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Estimating the Magnitude and Frequency of Floods at Ungaged Locations on Streams in Tennessee Through the 2013 Water Year

David E. Ladd, Paul A. Ensminger, T.A. Cohn, J.F. England, C.E. Berenbrock, R.R. Mason · U.S. Geological Survey

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Prepared in cooperation with the Tennessee Department of Transportation

Estimating the Magnitude and Frequency of Floods at Ungaged Locations on Streams in Tennessee Through the 2013 Water Year

INDIANA ILLINOIS

WEST VIRGINIA

KENTUCKY

MISSOURI

VIRGINIA

HA2 HA4

HA1

TENNESSEE

HA3

ARKANSAS

MISSISSIPPI

GEORGIA ALABAMA

Scientific Investigations Report 2024–5130

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

NORTH CAROLINA

SOUTH CAROLINA

Cover. See figure 1 on page 3.

Estimating the Magnitude and Frequency of Floods at Ungaged Locations on Streams in Tennessee Through the 2013 Water Year By David E. Ladd and Paul A. Ensminger

Prepared in cooperation with the Tennessee Department of Transportation

Scientific Investigations Report 2024–5130

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

U.S. Geological Survey, Reston, Virginia: 2025

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment—visit https://www.usgs.gov or call 1–888–392–8545. For an overview of USGS information products, including maps, imagery, and publications, visit https://store.usgs.gov/ or contact the store at 1–888–275–8747. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner. Suggested citation: Ladd, D.E., and Ensminger, P.A., 2025, Estimating the magnitude and frequency of floods at ungaged locations on streams in Tennessee through the 2013 water year: U.S. Geological Survey Scientific Investigations Report 2024–5130, 19 p., https://doi.org/10.3133/sir20245130. Associated data for this publication: Heal, E.N., Ladd, D.E., and Ensminger, P.A., 2025, Tabular and geospatial data for the magnitude and frequency of floods at ungaged locations on streams in Tennessee, through the 2013 water year: U.S. Geological Survey data release, https://doi.org/10.5066/P9IEJQB7. ISSN 2328-0328 (online)

iii

Contents Abstract��1 Introduction��1 Purpose and Scope��2 Description of the Study Area��2 Previous Investigations��2 Data Compilation��8 Streamgage Selection��8 Basin Delineation and Characteristics��8 Drainage Area��8 10–85 Channel Slope��9 Percent Imperviousness��9 2-Year Climate Factor��9 Annual Exceedance Probability Analysis��9 Expected Moments Algorithm��9 Multiple Grubbs-Beck Test for Detecting Potentially Influential Low Floods��10 Regional Skew��10 Development of Regional Regression Equations��10 Validation of Hydrologic Areas��10 Evaluation of Basin Characteristics��11 Generalized Least-Squares Regression��11 Applications of Methods��14 Streamgage Locations��14 Ungaged Locations on Gaged Streams��15 Accuracy and Limitations of GLS Equations��15 Summary��17 References Cited��17

Figures 1.

2. 3. 4. 5. 6.

Map showing streamgages, hydrologic areas, and the Tennessee StreamStats application boundary, Tennessee and parts of Alabama, Georgia, Kentucky, Mississippi, North Carolina, and Virginia��3 Map showing streamgages in hydrologic area 1, east Tennessee and parts of Alabama, Georgia, North Carolina, and Virginia��4 Map showing streamgages in hydrologic area 2, middle Tennessee and parts of Alabama, Kentucky, and Mississippi��5 Map showing streamgages in hydrologic area 3, middle Tennessee and part of Alabama��6 Map showing streamgages in hydrologic area 4, west Tennessee and parts of Kentucky and Mississippi��7 Scatter plot showing relation between residuals and predicted streamflows corresponding to the 0.01 annual exceedance probability using the generalized least-squares equations��13

Tables 1. 2. 3.

5. 6.

Physical and climatic basin characteristics used as explanatory variables in the generalized least-squares regression equations��8 Recurrence intervals that correspond with annual exceedance probabilities��9 Numbers of streamgages by State and hydrologic area used to estimate the magnitude and frequency of floods at ungaged location on streams in Tennessee through the 2013 water year��11 Final generalized least-squares regression equations for estimating streamflows corresponding to selected annual exceedance probabilities on rural streams in Tennessee and associated performance metrics��12 Drainage area generalized least-squares regression equations for estimating exponent b for equation 5��16 Ranges of physical and climatic basin characteristics for use in generalized least-squares regression equations, by hydrologic area��17

Conversion Factors U.S. customary units to International System of Units

Multiply

To obtain

Length inch (in.)

2.54

centimeter (cm)

foot (ft)

0.3048

meter (m)

mile (mi)

1.609

kilometer (km)

Area square mile (mi2)

259.0

square mile (mi2)

2.590

hectare (ha) square kilometer (km2)

Flow rate cubic foot per second (ft3/s)

0.02832

cubic meter per second (m3/s)

Hydraulic gradient foot per mile (ft/mi)

0.1894

meter per kilometer (m/km)

Datums 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).

Supplemental Information Water year A 1-year period from October 1 to September 30, defined by the year in which it ends; for example, the 2013 water year is the period October 1, 2012–September 30, 2013.

Abbreviations AEP

annual exceedance probability

CLIMFAC2YR

2-year recurrence-interval climate factor

CSL10_85

main-channel slope

DRNAREA

drainage area

EMA

expected moments algorithm

GIS

geographic information system

GLS

generalized least squares

hydrologic area

LC11IMP

percent imperviousness

LP3

log-Pearson type III

MGBT

Multiple Grubbs-Beck test

OLS

ordinary least squares

PILF

potentially influential low flood

RRE

regional regression equation

USGS

U.S. Geological Survey

WREG

weighted-multiple-linear regression computer program

Estimating the Magnitude and Frequency of Floods at Ungaged Locations on Streams in Tennessee Through the 2013 Water Year By David E. Ladd and Paul A. Ensminger

Abstract To improve estimates of the frequency of annual peak flows for ungaged locations on non-urban, unregulated streams in Tennessee, generalized least-squares multiple linear-regression techniques were used to relate annual peak flows from streamgages operated by the U.S. Geological Survey to physical, climatic, and land-use characteristics of their drainage basins. Geospatial data acquired since the previous study in 2003, annual peak-streamflow data through the 2013 water year, and Bulletin 17C methods for frequency analysis of annual peak-streamflow data were used in the study. Generalized least-squares regression equations were developed for four hydrologic areas with distinct hydrologic, geologic, and topographic characteristics. Drainage area was used as an explanatory variable in equations developed for each hydrologic area. In addition to drainage area, a 2-year recurrence-interval climate factor was used for hydrologic area 1, a 10–85 channel slope was used for hydrologic area 2, and percent imperviousness was used for hydrologic area 4. The regression equations can be used to estimate annual exceedance probability streamflows for ungaged locations on non-urban, unregulated streams in Tennessee. The term “unregulated” indicates that streamflow is not appreciably influenced by regulation from reservoirs or other impoundments. Average standard errors of prediction for the regression equations ranged from 44.4 to 51.4 percent for hydrologic area 1; 30.4 to 42.9 percent for hydrologic area 2; 35.7 to 42.6 percent for hydrologic area 3; and 32.5 to 47.4 percent for hydrologic area 4.

