Model Synthesis in Frequency 

Analysis of Missouri Floods 

CIRCULAR 708 

issouri State Highway Department and 

-- ....L..L...;l.~~..~..~..~..t:...l.l...l..ll.....J..L.i;J.L.~..:>U~..LU..o;U,Iun, Federal Highway Administration ~-~ 

Model Synthesis in 

Frequency Analysis of 

Missouri Floods 

By Leland D. Hauth 

GEOLOGICAL SURVEY CIRCULAR 708 

Prepared in cooperation with the Missouri State Highway Department and 

U.S. Department of Transportation, Federal Highway Administration 

1974 
United States Department of the Interior 

ROGERS C. B. MORTON, Secretary 

Geological Survey 

V. E. McKelvey, Director 

Free on application to the U.S. Geological Survey, National Center, Reston, Va. 22092 
CONTENTS 

Page 

Abstract ---------------------------------------------------------------- 1 

Introduction ------------------------------------------------------------ 1 

AcknowledgTnents --------------------------------------------------- 2 

Ddodel description -------------------------------------------------------- 3 

Nlethods of study -------------------------------------------------------- 4 

Model calibration -------------- _ _____ ____ _________ _ ___ _ _________ _____ 4 

Basic data ------------------------------------------------------ 4 

Parameter evaluation _______________ ------------- __ ------------- _ 4 

Synthesis of flood records -------------------------------------------- 7 

Processing rainfall records --------------------------------------- 7 

Processing evaporation records ----------------------------------- 7 

Defining synthetic flood frequencies ------------------------------------ 8 

Regional regression analysis ------------------------------------------ 9 

Discus~on -------------------------------------------------------------- 12 

Parameter evaluation ------------------------------------------------ 14 

Record synthesis ---------------------------------------------------- 14 

Regression analysis -------------------------------------------------- 15 

Summary and conclusions ------------------------------------------------ 15 

References -------------------------------------------------------------- 15 

ILLUSTRATIONS 

Page 

FIGURE 1. Map showing location of gaging stations where basic rainfallrunoff and precipitation data have been collected in Mis.souri __ 6 

2. Graph showing flood-frequency curves for Thompson Branch near 

Albany, Missouri ------------------------------------------ 10 

3. Graphical solution of the 2-year equation ----------------------- 13 

TABLES 

Page 

TABLE 1. The 10 model parameters and their applications in the modeling 

process --------------------------------------------------- 3 

2. Gaging stations for which the model was calibrated ------------- 5 

3. Summation of mod·el parameters ------------------------------- 8 

4. U.S. National Weather Service first-order precipitation stations 

used in synthesis of peak-flow data -------------------------- 8 

5. Station frequency data --------------------------------------- 11 

6. Summary of regression analyses ------------------------------- 12 

III 
FACTORS FOR CONVERTING ENGLISH UNITS TO 

INTERNATIONAL SYSTEM (SI) UNITS 

The following factors may be used to convert the English units published herein to 

the International System of Units (SI): 

Multiply English units 

Feet (ft) -------------------------

Square mile (mi2) -----------------

Miles (mi) -----------------------

Feet per mile (ft/mi) -------------

Cubic feet per second (fe/s) ______ _ 

By 

0.3048 

2.590 

1.609 

.189 

.02832 

IV 

To obtain Sf units 

Metres (m). 

Square kilometres (km2). 

Kilometres (km). 

Metres per kilometr.e (m/km). 

Cubic metres per second (m3 /s). 
Model Synthesis in Frequency Analysis of Missouri Floods 

Bv LELAl\'D D. HAuTH 

ABSTRACT 

Synthetic flood records for 43 small-stream sites 

aided in definition of techniques for estimating the 

magnitude and freqeuncy of floods in Missouri. The 

long-term synthetic flood records were generated by 

use of a digital computer model of the rainfall-runoff 

process. A relatively short period of concurrent rainfall and runoff data observed at each of the 43 sites 

was used to calibrate the model, and rainfall records 

covering from 66 to 78 years for four Missouri sites 

and pan-evaporation data were used to generate the 

synthetic records. 

Flood magnitude and frequency characteristics of 

both the synthetic records and observed long-term flood 

records available for 109 large-stream sites were used 

in a multiple-1egression analysis to define relations 

for estimating future flood characteristics at ungaged 

sites. That analysis indicated that drainage basin size 

and slope were the most useful estimating variables. 

It also indicated that a more complex regression model 

than the commonly used log-linear one was needed for 

the range of drainage basin sizes available in this 

study. 

INTRODUCTION 

This report describes technical aspects of a 

study to define methods for estimating the 

magnitude and frequency of future floods on 

Missouri streams. A significant part of the 

analytical procedure included using a rainfallrunoff model to extend short-term records of 

floods on small streams. The synthetic flood 

records generated by use of long-term climatic 

data in the model were then used with available flood records on large streams to define 

suggested relations for estimating future flood 

magnitudes at ungaged sites. A companion report by Hauth (1974) described the defined 

estimating relations and provided details on 

how to use the relations to obtain a design estimate, and two other reports by Hauth (1973a, 

1973b) presented basic data. The purpose of 

1 

this report is to document the analytical methods used. 

The economic design of drainage structures, 

the equitable regulation of flood-prone lands, 

and the establishment of realistic flood insurance rates require estimates of the magnitude 

and frequency of floods in the future. Many 

techniques have been proposed for estimating 

the magnitude and frequency characteristics 

of future floods, and the most reliable ones are 

based upon an assessment of past records of 

observed floods. In general, the longer the period of observed flood records at a site, the more 

reliable the future flood characteristics can be 

estimated at that site. To obtain estimates of 

future flood characteristics at ungaged sites, 

techniques are available for transfer of flood 

information from gaged sites. The accuracy of 

these estimates at ungaged sites also is related 

to the accuracy of the estimates at gaged sites 

and, therefore, to the length of observed flood 

records. 

Flood magnitudes have been observed systematically in Missouri by the U.S. Geological 

Survey and other Federal and State agencies 

since 1903. A large mass of flood data has been 

accumulated, but these long-term data observations have been confined primarily to the large 

streams. A study by Sandhaus and Skelton 

(1968) defined methods of estimating flood 

magnitude and frequency, but, because of the 

short period of records available on small 

streams, the technique was considered applicable only for estimates on streams draining 

more than 50 mP (130 km2 ). In a study of the 

adequacy of available Missouri streamflow records, Skelton and Homyk (1970) determined a 

need for a longer period of flood records on 

drainage areas of less than 50 mi2 (130 km2 ). 
One suggested alternative to awaiting the 

assen1bly of long-term flood records on small 

streams is to use available long-term climatic 

records in a rainfall-runoff model to estimate 

long-term flood records. The runoff process is 

a complex phenomena. A mathematical model 

allowing a reasonably accurate description of 

the process is also complex. It was not until the 

advent of the digital computer that the use of 

mathematical models describing the rainfallrunoff process became practical. 

The rainfall-runoff model used in this study 

w~s developed by the Geological Survey specifically for the purpose of modeling floodrunoff hydrographs from small watersheds. 

Details of its computational routine, necessary 

assumptions, and reliability have been described by Dawdy, Lichty, and Bergmann 

(1972). Primary characteristics of this model 

are that it is deterministic, structure imitating, 

digital, and relatively simple. Only 10 parameter values must be evaluated to calibrate the 

model to any specific drainage site. The parameters were conceived as real and measurable 

characteristics of the contributing watershed, 

but in practice the parameters are evaluated 

from a period of concurrent precipitation and 

runoff records for each site by an internal and 

automated optimization routine. 

