1247 8 8 4 0 . 1 6 7 0 . 6 0 2 0 . 5 4 0 1650 4 0 6 4 2 0 . 6 6 1 0 . 4 1 8 0 . 3 8 8 0 . 4 0 8 9 6 3 6 4 3 0 . 4 0 5 0 . 5 6 8 0 . 5 5 2 1818 4 0 6 1 3 0 . 2 9 2 0 . 1 8 4 0 . 0 4 1 0 . 6 9 4 329 2 3 5 0 . 0 9 1 0 . 3 4 3 0 . 4 8 1 9 5 1 15 3 1 -4 0 . 6 3 9 0 . 3 6 8 0 . 0 5 5 0 . 4 1 7 1268 1113 0 . 4 6 2 0 . 2 4 0 - 0 . 2 5 9 1965 4 5 6 2 5 0 . 8 4 0 0 . 5 6 2 0 . 2 2 4 0 . 6 5 3 835 7 2 8 0 . 1 6 9 0 . 2 9 9 0 . 3 4 3 1046 16 6 2 6 0 . 4 8 2 0 . 2 8 6 0 . 0 0 0 - 0 . 1 8 4 9 3 9 8 2 0 0 . 2 4 1 0 . 3 6 4 0 . 3 3 9 1 1 1 3 2 0 5 8 7 0 . 7 7 6 0 . 4 5 9 0 . 2 2 4 0 . 1 2 2 1694 1636 0 . 4 4 5 0 . 3 5 2 0 . 3 6 1 1 1 4 3 5 5 6 0 8 # 0 . 5 5 5 0 . 3 4 7 0 . 1 0 2 0 . 1 0 2 1880 1655 0 . 6 0 9 0 . 4 2 8 0 . 4 7 9 1683 8 0 6 1 9 0 . 5 8 6 0 . 3 8 8 0 . 1 6 3 0 . 3 2 6 1387 1159 0 . 5 2 7 0 . 4 0 5 0 . 4 5 1 2 0 7 7 + 30 5 9 10 0 . 5 0 8 0 . 3 0 6 0 . 1 6 7 0 . 3 3 3 1740 1423 0 . 5 0 4 0 . 3 7 7 0 . 4 5 8 1 9 6 7 7 0 6 6 n' 0 . 5 4 9 0 . 3 4 7 0 . 1 2 2 0 . 1 4 3 1709 1590 0 . 3 9 5 0 . 2 6 5 0 . 2 8 4 1148 30 6 5 12 0 . 4 7 6 0 . 3 1 7 0 . 1 0 2 0 . 4 0 8 1405 1207 0 . 4 6 0 0 . 5 1 2 0 . 4 3 4 1786 4 0 6 0 13 0 . 7 1 4 0 . 5 1 0 0 . 2 5 0 0 . 3 3 3 3 7 5 2 8 7 0 . 1 1 0 0 . 2 4 2 0 . 2 3 3 9 1 - 4 0 51 14 1 . 2 3 1 0 . 8 3 7 1 . 0 8 9 0 . 1 2 2 1012 9 3 8 0 . 3 8 1 0 . 4 9 4 0 . 5 2 6 6 6 1 100 3 8 15 0 . 9 3 7 0 . 5 8 1 0 . 5 8 3 0 . 2 5 0 1326 1201 0 . 5 9 4 0 . 4 1 7 0 . 4 4 5 5 5 4 140 3 9 16 1 . 5 1 0 1 . 0 2 0 1 . 5 1 0 + 0 . 3 0 6 1015 9 9 1 0 . 4 0 3 0 . 5 3 9 0 . 5 3 3 5 4 6 130 4 7 17 0 . 4 9 2 0 . 3 3 7 0 . 2 6 5 0 . 7 9 6 1408 1143 0 . 6 9 4 + 0 . 5 2 9 0 . 5 6 2 1169 120 6 2 18' 1 . 8 4 7 1 . 1 6 3 1 . 4 2 9 0 . 2 8 6 8 8 3 6 1 9 0 . 3 9 5 0 . 5 4 6 0 . 4 2 9 1 5 3 4 5 0 5 7 19 0 . 6 7 6 0 . 3 7 8 0 . 1 6 3 0 . 1 6 3 1905 1 8 1 4 0 . 4 9 6 0 . 5 1 3 0 . 5 1 3 1129 18 5 8 2 0 0 . 6 8 4 0 . 4 5 9 0 . 0 6 1 0 . 4 0 8 1945 1610 0 . 6 0 1 0 . 3 1 4 0 . 3 8 0 7 5 4 130 6 5 2 1 0 . 5 6 3 0 . 3 5 7 0 . 2 5 0 0 . 5 6 8 6 8 6 5 6 7 0 . 2 9 9 0 . 2 5 1 0 . 2 7 2 172 7 0 4 9 2 2 1 . 3 2 2 0 , -837 - O r 821 0 . 3 5 9 1175 - 1 1 2 8 - 0 . 3 7 6 0 . 2 8 6 0 . 2 9 7 4 2 6 120 3 4 2 3 0 . 7 2 0 0 . 3 9 8 0 . 2 9 1 0 . 5 2 1 9 2 0 8 3 2 0 . 3 4 2 0 . 3 7 1 0 . 3 6 5 3 4 1 9 0 6 0 2 4 # 0 . 6 3 1 0 . 4 5 9 0 . 1 9 1 0 . 3 8 3 2 0 3 - 1 7 4 - 0 . 0 6 4 0 . 2 8 1 0 . 2 7 0 9 6 7 16 5 8 2 5 0 . 0 4 3 - 0 . 0 3 1 - 0 . 0 0 0 - 0 . 0 4 1 - 3 6 6 2 6 2 0 . 0 4 2 0 . 3 0 9 0 . 2 3 3 1021 14 6 5 2 6 0 . 4 0 8 0 . 2 3 5 0 . 0 2 0 0 . 3 0 6 7 7 3 5 8 5 0 . 2 6 2 0 . 3 4 3 0 . 4 5 0 6 3 5 2 0 6 1 2 7 0 . 4 7 8 0 . 2 8 6 0 . 1 6 3 0 . 1 8 4 2 8 7 2 1 4 0 . 0 5 6 0 . 4 9 4 0 . 6 0 7 + 9 6 8 16 6 5 2 8 0 . 7 2 4 0 . 4 3 9 0 . 1 8 4 0 . 3 6 7 2 7 4 183 0 . 0 4 2 - 0 . 5 4 0 0 . 4 9 2 8 3 1 1 2 - 5 5 2 9 0 . 6 3 7 0 . 3 6 8 0 . 1 0 2 0 . 5 3 1 2 9 6 2 6 9 0 . 0 6 0 0 . 2 4 3 0 . 2 2 7 - 1201 15 6 0 3 0 0 . 8 1 9 0 . 4 8 0 0 . 6 5 3 0 . 2 2 4 2 1 2 2 1972+ 0 . 6 0 7 0 . 5 6 2 0 . 6 0 1 1 2 2 0 6 0 6 1 31' 0 . 2 9 0 0 . 2 5 5 0 . 1 0 6 0 . 5 5 3 1195 1 0 5 4 0 . 2 2 5 0 . 3 4 8 0 . 3 9 2 1480 2 3 6 8 + 3 2 0 . 6 5 7 0 . 4 5 9 0 . 1 0 4 0 . 5 2 1 3 3 8 3 2 9 0 . 1 0 1 0 . 5 6 2 0 . 5 5 9 1141 15 6 7 33 0 . 3 2 6 0 . 1 9 4 0 . 0 4 1 0 . 2 0 4 2 7 2 192 0 . 0 4 3 0 . 2 7 2 0 . 3 0 2 7 9 9 17 6 0 3 4 0 . 7 3 5 0 . 5 0 0 0 . 4 9 0 0 . 4 2 9 8 5 4 7 0 7 0 . 2 5 7 0 . 4 0 9 0 . 4 8 5 1 1 8 7 4 7 6 0 3 5 2 . 0 5 1 + 1 . 3 1 6 + 1 . 4 0 8 0 . 3 0 6 6 1 3 471 0 . 2 5 6 0 . 2 6 8 0 . 3 4 4 176 105 36 3 6 0 . 6 5 7 0 . 4 1 8 0 . 1 2 5 0 . 2 7 1 7 3 8 6 4 3 0 . 1 5 7 0 . 2 5 2 0 . 2 8 0 186 9 0 6 1 3 7 1 . 4 2 0 1 . 0 2 3 0 . 2 5 6 0 . 9 7 7 + 7 1 6 5 5 0 0 . 3 9 3 0 . 3 3 4 0 . 3 8 2 2 3 9 125 6 1 3 8 0 . 5 3 1 0 . 3 5 7 0 . 2 6 7 0 . 5 1 1 6 5 5 5 5 9 0 . 3 7 3 0 . 3 2 2 0 . 3 7 4 2 1 1 7 0 6 4 3 9 0 . 4 9 6 0 . 3 0 6 0 . 1 6 7 0 . 4 7 9 1012 8 6 0 0 . 5 8 2 0 . 3 8 6 0 . 4 3 0 4 0 0 7 0 6 4 4 0 0 . 6 6 1 0 . 3 9 8 0 . 2 8 6 0 . 2 8 6 4 9 5 4 6 4 0 . 2 0 0 0 . 3 4 7 0 . 3 4 9 182 7 0 6 4 41' 0 . 8 8 2 0 . 5 3 1 0 . 6 7 3 - 0 . 4 9 0 869 7 5 2 0 . 4 0 0 0 . 4 0 3 0 . 3 7 1 3 8 0 100 61 4 2 0 . 5 6 9 0 . 3 1 6 0 . 4 6 7 0 . 4 2 2 9 6 8 741 0 . 5 1 9 0 . 4 3 4 0 . 4 5 6 4 2 0 160+ 6 1 mean 0 . 7 2 5 0 . 4 6 5 0 . 3 6 0 0 . 3 7 0 9 7 8 8 3 8 0 . 3 2 9 0 . 3 9 0 0 . 4 0 8 9 3 7 6 0 5 8 s 0 . 4 0 1 . 0 . 2 6 5 0 . 4 1 3 0 . 1 9 5 5 2 6 4 8 8 0 . 1 8 9 0 . 1 0 9 0 . 1 0 7 5 6 6 42 9 unite km ^ km ' km ^ km m m m inches y -126-Appendix I l i a : Computer Program In this appendix, the FORTRAN IV program listing for the computer program used in this study (program GEOTRI) is g iven. Requirements for the input data are contained in comments at the beginning of the program l ist ing. -127-:ORTRAN IV G COMPILER MAIM 02-06-74 1 0 : 1 9 : 3 2 P A G E 0001 r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C P R O G R A M : G E O T ° I £ p ; [gon<;c; T I npTPP^ I N E S F L E T T E ™ n M p p P H p M P T gT C P A R A " C T F P S " V  C A N A L Y S I S O c T s I AMCLiL ^ 9 F A C E T S W H I C H A 3 0 R D X I ^ A T E T H E S U R F A C E * C P O S S I B L Y BfiS cn O N G R I D 0*.TA C W R I T T E N : D A V I D M 0 „ . M Ap-K T . . _ C E^3^ A P H Y , „ U » B . C » . i _1_9.I1  C C. T N P L ' T : E A C H J O B R E Q U I R E S ft J O B C A P O A N D A T I T L E C A R D , F O L L O W E D B Y C T H E Di T S T P | T ; e g : n ; T H E ,113,  C JOB C A R T : C O L 3-5 I T Y 3 F = 0 : P F E - T R I A N G U L A T E O D A T A . C = 1 : o p I P P A T A C = 9?..9..:._E.ND.0- . p U N ,...(F .DLL nWS_LAST..JOB) C 10 N E W 3 = 0 : R E A P NFrf P * T A P O I N T S C =1: U S E S A ' - ' E P O I N T S A S P R E V I O U S J O B C 15 » . ! = W T ^ Q : J F » n l R " p y f U T F N E W T C I A N G L E S  C =1: U S E S A M C T P . I A M G L E S A S P R c V I " i ! , ' S J C 8 C 16-27 X S C A L E : N U " B E P O c U E T P E S I N O N F M N I T I N C ' . : . X - ° n E C T I . O S ! , _ R C . P L ' J M N . S P A C I N G . F ^ R C O R 10 D A T A . D E F A U L T S T P i _ o I F M C T C S P E C I F I E D . ^ r io M A T : F J ? o 5 r. ? 8 - 3 ° Y S C & L E : A S A B O V E , C H P . Y-D I R c : T T Q\J n g  C - R O W S P A C I N G , D E F A U L T S T n X S C A L E * C 40-51 - S C A L E : M U M P E R O C M E T R E S IN " N E U N I T IN C : Z - 0 . H . . E C . T I O N o . 3 E = A ' J L T S . . . T ? . . . I , C C 56 . L A K E = 0 : A N A L Y Z E A L L T R I A N G L E S . C .. . =1: E X C L U D E L A K E A M D O C E A N T P I A N G L E So £ 6T I T R T ^ Q ; O M I T T P , I _ N ~ L E L I S T ,  ,.C =•-" . . L I S T P A R A M E T E R S F Q D E A C H T R I A N G L E . C 66 I _ H G = 0: U S E N W - S E D I A G O N A L S T O P R O D U C E ' C : T P J A N G L E S P P D M G R I D , . C =3: U S E N E - S W D I A G O N A L S , C 67-71 N'> : N U M B E R O C R O W S I N M A T R I X . . r_ 77-^6 N C t N U ' - ' B E P H F C O L U M N S I N M A T ? I X  C C T I T L E C A R T : T I T L E , M A X I M U M O F 76 C H A R A C T E R S . I C N O T I T L E IS C Q g S I P E P , A B L . A N < C A R P MUST Be J.MJ_EJ_T.ED_»_ C C DATA: . C P QE - T P I A N G M L A T F ^ D A T A ( I T V O E = 0 ) :  C - D A T A P O I N T S , I R R E Q U I R E D , E A C H C A * 0 H A V I N G P O I N T C N U M B E R A N D X , Y , A N D Z - C O - O R D I M A T E S . i F O R M A T A S W R I T T E N C LS.....U., 7.E.6. 