HYDROLOGIC STUDIES ON THE
CEDAR RIVER BASIN

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
DAVOOD HARIRI
1960

 

 

 

This is to certify that the

thesis entitled

Ibrdrologic Studies on the
Cedar River Basin

presented by

Davood Hariri

has been accepted towards fulfillment

of the requirements for

 

M.S. degree in Agricultural Engineering
Major professor
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L I B R A R Y
Michigan State
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HYDROLOGIC STUDIES ON THE
CEDAR RIVER BASIN

by
DAVOOD HARIRI

A THESIS

Submitted to the College of Agriculture of Michigan
State University of Agriculture and Applied
Science in partial fulfillment of the

requirements for the degree or
MASTER OF SCIENCE
Department of.mgricultura1 Engineering

1960

Approved (:f2;¢§;§gif:;;%é;€::Z//

 

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ACKNOWLEDGMENTS

The writer wishes to express his sincere thanks to
Professor Earnest H. Kidder, under whose guidance this study
was undertaken.

He is likewise grateful to Mr. Arlington Ash, chief,
and Mr. Sulo U. My assistant chief, of the United
States Geological Survey Surface water Branch in Michigan,
who were most helpful in collecting data.

Grateful acknowledgment is also due to Mr. C. H.
Hardison, chief, Hydrologic Studies Section, Surface Water,
United States Geological Survey, who has offered his personal

notes.

 

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TABLE OF CONTENTS

I. INTRODUCTION ...................................
1. Purpose of Study .........................

2. Review of Literature .....................

II. DESCRIPTION OF STUDY AREA ......................
1. Basin ....................................

Cedar River .............................

Grand River .............................

2. Gaging Station oooooooo00000000000000.0000

3. Climate .o................................
4. Geology ..................................
5. Soils ....................................
6. Land Use .................................

III. METHODOLOGY ....................................
1. Collection of Data ......................

2. Assumptions and Procedure ...............
Frequency Analysis of Stream Flow ......

Computing Weighed Average Pre!
cipitation o.0000000000000000000000000

Runoff-Precipitation Relation ..........

IV. DISCUSSION OF RESULTS .........................
Flood Frequency Analysis ....................
Drought Frequency Analysis ...d..............

Mean Flow Frequency Analysis ................

Annual Precipitation - Runoff Relation ......

Page

6

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16
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111

Table of Contents (continued)

VI.
VII.

Fall Precipitation - Runoff Relation ..........
Winter Precipitation - Runoff Relation ........
Spring Precipitation - Runoff Relation ........
Summer Precipitation - Runoff Relation ........
SUMMARY .........................................
CONCLUSION ......................................

REFERMCES OOOOOOOOOOOOOO0.0000000000000000000000

Page
58
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61
63
65
67

 

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Figure

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10a
10b
10c
11

Cedar River Map
Flood Frequency
Flood Frequency
Drought Frequen
Frequency Curve
Frequency Curve
Frequency Curve
Frequency Curve

Frequency Curve

LIST OF FIGURES

Curve (Gumbel's formula) ......
Curve (Hazen's formula) .......
cy Curve .......................
of Annual Mean Flow ...........
of Fall Mean Flow .............
of Winter Mean Flow ...........
of Spring Mean Flow ...........

Of Summer Mean Flow coco-.0000.

Classification Of Surface RunOff oooooooooooooo

Classification of Surface Runoff ..............

ClfiBSification 0f Surface Runoff 0.000.000.0000

Runoff-Precipitation Regression Line ..........

Page

25
26
29
32
33
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35
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Table

LIST OF TABLES

Annual and Quarterly Mean Discharges of the
Cedar River for Period of 1912-1919 and

1921-1930 00.0.0000...00.00.000.000...0000.0.0

The Annual Peak Floods on the Cedar River
for 47 Years 0.000000000000000000000000000.000

Annual Drought on the Cedar River for the
Period 01‘1932-1955 00000000000090...ooooooooo

The Limit Values for Classification Obtained
from Figures 5 "' 9 oooooooo00.000000000000000o

Weighed Precipitation for Cedar River Basin ..
Annual Carry-Over Precipitation for Cedar

River B58111 cocoonoooooooooooOOooooooooooooooo

Recurrence Interval of Floods on the Cedar

River 00000000000.000000000000...ooooooooooooo

Recurrence Interval of Droughts on the Cedar

River ooooooooooooooooooo00.000000000000000...

Classification of 47 Years of Surface Runoff
on Cedar River Basin ocooooooooooooooooooooooo

Page

17
22
27

31
39

45
50
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55

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I. INTRODUCTION

1 - Purpose of Study

There is a great need for information relating to hydro-
logic frequencies. Terraces, grassed waterways, gully control
structures, flood control projects, drainage projects, irri-
gation projects, the efficient design of coffer-dams, water-
way openings in bridges, highway and railway culverts, urban
storm sewers, stream control works, hydroelectric power
installations, water supply facilities, and many hydraulic
structures and projects are based on the frequency of certain
hydrologic events.

This study attempts to analyze the frequency of floods,
droughts, and annual and seasonal mean flows on the Cedar
River. The writer has classified forty-seven years of data
into eight classes, from extremely dry to extremely wet. He
also tries to establish a relationship between precipitation

and runoff on an annual and a seasonal basis.

2 - Review of Literature

Hydrology is that branch of physical geography dealing
with the waters of the earth with special reference to
properties, phenomena, and distribution. The prOblems of
hydrology were principally a matter of speculation until
recent years. Curiously enough, the greatest problem for the

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philosophers was that of describing the hydrologic cycle.

Such men as Homer, Plato, and Aristotle felt that rainfall

was quite inadequate to supply streamflow, and formulated
elaborate hypotheses to explain how the water welled up from
great subterranean caverns connected with the sea. Until the
fifteenth century A.D. it was considered heresy to question
the theories of the early philosophers. Da Vinci and Palissy
(1) expounded an outline of the hydrologic cycle quite similar

to present day conceptions. It was not until the seventeenth
century that Perreault supplied the necessary quantitative
proof by comparing measurements of rainfall and runoff for
the Seine River in France and showing that the runoff volume
was only about one-sixth of the rainfall. Perreault's find-
ings were soon supported by the English astronomer, Halley,
who demonstrated by experiments that the moisture evaporated
from the oceans was adequate for precipitation which resulted
in streamflow. Under the leadership of Torricelli, Pitot,
Bernoulli, de Chezy, Herschel, Venturi, and others, investi-
gations moved forward rapidly in the study of fluid flow.
Dams, bridges, levees, and canals were soon being designed by
hydraulic engineers with a fair knowledge of fluid mechanics
but with only the crudest sort of rules of thumb to estimate
the probable quantities of water to which these structures
might be subjected. As the number and size of structures
increased and the potential economic loss from structural or
Operational failure multiplied, the demand for a more adequate
basis for design increased, and modern hydrology found its

start.

