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Geological Society of America Bulletin

doi: 10.1130/0016-7606(1952)63[1117:HAAOET]2.0.CO;2 1952;63;1117-1142Geological Society of America Bulletin

ARTHUR N STRAHLER HYPSOMETRIC (AREA-ALTITUDE) ANALYSIS OF EROSIONAL TOPOGRAPHY

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BULLETIN OF THE GEOLOGICAL SOCIETY OF AMERICAVOL. 63. PP. 1117-1142. 23 FIGS.. 1 PL. NOVEMBER 1952

HYPSOMETRIC (AREA-ALTITUDE) ANALYSIS OP EROSIONAL TOPOG-RAPHY

BY ARTHUR N. STRAHLER

ABSTRACT

The percentage hypsometric curve (area-altitude curve) relates horizontal cross-sectional area of a drain-age basin to relative elevation above basin mouth. By use of dimensionless parameters, curves can be de-scribed and compared irrespective of true scale. Curves show distinctive differences both in sinuosity of formand in proportionate area below the curve, here termed the hypsometric integral. A simple three-variablefunction provides a satisfactory series of model curves to which most natural hypsometric curves can befitted. The hypsometric curve can be equated to a mean ground-slope curve if length of contour belt istaken into account.

Stages of youth, maturity, and old age in regions of homogeneous rock give a distinctive series of hyp-sometric forms, but mature and old stages give identical curves unless monadnock masses are present. It istherefore proposed that this terminology be replaced by one consisting of an inequilibrium stage, an equilib-rium stage, and a monadnock phase.

Detailed morphometric analysis of basins in five sample areas in the equilibrium stage show distinctive,though small, differences in hypsometric integrals and curve forms. In general, drainage basin height, slopesteepness, stream channel gradient, and drainage density show a good negative correlation with meanintegrals. Lithologic and structural differences between areas or recent minor uplifts may account for certaincurve differences. Regions of strong horizontal structural benching give a modified series of hypsometriccurves.

Practical applications of hypsometric analysis are foreseen in hydrology, soil erosion and sedimenta-tion studies, and military science.

CONTENTSTEXT Page

2. The percentage hypsometric function.. .. 1120age 3. Integration of the hypsometric function.. 1121Introduction 1118 4. Model hypsometric function 1122Principles of hypsometric analysis 1118 5. Family of curves for the value r = 0.1... 1122

Hypsometric curve in absolute units 1118 5. Comparison of several curve families. . . . 1123Percentage hypsometric curve 1119 7. Graphic solution of integrals and ex-Method of obtaining hypsometric data. . . . 1119 ponents 1124Integration of the hypsometric function. . . 1120 8. Small drainage basin in badlands, PerthA model hypsometric function 1121 Amboy, New Jersey 1125Inflection points and slopes 1123 9. Hypsometric curve of basin shown in Fig-Relation of hypsometric curve to ground ure 8 1126

slopes 1125 10. Hypothetical drainage basin 1126Geomorphic applications of hypsometric 11. Contour belt 1126

analysis. . . 1128 12. Correlation of mean ground slopes andThe geomorphic cycle 1128 adjusted slopes of hypsometric curve seg-Characteristics of the equilibrium stage.... 1130 ments 1127Relation of hypsometric forms to drainage 13. True mean-slope curve of basin shown in

forms 1136 Figure 8 1128Geologic factors affecting equilibrium forms. 1136 14. Inequilibrium (youthful) stage 1129Influence of horizontal structure 1139 15. Equilibrium (mature) stage 1130

Practical applications of hypsometric analysis. 1140 15. Monadnock phase 1131References cited 1141 17. Mean hypsometric curves of five areas

TT T T CTT> A TTY% re m ̂ * equilibrium stage 1132ILLUSTRATIONS ig Representative basins from five sample

Figure Page areas 11331. Figure of reference in percentage hypso- 19. Stream numbers and bifurcation ratios

metric analysis 1119 for five sample areas 11371117



1118 A. N. STRAHLER—ANALYSIS OF TOPOGRAPHY

Page20. Stream lengths and length ratios for five

sample areas ......................... 113721. Hypsometric curves of three basins in

Mesa Verde Region ................... 113922. Hypsometric curves of three basins near

Soissons, France ...................... 114023. Hypsometric curves of large drainage

basins ............................... 1140

FacingPlate page

*• Model hypsometric curves for five values°* r .................................. *"'

TABLESPage

\ Morphometric data for five sample areas. . 11342. Statistical data for mean integrals of five

areas ................................. 1135

INTRODUCTION

Topography produced by stream-channelerosion and associated processes of weathering,mass movement, and sheet runoff is extremelycomplex, both in the geometry of the formsthemselves and in the interrelations of the proc-esses which produce the forms. Although thefluvial-erosional landforms constitute the larg-est proportion of the earth's land surfaces andtherefore deserve intensive study, only in re-cent years have investigations moved from therather limited phase of simple visual observa-tion and generalized verbal descriptions to themore productive but vastly more refractoryphase of quantitative description and dynamicanalysis.

Dynamic-quantitative studies require, first,a thorough morphological analysis in orderthat the form elements of a landscape may beseparated, quantitatively described, and com-pared from region to region. Drainage networkcharacteristics and channel gradients, slopeprofile forms, declivities and lengths, drainagedensities, and hypsometric properties areamong the general classes of morphologicalinformation for which standardized measuresmust be set up so that the essential differencesand similarities between regions can be under-stood. Second, the topographic forms mustbe related quantitatively to the rates and in-tensities of the denudational processes. Theserelationships may take the form of empiricalequations derived by methods of mathematicalstatistics from the observational data, or de-duced mathematical models whose validity issustained by observed values.

The material in the present paper is merelyone very small part of the morphologicalanalysis. It concerns the investigation of hyp-sometric properties of small drainage basins—that is, area-altitude relationships and the

manner in which mass is distributed within adrainage basin.

Some parts of this paper represent work sup-ported by the Penrose Bequest, Project Grant525-48; but the greater part of the investigationwas supported by the Office of Naval Researchunder Contract N6 ONR 271, Task Order 30,Project No. NR 089-042.

The writer is greatly indebted to Dr. W. W.Rubey, Chairman of the National ResearchCouncil, and Dr. Luna B. Leopold, Water Re-sources Division of the U. S. Geological Survey,for critically reading the manuscript and mak-ing many suggestions for its clarification. Mr.James L. Lubkin of the Columbia School ofEngineering developed the model hypsometricfunction; Professor Robert Bechhofer and hisstaff of the Statistical Consulting Service ofColumbia University advised the author ontesting procedures.

PRINCIPLES OF HYPSOMETRIC ANALYSIS

Hypsometric Curve in Absolute Units

Hypsometric analysis is the study of the dis-tribution of ground surface area, or horizontalcross-sectional area, of a landmass with respectto elevation. The simplest form of hypsometriccurve (hypsographic curve) is that in absoluteunits of measure. On the ordinate is plottedelevation in feet or meters; on the abscissa thearea in square miles or kilometers lying abovea contour of given elevation. The areas usedare therefore those of horizontal slices of thetopography at any given level. This methodproduces a cumulative curve, any point onwhich expresses the total area (reduced tohorizontal projection) lying above that plane.

