Graphical Correlation of 

Gaging-Station Records

Manual of Hydrology: Part 1. General Surface-Water 

Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1541-C
Graphical Correlation of 

Gaging-Station Records

By JAMES K. SEARCY

Manual of Hydrology: Part 1. General Surface-Water 

Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1541-C

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.
CONTENTS

Page

Abstract.. -_-----------___-----__-_________--_____________________ 67

Introduction. _____________________________________________________ 67

Use of correlation_____----_____________________________________ 68

Hydrologic principles______________________________________________ 69

Factors affecting the relation between gaging-station records- _______ 70

Statistical principles, ______________________________________________ 71

Concepts and definitions________________________________________ 71

The graphical method._________________________________________ 76

Comparison of the graphical and numerical methods.._____________ 77

Simple correlation. ________________________________________________ 79

Graphical correlation with two variables._________________________ 80

Multiple correlation..______________________________________________ 84

Removing seasonal effect_______________________________________ 85

Graphical correlation with three or more variables.________________ 88

General case..____________________________________________ 89

Special case__-___________-__-__________-_-_---_-____----__ 96

Summary. ________________________________________________________ 96

Literature cited.__________________________________________________ 100

ILLUSTRATIONS

PLATE 1. Correlation of monthly discharge, White River and Greenwater River, Washington. _________________________ In pocket

FIGURE 9. Area in southern Mississippi. -_____-_____-________--_--_-

10. Correlation of monthly discharge, Leaf River and Bowie 

Creek, water years 1940-54- ___________________________

11. Correlation of monthly discharge, Leaf River and Bogue 

Chitto, water years 1940-54. ___-____._____---_-_-----_

12. Ratio, Y:YC , Bogue Chitto and Bowie Creek, water years

13. Correlation of monthly discharge, Leaf River and Bowie 

Creek (adjusted), water years 1940-54. _________________

14. Correlation of monthly discharge, Bowie Cieek and Leaf 

River basin between Hattiesburg and Collins, water years

79

81

90

93

97

98

in
IV CONTENTS

TABLES

Page 

TABLE 1. Monthly mean discharge, in cubic feet per second, of Leaf

River near Collins, Miss., drainage area, 752 square miles- 82

2. Monthly mean discharge, in cubic feet per second, of Bowie 

Creek near Hattiesburg, Miss., drainage area, 304 square 

miles. _______________________________________________ 83

3. Monthly mean discharge, in cubic feet per second, of White 

River at Greenwater, Wash., drainage area, 216 square 

miles. _______________________________________________ 86

4. Monthly mean discharge, in cubic feet per second, of Greenwater River at Greenwater, Wash., drainage area, 73.9 

square miles.___-_______-_-____-_-..-_-_--____-_--____ 86

5. Curve values (Ye) of monthly discharge, in cubic feet per 

second, of Bowie Creek near Hattiesburg, Miss., from 

figure 10--..----..-.-----.-.-----.-.-----.-.-------- 91

6. Y: Y0 ratio for Bowie Creek near Hattiesburg, Miss.----..-. 92

7. Monthly mean discharge, in cubic feet per second, of Bogue 

Chitto at Franklinton, La., drainage area, 985 square 

miles.._ _..--_.....-----.-.---_-.-.------..-------. 92

8. Curve values (Ya) of monthly discharge, in cubic feet per

second, of Bogue Chitto at Franklinton, La., from figure 11. 94

9. Y: Y0 ratio for Bogue Chitto at Franklinton, La ------------ 94

10. Y: Y0 ratio for Bowie Creek near Hattiesburg, Miss., from

figure 12------._------------------------------------ 95

11. Monthly discharge of Bowie Creek near Hattiesburg, Miss.,

adjusted for Bogue Chitto.---------------------------- 95

12. Monthly mean discharge, in cubic feet per second, of Leaf 

River at Hattiesburg, Miss., drainage area, 1,760 square 

miles..-----_--_------------_---_---_-----------_---_ 99

13. Monthly discharge, in cubic feet per second, Leaf River at

Hattiesburg, Miss, minus Leaf River near Collins, Miss__- 99
MANUAL OF HYDROLOGY: PART I. GENERAL SURFACE-WATER

TECHNIQUES

GRAPHICAL CORRELATION OF GAGING-STATION

RECORDS

By JAMES K. SEARCY

ABSTRACT

A gaging-station record is a sample of the rate of flow of a stream at a given 

site. This sample can be used to estimate the magnitude and distribution of 

future flows if the record is long enough to be representative of the long-term 

flow of the stream. The reliability of a short-term record for estimating future flow 

characteristics can be improved through correlation with a long-term record. 

Correlation can be either numerical or graphical, but graphical correlation of 

gaging-station records has several advantages.

The graphical correlation method is described in a step-by-step procedure with 

an illustrative problem of simple correlation, illustrative problems of three 

examples of multiple correlation removing seasonal effect and two examples 

of correlation of one record with two other records. Except in the problem on 

removal of seasonal effect, the same group of stations is used in the illustrative 

problems. The purpose of the problems is to illustrate the method not to 

show the improvement that can result from multiple correlation as compared 

with simple correlation.

Hydrologic factors determine whether a usable relation exists between gagingstation records. Statistics is only a tool for evaluating and using an existing 

relation, and the investigator must be guided by a knowledge of hydrology.

INTRODUCTION

Gaging-station records supply the basic data needed to evaluate 

and design water-resources developments. In addition to being 

a historical record of the streamflow that occurred at the gagingstation site, the streamflow record is a sample that often can be used 

to estimate the magnitude and distribution of future flows.

The reliability of the streamflow sample for estimating future flows 

at the gaging-station site depends upon how well the sample represents 

the flows that will occur during the life of a development. One 

important measure of representativeness is the length of the streamflow

67
68 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

sample. Frequently, streamflow data are needed for sites at which 

no gaging stations have been operated or for which the records are 

much too short to be representative of future flow.

