to I"' T CO?T

tl. S. G AL SURVSY

Double-Mass Curves

Manual of Hydrology: Part 1. General Surface-Water 

Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1541-B
Double-Mass Curves

By JAMES K. SEARCY and CLAYTON H. HARDISON 

With a section Fitting Curves to Cyclic Data

By WALTER B. LANGBEIN

Manual of Hydrology: Part 1. General Surface-Water 

Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1541-B

Methods and practices of the 

Geological Survey

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C. - Price 20 cents
CONTENTS

Page

Abstract.___________________________________________________________ 31

Introduction._______________________________________________________ 31

Explanation of the double-mass curve._________________________________ 33

Application to hydrologic data________________________________________ 34

Precipitation records.____________________________________________ 34

Checking consistency-___-______-_______-_____-_-_--_-----___ 35

Adjustment of precipitation records___________---_____________- 38

Missing record-____-_____________-_____-_--_----------_---_- 39

Streamflow records._____________________________________________ 40

Checking consistency.______-_-____-_--__--_____-_--_-_______ 40

Adjusting streamflow records________-_-_______-____-_-_____-_- 42

Sediment records._______________________________________________ 42

Precipitation-runoff relations..____________________________________ 44

Areal precipitation___________________________________________ 44

Effective precipitation._________________-_-.__-__--_---________ 44

Computed runoff- ___________________________________________ 47

Residual-mass curve._________-________-______-___-_------------_ 50

Statistical test for significance_____________-_-_____------_------------- 51

Effect of the degree of correlation between station records-___________ 57

Effect of the length of record______________________________________ 58

Summary. __________________________________________________________ 58

Fitting curves to cyclic data, by W. B. Langbein________________________ 59

Moving arc__________________-____-___-____-__-___----------_-_- 60

Parabolic arcs___________________-_____---____-------------_- 60

Quartic arcs______________________-_____________--_---_----_ 62

Double integration.___________-_-_____--__---_------------___-__- 63

Summary __ _____________________________________________________ 65

Literature cited________________________-__-_-_-________-------_----- 65

ILLUSTRATIONS

FIGURE 1. Double-mass curve of precipitation data__________________ 36

2. Double-mass curve of streamflow data___________________ 41

3. Cumulative sediment versus water discharge by 3-month increments for storm runoff and total runoff, Brandywine 

Creek at Wiltnington, Del., January 1947-Septetnber 1955 43

4. Relation of effective annual precipitation to annual runoff,

Colorado River near Grand Canyon, Ariz., 1921-46------ 48

5. Double-mass curve of measured runoff versus computed

runoff, Colorado River near Grand Canyon, Ariz., 1921-46- 49

6. Residual-mass curve of measured runoff minus computed

runoff, Colorado River near Grand Canyon, Ariz_ _______ 51

7. Illustration of smoothing by 5-point moving arc ____________ 61

8. Result of fitting a cycle graph by double integration------ 64

in
IV CONTENTS

TABLES

Page 

TABLE 1. Annual precipitation, in inches, for double-mass curve._______ 37

2. Annual runoff, in inches, for double-mass curve______________ 41

3. Correlation of annual precipitation and annual runoff, Colorado

River near Grand Canyon, Ariz__________________________ 46

4. Measured annual runoff and computed annual runoff, Colorado

River near Grand Canyon, Ariz__ ________________________ 50

5. Analysis of variance and covariance of data in table 2________ 54

6. Annual runoff, in inches, for double-mass curve with poor correlation between the station and the pattern-_____________ 57

7. Analysis of variance and co variance of data in table 6________ 58

8. Analysis of variance and covariance of data in table 6, after

adding 14 years of record________________________________ 58

9. Weights for w-point least-square parabolas___________________ 60

10. Illustration of use of 5-point moving parabolic arc ___________ 62

11. Weights for least-square quartic arcs__--_-_________________ 63

12. Example of double integration applied to monthly data._ __ _ _ ,. _ 64
MANUAL OF HYDROLOGY: PART 1, GENERAL SURFACEWATER TECHNIQUES

DOUBLE-MASS CURVES

By JAMES K. SEARCY and CLAYTON H. HARDISON

ABSTRACT

The double mass curve is used to check the consistency of many kinds of 

Jiydrologic data by comparing data for a single station with that of a pattern 

composed of the data from several other stations in the area. The double-mass 

curve can be used to adjust inconsistent precipitation data.

The graph of the cumulative data of one variable versus the cumulative data 

of a related variable is a straight line so long as the relation between the variables 

is a fixed ratio. Breaks in the double-mass curve of such variables are caused by 

changes in the relation between the variables. These changes may be due to 

changes in the method of data collection or to physical changes that affect the 

relation.

Applications of the double-mass curve to precipitation, streamflow, and sediment data, and to precipitation-runoff relations are described. A statistical test 

for significance of an apparent break in the slope of the double-mass curve is 

described by an example. Poor correlation between the variables can prevent 

detection of inconsistencies in a record, but an increase in the length of record 

tends to offset the effect of poor correlation.

The residual-mass curve, which is a modification of the double-mass curve, 

magnifies imperceptible breaks in the double-mass curve for detailed study.

Of the several methods of fitting a smooth curve to cyclic or periodic data, the 

moving-arc method and the double-integration method deserve greater use in 

hydrology. Both methods are described in this manual. The moving-arc method 

has general applicability, and the double integration method is useful in fitting 

a curve to cycles of sinusoidal form.

INTRODUCTION

Hydrologic data generally consist of a sequence of observations of 

some phase of the hydrologic cycle made at a particular site. The 

data may be a record of the discharge of a stream at a particular place, 

or it may be a record of the amount of rainfall caught in a particular 

rain gage. Although for most hydrologic purposes a long record is 

preferred to a short one, the user should recognize that the longer the 

record the greater the chance that there has been a change in the 

physical conditions of the basin or in the methods of data collection. 

If these are appreciable, the composite record would represent only a

31
32 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

nonexistent condition and not one that existed either before or after 

the change. Such a record is inconsistent.

An example of changes in physical conditions of the basin is a 

25-year gaging-station record that represents 10 years of runoff from 

an undeveloped drainage basin, 5 years during which a major irrigation 

or other development was being made, and 10 years of runoff after the 

development. This 25-year record would not represent the runoff 

under natural conditions or the future runoff under the existing 

developed conditions, even though it is an accurate record of the 

runoff during the 25 years.

An example of changes in methods of data collection is a record 

from a rain gage operated for several years at a particular location and 

later moved to a different location because of new observers. Records 

from the rain gage at the new location might indicate an increase or 

decrease in average rainfall even though no real change had taken 

place. Similar false changes may result from variation in exposure of 

a rain gage or alteration in equipment.

The use of a double-mass curve as described in this manual is a 

convenient way to check the consistency of a record. Such a check is 

one of the first steps in the analysis of a long record, except when the 

scarcity of other old records makes it infeasible. A double-mass curve 

is a plot on arithmetic cross-section paper of the cumulative figures of 

one variable against the cumulative figures of another variable, or 

against the cumulative computed values of the same variable for a 

concurrent period of time.

Most of the credit for the initial application of the double-mass 

curve to hydrologic data is due C. F. Merriam, retired, Pennsylvania 

Water & Power Co. The description of the double-mass curve is a 

revision of material prepared by W. B. Langbein and others, and used 

within the U.S. Geological Survey since 1948.

It should not be assumed that all inconsistencies shown by a doublemass curve represent inconsistencies due to change in methods of 

collecting the data. Further investigation should be made to find the 

reason for the inconsistency, and to appraise its probable effect on the 

data. Unless the inconsistency shown by the double-mass curve 

starts at the same time as a change in method of data collection, and 

unless the direction of the change shown by the curve could reasonably result from the change in method, no adjustment should be 

applied to the observed data, because the inconsistency could be due 

to other causes, such as works of man or vagaries of nature.

A considerable part of this manual is devoted to analyses of precipitation records, because once they have been adjusted for changes in 

methods of data collection, precipitation records are independent of 

the works of man and thus provide an index for evaluating changes
DOUBLE-MASS CURVES 33

in streamflow. Such an index is desirable because a decrease in 

annual runoff due to a decrease in the amount of annual precipitation 

is not as much of a problem as a decrease due to receiving less runoff 

for a given amount of precipitation. Sometimes the only way to 

make an analysis of trends in streamflow is to use precipitation data, 

because streamflow may be affected by works of man as well as by 

changes in methods of measurement.

With the coming use of weather modification, precipitation may 

become less of an index. Perhaps the double-mass curve technique 

will be useful for detecting artificially-induced changes in precipitation so that proper allowance can be made for such changes

EXPLANATION OF THE DOUBLE-MASS CURVE

The theory of the double-mass curve is based on the fact that a 

graph of the cumulation of one quantity against the cumulation of 

an other quantity during the same period will plot as a straight line 

so long as the data are proportional; the slope of the line will represent 

the constant of proportionality between the quantities. ,

A break in the slope of the double-mass curve means that a change / 

in the constant of proportionality between the two variables has 

occurred or perhaps that the proportionality is not a constant at all 

rates of cumulation. If the possibility of a variable ratio between 

the two quantities can be ignored, a break in the slope indicates the j 

time at which a change occurs in the relation between the two quan- M 

tities. The difference in the slope of the lines on either side of the 

break in the slope indicates the degree of change in the relation. 

