s Soutlrhwest Florida
7 Water Management District
j Es 1 P. O. BOX 457 BROOKSVILLE, FLORIDA 33512
... DERRILL McATEER, Chairman, Brooksville JOHN A. ANDERSON, Treasurer, St. Petersburg THOMAS VAN der VEER, Yankeetown
G 1EM S. C. BEXLEY, JR., Vice Chairman, Land O'Lakes HERMAN BEVILLE, Bushnell BOYCE A. WILLIAMS, Leesburg
EDWARD MEDARD, Secretary, Tampa PETER J. NEGRI, Ocala J. MASON WINES, Lakeland
Dale Twachtmann, Executive Director
July 18, 197Z /: \.:
|:!i JULl_ 1 f IIq
Mr. L. M. Blain 9J
Gibbons' Law Firm I --
Post Office Box 1363
Tampa, Florida 33601
Enclosed is information on the mean annual flood (U. S. Geological
Survey Water Supply Paper 1543-A).
I haven't had any success in trying to locate an extra copy of
Water Supply Paper 1541-A (General Introduction and Hydrologic
Very truly yours,
JAMES A. MANN
Chief, Permits Department
General Introduction and
By W. B. LANGBEIN and KATHLEEN T. ISERI
Manual of Hydrology: Part 1. General Surface-Water
GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1541-A
Methods and practices
of the Geological Survey '
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1960
By TATE DALRYMPLE
Manual of Hydrology: Part 3. Flood-Flow Techniques
GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1543-A
Methods and practices
of the Geological Survey
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
FLOOD-FREQUENCY ANALYSES 29
most probable value of T. It was also decided to make the test on
the 10-year flood because this is the longest recurrence interval for
which most records will give dependable estimates. The following
calculations are derived (T=10 years; y=2.25; and Y--=9.49.):
2 Fy= o-ri 2 y=12 -veryclosely
2e T- = 2X9.49V =6.33. [or 2 oy=2v'To=6.32]
A table giving values of y corresponding to T can be found in
Gumbel (1942; also Powell, 1943).
Values of y corresponding to T
[Columns headed TL and Tu respectively give the lower and upper limits of the chart for the
6.33 Lower limit Upper limit
v y -=2
(in years) -n+2 T
y- 2a TL y-2< Tv
5 2.25 2.84 -0.59 1.2 5.09 160
10 2.25 2.00 +. 25 1.85 4.25 70
20 2.25 1.42 .83 2.8 3.67 40
50 2.25 .90 1.35 4.4 3.15 24
100 2.25 .63 1.62 5.6 2.88 18
200 2.25 .45 1.80 6.5 2.70 1i A
500 2.25 28 1 97 7. 7 2.53 13
1000 2.25 .20 2.05 8.3 2.45 12
MEAN ANNUAL FLOOD
The mean annual flood for a gaging station is by definition, the
2.33-year flood from the graphic-frequency curve defined by points
which are referred to the same base-time period.
The magnitude of the mean annual flood may be affected by many
factors, which can be classed as either physiographic or meteorologic.
The problem is: Given a drainage basin of certain physical charac-
teristics, located in a region subject to certain meteorologic condi-
tions, what mean annual flood can be expected? The answer is ob-
tained by correlating the known mean annual floods of drainage areas
within a region with the known characteristics of the basin and the
The physiographic factors which may influence the mean annual
flood at a given point are: (a) The size of drainage area, (b) channel
storage (c) artificial or natural storage in lakes or ponds, (d) slope
of streams, (e) land slope, (f) stream density, (g) stream pattern,
(h) elevation, (i) aspect, (j) orographic position, (k) underlying
geology, (1) soil cover, (m) cultivation, and others.
30 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES
Practical methods of computing some of the important physical
factors are described by Langbein and others, (1947). Some of the
factors listed are fairly simple and can be expressed by a definite
figure. Others, such as geology, channel storage, or orographic pat-
tern are difficult to evaluate and have not yet been successfully used
in correlations with peak floods. Many of these factors are inter-
Development of the relations, if they exist, requires much work.
It also requires much work to compute them for ungaged areas. In
many cases, topographic maps are not available for computing topo-
Meteorologic factors are concerned with the magnitude and dis-
tribution pattern of the precipitation falling on a drainage area.
Some of the elements involved are: (a) Type of region, whether humid
or arid, (b) storm directions, (c) storm patterns, (d) storm volumes,
(e) precipitation intensities, (f) effect of snowmelt, (g) extent of
ice jams, and probably others.