Statistical correlation is a means by which short-term records can 

be adjusted to represent a long-term record. Correlation is the process 

of establishing a mutual relation between a variable and one or more 

related variables. For gaging-station records, the usual variables are 

the short-term gaging-station record and one or more long-term 

gaging-station records. The correlation is termed simple correlation 

if the relation is between two variables, and is termed multiple 

correlation if the relation is between one variable and two or more 

related variables.

Either type of correlation can be graphical or numerical (the two 

methods differ in procedure rather than in principle), but for gagingstation records, graphical correlation has decided advantages. When 

logarithms of the discharge are used, fitting a curved relation between 

gaging-station records by graphical methods is less laborious than 

fitting a curve by numerical methods.

Graphical methods for simple correlation and for three types of 

multiple correlation are explained in this report. These methods were 

developed principally within the Geological Survey by W. B. Langbein, 

C. H. Hardison, and others. The examples were selected to illustrate 

the methods and not to show the differences in the results of the 

various methods. The multiple correlation techniques presented here 

by no means exhaust the possibilities of multiple correlation. For 

example, the use of precipitation data in place of a third gaging-station 

record might prove best in some localities.

USE OF CORRELATION

The relation established between gaging-station records for a 

concurrent short period is sometimes used to obtain correlative 

estimates of monthly discharges at the short-term station corresponding to discharges at the long-term station outside the common period. 

Correlation is also useful for extending short-term gaging-station 

records to estimate flows of a given frequency and for analyzing 

storage-frequency. Often, the information on magnitude and frequency of low flows from a long-term station can be transferred 

directly to the short-term station through the curve of relation without 

synthesizing a table of discharges for the short-term station. Such 

estimates represent a discharge that has the same probability of 

occurrence as at the long-term station. The frequency data that are 

transferred directly have the same order of accuracy as if they had 

been computed from a table of correlative estimates for the same 

period.
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 69

Use of estimates derived from correlation are in some ways an 

unsatisfactory substitute for actual discharge records. For design 

purposes, however, a streamflow record, of any length, represents 

only a sample of the flow of the stream, and as such contains an 

inherent sampling error. This sampling error, which cannot be reduced 

by measuring the same sample more accurately, is often large in 

comparison to the errors that would be introduced by correlation 

with a longer record. Thus correlation with longer records can serve 

to reduce the sampling error and to provide a more reliable base from 

which to estimate the characteristics of future flow.

Langbein and Hardison (1955) demonstrated that the mean discharge determined from a short record can generally be improved by 

correlating it with a record more than 25 percent longer than the 

short record because the error introduced by the correlation is usually 

less than the decrease in sampling error accomplished by extension.

HYDROLOGIC PRINCIPLES

The runoff from a drainage basin depends upon the climate and 

upon the inherent characteristics of the basin. In areas of low relief, 

the climate of nearby drainage basins is much the same, but in mountainous areas the climate varies because of the positions of the mountain ranges and the differences in the altitude of the basins.

The principal drainage basin characteristics that affect the amount 

and distribution of runoff are location, size, shape, physiography, 

geology, soils, vegetative cover, and cultural developments. Numerous other basin characteristics have a lesser effect on the runoff 

from the basin. Any attempt to evaluate the extent to which one of 

these basin characteristics affects runoff and to relate the runoff of 

one basin to that of another by measuring the differences in their 

basin characteristics would be a tremendous undertaking.

The complex interrelations of climate and drainage basin characteristics are integrated in the flow of the stream, and their aggregate 

effect is measured directly at the stream-gaging station. The measured runoff from drainage basins obviously furnishes the best basis 

for comparing their runoff characteristics.

Relations between the flow at stream-gaging stations can be based 

on concurrent daily, weekly, monthly, or annual discharge. Generally, the relations are the same regardless of the time unit used, 

although the variation of the plotted points decreases as the time unit 

becomes longer. Selection of the time unit is a hydrologically 

arbitrary decision.

Monthly discharge, which is the unit used in many water-resources 

development studies, is a convenient time unit for establishing relations. Monthly discharges are published in the Geological Survey
70 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

Water-Supply Papers in the annual series of Surface Water Supply of 

the United States and are compiled from beginning of record through 

September 1950 in Water-Supply Papers 1301-1318. Monthly 

discharges are used in the illustrative problems of this chapter.

FACTORS AFFECTING THE RELATION BETWEEN GAGING-STATION

RECORDS

Although some characteristics of two nearby basins in areas of low 

relief may differ, their climates are generally similar. This bond of 

climate is the key to the correlation between gaging-station records. 

If the bond is missing, as, for example in attempting to correlate a 

mountain stream and a nearby desert stream, a usable correlation is 

impossible. The climate of the drainage basins need not be identical, 

but each basin should have the same chance of receiving a given 

rainfall.

The distance between the gaging stations adversely affects the 

degree of relation between gaging-station records, so that other factors 

being equal, the relation becomes less reliable as the distance between 

the stations increases. An exception to this is that records from two 

stations a considerable distance apart on the same stream are generally 

more closely related than are records from nearby streams.

The relative size of the drainage basins affects their relation, 

particularly in the time distribution of flows. The runoff of a large 

basin is generally the summation of high flows from some basins and 

low flows from others and is thus less variable than the runoff of most 

of the small, separate basins of which it is composed. Heavy rains 

near the end of a month substantially increase the monthly runoff 

from most small basins, whereas large ones receive little increase in 

runoff until the following month. This condition would cause the 

plotted points for two consecutive months to depart in opposite 

directions from a line of average relation. Therefore when basins 

differ greatly in size, the runoff from one is a poor indication of the 

concurrent runoff from the other basin. Correlation between stations 

whose drainage areas differ in size by more than tenfold should be 

avoided.

Differences in the surficial geology of drainage basins have a 

pronounced effect on low-flow relations. As compared with basins 

of low infiltration capacity, those of high infiltration capacity have 

little direct runoff from light rains but have higher sustained flows 

during dry periods. Direct runoff from rains that exceed the infiltration capacity of both drainage basins is less affected by the surficial 

geology and thus tends to become proportional to the size of the 

drainage areas.
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 71

A complete discussion of the factors that affect the relation between 

gaging stations is beyond the scope of this chapter. The purpose of 

this brief discussion is to emphasize that the hydrology of the two 

drainage basins is the basis of their relation not the statistics used. 

