Purpose and Scope

The purpose of this report is to present updated methods for estimating the magnitude and frequency of floods on rural streams in Georgia, South Carolina, and North Carolina. For this report, a rural basin is defined as a basin with less than 10 percent of the drainage area characterized by impervious surfaces during the period of record and the peak flows are not substantially regulated by flood-control, reservoir storage, or diversions at medium to high streamflows. The results presented in this report are based on flood‑frequency analyses of annual peak-flow data at streamgages through water year 2017. Following the landfall of Hurricane Florence across parts of North Carolina and South Carolina in September 2018 (during this study), data from selected streamgages where peak flows from this event ranked in the top five of annual peaks (as of water year 2018) also were included in this analysis. The data generated as part of this study were published separately in a data release by Kolb and others (2023).

This report describes the techniques and methods for computing flood‑frequency estimates for unregulated streamflows at the 50‑, 20‑, 10‑, 4‑, 2‑, 1‑, 0.5‑, and 0.2‑percent annual exceedance probability (AEP) for 807 rural streamgages in Georgia, South Carolina, and North Carolina with unregulated flow conditions (containing both redundant streamgages and those included in the regression analyses), which are provided in the associated data release by Kolb and others (2023, table 3). This data release also includes flood‑frequency estimates for 137 streamgages in Georgia, South Carolina, and North Carolina that were excluded from the regional regression analysis owing to redundancy, which is discussed later. In addition, flood‑frequency estimates published by Kolb and others (2023) for 131 rural streamgages from the surrounding States of Florida, Alabama, Tennessee, and Virginia that generally share a basin with or within about 50 miles (mi) of the borders of Georgia, South Carolina, or North Carolina were included in the regression analysis. In total, flood frequency estimates for 801 rural streamgages in the States of Georgia, South Carolina, North Carolina, Florida, Alabama, Tennessee, and Virginia were utilized in the regression analysis. This report describes the techniques used to develop regional regression equations for use in estimating the magnitude of peak streamflows for selected AEPs at ungaged sites in Georgia, South Carolina, and North Carolina. The regional equations are provided along with a discussion of the accuracy and limitations.

Flood‑frequency estimates also were computed for 72 streamgages with regulated conditions in Georgia, South Carolina, and North Carolina using annual peak-flow records through water year 2019. Procedures used to update the generalized (regional) skew for Georgia, South Carolina, and North Carolina are described in appendix 1.

Previous Studies

The USGS has completed numerous flood‑frequency studies throughout the southeastern United States. As additional years of annual peak-flow data are accumulated at streamgages, the streamgage flood‑frequency estimates and flood-prediction relations are commonly updated by the USGS on a 10‑ to 20‑year interval. For the most part, those studies addressed flood frequency in rural and urban areas separately. In addition, USGS flood‑frequency studies were historically completed by each State, which often led to differences in hydrologic regions at State boundaries. These differences caused some discontinuity and confusion as to which flood‑frequency techniques and results were most appropriate for drainage basins near or crossing State boundaries. In 2009, the USGS successfully applied a multistate approach for a rural flood‑frequency investigation in Georgia, South Carolina, and North Carolina, which resulted in a single set of regression equations that were applicable in all three States (Feaster and others, 2009; Gotvald and others, 2009; Weaver and others, 2009).

The focus of the following discussion will be on previous studies related to rural basins in Georgia, South Carolina, and North Carolina. For information related to previous flood‑frequency studies for urban basins, see the “Previous Studies” section in Feaster and others (2014).

Description of Study Area

The study area includes all of Georgia, South Carolina, and North Carolina, covering an area of about 142,500 mi² within seven U.S. Environmental Protection Agency (EPA) level III ecoregions—Southwestern Appalachians, Ridge and Valley, Blue Ridge, Piedmont, Southeastern Plains, Middle Atlantic Coastal Plain, and Southern Coastal Plain (fig. 1; U.S. Environmental Protection Agency, 2008). The ecoregions represent areas of general similarity in ecosystems and in the type, quality, and quantity of environmental resources. The ecoregions provide a spatial framework for the research, assessment, management, and monitoring of ecosystems and ecosystem components. Omernik (1987) and Griffith and others (2001, 2002) determined the ecoregions from an analysis of the spatial patterns and the composition of biotic and abiotic phenomena that include geology, physiography, vegetation, climate, soils, land use, wildlife, and hydrology. The Fall Line is the geological boundary separating the higher altitudes of the Southwestern Appalachians, Ridge and Valley, Blue Ridge, and Piedmont ecoregions from the low-lying Southeastern Plains, Middle Atlantic Coastal Plain, and Southern Coastal Plain ecoregions.

The Southwestern Appalachians ecoregion is composed of open, low mountains. The eastern boundary of this ecoregion, along the more abrupt escarpment where it meets the Ridge and Valley ecoregion, is relatively smooth and only slightly notched by small, eastward-flowing streams (Griffith and others, 2001, 2002). The Ridge and Valley ecoregion is composed of roughly parallel ridges and valleys with a variety of widths, heights, and geologic materials. Springs and caves are relatively numerous (as compared to other ecoregions), and present-day forests cover about 50 percent of the ecoregion. The Blue Ridge ecoregion varies from narrow ridges to hilly plateaus to more mountainous areas. The mostly forested slopes; high-gradient, cool, clear streams; and rugged terrain overlie primarily metamorphic rocks, with minor areas of igneous and sedimentary deposits. The Piedmont ecoregion is composed of a transitional area between the mostly mountainous ecoregions of the Appalachians to the northwest and the relatively flat Coastal Plain to the southeast. The Piedmont ecoregion is a complex mosaic of metamorphic and igneous rocks of Precambrian and Paleozoic age, with moderately dissected irregular plains and some hills. The soils tend to be finer textured than in the Coastal Plain ecoregions to the south. Once largely cultivated, much of this ecoregion has reverted to pine and hardwood forests, with increasing conversion to urban and suburban land cover (Omernik, 1987).