The absolute hypsometric curve has beenused in regional geomorphic studies to show thepresence of extensive summit flatness or terrac-



PRINCIPLES OF HYPSOMETRIC ANALYSIS 1119

ing, where the surfaces lies approximatelyhorizontal. Where these surfaces have a pro-nounced regional slope, they may not appearon the curve. Because a good topographic map,from which the hypsometric curve was pre-pared, will usually show these features, thejustification for an elaborate hypsometricprocess for interpreting geomorphic historyis doubtful.

For analysis of the form quality of erosionaltopography, use of absolute units is unsatis-factory because areas of different size and re-lief cannot be compared, and the slope of thecurve depends on the arbitrary selection ofscales. To overcome these difficulties, it is de-sirable to use dimensionless parameters inde-pendent of absolute scale of topographic fea-tures.

Percentage Hypsometric Curve

Hypsometric analysis, in general use forcalculation of hydrologic information (Lang-bein et al., 1947), takes a complete drainagebasin above a selected point on a main streamas the area of study. The present study of formqualities of erosional topography likewise usesnatural drainage basins, whether single orcomposite, on the assumption that the form ofeach drainage basin results from the interactionof slope-wasting and channel-deepening proc-esses within the limits of the drainage divide,and hence that each basin should be treatedas a unit.

Most drainage basins in homogeneous mate-rials are pear-shaped in outline, with lateraldivides converging to a clearly denned con-striction, or mouth (Horton, 1941, p. 303).For hypsometric study, a geometric unit ofreference consists of a solid bounded on thesides by the vertical projection of the basinperimeter and on the top and base by parallelplanes passing through the summit and mouthrespectively (Fig. 1). Although both of thesereference planes may be expected to changeas the basin is denuded, they are real pointswhich can always be determined.

The percentage hypsometric method usedin this investigation relates the area enclosedbetween a given contour and the upper (head-ward) segment of the basin perimeter to theheight of that contour above the basal plane.

The method has been used by Langbein (1947)for hydrologic investigations. Two ratios areinvolved (Fig. 1): (1) ratio of area betweenthe contour and the upper perimeter (Area a)to total drainage basin area (Area A), repre-

Mouth

Area aa

Area ^(entire basin) "

FIGURE 1.—FIGURE or REFERENCE IN PERCENTAGEHYPSOMETRIC ANALYSIS

Showing derivation of the dimensionless parame-ters used in Figure 2.

sented by the abscissa on the coordinate sys-tem. (2) Ratio of height of contour above base(h) to total height of basin (H), representedby values of the ordinate.

The resulting hypsometric curve (Fig. 2)permits the comparison of forms of basins ofdifferent sizes and elevations. It expressessimply the manner in which the volume lyingbeneath the ground surface is distributed frombase to top. The curve must always originatein the upper left-hand corner of the square(x = 0, y = 1) and reach the lower right-hand corner (x = 1, y = 0). It may, however,take any one of a variety of paths betweenthese points, depending upon the distributionof the landmass from base to top.

Method of Obtaining Hypsometric Data

Actual measurement and calculation of hyp-sometric data have been done by the writer in




the following steps: First, the drainage basinis selected and outlined. Selection of the basinis influenced by the purpose of the investiga-tion, which may call for a study of the first-order drainage basins or of composite basins

Y1.0

&S>O)^^c c2'*•>- oo^Q.O

0 0.5 1.0Proportion of total basin area

FIGURE 2.—THE PERCENTAGE HYPSOMETRICCURVE

whose trunk streams have an order of 3, 4, orhigher.1 Having made this decision, the operatordraws in the drainage divide on the map. Thedivide is carried down to the stream at its pointof junction with a stream of the same or higherorder.

With a polar planimeter, the operator meas-ures first the area of the entire basin, then theareas enclosed between each contour and theupper perimeter. Ratios are computed and willrange from 1.0 to 0.0. Where relief is strong andcontours closely crowded, every second orfifth contour is used, except near the summitwhere all available contours are used. Obviouslythe value of hypsometric analysis depends onuse of sufficiently accurate and large-scalemaps for the drainage basins involved. Wheretexture is fine and unit basins very small,

1 Stream orders have been defined by Horton(1945, p. 281-283), but the writer has followed asomewhat different system of determining orders:The smallest, or "finger-tip", channels constitutethe first-order segments. For the most part thesecarry wet-weather streams and are normally dry.A second-order segment is formed by the junctionof any two first-order streams; a third-order seg-ment is formed by the joining of any two second-order streams, etc. This method avoids the neces-sity of subjective decisions, inherent in Horton'smethod, and assures that there will be only onestream bearing the highest order number.

special field maps on a large scale must first besurveyed.

Height ratios are obtained by first determin-ing the total range between basin mouth andsummit point. The height of each measuredcontour above the mouth elevation is thendetermined and ratios to total basin heightcomputed. These will range from 0.0 to 1.0in inverse series to the area ratios.

The ratios are plotted on any convenientcross-section paper and the curve drawnsmoothly with the aid of a draftsman's curve.For purposes of comparison with model curvesillustrated in Plate 1, cross-section paper of10 divisions per % inch should be used, allottinga square 5 inches wide to the hypsometricgraph.

Integration of the Hypsometric Function

In order to calculate the volume of earthmaterial contained between the ground sur-face and the bottom and sides of the figureof reference (Fig. 1), the landmass may bethought of as consisting of horizontal slabs(Fig. 3). The total volume, V, consists of thesum of all slabs. The volume of one slab,AF, is obtained by multiplying the area of theslab, a, by its thickness, A/!. Following themathematical principle of integration, the en-tire volume may be stated by the expression

=/;summit eladh.

..aaeel

If we now divide both sides of this equationby H and A, which are constant terms,

summit el

y rsummitel^ / fr

ff3=Jbas..l A

This expresses the ratio of volume lying be-neath the surface, V, to the entire volume of the

reference figure, HA. Because —A

y, by our definition, then

v rw

x, anda.

II




Thus, if the hypsometric function, x =f(y), is integrated between the limits of x =0 and x = 1.0, a measure of landmass volumeremaining with respect to volume of the entire

performed to obtain information useful inhydrologic and other applications.

Inspection of a large number of hypsometriccurves has shown that the majority are s-

AV=QAh

Y

100%

-c0>

ot>cc

Vol =•summit elodh

'base el

Relative area 100%

Vol_

HAFIGURE 3.—INTEGRATION or THE HYPSOMETRIC FTTNCTION

And meaning of hypsometric integral.

reference solid is obtained. This integral ishere designated the hypsometric integral andis equivalent to the ratio of area under thehypsometric curve to the area of the entiresquare. It is expressed in percentage units andcan be obtained from any percentage hypso-metric curve by measuring the area under thecurve with a planimeter. Whether the integra-tion is of the function y = f(x) or x — j(y) isof no consequence. The latter function wasused in this explanation because the unit slabsof volume are thought of as being horizontal,rather than vertical.

As discussed elsewhere in this paper, boththe form of the hypsometric curve and the valueof the integral are important elements in topo-graphic form and show marked variations inregions differing in stage of development andgeologic structure.

A Model Hypsometric Function

It is desirable to find a relatively simple, yetflexible function which may be fitted to anynatural hypsometric curve. This is necessaryso that certain mathematical operations can be

shaped. An up-concavity is commonly presentin the upper part; a convexity in the lower part.Sinuosity varies greatly so that the slopes ofthe curves at their inflection points have a widerange. It is therefore necessary to use an equa-tion having two parameters, one to vary thehypsometric integral, the other to control thesinuosity.