Nevertheless statistics furnishes a powerful tool for organizing streamflow data to express useful relations.

STATISTICAL PRINCIPLES ; >

Few users of streamflow records are statisticians. Although 

many good statistics books are available, one is often confused in 

applying statistics to a streamflow problem because of the variety 

of symbols used by the different authors and the lack of reference 

to streamflow applications. This chapter explains an application 

of statistics to streamflow problems and the use of graphical correlation methods to determine the relation between gaging-station 

records; the explanation is not intended to be a treatise on graphical 

correlation. The technique described in this chapter is basically the 

same as that described by Langbein and Hardison (1955) and Somers 

(1954).

The statistical formulas given here are based on the premises 

that the data are normally distributed and that the observations are 

entirely random in sequence. (See "Concepts and definitions.") 

Consecutive daily streamflow data fit neither of these criteria, but the 

data can be normalized by a logarithmic transformation accomplished 

by either using the logarithm of the discharge or by plotting the discharge data on logarithmic paper. Streamflow is by nature nonrandom. Each daily discharge is affected by flows for previous days 

and in turn, affects succeeding daily flows. Monthly discharge is also 

nonrandom but to a lesser degree than daily discharge. The primary 

cause of nonrandomness in streamflow data is the lag in the rainfallrunoff relation (carryover effect). A secondary cause is the nonrandomness of precipitation and the tendency of wet and dry years 

to occur in groups rather than in random sequence.

In addition to being random, statistical data should be homogeneous 

(drawn from the same population). Streamflow data from the same 

stream would seem homogeneous by definition, but the nonhomogeneity may result from major developments that affect runoff, or 

occasionally by radical changes in data-collection methods. The 

double-mass curve discussed in a separate chapter is one means of 

testing the homogeneity of streamflow data.

CONCEPTS AND DEFINITIONS # " ^ r

Probability. Probability is the likelihood of the occurrence of any 

particular event expressed as a decimal fraction or as a percentage.

540886 O 60 2
72 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

A value of zero is assigned to an event that certainly will not occur, 

and a value of one to an event that is certain to occur. A probability 

of 0.50 means that an event is as likely to occur as not (a 50-50 

chance).

Individual. A unit of data is an individual. In this chapter the 

mean monthly flow is the unit or individual.

Population (universe}. Any set of individuals having a common 

measurable characteristic constitutes a population; for example, a 

long series of monthly flows of a stream unaffected by regulation, 

diversion, or major changes in land use.

Sample. A group of individuals taken from a population constitutes a sample. N is used to represent the number of individuals, 

or, in correlation, the number of pairs of individuals in the sample. 

For example, a 30-year record of flows at a gaging station is a sample 

from the life of the stream. A sample is taken for the purpose of 

making inferences about the population.

Random data. Random data are data in which the individuals 

in the sample were selected in such a manner that any individual in 

the population had an equal chance of being selected on the next 

draw.

Normal distribution. Many populations are distributed about 

the mean according to the Gaussian or normal distribution. In 

such a distribution 68.27 percent of the individuals lie within one 

standard deviation of the mean; 95.45 percent lie within two standard 

deviations; and almost all (99.73 percent) lie within 3 standard 

deviations of the mean.

The error formulas and most of the tests of significance are based 

on the assumption that the variable is distributed in the normal 

form. Unless departure from the normal form is severe the tests of 

significance and the error formulas are sufficiently accurate.

Standard deviation (s). The standard deviation (sometimes called 

the root mean square of the deviations) is a measure of the scatter 

of the items about the arithmetic mean of the sample and is computed by the formula

s 

where x=deviation of the item from the arithmetic mean (X) 

or x=X-~X

A^= number of items

Degrees of freedom (d.f.). A nebulous but important concept in 

statistics is that of degrees of freedom, but few textbooks attempt a 

definition. The number of degrees of freedom is, in general, the 

number of independent observations minus the number of constraints 

imposed on the system. The constraints imposed on the system are
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 73

called lost degrees of freedom (l.d.f.). Thus, we lose one degree 

of freedom in computing the standard deviation because one constraint (an average) has been placed on the data. For example, if the 

average of 10 numbers is 100, 9 of the numbers can have any value, 

but the 10th number can be only that which makes the average 100; 

we lose the freedom of choice on one item by specifying the average. 

The procedure for determining lost degrees of freedom will be shown 

in each problem.

Independent variable. The independent variable is the one from 

which another variable is to be estimated or predicted. It is plotted 

as the abscissa (J^-axis). The independent variable is generally the 

long-term station when extending streamflow records.

Dependent variable. The dependent variable is the one that 

is estimated or predicted from given values of the independent 

variable. It is plotted as the ordinate (F-axis). The dependent 

variable is generally the short-term station when extending streamflow re'cords.

Scatter diagram. Where comparative data (for example, concurrent 

monthly discharges) at two gaging stations are plotted against each 

other, the result is a scatter diagram. The pattern of the scatter 

indicates the degree of relation between the flow at the two stations. 

A narrow scatter band indicates a high degree of association between 

the two sets of data, and greater departures from the relation line 

indicate lesser degrees of association.

Plotting paper. Logarithmic paper (base 10) is used for plotting 

the relation between the flows at two gaging stations because for most 

data the relation on such paper can be approximated by one or more 

straight lines.

Line of regression. The line or equation expressing a linear relation 

between two variables is known as the line of regression. In the 

numerical method of analysis, the line of regression is generally fitted 

by the method of least squares. The line is termed the regression of 

Y on X if the squares of the Y residuals have been minimized or the 

regression of X on Y if the squares of the X residuals have been 

minimized.

Curve of relation. The graphically determined line of regression 

should preferably be called a curve of relation to disassociate it from a 

line fitted by the least squares method.

Equal-yield line. The equal-yield line is a line of relation (45° on 

logarithmic paper) between the flows at two stations, based on the 

assumption that the runoff at each station is proportional to its 

respective drainage area.

