Mtm _■_______---!

• going into suspension, but part of the samemt in the low velocitieskwater areas. Exampleseen noted close to riverere sand splays or dunes,t greatly different fromve been deposited on thejam bed. While it is notis paper to evaluate theof the other bed-load

manner, it is suggestedas to the applicability ofbed-load and total-load

sand-bed streams a greatterial is carried into sus--, it may be found practi-•ediment load of a water-erosion. Slope is a factorrally recognized bed-loadtree, 1955, p. 114]. Exten-ccssary to determine the

features as vegetativedumped rock dams, and

ould act as major dctcn-ice the slope,cd to gully erosion andere direct methods areheir quantities, estimatesust be based upon pastupon the best of the bed-therefrom.

:_xcesction of basic data on sedfcon Water Resource;;, ///.

/, pp. 53-62. October 1951.ee, C. EL, Computations ofNiobrara River near Cody,v. Paper 1357. 187 pp.. 1955.s-u. W. M.. Estimating bedphys. Union, 32, 121-12.5.ussion of future of reser-Civ. Eng., Ill, 1258-1262.

District, Tuisa, Oklalwnui\ 1. 1957; presented as pariitershed Erosion and Sed»"/-Seventh Annual Meeting.1. 1956; open for formal

58.)

Vo|. 38, No. 6 Transactions. American Geophysical Union December 1957

Quantitative Analysis of Watershed Geomorphology

Arthur N. Strahler

Abstract—Quantitative geomorphic methods developed within the past few years providemeans of measuring size and form properties of drainage basins. Two general classes of descriptive numbers are (1) linear scale measurements, whereby geometrically analogous unitsof topography can be compared as to size; and (2) dimensionless numbers, usually angles orratios of length measures, whereby the shapes of analogous units can be compared irrespective of scale.

Linear scale measurements include length of stream channels of given order, drainagedensity, constant of channel maintenance, basin perimeter, and relief. Surface and cross-sectional areas of basins are length products. If two drainage basins are geometrically similar,all corresponding length dimensions will be in a fixed ratio.

Dimensionless properties include stream order numbers, stream length and bifurcationratios, junction angles, maximum valley-side slopes, mean slopes of watershed surfaces,channel gradients, relief ratios, and hypsometric curve properties and integrals. If geometrical similarity exists in two drainage basins, all corresponding dimensionless numbers willbe identical, even though a vast size difference may exist. Dimensionless properties can becorrelated with hydrologic and sediment-yield data stated as mass or volume rates of flowper unit area, independent of total area of watershed.

Introduction—Until about ten years ago the..omorphologist operated almost entirely on aiescriptive basis and was primarily concerned with:he history of evolution of landforms as geologicaleatures. With the impetus given by Horton [1945],Bid under the growing realization that the classicaliescriptive analysis had very limited value in.radical engineering and military applications, afew geomorphologists began to attempt quantifica-:;ra of landform description.This paper reviews progress that has been made

n quantitative landform analysis as it applies tonormally developed watersheds in which runningvater and associated mass gravity movements are

; 'he chief agents of form development. The treatment cannot be comprehensive; several lines of•tudy must be omitted. Nevertheless, this paper

. may suggest what can be done by systematic ap-i proach to the problem of objective geometricalanalysis of a highly complex surface.Most of the work cited has been carried out at

I Columbia University over the past five years underj i contract with the Office of Naval Research,1 Geography Branch, Project NR 389-042 for the1 -tudy of basic principles of erosional topography.References cited below give detailed explanations"' techniques and provide numerous examples*^ken from field and map study.

j Dimensional analysis and geometrical similarity—i "e have attempted to base a system of quantita-

:ve geomorphology on dimensional analysis and

principles of scale-model similarity [Strahler, 1954a,p. 343; 1957]. Figure 1 illustrates the concept ofgeometrical similarity, with which we are primarily concerned in topographical description.Basins A and B are assumed to be geometricallysimilar, differing only in size. The-larger may bedesignated as the prototype, the smaller as themodel. All measurements of length between corresponding points in the two basins bear a fixedscale ratio, X. Thus, if oriented with respect to acommon center of similitude, the basin mouths Q'and Q are located at distances r' and r, respectively,from C; the ratio of r' to r is X. In short, all corresponding length measurements, whether theybe of basin perimeter, basin length or width, streamlength, or relief (h' and h in lower profile), are in afixed ratio, if similarity exists.