The Southeastern Plains ecoregion is composed of irregular plains with a mixture of cropland, pasture, woodland, and forest (Griffith and others, 2001, 2002). The sand, silt, and clay geology of this ecoregion contrasts with the older rocks of the Piedmont ecoregion. Altitudes and relief are greater than in the Southern Coastal Plain ecoregion but generally are less than in much of the Piedmont ecoregion. Streams have relatively low gradient (as compared to the Piedmont ecoregion) with sandy bottoms. The Southern Coastal Plain ecoregion consists of mostly flat plains, but it is a heterogeneous ecoregion containing barrier islands, coastal lagoons, marshes, and swampy lowlands along the Gulf and Atlantic coasts. This ecoregion is lower in altitude with less relief and wetter soils than the Southeastern Plains ecoregion. The Middle Atlantic Coastal Plain ecoregion consists of low-altitude flat plains, with many swamps, marshes, and estuaries. Unconsolidated sediments underlie the low terraces, marshes, dunes, barrier islands, and beaches. Poorly drained soils are common, and the ecoregion has a mix of coarse and finer textured soils compared to the mostly coarse soils in the majority of the Southeastern Plains ecoregion. The Middle Atlantic Coastal Plain ecoregion typically is lower, flatter, and more poorly drained than the Southern Coastal Plain ecoregion (Omernik, 1987).

The average annual precipitation for the study area generally ranges from 40 to 60 inches per year (in/yr). The southern portion of the Blue Ridge ecoregion receives up to or more than 80 in/yr of precipitation (PRISM Climate Group, 2015a). Precipitation in the study area is associated with the movement of warm and cold fronts from November through April and isolated summer thunderstorms from May through October. Occasionally, tropical storms or hurricanes that enter along the Atlantic and Gulf coasts produce unusually heavy amounts of rainfall. The mean annual air temperature in the study area ranges from 54 degrees Fahrenheit (°F) in northern North Carolina to 68 °F in southern Georgia, with variations as low as 46 °F in some of the higher Blue Ridge altitudes in western North Carolina (PRISM Climate Group, 2015b).

Physical and Climatic Basin Characteristics

Basin characteristics were selected for use as potential explanatory variables in the regression analyses based on the well-established theoretical and empirical relationships between these parameters and runoff characteristics and the ability to measure the basin characteristics in a geographic information system (GIS). For each of the 965 streamgages considered, 26 basin characteristics (such as drainage area, mean basin elevation, mean annual precipitation) were determined and considered as potential explanatory variables in the regression analyses (table 2 from Kolb and others, 2023).

Drainage-basin boundaries for this study were generated using the USGS StreamStats application (Ries and others, 2017) and used to determine the 26 basin characteristics. The underlying altitude data in the StreamStats program were generated using two different GIS data sources. In Georgia, StreamStats data were generated from National Elevation Dataset digital elevation models (DEMs) with 10‑meter (m) resolution (U.S. Geological Survey, 2014). In North Carolina, StreamStats data were generated from National Elevation Dataset DEMs with 30‑foot (ft) resolution (U.S. Geological Survey, 2014). The South Carolina StreamStats data were generated using 30‑ft resolution DEMs (Kolb and others, 2018) derived from light detection and ranging (lidar) data from the South Carolina Department of Natural Resources (2015). Boundary delineations were compared with NWIS drainage areas for QA/QC, as this was the first flood‑frequency study completed since the implementation of StreamStats for the South Atlantic Water Science Center (SAWSC). In the case of erroneous delineations (such as a missed culvert), the boundaries and drainage area values were improved using aerial imagery and DEMs. More information about the StreamStats applications in South Carolina, Georgia, and North Carolina is available in Feaster, Clark, and Kolb (2018), Gotvald and Musser (2015), and Weaver and others (2012), respectively.

Drainage-basin boundaries generated from StreamStats were compared to previously published drainage areas for the streamgages as a means of QA/QC. For most streamgages, the drainage areas agreed closely but for various streamgages, the drainage areas differed by more than 2 percent. In most cases where the difference exceeded this threshold (greater than 2 percent), the published drainage areas were determined manually from older topographic maps with 10‑ft contour intervals. Boundaries generated using StreamStats were considered more accurate than manual delineations. The streamgages with drainage area differences greater than 2 percent were revised to the StreamStats generated drainage-basin boundaries (U.S. Geological Survey, 2012).

Statistical Analysis of Trends in Annual Peak Flows

In this study, Kendall’s tau nonparametric test (Kendall, 1938) was used to determine statistical significance of monotonic trends in annual peak flow with time (Helsel and others, 2020). A trend was considered statistically significant for a probability value (p-value) less than or equal to 0.05. Kendall’s tau measures the degree of correspondence between two variables (for example, x and y). For this analysis, the x and y variables are water year and annual peak flow, respectively. A concordant pair results when both x and y variables increase or decrease; a discordant pair results when x increases and y decreases or x decreases and y increases. The number of concordant pairs and the number of discordant pairs were tallied for each streamgage considered here and the Kendall’s tau value (τ) was computed using the following equation:

The null hypothesis for this test is that there is no monotonic trend between the peak flows and time and the p-value of 0.05 indicates there is less than a 5‑percent chance of obtaining the sample result if the null hypothesis were true.

If the data indicate perfect positive correlation, then τ = 1; if there is perfect negative correlation, then τ = −1; and if there is no correlation between the pairs, then τ = 0. Therefore, a positive τ value is associated with an upward trend and a negative τ value is associated with a downward trend (Norton and others, 2014).

For hydrologic time-series data, Kendall’s tau test is best suited for analysis of long-term datasets. Although it can be applied to short time series, Kendall’s tau test may not provide information that is of practical importance, and care is needed to avoid misinterpreting the results. Tests applied to short time series may (1) fail to detect a statistically significant trend even though a large increase or decrease in flow has been measured, or (2) detect a statistically significant trend even though the trend is of no practical importance (Oki, 2004). Thus, long-term streamgaging data are better suited for trend assessments. The USGS typically considers 30 years of streamflow record as an appropriate threshold to designate long-term streamgages (U.S. Geological Survey, 2009).