0)_ _.__ C - L A S T D A T A P O I N T M U S T <*E c O L L C W E D BY A B L A N K r. A<? n C — T P I A N G L E S , I C F E D U I ' - E O , E A C H C A R D H A V I N G T P I A N G I . E C M t j M q c o t P p r ^ T MijM^rjc p c T H R E E V E R T I C E S , A N D " L " , . C W H I C H E Q U A L S \ I F T ^ c T = i A M G t . E I S O N A L A K E O 3 T H E C O C E A N , 7.F0 O T H E R W I S E . ( F O P M A T A S W R I T T E N , 515) £ - L A S _ T _ T R I A B L E M A 1 S I . B E . _ _ Q L L O W E D B Y _ _ A _ B_L A N < C A F D C G R I D D A T A ( I T Y ? E = ? 1: C - E A C H =0W O N A S r P A F . A T P C A P O , ( F O R M A T A S 4 ' I T T r . v ; 3 x , £ 1 5= c, 01 n • a L M K Z * R N T S ? M C L U P r P , S I N C F M ' ) " p r ? p p C R O W S ( N R ) I S S o r C I F I E O O N T H E J O B C A R D . C £1 A5_ J lQT£0 _AeDVE . . , L A S T . J ? B . I . _ F r L ! OWED. . . "Y_.A__C A.RD..W I . T H _ _ « o o ? " _ _ T . M . . C CCLU M NS 3 - 5 T O E N D T H E R U N , 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 F F O L L O W I N G R E S T R I CT\ " » y ^ ; - : : ^ 8 -ORTRAN IV G C Q M ° I L E R MAIN 0 2 - 0 6 - 7 4 : 1 0 : 1 9 : 3 2 PAGE 0 0 0 2 C -MAXIMUM Nl . iM?E c 0 C " " I V T C (\<p) i< CQO; C -MAXIMUM N i . l ' ^ E ' 0- T3 IA NGl.r S (NT ) T C iQOO; C —M ATP i x MAY NOT NAVE ' O R E THAN ?Q ROWS (NR) 3? ' 5 COLUMNS j _ _ ( • ' . ) :  C - I F D IMENS IONS ' ^ E I N C - E A S E 3 , K F F D IN MI \j D T H A T : C - (NR-1 ) * (VC-1 ) CANNOT E X C C F Q MO; £ _NT_ SHOULD f*E ? I U E N $ I O N TH 2 * N P » C " ~ " £********* ************************ *^ ************ ™ OOOl. D I M E N S I O N X ( 5 0 0 ) , Y ( ? 0 0 ) , I ( 500 ) , I T ( ? 0 0 0 ) , I V < 1 0 0 0 ) , J V ( 1 CO 0) ,KV (1 0 0 0 ) 1 , L V ( 1 0 0 0 > , T I T L E ( 1 3 ) , S U « ( 1 2 ) 0 0 0 2 CT 'MON. .X , Y , Z, I T fiy.j J V . K V . L V , T I T L E ,NP,NT, L A K E , I TR I 0 0 0 3 RAT-O,01 " ^ 5 3 2 9 3 0 0 0 4 J T ? S = 0 0 0 0 5 99 C A L L REAPER  0 0 0 6 I F ( L AK Eo EQ« 999) GO TO 999 0 0 0 7 A L A K E = 0 . 0 0 0 S N L = 0 : . , 0 0 0 9 Z M AX=-99^90„ 0 0 1 0 ZMIN=99999. 0 011 T?MAX=3 :  0 0 1 2 IZMIM=0 . 0 0 1 3 DO 3 1 = 1 , 1 2 001 4 . S 'J M ( I..) = 0 o . . . 0 0 : 5 3 CONT IN', jc 0 0 1 6 I PI I TRT , E 0 . 0) r,0 TO 94 001 7 W R I T E ( 6 • 1 52 ) ( T I T L E ( I ! ) t I T = ) , 1 9 )  0 0 1 8 152 F O R M A T ! • 1. • r l ° A 4 , / / ) 0 0 1 9 W R I T E ( 6 , 5 0 0 ) 0 0 2 0 5 0 0 F Q R" AT ( ' N O , CORNERS L 7.M ' t I PX t ' MAP T.rilS. 1 A R r A S L O P E ' ,/) 0 0 2 1 . 9 4 DO f- I = i , NT 0 0 7 ? I F ( Z ( ! V ( I n . L E o Z V f t X ) GO T_ 6 0 ; • 0 0 2 3 Z MAX= Z( I V ( I ) ) 0 0 2 4 I Z M » . X = I V ( I ) 002 5 5 0 I.EIOJ.V.U. M • LE ...Z MAX ) S O T ] 61 0 0 2 6 Z M A X = Z ( .)V ( I ) ) 0 0 2 7 I Z M A X = J V ( I ) 0 0 2 8 S i T c ( 7 t K V ( ! ) ) _ L E _ ZM-X ) G O T ' > 62  0 0 2 9 ZMAX = Z ( K V ( I ) ) 0 0 3 0 IZM/>x=K V( I I .0011 62 I F (.7 ( IV (I ) j , GEo Z M I N L ^ 0 . _ . T . : L 3 . 0 0 3 ? Z MIV •= Z ( IV (I )) • 0 0 3 3 " I Z M I N = I V < I ) ______ f>3 TP (7 1.IV(T))_r , E . 7 'TNH G " T 0 6 4 ; 0 0 3 5 ZMIV = Z ( J V ( I )) 0 0 3 6 I Z M T N = J V ( I ) . 0 0 3 7 ' 6„ I c (I (KV ( I ) ) . OG .ZM I.M..) ...G.0._I?...3_5. . . : _ . 003 3 ZMIN = Z ( K V ( I U 0 0 3 9 I ZM IN = KV( I ) Q 0 4 0 6 5 CONT IN') F 0041 VFC1 1 = X ( J V ( I ) ) - X ( I V ( I ) ) 0 0 4 2 V F C i 2=Y( J V ( ! ) )-Y( I V( I) ) Q043L V E C 1 3 = Z ( J V ( 11 J.- Z ( IV I I ) ) _ . O C - 4 V £ C 2 1 = X ( K V ( I) ) - X ( I V ( ! ) ) 0 0 4 5 V E C 2 2 = Y ( * V ( I ) ) - Y ( I VI I ) ) J3Q4_6 __FC? 3 =7 [ KV \ I ) . , ) .-LIV LI) ) : . O U T R A N I V G C O M P I L E R M A I N -129-0 2 - 0 6 - 7 4 1 0 : 1 9 : 3 2 P A G E O O P 1 0 0 4 7 0 Q 4 _ 0 0 4 9 0 0 5 0 0 0 5 1 J 3 _ 5 _ 2 _ C C V E C 1 A N D V E C 2 A R E V C C T P R S Z M = ( Z ( I V ( ! ) ) + Z ( J V < I ) ) + ? ( " V U - - T H E M = A M F L ; V A T I C N O F , * V C C 2 3 - v ? * . 3 - v ~ c 7 " > ZM I S _ y j _ = y _ F C i P N G 2 SI P E S P F T H E T ° I A N G L E ) ) ) / 3 . T H E T R I A N G L E v _ = V E C I 3 * V C C ? l . - V E C 5 1 " * V C C _ 3 V ? = V E C 1 1 * V E C 2 2 - V E . I 2"*V E C 2 1 v I j T H ~ C R c s S . . O O " T J c~. o E T H E TR [ A N G L C I F ( V 3 . G E . 0 . 0 ) G O T P 4 V I = ( - 1 . )*'P . _y F_C 1 A N P . . X E C ? • A N D I S . . T H P P = T H I G ? _ N A L .J O P * 3-0 0 5 4 H Q 5 . 5 . 0 0 5 6 OQ5 7 V 2 = < - 1 . ) * V 2 V 3 = ( - 1 . ) * V 3 S.P N T . I .N ' J E T H I S M A K E S T H F V E C T O R P O I N T V L = S Q R T ! V 1 * V ? + V 2 * V 2 + V 3 * V ? ) a T = V L / 2 - ' I J P ' 0 0 5 B D 0 5 9 _ 0 0 6 0 0 0 6 1 OF V I S T W I C E T H E T R U E A R E A OF T H E TP I A N G L E , AT T H E L E N G T H U 1 = V 1 / V L 'J2 = V Z / V L _ . L :  U 3 = V 3 / V L A = V 3 / 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 T H E 7 - C Q M p 0 M E N ! T O F V 0 0 6 2 0 0 6 3 C c r O I P = 1 . 5 7 0 7 9 6 3 - A F S I N ( U ? ) S = T A N ( 0 1 P ) _S S _ T _ H E . . _ . S L O P E . _ T A N G F . N T _ N 1 T E : I F O T H E R CHAR. A C T E R I S T ! THF A P p p p p ^ i A T F ST*TE'-1 ' W R I T E S T A T E M E N T C H A N G E C S P F I N D I V I D U A L T R I A N G L E S A R E D E S I : E M S S H O U L D B E T N S E R T E D H F ^ E A N D T H E D o cr. , : O R I E N T A T T P N P F T F I A M G L E . 0 0 6 4 0 0 6 5 0 0 6 6 . . 0 0 6 7 0 0 6 8 0 0 6 9 I F ( I T R I . F Q . Q ) ~,n T O 90 W R I ' E ' 6 , 2 0 0 ) I T I I ) , I V l I ) • J V ( ..2.00 _ F O R M A T (5.I.5.»_4F.i 5,o,.e_.L . 9 0 C O N T I N U E I F ( L V ( D o N E , 1 ) GO T O 7  A L AKE=ALAK£ + A  I ) , * V ( I ) , L V ( I ) , Z M , A , A T , S 0 0 7 0 0 0 7 1 - 0 0 7 2 0 0 7 ? N L = \ ' L + ! A L A K E I S T H E T O T A L A R E A p c _ T R T A N G L E S F A L L I N G O N . W A T E ? I c ( L A K E oE Go C) GO TO 7 I F ( I T R I . M E . 0 ) W R I T E ( 6 , 3 0 0 ) I = 0 R v A T ( 1 T R I A N G L E H ' , 1 3 , ' I L A K E S + O C E A N ; N L I S T H E N U M B E R PF S P N A L A K E OR T H E . O C E A N A N D I S E X C H ' P 0 0 7 4 00.7.5... 0 . 0 7 6 0 0 7 7 0 Q 7 P 1 E 0 * ) GO T O 6 ' ._ .S ' JM( D = S U M ( 1. ) + A S U M ( 2 1 - S U M ( 2 ) • A T S U M ( 3 ) = S U M ( 3 ) + Z M * A <j\)'A{ 4) = S U ' M 4 ) * S 0 0 7 9 O O B O 0 0 " 2 0 0 3 3 0 Q P 4 0 0 B 5 0 0 P 6 S ! J M ( _ ) = S L ' M ( 5 ) + S * A S U M ( 6 ) = S U ' M 6 ) + S * A T ..SUM! .7.) = SUM ( 7) +1)1 < ; I J M ( q ) = S U M ( P ) + J ? S U M ( 9 ) = S H ' M O ) 4.II-J v i M m ) = s u ' ' P Q ) » v i / ? o S U y < 1 1 ) = S U " M ) 1 ) + V . / 2 . S'JM( 1 2 ) = S ' J M ( 1 2 ) + V 3 / 2 < i__ONT...LNUE___ r C T H E F O L L O W I N G S E C T I C N C n " ° U T E S . G E O w r > R P H P M F T P T C PA°A«ETEPS ORTRAN IV G C O M ° I L E R MAIN - 1 3 0 -1 0 : 1 ° : 3 2 PAGE 0 0 0 4 0 0 ? 1 0 0 9 9 0 0 9 0 0 0 ° 1 I F ( L AKE) 1 3 , 1 9 , 1 8 13 AT0T = S U M ( 1 ) * A L A K E NT A-NT-ML GO TO ?0  03=52 0 0 9 3 0 0 ° 4 15 A T O T = S U M m N T A = N T ? 