Description of Study Area

The study area includes the State of Tennessee and parts of the surrounding States of Alabama, Georgia, Mississippi, North Carolina, and South Carolina that are within the Mississippi Valley Loess Plains, Southeastern Plains, Interior Plateaus, Southwestern Appalachians, Ridge and Valley, and Piedmont Level 3 (L3) ecoregions (U.S. Environmental Protection Agency, 2013; fig. 1). No urban streamgages were in the Mississippi Alluvial Plain, Central Appalachians, or Blue Ridge L3 ecoregions. Streamgages located in urban areas in the Middle Atlantic and Southern Coastal Plain L3 ecoregions in Georgia, North Carolina, and South Carolina were not considered because of the small number of available streamgages and because of tidal effects. The 136 streamgages selected for use in the study had a minimum of 10 percent developed imperviousness in their drainage basins, as indicated by data from the 2011 National Land Cover Database (NLCD) (Homer and others, 2015; Dewitz and U.S. Geological Survey, 2021) and a minimum of 10 years of annual peak streamflow data available through the 2022 water year. A water year is defined as the period October 1–September 30, named for the year in which it ends. Drainage areas for the streamgages ranged from 0.164 to 93.4 square miles (mi²) (refer to file “GLSresults.csv” in Wagner and Ladd, 2024b). Percent developed imperviousness (from 2011 NLCD; Homer and others, 2015; Dewitz and U.S. Geological Survey, 2021) in the drainage basins of the streamgages ranged from 10 to 79 percent. The percentage of developed land use (sum of classes 21–24 from 2011 NLCD) in the drainage basins of the streamgages ranged from 26 to 100 percent. The number of streamgages in each State was as follows: 21 in Alabama, 32 in Georgia, 9 in Mississippi, 38 in North Carolina, 16 in South Carolina, and 20 in Tennessee.

Previous Investigations

Several previous investigations have determined the magnitude and frequency of floods on urban streams in Tennessee and surrounding States (Wibben, 1976; Neely, 1984; Robbins, 1984; Feaster and others, 2014). Wibben (1976) used streamflow data from 14 basins in Davidson County, Tennessee, that ranged in size from 1.58 to 64 mi² and had impervious cover ranging from 3 to 37 percent of the basin area. Average record length was 11 years; records were extended using a digital model of the hydrologic system. Estimates of streamflow corresponding to the 2-, 5-, 10-, 25-, 50-, and 100-year recurrence intervals (50-, 20-, 10-, 4-, 2-, and 1-percent AEPs, respectively) were compared to those from regional equations for estimating peak-flows at ungaged locations in rural basins. In fully developed residential areas, flood peaks and lag times were not significantly different from those for undeveloped areas. However, the data were not sufficient to determine whether an increase in flood peaks would occur from extremely intensive development in extremely small basins.

Neely (1984) presented techniques for estimating the magnitude and frequency of peak streamflow and storm runoff of streams in urban areas of Memphis, Tennessee. Data from 27 streamgages with 8 or more years of annual peak streamflow record and drainage areas ranging from 0.04 to 19.4 mi² were used in the study; flood-frequency at each streamgage was computed using a rainfall-runoff model. Equations derived from regression analyses provided estimates of annual peak streamflow corresponding to recurrence intervals of 2–100 years (50- to 1-percent AEPs) for streams with drainage areas less than 20 mi². Statistically significant basin characteristics in the regression equations were drainage area and channel condition (paved or unpaved).

Robbins (1984) used 22 rainfall-runoff streamgages located in urban areas across Tennessee with drainage areas ranging from 0.21 to 24.3 mi² and impervious areas (based on aerial photographs) ranging from 4.7 to 74 percent of the streamgage basin areas. Flood-frequency estimates for each streamgage were computed using a rainfall-runoff model. Equations derived from regression analyses provided estimates of annual peak streamflow corresponding to recurrence intervals of 2–100 years (50- to 1-percent AEPs).