Introduction Transportation engineers require reliable estimates of the magnitude and frequency of peak streamflows to design bridges, culverts, and other structures near streams and rivers. Methods used to estimate the magnitude and frequency of peak streamflows at ungaged locations are commonly updated at the request of stakeholders to include recent streamflow data, incorporate statistical improvements in frequency

analysis, and implement technological improvements that increase the accuracy of basin and climatic characteristics that affect hydrologic response in streams. The U.S. Geological Survey (USGS) last published methods for estimating the magnitude and frequency of annual peak streamflow at ungaged locations on streams in Tennessee (Law and Tasker, 2003) using streamgages operated by the USGS having at least 10 years of observed annual peak streamflows through the 2000 water year. For this study, more than 10 additional years of streamflow record were available for many of the streamgages used to develop the predictive equations. During this period, previously known maximum flood peaks were exceeded at many USGS streamgages throughout middle and west Tennessee. Since the USGS last published methods for estimating magnitude and frequency of annual peak streamflow at ungaged locations on streams in Tennessee, updated guidelines for frequency analysis of annual peak streamflow by Federal agencies, referred to as “Bulletin 17C,” have been published (England and others, 2019). Bulletin 17C guidance incorporated in this analysis includes an updated regional skew coefficient for the study area; use of the expected moments algorithm (EMA) for fitting the log-Pearson type III (LP3) distribution to the annual peaks; and the application of the Multiple Grubbs-Beck test (MGBT) for identifying potentially influential low floods (PILFs) in the distribution of annual peak streamflows. In 2020, the USGS, in cooperation with the Tennessee Department of Transportation, updated the methods for predicting the magnitude and frequency of floods at ungaged locations for streams in Tennessee. Regression equations were developed to predict streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probabilities (AEPs). The new regression equations will be incorporated into the StreamStats web application (https:// streamstats.usgs.gov/ss/) (U.S. Geological Survey, 2019) to automate the process of predicting streamflows for selected AEPs for ungaged locations on streams in Tennessee. Geospatial data associated with the study, including drainage basins and basin characteristics, and the tabular data referenced in this report are available in Heal and others (2025).

Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year

Purpose and Scope The purposes of this report are to provide (1) updated estimates of streamflows corresponding to 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs for 454 streamgages operated by the USGS in the study area (figs. 1–5); and (2) updated regional regression equations (RREs) for estimating streamflows corresponding to AEPs at ungaged locations on streams within each of four hydrologic areas (HAs) (figs. 1–5). Basin characteristics derived from geographic information systems (GIS) and annual peak-streamflow records for 454 streamgages located in generally rural areas of Tennessee and in the adjacent States of Georgia, North Carolina, Virginia, Alabama, Kentucky, and Mississippi (fig. 1) were used to develop the RREs. Streamgages used to develop the RREs were required to have at least 10 years of observed annual peaks through water year 2013 (a water year is defined as the period October 1 through September 30, named for the year in which it ends), and those peak streamflows needed to not have been appreciably affected by urbanization. Of the 454 streamgages used for this study, 443 streamgages were used in the previous study and drained basins with 1 percent to about 30 percent total impervious area (Law and Tasker, 2003). Whereas most of the streamgages used in the current study have drainage basins with a relatively low percentage of urbanization, seven basins for streamgages in middle and east Tennessee (where the percentage of impervious area was not found to be useful in estimating annual peak streamflows) were determined to have greater than 20 percent impervious area. In west Tennessee, where percent imperviousness was found to be useful in estimating annual peak streamflows, only one streamgage used in the current study had a basin with greater than 10 percent impervious area. Of the 454 streamgages used in the current study, 430 streamgages had basins with less than 10 percent impervious area.

Description of the Study Area The description of this study area was originally presented in Law and Tasker (2003), and it is summarized herein. The diverse physiography and topography of Tennessee ranges from the lowlands of the Mississippi River Valley and Coastal Plain physiographic provinces to the hills of the Western Valley physiographic province and the Central Basin and Highland Rim physiographic sections, across the elevated and incised Cumberland Plateau physiographic section, to the mountains of the Valley and Ridge and Blue Ridge physiographic provinces (Fenneman and Johnson, 1946; Miller, 1974). Other than areas of anthropogenic disturbance such as quarries, land-surface elevations generally range from about 250 feet (ft) above the North American Vertical Datum of 1988 (NAVD 88) along the Mississippi River in west Tennessee to over 6,600 ft above NAVD 88 in the mountains of east Tennessee.

Geology in Tennessee is variable. West Tennessee is characterized by horizontal beds of unconsolidated sand, silt, clay, and gravel. Middle Tennessee is dominated by horizontal beds of karstic limestone. East Tennessee is characterized by folded beds of limestone and dolomite, and the mountains of east Tennessee are underlain by folded beds of complex metamorphic and igneous rock. The average annual precipitation in Tennessee varies from about 36 to 82 inches (in.), generally increasing from west to east (PRISM Climate Group, Oregon State University, 2014). Generally low average precipitation of 45 in. or less occurs in west Tennessee near the Mississippi River in the Coastal Plain physiographic province and in northern middle Tennessee in the Highland Rim and Nashville Basin sections of the Interior Low Plateaus physiographic province. Other areas of relatively low precipitation are in the northeastern part of the State in the Valley and Ridge physiographic province and the Blue Ridge physiographic province, where precipitation is variable. The highest average precipitation occurs in the eastern part of the State along the eastern boundary of the Appalachian Plateaus physiographic province and in the southeast part of the State along the Great Smoky Mountains in the Blue Ridge physiographic province (Fenneman and Johnson, 1946; Miller 1974). Annual peak streamflows typically occur during the winter and early spring (December through March) when frequent frontal storms yield widespread rains of high intensity on often already saturated ground. Localized peak-streamflow response of small watersheds occurs during the summer when thunderstorms produce intense downpours over relatively small geographic areas. In the fall, the remnants of hurricanes sometimes result in large rain events that can produce annual peak streamflows.