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Foster (6) adapted Pearson's theoretical frequency for
application to flood frequency curves. The use of logarithmic
probability paper was of great help.

Beard (1) states that the theory of the duration curve
is relatively simple. The occurrence of the future (those
that are "in store" for the future) are considered to con-
stitute a series, arranged in order of descending magnitude.
If there are an infinite number of occurrences in store, then
this series may be represented by a continuous curve. Further-
more, if it is considered that there is not a cycle or trend
in magnitude with respect to time, it may be assumed that
occurrence which happen within a given period have each been
selected at random from the store of occurrences. Lastly,
since the number of occurrences in store is considered to be
infinite, the selection of a finite number of occurrences
will have no effect on the distribution of magnitudes within
the remaining series of occurrences.

Since they have been chosen at random from the store
of occurrences, the occurrences of record (those to be used
in the analysis) may be used to estimate, by the laws of
prObability. With this curve defined, forecasts concerning
future occurrences may be made in plotting the duration curve.
the assumption was made that the recorded occurrences were
representative of any set of events that would occur in a
period equal to the period of record considered. This is
not true, however. The largest event may be twice as great

as would normally occur in the given period, and the second

 

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largest may be smaller than the normal second largest
occurrence.

For plotting position in frequency curve, Beard (1)
used the Foster-Hazen formula.

Gumbel (1,8), believes that the solutions proposed by
Board are too general and therefore inadequate for specific
problems. Gumbel, after a great deal of discussion, proposed
to use T = E_%;l. formula.

Extending stream flow record from the available pre-
cipitation data has been a problem for research for*many
years. As far as it is known, runoff is a residual of pre-
cipitation in that it is really P minus L. L represents
the evapotranspiration losses and thus varies with temper-
ature. The constant loss in any area would be a satisfactory
tool for estimating runoff from rainfall if itwere only the
average stream flow that was desired. To use the constant
loss method for annual runoff, the R.= P - C formula could
be used in which 0 is a constant for a given area. This
might be called the modern way as it was brought forth around
1935. The old fashion way was to compute runoff as a per-
centage of precipitation, or R»: KP. Neither of these methods
is the best available.

The R»: P‘- C method is weak in that it does not take
into account the fact that opportunity for evapotranspiration
during dry years is less than that during wet years. The
R.= KP method is weak in that there is a minimum annual pre-
cipitation below which the annual discharge would be
negligible. In 1945, Rice and Hardison (9) worked on this

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problem in Alabama. He found that for annual rainfall -
runoff data in Alabama that a formula, L = KP + G or i

R.= (1 - k)Ph C gave the best results. In order to have only
one variable factor to plot on a map as a geographic factor,‘
he used an average value of 0 equal to 10 inches and solved
for K.

In 1949, Carter and Williams (3) found that they could
improve the results by using-K as a constant equal to 0.5 and
letting G be a variable that varies from year to year depend-
ing on the effectiveness of the precipitation during the year.
The values of C to use would have to be determined from avail-
able gaging-station records in the area.

In 1945, Johnson and those working with him on a
water-resources report for Oklahoma (12) plotted precipitation
against runoff and found that correlation could be improved
by assigning a percentage weight to the previous year's pre-
cipitation and the remaining part of 100 per cent to the
current year's precipitation. The percentages varies from
station to station but were held constant at each station.

The purpose was to give some weight to carry over from last
year's precipitation in the form of ground water storage.

On the Cedar River the U. 3. Geological Survey has pre-
pared a flood frequency curve, but no other studies have been

carried on.

V4

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II. DESCRIPTION OF THE STUDY AREA
1 - Basin
Cedar River

The Cedar River has its source in Cedar Lake which is
located in southwestern Livingston County in Sections 28 and
29, Township 1 North, Range 3 East, or the Michigan.Meridian.

It flows generally northwestward through Livingston
County for approximately 18 miles and then continues westward
through Ingham County for 28 miles, where it enters the Grand
River within the city of Lansing. The river and its tribu-
taries drain an area of about 475 square miles, one-fourth of
which is drained by Sycamore Creek and its feeder streams.
These branches entering from the north are:

1. W01! Creek
2. Squaw Creek
3. Coon Creek
and branches entering from the south are:
1. Middle branch of the Cedar River
2. West branch of the Cedar River
3. Kalamink Creek
4. Dietz Creek
5. Doan Creek
6. Deer Creek
7. Sloan Creek
8. Herron Creek
9. Sycamore Creek

The river has an average gradient of 2.51 feet per mile,

with about one-half of the fall occurring within the uppermost

one-third of the river. Cedar Lake lies at an elevation of

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934 feet above sea level, and the confluence of the Cedar
River with the Grand River lies at 817 feet above sea level.
The stream pattern of the river and its tributaries is a
combination of trellis and dendritic types of drainage.

The river in its upper reaches is generally quite clear
following the initial spring floods, but quite often an
apparent reddish-brown color is observed which is caused
primarily by reflections from the bottom in the relatively
shallow water. The lower stretches of the river, however,
usually appear more turbid, largely because the deeper water
here accentuates this condition.

There are three artificial permanent impoundments on
the Cedar River (2). The most significant of these structures
is located at Williamston, and was originally constructed to
facilitate the operation of a sawmill. The original dam has
since been replaced by one which maintains a thirteen foot
head and aids in providing power for a private frozen food
and refrigeration plant.

The backwaters of this low dam extend upstream for
about two miles, but are contained, for the most part, within
a narrow belt extending a short distance out from the main
channel. The other two artificial low dams, one of which is
located at the Park in Okemos, the other on the Michigan
State University Campus in East Lansing, serve recreational
and aesthetic purposes. The East Lansing reservoir also

supplies cooling water for a campus power plant.

Grand River

A description of Grand River Basin at this point will
be pertinent in the understanding of the study. The Grand
River rises in the southern part of Jackson County, Michigan,
and flows in a general northerly and northwesterly direction.
It empties into Lake Michigan at Grand Haven. Its drainage
area, which comprises a rich agricultural region in the
south-central portion of Michigan includes extensive swamps
and marshes, but comparatively few lakes. At Grand Rapids
the stream passes over a limestone ledge, making a consider-
able fall, which has been developed for power purposes, and
at Grand Ledge a similar descent occurs over sandstone. The
Grand River receives a number of important tributaries,

notably, the Flat, Thornapple, Maple, Lookingglass, and Cedar.

2 - Gaging Station

A gaging station was established on August 30, 1902.

This gage was located at the highway bridge Just below the
Michigan State University Campus. The bed was sand and gravel,
fairly smooth and permanent. The gage was read twice each
day. This station was used until December 1903 to measure

the flow from a 358 square mile drainage area.

