Trend-Surface Analysis of 

the Basin and Range 

Province, and Some 

Geomorphic Implications 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 500-D 

Trend-Surface Analysis of 

the Basin and Range 

Province, and Some 

Geomorphic Implications 

By LAWRENCE K. LUSTIG 

THEORETICAL PAPERS IN THE HYDROLOGIC AND 

GEOMORPHIC SCIENCES 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 500-D 

A quantitative study of Basin and Range 

topography which provides a basis for 

considerations of the nature of a physiographic 

province, and of the relief age of ranges, the 

origin of pediments, and the development of 

mountain drainage systems in the Western 

United States 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1969 
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary 

GEOLOGICAL SURVEY 

William T. Pecora, Director 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 -Price 70 cents (paper cover) 
CONTENTS 

Page Trend-surface results-Continued Abstract __________________________________________ _ D1 Rangewidth __________________________________ _ Introduction ______________________________________ _ 2 Range height _____ ------------------------------ Acknowledgments _____________________________ _ 4 Range relief_ _____ - - - - - - - - - - - - - - - - - - --- - - --- - - - - Study area ____________________________________ _ 5 Range volume __________ - __ - - - - - - - - - --- - - - - - - - - - Quantitative methods _____ ------- __________________ _ 6 Cumulative length of trends ____________________ _ ~easurements _________________________________ _ 6 Cumulative deviation of trends __________________ _ 

Criteria and definitions _____________________ _ 6 Range width to length _________________________ _ Sources of error ____________________________ _ 8 Range width to height_ ________________________ _ 

Analytical techniques __________________________ _ 10 Range length to height _________________________ _ 

~anual contouring ________________________ _ 10 Summary of trend-surface results __________ -_-_- __ Relative-entropy function ___________________ _ 11 Geomorphic implications ______________ -------------- Fourier analysis ___________________________ _ 13 Physiographic provinces ______________________ - __ 

Trend-surface analysis ______________________ _ 17 Age of ranges ____________ - - - _ - - - ---- - - - - - - - - - - - Trend-surface results _______________________________ _ 21 Origin of pediments ______________________ - ___ - __ 

Area of ranges to total area _____________________ _ 22 · Drainage distinctions ___________________________ _ 

Range length _________________________________ _ 22 References ________________________________________ _ 

ILLUSTRATIONS 

FIGURE 1. Index map of the Western United States, showing the approximate boundary of the Basin and Range province 

Page 

D22 

22 

32 

32 

32 

42 

42 

51 

51 

51 

60 

60 

61 

64 

68 

69 

Page 

and the distribution of Precambrian and lower Paleozoic (Cambrian and Ordovician) outcrops- - ---- D 3 

2. Index map of the Western United States, showing the locations of the 1:250,000 topographic quadrangles 

that were used for this studY---------------------------------------------------------------- 5 

3. Diagrammatic model of an information source A that is composed of sources B and C, which yield equally 

likely outcomes nB and nc----- -------·------------------------------------------------------- 11 

4. Diagrammatic model of an information source that generates two groups of outcomes, nA and nB, which are not equally likely ______________________________________________________________ ------------ 12 

5. Diagram showing distribution of relative-entropy values for a three-component system ______ ------------- 13 

6. Graphs showing the Fourier expansion of the function y=x___________________________________________ 15 

7. ~ap of a synthetic double Fourier series representing a wave form containing four harmonics in both coordinate directions_________________________________________________________________________ 16 

8. Geometric relations of first-, second-, and third-degree equations for two _and three variables_____________ 19 

9-41. Computer printout maps: 

9. First-degree trend-surface map of area of ranges to total area _____________________ ------------- 23 

10. Second-degree trend-surface map of area of ranges to total area _______________ - _- _------------- 24 

11. Third-degree trend-surface map of area of ranges to total area ________________ ---_------------- 25 

12. First-degree trend-surface map of mean range length__________________________________________ 26 

13. Second-degree trend-surface map of mean range length________________________________________ 27 

14. Third-degree trend-surface map of mean range length ________________________ ----------------- 28 

15. First-degree trend-surface map of mean range width ___________________________ -_---_--------- 29 

16. Second-degree trend-surface map of mean range width________________________________________ 30 

17. Third-degree trend-surface map of mean ran.ge width_________________________________________ 31 

18. First-degree trend-surface map of mean range height__________________________________________ 33 

19. Second-degree trend-surface map of mean range height________________________________________ 34 

20. Third-degree trend-surface map of mean range height_________________________________________ 35 

21. First-degree trend-surface map of mean range relief_ _________________ - ___ --------------------- 36 

22. Second-degree trend-surface map of mean range relief_ ___________________ --------------------- 37 

23. Third-degree trend-surface map of mean range relief_ ________________________ ----------------- 38 

24. First-degree trend-surface map of mean range volume_________________________________________ 39 

25. Second-degree trend-surface map of mean range volume ______ ~- __________ --_------------------ 40 

III 
IV CONTENTS 

FIGURES 9-41. Computer printout maps-Continued Page 

TABLE 

26. Third-degree trend-surface map of mean range volume________________________________________ D41 

27. First-degree trend-surface map of cumulative length of trends__________________________________ 43 

28. Second-degree trend-surface map of cumulative length of trends_______________________________ 44 

29. Third-degree trend-surface map of cumulative length of trends_________________________________ 45 

30. First-degree trend-surface map of cumulative deviation of trends_______________________________ 46 

31. Second-degree trend-surface map of cumulative deviation of trends_____________________________ 47 

32. Third-degree trend-surface map of cumulative deviation of trends______________________________ 48 

33. First-degree trend-surface map of range width to length-------~------------------------------- 49 

34. Second-degree trend-surface map of range width to length_____________________________________ 50 

35. Third-degree trend-surface map of range width to length______________________________________ 52 

36. First-degree trend-surface map of range width to height_______________________________________ 53 

37. Second-degree trend-surface map of range width to height_____________________________________ 54 

38. Third-degree trend-surface map of range width to height______________________________________ 55 

39. First-degree trend-surface map of range length to height______________________________________ 56 

40. Second-degree trend-surface map of range length to height____________________________________ 57 

41. Third-degree trend-surface map of range length to height______________________________________ 58 

42. Idealized cross section of two ranges and an intervening basin in two different parts of the Basin and Range 

region~----------------------------------------------------------------------------------- 62 

43. Map showing volcanism, tectonism, and plutonism during the Eogene in the Western United States________ 63 

44. Map showing volcanism, tectonism, and plutonism during the Neogene in the Western United States______ 64 

45. Map of the United States, showing the distribution of recorded earthquake epicenters, as of 1957__________ 65 

46. The triangle of probabilities, showing the probability associated with any point on a two-dimensional random 

walk_____________________________________________________________________________________ 66 

TABLE 

Page 

1. Statistical measures associated with trend-surface results_____________________________________________ D 59 
THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE 

AND SOME GEOMORPHIC IMPLICATIONS 

By LAWRENCE K. LusTIG 

ABSTRACT 

The literature on the Basin and Range province provides many 

qualitative assessments of the size, shape, and arrangement of 

mountain ranges in the region, and it is well known that the 

topography is nonuniform within the province. This report represents an attempt to examine quantitatively the existing regional topographic variations. Eleven topographic parameters 

are used for this purpose, namely ( 1) area of ranges to total 

area, (2) range length, (3) range width, (4) range height, (5) 

range relief, (6) range volume, (7) cumulative length of trends, 

(8) cumulative deviation o·f trends, (9) range width to length, 

( 10) range width Ito height, and ( 11) range length to height. 

Measurements of these parameters were obtained from the 

46 (mapped at 1:250,000 scale) topographic quadrangles that 

cover the Basin and Range province. 

Four analytical methods of data treatment are describedmanual contouring, rel·ative-entropy function, Fourier analysis, 

and trend-surface analysis. The last method is used to analyze 

the topographic data, and trend-surface maps of first, second, 

and third degree are presented for each of the 11 parameters. 

The results show that each topographic parameter is not uniformly distributed throughout the region .but varies widely with 

respect to range of value on a given surface of best fit. Average 

values of parameters on the surfaces of best fit range from 100 

percent to 1,400 percent between the southern and the northern 

parts of the Basin and Range region. 

The analytical results indicate that three areas in the Basin 

and Range province exhibit sufficient topographic distinctiveness 

to warrant delineation. These are southwestern Arizona and 

southeastern Oalifornia, northwestern Nevada and adjacent 

parts of eastern California, and the eastern part of the entire 

region, which consists of southeastern Arizona, southwestern 

New Mexico, northeastern Nevada, and northwestern Utah. 

This topographic distinction is the first of four geomorphic 

implications that are discussed. The writer concludes that the 

question of whether these areas should be designated as sections within the Basin and Range province or as separate 

provinces in their own right involves two additional questions. 

The fi,rst of these is the nature of hierarchical classification systems in general and the magnitude of internal variance associated with increasing rank in the system. The second is a more 

specific question; namely, the magnitudes of within-province 

variance and between-province variance that can be considered 

to be average values applicable to any physiographic province. 

Until •these values and their respective ranges are assessed by 

quantitative means, it cannot be determined whether the withinprovince variance shown to exist in the Basin and Range province is normal or excessive for any province. Further work is 

indicated, but the results of this report suggest that any physiographic province, or finite area of the earth's surface, should be 

amenable to quantitative analysis by methods similar to those 

presented herein. 

Whether the three areas within the Basin and Range region 

are actually sections or separate provinces, however, the magnitudes of variance of the tof)Ographic parameters are clearly 

sufficient to suggest three additional geomorphic implications 

of a regional nature. 

The first of these, or second geomorphic implication, concerns 

the relief ages of ranges in the Basin and Range province. The 

variance of the regional topographic data that is indicated by 

the trend-su.rfa·ce maps implies that t'he topography reflects 

different erosional histories of .ranges and, hence, different relief 

ages of ranges. Relief ages are not identical with radiometric 

dates because radiometric dates do not generally coincide precisely with dates of orogenic activity throughout the Basin and 

Range region. The available geological data suggest that substantial topographic differences probably existed by the end 

of the Tertiary and that the nonuniform distribution of Quaternary block faulting in the region served to emphasi~e many 

of the preexisting topographic distinctions. The present regional 

topographic variations coincide well with the known distribution 

of historic earthquake activity, with the Quaternary block faulting, and with the regional distribution of Precambrian and 

lower p,aleozoic outcrops. The areas characterized by lower 

values of the area of ranges to total area ratio, range width, 

length, height, relief, and volume, and by higher V'alues of 

cumulative deviation of range trend from north, are areas which 

by implication include ranges that have the greatest relief ages 

and longest erosional histories. 

The known distributions of fans tand pediments in the Western Unrted States :alSIQ acco,rd well with the oopographic distinctions that are made. In gene~al, f,ans are more abundant 

in those a.reas characterized by 1arge average vaiues of vange 

width, length, height, relief, volume, 1and area of ranges to 

total .3Jrea, whereas pediments predominate in those ·areas where 

these values diminish. His argued that the various hypotheses 

of pediment formation dependent upon processes that are operative on existing pediments are not relevant because rthey focus 

upon the wrong land1lorm. Pediments ~a~re a natural and ine-vitable consequellC'e of mountain mass reduction thro11g1h time, 

and the on!ly real "pediment problem" is the question •of how 

this maSJS reduction is accomplished. The hypothesis offered in 

this repovt is that weathering predominates on the steep mountain fronts in interfluviai a.reas, which are probable loci of 

in~elberg formation, whereas fluvial pvocesses predominate in 

the drainage basins which are the loci of maximum mOUilltainmass reduction. According rto this view, pediments are sill1ply 

D1 
D2 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

a product of drainage-basin evolution in the moontains through 

time, as controlled by local base-level constraints. 

Finally, it i-s suggested that regional drainage distinctions 

that a.re compatible with the topographic, relief age, and fanpediment distinctions must also exist bec3.!use each is a function of time. It is argued that the IllUJIDbers of drainage systems 

and their order numbers are a function of the size, or mass, of 

a given range, in the absence of constraints of shape and lithology. As the range is reduced in mass through time, the average 

values of these variables should diminish. Drainage systems 

should accord with rthe steady sta:te that is pvedicled by 

stochastic models and by hydraulic principles in the older and 

smaller ranges of greater tectonic stability, but may not yet 

have achieved this state within the younger and l·arger ranges 

chal'lacterized by more recent tectonic activity. The trend.,surface maps of this report provide ·a basis :!lor the :future examioo.tion of such drainage distinctions within the Basin and Range 

province. 

INTRODUCTION 

The a.rea discussed in this report is commonly called 

the Basin and Range province of the Western United 

States. It extends from the Sierra Nevada on the west 

to the Wasatch Range on the east, and from the Mexican border on the south to approximately lat. 42° N., 

which is the northern boundary of Oalifornia, Nevada, 

and Utah. The boundary cannot be defined precisely, 

but that shown in figure 1 approxima.tely delineates the 

Basin and Range province. 

The term "Basin and Range province" has •appeared 

in a legion of geologic, geographic, hydrologic, and 

other reports, and the general intent of any user is 

widely understood. Nevertheless, some contradictions 

exist between the standard definition of the Basin and 

Range province ,and the facts that have been set forth 

or surmised by many who have worked in the region. 

The "Glossary of Geology and Related Sciences" 

(American Geological Institute, 1957, p. 26) defines 

"Basin and range landscape" as "Landscape consisting 

of fault-block mountains and intervening basins." The 

supplement to this volume (American Geological Institute, 1960, p. 50) expands upon this by defining any 

physiographic province as 'a "Region of similar structure and clima.te that has had a unified geomorphic 

history." By combination of definitions, then, the Basin 

and Range province is a region of fault-block mountains and intervening basins within which the geologic 

structure, climate, and geomorphic history are nearly 

uniform. 

The fact that Fenneman ( 1931) suggested nearly 40 

years ago that the Basin and Range province be divided 

into five sections clearly indicates that the nonuniform 

characteristics of the region have long been known. In 

discussing various distinctions, for example, Fenneman 

(1931, p. 328) stated: 

The Great Basin lies north of latitude 35°30'. In this the 

space taken by the mountains is about half the total * * *. 

South of this, in California and southwestern Arizona * * • 

ranges are smaller and perhaps older occupying perhaps onefifth of the space. 

In a similar vein, Lobeck stated in his geomorphology 

text (1939, p. 557) : 

In the southern part of the Basin and Range province, notably in ·southern California, southern Arizona, and New Mexico, 

the basin ranges have been almost annihilated by erosion. The 

presence of faults can only be. inferred. The ranges are more 

or less symmetrical in shape and reveal none of the diagnostic 

features cited as topographic evidence of block faulting. 

Finally, Eardley's work on the structural geology of 

North America includes the following summary remarks on the southern Arizona Rockies (1962 p. 425): 

A glance at the Geologic Map of the United States will show 

that the ranges of southern California and Arizona and southwestern New Mexico ·are smaller, more irregular in shape, less 

linear and parallel, and separated by relatively wider basins 

than those of western Utah and Nevada. Hence, the inclusion 

of the Sonoran Desert of Arizona in the Basin and Range province from a structural point of view must be made with reservations. The cri·sp boundaries imparted to ranges by block 

faulting are generally absent, and if the region is one of extensive block faulting, then the faults are older than those in 

Utah and Nevada, and erosion has beaten the fault scarps 

back considerable distances to form broad :flanking pediments. 

The essential burden of the foregoing statements is 

that substantial differences in topography and geologic 

structure are thought to exist between the.southern and 

northern portions of the Basin and Range province and, 

by inference, that the erosional history is, likewise, 

nonuniform. In addition, it has been stated by many 

authors that although the rocks of the region range in 

age from Precambrian to Quaternary, Precambrian 

outcrops are far more abundant in the south than elsewhere. The facts, therefore, seem to be at variance with 

the definition of a province, and it might be asked 

whether the broad similarities within the Basin and 

Range region outweigh the differences that exist. 

One purpose of this report is to examine the topographic differences that exist within the Basin and 

Range province in greater detail than previously attempted. It is noteworthy that the statements of the 

authors quoted above each contain some generalization 

or qualifying remarks that were necessary because of 

a lack of quantitative data at the time. Fenneman 

(1931) stated that the southern ranges occupy perhaps 

one-fifth of the total area; Lobeck (1939) stated that 

the southern ranges are more or less symmetrical in 

shape; and Eardley (1962) remarked that the size and 

shape and the degree of parallelism of these same ranges 

is also based on qualitative assessment to some extent. 

Each of these observations, however, is amenable to 

quantitative examination in terms of such questions as: 

How much smaller are the southern ranges~ Are the 
' '\ 

TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE 

--------. ----- _ CANADA / T---u'Nirnns"T.tl~ 

I I 

WASHINGTON i \ 

\..._--, __ -------1, 

OREGON 

( 

I 

( 

J 

l 

I 

) 

'\ 

'\_ 

I 

~/\ 

IDAHO 

MONTANA 

EXPLANATION 

.. 

Area of outcrop 

Boundary of Basin 

and Range province 

WYOMING 

D3 

-------, _____ -.......__ 

I 

I 

COLORADO 

\ --- ----------t------------c--

\ \ 

I 

I 

\ 

I 

I 

\ 

TEXAS 

FIGURE 1.-Index map of the Western United States, showing the approximate boundary of the Basin and Range province and the distribution 

of Precambrian and lower Paleozoic (Cambrian and Ordovician) outcrops. The province area shown in central New Mexico and the 

westernmost part of Texas was omitted from this study. Note the correspondence of the outcrop areas to areas of greatest relief age 

discussed in text and to those exhibiting distinctive values of topographic parameters on the trend-surface maps (figs. 9-41). 
D4 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

lengths, widths, and heights of these ranges uniformly 

proportional ~ What is the distribution of ranges, in 

areal percentage~ Do they occupy 20 percent of the 

southern part of the region and 50 percent of the northern part, as surmised by Fenneman ~ Do the trends of 

ranges resemble the often-cited "army of caterpillars" 

crawling along a north-south meridian~ The intent of 

this report is to answer these and other questions in 

quantitative terms. By such means, the regional generalizations that have been made can be more soundly 

based. 

