Engineering Geology Volume 1 issue 2 1965 Richard O. Stone; J. Dugundji -- A study of microrelief—its mapping, classification, and quantification by means of a fourier anal.pdf

VOLUME I NO. 2

SPECIAL ISSUE

A STUDY OF MICROREL[EF - -1TS MAPP ING, CLASSIF]~CATION, AND

QUANTIF ICAT ION BY MEANS OF A FOURIER ANALYS IS

RICHARD O. STONE AND J. DUGUNDJ!

CONTENTS

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Methods of data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Quantitative expression of microrelief . . . . . . . . . . . . . . . . . . . . . . 107 Data testing and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Classification of microrelief . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Eng. Geol., 1(2) (1965) 89-187

Engineering Geology-Elsevier Publishing Company, Amsterdam- Printed in The Netherlands

A STUDY OF MICRORELIEF-- ITS MAPPING, CLASSIFICATION, AND QUANTIFICATION BY MEANS OF A FOURIER ANALYSIS

RICHARD O STONE AND J. DUGUNDJI

Department of Geology, University of Southern California, Los Angeles, Calif. (U.S.A.) Department of Mathematics, University of Southern California, Los Angeles, Calif. (U.S.A.)

(Received August 8, 1964) (Resubmitted July 16, 1965)

SUMMARY

Microrelief is defined in this study as those surface irregularities measured from a fixed point of elevation that display internal differences of elevation of not more than 10 ft. or less than 3 inches and that manifest themselves within a distance of 4-64 linear ft.

Twenty-two California microterrains of different types were mapped, rang- ing from a wave-cut terrace to a boulder-free dry wash. Large scale topographic maps were prepared, generally using a horizontal scale of 1 inch equals l0 ft., and a series of radial profiles was constructed through each of the areas at intervals of 15 .

Quantitative expression of the microrelief was accomplished using these profiles to develop the terrain in a Fourier series, followed by separation of portions which had high frequencies. These were processed by a digital computer to determine (1) average changes in level as a profile or terrain is traversed (the simple relief factor M), (2) the average height of major relief features (the specific relief factor An), (3) average steepness of relief features encountered (the range of slope factor P), and (4) the extent to which there is periodicity, or repetition, as the curve or terrain is traversed (the structural homogeneity factor K). Further, from the computations for each terrain, an avoidance factor ~ (a measure of overall irregularity), is derived, as well as the cell length, CL, which delineates the distance from a given origin which must be traversed in order to encounter all significant features of the terrain.

Three general classes of microrelief terrains were established, based upon the factor (the product of the P and M components), as follows:

Class I, "gentle" microterrain, ~

92 R. O. STONE AND J. DUGUNDJ I

slopes or the range of relief is the most important consideration in the value of terrain roughness. The postscript N is used when neither P nor M dominates.

Further, the significance and geometric meaning of the other terrain factors derived were studied and evaluated, and it was found that the application of a Fourier analysis to terrain quantification, while not always conclusive, is sufficient- ly so to merit further testing and application.

INTRODUCTION

PuITose and scope

A principal consideration of any terrain study is the form of the ground surface. This is referred to as microrelief, surface roughness, surface "grain", or microgeometry. Characteristics of small-scale surface features are of concern to individ- uals in several scientific disciplines: the engineering geologist, the geomorpholo- gist, military geologists, soil scientists and land locomotion engineers. With the rapid expansion of knowledge regarding the sea floor and the lunar surface, the marine geologist and the selenologist have also become involved in the description and expression of microrelief.

References to microrelief in scientific literature are numerous but doubt and confusion exist as to what microrelief is, how to classify it, and how it may be expressed. As a consequence, a study was undertaken whose purposes were to perform large scale field mapping of selected microrelief areas, to devise a method for expressing microrelief in a quantitative geomorphological manner, and to present a classification of microrelief utilizing the quantitative data.

The study was made specifically for application to military terrain studies and for land locomotion research, but is applicable as well to other scientific areas of interest.

Definition

The first obligation of the study was to define the term "microrelief". The term is not new to the vocabulary of geology. It was apparently coined by LECONTE in 1877 to describe "hog wallows" or prairie mounds in California and Oregon, and was used by him to describe the configuration of a surface in which coalescent low mounds were 6-10 inches higher than the adjoining depressions. Subsequent to this application, microrelief has been used to describe relief features of the lunar surface, of the sea floor, of soil plots, and of small terrestrial features. A sample of the many situations and environments to which the term has been applied is presented in Table I.

It was apparent that the term "microrelief" must be specifically defined be-

Eng. Geol., 1(2) (1965) 89-187

A STUDY OF MICRORELIEF

TABLE I

LIST OF DEFINITIONS OF MICRORELIEF

Source Date Definition and application

93

LECONTE 1877

TERRY and STEVENSON 1957

DWORNIK et al. 1959

STRAHLER and KOONS 1959

SYTINSKAYA 1959

VAN LOPIK and KOLB 1959

SHIPEK 1961

GREEN 1962

SAUCIER and BROUGHTON 1962

MABBUTT 1963

Configuration of surface dotted by mounds 6-10 inches higher than adjoining depressions. Mounds, ridges, depressions or undulations on the sea floor. Lower limit 3 ft.; maximum limit 1040 ft. and in places 80 ft. Objects of surface irregularity less than 1 inch in height in a 7-ft. plot. Surface roughness involving measurements of height difference greater than 0.1 ft. Limits between 0.1 mm and 0.1 m for lunar irregularities. Surface geometry associated with terrain features exhibiting less than 10 ft. of relief. Microrelief on sea floor measured horizontally in meters and tens of meters and vertically in centimeters and meters. Microsurface (of the moon) is"bicycle smooth" and ranges in size from 1/~ to 1 cm. Surface configuration of terrain that exhibits relief of less than 10 ft. Patterns formed by "Wanderrie banks" which are 2-4 ft. high, 1/2 - 2 miles in length, and 60-1,200 ft. wide.

fore any progress could be made. A great many geologists, when confronted by the word, or when applying it, think in terms of small relief features, usually of limited extent. Some of the definitions and usages listed in Table I were derived directly from the effectiveness of equipment used in a given study, e.g., from the smallest variation delineated with standard echo-sounding equipment, or, in the case of the lunar application, on the basis of the sensitivity of certain scanning instruments. Other applications concerned with measurement of microrelief have not specified what it is.

"Microrelief", for the purposes of this study, was defined in terms of (1) existing military requirements, and (2) the necessity to provide sufficient detail in the definition to permit a mathematical evaluation.

Military terrain requirements This pro ject was under taken as an adjunct to previous microre l ie f work at the

Waterways Exper iment Station, notably that o f VAN LOPIK and KOLB (1958, 1959),

who stated that a landscape is def ined in terms of the plan-prof i le, slope occurrence,

and rel ief generated by the 10-ft. contour interval, i.e., having rel ief o f more than

Eng. Geol., 1(2) (1965) 89-187

94 R. O. STONE AND J. DUGUNDJ!

10 ft. Therefore, microrelief in this study is of necessity concerned with those features of terrain geometry having relief of less than 10 ft. The lower limit of relief was set at 3 inches. Smaller surface irregularities are considered by the military to be of little significance to either vehicular or foot travel. The height of microrelief features was not a problem in the mathematical analysis which follows because this factor was automatically taken care of by the mapping process; only those areas exhibiting less than 10 ft. of relief were considered.

Thus, from the military requirements standpoint alone~ microrelief was defined as differences of elevation of not more than 10 ft. or less than 3 inches.

Requirements Jbr analysis In the mathematical analysis that is applied later in the report there is no consideration of the slope of the surface on which the microrelief features occurred. The surface of study was considered parallel to the surface envelope generated by the 10-ft. contour interval. Therefore, microrelief features were defined as relief features on any surface, measured parallel to the surface, whether it was flat or inclined. It was necessary also, for the purposes of analysis, to define horizontal limitations for microrelief features. Values were selected which were intimately related to the analysis applied, as will be shown. (This was done, largely, to keep the number of harmonics derived from the surface profiles to a workable minimum.) These values were 4 and 64 ft. I f it should be desired to detect and express features that are developed in a horizontal distance of less than 4 ft., it is only required that more harmonics of the profile curves be used. It must be pointed out here that it is the "small" riding features on broad features, and not the broad features themselves, that determine microretief.