Information available in Missouri offered an 

ideal situation for testing the use of the rainfall-runoff modeling approach to defining floodfrequency estimating relations. Observations of 

concurrent rainfall and runoff began on a few 

small-stream sites as early as 1948, and the 

data collection program was enlarged and continued until, at the time of this study, an average of 10 years of observations were available 

for 43 sites that drain from 0.10 to 10 mi~ 

(0.26 to 25.9 km~) located in a variety of topographic settings. Continuous-recorder charts 

of precipitation averaging 70 years in length 

have been obtained by the National Weather 

Service at four areally dispersed sites, and 

sufficient pan-evaporation data were available 

to provide long-term climatic inputs needed for 

model synthesis. Long-term flood observations 

on larger streams draining up to 14,000 n1i~ 

(32,300 km:!) were already available for analysis. 

Flood-frequency characteristics defined from 

the 43 synthesized small-stream records and 

2 

the observed long-term records on large 

streams were used as dependent variables in a 

multiple-regression analysis to define relations 

for estimating flood characteristics at ungaged 

sites. The results of this analysis showed that 

drainage area and main-channel slope are the 

most effective indices for estimating floodflows 

in Missouri, but also indicated the inadequacy 

of the assumption of a simple linear relation 

between the logarithms of flood characteristics 

and the logarithms of basin characteristics. 

Such an assumption is commonly used nationwide and was found satisfactory for a limited 

range of basin sizes in Missouri. An alternate 

curvilinear model form suggested by Creager, 

Justin, and Hinds (1947) was found satisfactory for use with the broad range of basin 

sizes. 

The study described in this report is the first 

operational use of the rainfall-runoff model in 

defining flood-frequency estimating techniques. 

The experiences gained are documented as an 

aid to others who may be planning similar investigations. This study involves three distinct 

operations that are described first. They are as 

follows: 

1. Model calibration, or the evaluation of 

model parameters, for each of the 43 

small-stream sites. 

2. Synthesis of long-term flood-peak records. 

3. Regression of flood characteristics on basin 

characteristics. 

To remain within the scope of this study, and 

within bounds of the time and resources available, it was sometimes necessary to choose 

among alternative courses and to ignore some 

potential useful choices. Some of those potential useful choices where additional research 

may be beneficial are discussed in a subsequent 

section. 

The opinions, findings, and conclusions expressed in this report are not necessarily those 

of the Federal Highway Administration. 

ACKNOWLEDG~IENTS 

This report was prepared under the cooperative highway research program with the 

State of Missouri, Robert N. Hunter, Chief 

Engineer, and the U.S. Department of Transportation, Federal Highway Administration. 

Long-tern1 daily precipitation records and 

storm reductions to 5-nlinute increments were 
obtained from the National Oceanic and Atmospheric Administration (NOAA), Environmental Data Service, National Climatic Center, 

at Asheville, N.C. 

R. W. Lichty and P. H. Carrigan, Jr., hydrologists of the U.S. Geological Survey, provided valuable technical advice and assistance 

during the preparation of the report. 

l\IODEL DESCRIPTION 

The model estimates the flood hydrograph 

that a drainage basin will produce in response 

to a specific rainfall sequence. It utilizes a set 

of mathematical relations approximating the 

physical laws governing three components of 

the hydrologic cycle-antecedent soil moisture, 

infiltration, and surface-runoff routing. 

The antecedent soil-moisture component continually assesses the changes in soil moisture 

as a basis for determining the portion of subsequent rainfall that becomes surface runoff. 

The changes in moisture storage are assessed 

on a daily basis during nonstorm periods and 

on a unit-time basis during storm periods. For 

each site, a unit-time increment of 5, 10, or 15 

minutes is selected so that it provides a suitable degree of refinement in the computations 

and in the synthetic output hydrograph. 

The infiltration component of the model determines the portion of a unit-time rainfall 

observation that becomes surface runoff. In the 

surface-runoff component, this rainfall excess 

is then distributed in time to form the outflow 

hydrograph by using a modified Clark routing 

method. 

Ten parameter values control the operation 

of the model and must be evaluated to calibrate 

the model to a specific drainage site. These 

parameters are constants and limits in the 

mathematical relations, and table 1 lists them 

and describes the conceptual function of each. 

The model contains a trial-and-error, hillclimbing type of parameter optimization technique. The general method is that a set of 

assumed parameter values is first used to synthesize a flood record for a period of time when 

concurrent rainfall, pan-evaporation, and flood 

records are available. The optimization scheme 

compares the synthesized record with the observed flood record, then automatically changes 

one parameter value, repeats the computations, 

and compares the newly synthesized record 

with the observed record. The process of individually changing parameter values and computing a new synthesized record is repeated 

until the comparison of synthesized and observed records cannot be improved. The range 

of values any parameter can assume is limited 

by the investigator. 

Synthetic and observed flood records are 

compared by means of an objective function. 

This function is the sum of squares of logarithms of differences between observed and 

synthesized values of the peak flows, or of the 

storm volumes, or of a combination of the peak 

and volume. Thus, the parameter-evaluation 

TABLE 1.-The 10 model parameters and their applications in the modeling process 

Parameter 

EVC ----- RR ------ BMSM --- DRN 

PSP 

KSAT ___ _ 

RGF ____ _ 

Units 

Inches ----------

Inches ----------

Inches per hour _ 

Definition and application 

Antecedent moisture component 

Coefficient to convert pan evaporation to pot.ential evapotranspiration values. 

Proportion of daily rainfall that infilt1·ates the soil. 

Soil moisture storage volume at field capacity. 

Drainage parameter for redistributi.on of soil moisture (fraction of KSAT). 

Infiltration component 

Product of moisture deficit and suction at the wetted front for soil moisture at 

field capacity. 

The minimum (saturated) hydraulic conductivity used to determine infiltration 

rates. 

Ratio of the product of moisture deficit and suction at the wetted front for 

soil moisture at wilting point to that at field capacity. 

Surface runoff component (routim{) 

KSW ____ Hours ---------- Time characteristic for linear resorvoir routing. 

TC ------ Minutes ________ Lt-ngth of the ba~e of the triangular translation hydro·graph. 

TP/TC ___ ----------------- Ratio of time to peak to base length of the triangular translation hydrograph. 

3 
procedure develops a nonlinear least-square 

solution to the optimum set of parameter 

values. 

Once the parameters are evaluated for a specific drainage site, a long-term record of precipitation and pan evaporation may be put into 

the model to synthesize a flood record. 

Many assumptions are required to calibrate 

and use the rainfall-runoff model. Some of the 

more important ones are : 

1. Uniform distribution of rainfall over the 

drainage area. 

2. Accurate input data. 

3. Runoff unaffected by ice or snowmelt. 

4. No changes have destroyed the homogeneity 

of flood records during the period used in 

model calibration. 

5. Available long-term rainfall and evaporation records are applicable to the drainage basin. 

To the extent possible, measuring techniques 

and operational procedures are designed to 

minimize the effect of deviations from these 

assumptions. An obvious limitation is the adequacy of a single rain gage for representing 

the areal distribution of rainfall on a drainage 

basin. Most floodflows on small drainages in 

Missouri and other Midwestern States are produced by localized high-intensity thunderstorms. The variability of these storm rainfalls 

over a drainage basin is probably a function of 

the ratio of storm size and movement to size of 

drainage basin. To limit the effects of the assumption of uniform areal rainfall distribution, 

the model was calibrated only for basins draining less than 10 mi 2 (25.9 km2 ). 

~IETHODS OF STUDY 

The analysis included three distinct operations. The first was to evaluate the model 

parameters through a calibration process. The 

second was to synthesize a long series of annual 

flood peaks from long-term rainfall records and 

to define the flood-frequency curves. The last 

was to define by multiple-regression analysis 

the relations between the 2-, 5-, 10-, 25-, 50-, 

and 100-year flood magnitudes and drainagebasin characteristics as a basis for flood frequency estimates at ungaged sites. 

4 

MODEL CALIBRATION 

BASIC DATA 

Calibration of the rainfall-runoff model to 

any specific site requires concurrent data on 

streamflow, precipitation, and evaporation. 