A function2 which meets these requirementsfairly well is

ra

where a and d are constants, d always greaterthan a, and the exponent z, positive or zero(Fig. 4). All curves pass through A and B.The slope of the curve at its inflection point

depends on the ratio - , hereinafter designated

r. The general location of the curve dependsupon the exponent z.

2 The writer is indebted to Mr. James LeighLubkin of the School of Engineering of ColumbiaUniversity for developing this equation. It wasadapted from a somewhat similar equation usedby Hunter Rouse (1937, p. 536) to describe thedistribution of suspended load in a stream.




FIGURE 4.—MODEL HYPSOMETRIC FUNCTION

d-x= ̂ -0.1

Curve of inflection) points

'>H

.1 .Ex-a.3 .4 .5 .6 .7 .8 .9 l<0

(Proportion of total basin areaj

FIGURE 5.—FAMILY OF CURVES FOR THE VALUE,r = 0.1

For selected values of 2. Given alse are the inte-grals and the slope of each curve at its inflectionpoint. (Other curve families are given in Plate 1.)

abscissa as shown in Figure 4 should range from0 at x = a to 1.0 at x — d. This percentage, Ris therefore expressed as

Rd-a'

In subsequent illustrations of the model hypso-metric equation (Figs. 5, 6; PL 1) the abscissaappears scaled in terms of R.

To plot a family of model curves having oneparticular degree of sinuosity, a value of /is selected; curves within each family are therjobtained by using different values of the ex-ponent, z.3

As an illustration of a family of curves, thatparticular family in which r = 0.1 is givenin Figure 5. Curves for several values of z,ranging from 0.0625 to 2.0, are shown. Plate1 gives five families of curves and can be usedfor fitting of natural curves by inspection.Curves represented by this model functionhave the following characteristics (1) The curvesare s-shaped where z < 1, but are of simpleconcave-up form where z > 1. (2) Where z< 1, curves entering at A have a slope, where-as they are tangent to the vertical through thepoint B.

Decreasing the value of r increases the degreeof sinuosity of the curve, thereby reducing theslope of the curve in the region of inflection.This effect may be seen by studying individualcurves for the families r= 0.01, 0.05, 0.1, 0.25,and 0.5 (Fig. 6). For comparison, five curveswere selected whose integral is approximatelythe same.

It is not practical to obtain the hypsometricintegrals of theoretical curves by mathematicalprocedures, hence these were obtained by thewriter by planimeter measurement for allcurves plotted. On each model curve (PI. 1),the integral is given. The values are only ap-proximate, being subject to errors in measure-

3 For plotting, the following form of equation IIIcan be used:

In order to have a percentage scale on the IV y = —£— j 1 "abscissa, conforming with the percentage hyp- rJ L( — r) R + r Jsometric function as previously defined, a where r and R are as defined above. For a givenmodification of equation III is introduced. curve, r and z are constants; hence, by substitutingT. . , . , ,, . , . , a series of values of R ranging from 0 to 1.0. theIt is desired that the scale of values on the corresponding values of y may be obtained.




merit as well as errors in plotting the curves the second derivative of the function equal to 0.from which they were measured. For plotting, it is convenient to find the inflec-

FIGURE 6.—COMPARISON OP SEVERAL CTOVE FAMILIESShowing the effect of varying the value of r in the model hypsometric function. Integrals of these curves

are approximately the same.

Because one method of fitting model curvesto natural hypsometric curves involves thematching of integrals, it is desirable to have ameans of obtaining from a given integral theexponent, z, of a particular model curve whichpossesses that integral. A graphic solution isshown in Figure 7. Given an integral, measuredby planimeter from a natural hypsometriccurve, and having selected by inspection thecurve family whose value of r gives the closestfit as to shape, one can read the desired valueof z.

Inflection Points and Slopes

The point of inflection on any of the modelhypsometric curves where z is less than 1.0may be obtained by the usual method of setting

tion point in terms of R in Equation IV, asthe following equation:

V Jfc - 1 + * ~ 2r

2(1- r)

where Ri is the value of R at which the curveinflects. Inflection points and the curves onwhich they lie are shown on the graphs for theseveral values of r (Fig. 5; PI. 1).

Inflection points have morphological sig-nificance on hypsometric curves because theymark the level at which the rate of decrease ofmass upwards changes from an increasinglyrapid rate of decrease to a diminishing rate ofdecrease. Further investigation may provethis feature to be related to dynamic factors,such as the relative importance of sheet runoff




Equilibriumi .postage

(Maturity,Old Age )

.2 .3 .4 .5 .6 .7 .8 .9 1.0

IntegralFIGURE 7.—GRAPHIC SOLUTION OF INTEGRALS AND EXPONENTS

For curve families produced by five selected values of r. (See Plate 1 for further data.)

and creep at higher levels compared to channelerosion at lower levels.

While the position of the inflection point on anatural hypsometric curve is greatly affectedby chance irregularities of form not significantin the gross aspect of the drainage basin, theslope of the curve in the general region of theinflection can be expected to be a reliable formelement. Comparisons of the curve familiesshow that slope at the inflection point is steepwhere r has high values and diminishes as r

decreases. For the curve family r = 0.5, theslopes approach 80 per cent near the center ofthe diagram, while for the family r = 0.01 theyare reduced to about 30 per cent.

Hypsometric slope at the inflection point isthus a form characteristic which can be rapidlydetermined and used as one means of fittingnatural to model curves. When the slope of thenatural curve in the vicinity of its inflectionpoint has been measured, the curve can bematched to the family having a similar slope.




Then, by matching integrals, the particularvalue of z can be determined.

Precise values of slope at inflection pointscan be determined from Equation IV by takingthe first derivative of the function and sub-stituting for R the various values of these in-flection points already obtained. In view of thelabor of calculation involved and the fact thatexact values are not required for any uses ofhypsometric analysis thus far made, the slopeslisted opposite each integral on the graphswere determined by direct angular measure-ment from the graphs. These are, of course,subject to errors in the use of the protractor ona curve drawn through a number of plottedpoints.

Relation of Hypsometric Curve to Ground Slope

Characteristics of the hypsometric curve areclosely related to ground-slope characteristicsof a drainage basin. This is evident from thefact that steepening of slopes in the mid-sectionof a basin will be accompanied by a morerapid rate of change of elevation with respectto change of horizontal cross-sectional area ofthe basin. One might, at first thought, supposethat steep parts of the hypsometric curve wouldcoincide with belts of relatively steep slopes,gently sloping parts of the curve with gentleground slopes. Unfortunately the relationshipis not so simple. Figure 8 shows a small drain-age basin; Figure 9 is the corresponding hypso-metric curve. The curve has a gentle slopein the upper part, corresponding with a broaddivide area on the map. The steep interme-diate part of the hypsometric curve corre-sponds with steep valley wall slopes in the mid-section of the basin. But the very lowest partof the curve is steepest of all in the regioncorresponding to the mouth area of the basin,whereas the contours of the map show that theground slopes are less here than in the mid-section of the basin. The additional factor is,of course, the length of the belt between suc-cessive pairs of contours. ("Length" refers todistance along the contour.) Only if each con-tour belt is the same length can steepness ofground slope vary directly as steepness ofhypsometric curve. In Figure 10, all contourshave the same length, and the slope profile is

identical with the hypsometric curve. Ob-viously a drainage basin cannot fulfill this con-dition while narrowing to a mouth throughwhich all drainage is discharged by a narrow

87.0

90

99.70

100.9?