An 1additional purpose of this report is to demonstrate that surficial features can be quantified on a provincewide, or regional basis. For this reason, possible 

applications of analytical techniques other than trendsurface ,analysis are also discussed, and methods of 

map measurement and sources of error are described 

in some detail. 

Further, significant topographic differences are 

shown to exist within the Basin and Range province 

and to be of a m~agnitude sufficient to suggest that perhaps three provinces exist, rather than one. This implication, and certain aspects of the distribution of fans 

and pediments, the age of ranges, and the drainage systems in these ranges are discussed in the concluding 

section of this report. 

ACKNOWLEDGMENTS 

In 1964 it occurred to the writer that a quantitative 

~analysis of Basin and Range topography might profitably be undertaken. It was hypothesized that the results 

of such an ,analysis would be applicable to the -various 

questions stated in the section of this report entitled 

"Geomorphic Implications." In addition, no previous 

attempt had been made to treat an entire physiographic 

province hy an analytical method such as trend -surface 

analysis. At that time the writer assumed responsibility for the teaching of a course designated 'as "Quantitative Geomorphology" at the University of Arizona. 

This course, among other things, was intended to familiarize the students with the methods of analysis of 

topographic-map data. 

Of the several ways of accomplishing this end, one 

of the most common is to assign a dl'lainage basin for 

study by means of a standard Horton-type analysis. 

Other common alternatives include the quantitative description of drumlins, ·alluvi~al fans, beaches, or some 

other landform which has been previously studied and 

reported on in the literature. All such problems are 

useful ·as instructive devices, but the principal common 

failing is that the student virtually "knows" the answer in adv.ance. Although there can be no substitute 

for actual data gathering 1and analysis in order to .gain 

familiarity with the associated problems and procedures, such problems and procedures are clearly minimized if anything as well known as a Horton -type 

drainage analysis is used. From a purely pedagogic 

viewpoint, it would seem far better to set a problem that 

has not been undertaken previously .and to which the 

best approach, as well as the answer, is unknown. 

For this reason the writer's students were set the 

task of gathering the raw data on topographic parameters in the Basin and Range region. Mutual benefit 

was derived from the ensuing discussions of what 

parameters to measure and how these might best be 

defined. The students' many false starts proved far more 

instructive in the long run than any standard exercise 

possibly could have been, and, hopefully, these students 

also gained some useful insight to. the joys of research 

by attempting the unknown. 

Hence, the first ~acknowledgment here is to my students for their many man-hours of labor at the outset of 

this project ~and, in several instJances, for their stimulating discussions of subtle points and problems related 

to the data gathering that could not have been foreseen 

in advance. Their assistance is gratefully and collectively acknowledged. Specific mention is made of the 

contributions by Neil Jones, Frank Anderson, and 

George Richardson. 

The writer checked all results, compiled basic data 

on nine topographic parameters from several sheets that 

had not been assigned, computed mean values, and prepared these data for computer analysis. Because the 

computer system then a-vailable ~accepted only Fortran 

II programs, the standard trend-surface program 

available in Fortran IV language required conversion. 

Techniaa,l problems too numerous to cite here occurred, 

partly as a result of this conversion, and the ~assistance 

of Bart Cross was of inestimable value in obtaining 

satisfactory results. Computer time for the trend-surface analysis was generously provided by the University of Arizona and its Numerical Analysis Laboratory. 

Several previously published illustrations germane to 

this report have been reproduced, with some slight 

modifications. Permission to do so was kindly granted 

by the sevenal publishers involved. Acknowledgment is 

extended here to the Massachusetts Institute of Technology Press for the use of figures 3 and 4 in this 

report, to the Columbia University Press for figure 6, 

to the Kansas Geological Survey for figure 7, to the 

Geological Society of America for figures 43 and 44, 

and to the Princeton University Press for figure 45. 

Because of the somewhat varied nature of the topics 

treated in this report, the advice and opinions of several 

reviewers were sought. John Harbaugh, Thomas Maddock, Jr., Mark Melton, Richard Chorley, Robert Sharp, 

Stanley Schumm, James Gilluly, and William Bull each 

contributed helpful comments and suggestions on the 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D5 

manuscript. Additionally, Paul Damon provided an 

enlightening discussion of the chronology of events in 

the Basin and Range province, and, as previously stated, 

Bart Cross kindly resolved several computer-programing problems. The writer is extremely grateful to each 

of these individuals for the aid rendered. As often happens when opinions are sought from several quarters, 

however, some conflicting views were obtained. For this 

reason the writer necessarily accepts sole responsibility 

for the views set forth in this paper. 

126• 

. 120' 118' 116' 

48• 

~ 

46• ~ 

l_-__ ~ 

- ) 

STUDY AREA 

As previously stated, the Basin and Range province 

extends from the Sierra Nevada on the west to the 

Wasatch Range on the east and from the Mexican border on the south to the northern boundaries of California, Nevada, and Utah, approximately, as shown in figure 1. The locations of the topographic quadrangles of 

the study area are shown in figure 2. The total number 

of rna ps represented is 46, but because the boundaries of 

the Basin and Range region do not, of course, coincide 

104' 102' 

106' ---"' 

114' 112' 110' 108' , I I 

-

M v T 

\ 

44• ( l D A ~-N \ 

I 0 R E 

'---

42• 

40' 

38• 

36• 

34• 

30' 

( 0 N 

w y 0 M 1 N G 

l 

I 

-

l 

\ 

-

A 0 

FIGURm 2.-Index map of the Western United States, showing the loeations of the topographic quadrangles, the maps of which were used 

for this study. Mean values of all topographic parameters were plotted at the centers of the indicated quadrangle maps. These served 

as an orthogonal data grid for the trend-surface maps (figs. 9-41). 

326-328 Q-68--2 
D6 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

precisely with the quadrangle boundaries, certain variabies were analyzed using only 43 or 44 of the maps. 

Only one of the topographic quadrangle maps, namely 

that of Pocatello, Idaho (fig. 2) represents an area 

north of lat. 42° N. 

QUANTITATIVE METHODS 

The quantitative methods used in this study were of 

two types: those uesd in the actual assemblage of raw 

data from the topographic maps, and those used in the 

treatment of these data by analytical mapping procedures. Because the methods of study and the degree of 

reproducibility of results are obviously central to this 

report, these twin concerns are discussed in the sections 

entitled "Measurements," and "Analytical Techniques," 

respectively. 

MEASUREMENTS 

The first problem that arises in quantitative map 

study is what to measure. The parameters or variables 

selected must not only be measurable on maps of the 

scale selected but must be equally significant to the study 

under consideration. In this study the essential problem 

was to determine whether significant topographic differences exist in the Basin and Range region. Hence, the 

following parameters were selected for investigation: 

( 1) Area of ranges to total area, ( 2) range length, ( 3) 

range width, ( 4) range height, ( 5) range relief, ( 6) 

range volume, (7) range trend, and three derived ratios, 

namely (8) range width to length, (9) range width to 

height, and (10) range length to height. Each parameter reflects a topographic characteristic and is therefore 

relevant to the study. Subsequently, it became desirable 

to -apply two parameters for the description of range 

trend, and a total of 11 variables were ultimately 

examined. 

The second problem encountered in any such study is 

how to measure the selected parameters. This is far less 

straight forward than it might appear to the uninitiated. ·Each variable must be defined, and criteria must 

be established for its measurement. Regardless of the 

effort expended on these initial considerations, one often 

finds that seemingly satisfactory criteria cannot be applied with reproducible results once the work has begun. For this reason, the measurement procedures are 

partly trial and error by nature. This will be made clear 

in the following discussion of t:he criteria and definitions 

used in this study. 

CRITERIA AND DEFINLTIONS 

A rea of ranges to total cr:rea 

Three criteria were required to define the first topographic parameter, namely range area, in order to measure ratios of range area to total area. The ranges are, of 

course, represented by topographic-contour closures on 

the maps, and their locations can be seen at a glance. A 

slope criterion must arbitrarily be chosen at the outset, 

however, to provide a range boundary and to thus permit measurement of a given range area. Regardless of 

the slope value chosen, one is still confronted with the 

risk of including too great or too small a proportion of 

actual range in the nonrange area, and with the decision 

as to which is preferable. For example, if the topographic quadrangle maps of the southern part of the 

Basin and Range region are examined, one would be inclined to choose a slope value of about 200 feet per mile 

to define the range boundaries. However, the slopes of 

some alluvial fans in the northern and western parts of 

the Basin and Range region are as much as 800 feet per 

mile, and in such areas the application of a 200 feet per 

mile criterion would cause much basin are~:~, to be incorporated in the measured-range areas. Conversely, the application of 800 feet per mile in the southern areas 

would entirely eliminate some known bedrock outcrops 

of much gentler overall slope and thus yield too sma.Il a 

value for total range area. A compromise between these 

extremes was necessary, for the use of variable slope 

values would involve much subjectivity; no criteria 

could be devised that would permit reproducibility. 

Thus, for purposes of this study, a value of 500 feet per 

mile was chosen. A given range is therefore defined 

as an area including slopes that are equal to or greater 

than 500 feet per mile. It is not argued here that this 

slope criterion is everywhere the most satisfactory, and 

it should be noted that the u8e of another value would 

alter the range percentages given in this report. 

That this single criterion for the measurement of 

range area was necessary but not sufficient soon became 

apparent. In the southern part of the study area, for example, many of the ranges are divided into segments 

that are separated by passes or topographic lows. The 

application of the definition of a range given above to 

these areas would require that each such segment be considered a separate range. Because mean range area was 

of some concern, and because the establishment of range 

area affects the determined values of range width, length, 

and other variables to be difined later, a second criterion 

was necessary to avoid representing a single range of A 

square miles by three ranges with an average area of approximately A/3 square miles, for example. For these 

reasons, the definition of a given range was expanded by 

application of a second criterion, namely that a range include all apparent outcrops or topographic highs with 

slopes that are equal to or greater than 500 feet per mile 

and that are separated by no more than 2 miles. Note 

again that the value chosen for the separation distance 

is arbitrary, and the choice of another value would alter 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D7 

the magnitudes of the reported values o:f range area 

and other variables in certain places. 

Finally, it was necessary to apply a third criterion 

because o:f the occurrence o:f isolated knobs and hills of 

small size. Many volcanic plugs, :for example, are represented by topographic-contour closures around areas of 

perhaps 1-2 square miles. Application o:f the two criteria 

:for the definition o:f a range discussed above would require that each o:f these small :features be measured and 

computed as a separate range. Again, because mean 

values o:f several variables were o:f concern, the inclusion 

o:f these small :features would clearly produce erroneous 

results. The final criterion adopted to prevent this was 

that any range be greater than 4 square miles in area. 

These three criteria were applied to define and measure range areas on each of 44 topographic quadrangle 

maps. The total area of each map was then determined, 

and the area o:f ranges to total area, or percentage of 

range area, was calculated. All measurements o:f area 

were made with a compensating polar planimeter. 

Range length 

The length of a given range was initially defined as 

the distance between the points o:f intersection o:f the 

range area boundary and the longest contained straight 

line that could be constructed. This definition had the 

virtue o:f reproducibility, and, in addition, the trends of 

ranges could have been determined objectively by simply measuring the azimuth o:f a given length line. Unfortunately, the population o:f ranges that occurs in the 

Basin and Range region will not accommodate sueh a 

definition. Certain ranges mappoo in the northern part 

o:f the region are amoebalike in configuration and extend 

across parts o:f several! : 250,000 scale topographic gua.drangle maps. Even in the southern part o:f the region, 

horn -shaped ranges are not too uncommon, and their 

occurrence precludes the possiBility o:f representing 

range length meaningfully by the use o:f straight lines. 

The definition used in this study, therefore, was that 

the length o:f a given range is equal to the length o:f a 

line constructed along the midpoints of a range, as determined by the range area boundary. 

This definition does not permit precise reproducibility 

unless the number of midpoints and their spacing is 

specified. The degree of error thus introduced is negligible, however, compared with other factors to be later 

discussed in "Sources of Error." 

Ramge width 

In this report, the width of a given range is defined 

as the area of that range divided by its length, as the 

area and length are defined above. This definition, like 

all, is admittedly arbitrary and can only be considered 

to be an approximation o:f true range width. If one co~­

siders a range that is circular in plan view, the maximum error involved in so measuring range width can 

readily be calculated. The area of a circle ~ is ~qual to 

7('r2, where r is radius, or 7t'D2/4, where D IS diameter. 

According to the definition adopted here, however, 

range length L would equal D; therefore .AjL=7t'D2/4 

L='T('L/4, whi1ch would be the computed value of range 

width W. For a circle, however, width should equal 

length and it is obvious that W :f7rL/4; the magnitude of the error is equal to 1 - 7(' I 4, or about 21 percent for maximum width. If average width of the circle 

is ~~sidered then no error exists. Most ranges in the ' . Basin and Range region are elongate rather than circular and because many of them taper toward their ends 

and ~xhibit other irregularities, any alternate definition 

of range width will present some difficulties of application and will be equally arbitrary. The definition 

adopted here is simple to apply and yields results that 

are both reproducible and fairly consistent throughout 

the study area. 

Range height 

The height of a given range, as defined in this report, 

is the difference in elevation between the base of a range 

and its crest area. The use of this definition requires, in 

turn, the establishment of two criteria to permit 

measurement o:f base- and crest-area elevations. The 

base of a range will, in most places, occur at different 

elevations, and an average of these must be obtained. 

The mean base elevation was arbitrarily defined as the 

means of the elevations of ( 1) the two points of intersection o:f the range length line and the range area 

boundary, and ( 2) the eight points of intersection o:f 

:four lines that are perpendicular to the range length 

line and that divide it into equal segments with the range 

area boundary. Alternatively, the mean elevation of the 

base, or the range area boundary, can be determined 

by averaging the elevations of a large number of points, 

separated by fixed intervals, along its perimeter. But 

this would have been more time consuming, and in 

several tests the means o:f the 10 base-elevation points 

defined above were found to closely approximate the 

alternative values. 

The elevation of the crest area of a given range was 

determined :from the means of ( 1) the elevation of the 

highest peak, and ( 2) the elevations of points spaced 1 

mile apart along the divide, on either·side of the highest 

peak. In practice, oonsiderable difficulty was met in 

attempting to strictly :follow this, or any alternate, definition. Specifieation of elevation determinations at fixed 

intervals along the divides o:f ranges causes mean values 

for small ranges to be based on :fewer points than is true 
D8 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

of large ranges. Moreover, in many ranges the highest 

peak occurs near one end of the range rather than in the 

middle, and this prevents determination of values on 

both sides of the peak. Range height is, in fact, rather 

difficult to define satisfactorily in any manner. It seemed 

clear to the writer that any such definition should include the elevation difference in the zone occupied by the 

highest peak. Because this was accomplished, the range 

height comparisons to be later presented are deemed 

meaningful, but as with all the parameters discussed 

thus far, the use of another definition would und?ubtedly alter the absolute value of the height of any 

given range. 

Range relief 

Range heights, in addition to being difficult to define 

adequately, do not correspond very well to range relief 

because the base eleV'ations of ranges are quite disparate from average basin elevations. Hence, a separate 

parameter was defined to express range relief. It was 

defined as the difference in elevation between the crest 

height of a given range and the mean value of the lowest elevations in the two adjacent basins that can be obtained along two lines that intersect, and are perpendicular to, the range length line. This definition serves to 

restrict relief values to the immediate vicinity of the 

range, but these values are not identical with relief as it 

is commonly defined. The parameter is easily measurable, however, and provides a meaningful basis for comparison within the Basin and Range region. 

Range 1-'olume 

Range volume is a characteristic of considerable 

significance in light of one of the theses of this report 

namely that the ranges in the study 9.rea have bee~ 

subjected to periods of erosion of different. durations. 

The volume of a mountain block will, of course bear 

soi_Ue relationship to its areal extent, but the rel~tionship need not be everywhere the same. For this reason, 

an independent measure of range volume was sought. 

Volume is an exceedingly difficult topographic parameter ~o measu~e, and the approach used for purposes 

of this report IS based upon a geometric simplification. 

If one co?sider~ a mountain block to roughly approximat~ a t_riangle In cross section, the area of a given cross 

sectiOn I~ equ_al to twice that of a right triangle, or hb, 

where hIs hmght and b is the base of the right triangle. 

The area of such cross a section is therefore approximated by the product of range height and one-half the 

range width, as these measures previously were discussed. To obtain range volume, such cross sections must be cumulated along the length of a given 

range, and for this reason the ·approximation used 

for range volume in this report is 1f2 (range width X 

range height X Tange length). Although this would 

seem a fairly reasonable procedure, even granting 

that many ranges may deviate from the ideal, regular, triangular-solid model described here, one significant fact should be noted. Because range width 

was defined as range area divided by range length, 

the width-height-length product is actually equal 

to 1h (range area X range height). Thus, the range 

volume parameter is not truly independent of range 

area, and it involves range height which, as previously 

indicated, presented considerable measurement problems. The volume data given in this report are therefore 

considered to be valid only as first approximations. 

Range trend 

Range trend, as used in this report, refers to straightline escarpments that are not less than 1 inch (equalling 

approximately 4 miles) on the 1 : 250,000 seale topographic quadrangle maps used. Aside from this minimum length, the only additional criterion imposed was 

that the deviation of a given escarpment from the superimposed straight line not exceed one-tenth of an inch 

on the map. The bearings of each such escarpment trend 

were measured and weighted in accord with their 

length; a 2-inch trend length is weighted twice as heavily as a l-inch trend length, for example. 

The results were cumulated according to total lengths 

of trends and total deviations of trends from north 

for each of the 46 1:250,000 topographic quadrangle 

maps. Although the trends are measured from maps 

and the term "escarpment" is used above, it seems probable to the writer that these measures provide an adequate representation of large-scale structural trends 

throughout the Basin and Range province. 