Sufficient data were available for calibrating 

the model for 43 sites with drainage areas 

draining less than 10 mP (25.9 km2 ). Table 2 

lists in downstream order the 43 sites together 

with their drainage areas, caHbration period, 

time increment, and number of storms used in 

calibration, and figure 1 shows the location of 

the sites. 

In the extreme southeast part of the State, 

no rainfall-runoff data have been collected because of the inability to define stage-discharge 

relationships of small drainage areas. However, flood-frequency estimating relations presented later in this report are based upon information from this region for drainage areas 

greater than 10 mi 2 (25.9 km2 ) so that the 

estimating relationships have statewide applicability. 

All stage and rainfall records used in this 

study were recorded on strip charts. One rain 

gage was located in each basin. Two types of 

rain-gage records were used: (1) a tippingbucket gage that recorded to the nearest 0.1 

inch (2.5 mm) synchronously with the stagerecorder gage and on the same chart, and (2) 

a weighing-type gage placed somewhere within 

the basin, not necessarily at the stream gage, 

and recorded on a separate chart. 

PARAMETER EVALUATIO~ 

The parameter evaluation was accomplished 

in three phases. During phase one, the parameters shown in table 1 as controlling the volume of runoff (PSP, KSAT, DRN, RGF, 

BMSM, EVC, RR) were automatically adjusted until the volumes of observed runoff 

were reproduced with m1n1mum variance 

(minimum objective function). These parameters were then held constant through phase 

two, in which the parameters controlling the 

shape of the synthetic hydrograph (KSW, TC, 

TP /TC) were readjusted until the observed 

peaks were reproduced with minimum variance. In phase three, routing parameters were 

held constant and volume parameters were readjusted until the best fit of observed versus 
Station 

number 

05495100 __ _ 05497700 __ _ 05502700 __ _ 05503000 __ _ 06815550 __ _ 06816000 __ _ 06820000 __ _ 06821000 __ _ 06896180 __ _ 06896500 __ _ 06897700 __ _ 06902500 __ _ 06902800 __ _ 06907200 __ _ 06908300 __ _ 06909700 __ _ 06910250 __ _ 06918700 __ _ 06919200 __ _ 06921740 __ _ 0692.2700 __ _ 06925200 __ _ 06925300 __ _ 06926200 __ _ 06927100 __ _ 06928200 __ _ 06931500 __ _ 06935800 __ _ 07011200 __ _ 07011500 __ _ 07015500 __ _ 07017500 __ _ 07019100 __ _ 07035500 __ _ 07037700 __ _ 07053950 __ _ ()7054200 __ _ 07054300 __ _ 07063200 __ _ 07064300 __ _ 07064500 __ _ 07068200 __ _ 07185500 __ _ 

TABLE 2.-Gaging stat-ions for which the model 'Was calibrated 

Station name 

Big Branch Tributary nr Wayland, Mo ---------------

Bridge Creek nr Baring, Mo ------------------------- Easdale Branch nr Shelbyville, Mo ------------------- Oak Dale Branch nr Emden, Mo ---------------------- Staples Branch nr Burlington Junction, Mo - ---------

Mill Creek nr Oregon, Mo --------------------------- White Cloud Creek near Maryville, Mo --------------- Jenkins Branch nr Gower, Mo ----------------------- DeMoss Branch nr 8tranberry, Mo ------------------- Thompson Branch nr Albany, Mo --------------------- Grand River Tributary nr Utica, Mo ----------------- Hamilton Branch nr New Boston, Mo ----------------- Onion Branch nr St. Catherine, Mo ------------------- Shaver Creek Tributary nr Clifton City, Mo ----------- Trent Branch nr Waverly, Mo ----------------------- Petite Saline Creek Tributary nr Bellair, Mo __________ _ 

Traxler Branch nr olumbia, Mo ---------------------- Oak Grove Branch nr Brighton, Mo ------------------ Sac River Tributary nr Caplinger Mills, Mo ---------- Brushy Creek nr Blairstown, Mo --------------------- Chub Creek nr Lincoln, Mo --------------------------

Starks Creek nr Preston, Mo ------------------------ Prairie Branch nr Decaturville, Mo ------------------ Van Cleve Branch nr Meta, Mo ---------------------- Doane Branch nr Kingdom City, Mo ------------------ Laguey Branch nr Hazelgreen, Mo ------------------- Little Beaver Creek nr Rolla, Mo --------------------- Shotwell Creek nr Ellisville, Mo ---------------------- Love Creek nr Salem, Mo ---------------------------- Green Acre Branch nr Rolla, Mo --------------------- Lanes Fork nr Rolla, Mo ---------------------------- Dry Branch nr Bonne Terre, Mo --------------------- Murphy Branch nr Crystal City, Mo ----------------- Barnes Greek nr Fredericktown, Mo ------------------ Clark Creek nr Piedmont, Mo ------------------------ Ingentron Hollow nr Forsyth, Mo -------------------- Yandell Branch nr Kirbyville, Mo -------------------- Gray Branch nr Lutie, Mo ---------------------- ____ _ Pike Creek Tributary nr Poplar Bluff, Mo ___________ _ 

Fudge Hollow nr Licking, Mo ------------------------ Big Creek nr Yukon, Mo ---------------------------- North Prong Little Black River nr Hunter, Mo _______ _ 

Stahl Creek nr Miller, Mo ---------------------------

Drainage area 

(mi2 ) 

0.70 

2.54 

.71 

2.64 

.49 

4.90 

6.06 

2.72 

.38 

5.58 

1.44 

2.51 

1.04 

1.65 

.97 

.49 

.55 

1.30 

.14 

1.15 

2.86 

1.30 

1.48 

.75 

.54 

1.58 

6.41 

.81 

.89 

.62 

.22 

3.35 

.45 

4.03 

4.39 

.65 

.42 

.27 

.28 

1.72 

8.36 

1.23 

3.86 

Period of continuous 

records 

(yrs) 

1964-69 

1961-64 

1969-70 

1956-69 

1967-69 

1951-70 

1949-70 

1950-70 

1959-65 

1955-70 

1966-70 

1955-70 

1961-66 

1961-64 

1959-62 

1964-69 

1960-64 

1956-70 

1962-70 

1960-70 

1964-70 

1956-70 

1969-70 

1956-70 

1959-62 

1958-70 

1948-70 

1962-70 

1959-70 

1947-70 

1952-70 

1955-70 

1960-64 

1955-70 

1956-70 

1962-66 

1966-70 

1959-62 

1959-64 

1959-70 

1949-70 

1964-70 

1950-70 

Number Unit time of storms increment used in (min) calibration 

5 

5 

5 

15 

10 

15 

15 

15 

5 

15 

5 

15 

10 

5 

5 

5 

5 

10 

5 

10 

15 

15 

10 

10 

5 

15 

15 

10 

5 

5 

5 

15 

5 

15 

15 

5 

5 

5 

5 

5 

15 

5 

15 

16 

15 

18 

25 

8 

30 

28 

29 

12 

28 

24 

28 

11 

13 

15 

16 

14 

18 

12 

31 

24 

13 

18 

12 

13 

18 

14 

30 

9 

27 

19 

29 

8 

26 

23 

9 

12 

11 

11 

6 

29 

18 

25 

simulated peaks was achieved. Parameters determined in phase three were used in the synthesis of long-term peak discharges. 

Initial magnitudes of each parameter were 

selected on the basis of the watersheds' climate, 

geology, soil, and basin cover. Constraints were 

placed on the range in which some initial parameters were allowed to vary during calibration. For example, Kohler, N ordenson, and 

Baker (1959) computed an average panevaporation coefficient for Missouri of about 

0. 7; therefore, this parameter was constrained 

to keep its magnitude within a range of 0.5 to 

0.9. Initial magnitudes of routing parameters 

were determined from observed discharge hydrographs, and constraints were placed on 

these magnitudes to keep them within a range 

consonant with experience at each location. The 

range in which the remaining parameters were 

allowed to vary during calibration was set 

to allow the best fit of a plot of the final observed versus simulated peak discharge. 