100.4

10 Feet

FIGURE 8—SMALL DRAINAGE BASIN IN BADLANDS,PERTH AMBOV, NEW JERSEY

From a special large-scale topographical survey.

channel; a shortening of the length of contoursto a minimum approaching zero is required asthe drainage basin is followed to its mouth.At the upper end of the drainage basin, thecontours can maintain nearly equal lengthup to the divide (which may be horizontal),but normally the contour length diminisheshere, too, to approach zero on the highest peak.Thus the characteristic steepening of hypso-metric curves both at the lower and upper endsin mature topography is explained by the di-minishing contour lengths.

To relate hypsometric curve to ground slopeit is necessary to take contour length intoaccount. First, the length of each contourline is measured. For each belt of ground be-tween two successive contours the lengths ofthe upper and lower contours are added and thesum divided into two, giving a rough meanlength for the contour belt (Fig. 11). Next thearea of the contour belt is measured by planim-eter. Dividing area of the contour belt by meanlength gives a rough mean width (horizontaldistance) for the belt. Now, by dividing thecontour interval by the mean width we can




ACTUALGROUND H

SLOPE

°08'42^x42° 16'

57o\53°08'

57\53°085753°08'

HYPSOMETRIC ~>\44°09'CURVE -^ 3^40*03'

5703a\35044'

65°30\36°56'

75°\46°281

79°\420481

_ 840\32000''0 Tl .2 .3 .4 .5 .6 .7 .8 .9 1.0

Relat ive area -J-FIGURE 9. — HYPSOMETRIC CURVE OP BASIN SHOWN IN FIGURE 8

Showing relation between slope of segments of hypsometric curve and actual mean ground slopes ofcorresponding segments.

1.0FIGURE 10.—HYPOTHETICAL DRAINAGE BASIN

Upper contour^FIGURE 11.—CONTOUR BELT

Showing method of calculating mean length,width, and slope of contour belt.

determine the mean slope of the ground withinh

In which slope of hypsometric curve is identical this particular contour belt, for, tan a = — ,with ground-slope curve. w




X 1.0

UJ

IDO

O

Ei-iLl

Oo.xu_oUJoCO

0.0

^

0.0 0.5GROUND SLOPE

1.0( tan c

1.5

FIGURE 12.—CORRELATION OF MEAN GROUND SLOPES AND ADJUSTED SLOPES OFHYPSOMETRIC CURVE SEGMENTS

Basin 1 same as that in Figures 8 and 9.

wherea is angle of ground slope,h is contour intervalw is mean width of the belt measured in

horizontal projection.Values of mean slope angle for the basin

shown in Figure 8 are written directly on thehypsometric curve (Fig. 9) opposite the par-ticular segments to which they relate. Thecalculated mean slope figures compared withthe slope of the hypsometric curve showsrough correspondence only in the upper part.If, however, we correlate the mean groundslope figures with the contour map of thebasin, the slope angles vary as the spacing ofthe contours, being highest in the midsection,where slopes up to 53° are found.

Relationship of hypsometric curve to mean

ground slopes may be summarized by the fol-lowing equation, which takes into accountrelative length of each contour belt.

VI -J-J

K tana

where 9 — slope of hypsometric curvea = mean ground slope/ = contour length at given relative height

L = length of longest contour in basinK = a constant

To test the usefulness of this equation, thevalues of ground slope have been plotted againstcorresponding values of hypsometric curveslope for each contour interval of the drainagebasin (Fig. 12, Basin 1). Also plotted on Figure12 are corresponding data for a second drainage



1128

Q)0>

Co

95

LJ

90

A. N. STRAHLER—ANALYSIS OF TOPOGRAPHY

I2°08'

9l Mean slopebetween, successive

530 °o8 , one-foot

53° 08 contours..4 4° 09

40° 03'

Mean contour-belt width. (One-foot units)( cumulat ive)

FIGURE 13.—TRUE MEAN-SLOPE CURVE OE BASIN SHOWN IN FIGURE 8Abscissa and ordinate on same scale.

basin which is in the equilibrium (mature)stage of development and has a narrow divideridge crest. Note that the two curves, whichwere fitted by inspection, pass through theorigin but have markedly different slopes,which may be attributed to the difference instage of development of the two basins. Thetangent function is extremely sensitive to smallerrors of horizontal measurement, and, becausethe range of error in measurement from the mapis relatively large, the values are subject to con-siderable variation. Hence these correlationdiagrams should be thought of as only demon-strating the general validity of Equation VI.

A profile of the true mean ground slope(Fig. 13) is a cumulative plot of mean-slopeangles for each contour belt. This curve differsfrom the hypsometric curve of the samebasin (Fig. 9) in that the mean-slope curve isplotted with absolute values, the scale of feetbeing the same on both ordinate and abscissa.By use of this curve, ground slope distributionwith respect to height can be depicted fordirect visual analysis, inasmuch as the slope ofthe curve is the actual mean ground slope.

GEOMORPHIC APPLICATIONS OFHYPSOMETRIC ANALYSIS

The Geomorphic Cycle

The hypsometric curve exhibits its widestrange of forms in the sequence of drainagebasins commencing with early youth (inequi-librium stage), progressing through full matu-rity (equilibrium stage), and attaining tem-porarily the monadnock phase of old age.

A drainage basin in youth is shown in Figure14. It is from the Maryland coastal plain wherea large proportion of upland surface has notyet been transformed into valley-wall slopes.The hypsometric curve has a very high integral,79.5%, indicating that about four-fifths of thelandmass of the reference solid remains. De-spite the bold convexity of the curve throughits central and lower parts, the upper end hasthe concavity typical of nearly all normaldrainage basins, and shows that some reliefdoes exist in the broad divide areas.

Figure 15 represents a small drainage basinin fully mature topography of the Verdugo



GEOMORPHIC APPLICATIONS OF HYPSOMETRIC ANALYSIS 1129

TOO

50.795

0 .50 1.00

FIGURE 14.—INEQUILIBRTOM (YOUTHFUL) STAGEDrainage basin ofj Campbell Creek on the Maryland Coastal Plain (above) with its hypsometric

curve (below). From Yellow Tavern Quadrangle, Virginia, U. S. Geological Survey, 1:31,680.

Hills, southern California. Here divides arenarrow and no vestiges remain of an originalsurface. The hypsometric curve passes ap-proximately across the center of the diagram,with a hypsometric integral of 43%, and issmoothly s-shaped. This particular curve istypical of third- or fourth-order basins in rela-tively homogeneous rocks.

In late mature and old stages of topography,despite the attainment of low relief, the hypso-

metric curve shows no significant variationsfrom the mature form, and a low integral re-sults only where monadnocks remain. For ex-ample, a drainage basin in northern Alabamawhere low relief has developed on weak shalesand limestones, but with prominent monad-nock masses of sandstone which are outliersof a retreating escarpment, has a stronglyconcave hypsometric curve; the integral,17.6%, is unusually low (Fig. 16). After monad-




1080

FIGURE IS.—EQUILIBRIUM (MATURE) STAGEA small drainage basin in the Verdugo Hills,

near Burbank, California (above), correspondinghypsometric curve (below). From Sunland Quadran-gle, California, U. S. Geological Survey, 1:24,000.

nock masses are removed, the hypsometriccurve may be expected to revert to a middleposition with integrals in the general range of40% to 60%.