Log unit. A log unit is one cycle on logarithmic paper. Log units 

less than unity indicate the proportional part of a cycle. For example, 

one-tenth of the length of a cycle is 0.10 log unit.
74 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

Correlative estimates (Yc }. The curve of relation value for Y 

corresponding to a given value of X is a correlative estimate.

Residual (d). The vertical (Y} difference between the curve of 

relation and a plotted point is a residual (sometimes called error). 

It is the actual value of Y minus the calculated (estimated) value of 

Y, or d=Y Yc . The residual carries either a plus or a minus sign.

Standard error oj estimate (Se ). The standard error of estimate is a 

measure of the variation of the points about the curve of relation. 

It is similar to the standard deviation except that the curve of relation 

takes the place of the arithmetic mean. If the distribution of the 

scatter is normal, lines drawn a vertical distance equal to one standard 

error of estimate above and below the line or curve of regression and 

parallel to it, will encompass about two-thirds (68.27 percent) of the 

points. Parallel lines two standard errors of estimate above and 

below the line of regression will encompass 95.46 percent of the points, 

and lines three standard errors of estimate away will encompass 99.73 

percent of the points. In correlating gaging-station records, the 

standard error of estimate is conveniently expressed in log units, 

rather than in units of Y or in percentage of the mean.

For a straight-line graph the mathematical expression for the 

standard error of estimate is

T 2 v ' 

where

d=residual (Y Yc ) 

N= number of items

For any curve fitted to observed data, Se may be computed by the 

formula

l 

 V

N-l.d.f. 

where

l.d.j. =lost degrees of freedom or number of constraints placed on the 

system (2 degrees of freedom are lost by use of a straight line) 

Y = observed value of the dependent variable 

Yc = computed or curve value of the dependent variable

Coefficient oj correlation (r}. The size of the standard error of 

estimate is a measure of the closeness of the points to the curve of 

relation, but it does not show the improvement in the estimate of Y 

that results from using the correlation with X rather than estimating 

Y as the mean of the Y values. A measure of the usefulness of the 

curve of relation for estimating Y values may be obtained by relating 

the standard error of estimate (Se ) to the standard deviation of the Y 

values (« ). This relates the variation of the points about the curve 

of relation to the variation of the points about their arithmetic mean.
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 75

For a linear relation, such a measure is the coefficient of correlation 

(/ ) computed by the formula

(4)

The value of the coefficient of correlation ranges from 0 for no correlation to ± 1.00 for perfect correlation. A plus sign is assigned r if the 

line of regression slopes upward to the right and a minus sign if it 

slopes downward. A low coefficient of correlation does not always 

indicate poor correlation; it may result from fitting a straight line to a 

good relation that should be described by a curved line.

Index of correlation (p). For curvilinear relations the measure of 

correlation is called the index of correlation, or

p (5)

The similarity between r and p is apparent. The limits of p are 0 for 

no correlation to 1.00 for perfect correlation. No plus or minus sign 

can be assigned to p because the sign might vary in different parts of 

the curve.

Coefficient or index oj correlation oj gaging-station records. The coefficient, or index, of correlation furnishes a guide to the length of 

long-term record needed for improving the mean of a short-term 

record through correlation. Langbein and Hardison (1955) give 

criteria for determining the minimum ratio of the long-term to the 

short-term record for various values of the coefficient or index of correlation.

The usefulness of the correlation, as indicated by r or p, shows how 

much improvement in the estimates of Y results from using the correlation, as compared with estimating Y as the mean of all Y values. 

This concept is correct for many types of data, but as streamflow 

data usually have a definite seasonal pattern, for example, the most 

likely discharge for October would be the mean of the October discharges rather than the mean of all monthly discharges. Thus, the 

actual improvement in estimating Y from the curve of relation is not 

as great as the coefficient of correlation (computed by equation 4) 

indicates. A better measure of the improvement resulting from the 

correlation of gaging-station records is to compare the standard error 

of estimate (equation 3) with the standard deviations of the monthly 

discharge about their respective monthly means rather than about 

the mean of all the months (Langbein and Hardison, 1955, p. 1176-77 

of closure). The computed standard deviation about the respective
76 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES 

monthly means is adjusted for the 12 degrees of freedom lost (multiply / ^v"~

by */ AT 121 and substituted for the value of sy in the formula r or

/$A2 

1 ( - } - In the illustrative correlation of the section, "Removing seasonal effect," the index of correlation adjusted for monthly 

means is 0.94 as compared with an index of 0.98 using Se computed 

by formula 3. In some correlations the adjustment for deviations 

of the data about the monthly mean makes a great difference in the 

value of r or p.

THE GRAPHICAL METHOD

The first step in expressing the relation between pairs of associated 

measurements is to plot the measurement of one variable against the 

corresponding measurement of the other variable. Unless the variables 

are precise measurements of two quantities that bear a fixed relation 

to each other, the plot will present a "shotgun" pattern, technically 

known as a scatter diagram. The cluster of the points indicates the 

closeness of the relation or perhaps the accuracy of the measurements, 

whereas the shape of the pattern indicates the nature of the relation. 

A straight line is used if it fits the points fairly well the resulting 

relation is called linear. If the points lie on a curved line the relation 

is called curvilinear. The graphical method differs from the numerical method in the way in which the line or curve is fitted to the points. 

Neither method explains why the points scatter.

In graphical correlation, the shape of the relation line depends to 

a large extent upon the type of plotting paper used. Logarithmic 

paper is convenient for graphical correlation of gaging-station records 

because it tends to normalize the streamflow data and to transform 

the common curvilinear relation between gaging-station records to 

one or more straight lines. The Geological Survey uses 5-inch cycle 

logarithmic paper with either 3 by 2 cycles or 4 by 3 cycles. The 

5-inch cycle fits the 20-graduations-to-the-inch face of an engineers' 

scale so that direct readings can be made to hundredths of a log cycle.

Generally, the relation between two gaging-station records can be 

satisfactorily expressed on logarithmic paper by 2 straight lines, 1 

for the high-flow part of the relation and 1 for the low-flow part. 