Changes in the slope can be more accurately discerned if scales for 

the ordinate and the abscissa are chosen so that the general course 

of the curve is in a 45° direction relative to the axes.

In hydrologic studies, the use of the cumulations of two measured 

variables plotted as a double-mass curve may give indefinite results 

because we may be unable to say which of the variables caused a 

break in slope. To give more definite results, the cumulations of one 

of the variables can be plotted against the cumulations of a pattern 

composed of all similar records in a given area (Merriam, 1937). 

The pattern, which is composed of the average of many records, is 

less affected by an inconsistency in the record of any one station.

If a pattern is used, as in checking the consistency of precipitation 

records, enough stations should be included to insure that the average 

is not seriously affected by an inconsistency in the record for one of 

the stations. The number of stations that can be included in a 

pattern is sometimes limited by the criterion that the area in which 

the stations are located should be small enough to be influenced by 

the same general weather conditions. If less than 10 stations are
34 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

used in the pattern, each record should be tested for consistency by 

plotting it against the pattern, and those records that are inconsistent 

should be eliminated from the pattern.

Spurious breaks in the double-mass curve that should be recognized 

as such are caused by the inherent variability in hydrologic data. 

Most users recognize that the year-to-year breaks are due to chance 

and, thus, ignore any break that persists for less than 5 years. Breaks 

that persist for longer than 5 years are more subtle in that they may 

be due to chance or they may be due to a real change. Unless the 

time of the break coincides with a logical reason for the break, ^tatis^- 

tical methods should be used to evaluate the significance of the break.

APPLICATION TO HYDROLOGIC DATA

The methods for applying the double-mass curve technique to 

hydrologic data and the way the results are used vary somewhat with 

the type of data being analyzed. Therefore, the application of the 

double-mass curve to records of precipitation, runoff, sediment, and 

precipitation-runoff are treated separately in this manual even 

though this requires some repetition.

In using the double-mass curve, we assume that the relation 

between quantities X and Y can be expressed by a line having an 

equation of the form Y=bX, where b is the slope of the double-mass 

curve. This assumption is substantially correct for relations involving 

only precipitation data but is not true for many of the relations 

involving streamflow data or for relations between precipitation and 

streamflow.

The relations involving streamflow are usually much more complex 

than those expressed by the simple equation Y=bX. Among other 

forms of the relation are the straight line Y=a-\-bX, where a is the 

intercept of the straight line on the Faxis, and the curve Y=a-\-bX-\- 

cX2 , or Y=XJ> , where b and c are constants. If thejrelation is curviJinear or if it is a straight, .line with an intercept, spurious breaks 

pcc^U£jiJi__a_d£u^^ To avoid such spurious breaks, 

cumulations of Fare plotted against cumulations of Ye , the correlative 

estimates of Y corresponding to measured values of X. In other 

words, when the relation between hydrologic data is complex, the 

double-mass curve technique requires, as a first step, a definition of 

the relation between the variables. Once this relation is defined, 

values of Yc can be computed according to each recorded value of X.

PRECIPITATION BBCOBDS

Precipitation, unlike streamflow, is little affected by works of man. 

Moreover, records of precipitation (in the United States) are often 

longer than records of other hydrologic data. For these reasons,
DOUBLE-MASS CURVES 35

precipitation records are invaluable in hydrologic studies involving 

trends. Before precipitation records are used in such studies, they 

should be tested by the double-mass curve technique to ensure that any 

trends detected are due to meteorological causes and not to changes 

in gage location, in exposure, or in observational methods. If the 

changes detected are not due to meteorological causes, a precipitation 

record can usually be adjusted by coefficients determined from the 

double-mass curve.

Factors such as location and exposure affect the consistency with 

which a rain gage samples the rainfall in a particular area. The consistency is often affected when a gage is moved to the yard of a new 

observer. Even at the same location, the exposure of a precipitation gage can gradually change through the years, because of the 

growth of trees and other vegetation in the neighborhood or because 

of new buildings. Such changes sometimes go unnoted in the station 

history.

Differences in exposure sometimes occur even at a first-order U.S. 

Weather Bureau station. For example, moving a g£ge from one 

part of a roof to another can affect the catch of the gage greatly, even 

though the gage is moved only a few feet. Also, construction of tall 

buildings in the vicinity of the gage changes wind direction and 

affects thermal air currents, thus influencing the catch in the rain 

gage. Furthermore, when new post offices or Federal buildings are 

constructed, the rain gage often is moved from the roof of one building 

to that of another.

The double-mass curve technique should seldom be used for testing 

consistency of precipitation data in mountainous areas. The climate 

within a mountainous area changes with the difference in elevation, 

and the precipitation at two nearby stations differing greatly in 

elevation may be due to different meteorological events. Records 

from areas where the precipitation pattern for one season of the 

year differs greatly from that of another should be tested by doublemass curves prepared on a seasonal basis rather than a yearly basis.

In using the double-mass curve, some investigators prefer to start 

with the most recent data and to cumulate and to plot the data for 

previous years in reverse order to that shown in this report. Such 

a cumulation is convenient when records with different starting dates 

are to be tested for consistency, but it does not permit the addition 

of new data to the table or the graph.

CHECKING CONSISTENCY

The use of the double-mass curve for checking the consistency of 

precipitation records is explained by the following example in which 

the annual records of five precipitation stations, designated A, B,

534972 O 60 2
36 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

C, D, and E, in or near a given drainage basin are used. First the 

annual precipitation data for each year are tabulated and then cumulated in chronological order as in table 1. The mean of the cumulative precipitation shown in the last column of table 1 is the pattern 

for testing the individual station records. The cumulative precipitation for each station is then plotted against the cumulative precipitation of the pattern as shown in figure 1. For simplicity, the 

only double-mass curves shown in figure 1 are those for stations 

A and E.

The pattern used in figure 1 could be refined by testing each station 

against the pattern and eliminating from the average those stations, 

such as station E, whose double-mass curves show significant breaks 

in slope. If similar breaks occur at several stations, however, the

800

Recorded change 

^^in station location 

1930

o

100 200 300 400 500 600 

CUMULATIVE PRECIPITATION FOR PATTERN, IN INCHES

FIOURE 1. Double-mass curve of precipitation data.

700
CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO 

j H- [f>. p 

<| pi H- pi [f». O9 rf>. 1 HCO CO h- bO kf>- CO K| <l (M 00 p O> bO bO CO p bO 

p H- to jUk H- w j-* ^ p j-* as pi po po as po HbO p jf». Jf>- M p bO bO CO M p 00 CO pi p p 00 

O9 [f». tfk. h- O9 CO i bO p bO 
38 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

geographical location of the stations should be studied by plotting 

the stations on a map. If their grouping discloses a regional climatic 

anomaly, the breaks due to this anomaly do not necessarily indicate 

inconsistent records, but may indicate which stations should be 

grouped in a pattern.

The double-mass curve for station E shows a break in slope at the 

year 1930. The double-mass curve gives no clue as to the reason for 

the break but an examination of the station history reveals that, in 

1930, a new observer was hired and the gage was moved to the yard 

of the new observer some distance away. By convention breaks such 

as this one that have been definitely identified with changes in location 

are shown by a distinctive symbol. As station histories are sometimes incomplete, especially for the earlier years, breaks may be due 

to changes in gage location even though the station record does not 

indicate that these changes took place.

The double-mass curve for station A, which is virtually an unbroken 

straight line with a slope of 1.05, indicates that this record is consistent 

although the points scatter slightly on both sides of the line.

ADJUSTMENT OF PRECIPITATION RECORDS

When the double-mass curve of precipitation data from a particular 

station indicates a break in slope and the reason for the break is 

determined, the record for one set of conditions may be adjusted to 

what it would have been if it had been collected under the other set 

of conditions. The period of record to be adjusted depends upon the 

use that is to be made of the records. The investigator should be 

conservative when adjusting precipitation records. Underadjustment 

is preferable to overadjustment.

The theory of the double-mass curve suggests the method of adjusting an inconsistent record. For example, in figure 1, the doublemass curve of station E would plot as a straight line if the observation 

conditions for 1926-30 and 1931-42 were the same; in other words, 

the slope of the double-mass curve would be the same for the two 

periods. The observed data for 1926-30 are adjusted by multiplying 

them by the ratio of the slope of the double-mass curve for 1931-42 

to the slope for 1926-30, or

p _"a -p 

a~b,° 

where

Pa = adjusted precipitation 

P0 =observed precipitation

ba = slope of graph to which records are adjusted 

b o slope of graph at time P0 was observed
DOUBLE-MASS CURVES 39

Adjusted precipitation data for station E (1926-30) 

[b tt : & as 1.06:0.79]

Year

1926

1927-._.______

1928 _.