5 

An important step in the operational procedure of parameter evaluation was a screening 

process to identify and eliminate those storms 

having large errors in the recorded data. All 

storms were included in the first phase one and 

phase two computations, and the recorded 

storm precipitation was compared with the 

observed and simulated runoff hydrographs for 

each storm. Storms then were eliminated from 

subsequent calibration work if the runoff vol-
~155~0 ~6820ooo -- ---- -------r----05495100 

06896180 0 06896500 0 0 

~ 0 05497700 

'~,, 0 06816000 .. 

"'- 069025000 J ',? 006821;:8977000 069028000 ;: :: 

-., \ Cf.{; l w -13988 - (;:f"Co'~-, 'I 0 = ~ (F ~b ,, "=11.;____;;-J "'~~:=lf· --c> • 

1

-1 

- 06908300 ! W-93958 •'- I 

------------- o a·---... 

06909700-;-- 0 06927100 •• ~ .• ~ 

06910250 06935800 I ' 

0 06907200 

006921740 

06922700 

0 

06919200 

0 

06925200 

0 

0 

06925300 

--~\ -~ --=-------= ___ .-:---"' 

0069262.· . 

• 07015500 07017500 

06931500~ 0 

)f07011500 

006928200 )~ 07011200 

07019100 

• • 006918700 0 07064300 

07035500 

0 

r

: .o 0 

~ 0 ~ f 07064500 

1 07185500°"'\ Aj o ' PA,f? 

0 07037700 

0 

1 °• _/ W-13995 °• 

PA.Rr 

~ 

7 0 0 

07053950 

0 

07068200° 007063200 

I 070542000 07054300 

L-------------~----------------, 

0 50 100 

0 50 100 150 

150 MILES 

200 Kl LOM ETRES 

I " 

' ) 

[ __ 

EXPLANATION 

0 07054200 

Rainfall-runoff stations used in model calibration 

0 

Downstream order number 

U.S. National Weather Service precipitation station Major Basin divide 

35·~------~------------L-__________ _L __________ _L __________ ~-----------~-------------~ 

FIGURE 1.-Location of gaging stations where basic rainfall-runoff and precipitation data have been collected in Missouri. 

ume exceeded the precipitation, if the precipitation exceeded the runoff by an abnormal 

amount, or if other obvious data problems appeared. The evaluation of parameters by phase 

one and phase two was then repeated as many 

times as necessary on the reduced set of storms. 

Although the screening process was somewhat 

6 

subjective, the elimination of these stormsfor which recorded precipitation obviously was 

not representative of basin precipitation-was 

considered essential to establishment of the 

optimum set of parameter values for model 

calibration. 
A final set of parameter values was determined from the screened storm data by a phase 

three computation (readjustment of volume 

parameters holding phase two routing parameters constant). A plot was then prepared of 

observed peaks versus synthesized peaks and 

was used to graphically check for possible bias 

in calibrated model results. The average standard deviation of differences between observed 

and synthesized peaks was about 35 percent. 

Model calibration was completed for each 

station in downstream order (ascending station numbers). Experience gained early in the 

process allowed more efficient calibration of 

downstream stations and perhaps allowed 

parameters to be defined with more confidence 

and meaning. Table 3 lists the stations in 

downstream order and the best-fit parameter 

values. 

SYNTHESIS OF FLOOD RECORDS 

A calibrated model can be used to estimate 

the flood peak resulting from any selected 

storm rainfall and associated antecedent moisture conditions; therefore, a long record of 

rainfall and evaporation is needed to generate 

a long series of synthetic flood peaks. Four 

long-term, continuously recording rainfall 

gages have been operated in Missouri by the 

National Weather Service. These records provide the basic rainfall data for model input and 

are identified in table 4. 

One relatively long record of daily pan 

evaporation collected at Lakeside, Mo., near 

Lake of the Ozarks, was processed to provide 

needed evaporation inputs. 

PROCESSil\'G R.-\ll\'FALL RECORDS 

Model synthesis requires records of daily 

precipitation for nonstorm periods and unit 

precipitation for storm periods. Daily precipitation records for each of the four sites were 

available from the National Climatic Data 

Center, Asheville, N.C., and were processed 

and placed in computer storage. 

Unit precipitation information could be attained only by coding data from original recorder charts, a tedious, relatively time consuming, and therefore expensive, process. To 

provide a realistic basis for synthesizing an 

annual series flood record, unit data must be 

7 

available for all storms that might conceivably 

produce the largest peak flood during any 

water year (October 1 to September 30). To 

minimize costs, however, it was desirable to 

limit the number of storms selected. R. W. 

Lichty (written commun., 1973) has studfed 

the proble1n of identifying those storms likely 

to cause an annual maximun1 flood and the 

effect of the number of identified storms on the 

resulting frequency curves. For basins with lag 

times of about three-quarters of an hour, he 

concluded that three storms each year identified as the maximum daily or 2-day total rainfalls would produce a frequency curve not significantly different fron1 one based on unit data 

for a great many n1ore storms. For basins with 

lag tin1es of less than three-quarters of an 

hour, Lichty suggested that the number of 

storms identified for reduction to unit data 

should be increased to four or possibly five per 

year. 

A computer program supplied dates of the 

storms that would possibly cause the maximum 

annual flood peak by identifying the five largest daily or 2-day total rain eatches in each 

water year in each of the dai1y precipitation 

records. A reduced number of dates was then 

selected for the reduction of rainfall recorded 

charts to unit data by considering for each 

storm the antecedent rainfalls, the comparative 

total rainfalls, the hourly intensities, and similar available information. Table 4 shows the 

number of storms in each record for which the 

National Climatic Data Center, NOAA, of 

Asheville, N.C., prepared computer inputs of 5-

minute unit precipitation data. 

All the information on daily and unit precipitation for each of the four long-term raingage sites was presented in a report by Hauth 

(1973a). 

PROCESSIXG EVAPORA TIO~ RECORDS 

A daily evaporation record is required for 

model synthesis. The longest available record 

in Missouri is for the LakeBide, Mo., panevaporation gage located at Lake of the Ozarks, 

which began operation in February 1948. 

A computer program developed within the 

U.S. Geological Survey analyzed the available 

evaporation record to determine a harmonic 

(sine-cosine) function, which then generated a 
TABLE 3.-Summation of model parameters 

Station Parameters (see table 1 ) 