Of the 965 streamgages that were considered for use in the regional regression analysis, a Kendall’s tau test was performed using annual peak flows for streamgages with 30 or more years of record. The results of the trend analyses of the annual peak flows are shown in table 1 from Kolb and others (2023). Of the streamgages considered in table 1 from Kolb and others (2023), 495 contain 30 or more years of systematic streamflow peaks and are considered long-term streamgages with 332 of those long-term streamgages operating in 2017 (stations with combined records were not included in the trend analysis). Of the 332 long-term streamgages, 276 streamgages (83 percent) indicated no statistically significant trend (p-value >0.05) in annual peak flows, 45 streamgages (14 percent) indicated a statistically significant downward trend, and 11 streamgages (3 percent) indicated a statistically significant upward trend (fig. 3A).

For comparison of trends as record length increases (fig. 3A–D), an assessment of significant trends for the current long-term streamgages was done for four groups of stations: (1) from 30 to 49 years, (2) from 50 to 69 years, (3) from 70 to 89 years, and (4) 90 or more years of annual peak flows (fig. 4). As the data-record length increased, the percentage of streamgages with significant upward and downward trends remained relatively consistent. There are many streamgages with significant downward trends in the middle portion of eastern Georgia, as well as western and central South Carolina (fig. 3). The streamgages in these areas recorded large floods in the early part of the 20th century and multiple drought periods in the early part of the 21st century, which resulted in downward trends (see fig. 5 for an example of this result at USGS streamgage 02191300, site 452 on fig. 2). Significant upward trends also result in a small portion of southern Georgia and northern Florida (fig. 3). The streamgages in those areas recorded more frequent larger floods from water years 1985 to 2017, which resulted in the upward trends (see fig. 6 for an example of this result at USGS streamgage 02329000, site 687 on fig. 2). A comprehensive analysis of the causes of the trends in the annual peak flows is outside the scope of this report. Because of the lack of strong and consistent statistical evidence of significant long-term regional peak-flow trends throughout the study area, the traditional assumption of stationarity is used for this study with no adjustment for either upward or downward trends.

Flow Intervals and Perception Thresholds

The EMA method outlined in Bulletin 17C and briefly described above accommodates interval peak-flow data, which simplifies analysis of datasets containing censored observations, historic data, low outliers, and uncertain data points, whereas also providing enhanced confidence intervals for the estimated streamflows (Veilleux and others, 2014). The EMA methodology has been incorporated into the USGS peak-flow frequency analysis program, PeakFQ version 7.4 (Flynn and others, 2006; Veilleux and others, 2014; U.S. Geological Survey, 2022), and was used to compute the AEP flows for streamgages in this study. Within the EMA framework, flow intervals and perception thresholds are defined for each year in the annual peak-flow record of any streamgage. The published guidelines (Bulletin 17C) for determining flood flow frequency include many examples of flow intervals and perception thresholds as inputs to EMA to illustrate applications of the recommended techniques.

Flow intervals (defined with a lower and upper bound based on observations, written records, or physical evidence) are used to describe the peak-flow value. For most peak flows during the systematic period of record, the default lower and upper bounds of the streamflow interval both equal the observed peak flow, and for most years when no information has been recorded, the default lower and upper bounds are zero and infinity, respectively. If there is uncertainty in a peak-flow value for a given water year, the lower and upper bounds of the streamflow interval may be set to a range of probable streamflows.

Perception thresholds (lower and upper) identify the range of potentially measurable flood flows where the flow magnitude would have been measured had they occurred (England and others, 2019). Whereas a flow interval applies to a specific single occurrence of peak flow that results in a given water year, the range of streamflow specified in a perception interval is applicable over a given time range (usually a period of years). Generally, for annual peak flows recorded during the systematic period of record of a streamgage, the default perception thresholds range from zero to infinity. If the peak flow is unknown because the streamgage was discontinued or ceased operation, the perception thresholds are both set to infinity.

At some streamgages, flows can be determined only when water in the stream reaches a certain minimum measurable level. For example, it is possible that in some years, the water will not reach that minimum level or the bottom of a crest-stage gage (CSG); consequently, the lower perception threshold is the flow associated with the minimum measurable water level. For this study, years with measurable streamflow were assigned the default perception thresholds of zero to infinity. Alternatively, years when water did not reach the minimum level of measurement were generally assigned a lower perception threshold of either (1) the flow associated with the minimum measurable water level, or if that was not known, (2) one-half of the lowest flow associated with the systematic annual peak-flow record; upper thresholds were set to infinity. In some instances, historic peaks are documented as part of an annual peak-flow record; that is, a peak flow associated with a major flood event outside of the systematic period of record for the streamgage. For the ungaged years between the historic and systematic peaks, the lower perception threshold is typically set to a value the analyst determines would have been measured during minimum flow conditions, and the upper threshold is set to infinity. The flow interval and perception thresholds that were incorporated into the EMA analysis for each streamgage included in this study are provided in the associated data release (Kolb and others, 2023).

Occasionally, a streamgage site had a documented historic peak-gage height (record of flood height outside of the systematic period of record) with no associated peak flow. In these instances, a peak flow (or peak-flow range) was estimated based on a comparison to other peak-gage height and associated flow values at the site of interest.

In some instances, documented peak flows occurred outside of the systematic period of record, and were categorized as an opportunistic peak flow. Opportunistic peaks were measured based on factors other than the exceedance of a perception threshold and, thus, were not treated as historic peaks. Furthermore, these flows are not truly random as their sampling properties are unknown. Consequently, opportunistic peaks were not included in the flood‑frequency analyses because of the potential to bias the sample streamflow records.

Flow intervals used in the analyses of unregulated streamgage records are reported in table 5 from Kolb and others (2023). Perception thresholds used in the analyses of unregulated and regulated streamgage records are reported in tables 6 and 7 from Kolb and others (2023), respectively. Inputs to PeakFQ, which include all flow intervals and perceptions thresholds used in this study, are presented in Kolb and others (2023). The report files from the PeakFQ analyses also are available in Kolb and others (2023).