At the present time, the gaging station is located at
the southwest abutment of the Farm Lane Bridge, Michigan State
University Campus, three miles upstream from Sycamore Creek
and four miles upstream from thejJunction with the Grand

River. Its exact location is Latitude 42° 43' 40",

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Longitude 84° 28' 40" in SWi sec. 1ST, 4N R. 1 W. There are
a staff gage, a continuous recorder and a continuous recofdhfig
thermograph for limnological studies on this station. The
datum of the gage is 824.39 feet above mean sea level.

This station measures the runoff from about 355 square

miles of the basin.

3 - Climate

The study area is located within a climatic type
describedcby Koeppen (5) as Dbf. In this system of climatic
classification, the symbols Dbf indicate the following
conditions:

D - a humid microthermal climate; that is, one in which
the mean temperature of the coldest month is less than 32°F.
and the mean temperature of the warmest month is more than
50°F. Rather cold winters and large annual temperature
ranges are found in this type.

b - the mean temperature of the warmest month is less
than 71.6°F.

f - the location of the study area, about 100 miles to
the east of Lake Michigan and the prevailing westerly to
southwesterly winds bringing moisture from the lake, are
fictors important in determining this condition. Although
the precipitation is fairly evenly distributed throughout
the year, the period of maximum rainfall is in late spring
and early summer. The normal annual precipitation is about

30.5 inches, including melted snow. Snowfall generally

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averages about 45 inches, and this snow cover is generally
of sufficient duration to have a marked effect upon winter
temperatures. Sunlight falling upon thehsnow is reflected
to a great extent so that little of the solar energy is
effective in heating the ground or the atmosphere. The low
conductivity of snow tends to retard the flow of heat from
the ground below to replace that which is being lost. Winter
precipitation is largely cyclonic in origin. Maritime
tropical air masses from the Gulf of Mexico travel northward
up the Mississippi Valley, gradually being chilled by the
cold ground surface and by colder and denser continental
polar air which they over-ride. Summer precipitation ia.for
the most part, of a convectional nature, often falling from
cumulo-nimbus clouds as intense thunder showers. The same
maritime tropical gulf air masses which penetrate the area
set up conditions favorable for convection storms.

The mean annual temperature for the area is 46.8°F.,
with a winter mean of approximately 24°F. and a summer mean
of approximately 68.5°F. The frost-free season generally is
about 160 days and extends from the first week in May to the
first week in October.

Winds rarely attain high velocities in the area, evapo-
ration is generally low, and humidity is moderately high.

Seasonal weather is characterized by rapid and non-
periodic changes. During the winter, when the sun (and along
with it the storm belt) has moved south, weather conditions
are dominated by cyclones and anticyclones associated with

shifting polar and tropical air masses, and the fronts which

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develop along their'boundaries. An anticyclone made up of
cold polar or artic continental air descending from Arctic
Canada may produce a bitter "cold wave" over the area for
several days or weeks and then be replaced by warmer maritime

tropical air from the south.

4 - Geology

The rock strata which lie immediately beneath the
glacial drift in the area in which the study was undertaken
are of the Pottsville series of Pennsylvanian age. This
series varies in thickness from 0 to 535 feet, and includes
shales, coal beds, the Parma sandstone and a persistent,
shaly lime-bed called the Verne limestone.

The topography of the area is nearly level to rolling,
resulting from the cary phase of the late Wisconsin glaciation.
The drift is quite calcarious, containing large quantities
of limestone debris. Among the landforms which typify the
region are the characteristics of continental glaciation.
Between the recessional moraines which run in an east-west
direction in the area are found areas of gently undulating
ground moraine or till plain. A few eskers are found running
in a north-south direction, most of them in the southern
portion of the watershed. These mounds of water sorted
materials serve as a source of gravel for road construction
and concrete in the area. Numerous depressions, some of them

holding water, are found throughout the watershed. These

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are called pits if they are found in an outwash plain, or
kettles when they are found in morainic deposits, but they
are formed similarly in either case. Blocks of ice were left
buried beneath the drift as the glacier melted away and as
these ice blocks melted the overburden slumped into the
vacancies forming the pits or kettles.

In terms of the fluvial cycle, the Cedar River is in
the mature stage of valley development. Meanders are common
in the lower reaches of the river, and one oXbow has been

cut off about a mile downstream from Okemos.

5 - Soils

The soils of Ingham and Livingston Counties are derived
mainly from limy loam glacial till (13). The primary soil
series in the area are the Miami and the Conover, which are
essentially gray-brown podzols of good to intermediate drain-
age. The unweathered drift underlying the Miami soils is
alkaline and contains considerable amounts of limestone
debris.

More specifically, the soils in the immediate vicinity
of the Cedar River can be considered to be of three main
types:

1. Qgggggg‘£;n§,§§ggx,;2gg, This well-drained
alluvial soil is found along the lower reaches of the river
and extends upstream from the mouth almost to the city of
Williamston.

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2. Griffin loam, These alluvial deposits are more
poorly drained than the Genesee loam, and range from slightly
acid to alkaline in reaction. They are found from the
vicinity of Williamston upstream to the vicinity of Fowler-
ville. Here the soils along the river give way to Cariisle
muck.

3. Carligle muck. This organic soil type is found
from the proximity of Fowlerville upstream to Cedar Lake, the
source of the river. This soil is characteristically medium
acid to alkaline in reaction. It is generally rich in lime

and phosphorus but poor in potash.

6 - Land Use

Dairy and general farming predominate in the area as a
whole, with hogs, poultry, and sheep as minor enterprises (10).
Most of the crops grown are the feed crops of hay, pasture,
corn, and cats. The major cash crop is wheat, with sugar
beets and field beans important in areas with favorable soil
conditions. The major factors influencing the choice of
these particular enterprises are: (1) the relatively long
growing season; (2) the predominance of sandy loams, silt
loams, and loams of medium to high fertility; and (3) the
good market for whole milk in the area.

For the most part, land use practices and agricultural
techniques in the study area seem to be sound. Excessive
runoff during times of the year when the soil is capable of

absorbing water is not a problem. In the upper portions of

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the watershed, the river and its tributaries have, in many
places, been dredged to straighten and deepen the channel for
agricultural drainage purposes. The high incidence of wet
lands in the upper portion of the watershed was partially
responsible for this work which had marsh and swamp drainage
as the objective.

Wood lots, a few of them grazed quite heavily, dominate
the use of the land immediately adjacent to the river. A few
small fields of corn and other grain crops are found along
the river banks and other fields which have been allowed to
lie fallow for various periods are occasionally encountered.
In the wooded areas the main species of trees are the white
oak, elm, ash, soft maple, shagbark, hickory, and.basswood,
while the muck soils are dominated by tamarack, aspen, and

shrUbs of various kinds.

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.

III. METHODOLOGY
1 - Collection of Data

The data for stream flow since 1931, up to the present,
was obtained from water supply papers published by United
States Geological Survey.