The evaluation, treatment, and use of the meteorologic elements
are generally less certain than for the physiographic factors. The
difficulty lies in determining what precipitation figures to use. Total
annual precipitation has been used, but this is related only generally
to storm rainfall. Rainfall intensities would be more directly re-
lated to peak discharges, but intensities must be expressed by both
a definite period of time and a frequency of occurrence, as for exam-
ple, a 50-year, 5-minute intensity. The possible combinations are
many, and since this is only one of many other factors, both topo-
graphic and meteorologic, the selection of the best parameter be-
comes difficult. There is a great deal of opportunity for original
investigation in this field.
Many physiographic and meteorologic factors make demonstrating
significant correlation difficult except for those factors that are out-
Various combinations of previously listed factors have been used
in correlation with mean annual floods. One such combination is the
mean annual runoff. This is a general index of the amount of pre-
cipitation available, and also an indication of the runoff-inducing
characteristics of the basin. The mean annual rainfall is another
factor which has been used, although not as successfully as the runoff.
In the Illinois flood-frequency study by Mitchell (1954) the basin
lag was used. This is the time lag between the center of rainfall and
the center of runoff. This time lag represents the composite effect
- .-,. -;
FLOOD-FREQUENCY ANALYSES 31
of most or all of the topographic factors. It is therefore a very use-
ful figure, but, it cannot always be defined from known basin charac-
teristics. It must be measured during actual storms.
Another method equal to using a composite factor is that of divid-
ing the study region into several parts, called hydrologic areas.
Within each area a separate curve of mean annual flood is correlated
with the drainage area and perhaps some other significant factor.
In each of these areas such factors as rainfall, geology, and other
features probably have the same overall effect.
If satisfactory relations for several areas can be found by using
drainage area alone, it may not be worthwhile to include other factors
which would improve the correlation only slightly. The data are
generally limited, and the additional factors cause a loss of degrees
of freedom, so that no improvement results.
If, after all practicable factors have been considered in the corre-
lation, then the residuals from the average relation may be analyzed
for geographic location. A pattern may result which can be asso-
ciated with orographic effect, soils, or some other factor. If isograms
of the residuals are plotted, the use of a mean coefficient may improve
the accuracy of the mean annual flood.
SELECTION OF STATIONS
The first step in beginning a flood compilation is to list the gaging
stations. A record should be included if it is 5 or more years in
length, although generally recurrence intervals should not be com-
puted for records shorter than 10 years. Old, discontinued records
that otherwise qualify should be included. Storage or other artificial
factors which would tend to modify flood discharges significantly
should be a minimum. Exclude canals, ditches, and drains, in which
discharges are subject to substantial control by man. Always include
the total usable storage capacity in the basin above the gage.
If the records for 2 drainage areas on the same stream show differ-
ences in area of less than 25 percent, the 2 may be treated as 1 record.
If both records are for the same period of time, use the better one or
use both and give each a weight of one-half; if the records are for
different periods of time, combine them into one longer record. This
is not always done for the larger rivers, such as in the lower reaches
of the Missouri; see the treatment given to the Mississippi River
(Searcy, 1955, p. 12) and a later discussion on page 46.
Include stations maintained by other agencies, such as the U.S.
Weather Bureau; Corps of Engineers, Department of the Army;
U.S. Soil Conservation Service; State agencies; private agencies, such
-rn-:.,s~r~ r:-.-:~ir ~lra r~L11:l-s- 0~n -Ir ?L :
FLOOD-FREQUENCY ANALYSES 37
be numbered 1, the second largest 2, and so forth. These order
numbers are shown in column 6 of figure 9.
Compute recurrence intervals.-Next, compute recurrence intervals,
from the formula T=--(n 1), for each flood of record. Do not compute
recurrence intervals for the estimated figures. Record them as shown
in column 7 of figure 9.
The floods for 1933-50 have now been adjusted to the base period
PRELIMINARY FREQUENCY CURVE
Plot discharge against recurrence intervals for each gaging station.
Plot historical data in accordance with the discussion on page 17.
Plot on forms similar to those illustrated in figures 3 and 4 (Survey
form 9-179a and 9-179b).
Draw a frequency curve as a curve of visual best fit; do not com-
pute mathematically. Extend the curve as far as the data warrants.
A preliminary-frequency curve for Big Piney Run near Salisbury,
Pa., is shown as figure 11; this curve bends up at the outer end, but
if the data are plotted to a logarit-mic-discharge scale a straight-line
curve will be defined.