From the standpoint of hypsometric analy-sis, the development of the drainage basin in anormal fluvial cycle seems to consist of twomajor stages only; (1) an inequilibrium stageof early development, in which slope trans-formations are taking place rapidly as thedrainage system is expanded and ramified.(2) An equilibrium stage in which a stable

hypsometric curve is developed and maintainedin a steady state as relief slowly diminishes.The monadnock phase with abnormally lowhypsometric integral, when it does occur,can be regarded as transitory, because removalof the monadnock will result in restoration ofthe curve to the equilibrium form.

Figure 7 shows relations of hypsometricintegral, curve form, and stage of development.Values of z are plotted against hypsometricintegrals for each of five families of curvesrepresented by five values of r. From inspectionof many natural hypsometric curves and thecorresponding maps, the writer estimates thattransition from the inequilibrium (youthful)stage to the equilibrium (mature) stage corre-sponds roughly to a hypsometric integral of60%, but that where monadnocks becomeconspicuous features the integrals drop below35%. These two percentages have, therefore,been used as tentative boundaries of the stagesin Figure 7.

The hypsometric curve of the equilibriumstage is an expression of the attainment of asteady state in the processes of erosion andtransportation within the fluvial system andits contributing slopes (Strahler, 1950). In thisstate, a system of channel slopes and valley-wall slopes has been developed which is mostefficiently adapted to the reduction of the land-mass with available erosional forces, balancedagainst the resistive forces of cohesion main-tained by the bedrock, soil, and plant cover.The basins are no longer expanding in area;they are in contact with similar basins on allsides. The general similarity among hypso-metric curves of regions in the equilibriumstage, despite great differences in relief, drain-age density, climate, vegetation, soils, and li-thology, seems to show that the distribution ofmass with respect to height normally followsthe s-shaped model hypsometric curve with itsupper concavity and lower convexity.

Characteristics of the Equilibrium Stage

Five areas were selected which showed agreat range of relief, and for which excellentlarge-scale topographic maps and air photo-graphs were available. Within each area, sixbasins of the third or fourth order were out-lined and the hypsometric curves plotted foreach. A mean curve for each area was obtained




925

0 Feet 3000

620

1105

100

_H

.50

.176

.50 100

FIGURE 16.—MONADNOCK PHASEDrainage basin of Atwood Branch, Newburgh Quadrangle, Alabama (above) showing remnants

of retreating sandstone escarpment; corresponding hypsometric curve (below).

by plotting the arithmetic means of the or-dinates of the six individual basin curves atevery ten per cent division on the abscissa(Fig. 17). Figure 18 shows one drainage basinfrom each of the five areas; that basin was se-lected whose hypsometric curve most closely

follows the mean curve shown in Figure 17.In this way the reader can visualize the appear-ance of a drainage basin embodying the charac-teristics of the mean hypsometric curve. Table1 gives additional data relating to compositionof the drainage systems.




The five areas selected are all areas of den-dritic drainage, largely free from significant

1.0

.9

.8

.7

.6

.5

Cambrian Wissahickon schists of the PiedmontProvince in Virginia by the first area: moderate

ETo>.c .4

0>QC

0 Piedmont (Va.)(|) Gulf Coastal Plain (La.)(DOzark Plateau (III.)(DVerdugo Hills (Calif.)

.©Great Smoky Mts. (N.Car.)

.1 .2 .3 .4 .5Relative area

.6 .7 .8 .9 1.0Q_A

FIGURE 17.—MEAN HYPSOMETRIC CURVES OF FIVE AREAS IN THE EQUILIBRIUM STAGECurve 1: from Belmont Quadrangle, Virginia, U.S.A.M.S. 1:25,000. Curve 2: from Mittie Quadrangle,

Louisiana, U. S. Geological Survey, 1:24,000. Curve 3: Wolf Lake Quadrangle, Illinois, U. S. GeologicalSurvey, 1:24,000. Curve 4: La Crescenta, Glendale and Sunland Quadrangles, California, U. S. GeologicalSurvey, 1:24,000. Curve 5: Judson and Bryson Quadrangles, North Carolina, T.V.A., 1:24,000.

structural control. Long-continued fluvial ero-sion has removed all traces of flat interstreamuplands and it is assumed that the basins arestable in form and that the total regimen oferosion and transportation processes is in asteady state. In relief, lithology and rockstructure, vegetation, and climate, however,the five areas differ widely. Extremely lowrelief on weak Pliocene deposits of the Citro-nelle formation in western Louisiana is repre-sented by the second area; low relief on Pre-

relief developed on cherts and cherty limestonesof the Ozark Plateau province is exemplifiedby the third area. Extremely rugged terrain ofstrong relief and steep slopes on deeply weath-ered metasediments of the lower coastalranges of the Los Angeles region is seen in thefourth area; great relief with moderately steepslopes on deeply weathered Precambrian Wissa-hickon schists of the southern flank of the GreatSmoky Mountains in the fifth area.

Investigation of the five areas involved:



GEOMORPfflC APPLICATIONS OF HYPSOMETRIC ANALYSIS 1133

500 YDS 500 YDS

I. Piedmont (Va.)

500 YDS

2. Gulf Coastal Plain (La.)

3. Ozark Plateau (III.)

500 YDS

250 YDSi 1

4. VerdugoHills (Calif.)

5. Great Smoky Mts. (N.C.)

FIGURE 18.—REPRESENTATIVE BASINS FROM FIVE SAMPLE AREASFShowing the one drainage basin whose hypsometric curve most closely fits the sample mean curve of

Figure 17. Localities as described in Figure 17.

(1) analysis of the hypsometric curves, similar-ities and differences, and their degree of resem-blance to the model hypsometric function;(2) a comparison of hypsometric data withother categories of data, such as drainage net-work and slope characteristics. It was hopedthat significant differences in the hypsometriccurves could be correlated with significantdifferences in other drainage basin characteris-tics, and that this might provide clues to causa-tive factors determining the hypsometric prop-erties of mature topography.

The mean curves shown in Figure 17 haveappreciable differences both in hypsometricintegral and in form. The mean curves werefitted to the theoretical function by inspection,and the apparent best fits are described on thecurves and in Table 1 by values of r and z.All five curves were best described by the fam-ilies having r values 0.1 or 0.25 and we mayinfer that mature topography in relativelyhomogeneous materials tends to fall withinthis general range. Fit was very good for curves1 and 5, but was good only in the inflection




TABLE 1. MORPHOMETEIC DATA

L O C A L I T I E S INVESTIGATED

1. Piedmont

2.Gulf Coastal Plain

3.0zark Plateau

4.V«rdugo Hills

5. Great Smokies

Quadrangle

Belmont, Va.USAMS 1-25,000

Mittie, La.USGS 1=24,000

Wolf Lake, III.USGS l'24,000

Glendale.Sunland,Cat USGS l>24,000

Bryson Judson,N.C. USGSI-24jrjOO

STREAM NUMBERS

Total numberof Streams of

each order

!„, Z^.Zn.Eiv

141 34 6 0

96 27 8 (2)

198 38 10 (4)

201 38 9 0

389 87 24 6

Bifurcation

Ratios' FJ,

^ Z"s

4.15 5.67

355 3.37

5.21 330

5.29 4.22

4.47 3.62

STREAM LENGTHS

Mean length ofStream segments of

each order: Miles! • 1 iI, X2 43

0.234 0.345 1.130

0260 0.427 0.844

0.099 0.132 0368

0.062 0.116 0.295

0.115 0.185 0.269

LengthRatios

\ "%

1.47 327

1.65 1.97

1.33 2.79

1.87 254

1.61 1.60

zone and at one end in the other three. Allnatural hypsometric curves suffer from somedegree of misfitness at the lower end owing tothe development of a valley-bottom flat whichprevents the curve from approaching the valueof 1.0 on the abscissa as closely as on themodel curves.