Feaster and others (2014) used annual peak-flow data from 116 urban streamgages and 32 rural streamgages with drainage areas less than 1 mi², along with annual peak-flow data from an additional 340 rural streamgages in Georgia, South Carolina, and North Carolina. The Expected Moments Algorithm (EMA), which fits a log-Pearson type III (LP3) distribution to the logarithms of the annual peak-flow data from the streamgages, was used to compute at-site flood-frequency for the streamgages. GLS regression was used to generate regression equations for three hydrologic regions (Piedmont and Ridge and Valley, Sand Hills, and Coastal Plain). Annual peak-flow data from urban streamgages in Florida and New Jersey were used in the regression analysis for the Coastal Plain region, which allowed the applicability of the regression equations in that region to be expanded from 3.5 to 53.5 mi². Average standard error of prediction for the regression equations ranged from 25 percent for the 10-percent AEP for the Piedmont Ridge and Valley region to 73.3 percent for the 0.2-percent AEP for the Sand Hills region. Statistically significant basin characteristics in the regression models were drainage area, percent developed imperviousness from the 2006 NLCD (Fry and others, 2011; U.S. Geological Survey, 2011), percent developed land use in the streamgage basins from the 2006 NLCD, and the 24-hour, 50-year precipitation from the National Oceanic Atmospheric Administration’s Atlas 14 precipitation frequency estimates (National Oceanic and Atmospheric Administration, 2013).

Perception Thresholds

For continuous-record streamgages, perception thresholds for periods of gaged record were set to (0, infinity), meaning the entire range of streamflow could be observed (refer to “Data Representation” in England and others, 2018, p. 67–70); also refer to files “TNurbanFFreq.pkf” and “TNurbanFFreq.psf” in Wagner and Ladd, 2024a). For CSGs, perception thresholds for periods of gaged record were set to (minimum recordable streamflow, infinity). If the minimum recordable streamflow was unknown or unavailable, perception thresholds were set to (0, infinity). For all streamgages, perception thresholds for years or periods of missing record where historical information was unavailable were set to (infinity, infinity). If historical information was available, perception thresholds for missing years or historical periods were set to (value of historical peak, infinity), indicating that the flow is assumed to have been less than the magnitude of the historical peak. For annual peak streamflows assigned qualification code 4 (flow is less than indicated value; refer to table B.4 in appendix B.4 of Flynn and others, 2006), perception thresholds were set to (indicated value, infinity). In almost all cases, this occurred at CSGs during years when the gage height did not exceed the minimum recordable level. For annual peak streamflows assigned qualification code 8 (flow is greater than indicated value), in most cases perception thresholds were set to (0, infinity) because the entire range of streamflow could be observed that year. In cases where it was not known whether the entire range of streamflow was able to be observed, perception thresholds were set to (0, indicated value). For annual peak streamflows assigned qualification codes 4 and 8, associated flow intervals were also assigned to the affected annual peak streamflows.

Flow Intervals

Flow intervals were assigned to uncertain data points, including annual peak streamflows assigned qualification codes 4 or 8, in cases where the annual peak gage height was recorded but the annual peak streamflow was not determined or recorded, and for years when the annual peak streamflow was affected by backwater. In the latter case, the annual peak gage height is assigned a qualification code 1 (files “TNurbanFFreq.pkf” and “TNurbanFFreq.psf” in Wagner and Ladd, 2024a). For annual peak streamflows assigned qualification code 4, the associated flow interval was (0, indicated value). For annual peak streamflows assigned qualification code 8, the associated flow interval was (indicated value, infinity). For years when the annual peak gage height was recorded but the annual peak streamflow was not determined or recorded, the associated flow interval was determined from other annual peak streamflows in the dataset and assigned as either (0, selected value), (selected value, infinity), or (selected value, selected value), depending on whether the annual peak gage height could be determined to be less than another known annual peak, greater than another known annual peak, or between two known annual peaks, respectively. For years when the annual peak streamflow was affected by backwater, the associated flow interval was determined from other known annual peaks with similar annual peak gage heights and assigned as (0, selected value) based on the assumption that when affected by backwater, the flow is less than that normally associated with a given annual peak gage height.

Low Outliers

For 135 of 139 streamgages for which at-site flood frequency was computed, MGBT was used to screen for PILFs in the annual peak streamflow records (column “PILFmethod” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a). To help better fit the upper tail (above the median) of the frequency distribution, MGBT statistically determines whether annual peak streamflows are low outliers by using (1) an outward sweep from the median toward the smallest annual peak to determine if some break in the lower half of the data would suggest the sample is best treated as if it had a number of low outliers and (2) an inward sweep from the smallest annual peak toward the median (refer to appendix 6, “Potentially Influential Low Floods,” in England and others, 2018). MGBT detected and screened 1–14 low outliers at 33 streamgages (columns “PILFs” and “PILFthresh” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a). For 4 of the 139 streamgages (02173495, 02217274, 02218565, and 0357587728, column “Site_No” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a), analysts determined that MGBT did not sufficiently detect or screen low outliers to better fit the upper tail of the frequency distribution, and the low outlier threshold had to be manually specified (refer to values “FIXED” in column “PILFmethod” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a).

