COVER PHOTO: U.S. Geological Survey streamflow-gaging station 02162093, 

Smith Branch at Columbia, S.C. 

Photograph by James R. Douglas 
Estimating the Magnitude and Frequency 

of Floods in Small Urban Streams in 

South Carolina, 2001 

By Toby D. Feaster and Wladmir B. Guimaraes 

Prepared in cooperation with the 

South Carolina Department of Transportation 

Scientific Investigations Report 2004 – 5030 

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2004

For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal Center

Denver, CO 80225

For more information about the USGS and its products:

Telephone: 1-888-ASK-USGS

World Wide Web: http://www.usgs.gov/

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Feaster, T.D., and Guimaraes, W.B., 2004, Estimating the magnitude and frequency of floods in small urban streams in 

South Carolina, 2001: U.S. Geological Survey Scientific Investigations Report 2004–5030, accessed May 18, 2004, at 

http://pubs.water.usgs.gov/sir2004-5030
 iii 

Contents 

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1


Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2


Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2


Previous Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2


Peak-Flow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


Graphical Comparisons of Measured and Simulated Peak Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6


Statistical Comparisons of Measured and Simulated Peak Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


Kendall Tau Trend Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


F-test for Equality of Variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


Wilcoxon Rank Sum Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


Estimation of Flood Magnitude and Frequency at Gaged Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


Flood Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


Comparison of Selected Recurrence-Interval Flows Computed From Measured Data 


and Simulated Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


Conclusions from Comparisons of Measured and Simulated Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39


Estimation of Flood Magnitude and Frequency at Ungaged Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


Urban Basin Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


Ordinary Least-Squares Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


Generalized Least-Squares Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


Accuracy of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


Program for Computing Flood-Frequency Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


Example Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


Estimation of Flood Magnitude and Frequency at Ungaged Sites—Alternative Equations. . . . . . . . . . . . . . . . . . . 49


Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


Appendix: Generalized least-squares model description and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


Figures 

1.	 Location of streamflow-gaging stations with 10 or more years of record in urban areas 


of the Piedmont and upper and lower Coastal Plain physiographic provinces 


of South Carolina, Georgia, and North Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5


2.	 Comparison of measured and simulated peak flows at 13 urban streamflow-gaging 


stations in South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


3.	 Location of cities, rural streamflow-gaging stations used for comparison with urban 


streamflow-gaging stations, and physiographic provinces in South Carolina. . . . . . . . . . . . . . . . . . . . . . 13


4–8. Graphs showing:


4.	 Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001 


at 16 rural streamflow-gaging stations in South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
iv 

5.	 Box plots of the logarithms of the measured and simulated peak-flow data 


at 13 urban streamflow-gaging stations in South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


6.	 Box plots of the logarithms for two peak-flow periods (beginning of record 


to 1984 and 1985–2001) at 16 rural streamflow-gaging stations in South Carolina. . . . . . . . . . . . . 28


7.	 Comparison of the 2-, 25-, and 100-year recurrence-interval flows computed 


from the measured peak flows with those computed from the simulated peak flows . . . . . . . . 35


8.	 Comparison of 2-, 25-, and 100-year recurrence-interval flows from two


peak-flow periods (period of record and 1985–2001) at 16 rural 


streamflow-gaging stations in South Carolina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


9.	 Schematic of typical drainage basin shapes and subdivision into basin thirds for 


determination of basin development factor (BDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


10.	 Sensitivity of computed peak flows to changes in the three explanatory variables 


for selected peak-flow-frequency equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


11.	 Comparison of observed and estimated peak flows for the 2-, 25-, and 100-year 


recurrence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


Tables 

1.	 Streamflow- gaging stations in South Carolina, North Carolina, and Georgia with 


10 or more years of record used in the flood-frequency analysis for small urban 


streams in South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


2.	 National Weather Service rainfall stations used in Bohman’s (1992) synthesis of 


long-term flood-hydrograph data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6


3.	 Rural streamflow-gaging stations that were used to compare climatic conditions 


with similar periods at the 13 South Carolina urban streamflow-gaging stations, 


by physiographic province. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12


4.	 Summary of statistical trends in water-year peak flows for 13 urban streamflow-gaging 


stations in South Carolina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


5.	 Results of comparison testing of measured and simulated peak-flow data at 13 urban 


streamflow-gaging stations in South Carolina based on the Shapiro-Wilks test for 


normality and the F-test for equal variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


6.	 Results of comparison testing of peak-flow data from 16 rural streamflow-gaging stations 


in South Carolina for two periods (beginning of record to 1984 and 1985–2001) based 


on the Shapiro-Wilks test for normality and the F-test for equal variances. . . . . . . . . . . . . . . . . . . . . . . . . 22


7.	 Results of comparison testing of measured and simulated peak-flow data for 13 urban 


streamflow-gaging stations in South Carolina based on the Wilcoxon rank sum test . . . . . . . . . . . . . . 23


8.	 Results of comparison testing of peak-flow data from 16 rural streamflow-gaging


stations in South Carolina for two periods (beginning of record to 1984 and 1985–2001) 


based on the Wilcoxon rank sum test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27


9.	 Flood-frequency statistics for measured data from 20 urban streamflow-gaging stations 


in South Carolina, North Carolina, and Georgia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


10.	 Comparison of 2-, 25-, and 100-year flows computed from simulated peak flows 


(Bohman, 1992) with those computed from measured peak flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


11.	 Comparison of 2-, 25-, and 100-year recurrence-interval flows from two periods 


(period of record and 1985–2001) at 16 rural streamflow-gaging stations 


in South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


12.	 Selected basin characteristics for the 20 urban streams in South Carolina,


North Carolina, and Georgia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


13.	 South Carolina urban regional regression equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


14.	 Accuracy statistics for the urban regional regression equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

 v 

L 

15. Alternative urban regional regression equations for South Carolina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 

16. Accuracy statistics for the alternative urban regional regression equations. . . . . . . . . . . . . . . . . . . . . . . 55 

Conversion Factors, Datums, and Abbreviations 

and Acronyms 

Multiply By To obtain 

Length 

mile (mi) 1.609 kilometer (km) 

Area 

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate 

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s) 

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 


(NAVD 88).


Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).


Abbreviations and acronyms


BDF 

DA 

GIS 

GLS 

IA 

NWS 

OLS 

PRESS 

R2 

ROI 

S 

SCDOT 

USGS 

VIF 

basin development factor 

drainage area 

Geographic Information System 

generalized least squares 

impervious area 

length 

National Weather Service 

ordinary least squares 

prediction error sum of squares 

coefficient of determination 

region of influence 

slope 

South Carolina Department of Transportation 

U.S. Geological Survey 

variance inflation factor 

Estimating the Magnitude and Frequency of Floods in 

Small Urban Streams in South Carolina, 2001 

By Toby D. Feaster and Wladmir B. Guimaraes 

Abstract 

The magnitude and frequency of floods at 20 streamflowgaging stations on small, unregulated urban streams in or near 

South Carolina were estimated by fitting the measured wateryear peak flows to a log-Pearson Type-III distribution. The 

period of record (through September 30, 2001) for the measured 

water-year peak flows ranged from 11 to 25 years with a mean 

and median length of 16 years. The drainage areas of the 

streamflow-gaging stations ranged from 0.18 to 41 square 

miles. 

Based on the flood-frequency estimates from the 20 

streamflow-gaging stations (13 in South Carolina; 4 in North 

Carolina; and 3 in Georgia), generalized least-squares regression was used to develop regional regression equations. These 

equations can be used to estimate the 2-, 5-, 10-, 25-, 50-, 100-, 

200-, and 500-year recurrence-interval flows for small urban 

streams in the Piedmont, upper Coastal Plain, and lower Coastal 

Plain physiographic provinces of South Carolina. The most 

significant explanatory variables from this analysis were mainchannel length, percent impervious area, and basin development factor. Mean standard errors of prediction for the regression equations ranged from -25 to 33 percent for the 10-year 

recurrence-interval flows and from -35 to 54 percent for the 

100-year recurrence-interval flows. 

The U.S. Geological Survey has developed a Geographic 

Information System application called StreamStats that makes 

the process of computing streamflow statistics at ungaged sites 

faster and more consistent than manual methods. This application was developed in the Massachusetts District and ongoing 

work is being done in other districts to develop a similar application using streamflow statistics relative to those respective 

States. Considering the future possibility of implementing 

StreamStats in South Carolina, an alternative set of regional 

regression equations was developed using only main channel 

length and impervious area. This was done because no digital 

coverages are currently available for basin development factor 

and, therefore, it could not be included in the StreamStats application. The average mean standard error of prediction for the 

alternative equations was 2 to 5 percent larger than the standard 

errors for the equations that contained basin development factor. 

For the urban streamflow-gaging stations in South Carolina, measured water-year peak flows were compared with 

those from an earlier urban flood-frequency investigation. The 

peak flows from the earlier investigation were computed using 

a rainfall-runoff model. At many of the sites, graphical comparisons indicated that the variance of the measured data was much 

less than the variance of the simulated data. Several statistical 

tests were applied to compare the variances and the means of 

the measured and simulated data for each site. The results 

indicated that the variances were significantly different for 11 

of the 13 South Carolina streamflow-gaging stations. For one 

streamflow-gaging station, the test for normality, which is one 

of the assumptions of the data when comparing variances, indicated that neither the measured data nor the simulated data were 

distributed normally; therefore, the test for differences in the 

variances was not used for that streamflow-gaging station. 

Another statistical test was used to test for statistically significant differences in the means of the measured and simulated 

data. The results indicated that for 5 of the 13 urban streamflowgaging stations in South Carolina there was a statistically significant difference in the means of the two data sets. 

For comparison purposes and to test the hypothesis that 

there may have been climatic differences between the period in 

which the measured peak-flow data were measured and the 

period for which historic rainfall data were used to compute the 

simulated peak flows, 16 rural streamflow-gaging stations with 

long-term records were reviewed using similar techniques as 

those used for the measured and simulated data at the urban 

streamflow-gaging stations. For the rural sites, the period from 

1985 to 2001 was compared with the data measured from the 

beginning of record to 1984. Plots of the two periods at each 

rural site indicated no significant difference in the data. The 

statistical test for comparison of variances was applied to the 16 

rural streamflow-gaging stations, and the results showed that 

there was no statistically significant difference in the variances 

at 14 of the 16 streamflow-gaging stations. The statistical 

comparisons of the means for the two periods at the rural 

streamflow-gaging stations showed that there was no statistically significant difference at 12 of the 16 streamflow-gaging 

stations. Based on these comparisons, the differences between 

the measured and simulated urban water-year peak flows cannot be completely explained by climatic differences between 

the periods of record. 
2 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Introduction 

Knowledge of flood characteristics of streams is needed 

for the design of roadway drainage structures, the establishment 

of flood-insurance rates, and for other uses by urban planners 

and engineers. Because urbanization can produce significant 

changes in the flood-frequency characteristics of streams, rural 

basin flood-frequency relations are not applicable to urban 

streams. 

Urban flood-frequency equations were developed by 

Bohman (1992) for small urban streams in South Carolina using 

simulated peak-flow data from rainfall-runoff models. Recognizing the importance of measured data for comparison and verification of these equations, the U.S. Geological Survey 

(USGS), in cooperation with the South Carolina Department of 

Transportation (SCDOT), continued to collect data at many of 

the urban streamflow-gaging stations that were established during Bohman’s investigation. Comparisons of the measured and 

simulated peak-flow data indicated that there was enough of a 

significant difference in the two data sets to warrant updating 

the urban flood-frequency estimates using only the measured 

data. This investigation documents these data comparisons and 

updates the urban flood-frequency equations for South Carolina 

using measured data collected through the 2001 water year1

. 

There are several ways to continue improving the understanding of urban flood-frequency in South Carolina and to 

increase the confidence in future statistical analyses of wateryear peak flows. Hereafter in this report, “peak flow” refers to 

the maximum peak for the water year. One way is to expand the 

database used for estimating the magnitude and frequency of 

floods on small urban streams by continuing to collect streamflow data at existing urban streamflow-gaging stations, which 

will increase the length of record used in the analysis. Additionally, as funding is available and where appropriate, other 

streamflow-gaging stations used in the previous urban floodfrequency investigation could be considered for reactivation. It 

also may be worthwhile to review the geographical coverage of 

the urban streamflow network and consider the benefits of activating additional new urban streamflow-gaging stations, which 

will not only improve the geographical coverage of the State, 

but also increase the number of streamflow-gaging stations in 

the database. An extended monitoring network and database is 

likely to provide more accurate flood-frequency equations for 

use in design and planning. 

Purpose and Scope 

This report describes the comparison of peak flows measured through water year 2001 with peak flows computed by 

using a rainfall-runoff model during the previous urban floodfrequency investigation (Bohman, 1992). The flood-frequency 

estimates for 20 streamflow stations were updated using measured peak-flow data. Methods are presented for predicting the 

magnitude and frequency of floods in South Carolina at 

ungaged urban basins in the Piedmont, upper Coastal Plain, and 

lower Coastal Plain physiographic provinces. Statistics describing the uncertainty in the prediction equations are presented and 

the limitations of the equations also are discussed. 

Previous Investigations 

Speer and Gamble (1964) documented the earliest investigation of flood frequency of streams in South Carolina. They 

presented methods for estimating the magnitude of floods for 

selected recurrence intervals for rural streams in the South 

Atlantic slope basin, which extends from the James River in 

Virginia to the Savannah River along the South CarolinaGeorgia State boundary. Whetstone (1982) used multiple 

regression analyses to define the relation between flows and 

basin characteristics at recurrence intervals of 2, 5, 10, 25, 50, 

and 100 years for unregulated rural streams with drainage areas 

greater than 1.0 square mile (mi2). Sauer and others (1983) used 

data from 269 gaged basins in 56 cities in 31 states to develop 

flood-frequency relations for urban watersheds in the United 

States. Frequencies of peak flows were regionalized by Guimaraes and Bohman (1991) using generalized least-squares regression methods to define the relation of magnitude and frequency 

of flows to various basin characteristics on ungaged rural 

streams that were not affected significantly by regulation. 

Bohman (1992) described methods for determining peakflow frequency relations, flood hydrographs, average basin lag 

times, and runoff volumes associated with a given peak flow for 

ungaged urban basins by using data from 34 streamflow-gaging 

stations in 15 cities in South Carolina, North Carolina, and 

Georgia. A rainfall-runoff model was calibrated for 23 urban 

drainage basins in South Carolina. The model was then used to 

synthesize from 50 to 70 annual peaks, depending on the length 

of the long-term rainfall data from nearby National Weather 

Service stations. The logarithms of these peaks were fitted to a 

Pearson Type-III distribution to determine the frequency of 

peak discharges having recurrence intervals of 2, 5, 10, 25, 50, 

100, and 500 years at each streamflow-gaging station. The final 

step in analyzing these data was to develop regression equations 

that could be used to predict the magnitude and frequency of 

floods at ungaged urban sites in South Carolina. Detailed 

descriptions of the rainfall-runoff model calibration, the longterm simulation, and the regression analyses are provided in 

Bohman’s (1992) report. 

Feaster and Tasker (2002) used generalized least-squares 

regression to develop a set of predictive equations that can be 

used to estimate flows at the 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 

500-year recurrence intervals for rural ungaged basins in the 

1A water year is the 12-month period from October 1 to September 30 and is designated by the calendar year in which it ends. Thus, the 12-month period 

ending September 30, 2001, is the 2001 water year. 
Peak-Flow Data 3 

Blue Ridge, Piedmont, and Coastal Plain physiographic provinces of South Carolina. In addition, a region-of-influence 

(ROI) method also was developed to interactively estimate the 

recurrence-interval flows for rural ungaged basins. The predictive abilities of the regional regression equations were compared with the ROI methods for each physiographic province in 

South Carolina. The ROI method performed systematically better only in the Blue Ridge, which limits its usefulness only to 

that province. 