All five mean curves show a similar slope inthe inflection zone. This ranges from 0.52 to0.65 (27J^° to 33°), and may prove to be a com-mon characteristic of the mature or equilib-rium form, along with the tendency to resemblethe family of curves having values of r of 0.1to 0.25. Note also that the location of the in-flection point of the curve is generally higherfor the areas of low relief (Nos. 1-3) than in theareas of great relief (Nos. 4 and 5). Within anyone of the families of model curves, the inflec-tion point likewise moves down as the integraldiminishes, but in the five mean curves shownhere the inflection points all tend to be locatedhigher than in the model curves to which theywere fitted.

Because each of the mean curves representsa sample of only six basins, and the differences,while conspicuous on the graph, are not great,it might well prove that the differences be-tween integrals are not statistically significant,but might result from expectable variations

inherent in small samples despite the fact thatno real differences exist from one area to theother as regards the hypsometric characteris-tics. We must assume first that the samplingwas randomized. In actual fact, basins wereselected which appeared most representativeof the general facies of the area as a whole.None was discarded or added after data analy-sis was begun. At the time of selection thewriter was not aware of possible differences inhypsometric or other form characteristics whichmight later appear, nor did he have in mindany particular trend which he expected theanalysis to reveal. The selection, thereforewhile not mechanically randomized, is thoughtto be free of conscious prejudice.

Table 2 gives the sample mean, estimatedstandard deviation of the population (s), andstandard error of the mean (sj) for each sam-ple, consisting of the hypsometric integrals ofthe six individual basin curves. The table alsoshows the percentage probabilities of any twosamples being drawn from a population withthe same mean. The significance test is basedupon the I distribution, which is used for smallsamples. In this instance all tests involvedsamples of 6 and the table of t is entered underthe heading of 10 degrees of freedom. Theprobability stated is that representing the area



GEOMORPHIC APPLICATIONS OF HYPSOMETRIC ANALYSIS

JOR FIVE SAMPLE AREAS

1135

DRAINAGE DENSITY

TotalArea of6 basinsin eachlocality(sq. mi.)

7.47

9.58

226

0.77

5.14

TotalStream

Lengths

EL(miles)

5159

44.40

31.10

2028

72.71

Dd

3

£LA

6.90

4.64

13.78

26.17

14.16

Meanbasinheight

H (feet)

175.0

61.0

326.0

875.8

1880.2

CHANNELGRADIENTS

Mean StreamGradient of 3rdorder Streams'

% (tan) Degrees

0.0113

0.0033

0.0352

0.2246

0.1233

0-39'

0*10'

1*52'

12*40'

ro2'

GROUNDSLOPES

Valley-WallSlopes, Mean

value

Degrees %

9.9° .1745

3.4* .0594

28° 15' 537

44.7* .9896

41* 15' .867

HYPSOMETRIC CURVE

MeanSub-

surfaceIntegral

/

£968

5420

.4928

.4684

.4084

Best fit tomodel

hypso. function

P Z

025 .333

0.10 25

0.10 29

025 50

0.10 .40

Slope ofHypsometriccurve atinflectionpoint

%

.6009

5317

.5890

.6494

.5206

TABLE 2. STATISTICAL DATA FOR MEAN INTEGRALS or FIVE AREAS*MEAN

L O C A L I T Y INTEGRAL* S Sx P R O B A B I L I T Y

I.PIEDMONT, VA. 59.27 655 2.67^

2.GULF COASTAL 54.01 5.20 2.12-PLAIN , LA.

3 .0ZARK 4 8 9 1 567 2 3 1PLATEAU, ILL. 2'31

4 .VERDUGO 46.52 4.58 1.87-^H ILLS, CAL.

.16

.14

.44

.08

-,.016

.026

J.036

.005

.004

• COOI

5.GREAT SMOKY 40.61 588 240N. CAR.*Showing means, estimated standard deviation of the population, s, and standard error of the mean,

sf, as estimated from the sample. Probability figures refer to results of t tests of significance of differencein sample means of each pair indicated by bracket. Although integrals are here ranked in descending order,as in Figure 17, the probability figures have no relationship to ranking significance. Probability figure tellsonly the percentage of times that sample means drawn repeatedly from the same pair of areas will differby this amount or more through chance variations in sampling alone, assuming that no real difference inthe two population means actually exists.

under both tails of the t distribution curve,and hence tells the probability of obtainingdifferences of sample means as great as, orgreater than, the observed differences, withthe possibility of either mean being the larger.

Note that, in Table 2, no significant differenceis found between means of any two sampleswhose mean integrals differ by only 5 or 6%or less, but is present when the means differby 8% or more. While we cannot easily deter-




mine the significance of the ranking, or theprobability of rearrangements being likely tooccur in the ranking if similar samples of sixbasins were repeatedly drawn, we can perhapssafely infer that any two consecutive membersof the series might readily reverse their orderif another set of samples was taken, but thatit is most unlikely that one of the last two mem-bers of the series could switch places with thefirst two.

Relation of Hypsometric Forms to DrainageForms

It is not immediately apparent just why anytwo integrals of the mean hypsometric curvesshould differ significantly, or why they shouldfall into the general sequence which they take.In an effort to obtain clues to this problem,measurement was made of the stream numberand length characteristics, drainage density,slopes, relief, and stream gradients. Thesedata are tabulated in Table 1. A number of ob-servations relating to correlation, or lack ofcorrelation, among the various form factors ofthe topography are as follows:

In general, drainage basin height, slopesteepness, stream channel gradients and drain-age density show a good but negative correla-tion with the integral of the hypsometric curve.We may say that mature basins of low relief,gentle slopes, gentle stream gradients, and lowdrainage density tend to have relatively highintegrals; that areas of strong relief, steepslopes, steep stream gradients, and high drain-age density tend to give relatively low integralsin the average drainage basin of the third orfourth order. Table 1 bears this out well ifover-all trend of the series is considered, butthe values of areas 1 and 2 are in reverse order,as are the values of areas 4 and 5. As alreadystated, however, differences of integral in thesetwo pairs of samples are not significant (seeTable 2) and they might easily exchange posi-tions on the list if another sample were taken.What is significant is that Nos. 1 and 2 showvery much lower values of drainage density,basin height, slope steepness, and stream gra-dient than do Nos. 4 and 5, while No. 3 occu-pies an intermediate position in all cases.

No correlation seems to exist between hypso-metric integrals and either bifurcation ratios or

length ratios (Figs. 19, 20). Horton (1945, p.290) states that bifurcation ratios range fromabout 2 for flat or rolling country up to 3 or 4for mountainous regions. The writer's data,based on large-scale maps checked in the fieldor by stereoscopic study of air photographs,show not only considerably higher ratios, but acomplete lack of correlation of ratio with relief.Horton's data were taken from comparativelycrude, small-scale maps and he must haveomitted a large proportion of the stream chan-nels of first and second order which actuallyexist.