Previous Investigations Jenkins (1960), Patterson (1964), Speer and Gamble (1964), Randolph and Gamble (1976), Weaver and Gamble (1993), and Law and Tasker (2003) previously reported methods for estimating the magnitude and frequency of annual peak streamflows for rural streams in Tennessee. Randolph and Gamble (1976) delineated and statistically validated four HAs (fig. 1) similar to those used in subsequent studies. Weaver and Gamble (1993) used methods from Bulletin 17B (Interagency Advisory Committee on Water Data, 1982) and conducted the first study to use generalized least-squares (GLS) techniques for developing regional regression equations (Stedinger and Tasker, 1985). Law and Tasker (2003) presented regional regression equations and a region of influence method for predicting the magnitude and frequency of annual peak streamflows, in which a subset of streamgages is used to develop unique regression equations for estimating peak streamflows for AEPs at ungaged locations.

90˚

87˚30'

85˚

82˚30'

INDIANA ILLINOIS

WEST VIRGINIA

KENTUCKY

37˚30'

MISSOURI

VIRGINIA er

HA2

erl

HA4

HA1

Rive

TENNESSEE

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NORTH CAROLINA

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Ten n

35˚

SOUTH CAROLINA

ess

MISSISSIPPI

GEORGIA ALABAMA

0 0

25 25

50 50

100 MILES

100 KILOMETERS

EXPLANATION StreamStats (U.S. Geological Survey, 2019) application boundary for Tennessee Hydrologic area (HA) boundary

U.S. Geological Survey streamgage, by hydrologic area

Figure 1. Streamgages, hydrologic areas, and the Tennessee StreamStats application boundary, Tennessee and parts of Alabama, Georgia, Kentucky, Mississippi, North Carolina, and Virginia.

Introduction 3

HA1 HA2 HA3 HA4

84˚

82˚30'

81˚ WEST VIRGINIA

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KENTUCKY

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132 134

133 135 125 136

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128 126 127

89 88 39

123

122 124

84 76

VIRGINIA 119 82 64

36˚

ALABAMA

80 62

58 57 11

63 74 72 71 70 75 67 73 68 34 66 69 33 27

36 41 42 140 37 40 157 151 150 35 44 141 142 164 55 45 158 43 92 143 38 153 167 148 32 20 160 163 165166 93 46 94 18 30 26 152 144 47 95 17 54 31 29 159 162 154 50 19 147 24 16 25 13 99 52 49 156 145 146 98 15 161 168 53 51 48 155 14 97 23 169 112 114 22 149 117 96 116 115 170 21 171 113 172 108 104 107 198 193 190 103 110 191 105 210 196 106 192 111 102 HA1 109 194 100 181 211 197 201 180 101 213 177 199 4 189 208 176 195 187 214 212 200 175 186 179 188 207 205 184 215 185 209 202 5 174 183 204 2 203 1 3 182 178 206 173 6 7 8

TENNESSEE

12 10

NORTH CAROLINA

SOUTH CAROLINA

GEORGIA

0 0

30 30

60 MILES 60 KILOMETERS

EXPLANATION StreamStats (U.S. Geological Survey, 2019) application boundary for Tennessee Hydrologic area (HA) boundary

Figure 2. Streamgages in hydrologic area 1, east Tennessee and parts of Alabama, Georgia, North Carolina, and Virginia.

U.S. Geological Survey streamgage location and map number (Heal and others, 2025)

4 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year

85˚30'

87˚30'

85˚

37˚30'

82˚30'

KENTUCKY

ILLINOIS

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HA2

269 273 274 330 329

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316 327 326 317 318 324 325 312 314 311 323 296 313 308 295 310 297 309 307 298 306 305 299 304 293 302 303 301 300 MISSISSIPPI

258 257 261

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VIRGINIA

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251 247 242 239 238 248 241 240

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NORTH CAROLINA 288

SOUTH CAROLINA

282 284 286

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GEORGIA

283

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ALABAMA

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319 289 291 292

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256 262 259

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328 315

35˚

246 249

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225 224 223 222

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60 MILES 60 KILOMETERS

EXPLANATION StreamStats (U.S. Geological Survey, 2019) application boundary for Tennessee Hydrologic area (HA) boundary

279

U.S. Geological Survey streamgage location and map number (Heal and others, 2025)

Figure 3. Streamgages in hydrologic area 2, middle Tennessee and parts of Alabama, Kentucky, and Mississippi.

Introduction 5

6 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year 87˚

86˚

KENTUCKY

334

359 360

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ALABAMA

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GEORGIA

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20 MILES 20 KILOMETERS

Figure 4. Streamgages in hydrologic area 3, middle Tennessee and part of Alabama.

376

Hydrologic area (HA) boundary U.S. Geological Survey streamgage location and map number (Heal and others, 2025)

Introduction 7 90˚

88˚30' ILLINOIS

404

KENTUCKY

401 402

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414 428

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TENNESSEE

395 397

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ARKANSAS

398

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MISSISSIPPI

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20 MILES

20 KILOMETERS 453

StreamStats (U.S. Geological Survey, 2019) application boundary for Tennessee Hydrologic area (HA) boundary U.S. Geological Survey streamgage location and map number (Heal and others, 2025)

Figure 5. Streamgages in hydrologic area 4, west Tennessee and parts of Kentucky and Mississippi.

8 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year

Data Compilation A first step in conducting a flood-frequency analysis is the selection of candidate streamgages within the study area. The chosen streamgages must be reviewed for quality assurance and quality control prior to the computation of peak streamflows for selected AEPs. After the streamgages are selected and reviewed, basin characteristics for each streamgage must be calculated or acquired from other sources. For streamgages within a particular HA, the peak streamflows for AEPs are the dependent variables in the regional regression equations; the basin characteristics are the independent variables.

Streamgage Selection Streamgages within Tennessee and parts of Mississippi, Alabama, Georgia, North Carolina, Virginia, and Kentucky were considered for use in the study. USGS peak-streamflow data were acquired from the USGS National Water Information System (NWIS; USGS, 2016). Streamgages used in the study had at least 10 years of observed annual peaks through water year 2013 and were free from regulation by large dams and reservoirs. Annual peak-streamflow data from a final dataset of 454 streamgages (302 in Tennessee, 18 in Alabama, 19 in Georgia, 36 in Kentucky, 14 in Mississippi, 37 in North Carolina, and 28 in Virginia) were used in the flood-frequency analysis and to develop the RREs; of these 454 streamgages, 443 were used in the previous study (Law and Tasker, 2003).