For many streams east of the Mississippi River, the 2 lines intersect 

at from 1 to 2 times the average discharge. If the low-flow and the 

high-flow regression lines intersect at sharp angles, they should be 

connected by a transition curve. Drawing a transition curve is purely 

arbitrary but it is hard to conceive of a relation between two gagingstation records that changes abruptly at some specified discharge.
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 77

If the angle between the lines is very flat, a transition curve is unnecessary.

The relation between 2 records can sometimes be described by a 

single straight line and occasionally the relation between 2 records 

requires a third straight line at extremely low flows. In general, 

straight lines are drawn through the data and little weight is given 

to the position of points near the extreme upper or lower end of the 

line. Although the points sometimes seem to define sharply curved 

or even wavy lines, the line of relation for the succeeding period of 

equal length will often balance the irregularities of the first period 

and at the extremities define a hook in the opposite direction from 

that defined in the first period.

Where the high-flow relation line is poorly defined, the equal-yield 

(drainage-area ratio) line may be used as a guide in drawing the 

relation line. The high-flow relation line and the equal-yield line 

for most gaging-station correlations lie close together and are ab,out 

parallel. This is not true where the storage capacity (both surface 

and underground) within the two basins differs greatly.

The slope of the low-flow relation line is governed by the surficial 

geology of the two drainage basins. If the geology of the two basins 

were identical, no break would occur in the relation line. The 

streamflow data generally provide good definition of the low-flow 

relation line except at the extreme low end. When one of the streams 

ceases to flow or yields less at extreme low flows than the other streams, 

the low-flow relation line may break at extreme low flows and cross 

the axis of the better yielding stream.

Streams affected by snowmelt may have a well-defined relation 

during part of the year that differs from the relation during other 

periods of the year. Such a condition may occur because of differences in the aspect of the basins with respect to the sun, differences 

in forest cover, or differences in altitude that affect the rate of snowmelt. The procedure for correlating records where the rate of snowmelt differs between the basins is described in the section "Multiple 

correlation" under the subheading "Removing seasonal effect."

COMPARISON OF THE GRAPHICAL AND NUMERICAL METHODS

A principal difference between curve fitting by the graphical method, 

described here, and the numerical method of least squares is the 

effect of the remote points on the line of relation. By squaring the 

distance from the line of relation to the remote point, the method 

of least squares gives much more weight to the remote point than to 

one of the more numerous points near the regression line. This 

moves the line of relation toward the remote point and lessens the 

deviation of the point from the regression line while increasing the 

deviation of a point near to but on the other side of the line. As an
78 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

exceptional example, the line of relation might be moved to a position 

between where most of the points lie and where the few remote points 

lie. Thus, the use of the numerical method for correlation of gaging-station records sometimes biases most of the estimates. Using the mean 

of a group of points has the same objection but to a lesser degree.

The graphical method uses the median of groups of points to position the line of relation. Thus, the magnitude of a remote point has 

no influence on the position of the line of relation, except that it is 

above or below the mid value of the group. Generally a line of relation positioned by median points gives a better correlative estimate 

than a least-square line but where the conditions which produced the 

remote point recur, the median point line of relation gives a poorer 

estimate than the least-square line.

The graphical correlation procedure described in this manual gives 

a single line of relation between 2 gaging stations, whereas the numerical method gives 2 lines of best fit depending on which station is used 

as the dependent variable. Each of the two lines of best fit, called 

least-square lines, are regressed toward the mean of the dependent 

variable and the discharges estimated from the least-square line 

will have less variability than those of the sample used in the correlation. For most uses estimated discharge should maintain the original 

variability so that frequency presentations based on estimated data 

will not show streamflow to be less variable than it is simply because 

the coefficient of correlation was low. The single line of relation, or 

engineer line, is used in this manual to maintain the original variability 

of the sample.

Another advantage of the graphical method is that a curved relation 

can be defined graphically as easily as a linear relation, whereas 

fitting a curved line to data by the numerical method is complicated. 

A straight-line relation is seldom applicable throughout the range in 

discharge, because few gaging-station records have the same relation 

to another record at both high flows and low flows. The low flow, or 

dry-weather flow, comes from ground-water storage, whereas the high 

flow and the medium flow are principally direct runoff from precipitation. In order that the relation between two gaging-station records 

be linear throughout the range in flow, the effect of surficial geology 

on low flow must produce the same ratio between the records as the 

effect of other basin characteristics produces on high flow.

Neither the graphical method nor the numerical method is satisfactory for extending a curvilinear relation beyond the observed 

data. Extrapolation by either method is based on the dubious assumption that the curvature used to fit the extreme points will fit points 

that may lie outside the range of present data. Although the numeri-
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 79

cal method provides a single extension, the result is as questionable 

as a curve extended graphically by eye.

SIMPLE CORRELATION

The procedure for simple correlation of two gaging-station records 

is described in the following example, correlating two stations in 

southern Mississippi station A (Leaf River near Collins, drainage 

area, 752 square miles) and station C (Bowie Creek near Hattiesburg, 

drainage area, 304 square miles). In another section these stations, 

and two others in the vicinity (see fig. 9), are used to illustrate the

FIGURE 9. Area in southern Mississippi.

540886 O 60 3
80 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

multiple correlation procedures. Southern Mississippi has little 

variation in relief or in climate, and snowmelt is not a factor in runoff.

GRAPHICAL CORRELATION WITH TWO VARIABLES

1. Plot the selected unit (usually monthly discharge in cubic 

feet per second) of station A (table 1) against the concurrent unit of 

station C (table 2) on logarithmic paper with the independent variable 

on the X axis. (See fig. 10). For working with unfamiliar records, 

distinctive plotting symbols for each calendar month are recommended 

because this will reveal seasonal patterns in runoff (see "Removing 

seasonal effect").

2. Using vertical lines, divide the total distribution of the points 

in the X direction into 5 to 10 parts, or slices of equal width (8 slices 

are used in fig. 10).

3. Graphically determine the median of the points in each slice in 

in both the X and the Y directions. If the number of points in a 

slice is even, average (graphically) the middle two points; if odd, 

average the middle three points, giving the midpoint a weight of two. 