Derived ratios 

Three derived ratios, involving the lengths, heights, 

and widths of ranges were computed for comparative 

purposes in this report. They are range width to length, 

range width to height, and range length to height. Each 

ratio contains, of course, the uncertainties and approximations that are involved in measurement of its respective components. The principal merit of the derived 

ratios is that they provide information on the manner 

in which the respective components vary together 

throughout the region. 

SOURCES OF ERROR 

Four principal sources of error are involved in the 

assemblage of raw data. The first source is inherent in 

the measurement of the topographic parameters chosen. 

As indicated previously certain of the variables used 

are difficult to measure with consistency, are only crude 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D9 

approximations of the topographic property sought, or 

involve arbitrary criteria which directly affect the absolute magnitudes of the data obtained. 

The second general source of error is operator error, 

which undoubtedly comprises some component of the 

total variance. Although this is true of nearly all such 

studies, in this study the magnitude of the operatorerror factor may be greater because several individuals 

were involved in the initial compilation of map data. 

No attempt was made to determine the effects of this 

source of error. 

The third source of error arises from the fact tha,t a 

few of the 1 : 250,000 topographic quadrangle maps used 

are part of an older rna p series on which the contour 

interval is as much as 500 feet. This tended to produce 

illJaccuracies both in range delineation and in all those 

parameters which are dependent upon range boundaries 

for their measurement. Fortunately, these maps were 

few in number and of somewhat scruttered locations. For 

the latter reason, it was possible to correct several of the 

boundaries of ranges on maps which, in part, extended 

onto newer maps with a smaller contour interval. Nevertheless, some subjectivity was involved and should be 

noted as a possible source of error. The locations of the 

olderr maps generally coincide with the southernmost 

part of the study area. 

The fourth and last source of error arises from the 

areal-plotting procedures that were used-this is, the 

geographic locations at which the data on each variable 

were plotted prior to areal mapping. Ideally, each attribute of a given range would have been plotted at the 

centroid of that range in order to map its areal distribution throughout the region. This could not be accomplished, however, because suitable base maps are lacking; the various small-scale special-purpose maps of the 

Western States do not 'adequately reflect the topography 

that is apparent on the 1:250,000 topographic quadrangle maps used in this study. The best alternative, of 

course, would have been to plot 1the data at the centroids 

of ranges on the 46 1 : 250,000 topographic quadrangle 

maps and then have them all reduced to a single scalestable base map of suitable dimensions. Time and cost 

considerations mitigated against this choice, and the 

data were therefore plotted as mean values for each 

map on an orthogonal grid that corresponds to the coordinates of the cente.rs of the maps. This simplified 

plotting procedure led to nonuniform results for the 11 

variables considered. 

No particular problem arises with respeclt to the first 

parameter, namely area of ranges to total area. One 

would need to choose some unit area to compute this 

ratio in any case, and the procedure used simply means 

that the area of a 1 : 250,000 topographic quadrangle 

map is that unit area. On each map the ranges present 

were outlined by application of the range-boundary criteria previously discussed, the areas of these ranges were 

measured, and the total range area present was divided 

by the total area of that map. The latter values vary 

slighJtly with latitude, and the total area of each map 

was computed separately. The area of ranges to total 

area parameter cannot, therefore, even be considered 

as resulting, from an averaging procedure on each map; 

the unit area is the map area, and the value of the parameter could not be plotted in the range locations. 

The rtwo variables used to reflect range trendnamely cumulative length and cumulative deviation 

from north-could have been plotted at the centroids of 

the ranges measured, however. For these variables, plotting the mean values at the center of each map does represent an averaging procedure over the areas involved. 

The chief sources of error wi,th respect to plotting, 

however, a.rise from the application of this procedure to 

values of the other parameters, namely range width, 

length, and height, rthe derived ratios of these variables, 

and range volume and relief. Each of these is a property 

associated with a given range and because several of the 

ranges ~extend across map boundaries, ra problem of oon.:.. 

sidera;ble magnitude occurs. If one is concerned wi!th the 

area of ranges to total area on a given map, and a given 

range extends onto 'another map, then one simply measures the range area. present on eaeh map and assigns the 

proper values to the respective maps. If one is concerned 

with a parameter such as range length, however, then it 

is obviously not valid to assign portions of the total 

length of a given range to the maps on which those parts 

occur. This would result in an overall reduction of range 

length wherever this situation occurred. The most reasonable solution to this problem, with respect to range 

length and the other individual range ruttributes cited, 

was to assign the value of a given range variable to that 

map on which the largest part of the range was shown. 

Then, values were averaged for each map, and the mean 

value for each variable was plotted at the central position of each map as before. 

The procedure cited obviously leads to some errors, 

but these are not as great as one might suspect at first. 

Most of the very large ranges which do extend over 

several m·a ps are located in the northern part of the 

Basin and Range region. Commonly, in this part of the 

study area a range may trend roughly north-south across 

three 1: 250,000 topographic quadrangle maps, and the 

ends of it curve somewhat and thus occupy parts of two 

other maps. By use of the procedure cited above, the 

value of range length, width, or other attribute would 

be assigned to the central map of the three maps showing the north-south extent of that range (if the largest 
DlO THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

part of the range area were shown on that map) and the 

value would be incorporated into the mean value plotted 

for the map. Two facts should be noted as to the possible 

errors involved. First, if the centroids of ranges had 

been used for plotting locations, then almost every centroid location would occur somewhere within the boundaries of the map employed for plotting according to the 

procedure used in this report. In general, the maximum 

error in location is the distance from the center of a 

given map to one of its corners. And because these location errors are not all in the same direction across the 

region, they are partly compensatory; the avel"age location error is thought to be much less than this distance. 

The second fact is that more than one of these very large 

ranges cannot, in general, fall upon or occupy the same 

map by reason of space limitations. For this reason also, 

cumulation of the location errors does not occur. For 

example, if the attributes of a very large range are assigned to a single map, the values of length, width, 

height, volume, relief, and the derived ratios will, in 

general, all be high for that map. But this results because 

the magnitudes of these values are great and not because 

of the plotting procedures. The main point is that the 

locations of the various regional highs and lows discussed and .later shown in this report do not coincide 

precisely with geographic locations. They may be displaced by some fr:action of one-half the diagonal distance of a 1:250,000 topographic quadrangle map. 

Finally, some scale distortion is involved in the various trend maps presented later. The orthogonal grid 

used to plot mean values for the centers of the topographic maps consisted of points equally spaced in both 

coordinate directions. This procedure necessarily results 

in map elongation in a north-south direction because the 

topographic m1aps are essentially rectangles, rather than 

squares, and their long dimensions are oriented eastwest. The geographic locations of selected topographic 

maps are shown on the various trend maps to avoid the 

conceptual difficulty that this scale-distortion problem 

may present to the reader. 

ANALYTICAL TECHNIQUES 

Once the raw data ha.ve been obtained for a regional 

study by measurements, such as those previously described, some analytical technique must be selected to 

depict their areal distribution. The choice of technique 

is to some extent arbitrary and depends upon the assumptions made by the investigator, for a given set of 

data often can be represented by more than one, unique 

mathematical function. As an example, one might consider a set of data which represents the relation between 

stream discharge and channel slope in a downstream 

direction. In the absence of previous knowledge of the 

nature of this relationship, ~a given investigator might 

attempt to plot these two variables on arithmetic, semilogarithmic, and double-logarithmic graph paper, successively. He would then choose, in all probability, that 

graph which provided the best fit, calculate the regression expression for this graph, and present the relation 

between stream discharge and channel slope in terms of 

a linear equation, an exponential equation, or a power 

function, as might be appropriate. In reality, however, 

it is entirely possible that some other mathematical 

function more closely reflects the true relationship involved. The equation of the cycloid, for example, has 

never been used to represent the longitudinal profiles of river channels, despite the fact that it is the 

solution to the Bernoulli problem in the calculus of 

variations of the least travel-time between two points. 

Hence, a profile that corresponds to a cycloid path 

or some portion thereof might reflect the leastwork principle (Leapold and Langbein, 1962) that 

is operative in river systems under certain conditions. A given investigator therefore assumes the 

nature of the relationship between two variables by the 

very act of choosing the type of graph paper on which 

the data are plotted. In effect, he ultimately determines 

whether the relationship can be approximated by the 

mathematical function involved but not necessarily 

whether his choice of that function is the best among 

several alternative possibilities. 

The same reasoning that applied to two-dimensional 

graphic representation of data also applies to regional 

mapping, which is a three-dimensional problem. F;0rthis 

reason, four possible analytical techniques, each of 

which involves mapping, are. described and discussed 

below according to their applicability to the topographic 

data on the Basin and Range region. These techniques 

are (1) simple manual contouring, (2) the relativeentropy function, (3) Fourier analysis, and (4) trendsurface analysis. 

MANUAL CONTOURING 

Manual contouring is a time-honored method of depicting the areal distribution of a set of variables and is 

the simplest of the four methods discussed here. Because 

mean values of each of the 11 variables used in this 

study were plotted at the centers of the appropriate 

1:250,000 topographic quadrangle maps of the region 

(fig. 2), an orthogonal array of data values was available for each of these variables. Manual contouring of 

any such set of derived data values requires only the 

choice of an appropriate contour interval and the construction of contour lines of equal value that are plotted by interpolation between adjacent points in the data 

array. Every reader of this report is familiar with this 

method but the subjectivity involved is not, perhaps, 

often specifically cited. · 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE Dll 

The simplest presentation would consist of an orthogonal array of data values in which each particular 

horizontal or vertical line of points had the same value 

and was adjacent to another line of points of different 

value, the difference in value being equal to the selected 

contour interval. No interpolation between adjacent 

data points would be required, and the r.esulting contour 

map would consist solely of a series of equally spaced 

horizontal or vertical isopleths. The important point is 

that no subjectivity would be involved in the construction of the map; thus, maps produced by any number 

of investigators would necessarily be identical. 

The hypothetical ideal cited above seldom, if ever, 

actually occurs, however. The topographic data from 

the study reported here reflect considerable variation 

of each parameter across the entire study region. The 

construction of isopleths from such data cannot be objective because the map produced by a given investigator is not precisely reproducible. In geophysical or 

other studies in which the location of specific high or 

low values is of importance, the tendency for an investigator to produce as many contour closures as possible 

is inherent. If, on the other hand, the basic goal of the 

investigation is to determine regional trends or gradients, the opposite tendency will exist, namely to produce the minimum number of closures from the same 

set of data. Subconsciously or otherwise, two investigators can easily produce startlingly different maps from 

the same set of data by manual-contouring methods. 

The use of moving averages does not really eliminate 

this difficulty because it essentially represents an attempt 

to reduce one of the components of the total variance, 

namely local variance, and thus emphasizes regional 

trends or gradients. This is particularly true if successive moving averages are taken from the original 

data values. Regardless of the averaging scheme employed, the isopleth maps produced will always reflect 

some additional degree of subjectivity that is a function 

of map construction. For this reason, such maps were 

not constructed and employed for this study, despite the 

fact that discrimination of regional trends was an essential goal. 

RELATIVE-ENTROPY FUNCTION 

The mapping of variables by the manual-contouring 

method can be described as a univariate method because 

only the areal distribution of single variables, or perhaps some simple ratios, can be reflected by isopleth 

maps. In the study reported here, for example each of 

the 11 variables can be investigated on separate distribution maps of a given type; these would be classified as 

univariate maps. If, however, one wishes to treat the 

variables as parts of a multicomponent system, use of the 

relative-entropy function can, in certain problems, provide a quantitative and analytical technique. 

In thermodynamics, the entropy of a system is a measure of the free-energy variation, and this concept has 

been applied in a general way to open and closed geomorphic systems by Chorley (1962). An alternative consideration of entropy in terms of the summation of 

the logarithms of probabilities of the states of a system 

has been used by Leopold and Langbein (1962). This 

usage, dervied from statistical mechanics, is very close 

to the significance of entropy in the relative-entropy 

function, but a better understanding can be derived 

from information theory. 

As shown by Raisbeck (1963), if there is some information source, such as a transmitter or an experiment, which can yield n equally likely outcomes or 

messages, then the information associated with any 

given message, n, is log n. The reason for this can be 

seen if one considers a simple model (fig. 3) in which 

an information source, A, consists of two independent 

sources, Band 0. If the possible outcomes or messages 

from B and 0 are nB and no, and if these are equally 

likely, then the total number of possible outcomes, n.t, 

is equal to the product of nB and n 0 • The information 

transmitted by source A, however, must equal that 

transmitted by the sum of sources Band 0. Therefore, 

f ( nA) = f ( nB) + f (no) , and 

nA=nBno. 

One solution, then, is 

where dis some constant. 

(1) 

(2) 

(3) 

With this fact in mind, if one next considers a model 

in which the outcomes are not equally likely, the relationship of entropy to information content becomes 

clear. If an information source (fig. 4) generates n 

equally likely outcomes which can be segregated into 

two unequal groups, nA and nB, then if we ask whether 

·a given output or message will be in group nA or group 

Source A 

Source B 

Source C 

FIGURE 3.-Diagrammatic model of information source A that 

is composed of sources B and 0, which yield equally likely 

outcomes n 8 and n.c, respectively. (After Raisbeck, 1963, 

fig. 1.3.) 
D12 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

Source 

n equally 

likely outcomes 

r----

f-----

f-----

-

Group nA 

Group n 8 

FIGURE 4.-Diagrammatic model of an info·rmation s(}urce that generates two groups of outcomes, nA. and nn~ which are n(}t equally likely 

(Mter Raisbeck, 1963, fig. 1.4.) · 

nB, the probabilities of unequal outcomes must be inv~ti¥ated. The probability that a given message is 

within group nA is 

(4) 

and the probability that it is within group nB is 

(5) 

These expressions are simply statements of the fundamental definition of probability-that the probability 

of the occurrence of any given event is equal to the 

number of outcomes that will produce that event 

divided by the total number of possible outcomes. 

~he information associated with any of the equally 

likel~ messages within group nA is log nA, and that 

assoma.te~ with ~he messages within group nB is log 

nB. !his InformatiOn, however, can only be transmitted 

during a part of the total time n, namely during time 

nA/n for a message in group nA, and time nB/n for a 

message in group nB. Therefore, if information is defined as H, then 

(6) 

Because n can be taken as unity, and because nA/n 

and nB/n are equal to the probabilities PA and pB, 

respectively, 

or, in the most general form, 

N 

H = ~-Pt log Pt· i=l 

(7) 

(8) 

This expression, relating the information content of 

a source or system to the sum of the products of the 

probabilities and the logarithms of the probabilities of 

the components of that system, is essentially identical 

to the statistical definition of entropy and to the relative 

~ntropy function. The latter, as defined by Pelto (1954), 

IS 

(9) 

where Hr is relative entropy, Hm is maximum entropy, 

and the constant 100 is simply inserted to obtain percentage results directly. For a system that consists of 

any finite number of components, one need only convert 

the values of each of these components to percentages 

and substitute these percentages for probabilities to 

obtain the Pi log Pi products in the numerator of the relative-entropy function. Maximum entropy, Hm, is a 

function of the number of components in the system and 

occurs when the probabilities Pt-, p2, p 3, • • ., Pn are 

equal. For a three-component system, for example, 

H m = -3 ( 0.33 log 0.33) = 1.0986. (10) 

The applicability of the relative-entropy function to 

three-component facies mapping has been illustrated 

by Forgotson ( 1960) and a more general discussion has 

also been provided by Miller and Kahn ( 1962). The latter applied the relative-entropy function to mapping of 

an eight-component foraminiferal fauna and successfully revealed certain aspects of the areal distribution 

of facies that were not apparent from more conventional 

maps. Additional illustrations of the uses of entropyfunction mapping are not abundant in the literature, 

and to the writer's knowledge no attempt has been made 

to apply the technique to geomorphic problems. Nevertheless, it represents a potentially useful mapping technique and was considered as such during the conduct of 

the study reported here. 

If one wished to consider such topographic characteristics as the lengths, widths, and heights of ranges as 

components within systems of ranges, it would be entirely possible to convert the absolute values of these 

variables to percentages, compute relative-entropy val-
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D13 

ues for individual ranges or topographic maps, and produce a relative-entropy map. It is noteworthy, however, 

that in the context of information theory, entropy refers 

to the degree of intermixing of the components of a 

given system. That is, maximum entropy is equivalent 

to minimum information, and this occurs either when 

the degree of intermixing of end members is greatest 

or when the ,probabilities associated with each of the 

oomponents are equal. Conversely, maximum informa.-

tion is gained when entropy is a minimum, which occurs 

when all of the components or end members save one 

are equal to zero or are absent. Therefore, the relativeentropy function is a most efficient quantitative mapping tool when treating a set of components or variables, each of which ranges from 0 to 100 percent at some 

point within the system, or within the region studied. 

The components need not, of course, reflect the entire 

range from 0 to 100 percent, but a fairly wide range of 

values must be encompassed. Figure 5 shows the entropy distribution for a three-component syste.m. High 

entropy values dominate the central part of this triangular representation of the relative-ent~py distribution. For maximum information and low entropy 

values, two of the three components must tend toward 

zero; otherwise a corner of the total field will not be 

approached. 

The topographic variables that are discussed in this 

report do not lend themselves to the most efficient usage 

because any combination of components tends to oecupy 

high or moderate values of the relative-entropy distribution field. Like isopleth mapping, relative-entropy mapping was also considered and aiscarded in search of a 

means to extract the maximum information from the 

data by quantitative analytical techniques. 

A 

FIGURE 5.-Distrrbuti()n of relative-entropy values for a 

three-comp()nent system. Minimum inf()rmation ·is assodated with maximum entr()py values, at center of the 

field where the three components are equally abundant. 

Maximum information is associated with minimum 

entropy values at each of the three vertices. 

326-328 0-68--3 

FOURIER ANALYSIS 

Fourier analysis is another potentially useful quantitative mapping technique which is most applicable to 

regional data containing oscillatory characteristics. 

This is most easily understood from consideration of the 

two-dimentional graph-that is, the graph of any 

function in an x, y coordinate system. 