number PSP KSAT DRN RGF BMSM EVC RR KSW TC TP/TC 

05495100 --------- 4.17 0.051 0.63 16.6 4.73 0.70 0.84 1.22 87.5 0.51 

05497700 --------- 1.45 .028 .29 17.8 6.12 .67 .88 1.50 226 .20 

055027,00 --------- 1.62 .027 .16 4.86 37.8 .66 .84 .89 63.0 .93 

05503000 --------- 1.39 .037 .26 18.8 2.97 .59 .70 1.92 69.6 .40 

06815550 --------- 1.27 .030 .10 10.1 10.8 .67 .76 .76 42.0 .64 

06816000 --------- 3.85 .096 .22 10.6 4.77 .67 .79 .74 6·6.0 .49 

06820000 --------- 2.40 .050 .72 10.5 1.2·0 .76 .77 2.00 122 .55 

06821000 --------- 1.46 .067 .77 13.6 5.16 .72 .70 .87 52.0 .95 

06896180 --------- 2.19 .069 .96 8.50 4.84 .78 .66· .90 33.0 .20 

06896500 --------- 1.15 .070 .36 9.09 2.12 .62 .83 2.50 118 .80 

06897700 --------- 3.7 .040 .18 9.33 5.47 .66 .76 2.30 94.5 .74 

06902500 --------- .98 .020 .09 3.84 3.59 .63 .79 2.00 94.5 .84 

06902800 --------- 3.48 .030 .29 13.5 8.35 .74 .70 1.38 78.0 .98 

06907200 --------- 2.56 .040 .15 2.92 2.25 .67 .84 1.27 67.0 .87 

06908300 --------- 2.06 .070 .63 12.8 12.2 .71 .84 .34 54.0 .97 

06909700 --------- 8.64 .21 .76 4.43 3.40 .72 .90 .60 49.0 .84 

06910250 --------- 1.94 .050 .22 13.6 5.52 .71 .77 .35 55.0 .85 

06918700 --------- 1.99 .060 .31 27.2 4.12 .62 .68 1.23 26.0 .78 

06919200 --------- 2.52 .060 .84 14.8 2.43 .77 .84 .64 85.9 .36 

06921740 --------- 1.92 .050 .33 5.67 5.13 .70 .68 .80 30.0 .80 

06922700 --------- .60 .02 .17 2.62 14.8 .72 .73 1.69 19.2 .67 

06925200 --------- 2.57 .05 .50 12.2 4.8 .69 .85 2.31 126 .50 

06925300 --------- 4.48 .14 .51 15.8 7.14 .69 .74 1.36 34.2 .68 

06926200 --------- 2.07 .10 .91 12.3 37.0 .82 .81 .24 26.6 .93 

06927100 --------- 2.79 .050 .05 16.7 7.11 .70 .90 2.12 60.0 .34 

06928200 --------- 2.53 .030 .24 7.07 2.82 .61 .87 1.14 37.8 .59 

06931500 --------- 4.10 .050 .47 11.7 2.40 .52 .85 .84 75.0 .50 

06935800 --------- 2.35 .040 .36 17.4 3.00 .74 .69 .69 44.1 .96 

07011200 --------- 1.58 .030 .15 6.11 27.7 .79 .64 2.12 34.7 .75 

07011500 --------- 2.22 .050 .33 11.9 3.93 .77 .87 .49 32.3 .97 

07015500 --------- .68 .050 .09 5.70 8.66 .71 .94 .94 98.7 .67 

07017500 --------- 1.82 .060 .82 5.55 4.07 .63 .86 1.56 26.1 .40 

07019100 --------- 5.70 .12 .49 13.8 3.78 .73 .68 .40 24.3 .96 

07035500 --------- .84 .060 .13 16.4 4.29 .71 .74 .77 77.0 .86 

07037700 --------- 2.28 .060 .79 58.3 5.83 .64 .88 1.77 20.2 .55 

07053950 --------- 2.37 .070 .91 7.90 7.29 .80 .61 .54 66.0 .81 

07054200 --------- 5.81 .16 .48 6.06 3.47 .58 .51 .66 25.7 .90 

07054300 --------- 3.64 .10 .82 14.3 2.92 .63 .82 1.16 41.2 .88 

07063200 --------- 1.93 .060 .19 6.92 10.9 .80 .66 .63 26.1 .18 

07064300 --------- .17 .050 .36 14.0 27.4 .95 .72 .66 66.0 .62 

07064500 --------- 2.49 .040 .67 9.15 2.34 .63 .81 1.85 101 .62 

07068200 --------- 21.2 .080 .28 16.4 3.98 .64 .58 1.09 59.4 .96 

07185500 --------- 2.05 .050 .30 19.7 3.46 .71 .87 1.98 117 .72 

TABLE 4.-U.S. National Weathe·r Set·vice first-ordet· p1·ecipitation stat·i()ns used in 

synthesis of peak-flow data 

Station Period of Number 

number continuous of events 

(fig. 1) Station name record selected 

VV13988 __________ Kansas City VVSO ---------------- 1893-1970 217 

VV93963 __________ St. Louis vvso ------------------- 1893-1970 190 

VV93958 _______ ~-- Columbia Region vvso ------------ 1902-70 222 

VV13995 _______ ~-- Springfield VVSO ------------------ 1905-70 182 

synthetic daily evaporation record for the 1893 

to 1948 period. Experience has shown that 

model outputs are relatively insensitive to the 

clay-to-day variations in evaporation data and 

that an evaporation record n1ay be applicable 

to large regions. The partly synthesized evaporation record used in this study was considered 

8 

to be adequate for flood-peak synthesis at the 

43 sites considered. 

DEFINING SYNTHETIC FLOOD FREQUENCIES 

Four series of synthetic annual peaks were 

generated for each of the 43 sites by putting 

into the calibrated model the data from each of 
the four long-term rainfall gages and the 

evaporation gage. Each of the four series of 

peaks was then tested in pairs (Kansas City 

and St. Louis, Kansas City and Columbia, and 

so forth) for statistical distribution differences 

via the Cramer-von Mises Two-Sample Test as 

described by Conover (1971). In no instance 

could the hypothesis that the samples are identical be rejected at the 1-percent level of significance, and in only one station pairing could 

the hypothesis be rejected at the 5-percent 

level. These results indicate that there was no 

apparent advantage or reason to use one raingage record in preference to another and that 

an average of the four frequency relations 

would be justified. 

To define the average synthesized frequency 

curve, four individual curves were computed by 

mathematically fitting a Pearson Type III distribution to the logarithms of each of the four 

annual peak series using techniques recommended by the U.S. Water Resources Council 

(1967). The 2-, 5-, 10-, 25-, 50-, and 100-year 

recurrence interval peak magnitudes from 

each curve then were averaged to form the 

synthetic curve for each site. 

In addition to the synthesized records, the 

observed flood-peak record was available to aid 

in definition of the freqeuncy characteristics 

at each site. Observation and testing indicated 

that these relatively short records could be of 

value. Firstly, because the observed data are 

of sufficient length to define reliably the magnitude of flood peaks in the range of the 2- to 10-

year recurrence intervals. Secondly, because 

statistical analysis of the synthetic frequency 

curves indicates that the variance between 

curves decreases with increasing recurrence 

interval-that is, the model defines the extreme 

floods with more consistency than the smaller 

floods. A log-Pearson Type III distribution was 

also fitted to the observed flood series. 

A graphical weighting procedure was employed to combine the observed and synthesized 

frequency curves into a final curve. Both the 

average synthetic data curve and the observed 

data curve were plotted on a probability graph, 

and a curve was drawn from the 2-year observed value to merge smoothly with the average synthetic curve at a point generally greater 

than the 10-year recurrence interval. Figure 2 

9 

shows the relationships for Thompson Branch 

near Albany ( 06896500) and the final curve 

obtained by this graphical process. Each curve 

was judged individually in an effort to provide 

the best possible estimate of the frequency 

characteristics to use in the regression analysis. A summary of the flood characteristics for 

the 43 sites is given in table 5. 

REGIONAL REGRESSION ANALYSIS 

Flood data at a sample of gaged sites provide a basis for estimating flood characteristics 

at ungaged sites. The technique of transferring 

data used in this study defines from this sample some equations relating flood magnitudes 

to numerical indices describing characteristics 

of the contributing drainage basins. Separate 

equations are defined for selected points on the 

frequency relation, such as the 2-, 10-, and 50-

year recurrence interval peak magnitudes. 

Drainage-basin indices describe the size, shape, 

slope, soil characteristics, vegative cover, climate, and so forth. They are indices that can be 

evaluated from topographic, geologic, and climatic maps and from other readily available 

sources, and they logically can be expected to 

be related to the differences in floodflows at the 

various gaged sites. Planners and designers 

may then solve the defined equation to estimate the flood characteristics of an ungaged 

site by evaluating appropriate basin indices. 

Many recent studies throughout the Nation 

have defined flood-estimating equations based 

upon a model of the form: 

where 

Q1 =flood magnitude having a t-year recurrence interval, 

A, B, .. . N =drainage-basin indices, and 

b, b11 b'2, •... . b11 =constants for a given t. 