If a slice contains less than three points, do not determine a median 

for that slice. A simple method of giving the midpoint a weight of 

two is to average the midpoint and the point next above, then average 

the midpoint and the point next below; the average of the two averages is the median.

4. Repeat steps 2 and 3 for the Y direction. In practice, it is convenient to use different colors for the medians of the Y slices.

5. Plot the equal-yield line by drawing a dashed 45° line through 

the point obtained by plotting the drainage areas of stations A and C 

as used for discharge figures.

6. Draw straight lines (of relation) through the average of the 

median points giving reduced weight to the end points. The upper end 

of the relation is generally about parallel to the equal-yield line. The 

lower end is generally another straight line which can be connected 

to the upper line by a short transition curve.

7. Draw two lines each equidistant (vertically) from the curve of 

relation throughout its length, one above and one below the curve of 

relation, so that one-sixth of the points will lie above the upper curve 

and one-sixth of the points will lie below the lower curve. The upper 

and lower lines will seldom be at the same distance from the line 

of relation.
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50 60 70 80 90 100 200 300 500 1000

DISCHARGE, IN CUBIC FEET PER SECO

LEAF RIVER NEAR COLLINS, MISS., DRAINAGE AREA, 7

FIGURE 10. Correlation of monthly discharge, Leaf River and Bowie Cree
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84 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

8. The area between the upper and lower curves of step 7 contains 

two-thirds of the plotted points. By definition this band has a vertical 

spread of about two standard errors of estimate. Measure the vertical 

distance (using the 20 scale on 5-inch log cycle paper) between the 

upper and lower curves. One-half of this distance is an estimate of 

the unadjusted standard error of estimate in log units. If the log 

cycle of the paper used does not fit an engineers' scale, compute the 

standard error of estimate in log units by looking up the logarithm of 

the quotient of a discharge on the upper curve divided by the discharge vertically below o^ 'he lower curve or by converting the scaled 

distance to a decimal ratio of the log cycle distance.

9. Adjust the standard error of estimate by multiplying it by

V Six degrees of freedom (l.d.j.} are lost for the curve N-Ld.f.

described in step 6. The lost degrees of freedom are determined on 

the assumption that two points are needed to locate the upper line, 

two points to locate the lower line, and two points to locate the 

transition curve. A difference of several l.d.j. makes little difference 

in the result unless TV (the number of pairs of items) is small.

10. Convert the standard error of estimate first to a ratio and then 

to a percentage by looking up the antilog and the anticolog. A slide 

rule is convenient for making this conversion. For example: Assume 

that one-half of the distance between the curves is 0.10 log units. 

Set 0.10 on the L (log) scale and read 1.26 on the D scale. The ratio 

is 1.26 or +26 percent. Set the colog 0.90 on the L scale'and read 

the ratio 0.795 which corresponds to 20.5 percent on the D scale. 

The operation may be performed with a single setting by alining the 

ends of the scales and reading 0.795 on the CI scale.

11. Draw horizontal lines that exclude one-sixth of the points (in 

the Y direction) above the top line and one-sixth of the points below 

the bottom line. One-half of the vertical distance between the lines

multiplied by ~J =sy , the standard deviation of Y.

V

/ O \ 2 

1 ( 6 } 

If the relation is a single straight line, this formula gives the 

coefficient of correlation (r).

MULTIPLE CORRELATION

Multiple correlation is used to obtain better correlative estimates 

for the dependent variable than could be obtained by simple correlation alone. Multiple correlation is the use of more than two variables. 

The additional variables can be records from additional gaging 

stations, seasonal factors, or precipitation data.
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 85

This discussion of multiple correlation is limited to the removal of 

seasonal effect, to the use of records from two gaging stations for 

estimating the discharge of a third station, and to the use of records 

from two gaging stations (located on the same stream) for estimating 

the discharge of a tributary that enters a stream between the stations.

REMOVING SEASONAL EFFECT

The relation between the runoff from one basin and that from 

another basin may vary with the season of the year. This variation 

is called seasonal effect or monthly effect. The most common example 

of seasonal effect occurs when two streams receive runoff from snowmelt. A southern exposure of one basin might cause the bulk of the 

snowpack to melt in April, whereas a northern exposure of an adjacent basin might delay snowmelt until May. Differences in altitude 

may have a similar effect. During months when snowmelt is not a 

factor, the relation between the basins might remain fairly constant.

Seasonal effect might also be due to differences in vegetation. 

During the growing season profuse vegetation in one basin might 

consume large amounts of water so that the relation for that season 

differs from that of other seasons. Also, the use of large amounts of 

water from one stream during the summer could produce a seasonal 

effect.

Seasonal effect is indicated by the scatter diagram when the points 

for the same calendar month show a tendency to plot on the same 

side of and about the same distance from the curve. If the hydrologic differences are sufficiently stable from year to year, their average 

effect can be determined and removed by the following method, 

using two stations in western Washington:

1. Plot the selected unit (generally monthly) of discharge of 

station E (table 3) against the concurrent unit of station F (table 4) 

on logarithmic paper with the independent variable on the X axis 

(see pi. 1A). The symbols recommended aid in detecting seasonal 

effect and are essential to the application of the graphical method, 

but colors can be used instead of the ticks used in plate IA.

2. On any paper, draw about 16 vertical lines at equal intervals 

(pi. IB) for the seasonal adjustment curve. Each line represents a 

month, and the year overlaps about two months on each end. Draw 

a horizontal line to represent zero residual. Graduations on the 

seasonal adjustment curve are not used for plotting.

3. Draw a preliminary line of relation. The actual position of the 

line has little effect on the results but in some correlations the size of 

the adjustments are reduced by allowing for the expected adjusted 

position of the points.
86 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

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GRAPHICAL CORRELATION OF GAGJNG-STATION RECORDS 87

4. With a pair of dividers (or a scale), measure the vertical distance 

from the preliminary relation line to each October point. Transfer 

these distances to the October line of the seasonal adjustment curve. 

Mark the point above or below the zero residual line to match the 

relation of the plotted point to the preliminary relation line. Repeat 

this procedure with the points for each month. The advantage of 

using monthly symbols in step 1 is now evident.