A positive correlation is evident between theaverage length of the stream segments of anygiven order in each area and the correspondingmean hypsometric integrals. Figure 20, in whichmean stream lengths are plotted against ordernumbers, shows progressive decline in streamlength from left to right, in the same order asthat in which the integrals diminish. Althoughreversals occur in the trends of the first andsecond order lengths, the values for areas 1 and2 are always higher than those of areas 3, 4and 5.

Because length of stream segments tends tobecome less as drainage density increases, it isonly to be expected that the first two areas,whose texture is coarse, would have longerstream segments than the last three areas,whose texture is much finer. Now, since themean integrals decrease as drainage densityincreases, the effect is to give a positive correla-tion between mean stream segment lengthsand mean hypsometric integrals.

Geologic Factors A/ecting Equilibrium Forms

Turning from a purely quantitative analysisof the various categories of morphometric datato a qualitative approach, there are severaltopographic and geologic factors apparent tothe investigator to which he can attribute cer-tain of the differences in hypsometric curveforms.

The extreme members of the series (curves 1and 5, Fig. 17) are developed on essentiallysimilar types of rock, mapped as the Wissa-hickon schist. A / test of significance of dif-ference of sample mean integrals (Table 2)shows a probability less than .001, leading us todiscard the hypothesis that both samples have




1000

LJ

STREAM ORDERSFIGURE 19.—STREAM NUMBERS AND BIFURCATION RATIOS FOR FIVE SAMPLE AREAS

Fitted curve has slope of bifurcation ratio, n, whose mean value is given for each area. Number besideeach dot is order number.

0.04STREAM O.RDERS

FIGURE 20.—STREAM LENGTHS AND LENGTH RATIOS FOR FIVE SAMPLE AREASFitted curve has slope of length ratio, ri, averaged for each area. Number beside each dot is order number.




the same popalation mean. The hypothesisthat similarity of rock gives similarity of in-tegral is not sustained. Some other cause (orcauses) has produced a significant difference inthe mean hypsometric integrals.

Cause of the hypsometric curve differencesmay lie in the geomorphic histories of the twoareas. The Piedmont locality is thought to havebeen reduced to a peneplain, then dissectedinto a rolling topography of low relief in thepresent cycle. If so, the high integral (almost0.60) may be an expression of submaturity,with extensive divide areas as yet not entirelytransformed into the equilibrium slopes of themature stage. But neither field examinationnor map-air photograph study shows a distinc-tive unconsumed upland element, such as oneis accustomed to seeing, for example, on theMaryland coastal plain (Fig. 14) or in the olderdrift plains of the middle west where maturityis being approached. Instead, the divides arebroadly rounded and nothing suggests a com-posite topography involving two distinct cycles.The high integral of this hypsometric curvemay, however, mean that, following the at-tainment of an equilibrium system, an accelera-tion of stream corrasion associated with increas-ing relief set in, perhaps induced by regionalupwarping and an over-all steepening of gra-dient of east-flowing master streams. Do wehave here a manifestation of the Penckianprinciple of waxing development (aufsteigendeEntwicklung) ?

The basins in the south flank of the GreatSmoky Mountains produce a mean hypso-metric curve with an unusually low integral,about 0.40 (Fig. 17). The inflection point islocated low on the curve, and the upper two-thirds of the curve takes a broadly concaveform. The topographic maps show a noticeablesteepening of slopes above the level of 2800-3000 feet occurring at about 40%-50% ofthe basin height. The steepening of slopes withhigher elevation is not sharply denned, as in-structural benching found in a region of hori-zontal strata, but may be caused by differencesin rate of rock weathering at low and high alti-tudes. For example, if rate of alteration of thefeldspars and ferromagnesian minerals wereappreciably faster in the warmer temperaturesof the valleys, an opening out of the valleybottoms might perhaps be expected.

Among localities 2, 3, and 4, hypsometricdifferences are not strong. The curves of theOzark Plateau basins and those of the VerdugoHills basins are remarkably similar, with nosignificant difference statistically (Table 2),despite the fact that the Ozark Plateau is aregion of flat-lying Paleozoic chert and chertylimestone with an over-all uniformity of sum-mit levels, whereas the Verdugo Hills are partof a rugged, up-faulted mountain block con-sisting of metamorphosed sediments and in-trusive bodies. The Ozark curve departs fromthe theoretical function at the upper end, wherean excessive concavity is developed. This maybe an expression of the sapping of weaker for-mations from beneath more resistant beds nearthe summit, a condition which might be ex-pected in horizontal sedimentary strata.

The hypsometric curve of the Louisiana GulfCoastal Plain locality has a relatively highintegral, 0.54, but is otherwise quite conven-tional in appearance. Such small relief andfaint slopes prevail here that very little of valuecan be discerned from the topographic map orair photographs. The area is located withinthe belt assigned to the Montgomery Terraceof Sangamon age by Fisk (1939, p. 193) atelevations from 120 to 140 feet. The surfaceis underlain by the sandy Citronelle formation.The high integral might perhaps be explainedby a submature condition, in which insufficienttime has elapsed for attainment of full maturity.As in the Piedmont locality, however, nothingin the topography suggests remnants of aninitial surface not as yet completely consumed.The high integral may perhaps be a reflectionof slightly accelerated stream erosion rates asa result of recently accelerated southwardtilting of the region associated with epeirogenicuplifts (Fisk, 1939, p. 199) and might perhapsbe a manifestation of waxing development(aufsteigende Entwichlung). At the presentelementary stage of our investigations of thequantitative characteristics of erosional to-pography, we lack criteria for distinguishingamong hypsometric curve forms modified byepeirogenic crustal movements, those modifiedby rejuvenations induced by falling sea level,and those representing stages in attainmentof equilibrium under stable crustal and sealevel conditions.




Influence of Horizontal Structure

It is obvious that drainage basins developedin horizontally layered rocks, whether sedi-mentary strata or lavas, will have stronglymodified hypsometric curves if there aremarked differences in rock resistance on ascale which is large in proportion to the heightof the basin. In the region of cherts and chertylimestones of the Ozark Plateau Province,described above as one of the mature areas inapparently homogeneous materials, structuralbenching did not seem to produce any con-spicuous influence in the hypsometric form.Let us turn, then, to a contrasting example,where structural control is predominant: theregions of cliffs, buttes, and mesas of the south-ern Mesa Verde, located in northwestern NewMexico, within the Rattlesnake and ChimneyRock quadrangles.

Figure 21 compares three hypsometric curves.The first is of a drainage basin about 4 squaremiles in extent consisting of a deeply-incisedcanyon surrounded by a stripped structuralsurface of low relief. The canyon is cut intothe Mesa Verde sandstones and represents adeep re-entrant into the ragged escarpmentrising above a broad lowland of weak Mancosshales. As we might expect, the hypsometriccurve has a high integral, 68%, and resemblesthe curve of a youthful region in the inequi-librium stage of development, except for aconsiderable degree of relief in the upperpart of the basin, above the flattened part ofthe curve which represents the break fromcanyon walls to stripped surface. In the nor-mal curve of the young basin (Fig. 14), reliefon the interstream areas is much less, as wewould expect of an initial surface of deposition.

The second curve in Figure 21 shows anabnormally low integral, 33%. This basin isalmost entirely in Mancos shale, which ex-tends out from the base of the escarpment butincludes a small remnant of the Mesa Verdesandstone, Chimney Rock, rising strikinglyfrom the shale plain. This basin represents astage in retreat of a cliff line in which the re-sistant bed is all but completely removed.It is in virtually the same phase as the monad-nock phase of the normal cycle (Fig. 16).