Monthly records for the period 1921-1931, which were
computed with the correlation method between other close
basins by U. S. Geological Survey Surface Water Branch in
Washington, D. 0., were taken from the files of U. S. Geo-
logical Survey Surface Water Branch, Regional Office in
Lansing.

Fbr the period of 1912-1919, the stage of the river was
observed twice a day by U. S. Weather Bureau and published
in the U. S. Weather Bureau publications. The writer
established a relationship between U. S. Geological Survey
gage readings and U. S. Weather Bureau gage readings. He
computed the daily discharges for eight years from this re-
lationship.

The seasonal mean discharges for the period of 1912-
1919, and 1921-1930, are shown on Table 1.

The sources for precipitation data were U. S. Weather

Bureau climatological data publications.

D“

*4

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g a
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17

Table 1

Annual and Quarterly Mean Discharges of the
Cedar River for the Period of
1912-1919 and 1921-1930

Annual 005:}Dec. JZ::;§:r. A:£:}§Ene JE§;7§:pt.

Water Mean in Mean in Mean in Mean in Mean in
Leg}; c,f,s, c,f,s. c,f,s, g,f,s, 9.1.13,
1912 321 206 318 718 41
1913 311 135 715 374 20
1914 235 77 378 389 107

' 1915 169 48 484 107 351
1916 362 108 758 534 50
1917 200 33 244 419 105
1918 276 71 854 166 15
1919 259 169 282 543 42

I920?

1921 213 183 374 271 51
1922 280 182 344 457 61
1923 138 70 243 203 31
1924 174 74 219 355 40
1925 75 31 158 87 24
1926 214 171 258 374 55
1927 _186 188 296 228 32
1928 213 160 237 355 93
1929 321 181 395 533 57

1930 219 127 459 268 31

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18
2 - Assumptions and Procedures
Frequency Analysis of Stream Flow

Frequency may be defined as the number of flows of
a particular’magnitude as compared to the total number of all
such flows, whereas a frequency distribution is an arrangement
by magnitude of frequencies. For example, in a given 100-
year record there are 100 annual peak flows or floods. The
smallest of these flows is equalled or exceeded every year
and therefore can be designated the one-year flood. Similarly,
the fifth largest flood occurs on the average every twenty
years and becomes the twenty year flood; the highest flow,
the 100-year flood. These recurrence intervals may be de-

fined by this equation:

T=.I.1_'.':__1.
r m

where m is the order number when the values are arranged in
decreasing order and n is total number of events. It should
be noted that no chronological sequence is implied, but only
an average interval of occurrence.

In the following sections, the applicability of
frequency analysis to high flow or floods, to low flows or
droughts, and to all stream flow is shown.

Frequency analysis of flood flows - Flood frequencies
are developed for design of culverts, bridge waterways, and
spillways, for preliminary planning, and for economic studies.
Floods may be defined either as peak or volume floods, depend-

ing upon the purpose of the analysis. A peak flow is simply

nv

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19

the highest instantaneous discharge; whereas a volume flood
is that high rate of discharge persisting for a specified
duration, such as one day, three days, or ten day flood. To
develop a peak flood frequency curve, peak floods are
selected from a stream flow record. Floods are selected by
the highest flood in each water'year. A common problem upon
developing a flood frequency curve is the need to estimate
the magnitude of floods having recurrence intervals greater
than that of the length of record. The extreme curvature dis-
played by most frequency curves plotted on rectangular co-
ordinate paper makes extrapolation difficult; therefore,
plotting the curve on probability, log probability, semi-log
or log-leg paper is helpful in reducing the curvature.

Low flow - The low flows can be analyzed on a fre-
quency basis in an identical manner to that of flood flows.
Instead of selecting peak flows, the lowest recorded dis-
charges each year are employed. Droughts are defined as the
annual minima of discharges. Drought frequency curves have
important engineering applications for irrigation, industry,
and municipal water supply dependent upon stream flow, for
hydroelectric power developments, and for estimating storage
requirements.

Frequency analysis of all flows - Rather than
developing frequency curves from only small segments of stream
flow record, analysis of the entire period of record has
several engineering applications. Here, all of the mean

daily flows constitute a population.

e...

I.

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b . A ~ ~‘- A - n‘n- .
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2O

Plotting positions - In treating hydrologic data for
the study of their frequency of recurrence, two phases of
study are generally met. One is the frequency of recurrence
for the observed distributions, and the other one is the
corresponding frequency of recurrence for the theoretical
distributions of'best-fit. The former is usually required
for the purpose of plotting observed data and hence is called
the "plotting positions," a term used by Foster (7). The
latter is treated by mathematical theory of probability and
generally serves as a theoretical basis for interpreting the
observed phenomena. A correct determination of plotting
positions has been a moot question and has caused a great
deal of discussion. Many methods for computing plotting
positions have been proposed, but few of them deserve
theoretical explanation. An attempt is made here to give a
rational solution to the derivation of plotting position
formulas for T recurrence interval.

If it is assumed that n past observations were taken
from a certain unknown distribution of events, these n values
can be arranged in an order of descending magnitude in which
the rank, designated by m, of the largest value is equal to
one. According to Gumbel and von Schelling (4), it is
possible to compute the probability that an observed value
of any rank m which will be taken from the same distribution
in N future trials. It can be proved that the mean number
of exceedandes x for this m - the largest value to be

equalled or exceeded in N future trials is:

. v . .. I
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o
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1. ,.. .\ H.
v I
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u p a \ I

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p . e .
n
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a Q A a II ‘
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1 . . o .
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r . b I
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h
l,
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ID. a
u.

21

m
n + 1

 

For annual maximum values, the recurrence interval may be
defined as the time in years for N future trials that the
mth largest value in annual maxima will be equalled or ex-
ceeded once on the average. In other words, the recurrence
interval T is equal to N when the mean number of exceedance

i is equal to one. From above equation when i = 1,

 

1 = N m , and for T = N
n + 1

 

_ n + 1
T ' m

This indicates that the recurrence interval of an annual
maximum value is equal to the number of years of record plus
one and divided by the rank of the value. The above equation
is also used for computing the plotting positions of annual
minima, annual mean, and seasonal means.

In computing plotting position for annual minimum,
m is the order number when the values are arranged in in-
creasing plotting position formula, special plotting paper
which is used by U. 8. Geological Survey is adopted. For
annual maxima also, another plotting position which is known

as Hazen method, from the following formula:

F _ 100 (2m - 12 where
a “ n
F is the plotting postion in per cent

m8‘ is the rank number

n is the total number of observations
is computed and plotted on normal log paper.
On table 2 the peak flood is shown chronologically
with their rank numbers and plotting positions according to

Gumbel and Hazen formulas. The curves for these two methods

I
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5.!-

umfi “In.