1929___---_-._-

1930-__.__.___

Pa

32.85

28.08

33. 51

29. 58

23. 76

Pa

44.08

37.68

44. 96

39. 69

31. 88

The mass-curve technique is not suitable for adjusting daily precipitation or storm precipitation.

MISSING RECORD

Most analyses involving precipitation data are complicated by 

incomplete data for some of the years. For example, of the 128 

precipitation stations in the climatic summary for Kentucky, 1931 

through 1952, about 75 percent have 1 or more years for which a 

yearly total was not given. This percentage was computed without 

including incomplete years at the beginning or at the end of a record. 

Although records for one State selected at random cannot be considered typical, they do show that the problem of missing data is serious 

in a hydrologic analysis.

The practice of the U.S. Weather Bureau is to interpolate missing 

or incomplete monthly precipitation data by using the data from 

three adjacent stations (Paulus and Kohler, 1952). Where the normal 

annual precipitation at each of the three adjacent stations differs 

by less than 10 percent from that of the interpolation station (station 

with the incomplete record), the precipitation for the period of missing 

record is estimated as the straight average of the concurrent precipitation at the three adjacent stations. Where, as in a mountainous 

area, the normal annual precipitation at any of the three stations 

differs by more than 10 percent from that of the interpolation station, 

the normal-ratio method is used. In this method, the precipitation 

at each of the three stations is multiplied by the ratio of the normal 

annual precipitation at the interpolation station to the normal annual 

precipitation at each station. The weighted precipitation of the three 

stations is averaged to obtain the estimate for the interpolation station.

The double-mass curve can also be used to estimate missing precipation data, but the method is generally more laborious and no more 

accurate than the U.S. Weather Bureau method. To estimate the 

missing record at station A using the record of station B, multiply 

the data at B, corresponding to the missing period, by the ratio of 

the slope of the double-mass curve of station A to the slope of the 

double-mass curve for station B. Independent estimates are made
40 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

from several adjacent stations and the estimates are averaged to 

obtain the estimate for station A, sometimes weighting the individual 

estimates when they vary considerably. The estimator should not 

calculate from base estimates on distant long-term stations for which 

double-mass curves are available rather than using nearby short-term 

stations that might provide a more accurate estimate.

STREAMFLOW RECORDS

Streamflow records are among the most valuable of all hydrologic 

data because they represent an integration of all hydrologic factors. 

Furthermore, the flow of streams is a sensitive indicator of climatic 

variations, because runoff is the residual of precipitation after the 

demands for evapotranspiration are satisfied. With evapotranspiration losses fairly constant from year to year in a given area, variations 

in annual runoff are much greater in percentage than variations in 

annual precipitation.

Streamflow records to be used in any analyses involving the record 

as a whole should be checked for consistency, and the cause of any 

inconsistency investigated. Sometimes the record must be adjusted 

to consistent conditions. Inconsistency can be due to a change in 

the method of collecting streamflow data or to major changes in 

water use, storage, or evapotranspiration in the basin.

The double-mass curve can be used to check for inconsistencies in 

streamflow records in much the same manner as it is used to check 

for inconsistencies in precipitation records, but the assumption that a 

constant ratio exists between a given record and a group of records 

in the area may not be valid. In an analysis of annual discharge, 

such an assumption is usually justified, even although the ratio of 

monthly discharge is not constant, because annual discharge is composed largely of streamflow not aflected by variable basin conditions. 

If there is any question, a plot of the variables should be made, and, 

if the ratio is found to vary greatly with discharge, a double-mass 

curve of observed and computed values of the discharge should be used.

CHECKING CONSISTENCY

The method of using the double-mass curve to check the consistency 

of the streamflow record by comparing it with the records from a group 

of other stations in the same area is presented in the following example. 

In order to compute the group mean, or pattern, the streamflow data 

are first converted to a comparable basis, such as runoff in inches, in 

cubic feet per second per square mile, or in percent of mean flow; 

otherwise, the flow of a large stream in the group would have more 

effect on the pattern than would several small streams. In this 

example, annual runoff in inches is used as shown in table 2. The
DOUBLE-MASS CURVES 41

TABLE 2. Annual runoff, in inches, for double-mass curve

Water year

1921 _____

1922-..--.-.

1923-. ........

1924 

1925. - 

1926 

1927 . 

1928-.-..-..

1929 . . ..... 

1930 .

1931 

1932 ......

1933 

Stream A

Yearly

19.73 

15.80 

17.52 

16.58 

5.33 

16.45 

30.67 

21.22 

21.96 

19.34 

9.87 

24.81 

15.53

Cumulative

19.73 

35.53 

53.05 

69.63 

74.96 

91.41 

122.08 

143.30 

165.26 

184.60 

194.47 

219.28 

234.81

Pattern

Yearly

19.61 

12.29 

8.12 

14.39 

3.53 

13.80 

24.03 

12.40 

19.70 

18.10 

5.13 

18.30 

12.20

Cumulative

19.61 

31.90 

40.02 

54.41 

57.94 

71.74 

95.77 

108.17 

127.87 

145. 97 

151. 10 

169. 40 

181.60

Water Year

1934. 

1935 - 

1936 

1937

1938

1Q3Q

1940 -

1CL11

1942 

1943 

1944 

1945 

Stream A

Yearly

9.35 

32.75 

7.57 

19.72 

28.33 

15.04 

13.65 

17.42 

17.82 

9.41 

21.13 

37.85

Cumulative

244.16 

276.91 

284.66 

304.38 

332.71 

347. 75 

361.40 

378. 82 

396.64 

406.05 

427. 18 

465.03

Pattern

Yearly

7.94 

25.58 

4.06 

13.76 

28.64 

10.41 

10.68 

30.15 

21.60 

8.96 

20.01 

40.25

Cumulative

189.54 

215. 12 

219. 18 

232.94 

261.58 

271.99 

282.67 

312. 82 

334. 42 

343.38 

363.39 

403.64

data for the individual stations comprising the pattern are omitted 

from this table and only their averages are shown.

When the cumulative runoff from table 2 is plotted on figure 2, a 

decided break in slope is noted at the 1938 point. The mean annual 

runoff, in inches, of periods on both sides of the break is as follows:

Period Stream A Pattern

1921-38__---_______---_____.___ 18.48 14.53

1939-45-_-_________--__-_-.___. 18.90 20.29

The later period is shown to be somewhat wetter than the earlier 

period, but the increase is less marked on stream A than on the streams

1

o

z

^3

.5

Z) 

2 

.3 

O

500

400

300

200

100

100 200 300 400 500 600 

CUMULATIVE RUNOFF FOR PATTERN, IN INCHES

FIGURE 2. Double-mass curve of streamflow data.

700
42 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

comprising the pattern. Furthermore, during the earlier period the 

mean for stream A is higher than that of the pattern, and during the 

later period the relation is reversed. These contrasts suggest an 

inconsistency in the record for stream A.

The next step would be to examine the record for stream A and try 

to find a reason for the inconsistency. Sometimes the record should 

be recomputed or adjusted as explained in the next section.

If the break in the double-mass curve cannot be explained by error 

in the base data, it would be wise to make the statistical test for 

significance described in a later section. This test shows the likelihood of the break occurring by chance alone as compared with a real 

reason for the break. The statistical test for significance also furnishes 

a valuable guide to conclusions as to the effect on streamflow of known 

changes in stream regimen.

If the statistical test shows that the break is unlikely to have occurred by chance, further hydrologic analysis is required to find the 

reason for the break and to decide if the change in conditions is 

permanent or temporary.

ADJUSTING STREAMFLOW RECORDS

The double-mass curve, which is useful in detecting inconsistencies, 

should seldom be used to adjust streamflow records. Instead, the 

adjustment should be based on other analyses made after a probable 

source of the inconsistency has been found. If the inconsistency is 

due to a change in method, such as an improvement in the high water 

rating, the record would be recomputed on the basis of revised ratings. 

If the inconsistency is due to diversions out of the basin, the amount 

of the diversion would be estimated and added to the observed discharge. If a significant inconsistency is attributed to changes in land 

use or land management practices and if a hypothetical record for one 

of the periods is desired, estimates should usually be obtained through 

correlation methods. The reason for not adjusting streamflow 

records by the percentage shown by the break in the double-mass 

curve is that the adjustment likely varies throughout the range in 

discharge and, thus, would have a different average for a series of wet 

years than for a series of dry years.

SEDIMENT RECORDS

The double-mass curve is used for studying trends in sediment yield 

and in detecting the effect of watershed practices on sediment yield.