Potentially Influential Low Flows

Low-magnitude peaks that depart significantly from the sample of annual peak flows for a streamgage can result in a poor fit of the frequency curve at lower AEPs. When evaluating the sample of annual peak flows, attention must be given to annual peaks that are considered outliers. Referred to as potentially influential low flows (PILFs), these peak flows may appreciably affect the upper end of the peak-flow distribution, which tends to be most important for a flood‑frequency analysis.

The multiple Grubbs Beck test (MGBT; Cohn and others, 2013) is an option in the PeakFQ software to identify and censor PILFs. Censoring the PILFs typically results in improved agreement between the high end (where AEPs are small, such as the 0.2‑percent AEP) of the observed frequency distribution and the high end of the estimated frequency distribution. In some instances, censoring the PILFs may degrade the fit at the low end (where AEPs are large, such as the 50‑percent AEP) of the frequency distribution. As recommended in Bulletin 17C guidelines, for instances when PILFs were identified by means of the MGBT, a careful analysis of the PILFs was conducted by applying local knowledge of the watershed and hydrologic considerations. This analysis was used to determine whether the use of the MGBT was appropriate for the identification of PILFs.

Regional Skew Coefficient

The regional skew coefficient is associated with a defined region and is derived from an analysis of skew coefficients for streamgages with longer annual peak-flow records within the defined region. The skew coefficient measures the asymmetry of the probability distribution of a set of annual peak flows. The skew coefficient is zero when the mean of the annual series equals the median and the mode; positive when the mean exceeds the median, which in turn exceeds the mode; and negative when the mean is less than the median, which, in turn, is less than the mode (fig. 7). The skew coefficient is strongly affected by the presence of outliers. Large positive skews typically are the result of high outliers, and large negative skews typically are the result of low outliers. The streamgage skew coefficient, which is calculated by using the annual peak-flow record for a streamgage, is sensitive to extreme hydrologic events; therefore, the streamgage skew coefficient for short records may not provide an accurate estimate of the true skew coefficient.

Bulletin 17C recommends the skew coefficient used in defining the probability distribution be a weighted average of the streamgage skew coefficient and a regional skew coefficient that reflects regional and long-term (decadal) conditions (England and others, 2019). As part of this study, the regional skew was updated using Bayesian WLS/Bayesian GLS methodology, and the procedures and results are presented in appendix 1. Flood‑frequency estimates for streamgages with unregulated flow were computed using a weighted average of the streamgage skew and the regional skew. The following equation shows how the weighted skew coefficient for a given site is computed:

Georgia

Peak-flow data used to derive flood‑frequency estimates for streamgages in Georgia were obtained from the USGS NWIS database (U.S. Geological Survey, 2019). For certain regulated streamgages in Georgia, valuable flood information observed in the unregulated record were incorporated into the regulated record before the flood‑frequency analysis was performed. Historic floods and major floods from the unregulated period were adjusted (reduction in magnitude) to reflect regulated conditions associated with dams and reservoirs. The adjustments to Savannah River peak flows from pre- to post-regulation are based on previously applied methods described in Sanders and others (1990).

Continuous peak flows were assessed at streamgage 02197000, Savannah River at Augusta, Ga. (site 475 on fig. 9), for water years 1875 through 2019. It also has historic peak flows available for water years 1796, 1840, 1852, 1864, and 1865. Although there are various smaller reservoirs in the basin, the first major reservoir built was the J. Strom Thurmond Reservoir in 1952. Lake Hartwell, which is upstream from Thurmond Reservoir, was completed in 1962. Richard B. Russell Reservoir, which is between Hartwell and Thurmond Reservoirs, was completed in 1986. The reservoir system was built for the purposes of flood control and providing hydroelectricity, but also provides water supply and recreation. The slope of the SMC for the peak flows shows a significant shift about 1952 but has been relatively stable since then. Because Thurmond Reservoir has the largest maximum storage capacity of the reservoirs discussed above and is the furthest downstream reservoir in the system, the other reservoirs that came online after Thurmond did not appreciably alter the peak-flow patterns at streamgage 02197000.

Sanders and others (1990) adjusted the historic, unregulated peak flows measured at streamgage 02197000 to account for regulation conditions present at that time. As previously noted, the SMC analysis indicated that peak-flow patterns have been relatively stable at streamgage 02197000 since about 1952. The historic peak flows for 1796, 1840, 1852, and 1865, along with other major floods in 1888, 1908, 1929, 1930, 1936, and 1940 at streamgage 02197000 were adjusted for regulation and combined with the regulated peaks from 1953 through 2019 at the site to derive the AEP estimates for the regulated period (table 8 from Kolb and others, 2023). An estimated peak of 252,000 ft³/s was set as the perception threshold in the EMA analysis at streamgage 02197000 for the period 1796–1952. It should be noted that the historic peak flows shown in Sanders and others (1990) for water years 1796, 1840, 1852, and 1865 were revised in water year 1994 (Cooney and others, 1995).

Streamgage 02197500, Savannah River at Burtons Ferry Bridge near Millhaven, Ga. (site 479 on fig. 9), has unregulated peak flows for 1930 and 1940 through 1951. The streamgage record indicates regulated peaks from 1953 through 1970 and from 1983 through 2019. To be consistent with the regulated flood‑frequency analysis at streamgage 02197000, the Maintenance of Variance Extension, Type 1 (MOVE.1) method of correlation analysis (Hirsch, 1982) was used to correlate the concurrent unregulated period of record of 1930 and 1940 through 1951 at streamgages 02197000 and 02197500, with a correlation coefficient of 0.99. From that relation, the unregulated peaks for 1796, 1840, 1852, 1865, 1888, 1908, 1930, 1936, and 1940 were estimated at streamgage 02197500. It is worth noting that the measured water year 1930 peak at streamgage 02197500 was 220,000 ft³/s and the MOVE.1 estimated peak was 216,000 ft³/s, a difference of less than 2 percent. At streamgage 02197000, the mean ratio of the regulated to unregulated peak flows from Sanders and others (1990) for water years 1796, 1840, 1852, 1865, 1888, 1908, 1930, 1936, and 1940 was 0.50. At streamgage 02197500, the MOVE.1 unregulated peak flows for those same water years were adjusted to account for regulation by multiplying the unregulated peak flows by 0.50. Those adjusted peaks, along with the measured regulated peaks from 1953 through 1970 and from 1983 through 2019, were included in the current flood‑frequency analysis for streamgage 02197500 (table 8 from Kolb and others, 2023). An estimated peak of 124,000 ft³/s was set as the perception threshold in the EMA analysis at streamgage 02197500 for the period 1796–1952.