I"

water
Year

1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933

Peak

Annual
Flood

1n Cafes.

4430
3000
2900
3260
5050
2600
5540
3340
5000
1100
2080
1950
1800

900
2520

800
1160
1840
2180

350
1160
2400

22

Table 2

Rank
No,

5
12
14
1o

3
17

2

8

7
50
29
31
33
43
19
46
39
32
28
47
38
22

Recurrence
Interval
in Years

Gumble

9.6
4
3.42
5.8

16

2.82

24

1.26
2.18

The Annual Peak Floods on the Cedar River
for 47 Years

Recurrence
Interval
in Years

Hagen
10.4

4.08
3.48
4.95
18.85
2.85
31.3
6.25
7.25
1.19
1.65
1.54
1.45
1.11
2.54
1.03
1.22
1.49
1.71
1.01
1.25
2.19

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23
Table 2 (continued)

Peak Recurrence Recurrence

Annual Interval Interval
Water Flood Rank in Years in Years
'1g§;__, 1n o,f.e, _§2; Qumble Hagen
1934 1340 37 1.29 1.29
1935 1800 34 1.41 1.40
1935 1020 41 1.17 1.16
1937 2330 24 2 2
1938 £020 6 8 8.55
1939 1750 35 1.37 1.37
1940 1970 30 1.60 1.59
1941 1020 42 1.19 1.14
1942 2660 16 3 3
1943 2490 20 2.4 2.4
1944 2320 25 1.92 1.92
1945 2930 13 3.69 3.76
1946 2320 26 1.84 1.86
1947 5920 1 48 91
1948 4960 4 12 13.5
1949 2580 18 2.66 2.69
1950 3170 11 4.38 4.50
1951 2240 27 1.78 1.78
1952 2740 15 3.2 3.24
1953 878 44 1.09 1.08
1954 2490 21 2.28 2.29
1955 1440 36 1.33 1.33
1956 3310 9 5-33 5.52
1957 2400 23 2.06 2.08
1958 878 45 1.06 1.06

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24

are shown on figures 2iand 3.

For the analysis of frequency of droughts, the five
consecutive days with lowest flow in each water year’were
chosen and the average of the five days was used to represent
the lowest flow of the water year. This analysis was done
for twenty-five years (1932-1958). These data are listed in
table 3, chronologically with rank numbers and plotting
positions; the drought frequency curve is illustrated on
figure 6. This curve was plotted on special probability
paper which is used by U. 8. Geological Survey for droughts
frequency curves. For the mean annual and four mean seasonal
frequency analysis, forty-seven years of stream flow was
considered. Fbr each season (spring, from April until July;
summer, from July until Octdber; fall, from OctOber until
January; winter, from January until April) the average of the
three months was used. These data were ranked and plotting
positions were determined from Gumble's formula. Five curves
were plotted for annual and four seasonal frequency distri-
bution. It was assumed that the median value has a re-
currence interval of two years. On the average, out of
every two years, one is drier and one is wetter than the
median value. The curves have each two branches. The upper
branch indicates the values higher (wetter) than normal
(median) and the lower one those drier than normal. The ob-
servations were classed in groups according to their
recurrence. values with a recurrence smaller than three

years were called "normal.” On the average, out of every

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26

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27

Table 3

Annual Drought on the Cedar River
Period 1932-1956

Average Recurrence
23;: D... 2.5 23‘3", 3’15: “2°23: 1"
1932 June 24-28 17.4 13 2
1933 July 22-26 8.4 5 5.2
1934 July 17-21 3.8 1 26
1935 August 12-16 19.2 15 1.67
1936 August 3-7 6.4 ’ 2 13.0
1937 Dec. 3-7 18.8 14 1.86
1938 July 26-30 7.8 3 8.66
1939 Sept. 26-30 10.1 7 3.71
1940 August 15-19 17 12 2.16
1941 Sept. 14-18 11.1 8 3.25
1942 Sept. 15-20 28.2 21 1.24
1943 Oct. 2-6 55.6 25 1.04
1944 Sept. 11-15 31.2 23 1.13
1945 Oct. 29-Nov. 2 20.2 17 1.53
1946 Sept. 14~18 9.1 6 4.33
1947 Oct. 4-8 15.2 10 2.6
1948 Sept. 1-5 23.8 18 1.44
1949 Sept. 4-8 16.4 11 2.36
1950 August 21-25 37.4 24 1.07
1951 Sept. 21-25 27.6 20 1.3
1952 Sept. 25-30 28.4 22 1.18

1953 Sept. 26-30 25.6 19 1.37

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Table 3 (continued)

water
lea; Date
1954 Sept. 6-10

1956 Date 1‘5

28

Average
of 5 days

in c.f.s.

12.2
8.2
19.6

Recurrence
Interval in
Years

 

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Flow in c.f.s.

 

 

 

 

 

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30

three years one is wetter than the upper limit of the
"normal" class, one is within the "normal" class and one is
drier than the lower limit of the "normal" class. Within the
group "normal” the values wetter than median were marked with
+ and those drier with - sign. Values with a recurrence
between three and ten years were classed "wet" or "dry."

Very wet and very dry have a recurrence between ten
and fifty years. The extreme values more exceptional than
once in fifty years were called "extremely wet" or "extremely
dry.” The limits obtained by this assumption from the five

frequency curves on figures 5 - 9 are shown in table 4.

 

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31

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35

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37

b - Computing Weighed Average Precipitation
for the Basin

In the Cedar basin only the East Lansing precipitation
gaging station had continuous records. Three other gaging
stations (Jackson, Milford, and Owosso), which had continuous
data and were relatively close to the basin, were also
applied.

Twenty-seven years of data were used for this study.
The following procedure for weighing the mean precipitation
of the basin was applied. A table showing runoff and pre-
cipitation in inches was prepared. Then the items in each
column were ranked in descending order. The differences of
the ranks of each precipitation station and runoff were
squared. The sum of the squares were used as relative indi-
cations of correspondence between precipitation and runoff.
The smaller the total, the higher the correspondence.

The following formula for weighing each precipitation

station was used:

PJ

3..
E
a
45+

Where P' is the weighed precipitation, d is difference in
rank, and P is the precipitation at each station.

= 1 ‘ 1

D EdeL +511?! +Sdo "' 213‘}
The sample form is shown in table 5 for annual precipitation.
The annual runoff in inches is written in the first column

for twenty-seven years, from 1932 until 1959; and in the

other four successive columns, the amount of precipitation

.

V
W
t
.

l

.

.

s ‘

Q I
t.
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38

for the same twenty-seven years at the East Lansing, Milford,
Owosso, and Jackson stations are shown.

The upper left superscripts are the rank numbers for
each column, and the upper right superscripts are the squares
of the differences between runoff rank and precipitation rank.
The sum of these squares are in the bottoms of the columns.
The last column is the weighed average precipitation for the
basin.