5. Draw horizontal lines on each monthly line so as to exclude onesixth of the points above the top line and one-sixth of the points below 

the bottom line. The distance between the lines is, by definition, two 

standard deviations.

6. Graphically deteimine the median value for each month by averaging the middle two points, if the number of points is even, or by 

averaging the middle three points, giving the midpoint a weight of 

two, if the number of points is odd. Transfer the median and standard 

deviation lines of the last two months to the opposite end to facilitate 

drawing the seasonal adjustment curve.

7. Draw a smooth curve to average the median values, but keep the 

curve within the standard deviation lines. The curve should contain 

as few bends as possible; for example, the curve from January through 

April (pi. 15). A mathematical method of drawing this is described 

in Water-Supply Paper 1541-B, "Double-Mass Curves," by Searcy 

and Hardison under the section "Fitting curves to cyclic data," by 

Langbein.

8. The distance from the zero residual line to the curve is the adjustment for the month. Set the adjustment for October on the dividers 

and in plate 1A apply this adjustment to the points for all Octobers. 

This adjustment will result in moving the observed points down if the 

indicated adjustment lies above the zero residual line and moving them 

up if it lies below. Repeat this procedure for all months. The adjusted value is marked by a solid circle, usually on the same sheet with 

the preliminary line. To reduce clutter, the preliminary regression 

line could be erased. For clarity, the adjusted points in the example 

have been plotted as a separate graph. (See pi. 1.)

9. Repeat the graphical correlation procedure for the two variables 

given in the section "Simple correlation," using the adjusted position 

of the points.

In computing Se , 6 degrees of freedom are lost: 2 for the relation 

line and 4 for the adjustment curve. The degrees of freedom lost in 

the adjustment curve are determined by spotting the points necessary
88 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

to approximately reproduce the curve. In plate IB the curve could 

be redrawn from points on the curve on the October, February, June, 

and August lines.

Compute the index of correlation (p) by formula 5 and, for comparison, compute an adjusted index using the standard deviation of 

the Y values as determined by averaging the standard deviations of 

the monthly discharges around their respective mean monthly discharges. The procedure is explained in the section "Concepts and 

definitions" under "Coefficient or index of correlation for gagingstation records." In the illustrative problem (pi. 1) the adjusted 

index of correlation is 0.94 as compared with an index of 0.98 computed from the standard deviation of the Y values around the mean 

of all the Y values.

For added refinement, the line of regression obtained from the adjusted points (pi. 1) can be used as the preliminary line of regression 

in step 3 and steps 4-9 repeated.

Use. To use the adjusted relation curve for estimating a value 

for Greenwater River corresponding to a given discharge of White 

River 

(a) Enter the curve of step 9 (pi. 1) with the value of White River 

and obtain the Yc value of Greenwater River.

(b) Obtain the appropriate adjustment to Yc from the seasonal 

adjustment curve (pi. IB).

(c) Add, graphically, the plus or minus adjustment to Yc to obtain 

the estimated actual point for Greenwater River. An alternate procedure is to use the adjustments as ratios and multiply the Yc discharge 

by the appropriate monthly ratio.

GRAPHICAL CORRELATION WITH THREE OR MORE VARIABLES

When the standard error of estimate of simple correlation is too 

great, using more stations in the correlation will often reduce the 

magnitude of the standard error of estimate. The general case will be 

described by an example, correlating Bowie Creek near Hattiesburg, 

Miss, (station (7), as the dependent variable, with Leaf River near 

Collins, Miss, (station A), and Bogue Chitto at Franklin ton, La. 

(station B), as the two independent variables. (See fig. 9.)

On figure 9, station D is shown on the Leaf River at Hattiesburg, 

Miss., downstream from the confluence of Leaf River and Bowie 

Creek. Bowie Creek supplies much of the difference in flow between 

gaging stations D and A. Thus, a correlation of the Bowie Creek
GRAPHICAL CORRELATION OF GAGJNG-STATION RECORDS 89

discharge with the difference in discharges between the Leaf River 

stations D and A (see section "Special case") would be expected to 

give better results than a correlation of the Bowie Creek discharge with 

the discharge from two streams in nearby basins. (See section 

"General case.") This is demonstrated by comparing the results 

obtained from the general and the special case.

Other combinations of records from the same stream system could 

be used; for example, discharge for Leaf River at Hattiesburg could 

be correlated with the sum of the Collins station and Bowie Creek 

discharges, or the sum of the Collins station and Bowie Creek discharges could be averaged with those of a downstream station, Leaf 

River at McClain, and the average used to correlate with the Hattiesburg discharge. Addition of other stations does not necessarily 

improve the correlation. On this particular stream, averaging the 

records for the downstream station at McClain with the sum of records 

of the two upstream stations would probably give poorer results than 

using the sum of the two upstream stations alone. Actual correlation 

by both combinations is the only way to tell which combination gives 

the best result.

GENERAL CASE

Records of two long-term stations, A and B, are correlated with the 

record of a short-term station, C, as follows:

1. Correlate station A, (Leaf River near Collins, Miss., table 1) as 

abscissa, with station C (Bowie Creek near Hattiesburg, Miss., table 2). 

This is the correlation of figure 10.

2. Obtain the curve value (Yc ) for station C (Bowie Creek) corresponding to each value of station A (Leaf River near Collins, table 1). 

(See table 5.)

3. Compute the station C (Bowie Creek) ratio, Y: Yc , from tables 

2 and 5. (See table 6.)

4. Correlate station A (Leaf River near Oollins, table 1) as abscissa 

with station B (Bogue Chitto at Franklinton, La., table 7). (See 

fig. 11.)

5. Obtain the curve value (Yc) for station B (Bogue Chitto) corresponding to each value of station A (Leaf River near Collins, table 

1). (See table 8.)
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DISCHARGE, IN CUBIC FEET PER SECOND 

LEAF RIVER NEAR COLLINS, MISS., DRAINAGE AREA, 752 SQU

FIGURE 11. Correlation of monthly discharge, Leaf River and Bogue Chit
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GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 93

6. Compute the station B (Bogue Chitto) ratio, Y: Yc , from tables 

7 and 8. (See table 9.)