0 C O M X ! N _ M E C c c : A , > M I : S U X ( I ) I S THE ANALYSIS A<?EA, WHICH S I S AT OT I S THE T 0 T A L MAY E X C L U D E LAKES, NT I S THE' T O T A L N'_P">EC O c ^ T A N G L E S , NTA t H E NilMQER F0? ALALY 0 0 9 5 0 D 9 6 00 9 7_ 0 0 9 8 0 0 9 9 0 1 0 0 ° L A K E = ! 0 ' 0 . * A L A K E / A T O T 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 )  0 1 0 1 J_ 0 2_ 0 1 0 3 3 1 0 4 S M 3 = S U M ( 6 ) / S U M { ? ) T H E S M ' S A R E M E A N S L O P E S : SM I . _ MVWFT G H T E O ; S M 2 0 0 W T D f?Y A ; S M 3 . on v _H._I_ AX.r £H_N . . . AT... H Y ? S = ( Z'*~ 7. M I V ) / H H Y P S I S T H E H Y P S O M E T R I C I N T E G R A L AR AT I O = S U M 1 2 ) / S J * ( 1 )  0 1 0 5 C _r_ C C c A M T ? I - S U M M ) / T M A M T C I IS T H E M E A N A R E A OF T H E T R I A N G L E S A N A L Y Z E D T H I S SECTION IS FOR (UNWEIGHTED) VECTOR TO W = 1GMTED AMAI.YSI VECTORIAL A N A L Y S I S . V A R I A B L E S RELATED TO ANALYST^ INCLUDE "V" IN NAMES, THOSE P E L A TNCLJPE "V" 'o UN I T T C 0 0 1 0 6 3 1 0 7 0 1 0 8 C C __ R = VECT3R L E N G T H , K = D R E C ! S I C ' N P A R A M E T E R , K. T =J 0 0 / K , L = 7 C G T O R S I R ! ( - ) , RF = RO:.'GHNESS F A C T O R , 0= OR I E NT ATT ON, D = O I P OF MEAN V E C T O R R U = S O R T ( S U M ( 7)**2 * S U M ( 3 )**2 + S U V ( ° ) * * 2 ) RV = S0RT(SUM ( 1 3 ) **2 + S'JM ( 11 ) **2+?UM( 1 2 1 * * 2 ) ! jK> ( TN-1 . ) / ( T N-RU)  0109 0110 J? 1 JJL 0112 0113 U K I = 1 0 0 . / U K U L = 3 0 0 . * P U / T N _ V L =< 0 0 . * R V / S U M ( 2 ) '  R F U = 1 0 0 . - U L R F V = 1 0 0 . - V L U L : V E C T O R S T R E N G T H , U N I T V I C T O R S : VI. WEIGHTED BY A T e 0114 0115 .01L6_ ,0117 OU 8 O ' .i q A U = S U M ( 7 ) / R U 9 U = S U M ( P . ) / R U __J=FUM( ?.iyp.u_ A V = S U M ( 1 0 ) / R V B V = S U M ( 1 1 ) / < ? V C V = C U M P 2 ) / R V 0 1 2 0 0 1 2 1 _!Z2_ 0 1 2 3 0 1 2 4 012 5 I F ( P U . N E . O . O ) GO I F ( A ' j ) l l , 1 2 , 1 3 _1L_0U-? TO. GO TO 2 4 12 O U = 0 . 0 GO T O ?4  TO 1 0 0 1 2 6 01 2 7 _ _ 2 8 . . 0 1 2 9 013 0 01 3 1 13 0 U = ° 0 . GO TO 2 4 ___._.._ NT I NUE. 3 U = A T A N ( A U / B U ) / P A O I P C U ) 2 3 , 2 5 , 2 5 ?5 TF ("111 3"*, ? 4 . " > 4 -131-"ORTRAN IV G COMPILER M A I N 0 2 - 0 6 - 7 4 1 0 : 1 9 : 3 2 P A G E 0 0 0 5 0 1 3 ? 0 1 3 3 0 1 3 4 01 3 5 0 1 3 6 0 1 3 7 _ 0 I _ 5 L 0 1 3 9 0 1 4 0 0 1 4 1 23 I C ( Q U > 3 3 , 3 2 , 3 2 3? 0U = ? 8 0 . + O U 33 0 U = O U - t-l Q 0 . ?4 C O N T T M H C OiJ=A"» ^ TN( GUI / ' i D I c ( R V . N E . O o O ) GO TO 14 I.~(A.Vt 1 5 , 1 6 , 17 15 0 V = 2 7 0 o GO TO 4 4 16 py=o, 0 . • 0 1 4 2 0 1 " - 3 _Qi.4_4_ 0 1 . 4 5 0 1 4 6 0 1 4 3 0 1 4 9 J X L ? J _ 01 5 1 01 5 2 01 r,o TO ^ 4 17 3 V = ^ 0 . _n . _ T 0 . . 4 4 14 C O N T I N U E O V = A T A N { A V / B V ) / ° A 0 45 I F ( ? V ) 5 3 , 4 4 , _ 4 43 I c ( « V ) f 3 , 5 2 , 5 2 _ 5 . „ . 0 V = ^ V « - 1 8 0 , 53 O V = . V + l t » 0 . 44 CONTINUE Q V - A P . S I M ( C V ) 1 ° A n 0 1 5 4 0 1 5 5 0 1 ^ 6 C c THE c O L L O W I N G S E C T I O N PRINTS OUT RESULTS W R I T E ( 6 , 1 5 2 ) ( T T ~ L C ( I T ) , 1 1 = 1 , 1 9 ) WRITE ( 6 ,6 03 ) *!° , N T , A T O T , PL A < E »NT A , S I !M ( i ),AMTR I 6 3 3 = O R M A T ( « GENgoft t : ' , / / , ' M I I M R F P OF POINTS = ' , 1 0 X , 1 5 , / , ' \ " v a <ro 0 1 5 7 0 1 5 8 0 1 5 9 Ql 6 Q XLL6_1_ 0 1 6 2 0 1 6 3 ._16.4_ 0 1 6 5 0 1 6 6 - - , - / • • • T_> T AL * 3 E A = »,-i=.15-, / , ' D t - C F V T L ' = . ' A V A L Y S I S = ' , 1 0 X , 1 5 , / , ' A R F A F O R A ' E A M T O T A N G L E 4 C C A = • , E l 5 . 2 , / / ) _ _ _ _ _ M R I T E ( 6 , 6 01 ) 7 M I N , ! 7 M I M , 7.M A X , I Z 7 4 X , H , H YP S , S M 1 , S M 2 ,"sM 2 , A R ATf'o""" 6 0 1 F O R M A T . ' G E 0 M 0 ' D H C M E T R Y : ' , / / , * M I N I M ' J M E L E V A T I O N = ' , c 8 _ 2 , ' L O 9CATED AT P O T NT NU^B " ' , I 4 , / , ' " i x i m . ' M E L ^ V A T ^ O N = ' , f q - > 2 , ' '. 0 r 1 O c TR IAN Gl-F S = , } 0 X , ! 2AKES SEA = ' , 3 X , F ? . . 3 N A L Y S I S _ = «_, E 1 5_ P , / , • 1ATE0 A T O Q I N T M JMf} E~ • , ! 4 , / , ' L O C A L P E L T F F = « , F P S 2 , 1 O X , / , 2 H Y P S OMETR. I C I N T E G R A L = ' , F ? „ 5 , / , ' M C A N S L O P E : ' , / , • J N W E I G H T F O - P . ? , ? , / , ' . >'F I GH T : 0 BY MAP AREA , = P 0 5,7", •ARE A RATIO c 9 . 5 , / , ' „ . W E I ; = ' , F 8 , 5 , / / ) HT E I 4Y T p U F AREA W R I T E ( 6 , 6 0 2 ) O U , n j , U L , U K , J K l , P F U , O V , O V , V L , » R V 6 - ) 7 Fnc.»j)_T(» yc CTQR ANALYS I S : ' , / / , ' U N W E I GHT=0 ( U N I T VECTORS ) : ' , / , ' I E N T A T I O N = ' , F ? o 2 , / , » OTP = ' , F 8 o 2 , / , ' L I ? ) 2 = ' , F 3 . 2 , / , ' K = ' , F 3 . 2 , / , ' J 0 0 / K 3 F 3 . ? , / . • ROUGHNESS. FA CT0.'_5' , F f l , 2 , //_, • VE T G.1TED_ BY T " J F._AP F,A : • , / 4 O R I E N T A T I O N = ' , F 8 . 2 , / , ' ^ 1 P = » , F e . 2 , 7 t * L ( « ) 5 = ' , F 8 . 2 , / , ' ROUGHNESS FACTOR = ' , F 8 . 2 > j QQ<; _ J flRC » i  n e T GO T 0 9 9 9 9 9 WR I T E ( 6 , ° 0 0 ) JO^S 033 FORMAT ( M ' , "=N"i OF... P I IN- __ THI 5. ST 0 ° END .D_UN_I.'-'.CJL_UDFO__j I 3 i l . .J0 .3S. . ' )_._ TOTAL MEMORY REQUIREMENTS 0 0 1 3 A 5 BYTES X f lMJ - lLF . ULME = !_,___. SE C fJ.N I S - 1 3 2 -:0.R TR AN IV G COM PI L E R R E A D E R 02-06 - 7 4 10:19:35 P A G E 0001 0001 S'B'O'JTINE RE I D E C 0002 D H E N S I O N X ( 5 0 D ) , Y ( 5 0 3 ) , 7( 5 0 0 > , I T ( 1 Q O O ) , I V ( 1 ODD) , 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 M f ? 0 , 2 0 ) 0003 t O M v , n \ j X, Y , 7., I T , T V , J V , KV , L V» T I T l . F , V O T M T , L A * C , I T R I 0004 R = A ^ ( 5 , I C O ) I T Y = E , N F W " , N E . ^ T , X S C A L E , Y S C A L F , Z S C A L F , L A K E , I T P I , I D I A G , V P I ,NC 0005 1 00 C 0 R * - * A T ( _ I 5 » 3E 1 2 , 5 , 5 1 5 ) 0006 I p ( T T Y P E , N E . 9 3 0 ) G O T O I 0007 L» <" =opo 000* 3 = T U R N 0009 1 R E A D ( 5 , 1 0 3 ) ( T I T L C ( I ) , 1 = 7 , 1 9 ) 0010 103 C O R ' J A T ( i o > 4 ) 0011 I C ( x S O A L E „ F o, 0, 0 ) X SC A L F =' , 0 0012 I C ( Y S C A L E _ E O - , 0 , 0 ) YS C . A L E = X S C AL E 0013 I F ( 7 S C A L E . E C . 0 , 0 ) Z S C A L E = 1.0 00 T 4 I F ( I T Y P E . F O . l ) G O T R I =0 00' 5 I F { > ' E . y P O F 0 . 1 ) G O T O 3 0016 NP=o 0 0 1 7 •> R E A 0 ( 5 , 7 0 1 ) I , X ( ! ) , Y ( I) , 7 ( I ) 0 0 1 8 101 F O R M A T ! 15 , ? , F 6 o 0 ) 0019 I F ( T , 5 0 . 0 ) G O T O 3 0020 X ( I ) = X ( I ) * X S C A ! F 0021 Y ( I ) = Y ( T ) * Y S C A L E 0022 Z< I ) = Z ( T ) * Z S C A L E 0023 NP=N'P+1 0024- GO TO 2 00 2 5 3 I F M E W T . E Q . U GC TO 5 5 0 0 2 6 NT = 1 0027 4 RE A D ( 5 , 1 0 2 ) I T ( M T ) , I v ( N T ) , J V ( N T ) , K V ( N T » , 1 V ( N T ) 0023 102 "F0R^ ATr5T5 :) 002<3 I F ( I T ( N T ) «.EOoOJ G O TO 5 0030 NT=NT«-1 0031 . GO T0 4 003? 5 \jT=VT_i 0033 55 W R I T F ( 6 , 2 0 0 ) ( T I T L E ( I ) , 1 = 1 , 1 ^ ) 0034 200 FO R U A T ( ' 7 ' , 1 ° A 4 , / ) 00*5 W R I T F ( 6 ,201) N», NT 0036 201 C O R V A T ( • T H I S DATA S E T W A S T R I A N G U L A T E D MAN ( A L L Y ; i f " ' ? : O N T A T N S ~ " » ,T3 1,' P O I N T S . AND ' , 1 3 , ' T ' I A M G L E S , ' , / / ' O P T I O N S : • I 003 7 I F ( M EW° « E 0,1) W R I T = ( 6 , ? 0 Z ) 0038 2 02 F D R « A T ( ' - U S E S S A ^ E P O I N T S AS P R E C F D I N G DATA S E T ' ) 0039 I F ( V E W T . F O . 