Definition to be applied From the foregoing discussion the definition that follows is applied throughout the report: microrelief features are small-scale landforms, measured from a fixed point of elevation, which display internal differences of elevation of not more than 10 ft. or less than 3 inches and which manifest themselves within a distance of 4-64

linear ft. A precise mathematical definition for microrelief is given in a later section

of the report.

Areas of investigation

Field mapping of microrelief was undertaken in semi-arid and arid portions of the Mojave and Colorado Deserts of California. In addition, microrelief features were mapped along more humid regions bordering the Colorado River and along the coastline of southern California.

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A STUDY OF MICRORELIEF 95

METHODS OF DATA COLLECTION

General statement

In order to collect data it was necessary to determine the nature of surface roughness by means of field surveys. On ordinary topographic maps the horizontal scale is small and the contour interval relatively large, and hence microrelief features simply are not in evidence. Areas of surface roughness which might prohibit vehicular traverses appear as flat surfaces. Consequently, to hope to understand the nature of microrelief and to possibly relate slope, relief, slope occurrence, homogeneity, and other parameters of surface geometry, detailed mapping and preparation of large scale topographic maps was performed. It is hoped that the maps herein will be valuable for further research.

Microrelief features were mapped by several methods in the course of the project work, and procedures were adapted to the purpose for which the data were to be used. These methods, together with some utilized by other investigators to determine the character and extent of microrelief are examined, and specific applications and recommendations are indicated.

FieM mapping

Selected microrelief type areas Twenty-two areas, consisting of different types of microrelief in the southern California area, were selected for mapping. The sites chosen by the surveyors in all cases exhibited relief within the limits of the definition. A limited but character- istic portion of each site was surveyed. Involved in the selection process was consideration of the different geologic agents and processes involved in the develop- ment of the surface. In this manner the need for variety and adequate geomorphological coverage was satisfied.

In mapping specific areas of microrelief such as miniature badlands, a wadi course, a fault scarp, flood plain mounds, a seif dune, and random boulders on a desert pavement, large scale maps of typical areas were prepared so that some idea of the nature of the microrelief could be formed. Two examples of contour maps prepared from the field surveys are shown as Fig.1 and 2 and a list of all selected areas surveyed is presented in Table II and III.

Surveys were made either by the plane table, alidade, and stadia rod method or with a Zeiss level and graduated rod. The scale of mapping varied with the type of feature surveyed. In most cases a horizontal scale of 1 inch equals 10 ft. was applied, but in a few instances scales of 1 inch to 20 ft. and 1 inch to 4 ft. were used- The x-y-coordinates of a series of points were determined and plotted on the plane table sheet as the survey progressed, and a rough topographic map was prepared in the field. Survey points were selected at breaks in slope, at the midpoint, and at

Eng. Geol., 1(2) (1965) 89-187

96 R.O. STONE AND J. DUGUNDJI

~o c~ lo 20 30

FEET

Fig.l .~Microterrain~map of pahoehoe lava flow, Pisgah Crater, California. Contour interval 0.5 ft.

TABLE II

MICRORELIEF AREAS MAPPED BUT NOT USED IN QUANTITATIVE ANALYSIS

Microrelief area Horizontal scale Contour interval (inch to ft.) (ft.)

Boulder-strewn wadi 1 : 10 0.5 Longitudinal dune 1 : 10 0.5 Vegetation clumps 1 : 10 0.5 Pahoehoe lava flow 1 : 10 0.5 Wave cut platform 1 : 20 1 Selenite buttes 1 : 10 1

Eng. Geol., 1(2) (1965) 89 187

\

I 0 . __ 0 I0 20 30 40

FEET

Fig.2. Microrelief terrain of wave-cut platform, Laguna Beach, California. Contour interval 1 It.

TABLE Il I

MICRORELIEF AREAS USED IN ANALYSIS

Microrelief area Ray sequence Horizontal scale Contour interval (inch to ft.) (ft.)

Boulder-free wadi 260-266 1 : 10 1 Boulders on desert pavement 300-306 1 : 10 0.25 Microbadlands 320-326 1 : 20 1 Salt polygons 340-346 1 : 5 0.25 Floodplain mounds 360-366 1 : 10 0.25 Floodplain ridges 400-406 1 : 10 0.25 Drainage channels 420-426 1 : 10 0.50 Fault scarp 440-446 1 : 10 1 Sand sheet 460-466 1 : I0 0.50 Vegetation mounds 480-486 1 : 10 0.50 Complex dunes 500-506 1 : 10 0.50 Gas pits on playa 520-526 1 ." 5 0.25 Turret dunes 540-546 1 : 20 1 Playa surface 560-566 1 : 100 0.10 Pleistocene lake terrace 580-586 1 : 10 1 Incised pediment 620-626 1 : 10 1

Eng. GeoL, 1(2) (1965) 89-187

98 R. O. STONE AND J. DUGUNDJI

the tops of slopes, and at all other points necessary to give sufficient control to prepare a topographic map. (The density of points is controlled by the nature of the microrelief; obviously a higher density is necessary for an area of microbadlands than for a simple undulatory dissection by a shallow dry wash.) Finished of- rice maps were drafted and contoured at an interval appropriate for the amount of relief exhibited on the map. Usually a contour interval of 1 ft. was used, but several of the maps were made with intervals of 2 ft., and, in one case, 3 inches.

Other field mapping methods Different procedures have been undertaken by otheri nvestigators and have def- inite application. DWORNIK et al. (1959) in their work with small microrelief features obtained a series of profiles with a mechanical instrument that measured elevations at l-inch intervals along a 7-ft. traverse. This was augmented by a Zeiss P-I 0 aerial camera to obtain stereoscopic pairs of photographs which could later be interpreted in the office. This method is applicable to such small microrelief features as ripple marks, pebble surfaces, and small vegetation tufts, but the 7-ft. horizontal limitation and the modicum of relief that can be determined, obviates its use for the more common microrelief features.

A cart method of measuring runway elevations using a light beam was applied by GmMES (1957) but it is doubtful that it would be useful in measuring larger microrelief features or would be a convenient instrument under field conditions.

Two very promising methods of rapid accumulation of microgeometry data are the use of large scale aerial photography and the utilization of a continuous ride terrain geometry vehicle.

The volcanic terrain of the Pisgah Crater of the Mojave Desert was minutely studied by the Space and Information Systems Division of North American Avia- tion by means of a series of photographs prepared by the Aero Service Corporation of Salt Lake City. Four areas 1,000 m on a side were flown and topographic maps were prepared from the photographic data, at a horizontal scale of 1 inch equals 30 ft. These were contoured, using photogrammetric methods, at a 25-cm interval. Detail is excellent, and from such maps detailed profiles can easily be constructed. Photogrammetric methods are fast and particularly applicable to inaccessible areas.

Perhaps the most promising method for rapid data collection of microrelief was devised by the Land Locomotion Laboratory of the Detroit Arsenal. New terrain geometry equipment, designed around the slope-integration method of terrain analysis, was mounted on, first, an articulated vehicle and, subsequently, on a M-113 Armored Personnel Carrier. Equipment for measuring soil strength is also incorporated. As the vehicle traverses rough terrain, x- and y-coordinates of the terrain are gathered, edited, amplified, and digitized. The end product is a magnetic tape suitable for use on a digital computer.

An even newer device is one developed by the Aero Service Division of

Eng. Geol., 1(2) (1965) 89 187


Litton Industries (ANONYMOUS, 1965). Utilizing an airborne terrain profiler with a laser, an accuracy of better than 1 ft. is obtained in an aircraft flying 250 miles/h at an altitude of 1,000 ft.

Terrain profiles

Fan-shaped radial profiles The fan-shaped radial profiles used in the mathematical analysis were developed from the field maps whose preparation is discussed above. This was accomplished by extending lines at 15 intervals from the lower right hand portion of the finished map, giving seven profiles (at 0 , 15 , 30 , 45 , 60 , 75 , and 90 ) for each area of microrelief selected (refer to Fig.3-8). The profiles were plotted on 1-inch graph paper subdivided with 0.1 inch squares and each was labeled to indicate its position in the fan-like spread. This system of course has several inherent difficulties and the possibilities of utilizing either a iectangular grid system or a polar coordinate system were considered.