Trends in Annual Peak Streamflow

Annual peak-flow data were examined for trends using the Mann-Kendall test, which is implemented in version 7.4.1 of the PeakFQ software (Flynn and others, 2006; Veilleux and others, 2014; supplemental information provided at https://www.usgs.gov/tools/peakfq and https://water.usgs.gov/water-resources/legacy-software/). Seventeen of the 139 streamgages for which at-site flood frequency was computed exhibited statistically significant trends for which the p-value associated with the Mann-Kendall test was less than or equal to 0.05 (column “p-value” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a). All but 2 of the 17 streamgages (USGS site numbers 0208735012 and 02135518; U.S. Geological Survey, 2024) had positive trends. The annual peak-flow records of the streamgages with statistically significant trends were examined, but those trends could not be attributed to anthropogenic activity. The absence of trends in the annual peak streamflow records of most streamgages in the study area indicates the observed trends are also not related to a regional trend in climate. The trends appear to be the result of the available or selected periods of record used in the analysis. Some streamgages with positive trends have periods of record that either began in the late 1980s, which was a generally dry period across the region, or had large annual peak streamflows late in their periods of record. Other streamgages with positive trends were affected by Hurricanes Matthew (October 2016) and Florence (September 2017), which resulted in large annual peak streamflows late in their periods of record. Because only 17 streamgages exhibited statistically significant trends, and because those trends could not be attributed to anthropogenic activity or increasing urbanization in the respective drainage basins, no streamgages were removed from the study for trends.

Exploratory Regression

A total of 139 streamgages for which at-site frequency analysis was conducted (Wagner and Ladd, 2024a) were considered for use in developing regression equations for estimating streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs at ungaged locations on streams in the study area. All-subsets, ordinary least-squares (OLS) regression was used to relate the basin characteristics to the selected AEPs and test the statistical significance and predictive power of all possible combinations of basin characteristics. All subsets regression was conducted using the allReg() function in the smwrStats package (Lorenz and DeCicco, 2017) in version 4.1.1 of R software (R Core Team, 2021). The best three 1–4 variable models for each AEP, based on the adjusted coefficient of determination (adjusted R-squared), were identified as candidate models. These candidate models were then further tested using the multReg() function in the smwrStats package to output the full suite of performance metrics for the OLS regression models. Visual examination of plots relating drainage area to estimates of streamflows corresponding to the selected AEPs indicated that two streamgages with drainage areas greater than 100 mi² (USGS site numbers 02087324 and 02207120) and one other streamgage (USGS site number 02169570) were obvious outliers (U.S. Geological Survey, 2024). A decision was made to limit the drainage area of streamgages used in the study to less than 100 mi² and remove streamgages 02087324 and 02207120 from the regression analysis. Streamgage 02169570 was investigated and found to be downstream from a reservoir, although the annual peak streamflow data were not coded as regulated. This streamgage was also removed from the regression analysis, leaving 136 streamgages for use in the final regression model.

Exploratory regression indicated that the base-10 logarithm of drainage area, the percentages of the streamgage basins in developed land use (the sum of developed land use classes 21–24 from 2011 NLCD), and the percentages of the streamgage basins in the Piedmont and Ridge and Valley L3 ecoregions were the best combination of basin characteristics for predicting the selected AEPs by means of the adjusted R-squared values. Multicollinearity of the final selection of basin characteristics was evaluated by means of the variance inflation factor, which was desired to be less than 5; the variance inflation factors for all four characteristics were less than 2.

Generalized Least-Squares Regression

After the covariates were determined using exploratory OLS regression, GLS regression was used to generate the final regression models. Because there is often a high degree of similarity among streamflow statistics and basin characteristics (response variables) from neighboring streamgages, streamflow statistics and response variables cannot be assumed to be independent (Farmer and others, 2019). In addition to the length of the annual peak-flow records and variances of the streamflow estimates, GLS regression accounts for both correlated streamflows and time-sampling errors by incorporating the concurrent record lengths and the estimated cross-correlation of the time series of streamflows for pairs of streamgages used in the regression.

GLS regression was conducted using version 3.0 of USGS weighted regression software (Eng and others, 2009; Farmer, 2021) in R software (R Core Team, 2021). To account for the cross-correlation of streamflows between pairs of streamgages, the weighted regression software incorporates a correlation smoothing function that is developed by the user. Streamgage pairs with 35 or more years of concurrent annual peak-flow record and 200 kilometers (km) or more distance between the streamgages exhibited a relatively even spread of correlation around zero (fig. 2). The alpha and theta values that define the correlation smoothing function used in the study were adjusted to make the correlation smoothing function range from a value of 1 at 0 km to a value of 0 at distances greater than 200 km.