THE MEAN ANNUAL FLOOD
The mean used for each frequency distribution is the graphical
mean determined by the intersection of the visually best fitting fre-
quency line with the line corresponding to the 2.33-year recurrence
interval. The graphical mean is more stable and dependable than
an arithmetic mean. This method of determining the mean gives
greater weight to the medium floods than to the extreme floods with
large sampling errors, and for this reason is not influenced adversely
4 = ...
1 01 1.1 2 5 10 20 50 100
RECURRENCE INTERVAL, IN YEARS
FIouRE 1.---Prelminary frequency curve, Big Piney Run near Salisbury, Pa.
1a. 01 1125 1 05 0
REUREC INEVL I ER
-38 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES
by the chance inclusion or exclusion of a major flood, as is the arith-
In determining the mean annual flood, the curve should be care-
fully fitted to the points plotting near the 2.33-year line. In the
example, the mean annual flood for Big Piney Run is 1,060 cfs ac-
cording to the curve of figure 11. (The arithmetic mean, computed
from the 18 years of record, is 1,540 cfs.)
The test setup requires a study of the 10-year floods as estimated
at each station. Each 10-year flood should be divided by the
mean flood to get the 10-year ratio, and an average of these ratios
should be obtained. Tabulate for each station the length of record
in years and the recurrence interval corresponding to a discharge
equal to the average flood ratio times the mean flood. Data should
be tabulated as shown in the following table:
Data for homogeneity test, base period 1914-50
Drainage Mean 10-year T for Period
area annual flood, Ratio Q2.s3 Q of of record,
No. Stream (square flood, Q1o Qio X1.70 column 7 adjusted
miles) Q3.s3 (cfs) 2 (cfs) (years) (years)
1 2 3 4 5 6 7 9
I Stony Creek--...----. 451 10,800 22,400 2.07 18, 700 7 37
2 Conemaugh River-....- 715 19,000 31,500 1.66 32,300 11 25
3 Conemaugh River.--... 1,358 34,000 54,000 1.59 57,800 14 24
4 Kiskiminetas River-... 1,723 42,000 65,000 1.55 71,400 16 32
5 Little Conemaugh
River--.--------- 183 6,250 9,800 1.57 10,600 13 24
6 Blacklick Creek..-----. 390 11,500 18,800 1.64 19,600 12 37
7 Loyalhanna Creek.--... 168 6,000 9,600 1.60 10,200 13 24
8 Loyalhabna Creek-..... 265 7,400 11,800 60 12,600 13 28
9 Youghiogheny River... 1,062 26,300 43, 100 1.64 44,700 11 33
10 Youghiogheny River_.. 1,362 31,500 58,100 1.84 53,500 8 33
11 Youghiogheny River-.. 1,715 38,200 62.500 1.64 65,000 12 33
12 Casselman River.- 382 11,200 19,200 1.71 19,000 10 36
13 Big Piney Run-.--..... 24.5 1,060 2,260 2.13 1,800 6 27
14 Laurel Hill Creek---- 121 4, 850 7,400 1.53 8.250 17 37
15 Oreen Lick Run.------. 3.07 230 500 1.79 476 8 30
Average ratio----....---..-------..............----. -----------------. 1.70
The adjusted period of record, column 9, is the number of years of
actual record plus one-half the number of years the record was
A set of test curves has been devised that shows within what range
of recurrence intervals an estimate of a 10-year flood should be for a
specified length of record; a range of 2 standard deviations is allowed.
It is appropriate to base the test on the 10-year flood, because this is
the longest recurrence interval for which many records will give
dependable estimates. The test curves may be drawn on a standard
chart, such as Survey form 9--179a, or a chart prepared from data
---- r.--..C ~L~ .. .I~rr r~--~C C~~~jlLI) ~ '4%h
54 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUE*
floods; three 333-year floods; and so on occur throughout the entire
range. They are in random order, and short consecutive periods of
this array are similar to what might be found in short periods of
record in nature. Analysis of the frequency curves computed from
short periods may be compared with the known characteristics of the
basic long-term frequency curve. Since the order of the items was
established by random drawing, any differences from the basic curve
are due to chance alone.
MEAN ANNUAL FLOOD
According to the theory of largest values (Gumbel, 1945a), the
mean of all the annual floods has a value corresponding to the fldod
of 2.33-year recurrence interval. The arithmetic mean of the 1,000
items shown in column 5 of the preceding table is 3,064, which is
the same as the value at 2.33-years on the base curve of figure 19.