Other computations for annual precipitation are:

2 ._
tdL ‘ 1524 m; = .00065
2am? - .1116
2 ..
2‘11) " ‘73‘ £757 = .00058
53%? = .00066

D a .00065 + .00090 + .00058 + .00066 = .00279
The weight for each station which was derived from this
fo rmula:

1
weight 2:de2 are:
D

East Lansing .22
Milford 032
Owosso .24
Jackson .22

' Runoff - Precipitation Relation

The runoff during a given period, such as a month,

may be expressed as a function of the net rainfall after

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42

losses, abstractions, and of the carry-over from previous
runoff. Algebraically this statement might be written as
follows:

R6 = a(Po - Lb) + KRl
where Re is the runoff during a given period; L0 is the
abstraction and losses from the rainfall, El is antecedent
runoff and a, and K are constants. However no is affected
by antecedent precipitation. This effect may be expressed
as:

L0 = b - cPl
where P1 is antecedent precipitation, and b and c are seasonal
constants.

Antecedent runoff R is also correlated with antecedent
precipitation in the same way as R0, as follows:

R:= a (P - L) + k R?
where L may be expressed:

L = b - cP2 and so on. Collecting all terms,
there results: _

R = aPo + (c + ak)P1 + k(c + ak)P2 + k2(c + ak)P3+...

kn (c + ak)Pfi
since the value of k is less than unity, the value of kn may
be decreased to as small a value as may be desired, so that
the last term may be neglected.

The basic principle employed is: That stream flow
during a given.month is the resultant of all weather, currant
and past to the beginning of time.

Since precipitation is the dominant climatic force

OC

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43

this relationship may be expressed as follows:

so = k + UOPO + 01?, + UQPQ + U3P3 + ----
where Ro is the runoff during interval 0; Po is precipitation
during the interval current with R0; P1, P2, P3, are the
precipitation amounts during successive antecedent intervals,

U

0’ U1, U2, etc., are coefficients and k is constant.

This is the form of equation adopted by Hayford and
Fblse. However, it is not the only form of equation avail-
able; others involving product-terms, or power terms might
be devised to give better agreement with this or that theory.
The advantage of the above expression is its comparative
simplicity.

Methods of solution - The solution requires evaluation
of the several coefficients, Uo ----- n. There are two
means of getting the solution:

51 Graphical method
2 Analytical method

The graphical method is convenient where the correlation is
high and converges quickly on the answer. An analytical
method seems most applicable where the form of the results
cannot be anticipated or the correlation is poor. A combi-
nation analytical and graphical method was used.

In this basin, annual and four seasonal relations
were studied. For this reason only U5 and U1 were computed.
The procedure, which can be called trial and error, is as
follows: Fbr the first trial .9 and .1 were used for U0
and U1, respectively; in the second trial .8 and .2 were

used for U6 and U1. This order of decreasing U0 and

 

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44

increasing U1 was continued further.

The computed data using the above procedure was pre-
pared in tables. Table 6 exemplifies the derivation for annual
carryover precipitation. The items in each column were ranked
in descending order (highest as one). The differences in
the ranks of precipitation and runoff were squared and
totaled for each column. The column which had the smallest
sum of the squares was chosen to have the best correlation
with the runoff.

In table 6 the upper left superscripts are rank
numbers, and the upper right superscripts are squares of
differences in rank. The sums of the squares, revealing
differences in rank, are shown at the bottom of their
respective columns.

To find the correlation, the runoff is assumed to be
a function of the precipitation. That means precipitation
is an independent variable, x, runoff is a dependent vari-

able, y, and the regression coefficient bxy is:

§§UL:;&§§J_1§IJ.
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y = y +‘b(x - i) where
x is mean of x's
5; is mean of y's
and the correlation coefficient is:

r =‘b éE. where
3y

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Sx is the variance of x's
Sy is the variance of y's
The standardeerror of estimate was computed from the fOIlow-

ing equation:

- N - 1 r2 2
x — 1 - S
5y \jN_2( )y
where ng is standard error of estimate
S y is variance of the y's
r is correlation coefficient
N is number of observations

 

 

This study was carried on for the annual water year
from October 1 until September 30 and four seasons (spring,
from April until July; summer, from July until October; fall,
from October until January; and winter, from January until
April). The annual precipitation and runoff relation study
detail computations are illustrated here. In table 6 the
first column is runoff in inches. In the second column the
carry-over precipitation was computed by using .9 and .1 for
U6 and U1, respectively; the third column has values of
U6 = .8 and U, = .2; and in the fourth column Uo = .67 and
U, = .33.

The upper left superscripts are rank numbers and
upper right superscripts are the squares of the differences
between runoff rank and precipitation rank. The sum of the
squares are at the foot of the respective columns.

The fourth column has the least of the squares of
the differences between runoff and precipitation rank. The
U6 = .67 and U1 = .33 was chosen for annual carry-over co-
efficients, and column four shows the annual carry-over

precipitation Pc. The other three columns are the squares

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49

of the'runoff, squares of the carry-over precipitation, and
product of the runoff and precipitation. The rest of the
computations are:

i = 7.63
= 31.68

MI

The coefficient of regression was bxy = 1.075; the equation
of regression line was:

y = 7.63 + 1.075 (x - 31.68)
The standard deviation of precipitation was Sx = 2.92; the
standard deviation of runoff was Sy = 3.56; the coefficient

of correlation was r = .88; standard error of estimate was

Syx = 1.730

IV. DISCUSSION OF RESULTS
Flood Frequency Analysis

The data obtained from the flood frequency curves I
plotted from Hazen's (figure 2) and Gumbel's (figure 3)

formulas are shown in table 7. ;

 

 

Recurrence Gumbelfs Hazen's
Interval in Formula - Formula -
Years glow in c.;;§; Flow in cg£LgL
2 2250 2100
3 2850 2650
S 3500 3300
10 4320 4200
50 6160 6200
100 6900 7200

Table 7. Recurrence Interval of Floods

This table, prepared in accordance with Gumbel's and
Hazen's methods, shows that there is not a great difference
in the magnitude of flood calculated by the two methods.
During the forty-seven year period, the highest peak flood
of about 5920 c.f.s. was recorded in 1947.

Drought Frequency Analysis

Data obtained from figure 4, five day annual droughts

:frequency curve, are shown in table 8.

1".“ ,' )
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Mean Five Day

Recurrence Interval, Minimum Flow,
._____$E_!£§£§_____. ._J£L£hlh§m__.

2 15.5

3 11.5

8.5

10 6.0

50 3.0

100 2.2

Table 8. Recurrence Interval of Droughts
on the Cedar River

Mean Flow Frequency Analysis

The annual and seasonal frequency curves of mean flow
for forty-seven years are shown graphically in fugures 10a,
10b, and 10c.