In GLS regression, all basin characteristics were statistically significant (p-value < 0.05) in the regression models for all AEPs except for the percentage of streamgage basins in developed land use, which was not statistically significant (p-value = 0.07) in the regression model for the 0.2-percent AEP (500-year flood). Regression coefficients ranged from 0.6208 to 0.6518 for drainage area, from 0.0024 to 0.0062 for percentage of developed land use in the streamgage basins, from −0.0027 to −0.0020 for the percentage of the streamgage basins in the Piedmont L3 ecoregion, and from −0.0025 to −0.0021 for the percentage of the streamgage basins in the Ridge and Valley L3 ecoregion (Wagner and Ladd, 2024b). The positive coefficients for drainage area and the percentage of streamgage basins in developed land use are consistent with the assumption that streamflow should increase with drainage area and developed land use because of greater coverage by impervious surfaces and rapid delivery of runoff to streams via storm drain networks. The coefficient for the percentage of streamgage basins in developed land use was greatest for the 50-percent AEP and decreased with AEP. The coefficients for the percentages of the streamgage basins in the Piedmont and Ridge and Valley L3 ecoregions are negative because the annual peak streamflows were generally lesser in magnitude for the streamgages in these ecoregions used in this study when compared to streamgages with equivalent drainage areas in the other L3 ecoregions. The negative coefficients are small and have no effect on the predictions of streamflow for streamgages outside these ecoregions.

The relation between observed (at-site) and predicted (regression equation) streamflows corresponding to the selected AEPs was generally good but showed greater scatter for the 2-, 1-, 0.5-, and 0.2-percent AEPs (100-, 200-, and 500-year floods, respectively, fig. 3) than for larger AEPs. The greater scatter in this relation for the smaller AEPs is not surprising, given the large uncertainty associated with estimating small AEPs for streamgages with short record lengths and relatively high positive skews of the LP3 statistical distribution, which were numerous in the study (fig. 4). Record lengths used in the study ranged from 10 to 76 years, with a median of 26 years and third quartile of 37 years, whereas the at-site skews ranged from −1.227 to 3.592, with a median of 0.153 and third quartile of 0.6025 (columns “HistPeriod” and “AtSiteSkew” in file “TNURBANFFREQ.csv” in Wagner and Ladd, 2024a).

Depending on AEP, 12–13 streamgages had high leverage (greater than 0.0735; eqs. 40 and 42 in Eng and others, 2009) and 10–16 streamgages had high influence (greater than 0.0294; eqs. 43 and 44 in Eng and others, 2009) on the regression models (columns beginning with “Lev” and “Inf” for each corresponding AEP in file “GLSresults.csv” in Wagner and Ladd, 2024b). Streamgages with high leverage and influence on the regression models were investigated, but no reasons were found to remove them from the regression models.

Accuracy and Limitations of Regional Regression Equations

The regression equations are applicable at locations on urban streams that have basin characteristics within the range of those used to develop the equations. The methods described in this report do not apply to locations on streams that are substantially affected by regulation from upstream impoundments or other man-made structures or that are subject to tidal effects. The accuracy of the regression equations is not known for locations outside the study area or that have basin characteristics outside the following ranges used to develop the equations: 10 percent or greater developed imperviousness in the basin (LC11IMP; refer to file “GLSresults.csv” in Wagner and Ladd, 2024b); 0.164–93.4 mi² of drainage area (DRNAREA); 26–100 percent developed land use in the basin (LC11DEV); 0–100 percent of the basin in the Piedmont L3 ecoregion (L3_PIEDMNT); and 0–100 percent of the basin in the Ridge and Valley L3 ecoregion (L3_RDGVLY). The regression equations are not valid in the Middle Atlantic or Southern Coastal Plain Level 3 ecoregions in North Carolina, South Carolina, Georgia, Mississippi, and Alabama; no urban streamgages in these ecoregions were used to develop the equations because of the small number of available urban streamgages and potential tidal effects.

The accuracy of the GLS regression model is expressed as the residual mean squared error (MSE; eq. 31 in Eng and others, 2009, p. 5), which is the sum of the model and time-sampling errors, and the model error variance (MEV), which is the same as the MSE only if the time-sampling error variance is zero. MSE ranged from 0.025 for the 50- and 20-percent AEPs (2- and 5-year floods) to 0.104 for the 0.2-percent AEP (500-year flood; column “MSE” in file “RegressionEquations.csv” in Wagner and Ladd, 2024b). The time-sampling error variance was not zero in this study, and the MEV ranged from 0.022 for the 20- and 10-percent AEPs (5- and 10-year floods) to 0.047 for the 0.2-percent AEP (500-year flood; column “MEV” in file “RegressionEquations.csv” in Wagner and Ladd, 2024b).