SHORT PERIODS OF RECORD
The base data were divided into "records" of various lengths for
purposes of analysis; for example, in constituting twenty 50-year
"records" the first 50 on the random list was the first "record" and
so on. Division was as follows:
1. One hundred consecutive 10-year periods:
Items 1-10, 11-20, 21-30-991-1,000.
2. Forty consecutive 25-year periods:
Items 1-25, 26-50-976-1,000.
3. Twenty consecutive 50-year periods:
Items 1-50, 51-100-951-1,000.
4. Ten consecutive 100-year periods:
Items 1-100, 101-200-901-1,000.
Each of the 170 periods was analyzed separately, as though it were
an independent record of 10, 25, 50, or 100 years. Recurrence inter-
vals were computed and plotted and individual frequency curves were
drawn for each.
DRAWING OF FREQUENCY CURVES
In drawing the frequency curves, the same procedures were fol-
lowed as when actual station records are being studied:
1. The curves were drawn by eye as the graphical curves of best fit.
2. They were drawn as straight lines or curves depending on the trend of the
3. The curves were drawn to average rather than to follow individual points,
the object being to avoid sharp breaks or bends.
4. The curves were kept as close as was reasonably possible to the plotted
points in the vicinity of the 2.33-year line.
-C- ,., ~,*~ I ..* ..- .ur.. rC~ -L -9 F- "- ~l-*P*l ~ C---rL~~-~ l -CI --iT-
56 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES
investigators advocate drawing a straight line through the lower
points, giving little weight to the upper points. Where all points
indicate definite curvature throughout the range, this procedure be-
comes wholly arbitrary and subjective. The greatest objection to the
drawing of straight-line graphs, whether computed or by eye, is that
it is first necessary to support the belief that they should be straight
lines. There is no firm basis for rigid adherent to straight lines.
The individual frequency curves for 10-, 25-, 50-, and 100-year
periods are shown as figures 20 to 23, respectively. All those based
on a common length of period have been plotted together to show
graphically the extent of variation obtained.
ARITHMETIC VERSUS GRAPHICAL MEAN ANNUAL FLOOD
The mean of the annual floods, or "mean annual flood," is an ex-
tremely important factor, used in correlation studies and in regional
flood-frequency compilations. It is desirable to determine it from
fairly short-term records.
In short-term records particularly, the arithmetic mean is affected
considerably by the chance inclusion of one or more major floods.
The Geological Survey uses the graphical mean annual flood to avoid
this adverse condition. The graphical mean is the value determined
by the intersection of the visually best fitting frequency curve with
the mean line (the line corresponding to the 2.33-year recurrence in-
terval). The graphical mean is more stable and dependable than
the arithmetic mean. This method of determining the mean gives
greater weight to the medium floods than to the extreme floods with r
large sampling errors. The resulting figure is no longer the mean
of the annual floods, but is the "mean annual flood" by definition.
This investigation makes possible a practical comparison between
values of arithmetic and graphical means, and a demonstration of
the variation in the value of the mean with the length of the record.
The average of the arithmetic means of all the short-term periods
is 3,064. The averages of the graphical means, as obtained from the
individual short periods, vary as follows:
Length of Average of
record Number of graphical
(years) records means
10 100 3, 100
25 40 3, 110
50 20 3, 100
100 10 3, 080
These are within small percentages (less than 1.5 percent) of the
true values, so that no gross errors are involved in using the graphi-
cal mean. .
FLOOD-FREQUENCY ANALYSES 61
RELIABILITY OF MEAN ANNUAL-FLOOD VALUES
In statistical studies, the most favorable expectancy during 95 per-
cent of the time is commonly used as the criterion for dependable
The extent of variation from the true mean of the mean annual
floods (determined graphically) in this study, during 95 percent of
the time, was 28 percent for 10-year periods, 14 percent for 25-year
periods, 12 percent for 50-year periods, and about 5 percent for 100-
These percentages cannot be assumed to apply generally, because
they will vary with the slope of the base-frequency curve. They are
a general indication of what may be expected from short- and long-
term records. The 10-year records will give a much wider range
in determining the mean.
It is desirable to have some means of allowing for the range in
value due to chance, and a method is hereby outlined which gives
consideration to the length of the record from which the mean annual
flood has been determined.