Peak-Flow Data 

The peak flows collected at USGS streamflow-gaging stations are the empirical basis for estimating specific recurrenceinterval flows for this investigation. As recommended in Bulletin 17B (Hydrology Subcommittee of the Interagency Advisory 

Committee on Water Data, 1982), only streamflow-gaging 

stations with at least 10 years of measured peak flows were used 

to develop flood-frequency estimates. Of the 34 streamflowgaging stations used in the Bohman (1992) investigation, 

streamflow-gaging stations with sufficient lengths of record 

to be included in this investigation included 16 of the 23 

South Carolina streamflow-gaging stations, 3 of the 7 Georgia 

streamflow-gaging stations, and all 4 North Carolina 

streamflow-gaging stations (table 1; fig. 1). After reviewing 

the peak-flow data, data from 3 of the 16 South Carolina 

streamflow-gaging stations were excluded from the analysis 

for reasons explained later. 

At continuous-record streamflow-gaging stations, the 

water-surface elevation, or stage, of the stream is recorded at 

fixed intervals typically ranging from 5 to 60 minutes. At creststage, partial-record streamflow-gaging stations, only the crest 

(highest) stages that occur between site visits, usually 6 to 8 

weeks, are recorded. An attempt is made to measure streamflow 

throughout the range of recorded stages. If this is possible, a 

relation between stage and streamflow is developed for the 

gaged site. Using this stage-streamflow relation, or rating, 

streamflows for recorded stages are estimated. Because stream 

channels are dynamic, periodic streamflow measurements are 

made to verify that the hydraulic conditions at the site remain 

stable. If the measurements indicate that conditions have 

changed, additional data are collected and used to make adjustments to the stage-streamflow relation. At some crest-stage 

sites, indirect flow-computation methods are used to develop a 

theoretical rating. This method has been used extensively to 

compute streamflows for small drainage areas, which are typical of urban streams (Bodhaine, 1968). 

Initial reviews of the peak-flow data for the South Carolina 

streamflow-gaging stations included comparing the peak flows 

Table 1. Streamflow-gaging stations in South Carolina, North Carolina, and Georgia with 10 or more years of record used 

in the flood-frequency analysis for small urban streams in South Carolina. 

[mi2, square miles] 

Station 

) 

Station name 

2

) record Longitude Description 

number 

(fig. 1

Location Drainage 

area 

(mi

Period of 

Latitude 

South Carolina 

02110740 Midway Swash at Myrtle 33° 39'44'' 78° 55'25'' Horry County, at culvert on 0.80 1987–2001 

Beach, S.C. U.S. Highway 17 

02131130 Gully Branch at Florence, S.C. 34° 53'00'' 79° 46'12'' Florence County, at culvert on 1.92 1985–2001 

Cherokee Road 

02135518 Turkey Creek at Sumter, S.C. 33° 55'13'' 80° 19'43'' Sumter County, at culvert on 2.20 1987–2001 

East Liberty Street 

02145940 Little Dutchman Creek tribu- 34° 58'34'' 81° 01'02'' York County, at culvert on 3.50 1986–97 

tary at Rock Hill, S.C. Celanese Road 

02159785 Fairforest Creek tributary at 34° 57'10'' 81° 57'57'' Spartanburg County, at culvert .52 1987–2001 

Spartanburg, S.C. on Secondary Road 485 

02162093 Smith Branch at Columbia, 34° 01'38'' 81° 02'31'' Richland County, at culvert on 5.49 1977–2001 

S.C. North Main Street 

02164011 Brushy Creek (Reedy River 34° 49'25'' 82° 24'26'' Greenville County, at culvert 3.02 1985–2001 

tributary) at Greenville, S.C. on Grove Road 
4 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 1. Streamflow-gaging stations in South Carolina, North Carolina, and Georgia with 10 or more years of record used 

in the flood-frequency analysis for small urban streams in South Carolina. — Continued 

[mi2, square miles] 

Station 

) 

Station name 

2

) record Longitude Description 

number 

(fig. 1

Location Drainage 

area 

(mi

Period of 

Latitude 

South Carolina (Continued) 

02166975 Sample Branch at Greenwood, 34° 12'56'' 82° 09'20'' Greenwood County, at culvert 1.16 1986–2001 

S.C. on U.S. Highway 178 bypass 

02167020 Crane Creek tributary at 34° 03'02'' 81° 02'05'' Richland County, at culvert on .28 1986–2001 

Columbia, S.C. Carola Street 

02168845 Saluda River tributary at 34° 02'26'' 81° 08'29'' Richland County, at culvert on .45 1986–96 

Columbia, S.C. Bush River Road 

02169568 Pen Branch at Columbia, S.C. 34° 00'46'' 80° 58'56'' Richland County, at culvert on 2.26 1986–2001 

Brentwood Street 

02173491 Hess Branch at Orangeburg, 33° 30'12'' 80° 52'41'' Orangeburg County, at culvert .45 1987–2001 

S.C. on Middleton Road 

02176380 Coosawhatchie River tributary 32° 59'53'' 81° 19'01'' Allendale County, at culvert 2.06 1986–2001 

at Allendale, S.C. on Secondary Road 129 

North Carolina 

02146300 Irwin Creek near Charlotte, 35° 11'52'' 80° 54'16'' Mecklenburg County, on left 30.5 1963–77 

N.C. bank at city of Charlotte 

sewage-disposal plant 

02146500 Little Sugar Creek near 35° 09'13'' 80° 51'18'' Mecklenburg County, on right 41.0 1962–77 

Charlotte, N.C. bank upstream from bridge 

on Tyvola Road at city of 

Charlotte sewage-disposal 

plant 

02146600 McAlpine Creek at Sardis Road 35° 08'16'' 80° 45'03'' Mecklenburg County, near left 38.3 1962–77 

near Charlotte, N.C. bank on downstream end of 

bridge pier at Sardis Road 

(Secondary Road 3356) 

02146700 McMullen Creek at Sharon 35° 08'27'' 80° 49'12'' Mecklenburg County, on left 6.98 1963–77 

View Road near Charlotte, bank downstream of culvert 

N.C. wingwall at Sharon View 

Road (Secondary Road 

3673) 

Georgia 

02196760 Rocky Creek tributary at 33° 27'07'' 82° 02'57'' Richmond County, at culvert 1.56 1979–96 

Augusta, Ga. on U.S. Highways 78 and 

278 

02203543 Wilshire Canal near Savannah, 31° 59'27'' 81° 08'15'' Chatham County, at culvert on .95 1979–96 

Ga. Tibet Avenue 

02203544 Wilshire Canal tributary near 31° 58'25'' 81° 08'20'' Chatham County, at culvert on .18 1979–96 

Savannah, Ga. Windsor Road 
Figure 1. Location of streamflow-gaging stations with 10 or more years of record in urban areas of the Piedmont and upper and low

physiographic provinces of South Carolina, Georgia, and North Carolina.
6 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

listed in the respective station peak-flow file with those 

in the station’s water-year analysis file. Part of this review 

included a visual inspection of the peak-flow data plotted by 

water year. Along with the visual inspection, a Kendall tau 

statistic was used to assess the homogeneity of the record at 

each streamflow-gaging station. 

According to Rantz and others (1982), a rating should not 

be extended more than two times the maximum measured flow 

unless no other options are available. Therefore, at streamflowgaging stations with established ratings, the flow-measurement 

files were reviewed to determine if the ratings had been excessively extended. The peak flows were plotted against water year 

along with the maximum measured flow and the flow equal to 

two times the maximum measured flow. This plot was used to 

review peaks that may have been estimated from an excessive 

stage-flow rating extension, but not updated if and when the rating was later defined by greater flow measurements. 

From these initial reviews, three of the South Carolina 

streamflow-gaging stations were excluded from the analysis: 

station 02160325 (Brushy Creek near Greenville, S.C.), station 

02169505 (Rocky Branch at Columbia, S.C.), and station 

02173495 (Sunnyside Canal at Orangeburg, S.C.). At station 

02160325 (Brushy Creek near Greenville), 8 of the 15 peak 

flows exceeded the station rating by more than two times the 

maximum measured flow. At station 02169505 (Rocky Branch 

at Columbia), the stage-flow relation is incomplete at high 

stages because physical conditions prevent the flow from being 

computed. At station 02173495 (Sunnyside Canal at Orangeburg), a trend was detected in the measured data. An exposed 

pipeline located downstream from this gage often catches 

debris, which may produce backwater at the gage. As a result of 

the data trend and problems noted during gage inspections, this 

station was excluded from the analysis. 

Graphical Comparisons of Measured and Simulated 

Peak Flows 

For each of the 13 South Carolina streamflow-gaging stations (fig. 1; table 1), a comparison was made between the measured peak-flow data and the simulated peak-flow data generated by using a rainfall-runoff model (Bohman, 1992). A 

graphical comparison was made for each streamflow-gaging 

station by plotting the two sets of data together by water year 

(fig. 2). At most of the 13 streamflow-gaging stations, the plots 

show a considerable difference in the variance of the measured 

and simulated peaks. Note in figure 2 for streamflow-gaging 

stations 02159785, 02166975, 02173491, and 02176380 that 

two sets of simulated data are given. Bohman (1992) noted that 

the rainfall-runoff model showed little sensitivity of volumes 

and peaks to the evaporation data sets used to synthesize longterm hydrographs but was sensitive to the long-term rainfall 

record chosen for a basin. Usually, data from the closest longterm rainfall and evaporation stations were used to synthesize 

the long-term hydrographs. Bohman (1992) noted that “Even in 

cases where both long-term rainfall stations seemed to be 

located in physiographically and meteorologically similar settings, substantially different results were obtained when each 

rainfall-data set was applied to the calibrated basin models.” 

Therefore, for study basins located between rainfall stations 

where such disparity in results occurred, the flood-frequency 

estimates were interpolated by weighting the results inversely 

proportional to the distance between the site and the two rainfall 

stations. 

In the previous investigation by Bohman (1992), five 

National Weather Service rainfall stations were used in the synthesis of long-term flood-hydrograph data, with periods of 

record ranging from 49 to 89 years (table 2). 

Table 2. National Weather Service rainfall stations used in Bohman’s (1992) 

synthesis of long-term flood-hydrograph data. 

Station number Location of record 

Number of years Period of record 

320800081120050 Savannah, Ga. 89 1898–1987 

332200081580050 Augusta, Ga. 72 1902–73 

340000081030001 Columbia, S.C. 53 1901–53 

345000082240001 Greenville-Spartanburg, S.C. 49 1918–71 

351400080560001 Charlotte, N.C. 68 1901–69 
Peak-Flow Data 7 

Figure 2. Comparison of measured and simulated peak flows at 13 urban streamflow-gaging stations in South Carolina. 
8 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 2. (Continued) Comparison of measured and simulated peak flows at 13 urban streamflow-gaging stations in South Carolina. 
Peak-Flow Data 9 

Figure 2. (Continued) Comparison of measured and simulated peak flows at 13 urban streamflow-gaging stations in South Carolina. 
10 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 2. (Continued) Comparison of measured and simulated peak flows at 13 urban streamflow-gaging stations in South Carolina. 
Peak-Flow Data 11 

Figure 2. (Continued) Comparison of measured and simulated peak flows at 13 urban streamflow-gaging stations in South 

Carolina. 

To determine if the variance differences between the measured peak-flow data and the simulated peak-flow data could be 

related to climatic differences in the collection periods of measured data and at the long-term raingages, a review was made of 

16 rural streamflow-gaging stations that were included in the 

South Carolina rural flood-frequency analysis (Feaster and 

Tasker, 2002). These streamflow-gaging stations all have systematic record lengths through water year 2001 ranging from 42 

to 74 years (table 3; fig. 3). 

The peak-flow data for the rural streamflow-gaging stations were plotted for two periods—from the beginning of the 

record to 1984 and from 1985 to 2001 (fig. 4). The 1985–2001 

period was chosen because 1985 was the earliest beginning year 

for 12 of the 13 South Carolina urban streamflow-gaging stations. Station 02162093 (Smith Branch at Columbia, S.C.) is the 

only continuous-record streamflow-gaging station of the 13 

South Carolina urban streamflow-gaging stations with the 

period of record beginning in 1977. The plots of the peak flows 

at the rural streamflow-gaging stations do not show significant 

differences between the two periods, which suggests that the 

differences in the variances of the simulated and measured data 

at the 13 South Carolina urban streamflow-gaging stations are 

not related to climatic differences. 
12 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 3. Rural streamflow-gaging stations that were used to compare climatic conditions with similar periods at the 

13 South Carolina urban streamflow-gaging stations, by physiographic province. 

number 

(fig. 3) 

Number of Period of 

Piedmont 

Station 

Station name Drainage area 

(square miles) years of 

record record 

02147500 Rocky Creek at Great Falls, S.C. 194 45 1952–2001 

02154500 North Pacolet River at Fingerville, S.C. 116 71 1931–2001 

02160000 Fairforest Creek near Union, S.C. 183 60 1940–2001 

02162500 Saluda River near Greenville, S.C. 295 57 1942–2001 

02163500 Saluda River near Ware Shoals, S.C. 581 63 1939–2001 

02165000 Reedy River near Ware Shoals, S.C. 236 61 1940–2001 

02192500 Little River near Mt. Carmel, S.C. 217 59 1940–2001 

02196000 Stevens Creek near Modoc, S.C. 545 58 1940–2001 

02130900 

02132500 

02173000 

02173500 

02174000 

Upper Coastal 

Black Creek near McBee, S.C. 

Little Pee Dee River near Dillon, S.C. 

South Fork Edisto River near Denmark, S.C. 

North Fork Edisto River at Orangeburg, S.C. 

Edisto River near Branchville, S.C. 

Plain

108 

524 

720 

683 

1,720 

42 

61 

69 

63 

56 

1960–2001 

1940–2001 

1932–2001 

1939–2001 

1946–2001 

Lower Coastal Plain 

02110500 Waccamaw River near Longs, S.C. 1,110 51 1951–2001 

02136000 Black River at Kingstree, S.C. 1,250 74 1928–2001 

02176500 Coosawhatchie River near Hampton, S.C. 203 51 1952–2001 
Figure 3. Location of cities, rural streamflow-gaging stations used for comparison with urban streamflow-gaging stations, and phySouth Carolina.
14 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 4. Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001) at 16 rural streamflow-gaging stations 

in South Carolina. 
Peak-Flow Data 15 

Figure 4. (Continued) Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001) at 16 rural streamflowgaging stations in South Carolina. 
16 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 4. (Continued) Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001) at 16 rural streamflowgaging stations in South Carolina. 
Peak-Flow Data 17 

Figure 4. (Continued) Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001) at 16 rural streamflowgaging stations in South Carolina. 
18 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 4. (Continued) Comparison of two peak-flow periods (beginning of record to 1984 and 1985–2001) at 16 rural streamflowgaging stations in South Carolina. 
Peak-Flow Data 19 

Figure 4. (Continued) Comparison of two peak-flow periods (beginning of record to 1984 and 1985 –2001) at 16 rural 

streamflow-gaging stations in South Carolina. 

Statistical Comparisons of Measured and Simulated 

Peak Flows 

Along with the graphical comparisons of the simulated 

and measured peak flows at the 13 South Carolina urban 

streamflow-gaging stations, several statistical tests also were 

used to compare the data. The statistical analyses and computations were made using procedures defined by the SAS Institute, 

Inc. (1990). The statistical analyses were performed for a 

p-value of 0.05. The p-value, also known as the level of significance, is the probability of obtaining the computed test statistic, or one even less likely, when the null hypothesis is true. The 

null hypothesis is what is assumed to be true about the data until 

indicated otherwise. It usually states the “null” situation—no 

difference between groups (Helsel and Hirsch, 1995). In the 

case of equal variances, the null hypothesis is that the variances 

between the measured and simulated data are equal. Thus for 

p-values of less than 0.05, the null hypothesis is rejected, and it 

is reported that there is a statistically significant difference in 

the variances at the 0.05 level. In other words, there is less than 

a 5-percent chance that the variances are equal. 

Kendall Tau Trend Analysis 

One of the assumptions used in the flood-frequency analysis is that the watershed of a streamflow-gaging station does not 

change significantly through the data-collection period; as such, 

the peak-flow characteristics at each streamflow-gaging station 

are homogeneous or do not significantly change over time. The 

Kendall tau statistical test was chosen to assess the homogeneity of the record at each streamflow-gaging station. The Kendall 

tau trend analysis was used to determine if a trend exists in the 

data by measuring the correlation of the peak flow and years 

(time). The Kendall tau test is based on a ranking system and 

measures the strength of the monotonic relation between two 

variables. In a monotonic relation, successive values in a 

sequence either consistently increase or decrease but do not 

oscillate in relative value. Being rank based, the Kendall tau statistical test is resistant to the effect of a small number of unusual 

values (Helsel and Hirsch, 1995). 