The overall accuracy of predictions from the regression equations is expressed in terms of the coefficient of determination, or R-squared value, the average variance of prediction (AVP), and standard error of prediction (Sp). The R-squared value represents the proportion of the variation in the dependent variable (streamflow estimates for the selected AEPs) that is explained by the basin characteristics in the regression model. For GLS regression, the pseudo R-squared value is reported (column “R2pseudo” in file “RegressionEquations.csv” in Wagner and Ladd, 2024b), which is based on the variability in streamflow explained by the regression after removing the effect of the time-sampling error (eq. 39 in Eng and others, 2009, p. 6). Pseudo R-squared values ranged from 0.86, or 86 percent, for the 50- and 20-percent AEPs (2- and 5-year floods) to 0.71, or 71 percent, for the 0.2-percent AEP (500-year flood). The AVP (in base-10 logarithmic units; eq. 32 in Eng and others, 2009, p. 5) ranged from 0.023 for the 20- and 10-percent AEPs to 0.05 for the 0.2-percent AEP (column “AVP” in file “RegressionEquations.csv” in Wagner and Ladd, 2024b). Because the base-10 logarithms of the streamflow estimates were used to develop the regression equations, the AVP can be reported as a percentage of the predicted value, known as the standard error of prediction (eq. 33 in Eng and others, 2009, p. 5). The standard error of prediction ranged from 35.8 percent for the 20-percent AEP (5-year flood) to 55.4 percent for the 0.2-percent AEP (500-year flood; column “AVP” in file “RegressionEquations.csv” in Wagner and Ladd, 2024b). The standard errors of prediction were similar to those for the rural regression equations for Tennessee, which ranged from 30.44 to 51.4 percent, depending on flood region (table 7 in Ladd and Ensminger, 2025).

Six streamgages in Tennessee, 03491544, 03535400, 03536450, 03536550, 03538235, and 07032200, that were used to develop the urban equations were also used to develop the rural equations. For the selected AEPs, the predictions using the urban equations were greater than those using the rural equations, with two exceptions. First, for streamgage 07032200, the predictions from the urban equations were less than those predicted using the rural equations. Second, for streamgage 03535400, the predictions for the 50-, 20-, 10-, and 1-percent AEPs were greater using the urban equations and the predictions for the 4-, 2-, 0.5-, and 0.2-percent AEPs were greater using the rural equations. For locations in Tennessee having 10 percent or greater developed imperviousness in their drainage basins, the urban regression equations in this report should be used to predict streamflows; for locations with less than 10 percent developed imperviousness, the rural equations should be used (refer to table 4 in Ladd and Ensminger, 2025).

Gaged Locations

For streamgages with short record lengths, the uncertainty of at-site estimates of streamflows corresponding to various AEPs can be reduced by weighting those estimates with predictions from the regression equations (refer to appendix 9, “Weighting of Independent Estimates,” in England and others, 2018). If the two are assumed to be independent and unbiased and are weighted in inverse proportion to their associated variances, the variance of the weighted estimate will be less than the variance of either of the independent estimates. A weighted estimate of streamflow corresponding to a selected AEP can be computed using equation 9-2 in England and others (2018):

A 95-percent confidence interval for the weighted estimate (in cubic feet per second; refer to eq. 8 in Wagner and others, 2016) can be calculated as

The variance of the weighted estimate for streamgage i (in base-10 logarithmic units) can be calculated as

Ungaged Locations on Gaged Streams

If the drainage area at an ungaged location on a stream is within 50 percent of the drainage area at a streamgage (drainage area ratio is more than 0.5 or less than 1.5) on the same stream, streamflows corresponding to selected AEPs at the ungaged location can be computed from the streamgage data using a drainage area ratio method (refer to eq. 13 in Wagner and others, 2016). The drainage area ratio can be calculated by using the following equation:

If the drainage area at the ungaged location differs by more than 50 percent from the drainage area of the streamgage, the predicted streamflow from the regression equations for the ungaged location should be used. If an ungaged location is between two streamgages on the same stream, the streamgage with the longest period of record or that yields the smallest drainage area ratio should be used.

Locations on Ungaged Streams

For locations on ungaged streams, streamflows corresponding to selected AEPs should be determined using the regression equations. The USGS StreamStats application (U.S. Geological Survey, 2019; https://streamstats.usgs.gov/ss/) provides a platform for delineating drainage basins, computing basin characteristics, and estimating streamflows corresponding to selected AEPs at ungaged locations using the regression equations.