At streamgage 02198500, Savannah River near Clyo, Ga. (site 487 on fig. 9), unregulated peak flows were available for 1925 through 1951 and regulated peak flows from 1952 through 2019. A MOVE.1 correlation was done using the concurrent unregulated peaks from streamgages 02198500 and 02197000 for water years 1925 through 1950, with a correlation coefficient of 0.96. The MOVE.1 relation was used to estimate the unregulated flows at streamgage 02198500 for water years 1796, 1840, 1852, 1865, 1888, 1908, 1930, 1936, and 1940. As was done at streamgage 02197500, the unregulated peak flows for those years were converted to regulated peak flows by multiplying these flows by the mean ratio (0.50) of the regulated and unregulated peak flows at streamgage 02197000 from Sanders and others (1990). A flood‑frequency analysis was done using these regulated peak flows combined with the measured regulated peak flows from 1953 to 2019 (table 8 from Kolb and others, 2023). An estimated peak of 118,000 ft³/s was set as the perception threshold in the EMA analysis at streamgage 02198500 for the period 1796–1952.

South Carolina

Peak-flow data used to derive flood‑frequency estimates for streamgages in South Carolina were obtained from the USGS NWIS (U.S. Geological Survey, 2019). For certain regulated streamgages, the data were then adjusted or altered. Where deemed helpful, information also is included concerning the EMA perception thresholds included in the flood‑frequency analysis.

Catawba and Wateree Rivers

Completed in 1963, Lake Norman was the last major reservoir constructed in the Catawba and Wateree River Basin (table 8 from Kolb and others, 2023). Prior to the construction of the large reservoirs currently in the Basin, two major floods were recorded in 1908 (366,000 ft³/s) and in 1916 (400,000 ft³/s) at streamgage 02148000, Wateree River near Camden, S.C. (site 352 on fig. 9). To adjust the 1908 and 1916 peak flows to reflect current regulated conditions for the flood‑frequency analysis, a correlation analysis was completed using streamgage 02161000, Broad River at Alston, S.C. (site 393 on fig. 2), as an index (or predictor) streamgage. To estimate the 1908 and 1916 peak flows at streamgage 02161000, a MOVE.1 correlation analysis was done with streamgage 02169500, Congaree River at Columbia, S.C. (site 408 on fig. 9), using concurrent unregulated peaks from 1897 through 1907 and from 1926 through 1928. The correlation coefficient for those concurrent peaks is 0.98.

An SMC analysis of the peaks at streamgage 02148000 showed that the peak-flow patterns have been relatively stable since about 1954. A double-mass curve analysis showed that the relation between peak flows at streamgages 02161000 and 02148000 also has been relatively stable from 1954 through 2019. A double-mass curve is a graph of the culmulation of one quantity against the cumulation of another quantity for the same period (Searcy and Hardison, 1960). If the data are proportional, the slope of the line of the double-mass curve will be consistent. A change in the slope would indicate a change in the proportionality between the two quantities. Therefore, a MOVE.1 analysis was done using the concurrent peaks at streamgages 02148000 and 02161000 from 1954 through 2019. The correlation coefficient for those peaks was 0.75. For comparison, an ordinary least squares (OLS) regression also was performed, and the regression line was compared with the MOVE.1 line. For 1954 through 2019, the largest peak flow measured at streamgage 02161000 was 146,000 ft³/s in October 1976. The estimated 1908 peak flow at streamgage 02161000 was 251,000 ft³/s and, therefore, an extrapolation of the MOVE.1 and OLS lines was necessary to estimate the 1908 peak at streamgage 02148000 to reflect regulated conditions. For the largest flows in the correlation analysis, there was some divergence between the OLS and MOVE.1 lines with the OLS line being more reflective of the slope of the largest peak flows used in the analysis (fig. 10). Therefore, the 1908 peak flow for streamgage 02148000 under regulated conditions was estimated by taking the mean of the OLS and MOVE.1 estimates, which resulted in a peak flow of 218,000 ft³/s.

As noted earlier at streamgage 02148000, the 1908 flood (366,000 ft³/s) was lower than the 1916 flood (400,000 ft³/s). At streamgage 02161000, neither the 1908 nor 1916 peak flows were measured; however, nearby streamgage data indicate that the 1908 flood would be the peak of record and the 1916 flood would be lower. Consequently, if the 1916 flood at streamgage 02148000 under regulated conditions was estimated directly from the MOVE.1 and OLS relations, the peak flow would have been lower than the 1908 flood. To overcome this discrepancy, the 1916 peak flow at streamgage 02148000 under (estimated) regulated conditions was derived by multiplying the estimated regulated flood for 1908 by the ratio of the unregulated 1916 and 1908 peaks: 218,000 × (400,000/366,000) = 238,000 ft³/s. For the EMA analysis, lower and upper perception thresholds of 238,000 ft³/s and infinity, respectively, were used for the periods from 1886 through 1907, from 1909 through 1915, and from 1917 through 1953, reflecting the range of flows that would have been measured had these flows resulted during those periods (England and others, 2019).

The peak flows at streamgage 02146000, Catawba River near Rock Hill, S.C. (site 344 on fig. 9), include unregulated peak flows from 1896 through 1903 and regulated flows from 1942–2019. Streamgage 02147000, Catawba River below Catawba, S.C., has a peak-flow record from 1968 through 1991. Streamgage 02147020, Catawba River below Catawba, S.C. (site 348 on fig. 9), has a peak-flow record from 1993 through 2019. The difference in the drainage area between streamgages 02147000 and 02147020 is less than 1 percent. Therefore, for the flood‑frequency analysis, the peak flows at these two streamgages were combined and are hereafter referred to as streamgage 02147020.