Basin Delineation and Characteristics Of the 454 candidate streamgages used in the study, 441 are within the StreamStats application boundary for Tennessee (fig. 1). GIS-based methods similar to those incorporated in

the StreamStats application (USGS, 2019) for Tennessee were used to delineate drainage basins and determine drainage area (DRNAREA), main-channel slope (CSL10_85), and 2-year recurrence-interval climate factor (CLIMFAC2YR) (table 1) for streamgages within the StreamStats application boundary for Tennessee. Calculated DRNAREA values for the streamgages were compared to published values available from NWIS (USGS, 2016) for validation. Geospatial- and StreamStats-based delineations for streamgages 07030240 and 02348600 produced basins that were inaccurate and required manual correction before basin characteristics could be computed. For 13 streamgages located outside of the StreamStats application boundary for Tennessee (fig. 1), delineation of drainage basins and calculation of DRNAREA were accomplished by using StreamStats (USGS, 2019), and CSL10_85 and CLIMFAC2YR values were obtained from Law and Tasker (2003). For 450 of the 454 streamgages in the study area, percent imperviousness (LC11IMP) (table 1) was calculated by using National Land Cover Database Percent Developed Imperviousness data, 2011 edition (USGS, 2014). For the four streamgages in North Carolina that are outside the StreamStats application boundary for Tennessee (fig. 1), LC11IMP was obtained from StreamStats.

Drainage Area Drainage area (DRNAREA) is the surface area of the drainage basin of a streamgage, in square miles. For the 441 streamgages within the Tennessee StreamStats application boundary (fig. 1), drainage basins were delineated by using flow-direction grids derived from the 1/3 arc-second (approximately 10-meter [m]) National Elevation Dataset (USGS, variously dated). For the remaining 13 streamgages which lie outside of the Tennessee StreamStats application boundary (fig. 1), delineations of drainage basins and computations of DRNAREA were accomplished by using StreamStats.

Table 1. Physical and climatic basin characteristics used as explanatory variables in the generalized least-squares regression equations. Basin characteristic

Unit of measure

Definition Physical characteristics

DRNAREA

Square miles

Drainage area is the basin area for the location of interest.

CSL10_85

Feet per mile

The 10–85 channel slope is the change in elevation measured between points along the main channel at 10 and 85 percent of the stream length upstream from the streamgage of interest, divided by the stream-wise distance between the points.

LC11IMP

Percent

Imperviousness is the percentage of area beneath which precipitation is unable to infiltrate into the soil within the basin of a location of interest. Climatic characteristic

CLIMFAC2YR

Dimensionless

2-year recurrence-interval climate factor, describing spatial variability in the peak-streamflow frequency of small basins

Annual Exceedance Probability Analysis 9

10–85 Channel Slope The 10–85 channel slope (CSL10_85), in feet per mile, is the change in elevation between points along the main channel at 10 and 85 percent of the stream length upstream from the streamgage, divided by the stream-wise distance between the points. Ideally, the stream length used for determining points at 10 and 85 percent upstream from the streamgage is measured from the outlet to the basin divide. The further the headwaters of the main channel are from the watershed divide, the more subjective the manual computation of stream length can become. A GIS method was used to determine the stream length by identifying the longest flow path upstream from the site of interest; the longest flow path does not always conform to the main-channel path from the streamgage to the basin divide.

Percent Imperviousness Percent imperviousness is the percentage of surface area of a drainage basin through which precipitation is unable to infiltrate into the soil. Impervious surface can increase storm runoff to streams. LC11IMP was calculated for the basins of 450 streamgages in this study by using the National Land Cover Database 2011 Percent Developed Imperviousness dataset (USGS, 2014). For the remaining four streamgages located in North Carolina, LC11IMP was obtained from StreamStats (USGS, 2019). Conceptually, LC11IMP is relatively high in drainage basins of urban streams, where much of the surface area is paved. Even though most of the basins for streamgages used in this study had relatively low values of LC11IMP, seven of the basins had LC11IMP values greater than 20 percent. These basins with LC11IMP values greater than 20 percent are in HAs 1–3, where LC11IMP was not determined to be a statistically useful explanatory variable in prediction of streamflows for AEPs. Within HA4, only streamgage 07032200 had an LC11IMP value (13.7 percent) greater than 10 percent.

2-Year Climate Factor Lichty and Karlinger (1990) describe a method for integrating long-term rainfall and pan evaporation data into a climate factor to help describe the spatial variability in the frequency of annual peak streamflow of small basins. Climate factors for 2-, 25-, and 100-year recurrence-interval floods (streamflows for 0.5, 0.04, and 0.01 AEPs, respectively) were developed using data from 71 long-term rainfall stations and regression analysis relating peak-streamflow estimates from 50 rainfall-runoff models to basin characteristics. The climate factors, which describe the effects of climate variability on peak streamflows, were developed and regionalized by kriging within a large part of the United States. Law and Tasker (2003) used only the 2-year recurrence interval climate factor

(CLIMFAC2YR) for predicting peak-streamflow-frequency magnitudes within the study area, and CLIMFAC2YR is the only climate variable used in this study.

Annual Exceedance Probability Analysis An AEP is the probability that a streamflow of a given magnitude will be met or exceeded in a given year. For this study, streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs were computed. The previous study of the magnitude and frequency of flooding in Tennessee (Law and Tasker, 2003) reported flood recurrence intervals, such as the 100-year flood. An annual recurrence interval is the reciprocal of an AEP for a given flood magnitude; for example, a flood having an AEP of 1 percent (0.01, or 1/100) is equivalent to the 100-year recurrence interval (table 2). AEPs corresponding to annual peak streamflows for the 454 streamgages in the study area were computed by using the USGS program PeakFQ, version 7.0.29368 (Flynn and others, 2006; Feaster and others, 2014; Veilleux and others, 2014). The PeakFQ program was used to fit a log-Pearson type III distribution to base-10 logarithms of annual peak streamflows per Bulletin 17C guidelines (England and others, 2019), which include the use of EMA (Cohn and others, 1997) and MGBT (Cohn and others, 2013) for identifying PILFs during the fitting process. The LP3 distribution is a three-moment distribution that requires estimating the population mean, standard deviation, and skew coefficient.

Expected Moments Algorithm The EMA method (Cohn and others, 1997) for estimating streamflows corresponding to AEPs incorporates streamflow intervals and perception thresholds that describe peak-streamflow knowledge for every year in the period of record for a streamgage, including the systematic peak Table 2. Recurrence intervals that correspond with annual exceedance probabilities. Annual exceedance probability

Recurrence interval (in years)

0.5

0.2

0.1

0.04

0.02

0.01

100

0.005

200

0.002

500

10 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year streamflows, historical peak streamflows, and gaps in the peak-streamflow record (Southard and Veilleux, 2014). Using streamflow intervals and perception thresholds, the EMA can accommodate censored peak-streamflow data (streamflows that are known only to be greater than or less than some value) and interval peak-streamflow data (streamflows that are known only to be greater than some value and less than another value).

station skews were used as the weights. The weighted mean regional skew for the study area was estimated to be a constant value of 0.10 with a mean square error of 0.132.