All corresponding angles are equal in prototypeand model (Fig. 1). This applies to stream junctionangles a' and a, and to ground slope angles $' and/3. Angles are dimensionless properties; hence thegeneralization ■ that in two geometrically similarsystems all corresponding dimensionless numbers or products describing the geometry must beequal.

Studies of actual drainage basins in differingenvironments show that in many comparisons inhomogeneous rock masses, geometrical similarityis closely approximated when mean values areconsidered, whereas in other comparisons, wheregeologic inhomogeneity exists, similarity is def-

.!••

05 MILE

order bo./.

designating stream orders, 1954a, p. 344)

etry through use of order3 in drainage-network anal-)f stream segments of each1 by analysis of the way inearn segments change with

'orlon's [1945, p. 291] law ofthat the numbers of stream

r form an inverse geometricnumber. This is generally

ted data [Strahler, 1952, p.p. 603] and is convenientlyFigure 3. A regression ofof streams of each orderorder (abscissa) generally

plot with very little scatterthough the function relating.ed only for integer values of.le, a regression line is fitted;or regression coefficient b isthm of b is equivalent toratio rb and in this case hass means that on the averagee-half times as many streamsnext higher order.tt the bifurcation ratio wouldmensionless number for ex-a drainage system. Actually• stable and shows a smalli region to region or environ-lt, except where powerfulinate. Coates [1956, Table 3]

-o.o-o 2

ORDER,

F,G. 3 - Regression of number of stream segments onstream order; data from Smith (1953, Plate

found bifurcation ratios of first-order to second-order streams to range from 4.0 to 5.1; ratios ofsecond-order to third-order streams to range from2.8 to 4.9. These values differ little from Strahler's;i952, p. 1134].

Frequency distribution of stream lengths—Lengthof stream channel is a dimensional property whichcan be used to reveal the scale of units comprisingdie drainage network. One method of—lengthanalysis is the measurement of length of_____hsegment of channel of a given stream order. For a■_iven watershed these lengths can be studied byfrequency distribution analysis [Schumm, 1956,p. 607]. Stream lengths are strongly skewed right,but this may be largely corrected by use of logarithm of length. Arithmetic mean, estimatedpopulation variance, and standard deviation serveas standards of description whereby differentdrainage nets can be compared and their differences tested statistically [Strahler, 1954b].

Relation of stream length to stream order—Stillanother means of evaluating length relationshipsin a drainage network is to relate stream length tostream order. A regression of logarithm of totalstream length for each order on logarithm oforder may be plotted (Fig. 4). Again, the functionis defined' only for integer values of order. Severalsuch plots of length data made to date seem toyield consistently good fits to a straight line, butthe general applicability of the function is not yetestablished, as in the case of the law of streamnumbers.

The slope of the regression line b (Fig. 4) is theexponent in a power function relating the twovariables. Marked differences observed in theexponent suggest that it may prove a useful

measure of the changing length of channel segments as order changes. Because this is a nonlinear variation, the assumption is implicit thatgeometrical similarity is not preserved with increasing order of magnitude of drainage basin.

Drainage basin areas—Area of a given watershedor drainage basin, a property of the square oflength, is a prime determinant of total runoff orsediment yield and is normally eliminated as avariable by reduction to unit area, as in annualsediment loss in acre-feet per square mile. In orderto compare drainage basin areas in a meaningfulway, it is necessary to compare basins of the sameorder of magnitude. Thus, if we measure the areasof drainage basins of the second order, we aremeasuring corresponding elements of the systems.If approximate geometrical similarity exists, thearea measurements will then be indicators of thesize of the landform units, because areas of similarforms are related as the square of the scale ratio.

Basin area increases exponentially with streamorder, as stated in a law of areas [Schumm, 1956, p.606], paraphrasing Horton's law of stream lengths.