From the peak-flow record at streamgage 02148000, which is downstream from streamgage 02146000, and other historical records (American Meteorological Society, 2021), the 1908 and 1916 floods would have also been floods of record at streamgages 02146000, Catawba River near Rock Hill, S.C., and 02147020, Catawba River below Catawba, S.C. Therefore, techniques like those used to estimate the magnitude of the 1908 and 1916 peak flows at streamgage 02148000, under current regulated conditions, were used to estimate those peak flows at streamgages 02146000 and 02147020.

As was performed for streamgage 02148000, both MOVE.1 and OLS analyses were performed using concurrent periods of record at streamgages 02146000 and 02147020. For streamgage 02146000, the SMC analysis indicated that peak-flow patterns have been stable since about 1964. A double-mass curve analysis of the peak flows from 1964 through 2019 for streamgages 02146000 and 02161000 also indicated a stable relation. Therefore, the concurrent peaks from 1964 through 2019 were used in the correlation analyses to estimate the 1908 peak at streamgage 02146000 for current regulation conditions. Like streamgage 02148000, extrapolation of the MOVE.1 analysis and OLS analysis interpolation lines was necessary to estimate the 1908 peak, and the mean of the two estimates (137,000 ft³/s) was used. The 1916 peak adjusted for current regulation conditions was estimated by multiplying the 1908 peak by the ratio of the unregulated 1916 and 1908 peaks at streamgage 02148000: 137,000 × (400,000/366,000) = 150,000 ft³/s. The 1908 and 1916 peaks adjusted for current regulation were included in the EMA analysis as historic peaks. Lower and upper perception thresholds of 150,000 ft³/s and infinity, respectively, were used from 1897 through 1907, from 1909 through 1915, and from 1917 through 1963, indicating the range of regulated flows that would have been measured had they resulted during those periods (England and others, 2019).

For streamgage 02147020, the SMC analysis indicated that peak-flow patterns have been stable for the period of record from 1968 through 1991 and from 1993 through 2019. MOVE.1 and OLS correlation analyses were done using the concurrent peak flows from streamgages 02147020 and 02161000. Like streamgages 02148000 and 02146000, extrapolation of the MOVE.1 and OLS lines was necessary to estimate the 1908 peak flow, and the mean of the two estimates (176,000 ft³/s) was used. The 1916 peak adjusted for current regulation conditions was estimated by multiplying the 1908 peak by the ratio of the unregulated 1916 and 1908 peaks at streamgage 2148000: 176,000 × (400,000/366,000) = 192,000 ft³/s. The 1908 and 1916 peaks adjusted for current regulation conditions were included in the EMA analysis for streamgage 02147020 as historic flow peaks. Lower and upper perception thresholds of 192,000 ft³/s and infinity, respectively, were used for the period from 1909 through 1915 and from 1917 through 1967 indicating the range of regulated flows that would have been measured had these flows resulted during those periods (England and others, 2019).

North Carolina

Weaver and others (2009) published streamgage flood‑frequency estimates for 49 streamgages in North Carolina known or considered to have regulated and (or) channelized periods of record through water year 2006. Updated flood‑frequency estimates were computed for 33 of these 49 streamgages using the regulated periods of record through water year 2019. The 33 sites include 31 streamgages where the flood‑frequency estimates were previously published in Weaver and others (2009) and 2 streamgages (02091814, Neuse River near Fort Barnwell, N.C. [site 150 on fig. 9], and 0351706800, Cheoah River near Bearpen Gap near Tapoco, N.C. [site 983 on fig. 9]) previously lacking sufficient record to derive flood‑frequency estimates.

Of the 49 streamgages, 18 streamgages with previously published flood‑frequency estimates (Weaver and others, 2009) were not updated during this study because of the following reasons:

Tests for Redundancy

Redundancy results when the drainage basins of two streamgages are nested one within another and similarly sized, leading to concurrent streamflow records. When this redundancy results, the two streamgages nearly have the same hydrologic response to a given storm event and, thus, effectively represent only one spatial observation (Gruber and Stedinger, 2008). To determine if two streamgages provided potentially redundant information, the following three types of information were considered: (1) the distance between basin centroids (a basin centroid is the location of the point within a drainage basin that represents the geometric center of the basin), (2) the ratios of the basin drainage areas, and (3) whether the streamgage records were concurrent.

A standardized distance was used, in part, to determine the likelihood that the streamgages provide redundant information. The standardized distance between two streamgages (i and j), SD_ij, is defined as follows:

Along with the standardized distance, a drainage-area ratio was used to determine if the drainages associated with streamgages were sufficiently similar in location and size to conclude that the streamgages may provide redundant information for the purposes of developing a regional hydrologic model. The drainage-area ratio, DAR, is defined as follows:

Previous studies have suggested that screening thresholds of standardized distance less than or equal to 0.50 miles combined with a drainage-area ratio less than or equal to 5 are appropriate for identifying sites with potentially redundant information (Veilleux, 2009; Mastin and others, 2016); consequently, those screening thresholds were adopted for this study. All possible combinations of streamgage pairs from the 965 streamgages with 10 or more years of unregulated record were considered and, if deemed potentially redundant, one streamgage from the pair was removed from the regression dataset (table 1 from Kolb and others, 2023). If the peak-flow record of the streamgage with the shorter period of record from the redundant pair was within the period of record of the streamgage with the longer peak-flow record, the streamgage with the shorter peak-flow record was removed from the analysis. However, if the peak-flow record of the streamgage with the shorter record was outside of the period of record of the longer peak-flow record streamgage, then both streamgages were used in the analysis. In some cases, if only a portion of the period of record for both streamgages had a substantial overlap (10 or more years), then the record for the longer record streamgage was extended by using the MOVE.1 method of correlation analysis (Hirsch, 1982), and the streamgage with the shorter record was removed from the analysis. Three streamgages for which the MOVE.1 method was used for record extension are listed in table 1 from Kolb and others (2023). Of the 965 streamgages with 10 or more years of unregulated record, 164 (about 17 percent) were removed from the regression dataset because of potential redundancy, leaving a regression dataset composed of data for 801 streamgages.