The third curve, intermediate between thefirst and second, represents a basin entirely

underlain by the Mancos shale, well out beyondthe limits of the escarpment. Here no ves-tiges remain of the overlying resistant formationand the basin is in a virtually homogeneous

[} Plateau with deepcanyons

Butte risingabove shale

lowland

.2 .3 .4 .5 .6 .7 .8 .9 1.0Relative area -7-A

FIGURE 21.—HYPSOMETRIC CURVES or THREEBASINS IN MESA VERDE REGION, NORTH-

WESTERN NEW MEXICOFrom Chimney Rock Quadrangle, New Mexico,

U. S. Geological Survey, 1:62,500.

weak material. Here, as is normal in the equi-librium stage, the integral is close to 50% andthe curve has a smooth, s-shaped form which iswell described by the model hypsometric func-tion with the values r = O.OS, z = .275.

To summarize the effect of massive, resistanthorizontal strata of an erosional escarpmentupon the hypsometric function: a high integralcharacterizes the early phases of developmentin the zone of canyon dissection close to thecliffs; the integral drops to low values as theproportion of basin of low relief on weak rockincreases and the remnants of resistant rockdiminish; and finally, when the basin is entirelyin weak rock, the curve reverts to the normalform of the equilibrium stage.

A good example of the modified hypsometriccurve resulting from the presence of a massive,resistant formation above a weaker rock isfound in the dissected plateau near Soissons,France, north and south of the Aisne River.There the Tertiary chalk forms an extensiveinterstream upland surface at 170-200 meterselevation. The Aisne and its immediate tribu-taries have cut into weak sands and clays be-



114

1.0

.8hH

.6

4

.2

0

FIG

0 A. N. STRAHLER— ANA

^

^^_

^^5

^\ ^N

\^\ \

YV\

X&

^X

•

—

3 .2 .4 a .6 .8 l.(A

CTRE 22. — HYPSOMETRIC CURVES OF THREEBASINS NEAR SOISSONS, FRANCE

Showing influence of a resistant chalk formationupon curve form. From Soissons Quadrangle,France, 1:50,000.

.2 4 .6 .8Relative area ^

1.0

FIGURE 23.—HYPSOMETRIC CURVES or LARGEDRAINAGE BASINS

From Langbein (1947). Values of r and z, addedby writer, were fitted by inspection.

neath the chalk, giving the drainage basinssteep inner slopes but very gentle slopes on theextensive divides. Curves of three third-orderbasins ranging from 14-26 square kilometersin area differ slightly in integral, but are re-markably alike in form (Fig. 22). Note that theresistant chalk produces a high integral and a

pronounced convexity in the upper third of thecurve. This curve has a double inflection anddoes not fit the model hypsometric function.

PRACTICAL APPLICATIONS OF HYPSOMETRICANALYSIS

The hypsometric analysis of drainage basinshas several applications, both hydrologic andtopographic. Langbein (1947) applied the per-centage hypsometric curve to a number ofNew England drainage basins (Fig. 23) of amuch larger order of size than those analyzedhere, but the curves have basically similarforms and can be described by the modelhypsometric function. On Figure 23 the valuesof r and z are given for the best fit. Fit rangesfrom fair to excellent, and the results are satis-factory considering that most of these basinslie in a glaciated area combined with complexstructure.

Referring to practical value of hypsometricdata in hydrology, Langbein states (1947,p. 141):

"For example, snow surveys generally show anincrease in depth of cover and water equivalentwith increase in altitude; the area-altitude relationprovides a means for estimating the mean depthof snow or its water equivalent over a drainagebasin. Barrows (1933) describes a significant varia-tion in annual precipitation and runoff in the Con-necticut River Basin with respect to altitude. Theobvious variation in temperature with change inaltitude is further indication of the utility of thearea-altitude distribution curve."

Another application might be found in thecalculation of sediment load derived from asmall drainage basin in relation to slope. Be-cause the hypsometric function combines thevalue of slope and surface area at any elevationof the basin, it might help obtain more precisecalculations of expected source of maximumsediment derived from surface runoff in atypical basin of a given order of magnitude.

Dr. Luna B. Leopold (personal communi-cation) has applied the hypsometric method toanalysis of the relationship of vegetative coverto the areal distribution of surface exposed toerosion in the Rio Puerco watershed, NewMexico. Because of distinctive vertical zoningof grassland, woodland, and forest, the relative



PRACTICAL APPLICATIONS OF HYPSOMETRIC ANALYSIS 1141

surface areas underlain by each vegetativetype can be described by the hypsometricfunction, which can thus be used as a basisfor calculation. Furthermore, because rainfallincreases with elevation, the hypsometricfunction can be used to calculate the total areasubject to a given amount of rainfall.

A military application of the hypsometricmethod is foreseen in the use of the hypsometricintegral as a term descriptive of the characterof the terrain in quantitative terms. A highintegral, such as that in Figure 14 would indi-cate extensive interstream areas of low relief,suitable to the rapid movement of mechanizedforces, but with the valleys forming small nar-row pockets suitable for defense and not read-ily observed from outside. A medium integralwould indicate that the land surface was almostentirely in slope, which might be steep in agiven region, and lacking in extensive belts ofeasy trafncability, either in the valley floorsor along the divides. A very low integral wouldmean the development of extensive intercon-nected valley floors adapted to rapid move-ment, but with isolated hill summits whichwould offer defense positions with wide visibil-ity. Obviously these terrain characteristicscan be seen at a glance from any contourtopographic map, and hypsometric analysiswould be of value only in quantitative calcula-tions using empirical formulas in which each

aspect of the terrain is given a numerical state-ment.

Planning of soil erosion control measures andland utilization may profit from topographicanalysis in which such terrain elements ashypsometric qualities, slope steepness, anddrainage density are quantitatively stated.

REFERENCES CITED

Barrows, H. K. (1933) Precipitation and runoff andaltitude relations for Connecticut River, Am.Geophys. Union, Tr., 14th Ann. Meeting, p.396-406.

Fisk, H. N. (1939) Depositional terrace slopes inLouisiana, Jour. Geomorph., vol. 2, p. 181-200.

Horton, R. E. (1941) Sheet erosion—present andpast, Am. Geophys. Union, Tr., Symposiumon dynamics of land erosion, 1941, p. 299-305.

(1945) Erosional development of streams andtheir drainage basins; hydrophysical approachto quantitative morphology, Geol. Soc. Am.,Bull., vol. 56, p. 275-370.

Langbein, W. B. et al. (1947) Topographic charac-teristics of drainage basins, U. S. Geol. Survey,W.-S. Paper 968-C, p. 125-157.

Rouse, Hunter (1937) Modern conceptions of themechanics of fluid turbulence, Am. Soc. Civ.Eng. Tr., vol. 109, p. 523-543.

Strahler, A. N. (1950) Equilibrium theory of ero-sional slopes approached by frequency distribu-tion analysis, Am. Jour. Sci., vol. 248, p. 673-696, 800-814.

DEPARTMENT OF GEOLOGY, COLUMBIA UNIVERSITY,NEW YORK 27, N. Y.

MANUSCRIPT RECEIVED BY THE SECRETARY OFTHE SOCIETY, DECEMBER 12, 1951.

PROJECT GRANT 525-48.



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