Rectangular grid system It is a fairly common practice to present the ground profiles of a specified area along the lines of a rectangular grid superimposed on the area. There are several disadvantages in this system, among which are:

(a) Unless the axes of the grid are carefully chosen, any "grain" which the surface may have (such as that developed by prevailing winds or parallel channels) is diffficult to determine from the profiles.

(b) If there are isolated irregular features of importance (such as mounds) the grid lines may miss most of them. Therefore, both the grid spacing and the grid orientation must be adapted to the ground being cross-sectioned in order to give a representative picture of the terrain, and this can be done in a subjective fashion at best.

(c) From a study of the profiles, it is difficult to determine the elevation of a specified point of the ground which is not on one of the grid lines. It is also difficult to determine the appearance of a cross-section along a line which is not one of the grid lines.

(d) There is a large amount of physical work in plotting an area by this method. Assuming an area of 100 100 yards, and that the grid square has 10-yard sides, 12 12 =- 24 profiles must be drawn.

(e) An untrained observer, standing on the ground, has a difficult time visual- izing a straight line on the ground which does not pass through him. Even though he is equipped with profile charts of the area, it is not easy to determine which profile corresponds to which cross-section.

On the basis of these observations it was decided neither to undertake nor recommend the use of a grid system to establish profile lines in microrelief areas.

Eng. Geol., 1(2) (1965) 89-187


"-4 G

C) o~ ' )00.5

- I001.0

~0 ~o

"0 |

9

\ #

%' i 0', /

o , ,~ '

I 0 0 5 I0

FEET ?OF'/L E ORIGIN

Fig.3. Microterrain contour map of boulder-strewn wadi in Mojave Desert, California. Contour interval 0.5 ft.

Eng. Geol., 1(2) (1965) 89-187


Fig.4. Radial profile system, depicted on Fig.3. Individual profiles, made at 15 intervals, are shown.

Polar plotting In the case of polar plotting many of the objections raised against the rectangular grid system become less pronounced, as shown by the following:

(a) In dealing with areas, the distance from the observer is in many respects a natural parameter; this has been recognized, for example, in the widespread change to P.P.I. (Plan Position Indicator) presentations for most radar purposes.

(b) There is no problem of axis orientation. (c) The observer at the origin has a sound concept of how the ground be-

haves as he moves away. Since all profiles go through his vantage point, no difficulties in visualization are involved. He knows the obstacles he will encounter in any direction.

(d) Though a rectangular plot has equal accuracy in each grid section, the polar plot has extremely high accuracy close to the observer. This accuracy de- creases as one moves from the origin. For example, if an area is plotted rectangu- larly at a grid spacing ofg units, and again with polar coordinates at 15 intervals, then within a circle of radius r = 3.8 g, centered at the polar coordinate origin, the polar plot will be the more accurate (i.e., it will give much more ground detail and will miss fewer features).

Eng. GeoL, 1(2) (1965) 89-187


I0 0 5 I0

FEET

Fig.5. Contour map of a portion of a longitudinal dune, Superstition Hills, California. Contour interval 0,5 ft.

Eng. Geol., 1(2) (1965) 89 187


Fig.6. Profiles at intervals of 15 , depicted on Fig.5.

(e) Polar plots are more economical because fewer cross-sections are required to achieve a specified accuracy. For example: consider a 100 x 100 yard area plotted by means of a rectangular grid of spacing 10 yards. It has been shown that 24 profiles are required. If one takes a polar plot with the center of the area as polar origin, plotting profiles along the eighteen diameters spaced 10 apart will yield accuracy equal to that of the rectangular grid at the edges, and much greater accuracy close to the center. In other words, with the eighteen polar profiles one will never show less detail than with 24 rectangular grid profiles, and, in fact, will have more detail as one moves away from the edges of the area toward the center.

(f) The part of the ground not on one of the profiles can easily be interpolated in a polar pattern. For example, assume that the plots of the profiles along the lines are l, i = 0.1 given. Let hi(r) be the elevations along the diameter, where r is the distance from the origin. Then the profile along a ray (9, (9o -< 0 _< (9~ can be interpolated linearly to give:

h(r) = ho(r) + t-~-~ hffr) - - ho(r)

Eng. Geol., 1(2) (1965) 89-187


""4 I

io o io 20 3o 40 PROFILE

' OR IG IN FEET

Fig.7. Microterrain map of sediment accumulation around vegetation clumps, Mojave Desert, California. Contour interval 0.5 ft.

Observe that (-0)/(1-0) is a fixed ratio, once O is known. For example, if the plot along O0 = 30 and 01 = 45 is known, the plot along the 40 = O line

can be interpolated as follows:

Eng. Geol., 1(2) (1965) 89-187


0

15"

\ \ \ \ \ \ \ \ \ \ \ \ ~ - ~ \ \ \ \ \ \ \ \ \ \ \ \ \ -,,

30

~ \ \ \ \ \ \ \ \ \ \ \1 45

60

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~1 75

- . \ \ \ \ \ \ \ \ ~ ' ~ ' ~ - ~ ~ ~ - - \ \ \ \ \ \ \ \1 90

Fig.8. Radial profiles, depicted on Fig.7.

,0{ } h(r) - - ho(r) + ~ hi(r) - - ho(r)

Thus if h0(10) = 10 and hi (10) ---- 25, then at 10 units away along the 40 line,

h(10) = 10 + 10/152(5-10) = 10 + 10 = 20. Clearly, l inear interpolat ion is less

accurate the further away one goes from the origin; and the closer 190 and 191 are, the longer the r for which accuracy is achieved.

(g) The straight line distance between two objects is not much harder to

calculate than in rectangular grids when polar coordinates are used. I f 19 is the central angle _< 180 subtended by these two objects, their distance apart is d 2 =

rl 2 + r22 - - 2rlr2 cos 19, where r l and rs are their distances from the polar origin. Thus, if Fig.9 is given, thend 2 = 64 + 4 - - 2 .16 cos 60 = 68 - - 2 16 1/2 = 52 and d ----- ~/52 = 7.2.

(h) The distance from the origin as one moves along a straight line jo in ing two objects is as complicated in polar coordinates as it is in rectangular coordinates. Given Fig.10, then:

x r e sin 19

r sin (9' + ~ sin ( -- ')

Eng. Geol., 1(2) (1965) 89--187

106 R. O. STONE AND J. DUGUNDJ1

_ _ o _ \ - / ; y

o o

Fig.9. Fig.10.

For example, if r 8, ~ = 2, 6) = 60 , then the distance x along the line from the line r (6)' == 30 ) is:

8 2 sin 60 X ~ ~---

8s in30 + 2s in(60 - 30 )

16.1/2~/3 = 2.77

8 " 1/2 t 2 " 1/2

(i) There is only one subjective feature (rather than the two for rectangular grids) involved in polar plots, and this is the location of the origin of the area considered. This should be any accessible point close to the subject matter of main interest.

The problem of locating the origin of a coordinate system on a map or in the field has been studied throughout the program. The observer must decide subjec- tively what area is to be mapped, how much of it should be included, and what scale to use. This makes the subsequent application of a random number system or some similar device of little true value. Further, emphasis must be placed on those specific features of the terrain which influenced the observer's choice of the area. This is best accomplished by placing the profile origin in an accessible position from which the main points of interest will be covered. It is concluded that there is no way to avoid some experienced judgement in the collection of data con- cerning surface roughness characteristics.

Recommendations on data collection

On the basis of the mapping of selected microrelief features and the application of data to the quantitative expression system, the following recommendations on data-collection procedures are made:

(1) Areas to be mapped should be selected by an observer who is familiar with the microrelief concept.

(2) Field mapping should be undertaken with a transit or optical level and

graduated rod. (3) A radial profile system should be used to encompass the area under con-

sideration. Standardization is desirable. Twelve profiles spaced 15 apart will cover a region adequately if a profile radius of 160 ft. is used.

(4) The point of origin of the profile should be at an accessible point and in such a way that the characteristics of the area will be covered.

Eng. Geol., 1(2) (1965) 89-187


(5) Readings of x- and y-coordinates should be made at 2-ft. intervals from the point of origin. On some surfaces (e.g., dry lake surfaces, desert flat, desert pavement, wave terraces) multiples of 2 ft. are more practical.

(6) Horizontal scales must of necessity vary with the type of feature being mapped, but 1 inch equal 10 ft. is recommended for use whenever possible.