Schumm [1956, p. 607] has shown histograms ofthe areas of basins of the first and second ordersand of patches of ground surface too small to havechannels of their own. Basin area distributions arestrongly skewed, but this is largely corrected byuse of log of area. Area is measured by planimeterfrom a topographic map, hence represents projected, rather than true surface area. Estimationof true surface area has been attempted wheresurface slope is known [Strahler, 1956a, p. 579].

-V-v Pleistocene sediments,/*—■ Coast Ranges, Souther

California

CDenDCoa

Igneous, metamorphiccomplex, Coast Ranges,S. California

rV«~ Carboniferous1 Appalachian Plateau,

Pennsylvania

sandstones

/V* number ofcontour ere nutationsW

' Contourwith most

(crenulatfons

P- length of perimeter. i ( i ■ . i i ■- i 1 i i L _ i0

T e x t u r e r a t i o , T * j r100 1000

Fig. 5 - Definitions of drainage density and texture ratio (Strahler, 1954a, p. 348)

Drainage density and texture ratio—An importantindicator of the linear scale of landform elementsin a drainage basin is drainage density, defined byHorton [1945, p. 283]. The upper left-hand corne'rof Figure 5 shows the definition of drainage densityas the sum of the channel lengths divided bybasin area. Division of length by area thus yields anumber with the dimension of inverse of length.In general, then, as the drainage density numberincreases, the size of individual drainage units,such as the first-order drainage basin, decreasesproportionately.

Figure 5 shows the relation between drainagedensity and a related index, the texture ratio,defined by Smith [1950]. Because the contour

inflections on a good topographic map indicate theexistence of channels too small to be shown bystream symbols, their frequency is a measure ofcloseness of channel spacing and hence also correlates with drainage density.

Drainage density is scaled logarithmically onthe ordinate of Figure 5. The grouped points in thelower left-hand corner of the graph representbasins in resistant, massive sandstones. Here thestreams are widely spaced and density is low. Thenext group of points encountered represents typicaldensities in deeply weathered igneous and metamorphic rocks of the California coast ranges. Inthe extreme upper right are points for badland?.where drainage density is from 200 to 900 miles of

QUANTITA1

channels per square mile [,\>)56, p. 612].Because of its wide ratio

density is a number of p.landform scale analysis,sediment yield would show a jship with drainage density.^the relation of drainage depredicting the morphologicpected when ground surfaby land use, has been outlir

Constant of channel[1956, p. 607] has useddensity as a property termemaintenance. In Figure 6area (ordinate) is treated asof total stream channel lenjlength is cumulative for a gilall lesser orders; it is thusin a watershed of given ordeis projected to the horizon,true lengths would be obtcorrection for slope.

An individual plotted pointjsents a given stream ordernumbered 1 through 5. Usingexamples given by Schumm,dose to a straight line of 45ctionship is treated as linearhere on log-log paper. If the kcept is read at log stream lclog of this intercept is taken,!

LOG

BASINAREAFT.*

l09..A«o

logAQ=logC +1A__= __:___

______ _ _L-*U_ °d

logC2.6..250

0.94

_y

c°=_ AreoLength]

- 1 0 1 2 3LOG TOTAL STREAM

CUMULATIVEFig. 6 - Constant of channelI

Data replotted on logarithmSchumm (1956, p.

Badlands,Perth Amboy,New Jersey

Badlands,* Petrified Forest,Arizona

nents, -outhern

'« number ofour crenulotions

1000

ler, 1954a, p. 348)

topographic map indicate theIs too small to be shown by.-ir frequency is a measure otspacing and hence also corre-lensity.is scaled logarithmically on

re 5. The grouped points in therner of the graph representmassive sandstones. Here the-paced and density is low. Theencountered represents typicalweathered igneous and met*he California coast ranges. Inright are points for badlands,„ty is from 200 to 900 miles of

QUANTITATIVE ANALYSIS OF WATERSHED GEOMORPHOLOGY 917

channels per square mile [Smith, 1953; Schumm,1956, p. 612].