1) WP.ITE( 6, 2 0 3 ) 004.0 20"* F_RMAT(«. .-USES. ..SA'-'E T P I A N . G L ES AS P R E C E D I N G D A T A S E T ' ) 0041 I F ( X S C A L E . N E . l . O ) W R I T E ( 6 , i ? 0 4 ) X S C A L E - - - -0042- 204 F O R « A T ( » - X S C 4 L E = • , E 1 2 . 5) 0 0 4 3 I - E ( Y S C A L E , N E , 1 , 0 ) WRlTP(f,,?r>5) Y S C A L = 0044 205 F O R M A T ( * - YSC A L E = ' , E 1 . 2, 5) 0045 IF ( 7 S C A L E . M E . 1,0 ) W R r T F ( 6, 206) Z S C A I E .004.6 20 6. F O R M A T ( • . . - Z S C A L E - ' , F I ; „ 5 | 0047 I F ( L A K . E Q . l ) W3 I T _ ( A , ? _ 7 ) 0048 207 FO R W A T ( * - L A K E S A R E E X C L J P E D F 3 D U T H E A N A L Y S I S ' ) 0 0 4 P p= T"PN 0050 50 IF(NEWP.EO.1) G O TO 61 0051 DO 6 0 I = 1 , N » 0052 .60 ._ — R.E AD I 5 ,15 D) ( A LT (.1 • J ) , J=l, NC ) 0053 150 F O R M A T ( 3 X , 1 5 C 5 . 0 ) 0054 NP=NP.*NC 0055 NR. 1= MR-7 OP TR AN IV G COMPILER READER -133-02-06-74 10:19:35 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 0 Q 7 7 0078 0079 0081 0082 008 3 0034 0085 _036_ 0087 008 8 0039 0090 0091 _00.?__ 0093 0094 0095 0096 0397 ___L8_ 0099 0130 0 1 0 7 . 0102 0103 0104 31Q5_ 0106 0107 0 1 0 9 0 1 1 0 PAGE 0002 51 61 54 301 NC1=NC-1 N T = 2 *NR1*NCI 00 c l 1 = 1 , N R 00 51 J=liNC 10= ( I-I )"NC«-J IDMf _,.)) = i o _X'.J? _! = c - OA T.I.Jt *v SC ALE Y{ 10 } = CL0 AT (NP-! )"Y SCALE" Z( I 3 ) = ALT ( I , J ) -*ZSC ALE CONT INiIE CONTINUE INT = 1 DO 5 2 1 = 1 00 52 J=1,NC1 1 c(TDIAG) 53, 53, 5^ IT( I N T ) _ T : ' ) T I VUNT ) = IGM( I , j ) JV(INT)=IOM(i+i,j) _ YJ INT ) = 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 ».  5 2 C O N T I N U F I N T = I N T - 1 H P I T E { 6 , ? 0 0 . ) ( . T I T I . E ( . I ) , 1 = 1,19) I F ( I N T , N E . N T ) W R I T E ( 6, 300) I N T E N T 303 F O R M A T ( • I N T = • , I ' , B U T N T = • , I 4,/) W R I T=(6.301) N R , M : . N Q , M T 3 0 2 FORM AT( ' THIS DATA SET IS P.-VS^ O ON A ',13,' BY 1R IX; •, /, ' IT CONTAINS i , n , i POINTS AMn _2.0NS: '.).. -._•.: _ IS(VEWPoEO.1) WPITE(6, 202) IF(NEWT.EQ.l) WPITEI6,20?) _-J-EJ_X_SCALE, EO. YSCAl E ) WD IT F( 6 , ? 02 ) XSCALF FORMAT.' -SOUA^E GPIO. G'TO S P /* C I NG = • , F1 2, ',13,' ALT IT'JOE M . T TRIANGlFS',//,»0PTI ,_, „„ - - .5, • MFTPFS') IF (XSCALF.NE.YSCALE ) W RITE ( 6 , ?0 : ! J XSCALF, YS~ A L p -3^-3 rOR>VAL(.L_._.r_COL.UMN..SfiACIMjG=' , El 2, 5, • V«=TRFS ; ~R_tw SPAC ING=•,c1?_ <=. • 304 1MET-ES * ) IF(IOIAG.EO.O) WPITE(6,?04) P Q ^ " > T ( „. -USEO N3RTHWPST-SPUT(-FA.ST 01 AGONAL S ' ) - 1 3 4 - • • . •• , :QRTPAN IV G COMPILER REACE° 02-06-74 1,0:19:35 : •'' PAGE 0003 0111 IF (10 I AG. EO.n WPITF(6,'05» 0112 305 FORMAT ( • -'JSEO NO-. TMC * S T- F 3 UTHWE S T ! A G 0 NA L S * ) 0113 IP(ZSCALE.NE.l.O) W ' I' E ( 6, 2 0 6 ) Z S C AL F 0114 R c TURN  0115 END TOTAL MEMORY REQUIREMENTS 001B2C BYTES COMPILE TIME = 0.5 SECONDS -135-Appendix 1Mb: Triangular Data-sets Maps of a l l the triangular data-sets analyzed in this study are g iven, wi th the exception of sample 11a which was il lustrated in Figure 6 . 1 . These maps are al l at the scale of 1:50,000; 1:250,000 scale maps of the same areas are given in insets. -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 l ie: 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 tit les of these output sheets are self-explanatory . . _ _ -J_45-I L L E C 11 L E W A t T ( S A ^ L E 3 ) GENERAL: A i J M ' ^ F K CF P O I N T S _= 9_0_ NUMBER OF TR I ANGLE S= IS 3 TOTAL AREA = 0 o 4 3 C 9 92 9 6 ~ 03 PERCE NT_ L..KI. S.. + ..SEA=_ 0 .3 NUMBER FOR ANALYSIS= i f 3 AREA FOR ANALYSIS = 0, 439 V->29 6S 08 .HE__N TRIANGLE AREA = Oo3202S661E 06 G E Q M r ' R f M H G M F X R Y _ L MINIM'JM ELEVATICN = 377 . 82 LOCATED AT POINT DUMBER 83 MA XI MUM E LE VATICM _ 2 74 3 ,20 LOCATED AT POINT NUMBER 53_ "LOCAL RELlFF ' = m 6 5 o ' 3 3 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 VECTORS ) : ORIENTATION = 102 ,50 DIP = 3 1 , 7 0 = 3 6 . 9 1 _ K. ~ ~ = 7 . 59 100/K = 13 .18 ROUGHNESS FACTOR^ 13 .09 WEIGHTED BY TRUE AREA: 0IR I Fj>l TAT ION 115 ,53 " 'DIP • = 33 .18" L U ) = 3 6 . 2 0 ROUGHNESS F ACTQR= 13 .30 . - 146 -PTARM I G A N C P E E K ( S A ' 1 P L E 1 1 A ) : A L L P 0 1 M 1 S G E N E R A L : N U _ 3 J _ J J E _ P O I N T S = KiJM rj P R 0 F " T R I A~\) G L E S ~ " ' ~ 1 3 6 T O T A L A R E A = 0 . 4 3 ° 9 Q 3 2 3 E 0 8 P E R C E N T L A K - S__ + G E \ - - _ 0 , 0 N U M B E R F O R A N A L Y S I S = 1 3 6 AREA F O R A N A L Y S I S = 0 , 4 3 C 9 9 3 2 3 E 0 3 W E A N T R I A N G L E A R E A = 0 , 3 6 0 . 8 9 1 2 6 0 6 G E O M O R P H O M E T R Y : MINIMUM E L E V A T I C ' N = 393.00 LOCATE 0 AT P O I N T N U M B E R 79 M A X I M U M E LEV ATI r N = 2 4 0 4 , 0 0 L O C A T E D AT_ P O I N T NUMBER 68 L O C A L R E L I E ' F " " = l ^ o 9 . 0 0 " " ' H Y P S O M E T R I C I N T E G R A L = 0 , 2 6 3 4 1 MEAN S L O P E : U N W E I G H T E D " " = 0 , 4 1 4 5 3 W E I G H T E D B Y M A P A \E A = 0 , 3 5 5 3 7 W E I G H T E D B Y T R U E A R E A = 0 3 3 ? 8 C 0 A R E A R A T I O ' = I , C 8 9 1 7 ~ VECTOR 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 = 3 1 , 8 6 D I P = 8 5 , 7 6 H%)__ = 9 0 , 9 7 'k' ~ ' = 1 1 . 0 0 100 / K = 3 , 0 9 R O U G H N E S S FACTOR= 9 . 0 3 W E I G H T E D B Y T R U E A R E A : _0 R I E N J £ n O N_ ^_ 6 0o35 " DTP~ •' • = a y 0 e 7 L I S ) = 9 1 , 3 8 R O U G H N E S S E A C T O R - 3 , 1 2 - 1 4 7 - _ _ _ _ _ P f A i > • 1 1 0 A N C P S V H ; L E 11 8 ) : F R 0 M " l : ? 5 0 , 0 0 0 •• ' •AP G E N E R A L : N U M B E R 0 F P Q I ' l T S _f 2 _ 9 N U M B E R CP T R " l V > I G L S S = " 4 1 T O T A L i R E A = 0 „ 4 ) r > 9 9 7 7 6 E O S P e R C E N J . _ L f e K E S . . . t _S = A = _ . 0.0 N U M B E R F O R A M A L Y S I S = 4 1 A Q E A, F O P A N A L Y S I S = O 0 4 '.<"• 9 9 7 7 6 E 0 3 M E A N T R I A N G L E A R E A ~ 0 . 1 1 9 , 3 1 6 0 E 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 3 6 2 o . 1 0 L O C A T E D A T P O I N T N U M 3 E P 1 9 L O C A L R E L I E F = 1 6 6 L . 1 6 H Y P S O M E T R I C I N T E G R A L = 0 . 2 6 5 3 0 M E A N S L O P E : U N W E I G H T E D " ~ " = 0 . 3 7 3 8 3 W E I G H T E D R Y M A ° - A R E A - 0 » 3 7 8 4 3 W E I G H T E D B Y T R U E A R E A = 0 , 3 9 3 6 4 A R E A R A T I O = 1 . 0 8 3 2 2 I i i 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 4 J . 4 3 D I P 8 7 , 2 9 L ( % ) _ 9 2 . 2 4 K ~ • ' " = 1 2 . 5 7 100/K = 7 0 9 r > R O U G H N E S S F A C T O R = 7 « 7 6 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N 1 3 . 3 6 " D I P - - - 8.3 c/+6 Lit) = 9 1 . 9 3 R O U G H N E S S F A C T O R = 8 o 0 7 PTARMIGAN C.K ?f_K ( S A' 1 •' L r J l C l -148 - _ A "OR )X'„ SAM" POINTS AS CUT 1 :5 0 ,00 GENERAL (VUM3F P. OF POINTS ? 9 NUMBER OF TP I ANGLES= 4 3. TOTAL A R F A = 0 o 4 39 99 64 0?: 0 8 PERCENT LAKES + SE\=_ 0 o 0 NUMBER FOR ANAL YSI S= " 4 1 AREA FOR ANALYSIS = 0, 48999 B4GE 0 3 MEAN TRIANGLE AREA = 0,3. 1951 13CE C7 GEOMOR°H 0 M E T R Y : MINIMUM EI.EVATIC.-. MAXIMUM ELEVATION LOCAL RELIEF = 1709 ,32 HYPSOMETRIC INTEGRAL = 0 ,26517 V E AN SLOPE: U N W E I G H T E D " = 0 o 3 7 6 ? 0 WEIGHTED BY MAP A <E A = 0 . 