(7) For areas of unusual microrelief contour maps should be prepared for comparison with subsequent computer work.

(8) In order to save several steps in processing the data through computers, the x-y-coordinates determined in the field should be tabulated and submitted for analysis. However, it is suggested that profiles be constructed from these data for comparative purposes once the processed data have been received.

QUANTITATIVE EXPRESSION OF MICRORELIEF

General statement

The principal obligation of the study was to express microterrain characteristics in quantitative or semiquantitative terms. A variety of approaches was investi- gated and several were discarded which were attempted. These included: expressing the geometry of small-scale land forms as a series of number or degree ranges, graphical expression, several contour line interpolation or extrapolation systems, the application of Chebyschev polynomials, and the use of power spectra or spec- tral density analyses. The latter approach has been applied to airfield-runway roughness problems by such workers as GRIMES (1957), and HOUBOLT (1961), and to terrain roughness in studies related to land-locomotion problems by BOGDA- NOFF and KOZIN (1959, 1962), and KOZIN and BO~DANOFF (1960). These applications are allied to, but materially different from, what is attempted in this work. It was decided that the device that held the most promise was the use of a Fourier analysis of terrain profiles drawn through microrelief areas.

This study is concerned solely with the quantitative expression of known profiles, no attempt is made either to predict terrain or to determine the probability of adjoining terrains being similar or dissimilar. The problem is to express mathematically the nature of prepared profiles of mapped terrains, and not to give a statis- tical description of the terrain.

Fourier analysis of land surfaces

Introduction Let g(t), 0


ly vague term--it is widely known that what appears "rough" to one person frequently seems "smooth" to another--the main problem is two-fold:

(1) To present a precise, clear-cut definition of what is meant by the "roughness" of a given curve or terrain.

(2) To devise some numerical expression for "roughness" based on the definition adopted in (1).

Naturally, it is required that the definition adopted in (1) be harmonious with intuition if possible; (2) would then permit an objective comparison of profiles (and of terrains) with respect to "roughness".

To obtain a better idea of what the concept "roughness" involves, and how it should be defined, it is best to start with some simple observations.

(1) Of two sine waves having the same wavelength, the one with the larger amplitude would be called "rougher" by most people. Thus, the amplitude of any oscillations must be involved in a workable definition of "roughness".

(2) Of two sine waves having the same amplitude, the one with the shortest wavelength would be called "rougher". Thus, the steepness of any oscillations must by involved in the concept of "roughness".

(3) The profile of a washboard terrain is usually conceded to be smoother than that of one having the same number of ridges, but irregularly spaced. Thus, the periodic repetition of prominent features must enter into any good "roughness" definition.

The crucial fact about these three simple observations is that the quantities (or factors) involved are independent, i.e., a curve can be "rougher" than another in one respect, but not in another. For example, a sine wave having a wavelength of 10 ft. and an amplitude of 1 ft. is rougher relative to amplitude (see (l) above) than a sine wave of wavelength 2 ft. and amplitude 0.5 ft., whereas the second curve is rougher than the first relative to steepness (see (2) above). This, and other examples indicate that "roughness" is composed of at least the three independent factors listed above: if it is desired to express the "roughness" of a given curve (or terrain), it is necessary to specify the extent to which the curve possesses each of these three factors. Thus, to obtain any comprehensive numerical description of "roughness", the idea that this can be done by giving a single number must be abandoned and one is forced to conclude that "roughness" is, in mathematical terminology, a vector, rather than a scalar quantity. Stated in another way, "'roughness" is not a single elementary property. To indicate the "roughness" of a given curve (or terrain) the amount of each elementary property present in the curve must be specified, and it is this list of numbers (the components of the "roughness vector") that serve to give a picture of the nature and kind of roughness the curve has. In short, no curve is "rougher" than another; it is only "rougher" than another with respect to some property.

This being admitted, the problem of defining "roughness" reduces to deter- mining what the independent measurable constituents of "roughness" are, and

Eng. Geol., 1(2) (1965) 89-187


measuring them. Mathematically it is necessary to identify what the components of "roughness" are. In this report, four axes or components have been isolated. It is not proposed that these four are all the components of "roughness" (i.e., that "roughness" is a vector in four-dimensional space). More components will have to be determined, and a method for measuring them objectively will have to be specified, in order to obtain a complete description of "roughness". It is asserted, however, that the four components isolated do in fact suffice to convey a precise idea of "roughness" that is useful in many circumstances. The components or axes determined are: average changes in level as a profile or terrain is traversed (simple relief factor M), average height of major relief feature (specific relief factor A n), average steepness of relief features encountered, as the curve (or terrain) is traversed (slope factor P), and extent to which there is periodicity, or repetition, as the curve or terrain is traversed (structural homogeneity factor K).

Further, from the roughness vector of each terrain two quantities can be determined; the avoidance factor Q, which indicates the difficulty one encounters in traversing the terrain, and the cell length CL, which indicates how far one must progress from a given origin in order to encounter "all" the significant features of the terrain being studied.

One incidental advantage in defining "roughness" as a vector is that this permits comparisons of terrains with respect to a particular aspect of "roughness" most relevant to a given consideration. For example, to determine if a terrain is easily traversable on foot involves mainly component M, whereas to see if it is traversable by a heavy vehicle involves instead component P.

To determine the relevant components of a curve's (or terrain's) roughness vector, a Fourier analysis of the given curve is relied upon. From this viewpoint, "roughness" is caused by a mixed and significant high-frequency content: it is the high frequencies that cause a curve to have sharp corners, steep slopes, and irregularities. This, however, raises this question: how high must the frequency be in order that it be judged "high"? This clearly depends only on the horizontal extent of the relief features of interest.

In this project only microrelief features are under consideration. Even before stating what a "high" frequency is, the first question to be considered is: what is a microrelief feature ? For example, is a sine wave of wavelength 3,000 ft. and amplitude 1 ft. a microrelief feature? Is a sine wave of wavelength 30 ft. and amplitude 5 ft. a microrelief feature? Unfortunately, all the definitions and descriptions presented on Table I explain only what a microrelief map of a terrain is, namely the recording of all variations of, say, 1 ft. None of these definitions can be used to decide whether or not the Alps constitute a microrelieffeature on the map of Europe.

Thus, some definition of what constitutes a microrelief feature must be made. On the basis of a study of the physical evidence in this project, a definition has been adopted that intuitively amounts to the following: a microrelief feature on a curve is any feature that manifests itself with a horizontal distance of not more

Eng. GeoL, 1(2) (1965) 89-187

1 l0 R. O. STONE AND J. DUGUNDJI

than 64 ft. Alternatively stated, any "bump" on a surface that is more than 64 ft. long is not a microrelief feature. For example, a sine curve of wavelength 3,000 ft. and amplitude 1 ft. is not a microrelief feature, according to our definition, whereas one of 20 ft. wavelength and amplitude 1 ft. is. As a further example, if a sine wave of wavelength 20 ft. is superimposed on a sine wave of wavelength 3,000 ft., the first sine curve is the microrelief of the combined curve.

Mathematically, this notion can be formulated precisely: the microrelief of a given curve is that portion of its Fourier series expansion containing wavelength of < 64 ft. For an actual smface, this definition can be seen to imply that any "protuberance" or depression which has some lateral extent c: 64 ft. is a microrelief feature, regardless of how far it extends in other directions. For example, the furrows in a plowed field will be microrelief features whenever they are spaced -- 64 ft. apart, even though each furrow itself may be 3,000 ft. long.

Observe that the definition of "microrelief feature" does not put any limit on the height of the feature. A sine wave of wavelength 20 ft. is a microrelief feature, whether its height is 1 ft. or 1,000 ft. Mathematical considerations do not require that any limitation on the height of the feature be imposed. (In the work undertaken in this report an arbitrary height limit of 10 ft. was imposed by the selection of the microterrain area mapped. Only areas exhibiting less than 10 ft. of relief were considered, a condition imposed by military terrain requirements.)

The relation of this concept of microrelief feature to that of microrelief mapping as delineated by other investigators (Table I) is this: No matter what scale the terrain analyst uses to map a given terrain, the Fourier analysis indicated below will give him the "microrelief features" of his map, i.e., the extent to which he will encounter "bumps" of length _< 64 ft. on his map. I f the scale used is gross, the mathematical analysis will yield few microrelief features; the finer the scale used, the more microrelief features the analysis will yield. Thus, it can be seen why those writers so strongly specify the scale of the plot: though any scale used will give some microrelief features, the finer the scale the more microrelief features are shown. It should be noted, however, that the authors do not know of any way to determine in advance how fine a scale should be used in mapping a given terrain. This appears to be a matter for the field investigator to decide, and evidently depends on the microrelief features of interest to him.