The annual variations are shown with bold lines and

the seasonal variations are shown with hatching.

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Table 9

Classification of 47 Years of Surface Runoff
dn..:Cedar River Basin

322;: Annual Fall Winter Spring Summer
1912 Very wet Very wet +Norma1 Ext. wet -Nomal
1913 Wet +Norma1 Very wet +Normal Dry

1914 «manual Normal +Nonnal +Normal Very wet
1915 -Normal Very dry Very wet Very dry -Nomal
1916 Very wet +Normal Very wet Wet +Normal
1917 +Normal Dry -Normal Wet Very wet
1918 Wet -Nomal Ext. wet Dry Very dry
1919 Wet Very wet -Nomal Wet -Normal
1 921 +Nonnal Wet +No rmal Dry Normal

1 922 Wet Wet +No rmal Wet +Norma1
1 923 Dry -No mal -Normal Dry Dry

1924 -Nonnal ~Normal -Nonnal +Normal -Normal

1925 Very dry Very dry Dry Very dry Dry

1 926 +Nonnal +No rmal -Normal +Normal +No rmal

1 927 -Normal Wet 3Nomal -Noma1 Dry

1928 +Nonnal Wet -Normal +Nomal Wet

1929 Very wet Wet +Norma1 Very wet Wet

1 930 +No rmal +Normal Wet -Normal Dry

1931 Very dry Dry Very dry Very dry Dry

1 932 Dry -Norma1 Dry -Normal +Norma1
1 933 +No rmal Wet -Normal +Norma1 Dry

1934 Very dry +Normal Very dry Dry Ext. dry
1935 Dry Very dry -Nonnal Dry -Noma1
1936 Very dry -Normal +Normal Dry Very dry

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l v ~ 1 t
I
.' . ‘
f
\C
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Q‘ I
_ ~ I
-..
. 1 “.
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l 1 "
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1
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a .
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7
n
1
.

Table 9 (continued)

Water
Year

1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959

Annual

+Normal
Normal
Dry

Very dry
Dry
-Nbrmal
Very wet
-Norma1
+Normal
Normal
Wet

Very wet
-Norma1
Very wet
Wet

Wet

Dry
Normal
-Normal
Wet

-NOrma1

Dry

Fall

Dry

Dry

Very dry
Very dry
+Normal
+Norma1
Very wet

+Normal

DRY

. Wet

Dr?
Em. wet
-Normal
+Norma1
Wet

Wet
Normal
Dry

Wet
-NOrma1
Dry

Wet

-Normal

56

Winter

Very dry
Wet

Dry

Very dry
-Normal
+Norma1
Wet
-Normal
Very dry
Wet

Very dry
Wet
+Norma1
Very wet
Wet

Wet

Dry
+Normal
+Normal
+Normal
Dry

DPT

+Normal

..S.2.r_isa.
Wet

Dry
eNormal
Dry

Dry

Dryr

Wet
-Normal
+Normal
Dry

Wet

Wet

Wet

Wet
+Normal
+Normal
~Normal
-Normal
Dry

Very wet
Nbrmal
Very dry
Wet

Summer
eNormal
Dry
Very dry
Wet
Very dry
-Normal
Wet
Dry
+Normal
Dry
Very wet
-Normal
+Normal
Wet
-Normal
-Normal
Normal
+Norma1
Dr?

Wet
Very wet
Wet

 

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57

The years are taken as water years starting from
October 1 until September 30, the seasons are fall, winter,
spring, and summer, respectively. This classification is
also shown in table 9.

The basic assumptions for this classification were
discussed in the previous chapter, under methodology. This
classification indicates that chronologically there was a ’
general frequency of ten years wet, six years near normal, F’I
two years wet, thirteen years dry, ten years wet, six years ‘5
dry in this basin. This does not show a pattern of rotation. [9
The variations of the seasonal runoff were not regular. The ‘
runoff through the year was not well distributed.

There was only one extremely wet fall season, in 1948.
An extremely wet winter occurred in 1918, and in the spring
season of 1912 it was also extremely wet. The summer of
1934 was extremely dry.

The annual runoff for recent years seemed to be

drier than normal, Whereas summer runoff was wet.

Annual Precipitation and Runoff Relation

The weighing formula for weighed mean annual
precipitation for the basin was:
P = .22PL + .32PM + .24PO + .22PJ
The carry-over relation was:
Pc = .67Po + .33P1
where

 

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PWA _
PL =
PM =
P0 =
PJ 2
PC =
P0 =
P1 =

If we assume:

58

- Weighed average precipitation for the basin

Precipitation in East Lansing

Precipitation in Milford

Precipitation in Owosso

Precipitation in Jackson

Carry-over precipitation for the basin
Current year's weighed average precipitation

Previous year's weighed average precipitation

X as precipitation, and

Y as runoff

Mean of the runoff for 27 years was i = 7.63 inches

Mean of the precipitation ferr27 years was i = 31.68 inches

The coefficient of regression was bxy = 1.075

The equation of regression line was:

2:70

63 + 1.075 (x - 31.68)

The standard deviation of precipitation was Sx = 2.92 inches

The standard deviation of runoff was Sy = 3.58 inches

The coefficient of correlation was r = .88

The standard error of estimate was Sxy = 1.73 inches

The regression line is illustrated in figure 11.

Fall (October through December)
Precipitation and Runoff Relation

The weighing formula for weighed average of fall

precipitation for the basin was:

Pw.A.

'3 .2421, + .30 PM + .24 P0 + .22 P;

 

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Regression Line for Annual Precipitation and

Runoff on the Cedar River Basin

:Fienxrei"10

6O

carry-over relation was:
Pc = .6 Po + .4 P1
equation of the regression line was:
I = 1.01 + .371 (x - 7.40)
coefficient of correlation was r = .89
standard deviation of precipitation was Sx = 1.66 inches
standard deviation of the runoff was Sy = .69 inch

standard error of estimate was Sxy = .36 inch

Winter (January through March)
Precipitation and Runoff Relation

The weighing formula for weighed average of winter

precipitation for the basin was:

The

ngA. ‘ .24 PL + .25 PM + .30 Po + .20 PJ
carry-over relation was:

c
equation of regression line was:

I = 3.0 + .724 (x - 6.32)
coefficient of correlation was r = .71
standard deviation of the precipitation was Sx = 1.56 inch
standard deviation of the runoff was Sy = 1.57 inches

standard error of estimate was Sxy = 1.11 inches

Spring (April through June)
Precipitation and Runoff Relation

The weighing formula for weighed average of spring

precipitation for the basin was:

 

 

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61

quA. = .22 PL + .28 PM + .32 P0 + .18 PJ
carry-over relation was:
Po = P0
equation of regression line was:
I = 2.97 + .463 (x - 10.35)
coefficient of correlation was r = .76
standard deviation of the precipitation was Sx = 2.95 inch I
standard deviation of the runoff was sy = 1.78 inches g“

standard error of estimate was Sxy = 1.17 inches (
[

Summer (July through September)
Precipitation and Runoff Relation

The weighing formula for weighed average of summer

precipitation for the basin was:

Pwer = .23 PL 1' .36 PM + .23 P0 4’ .18 PJ
carry-over relation was:

P = .8 PC + .2 P1

c
equation of regression was:

I = .57 + .141 (x - 8.89)
coefficient of correlation was r = .70
standard deviation of the precipitation was 81 = 2.01 inches
standard deviation of the runoff was Sy = .40 inch

standard of error of estimate was Sxy = .26 inch

This study shows that in the Cedar River Basin, 24

per cent of the annual precipitation turns out into the run-

off. This percentage varies greatly in differentiieasons.