Measures of monotonic correlation (τ ) are characterized 

by being dimensionless and scaled between positive one and 

negative one. When the two variables are not correlated, τ 

equals zero. When one variable increases with the increase of 

the other variable, τ is a positive number, and when the two 

variables vary in opposite directions, τ is negative. When one 

variable is a measure of time or location, correlation becomes a 

test for temporal or spatial trend. The significance of the correlation is evaluated by forming a null hypothesis that the coefficient is zero against the alternative that it is nonzero, then computing the probability of rejecting the null hypothesis (Helsel 

and Hirsch, 1995). The probability of rejecting the null hypothesis was computed to a 0.05 level of significance. 

Results of the trend analysis of the relation between peak 

flows and time are shown in table 4. Both the measured and 

simulated data were analyzed. A trend in the simulated data 

would indicate a significant change in rainfall patterns over 

time. A trend in the measured data, in the absence of a climatic 

trend, would indicate a change in urbanization over time. For 

the 0.05 level of significance, there were no statistically significant trends in either the measured or the simulated data 

(table 4), although two streamflow-gaging stations were considered borderline cases with p-values at the 0.05 level (simulated data at station 02135518 and measured data at station 

02176380). 
20 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 4. Summary of statistical trends in water-year peak flows F-test for Equality of Variances 

for 13 urban streamflow-gaging stations in South Carolina. 

record 

Kendall-tau 

p-value 

Simulated data 

Station number 

(fig. 1) 

Period of 

value 

The F-test was chosen to test the equality of variances for 

the simulated and measured peak-flow data at the 0.05 level of 

significance. A random variable that consists of the ratio of two 

sample variances has an F distribution if the two samples are 

independent and from normal populations with equal population 

02110740 1898–1986 0.05 0.53 variances (Iman and Conover, 1983). Therefore, before testing 

for equal variances, a univariate procedure was used to deter- 02131130 1901–53 -.16 .09 mine if all distributions were normal according to the Shapiro02135518 1901–53 -.19 .05 Wilks statistic (SAS Institute, Inc., 1990). All peak-flow data 

were transformed to logarithmic units before conducting the 

02145940 1902–69 -.05 .57 statistical analysis and computations. For 8 of the 13 urban 

streamflow-gaging stations, the Shapiro-Wilks statistic showed 02159785 1918–71 .01 .90 that the logarithms of the measured and simulated peak flows 

02162093 1902–53 -.18 .06 were normally distributed (table 5). For seven of the eight 

streamflow-gaging stations with normal distributions, the F-test 

02164011 1918–71 .03 .74 indicated that there was a statistically significant difference in 

02166975 1902–74 .01 .92 the variances of the measured and simulated data. For station 

02169568, the F-test indicated that there was no statistically sig- 02167020 1901–53 -.17 .07 nificant difference in the variances at the 0.05 level of significance. At three of the streamflow-gaging stations where the 02168845 1901–53 -.14 .13 

Shapiro-Wilks statistical test indicated that the peaks were not 

02169568 1901–53 -.18 .06 normally distributed at the 0.05 level of significance, the pvalues were not much lower than 0.05 (0.04, 0.02, and 0.01). 02173491 1902–74 .01 .92 Although the F-test is not technically correct for data sets that 

02176380 1903–73 .02 .86 are not normally distributed, the F-test was still used as an indicator for the three streamflow-gaging stations and showed that a Measured data statistically significant difference in the variances of the mea02110740 1987–2001 0.35 0.07 sured and simulated data occurred at all three streamflowgaging stations. For station 02167020, the Shapiro-Wilks statis- 02131130 1985–2001 -.14 .43 tical test indicated that neither the measured nor the simulated 

02135518 1986–2001 -.10 .59 data were normally distributed; therefore, the F-test was not 

used to compare the variances. It is clear from the plot of the 

02145940 1986–97 -.02 .94 measured and simulated peak-flow data, however, that there is a 

significant difference in the variances (fig. 2). 02159785 1987–2001 .13 .51 For comparison purposes, similar statistics were computed 

02162093 1977–2001 .01 .94 for the two periods at the 16 rural streamflow-gaging stations 

listed in table 3 and shown in figures 3 and 4. The Shapiro- 02164011 1985–2001 .30 .09 Wilks statistic indicated that the logarithms of the peak-flow 

02166975 1986–2001 .13 .50 data at 14 of the 16 streamflow-gaging stations were distributed 

normally for the two periods (table 6). The Shapiro-Wilks statis- 02167020 1986–2001 .06 .75 tic indicated that the peak-flow data at station 02192500 for the 

1985–2001 period were not distributed normally. A series of 02168845 1986–96 -.34 .11 

peak flows at a streamflow-gaging station may include low or 

02169568 1986–2001 .13 .50 high outliers, which are data points that depart significantly 

from the range of the remaining data. Based on the flood- 02173491 1986–2001 .13 .50 frequency analysis at station 02192500, the 1988 peak flow was 

02176380 1986–2001 -.37 .05 a low outlier. Therefore, the Shapiro-Wilks test was conducted 

with the 1988 peak flow excluded, and the test statistic indicated 
Table 5. Results of comparison testing of measured and 

simulated peak-flow data at 13 urban streamflow-gaging stations 

in South Carolina based on the Shapiro-Wilks test for normality and 

the F-test for equal variances. 

[Values in parentheses are p-values at the 0.05 level of significance; <, less 

than] 

number 

( ) 

Are the 

logarithms of 

distributed? 

Are the 

logarithms of 

the simulated 

distributed? 

statistically 

significant 

difference in 

Piedmont 

Station 

fig. 1

the measured 

data normally data normally 

Is there a 

variances? 

02145940 yes (0.46) yes (0.15) yes (0.005) 

02159785 yes (0.54) ayes, yes ayes, no 

(0.18, 0.09) (0.04, 0.08) 

02164011 yes (0.27) yes (0.50) yes (0.001) 

02166975 yes (0.73) ayes, yes ayes, yes 

(0.34, 0.28) (0.001, 

0.001) 

02168845 no (0.04) yes (0.69) byes (0.0002) 

Upper Coastal Plain 

02162093 yes (0.64) yes (0.12) yes (0.001) 

02167020 no (0.002) no (0.03) did not compute 

02169568 yes (0.31) yes (0.71) no (1.00) 

02173491 no (0.02) ayes, yes a, byes, no 

(0.25, 0.30) (0.04, 0.01)) 

Lower Coastal Plain 

02110740 yes (0.80) yes (0.31) yes (0.02) 

02131130 no (0.01) yes (0.18) byes (<0.0001) 

02135518 yes (0.32) yes (0.92) yes (0.003) 

02176380 yes (0.35) ano, no a, byes, yes 

(0.003, 0.01) (0.02, 0.002) 

a

Two sets of peak flows were synthesized using rainfall data from two 

different long-term rainfall gages. 

b

Because the Shapiro-Wilks test indicated that one of the data sets was not 

normally distributed, the F-test is not technically valid but was still used as an 

indicator. 

Peak-Flow Data 21 

that the logarithms of the peak data were distributed normally. 

In addition, the Shapiro-Wilks statistic indicated that the logarithms of the peak-flow data at station 02173000 from the 

beginning of the record to water year 1984 were not distributed 

normally. Based on the flood-frequency analysis at station 

02173000, there were two large peaks that exceeded the highoutlier threshold. When the Shapiro-Wilks test was conducted 

excluding those two peaks, the logarithms of the data from the 

beginning of the record to 1984 were distributed normally. 

Therefore, the F-test also was conducted on stations 02192500 

and 02173000. The F-test indicated that there was no statistically significant difference in the variances for the 1985–2001 

period or for the beginning of record to 1984 period for 14 of the 

16 rural streamflow-gaging stations. For the 1985–2001 period 

at station 02130900, the log-Pearson Type-III analysis indicated 

that the 1991 peak was a high outlier. Consequently, the F-test 

was run excluding the 1991 peak, and the results indicated that 

there was no statistically significant difference in the period 

from 1985 to 2001 and the beginning of record to 1984. The 

F-test indicated that there was a statistically significant difference in the variances in the peak-flow data for the two periods 

at station 02160000. However, the plot of the data in figure 4 

indicates that the variances do not appear to be drastically different. 

Overall, the F-test indicated that there is a statistically 

significant difference in the variances at 11 of the 13 urban 

streamflow-gaging stations. The F-test indicated no significant 

difference at one of the urban streamflow-gaging stations. The 

remaining streamflow-gaging station was not analyzed because 

the Shapiro-Wilks statistic indicated that the data were not distributed normally. However, for the rural streamflow-gaging 

stations, the F-test indicated that there were no statistically significant differences in the variances at 14 of the 16 streamflowgaging stations. At station 02130900, a statistically significant 

difference in the variances of the two periods was present; 

however, when analyzed without the 1991 peak, which was a 

high-outlier in the log-Pearson analysis, there was no significant 

difference in the variances. Consequently, this comparison 

suggests that differences in the variances between the measured 

and simulated peak-flow data at the urban streamflow-gaging 

stations cannot be solely attributed to changes in climatic 

conditions. 
22 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 6. Results of comparison testing of peak-flow data from 16 rural streamflow-gaging stations in South 

Carolina for two periods (beginning of record to 1984 and 1985–2001) based on the Shapiro-Wilks test for 

normality and the F-test for equal variances. 

[mi2, square miles; Values in parentheses are p-values at the 0.05 level of significance] 

Station 

number 

( ) 

area 

(mi2

) 

the data from 

normally 

the data from 

the beginning of 

normally 

statistically 

significant 

difference in 

fig. 3

Drainage Period of 

measured data 

Are the 

logarithms of 

1985 to 2001 

distributed? 

Are the 

logarithms of 

record to 1984 

distributed? 

Is there a 

variances? 

Piedmont 

02147500 194 1952–2001 

02154500 116 1931–2001 

02160000 183 1940–2001 

02162500 295 1942–2001 

02163500 581 1939–2001 

02165000 236 1940–2001 

02192500 217 1940–2001 

02196000 545 1940–2001 

yes (0.69) 

yes (0.28) 

yes (0.26) 

yes (0.12) 

yes (0.25) 

yes (0.51) 

no (0.02) 

yes (0.99) 

yes (0.07) no (0.20) 

yes (0.92) no (0.19) 

yes (0.28) yes (0.002) 

yes (1.00) no (0.11) 

yes (0.09) no (0.31) 

yes (0.60) no (0.92) 

yes (0.87) no (0.052) 

yes (0.70) no (0.43) 

Upper Coastal Plain 

02130900 108 1960–2001 

02132500 524 1940–2001 

02173000 720 1932–2001 

02173500 683 1939–2001 

02174000 1,720 1946–2001 

yes (0.17) 

yes (0.63) 

yes (0.46) 

yes (0.12) 

yes (0.44) 

yes (0.72) 

yes (0.57) 

no (0.0004) 

yes (0.62) 

yes (0.81) 

yes (0.002) 

no (0.94) 

no (0.89) 

no (0.78) 

no (0.57) 

Lower Coastal Plain 

02110500 1,110 1951–2001 yes (0.98) yes (0.55) no (0.14) 

02136000 1,250 1928–2001 yes (0.29) yes (0.37) no (0.39) 

02176500 203 1952–2001 yes (0.91) yes (0.58) no (0.23) 
Wilcoxon Rank Sum Test 

The Wilcoxon rank sum test, also know as the MannWhitney or Wilcoxon-Mann-Whitney test, was used to check 

for statistically significant differences in the means of the measured and simulated peak-flow data. The Wilcoxon test is similar to the t-test except that it is applied to the ranks of the data 

rather than to the data values. In addition, the t-test assumes that 

both groups of data are distributed normally and that the variances are the same (Helsel and Hirsch, 1995). The Wilcoxon test 

makes no such assumptions about how the data are distributed 

nor does it require that the groups have the same variance. The 

statistical procedure used to perform the computations was 

NPAR1WAY (SAS Institute, Inc., 1990). 

For the urban streamflow-gaging stations in South Carolina, the Wilcoxon test indicated that there was no statistically 

significant difference in the mean values of the measured and 

simulated peak-flow data at 8 of the 13 streamflow-gaging stations (about 62 percent) at the 0.05 level of significance 

(table 7). As can be seen in table 7, there appears to be no bias 

with respect to physiographic province or drainage-area size. 

The logarithmic data also are presented in box plots (fig. 5). 

Although there is no statistically significant difference in the 

mean values at 8 of the 13 urban streamflow-gaging stations, 

once again it is clear that there is a significant difference in the 

variance of the two groups. 

For comparison with the urban streamflow-gaging stations, 

the 16 rural streamflow-gaging stations used in the comparison 

of variances also were analyzed using the Wilcoxon rank sum 

test. As with the comparison of variances, the period from 1985 

to 2001 was compared with the period from the beginning of 

record to 1984. The Wilcoxon test indicated that 12 of the 16 

rural streamflow-gaging stations had no statistically significant 

difference in mean peak-flow values at the 0.05 level of significance (table 8; fig. 6). Of the four streamflow-gaging stations 

that had a significant difference between the periods, station 

02174000 had no significant difference when analyzed without 

the historic peak that occurred in water year 1928. Consequently, 81 percent of the rural streamflow-gaging stations had 

no statistically significant difference in mean peak flows from 

1985 to 2001 and from the beginning of station record to 1984. 

Overall, the results of the Wilcoxon test of the peak-flow 

data for the urban and rural streamflow-gaging stations were 

somewhat similar, with 62 percent of the urban streamflowgaging stations and 81 percent of the rural streamflow-gaging 

stations showing no statistically significant difference in the 

mean values. At 31 percent of the urban streamflow-gaging 

stations, the mean peak flows for the measured data were greater 

than the mean peak flows for the simulated data, with the average difference being about 18 percent. For 25 percent of the 

rural streamflow-gaging stations, the mean peak flows for 

1984–2001 were greater than the mean peak flows for the beginning of record to the 1984 period, with the average difference 

Peak-Flow Data 23 

Table 7. Results of comparison testing of measured and 

simulated peak-flow data from 13 urban streamflow-gaging 

stations in South Carolina based on the Wilcoxon rank 

sum test. 

[Values in parentheses are p-values at the 0.05 level of significance] 

number 

(fig. 1) ) the means? 

Piedmont 

Station Drainage area 

(in square 

miles

Does the Wilcoxon test 

indicate a statistically 

significant difference in 

02145940 3.50 no (0.14) 

02159785 .52 ano, no 

(0.32, 0.39) 

02164011 3.02 no (0.22) 

02166975 1.16 a

yes, yes 

(0.0001, 0.001) 

02168845 .45 yes (0.01) 

Upper Coastal Plain 

02162093 5.49 no (0.35) 

02167020 .28 no (0.27) 

02169568 2.26 yes (0.0004) 

02173491 .45 a

yes, yes 

(0.0498, 0.002) 

Lower Coastal Plain 

02110740 0.80 no (0.43) 

02131130 1.92 no (0.36) 

02135518 2.20 no (0.46) 

02176380 2.06 a

no, yes 

(0.49, 0.006) 

a

Two sets of peak flows were synthesized for this station using rainfall data from two different long-term rainfall gages. 

being about 12 percent. Consequently, with respect to mean 

peak flows, the simulated and measured data at the urban 

streamflow-gaging stations had similar bias as the rural 

streamflow-gaging stations for the two periods compared. Four 

of the rural streamflow-gaging stations had peak-flow values 

that were high outliers based on the log-Pearson analysis— 

stations 02130900, 02173000, 02173500, and 02174000. The 

high outliers were excluded in the computations of the mean 

peak-flow values at these streamflow-gaging stations. 
24 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 5. Box plots of the logarithms of the measured and simulated peak-flow data at 13 urban streamflow-gaging stations in 

South Carolina. 
Peak-Flow Data 25 

Figure 5. (Continued) Box plots of the logarithms of the measured and simulated peak-flow data at 13 urban streamflow-gaging 

stations in South Carolina. 
26 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 5. (Continued) Box plots of the logarithms of the measured and simulated peak-flow data at 13 urban streamflow-gaging 

stations in South Carolina. 
Peak-Flow Data 27 

Figure 5. (Continued) Box plots of the logarithms of the 

measured and simulated peak-flow data at 13 urban 

streamflow-gaging stations in South Carolina. 