According to the mathematical definition given above, the microrelief information of a given profile is obtained by developing the curve of that profile in a Fourier series, and using only that part of the Fourier series containing the wavelength 2~- 64 ft.; this part of the Fourier series is termed the "microrelief packet" associated with the given wave. Once this packet is obtained, the given profile is of no further interest for microrelief work; the packet contains all the microrelief information. It is this packet that is processed to determine the "roughness value" of the given curve.

Eng. Geol., 1(2) (1965) 89-187


J \ \

~ f J

Y

-L 0 L

Fig. 11.

\ " s " '~ l P

\,, e S

21 t

Processing the curve for the microrelief packet Lety = g(t), 0 _< t _< L, be the given curve (Fig.11). This curve is first extended to the entire t-axis by first reflecting it across the y-axis and then extending by periodicity, to obtain a curve--still denoted by g(t)-- that is periodic of period 2L.

It is desired to have a curve defined on the entire t-axis because this simpli- fies the analysis considerably. This method was chosen for extending the curve (rather than, for example, simply shifting g(t) over by L units) because the extension will not have any breaks or discontinuities, so that in the proposed analysis the high frequencies will essentially be due to g(t) itself, rather than to breaks that have been artificially created in extending g(t). Finally, the resulting function g(t), being even--that is g(t) = g( - - t ) - - has a pure cosine Fourier series:

O0

g(t) = Ao + ~, an cos 1

where: L

o and:

n7~ t ( -~

1 12 R. O. STONE AND J. DUGUNDJI

Fig.12.

~y 8

- t

(i.e., b -- 2) are to be computed, this requires that n = 1,000/2 -- 500 harmonics be calculated, which is not an easy task. Since, as stated in the introductory portions of this section, it is precisely in the high frequency terms that we are interest- ed, it is therefore necessary to devise some method of obtaining quickly the ampli- tudes of the high frequency terms, or at least a certain proportion of them.

The attitude was adopted that it is desirable to compute no more than sixteen harmonics and still come down to the harmonic of wavelength 4 ft. This is done as follows:

(a) Divide the given curve into lengths l--nO 16 2 -- 32 ft. (any residual part of the curve is discarded).

(b) For each piece of length I = 32 ft., compute the first sixteen harmonics:

at( l) , . . . , a16 (1)

al(2)~... , a16 (2)

a l (K), . . . . a16 (K)

where KI

A STUDY OF MICRORELIEF l 13

Proof:

l

a8 (1) = 2 f g(t)cosS~tdt l J l

0

21 2l 2 ~ . . sz~(t--l). 2 r sztt

as(~) -- ~- Ig ( t )cos - - --clt = (--1) s-, I g ( t )cos~dt td t td t

1 l

Adding, gives:

a8 m

a~(a) +. . . +( - - 1)s(K-1)as(K)

K

Kl

2 J~'g(t)cs sKztt dt Kl , KI

0

KI 1 2 {" sm

. . . . . . |g(t)cos dt K l J l

0

showing that a8 = asK, the amplitude of the harmonic with wavelength 2~t/(sK~r/ Kl) = 21/s = 64/s, and the proof is complete.

With this matter settled, only the "high" frequencies will be considered, as explained earlier. In other words: given the ground profile g(t), the coefficients an are calculated, and hencefolth consideration is given only to the curve:

16 nTc u(t) = ~anCOSWntl ( - -~


accurately computed; and since in this case the computations were based on the value ofg(t) at 2-ft. intervals, this accounts for using only wavelengths 2 such that 4 2 64 in our "associated microrelief packet".

(b) Another way to state that interest is in the high-frequency content of the ground profile g(t) only is to say that examination is made only of the high end of the frequency spectrum of g(t) (which, because g(t) is periodic, is a line spectrum). Because of the inherent limitation in (a), we therefore must examine this spectrum only in a band, as follows in Fig. 13. Note that the "microrelief packet" is built out of this by considering the spectrum G(w) ofg(t) at the regularly spaced frequencies w~0.1, 0.2 . . . . , 1.6.

IG(w>l Range of interest

I 27r 64

l \\ 2'r.16 64

W = - - 7x

Fig.13.

(c) If the ground profile to be analyzed is defined for 0 ~ t _< 32K, the entire Fourier series for this curve is:

O

A STUDY OF MICROREL1EF 115

behaved in between the lines that have been computed (which are fairly close together), and (2) since this is a pilot study to determine the efficacy of the ideas suggested, it is felt that no great harm will result from the neglect of the inter- mediate frequencies. The countervailing advantage for the procedure presented is that, even though the length of the curves analyzed may differ, their content of wavelengths 64/s ft. (s= 1,. . . 16) is always obtained, so that this standardization permits the comparison of profile "roughness", irrespective of the lengths of the curves compared, and also allows us to pass quickly to a two-dimensional analysis of the terrain.

Elementary high frequency features It is first noted that:

L L

f cos~ontcosogstdt = f sin~ntsin~ostdt = 0

-L -L

I fn~s :

L L

f cos2~ontdt = f sinZ~ontdt =- L -L -L L

f coso)ntsino~stdt = 0

-L

for all n, s because the family {cosmnt, sine)nt In = 1,2 . . . . } is orthogonal on the interval - -L _< t ~ +L (recall L --~ 32K and O~n ~-- mr/32). 16

(A) Variation in height of microrelief packet, M. Let u(t) = ~ancoscont be a microrelief packet. The above formulas immediately give: 1

L L if u( t)dt ~- ancos~ontdt = 0 2L 2L

-L -L

and:

L L

1 f l l f : 1 2t u( t) ] '2dt = 2-1-. anascos~ontCOSCostdt = --2- ~a , z -L -L

To obtain an interpretation of these numbers, consider the closed interval - -L _

[ [6 R. O. STONE AND J. DUGUNDJI

t 2 ~an 2. Thus, 7~an measures the dispersion of the values of u(t) around its average value, 0: it indicates whether or not the values of u(t) are clustered close to zero. Alternatively stated, ~a~ ') measures the probable range of values of u(t): for using Chebyschev's inequality, that probability {tu( t )>c} Var(u)/c '2 it is found that:

i/~a~,'~ 1 probability {t i u(t)[ > 5~/~a, ~} 25~a,/~ = 50 :- 0.02

which says that except possibly for intervals of total length :-0.04L the high-frequency packet associated with the given ground terrain is confined to a band of width 10~/~an z centered on the t-axis. Thus ~a~, z (or simply ~a,~ 2) indicates the largest change in level that can be expected as the curve is traversed. Conse- quently, M ~a, z ~ simple microrelief factor --- expected range of heights of prominent microrelieffeatures. If ~a , 2 - 0, then each an -- 0 so that the high- frequency packet u(t) _=_ O, i.e., it is flat, the ground profile then has no microrelief features. The larger the value ~an ~, the taller the microrelief features that can be expected.

(B) Specific microrelief factor, A~, It has already been noted that, due to both data and computer limitations, only the first ten of the Fourier coefficients a,~ i ::s ~ : 16 that are obtained are fairly reliable.

we define: A~,-- max.{(~ia8 l]as[-:0.1, l :s : :10) , (max.[a,s] 1 :s :10)} specific microrelief factor expected height of the tallest mierofeature.

The reason for this interpretation is: if the terms involved in calculating A,, are an1 . . . . . anK, then the predominant part of the microrelief packet is ~a~zcos (oit; thus an upper limit for the maximal amplitude of the microrelief is approxi- mately A~.. In particular a larger A ~ would indicate roughly that a taller microrelief feature can be present; in conjunction with the component M, An can help decide whether or not there are a few isolated microfeatures or simply one large one.

A brief explanation is required regarding the determination of the A~, component. The derived a8 values are examined for a terrain. All values greater than 0.10 are summed to determine the An value. If there are no values larger than 0.10 amongst the a8 terms, then the largest value is used to designate A,~.