3 6694 WEIGHTED RY TRUE AREA = 0 .38063 6 9 4 , 94 LOCATF 0 AT POINT NUMBER 17 2404 ,26 LOCATED AT POINT NUMBER 19 AREA RATIO l . o a 2 a i 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 DI P LiZ) ROUGHNESS FACTOR: "38,0 5" 92 ,41 7 ,59 - 1 4 9 -P T A R M I G A N C R E E K ( S A M P L E lij>) : \LL P O I N T S G E N E R A L : N U M B E R O F P O I N T S N U M B > ; R O E TRI A N G L E S = " " ' " " ' 1 4 3 TOT A L A R E A = 0 . 4 8 0 9 92 4 3 6 0 8 PERCENT L A K E S + S E A _ . 0 . 1 5 N U M 3 E k ' E 0 R A N A L Y S I S = 1 4 3 A R E A F O R A N A L Y S I S = 0 , 4 3 9 ' " > 9 2 4 8 E 0 8 M E A N T R I A N G L E A R E A = 0 , 3 4 2 6 5 2 0 6 E 0 6 G E O M O R P H C M E T R Y M I N I M U M M A X I M U M E L E V A T I O N E L E V A T I O N 6 9 4 o " 5 4 L O C A T E 0 A T P O I N T N U M B E R 3 5 2 4 0 4 o 2 4 L O C A T E D A T P O I N T N U M B E R 7 0 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 S L O P E : U N W E f 3 H T E 0 ' W E I G H T E D B Y W E I G H T E D B Y I N T ! AL M A P A R E A T R U E A R E A 1 7 0 < 3 0 3 2 0 , 2 6 1 6 5 0 , 3 3 1 1 6 D . 3 7 0 7 0 G o 3 9 0 5 3 A R E A R A T I O 1 , 0 9 1 4 6 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 ) : O R I E N T A T I O N D I P L ( % ) = K ~ ' = 100/K R O U G H N E S S F A C T O R ^ 4 1 o 0 5 8 4 , 6 4 9 2 , 0 5 1 2 8 , 4 9 CI W E I G H T E D BY T R U E A R E A : O R I E N T A T I O N = 4 9 , 6 5 D I P L ( % ) R O U G H N E S S F A C T O R : 3 7 . 3 3 9 . 1 . 6 9 3 , 3 1 - 1 5 0 -MANN f N G P A R K ( S A M PL 11 l«) ) : A - 1 : .C ,CC'J S C A L E " H I G H R r SC L L T I E N " G E N E R A L : S LUMEiLJJ QE P C I i l l S . = 115. f NUMBER C F T R I A N G L E S = 2 C 7 i T O T A L A R E A = 0 . 4 3 « 9 9 C 7 2 E C 8 I .... P E R C E N T . L A K E S .+ . .SEA-.. . . . 0 . 0 i NUM13ER FOR AN AL Y 3 I S= 2 C 7 A R E A FOR A N A L Y S I S = 0 . 4 3 ° 9 9 C 7 2F C 8 ! M E A N T R I A N G L E A R E A •= 0 . 23 6 71 C4_t_F_ 06 .._ GEC.M0R_PH CM£I_R_Y_:_ _. _ M I N I M U M E L E V A T I O N = 1 0 5 1 . 6 6 L O C A T E D A T P O I N T NUMBER 2 M A X I M U M E L E V A T I O N - 19 3 5 . ^ 3 LJJ^I J:I.._AT_ j _ 0 j N T N U M B E R __15 L O C A L R E L I E F = . 7 7. S 2 H Y P S O M E T R I C I N T E G R A L = 0 . . < 2 2 6 . .NEA.N..S.LC.P.E .... . U N W E I G H T E D = 0 . 3 £ 9 . 2 W E I G H T E D BY MAP A R E A = 0 . 3 9 3 15 W E I G H T E D BY T R U E A R E A = 0 . 3 . 7 4 7 A R E A R A T I O - 1 . C 7 9 S 4 V E C T O R A N A L Y S I S : U N W E I G H T E C ( U N I T V E C T O R S ) : O R I E N T A T I O N D IP L m 2 4 . 1 6 8 9 . 2 7 9 2 . 7 5 K 1 0 0 / K R O U G H N E S S F A C T O R -1 3 . 72 7 . 2 9 7 . 2 5 W E I G H T E D BY T R U E A R E A : O R I E N T A T I O N = 3 5 1 , 0 1 DI P L ( ? ) R O U G H N E S S F AC TOR -3 9 . 5 2 9 2 . 6 1 7 . 3 9 •151 M A N N I N G P A R K I S A M P L E 1 8 ) : M— l : ? 5 C , C C 0 S C A L E M A P G E N E R A L : N U M B E R CF P O I N T S i l _ 5 . N U M B E R E E T R I A N G L E S ^ 22 T O T A L A R E A = 0 . 4 3 9 9 9 9 C H E C 8 P E R C E N T ....L A K E S . + ...S E \ = 0 , 0 . . N U M B E R F O R A N A L Y S I S - 3 2 A R E A F O R A N A L Y S I S = 0 . 4 R 9 5 c c c 4 E C 8 _ M E A N T R I A N G L E A R E A = 0 . 1 3 3 1 2 4 7 0 E C 7 - G E C M . G R £ H C M E 0 L ! 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 A T P O I N T M J M P E R N A X I M U M E L E V A T I O N - 1 9 C 5 . C O L O C A T E 0 A T P O IN T N U M B E R _ L O C A L R E L I E F = 8 3 8 . 2 0 H Y P S O M E T R I C I N T E G R A L = 0 . 5 1 7 3 1 _ . f ' E . A N . „ S . L O P . E j L . _ UNWEIGH TEO = 0 . 2 6 0 1 5 W E I G H T E D B Y M A P A R E A •= C . 2 r : 5 1 9 W E I G H T E D B Y T R U E A R E A = 0 . 2 T 7 2 6 A R E A R A T I O = 1 . 0 3 5 8 2 J V E C T O R A N A L Y S I S : i U N W E I GHTE 0 ( U N 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 = 2 6 . 9 7 1 0 0 / K •= 3 . 7 1 A O U GHNESS F A C T O R = 3 . 5 9 W E I G H T E D BY T R U E A R E A : ORIENTATION = 3 7 . 2 9 D I P = 3 9 . 3 9 L I S ) - 9 6 . 5 5 R O U G H N E S S F A C T O R = 3 . 4 5 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 , A P P R O X . S A M F P > ' . N T S A S 1 8 6 G E N E R A L J>JJjM.'3_E R O F P Q I v| T S N U M B E R C F T R I A N G L E S ^ 2 2 T O T A L A R E A = 0 . 4 3 9 9 < > 9 C 4 E C 3 PERCENT . L_AK ES.__+_ S E A " C O N U M BE R FOR A N A [ Y S I S = " " ' 3 2 A R E A F O R A N A L Y S I S = 0 . 4 3 9 9 9 9 0 4 E C S M E A N T R I A N G L E A R E A = 0 . 1 5 3 1 2 A 7 C E C 7 .-GE0.MJBPH0Mr3T.B_Y_ M I N I M U M J i A X j _ _ _ U M _ E L E V A T I O N E L E V A T I O N L M W E I G H T E D W E I G H T E D B Y W E I G H T E D B Y 1 C 6 6 . 3 0 1 8 3 9 . 7 6 L O C A L R E L I E F = 8 2 2 . 9 6 H Y P S O M E T R I C I N T E G R A L •= 0 . 5 4 3 . 2 X E A J S L S k Q P f : M A P A ^ F A T R U E A R E A = 0 . 3 C E 5 2 - 0 . 2 9 5 4 6 = 0 . 2 9 8 5 2 A R E A R A T I C 1 . C 4 7 4 6 L O C A T E C L O C A T E D A T Al. P 0 I N T P O I N T N O M E E R N U M B F R 2 3 V E C T O R A N A L Y S I S U N W E I G H T E C ( 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 . L U L L 3 5 5 . 5 1 8 3 . 7 4 _ _ 9 5 _ . 1 8 _ K 1 0 0/K R O U G H N E S S F A C T O R -2 0 . 1 1 4 . 9 7 4 . 3 2 W E I G H T E D B Y T R U E A R E A O R I E N T A T I O N = D I P L(*) R O U G H N E S S E A C T O R -1 0 - 6 JL 8 9 . 3 2 9 5 . 4 3 - 1 5 3 -T A T L A L A K E ( S A M P L E 2 4 ) : ' i N C L U O T N G L A K E S G E N E R A L : N U M B E R O F P O I N T S j 4 ? N U M B E R O F T R I A N G L E S = " 2 5 0 T O T A L A R E A = 0 o 4 3 9 9 B 9 2 n E 0 8 R E K C E N T . L A K E S . ± _ . S E A = _ . . 3 . 7 6 N U M B E R F U R A N A L Y S I S ^ 2 5 0 A R E A F O R A N A L Y S I S = 0 . 4 3 9 9 3 9 2 S E C 8 M E A N T R I A N G L E A R E A = 0 . 1 9 5 5 9 5 6 9 E 0 6 . G E O M O R P H O M E T R Y : M I N I M U M M A X I M U M E L E V A T I O N E L E V A T I O N 9 0 9 c S 3 1 1 1 2 . 5 2 A R E A R A T I O = 1 . 0 0 2 ' » 6 L O C A T E O L O C A T F 0 L O C A L R E L I E F = 2 0 2 . 6 9 H Y P S O M E T R I C I N T E G R A L = 0 . 2 6 7 3 9 M E A N S L O P E : U N W E I G H T E D " ' " = ' 0 . C 7 4 7 0 W E I G H T E D B Y M A P A R E A = 0 . 0 4 7 7 2 W E I G H T E D B Y T R U E A R E A = 0 . 0 4 3 C 3 A T A T P O I N T P O I N T N U M B E R N U M B E R 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 ) : O R I E N T A T I O N D I P I C S ) 4 5 . 5 2 8 9 . 6 4 9 9 . 4 1 K " = 1 6 7 . 9 9 J.OO/K = 0 . 6 0 R O U G H N E S S E A C T O R = 0 . 5 9 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N 3 4 1 . 5 9 D I P " = """ 8 9 . T 7 L ( 3 5 ) = 9 9 . 7 6 R O U G H N E S S F A C T O R = 0 . 2 4 .. - 1 5 4 -TATLA LAKE (SAMPLE 2 4) : EXCLUD ING LAKES GENERAL: NUMBER OF POINTS • \.