Multiple Grubbs-Beck Test for Detecting Potentially Influential Low Floods

Streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs were related to basin characteristics by using multiple linear regression techniques. Ordinary least-squares (OLS) regression techniques were used to validate previously defined HAs (Randolph and Gamble, 1976; Law and Tasker, 2003) and to determine the most useful significant values for basin characteristics for use in predicting streamflows corresponding to AEPs within each HA. Once the most useful basin characteristics were determined, final regression equations were developed by using GLS regression techniques (Tasker and Stedinger, 1989).

The distribution of annual peak streamflows for a streamgage can contain one or more low outliers, which are small observations that deleteriously affect the upper-tail results (decreasing AEP) (Interagency Advisory Committee on Water Data, 1982; Cohn and others, 2013). For example, during drier years, some small annual peak streamflows may not represent floods or whole basin response to precipitation inputs. These small annual peak streamflows, or PILFs, are small observations that could potentially exert a large influence on the fitted flood-frequency curve (Cohn and others, 2013; England and others, 2019). PILFs can result from physical processes different from those associated with large floods and should not affect the part of the flood-frequency curve representing large floods (Cohn and others, 2013). The MGBT used with EMA provides a standard approach for detecting multiple PILFs. The use of the MGBT can improve the fit of the flood-frequency curve for large floods (Cohn and others, 2013; Southard and Veilleux, 2014).

Regional Skew The skew coefficient is a measure of asymmetry of the distribution of a set of annual peak streamflows about the median. If the mean of a distribution of annual peak streamflows is equal to the median, the distribution has zero skew. One or more extremely high annual peak streamflows in a distribution can result in a positive skew, when the mean of the distribution exceeds the median. One or more extremely low annual peak streamflows in a distribution can result in a negative skew, when the mean of the distribution is less than the median. The skew coefficient estimated from a series of annual peak streamflows at a given streamgage (the station skew) may not provide an accurate estimate of the population skew. England and others (2019) suggested that the station skew be weighted with a regional skew determined by analysis of skew of annual peak-streamflow data from selected long-term streamgages in the study area. For this study, a regional skew was determined for input to the weighted skew computation. Station skew coefficients and their associated mean square errors for 214 streamgages in the study area with at least 30 years of record were used to compute a weighted mean skew. The inverses of the mean square errors of the

Development of Regional Regression Equations

Validation of Hydrologic Areas Parts of eight physiographic divisions defined by Fenneman and Johnson (1946) and Miller (1974) were combined to create four HAs (HAs 1–4), which had been previously defined by Randolph and Gamble (1976). The boundaries of these HAs were modified slightly for use in the most recent low-flow and flow duration study for Tennessee (Law and others, 2009), and those modifications were adopted for the current peak-streamflow study. Each HA has distinct hydrologic, geologic, and topographic characteristics (Law and Tasker, 2003). HA1 covers most of east Tennessee (fig. 1), including most of the Cumberland Plateau physiographic section and the entirety of the parts of the Valley and Ridge and Blue Ridge physiographic provinces that are in Tennessee. HA2 includes almost all the Highland Rim and parts of the Cumberland Plateau physiographic sections and the Western Valley physiographic province. HA3 is predominantly in the Central Basin physiographic section. HA4 includes the Coastal Plain physiographic province and the western part of the Western Valley physiographic province (Fenneman and Johnson, 1946; Miller, 1974). Of the 454 streamgages used in this study, HA1 contains 216, HA2 contains 114, HA3 contains 63, and HA4 contains 61 streamgages (table 3). To determine the validity of the previously defined HAs for use in this study, a Wilcoxon signed-rank test (Tasker, 1982) was performed on the residuals of a single-variable OLS regression equation developed using all 454 streamgages in the study, with the base-10 logarithm of DRNAREA as the explanatory variable and the base-10 logarithm of the streamflow corresponding to the 0.01 AEP as the dependent variable. At a significance level of 0.05 (p≤0.05), the residuals from HAs 1–3 were determined to be a statistically different population than the residuals from all the streamgages. The residuals from HA4 were not found to be significantly

Development of Regional Regression Equations 11 Table 3. Numbers of streamgages by State and hydrologic area used to estimate the magnitude and frequency of floods at ungaged location on streams in Tennessee through the 2013 water year. Number of streamgages by hydrologic area (fig. 1)

State 1

Total streamgages by State

Alabama

Georgia Kentucky Mississippi North Carolina

Tennessee

130

302

Virginia

Total streamgages by hydrologic area

216

114

454

different from the residuals of all the streamgages, but the residuals from HA4 were determined to be a statistically different population than the residuals from each of the other regions (HAs 1–3) when tested separately at a significance level of 0.05.

Evaluation of Basin Characteristics Exploratory regression techniques can be used to iterate through all unique combinations of explanatory variables (basin characteristics) to determine the optimal combination that explains the independent variables (streamflows corresponding to the AEPs). Exploratory OLS regression was used to determine which basin characteristics were statistically significant in explaining streamflows corresponding to the selected AEPs, and therefore would be used in the development of the final GLS regression equations for each HA. For a subset of AEPs, unique combinations of DRNAREA, CSL10_85, CLIMFAC2YR, and LC11IMP were evaluated. All basin characteristics and AEPs were logarithmically transformed using base-10 logarithms, and 1 was added to LC11IMP prior to the transformation. Within each HA, the combination of explanatory variables with the highest adjusted R-squared value and for which all variables were significant at the 95-percent confidence level (p≤0.05) was selected from the suite of possible combinations. The selected basin characteristics were then used to develop the final GLS regression equations for this study.

Generalized Least-Squares Regression GLS regression techniques, which can account for differences in streamflow record length, variability of the streamflow record, and cross correlation of concurrent streamflow at different streamgages, were used to generate the final regression equations and measures of accuracy. Particularly when these issues affect streamflow data used in the analysis, GLS regression techniques produce more

accurate predictions and better estimates of accuracy than OLS techniques (Stedinger and Tasker, 1985). Using GLS regression, streamflows corresponding to selected AEPs were estimated using Bulletin 17C (England and others, 2019) methods. The GLS RREs were determined by using the weighted-multiple-linear regression (WREG) computer program (Eng and others, 2009; U.S. Geological Survey, 2013). GLS RREs predicting streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs were developed for HAs 1–4 (fig. 1; table 4). Explanatory variables used in the final equations were DRNAREA and CLIMFAC2YR for HA1; DRNAREA and CSL10_85 for HA2; DRNAREA for HA3; and DRNAREA and LC11IMP for HA4 (table 4). Because LC11IMP is expressed as a percentage of basin area and is zero at many locations, 1 was added to the values of LC11IMP before the logarithmic transformation. Other than the explanatory variable CSL10_85 used in equations predicting streamflows corresponding to the 0.005 and 0.002 (0.5- and 0.2-percent) AEPs for HA2, all basin characteristics used in the final GLS regression equations (table 4) were statistically significant (p≤0.05). For HA2, DRNAREA and CSL10_85 were statistically significant in predicting streamflows corresponding to the 0.5, 0.2, 0.1, 0.04, 0.02, and 0.01 (50-, 20-, 10-, 4-, 2-, and 1-percent) AEPs. For the 0.005 and 0.002 AEPs, CSL10_85 was not statistically significant when DRNAREA and CSL10_85 were used as the only explanatory variables. To avoid the potential for predictions of streamflows corresponding to greater AEPs to exceed the predictions for the 0.005 and 0.002 AEPs, DRNAREA and CSL10_85 were used as explanatory variables in all equations for HA2. Average standard errors of prediction for the regression equations ranged from 44.4 to 51.4 percent for HA1, 30.4 to 42.9 percent for HA2, 35.7 to 42.6 percent for HA3, and 32.5 to 47.4 percent for HA4 (table 4). The relation between residuals and predicted streamflow corresponding to the 0.01 (or 1-percent) AEP using the GLS equations is shown in figure 6.