(C) Variation in slope of the microrelief packet, P. We have u'(t) \" - / _an (,,J =- -Jr/32~nansino)nt. Regarding now u' as a random variable on the interval -- L s t . ~ L and proceeding as in (a), it is found that E(u') -- 0 and Var (u') == (1/2)ffz'e/322)~n2an z. Since (1/2)(~z/32 z) is a factor independent of the curve being considered, it is concluded as in (a) that: P ~n2an 2 slope factor expected range of slopes of' prominent microrelief features. If ~n2an 2 : 0, then u(t) 0 i.e., there is no microrelief. The larger value ~nZan 2, the steeper the microrelief slopes.

Eng. Geol., [(2) (1965) 89--187

A STUDY OF MICRORELIEF 1 l 7

Correlation techniques It is defined that:

L

-L

which measures the amount by which the high-frequency packet differs from itself

when shifted to the left v units. Then:

L L L 1

D(z) = _ l f [ u(t +z) 2at +12L 3 (] u(t) l Zdt -~ f u(t +z)u(t)dt -L -L -L

Since we are integrating over an entire period, it follows that:

L L

f l u(t+z) i eat = f I u(t)[ zdt -L -L

so that:

L L

D(z) = -~ f [u(t)12dt--@S u(t +z)u(t)dt - L -L

Defining the autocorrelation function by:

L

c(z) = _1 f u(t)u(t+z)dt L J

-L

we have D(z) = c(o)--c(z). The function c(z) has the following properties.

Theorem 1. (a) c(o) >>_ O, but c(z) can become negative for some z. (b) I c(z) [ -< c(o), for all v. (c) c(z) -- c(--z), for all z. Proof (a) is obvious. (b) By Schwarz's inequality:

L L 1/2

I ' J VL VL LL-J -L -L -L

so that:

I c(z) [ -


(c): L L-r L

f u(t)u(t--z)dl = f u(~+r)u(~)d~ ~ j'u(~q r)u(~e)d~ e - L -L-r -L

the latter equality following because integration takes place over an entire period. For the difference function D(z) - c(o)--c(t) it is found at once that:

Corollary 1: (a) D(o) = 0, and D(r) ~ 0 for all z. (b) ! D(r) 2c(o), for all r. (c) D(r) = D(-- r), for all r.

Furthermore, the c(z) can be computed explicitly for the microrelief packet.

Theorem 2. Let u(t) = ~ancosont be the microrelief packet associated with a given ground terrain. Then its autocorrelation is:

C('t')--Zan2COSO)n'r

Proof: We have: L L

f c(T) ..... ~ ~u(t+r)u(t)dt .... 1 ~asanCOSO)ntcosos(t+r)dt c j -L -L

Noting that:

coso~s(t - r ) = cos~odcoso)sr-- sinoodsin,osz

so that expressions of form:

COSO)ntCOS(Ost; COS~Ontsin(ost

must be integrated, the formulas at the beginning of the section on elementary high-frequency features establish the theorem.

Proceeding now as in the preceding section (A), both c(r) and D(~) can be regarded as random variables in the probability space - -L~ t-~ +L , and the following is obtained:

Theorem 3. (a) E(c) = 0; Var (c) = ~a#. (b) E(D) = c(o) -- ~an2; Var (D) -:: 1~ a 4 ~/ . n- P roo f : for E(c) and Var(c) the calculations are straightforward, as before, and those for E(D), Var(D) are then immediate consequences of these results and the for- mula D(-c) = c(o)--c('O.

The form of c('v) given in theorem 2 is fairly difficult to handle algebraically. A simple approximation will be derived: from the expression for c(r) we have (as-

Eng. Geol., 1(2) (1965) 89-187

A STUDY OF MICRORELIEF l 19

suming c(o)~O) that maximum c(z) = c(o)>0, that c'(o) = 0, and that c"(o)_c(o)}_< {c(o)} 2 - {~an2} ~

Now, if [D(z)-c(o)l->c(o), it follows from the corollary to theorem 1 that either D(z) = 0 or D(z-) = 2 c(o). If D(z) = 0, the microrelief packet coincides with itself

Eng. Geol., 1(2) (1965) 89-187


when shifted r units to the left; if D(r) =- 2c(o), that is c(r) . . . . c(o), this means that the z-shifted curve is negatively correlated with the given curve. Thus, K : }. { ~an4/( ~anZ) 2} is the structural similariO, factor; it indicates the tendency of the microrelieffeatures to be repeated.

The smaller Kis, the more diverse the microrelief features are, either in spacing or in form. Observe that (~aj~) 2 - ~a~ 4 i Q, where Q-=-0, so we always have 0 ~K~; Note also that if Var c 0, that is, ~_an 4 ~- O, then each ar~ = 0 so that Kreduces to 0/0, which is indeterminate. Thus for a profile with no microrelief, K is undefined.

A voidance.factor o. Let rp be the time it takes for the autocorrelation of the microrelief packet to drop off to zero, i.e., to become completely uncorrelated with itself. The average rate of decorrelation, ~, would then indicate the overall irregularity of the microrelief, and is called the avoidance factor.

Theorem 4 is used to compute rp: from 0=-e(o) - ([ c"(o) rp 2) we obtain r, -- V'{2c(o)/ic"(o)! }. The amount of decorrelation per unit time is therefore ~; = c(o)/rp and is given by:

2 32 " v"2 /(anz)'( nzan2)

Since ~/32 1/~/2 is a constant factor, it can be neglected, as also can the square root operation. Thus, ~ (~an2)(~n2an 2) -- avoidance factor which indicates the overall irregularity in the microrelieffeatures. Clearly o = 0 if there is no microrelief; the larger o is, the more irregular the microrelief features are. I f one were forced to describe "roughness" by a single number, is the most informative of the factors we obtained. For two-dimensional terrains, ~ indicates, intuitively, how difficult it is to traverse the terrain.

Cell length CL. The "cell length" of a terrain profile is the length of curve one must traverse, from a given origin, in order to observe all the significant microrelief features. Selection of "significant" features is highly subjective, however. There- fore, it is desirable to express this concept in mathematical terms and thereby obtain a precise formulation. To do this, one can begin with the considerations presented below.

Consider the discrepancy function D(r); its average value is c(o), and it is assumed that c(o)#O, so that c( r )~0. D(r) is periodic, with D(o) ~ D(L) = O. Now let re, be the last time in the interval 0 ~ z ~- L that D(r) -- c(o), its average value. After time rG, the discrepancy D(r) between the curve and its shifted self falls off and goes to zero, so it can be quite reasonably said that the main disturb- ing features of the curve have been observed; at least from re; on, the discrepancy is always smaller than average and goes to zero. This broadly corresponds

Eng. Geol., 1(2) (1965) 89-187


to the intuitive idea of "cell length", so we will define 0

122

L cs ~2 t

___o ~ 3 i

length

Fig.15.

R. O. STONE AND J. DUGUNDJI

21T cos ~ t

~ ', 32 cell length

Fig.16.

k" t

Since z~ ~ L--T0, we have:

/ / ~a,,2 \ CL ~ L / l - - /---:~ ~

\ ~x/~nau ,/

= cell length: 0


(3) Expected range of slopes of prominent microrelief features:

P = ~n2an 2

(4) Tendency of microrelief features to be repetitive in form and/or spacing:

~ an 4 K - :(~n2)2

(5) Overall irregularity of microrelief features, the avoidance value:

e = (~an2)(~ n2an2)

(6) Cell length; the length of curve that must be traversed to experience all "significant" features:

CL=L(1-- / 2N/ ~n2an2 ]

Extensions to actual terrains It is assumed that the profiles of radial cross-sections of the ground terrain all originate at a given point, as indicated in Fig. 17. It is the purpose now to extend

Fig.17.

the concepts for linear profiles to the actual terrain. Again the microrelief packet associated with the ground profile along each

radial line is obtained, and using polar coordinates, it can be written that:

16 u(r,@) = ~as(O)coso~sr

S=l

where as(O) is the 64/s harmonic of the profile along the @ line, and r is radial distance from the origin.