7. On logarithmic paper, plot the ratios (from step 6) representing 

station B deviations as the abscissa against ratios (from step 3) representing station C deviations as the ordinate. (See fig. 12.) Draw 

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8. Select Y: Yc ratios, from the curve of step 7 (fig. 12), for station 

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8). These data are tabulated in table 10.

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(table 10) selected from the curve. The adjusted discharges are 

tabulated in table 11.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0

BOGUE CHITTO AT FRANKLINTON, LA. DRAINAGE AREA, 985 SQUARE MILES

FIGURE 12. Ratio Y: Ye Bogue Chitto and Bowie Creek, water years 1940-54.
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96 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

10. Correlate adjusted station C discharges (table 11) with station 

A discharges (table 1). (See fig. 13.) The lost degrees of freedom 

are eight six for the correlation plus two for the Y:YC curve.

Use. To use the relation curve for estimating station C, given 

the discharge for station A and station B:

(a) Enter the curve of step 4 (fig. 11) with the value of station A 

(Leaf River near Collins) and obtain the Yc value for station B (Bogue 

Chitto).

(b) Compute Y:YC for station B.

(c) Enter the ratio curve of step 7 (fig. 12) with the ratio of step (b) 

and read the Y:YC value for station C (Bowie Creek).

(d) Read the adjusted discharge of station C (Bowie Creek) from 

the curve of step 10 (fig. 13).

(e) Multiply the adjusted discharge of station C (step (d)) by the 

Y:YC ratio from step (c) to obtain the discharge of station C.

SPECIAL, CASE

One of a variety of special cases is illustrated by the following 

example: Gaging stations are located on the main stem of the Leaf 

River above and below the tributary, for which correlative estimates 

are desired. (See fig. 9.) In such a situation, the difference between 

the discharge of the upstream station and of the downstream station 

would furnish the best independent variable for correlating with the 

dependent station record.

The monthly discharge of upstream station A, Leaf River near 

Collins (table 1), is subtracted from the records of downstream 

station D at Hattiesburg (table 12), with the results tabulated in table 

13.

The differences in monthly discharge (table 13) are correlated with 

the monthly discharges of Bowie Creek (table 2). The methods of 

simple correlation are used for this purpose. The results are shown

in figure 14.

SUMMARY

Gaging-station records are the best basis for estimating the magnitude and the distribution of future flows. Often, only a short-term 

record is available at the site of a proposed water resources development. Such a record is less likely to represent the future flow of a 

stream than a long-term record, but through correlation it can be 

adjusted to represent the long-term period of a nearby long-term 

station.
400 IS

50 60 70 80 90100 200 300 500 1000

DISCHARGE, IN CUBIC FEET PER SECOND

LEAF RIVER NEAR COLLINS, MISS., DRAINAGE AREA, 752 

FIGURE 13. Correlation of monthly discharge, Leaf River and Bowie Creek (ad
&(adjusted) ^nnfilt/180 = o.062 = + 15 P*rcen

2S.=0122 180-6 -13.5 perc

= 0.061

200 500 1000 200

DISCHARGE, IN CUBIC FEET PER SECOND

LEAF RIVER, MISS. (HATTIESBURG MINUS COLLINS), DRAINAGE AR

FIGURE 14. Correlation of monthly discharge, Bowie Creek and Leaf River basin between H
GRAPHICAL CORRELATION OF GAGING-STATION RECORDS 99

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100 HYDROLOGY: GENERAL SURFACE-WATER TECHNIQUES

Graphical correlation is a practical method of establishing a relation 

between a long-term record and a short-term record. The relation 

established for a concurrent short-term period is used to adjust the 

short-term record to that of the long-term period. When the relation between stations is good, the short-term record is improved by 

correlating it with records that are more than 25 percent longer than 

the short-term record; as the degree of relation decreases, increased 

length of the long-term record is required to improve the short-term 

record; and if the relation becomes poor no amount of long-term record 

can improve the short-term record. The degree of correlation is 

measured by the index or coefficient of correlation. (See p. 75.)

Simple correlation is the establishment of a relation between two 

variables (in this example gaging-station r ecords); multiple correlation 

is the establishment of a relation between a dependent variable (the 

record to be estimated) and two or more independent variables 

(generally long-term records).

Graphical correlation of streamflow data is accomplished on logarithmic paper because the range in data can be plotted without 

undesirable condensation of the scale, because the relations between 

gaging-station records tend to plot as one or more straight lines, and 

because the residuals from logarithmic relations tend to be distributed 

according to the normal curve.

A relation exists between gaging-station records because of a common climate, and the shape of the relation is fixed by the hydrology 

of the drainage basins. Statistics furnishes only a useful method for 

defining an existing relation, it cannot provide the relation.

LITERATURE CITED

Langbein, W. B., and Hardison, C. H., 1955, Extending streamflow data: Am.

Soc. Civil Engineers Proc., Paper 826. 

Somers, W. P., 1954, Statistical analysis of streamflow relationships, 22d Western

Snow Conf., 1954: Salt Lake City, Proc., p. 26-32.

O
UNITED STATES DEPARTMENT OF THE INTERIOR 

GEOLOGICAL SURVEY

WATER-SUPPLY PAPER 1541 

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500 1000 2000 3000

DISCHARGE, IN CUBIC FEET PER SECOND

WHITE RIVER AT GREENWATER, WASH., DRAINAGE AREA, 216 SQUARE MILES 

DISCHARGE SHOWING SEASONAL EFFECT

WASH., D GREENWATER, DISCHARGE, CUBIC IN 

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500 1000 2000 30

DISCHARGE, IN CUBIC FEET PER SECOND

WHITE RIVER AT GREENWATER, WASH., DRAINAGE AREA, 216 SQUARE MILES 

DISCHARGE AFTER ADJUSTING FOR SEASONAL EFFECT

CORRELATION OF MONTHLY DISCHARGE, WHITE RIVER AND GREENWATER RIVER, WASHINGTON, WATER YEARS 1947-53

540886 O -60 (In pocket)