12 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year Table 4. Final generalized least-squares regression equations for estimating streamflows corresponding to selected annual exceedance probabilities on rural streams in Tennessee and associated performance metrics. [AEP, annual exceedance probability; RI, recurrence interval; GLS, generalized least squares; ft3/s, cubic foot per second; Pseudo-R2, pseudo coefficient of determination; AVP, average variance of prediction; log10(ft3/s), base-10 logarithm of cubic feet per second; %, percent; Q, annual exceedance probability streamflow; DRNAREA, drainage area, in square miles; CLIMFAC2YR, 2-year recurrence-interval climate factor; CSL10_85, 10 and 85 percent channel slope, in feet per mile; LC11IMP, impervious area, in percent]

AEP

RI, in years

GLS equations, in ft3/s

Pseudo-R2

AVP, in squared log10 (ft3/s)

Average standard error of prediction, in percent

Hydrologic area 1 (fig. 2)—216 streamgages 0.5

Q50% = 4.86 DRNAREA0.735 CLIMFAC2YR4.068

92.3

0.04

46.7

0.2

Q20% = 9.55 DRNAREA0.726 CLIMFAC2YR3.844

92.7

0.03

44.4

0.1

Q10%

= 14.1 DRNAREA0.721 CLIMFAC2YR3.687

92.5

0.03

45.0

0.04

Q4%

= 21.8 DRNAREA0.716 CLIMFAC2YR3.486

92.1

0.04

45.7

0.02

Q2% = 29.0 DRNAREA0.712 CLIMFAC2YR3.350

91.6

0.04

47.3

0.01

100

Q1%

= 37.8 DRNAREA0.708 CLIMFAC2YR3.219

91.0

0.04

48.9

0.005

200

Q0.5% = 48.5 DRNAREA0.705 CLIMFAC2YR3.090

90.7

0.04

49.7

0.002

500

= 65.8 DRNAREA0.701 CLIMFAC2YR2.928

90.0

0.04

51.4

Q0.2%

Hydrologic area 2 (fig. 3)—114 streamgages 0.5

Q50% =84.9 DRNAREA0.796 CSL10_850.196

95.4

0.02

33.2

0.2

Q20%

= 155 DRNAREA0.776 CSL10_850.182

95.9

0.02

30.4

0.1

Q10% = 214 DRNAREA0.766 CSL10_850.172

95.6

0.02

31.1

0.04

Q4%

= 306 DRNAREA0.753 CSL10_850.158

94.4

0.02

35.1

0.02

Q2% = 388 DRNAREA0.745 CSL10_850.148

93.9

0.02

36.7

0.01

100

Q1%

= 483 DRNAREA0.736 CSL10_850.139

93.0

0.03

39.0

0.005

200

Q0.5% = 589 DRNAREA0.729 CSL10_850.130

92.5

0.03

40.5

0.002

500

= 753 DRNAREA0.719 CSL10_850.118

91.6

0.03

42.9

0.5

Q50% = 348 DRNAREA0.659

93.7

0.03

37.7

0.2

Q20%

= 540 DRNAREA0.651

94.2

0.02

35.7

0.1

Q10% = 676 DRNAREA0.648

93.9

0.02

36.5

0.04

Q4%

= 853 DRNAREA0.645

93.6

0.02

37.3

0.02

Q2% = 989 DRNAREA0.643

93.1

0.03

39.0

0.01

100

Q1%

= 1,127 DRNAREA0.642

92.8

0.03

39.9

0.005

200

Q0.5% = 1,268 DRNAREA0.641

92.5

0.03

40.9

0.002

500

= 1,459 DRNAREA0.640

91.9

0.03

42.6

0.5

Q50% =348 DRNAREA0.527 (LC11IMP+1)0.271

91.2

0.02

34.2

0.2

Q20%

=498 DRNAREA0.542 (LC11IMP+1)0.285

92.4

0.02

32.5

0.1

Q10% = 600 DRNAREA0.550 (LC11IMP+1)0.290

92.1

0.02

33.7

0.04

Q4%

= 724 DRNAREA0.560 (LC11IMP+1)0.294

91.0

0.02

36.8

0.02

Q2% = 817 DRNAREA0.566 (LC11IMP+1)0.297

89.9

0.03

39.6

0.01

100

Q1%

= 908 DRNAREA0.572 (LC11IMP+1)0.299

89.1

0.03

41.9

0.005

200

Q0.5%

= 1,002 DRNAREA0.578 (LC11IMP+1)0.300

88.2

0.03

44.3

0.002

500

Q0.2% = 1,127 DRNAREA0.584 (LC11IMP+1)0.300

87.0

0.04

47.4

Q0.2%

Hydrologic area 3 (fig. 4)—63 streamgages

Q0.2%

Hydrologic area 4 (fig. 5)—61 streamgages

Residual, in base-10 logarithm of cubic feet per second

Development of Regional Regression Equations 13 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 −0.05 −0.10 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40 −0.45 −0.50 −0.55 −0.60 −0.65 −0.70

EXPLANATION U.S. Geological Survey streamgage, by hydrologic area (HA) (fig. 1) HA1 HA3 HA2 HA4

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Streamflow corresponding to the 0.01 annual exceedance probability, in base-10 logarithm of cubic feet per second, estimated by generalized least-squares regression equations

Figure 6. Relation between residuals and predicted streamflows corresponding to the 0.01 annual exceedance probability using the generalized least-squares equations.

14 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year

Applications of Methods RREs are statistical models that must be interpreted and applied within the limits of the data used to develop them and with the understanding that the results are best-fit estimates with an associated variance. The RREs are not valid at locations where dams, flood-control structures, channelization, diversion, or backwater have an appreciable effect on annual peak streamflow. Applications of the equations for estimating streamflows corresponding to selected AEPs differ for gaged locations on streams, ungaged locations on gaged streams, and locations on ungaged streams. EMA estimates for gaged locations and GLS estimates for ungaged locations can be used to determine weighted estimates for gaged locations and ungaged locations near streamgages. EMA, GLS, and weighted estimates for streamgages used in the study are available in tables A and B of Heal and others (2025).