Because of its wide ratio of variation, drainagedensity is a number of primary importance inlandform scale analysis. One might expect thatsediment yield would show a close positive relationship with drainage density. A rational theory ofthe relation of drainage density to erosion intensity,predicting the morphological changes to be expected when ground surface resistance is loweredbv land use, has been outlined by Strahler [19566].

Constant of channel maintenance—Schumm[1956, p. 607] has used the inverse of drainagedensity as a property termed constant of channelmaintenance. In Figure 6 the logarithm of basinarea (ordinate) is treated as a function of logarithmof total stream channel length (abscissa). Streamlength is cumulative for a given order and includesall lesser orders; it is thus the total channel lengthin a watershed of given order. Length in this caseis projected to the horizontal plane of the map;true lengths would be obtained by applying acorrection for slope.

An individual plotted point on the graph represents a given stream order in the watershed, asnumbered 1 through 5. Using data of the threeexamples given by Schumm, the sets of points falldose to a straight line of 45° slope; thus the relationship is treated as linear even though plottedhere on log-log paper. If the logarithm of the intercept is read at log stream length = 0, and the anti-log of this intercept is taken, we obtain the con-

L0G

BASINAREAFT.'

loq,.A__

10

9

8

7

6

5

4

3

2

I

0

log A0= logC +1 logIL__

C>

0 9 4

Per th Amboy 8 .7Chileno Can. 316

/ d i e 4 2 5

--Area ______ iu "Length" L

O I 2 3 4 5 6 7LOG TOTAL STREAM LENGTH, FT,

C U M U L AT I V E l o g i o £ L _ ,

Fig. 6 - Constant of channel maintenance, C.Data replotted on logarithmic scales from

Schumm (1956, p. 606)

stant of channel maintenance C which is actuallythe slope of a linear regression of area on length.

The value of C = 8.7 in the Perth Amboy badlands means that on the average 8.7 sq ft of surfaceare required to maintain each foot of channellength. In the second example, Chileno Canyon inthe California San Gabriel Mountains, 316 sq ft ofsurface are required to maintain one foot of channellength.

The constant of channel maintenance, with thedimensions of length, is thus a useful means ofindicating the relative size of landform units in adrainage basin and has, moreover, a specificgenetic connotation.

Maximum valley side slopes—Leaving now thedrainage network and what might be classified asplanimetric or areal aspects of drainage basins, weturn to slope of the ground surface. This bringsinto consideration the aspect of relief in drainagebasin geometry. One significant indicator of theover-all steepness of slopes in a watershed is themaximum valley-side slope, measured at intervalsalong the valley walls on the steepest parts of thecontour orthogonals running from divides toadjacent stream channels.

Maximum valley-side slope has been sampledby several investigators in a wide variety ofgeological and climatic environments [Strahler,1950; Smith, 1953; Miller, 1953; Schumm, 1956;Coates, 1956; Mellon, 1957]. Within-area varianceis relatively small compared with between-areadifferences. This slope statistic would thereforeseem to be a valuable one which might relateclosely to sediment production.

Mean slope curve—Another means of assessingthe slope properties of a drainage basin is throughthe mean slope curve [Strahler, 1952, p. 1125-1128]. This requires the use of a good contourtopographic map. The problem is to estimate theaverage, or mean slope of the belt of ground surfacelying between successive contours. This may bedone by measuring the area of each contour beltwith a planimeter and dividing this area by thelength of the contour belt to yield a mean width.The mean slope will then be that angle whosetangent is the contour interval divided by themean belt width. Mean slope of each contourinterval is plotted from summit point to basinmouth. Curves of this type will differ from regionto region, depending upon geologic structure andthe stage of development of the drainage system.If the mean slope for each contour belt is weightedfor per cent of total basin surface area, it is possible

918 ARTHUR N. STRAHLER

to arrive at a mean slope value for the surface ofthe watershed as a whole.