±Z NUMBER CF" 1'RIANGLE S = " 250 TOTAL AREA = 0, 4 39° 9C C 8E 0 5 PERCENJ L A K E S +_ 5CA.. . . __ 3, 76 N U M3 E R F 0 R A MALY S I S= 2 3 8 AREA FOR A N A L Y S I S = 0, 47159CC8E 0 3 MEAN T R I A N G L E AREA = 0 O 1 9 3 1 4 7 C 6 E 06 GEOMORPHOMETRY MINI MUM MAXIMUM EL E V A T I O N E L E V A T I O N 9 0 9 0 3 3 1 1 1 2 , 5 2 L O C A L R E L I E F = 20 2,69 H Y P S O M E T R I C INTEGRAL = 0 , 2 7 5 3 0 M E A N SLOPE: UNWE T GHTED * = 0 , C 7 3 46 WEIGHTED B Y MAP AREA = 0,045 58 - J l L l i L L T ED B Y TR i JE AR E A = 0 , C 49 90 LOCATED LOCATED AT AT P 0 I N T "01 NT NUMBER NUMBER 1 7 '6 AREA P A T I O 1,0 02 55 VECTOR A N A L Y S I S : UNWEIGHTED(UNIT VECT0 R S) OR I EN TAT ION DIP L m K 1 0 0 / K R O U G H N E S S 4 5 , 5 2 3 9 » 6 3 9 9 , 3 3 F ACTOR: 1 5 9 , 9 2 0,63 0,62 WEIGHTED BY TRUE ORIENTATION ARE A: = 3 41,5 9 D I P = 3 9 , 7 6 L.S) = 9 9 , 7 5 ROUGHNESS F AC TO R= 0,2 5 ! G U I T E 7. L I L A K E ( S A M ' L E E I ) -155 -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 C E P U I N T S 1 1 4 N U M B E R O F T R I A N G L E S 1 T O T A L A R E A P E R C E N T _ L A K E S _ _ j S E A = 3, N U M B E R 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 = 0 , M E A N T R I A N G L E A R E A = 0 , 19 6 4 8 9 9 9 1 3 6 E O S 4 o 7 8 1 9 8 4 8 9 9 9 1 3 6 E 0 3 2 4 7 4 7 C 3 7 E 0 6 G E O M O R P H O M E T R Y : M I M I M U M M A X I M U M E L E V A T I O N E L E V A T I O N 1 0 6 3 0 7 5 2 2 ~> e. , 5 , 7 L O C A T E D L O C A T E D A T A T P O I N T P O I N T L O C A L R E L I E F = 1 1 9 40 3 2 H Y P S O M E T R I C I N T E G R A L = 0 .3 3 7 23 M E A N S L O P E : U N W E I G H T E D " " ~ ~ = 0 , 27 5 1 4 W E I G H T E D B Y M A P A R E A - 0 o 2 G 2 5 9 W E I G H T E D B Y T R U E A R E A = 0 , 2 1 5 9 0 NI.J M B E R N U M B E R 2 7 2 A R E A R A T I O l o 0 3 3 3 1 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 ) O R I E N T A T I O N D I P L ( % ) 7 . 9 2 8.3, 2 9 9 5 , 6 5 K = 22,88 100/K •= 4.37 R O U G H N E S S F A C T O R - 4 . 3 5 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N = 5 , S 3 D I P L ( % ) R O U G H N E S S F A C T O R -8 5 . 4 9 9 7 , 0 8 2 . 9 2 GH i TF 7 I. [ I.AK E ( S V -1 H. E i 1 ) -156 -E X C L U U I N'' LAKES GENERAL: NUMRcf QF POINTS 1 1 4 N U M B E R OF T R I A N G L E S - 1 9 3 T O T A L A R E A - 0 0 4 3 9 9 9 I 5 2 E 0 3 P F R C E N T L AK E S + _SE A = NU V -3E"R " F O R " AN 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 AREA = 4, 7 8 18 5 0.4<!6554CcE 08 0 . 2 5U913 7 E 06 G E O M O R P H O M E T R Y : M I N I M U M M A X I M U M ELEVATION ELEVATION 1 0 6 3 0 7 5 2 2 5 E o 5 7 A R E A R A T I O l o 0 3 4 9 3 L O C A T E D L O C A T E D AT AT L O C A L R E L I E F = 1 1 9 4 . S 2 H Y P S O M E T R I C I N T E G R A L = 0 . 3 5 4 1 3 M E A N S L O P E : UNWErGHT E D " " " ' " " ~ = 0 . 2 9 4 4 8 W E I G H T E D 3 Y MAP A R E A - 0 . 2 1 2 7 7 W E I G H T E D BY T R U E A R E A = 0 . 2 2 6 3 8 P O I N T P O I N T N U M B E R N U M B E R 2 7 2 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 ) : OR I EN T AT I O N D I P L ( % > 7 . 0 2 8 2 . 7 9 9 5 . 4 0 K = 2 1 . 6 1 100 / K - 4 . 6 3 R O U G H N E S S F A C T O R - 4 . 6 0 W E I G H T E D BY T R U E A R E A : 0 R_I EM T A T I O N = 5 . 8 3 " D I D " ~ " = 3 5 , 2 7 - 9 6 . 9 5 R O U G H N E S S F A C T O R - 3 . 0 5 O C N A K I V E P . ( S A M P L E 4 1. ) G E N E R A L : -10/-N U M B E R GIF P H I N T S N U M B E R n E T R T A N G L E S 1 T O T A L A R E A p E R C E N T L A K E S . *.._S.:.-.V N U M B E R F O R A N A L Y S I S r A R E A F Q » ? A N A L Y S I S --M E A N T P I A N G L E A R E A = . 1 3 . 8 . 2 4 0 0 o 4 3 9 c B 0 6 C E O B _ Q . Q 2 4 0 0 . 4 8 9 9 8 * 6 3 E 0 8 0 . 2 0 4 1 6 2 3 I E 0 6 - G E . 0 M J 3 R f H_0 M E T R Y_ M I N I M U M M A X I M U M . E L E V A T I O N E L E V A T I O N 3 C o 4 3 3 9 9 o 1 6 L O C A T E D L O C A T E D A T A T POINT POINT N U M B E R 1 3 1 N U M B E R 7 3 L O C A L R E L I E F . = 8 6 8 . 5 3 H Y P S O M E T R I C I N T E G R A L = 0 . 4 C 3 3 0 M E A N S L O P E : U N W E I G H T E D * = 0 . 3 9 1 6 4 W E I G H T E D B Y M A P A R E A = 0 . 3 8 0 9 0 W E I G H T E D B Y T R U E U 2 A = 0 . 3 9 1 2 2 A R E A R A T I O 1 . C 8 2 6 0 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 ) : O R I E N T A T I O N D I P L ( S) 2 2 0 . 1 7 3 8 . 0 9 9 2 . 4 2 K 100/K R O U G H N E S S F A C T O R ^ 1 3 . 1 4 7.61 7 . 5 3 W E I G H T E D B Y T R U E A R E A : ' O R I E N T A T I O N = 2 6 0 . 6 9 DI P L I %) R O U G H N E S S F A C T O R ^ 8 3 . " 7 2 9 2 . 3 9 7 . 6 1 I L L E C I L L E W A E T ( S A M P L E ^ 3 ) G P.To , N W - S E 7)1 'AGON A'fs G E N E R A L N U M B E R C F P O I N T S 2 2 5 N U M B E R GF T R I A N G L E S = 3 9 2 T O T A L A R E A = 0 . 4 3 9 9 7 9 3 6 E 0 8 P E R C E N T L A K E S _ _ SEA = _ _ 0 O 0 NUMBER" F O R " A~NALYS"I 'S= " 3 9 2 " A R E A F O R A N A L Y S I S = 0 . 4 3 9 - 9 7 9 3 6 E 0 8 MEAN T R I A N G L E A R E A = 0 . 1 2 4 9 9 4 6 9 E 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 = 3 9 0 . 0 2 L O C A T E D A T P O I N T N U M B E R 2 0 7 = 2 7 4 3 o 2 0 L O C A T f D A T P O I N T N U M B E R 9 2 L O C A L R E L I E F = 1 8 5 3 . 1 8 H Y P S O M E T R I C I N T E G R A L = 0 . 4 3 5 3 6 MEAN S L O P E : U N W E I G H T E D = 0 . 5 2 3 4 4 W E I G H T E D BY MAP A R E A - 0 . 5 2 3 4 5 W E I G H T E D B Y T R U E A R E A = 0 . 5 2 9 0 0 A R E A R A T I O = 1 . 1 4 3 C 7 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 ) : O R I E N T A T I O N = 1 1 4 . 9 4 D I P = 8 3 . 1 7 Lit) _= § 8 . 6 4 _ K " " = ~ 3 . 7 8 1 0 0 / K - 1 1 . 3 9 R O U G H N E S S F A C T O R = 1 1 . 3 6 0 7 W E I G H T E D BY T R U E A R E A : OR I E N T A T_I_0N = _ 1 _ 1 5 , D I P " =" " 3 3 . 2 3 l i t ) = 8 3 . 1 0 R O U G H N E S S F A C T O R = 1 1 . 9 0 . - 1 5 9 -I L L E C I L L E W A E T ( 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 M B E R O F P O I N T S - 111 N U M B E R O F T R I A N G L E S - 3 9 2 T O T A L A R E A = 0 o 4 - 3 9 9 7 9 3 6 E 0 8 P E R C E N T L A K E S _ + _ . S E A - _ ...... 0 » 0 N U M B E R F 0 R A N A L Y S I S - 3 9 2 A R E A F O R A N A L Y S I S - 0 . 4 8 9 9 7 9 3 6 E 0 8 M E A N J R i A N G L E _ A R _ E A - 0 . 1 2 4 9 9 4 6 9 E 0 6 G E O M O R P H C M E J . R Y : M I N I M U M E L E V A T I O N - 8 9 0 . 0 2 L O C A T E D A T P O I N T N U M B E R 2 0 7 M A X I M U M E L E V A T I O N = 2 7 4 3 . 2 0 L O C A T E D A T P O I N T N U M B E R 9 2 L O C A L R E L I E F = 1 8 5 3 . 1 3 H Y P S O M E T R I C I N T E G R A L = 0 . 4 3 5 7 6 N E A N S L O P E : U N W E I G H T E D " • " = 0 . 5 1 8 4 3 W E I G H T E D B Y M A P A R E A - 0 . 5 1 8 4 4 _ W E I G H T E D B Y T R U E A R E A = 0 . 5 3 6 5 4 A R E A R A T I O = 1 . 1 4 2 3 7 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 ) O R I E N T A T I O N = 1 1 3 . 0 5 D I P = 8 3 . 2 2 L m _ - 83.77 K ~ " " " " " " = ' ' 8 . 3 8 ' 1 0 0 / K - 1 1 . 2 6 R O U G H N E S S F A C T O R - 1 1 . 2 3 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N _ _= 1 1 5 ^ 0 7 _ D I P " ~ ~ = 8 3 . 2 3 L ( 5 g ) = 8 8 . 1 5 R O U G H N E S S F A C T O R - 1 1 . 8 5 _ - 1 6 0 -f P T A R M I G A N CREEK (3V-1PLE 11 T? GRT6 , ~ T : ^ 0 . OI&^HNW-SF >^TAC('NTLS ! GENERAL: ! > NUMBER CF POINTS ?2 5_ ( NUMBER OF TRI ANGLE S= 392 i TOTAL AREA = 0.43997936E 03 I P ERCENJ . LAKES.* SEA= ... .0 .0 I 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 L R U MINIMUM ELEVATION 6 9 4 . 9 4 L O C A T E D A T P O I N T N U M B E R 22 5 MAXIMUM ELEV ATI O N — 2 3 0 l o 24 L O C A T E D AT P O I N T N U M B E R 1. ° o LOCAL RELIEF — 1 6 C 6 . 