Figure 3, from Guy's (1957) report, is a double-mass curve of cumulative sediment plotted against the cumulative runoff. The slope of 

the double-mass curve defines the mean concentration during the 

period. The double-mass curve for storm runoff and sediment dis-
sr

CD

CD 

P

O O

£ sr 

3 85

i^ CTQ 5^ ^

£T. o

O P

S3 S3

O Q- 

-* CD 

P M

?3

tO CD

§a 

gs0 oo

o

^ co

^1.

O CTQ 

pi Sj

i;5" S3

O , 

p ^

o- 2:

CD ST 

§* »

w ST

& CD 

CD g,

O"

Hj p

s -

F? P

11rtj M

Spf 

s °

« ! 

of

i-9 09 

fl> CD

ll

| S 

P 8.

o a

c? e 

 O P

IB 

!i 

s?

o

c 

s

Im

CO

m 

o

z

H

O

33

O

CUMULATIVE TOTAL RUNOFF, IN THOUSANDS OF C

o o o o o 

X

o

I

CO

o 

-n

O 

Z 

CO

\

"^J

XQ,

s

Sj

X NO

V^S

s. ^^**

U1 

VO 

00

 o 

 o 

3

i »

00

vo

'1

k_

X

00 

(71

 o 

 o

w

\,

x^

O \3

K

^

i «

§

 o 

 o

\

Nxj

1 o 

 o

\ % 

^Oo

V

V

^

c-, iji i-; j-* ro 

8 s S 8 

CUMULATIVE STORM RUNOFF, IN THOUSANDS OF 
44 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

The double-mass curve of total runoff and sediment discharge is a 

series of lines differing in slope. The variable relation between total 

runoff and sediment discharge is understandable because, during the 

period of the study, storm-runoff periods accounted for 89 percent of 

the sediment but for only 19 percent of the total runoff.

PRECIPITATION-RUNOFF RELATIONS

When the double-mass curve is used to study trends or possible 

changes in precipitation-runoff relations (Oltman and Tracy, 1951) 

or when it is used with precipitation to check the consistency of streamflow records (Merriam, 1937), the cumulative measured runoff should 

be plotted against the cumulative computed runoff, taken from a 

precipitation-runoff relation. A double-mass curve of cumulative 

measured runoff and cumulative precipitation should not be used 

because the relation between precipitation and runoff is seldom a 

constant ratio even during a period when there was no change in the 

relation. As explained earlier in this report a variable ratio would 

violate the theory of the double-mass curve.

AKEAL, PRECIPITATION

The average amount of precipitation on a particular river basin is 

obtained by averaging the precipitation measured at the rain gages 

in and near the basin. Methods used to compute the average rainfall 

on the basin are:

1. Using the catch of a single gage for small areas (1 square mil3 or less).

2. Computing the arithmetic mean of the rain-gage records.

3. Weighting the precipitation at individual stations by the Thiessen method. 

(Horton, 1924.)

4. Drawing isohyetal lines and computing a weighed average.

The application of the various methods is discussed in WaterSupply Paper 846 (Williams and others, 1940) and in standard texts 

on hydrology.

Precipitation in a given area generally increases with the altitude, 

therefore the average precipitation on a basin with a wide range in 

elevation should be adjusted (Williams and others, 1940) for this 

increase.

EFFECTIVE PRECIPITATION

The amount of precipitation that fell the previous year is one of the 

factors that affect the relation between precipitation and runoff and 

cause the points on a graph of annual precipitation plotted against 

annual runoff to assume a "shotgun" pattern. Generally the scatter 

of the points can be reduced by plotting an effective precipitation 

instead of the observed precipitation.

Use of an effective precipitation is one way of making allowance for 

the variable amount of water carried over from year to year as ground
DOUBLE-MASS CURVES 45

water storage in the basin. The effective precipitation (Pe) commonly used is that proportion of the current year's precipitation (P0) 

and the proportion of the preceding year's precipitation (Pi) that 

furnishes the current year's runoff, or

Pe=aP0+bPl

When the carryover effect extends beyond the previous year, portions 

of additional years (P2 . . Pn) with their coefficients are added to the 

effective precipitation formula. The sum of the coefficients, a, 6, c, 

etc., must equal unity. The coefficients a and 6 can be determined 

by rank correlation, as shown in table 3.

In table 3, the yearly runoff (column 3) is assigned a number according to rank beginning with the highest runoff as number 1 as shown in 

column 2. When two values are identical, they are both assigned 

the average of the two order numbers they would have if they were 

slightly different from each other. The yearly runoffs are cumulated 

in column 4.

The yearly average precipitation (P0) over the basin is tabulated 

in column 6 and order numbers are assigned as was done for the 

yearly runoff and are shown in column 5. The difference in rank or 

order number between the precipitation and the runoff is squared and 

noted in column 7. The sum of the squares of the difference in rank 

is obtained.

A formula for effective precipitation is assumed and values of 

effective precipitation are computed. In the first trial, tabulated in 

columns 8-10, the following formula is assumed:

The values of effective precipitation are assigned order numbers and 

the difference in rank between these numbers and the rank of the 

runoff for the same year are squared. The sum of the squares of the 

difference in rank, 515, is compared with that for the current year's 

precipitation, 775. In the first trial the effective precipitation agrees 

with the runoff better than the current year's precipitation as shown 

by a lower sum of squares of the difference in rank.

A second assumption for effective precipitation is made where:

Pe=0.6P0 +0.4P1 .

The procedure of the foregoing paragraph is repeated; the sum of 

squares of the difference in rank, 556, is found to be greater than that 

of the first trial.
TABLE 3. Correlation of annual precipitation and annual runoff, Colorado 

[The yearly runoff for 1921 and for 1922 is estimated. Columns 2, 5,8,11, and 14 are assigned numbers according 

Water 

year

1

1920---...

1921......

1922... ... 

1923......

1924__.._. 

1925......

1926-.. 

1927...... 

1928...... 

1929..--.

1930-.--.. 

1931.. .

1932... 

1933-. .

1934......

1935-.--.

1936.-.-. 

1937.. .

1938--. 

1939_- _

1940-. .

1941-- 

1942......

1943. .

1944- 

1945-_ _

1946.. .

Total...

Average...

Runoff (millions of 

acre-feet)

Relative 

rank

2

1 

3 

6 

14 

18

11 

24.5 

*9.5 

2 

13

25

8 

21 

26 

20

16 

15 

»9.5 

22 

24

7 

»4.5 

19 

12 

17 

23

Yearly

(Y)

3

21.27 

17.84 

17.05 

13.01 

11.74

14.42 

17.26

15; 63

19.43 

13.42

6.74 

15.97 

10.01 

.4.66 

10. 22

12.32 

12.41 

15.63 

9.62 

7.44

16.94 

17.26 

11.43 

13.53 

11.87 

9.09

346.21 

13.32

Cumulative

4

21.27 

39.11 

56.16 

69.17 

80.91

95.33 

112.59 

128.22 

147.65 

161.07

167. 79 

183.76 

193.77 

198.43 

208.65

220.97 

233.38 

249.01 

258.63 

266.07

283.01 

300.27 

311. 70 

325.23 

337.10 

346.19

Precipitation (inches)

Original data

Relative 

rank

5

4 

8 

5 

23 

6

13 

1 

18 

2 

'9.5

26 

12 

22 

25 

15

16 

19 

11 

24 

21

3 

14 

7 

17 

29.5 

20

Yearly 

average 

precipitation

6

16.68 

16.52 

14.16 

16.30 

11.56 

15.15

13.78 

20.36 

12.49 

18.43 

14.00

9.94 

13.85 

12.12 

10.09 

13.12

12.72 

12.47 

13.99 

11.14 

12.31

16.94 

13.65 

14.63 

12.51 

14.00 

12.39

Column 

(5-2) *

7

9 

25 

1 

81 

144

4 

12 

72 

0 

12

1 

16 

1 

1 

25

0 

16 

2 

4 

9

16 

90 

144 

25 

56 

9

775

Effective precipitation assu

First assumption

Relative 

rank

8

3

7 

5 

2 20. 5 

2 8.5

12 

1 

11 

2 

6

25 

15 

22 

26 

a 20. 5

17 

19 

14 

24 

23

4 

10 

2 8. 5 

16 

13 

18

0.8P. 

+ 

0.2Pi

9

16.55 

14.63 

15.87 

12.51 

14.43

14.05 

19.04 

14.06 

17.24 

14.89

10.55 

13.07 

12.47 

10.50 

12.51

12.80 

12.52 

13.69 

11.71 

12.08

16.01 

14.31 

14.43 

12.93 

13.70 

12.71

Column 

(8-2) *

10

4 

16 

1 

42 

90

1 

12 

2 

0 

49

0 

49 

1 

0 

0

1 

16 

20 

4 

1

9 

30 

110 

16 

16 

25

515

Second assumption

Relative 

rank

11

2 

7 

6 

13 

12

10 

1 

5 

3 

4

25 

21 

19 

26 

23

18 

20 

15 

22 

24

8 

9 

11 

16 

14 

17

0.6P. 