Table 8. Results of comparison testing of peak-flow data from 16 rural streamflow-gaging stations in South 

Carolina for two periods (beginning of record to 1984 and 1985 – 2001) based on the Wilcoxon rank sum test. 

[Values in parentheses are p-values at the 0.05 level of significance] 

) 

Piedmont 

Station number 

(fig. 3

Drainage area 

(in square miles) 

Does the Wilcoxon test indicate a statistically 

significant difference in the means? 

02147500 194 no (0.055) 

02154500 116 no (0.17) 

02160000 183 no (0.27) 

02162500 295 no (0.14) 

02163500 581 yes (0.02) 

02165000 236 no (0.053) 

02192500 217 no (0.36) 

02196000 545 no (0.10) 

Upper Coastal Plain 

02130900 108 no (0.15) 

02132500 524 no (0.20) 

02173000 720 yes (0.01) 

02173500 683 yes (0.01) 

02174000 1,720 a

yes, no (0.04, 0.052) 

Lower Coastal Plain 

02110500 1,110 no (0.34) 

02136000 1,250 no (0.23) 

02176500 203 no (0.31) 

a

Analytical results exclude the 1928 peak, which was a high outlier. 
28 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 6. Box plots of the logarithms for two peak-flow periods (beginning of record to 1984 and 1985 –2001) at 16 rural streamflowgaging stations in South Carolina. 
Peak-Flow Data 29 

Figure 6. (Continued) Box plots of the logarithms for two peak-flow periods (beginning of record to 1984 and 1985 –2001) at 16 rural 

streamflow-gaging stations in South Carolina. 
30 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001


Figure 6. (Continued) Box plots of the logarithms for two peak-flow periods (beginning of record to 1984 and 1985 –2001) at 16 rural 

streamflow-gaging stations in South Carolina. 
Peak-Flow Data 31 

Figure 6. (Continued) Box plots of the logarithms for two peak-flow periods (beginning of record to 1984 and 1985 –2001) at 16 rural 

streamflow-gaging stations in South Carolina. 
32 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Estimation of Flood Magnitude and 

Frequency at Gaged Sites 

A frequency analysis of peak flows at a streamflow-gaging 

station provides an estimate of the flood magnitude and frequency at that specific site. The estimates typically are presented as a set of exceedance probabilities or, alternatively, 

recurrence intervals along with the associated flows. Exceedance probability is defined as the probability of exceeding a 

specified flow in a 1-year period and is expressed as a decimal 

fraction less than 1.0 or as a percentage less than 100. A flow 

with an exceedance probability of 0.01 has a 1-percent chance 

of being exceeded in any given year. Recurrence interval is 

defined as the number of years, on average, during which the 

specified flow is expected to be exceeded one time. A flow with 

a 100-year recurrence interval is one that, on average, will be 

exceeded once every 100 years. However, a flood with a 100-

year recurrence can occur more frequently than once every 100 

years and could occur more than once in a given year. Recurrence interval and exceedance probability are mathematically 

inverse; therefore, a flow with an exceedance probability of 

0.01 has a recurrence interval of 1/0.01 or 100 years. 

Flood Frequency 

Flood-frequency estimates at gaged sites can be computed 

by fitting the peak flows to a known statistical distribution. This 

investigation followed the guidelines and computational methods described in Bulletin 17B of the Hydrology Subcommittee 

of the Interagency Advisory Committee on Water Data (1982). 

The flood-frequency estimates were computed by fitting the 

logarithms (base 10) of the peak flows to a Pearson Type-III 

distribution. The equation for fitting the log-Pearson Type-III 

distribution to a series of peak flows is as follows: 

(logQT = X KS + ), (1) 

where 

QT is the T-year recurrence-interval flow, in cubic feet 

per second, 

X is the mean of the log-transformed peak flow, 

K is a factor dependent on recurrence interval and the 

skew coefficient of the log-transformed peak 

flow, and 

S is the standard deviation of the log-transformed 

peak flow. 

A series of peak flows at a streamflow-gaging station 

may include low or high outliers, which are data points that 

depart significantly from the trend of the remaining data. The 

streamflow-gaging station record also may include information 

about maximum peak flows that occurred outside the period of 

regularly collected, or systematic, record. These peak flows are 

known as historic peaks and are often the maximum peak flows 

known to have occurred during an extended period of time 

beyond the period of collected record. Bulletin 17B (1982) provides guidelines for detecting and interpreting low and high outliers and historic data points and provides computational methods for making appropriate corrections to the distribution to 

account for their presence. In some cases, low or high outliers 

may be excluded from the record; therefore, the number of systematic peaks may not be equal to the number of years in the 

period of record. 

The USGS computer program PEAKFQ (W.O. Thomas 

and others, U.S. Geological Survey, written commun., January 

1998) was used to compute the relation between flood magnitude and probability of occurrence at each of the 20 urban 

streamflow-gaging stations in South Carolina (13), North Carolina (4), and Georgia (3; fig. 1). PEAKFQ includes the features 

described in Bulletin 17B (1982) but requires the user to exercise judgment when providing data on historic peaks, specifying screening levels for outliers, and interpreting the appropriateness of the resultant frequency curve to the measured data 

set. In the previous urban flood-frequency investigation 

(Bohman, 1992) and in a similar investigation in North Carolina 

(Robbins and Pope, 1996), the authors used the flood-frequency 

estimates from Sauer and others (1983) for the four streamflowgaging stations in Charlotte, N.C. (02146300, 02146500, 

02146600, and 02146700; fig. 1). Because there were no recommended or generally accepted procedures available for estimating skew coefficients for urban areas, Sauer and others 

(1983) defined an average skew value for each city or metropolitan area. The assigned city skew coefficients were then 

weighted with skew coefficients computed from actual floodpeak records according to published guidelines (Water 

Resources Council, 1977). For consistency with the other 

streamflow-gaging stations in this investigation, the floodfrequency estimates for the four Charlotte, N.C., streamflowgaging stations were computed using their respective station 

skew (table 9). 

Comparison of Selected Recurrence-Interval Flows 

Computed From Measured Data and Simulated Data 

One purpose of this investigation was to compare the 

selected recurrence-interval (T-year) flows computed by using 

the rainfall-runoff model data from the previous investigation 

(Bohman, 1992) with those computed from the measured data. 

If the comparison showed that there was no statistically significant difference in the T-year flows, the data sets could be combined resulting in a longer period of record for the frequency 

analysis. Based on comparisons of the measured and simulated 

peak flows presented earlier in this report, it was obvious that 

the data sets should not be combined. As shown in equation 1, 

the standard deviation of the peak flows is an important part of 

the equation for computing the T-year recurrence-interval 

flows. Because the variance at each streamflow-gaging station 

is directly related to the standard deviation, it is apparent that 

the simulated data will predict T-year recurrence-interval flows 
Estimation of Flood Magnitude and Frequency at Gaged Sites 33 

Table 9. Flood-frequency statistics for measured data from 20 urban streamflow-gaging stations in South Carolina, North Carolina, and 

Georgia. 

[L, lower Coastal Plain; U, upper Coastal Plain; P, Piedmont] 

Station 

number (in square 

miles) 

record 

peak flows 

Mean 

(log) (log) 

Skew of 2-year 25-year (fig. 1) 

Physiographic 

province 

Drainage 

area Period of 

Statistical data for water-year Recurrence-interval flow 

(in cubic feet per second) 

Standard 

deviation logarithms 100-year 

Town of Allendale, S.C. 

02176380 L 2.06 1986–2001 1.972 0.196 0.807 88.3 231 348 

City of Augusta, Ga. 

02196760 U 1.56 1979–96 2.554 0.194 0.530 345 844 1,200 

City of Columbia, S.C. 

02162093 U 5.49 1977–2001 3.131 0.117 -0.075 1,360 2,150 2,490 

02167020 U .28 1986–2001 2.250 .105 2.069 165 305 430 

02168845 P .45 1986–96 2.194 .084 1.585 149 237 299 

02169568 U 2.26 1986–2001 2.840 .239 .757 646 2,060 3,340 

City of Charlotte, N.C. 

02146300 P 30.5 1963–77 3.493 0.194 1.011 2,890 7,770 12,100 

02146500 P 41.0 1962–77 3.636 .161 .158 4,320 8,490 10,800 

02146600 P 38.3 1962–77 3.431 .188 -.509 2,800 5,310 7,230 

02146700 P 6.98 1963–77 2.967 .153 -.431 950 1,620 1,870 

City of Florence, S.C. 

02131130 L 1.92 1985–2001 2.772 0.083 -0.857 608 777 817 

City of Greenville, S.C. 

02164011 P 3.02 1985–2001 2.983 0.107 -0.614 986 1,400 1,530 

City of Greenwood, S.C. 

02166975 P 1.16 1986–2001 2.290 0.097 -0.454 198 278 304 

City of Myrtle Beach, S.C. 

02110740 L .80 1987–2001 2.474 0.142 0.222 294 542 674 

City of Orangeburg, S.C. 

02173491 U .45 1987–2001 2.327 0.134 -0.064 213 362 430 

City of Rock Hill, S.C. 

02145940 P 3.50 1986–97 2.940 0.076 -0.278 879 1,160 1,260 

City of Savannah, Ga. 

02203543 L .95 1979–96 2.445 0.178 1.409 254 666 1,060 

02203544 L .18 1979–96 1.911 .108 -.409 82.8 121 134 

City of Spartanburg, S.C. 

02159785 P .52 1987–2001 2.155 0.177 -0.510 148 270 316 

City of Sumter, S.C. 

02135518 L 2.20 1986–2001 2.537 0.155 -1.302 370 538 588 
34 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

greater than those computed from the measured data. Several 

comparisons were made to evaluate the magnitude of these differences. 

A comparison was made of the 2-, 25-, and 100-year 

recurrence-interval flows computed from the simulated data 

with those computed from the measured data. The three 

streamflow-gaging stations in Georgia (02196760, 02203543, 

and 02203544; fig. 1) were included in these comparisons 

because the recurrence-interval flows computed from the simulated data were readily available (Bohman, 1992). The Georgia 

streamflow-gaging stations were not included in earlier 

comparisons of peak flows because the modeling for these 

streamflow-gaging stations was done by the USGS Georgia 

District for the Bohman (1992) investigation, and the data were 

not readily available. The percent difference between the measured flows and simulated flows was computed as follows: 

Percent difference=((simulatedQT – measuredQT) (2) 

⁄ measuredQT) × 100. 

where 

simulated QT is the T-year recurrence-interval flow 

computed from the simulated peak-flow data, in 

cubic feet per second, and 

measured QT is the T-year recurrence-interval flow 

computed from the measured peak-flow data, in 

cubic feet per second. 

The result from the percent difference computation represents 

the percentage by which the measured value would have to be 

changed to obtain the simulated value. Consequently, a positive 

percent difference indicates the simulated value is greater than 

the measured value, and a negative percent difference indicates 

the simulated value is less than the measured value. 

As shown in figure 7, the 2-year recurrence-interval flows 

compared well with the data, remaining relatively close to the 

line of equality throughout the range of values. The percent differences ranged from -41.5 to 45.5 percent, with a mean difference of -1.5 percent and a median difference of -0.4 percent 

(table 10). For the 25-year recurrence interval, the percent differences ranged from -53.9 to 158 percent, with a mean difference of 56.4 percent and a median difference of 56.3 percent. 

For the 100-year recurrence interval, the percent differences 

ranged from -62.3 to 222 percent, with a mean difference of 

78.1 percent and a median difference of 72.9 percent. As the 

percent differences indicate, the recurrence-interval flows computed from the simulated data in most cases are greater than 

those computed from the measured data. The mean and median 

percent differences were about the same, indicating that there is 

no significant skew in the data and that it is fairly uniform about 

the mean. 

According to the values shown in figure 7, as the recurrence interval increased, the data plotted farther away from the 

line of equality indicating that the recurrence-interval flows 

computed from the simulated peaks were increasingly greater 

than those computed from the measured peaks. Two 

streamflow-gaging stations in the upper Coastal Plain had 

greater recurrence-interval flows from the measured data than 

from the simulated data. Station 02169568, Pen Branch at 

Columbia, S.C., had differences of -41.5, -53.9, and 

-62.3 percent for the 2-, 25-, and 100-year recurrence-interval 

flows, respectively. Station 02196760, Rocky Creek tributary at 

Augusta, Ga., had differences of -18.8, -9.4, and -9.2 percent for 

the 2-, 25-, and 100-year recurrence-interval flows, respectively 

(fig. 1; table 10). 

For the 16 rural streamflow-gaging stations with long-term 

records, the 2-, 25-, and 100-year recurrence-interval flows for 

the 1985–2001 period were compared with flows from the total 

period of record (table 11; fig. 8). A positive percent difference 

indicates that the 1985–2001 QT value is greater than the total 

period of record QT value, and a negative percent difference 

indicates that the 1985–2001 QT value is less than the total 

period of record QT value. The differences for the 2-year recurrence interval ranged from -23.6 to 13.0 percent, with a mean 

difference of -11.6 percent and a median difference of -14.3 percent. Overall, the 2-year flows for the 1985–2001 period were 

less than those for the entire period of record. This can probably 

be attributed to the severe drought that occurred in South 

Carolina during 1998–2001. The differences for the 25-year 

recurrence-interval flows ranged from -26.1 to 44.3 percent, 

with the mean and median differences of 0.5 and -3.9 percent, 

respectively. The differences for the 100-year recurrenceinterval flows ranged from -31.4 to 77.9 percent, with the mean 

and median differences of 7.0 and 1.0 percent, respectively. The 

plots in figure 8 indicate that the data are well distributed about 

the line of equality and that there is no significant deviation 

between the recurrence-interval flows for the 1985–2001 

period and those for the total period of record. 
Estimation of Flood Magnitude and Frequency at Gaged Sites 35

Figure 7. Comparison of the 2-, 25-, and 100-year recurrence-interval flows computed from the measured peak flows with 

those computed from the simulated peak flows.
36 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 10. Comparison of 2-, 25-, and 100-year flows computed from simulated peak flows (Bohman, 1992) with those 

computed from measured peak flows. 

[ft3/s, cubic feet per second; yr, year] 

number 

Flows computed from 

simulated peak flows

3

/s) Period of (ft3 Station /s)

(fig. 1) 

(ft

record 

Flows computed from 

measured peak flows Percent difference 

Recurrence interval Recurrence interval Recurrence interval 

2-yr 25-yr 100-yr 2-yr 25-yr 100-yr 2-yr 25-yr 100-yr 

Piedmont 

02145940 966 2,020 2,450 1986–1997 879 1,160 1,260 9.9 74.1 94.4 

02159785 154 460 635 1987–2001 148 270 316 4.1 70.4 101 

02164011 1,050 2,820 3,860 1985–2001 986 1,400 1,530 6.5 101 152 

02166975 288 718 980 1986–2001 198 278 304 45.5 158 222 

02168845 109 307 412 1986–96 149 237 299 -26.8 29.5 37.8 

Upper Coastal Plain 

02162093 1,400 3,010 3,660 1977–2001 1,360 2,150 2,490 2.9 40.0 47.0 

02167020 161 396 491 1986–2001 165 305 430 -2.4 29.8 14.2 

02169568 378 949 1,260 1986–2001 646 2,060 3,340 -41.5 -53.9 -62.3 

02173491 148 362 470 1987–2001 213 362 430 -30.5 0.0 9.3 

02196760 280 765 1,090 1979–96 345 844 1,200 -18.8 -9.4 -9.2 

Lower Coastal Plain 

02110740 296 771 1,020 1987–2001 294 542 674 0.7 42.3 51.3 

02131130 555 1,560 2,050 1985–2001 608 777 817 -8.7 101 151 

02135518 334 1,060 1,490 1987–2001 353 539 615 -5.4 96.7 142 

02176380 107 497 958 1986–2001 88.3 231 348 21.2 115 175 

02203543 250 706 965 1979–96 254 666 1,060 -1.6 6.0 -9.0 

02203544 101 244 310 1979–96 83.0 121 134 21.7 102 131 

Mean: -1.5 56.4 78.1 

Median: -0.4 56.3 72.9 
Estimation of Flood Magnitude and Frequency at Gaged Sites 37 

Table 11. Comparison of 2-, 25-, and 100-year recurrence-interval flows from two periods (period of record and 1985 –2001) 

at 16 rural streamflow-gaging stations in South Carolina. 