Exploratory Regression Analysis

For the exploratory regression analysis, OLS regression techniques were used to determine the best regression models for all combinations of basin characteristics and testing of the hydrologic regions that define the study area (fig. 2). In OLS regression, linear relations between the explanatory and response variables are necessary; thus, variables sometimes must be transformed to create linear relations. For example, the relation between arithmetic values of basin drainage area and P-percent AEP streamflow (such as the 1‑percent AEP streamflow) typically is curvilinear; however, the relation between the logarithms of drainage area and the logarithms of P-percent AEP streamflow normally is linear. Homoscedasticity (a constant variance in the response variable over the range of the explanatory variables) about the regression line and normality of the residuals is another assumption for OLS regression. Transformation of the P-percent streamflow and the explanatory variables to logarithms often enhances the homoscedasticity of the data about the regression line (Farmer and others, 2019). Homoscedasticity and normality of residuals were examined in residual plots. Additionally, residuals, which are the difference between the observed and predicted values, were mapped to assess the geographical distribution in the uncertainty of the predictions. If the geographic distribution of residuals shows clustering of positive and (or) negative values, that might suggest having subregions could reduce the uncertainty. Multicollinearity, which is a situation where two or more independent variables are highly correlated with strong linear dependence, was also assessed by the variance inflation factor. A variance inflation factor greater than 10 indicates highly correlated explanatory variables and warrants additional investigation (Montgomery and others, 2012), and a variance inflation factor of less than 5 is preferred (Farmer and others, 2019).

Initial OLS regressions were done for the entire study area for the 1‑ and 10‑percent AEP streamflows using only drainage area as the independent variable. Mapping of the residuals indicated clear regional differences, as was expected based on previous flood‑frequency regression analyses. After determining any clear regional differences, numerous potential regions were tested using the EPA level III and IV ecoregions (Omernik, 1987) and the previous hydrologic regions from Feaster and others (2009), Gotvald and others (2009), and Weaver and others (2009), along with variations of those hydrologic regions. From these exploratory analyses, it was concluded that the hydrologic regions used in the previous regional flood‑frequency analyses for the study area were still appropriate.

Two regions within the total study area (Georgia, South Carolina, and North Carolina) have been identified with flood characteristics that are difficult to define. The first region contains the Okefenokee Swamp in southeastern Georgia (fig. 2). This region is undefined because there are no streamgages to define the magnitude and frequency of floods for the basins that drain into the swamp. Feaster and others (2009) identified a second undefined region in the Upper Three Runs River Basin near the midwestern boundary of South Carolina. Although the area includes two streamgages (02197300 and 02197310, sites 476 and 477 on fig. 2), its flood characteristics are not well defined by the two streamgages. Large sand deposits at the upper end of this basin seem to affect rainfall runoff more than other regions of the Sand Hills. As part of the exploratory analysis, selected AEP streamflows were plotted with drainage area by hydrologic region. On that plot, the AEP streamflows for the Upper Three Runs streamgages tend to plot low on the data cloud of streamgages in the Sand Hills region but are not substantially outside the cloud. Therefore, it was decided that the regression equations for the Sand Hills (hydrologic region 3; see fig. 2) are appropriate for the Upper Three Runs River Basin considered in this study. However, users of the regional regression equations should be aware that these equations will likely tend to over predict AEP streamflows in this area and, where possible and appropriate, weighted AEP streamflows should be used (table 5 from Kolb and others, 2023).

All-possible-subsets regression methods were tested using the candidate explanatory variables shown in table 2 from Kolb and others (2023). The final explanatory variables for the exploratory regression analysis were selected based on primarily five factors, including (1) standard error of the estimate, (2) Mallow’s Cp statistic, (3) statistical significance of the explanatory variables, (4) coefficient of determination (R²), and (5) ease of measurement of explanatory variables (Farmer and others, 2019). Results of the all-possible subsets analysis indicated that drainage area, percentage of hydrologic regions 1, 3, 4, and 5, along with the cross product of drainage area and percentage of hydrologic region 2 were the top candidate explanatory variables. The statistical significance of the cross product of drainage area and percentage of hydrologic region 2 indicates an appreciably different slope for the regression line for hydrologic region 2 as compared to those for the other hydrologic regions.

Regional Regression Equations

Generalized least squares (GLS) regression methods, as described by Stedinger and Tasker (1985, 1986), were used to determine the final regional P-percent AEP flow regression equations using the weighted-multiple-linear regression (WREG) program version 3.0 written in statistical software R (R Core Team, 2020; Farmer, 2021). Stedinger and Tasker (1985, 1986) found that GLS regression equations are more accurate and provide a better estimate of the accuracy of the equations than OLS regression equations when annual peak-flow records at streamgages are of different and widely varying lengths and when concurrent flows at different streamgages are correlated. GLS regression techniques give less weight to streamgages with shorter periods of record than streamgages with longer periods of record. Less weight also is given to streamgages where concurrent peak flows are correlated because of the geographic proximity with other streamgages (Hodgkins, 1999).

For both the OLS and GLS regression analyses, regression diagnostics were computed and reviewed to assess potential problems with the regression models. Along with reviewing the residuals in terms of being randomly distributed around zero and assessing the geographical distribution, regression diagnostics also were reviewed to assess high leverage and high influence metrics. The leverage metric measures how far away the values of independent variables at one streamgage are compared to the values of the same variables at all other streamgages. The influence metric indicates whether the data at a streamgage had a high influence on the estimated regression metric values (Eng and others, 2009; Farmer and others, 2019). A streamgage may have a high leverage metric indicating that its independent variables are substantially different from those at all other streamgages, but the same streamgage may not have a high influence on the regression metrics. Conversely, a streamgage with a high influence may not have a high leverage metric. Sometimes, measurement or transposing errors in reported values of some independent variables may produce high leverage or influence metrics. Streamgages with high influence or leverage metrics were given additional review to determine if such errors had been made or if the streamgage should be excluded for other reasons. Brief notes are included in table 4 from Kolb and others (2023) for the streamgages that were excluded based on reviews of regression diagnostics. For the final regression analyses, 801 streamgages were included, and the distribution by State is shown in table 1 (see also fig. 2). For those 801 streamgages, the distribution of the systematic peak-flow record lengths is shown in figure 11.