3 0 HYPSOMETRIC INTEGRAL 0 , 2 7 9 2 2 MEAN SLOPE: UNWEIGHTED — 'bo~3 4 3S7 WEIGHTEO BY MAP AREA 0 . 3 43 6 9 WEIGHTED BY T R U E AREA 0 . 3 6 0 39 AREA RATIO = 1 . 0 78 58 VECTOR ANALYSIS: — — -UNWEIGHTEDIUNIT VECTORS): ORIENTATION = 5 3 . 9 2 DIP = 8 8 . 4 9 l(%) = 9 3 , 2 4 K = 14.75 100/K = 6.78 ROUGHNESS FACTOR= 6.76 W E I G H T E D BY T R U E A R E A : O R I E N T j A T _ I O N = 5 5 _ . _ 6 2 " D l P " " ~ ' ="87.94 LIZ) = 92.77 R O U G H N E S S F A C T O R = 7.23 V. - 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 N U M B E R O F P O I N T S 2 2 5 f NUM1 3ER O F T R I A N G L E S - 3 9 2 T O T A L A R E A 0 . 4 3 9 9 7 9 3 5 E 0 8 P E R C E N T L A K E S +• S E A - 0 • o. N U M B E R F O R A N A L Y S I S - 3 9 2 A R E A F O R A N A L Y S I S - 0 . 4 3 9 9 7 9 3 6 E 0 8 M E A N T R I A N G L E A R E A = 0 . 1 2 4 9 9 4 6 9 E 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 = 6 9 4 . 9 4 L O C A T E D AT POINT N U M B E R 2 2 5 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 A T POINT N U M P F R 1 ° 9 L O C A L R E L I E F = 1 6 0 6 . 3 0 H Y P S O M E T R I C I N T E G R A L = 0 . 2 8 0 5 4 M E A N S L O P E : U N W E I G H T E O - 0 , 3 5 8 3 6 W E I G H T E D B Y M A P A R E A - 0 . 3 5 8 3 7 W E I G H T E D B Y T R U E A R E A = 0 . 3 7 1 5 2 A R E A R A T I O = 1 . C 7 9 3 2 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 ) : O R I E N T A T I O N 5 8 . 6 4 D I P 8 3 . 3 9 L m 9 3 . 0 7 K 1 4 . 4 0 100/K 6 . 9 5 R O U G H N E S S F A C T O R - 6 . 9 3 W E I G H T E D B Y T R U E 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 * L U ) 9 2 , 7 1 R O U G H N E S S F A C T O R - 7 , 2 9 i • V. _ - 1 6 2 -"PTARMI G A N ~CR E E K " ( S V I P ' t E T l V T OR I 6 7 " I : 2 50,"o 0 0 ~ ~~NW^S E D f > ^ G 0 > « A l ' . " S G E N E R A L N U M B E R O F P O I N T S 2 2 5 N U M B E R O F T R I A N G L E S 1 T O T A L A R E A P E R C E _ N J L A K E S , +__S_E A j N U M B E R F O R A N A L Y ST S= A R E A F O R A N A L Y S I S = M E A N T R I A N G L E A R E A = 3 9 2 0 . 4 3 9 9 7 9 3 6 E 0 8 0 • 0 " 3 9 2 3 . 4 8 9 9 7 9 3 6 E 0 3 0 . 1 2 4 9 9 4 6 9 E 0 6 G E O M O R P H G 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 7 0 l o C 4 2 3 6 2 o 2 0 L O C A T E D L O C A T E D AT AT P O I N T P U I N T N U M B E R N U M B E R 2 2 5 1 9 9 L O C A L R E L I E F = 1 6 6 1 . 1 6 H Y P S O M E T R I C I N T E G R A L = 0 . 2 7 4 6 4 V E A N S L O P E : ____ _ U N WE I G H T E D " " " - — W E I G H T E D B Y M A P A R E A •= 0 . 3 5 4 6 0 W E I G H T E D R Y T R U E A R E A = 0 . 3 7 2 7 5 A R E A R A T I O 1 . C 8 1 9 5 V E C T O R 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 = 5 3 . 0 1 O I P = 8 8 . 6 5 _ _ = 9 3 . 0 1 K " ~ = 1 4 . 2 6 1 0 0 / K = 7 . 0 1 R O U G H N E S S F A C T O R = 6 . 9 9 W E I G H T E D B Y T R U E A R E A : 0 R I E N T A T I O N = _ _ 5 1 .JU D I P - ~ ~ ' 8 3 . 0 4 L ( ^ ) = 9 2 . 4 8 R O U G H N E S S F A C T O R = 7 . 5 2 P T A R M I G A N C R E E K ( S A M P L E 1 1 . - 1 6 3 - _._ ... .. 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 : . N U M B E R O F P O I N T S 221. N U M B E R O F T R I A N G L E S -T O T A L A R E A P E R C E N T L A K E S S E A ; N U M B E R 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 = _ M E A N T R I A N G L E A R E A ; 3 9 2 0 . 4 3 9 9 7 9 3 6 E 0 8 O o O 3 9 2 0 . 4 3 9 9 7 9 3 6 E 0 8 0 . 1 2 4 9 9 4 6 9 E 0 6 G E O M q 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 7 0 l o G 4 2 3 6 2 . 2 0 L O C A T E D L O C A T E D A T AT P O I N T P O I N T N U M B E R N U M B E R 225 199 L O C A L 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 M E A N S L O P E _ : ' U N W E I G H T E D W E I G H T E D B Y M A P A R E A W E I G H T E D B Y T R U E A R E A 1 6 6 1 o 1 6 0 . 2 7 5 3 9 0 7 3 6 7 6 3" 0 . 3 6 7 6 9 0 . 3 8 2 2 9 A R E A R A T I O = 1 . 0 8 2 7 4 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 ) : O R I E N T A T I O N = 5 3 . 3 0 D I P - 88.52 Li%) _____ _ = 9 2 .83 K ~ = 1 3 . 9 2 1 0 0 / K = 7 . 1 9 R O U G H N E S S F A C T O R - 7 . 1 7 W E I G H T E D BY T R U E A R E A : O R I E N T A T I O N _ = 5 1 * 0 1 _ D I P ~ " " = " 8 3 . 0 4 l i t ) = 9 2 . 4 1 R O U G H N E S S F A C T O R - 7 . 5 9 M A N N I N G P A R K ( S A M P L E 1 3 ) . - 1 6 4 - ........ i n , N W - S E D I A G O N A L S G E N E R A L : N U M B E R O F P O I N T S 2 2 5 N U M B E R C F T R I A N G L E S ^ T O T A L A R E A P E R C E N T L A K E S _ - t - _ S E A = N U M B E R 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 = M E A N T 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 O o O 3 9 2 0 , 4 3 9 9 7 9 3 6 E 0 8 0 , 1 2 4 9 9 4 6 9 E 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 _ 1 0 6 6 o 3 0 L O C A T E D A T 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 8 9 o 7 6 L O C A T E D A T " H I N T N U M B E R 8 L O C A L R E L I E F - = 8 2 2 o 9 6 H Y P S O M E T R I C I N T E G R A L = 0 , 5 6 5 8 9 M E A N S L O P E : U N W E I G H T E D — 0 , 3 3 1 0 3 " W E I G H T E D B Y M A P A R E A 0 , 3 2 1 0 9 W E I G H T E D B Y T R U E A E A = 0 , 3 3 3 4 5 A R E A R A T I O — 1 , 0 6 3 4 7 V E C T O R A N A L Y S I S : — — - • • — — - -U N W E I G H T E O f U N I T V E C T O R S ) : O R I E N T A T I O N - 343,67 D I P = 8 9 , 3 9 L(%) = 94,25 K " = 17,33 100 /K = 5,77 R O U G H N E S S F A C T O R - 5,75 W E I G H T E D B Y T R U E A R E A : 0R I E N T A T I O N - 3_50, 60 D I P " " = " " " " L ( % ) R O U G H N E S S F A C T O R -8 9 , 3 4 9 4 , 0 4 5 , 9 6 V. f G E N E R A L : N U M 3 E R O F P 0 I N T _ _ S _ _ _ = 2 2 _ 5 N U M 3 E R O F T R I A N G L E S = " ' ~ " " 3 9 2 T O T A L A R E A = 0 . 4 3 9 9 7 9 3 6 E 0 8 P E R C E N T L A K E S + _ S E A = _ 0 o 0 N U M B E R " F O R ' A N A L Y S I S = " 3 9 2 A R E A F O R A N A L Y S I S . = 0 . 4 3 9 9 7 9 3 6 E 0 8 M E A N T R I A N G L E A R E A = 0 0 1 2 4 - 9 9 4 6 9 E 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 1 0 6 6 . 3 0 L O C A T E D A T P O I N T N U M B E R 2 1 8 3 9 „ 7 6 L O C A T E D A T P O I N T N U M B E R 8 L O C A L R E L I E F 3 2 2 o 9 6 H Y P S O M E T R I C I N T E G R A L = 0 . 5 6 6 9 1 M E A N S L O P E : U N W E I G H T E D 0 . 3 3 4 7 3 W E I G H T E D B Y M A P A R E A 0 , 3 3 4 7 5 W E I G H T E D B Y T R U E A R E A — 0 . 3 4 1 3 5 A R E A R A T I O 1 . 0 6 3 6 1 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 ) 3 5 1 . 7 2 8 9 , 3 6 9 4 . 2 1 O R I E N T A T I O N D I P Li%) K = " ~ 1 7 . 2 3 1 0 0 / K = 5 , 8 0 R O U G H N E S S F A C T O R = 5 . 7 9 W E I G H T E O B Y T R U E A R E A : O R I E N T A T I O N = 3 5 0 . 6 0 " D I P - ~ _ ~ ~ " = " 8 9 . 3 4 Ll%) •= 9 4 . 0 3 R O U G H N E S S F A C T O R = 5 . 9 7 V. . .._ -166 -T A T L A L A K E ( S A M P L E 2 4 ) : O R 1 0 N W - S E O I - A O O N A L S G E N E R A L N U M B E R O F P O I N T S N U M B E R O F T P I A N G L E S = T O T A L A R E A P E R C E N T L A X E S t S E \ = N U M B E R 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 = M E A N T R I A N G L E A R E A = 3 9 2 0 , 4 3 9 R 7 9 3 6 E O B 0 . 9 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 = J O 5 . 