To find how the terrain varies with distance from the origin, the 19 is integrated out to obtain:

n/2

u(r)= y(~fagO)dO)cos~o,r o

Assuming that as(19)for =0 , 19-----15 , 19----30 , 19 =45 , 19----60 o, 19__75 o,

Eng. Geol., 1(2) (1965) 89-187


TABLE IV

SAMPLE OF TABULATED DATA FROM TERRAIN PROFILES DERIVED BY UT IL IZ ING A BENSON- -LEHNFR

AUTOMATIC PLOTTER. DATA FOR MAP 400, FLOOD-PLA IN R IDGES

x y x

400 401 402 403 404 405 406

2 1 2 7 6 13 20 27 2 4 6 22 39 50 62 73 4 6 2 10 48 63 62 68 79 6 8 13 31 62 58 44 32 35 8

10 20 50 61 44 2- 28- 21- 10 12 34 56 60 4-- 45 48- 34- 12 14 49 59 45 35- 58- 36- 27 14 16 57 59 3- 52- 47- 27 7 16 18 59 55 23 51- 19 16- 10 18 20 6l 55 49- 33- I - 30- 24- 20 22 68 58 57- 32 29- 41- 39- 22 24 72 53 51- 13 42- 56- 50- 24 26 76 48 45- 22- 56- 62 63- 26 28 76 49 11 32- 65- 65- 66- 28 30 71 46 18 40- 55- 66- 60- 30 32 75 30 10- 49- 31- 47- 38- 32 34 80 10- 29 51- 48- 60- 59- 34 36 78 46- 37- 2 52- 39- 38- 36 38 77 64- 45- 14- 63- 39- 37- 38 40 78 60- 53- 31 22- 45- 50- 40 42 74 1 54- 44- 45- 62- 57-- 42 44 79 24 50- 51- 57- 64- 19- 44 46 73 24 42 60 59- 23- 55- 46 48 70 1 66- 56- 71- 48 50 69 21- 37- 1- 70- 71- 123- 50 52 64 36- 21- 47- 54- 109- 162- 52 54 62 47- 30- 55- 26 140- 160- 54 56 58 50- 37- 60- 9 - 161 145- 56 58 60 50- 47- 63- 60- 173- 128- 58 60 60 45- 53- 72- 79- 177- 109- 60 62 61 50- 48- 70- 99- 165- 91- 62 64 60 54- 48- 126- 136- 27- 64 66 60 51- 22 143- 119- 3 66 68 66 3 14- 32- 126- 100- 47 68 70 58 22- 37- 30- 101- 78- 60 70 72 36 35- 48- 47- 118- 44- 32 72 74 14 37- 58- 69- 125- 12 74 76 1- 39- 67- 79- 114- 76 78 17- 45- 75- 95- 84- 78 80 36- 50- 84- 122- 43- 80 82 49- 53- 72- 128- 15- 82 84 52- 55- 48- 122- 84 86 50- 61- 29 106- 86 88 44- 56- 20 80- 88 90 41- 22 27- 23- 90 92 33- 64 70- 46- 92

Eng. Geol. 1(2) (1965) 89-187


]'ABLE IV (continued)

125

x y x

400 401 402 403 404 405 406

94 27- 21 79- 85- 94 96 23- 27- 33- 101- 96 98 22- 45- 50 86- 98

100 22- 52- 42 63- 100 102 27- 59- 16 13- 102 104 42- 63- 3 104 106 61- 23- 106 108 61- 56- 108 110 75- 110 112 77- 112 114 79- 114 116 82- 116 118 80- 118 120 83- 120

O=90 , i.e., seven different angles, the integral can be evaluated by using, say, the trapezoidal rule. For this, then:

n/2 2 f 1 Fas(O ) _ o- o- as(90)-[

As = r as(O)dO = ~[ -~- -+as(15 )+. . .+as(75 ) - ?~J

0

showing:

16 u(r) = ~ Ascosogsr

1

which is called the microrelief packet of the terrain associated with distance from the origin.

Working now with u(r) as with profiles in the foregoing sections, the quantities, 9, P, M, An, K, and CL are computed using the coefficients of u(r), and they give the average behaviour of the ground terrain; ~, P, M, and An are the components of the "roughness vector" of the terrain, and they have the same interpre- tations as before. In the case of the cell-length determination, whenever the lengths of the rays are not all the same, the "effective" length, L, used in computing the cell length of the terrain is taken to be the average of the lengths of the individual rays.

DATA PROCESSING

All of the profiles from each of the selected microrelief areas were submitted for

Eng. Geol., 1(2) (1965) 89-187


\ \ \

ORIGINAL PROFILES

REPLOTTED PROFILES

Fig.18. Original profiles from floodplain ridge terrain (400), and replotted curves used to check accuracy of Benson-Lehner automatic plotter data.

Eng. Geol., 1(2) (1965) 89-187


\

\\~\\ \ $5\c4~)

ORIGINAL PROFILES

127

.~ 60(464) X, "~ "N, "~ N I

% ~ 75(465) "X,

REPLOTTED PROFILES

Fig. 19. Original terrain profiles of map 460, and replotted profiles derived from Benson-Lehner data. Vertical exaggeration in replotted curves does not affect computations.

Eng. Geol., 1(2) (1965) 89-187

TA

BL

E

V

TY

PIC

AL

D

AT

A

SH

EE

T D

ER

IVE

D F

RO

M E

XT

RA

CT

ION

O

F F

OU

RIE

R C

OE

FF

ICIE

NT

S F

RO

M T

ER

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130 R. O. STONE AND J. DUGUNDJ I

programming and analysis to the Computer Center at the Waterways Experiment Station. A Benson-Lehner automatic plotter was used to determine the x- and y- coordinates of the profiles at 2-ft. intervals. The length (in ft.) of profile analyzed was always a multiple of 64 (32, 64, 96, 128, etc.), and any portion of the curve extending between such lengths was ignored. An example of the results of one of the automatic plots is shown on Table IV. The left hand column indicates the horizontal distance along the profile at 2-ft. intervals and the right hand column the y-value or "height" above the origin, to two decimal digits. The x-y-coordinates of the terrain profiles derived by the automatic plotter were used to replot the curves from which the coordinates were derived in order to check the accuracy of the work. Two sets of original profile curves and the replotted curves are shown on Fig.18 and 19.

The coordinate data were processed by a high speed computer in order to determine the sixteen Fourier coefficients corresponding to wave lengths of 128 down to 4. Similarly, these harmonics were determined for the area represented by each set of seven profile curves. On the data sheet from the computer (Table V), the G-value of the profile ray, curve number, area values, Fourier coefficients, and

Field of SUrvey ] Microre/ief Area

Preparation of Contour Mop I

1 Construction of Radial Profiles [ X, Y coordinates [

| ol 2~ntervo/s along I /1 radial profiles, | d,'roctty from I

/ L f,'eld mopping I Replol of Profiles ~ Oonson -- Lehner (chock) Automatic Plotter I

X~ Y coordinates I / I 4eterminedl /

(PROF/L E DA TA )

High FrequenCy L

I Components ,1~ ] Components]

Vector / 2nd derivative o f [ /

J High Frequency L component

A' Ai'TA' J H/'gh Frequency]

J [ Camp . . . . ts I

CcmPo ~nenls~ / / ~oo~hness I".

t ~ectr I ~ 1 ~ leod derJ'votiveof I

~ H i g h Frequency ] Component ]

Fig.20. Flow sheet showing data collection and data processing procedures.

Eng. Geol., 1(2) (1965) 89-187


the second derivative of the Fourier coefficients are tabulated. From these data, the terrain components ~, P, M, K, CL, and A n 2 were computed, using the equations presented in the foregoing section.

A flow chart showing the procedure of processing terrain data is presented in Fig.20.

It must be mentioned in this section that for two of the microterrain areas, turret dunes (540), and the playa-lake surface (560), it was necessary to make ad- justments to the computed values. Coordinates along the profiles were determined using a horizontal scale other than the one at which the mapping was undertaken and hence the values derived varied from those expected. A reanalysis of the mathematical system was made and it was determined that for the 540 terrain, true P and M values were 0.25 of the computed value, and true ~ could be ascer- tained by multiplying by a factor of 0.0625. In the case of the 560 map where the true horizontal scale is 1 inch equals 100 ft. and computations were made at a scale of one-tenth of this, a close approximation was made by applying a factor of 0.01 to the P and M values and 0.0001 to the 9 values.

Correction of the values for the playa surface is necessary of course, but even prior to correction this terrain exhibited the smallest terrain characteristics in all three categories. However, the turret-dune terrain, as will be seen, presented difficulties in some of the evaluations and it may be best to consider the corrected values for the area questionable.