[ft3/s, cubic feet per second; yr, year] 

number 

(fig. 3) 

3

/s) Period of (ft3 Station /s)

Flows computed from peak flows 

for the total period of record 

(ft

record 

Flows computed from peak flows 

for 1985 –2001 Percent difference 

Recurrence interval Recurrence interval Recurrence interval 

2-yr 25-yr 100-yr 2-yr 25-yr 100-yr 2-yr 25-yr 100-yr 

Piedmont 

02147500 6,340 17,200 24,500 1952–2001 5,150 15,600 22,700 -18.8 -9.3 -7.4 

02154500 2,930 8,390 11,600 1931–2001 2,430 8,390 12,500 -17.1 0.0 7.8 

02160000 3,850 9,010 11,300 1940–2001 3,070 13,000 20,100 -20.3 44.3 77.9 

02162500 4,490 9,680 12,300 1942–2001 3,880 10,300 14,200 -13.6 6.4 15.4 

02163500 8,850 19,700 24,700 1939–2001 6,760 18,800 26,700 -23.6 -4.6 8.1 

02165000 

02192500 

02196000 

4,160 

4,800 

12,300 

10,100 

12,300 

27,400 

13,300 

16,700 

35,000 

1940–2001 

1940–2001 

1940–2001 

3,380 

4,650 

10,800 

8,630 

11,900 

22,900 

11,900 

16,400 

29,600 

-18.8 

-3.1 

-12.2 

-14.6 

-3.2 

-16.4 

-10.5 

-1.8 

-15.4 

Upper Coastal Plain 

02130900 754 1,970 2,870 1960–2001 706 2,600 4,290 -6.4 32.0 49.5 

02132500 2,610 6,670 9,250 1940–2001 2,950 7,140 9,600 13.0 7.0 3.8 

02173000 2,470 6,040 8,360 1932–2001 1,930 4,870 6,700 -21.9 -19.4 -19.9 

02173500 2,460 6,360 9,070 1939–2001 1,960 4,870 6,830 -20.3 -23.4 -24.7 

02174000 5,610 13,500 18,300 1946–2001 4,770 

Lower Coastal Plain 

11,000 14,900 -15.0 -18.5 -18.6 

02110500 6,010 16,500 23,200 1951–2001 6,370 22,500 35,100 6.0 36.4 51.3 

02136000 5,540 24,100 41,100 1928–2001 5,040 17,800 28,200 -9.0 -26.1 -31.4 

02176500 1,660 5,380 8,100 1951–2001 1,580 6,320 10,300 

Mean: 

Median: 

-4.8 17.5 27.2 

-11.6 0.5 7.0 

-14.3 -3.9 1.0 
38 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001

Figure 8. Comparison of 2-, 25-, and 100-year recurrence-interval flows from two peak-flow periods (period of record and 

1985 –2001) at 16 rural streamflow-gaging stations in South Carolina.
Estimation of Flood Magnitude and Frequency at Gaged Sites 39 

Conclusions From Comparisons of Measured and 

Simulated Data 

Comparisons of the measured and simulated data at the 

urban streamflow-gaging stations and of data from two periods 

at the rural streamflow-gaging stations indicate that there is no 

evidence of significant climatic trends in the peak-flow data. 

The comparisons also indicate a significant difference in the 

variances of the simulated and measured urban data that was not 

evident in the two periods used to compare the rural data. The 

Wilcoxon rank sum test results for the simulated and measured 

urban data were similar to the results for the rural data for the 

two periods analyzed, which suggests that the average simulations from the rainfall-runoff model were similar to the average 

measured data. However, the results from comparisons of 

recurrence-interval flows computed from the simulated urban 

data with those computed from the measured urban data indicate that the mean difference for the 25- and 100-year 

recurrence-interval flows were 56 and 78 percent, respectively 

(table 10). The mean difference for the 2-year recurrenceinterval was only -1.5 percent. For the rural streamflow-gaging 

stations, the mean differences in the 25- and 100-year 

recurrence-interval flows for the two periods were 0.5 and 

7.0 percent, respectively (table 11). 

Revisiting the calibration of and assumptions for the 

rainfall-runoff model used in the Bohman (1992) investigation 

is beyond the scope of this report. However, several possibilities have been considered regarding the significant differences 

between the simulated peak-flow data and the measured peakflow data. 

The data-collection period for the Bohman (1992) investigation was from 1983 to 1990, but 1986 was considered a major 

period of drought in South Carolina (Waters, 2003). The maximum peak flows from a data report documenting the hydrographs used to calibrate the rainfall-runoff model (Logan and 

others, 1995) were compared with the log-Pearson results computed for the current investigation for the 13 urban streamflowgaging stations in South Carolina. Six of the 13 streamflowgaging stations had maximum peak flows with recurrence intervals less than 2 years. Four of the 13 streamflow-gaging stations 

had maximum peak flows with recurrence intervals between 2 

and 5 years, and the three remaining streamflow-gaging stations 

had maximum peak flows with recurrence intervals of approximately 6, 12, and 18 years. Consequently, for about half of the 

South Carolina streamflow-gaging stations included in the current investigation, the rainfall-runoff models from the previous 

investigation were calibrated to hydrographs with peak flows 

less than a 2-year recurrence interval, and 77 percent were calibrated with peak flows less than a 5-year recurrence interval. It 

is interesting to note that the maximum peak flows at station 

02169568 (Pen Branch at Columbia, S.C.; fig. 1) had the longest recurrence interval of approximately 18 years. As previously 

noted, station 02169568 is one of only two streamflow-gaging 

stations where the measured data were consistently greater than 

the simulated data. Although a few “large” flood events did 

occur during the 4- or 5-year data-collection period, a total of 30 

to 45 events were utilized to calibrate the model parameters for 

each basin. Though the larger events undoubtedly carried more 

weight in the subjective calibration process, the results are probably most representative of flooding that could be expected to 

occur many times each year. 

A related factor that may account for some of the differences between the simulated and measured data is storage. The 

drainage areas of the 13 urban streamflow-gaging stations in 

South Carolina range from 0.28 to 5.49 mi2

 (table 1). It would 

not be unreasonable to assume that the temporary storage characteristics for these small basins would be significantly different between small and large storm events. Small storm events 

can be expected to move water through the basin without significant storage; however, large storm events likely cause more 

water to be stored behind roadways and upstream from clogged 

culverts and storm drains. This tends to reduce the peaks and 

decrease the variance. However, because the rainfall-runoff 

models were calibrated to small storm events, the routing 

parameters in the models will be biased toward little or no storage. Therefore, for large storm events (longer recurrence intervals), the models will route the flows downstream as if for a 

small flood with little or no storage, resulting in greater simulated flows than those actually measured. 

A third possibility for the significant difference between 

the measured and simulated peak-flow data is the rainfallrunoff modeling assumption of spatially uniform rainfall. The 

rainfall-runoff models were calibrated based on storm hydrographs and rainfall collected at each streamflow-gaging station. 

The calibrated models then were used to synthesize long-term 

flow data at a streamflow-gaging station by using long-term 

rainfall data from a National Weather Service (NWS) rainfall 

station (table 2) with the assumption that rainfall patterns are 

similar at the two locations. Station 02159785 was one of several streamflow-gaging stations where the long-term synthesized streamflow data were generated by using long-term rainfall data from two NWS stations and then weighting the results 

to obtain the final synthesized record. As shown in figure 2, the 

peak flow at times was significantly different depending on 

which long-term rainfall-gaging station was used in the rainfallrunoff modeling. During an investigation in Mecklenburg 

County, North Carolina, from July 1995 through June 1997, 

precipitation data were collected at 46 sites (Robinson and others, 1998). The results of the investigation indicated that the distribution of annual rainfall in parts of Mecklenburg County 

(528 mi2) ranged from 35 to 50 inches. In addition, the distribution of recurrence intervals for a 24-hour rainfall duration in the 

city of Charlotte for the storm of August 26–27, 1995, based on 

24 rainfall-gaging stations, ranged from less than 2 years to 

more than 100 years (Hershfield, 1961; Hazell and Bales, 

1997). Consequently, the investigation showed that rainfall patterns can be significantly different over the same time period in 

a single county. For the Bohman (1992) investigation, simulated annual peak flows were synthesized using the largest three 

to five rainfall events (in total inches) each year. Unlike the 

calibration phase, no attempt was made to verify the areal 
40 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

uniformity of the events that were selected for long-term simulation. Point rainfall extremes, as shown in the 1995 Charlotte 

example, can vary greatly and the assumption that these large 

rainfall amounts occurred evenly over the entire basin may be 

erroneous and could result in unusually high peaks, which 

would further translate to overestimation of flood magnitude 

and frequency values. 

Inman (1997) compared the 2-, 25-, and 100-year 

recurrence-interval peak flows computed from measured data 

with urban flood-frequency estimating equations for Georgia 

(Inman, 1995) and found that 

“The flood-frequency data computed from the statewide regression equations are higher than the floodfrequency data computed from observed data for the 

2-year flood at 15 urban stations and are equal at 

one urban station; higher for the 25-year flood at 20 

stations and equal at one station; and higher for the 

100-year flood at 22 stations * * *. Therefore, the 

peak flows computed with the statewide estimating 

equations generally are higher than those computed 

using the observed data.” 

The mean residuals in the Inman (1997) investigation for the 

2-, 25-, and 100-year floods estimated from the regional regression equations were 13.5, 19.9, and 22.4 percent higher, respectively, than the mean residuals of the corresponding flows 

computed from the same 20 years of simulated annual peak 

flows. Although the t-test indicated that the differences were 

statistically significant in all cases, the differences were within 

the range of standard error of prediction for the statewide 

regression equations (Inman, 1995); therefore, despite the 

apparent bias, the Georgia urban regression equations were not 

updated using observed data. 

Estimation of Flood Magnitude and 

Frequency at Ungaged Sites 

For small urban streams for which gaged data are not 

available, regionalization methods can be used to transfer 

flood-characteristic information from gaged to ungaged sites. 

Regionalization methods typically define relations between 

flood-frequency characteristics and explanatory drainage basin 

variables for gaged streams that have similar characteristics in 

a specific class or region. For this investigation, ordinary leastsquares regression was used as an exploratory tool to select the 

preliminary regression model by using numerous combinations 

from a set of explanatory variables. A qualitative explanatory 

variable, which indicated location by physiographic province, 

also was tested. After the initial model was chosen, generalized 

least-squares regression was used to define a set of predictive 

equations relating peak flows for the 2-, 5-, 10-, 25-, 50-, 100-, 

200-, and 500-year recurrence intervals to selected basin characteristics. 

Urban Basin Characteristics 

Several urban basin characteristics describing basin size 

and shape, indices of urban development and channel improvement, climate, and rural or background flood characteristics 

were included in this investigation (table 12). All of the basin 

characteristics used in the preliminary regional analysis were 

obtained from the database used by Bohman (1992) in the previous urban flood-frequency investigation except for the rural 

peak-flow characteristics, which were computed with regression equations developed by Feaster and Tasker (2002). 

Characteristics describing basin size and shape included 

drainage area (DA), measured in square miles and determined 

from USGS topographic maps, and(or) storm-sewer maps 

obtained from city engineering or public works departments 

(Logan and others, 1995). Main-channel length (L) was measured in miles from the streamflow-gaging station upstream 

along the channel to the basin divide. Channel slope (S) was 

measured in feet per mile and computed as the difference in elevation between the 10- and 85-percent points along the stream 

channel divided by the length between those two points. 

Urban development and channel improvement were characterized by percent impervious area (IA) and basin development factor (BDF). Impervious area is a dimensionless value 

determined by overlaying a grid on basin maps, delineating the 

impervious areas, such as roads, parking lots, and rooftops, and 

determining the percent of grid cells that constitute areas impervious to the infiltration of rainfall. According to Cochran 

(1963), a minimum of 200 points, or grid intersections, for each 

area or subbasin will provide a confidence level of 0.10. Grid 

intersections over points on buildings, streets, and parking lots 

were counted as impervious surface points. Grid intersections 

located over forests, lawns, unpaved industrial yards, and so 

forth, were treated as pervious surface points. An estimate of the 

percent of total impervious area was determined by dividing the 

impervious points by the total number of grid intersections. 

Three counts of at least 200 points for each subbasin were 

obtained with different orientations of the grid network, and the 

results were averaged for the final value. 

The BDF can be determined by using the methods 

described in the following excerpt from Sauer and others (1983, 

p. 8).

“The most significant index of urbanization that 

resulted from this study is a basin development factor 

(BDF), which provides a measure of the efficiency of 

the drainage system. This parameter * * * can be 

easily determined from drainage maps and field 

inspections of the drainage basin. The basin is first 

divided into thirds * * *. Then, within each third, 

four aspects of the drainage system are evaluated 

and each assigned a code as follows: 

1.	 Channel improvements.—If channel improvements such as straightening, enlarging, deepening, and clearing are prevalent for the main 

drainage channels and principal tributaries 
Table 12. Selected basin characteristics for the 20 urban streams in South Carolina, North Carolina, and Georgia [DA, drainage area; mi2, square miles; L, main-channel length; mi, miles; S, slope; ft/mi, feet per mile; L/S0.5, a ratio, where L and S have been defined previously; IA, imp

development factor; RI2,2, 2-year, 2-hour rainfall amount; in., inches; RQ, rural equivalent peak flow for 2-, 5-, 10-, 25-, 50-, and 100-year flows; ft3/s, cubic feet per secon

Station 

number 

) 

DA 

(mi2) 

L S 

( ) 

0.5 IA RI2,2 

(in.) 

RQ2 

3/s) 3/s) 3/s) 

RQ25 

3/s) 

(fig. 1 (mi) ft/mi L/S (percent) BDF (ft RQ5 (ft RQ10 (ft (ft 

02110740 0.80 1.40 9.2 0.462 23.0 7 2.24 aDA was outside of rural-regression analysis limi02131130 1.92 1.97 29.1 .365 31.0 9 2.24 79.7 148 207 299 02135518 2.20 1.79 20.2 .398 25.0 8 2.19 87.2 162 226 326 02145940 3.50 2.90 46.8 .424 19.0 6 1.95 260 448 589 784 02146300 30.5 11.2 13.7 3.026 20.0 9 1.90 1,490 2,450 3,210 4,320 02146500 41.0 11.0 13.1 3.039 22.0 9 1.90 1,830 2,990 3,900 5,220 02146600 38.3 8.72 12.2 2.496 10.0 7 1.90 1,740 2,860 3,730 5,000 02146700 6.98 5.20 20.9 1.137 12.0 9 1.90 528 902 1,210 1,670 02159785 .52 1.75 112 .165 14.0 0 2.00 71.2 128 172 233 02162093 5.49 3.37 30.6 .609 34.0 8 2.10 76.6 117 146 186 02164011 3.02 3.01 48.7 .431 34.0 6 2.15 235 407 536 714 02166975 1.16 1.63 47.2 .237 24.0 6 2.04 123 217 288 388 02167020 .28 .69 196 .049 40.0 8 2.10 aDA was outside of rural-regression analysis limit02168845 .45 1.07 120 .098 23.0 9 2.10 64.6 117 156 212 02169568 2.26 2.30 55.5 .309 29.0 10 2.10 aDA was outside of rural-regression analysis limit02173491 .45 1.17 82.0 .129 29.0 7 2.20 aDA was outside of rural-regression analysis limit02176380 2.06 2.02 24.3 .410 13.0 2 2.28 83.5 155 217 313 02196760 1.56 2.07 111 .196 23.0 8 2.16 aDA was outside of rural-regression analysis limit02203543 .95 1.78 13.0 .494 29.7 9 2.56 aDA was outside of rural-regression analysis limit02203544 .18 .51 29.9 .093 25.9 9 2.56 aDA was outside of rural-regression analysis limit

aFor more details, see “Limitations” section on p. 24 of Feaster and Tasker (2002). 
42 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Figure 9. Schematic of typical drainage basin shapes and subdivision into basin thirds for determination of 

basin development factor (BDF; from Sauer and others, 1983, fig. 2, p. 7). 

2. 