First, it should be recalled that specific functions can 

be approximated by expansion of terms in a Taylor or 

Maclaurin series. For example, 

(11) 

(12) 

In general, if Y1=ao+a1x, Y2=a2x2, Ya=aax3

, and 

Yn=anxn, then the Taylor expansion of y(x) is 

This means that the graph of a function y(x) can be 

approximated by the graphs of y 1, which is a straight 

line, y 2, which is a quadratic parabola, y3, which is a 

cubic parabola, and other functions of successively 

higher order. The greater the number of terms that are 

considered the better the approximation to the original 

function, y(x). 

If the function is periodic, however, then it may best 

be approximated by a series of terms, each of which is 

also periodic. This was first shown by Fourier in 1807, 

who assumed that a function, y= f(x), could be approximated by an expansion into sine and cosine terms, 

namely 

1 

f(x) =2 ao+al cos x+a2 cos 2x+aa cos 3x . 

+b1 sin x+b2 sin 2x+ba sin 3x+ ... ,. (14) 

or that, 

1 CX> CX> 

j (x) = 2a0+ ~an cos nx+ ~ bn sin nx. {15) 

The terms in this expansion of f(x) therefore make 

n complete oscillations in the interval -7r to +1r and 

the problem involved is finding the values of ao and 

the coefficients a1, a2, as, ... , an, and b~, b2, ba, ... , 

bn. The problem is simplified by certain properties of 

the Fourier series which are termed "orthogonality 

conditions." These ares:1f'cos mx cos nx dx= s:1f'sin mx sin nx dx, which 

=0 (when m ~n), 

=1r (when m=n), and (16) 
D14 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

J:r cos mx sin nx dx=O, (m, n are integers). (17) Because (m =;t.n), J:r cos Ox cos nx dx=O, and because of 

The proofs of these orthogonality conditions are very 

simple and are based upon trigonometric identities. 

For example: 

when (m=;t.n), 

cos mx cos nx=l cos(m+n)x+! cos(m-n)x; (18) 

therefore, 

f

'lr 1f11" -r cos mx cos nx dx=2 -r cos ( m +n) x dx 

1f'lr + 2 -rcos (m-n)x dx (19) 

1 ]11" 1 ]11" 

2(m+n) sin (m+n)x -r + 2(m-n) sin (m-n)x -r 

(20) 

1 1 

2(m+n) (0)+2(m-n) (O)=O. <

21) 

When (m=n), 

cos mx cos nx=cos2 mx=!(l+cos 2mx); (22) 

therefore, 

f

11' 1f11" -r cos mx cos nx dx=2 -r (1 +cos 2mx)dx, which 

(23) 

(24) 

1 ]11" 1 ]11" =2 x + 4

- sin 2mx ' and -r m -11' (25) 

(26) 

The third orthogonality condition ( eq 17) can be 

proven by similar methods, and the use of these relations permits determination of a0, the remaining coefficients, and the general simplification of the Fourier series. 

To determine a0, the original expansion of f(x) is multiplied by cos Ox, which = 1, namely 

(15) 

J:11'j(x)dx)=~aoJ:r cos Oxdx+ t.anf_r'll" cos Ox cosnx dx 

+ t. bnf:r cos Ox sin nxdx. (27) 

condition (17), f-11'11' cos Ox sin nx dx=O, therefore, 

f

11' 1 fr 1 f11' -11'j(x)dx)=2ao -r cos Ox dx=2ao -11' dx, and 

1f11" ao=; -r j(x)dx. 

By similar means it can be shown that 

1f11' an=; -11' j(x) cos nx dx, and 

bn=- j(x) sin nx dx, 1f11' 

7r -r 

and that for even functions of f(x) 

bn=O, and 

an=-2i'll" f(x) cos nx dx, 7r 0 

whereas for odd functions of j(x) 

an=O, and 

bn=-2ir j (x) sin nx dx. 7r 0 

(28) 

(29) 

(30) 

(31) 

(32) 

(33) 

(34) 

(35) 

The foregoing background material is sufficient to 

permit an illustration of the application of Fourier 

analysis to the two-dimensional graph. If one chooses 

to approximate one of the least likely mathematical 

functions by a summation of periodic terms, namely 

y=w, which is the graph of a straight line, the procedure is as follows: 

First, because y= x is an odd function, relations 34 

and 35 hold, namely 

bn=-2i11' f(x) sin nx dx. 7r 0 

Substituting for j(x), 

2ir · bn=- x sin nx dx, and 7r 0 

=-2 x(-d cos nx), i11' 

1rn o 

the solution of which is 

2 

--cos n1r 

n ' 

(34) 

(35) 

(36) 

(37) 

(38) 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D15 

and because cos n1r= + 1 when n is even and = -1 

when n is odd, 

(39) 

Therefore, for ( -11"<x<11"), 

"" sin nx f(x)=X=2 ~ ( -1)n-l __ , n=l n 

(40) 

and the Fourier expansion of y= x for n= 1 through 

n=4 is, respectively, 

f(x)=2 sin x, (41) 

( . sin 2x) f(x)=2 sm x-- 2

- ' (42) 

f( ) 2( . sin 2x+sin 3x) d X= Sill X--- -- 'an 2 3 (43) 

j(x)=2(sin x-sin 2x +sin 3x _sin 4x)· 

2 3 4 (44) 

The graphic approximation to y = x that is achieved by 

this Fourier expansion is shown in figure 6. It is obvious 

that the approximation improves as successively higher 

order terms are considered and, also, that if the function 

treated was by nature periodic, instead of the straight 

line y=w, that a good fit certainly oould be obtained. In 

y 

1r --------

1ry --------

3 terms 

FIGURE 6.-Jl'ourier expansion of the function, y=w. Note the improvement of approximation to this straight-line function by the successive 

addition of terms. (After Salvadori and Miller, 1953, fig. 6.2.) 

many practical applications of Fourier analysis there 

are two notable departures from the ideal case treated 

above. First, the functional interval involved is not the 

Fourier interval of +1r to -1r, or the half interval 

0 to 1r. This is readily overcome by modification. of the 

basic relationships to 

1 N n1r N .nr 

j(x) =2 ao+ ~an cos L x+ ~ bn SID L x, (45) 

1 f2L n1r 

an=L Jo j(x) cosT x dx, and (46) 

1 f2L . n1r 

bn=L Jo j(x) sm L X dx, (47) 

where L simply represents the fact that some finite 

interval ( + L to - L) is involved and that the function 

has a period equal to 2L. The second difficulty is that the 

form of the function f ( w) may not be known thus pre· 

eluding the possibility of analytical solution of the 

required integra;Is in order to obtain the coefficients. A 

large number of numerical integration schemes are 

available, however. Salvadori and Miller (1953} described the Runge and selected -ordinate method and 

Harbaugh and Preston ( 1965) discussed the problem 

and cite several appropriate papers and texts that present solutions, including the use of optical and mechanical devises. Basically, what is involved in any of the 

numerical integration schemes is the use of data points 

from the graph of a function in the ·absence of knowledge of the function proper. That is, for each point 

used, 

1 N n1r N • n1r 

Yt=- a0+ ~a,_ cos -L Xt + ~ bn sin -L Xi' ( 48) 2 n=l n=l 

where w,, y;, represent the coordinates of any point on 

the curve. 

With respect to quantitative mapping by Fourier 

analysis, one simply extends the two-dimensional treatment above to the third dimension. That is, instead of 

treating the oscillatory character of a plotted curve in 

an w,y coordinate system, the problem involved is fitting 

an oscillatory surface to data on some parameter, z, 

which is a function of two variables w and y. In the 

Basin and Range region under consideration, for example, any of the 11 parameters previously ~iscus~d 

represent a series of z values in the third dimensiOn 

which are functions of geographic location or an w,y 

coordinate system on the ground. If the z values of these 

parameters oscillate with some fundamental periodicity 

in mutually perpendicular direotions, such as northsouth and east-west, a double Fourier analysis will serve 

to reveal the harmonics involved, and a map of the 

surface of best fit can be obtained. Assuming that the 
D16 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

fundamental periods involved are 2L in the x-coordinate direction and 2H in they-coordinate direction, the 

double Fourier series to be used to approximate the 

surface of best fit to a given set of z values is 

m=M n=N [ 1rm n z~j(x, y) = ~ ~ >.m, n am,n cos L x cos~ y 

+b . 1rm 1rn 1rm . 1rn m.n Sill LX COSH y+cm,n COS LX Sill H y 

+d . 1rm . 1rn J m,nSill LX Sill H y ' 

where 

'Am,n=%, for m=n=O, l 'Am,n=~, for m=O, n>O; or m>O, n=O, and 

'Am,n=l, for m>O, n>O. 

(49) 

(50) 

And the coefficients of the sine and cosine terms in the 

double Fourier expansion can be determined from the 

following double integrals: 

1 JH JL m7r n7r am,n=LH -H -Lj(x,y) cos LX cosH y dxdy, (51) 

1 JHJL . m7r n7r bm,n= LH -H -Lj(x,y) Sill LX COSH y dxdy, (52) 

1 fHfL m7r • 'f1.7r Cm,n=LH -H -Lj(x,y) cosyx Sill Hydxdy, and (53) 

1 Jn JL . m7r . n'lr d m, n= LH -H -L j(x,y) Sill LX Sill H y dxdy. (54) 

The analogy of the double Fourier expansion series 

and the expressions that define the coefficients, with the 

two-dimensional or single series and its coefficient equations, is readily apparent. As before, the integral equa-

+15 

t 

y 0 

~ 

tions that define the coefficients cannot be determined 

analytically unless the function I ( x,y) is known. In a 

practical problem, such as this Basin and Range study, 

one virtually has at hand sets of z values at data points 

in an x,y coordinate system, and the form of the surfa.ce, 

or the I ( x,y) function, is not known. For this reason 

one must again resort to numerical integration or to 

other available schemes. Harbaugh and Preston (1965) 

and Preston and Harbaugh ( 1965) presented numerical 

summation equations that can be used for the requisite 

coefficient determination, for exa,mple. They also presented some examples of the application of double 

Fourier analysis. 

The applications involve the areal variation of calcium carbonate in a magnesite deposit and an analysis 

of geological structure in Kansas. The paper by Harbaugh and Preston (1965), however, also contains a 

discussion of other possible applications, among which 

was the analysis of harmonics of topographic surfaces. · 

To the writer's knowledge, the sole application of 

Fourier analysis to landform topography was that presented in a military-contract report on microrelief in 

Thailand. Harbaugh (written commun., 1967) cited a 

second example which consists of the application of 

single Fourier series, radiating from a common point, 

to surface topography. The suggestion that Fourier 

analysis is a ,potentially useful quantitative mapping 

technique in geomorphic investigations is therefore 

worthy of reemphasis here because it must be classified 

as seldom used for this purvose 

Figure 7, a reproduction of an illustration that appears in both papers by Ha~baugh and Preston cited 

-15 -~. ~--------~--------~----~----~--~----._--------~------~--------~----~ 

-12 0 +12 

FIGURE 7.-Synthetic double Fourier series representing a wave form containing four harmonics in both coorddnate directious. (After Preston and 

Harbaugh, 1965, fig. 2.) 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D17 

above, is a map resulting from the synthetic generation 

of a double Fourier series containing terms as high as 

order m=n=4. In other words, the map represents a 

wave form containing four harmonics in both x- and 

y-coordinate directions. This can be considered as an 

ideal representation, and the symmetrical character of 

the doubly oscillating surface is apparent. The distortion present is the result of unequal grid spacing in both 

directions. The trend -surface maps of Basin and Range 

topography that are presented later in this report (figs. 

9-41) clearly show that regional high and low values 

of the 11 chosen parameters exist within the area and 

that any surface of best fit to the z values probably 

has some underlying oscillatory or harmonic components in both the north -south and the east-west directions. A double Fourier analysis of these topographic 

variations would by no means yield a map as simple as' 

that shown in figure 7, but the writer firmly believes that 

. the region would be amenable to analysis by this method. 

Two basic problems prevented a double Fourier 

analysis of the data on the Basin and Range region. 

First, there is no standard method for orienting the 

x, y coordinate system with respect to regional topography. Although it was stated above that generally regional highs and lows of topographic parameters are 

present in both the north-south and the east-west directions, the first-degree trend -surface maps to be later 

discussed clearly show that the regional "grain" is 

oriented northwestward or northeastward, rather than 

due north in most places, and that similar departures 

from due east or west exist. This suggests that in any 

future studies of regional topography by double 

Fourier analysis it might be ·most profitable to obtain 

first-degree trend-surface maps at the outset to orient 

a coordinate system for Fourier analysis perpendicular 

and parallel to the regional gradients. Such orientation 

would at least prevent the arbitrary selection of possibly 

unfortunate coordinate directions. 

The second problem cannot be overcome so readily, 

given the data contained in this report, however. Because the Basin and Range region is irregular in plan 

(fig. 1), there is a corresponding irregularity in the 

numbers of topographic maps of the region, the irregul~arity varying according to the direction of traverse. 

As shown in figure 2, for example, at the narrow "waist" 

of the region only two maps were used for this study, 

namely Death Valley •and Las Vegas, whereas in a 

north-south direction a maximum of 11 maps exist 

along the Pocatello-Sonoyta meridian. Because each 

map represents a single data point, •as previously discussed, the irregularity of this available grid would 

be too great to yield valid regional results, particularly 

in an east-west direction. 

A double Fourier analysis is merely a combination 

of single Fourier ·analyses in two mutually perpendicular directions. This is shown by equation 49, for example. Although there are eight times the number of 

sine and cosine terms in a double Fourier series, the 

results of •a double analysis are nevertheless dependent 

upon the validity of the two components. This does not 

necessarily mean that one must deal with a square data 

array, but if less than some minimum number of data 

points are available in one of the coordinate directions, the results may well be meaningless. Considering 

again the fact that a single Fourier series expansion is 

used to approximate some function y= f(x) in two dimensions, it is obvious that some oscillatory characteristics can always emerge if the basis for the graph of 

the function is only two or three points. Some number 

of harmonics may well underlie the regional variation 

of topographic parameters in the Basin and Range 

a.rea, but valid appraisal of this possibility requires 

the availability of a greater number of data points in 

an east-west direction than is afforded by use of the 

1:250,000 topographic quadrangle maps. Quantitative 

mapping of Basin .and Range topography by means of 

double Fourier analysis is therefore strongly recommended as a future project, with the provision that the 

data analyzed be plotted ·at the locations of range 

centroids. The use of maps of larger seale than 1 : 250,000 

would not eliminate the problem raised here concerning numbers of data points because the plotting of mean 

z values at the eenters of larger scale maps would inerease location errors intolerably. If range centroids 

are used for z-value locations, the resulting irregular 

data ·array can he analyzed by means of the Fortran IV 

computer program for double Fourier alllalysis recently 

provided by James ( 1966) . 

TREND-SURF ACE ANALYSIS 

Trend-surface analysis can be considered to embrace 

many forms of least-squares analysis and, in this sense, 

the fitting of double Fourier series to a set of data is 

actually one form of trend-surface analysis. In common 

usage, however, trend-surface analysis implies leastsquares fitting of polynomial surfaces, and Fourier 

analysis and trend-surface analysis are thus distinguished in this report. 

Like the application of single and double Fourier 

analysis previously described, trend -surface analysis 

is a potentially useful quantitative analytical technique 

that may be most readily understood by consideration 

of a two-dimensional analogy. If one is confronted 

with a set of data that approximates a linear distribution on arithmetic-graph paper, three choices of presentation are open. The investigator can ( 1) simply plot 

the data as a scatter di~agram, ( 2) construct ·a straight 
Dl8 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPIDC SCIENCES 

line through the data points by visual means, or (3) 

construct the line of best fit through the data points by 

the statistical procedure of the least-squares method. 

Choices (2) 'and ( 3) .above 'both yield simUar results; a linear equation of the form 

y=mm+b (55) 

is obtained, where m is the slope of the fitted line, and 

b is the intercept on the y axis. There is, however, an 

extremely important difference between construction 

of a fitted straight line by visual means and that by the 

least-squares method. The least-squares method, as is 

well known, permits the construction of a unique 

straight line, ~about which the sum of the squares of the 

departures of the ah,yi observed data values .and the 

corresponding computed VJalues for that line is a minimum. Because this method can yield only one line of 

best fit for a given set of data, it is objective. Construction of such .a line by visual means must, in contrast, 

be subjective because different investigators may well 

produce lines of different slopes .and intercept values. 

The analogy to the three-dimensional case, where one 

attempts to map the distribution of z values which are 

a function of w,y coordinates, should be obvious. One is 

attempting, in fact, to map a surface of best fit to the 

z values in three-dimensional space. This can be accomplished by simple manual contouring, but this, in 

turn, is analogolus to the visual contruction of straight 

lines in the two-dimensional case. As noted above, and 

as previously mentioned with respect to manual contouring in general (p. DlO), this method is subjec:tive 

and is not necessarily reproducible. Alternatively, one 

can produce a three-dimensional surface of best fit to a 

given set of z values by mathematical techniques which 

will yield a unique surface that is therefore both reproducible and objective. One such technique is the method 

of trend -surface analysis, by means of which the Basin 

and Range data treated in this report were mapped. 

If we return to two-dimensional examples for a moment, a straight line represents an equation of the first 

degree which, for reasons of consistency that will 

become apparent, can be written in the form 

Y=A+Bx. (56) 

In this equation, the coefficient A is the zeroth-degree 

term, and Bx is the first-degree term. An equation of 

the second degree in an x,y coordinate system is a 

parabola whose equation is of the form 

Y=A+Bx+Cr. (57) 

It can be noted that equation 57 differs from an equation 

of the first degree (equation 56) by reason of the addition 

of a quadratic term, namely Or. Similarly, an equation 

of the third degree in an x,y coorinate system is a 

sinusoidal curve whose equation is of the form 

Y=A+Bx+Cx2+Dr. (58) 

This equation differs from that of a quadratic parabola 

(eq 57) by addition of the cubic term Dr. Equations of 

any given higher degree can similarly be written, 

requiring only the addition of an appropriate number 

of terms of successively increasing order. Because this 

report makes use of trend surfaces that are not of a 

degree higher than the third, however, these twodimensional or two-variable expressions (eq 56-58) 

will suffice for illustrative purposes. 