"WORKING RANGE" OF MEAN ANNUAL FLOODS
Based on the theory of extreme values, it is possible to compute,
for conditions existing 95 percent of the time, and for any length of
record, the range in recurrence intervals which would be found for
the 2.33-year flood. The theoretical limiting curves for these values
are shown as figure 24.
These curves are generally applicable. The extent to which results
of this 1,000-year frequency study conform with the theoretical
curves is shown by the plotted points on figure 24. These points rep-
resent the apparent recurrence intervals corresponding to the known
mean annual flood of 3,064, as determined from the 170 individual
frequency curves developed in the study. Theoretically, 5 percent
of these points, or 8.5 points, should be outside the limits; actually,
7 of the points lie outside.
In regional flood-frequency studies, the relation between the mean
annual flood and the drainage area (or other basin characteristics)
is required and can be determined graphically by plotting one against
the other. Instead of plotting a point, a range line may be plotted
using the chance range in the mean annual flood based on the length
of record. An average curve passing through all the range lines
may then be considered as the best determination of the relation
The method of determining the chance or working range of the
mean annual flood is illustrated in figure 25. Curve A is an assumed
frequency curve, based on a presumed record of 10 years. The curves
--- ~ __._ _~._
62 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES
--- -- --- -- ---.--,---
_- _,_-_ _ __- i r -
= = = = = = =-__ _ -_ ^
-- y -- -- -- "
- - -.-- -- I ...
SIV3A NI 'OQOO33 -O HI1N31
ts~ cZ k
... ...... 'I -. r 1
FLOOD-FREQUENCY ANALYSES 63
5000 - -
RECURRENCE INTERVAL, IN YEARS
FIGUaR 25.-Computing range for mean annual flood.
of figure 25 show that the recurrence interval for a mean annual flood
based on 10 years of record might have an apparent recurrence inter-
val of between 1.30 and 5.20 years (95 percent of the time). The
mean annual flood, determined from curve A, is 3,000 cfs. Curves
B and C are drawn parallel to curve A passing through 3,000 cfs at
recurrence intervals of 1.30 and 5.20 years. These represent the ex-
treme positions which curve A might take due to the chance varia-
tion in 10 years of record. (It is assumed that the slope would be
parallel within these short segments.) These curves intersect the
2.33-year line at 2,100 and 3,800 cfs; this range is interpreted as
showing that the actual value of the mean annual flood might, due
to chance variation, lie anywhere between 2,100 and 3,800 cfs. The
mean annual flood for this station record is plotted as a range line
between these limits, against any other desired factor.
RELIABILITY OF FLOOD MAGNITUDES
A study was made of the maximum and minimum values of the
10-, 25-, 50-, and 100-year floods as determined graphically from rec-
ords of various lengths, for 80, 95, and 100 percent of the time.
These results show, for example, that (19 out of 20 times) for
these data, any 39-year record can define the 50-year flood within 25
percent of the true, long-term value.
.. .. .. ,, .. ..
64 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES
Lengths of record necessary to come within 25 percent of the correct value 95 or 80
percent of the time
Length of record
Magnitude of flood __
95 percent 80 percent
of the time of the time
2.33----------- --- -12 -
10., ------ 18 8
25---------------- 31 12
50---------------- 39 15
Lengths of record necessary to come within 10 percent of the correct value 95 or 80
percent of the time
Length of record
Magnitude of flood ___
95 percent 80 percent
of the time of the time
2.33-------------- 40 25
10---------------- 90 38
25 -------------- 105 75
50 ---------------- 110 90
100 -------------- 115 100
In recent years, many crest-stage gaging stations have been estab-
lished for the purpose of defining the flood potential of a region.
The mean annual flood can be determined within 25 percent (95 per-
cent of the time) by a 12-year record of such stations.
The results of this study are a general qualitative indication of
long- and short-term records. They show, for example, that if we
do not demand too great a degree of accuracy, the 50- and 100-year
floods may be determined from the 40- or 50-year records which are
It seems that the individual short-term station record is perhaps
less reliable than we have generally considered it, but the individual
long-term record is surprisingly dependable.
For less than the 100-year flood, a longer record than the period
of the desired flood is necessary, for the result to be within 10 percent
of the correct answer. This is increasingly true for the short-term
floods. The correct answer for short-term floods probably would be
10 percent or less. In general, we should feel content if we are rea-
sonably sure of predicting a given flood within 25 percent.
The figures shown cannot be expressed as percentages of the cor-
rect values and applied generally. The percentages are dependent
on the slope of the individual frequency curve in question.