The percentage of this value for fall season is only 14;

 

 

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62

for winter, because of the frozen ground, this percentage
goes up as high as 48. During the spring season, because
of the frequent heavy thundershowers and low evaporation,
this valué’is 39 per cent; whereas, in the summertime the
percentage is as low as 6.5.

The runoff-precipitation analysis also illustrates
that the carry-over precipitation amount should always be
taken into account. This percentagevvalue of the carry-over
for annual precipitation in this basin is about 33, which
shows the ground water has a great influence in stream flow.
For the fall and winter seasons, the percentage values of
carry-over are about 40; whereas, this amount for summer is

20, and for spring is zero.

 

 

 

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V'. SUMMARY

A study on the Cedar River Basin was made to find
the frequency of the flood, drought, and mean flow; and to
establish a relationship between precipitation and runoff. l

Most of the data used in this study was obtained

-u — .:.
.‘

from official publications of the U. 8. Geological Survey
and U. 5. Weather Bureau. Eight years of daily discharges ‘W
from 1912-1918 were computed by the author. I
A flood frequency analysis was derived by Hazen's
and Gumbel's methods. The 100 year flood by Hazen's and
Gumbel's procedure for East Lansing was 7200 and 6900 cubic
feet per second, respectively. The 100 year drought ob-
tained from mean five day of annual minimum flow was 2.2
cubic feet per second.
The mean.flow frequency analysis did not show a
regular pattern of cycles of wet and dry years. There was
no extreme annual mean flow, but seasonal variations had a
large range and there were several extremely wet and dry
seasons. '
The precipitation and runoff study shows that 24 per
cent of the annual precipitation of the basin goes to the
Cedar River.
To establish the precipitation-runoff relationship,

the annual and four seasonal precipitation of the four

64

stations, East Lansing, Milford, Owosso, and Jackson were
weighed to obtain a mean annual and mean seasonal precipi-
tation for the basin. It was assumed that because of the
contribution of the groundwater to the runoff, a time lag
for precipitation should be taken into account. A percent-
age of the previous years or previous seasodh precipitation
was considered to secure annual or seasonal carry-over
precipitation values. The regression line equations between
the annual and four seasonal precipitation and runoff were
established.

The precipitation and runoff in the basin had a
relatively high correlation. The correlation coefficient was
0.88 for the annual relationship, 0.89 for fall, 0.71 for

winter, 0.76 for spring, and 0.70 for summer.

1
no
.
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VI. CONCLUSION

The frequency analysis shows that the flow of the
river is usually highest during the late winter when a combi-
nation of melting snow, rather heavy rains, and not yet
thawed ground occasionally brings about serious floods. The
lowest flows are found in the late summer prior to fall
precipitation. The 50 and 100 year peak floods are about
6200 and 7200 cubic feet per second, respectively. The 50
and 100 year five days mean droughts are 3 and 2.2 cubic
feet per second, respectively. The annual median flow of
195 cubic feet per second, has a two years recurrence
interval.

By using the results of the precipitation-runoff
relationship, missing runoff data for a certain year or
season on the Cedar River could be computed. The securing
of missing data could be accomplished by the following
procedure:

1. The mean precipitation for the basin, for a

certain year or season which runoff data
is needed, should be weighed by using the
precipitation of the four stations, East
Lansing, Milford, Owosso, and Jackson, in

the proper equation.

 

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2.

66

The weighed mean precipitation or the basin
should be put in the carry-over precipitation
formula to obtain carry-over precipitation.
If the carry-over precipitation value is
substituted for x in the regression line
equation, the value of y will be obtained

for runoff.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

VII. REFERENCES

Beard, L. R. (1943). Statistical Analysis ip,gzdrologp,
Transactions American Society of Civil Engineers, V.

108, pp. 1110-1160.

Brehmer, Morris L. (1956). ;g Biological and Chgpicgl \

Survey p£_the Red Cedar giver pp the Vicipity g; t
Williamston, Michigan. M. S. Thesis, Department of H
Fisheries and Wildlife, Michigan State university. a

Garter, R. w. and M. R. Williams (1949). water L

Resources and gxdrolpgp g; Southeastern Alabama,
Geological Survey of Alabama Special Report 2.

Chow, v. T. (1953). Ereguencz gpalzsis g; Hydrologic
Data with Special Application pp_Rainfal;

Intensities, University of Illinois Eng. Exp. Sta.
Bul.

Finch, Vermor C. (1957). Phxsicp; Elm entg of Qgpg_appz,
4th ed., McGraw Hill Book Co., New York, €41 pp.

Foster, H. A. (1924). Theoretical Freguencp Cungs
Egg Their Application pp_§pg;pggpgpg,Problems,
Transactions American Society of Civil Engineers,
V. 87, pp. 142-203.

Foster, H. A. (1934). Duration Cu es, Transactions
American Society of Civil Engineers, V. 99. Pp.
1213-1235.

Gumbel, E. J. (1954). Statistical Theog§ 9;,ppppgppp,
PrOCI Amer. SOC. of CiVil Engin.g V. 0, September,
No. 439. 19 pp.

Hardison, C. H. and E. B. Rice (1945). Natural water
Lpsses from Selected Dppinage Basins in Alabama,
Geological Survey of Alabama Bulletin 56.

Hill, E. B. and R. G. MaWby (1954). szgg of E§__ggg
in Michi an, Michigan State College Agri. Exp.
Sta. SpeC. Enll. NO. 206, 80 pp.

Linsley, R. K., M. A. Kohler, and J. L. H. Paulhus

(1949). Applied Hydrology, McGraw-Hill Book Co.,
New Ybrk.

 

 

:0

9 out.

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(12) (1945). Oklahoma water, Oklahoma Planning on
Research Board, Water Resources, U. S. Geological
Survey.

(13) Whiteside, E- P. (1959). 50118 9; M1 hi an, 14101118811
State University, Agri. Exp. Sta. Spec. Bull. No.
402, 52 pp.

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