Slope maps—Another means of determiningslope conditions over an entire ground surface of awatershed is through the slope map [Strahler,1956a]. (1) A good topographic map is taken. (2)On this map the slope of a short segment of linenormal to the trend of the contours is determinedat a large number of points. These may be recordedas tangents or sines, depending upon the kind ofmap desired. (3) These readings are contouredwith lines of equal slope, here called isotangents.(4) The areas between successive isotangents aremeasured with a planimeter and the areas summedfor each slope class. (5) This yields a slope frequency percentage distribution. Because theentire ground surface has been analyzed, the mean,standard deviation, and variance are treated as'population parameters, at least for purposes ofcomparison with small samples taken at randomfrom the same area.

Lines of equal sine of slope, or isosines, mayalso be drawn. The interval between isosines onthe map becomes the statistical class on the histogram. Sine values are designated as g valuesbecause the sine of slope represents that proportionof the acceleration of gravity acting in a down-slope direction parallel with the ground surface.

Rapid slope sampling—-The construction ofslope maps and their areal measurement is extremely time-consuming. Experiments have shownthat essentially the same information can beachieved by random point sampling [Strahler,1956a, p. 589-595]. Both random coordinate-sampling and grid sampling have been tried. Inthe random-coordinate method a sample square isscaled in 100 length units per side. From a table ofrandom numbers the coordinates of sample pointsare drawn for whatever sample size is desired. Thegrid method does much the same thing, but is notflexible as to sample size.

Point samples, which are easy to take, werecompared with the frequency distribution measured from a slope map. Noteworthy is the ex-,tremely close agreement in means and variances,and even in the form of the frequency distributions,'including a marked skewness. Tests of samplevariance and mean are discussed by Strahler[1956a].

Chapman [1952] has developed a method ofanalyzing both azimuth and angle of slope fromcontour topographic maps. Although based onpetrofabric methods and designed largely for use

in geological analysis of terrain, the methrv. •be applied to a watershed as a means of Z?'both slope steepness and orientation si_^

Relief ralio-Schumm [1956, p. 612] has dev__4and applied a simple statistic, the rdief^defined as the ratio between total basin relief Siis, difference in elevation of basin mouth 2summit) and basin length, measured as the Ion!!dimension of the drainage basin. In a general«the relief ratio indicates overall slope of the w_2'shed surface. It is a dimensionless number r__Scorrelated with other measures that do not d«aon total drainage basin dimensions. Relief ra_I_simple to compute and can often be obtain*

llckin detaUed inf°rmati0n on toPogniptybSchumm [1954] has plotted mean annual sedi

ment loss in acre feet per square mile as a functionof the relief ratio for a variety of small drama-basins in the Colorado Plateau province [Fig TThe significant regression with small scattersuggests that relief ratio may prove useful inestimating sediment yield if the parameters for igiven climatic province are once established

Hypsometric -*-iy_i_-Hypsometric analysis, orthe relation of horizontal cross-sectional drainai.basin area to elevation, was developed in itsmodern dimensionless form by Langbein and othen11947]. Whereas he applied it to rather large water-sheds, it has since been applied to small drainagebasins of low order to determine how the mass a

SEDIMENTLOSS

(ACRE FTPER Ml*)

Fig. 7 - Regression of sediment loss onrelief ratio, after Schumm (1954,

p. 218)

distributed witla, 1952; Mi1956).

Figure 8 ilkdimensionlessdrainagea horizontalthe relativecontour h tothe ratio ofentire basin

tfoi lSummit

Area

Mode

\

*-<_-».

analysis of terrain, the method might) a watershed as a means of assessingsteepness and orientation simultane-

)—Schumm [1956, p. 612] has deviseda simple statistic, the relief ratio,

e ratio between total basin relief (that_ in elevation of basin mouth andi basin length, measured as the longestt the drainage basin. In a general way,io indicates overall slope of the water-. It is a dimensionless number, readilyith other measures that do not depend.inage basin dimensions. Relief ratio is:ompute and can often be obtained.iled information on topography is

[1954] has plotted mean annual sedi-acre feet per square mile as a function

:' ratio for a variety of small drainageic Colorado Plateau province [Fig. 7].xant regression with small scatteriat relief ratio may prove useful insediment yield if the parameters for atic province are once established.trie analysis—Hypsometric analysis, ori of horizontal cross-sectional drainage

to elevation, was developed in itslensionless form by Langbein and othersireas he applied it to rather large water-is since been applied to small drainage>w order to determine how the mass is

QUANTITATIVE ANALYSIS OF WATERSHED GEOMORPHOLOGY 919

7 - Regression of sediment loss onrelief ratio, after Schumm (1954,

p. 218)

distributed within a basin from base to top [Strah-/tT) 1952; Miller, 1953; Schumm, 1956; Coates,1956].