Constants are evaluated most easily by multiple-regression analysis when the variables are 

transformed to logarithms to form a linear 

relation: 

log Q,=log b+b1log A+b'2log B+ .... bn log N. 

Regression analysis techniques also provide a 

measure of the accuracy of the equation, known 

as the standard error of estimate, and test the 

significance of the b coefficient of each independent variable. Only those independent vari-
0 

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0.... 0::: 

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0.... LLJ 

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1..1... 

LLJ ~ 

S2 1000 S2 CD CD 

::J 800 ::J 

u u 

~ 600 ~ 

w w (!) EXPLANATION (!) 

0::: 0::: 

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u Synthesized data 10 u C/) C/) 

0 ~ 0 

::!C Observed data (1956-70) ::!C 

< < LLJ 200 LLJ 

0.... 0.... 

2 3 4 5 6 7 8 9 10 20 30 40 50 100 

RECURRENCE INTERVAL, IN YEARS 

FIGURE 2.-Flood-frequen~y curves for Thompson Branch near Albany, Missouri. 

abies are retained in an estimating equation 

that have a coefficient that is significantly different from zero at the 95-percent confidence 

level. 

Sandha us and Skelton ( 1968) previously 

used the log-linear model form to define floodestin1ating relations for Missouri streams. 

They provided separate relations for estimating magnitudes of floods having six selected 

recurrence intervals ranging from 1.2 to 50 

years. They found only the coefficients of 

drainage-area size and main-channel slope to 

be significant, and they found the assumption 

of a linear relation between the logarithms of 

the variables to be acceptable. Standard errors 

of estimate ranged fron1 50.7 percent for the 

sn1aller floods to 36.9 percent for the larger 

floods. Because of the lin1ited data on small 

streams, the relation for estimating floods 

larger than 25-year recurrence interval was 

10 

not considered applicable to streams draining 

less than 50 mi:! (130 km:!). 

In this study, data for the 43 small-stream 

sites were analyzed by multiple regression 

m~ing the log-linear model form. Separate relations were defined for recurrence intervals of 

2, 5, 10, 25, 50, and 100 years. Drainage-basin 

indices considered as independent variables 

were: 

Drainage area C4), in square miles, as planimetered on the best available topographic 

maps. 

Main-channel slope (S), in feet per mile, between points 10 and 85 percent of the distance upstream from the gage to the basin 

border. 

Main-channel length ( L) , in miles, fron1 the 

gaging station to the basin divide. 

Mean-basin elevation (E), in feet above mean 

sea level. 
TABLE 5.-Station frequency data 

Station 

number 

05495100 __ 05497700 __ 05502700 __ 05503000 __ 06815550 __ 

Station name 

Big Branch Tributary nr Wayland, Mo ______ 

Bridge Creek nr Baring, Mo ---------------- Easdale Branch nr Shelbyville, Mo ---------- Oak Dale Branch nr Emden, Mo ------------- Staples Branch nr Burlington Jet., Mo -------

06816000 __ Mill Creek nr Oregon, Mo ------------------- 06820000 __ White Cloud Creek nr Maryville, Mo ---------

06821000 __ Jenkins Branch nr Gower, Mo --------------- 06896180 __ DeMoss Branch nr Stanberry, Mo ----------- 06896500 __ Thompson Branch nr Albany, Mo ------------ 06897700 __ Grand River Tributary nr Utica, Mo --------- 06902500 __ Hamilton Branch nr New Boston, Mo -------- 06902800 __ Union Branch nr St. Catherine, Mo ---------- 069·07200 __ Shaver Creek Tributary nr Clifton City, Mo __ 

06908300 __ Trent Branch nr Waverly, Mo --------------- 06909700 __ Petite Saline Creek Tributary nr Bellair, Mo __ 

06910250 __ Traxler Branch nr Columbia, Mo ------------ 06918700 __ Oak Grove Branch nr Brighton, Mo ---------- 06919200 __ Sac River Tributary nr Caplinger Mills, Mo __ 

06921740 __ Brushy Creek nr Blairstown, Mo ------------ 06922700 __ Chub Creek nr Lincoln. Mo ------------------ 06925200 __ Starks Creek nr Preston, Mo ---------------- 06925300 __ Prairie Branch nr Decaturville, Mo ---------- 06926200 __ Van Cleve Branch nr Meta, Mo -------------- 06927100 __ Doone Branch nr Kingdom City, Mo --------- 06928200 __ LaquE>y Branch nr Hazelgreen, Mo -----------

06931500 __ Little Beaver Creek nr Rolla, Mo ------------ 06935800 __ Shotwell Creek nr Ellil"-ville, Mo ------------- 07011200 __ Love Creek nr Salem, Mo ------------------- 07011500 __ Green Acre Branch nr Rolla, Mo ------------ 07015500 __ Lanes Fork nr Rolla, Mo -------------------- 07017500 __ Dry Branch nr Bonne Terre, Mo ------------- 07019100 __ Murphy Branch nr Crystal City, Mo _________ 

07035500 __ Barnes Creek nr Fredericktown, Mo --------- 07037700 __ Clark Creek nr Piedmont, Mo --------------- 07053950 __ Ingenthron Hollow nr Forsvth, Mo ----------

07054200 __ Yandell Branch nr Kirbyville, Mo ----------- 07054300 __ Gray Branch nr Lutie, Mo ------------------ 07063200 __ Pike Creek Tributary nr Poplar Bluff, Mo ____ 

07064300 __ Fudge Hollow nr Licking, Mo --------------- 07064500 __ Big Creek nr Yukon, Mo -------------------- 07068200 __ North Prong Little Black River nr Hunter, Mo. 

07185500 __ Stahl Creek nr Miller, Mo ------------------

Drainage area 

(mi2 ) 

0.70 

2.54 

.71 

2.64 

.49 

4.90 

6.06 

2.72 

.38 

5.58 

1.44 

2.51 

1.04 

1.65 

.97 

.49 

.55 

1.30 

.14 

1.15 

2.86 

1.30 

1.48 

.75 

.54 

1.58 

6.41 

.81 

.89 

.62 

.22 

3.35 

.45 

4.03 

4.39 

.65 

.42 

.27 

.28 

1.72 

8.36 

1.23 

3.86 

Magnitude of flood, in cubic feet per second, for indicated recurrence interyaJ, in years 

2 5 10 25 50 100 

130 239 317 416 489 561 

399 800 860 1,030 1,460 1,650 

399 495 577 689 771 852 

693 993 1,190 1,450 1,660 1,870 

142 316 435 525 586 645 

740 1,600 2,320 3,350 4,190 5,320 

729 1,680 2,260 2,820 3,220 3,580 

638 1,350 1,700 2,120 2,420 2,700 

145 207 248 305 345 383 

738 1,490 2,020 2,520 2,880 3,220 

392 485 560 665 761 854 

576 941 1,210 1,580 1,780 1,980 

231 390 494 623 718 809 

480 887 1,070 1,290 1,450 1,610 

339 670 890 1,140 1,290 1,430 

136 191 242 325 408 504 

277 456 564 678 755 825 

182 392 567 820 1,020 1,160 

47.0 70.0 89.0 112 127 142 

498 768 959 1,140 1,280 1,400 

766 1,360 1,730 2,160 2,430 2,700 

822 1,270 1,530 1,840 2,060 2,330 

190 310 425 633 850 1,110 

255 545 722 947 1,110 1,280 

111 185 233 293 335 376 

409 860 1,100 1,320 1,490 1,640 

1,240 2,340 3,430 5,280 6,060 6,800 

365 520 617 755 848 933 

121 265 370 505 598 671 

306 509 616 737 818 890 

71.5 121 152 193 218 240 

660 1,290 1,700 2,220 2,550 2,820 

130 221 300 435 572 732 

929 2,040 2,850 3,560 4,040 4,510 

753 1,230 1,680 2,400 2,940 3,500 

197 360 481 598 679 757 

65.1 158 224 318 394 474 

50 97.5 132 178 212 245 

124 210 252 305 342 376 

100 360 600 970 1,300 1,620 

1,750 2,950 3,650 4,510 5,120 5,710 

130 245 355 565 795 1,100 

637 1,120 1,440 1,820 2,090 2,340 

Forest cover (F), expressed as the proportional part of the drainage area covered 

by forests as shown on the topographic 

map. 