DATA TESTING AND CONCLUSIONS

Introduction

In order to determine the applicability of the terrain components derived by the analysis it was necessary to provide some means of testing both the individual ray data and the area data from the terrain. The roughness components for each ray of a sequence were ordered, from high to low value, and then a visual ranking of components was made. In the case of the area data, the terrain components were tabulated and comparisons of the roughness vectors made numerically and visually among the various terrain types.

It is agreed that visual scanning of curves is not completely meaningful, and in the case of Kappa, the measure of structural similarity, reliable visual testing could not be made. Examination of curves for roughness vectors may lead to observer bias and error, but on the other hand permits to some degree the differen- tiation of the most irregular curve, the widest variety of slopes, and the tallest feature. It also provides a reliable view of the profiles exhibiting the greatest and least amount of surface roughness. In fact, in ray testing it was found that there was seldom any doubt about the two individual extremes and usually the two mere-

Eng. Geol., 1(2) (1965) 89-187

l'-J

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TABLE VI

VISUAL AND COMPUTED RANKINGS OF PROFILE SEQUENCES FOR FLOODPLAIN RIDGE TERRAIN

Terrain component 1 2 3 4 5 6 7

o Visual ranking 404 406 405 402 403 401 Computed ranking 404 406 403 405 402 401 and value (0.435) (0.319) (0.152) (0.113) (0.027) (0.005)

P

M

An

400 400 (0.00016)

Visual ranking 404 403 406 402 405 401 400 Computed ranking 404 406 403 405 401 402 400 and value (3.425) (3.160) (1.797) (1.794) (0.931) (0.752) (0.040)



Eng. Geol., 1(2) (1965) 89-187


%

,3~0o

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I0 0 I0 I I

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I

Fig.23. Prepared map for area 480, vegetation mounds, Devil's Cornfield, Death Valley, Califor- nia. Contour interval 0.25 ft.

Eng. Geol., 1(2) (1965) 89-187


Effective Roy Lel~lh

- - \ \ \ \ \

Fig.24. Ray profile sequence for area 480 (Fig.23).

TABLE VII

VISUAL AND COMPUTED RANKINGS OF PROFILE SEQUENCES FOR TERRAIN 4807 VEGETATION MOUNDS

Terrain component 1 2 3 4 5 6 7

P

M

An

Visual ranking 480 486 482 485 484 Computed ranking 480 482 486 485 484 and value (0.043) (0.007) (0.003) (0.002) (0.002)

483 481 483 481 (0.00094) (0.0013)




Eng. Geol., 1(2) (1965) 89-187

136 R.O. STONE AND J. DUGUNDJ I

bers at each end of the roughness ranking could be determined.

Testing of individual profiles

All of the profile data were examined and the values visually compared with the configuration of the original profiles. The study will be illustrated with rays from four microterrains. Tabulated data from the remaining profiles, and the maps from which they were derived are presented in the Appendix. The ray study encompassed the following examples: Floodplain Ridges (400-406), Vegetation Mounds (480- 486), Complex Dune Area (500-506), and the Incised Pediment (620-626). These were selected for the variety of features displayed. Some of the remaining sets of profiles do not correlate with the derived data as well as those chosen, others correlate equally well.

The computed values were utilized to rank each curve in terms of its 0, P, M, and A~, values; always from the largest to the smallest value. Each of the profiles was then scanned to attempt to determine visually the relative rank of the terrain components, and the curves were again individually ranked for each roughness vector.

Floodplain ridges (400-406) The microterrain along a portion of the floodplain of the Colorado River (Fig. 21 and 22) is characterized by a series of low, longitudinal ridges, sub-parallel in arrangement. This sequence of curves proved relatively easy to analyze by eye. What were visually the roughest terrain profiles--404, 406, and 403--have high numerical values for all four of the terrain components, and the curves exhibiting the least microrelief variations-402, 401, and 400-- consistently show low microrelief values.

End members of the ray sequence are 406 and 400. The former consists of four jagged highs separated by irregular lows along its 64-ft. length. A small rise, followed by a broad flat area and a gentle dip typifies curve 400, the smoothest of the seven profiles. The relationship between these curves is easily discerned both from the quantitative values and by examination.

By means of averaging the position rankings for the four values a final ranking of the profiles based on the calculated values is: 404, 406, 403, 405, 401, 402, 400. The overall ranking by averaging the visual rankings of each terrain component was similar: 405 exchanging places with 403, and 402 with 401. A listing showing curve rank by computed values, the computed values, and the visual rankings is shown in Table VI,

Vegetation mounds (480-486) These microrelief features are a series of small mounds up to 1-1.5 ft. in height around the base of vegetation clumps in Death Valley. This area, Fig.23 and 24,

Eng. Geol., 1(2) (1965) 89-187


was selected for ray testing because of the numerous "bumps" present and their apparent random distribution. A reasonable check was obtained between the derived values and the observed rank for this series of profiles.

In the profile spread, curve 481 does not cross any obstructions and curve 483 only the lower edges of two mounds, whereas curves 480, 482, and 486 pass over three or four of the mounds. These two groupings represent the extremes of the terrain and this is borne out by the derived terrain components.

In Table VII, the computed and observed values for the seven profiles are shown. Close agreement was reached for all ~, P, and M values; minor variations in ranking occur generally where the numerical difference between the two curves is minute. Difficulty was encountered is assessing the An component in the rougher or higher-value end of the scale. This was especially true in the case of curve 480, which was ranked fourth by observation yet had the highest computed value. At this time the apparent anomaly cannot be explained.

The ranking of this ray series as derived from computer data is: 480, 482, 486, 485, 484, 483, and 481. A nearly identical overall ranking was obtained by inspection, 482 and 486 being considered of equal rank and the "roughest" rating of 480 not affected by its lower visual ranking in the An category.

Complex dunes (500-506) The irregular surface produced by an eolian deposition and erosion was selected as the third test series. This group of profiles (Fig.25 and 26) consists of four curves which show pronounced microvariations: 504, 505, 506, and 503. The first three of these curves pass over the high portion of the dune area and the fourth over lower portions of the major dune masses. Profiles 500 and 501 only cover the marginal sandy areas and 502 an edge of the dune accumulation.

Intuitively and numerically curve 504 is the roughest and curve 500 the smoothest. The former has the largest terrain-geometry values for all components except slope variation where it ranks second to curve 505. This is due both to the termi- nation of the processed course of 504 before the full sweep of the high is completed and to the steep slope on the extreme end of profile 505. Very close correlation was found between computed and observed values for all components in this terrain, perhaps due to distinct grouping of curves described in the previous paragraph.

All computed and estimated ranks are shown in Table VIII, and the final ranking, computed and visual, for the profiles is: 504, 505, 506, 502, 503, 501,500.

Incised pediment (620-626) The final ray sequence tested in detail covers a terrain consisting of residual rock knobs rising above a sloping surface in which shallow dissection channels have been carved (Fig.27 and 28). End members of the roughness sequence are clear-cut, numerically and visually. Profile 625, consisting of a pronounced high, three lows, and a rise has maximum values for all components except the range of slopes value,

Eng. GeoL, 1(2) (1965) 89-187


TABLE VIII

VISUAL AND COMPUTED RANKINGS OF PROFILE SEQUENCES FOR TERRAIN 500~ COMPLEX DUNES

m

Terrain component

P

M

An

1 2 3 4 5 6 7





Fig.25. Legend see p. 139.

I0 0 ~ ~0

FEET

Eng. Geol., 1(2) (1965) 89--187


zsoc osl \ \ \ \ \N

Fig.26. Radial profiles of area of Fig.25.

139

Fig.25. Contour map of complex dune area, Saratoga Springs, California. Contour interval 0.5 ft.

Eng. Geol., 1(2) (1965) 89-187


FEET

J

Fig.27. Microterrain contour map of incised pediment area, Old Woman Springs, California. Contour interval 1 ft.

TABLE IX

VISUAL AND COMPUTED RANKINGS OF PROFILE SEQUENCES FOR TERRAIN MAP 620, INCISED PEDIMENT

Terrain component 1 2 3 4

o Visual ranking 624 625 626 620 Computed ranking 625 624

Documents

Engineering Geology Volume 1 issue 2 1965 Richard O. Stone; J. Dugundji -- A study of microrelief—its mapping, classification, and quantification by means of a fourier anal.pdf