+ 

0.4Pi

12

16.58 

15.10 

15.44 

13.46 

13.71

14.33 

17.73 

15.64 

16.06 

15.77

11.56 

12.29 

12.81 

10.90 

11.91

12.88 

12.57 

13.38 

12.28 

11.84

15.09 

14.97 

14.24 

13.36 

13.40 

13.03

Column 

(11-2)

13

1

16 

0 

1 

36

1 

12 

20 

1 

81

0 

169 

4 

0 

9

4 

25 

30 

0 

0

1 

20 

64 

16 

9 

36

556

i PI is the preceding year's precipitation and P. is the current year's precipitation in the effective precipitation a

J When two values are identical, they are both assigned tne average of the two order numbers they would have 
DOUBLE-MASS CURVES 47

A third assumption for effective precipitation is made where : 

Pe=0.7P0 +0.3P,.

The sum of squares of the difference in rank, 480, is the lowest of the 

four trials. The relation

Pe=0.7P0+0.3P!

is thus adopted as the effective precipitation that will correlate best 

with the yearly runoff. It is noted in table 3 that the sums of squares 

decrease with increasing percentages of the previous year's precipitation until a low point, hi this example 0.3Pi, is reached. Beyond this 

point, the sums of squares increase with increasing percentages of the 

preceding year's precipitation.

COMPUTED RUNOFF

Computed runoff as used in this report is the runoff taken from 

a curve of relation of precipitation and runoff. It represents the 

most probable amount of runoff for a given amount of effective precipitation under conditions such as existed during the period used to 

define the curve.

In figure 4 is an example of a curve used to obtain computed 

runoff. The yearly values of effective annual precipitation are 

plotted against the annual runoff, and a curve is drawn to average 

the shotgun pattern of plotted points. In this example the points 

can be averaged by a straight line.

One form of the equation of a straight line is:

Y=a+bX 

in which, for the least squares method :

a (the intercept on the Y-axis) =Y bX

* m, i t n, r \ b (the slope of the lme)=

X=An annual effective precipitation value

F=An annual runoff value 

and

S=sum of

X, F= average of the X's or of the F's 

A/r=number of pairs of items
48 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

30

cc.

O 

V)

z

O

20

10

10 20 

ANNUAL EFFECTIVE PRECIPITATION, IN INCHES

30

FIGURE 4. Relation of effective annual precipitation to annual runoff, Colorado River near Grand Canyon, Arlz., 1921-46.

Using this formula for the data in figure 4 and substituting the 

values from columns 18 to 20 of table 3:

,, 4,946.52-26(12.84)(13.32)^153.45 

5,064.48-26(13.84)2 84.29

0=13.32-1.82(13.84) = -11.87

Thus F= 11.87+1.82^

The computed runoff given by Y is usually indicated by the symbol 

Yc or Rc .

If the points in figure 4 had indicated a curved relation, a line 

would not have been fitted by least squares; instead, a smooth curve 

to average the points would have been drawn.

The computed runoff is the runoff expected from a given amount 

of precipitation if there is no change in the precipitation-runoff rela-
DOUBLE-MASS CURVES 49

tion during the period. The double-mass curve of measured runoff 

versus computed runoff thus would reveal changes in the precipitation-runoff relation as well as possible inconsistencies in the records.

In the example, an expected annual runoff (Ye) is computed for 

each year from the relation Yc= 11.87 + 1.82^, where X is the 

effective annual precipitation. The expected annual runoff is listed 

in column 4 of table 4 and cumulated in column 5. The annual and 

cumulated measured runoff is transferred from table 3. The doublemass curve of measured runoff versus computed runoff is plotted 

in figure 5.

The double-mass surves for the period 1921-26 and for the period 

1927-46 are about parallel but are offset between 1926 and 1927. The 

offset is explained by the large difference between the measured and 

the expected runoff in a single year, 1927, and should not be interpreted 

as a break.

300

1945,.

200

1935

1930

100

100 200 300 

CUMULATIVE MEASURED RUNOFF, IN MILLIONS OF ACRE-FEET

FIOURE 5. Double-mass curve of measured runoff versus computed runoff, Colorado River near Orand

Canyon, Ariz., 1921-46.
50 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

TABLE 4. Measured annual runoff and computed annual runoff, Colorado River

near Grand Canyon, Ariz.

[Y for 1921-22 is estimated]

Water Year

1921 

1922 .. ~ ...........

1923. ........

1924...... ..... ........ .. ..

1925 

1926 - 

1927 

1928 

1929 

1930 

1931 

1932 ...

1933 

1934 

1935 

1936 

1937 

1938 

1939 

1940 

1941 

1942 

1943 

1944 

1945 

1946 

Runoff (millions of acre-feet)

Measured

y

21.27 

17.84 

17.05 

13.01 

11.74 

14.42 

17.26 

15.63 

19.43 

13.42 

6.74 

15.97 

10.01 

4.66 

10.22 

12 32 

12.41 

16.53 

9.62 

7.44 

16.94 

17.26 

11.43 

13.53 

11.87 

9.09

Cumulative

21.27 

39.11 

56.16 

69.17 

80.91 

95.33 

112.59 

128.22 

147. 65 

161.07 

167.79 

183.76 

193. 77 

198.43 

208.65 

220.97 

233.38 

249 01 

258.63 

266.07 

283.01 

300.27 

311. 70 

325.23 

337. 10 

346.19

Expected

y.

18.29 

15.19 

16.63 

11.75 

13.74 

13.96 

21.60 

15.16 

18.43 

16.03 

8.44 

11.21 

11.13 

7.60 

10.35 

11.50 

10.95 

12.75 

9.97 

9.90 

16.43 

14.77 

14.23 

12.06 

12.79 

11.55

Cumulative

18.29 

33.48 

50.11 

61.86 

75.60 

89.56 

111. 16 

126.32 

144.75 

160.78 

169.22 

180.43 

191.56 

199.16 

209.51 

221. 01 

231.96 

244.71 

254.68 

264 58 

281.01 

295.78 

310. 01 

322.07 

334 86 

346. 41

Residual

y-y«

+2.98 

+2.65 

+.42 

+1.26 

-2.00 

+.46 

-4.34 

+.47 

+1.00 

-2.61 

-1.70 

+4.76 

-1.12 

-2.94 

-.13 

+.82 

+1.46 

+2.88 

-.35 

-2.46 

+.51 

+2.49 

-2.80 

+1.47 

-.92 

-2.46

Cumulative

+2.98 

+5.63 

+6.05 

+7.31 

+5.31 

+5.77 

+1.43 

+1.90 

+2.90 

+.29 

-1.41 

+3.35 

+2.23 

-.71 

-.84 

-.02 

+1.44 

+4.32 

+3.97 

+1.51 

+2.02 

+4.51 

+1.71 

+3 18 

+2.26 -.20

When a break in the double-mass curve occurs, the statistical test 

for significance, described in a later section, furnishes a valuable guide 

to the decision as to whether a change has occurred in the precipitation-runoff relation.

RESIDUAL-MASS CURVE

Minor breaks that are hidden by the smoothing of a double-mass 

curve can be magnified for detailed study by using a residual-mass 

curve. The example of a residual-mass curve in figure 6 is obtained 

by subtracting the computed runoff from the measured runoff (table 

4, col. 6) and cumulating the residuals (table 4, col. 7) and plotting 

them against the year of occurrence.

The residual-mass curve has the following characteristics:

1. Changes in the relation between two variables are magnified.

2. A break in the double-mass curve corresponds to a maximum or a 

minimum point on the residual-mass curve.

Figure 6 reveals many minor breaks in the double-mass curve but, 

in general, the residuals increase during the period 1921-26 and fluctuate without a particular trend after 1927. This conclusion corresponds with the two lines on the double-mass curve and supports the 

conclusion that there has been no major change in the relation between 

rainfall and runoff during the period 1921-46.
DOUBLE-MASS CURVES 51

o

1920 1925 1930 1935 

WATER YEAR

1940 1945

FIGURE 6. Residual-mass curve of measured runoff minus computed runoff, Colorado River near Grand

Canyon, Ariz.

STATISTICAL TEST FOR SIGNIFICANCE

A statistical test of significance of an apparent break in slope of a 

double-mass curve is usually the last step of a double-mass-curve 

analysis. Breaks in the double-mass curves that are due to changes 

in method of computation or to known diversion of water, not returned 

to the stream, will have been removed so that the remaining breaks 

are due either to chance or to a change in the relation, but we do 

not know which. Statistical methods are used to help answer this 

question.

The significance of an apparent break in the double-mass curve can 

be determined by an analysis-of-variance test (covariance in this 

example). Variance sometimes called the mean square, is a measure 

of the scatter of individual values about the mean of the values and 

is computed as the sum of the squares of the differences between 

the mean and the individual values divided by one less than the number of values. Covariance is a mean of the cross product of the 

deviations in the same way as variance is a mean square, and is used 

in computing the variance about a line of regression. Details of the 

analysis-of-variance method are found in textbooks on statistics, and 

only the application of the F test (variance-ratio test) to the data in 

table 2 is discussed here. For the purpose of this example it is 

assumed that the difference in slope that occurred in 1938 cannot be 

removed by recomputing the records or by adjusting for known 

diversions. A graphical method for evaluating the significance of 

breaks in the double-mass curve is given by Weiss and Wilson (1953).