Figure 8 illustrates the definition of the twodimensionless variables involved. Taking thedrainage basin to be bounded by vertical sides anda horizontal base plane passing through the mouth,the relative height is the ratio of height of a givencontour h to total basin height H. Relative area isthe ratio of horizontal cross-sectional area a toentire basin area A. The percentage hypsometric

curve is a plot of the continuous function relatingrelative height y to relative area x.

As the lower right-hand diagram of Figure 8shows, the shape of the hypsometric curve variesin early geologic stages of development of thedrainage basin, but once having attained anequilibrium, or mature stage (middle curve ongraph), tends to vary little thereafter. Severaldimensionless attributes of the hypsometric curveare measurable and can be used for comparativepurposes. These include the integral, or relative

Summitplane ^ J^ 1.0

<=l*.8

_=_£_\6

Percentage hypsometriccurve:

Mouth.4

Area a Area A(entire basin)

Y Model hypsometricfunction:

°.2OJ

CC0

I I I

\ H- x*, x\£&r.

1

■ * i .

1 1 1 1

0 . 2 . 4 . 6 . 8 1 . 0Relative area, -j-

Characteristic curvesof erosion cycle*.

Inequilibrium(young)

stageEquilibrium >^

(mature)stage

Monadnockphase

0 Relative area I.Fig. 8 - Method of hypsometric analysis (Strahler, 1954a. p. 353)

_________■

920 ARTHUR N. STRAHLER

area lying below the curve, the slope of the curveat its inflection point, and the degree of sinuosityof the curve. Many hypsometric curves seem to beclosely fitted by the model function shown in thelower left corner of Figure 8, although no rationalor mechanical basis is known for the function.

Now that the hypsometric curves have beenplotted for hundreds of small basins in a widevariety of regions and conditions, it is possible toobserve the extent to which variation occurs.Generally the curve properties tend to be stable inhomogeneous rock masses and to adhere generallyto the same curve family for a given geologic andclimatic combination.

Conclusion—This paper has reviewed briefly avariety of geometrical properties, some of lengthdimension or its products, others dimensionless,which may be applied to the systematic description of drainage basins developed by normalprocesses of water erosion. Among the morphological aspects not mentioned are stream profilesand the geometry of stream channels. These, too,are subject to orderly treatment along the linessuggested. The examples of quantitative methodspresented above are intended to show that, complex as a landscape may be, it is amenable toquantitative statement if systematically brokendown into component form elements. Just which ofthese measurements or indices will prove mostuseful in explaining variance in hydrologicalproperties of a watershed and in the rates oferosion and sediment production remains to beseen when they are introduced into multivariateanalysis. Already there are definite indications ofthe usefulness of certain of the measures and it isonly a matter of continuing the development ofanalytical methods until the most important geomorphic variables are isolated.

References

Chapman, C. A., A new quantitative method of topographic analysis, Amer. J. Sci., 250, 428-452,1952.

Coates, D. R., Quantitative geomorphology of smalldrainage basins in southern Indiana, Of. Nav. Res.Proj. NR 389-042, Tech. Rep. 10 (Columbia Univ.Ph.D. dissertation), 57 pp., 1956.

Morton, R. E., Erosional development of streams andtheir drainage basins; hydrophysical approach toquantitative morphology, Bui. Geol. Soc. Amer., 56,275-370, 1945.

Langbein, VV. B., and others, Topographic characteristics of drainage basins, U. S. Geol. Suro. Water-Supply Paper 968-C, 157 pp., 1947.