(those that drain directly into the main channel), 

then a code of 1 is assigned. Any or all of these 

improvements would qualify for a code of 1. To be 

considered prevalent, at least 50 percent of the 

main drainage channels and principal tributaries 

must be improved to some degree over natural 

conditions. If channel improvements are not prevalent, then a code of zero is assigned. 

Channel linings.—If more than 50 percent of the 

pal tributaries has been lined with an impervious 

material, such as concrete, then a code of 1 is 

assigned to this aspect. If less than 50 percent of 

these channels is lined, then a code of zero is 

assigned. The presence of channel linings would 

obviously indicate the presence of channel 

improvements as well. Therefore, this is an added 

factor and indicates a more highly developed 

drainage system. 

length of the main drainage channels and princi-
3.	 Storm drains, or storm sewers.—Storm drains are 

defined as enclosed drainage structures (usually 

pipes), frequently used on the secondary tributaries where the drainage is received directly from 

streets or parking lots. Many of these drains 

empty into open channels; however, in some 

basins they empty into channels enclosed as box 

or pipe culverts. When more than 50 percent of 

the secondary tributaries within a subarea (third) 

consists of storm drains, then a code of 1 is 

assigned to this aspect; if less than 50 percent of 

the secondary tributaries consists of storm 

drains, then a code of zero is assigned. It should 

be noted that if 50 percent or more of the main 

drainage channels and principal tributaries are 

enclosed, then the aspects of channel improvements and channel linings would also be assigned 

a code of 1. 

4.	 Curb-and-gutter streets.—If more than 50 percent 

of a subarea (third) is urbanized (covered by residential, commercial, and/or industrial development), and if more than 50 percent of the streets 

and highways in the subarea are constructed with 

curbs and gutters, then a code of 1 would be 

assigned to this aspect. Otherwise, it would 

receive a code of zero. Drainage from curb-andgutter streets frequently empties into storm 

drains. 

The above guidelines for determining the various 

drainage-system codes are not intended to be precise 

measurements. A certain amount of subjectivity will 

necessarily be involved. Field checking should be 

performed to obtain the best estimate. The basin 

development factor (BDF) is the sum of the assigned 

codes; therefore, with three subareas (thirds) per 

basin, and four drainage aspects to which codes are 

assigned in each subarea, the maximum value for a 

fully developed drainage system would be 12. Conversely, if the drainage system were totally undeveloped, then a BDF of zero would result. Such a 

condition does not necessarily mean that the basin is 

unaffected by urbanization. In fact, a basin could be 

partially urbanized, have some impervious area, 

have some improvement of secondary tributaries, and 

still have an assigned BDF of zero . . . such a condition still frequently causes peak discharges to 

increase. 

The BDF is a fairly easy index to estimate for an 

existing urban basin. The 50-percent guideline will 

usually not be difficult to evaluate because many 

urban areas tend to use the same design criteria, and 

therefore have similar drainage aspects, throughout. 

Also, the BDF is convenient for projecting future 

development. Obviously, full development and maxiEstimation of Flood Magnitude and Frequency at Ungaged Sites 43 

mum urban effects on peaks would occur when 

BDF = 12. Projections of full development or intermediate stages of development can usually be 

obtained from city engineers.” 

One of the main assumptions in a regional flood-frequency 

analysis is that the data are from stable basins. The urban 

streamflow-gaging stations used in the previous investigation 

were selected partly based on their locations in older, wellestablished urban areas (L.R. Bohman, U.S. Geological Survey, 

written commun., 2000). In order to test the assumption of basin 

stability, seven of the urban streamflow-gaging stations were 

examined to determine if significant changes had occurred in 

land use and(or) urbanization. The seven streamflow-gaging 

stations are listed in the following table. 

Station 

number Station name 

02145940 Little Dutchman Creek tributary at Rock Hill, S.C. 

02146100 Manchester Creek tributary at Rock Hill, S.C. 

02159785 Fairforest Creek tributary at Spartanburg, S.C. 

02160325 Brushy Creek (Reedy River tributary) at Greenville, 

S.C. 

02166975 Sample Branch at Greenwood, S.C. 

02168845 Saluda River tributary at Columbia, S.C. 

02174240 Middle Pen Branch at Orangeburg, S.C. 

Although several of these streamflow-gaging stations were not 

included in the regional analysis for reasons other than significant basin changes, these were determined to be the best indicator sites or sites that would have the highest probability of 

significant changes with regard to land use. 

The seven urban basins were examined for land-use 

changes by comparing aerial photographs from 1986 with aerial 

photographs from 1999. Newly developed areas were delineated 

on the 1999 photographs. If the newly developed areas were 

deemed to be significant, the drainage area of the new development was computed and compared to the total drainage area of 

the basin. The average percent impervious area was then computed based on information provided by the Soil Conservation 

Service (1986). Of the sites reviewed, the impervious area 

changes ranged from 0 to 4 percent. Consequently, it was 

concluded that no significant changes had occurred in the 

impervious area percentages computed during the previous 

investigation and that the same values could be used. 

Climatic measurements included in the regional analysis 

were the cumulative 2-hour rainfall corresponding to the 2-year 

recurrence interval (RI2,2) as determined from the NWS, formerly known as the U.S. Weather Bureau (1961). The rural 

flood characteristics were provided as the rural flood-frequency 

values for the 2-, 5-, 10-, 25-, 50-, and 100-year recurrenceinterval flows. Measures of the rural flood characteristics were 

derived from regression equations developed by Feaster and 
44 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Tasker (2002) for the Blue Ridge, Piedmont, upper Coastal 

Plain, and lower Coastal Plain physiographic provinces of 

South Carolina. 

Ordinary Least-Squares Analysis 

Ordinary least-squares (OLS) regression techniques were 

used to select the explanatory variables for the preliminary 

models. In OLS regression, linear relations between the explanatory and response variables are necessary; consequently, variables must often be transformed. For example, the relation 

between drainage area and peak flow typically is not linear; 

however, the relation between the logarithms of drainage area 

and the logarithms of peak flows often is linear. Homoscedasticity (a constant variance in the response variable over the 

range of the explanatory variables) about the regression line and 

normality of residuals also are requirements for OLS regression. Transformation of the flow data and certain other variables 

to logarithms often enhances the homoscedasticity of the data 

about the regression line. Linearity, homoscedasticity, and normality of residuals were examined in residual plots. 

The hydrologic model used in the regression analysis is of 

the form: 

QT = aAbBcCd. . . , (3) 

where 

QT is the response variable, a flood magnitude having 

T-year recurrence interval, in cubic feet per second; 

A, B, C are explanatory variables (basin characteristics); 

and 

a, b, c, d are regression coefficients. 

If the response and explanatory variables are logarithmically 

transformed, the hydrologic model has the following linear 

form: 

Log QT = log a + b (log A) + c (log B) + d (log C) + . . . , (4) 

where the variables are previously defined. A combination of 

the arithmetic and logarithmic relation was used in this investigation because the logarithmic transformation of IA and BDF 

did not improve the linear relation with QT. 

Ordinary least-squares regression of all possible subsets 

was used to determine the best combination of explanatory variables to use in the final regression equations. Cross products, 

such as DA × L, also were computed using combinations of the 

explanatory variables and were included in the OLS regression. 

The use of cross products allows the regression lines to converge or diverge, thereby decreasing or increasing the effect of 

one variable with the effect of another variable. For example, 

the percentage of IA may decrease with increasing DA. None of 

the cross products were found to be statistically significant in 

the regression. The best combination of the variables was chosen on the basis of Mallow’s Cp statistic, the adjusted coefficient of determination (R2), and the statistical significance of 

the explanatory variables. The best combination of explanatory 

variables that were consistent for all response variables (the 

base-10 logarithms of the T-year peak flows; T = 2, 5, 10, 25, 

50, 100, 200, and 500) was chosen for the final models and consisted of L, IA, and BDF. 

Throughout the years, USGS regionalization studies for 

flood frequency have resulted in DA being the most significant 

explanatory variable. For this investigation, L was determined 

to be the most significant explanatory variable although not 

drastically different from the statistical significance of DA. 

Usually, DA and L are strongly related, which was the case in 

this investigation; therefore, it would not be appropriate to 

include both variables in the regression model because of 

multicollinearity, which occurs when the explanatory variables 

are correlated. 

Often regional regression analysis for urban streams 

results in an equation that includes the equivalent rural 

recurrence-interval flow. In such cases, the equivalent rural 

recurrence-interval flow is viewed as an intercept value for the 

urban recurrence-interval flow as IA and(or) other characteristics related to urbanization approach zero. In this investigation, 

the equivalent rural recurrence-interval flows were not determined to be part of the most significant basin characteristics. 

For 7 of the 20 (35 percent) urban streamflow-gaging stations 

included in this investigation, the DA size was outside the limits 

used for the rural flood-frequency analysis (Feaster and Tasker, 

2002). Consequently, even if the equivalent rural recurrenceinterval flows had been determined to be statistically significant, rural values that were computed outside the limits of the 

rural-regression analysis would be considered unreliable. 

Regression diagnostic tools were used to test the adequacy 

of the OLS regressions. Multicollinearity was measured by the 

variance inflation factor (VIF). There were no problems with 

multicollinearity. The influence of individual streamflowgaging stations was measured by Cook’s D and DFFITS (Helsel 

and Hirsch, 1995). Any streamflow-gaging stations that were 

noted as having a high influence were further reviewed for 

potential problems with the data. Regression residuals (for the 

100-year peak flows) also were plotted and inspected for geographical patterns of bias. No distinct patterns were apparent in 

the mapped residuals. In addition, “qualitative variables” for the 

Piedmont, upper Coastal Plain, and lower Coastal Plain physiographic provinces were included in the regression analysis to 

detect geographical bias (Feaster and Tasker, 2002). The 

regression analysis also indicated no statistically significant difference among the three physiographic provinces. Therefore, 

the urban flood-frequency equations may be used to estimate 

flood magnitude and frequency in the Piedmont, upper Coastal 

Plain, and lower Coastal Plain physiographic provinces of 

South Carolina. 

Generalized Least-Squares Analysis 

Generalized least-squares (GLS) regression, as described 

by Stedinger and Tasker (1985), was used to compute the final 
Estimation of Flood Magnitude and Frequency at Ungaged Sites 45 

coefficients and the measures of accuracy for the regression 

equations using the USGS computer program GLSNET (G.D. 

Tasker, K.M. Flynn, A.M. Lumb, and W.O. Thomas, Jr., U.S. 

Geological Survey, written commun., 1995). Stedinger and 

Tasker (1985) found that GLS regression equations are more 

accurate and provide a better estimate of the accuracy of the 

equations than OLS regression equations when streamflow 

records at streamflow-gaging stations are of different and widely 

varying lengths and when concurrent flows at different 

streamflow-gaging stations are correlated. GLS regression 

techniques give less weight to streamflow-gaging stations that 

have shorter periods of record than streamflow-gaging stations 

with longer periods of record. Less weight also is given to 

streamflow-gaging stations where concurrent peak flows are correlated with other streamflow-gaging stations (Hodgkins, 1999). 

Table 13 lists the peak-flow regression equations for recurrence 

intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years that resulted 

from the GLS regression analysis for South Carolina. 

Table 13. South Carolina urban regional regression equations. 

[L, main-channel length, in miles; IA, impervious area, in percent; BDF, basin 

development factor, dimensionless] 

Urban floodrecurrence 

interval 

(years) 

Urban regional regression equations 

2 34.8 L1.40 100.0158IA 100.0319BDF 

5 48.8 L1.43 100.0144IA 100.0324BDF 

10 57.1 L1.45 100.0138IA 100.0337BDF 

25 65.7 L1.47 100.0131IA 100.0356BDF 

50 71.0 L1.48 100.0127IA 100.0369BDF 

100 75.6 L1.50 100.0124IA 100.0384BDF 

200 79.6 L1.51 100.0123IA 100.0398BDF 

500 84.2 L1.52 100.0122IA 100.0419BDF 

Accuracy of the Method 

The regional regression analyses for South Carolina 

resulted in the development of a set of equations that allow the 

user to estimate the 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year 

recurrence-interval flow at an ungaged, unregulated stream in an 

urban drainage basin (table 13). When applying these equations, 

users should not interpret these empirical results as exact. These 

regression equations are statistical models that should be interpreted and applied within the limits of the data and with the 

understanding that the results are best-fit estimates with an associated scatter or variance. 

One measure of how well the regression equations estimate 

peak flows at an ungaged site is the standard error of prediction 

(Sp). The Sp is the square root of the mean square error of prediction (MSEp). The MSEp is the sum of two components—the 

mean square error resulting from the model, γ 2

, and the sampling mean square error, MSEs,i

, which results from estimating 

the model parameters from samples of the population. The mean 

square model error, γ 2

, is a characteristic of the model, constant 

for all sites, and cannot be reduced by additional data collection. 

The mean square sample error, MSEs,i

, for a given site, however, 

depends on the values of the explanatory variables; in this case, 

L, IA, and BDF were used to develop the flow estimate at the 

site. Consequently, the sampling error can be reduced by additional data collection at existing streamflow-gaging stations, or 

by installing new streamflow-gaging stations in the same physiographic province, or by some combination of both. The standard error of prediction for a site, i, is computed as: 

Sp i = (γ 2 + MSEs i)

0.5, (5) , , 

(variables previously defined) and varies from site to site. 

Assuming the explanatory variables for the gaged sites in the 

regression are a representative sample of all sites in the region, 

the average accuracy of prediction for the regression model can 

be determined by computing the average standard error of prediction: 

⎧

γ 

2 1 n ⎫

0.5 

Sp = ⎨ + -- ∑ MSEs i⎬ , (6) 

⎩ n , 

i = 1 ⎭

where n is the number of observations and all other variables 

have been defined previously. 

The standard error of the model (SE(model)) can be converted from log (base 10) units to error percentage by using the 

transformation formulas, 

+ PercentSE(model) = 100 10γ [ – 1] (7)

and 

– PercentSEmodel = 100 10–γ [ – 1], (8)

where the variables have been defined previously. 

Similarly, the average standard error of prediction (Sp) can 

be transformed to positive or negative error percentage by substituting Sp

2 for γ 2 in equations 7 and 8, respectively. Computation of Sp,i for a given ungaged site, i, involves fairly complex 

matrix algebra; therefore, a computer program that computes 

the standard error of prediction for any study site has been 

developed. Details of the GLS regression method and calculation of prediction errors and intervals are provided in the 

Appendix. 

Another overall measure of how well regression equations 

can be used to estimate flood peaks when applied to ungaged 

basins is the PRESS (prediction error sum of squares) (Helsel 

and Hirsch, 1995) statistic. The PRESS statistic is a validationtype statistic. To compute the PRESS statistic, one streamflow-
46 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

gaging station is removed from the streamflow-gaging stations 

used to develop the regression equation, then the value of the 

one removed is predicted. The difference between the predicted 

value from the regression equation and the measured peak flow 

at the streamflow-gaging station is computed. The streamflowgaging station that is removed is then changed and the above 

process repeated until every streamflow-gaging station has been 

left out once. The prediction errors are then squared and 

summed (Helsel and Hirsch, 1995). PRESS/n is analogous to 

the average variance of prediction, and the square root of 

PRESS/n is analogous to the average standard error of prediction. Values of the square root of PRESS/n close to the values 

of the average standard error of prediction provide some measure of validation of the regression equations (Hodgkins, 1999). 

A third measure of the overall accuracy of the regression 

equations is the average equivalent years of record. This measure represents the average number of years of peak-flow data 

needed to provide an estimate by using log-Pearson Type-III 

techniques that would be equal in accuracy to an estimate made 

by using regional methods (table 14). The average equivalent 

years of record is a function of the accuracy of the regression 

equations, the recurrence interval, and the average variance and 

skew of the peak flows at streamflow-gaging stations (Hardison, 1971). 