Since the double-mass curve between stream A and the pattern 

seems to indicate that a change in the relation between that stream
52 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

and the pattern occurred in 1938, an analysis of variance test is applied 

to the data for the periods 1921-38 and 1939-45. The statistics used 

in this test is F, the ratio of the among-periods variance to the withinperiods variance. The larger the apparent shift in the relation the 

larger this ratio becomes.

Most collections of statistical tables contain a table of the F distribution to use in testing for equality of variance. The test consists of 

computing F, the variance ratio, from the data being tested and 

comparing the computed value with the tabulated value of the F 

distribution for the level of significance selected (generally 5 percent). 

The chances of the computed value of F equaling the tabular value 

at the 5-percent point by chance alone are 1 in 20. When, as in this 

example, the computed value of F exceeds the tabular value for the 

5-percent level, the probability of obtaining such a result by chance 

alone becomes less and the result is significant at the 5-percent level 

(sometimes stated as the 95-percent level). As applied to the data 

in table 2, a computed value of F that exceeds the value tabulated 

for the 5-percent level would support the presumption that the break 

in the double-mass curve is caused by a change in the relation between 

station A and the pattern and not by the vagaries of sampling. The 

test does not prove that the relation is different in the two periods, 

but only that a break this great would be unlikely to occur were there 

really no difference in the periods.

The data in table 2 are analyzed by the variance-ratio test (F test) 

in the following example in which these symbols are used: 

N=25 years total period (1921-45). 

N!=IS years period 1 (1921-38). 

N2 =7 years period 2 (1939-45). 

X=Au annual runoff for the pattern. 

Xl =An annual runoff during period 1 for the pattern. 

-Xr2 =An annual runoff during period 2 for the pattern. 

F=An annual runoff for station A. 

Yl =An annual runoff during period 1 for station A. 

F2 = An annual runoff during period 2 for station A. 

S=Total for period. 

x, x^ z2 =Deviations difference between an -X" value and the period

mean. 

y} yl} 7/2=Deviations difference between a Y value and the period

mean.

or -TT^-T= Variance of X or Y. -1 "l N-l

=Covariance of X and F. (d.f.)
DOUBLE-MASS CURVES 53

d= Residual the vertical (F) difference between the line through

the plotted points and the plotted point.

d.J. = Degrees of freedom. In general, the degrees of freedom equal 

the number of items minus the number of constraints imposed 

on the system. Degrees of freedom for this example are discussed in the explanation of table 5.

In this example, shortcut statistical methods are used in the special 

application of the F-test to a specific problem that of determining the 

significance of a break in the slope of a double-mass curve. The shortcut methods obscure what is being done but gives an example that can 

be followed easily as a model when data of the same type are to be 

tested.

The annual runoff data in table 2 are prepared for the .P-test as 

follows : 

Step 1

Add the X and Y values for period 1 and period 2 and for the entire 

period.

1 = 261.58 SFi = 332.71 

6 SF2 =132.32 

4 SF=465.03 

Step 2

Square the individual X and F values and obtain the sum of squares 

by period and for the entire period.

SJ£i2 =4,711.9 SFi2 =7,132.6 

2^,2=3,698.8 SF22 =3,001.2 

SJ£2=8,410.7 SF2= 10,133. 8 

Step 3

Square the total for each period and divide by the number of years 

in the period.

2+N= 6,517.0

Step 4

Multiply the X and corresponding F values and sum the products 

for each period, and for the entire period.

i = 5,710.8 

2 = 3,243.1

Step 5

Multiply the total X for each period by the total F for that period 

and divide by the number of years in the period.

-*-JV= 7,508.2
54 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

The analysis of variance and covariance is summarized in table 5. 

TABLE 5. Analysis of variance and covariance of data in table 2.

(a) Among periods. _____

(b) Within periods.. ... ... .

(c) Total..................

1

Degrees 

of freedom

1

23

24

2

Sum of

X*

167.3

1, 726. 4

1, 893. 7

3

Sum of V2

0.9

1, 482. 8

1,483.7

4

Sum of 

xy

12.1

1, 433. 6

1, 445. 7

5

Degrees 

of freedom

Ni=l

Ni=22

23

6

Sum of 

^

87.7

292.3

380.0

7

Mean 

^

87.7

13.3

87 7 F=-^=6.59; from table for Nt=l and N2=22, F=4.30 at 5 percent lo.o 

level and 7.94 at 1-percent level.

The items in table 5 are computed as follows:

Column 1 (degrees of freedom)

(a) Among periods:

The degrees of freedom among periods=the number of 

periods (2) minus 1 = 1.

(b) Within periods:

The degrees of freedom within periods = the number of 

items (25) minus 2 (a mean for period 1 and a mean for 

period 2 has been computed) =23.

(c) Total:

The degrees of freedom for the total equal the number of 

items (25) minus 1 (lost through computing a mean for 

the total period) =24.

Column 2 (sum of x2 )

(a) Among periods:

N2 N

From step (3) :

3,801.3+2,883.0-6,517.0=167.3 

(b) Within periods:

From steps (2) and (3):

8,410.7- (3,801 .3+2,883.0) = 1,726.4
DOUBLE-MASS CURVES 55

(c) Total:

From steps (2) and (3) :

8,410.7-6,517.0=1,893.7

Column 3 (sum of y2 )

(a) Among periods:

^ , 2-'y ~ "*"

AT, ' N2 N

From step (3):

6,149.8+2,501.2-8,650.1=0.9 

(b) Within periods:

r/^r^v ,

lj-+

I N, -r N2 J

From steps (2) and (3):

10,133.8-(6,H9.8+2,501.2) = 1,482.8 

(c) Total:

From steps (2) and (3):

10,133.8-8,650.1 = 1,483.7

Column 4 (sum of xy)

(a) Among periods:

From step (5):

4,835.0 + 2,685.3-7,508.2 = 12.1
56 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

(b) Within periods :

From steps (4) and (5) :

8,953.9- (4,835.0+2,685.3) = 1,433.6 

(c) Total:

From steps (4) and (5) :

8,953.9-7,508.2=1,445.7

The analysis of variance and covariance in table 5 (columns 5-7) 

compares the mean of the sum of the squares of the residuals (sum of 

d?) from the regression line through all the items for the total period 

with the mean of the sum of the squares of the residuals from separate 

but parallel regression lines through the points for each part of the 

period. 

Column 5 (degrees of freedom)

One additional degree of freedom is lost from (b) Within periods, 

and from (c) Total, by reason of computing a slope of the 

regression line. 

Column 6 (sum of d2 )

The sum of squares of the residuals is computed by the formula:

(c) Total:

/"i ..

2= 1,483.7-^1=380.0

The total dispersion of the Y values about the means for the 

total period minus the variance due to the regression is 

computed from the formula given for Sd2 using the values 

in columns 2-4, table 5. 

(b) Within periods:

2= 1,482.8 -
DOUBLE-MASS CURVES 57

The sum of squares of the residuals about a regression line 

through the dispersion of the points about their period 

means is computed from the formula given for 2d2 using 

the values in columns 2-4, table 5. 

(a) Among periods:

The reduction in the sum of squares of the residuals between 

the "Total" and the "Within periods" line is attributable 

to the dispersion of the means, or 380.0 292.3 = 87.7. 

F (6.59) is computed by dividing the among-periods mean

d2 (87.7), by the within-periods mean d? (13.3). 

Interpretation of the test. The computed value of F (6.59) exceeds 

the tabular value of F (4.30) at the 5-percent level. Therefore, the 

break in the double-mass curve is significant at the 5-percent level. 

Thus, from a purely statistical evaluation, the break in the curve 

between 1938 and 1939 can be attributed to a change in the relation 

between station A and the pattern.

EFFECT OF THE DEGREE OF CORRELATION BETWEEN STATION

RECORDS

Poor correlation between the station being tested and the pattern 

can obscure the detection of significant differences between periods. 

To illustrate this point, the data in table 2 have been changed to 

decrease the correlation between station A and the pattern without 

changing the means for the two periods. The changed data are listed 

in table 6.

TABLE 6. Annual runoff, in inches, for double-mass curve with poor correlation 

between the station and the pattern

Water year

1921. ................ . ....

1922.........................

1923..-.-... ......... ---

1924.......... _ .... .....

1925---. 

1926.... ...................

1927-------.------. . .....

1928... .-._- ......

1929-. -_ .. _. ..... .....

1930.-------------.........

1931.... ...... .............

1932................ .....