Standard errors ranged from 25.4 to 42.6 percent, indicating that the assumption of a linear 

relation between logarithms of the variables 

was reasonable. 

Mean-annual precipitation (P), in inches. 

The maximum 24-hour rainfall, (/24.2), in 

inches, having a recurrence interval of 2 

years (2-year, 24-hour rainfall) from U.S. 

Weather Bureau Technical Paper 40 

(1961). 

Soil infiltration index (Si), in inches, determined by the Soil Conservation Service. 

Upon testing all independent variables, only 

the coefficient to drainage area remained significant at the 95-percent confidence level. 

11 

It had been anticipated that equations defined from data for the 43 small-stream sites 

would be useful for estimating flood characteristics of ungaged sites draining less than 10 

mi~ (25.9 km2 ) and that the Sandhaus and 

Skelton relations would be used for estimates 

on larger streams. Comparison of the estimates 

obtained by the two sets of equations showed, 

however, that there was a discontinuity for 

sites having drainage areas of sizes between 

10 and 100 mP (25.9 and 259 km2 ). No simple, 
yet logical, system of using the two sets of 

equations was apparent, and their utility for 

obtaining design discharge estimates was rejected. 

Recognizing that a continuous relation 

should exist between flood and basin characteristics for the complete range of basin sizes, 

data for a sample of large streams were included with the small-streams data for a combined analysis. Flood characteristics were 

evaluated for 109 large-stream sites where 

floodflows are considered to be virtually natural 

and a sufficient number of annual peaks had 

been recorded to adequately define the 50- to 

100-year flood. Frequency then was analyzed 

by fitting a Pearson Type III distribution to 

the logarithms of annual peaks using a regional 

average coefficient of skew of - 0.5, as suggested by Hardison (1974). Data on the size 

and main-channel slope of the large-stream 

sites were available from the Sandhaus and 

Skelton study (1968). 

Definition of estimating equations based 

upon the combined large- and small-stream 

data indicated that the assumption of a linear 

relation between the logarithms of the variables was inadequate. The defined log-linear 

relations tended to overestimate flood magnitudes at intermediate size sites. Alternative 

curvilinear models and variable transformations were therefore tested on the data. 

The most satisfactory model found was one 

suggested by Creager, Justin, and Hinds 

(1947) : b Qt = bA blA. 2Sba, 

where A is drainage-basin size and S is mainchannel slope. By logarithmic transformation 

of the variables the model becomes : 

b2 log Qt=log b + b1 A log A + b3log S. 

Although the constants in this relation cannot 

be evaluated directly by linear multiple-regression analysis, repeated trial and error solutions 

indicated that when b2 = - 0.02, multiple-regression analysis to determine remaining constants provided relations having a minimum 

standard error of estimate. 

Table 6 summarizes the final set of best-fit 

equations. To simplify use of these equations, 

12 

Hauth (1974) provided graphical solutions, 

one of which is shown in figure 3 as an example. 

TABLE 6.-Summary of regression analyses 

Standard Recurrence error of 

interval of estimate 

flood peak Equation (percent) 

2-year --------- Q=53.5A o.ssu -o.02 SQ.a5s 38.6 

5-year --------- Q=G4:0A o.856A. -o.02 so.4so 34.7 

10-year --------- -0.02 Q=67.6A o oo~A so.soo 34.5 

25-year --------- -0.02 Q=73. 7 A 0.9MA so.542 3.5.0 

50-year --------- -0.02 Q=79.8A o.926A so.5110 3.3.3 

10.0-year --------- Q=85.1A o.rou -o.02 so.s76 33.3 

The defined relations are considered useful 

for providing design flood estimates at ungaged 

sites in Missouri having characteristics within 

the range of experience in the sample datathat is, they are useful at sites where floodflow 

is virtually natural, where drainage areas 

range from 0.1 to 14,000 mi 2 (0.26 to 32,300 

km2 ), and where main channels have slopes 

between 1 and 300 ft/mi (0.19 and 56.7 m/km). 

The relations should not be used at sites where 

ftoodflows are significantly affected either by 

the works of man, such as the development of 

urban centers, reservoirs, and improved channel, or by backwater of larger streams, as 

might be expected near the mouth on tributary 

channels. The relations also are not useful for 

main-stream sites on the Mississippi and Missouri Rivers, but alternate design estimating 

techniques previously have been provided for 

those rivers and for some regulated streams by 

Patterson and Gamble (1968), Patterson 

(1964, 1966), and Matthai (1969). 

DISCUSSION 

This first operational use of the rainfallrunoff model proved the practicality of record 

extension for use in regional frequency analysis. Valuable experience was gained, but many 

questions and problems arose. A general discussion of some of the more important observations is presented here. Many of the techniques 

used in this study were necessarily selected on 

an intuitive basis and perhaps will be useful in 

future studies, yet it is hoped that this discus-
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EXPLANATION 

2-year frequency 

-002 

Q = 53.5A0851A SU356 

100 1000 10,000 

DISCHARGE, IN CUBIC FEET PER SECOND 

FIGURE 3.-Graphical solution of the 2-year equation. 

13 

100,000 
sion also can suggest areas of beneficial future 

research. 

PARAMETER EVALUATION 

Parameter evaluation was the most complex 

operation in the study and the one requiring 

the most arbitrary judgments. Development of 

improved guidelines for parameter evaluation 

could be of significant benefit. 

Information on the probable magnitude and 

possible range of parameter values is needed 

for efficiency of calibration. At first it was 

difficult to estimate initial parameter values, 

but as experience was gained the efficiency in 

parameter evaluation increased. The data 

shown in table 3 will be of value to subsequent 

analysts in areas near Missouri, but it would 

be useful to know the variation in parameter 

values is associated with basin characteristics. 

A relatively simple approach would be to use 

the values in table 3 as dependent variables in 

a multiple-regression analysis. Results would 

not only aid the efficiency of parameter evaluation at short-record sites, but conceivably could 

allow model calibration for ungaged sites. 

Information on the sensitivity of output results to changes in parameter values would also 

assist future investigators. For example, the 

routing parameter TP /TC was observed during this investigation to exert little influence 

on output results. Although no detailed sensitivity investigation was undertaken, on the 

basis of experience with the earlier calibration 

this parameter was virtually ignored during 

latter parts of the study. More detailed study 

might indicate no need to use the parameter 

value TP lTC in any model calibration and 

might set relative priorities for calibration of 

other parameters. 

A knowledge of the amount of information 

needed for effective model calibration would be 

of great importance to other stream flood investigations. In this study, from 8 to 31 storms 

were used. Research studies might test the 

effectiveness of lesser calibration data as a 

basis for knowing how long data collection progams should be operated. A related study of 

criteria to judge the effectiveness of individual 

storms and number of individual storms during the data collection period also would be 

helpful. 

14 

Lacking any technical basis, the storm 

screening process used in this study to include 

or to omit storms in the calibration process was 

judged necessary to obtain the most effective 

set of model parameters. This screening process required considerable effort in recalibration 

on a reduced number of storm events. It would 

be interesting to test how much differently the 

results might be if there had been no screening. 

The three-phase calibration scheme in which 

storage and routing parameters were calibrated separately was devised for efficiency on 

a rational basis. Alternate schemes are available and might have been selected. They ranged 

in complexity from direct calibration in a 

single operation to calibration of parameters in 

multiphase operations where alternate parameter groups are identified. Research studies 

might suggest the most effective scheme. 