Sensitivity Analysis 

The sensitivity of the equations to errors in the explanatory 

variables, L, IA, and BDF, was evaluated by changing each 

explanatory variable by 10-percent increments from -50 to 

+50 percent, while holding the other variables constant. For the 

base computations, the mean values for each explanatory 

variable were substituted in the equations for the 2-, 25-, and 

100-year recurrence-interval floods. The results are shown in 

figure 10. For the -50 to +50 percent change in L, the percent 

change in the peak flows ranged from -62 percent to 84 percent, 

respectively. For the -50 to +50 percent change in IA, the 

change in peak flows ranged from -30 to +42 percent, respectively. For the -50 to +50 percent change in BDF, the change in 

peak flows ranged from -24 to +38 percent, respectively. From 

these percentages and the graphs shown in figure 10, L is the 

most significant explanatory variable. From the OLS regression 

of the 100-year peak flow, R2

 was 0.94 with all three explanatory variables included. When only L was included, the R2

 was 

0.83, indicating that 83 percent of the variation in the 100-year 

peak flow can be explained by L. For IA, the percent change in 

the peak flows relative to the percent change in the IA decreased 

when going from the 2-year flood to the 100-year flood. As the 

sensitivity analysis suggests, it is reasonable to assume that as 

the magnitude of a flood increases, the effects of the IA 

decrease. 

Limitations 

The multiple-regression equations developed in this report 

for estimating flood magnitude and frequency are applicable to 

sites on small urban streams in South Carolina that have basin 

characteristics within the following range: L, 0.51 to 11.2 mi; 

IA, 10 to 40 percent; and BDF, 0 to 10. Applying the equations 

to sites on streams having basin characteristics outside of the 

Table 14. Accuracy statistics for the urban regional regression equations. 

[PRESS, prediction error sum of squares; n, number of sites] 

Urban floodinterval (years) (percent) 

of prediction 

(percent) 

(percent) recurrence 

Mean standard 

error of 

prediction 

Average mean 

standard error (PRESS/n)0.5 Average 

(PRESS/n)0.5 

(percent) 

Average 

equivalent 

years of record 

2 -30 to 44 + 37 -32 to 47 + 39 0.8 

5 -27 to 36 + 31 -29 to 40 + 34 1.7 

10 -25 to 33 + 29 -27 to 37 + 32 2.8 

25 -25 to 33 + 29 -27 to 37 + 32 3.9 

50 -27 to 37 + 32 -29 to 41 + 35 4.3 

100 -29 to 41 + 35 -32 to 47 + 39 4.2 

200 -32 to 47 + 39 -35 to 55 + 45 4.0 

500 -36 to 57 + 46 -40 to 68 + 54 3.6 
Estimation of Flood Magnitude and Frequency at Ungaged Sites 47 

Figure 10. Sensitivity of computed peak flows to changes in the three explanatory variables for selected peak-flow-frequency 

equations. 
48 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

range of those used in this investigation may result in prediction 

errors that are considerably greater than those suggested by the 

standard error of prediction percentages listed in table 14. 

The equations may not apply to urban streams where temporary in-channel storage or detention storage significantly 

affects the magnitude of peak flows. Detention storage, for this 

report, is defined as the storage occurring in planned areas, such 

as ponds upstream from dams, or in unplanned detention areas, 

such as upstream from highway and railroad embankments. 

Detention storage typically reduces the peak outflow from the 

detention area as compared to the peak inflow. The investigation sites chosen probably reflect average storage conditions 

and negligible permanent storage (Bohman, 1992). 

The length of record for the streamflow-gaging stations 

used in this investigation ranged from 11 to 25 years, with the 

mean and median being 16 years. Although the Wilcoxon rank 

sum test for 12 of 16 rural streamflow-gaging stations with 

long-term record showed that there was no statistically significant difference in the means of the peak-flow data for the periods from the beginning of record to 1984 and 1985–2001, it is 

important to understand that there is greater uncertainty in the 

peak-flow predictions at the higher recurrence intervals. Previous investigators (Hydrology Subcommittee of the Interagency 

Advisory Committee on Water Data, 1982) suggest that a 

streamflow-gaging station should have at least 10 years of 

record to warrant statistical analysis; other investigators suggest 

that estimates of frequencies of floods should be no greater than 

twice the length of record (Viessman and Lewis, 1996). However, water-resource managers often must tolerate limitations 

and uncertainty associated with estimating equations because 

long-term estimates of the magnitude and frequency of floods 

are needed at locations where no long-term measured data are 

available. 

In some cases, a comparison of the recurrence-interval 

flows computed for a small urban stream using the equations in 

table 13 may result in values that are less than those obtained by 

using the equations for a rural stream of the same size. This 

most often occurs for higher recurrence-interval flows. Sometimes this condition is caused by time-sampling errors in the 

data and(or) errors in the statistical model; however, research 

also indicates that factors such as detention storage and location 

of urbanization can reduce peak flows (Sauer and others, 1983). 

For this investigation, 5 of the 20 streamflow-gaging stations 

had 100-year peak flows that were lower when computed with 

the rural equation than when computed with the urban equation. 

The ratio of urban to rural 100-year peak flows for these five 

sites ranged from 0.76 to 0.99 with an average of 0.83. When 

this occurs, it is left to the discretion of the user, based on hydrologic judgment and knowledge of the area, to decide which 

computed peak flow to use. 

Program For Computing Flood-Frequency Estimates 

A computer program that was developed to estimate the 

magnitude and frequency of floods at rural ungaged sites in 

South Carolina (Feaster and Tasker, 2002) was modified for this 

investigation to allow the user to also compute the magnitude 

and frequency of floods at small, urban ungaged sites in South 

Carolina. The program produces an on-screen summary of 

results and generates an output file containing the results of the 

frequency estimates for a site. For each site of interest, the output file contains flood-magnitude predictions, standard error of 

the predictions, and 90-percent prediction intervals for the 2-, 

5-, 10-, 25-, 50-, 100-, 200-, and 500-year recurrence-interval 

flows. 

The computer program and necessary data files can be 

downloaded, using an Internet browser to a personal computer 

from the USGS Web site http://sc.water.usgs.gov/SCFFREQ/. 

From this site, the compressed (or zipped) file named scff.zip 

can be downloaded to a designated directory on a personal 

computer. The following six files then can be extracted: 

(1) scff.for—the program source code; (2) scff.exe—the 

executable file; (3) sc.cmn—a common block file used with 

the source code; and (4) sc115.txt, (5) scroi.cr, and (6) scroi.rl, 

which are data files for the region-of-influence method. The 

region-of-influence method is an optional method for computing rural flood-frequency estimates in the Blue Ridge physiographic province of South Carolina (Feaster and Tasker, 2002). 

Once the files have been extracted, the program can be run by 

opening a DOS window and typing “scff.” 

Example Computation 

Flood-frequency estimates at specific recurrence intervals 

can be made for small urban drainage basins located in the Piedmont and upper and lower Coastal Plain physiographic provinces of South Carolina. For the site of interest, determine the 

L, IA, and BDF as described earlier in this report. Verify that 

the basin characteristics are within the range of those described 

in the “Limitations” section of this report and apply the appropriate equation from table 13. 

The following example can be used to estimate the 100-

year recurrence-interval flow for a site having the following 

characteristics and located in the upper Coastal Plain of South 

Carolina: L = 1.79 mi2

, IA = 25.0 percent, BDF = 8. The basin 

characteristics are within the limitations of the regression analysis; therefore, the 100-year recurrence-interval flow (UQ100) is 

computed from the following equation in table 13: 

UQ100 = 75.6(L1.50)(100.0124IA)(100.0384BDF) 

UQ100 = 75.6(1.791.50)(100.0124*25.0)(100.0384*8) 

UQ100 = 750 ft3/s. 

If the computer program is used, the results will appear on the 

screen and in an output file as follows: 
Estimation of Flood Magnitude and Frequency at Ungaged Sites — Alternative Equations 49 

The abbreviations used in the output file are RI, recurrence 

interval; SE, standard error; and cfs, cubic feet per second. 

Estimation of Flood Magnitude and 

Frequency at Ungaged Sites—Alternative 

Equations 

Of the three basin characteristics that are included in the 

regressions equations in table 13, L was determined to be the 

most statistically significant variable followed by IA and BDF. 

When BDF was not included in the GLS analysis, the average 

mean standard error of prediction for the recurrence-interval 

flows increased from 2 to 5 percent with the average increase 

being 4 percent. It was concluded that the improvement in the 

standard error warranted including BDF in the regression equations. However, BDF is probably the most time-consuming 

characteristic to obtain because it requires a field visit to the 

basin of interest, whereas, L and IA can often be determined 

from maps or digital data. 

The USGS Massachusetts District has developed a Geographic Information System (GIS) application called StreamStats to make the process of computing streamflow statistics at 

ungaged sites faster and more consistent than manual methods 

(Ries and others, 2000). The USGS is currently working on a 

new prototype of StreamStats in several other States. The new 

prototype has been designed so that it can be implemented in 

any state. Currently, no GIS coverages for basin characteristics, 

such as BDF, are available. Consequently, flood-frequency 

equations that include BDF would not be compatible with 

StreamStats. Considering the future possibilities of implementing StreamStats in South Carolina, a second set of regression 

equations was developed that only includes L and IA (table 15). 

The accuracy statistics for the alternative equations are shown in 

table 16. In figure 11, the 3-parameter (L, IA, and BDF) equations and the 2-parameter (L and IA) equations for the 2-, 25-, 

and 100-year recurrence-interval flows are compared with the 

observed peak flows at the 20 urban stations included in this 

investigation. 

Table 15. Alternative urban regional regression 

equations for South Carolina. 

[L, main-channel length, in miles; IA, impervious area, in percent] 

Urban floodrecurrence 

interval (years) 

Alternative urban regional 

regression equation 

2 41.6 L1.47 100.0213IA 

5 58.8 L1.50 100.0198IA 

10 69.9 L1.51 100.0192IA 

25 82.3 L1.53 100.0187IA 

50 90.3 L1.55 100.0185IA 

100 97.2 L1.56 100.0185IA 

200 103 L1.58 100.0185IA 

500 111 L1.60 100.0187IA 
50 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Table 16. Accuracy statistics for the alternative urban regional regression equations. 

[PRESS, prediction error sum of squares; n, number of sites] 

Urban flood-

(percent) 

Average mean 

of prediction 

(percent) 

(percent) (percent) 

recurrence equivalent

interval (years) 

Mean standard 

error of 

prediction 

standard error (PRESS/n)0.5 Average 

(PRESS/n)0.5 

Average 

years of record 

2 -32 to 47 + 39 -33 to 50 + 42 0.7 

5 -29 to 41 + 35 -31 to 45 + 38 1.4 

10 -28 to 39 + 33 -30 to 43 + 36 2.1 

25 -28 to 39 + 34 -30 to 43 + 37 3.0 

50 -30 to 42 + 36 -32 to 47 + 39 3.3 

100 -32 to 47 + 39 -34 to 52 + 43 3.4 

200 -34 to 52 + 43 -37 to 60 + 49 3.3 

500 -38 to 62 + 50 -42 to 73 + 57 3.1 
Estimation of Flood Magnitude and Frequency at Ungaged Sites — Alternative Equations 51

Figure 11. Comparison of observed and estimated peak flows for the 2-, 25-, and 100-year recurrence intervals.
52 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Summary 

Knowledge of flood characteristics of streams is needed 

for designing roadway drainage structures, establishing floodinsurance rates, and for other uses by urban planners and engineers. Because urbanization can produce significant changes in 

the flood-frequency characteristics of streams, flood-frequency 

relations for rural streams are not applicable to urban streams. 

Methods for estimating the magnitude and frequency of floods 

on small urban streams in South Carolina were developed by 

using data from 20 streamflow-gaging stations in South 

Carolina, North Carolina, and Georgia. 

The magnitude and frequency of floods at the 20 urban 

streamflow-gaging stations were estimated by fitting the logarithm of the maximum peak flows to a Pearson Type-III distribution. Ordinary least-squares regression was used to select the 

explanatory variables for the preliminary statistical models to 

estimate the 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year 

recurrence-interval floods at ungaged urban streams. The best 

combination of variables was chosen based on Mallow’s Cp, the 

adjusted coefficient of determination, and the statistical significance of the explanatory variables. Based on the results of the 

preliminary regression, generalized least-squares regression 

was used to compute the final coefficients and the measures of 

accuracy for the regional regression equations. The explanatory 

variables included in the final model were main-channel length, 

in miles, percent impervious area, and basin development factor.

 Several measures of how well the regression equations 

estimate the specified recurrence-interval floods at an ungaged 

site were used; these included the standard error of prediction, 

the PRESS (prediction error sum of squares) statistic, and the 

average equivalent years of record. The mean standard error of 

prediction for the urban regional regression equations ranged 

from -25 to +33 percent for the 10- and 25-year recurrenceinterval floods and -36 to +57 percent for the 500-year 

recurrence-interval flood. The square root of the PRESS/n, 

where n is the number of streamflow-gaging stations in the 

regression, is analogous to the average standard error of 

prediction and provides some measure of validation of the 

regression equations. For this investigation, the square root of 

the PRESS/n values ranged from -27 to +37 percent for the 

10- and 25-year recurrence-interval floods and -40 to 

+68 percent for the 500-year recurrence-interval floods. The 

average equivalent years of record represents the average number of years of peak-flow data needed to provide an estimate 

using log-Pearson Type-III techniques that would be equal in 

accuracy to an estimate made by using regional methods. For 

this investigation, the average equivalent year of record ranged 

from 0.8 years for the 2-year recurrence-interval floods to 

4.3 years for the 50-year recurrence-interval flood. 

A comparison was made of measured peak flows with 

those that were computed using a rainfall-runoff model during 

a previous urban flood-frequency investigation. From a graphical comparison of these two data sets, an apparent difference in 

the variances was observed. To statistically compare the data 

sets, the Kendall tau trend test was used to test for trends over 

time, the F-test was used to test the equality of variances, and 

the Wilcoxon rank sum test was used to test for differences in 

the means. From the Kendall tau test, there were no statistically 

significant trends in either the measured or the simulated data. 

Results from the F-test indicated that there were statistically 

significant variances in the equality of variances at 85 percent 

of the South Carolina urban streamflow-gaging stations. The 

Wilcoxon rank sum test indicated that there was a statistically 

significant difference in the means at 38 percent of the South 

Carolina urban streamflow-gaging stations. 

To test the hypothesis that climatic differences may have 

occurred between the period in which the peak-flow data were 

measured and the historic period in which rainfall data were 

used to compute the simulated peak flows, similar statistical 

tests were applied on long-term data collected at 16 rural 

streamflow-gaging stations in South Carolina. The periods of 

record at these rural streamflow-gaging stations ranged from 42 

to 74 years with a mean length of 58.5 years and a median 

length of 59.5 years. For the rural sites, the period from 1985 to 

2001 was compared with the data measured from the beginning 

of record to 1984. Plots of the two periods at each rural site indicated that there were no significant differences in the data. The 

statistical test for comparison of variances was applied to the 16 

rural streamflow-gaging stations, and the results indicated that 

there were no statistically significant differences in the equality 

of variances at 88 percent of the sites. The statistical comparisons of the means of the two periods for the rural streamflowgaging stations indicated that there was no statistically significant difference at 75 percent of the streamflow-gaging stations. 

For the urban streamflow-gaging stations, results of the 

Wilcoxon test indicated no statistically significant difference 

between the average peak-flow simulations from the rainfallrunoff model and the average measured peak-flow data at 

approximately 60 percent of the urban streamflow-gaging 

stations. For the rural streamflow-gaging stations, the test indicated no statistically significant difference at 75 percent of the 

rural streamflow-gaging stations for the two periods compared. 

This indicates that the rainfall-runoff model reasonably simulated the average peak flows. However, the results from the 

comparisons of recurrence-interval flows computed from the 

simulated urban data with those computed from the measured 

urban data indicated that the mean differences for the 25- and 

100-year recurrence-interval flows were 56 and 78 percent, 

respectively. The difference for the 2-year recurrence-interval 

was only -1.5 percent. This indicates that the significant differences in the variances of the simulated and modeled data at the 

urban streamflow-gaging stations strongly affects the estimated 

recurrence-interval flows, especially at higher recurrence intervals. For the rural streamflow-gaging stations, the differences in 

the 25- and 100-year recurrence-interval flows for the two periods were 0.5 and 7.0 percent, respectively. These findings indicate that there is no significant climatic trend that explains the 

differences between the measured and simulated urban peak 

flows. A more thorough review of the differences would require 
revisiting the assumptions and calibrations of the rainfall-runoff 

models from the previous urban flood-frequency investigation, 

which is beyond the scope of this investigation. 

Selected References 

Becker, L.D., 1986, Techniques for estimating flood-peak discharges from urban basins in Missouri: U.S. Geological 

Survey Water-Resources Investigations Report 86-4322, 

38 p. 