Streamgage Locations Accuracy of at-site estimates of streamflows corresponding to AEPs computed for a streamgage can be increased by weighting the at-site estimates with estimates computed by using the GLS equations. If the at-site and GLS estimates of a given AEP are weighted inversely proportional to their associated variances, the variance of the weighted estimate will be less than the variances of either the at-site or the GLS estimate. The weighted estimate is computed by using the following equation (England and others, 2019):

V Pp  (g)r * log10 Qp  (g)s + V Pp  (g)s * log10 Qp  (g)r log10 Qp  (g)w = __________________________________             , V Pp  (g)r + V Pp  (g)s where

(1)

Qp(g)w

is the weighted estimate of streamflow corresponding to the selected p-percent AEP at streamgage (g), in cubic feet per second;

VPp(g)r

is the variance of prediction of streamflow corresponding to the p-percent AEP at streamgage (g) from the GLS regression model, in log base-10 units;

Qp(g)s

is the estimate of streamflow corresponding to the p-percent AEP at streamgage (g) from the EMA analysis, in cubic feet per second;

VPp(g)s

is the variance of prediction of streamflow corresponding to the p-percent AEP at streamgage (g) from the EMA analysis, in log base-10 units; and

Qp(g)r

is the estimate of streamflow corresponding to the p-percent AEP at streamgage (g) from the GLS regression model, in cubic feet per second.

The variance of prediction of the weighted estimate for streamgage (g) (in log base-10 units) can be computed as:

V Pp  (g)s *V Pp  (g)r _______________ V Pp  (g)w =       , V Pp  (g)s + V Pp  (g)r where VPp(g)w

(2)

is the variance of prediction of the weighted estimate of streamflow corresponding to the p-percent AEP at streamgage (g), in log base-10 units.

A 95-percent confidence interval for the weighted estimate (in cubic feet per second) can be computed as: _

95 %CI upper,i = 10 (log10Qp  g w +1.96√VPp  g w ), ( )

( )

95 %CI lower,i = 10 (log10Qp  g w −1.96√VPp  g w ), ( )

where 95 %CI upper,i 95 %CI lower,i

( )

is the upper 95-percent confidence interval for the weighted estimate of the streamflow corresponding to the selected AEP at streamgage (g), and is the lower 95-percent confidence interval for the weighted estimate of the streamflow corresponding to the selected AEP at streamgage (g).

(3) (4)

Applications of Methods 15 │ΔA│

Ungaged Locations on Gaged Streams For ungaged locations, streamflows corresponding to selected AEPs are estimated by using the appropriate GLS equation (table 4). Streamflows at ungaged locations on gaged streams can be improved by applying a drainage area ratio. The following procedure is recommended if the drainage area at the ungaged location is within 50 percent of the drainage area at the gaged location (drainage area ratio is more than 0.5 or less than 1.5) (Ries and Dillow, 2006):

DRNARE A(  u) Qp(u) = ___________    Q , (DRNARE A(  g)) p(g)w

DRNAREA(g)

is the drainage area at the streamgage location, in square miles;

Qp(u)r

is the GLS estimate of streamflow at the ungaged location corresponding to the p-percent AEP, in cubic feet per second; and

Qp(u)

is the estimate of streamflow corresponding to the p-percent AEP at the ungaged location after transferring the weighted estimate from the gaged location, u, in cubic feet per second.

where

Qp(u)

(5)

is the estimate of streamflow corresponding to the p-percent AEP at the ungaged location after transferring the weighted estimate from the gaged location, u, in cubic feet per second;

DRNAREA(u)

is the drainage area at the ungaged location, in square miles;

DRNAREA(g)

is the drainage area at the streamgage location, in square miles;

b Qp(g)w

is the exponent of drainage area from the appropriate GLS equation (table 5); and is the weighted estimate of streamflow corresponding to the p-percent AEP at the streamgage location, in cubic feet per second.

The weighted flood estimate at the ungaged location can be computed by using the following equation: 2|ΔA| 2|ΔA| ___________ ___________ Qp  (u)w =    Qp  (u)r + 1 −    Qp  (u), (6) DRNARE A(  g)) (DRNARE A(  g)) (

where

Qp(u)w

is the weighted estimate of streamflow at the ungaged location for the p-percent AEP, in cubic feet per second;

is the absolute difference in drainage areas between the ungaged location (DRNAREA(u)) and the streamgage (DRNAREA(g)) location, in square miles;

If the drainage area of an ungaged location differs by more than 50 percent from that of the streamgage location, the GLS estimate becomes considerably more uncertain. If an ungaged location is between two streamgages on the same stream, the streamgage with the closest drainage area ratio and longest period of record is suggested for use (Sauer, 1964).

Accuracy and Limitations of GLS Equations Accuracy and limitations of the GLS equations are affected by several factors. The equations are applicable at locations on streams in Tennessee where annual peak streamflows are not substantially affected by regulation, diversion, channelization, urbanization, or backwater. The equations are most applicable when estimating streamflow for basins having characteristics (explanatory variables) within the ranges of those used to develop the GLS equations (table 6); using explanatory variables outside of these ranges may result in prediction errors greater than expected (table 4). The GLS equations are intended for use with basin characteristics computed using the methods described in this report. The StreamStats application (USGS, 2019) for Tennessee contains the same GIS layers and measurement methods used to calculate basin characteristics and estimates streamflows corresponding to various AEPs by using the GLS equations developed for this study.

16 Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year Table 5. Drainage area generalized least-squares regression equations for estimating exponent b for equation 5 (Ries and Dillow, 2006). [AEP, annual exceedance probability; RI, recurrence interval; GLS, generalized least squares; ft3/s, cubic foot per second; Pseudo-R2, pseudo coefficient of determination; AVP, average variance of prediction; log10(ft3/s), base-10 logarithm of cubic feet per second; %, percent; Q, annual exceedance probability streamflow; DRNAREA, drainage area, in square miles]

AEP

RI, in years

GLS equations, in ft3/s

Exponent b

Pseudo-R2

AVP, in squared log10 (ft3/s)