RECORD SYNTHESIS 

Experience in synthesizing records by using 

long-term climatic data in a calibrated model 

defined some unexpected problems. 

The n1ost perplexing problem was the inability to distinguish between the frequency curves 

defined from the four long-term precipitation 

records. Additional study to define whether any 

significant difference exists within the rainfall 

records themselves would be of interest. Such 

a study might suggest which available longterm rainfall records in an area are effective 

for model synthesis and, therefore, the ones 

which should be reduced for model use. 

Techniques of combining frequency curves 

should be investigated. Intuitively, observed 

records should have some weight in defining 

the middle range of an annual series frequency 

curve. Experience with the model-synthesized 

records indicates that larger floods may be 

more reliably estimated than the average-size 

floods. Some technical basis is needed for combining and weighting the observed and synthesized data in freqeuncy analysis. 

One other practical problem that could benefit from additional study and could improve 

operational guidelines is the selection of storn1s 

to be reduced to unit data. Perhaps investigation of the unused unit data in the four Missouri long-tern1 rainfall-runoff records could 

identify criteria and thereby save the data-
reduction effort necessary in similar future 

studies. 

REGRESSION ANALYSIS 

The nonlinearity of the relation between the 

logarithms of the floods and basin characteristics was surprising. Perhaps alternate model 

forms would be as effective or more effective 

than the one used, but of more interest is the 

reality of the nonlinear model form. Is it a 

relation that would be defined if long-term 

records were available at the small-stream 

sites, or is it a form that results from some 

idiosyncrasy in the rainfall-runoff model? For 

example, it is conceivable that the use of a 

single rain-gage record with the necessary assumption of uniform areal precipitaion may 

tend to overestimate the flood magnitudes in 

the larger basins (approximately 10 miZ or 26 

km2 ), thereby causing the apparent curvilinear 

regression. 

The standard errors of estimate of the regression relations defined in this study are 

considerably improved over the standard errors 

of relations previously defined for Missouri. It 

must be noted, however, that these newly defined relations are based upon synthesized flood 

records that are highly correlated among themselves. This cross-correlation will reduce significantly the time sampling errors inherent in 

observed records; but it remains to be shown 

whether or not the improved standard errors 

actually indicate a more reliable estimate of 

future floodflows. 

SUMMARY AND CONCLUSIONS 

This study has demonstrated the utility and 

practicality of using a rainfall-runoff model of 

small-stream floods to define techniques of estimating flood magnitude and frequency at ungaged sites. Short-duration records of concurrent rainfall and runoff defined parameters 

that are used in a mathematical model and that 

define each basin's response to rainfall. Longterm climatic data then were put into the model 

to generate long-term peak-flow records. Results obtained from this process indicate that 

there is a rational consistency of parameter 

values from basin to basin throughout the 

State. The average standard error of estimate 

from all calibrations, which in general indi15 

cates the ability to mathematically define the 

basin response, was about 35 percent. 

Synthetic annual peak discharges were obtained at each gaging station by transferring 

long-term rainfall data (an average of 70 

years) from four locations. Statistical tests indicated that the synthetic peaks based upon the 

four different rainfall records were identically 

distributed and that a simple averaging of synthetic frequency curves was justified. An examination of available short-term rainfall records throughout the State shows a few rainfall 

intensities to be greater than those from the 

four long-term records and demonstrates the 

time sampling error inherent in hydrologic 

data. Extreme storms such as these can cause 

extreme runoff, and the recurrence intervals for 

these floods are difficult to define within a short 

observed record. However, by using a longer 

period of rainfall with the calibrated model, a 

long series of annual peaks were generated as 

improved time samples for definition of frequency curves at each site. 

Simple regression of the logarithmic transforms of streamflow characteristics versus 

logarithmic transforms of drainage area 

proved to be linear for basins used in the synthesis. However, when data for small and 

large basins were combined, a nonlinearity 

occurred. Linearity of regression was then obb A-o·02 b 

tained by the model equation Qt = bA 1 S 3

• 

A map plot of residuals suggests the statewide 

applicability of the equations. Equations defined in this study are based upon more data 

and a more rigorous analysis than those presented in the report by Sandhaus and Skelton 

(1968). 

REFERENCES 

Benson, M. A., 1962, Factors influencing the occurrence 

of floods in a humid region of diverse terrain: U.S. 

GeoL Survey Water-Supply Paper 1580-B, 64 p. 

--1964, Factors affecting the occurrence of flood.s in 

the Southwest: U.S. Geol. Survey WaJter-Supply 

Paper 1580-D, 70 p. 

Conover, W. J., 1971, Practical nonparametric statistics New York, John Wiley, p. 314-316. 

Creager, W. P., Justin, J. D., Hinds, Julian, 1947, Engineering for dams, volume I: New York, John 

Wiley, Ji. 125-126. 

Dawdy, D. E., Lichty, R. W., and Bergmann, J. M:, 

1972 A rainfall-runoff s.imulation model for estimati~n of flood peaks for small drainage basins.: 

U.S. GeoL Survey Prof. Paper 506-B, 28 p. 
Hardison, C. H., 1974, Generalized skew coefficients of 

annual floods in the United States and their application: Water Resources Re·search, v. 10, no. 4. 

(In press.) 

Hauth, L. D., 1973a, Selected storm events in 5-minute 

incr·ements from Missouri rainfall stations at Kansas City, St. Louis, Springfield, and Columbia, 

Missouri: U.S. Geol. Survey open-file rept., 191 p., 

Rolla, Mo. 

--- 1973b, Rainfall-runoff data for small drainage 

areas of Missouri: U.S. Geol. Survey open-file 

rept., Rolla, Mo. 

--- 1974, A technique for estimating the magnitude 

and frequency of Missouri floods: U.S. Geol. Survey open-file rept., Rolla, Mo. 

Kohler, M. A., Nordenson, T. J., and Baker, D. R., 

1959, Evaporation maps for the United States: 

U.S. Weather Bureau Tech. Paper no. 37, 13 p. 

Matthai, H. F., 1969, Magnitude and frequency of 

floods in the United States, Part 6B, Missouri 

River below Sioux City, Iowa: U.S. Geol. Survey 

Water-Supply Paper 1680, 491 p. 

Patterson, J. L., 1966, Magnitude and frequency of 

floods in the United States, Part 6A, Missouri 

River abo;ve Sioux City, Iowa: U.S. Geol. Survey 

Water-Supply Paper 1679, 471 p. 

--- 1964, Magnitude and frequency of floods in the 

United States, Part 7, Lower Mississippi River 

basin: U.S. Geol. Survey Water-Supply Paper 

1681, 6·36 p. 

Patterson, J. L., and Gamble, C. R., 1968, Magnitude 

and frequency of floods in the United 8tates, Part 

5, Hudson Bay and upper Mississippi River basins: 

U.S. Geol. Survey Wate,r-Supply Paper 1678, 546 p. 

Sandhaus, E. H., and Skelton, John, 1968, Magnitude 

and Frequency of Missouri floods: Missouri Geol. 

Survey and Water Resources rept. 23, 276 p. 

Searcy, J. K., 1955, Floods in Missouri, magnitude and 

frequency: U.S. Geol. Survey Circ. 370, 126 p. 

Skelton, John, and Homyk, Anthony, 1970, A proposed 

streamflow data program for Missouri: U.S. Geol. 

Survey open-file rept., 77 p. 

U.S. Water Resources Council, 1967, A uniform technique for determining flood flow frequencies: U.S. 

Water Resources Council Bull. 15, 15 p. 

U.S. Weather Bureau, 1961, Rainfall frequency atlas 

of the United States: U.S. Weather Bureau Tech. 

Paper 40, 115 p. 

*u.s. GOVERNMENT PRINTING OFFICE: 1974 0-585-468/37 

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