1933 -- -- ---_

Stream A 

Y

21.73

13.80

19.52

H *&

7.33

14 45

32.67

19.22

23.96

17.34

11.87

22.81

17.53

Pattern 

X

17.61

14.29

6.12

16.39

1.53

1 f\ Sft

22.03

14.40

17.70

3.13

10.20

Water year

1O14

1935.. 

1936... 

1937

1938--- 

1Q3Q

1QA[\

1941.. 

1942-. 

1943.- -

1944-- 

1945... -

Stream A 

Y

7.35

34.75

5.75

21.72

26.33

17.04

11.65

19.42

15.82

11.41

19.13

37.85

Pattern 

X

9.94

23.58

6.06

11.76

30.64

8.41

12.68

28.15

23.60

6.96

22.01

40.25

The analysis of variance and covariance in table 7 was computed 

in the manner described for table 5 in the preceding section. It will 

be noted that the computed F (2.33) is less than the tabular value 

(4.30) at the 5-percent level. Thus the F test indicates that the 

break is not significant. As shown in the following section, the effect
58 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES 

TABLE 7. Analysis of variance and covariance of data in table 6

Total-. - - 

Degree of 

freedom

1

23

24

Sum of

X*

1, 826. 4

1, 993. 7

Sum of

V1

0.9

1, 589. 7

1, 590. 6

Sum of 

xy

12.2

1, 330. 0

1,342.2

Degree of 

freedom

2V,=1

2V2=22

23

Sum of 

d»

65.8

621.2

687.0

Mean d»

65.8

28.2

JP=:=2.33; from table for JV, = 1 and N2 =22, F=4.30 at 5-percent

Zo.Z

level and 7.94 at 1-percent level.

of poor correlation can be overcome by extending the length of 

record.

EFFECT OF THE LENGTH OF RECORD

An increase in the length of record tends to offset the effect of 

variability of data. The effect of increasing the length of record can 

be shown by adding to the second period two 7-year periods, both 

identical with the data for 1939-45 (table 6). The analysis of the 

data with the 14 added years is given in table 8.

TABLE 8. of variance and covariance of data in table 6, after adding 14 

years of record

Tot&l-.. .................

Degrees of 

freedom

37

38

Sum of

X*

321.7

3, 528. 1

3,849.8

Sum of V1

3.9

2, 551. 8

2, 555. 7

Sum of 

xy

24.4

2, 429. 3

2. 453. 7

Degrees of 

freedom

Ni- 1

2V?=36

37

Sum of 

ffi

112.7

879.1

991.8

Mean d»

112.7

24.4

112 7 ^=_-i-=4.62; from table for Ni = l and N2=36, F=4.11 at 5-percent

level and 7.39 at 1-percent level.

The computed F (4.62) exceeds the tabular value (4.11) at the 5- 

percent level. Thus we can state that the break in the double-mass 

curve is significant.

By comparing the result of the analysis of the variable data (table 

7) with that of the less variable data (table 5), it is evident that 

conclusions drawn from short periods of high variability are more 

likely to be in error than conclusions based on less variable data for 

the same length of period. With data of high variability (table 7) 

the length of period must be greater (table 8) to arrive at accurate 

conclusions.

SUMMARY

Although the double-mass curve is a valuable tool in hydrologic 

studies, a good understanding of its theory and application to various
DOUBLE-MASS CURVES 59

types of hydrologic problems is necessary if misuse is to be avoided.

What may be a good practice when working with precipitation 

records may be a bad practice when working with streamflow records 

or precipitation-runoff relations. Furthermore, if the variables do 

not bear a fairly constant ratio to each other throughout the range 

of data, the double-mass curve should be based on observed data 

and data computed from a curve of relation rather than on a 

cumulation of the variables themselves.

For precipitation records, the double-mass curve can be used to 

detect changes in the consistency of precipitation records and to 

determine the amount of adjustment to be applied to make them 

consistent. Precautions should be taken to see that the record is 

not being tested against records that are in themselves inconsistent. 

A plausible cause for any apparent inconsistency should be found 

before a record is said to be inconsistent. If the cause of the inconsistency is in doubt, adjustments should not be made unless the 

inconsistency is statistically significant.

For streamflow records, the double-mass curve can be used to 

detect possible changes in consistency but should not be used to 

determine the amount of adjustment. The date of the possible 

change in consistency is a clue to possible changes in method of 

collecting the records or to possible diversions from or to the basin. 

If the change is due to a change in methods the records should be 

recomputed from the adjusted basic data, and if the change is due to 

diversions, adjustments for diversion should be applied. Streamflow 

records should seldom be adjusted by the application of a correction 

factor.

For precipitation-runoff relations, the precipitation records being 

used should first be tested for consistency by the double-mass curve 

method and the runoff records should be examined for changes in 

location and method. The records should be made as consistent as 

possible, and then be plotted against each other to define a relation. 

The double-mass curve should be based on observed and computed 

values of runoff, and should be analysed statistically before a break 

in slope is said to be significant.

FITTING CURVES TO CYCLIC DATA

By W. B. LANGBEIN

A common problem in hydrology is to fit a smooth curve to cyclic 

or periodic data. This section describes two methods that are 

applicable to those problems where the length or period of the cycle 

is known beforehand as, for example, a day, a year, or a meander 

length. Curve-fitting by freehand drawing offers a simple and direct 

course where the data are closely aligned and a fit is clearly evident.
MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

In many problems, however, a satisfactory fit is far from obvious, 

and analytical methods may be necessary. Among the analytical 

methods available for curve fitting are:

1. Moving arithmetic average.

2. Moving arc.

3. Fitting polynomial by least squares.

4. Double integration.

5. Fitting Fourier (sine and cosine) series.

Of these methods, only the moving arithmetic average is in common 

use in hydrologic practice. The moving arithmetic average method 

will greatly attenuate and modify the curve of data that conform, 

for example, to a simple sine curve. In principle, the moving average 

fits a series of straight lines to a set of data. If the graph of the data 

is a curve, a series of straight lines does a poor job of fitting the 

plotted points. Fitting curves by the polynomial and by the Fourier 

series methods is exceedingly complex and, in view of the inaccuracies 

inherent in hydrologic data, it is not practical for most hydrologic 

purposes. For these reasons the moving arc and the double integration methods deserve consideration in fitting curves to hydrologic 

data. Furthermore, these methods are theoretically sound and

eminently practical.

MOVING ABC

PARABOLIC ARCS

Moving arcs, which successfully follow a curve, are described in 

the actuarial literature (for example, Whittaker and Robinson (1932), 

and Sasuly (1934)). Fitting moving parabolic arcs seems simplest. 

The respective weights for parabolic arcs of various lengths are 

given in table 9.

TABLE 9. Weights for n-point least-sauare parabolas 

[Weights are symmetrical about central values in boldface. Whittaker and Robinson (1932, p. 295)]

Number of points, n

5

-3

12

17

12 -3

35

7

-2

3

6

7

6

3

-2

21

9

-21

14

39

54

59

54

39

14

-21

231

11

-36

9

44

69

84

89

84

69

44

9

-36

429

13

-11

0

9

16

21

24

25

24

21

16

9

0

-11

143

15

-78

-13

42

87

122

147

162

167

162

147

122

87

42

etc.

1,105

17

-21

-6

7

18

27

34

39

42

43

42

39

34

27

etc.

323

19

-136

-51

24

89

144

189

224

249

264

269

264

249

224

etc.

2,261

21

-171

-76

9

84

149

204

249

284

309

324

329

324

309

etc.

3,059
DOUBLE-MASS CURVES 61

To illustrate how the above weights can follow a curve, consider 

the following values from a sine curve:

0

. 500

. 866

1. 000

. 866

. 500

0

A straight arithmetical average of 5 points centered on the 4th 

item above, gives a value of 0.75 compared with 1.000 for the known 

points. Applying the 5-point parabolic weights gives 0.995 as a 

result.

Table 10 illustrates the application of a 5-point moving parabolic 

arc to the residual monthly deviations from a correlation of the flows 

of two streams in Utah. In this problem, the smoothed values of 

the monthly deviations are probably superior to the deviations as 

originally computed, because logically there should be a uniform 

variation in these deviations among the months. It is desired, however, not to destroy the intrinsic character of the variations.

The procedure is as follows: The monthly average deviations as 

given are listed under y in table 10 and plotted on figure 7. Each

0.3

0.2

GO

z 

g 0.1

<

-0.1

-0.2

JAN. FEB. MAR. APR. MAY JUNE JULY AUG. SEPT. OCT. NOV. DEC. JAN. 

FIGURE 7. Illustration of smoothing by 5-point moving arc.
62 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

TABLE 10. Illustration of use of 5-point moving parabolic arc

Month

January ........

April...........

July............