To understand trend surfaces one must consider the 

three-dimensional surfaces that correspond to first-, 

second-, and third-degree equations in an x,y,z coordinate system. A first-degree surface is a plane whose 

equation is linear and is of the form 

Z=A+Bx+Cy. (59) 

A second-degree surface is a paraboloid, and (like the 

two-dimensional equivalent (eq 57)) its equation contains the zeroth- and first-degree terms of a first-degree 

or linear surface (eq 59) and the addition of appropriate 

quadratic terms, namelyA third-degree surface is oscillatory or doubly sinusoidal 

in space, and its equation consists of the linear and 

quadratic terms contained above (eq 60) and four 

additional cubic terms, namelyZ=A + Bm+Oy+Dw2+Ewy+Fy2 

+ Gw + H w2y+ I W!f2 + J '!/. (61} 

As in the two-variable case, one can write an equation 

representing a three-dimensional surface of any given 

degree by merely adding appropriate terms of higher 

degree. The geometric relations of first-, second-, and 

third-degree equations for two and three variables are 

shown in figure 8. 

To find the equation of the surface of best fit of any 

given degree, the coefficients of the appropriate polynomial expression must be determined and, like the two 

VJariable case, the departures of the observed z values 

from the corresponding computed z values on the fitted 

surface must he minimized by a least-squares procedure. 

One can obtain the required coefficients by the manual 

solution of simple matrices, .as described by Krunrbein 

and Graybill (1965}, or by relaxation or other successive-approxim·ation procedures. In fact, an actual trend-
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D19 

1st Degree 2d Degree 3d Degree 

2 variables y / y (\ y f 

X X X 

Y=A+Bx Y=A+Bx+Cx 2 Y =A+ Bx + Cx 2 + Dx3 

y y 

Z=A+Bx+Cy Z =A+ Bx+ Cy + Dx2 + Exy + Fy 2 Z=A + Bx+ Cy+ Dx2 + Exy+ Fy 2 

+Gx3 +Hx2 y+lxy2 +Jy3 

FIGURE 8.-Geometric relations of first-, second-, and third-degree equations for two and three variables. 

surface map can be produced by taking successive moving averages of sets of mutually perpendicular profiles 

across the data arra.y (Krumbein, 1956). This method 

is simple, but it involves some subjectivity in the 

smoothing of the profiles and does not provide the equation of 'a given surface. Moreover, it is time consuming. 

The most rapid method of generating the surfaces of 

best fit and their corresponding equations is to use computers. O'Leary, Lippert, .and Spitz (1966) recently 

provided a program in Fortran IV which will compute 

and plot trend surfaces of first through sixth degree. 

In a.ddition to the contoured trend surfa.ces, the output 

includes the equations of each surface of best fit, a tabular summary of z values for each surface of a given 

degree, and the corresponding departures, or residuals, 

from the z observations, the ooefficient matrix, plots of 

original data, plots of residuals for each surface of a 

given degree, and several useful sta.tistical-error 

measures. 

The latter include the total variance V, which is the 

sum of the squares of the departures of the z values, zi, 

from the mean observed z value, namely 

(62) 

The zi represents original data, and this statistic is 

identical with that involved in calculating the deviation 

of x values about their mean, x, in a two-variable data 

system. The total variance V is therefore a constant 

value for a set of data and is not related to the degree 

of a given surface that may be fitted to the data. 

The variance that is not explained by the surface is, 

therefore, defined as 

(63) 

where zi<o> refers to an observed z value, or the original 

data, and zi<c> refers to a corresponding computed z 
D20 THEORETICAL PAPER,S IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

value _on a surface of best fit of a given degree. In a 

two-variable system the analogy to this statistic is the 

deviation or scatter of data values about the leastsquares regression line of best fit. Some points will 

occur above the line, and some below, and because the 

arithmetic sum of these departures will equal zero, the 

departures must be squared. The reasoning is precisely 

the same with respect to z values. Some will occur above 

the surface of best fit, and some will occur below. 

Squaring the departures and taking their sum provides 

a measure of the scatter about a surface of best fit of a 

given degree and, therefore, provides information on 

the unexplained variance. 

The variance that is explained by a given surface, E, 

is simply the difference between the total variance and 

the unexplained variance. Therefore, 

E=V-U. (64) 

If the variance that is explained by a given surface 

is divided by the total variance, it provides the variance 

that is explained as a percentage. This is defined as 

the coefficient of determination, namely, 

(65) 

The coefficient of correlation is the square root of 

the coefficient of determination, or 

R=DtJ2. (66) 

Finally, a measure of the standard deviation of the 

data for each surface of a given degree is also generated 

by this computer program. The statistic, or error 

measure, is again analogous to the corresponding 

statistic for a set of data in a two-variable system. 

That is, 

(67) 

The program discussed above (O'Leary and others, 

1966) was modified and rewritten in Fortran II because 

Fortran II machine language was required by the computer system available at the time the Basin and Range 

study was made. The computer output corresponds 

exactly to that described above, except that trend surfaces and associated data of degrees higher than third 

were not generated. In a subsequent section of this report, the trend-surface maps and certain error measures 

for first-, second-, and third-degree surfaces of best fit 

are presented for each of the 11 variables used in this 

study. Before considering these results, however, a few 

additional remarks should be made about trend -surface 

mapping and the previous work done in this field. 

With respect to previous work, in recent years Merriam and Lippert (1964, 1966), Merriam and Harbaugh 

(1964), and Merriam and Sneath (1966) have applied 

trend -surface analysis to geological structure problems 

in l{ansas. Harbaugh (1964a, b) has treated oil-field 

and facies data in both three- and four-component systems; computer programs other than that discussed 

above have been written by Harbaugh (1963) and by 

Sampson and Davis (1966a, b); and Spitz (1966), 

among others, has treated the problem of generating 

orthogonal polynomials for an irregularly spaced array 

of original data points. Read and Merriam (1966) investigated the thickness of stratigraphic units in Scotland by trend -surface analysis, and a classic petrological and mineralogical study of the Donegal granite by 

trend-surface methods was reported by Whitten in 1959. 

Since that time, Whitten ( 1961, 1962) has obtained similar results which delineated previously obscure regional 

trends in other European granite bodies. The general 

application of trend -surface analysis to sedimentary 

environment studies was discussed by Miller in 1956. 

All the reports cited here contain additional references 

to previous studies that involved trend-surface analysis, 

and many pertinent papers are also cited in the texts by 

Miller and Kahn (1962) and Krumbein and Graybill 

(1965), each of which provide a good general discussion 

of the method. The total body of literature on the subject 

suggests that applications have been directed primarily 

toward problems of stratigraphy and sedimentation, 

structure and petrology, and geochemistry and ore 

deposits. Surficial deposits, such as soils and sands, have 

been treated (Chorley, 1964), but regional topographic 

variations and general applications to goemorphology 

have not been studied previously by trend-surface 

analysis. 

Certain of the work on surficial sands and soils well 

illustrates a fundamental point concerning trend analyses in general, however, and this should be emphasized 

here. Chorley, Stoddart, Haggett, and Slaymaker (1966) 

wished to investigate the origins of surficial sands in 

an area of 1,000 square kilometers in eastern England. 

Accordingly, they undertook a trend-surface analysis 

of sand size, where z values were sand size, in mm, and 

m,y coordinates represented the geographic locations of 

sampling stations. This analysis revealed a regional 

trend, consisting of highs to the northeast and a dominant low to the southwest, but the third-degree trend 

surface of best fit explained only 21.5 percent of the total 

variance. The authors therefore designed a hierarchical, 

or nested, sampling technique on six levels which utilized 

the knowledge of regional trends, and by an analysis 

of variance they examined the parts of the unexplained 

total variance that could be attributed to local variance, 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D21 

corresponding to each of these six levels of sampling. 

This technique enabled them to distinguish several possible environmental factors that, in combination, could 

account for the overall variability of sand size. 

The fundamental point is that in any study of natural 

phenomena which are not constant in a given region, the 

total variance is commonly a function of variability at 

several levels. Trend -surface analysis is a means of distinguishing regional variation, or the part of the total 

variance that is attributable to the highest level of 

variability. In a study of range heights in the Basin and 

Range region, for example, one would not expect to find 

that all ranges at a given latitude are of precisely the 

same height nor that a uniform decrease in height occurs 

with latitudinal variation. If this were true, every z 

value for range height would fall precisely on a plane 

of best fit, or first-degree surface, which had a uniform 

dip. As previously stated, one seldom, if ever, obtains a 

perfect fit to either least-squares regression lines or regional trend surfaces. The departures of the observed 

values from the computed values on the line or surface 

represent the unexplained variance V ( 63) . In trend surfaces, this unexplained variance is attributable to more 

localized causes of variability. If one is interested primarily in the question of whether regional variation 

exists and is persistent, then trend -surface mapping is a 

quantitative analytical technique that can provide the 

answer. If one wishes to examine all possible causes of 

the total variance, then one must engage in a secondary 

analysis of the unexplained variance revealed by regional trend analysis. In the study reported here, the 

primary goal was to determine whether the magnitudes 

of regional variation were compatible with the hypothesis that significant topographic differences exist 

within the Basin and Range region. Trend-surface 

analysis showed that this was true, and there was no 

reason to examine the various sublevels which would 

account for the residual unexplained variance for each 

topographic parameter. Returning to range heights, for 

example, within any given subregion that exhibits predominantly high or low values of range height, some 

ranges will obviously depart from these general subregional values. Thi~ constitutes a part of the total 

variance of range heights that may be ascribed to a lower 

level of variability, namely the individua] range. 

Troeh (1965) has argued that the complexities of a 

series of hills and valleys are so great that it is impossible to describe them by a three-dimensional equation. 

This should not be construed as ·a denigration of trend 

surface techniques, however, because again the ultimate 

goals must be kept in mind. If the goal is to describe 

some specific landform, such as the alluvial fans de326-328 0-68--4 

scribed by Troeh ( 1965) , the drumlins by Chorley 

( 1959), or the headland -bay beaches by Yasso ( 1965), 

then of course one attempts to fit some specific descriptive function to the landform in question. If, however, 

one is concerned with regional variation among groups 

of landforms, one of the most appropriate means of· 

separating the relatively large scale systematic variations from more localized and smaller scale variations 

is indeed the method of trend-surface analysis. 

With this background, the results of the trend-surface 

analysis of each of the 11 topographic parameters within the Basin and Range region can now be considered. 

TREND-SURF ACE RESULTS 

The trend-surface results presented here consist of 

contour maps of the first-, second-, and third-degree 

surfaces of best fit for each of the 11 topographic 

parameters used in this study. The statistics associated 

with these results are presented in the section entitled 

"Summary of Trend-Surface Results" (p. D51); however, a few general points previously covered are restated here for convenience. 

First, the criteria employed for measurement of any 

given variable are ·arbi,tra.ry and a choice of different 

criteria would result in different absolute z-values than 

those determined for this report. Second, of the four 

sources of error mentioned previously (p. DS), the most 

important is that which results from plotting procedures. Because mean z-values of a given parameter were 

plotted at the centers of each of the 1 : 250,000 topographic quadrangle maps used (fig. 2), instead of plotting the population of z-values at the centroids of each 

of the ranges that make up this population, some dislocation of high and low regional values occurs. The dislocation has no effect on the area of ranges to total area 

ratio and the range trend parameters ; nonetheless, the 

average plotting error is thought to be less than one-half 

of the diagonal distance on any given 1 : 250,000 topographic quadrangle map. Third, the previous discussion 

of trend-surface analysis (p. D17) indicates that the 

maps of the first-, second-, and third-degree surface of 

best fit for each of the topographic parameters represent 

planes, paraboloids, and oscillatory surfaces of best fit, 

respectively, to the z-values that were obtained. (See 

eq 59-61; fig. 8.) Finally, this previous discussion emphasized that the trend-surface results generally 

reflect the part of the total variance that is attributable 

to regional causes. For each of the topographic parameters discussed here, there is some fraction of the total 

variance that is attribut·able to more localized causes, 

and no effort has been made to detennine the latter. 

Regional variation, alone, is of concern here. 
D22 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPIDC SCIENCES 

AREA OF RANGES TO TOTAL AREA 

Area. of. ranges to total area is, perhaps, one of the 

most significant of the topographic parameters used. 

Fenneman, Lobeck, and Eardley, each of whom is 

quoted on page D2 of this report, clearly invoked this 

parameter in discussing regional differences within 

the Basin and Range area (fig. 1). Moreover, Fenneman's statement includes the estimate that the ratio of 

area of ranges to total area is 50 percent in the northern 

part of the region and 20 percent in the southern part 

of the reg~on. Alth~ugh the absolute values of any parameter will vary with the measurement criteria used it . ' Is n~teworthy that values of 50 percent or more were 

obta~ned for only two topographic maps, namely 

Mar1 posa and Death Valley in the north and that values 

of this parameter in the southern part of the region are 

less than 10 percent in the area covered by the Santa 

Ana-Salton Sea-Ajo-Sonoyta maps (fig. 2). 

The trend-surface maps reflect, of course, the absolute 

values used in their preparation. Figure 9 is the firstdegree trend-surface map of the area of ranges to total 

a~. The values of this parameter on the plane of best 

fit Increase to t~e northwest. The regional values range 

from 3 percent In the south to 39 percent in the extreme 

~ort~west corner of the map; thus, the regional variation IS 13-fold for this parameter. 

The second-d~gree trend-surface map (fig. 10) shows 

th~t ~ paraboloid of best fit to the regional data has 

a Similar northwest-southeast orientation. This seconddegr~e surface may be likened to the part of a tablespoon 

that Is concave upward, with its bowl in the northwest 

and tip in _the southeast. The margins of the paraboloid 

(or the rim of the hypothetical tablespoon) extend 

through the Mohave Desert area to the southwest and 

the Salt Lake desert area to the northeast, where comparable map values occur. In other words, the northeastern part of the Basin and Range region exhibits 

greater similarity to the southern part of this region 

than to the northern part, on the basis of the variation 

of this topographic parameter. 

The third -degree trend -surface rna p of the area of 

r~nges to total area ratio (fig. 11) reinforces this suggestion to some extent. Edge effects are troublesome along 

the east margins of all the maps presented because of a 

~ack of data points (fig. 2) ; nevertheless, a large region 

In the no~theast can be seen to exhibit values comparable 

to those In the south, and a regional high occurs in the 

northwest. Trend-surface maps of several parameters 

confirm. these regional distinctions. The regional high 

shown Ill figure 11 delineates an area of central and 

western Nevada and eastern California within the 

Basin and Range province, which is distin~tly different 

from the rest of the region. The precise location of the 

contour closure is not always the same on the maps of 

the several variables presented but it is invariably in 

the northwestern part of the region; the total migration 

distance of the closure on the several maps is not great. 

RANGE LENGTH 

The first-degree trend-surface map of range length is 

shown in figure 12. The orientation of the plane of best 

fit is seen to be more nearly north-south than was true 

for the area of ranges to total area parameter, and the 

average lengths of ranges vary from 8 miles in the south 

to more than 30 miles in the north. Figure 13 shows the 

second -degree surface or paraboloid of best fit. The axis 

of this surface is oriented northwest-southeast to some 

extent. As before (fig. 10), comparable values of this 

topographic parameter occur in the northeastern and 

southern parts of the region; the dominant high is to the 

north west. The third-degree trend-surface map of range 

length (fig. 14) clearly shows the location of this regional high, the innermost contour closure of which ris 

north of that shown in figure 11 for the area of ranges to 

total area parameter, but which reflects the same 

regional distinction. The northern and southern parts of 

the region differ markedly with respect to range length, 

and, if edge effects are disregarded, the northeastern 

part of the region more closely resembles the southern 

than the northwestern part of the region. 

RANGE WIDTH 

The.regional characteristics of range width appear to 

be similar on the basis of the first-degree trend-surface 

map (fig. 15). Again, a plane of best fit trends slightly 

northwest, and average values increase in this direction; 

in this example the increase is from 2.2 miles in the 

southeast to 5.2 miles in the northwest, or more than 

100 percent. The second-degree (fig. 16) and thirddegree (fig. 17) trend-surface maps of range width, 

however, reveal ·that this parameter inc~ases in value 

both to the northeast and to the northwest. Of these two 

trends, the northeastward increase in range width is 

less real than i·t appears because there is far more variability in the data from this area. That is, the majority 

of the ranges to the north west exhibit large values of 

range width, whereas a lesser percentage of exceptionally wide ranges in the northeast have affected the 

mean values in a similar manner because, in general, 

the ranges are less abundant in that area. It can again 

be said, however, that the northern and southern parts 

of the Basin and Range region are disparate, with 

respect to this topographic parameter. 

RANGE HEIGHT 

As indicated in the discussion of the criteria used 

for the measurement of range height (p. D7), the range 
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D32 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPIDC SCIENCES 

height parameter is difficult to measure in an entirely 

satisfactory manner. The trend -surface results for range 

height are therefore subject to some question, but the 

maps are reasonably consistent with those of range width 

(figs. 15-17). Figure 18 shows the first-degree trendsurface map of range height, and the plane of best fit is 

oriented to the northeast; values on this surface range 

from 1,200 feet in the southwest to more than 2,100 feet 

in the northeast. The second-degree trend-surface map 

(fig. 19) shows a paraboloid of best fit whose axis is oriented approximately north-south, and the third-degree 

trend-surface map (fig. 20) deviates only slightly from 

this general configuration. Range height, like range 

width (fig.17), increases both to the northeast and to the 

northwest. As stated in connection with range width, 

however, greater uniformity, or persistence of values, 

prevails in the northwest, and the large area occupied 

by the 1,900-foot and 2,000-foot contours attests to 

this fact. The variation of range height on this thirddegree surface of best fit (fig. 20) is approximately 100 

percent between the northern and southern parts of the 

Basin and Range region. 