Bodhaine, G.L., 1968, Measurement of peak discharge at culverts by indirect methods: U.S. Geological Survey Techniques of Water-Resources Investigations book 3, chap. A3, 

60 p. 

Bohman, L.R., 1992, Determination of flood hydrographs for 

streams in South Carolina —Volume 2. Estimation of peakdischarge frequency, runoff volumes, and flood hydrographs 

for urban watersheds: U.S. Geological Survey WaterResources Investigations Report 92-4040, 79 p. 

Cochran, W.G., 1963, Sampling techniques: New York, John 

Wiley, p. 71–86. 

Feaster, T.D., and Tasker, G.D., 2002, Techniques for estimating the magnitude and frequency of floods in rural basins of 

South Carolina, 1999: U.S. Geological Survey WaterResources Investigations Report 02-4140, 34 p. 

Guimaraes, W.B., and Bohman, L.R., 1991, Techniques for 

estimating magnitude and frequency of floods in South Carolina, 1988: U.S. Geological Survey Water-Resources Investigations Report 91-157, 174 p. 

Hardison, C.H., 1971, Prediction error of regression estimates 

of streamflow characteristics at ungaged sites: U.S. Geological Survey Professional Paper 750-C, p. 228–236. 

Hazell, W.F., and Bales, J.D., 1997, Real-time rainfall measurement in the city of Charlotte and Mecklenburg County, North 

Carolina: U.S. Geological Survey Fact Sheet FS-052-97, 4 p. 

Heath, R.C., 1977, Water resources data for North Carolina, 

water year 1976: U.S. Geological Survey Water-Data Report 

NC-76-1, 411 p. 

Helsel, R.M, and Hirsch, D.R., 1995, Studies in environmental 

science 49—Statistical methods in water resources: Amsterdam, Elsevier Science, 529 p. 

Hershfield, D.M., 1961, Rainfall frequency atlas of the United 

States: Washington, D.C., Department of Commerce, 

Weather Bureau, Technical Paper No. 40, 114 p. 

Hodgkins, G.A., 1999, Estimating the magnitude of peak flows 

for streams in Maine for selected recurrence intervals: U.S. 

Geological Survey Water-Resources Investigations Report 

99-4008, 45 p. 

Hydrology Subcommittee of the Interagency Advisory Committee on Water Data, 1982, Guidelines for determining 

flood frequency: U.S. Geological Survey Bulletin 17B, 

Office of Water Data Collection, Reston, Va., 183 p. 

Iman, R.L., and Conover, W.J., 1983, A modern approach to 

statistics: New York, John Wiley & Sons, 497 p. 

Selected References 53 

Inman, E.J., 1995, Flood-frequency relations for urban streams 

in Georgia—1994 update: U.S. Geological Survey WaterResources Investigations Report 95-4017, 27 p. 

Inman, E.J., 1997, Comparison of the 2-, 25-, and 100-year 

recurrence interval floods computed from observed data with 

the 1995 urban flood-frequency estimating equations for 

Georgia: U.S. Geological Survey Water-Resources Investigations Report 97-4118, 14 p. 

Landers, M.N., and Wilson, K.V., Jr., 1991, Flood characteristics of Mississippi streams: U.S. Geological Survey WaterResources Investigations Report 91-4037, 82 p. 

Logan, S.W., Eckenwiler, M.R., and Bohman, L.R., 1995, 

Selected hydrologic data for urban watersheds in South 

Carolina, 1983–90: U.S. Geological Survey Open-File 

Report 95-351, 434 p. 

Ragland, B.C., Walters, D.A., Cartano, G.D., and Taylor, J.E., 

2003, Water resources data, North Carolina, water year 2002: 

U.S. Geological Survey Water-Data Report NC-02-1B, 


645 p.


Rantz, S.E., and others, 1982, Measurement and computation of 

streamflow—Volume 2. Computation of discharge: U.S. 

Geological Survey Water-Supply Paper 2175, 347 p. 

Rasmussen, T.J., and Perry, C.A., 2001, Trends in peak flows of 

selected streams in Kansas: U.S. Geological Survey WaterResources Investigations Report 01-4203, 62 p. 

Ries, K.G., III, and Friesz, P.J., 2000, Methods for estimating 

low-flow statistics for Massachusetts streams: U.S. Geological Survey Water-Resources Investigations Report 00-4135, 

87 p. 

Ries, K.G., III, Steeves, P.A., Freeman, A., and Singh, R., 2000, 

Obtaining streamflow statistics for Massachusetts streams on 

the World Wide Web: U.S. Geological Survey Fact Sheet 

104-00, 4 p. 

Robbins, J.C., and Pope, B.F., 1996, Estimation of floodfrequency characteristics of small urban streams in North 

Carolina: U.S. Geological Survey Water-Resources Investigations Report 96-4084, 21 p. 

Robinson, J.B., Hazell, W.F., and Garrett, R.G., 1998, Precipitation, streamflow, and water-quality data from selected sites 

in the city of Charlotte and Mecklenburg County, North 

Carolina, 1995–97: U.S. Geological Survey Open-File 

Report 98-67, 220 p. 

SAS Institute, Inc., 1990, SAS procedures guide: Cary, N.C., 

705 p. 

Sauer, V.B., Thomas, W.O., Stricker, V.A., and Wilson, K.V., 

1983, Flood characteristics of urban watersheds in the United 

States: U.S. Geological Survey Water-Supply Paper 2207, 

63 p. 

Soil Conservation Service, 1986, Urban hydrology for small 

watersheds (2d ed.): U.S. Department of Agriculture, Soil 

Conservation Service Technical Release No. 55. 

Speer, P.R., and Gamble, C.R., 1964, Magnitude and frequency 

of floods in the United States, Part 2-A, South Atlantic Slope 

basins, James River to Savannah River: U.S. Geological 

Survey Water-Supply Paper 1673, 329 p. 
54 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001 

Stamey, T.C., and Hess, G.W., 1993, Techniques for estimating 

magnitude and frequency of floods in rural basins of Georgia: 

U.S. Geological Survey Water-Resources Investigations 

Report 93-4016, 75 p. 

Stedinger, J.R., and Tasker, G.D., 1985, Regional hydrologic 

analysis 1. Ordinary, weighted, and generalized least square 

compared: Water-Resources Research, v. 21, no. 9, p. 1421– 

1432. 

Stedinger, J.R., and Tasker, G.D., 1986, Correction to regional 

hydrologic analysis 1. Ordinary, weighted, and generalized 

least square compared: Water-Resources Research, v. 22, 

no. 5, p. 844. 

U.S. Weather Bureau, 1961, Rainfall-frequency atlas of the 

United States: U.S. Weather Bureau Technical Paper 40, 

115 p. 

Viessman, W., Jr., and Lewis, G.L., 1996, Introduction to 

hydrology (4th ed.): New York, Harper Collins College 

Publ., 760 p. 

Water Resources Council, 1977, Guidelines for determining 

flood flow frequency: U.S. Water Resources Council Bulletin 17A, 26 p. 

Waters, K.E., 2003, Ground-water levels in South Carolina—A 

compilation of historical water-level data: South Carolina 

Department of Natural Resources Water Resources Report, 

18 p. 

Whetstone, B.H., 1982, Techniques for estimating magnitude 

and frequency of floods in South Carolina: U.S. Geological 

Survey Water-Resources Investigations Report 82-1, 78 p. 
Appendix— Generalized Least-Squares Model Description and Assumptions 55

Appendix—Generalized Least-Squares Model Description and Assumptions

A streamflow characteristic, such as the logarithm of the 50-year peak flow, is estimated at each gaged site, 

(1)

 where is the true (but unknown) log of the 50-year peak and ηi is a random error. If yi is an unbiased estimate of ψi, then ηi

(sometimes called time sampling error) has a mean of zero and a variance that is a function of how many years of data are available 

for the site and the standard deviation of water-year peaks. In addition, there are k basin characteristics, such as log of drainage 

area, that are measured with negligible error.

Assuming that (within the region defined by the basin characteristics at the n stations) ψ is approximately linearly related to 

the basin characteristics (x’s), then the model formulation can be written as:

(2)

where εi is a model error assumed uncorrelated from observation to observation, with mean zero and constant variance, γ 2. 

Substituting into equation 1,

(3)

In matrix notation:

(4)

where

(5)

where E[υ]=0, and E[υ υ T]=Λ . Now the GLS estimator of β is:

(6)

The problem with this estimator is that Λ is unknown and must be estimated from the data. In ordinary least-squares 

regression, Λ is estimated as , which would be a good estimate if all stations in that region have approximately the same lengths 

of record, or if the variance of is small relative to the variance of at every station in the region.

Because this assumption may be hard to justify, a better estimate of Λ is attempted. Denote this estimated covariance matrix 

, and the GLS estimator, b, will be referred to as an Estimated Generalized Least Squares (EGLS) estimator.

 EGLS Regression

An example illustrates how is estimated. Suppose that yi is the log of the 50-year peak flow estimated from mi years of 

record and that the water-year peaks follow a log-Pearson Type-III (LPIII) distribution at all sites. Further, to minimize notation, 

assume that the skew coefficient at all sites is zero. The elements of would be given by:

yi Ψi ηi = + ,

Ψ i

Ψi β0 β1x1i β2x + 2i... βkxki ε i = + + + (i=1, 2, ..., n; n>k),

yi β0 β1x1i β2x2i ... βkxki ηi ε i = + + + + + + .

Y = Xβ υ + ,

Y = 

y1

y2

...

yn

 X = 

1 x11 x21 ...xk1

1 x12 x22 ...xk2

...... .........

1 x1n x2n ...xk n

 β

β0

β1

...

βk

 υ

ε 1 + η1

ε 2 + η2

...

ε n + ηn

= = ,

b = (XTΛ –1X)–1XTΛ –1Y .

σ

2

I

ηi ε i

Λ

ˆ

Λ

ˆ

Λ

ˆ
56 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001

(7)

In this equation, K (LPIII standard deviate for zero skewness and 50-year recurrence interval), mi (record length at station i), 

mj

 (record length at station j), and mij (concurrent record length for stations i and j) are known, but (standard deviation of wateryear peaks at station i), (cross correlation of water-year peaks at stations i and j), and γ 2 (variance of model error) must be 

estimated from the data. Furthermore, (the sample estimate of ) cannot be used as an estimate of without introducing bias, 

and the use of (sample cross correlations) for often causes numerical problems. Therefore, we estimate and as 

follows.

The standard deviation of water-year peaks, , is estimated from a regional regression of the form:

(8)

By estimating the standard deviations, si, that enter into equation 7 with equation 8, we are assured that the rows of the Λ

matrix are not correlated with the observed dependent variable Y. This quality is necessary for the estimates of β to be unbiased.

The cross correlation coefficient, ρij, is estimated by developing an empirical relation between sample cross correlations, rij, 

and distance between stations of the form:

(9)

Estimating the cross correlations in this manner assures us that the matrix Λ will be positive definite. Figure 1 below shows a 

smooth curve with = 0.9812 and α = 0.00412 based on data from Illinois. This curve was developed by running the GLSNET 

program that will be described later.

λij

γ 2 σ2

i 1 0.5K2 ( ) +

mi

+ ----------------------------------for i j ( ) =

or

ρijσi

σjmij 1 0.5K2 ( ) +

mi

mj

---------------------------------------------------for i j ( ) ≠

 .

⎝

⎜

⎜

⎜

⎜

⎜

⎜

⎛

=

σi

ρij

si σi σi

rij ρij σi ρij

σi

ln si ( ) b0 b1x1i b2x2i ... bkxki = + + ++ .

rij Θ

dij

αdij + 1 -------------------

= .

Θ

Figure 1. Relation between cross correlation and distance.
Appendix— Generalized Least-Squares Model Description and Assumptions 57

Now the only parameters left to find in the EGLS model are the regression coefficients, b, and variance of the model error, 

γ 2

. The model error variance, γ 2, and regression coefficients, b, are found by iteratively searching for the best non-negative 

solution to the equation:

(10)

The GLSNET/AIDE package leads one through the development of equations 8 and 9 in preparation for the estimation of the 

GLS regression coefficients.

Reporting Results and Errors

The predicted response at ungaged site k with basin characteristics xk =(1, xk,1, xk,2, ..., xk,p) is:

(11)

The standard error of the prediction in OLS regression is:

(12)

In GLS regression, the standard error of prediction is:

(13)

The S( ) is a function of x and the computed standard error of a prediction in percent also will be a function of x.

Standard Errors in Percent

When a standard error or average prediction error in log units follows a normal distribution, the error may be expressed in 

percent of the predicted value in cubic feet per second. Denote σ as the standard error in log (base 10) units, Scfs as the standard 

error in cubic feet per second, and E(q|xk) as the predicted value of q, in cubic feet per second, given xk, and xk =(1, xk,1, xk,2, ..., 

xk,p) is a vector of basin characteristics. The standard error in percent, Spercent, is given by:

 (14)

(Aitcheson and Brown, 1957).

Sometimes it is said in OLS that two-thirds of the points lie within one standard error of estimate of the regression function.

This is true for the log unit standard error of estimate, σ , but it generally is not correct for Spercent. This is true because the errors 

in log space are symmetrically distributed under the assumption of normality of the log errors, but the errors in cubic feet per 

second are skewed. One can, however, calculate +percent and -percent errors with the following formulas:

(15)

and

(16)

E Y X ( ) – β TΛ –1 { } ( ) Y X– β = n k – – 1 .

y

ˆ

k xk = b.

S yˆ() σ k

2 1 xk(X' X)–1

x' + k { } [ ]

0.5 = .

S yˆ( )k γ

ˆ 2

x + kX'Λ

ˆ –1

x'k = .

y

ˆ

k

Spercent 100

Scfs

Eqxk ( )

------------------ 100 e5.302σ 2

= = ( ) – 1

Splus 100 10σ = ( ) – 1 ; 

Sminus 100 10–σ = ( ) – 1 .
58 Estimating the Magnitude and Frequency of Floods in Small Urban Streams in South Carolina, 2001

The three formulas above apply not only to the standard error of estimate for a regression, but they also apply to the standard

error of the model, , in GLS regression, the average prediction error, and standard error of a prediction in both OLS and GLS.

Average Prediction Error (APE)

One overall measure of how good the regression model is for prediction is the average prediction error (Hardison, 1971), 

where the average is taken over prediction sites with X variables identical to the observed data. This measure assumes the observed 

data have been collected at a representative set of sites in the region. It is computed as:

(17)

The first term in the brackets on the right side of equation 17 represents an estimate of the average squared model error for 

the n sites and the second term inside the brackets is an estimate of the average squared error due to estimating true model 

parameters from a sample of data.

Prediction Interval

Users of the regression model are probably more interested in a measure of error in a particular prediction rather than an 

average prediction. A good measure of the error of a particular prediction is the confidence interval of a prediction, or prediction 

interval. Let x0 represent the usual row vector of basin characteristics at a prediction site. As usual x0 is augmented by a 1 as the 

first element. The predicted value is . A 100(1-α) prediction interval would be:

(18)

where

(19)

where tα/2, n–p' is the critical value from a t-distribution for n–p' degrees of freedom.

If a log transform had been made so that y0 = log10(q0), then the prediction interval would be:

(20)

Selected References to the Appendix

Acreman, M.C., and Wiltshire, S.E., 1987, Identification of regions for regional flood frequency analysis [abs.]: EOS, v. 68, no. 44.

Aitcheson, J., and Brown, J.A.C., 1957, The log-normal distribution: London, Cambridge University Press.

Hardison, C.H., 1971, Prediction error of regression estimates of streamflow characteristics at ungaged sites: U.S. Geological

Survey Professional Paper 750-C, p. 228–236.

γ

ˆ

APE γ

ˆ 2

i

n

------- xi X 'Λ

ˆ –1

( ) X –1x'

i

n ---------------------------------------

i = 1

n

+ ∑

i = 1

n

∑ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ 1 2/

= .

y0

ˆ x0 = b

y0

ˆ T y0 y0ˆ – ≤ ≤ + T ,

T tα

2

--, n p – ' γ

ˆ 2

0 x0 X'Λ

ˆ –1

( ) X –1x' + 0 = ( ) ,

10y0

ˆ – T

q0 10y0

ˆ + T

≤ ≤ .