3 3 L O C A T E D A T P O I N T N U M B E R 3 0 M A X I M U M E L E V A T 1 0 N = 1 0 9 7 , 2 8 j E j O j C A T E D A T P O I N T N U M B E R l_ L O C A L R E L I E F " - 1 8 7 , 4 5 H Y P S O M E T R I C I N T E G R A L = 0 . 2 9 8 1 0 M E A N SLOPE:_ _ U N W E I G H T E D " " " " " ' " " ' " ' " = " 0 . ' 0 4 0 8 1 " W E I G H T E D B Y M A P A R E A = 0 , 0 4 0 8 1 W E I G H T E D B Y T R U E A R E A = 0 , 0 4 C 8 7 ' A R E A R A T I O = l . O C l l 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 ) : O R I E N T A T I O N = 3 4 3 . 3 8 D I P 3 9 . 3 0 L ( %) 9 9 . 8 7 K = 7 4 0 . 0 8 1 0 0 / K 0 . 1 4 R O U G H N E S S F A C T O R = 0 . 1 3 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N _34_8o 5 9 D I P ~ " =' " 3 9 . 7 9 = 9 9 . 6 7 R O U G H N E S S F A C T O R 3 0 . 1 3 ; - 1 6 7 -T A T L A L A K E ( S A M P L E 2 4 ) : G R I D N E - S W D I A G O N A L S I G E N E R A L : v N J J M B E R O F P O I N T S = 2 2 5 . N U M B E R O F T R I A N G L E S - 3 9 2 i T O T A L A R E A = 0 , 4 S 9 9 7 9 3 6 E . 0 8 P E R C E N T L A K E S + _ S E A _ = _ _ 0 . 0 . I ' N U M B E R F O R A N A L Y S I S - 3 9 2 A R E A F O R A N A L Y S I S = 0 . 4 3 ' - 9 7 9 . 3 6 F 0 8 M E A N T R I A N G L E A R E A = 0 „ 1 2 4 9 9 4 6 9 E 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 9 0 9 c 8 3 L O C A T E D AT r > 0 1 NT N U M B E R 3 0 M A X I M U M E L E V A T I O N 1 0 9 7 . 2 8 L O C A T E D AT ° Q I NT N U M B E R 1 L O C A L R E L I E F zz 3 . 8 7 . 4 5 H Y P S O M E T R I C I N T E G R A L = 0 . 2 9 6 3 0 M E A N S L O P E : U N W E I G H T E D zz " O . C 3 9 3 5 W E I G H T E D B Y M A P A R E A = 0.0 3 9 3 5 W E I G H T E D B Y T R U E A R E A zz 0 . 0 3 9 4 1 A R E A R A T I O — 1.001-30 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 ) : O R I E N T A T I O N = 3 4 3 . 3 3 D I P = 8 9 . 7 9 L ( 2 ) = 9 9 . 8 7 K ~ ~ - 7 4 3 . 5 2 100/K = 0.13 R O U G H N E S S F A C T O R - 0.13 W E I G H T E D B Y T R U E A R E A : O R I E N T A T I O N - 3 4 8 . 5 9 D I P = 3 9 . 7 9 LiZ) - 9 9 . 8 7 R O U G H N E S S F A C T O R - 0 . 1 3 v. ( G H I TF.TL"i L A K E " ( " S A M P T ' E " 3 1 ) ~ : ~ ~ G R i 0, " NW- sTTTTlA G O K A L S G E N E R A L : N U M B E R O F P O I N T S = 22_5 N U M B E R " O F T R I A N G L E ' S - " " ~ 3 9 2 T O T A L A R E A = 0 . 4 8 9 9 7 9 3 6 E 0 8 P E R C E N T L A K E S + S E A = _ _ 0 . 0 N U M B E R F O R A N A L Y S I S - ' ~ ' 3 9 2 A R E A F O R A N A L Y S I S = 0 . 4 3 9 9 7 9 3 6 E 0 8 M E A N T R I A N G L E A R E A - 0 . > 4 9 9 4 6 9 E 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 _ 1 0 6 3 . 7 5 L O C A T E D A T P O I N T N U M B E R 1 7 M A X I M U M E L E V A T I O N zz 2 2 5 5 , 5 2 L O C A T E D A T P O I N T N U M B E R 1 5 2 L O C A L R E L I E F = 1 1 9 1 . 7 7 H Y P S O M E T R I C I N T E G R A L zz 0 . 3 3 3 9 5 M E A N S L O P E : U N W E I G H T E D — 0 . 1 8 6 8 5 W E I G H T E D B Y M A P A R E A = 0 . 1 8 6 8 6 W E I G H T E D B Y T R U E A R E A .zz 0 . 1 9 4 3 6 A R E A R A T I O zz 1 . 0 28 03 VECTOR 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 ) : O R I E N T A T I O N - 2 . 4 4 DIP = 8 5 . 8 9 L( %) 9 7 . 7 3 K ' = "43,96 1 0 0 / K = 2 . 2 7 ROUGHNESS F A C T O R - 2 . 2 7 W E I G H T E D B Y T R U E A R E A : n R I E N T j A J ) _ O N _ = 2 . 2 6 . ^ - p - —-- 5 g Lit) = 9 7 , 5 6 R O U G H N E S S F A C T O R - 2 . 4 4 G H I T E Z L I LAKE - 1 6 9 -3 1 ) : GRTOY NE-'SW" DTAGONALS" G E N E R A L : N U M B E R O F P O I N T S 2 2 5 N U M B E R O F T R I A N G L E S -T O T A L A R E A P E R C E N T _ L A K E S + _ S E A = N U M B E R 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 = M E A N T R I A N G L E A R E A = 3 9 2 G o 4 3 9 9 7 9 3 6 E 0 3 _ ) o 0 3 9 2 G o 4 3 9 9 7 r o 6 E 0 8 0 . 1 2 4 9 9 4 6 9 E 0 6 G E O M O R P H O M E T R Y : MINIMUM MAXIMUM E L E V A T I O N E L E V A T I O N 1 0 6 3 o 7 5 LOCATED AT POINT NUMBER 1 7 2 2 5 5 . 5 2 LOCATED AT POINT NUMBER 152 L O C A L 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 M E A N S L O P E : U N W E I G H T E D W E I G H T E D B Y W E I G H T E D B Y M A P A R E A T R U E A R E A 1 1 9 1 . 7 7 0 . 3 3 4 3 1 0 . 1 8 4 7 4 0 . 1 8 4 7 5 0 . 1 9 2 4 0 A R E A R A T I O 1 . 0 2 8 0 1 V E C T O R 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 ) _ = 9 7 . 7 4 K " * ~ = 4 4 o " G 3 100/K = , 2 . 2 7 R O U G H N E S S F A C T O R * 2 . 2 6 W E I G H T E D BY T R U E O R I E N T A T I O N _ " D I P L m A R E A : _= 2.26 - 35Y53 9 7 . 5 7 R O U G H N E S S F A C T O R ^ 2 . 4 3 _ _ - l / U -0 C 1 N A R I V E R ( S A M P L E 4 1 ) : G R I D , N W - S E D I A G G N ' A T S G E N E R A L : V, N U M R E R O F P O I N T S =_ . ? _ 2 5 ( " ' N U M B E R O F T R I A N G L E S * 3 9 2 T O T A L A R E A = 0 . 4 3 9 9 7 9 2 6 E 0 3 ' _ P E R C E N T L A K E S +•_ S . E ; \ = 0 . 0 j N U M B E R F O R A N A L Y S I S * 3 0 2 : A R E A F O R A N A L Y S I S = 0 . * 8 9 9 7 9 3 6 E 0 3 ; M E A N _ T R _ I A N G L E A R E A = 0 . 1 2 4 9 9 4 6 9 E 0 6 G E 0 M O R P_HC M E X R Y J M I N I M U M E L E V A T I O N = 3 0 . 4 8 L O C A T E D A T P O I N T N U M B E R 2 2 5 M A X I M U M E L E V A T I I N = 8 5 3 . 4 4 L O C A T E D A T P O I N T N U M B E R 1 1 3 L O C A L R E L I E F 3 2 2 o 9 6 H Y P S O M E T R I C I N T E G R A L = 0 . 4 1 9 1 4 M E A N S L O P E : U N W E I G H T E O * " 0 7 3 2 4 4 4 " W E I G H T E D B Y M A P A R E A = 0 . 3 2 4 4 5 W E I G H T E D B Y T R U E - A R E A - 0 . 3 3 4 9 5 A R E A R A T I O = 1 . 0 6 5 2 4 V E C T O R A N A L Y S I S : U N W E I G H T E C I U N I f V E C T O R S ) : O R I E N T A T I O N D I P * 11%) K ' " " " ' " ~ " " " ~ = 1 0 0 / K R O U G H N E S S F A C T O R * W E I G H T E D B Y T R U E A R E A : O R I E j N T A T I Q N _ = _ 2 6 3 _ . 8 0 D I P ' ~ ~ ~ " ~ = " 8 3 . 3 0 ' 11%) = 9 3 . 9 0 R O U G H N E S S F A C T O R * 6 . 1 0 2 6 6 . 8 0 8 8 . 7 7 _ 9 4 ° 2 0 1 7 . 2 0 5 . 8 1 5 . 8 0 OON A R I V E R ( S A M P L E 4 1 ) : O R I ! . ) , - 1 7 1 -N E - S W D I A G O N A L S G E N E R A L NUMBER O F P O I N T S 2 2 5 0, NUMBER O F T R I A N G L E S = T O T A L A R E A P E R C E N T L A K E S_ + _SEA_=_ _ N U M BE R " F O R "A N A L Y S I 3 = " " A R E A FOR A N A L Y S I S = 0 MEAN T R I A N G L E A R E A = 0 3 9 2 4 3 9 9 7 9 3 6 E OS 0 . 0 39 2' 4 8 C 9 7 9 3 6 E 0 3 1 2 4 9 9 4 6 9 E 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 MAXIMUM E L E V A T I O N 3C< 95 3, A3 44 L O C A T E D L O C A T E D A T A T P O ! NT P O I N T NUMBER 1 1 3 L O C A L 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 MEAN 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 A R E A W E I G H T E D BY T R U E A R E A 3 2 2 . 9 6 0 . 4 2 C C 2 0 . 3 26 32 0 , 3 2 6 3 4 0 . 3 . 3 6 3 8 A R E A R A T I O 1 . 0 6 5 33 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 ) : O R I E N T A T I O N = 2 6 3 . 9 2 D I P 8 8 . 3 0 LIZ) 9 4 . 1 8 K 1 7 . 1 3 1 0 0 / K 5 . 8 4 R O U G H N E S S F A C T O R * 5 . 8 2 W E I G H T E D BY T R U E A R E A : O R I E N T A T I O N =_ 26_3_.3_0 DI P " • = . 3 8 . 3 0 11%) = 9 3 . 8 9 R O U G H N E S S F A C T O R * 6 . 1 1 

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