RANGE RELIEF 

In measuring range relief an essential distinction is 

that whereas range height involves the difference in elevation between the crest and base areas of a given range, 

the range relief is measured from the lowest points 

in the basins adja·cent to a given range (p. DS). For this 

reason, the absolute values obtained for range relief 

in this study, and the average values on the trend surfaces of best fit to the regional data, better accord with 

most workers' concepts of the "size" of the ranges than 

do the previously discussed val Uffi of range height. The 

first-degree trend-surface map of range relief (fig. 21) 

shows that, like range height, this parameter increases 

from southwest to northeast and that on the plane of 

best fit the magnitude of this increase is approximately 

100 percent. The second-degree trend-surface map of 

range relief (fig. 22) reveals a regional high to the north 

and to the west and reveals far a more rapid decrease 

of values to the southwest than to the southeast. The 

third-degree trend-surface map (fig. 23) depicts a complex hyperbolic surface of best fit. In the southern part 

of the region, a prominent low is centered about AjoSonoyta (fig. 2), and values increase eastward into New 

Mexico, westward to the Mohave Desert, and continuously northward toward the dominant high with a 

contour-closure value of 4,000 feet in N eveda. This 

third-degree surface clearly shows a marked difference 

in range relief in the northern and southern parts of the 

Basin and Range region. Moreover, the decrease to the 

northeast, although partly induced by edge effects, tends 

to support the area of ranges to total area ratio and the 

range length parameter because the northeastern values 

of range relief are more closely comparable to values 

in the south than in the north. 

RANGE VOLUME 

Range volume was chosen as a descriptive topographic parameter for several reasons, but it is obvious 

that the regional variation in range volume should be 

ro~ghly compatible with the regional variation in the 

area of ranges to total area ratio. This expectation is 

sustained by the trend-surface maps of range volume. 

The first-degree surface of best fit is shown in figure 

24. The plane exhibits a regional trend to the northwest, 

and values on the surface range from less than 2.5 cubic 

miles in the southeast to more than 32.5 cubic miles in 

the northwest. This range of values, namely 13-14 times 

the minimum volumes obtained is closely comparable to 

the 13-fold increase in the area of ranges to total area 

ratio (fig. 9). The second-degree trend-surface map (fig. 

25) shows that the values on this surface of best fit to 

the range volume data follow a.n arcuate path from the 

high in the northwest, through the central part of the 

region, to the southwest, where they decrease to less than 

2.5 cubic miles in southern California and southwestern 

Arizona. The third-degree trend-surface map (fig. 26) 

reveals the persistent high in the north west that is accentuated by the 35-cubic-mile contour closure and by 

the marked trough to the south. This map is very similar 

to the third-degree trend-surface map of area of ranges 

to total area (fig. 11) in this regard. The chief difference 

occurs in the northeast; range volume attains maximum 

values in the northeast corner of the area shown in figure 

26. This is partly due to edge effects, but it also reflects 

the fact that high width (fig. 17) and height (fig. 20) 

values exist in this area; both parameters are involved 

in the computation of range volume. 

The third-degree trend-surface map of range volume 

(fig. 26) shows that regional differences clearly exist 

within the Basin and Range area, with respect to this 

parameter, and that, in addition to a lesser mean range 

area in the south, there is a lesser mean range volume. 

The implications of this fact will be discussed later in 

this report. 

CUMULATIVE LENGTH OF TRENDS 

Cumulative length is the first of two parameters that 

were devised to represent escarpment trends in the Basin 

and Range region. Although somewhat variable because 

of the measurement criteria used ( p. DS), maximum 

values of comulative length of trends will tend to occur 

in areas where topographic escarpments are longest, 

whereas minimum values will tend to be associated with 

areas containing the shortest escarpments. Accordingly, 
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D42 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

the first-degree trend -surface map of this parameter 

(fig. 27) shows that longer escarpments and greater 

cumulative lengths occur in the western and northwestern parts of the region. The range of values on this plane 

of best fit, from less than 14 miles in the southeast to 

more than 77 miles in the northwest, or approximately 

six times the :rriinimum value, is greater than the range 

of values associated with range width, height, and relief, and less than that as80Qiated with range area and 

volume. · 

The second-degree trend-surface map oi cumulative 

length of trends (fig. 28) provides better definition of 

the regional variation of this parameter. The paraboloid 

of best fit can be seen to trend southeast-northwest and . . ' minimum values are attained to the northeast, southeast, and southwest. The third-degree trend-surface 

map (fig. 29) reinforces this impression and provides 

further definition of regional variation. The pattern 

is similar to that obtained for the area ratio (fig. 11), 

range length (fig. 14), range relief (fig. 23) and range 

volume (fig. 26) parameters. The pattern tends to support the idea that the area occupied by the 91-mile 

cumulative-length high in the northwest (fig. 29) is 

one containing longer, larger, and more massive ranges 

than generally occur elsewhere in the region. Thus, the 

trend-surface maps of cumulative length of trends also 

suggest that marked topographic differences exist within the Basin and Range region. 

CUMULATIVE DEVIATION OF TRENDS 

In addition to the lengths of trends of escarpments 

and the regional variation of cumulative length it is 

obviously useful to specify the orientation of' such 

trends. The parameter expressing the cumulative deviation of trends from due north serves to accomplish this 

purpose. Maximum values of this parameter in a given 

area reflect maximum deviation from the north-south 

direction within that area. Figure 30 shows the firstdegree trend-surface map of cumulative deviation of 

trends. The values on the plane of best fit increase from 

less than 12° in the northeast to 42° in the northwest. 

Because the qualitative assessment of range trends 

in terms of "An army of caterpillars marching to Mexico 

* * * would necessarily coincide with a cumulative 

deviation of 0° these quantitative results obviously render specious such a description. 

The second-degree trend-surface map of cumulative 

deviation of trends (fig. 31) provides further definition 

of the regional variations. Minimum deviation occurs 

in the north-central part of the region, and the values 

on the paraboloid of best fit increase to the south southeast, and southwest. Several additional refinements ' appear on the third-degree trend-surface map for this 

parameter (fig. 32). The high values to the southwest 

are misleading because they are produced, in part, by 

the inclusion in the average of one or two topographic 

maps in the southwest which were dominated by a single 

range trending nearly east-west. Similarly, the decrease in cumulative deviation to zero in the northeast 

is more apparent than real. But the 15° contour closure 

in the northwest is quite real, and the coincidence of 

location of this regional low and the several regional 

high values for parameters previously discussed will 

be apparent to the reader. Also, the general range of 

values from about 30° to 40° across the southern part 

of the Basin and Range region reflects the topographic 

conditions rather faithfully. This third-degree map indicates that ranges in the northern part of the Basin 

and Range region deviate from due norlh much less 

than do the ranges in.the southern part. In other words, 

consideration of the cumulative deviation of trends, like 

the previous parameters discussed, suggests that very 

substantial quantitative differences in topography exist 

in the Basin and Range region. 

RANGE WIDTH TO LENGTH 

Range width to length is the first of three derived 

ratios that were used to further characterize regional 

topography in the Basin and Range region. Any such 

ratio obviously serves to compound the sins of which 

its component variables may be guilty. That is, if either 

width or length values are suspect for some reason in 

any given local area, a ratio of these two variables 

will not obviate such difficulty. The width to length 

ratio, like the other two derived ratios yet to be discussed, should merely be considered as an alternative 

means of representing regional topographic variations. 

They are not independent of their individual components, but these ratios do reveal the manner in which 

a given variable changes in relation to a second variable throughout the region. 

Figure 33 shows the first-d-egree trend-surface map of 

range width to length. From previous know ledge of the 

general distribution of range width (fig. 15) and length 

(fig. 12) values, one would anticipate this outcome. In 

general, ranges tend to increase both in length and in 

width from south to north, but as shown by the plane 

of best fit to the data, the magnitude of the increase of 

range length is greater. For this reason range width to 

length ratios decrease northward. The range of values 

on this first-order surface of best fit (fig. 33) is from 0.12 

to 0.24, or 100 percent. This range of values is similar 

to the minimum range for most of the parameters considered thus far. 

The second -degree trend -surface map of range width 

to length (fig. 34) shows that the paraboloid of best 
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-1l.OGOOOO 

Y-VALUE = -11.00 + 0.1048 X (SCALE VALUE} 

X-SCALE IS VERTICAL 

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TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D51 

fit has a slight northwest-southeast orientation. A regional low exists in the central and northwestern parts 

of the Basin and Range region., where the range lengths 

are great, and the high values of the width to length 

parameter occur elsewhere. This is particularly true in 

the Mohave Desert to the southwest and along the eastern margin of the region, where values as large as 0.30 

or more occur. The third-degree trend-surface map of 

this topographic parameter (fig. 35) shows an accentuation of these same regional trends. The north-central 

and northwestern parts of the Basin and Range region 

clearly differ from the rest of the area on the basis of 

width to length ratios. 

RANGE WIDTH TO HEIGHT 

Range width to height is the second of the three derived topographic parameters considered here. An increase in this ratio can be caused in several waysnamely by an increase in width, by a decrease in height, 

or by some combination of effects, such as an increase 

in both variables whereby width increases at a greater 

rate than height. The genera.! distribution of range 

width (fig. 15) and range height (fig. 18) values on 

surfaces of best fit in the Basin and Range region indicates that both the widths and heights of ranges tend 

to increase from south to north. The first-degree trendsurface map of range width to height (fig. 36) shows, 

however, that this ratio increases from 8.8 in the southeast to more than 13.2 in the northwest, on the plane of 

best fit. The second-degree trend-surface map (fig. 37) 

of this parameter more clearly delineates the regional 

high to the northwest and the widespread low in the 

southern part of the region, which decreases to a minimum value in the southwest. The range in values on the 

paraboloid of best fit is greate.r than 100 percent of the 

minimum, and the results clearly demonstrate that the 

ranges in the northwest are distinctive because the range 

widths increase more rapidly with range heights than 

elsewhere in the region. 

The third-degree trend-surface map of range width 

to height ratios (fig. 38) is more complex. The trough 

of regional low values in the northeast on the seconddegree map (fig. 37) has migrated to the north-central 

region, but the northwest regional high is still in evidence. South of the central fielld, which is characterized 

by range width to height values of about 12: 1, a 

general decrease in this ratio is apparent. The extreme 

low to the southwest, like the high to the northeast, is 

partly induced by edge effects and by peculiarities of 

one or two sheets which gave somewhat anomalous 

z-values. The general regional trends are very real, 

however, and this map of the third-degree surface of 

best fit (fig. 38) indicates that substantial regional differences exist in the width to height parameter. 

RANGE LENGTH TO HEIGHT 

The last of the three derived topographic ratios, and 

the last of the 11 parameters considered is range length 

to height. As was true of range width to length and 

range width to height ratios, the regional variation in 

range length to height values is caused by the fact that, 

although both component variables increase from south 

to north (figs. 12, 18), they do not increase uniformly. 

The magnitude of the increase in average range le~h 

in a northwa.rd direction is much greater than themcrease in height, and ma.ximum length to height ratios 

therefore occur in the northern part of the region. 

The northward increase in average range length is 

shown by the first-degree trend -surface ma.p of range 

length to height (fig. 39); values on the pl,ane of best 

fit range from less than 40 in the southeast to more than 

7 5 in the north west. Figure 40 shows the second -degree 

paraboloid of best fit which accentuates a larg~ re~ion~l 

high in the northwestern part of the area. This high IS 

further emphasized on the third-degree trend-surface 

map of range length to height (fig. 41); this part differs 

markedly from the rest of the Basin and Range region 

on the basis of length to height ratios. Some rather 

long ranges exist in eastern Arizona and wes~rn New 

Mexico, and elsewhere along the eastern margin of the 

region, and the crest of values from 30 to 50 on the 

third -degree map reflect this fact. The southern and 

particularly the southwestern part of the.area, howeve;r, 

is generally represented by values of this topographic 

parameter that are one-half the values in the northern 

part. 

SUMMARY OF TREND-SURFACE RESULTS 

The statistical measures that are associated with each 

of the trend-surface maps discussed above include: the 

total Vlariance, V; the variance not explained by the 

surface U · the variance explained by the surface, E; 

' ' ffi' f the coefficient of determination, D ; the coo Cient o 

correlation, R; and the standard deviation, u. These 

statistics are defined by equations 62 through 67, respectively, but for purposes of discussion here only the total 

variance, V, the explained variance E, and the correlation coefficient R, need he considered. These measures are incorporated in table 1 for the first-, second-, 

and third-degree trend surfaces of each of the 11 

topographic parameters that were treated. 

The E /V ratios are equivalent to the percentage of 

total variance that is explained by a surface of given 

degree for each listed parameter. These percentages and 

the associated correlation coefficients permit some summary generalizations to be made. In terms of either 

E JV ratios or the values of the correlation coefficient 

associated with the third-degree trend surface (table 1), 
' RANGE WIDTH TO LENGTH - BASIN AND RANGE PROVENCE ~ 

q 

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X-SCALE IS VERTICAL 

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0.02 

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TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D59 

TABLE !.-Statistical measures associated with trend-surface results 

[E, explained variance; V, total variance; R, correlation coefficient] 