Maxwell, J. C, The bifurcation ratio in Horton'slaw of stream numbers, (abstract), Trans. AmerGeophys. Union, 36, 520, 1955.

Melton, M. A., An analysis of the relations antonielements of climate, surface properties, and geomorphology, Of. Nav. Res. Proj. NR 389-042, Tech.Rep. 11 (Columbia Univ. Ph.D. dissertation),102 pp., 1957.

Miller, V. C, A quantitative geomorphic study ofdrainage basin characteristics in the Clinch Mountainarea, Virginia and Tennessee, Of. Nav. Res. Proj.NR 389-042, Tech. Rep. 3 (Columbia Univ. Ph.D.dissertation), 30 pp., 1953.

Schumm, S. A., The relation of drainage basin relief tosediment loss, Pub. International Association ofHydrology, IUGG, Tenth Gen. Assembly, Rome,19'54,1, 216-219, 1954.

Schumm, S. A., Evolution of drainage systems andslopes in badlands at Perth Amboy, New Jersey,Bui. Geol. Soc. Amer., 67, 597-646, 1956.

Smith, K. G., Standards for grading texture of erosional topography, Amer. J. Sci., 248, 655-668,1950.

Smith, K. G., Erosional processes and landforms inBadlands National Monument, South Dakota, Of.Nav. Res. Proj. NR 389-042, Tech. Rep. 4 (Columbia Univ. Ph.D. dissertation), 128 pp., 1953.

Strahler, A. N., Equilibrium theory of eroskoilslopes approached bv frequency distribution analysis, Amer. J. Sci., 248, 673-696, 800-814, 1950.

Strahler, A. N., Hypsometric (area-altitude) analysis of erosional topography, Bid. Geol. Soc. Anetn63, 1117-1142, 1952.

Strahler, A. N., Quantitative geomorphology oferosional landscapes, C.-R. 19th Intern. Geol. Conf,Algiers, 1952, sec. 13, pt. 3, pp. 341-354, 1954a.

Strahler, A. N., Statistical analysis in geomorphicresearch, /. Geol., 62, 1-25, 1954b.

Strahler, A. N., Quantitative sloDC ana'ys55. s,i-Geol. Soc. Amer., 67, 571-596, 1956a.

Strahler, A. N., The nature of induced erosion andaggradation, pp. 621-638, Wenner-Gren SymposiumVolume, Man's role in changing the face oftheEont,Univ. Chicago Press, Chicago, 111., 1193 pp., 1956b.

Strahler, A. N., Dimensional analysis in geomm-phology, Of. Nav. Res. Proj. NR 389-042, Ted-Rep. 7, Dept. Geol., Columbia Univ., N. Y., 43 pp.,1957.

Department of Geology, Columbia University, New I «•*27, N. Y.

(Manuscript received April 1, 1957; presented as pariof the Symposium on Watershed Erosion and -eminent Yields at the Thirty-Seventh Annual Meet**Washington, D.C., May 1, 1956; open for formal ducussion until May 1, 1958.)

Vol. 38, No. 6

Relating;

Abstract—The yield jinherent watershedof vegetation, andflow which produce;sediment measuringsources of variationstudy of the yieldsediment yields. Such!sediment, to evaluatecriteria for design ofwhich multiple regreThe studies are discusfunctions; and the effenificant variables.

Introduction—Many hydrol^erate within watersheds, andvariable 'material,' with highl]As a consequence, we haveworking out the end results,yield, of all these processes operiof a watershed. Also, when weyield, we have equal difficultyshare each process contributneed to predict sediment yielcreservoirs and channels; weprocesses and the contributiorparts of watersheds in order to 1_sources and evaluate how effetlthose sources would be in redur.

Several research workers hav.pie regression analysis offersdifficulty. This paper reviewsto see how multiple regressicuseful and some ways to makeuseful.

Why is multiple regressiorsediment yield studies? It tewant to know:: how theof watersheds contribute tohow well we can predict the ;watershed by study of theerrors in each are evaluated in!Hiving us a measure of how gcand some clues as to where toiment.

The studies which this papecusses were based on the hypotyield from whole watersheds is