Average standard error of prediction, in percent

Hydrologic area 1 (fig. 2)—216 streamgages 0.5

Q50% = 127 DRNAREA0.732

0.732

91.6

0.04

48.9

0.2

Q20% = 207 DRNAREA0.725

0.725

92.3

0.04

45.8

0.1

Q10%

= 269 DRNAREA0.721

0.721

92.0

0.04

46.4

0.04

Q4% = 355 DRNAREA0.716

0.716

91.7

0.04

47.0

0.02

Q2%

= 424 DRNAREA0.712

0.712

91.3

0.04

47.8

0.01

100

Q1% = 497 DRNAREA0.709

0.709

90.8

0.04

49.3

0.005

200

Q0.5%

= 574 DRNAREA0.706

0.706

90.4

0.04

50.1

0.002

500

Q0.2%

= 684 DRNAREA0.703

0.703

89.8

0.04

51.8

Hydrologic area 2 (fig. 3)—114 streamgages 0.5

Q50% = 201 DRNAREA0.716

0.716

94.7

0.04

35.7

0.2

Q20% = 344 DRNAREA0.702

0.702

95.4

0.04

32.1

0.1

Q10%

= 456 DRNAREA0.695

0.695

94.9

0.04

33.4

0.04

Q4% = 628 DRNAREA0.689

0.689

94.1

0.04

35.7

0.02

Q2%

= 748 DRNAREA0.684

0.684

93.4

0.04

37.9

0.01

100

Q1%

= 889 DRNAREA0.680

0.680

92.8

0.04

39.4

0.005

200

Q0.5% = 1,045 DRNAREA0.676

0.676

92.3

0.04

40.9

0.002

500

= 1,271 DRNAREA0.671

0.671

91.4

0.04

43.2

Q0.2%

Hydrologic area 3 (fig. 4)—63 streamgages 0.5

Q50% = 348 DRNAREA0.659

0.659

93.7

0.03

37.7

0.2

Q20% = 528 DRNAREA0.651

0.651

94.2

0.02

35.7

0.1

Q10%

= 676 DRNAREA0.648

0.648

93.9

0.02

36.5

0.04

Q4%

= 853 DRNAREA0.645

0.645

93.6

0.02

37.3

0.02

Q2% = 989 DRNAREA0.64

0.643

93.1

0.03

39.0

0.01

100

Q1%

= 1,127 DRNAREA0.642

0.642

92.8

0.03

39.9

0.005

200

Q0.5% = 1,268 DRNAREA0.641

0.641

92.5

0.03

40.9

0.002

500

= 1,459 DRNAREA0.640

0.640

91.9

0.03

42.6

Q0.2%

Hydrologic area 4 (fig. 5)—61 streamgages 0.5

Q50% = 439 DRNAREA0.525

0.525

90.0

0.02

36.3

0.2

Q20%

= 640 DRNAREA0.539

0.539

91.2

0.02

34.8

0.1

Q10% = 775 DRNAREA0.547

0.547

90.6

0.02

36.5

0.04

Q4%

= 944 DRNAREA0.556

0.556

89.8

0.03

39.0

0.02

Q2% = 1,072 DRNAREA0.562

0.562

88.9

0.03

41.2

0.01

100

Q1%

= 1,194 DRNAREA0.568

0.568

87.8

0.03

44.0

0.005

200

Q0.5% = 1,321 DRNAREA0.573

0.573

86.9

0.04

46.3

0.002

500

= 1,489 DRNAREA0.579

0.579

85.9

0.04

48.9

Q0.2%

References Cited 17 Table 6. Ranges of physical and climatic basin characteristics for use in generalized least-squares regression equations, by hydrologic area. [DRNAREA, drainage area; mi2, square miles; CSL10_85, 10 and 85 percent channel slope; ft/mi, feet per mile; LC11IMP, imperviousness; CLIMFAC2YR, 2-year recurrence-interval climate factor; N/A, not applicable]

Hydrologic area (fig. 1)

Number of streamgages used

Range of basin characteristic DRNAREA, in mi2

CSL10_85, in ft/mi

LC11IMP, in percent

CLIMFAC2YR, dimensionless

216

0.20–21,400

N/A

2.05–2.42

114

0.42–2,560

1.74–356

N/A

0.17–2,050

N/A

0.95–2,310

N/A

0.26–13.7

N/A

Summary Methods for estimating the magnitude and frequency of flooding at ungaged locations on rural streams in Tennessee were last updated in 2003. Since then, many years of additional annual peak-streamflow data have been collected, more geospatial data for calculation of explanatory variables have become available, and improved statistical methods have been developed for flood-frequency analysis. Thus, the U.S. Geological Survey (USGS), in cooperation with the Tennessee Department of Transportation, developed multiple linear regression equations to estimate streamflows corresponding to selected annual exceedance probabilities (AEPs) at ungaged locations on streams in Tennessee. Version 7.0.29368 of USGS PeakFQ software was used to conduct frequency analyses of annual peak streamflows for 454 streamgages operated by the USGS in Tennessee and surrounding States. Streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs (the 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year floods, respectively) were estimated by using the expected moments algorithm in PeakFQ. Four hydrologic areas used in previous flood-frequency studies in Tennessee were validated for statistical independence. For each hydrologic area, generalized least squares (GLS) regression equations were developed, incorporating combinations of drainage area (DRNAREA), 10–85 channel slope (CSL10_85), 2-year recurrence-interval climate factor (CLIMFAC2YR), and percent imperviousness (LC11IMP) as explanatory variables. Statistically significant (p≤0.05) explanatory variables used in the equations were DRNAREA and CLIMFAC2YR for HA1; DRNAREA and CSL10_85 for HA2; DRNAREA for HA3; and DRNAREA and LC11IMP for HA4. The final GLS regression models were generated using the USGS weighted-multiple-linear regression (WREG) program, resulting in standard errors of prediction ranging from 30.4 to 51.4 percent. The GLS regression equations are applicable at locations on streams in Tennessee where annual peak streamflows are not substantially affected by regulation, diversion, channelization, backwater, or urbanization. The

applicability and accuracy of the GLS regression models depend on the basin characteristics measured for an ungaged location on a stream being within range of those used to develop the equations. Methods are presented for using the GLS regression equations to estimate streamflows corresponding to selected AEPs at ungaged locations on both gaged and ungaged streams and to calculate weighted at-site estimates of streamflows corresponding to AEPs at streamgage locations. The final GLS regression equations and methods for estimating streamflows corresponding to selected AEPs have been incorporated in the USGS StreamStats application for Tennessee.

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For more information about this publication, contact Director, Lower Mississippi-Gulf Water Science Center U.S. Geological Survey 640 Grassmere Park, Suite 100 Nashville, TN 37211 For additional information, visit https://www.usgs.gov/centers/lmg-water/ Publishing support provided by Lafayette Publishing Service Center

Ladd and Ensminger—Estimating the Magnitude and Frequency of Floods at Ungaged Streams, Tenn., Through the 2013 Water Year—SIR 2024–5130

ISSN 2328-0328 (online) https://doi.org/10.3133/sir20245130

Estimating the Magnitude and Frequency of Floods at Ungaged Locations on Streams in Tennessee Through the 2013 Water Year

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