Parameter 

First degree 

E[V R 

Area of ranges to total area _________ 37 0.608 

~~~ gr~~~~~~=~==~~~=~=~~~~~== 34 . 586 41 . 639 30 . 545 27 • 516 27 . 518 Cumulative length of trends ________ 29 . 540 Cumulative deviation of trends _____ 37 . 609 Range width to length _____________ 12 .337 Range width to height_ __ 20 .452 Range length to height_ __ ~~~~~~~~~~ 19 .433 

Second degree 

E/V R 

60 0. 779 39 .625 54 . 738 44 . 664 31 • 557 29 . 540 56 . 751 45 . 672 26 .509 34 . 586 32 . 563 

Third degree 

E[V 

71 

51 

56 

58 

58 

36 

74 

66 

37 

45 

37 

R 

0.842 

• 708 

. 748 

. 759 

. 758 

.602 

. 858 

.812 

.611 

.674 

.607 

a relative ranking of the topographic parameters is 

possible. 

The R values associated with the third-degree trend 

surfaces permit a tripartite division to be made. In the 

first category, the area of ranges to total area and 

cumulative length and cumulative deviation of t;ends 

each yield a correlation coefficient greater than 0.800~ A second category includes range length width height 

d 1. f ' ' ' an re Ie , each of which is associated with a correlation coefficient greater than 0.700 but less than 0.800. 

And last, range volume and the three derived ratios 

namely ran~e width to length, width to height, and 

length to height, each are associated with correlation 

coefficients that range from 0.600 to 0.700. In terms of 

percentage explanation by the third -degree surfaces, 

the values for these three categories are approximately 

66-7 4 yercent, 50-58 percent, and 36-45 percent, 

respectively. 

To designate these values as rankings of significance 

would be somewhat misleading because each of the 

t?pographic parameters does exhibit regional variation to some degree, or percentage. The results cited 

simply indicate that local variations are relatively 

greater for certain topographic parameters than for 

others. Howarth .(1967) has shown, however, that percentage explanatiOns of approximately 6 12 and 16 

t b . ' ' percen can e considered as the lower limits of reliability at the 95-percent confidence level for trend surfac~s of first, second, and third degree, respectively. 

This was determined by fitting trend surfaces to randomly generated data and analyzing the resulting sums 

of squares. Because all the percentages of explanation 

(table 1) are greater than these limiting values, the 

results reported here can be considered to be valid. Also 

there is a definite improvement in e;v ratios and .R 

values (table 1) for successively higher degree surfaces for each parameter. This might be expected because, as previously stated in this report, a first-degree 

surface, or plane of best fit, for example will never 

explain a large percentage of the total variance, unless 

the z values for a given parameter are uni~formly distributed. That is, only if ranges were of the same dimensions along a given parallel of latitude, for example, 

and were to exhibit a uniform decrease in these dimensions from one parallel to the next would a plane o:f 

best fit closely coincide with the distribution of z values. 

From this point of view, and from know ledge of the 

results shown in table 1, there is no question that trend 

surfaces of degree higher than the third would satisfactorily explain a greater percentage of the total 

variance as regional variance than is indicated above. 

This statement is based upon the fact that the percentage variance that is explained by the third -degree 

surfaces represents a considerable improvement beyond 

the percentage variance that is explained by the seconddegree surfaces for several·p·arameters. This improvement in goodness of fit is 27 percent for range relief, 

21 percent for the cumulative deviation of trends, 18 

percent for the cumulative length of trends, 14 percent 

for range height, and 11 percent for range length, range 

width to length, range width to height, and area of 

ranges to total area. These data suggest that the percentage variance explained would increase further if 

trend surfaces of degree higher than the third were 

fitted to the z values of these parameters. This is probably not true for range width, range volume, and range 

length to height. The increase in percentage explanation for these parameters, from the second- to thirddegree surfaces, is 7 percent or less, and the law of 

diminishing returns would seem to apply. 

In summary, the trend -surface results indicate that 

each of the 11 topographic parameters exhibit marked 

regional variations within the Basin and Range area, 

and these variations have been shown quantitatively 

on the several trend-surface maps. A fairly large percentage of the total variance remains unexplained by 

the third-degree surface; this can be attributable to 

local variations, but the regional differences indicated 

are real. In the discussions of trend -surface results for 

each of the topographic variables, it was noted that 

the range of values on the surfaces of best fit was at 

least 100 percent for most of the parameters, and as 

great as 1,300-1,400 percent for range volume and the 

area ratio, and that this range of values reflected the 

general difference between the northern and southern 

parts of the Basin and Range area. Clearly, the magnitudes of regional differences are not minor. The range 

of values cited should not be construed as the range between extreme z values within the region. The extreme 

z values never occur on the surfaces of best fit, and the 

magnitude of the range of extreme values would be 

substantially greater than any percentage indicated. 

In addition to this general distinction between the 

northern and southern parts of the Basin and Range 

region, several other observations were previously made. 
D·60 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

The persistent contour closures or regional high values 

that recur to the north west on nearly all the trend -surface maps clearly suggest that this area, which embraces central and western Nevada and parts of eastern 

California, is topographically different from the rest 

of the region. It is set apart by reason of the much 

greater lengths, widths, heights, areas, and volumes of 

the ranges than those which occur elsewhere and because of the more nearly north-south orientation of the 

ranges. The topographic characteristics of southern 

California and southwestern Arizona form the other 

extreme, with respect to the sizes and attitudes of 

ranges. In several respects the Salt Lake desert area 

resembles this southern region; at least it resembles 

this area more closely than it does the central and western Nevada area. As a summary generalization, however, the northeastern part of the Basin and Range area 

may be grouped with eastern Arizona and western New 

Mexico. Recourse to the trend-surface maps of the several topographic parameters supports this intermediate 

grouping and, indeed, supports this three-fold division 

in general. 

Finally, the regional distinctions drawn from the 

trend-surface data are simply that, regional distinctions. It is by no means claimed that every range in the 

northwestern area delineated here is of great dimensions, nor that every range in the southwestern area is 

of small dimensions. There are indeed some ranges in 

any of these areas that depart from the mean size and 

from the other range attributes that have been defined. 

These exceptional ranges are one of the causes of the 

existence of lower level, or more local, variance components. That the distribution of differences of the average 

topographic attributes can be and has been demonstrated to be regional in extent is significant, and these 

regional differences give rise to several geomorphic 

implications which will be discussed below. 

GEOMORPHIC IMPLICATIONS 

Quantitative methods and statistical analysis of data 

in general basically answer the question "what~," not 

"why~" This is obviously true of the Basin and Range 

topographic data that have been presented in this repo~t. Regional topographic distinctions have been sustained on the basis of quantitative data ra;ther than by 

qualitative assessments, and the distinction of regions 

within the Basin and Range area has been accomplished by the objective method of trend-surface 

analysis. All of this, however, simply tells us what the 

topography of the area is like, within relatively narrow limits of error. To attempt to explain why the topographic variations occur is virtually to enter the 

realm of inference and deduetion. Nonetheless, the 

writer believes that some implications do follow from 

the data that have been presented. The four topics most 

closely related to this analysis of Basin and Range 

topography are ( 1) the nature and boundaries of 

physiographic provinces in general, _(2) the age. of the 

ranges within the region, ( 3) the origin of p~di~ents, 

and ( 4) the drainage distinctions that m~y exist In the 

ranges; they are discussed in the order given. 

PHYSIOGRAPHIC PROVINCES 

At the outset of this report the American Geological 

Institute ( 1957, 1960) definition of a physiograp~ic 

province was given; a province is said to be a regi~n 

within which the structure, climate, and geomorphic 

history are similar or uniform. Thornbury ( 196~) provided a fairly extensive introductory chapter to his book 

on regional geomorphology which treated the subject of 

physiographic provinces, their definition, and the mea~s 

of recognition of "geomorphic units." Much of this 

material is pertinent here and is summarized briefly, 

but for additional details the reader should consult 

Thornbury (1965). In Thornbury's introductory chapter Bowman (1911) is quoted as defining a province as 

"a 'tract in which the topographic expression is in the 

main uniform." Fenneman (1928) stated that "all 

orders of (geomorphic) divisions rest ultimately on 

existing differences in topography and elevation." 

Hinds ( 1952) said that geomorphic provinces were 

characterized, among other things, by "more or less 

unifor~ relief features or combination of features 

throughout its area." The definition favored by Thornbury (1965) is "A physiographic unit is an area or 

division of the land in which the topographic elements 

of altitude, relief, and type of landforms are characteristic throughout and as such is set apart or contrasted 

with other areas or units with different sets of characteristic topographic elements." In addition, most of the 

physiographic maps of individual States invoke topographic criteria in some way to justify the province or 

division boundaries shown. 

Clearly, all classifications of physiographic provinces 

or geomorphic units are based primarily, if not solely, 

on topographic characteristics. This fact has been obscured occasionally by reference to the "unified geomorphic history" that is thought to prevail within a 

given province. Thornbury (1965, p. 9) stated, however, 

that "given a certain geologic framework, the topographic condition or expression of an area is largely 

determined by ·its geomorphic history." This means, 

essentially, that similarity of geomorphic history is 

inferred, if similarity of topography exists. 

Obviously, then, physiographic provinces and their 

boundaries can best be determined from quantitative 

analysis of topographic parameters in a given region. 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D61 

Such province classification or boundary delineation, by 

methods akin to those described in this report, or others, 

must remain somewhat arbitrary. That is, some arbitrary choice of value for some parameter or combination 

of parameters must be made to fix any required boundary. The chief virtues of such procedure, however, are 

objectivity, reproducibility, and the facilitation of quantitative comparison of within-province variation and between-province variation. 

Only within-province variation has been treated in 

this report. As previously stated, this writer would tend 

to define three physiographic provinces within the Basin 

and Range region on the basis of the quantitative data 

obtained. Others might prefer to term these three areas 

"sections," as did Fenneman (1931), who described five 

subdivisions in the Basin and Range province. The discrepancy in numbers of subdivisions is not at issue here; 

rather, the choice between the use of the terms "section" 

and "province." 

The choice is not merely a matter of semantics but 

one that basically involves the very nature of classifitions and of hierarchical systems in general. In any such 

system, the within -class variance will inevitably increase 

with an increase in order or rank. Therefore, the total 

internal variance at the highest levels, such as within the 

kingdom ranking of the zoologist, will be large indeed. 

But beyond some critical magnitude of total internal 

variance, the utility of a given classification may be 

destroyed. 

To classify physiographic provinces and sections one 

basically needs to know the magnitudes of internal topographic variance allowable within each rank. The present writer has shown that average values of such 

parameters as range volume and the area of ranges to 

total area ratio exhibit a variation that is as much as 

1,300 to 1,400 percent. The rest of the parameters exhibit a minimum variation of 100 percent. The question 

to be answered is whether such topographic variance is 

common to the province-level rank. If so, then the areas 

discussed in this report can be designated as "sections" 

in a hierarchical scheme. If this variance is excessive 

however, the Basin and Range province as we know it 

should be subdivided into several provinces. 

The answer to the question posed is beyond the 

scope of this report. There are no quantitative treatments of physiographic provinces that would enable 

one to state the average magnitude of topographic variance >associ'ated with the province rank 1and thereby to 

determine whether the known variance in the Basin 

and Range region is excessive. The writer recommends, 

therefore, that a province-by-province evaluation be 

undertaken to determine average values of witliin-province variation and between-province variation. In addition to the problem of proper subdivision of the Basin 

and Range region, several other questions of interest 

might be resolved. 

To examine the degree of topographic simila.rity 

between the folded mountains of the Appalachian region, for eXJample, and the Coast Ranges of Oalifornia 

would be useful. Despite known differences in the geological histories of these two regions, the landform map 

of the National Atlas Series (U.S. Geological Survey, 

1966) indicates that topographic similarity exists. 

Quantitative eXJamination of these two :areas might 

therefore lead to some assessment of the validity of the 

inference that simiLarity of topography implies similarity of geomorphic history. In 'another vein, the 

Basin and Range province of North America might 

be compared to its counterparts elsewhere in the world. 

The Dasht-e-Lut and Dasht-e-Kavir of Iran, or the 

basin and range regions of Afgh.ianistan ·and West 

Pakistan, are 1among the more obvious choices for such 

an undertaking. This exercise would have more direct 

bearing upon the range of topographic values that exist 

within such .areas. 

In summ·ary, further work is indicated, but the results of this report suggest that any physiographic province or finite area of the earth's surface is amenable 

to quantitative examination. One of the desirable goals 

of such efforts should be the estrublishment of quantitative standards of within-province and between-province 

variation, as well as the objective delineation of province boundaries. 

AGE OF RANGES 

The treatment of topographic data presented in this 

report supports the view of many workers that the 

ranges in the Basin and Range province are probably 

of different ages. This conclusion is indirectly implied 

by the topographic data, but it is supported by many 

other lines of evidence, some of which ·are discussed 

here. First, it should be recalled that the ranges in central and western Ne~ada and in part of eastern California are longer, wider, higher, and more closely 

spaced than are those of southwestern Arizona and 

southeastern California. An idealized cross section 

through a basin 1and two adjacent ranges in each of 

these two gener.al categories of regions would resemble 

that shown in figure 42. The question is why this difference exists. 

Among the possible explanations, three are most 

prominent. First, the difference in topographic relief 

and range spacing could be attributable to differences 

in depth of basin fill. Figure 42 shows that ·an increase 

in depth of basin fill would, for the l'arger ranges, produce topographic expression siinilar to that presently 

exhibited by the smaller ranges. Essentially, burial of 
D62 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

FIGURE 42.-Idealized cross section of two ranges and an 

intervening basin in two di:fferen.t parts of the Basin and 

R.ange .region. Upper part of sketch shows ranges· of large 

dimensiOn and close spacing which occur in north-central 

and ~orthwestern Nevada and eastern California, where 

alluvial fans predominate. Lower part of sketch shows 

ranges of small dimension and wide spacing which occur in 

sou~hwestern Arizona and southeastern California, where 

pediments predominate. Dashed line in upper part of sketch 

illustrates how these topographic differences could be 

achieved by varying depths of basin fill. Area between 

ranges in lower part of sketch, however, is known to contain 

surface or near-surface pediments, and the depth-of-:fill 

hypothesis is therefore untenable. 

the 11arger ranges by alluvium would leave only their 

tops exposed, 'and the apparent range spacing and size 

would change ~accordingly. Aside from consideration 

of the ~ange e.rosi.on necessary to accomplish this end, 

the bas~c questwn.Is whether the existing data on depths 

of ~ll In the ~asin and Range area will support such 

an Interpret:Jatwn. Such data are very sparse, as indicated by the recent basement map of North America 

( A~erican Association of Petroleum Geologist and 

United States Geological Survey, 1967), which is 

largely blank over the part of the Western United 

States considered here. The va:vious geophysical surveys that have been made in the northern rand southern 

p~rts of the Basin and Range area, however, will cert~Inly not sustain the idea that depths of fill differ drastiCally. Moreover, as is well known, the southern basins 

have a history ~f through-flowing drainage, whereas 

the northern basins do not, and this has permitted the 

transportation of sediment out of the southern basins 

a_t various times. This ralso mitig,ates against the suggestion that the smaller and more widely spaced ranges 

are a consequence of greater depths of basin fill. The 

depth-of-fill explranation seems therefore to be untenable on a regional basis and mray be disregarded. 

A s~ond possibility is that the differences in topographic rattributes are related to basic structural differences. Speci~call;y-, the spacing of faults and the magnitudes of uphft differed In the several parts of the Basin 

a~d Range area ~o such an extent that the topographic 

differences are directly rel,ated to structural conditions. 

The rather voluminous geological literature on the Basin 

and Range area clearly suggests that this hypothesis is 

equally improbable. In general, faults and fault scarps 

are more commonly observed in the northern part of the 

region, and they are in zones that are reasonably close 

to the mountain blocks. This is not true of the southern 

part of the region, where broad bedrock pediments are 

far more common, and evidence of faulting is obscure. 

The argument would require rthe existence of fault 

· planes between the pediments and the ranges which they 

abut; faults in such locations have not been observed. If 

this second hypothesis is disregarded,.the remaining alterruative is that the ages of the ranges generally differ 

in the Basin and Range area. 

Reference to the "age" of ranges in the context of this 

discussion means relief age or the time span since topographic expression was achieved. Clearly, a given relief 

age need not be coincident with the radiometric age that 

may exist for the plutonic or volcanic rocks in a given 

range. In fact, in this oontex,t a radiometric age can only 

serve as a limiting value for the relief age of a range. 

That is, the radiometric age must be greater, for it represents either the time of crystallization of a specific 

part of a given magma or the time of extrusion of a 

given lava. Damon and Mauger (1966) and Damon 

(1967) have discussed the chronology of events in the 

Basin and Range province in such a way that the question of the relief ages of ranges is somewhat obscured. 

Some discussion of this point is therefore warranted. 

Basically, these authors presented potassium-argon 

dates for volcanic and hypabyssal plutonic rocks that 

have a bimodal distribution in time. One peak of the distribution occurs 60-70 million years ago and this represents Late Cretaceous or Laramide time. The seoond 

peak occurs 20-30 million years ago, in Oligocene to 

Miocene, or late Eogene to early Neogene time. They 

generally concluded that these radiometric-age peaks 

are coincident with times of orogeny throughout the 

Basin and Range province and related the two time 

pe~ks to the classical pulse-of-the-earth hypothesis, 

whiCh demands synchroneity of magmatism and orogeny. Whether such synchroneity exists depends upon 

the scale of evenrts that is considered ,and upon whether 

the equation of magmatism with orogenic activity is 

actually justified. 

R·elated to the first of these factors, the scale of events, 

each of the radiometric age-distribution peaks is associated with a chronological range of values of considerable magnitude. Thus, the Laramide magmatism 

(Damon and Mauger, 1966; Damon, 1967) occurred between 50 and 90 million years ago, and the Tertiary 

episode occurred between 5 and 45 million years ago. 

The bimodal distribution of ages is not questioned here, 

but the range of values about each mode-namely 40 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D63 

million years-indicates why .the scale of events under · 

consideration is important. These radiometric results 

can be interpreted broadly 1as indicative of two general 

magmatic episodes that occurred throughout .the Basin 

and Range province. Considered in detail, however, the 

rocks associated with either episode can obviously differ 

drastically in absolute age. The potassium-argon data 

for Laramide plutons between Nogales, Mexico, and a 

point approximately 100 miles to the north, for example, 

show that even in .this small part of southern Arizona 

the absolute ages of rocks range from 59 million years 

to 75 million years (Damon, 1967). In general, if the 

potassium-argon daJtes are accepted as limiting dates of 

actual orogenic ·activity, the relief ages of ranges in the 

Basin 1and Range province may differ by as much as 40 

million years for either of the two magmatic episodes. 

This span of time is certainly sufficient for substantial 

modification of the older ranges to have occurred by 

erosion. 

The second factor, namely the equation of magmatism 

with orogenic activity, has been discussed at length by 

Gilluly ( 1965), who reviewed the available data on 

volcanism, plutonism, and tectonism in the Western 

United States. Basically, Gilluly concluded that plutonism and orogenic activity have not been synchronous 

throughout the region and, in support of the view expressed here, that radiometric age determinations do not 

necessarily date orogenies. Figures 6 and 7 from GilJuly's paper are reproduced here 3JS figures 43 and 

44, respectively. In absolute age, Eogene corresponds 

to a time span ranging from 30 to 70 million years ago 

in Gilluly's usage. Hence, figure 43 is referable to conditions in the Western United States during a substantial 

part of the Laramide magmatic episode (50 to 90 million 

years ago) in Damon's (1967) view. This depiction of 

Eogene conditions (fig. 43) shows that putonism and 

volcanism, or magmatism in general, are not everywhere 

coincident with orogenic activity. Evidence for tectonism over most of central and western Nevada is lacking, 

despite the occurrence of siliceous volcanic rocks in this 

area. Orogenic activity became more pronounced in 