August . .. 

y

-0.07 -.05

_ fl<l

+ no

4- 91

+.22

+ no

-.06

-.12

-.13

-.11

-.08

-.07

-.05

-.04

Product of

-3

+0.21

[+.15]

+.12

-.24

-.63

[-.66]

-.24

+ 1Q

1 nff

+.39

+.33

+.24

J- 91

+.15

+.12

y times indie

+12

-0.84

 60

r_ 401

J- Qfi

T_1_O K91

i o CA

+ QA

-.72

_ j 44

-1.56

-1.32

_ Qfi

 84

 60

-.48

ated weight

+17

-1.19 -.85

-.68

m Qfil

+3.57

+3.74

+1.36

-1.02

 9 ftl

-2.21

-1.87

-1.36

-1.19 -.85

-.68

5-point

total

-0.74

+2.89

+7.05

+7.16

+3.01

-1.77

-4.23

-4.55

-3.82

-2.98

-2.30

-2.17

5-point

average

-0.021

+.083

+.20

+.205

+.086

-.050

 19

-.13

-.109

-.085

-.066

-.062

Month

April

July

value of y is then multiplied successively by each weight; the value 

for January, 0.07, is multiplied by 3, by 12, and by 17. To 

obtain the 5-point total for April, for example, the weighted values 

for February, March, April, May, and June (bracketed figures in 

table 10) are totaled: + 0.15-0.48+1.36+2.52-0.66=+2.89. The 

5-point average is found by dividing +2.89 by the algebraic sum of 

the weights, 35, to yield +0.083.

In using the moving arc, how many points are to be used should 

be decided. A short arc will provide close local fits, but will provide 

little averaging. On the other hand, long arcs may tend to alter the 

intrinsic character of the cycle. A useful compromise would be to 

use at least one-fourth but no more than one-half the number of points 

available for defining the cycle. An odd number of items in the arc 

is advised because it provides a definite central value. To provide 

sufficient averaging of random components, at least 5 points should 

be used. This requirement in turn, necessitates at least 10 points 

in the cycle for use of a parabolic arc.

QTJARTIC ARCS

If there are fewer than 10 points in the cycle, the 5-point parabola 

will not make a satisfactory fit. In this situation, a 7- or 9-point 

quartic arc should be used with the weights given in table 11. The 

quartic arcs could be used in place of the parabolic arcs for all problems, 

although this entails more work.

The arithmetic work can be simplified by rounding off the central 

item in table 11 by subtracting or adding one as required. The 

weights are then divisible by 5. For example, approximate weights 

for a 7-point quartic arc are 1, 6, 15, 26, 15, 6, and 1. Their sum 

is 46.
DOUBLE-MASS CURVES 

TABLE 11. Weights for least square quartic arcs

63

7 point

+ 5

-30

+ 75

+ 131

+ 75 -30

+ 5

+ 231

9 point

+ 15

-55

+ 30

+ 135

+ 179

+ 135

+ 30 -55

+ 15

+ 429

11 point

+ 18

-45

-10

+ 60

+ 120

+ 143

+ 120

+ 60

-10

-45

+ 18

+ 429

DOUBLE INTEGRATION

Double integration can be used where the cyclic data approximate 

the simple form:

sin

P

(1)

in which a represents the mean, 6 the amplitude, and p the length of 

the cycle; x represents the position in the cycle, and e the random component. If a, the mean, is subtracted from each of these values, and 

the progressive totals, or first integral, of these differences are taken, 

the result is a series of the form:

bp 2irx . ^- cos ±Se 

2x p

A second integration will give a series of the form:

-<0sin

P

±SSe

(2)

(3)

The term SSe will, in general, be small, since the positive chance 

variations in e will tend to balance the negative terms. The second 

integration, though opposed in sign, is in phase with the original data 

and substantially all random variations will have been averaged out. 

The second integration provides adequate smoothing and so the integrating process need go no further. Although minor cyclic components tend to be averaged out, the second integration may depart 

appreciably from a true sine curve if the original data so require.

The double integration method is, therefore, peculiarly adapted to 

those problems where a sinusoid may be considered a close approximation of the curve sought. The process is illustrated by the following
64 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

V)

UJ

5 1-5

Z

z

z

0

< 1.0 

El

z

UJ

n

-^

/

'*

/

/

/ 

/

/

/"

/

^N

\

\

.\\

\

SV

JAN. FEB. MAR. APR. MAY JUNE JULY AUG. SEPT. OCT. NOV. DEC. 

FIGURE 8. Result of fitting a cycle graph by double integration.

TABLE 12. Example of double integration applied to monthly data

[Note that entries under the 1st integration, representing sums to the end of each month, are moved down 

one line; thus they are on a line midway between the months. Entries under the 2d integration are again 

moved down one line; thus they are on a line even with the months]

Month

April . 

June __ ______

July.. ............

Computed 

evapotranspiration

1

0.18

.27

.82

.80

.68

1.88

1.72

1.40

.88

.44

.48

.18

.81

Deviation 

from 

mean

2

-0.63

-.54

+.01

-.01

-.13

+1.07

J- 01

+.59

+.07

-.37

-.33

-.63

1st integration

3

-0.63

-1.17

-1.16

-1.17

1 ^Ift

-.23

+ fift

i_i 07

+1 34

J- 07

+ RA

+.01

-.06

Deviation 

from 

mean

4

-0.57

-1.11

-1.10

-1.11

-1.24

-.17

+.74

+1.33

-4-1 4(\

+1.03

+.71

+.08

0

2d integration

5

-0.57

-1.68

-2.78

-3.89

-5.13

-5.30

-4.56

-3.22

-1.82

-.79

-.08

0

-2.48

Deviation 

from 

mean

6

+1.91

+.80

-.30

-1.41

-2.65

-2.82

-2.08

-.74

+.66

+1.69

+2.40

+2.48

0

Devation -(12-s2ir2)

7

-0.52

-.22

+.08

+.39

+.73

+.77

+.57

+.20

-.18

-.46

-.66

-.68

0

Smoothed 

values

8

0.29

.59

.89

1.20

1.54

1.58

1.38

1.01

.63

.35

.15

.13

.81
DOUBLE-MASS CURVES 65

computations. The data shown on figure 8 and in table 12 are for 

monthly evapotranspiration from ground water as determined from 

a hydrologic budget for Beaverdam Creek in eastern Maryland. 

Evapotranspiration follows a sinusoidal variation from month to 

month through the course of the year. The erratic variations in the 

given data represent errors that arise because the figures are the 

residuals between relatively large quantities.

The given values of computed evapotranspiration are listed in the 

first numbered column in table 12. The deviations from the mean of 

the data are listed in the second column. The third column shows 

the cumulative summation of the deviations, and completes the first 

integration. The process is repeated in the fourth and fifth columns, 

and a second integration is made as shown in the fifth column. Deviations from the mean of the second integration are shown in the sixth 

column. These figures are divided by the constant (12-r-2ir) 2 in 

which the numerator 12 corresponds to p, the number of items in the 

cycle. The divisor (12-f-2ir) 2 converts the data for equation (3) to 

that for equation (1). The quotients listed in column 7 represent the 

smoothed deviations from the mean of the data, in this example, 

0.81. Adding these to the mean gives the final smoothed results, 

shown in column 8 and on figure 8, which represent a reasonable 

interpretation of the original data.

SUMMARY

The moving arc methods are quite general in application. Of the 

moving arc methods, the parabolic arcs have a slight advantage of 

simpler arithmetic. However, when parabolic weights extend beyond 

a reversal in curvature, quartic weights should be used instead. The 

double integration method is a simple means of fitting a curve when 

the cycle is of a sinusoidal form.

LITERATURE CITED

Guy, H. P., 1957, The trend of suspended-sediment discharge of the Brandywine 

Creek at Wilmington, Del., 1947-1955: U.S. Geol. Survey open-file report, 

p. 14, fig. 3.

Horton, R. E., 1924, Determining the mean precipitation of a drainage basin: 

New England Water Works Assoc., v. 38, p. 1-47.

Merriam, C. F., 1937, A comprehensive study of the rainfall on the Susquehanna 

valley: Am. Geophys. Union Trans., pt. 2, p. 471-476.

Oltman, R. E., and Tracy, H. J., 1951, Trends in climate and in precipitationrunoff relation in Missouri River basin: U.S. Geol. Survey Circ. 98, 113 p.

Paulhus, J. L. H., and Kohler, M. A., 1952, Interpolation of missing precipitation records: U.S. Weather Bur. Monthly Weather Rev., v. 80, p. 129-133.

Sasuly, Max, 1934, Trend analysis of statistics: Washington D.C., Brookings 

Institute.
66 MANUAL OF HYDROLOGY, GENERAL SURFACE-WATER TECHNIQUES

Weiss, L. L., and Wilson, W. T., 1953, Evaluation of significance of slope changes 

in double-mass curves: Am. Geophys. Union Trans., v. 33, p. 893-896.

Whittaker, E. T., and Robinson, G., 1932, The calculus of observations: London, 

Blackie & Son.

Williams, G. R., and others, 1940, Natural water loss in selected drainage basins: 

U.S. Geol. Survey Water-Supply Paper 846, p. 6-11.

o