Nevada during N oogene .time (fig. 44), but this coincided with diminished tectonism in southern Arizona. 

The data on which these maps are based would appear 

to indicate that the synchroneity of magmatism and 

tectonism remains at least debatable, if not, indeed, in 

the category of unproven assumptions. 

If relief ages of ranges in the Basin and Range 

province differed by as much as 40 million years because 

of nonuniform tectonic activity during Late Cretaceous 

to Tertiary time, then pre-Quaternary topography 

must have been nonuniformly distributed. Many authors have stated that the present relief features of the 

- - Quartz diorite line of Moore (1959) 

-- Average east border of eugeosyncline 

6 Orogenic activities 

II II Volcanics, andesite, and more siliceous rocks 

• Pluton, Laramide 

FIGURE 43.-Volcanism, tectonism, and plutonism during the Eogene in 

the Western United States. (After Gilluly, 1965, fig. 6.) 

Basin and Range province are closely related to ~he 

block faulting of Quarternary time that was superimposed on preexisting topography. Because Quarte~nary 

tectonic activity was also distributed in a nonuniform 

manner, the preexisting topographic differen~es were 

accentuated in many places. King ( 1965) provided the 

most recent review article pertinent to this subject, 

which relies to some extent upon the distribution of 

recorded earthquake epicenters in the United States, as 

well as upon field evidence. King (1965, ~g. 2) co~­

toured the epicenter distribution; his map IS shown In 

figure 45. 

This map of epicenter distribution in the Western 

United States (fig. 45) clearly shows their nonuniform 

occurrence. The greatest concentrations occur in the 

northwestern part of the study area, and along the 

east margin of the Basin and Range area. Quiescence, 

or stability, is indicated for southern Arizona, eastern 

Nevada and westernmost Utah. King (1965) stated that 

' . al this p31ttern closely coincides with the k~own re~wn. 

distribution of Quaternary block faulting, which IS 

most extensive in western Nevada, eastern California, 
D64 THEORETICAL PAPER<S IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

-------- ----, 

I 

I 

I 

I ~ -1-<1++---- ---i 

6 6 1 

- - Quartz diorite line of Moore (1959) 

-- Average east border of eugeosyncline 

t::. Orogenic activities 

1111/1 Volcanics, andesite,and more siliceous rocks 

• Pluton, middle Tertiary 

FIGURE 44.-Volcanism, tectonism, and plutonism during the Neogene tn 

the Western United States. (After Gilluly, 1965, fig. 7.) 

and in western Utah and New Mexico. The quantitative 

topographic data and trend-surface maps (figs. 9-41) 

presented in this report also coincide notably well with 

these regional structural patterns. 

As a final point bearing upon the relief age of ranges 

in the Basin and Range area, figure 1 is instructive. 

Although highly generalized and locally inaccurate because of the small scale of the compilation (1: 7,500,-

000), this map (fig. 1) clearly shows that the distribution of Precambrian and lower Paleozoic rocks in the 

Basin and Range area is nonuniform. It can be seen that 

outcrops of rocks of these ages occur primarily in the 

southern part of the region and in eastern Nevada and 

western Utah. These are precisely the areas which correspond to minimai recorded ea~thquake activity (fig. 

45) and Quaternary block faulting, and the logic of 

Occam's razor would suggest that the relief ages of 

ranges in these areas are greatest. That is these stable . ' regiOns have been subjected .to the longest time spans of 

erosion; hence, the oldest rocks are exposed within the 

ranges that exist. 

In summary, the geomorphic implication that can 

most reasonably be drawn from the trend-surface maps 

of Basin and Range topogra~hy is that the relief ages 

of ranges in the region are different. Differing time 

spans of erosion is the simplest explanation for the variance of length, width, height, area, volume, relief, and 

trend of ranges that was noted previousiy, and this 

explanation accords well with the known geological 

history of the region. Because durations of topographic 

ex;pression and of consequent erosional periods have 

varied within the region, a third inference can be drawn 

from the topographic data. This concerns the origin 

and distribution of pediments, which are discussed 

below. 

ORIGIN OF PEDIMENTS 

The subject of pediments and theories of their origin 

occupies a vast number of papers in the literature on 

the Basin and Range area and other parts of the world. 

Most of these papers have been cited elsewhere (Lustig, 

1967) and will not be specifically discussed here, but a 

recent review of the pediment literature by Hadley 

( 1967), which appeared after the text of this report was 

written, is worthy of note. Several of the better-known 

theories of pediment formation are probably incorrect. 

As an example of this generalization, consider the often -cited claims that sheetflooding is the process responsible for pediment production. Such claims are 

based solely on two or three reports of early vintage 

which described wagons overtaken by vast "walls of 

water" in the Western United States. The wagons in 

question were all traversing roads that bordered the 

lower reaches of a pediment surface, in low areas where 

the effect of channelized flows on the pediment surface 

combined to produce high water. This writer has never 

dbserved an unconfined "wall of water" on a gently 

sloping desert surface and is unacquainted with any 

who have observed such a phenomenon. Moreover, to 

attribute the origin of bedrock surfaces to sheetflows is 

to confuse cause and effect, even supposing that the 

shallower channels on these surfaces are occasionally 

overtopped during intense runoff events. The surfaces 

must necessarily predate the runoff events in question. 

Lateral erosion is another frequently offered explanation. This hypothesis primarily treats the migration of channels that exist on pediment surfaces. No 

doubt, channels will migrate with time on these surfaces, and some erosion will therefore be accomplished, 

but the part of any pediment that abuts the mountain 

front cannot be explained in this manner. The hypothesis virtually requires that streams emerge from a given 

mountain range and, on occasion, turn sharply to one 

side or the other to "trim back" the mountain front in 

interfluvial areas. Such stream paths, nearly perpendic-
10Q_0 

__ 90° 

FIGURE 45.-Distribution of recorded earthquake epicenters, as of 1957, ln the United States. (After King, 1965, fig. 2.) 

@ 

,Epicenter of major 

earthquake 

(magnitude greater 

than about 7.0) 

25° 

8 

l;lj 

t:;j 

z 

t::l 

I 

rn 

q 

l;lj 

l'%j 

> 0 

t:;j 

~ > t"4 

~ 

rn 

~ 

rn 

0 

l'%j 

8 

~ t:;j 

tl:l 

> rn 

~ 

z 

~ t::l 

l;lj 

~ 0 

t:;j 

I'd 

l;lj 

0 

< z 

~ 

tj 

~ 

01 
D66 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

ular to a sloping surface, would defy the laws of grav~ 

ity and have not been observed except those in areas 

where drastic tilting has occurred. This is not surprising, because knowlege of the triangle of probabilities 

(fig. 46) suggests that the least probable path in a twodimensional consideration is precisely the right-angle 

path required of a stream emerging from a given range. 

This triangle can be constructed by moving a marker 

to either the left or the right of any given origin in 

successive steps. Each given probability is equal to onehalf the sum of the two numbers above it, and the probability at any point can be computed from the following general formula : 

where Znk is the probability that the marker will be at 

point k after n trials. Thus the path of greatest probability coincides with the central portion of the triangle 

(fig. 46). In the absence of constraints the threedimensional natural condition would most assuredly be 

best represented by a stream that emerges from the 

mountains and flows downslope to the basin floor, or 

pediment margin, with minimum deviation from a central path. Hence, lateral planation undoubtedly occurs 

1 

2 

1 

4 

1 3 

8 8 

1 4 

16 16 

1 5 10 

32 32 32 

1 6 15 

64 64 64 

1 7 21 35 

128 128 128 128 

1 8 28 56 

2 

4 

6 

16 

on pediment surfaces, although quantit~tive data indicating the effects of the process are lacking, but it cannot 

account for the existence of pediments as such. 

It has also been proposed that pediments result from 

parallel slope retreat of the mountain front. Slope retreat of this type is itself a topic that ·occupies much 

literature, and the more ardent champions of this process argue for parallel retreat of cliffs on a continental 

scale. The writer's observations in .several deserts of the 

world suggest that few, if any, escarpments are not 

dissected by prominent drainage systems. This is true 

even in the driest regions, such as the Namib Desert in 

South West Africa or the Tuwaiq escarpment of central 

Arabia. The precipitation is about 2 inches per year in 

both areas. The existence of drainage basins in .the 

mountain ranges is of central importance to the various 

pediment arguments. These basins are the loci of the 

most effective erosional processes that operate on mountain ranges in the Basin and Range region or elsewhere. 

The mountain fronts may well retain some characteristic slope angle that reflects rock strength, structure, 

weathering characteristics, and other variables, and 

they may retreat at this angle. This does not prove, 

however, that ranges are primarily reduced by parallel 

retreat of escarpments. This point will be pursued later. 

1 

2 

1 

4 

3 1 

8 8 

4 1 

16 16 

10 5 1 

32 32 32 

20 15 6 1 

64 64 64 64 

35 21 7 1 

128 128 128 128 

70 56 28 8 1 

256 256 256 256 256 256 256 256 256 

" ~ " ~ J/ ~ Jl' ~ J/ ~ Jl' ~ Jl' ~ Jl' ~ J/ ~ 

FIGURE 46.-The triangle of probabilities, showing the probability associated with any point on a two-dimensional random walk. The origin 

1s represented by the apex of the triangle, which has a probability of 1. 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D67 

Although additional arguments presented in literature concern the origin of pediments, the above examples 

will suffice here. The many discussions of pediment surfaces have focused upon the wrong landform in the 

writer's opinion. There is no question that processes of 

subaerial and suballuvial weat-hering occur on pediments 

today, nor that fluvial erosion also occurs. A pediment 

must exist prior to the onset of these processes, however, 

and in this sense the origin of pediments resides in the 

adjacent mountain mass and its reduction through time. 

Even if quantitative data on the rates and intensities of 

processes that act upon pediments were obtained, these 

would have little bearing upon the origin of bedrock 

surfaces. 

To say that pediments result from the reduction of adjacent mountains through time is to give voice to a 

seemingly obvious and intuitive argument. But this 

point has often been insufficiently stressed in the past. 

Each range has some local base level in an adjacent 

basin, and it cannot be eroded to an elevation below this 

level. Given stability for a sufficient period of time, the 

consequences of mountain reduction must inevitably include the production of a pediment, whether in arid or 

nonarid regions. The nature of the surface produced may 

vary, and it may be mantled by, or free of, alluvium. 

However, it simply represents an area that was formerly 

occupied by a mountain or other bedrock topographic 

high. The only real "pediment problem" is how the reduction or elimination of mountain mass occurs. This 

question will be considered here, admittedly in a deductive manner, and the relationship of the topographic 

data to the pediment question will subsequently be 

discussed. 

First, it is significant to note that the "sharp break in 

slope" between the pediment surface and the mountain 

front, which has always been emphasized in the literature, exists only in interfluvial areas. Observation shows 

that the course of any master stream channel from a 

given drainage basin in the mountains onto the pediment 

surface and thence to the basin floor below has no sharp 

break in slope. In the absence of constraints, such as 

recent structural disturbances, any such stream channel 

will exhibit a relatively smooth, concave upward, longitudinal profile that accords with the local hydraulic 

geometry. There are no hydraulic anomalies in nature, 

and none exists at the "mountain front." The interfluvial 

areas, however, generally do exhibit a marked change 

in slope, at least within a narrow zone parallel to the 

mountain front. The reason for the existence of such a 

zone is precisely that it is an interfluvial area; the dominant process that operates on the mountain front is not 

fluvial. 

Qualitatively, it can be argued that two basic processes 

are operative on a given mountain mass. The steep slopes 

of the mountain front are interfluvial areas that are subject to weathering. Runoff on these steeply sloping surfaces is of short duration and is not concentrated. The 

runoff serves largely to remove the finer weathered 

debris that is transportable. Larger particles generally 

remain in place until th~y are reduced in size by weathering. The rates of mountain-front retreat are basically 

unknown, but by any reasonable assessment they are 

slow in relation to rates of processes that are operative 

in drainage basins. This is clearly true because the headwater region of any given drainage basin also consists 

of steep walls that are virtually indentical to those of 

the mountain front in interfluvial areas. In these 

headwater regions the same processes of weathering and 

of removal of debris occur. Hence, the rates of retreat 

of the bounding walls in the headwaters of drainage 

basins must be at least as great as the rate of retreat of 

the mountain front in the interfluvial areas. Also, however, the drainage basins represent the only parts of 

any mountain range that are subjected to concentration 

of flow and to its erosional effects, and these basins must 

therefore be the principal loci of mountain-mass 

reduction. 

The entire argument may be summarized by stating 

that two processes are operative on mountain ranges, 

namely "A" and "B," and that process "B" is rapid relative to "A." If "A" operates on the mountain front, 

whereas both "A" and "B" operate in drainage basins 

then the bulk of range reduction must occur in the latter 

areas. Moreover, it is logical to further infer, as a corollary, that the interfluvial areas, or parts of the mountain front proper, must be "left behind" with the passage of time. These areas, which are often described as 

"triangular facets" in the literature, are possible loci 

of inselberg production. The latter landforms are mountain residuals which are left on the pediment surfaces, as 

these surfaces are produced by drainage-basin evolution 

through time. 

The foregoing hypothesis appears to have the support 

of logic, but because quantitative data are lacking it can 

be considered no more than this, namely a hypothesis. 

Nevertheless, some tests of its merit are possible, and in 

this sense the topographic trend-surface data of this 

report are related to the pediment question. 

If, as has been argued here, the origin of pediments 

is a direct consequence of mountain-mass reduction, the 

average mountain mass should be significantly smaller 

in certain areas that are characterized by the existence 

of pediments. The trend-~rface results of this report 

have clearly indicated that regional differences do exist 

with respect to each of the 11 topogra;phic parameters 
D68 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES 

used. The lengths, widths, heights, relief, areas, and 

volumes of ranges (figs. 9-41) are, on the average, much 

smaller in southwestern Arizona and southeastern California than in the north-central and northwestern parts 

of the Basin and Range area.; western New Mexico, 

northwestern Utah, and northeastern Nevada occupy 

an intermediate status with respect to most of these 

characteristics. As djscussed in the section "Age of 

Ranges," the areas occupied by smaller ranges are believed to be those where the ranges are of greatest relief 

age and, hence, where the ranges have longer erosional 

histories. It is also well known that the areas occupied 

by significantly smaller ranges, as delineated by the 

trend-surface maps, are indeed those areas within the 

Basin and Range region which are characterized by the 

existence of pediments. In those areas in which the 

larger ranges occur, alluvial fans are far more characteristic. The generalized cross section of an ideal basin 

and two adjacent ranges (fig. 42) indicates why this ie 

true. Reduction of the larger ranges, shown in the upper 

part of the figure, to the dimensions of the smaller 

ranges, shown below, must be accompanied by the production of a residual surface surrounding the ranges, . 

the elevation and nature of which is primarily a function of local base level. This should not he construed 

as an argument that no pediments exist in the northwestern part of the Basin and Range region nor that no 

alluvial fans exist in the southwestern part of the region. As indicated for the several topographic para.meters, some part of the total variance always arises from 

local components. But there is a significant regional 

variation in topography, and this eoincides with the 

regional variation in the distribution of fans and 

pediments. 

This coincidence does not necessarily prove that the 

hypothesis for pediment formation offered here is correct. It simply shows that mountain-range reduction 

and the occurrence of pediments go hand in hand. If 

pediment production is accomplished by drainage basin 

evolution through time, however, some regional drainage distinctions may exist within the Basin and Range 

province. The question of whether such distinctions 

exist and their nature constitutes a final geomorphic 

implication. 

DRAINAGE DISTINCTIONS 

A considerable body of literature supports the idea 

that drainage evolution, in the absence of constraints, 

accords with stochastic processes. The several randomwalk models of Leopold and Langbein ( 1962), Schenk 

(1963), and Scheidegger (1967), among others, and 

the mathematical analysis of aspects of Horton's laws 

( 1945) by Shreve ( 1966), and by W oldenberg ( 1966) 

dearly suggest that in areas of homogeneous rock type 

and structural stability, the drainage pattern that will 

develop is dendritic and, moreover, that this is the 

most probable pattern in nature. Combined with the far 

greater amount of information on the hydraulic 

geometry of river channels, which includes da:ta on the 

profiles of drainage systems as well as their areal pattern, the status of knowledge of drainage basins and 

their contained drainage networks is known to be at 

least one order of magnitude in advance of the period 

when drainage was characterized as "youthful," 

"mature," and "old." 'The present writer believes, however, that so mueh effort has been devoted to establishing the concept that drainage systems reflect the 

most probable steady-state conditions that insufficient 

attention has been given to the study of the effects of 

time and whether these effects produce measurable 

drainage distinctions. 

The drainage patterns that occur in the mountain 

ranges of the Basin and Range region have not been 

imposed by hydrologic input upon fuHblown mountain 

blocks with planar sides, as so often depicted in older 

geomorphology texts and in the block diagrams of 

random-walk papers. Although the rates of orogeny are 

probably greater than the rates of denudation in the 

region since the Tertiary (Schumm, 1963), the existing 

drainage patterns must have begun to evolve at the very 

outset of the development of topographic relief. This 

means that complete equilibrium between the drainage 

systems and their respeotive host ranges may not yet be 

achieved in the more recently uplifted ranges, although 

these systems rna y be closer to mean steady -state conditions than might be suspected. In those ranges which 

have undergone long periods of stability, the drainage 

systems should be well adjusted and should reflect all 

of the attributes that are predicted by the stochastic 

models and by the hydraulic-geometry relations. Some 

measurable drainage distinotions should exist on this 

basis alone, ho\Vever. That is, those parts of the Basin 

and Range region which are characterized by larger 

area of ranges to total area ratios, and by greater average widths, lengths, heights, volumes, and relief of 

ranges, might be predicted to be areas that are also 

characterized by some degree of departure from 

steady -state drainage systems. 

Aside from this possibility, however, the effects of 

time may influence drainage systems in another way. 

Given a mountain block of finite size and homogeneous 

lithology, under specified hydrologic conditions there 

will be some finite number of dra,inage systems that can 

develop. Although the number of drainage systems that 

can coexist has never been specified, it clearly must be a 

:function of mountain mass or the size and shape of a 
TREND-SURFACE ANALYSIS OF THE BASIN AND RANGE PROVINCE D69 

given range. W. B. Langbein (written commun., 1966) 

used random-walk models to investigate the drainage 

systems that will develop in square and rootangular 

areas in the absence of oontraints. For a square area the 

largest stream system drained 21 percent of the total 

area, whereas the largest stream drained only 17 percent 

of the total area of a rectangle with a 2: 1 side ratio. 

These results suggest that the problem is also amenable 

to testing by simple counts of the number of drainage 

systems in ranges of similar size, shape, lithology, and 

erosional history. The important question is, however, 

if this is true, then what effect has the passage of time 

on the numbers and the nature of drainage systems in a 

given range~ 

The writer has stated previously that mountain reduction is basically accomplished by the enlargement of 

drainage systems through time and that this results in 

pediment formation. But if the number of drainage systems is a function of the size of a range, this number 

must diminish as the mass of the range is reduced 

through time. Again, it might be predicted that those 

parts of the Basin a.nd Range region characterized by 

different values of the 11 topogra.phic parameters discussed will also be characterized by different values of 

mean number of drainage systems. 

Finally, as the mass of a given range is reduced beyond some critical value, the nature of these drainage 

systems, as well as their number may be altered in some 

di~cernable manner. Clearly, the order numbers of the 

drainages must diminish, but, in addition, in a region 

characterized to some extent by orographic controls on 

precipitation, the hydrologic input will necessarily 

diminish, and weathering-rather than fluvial 

processe&-will attain the dominant role in final mas~ 

r~uction. Qualitative observation of many long-stable, 

reSidual ranges in southern Arizona and southeastern 

California supports this view; the lack of any prominent drainage network on the slopes of these low, narrow, and commonly linear ridges is characteristic. Thus, 

although this writer dislikes the use of the word 

''cycle," because of the many unsound or incorrect implications associated with it in geomorphology, the evolution of drainage systems. in the Basin and Range region 

may well be termed "cyclic" in a restricted sense. 

Although there is undeniably an adjustment of process 

and form through time, predictable on the basis of the 

laws of probability and hydraulics, the evolution of 

drainage systems is intimately related to mountain mass 

and its ultimate reduction. The number of drainage systems and their order must diminish through time, and 

the relative roles of fluvial erosion and weathering 

must be reversed as the drainage systems diminish 

through time. 

A search for quantitative regional drainage distinctions is therefore worthy of future efforts in the Basin 

and Range province. The writer suggests that these distinctions will probably coincide with the distribution of 

fans and pediments and with the regional topographic 

differences discussed in this report. 

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