The final set of regression equations for estimating peak flows at the selected AEPs are listed in table 2. The equations allow for the computation of AEP streamflows for unregulated, ungaged rural basins that drain from one or more of the five hydrologic regions (fig. 2), with hydrologic region 4 being the “base” region, which means that at an ungaged location when basin percentages of hydrologic regions 1, 2, 3, and 5 are zero, the site is located 100 percent within hydrologic region 4. Including the percentage of hydrologic region allows for a smooth transition in flood‑frequency estimates among the hydrologic regions. For basins that are 100‑percent contained within one of the five hydrologic regions, the equations are reduced to a simpler form as shown in table 3. Plots of the observed and predicted 10‑ and 1‑percent AEP streamflows are shown in figure 12A and B. The plots indicate a reasonable scatter about the line of equality throughout the range of streamflows. A data release by Weaver and others (2023) provides a model archive of the inputs and outputs for (1) the at-site flood‑frequency statistics and (2) the regression models developed to allow for estimation of flood‑frequency statistics at ungaged stream locations in the study area.

In the simplified equations that represent basins draining 100 percent from a single hydrologic region as a function of drainage area, the regression constant represents the intercept of the regression line, which for logarithmic space results when the drainage area is equal to one. The coefficient for drainage area represents the slope of the regression line. Thus, as can be seen by the equations in table 3, the regression lines for hydrologic regions 1, 3, 4, and 5 have the same slope but different intercepts (fig. 13A). The use of percentage of hydrologic regions as independent variables in the regression equations allows for a smooth transition in the AEP estimates for basins that do not lie wholly within one hydrologic region. The cross product of drainage area and percentage of hydrologic region 2 acts as a “slope adjustment factor” for hydrologic region 2 accounting for the difference in the slope of the regression line from that of the other four hydrologic regions. An example of the transition from a site located 100 percent in hydrologic region 1, represented by the “base” slope (for example, 0.604 for the 1‑percent AEP shown in table 3), to a site located 100 percent in hydrologic region 2 is shown in figure 13B.

Accuracy and Limitations

Regression equations are statistical models that must 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. Errors in the model (that is, differences between the predicted and observed values) can be examined to determine parameters that describe the accuracy of a regression equation, which depends on both the model error and the sampling error. Model error measures the capacity of a set of explanatory variables to estimate the values of peak-flow characteristics calculated from the streamgage records used to develop the regression equation. The model error depends on the number and predictive power of the explanatory variables in a regression equation. Sampling error measures the capacity of a finite number of streamgages with a finite number of recorded annual peak flows to describe the true characteristics of the entire peak-flow record for a streamgage. The sampling error depends on the number and record length of streamgages used in the analysis and decreases as the number of streamgages and record lengths increase. A measure of the uncertainty in a regression-equation estimate for a site, i, is the variance of prediction, V_p,i. The V_p,i is the sum of the model-error variance and sampling-error variance and is computed using the following equation:

Assuming that the explanatory variables for the streamgages in a regression analysis are representative of all streamgages in the hydrologic region, the average accuracy of prediction for a regression equation can be determined by computing the average variance of prediction, AVP, for n number of streamgages:

A more traditional measure of the accuracy of P-percent AEP streamflow regression equations is the standard error of prediction, S_p, which is simply the square root of the variance of prediction. The average standard error of prediction for a regression equation can be computed in percent error using AVP, in log units, and the following transformation formula:

The S_p,ave is a measure of the average accuracy of the regression equations when predicting flood estimates for ungaged sites, which is the most common application of the regression equations. There is about a 68‑percent probability that the true AEP streamflow at an ungaged location will be between plus or minus the S_p,ave of the regression estimate (Hodgkins, 1999).

A measure of the proportion of the variation in the response variable explained by the explanatory variables in OLS regressions is the coefficient of determination, R² (Montgomery and others, 2012). For GLS regressions, a more appropriate performance metric than R² is the pseudo coefficient of determination, pseudo R², described by Griffis and Stedinger (2007). Unlike the R² metric, pseudo R² is based on the variability in the response variable explained by the regression after removing the effect of the time-sampling error. The pseudo R² is computed using the following equation:

The average variance of prediction, average standard error of prediction, and pseudo R² for the final set of regional regression equations are listed in table 4.

Users of the regression models given above may be interested in a measure of uncertainty at a particular ungaged site as opposed to the uncertainty statistics based on streamgage data used to generate the regression models. One such measure of uncertainty at a particular ungaged site is the confidence interval of a prediction, or prediction interval. Prediction interval is the minimum and maximum value between which a stated probability that the true value of the response variable is present. Tasker and Driver (1988) determined that a 100 (1–α) prediction interval for the true value of a streamflow statistic for an ungaged site from the regression equation can be computed as follows:

The values for γ² and U are presented in table 9 from Kolb and others (2023).

The following limitations should be considered when using the final regional regression equations for Georgia, South Carolina, and North Carolina:

Comparison of Regression Results with Previous Rural Flood‑frequency Study

Flood‑frequency estimates at streamgages and regional flood‑frequency equations developed from those streamgage estimates contain uncertainty based on numerous factors, such as length of streamgage record, hydrologic conditions represented by the streamgage records, number of streamgages included in the regionalization, and range of the basin characteristics included in the regionalization. For flood‑frequency estimates computed from peak flows at specific streamgages, Benson and Carter (1973) showed that the standard error tends to be reduced as the period of record increases. Dalrymple (1960) generated 1,000 peak-flow events and then divided the events into various consecutive periods of record including one hundred 10‑year periods, forty 25‑year periods, twenty 50‑year periods and ten 100‑year periods. For each set of data, flood‑frequency curves were then drawn on the same graph for each period resulting in frequency plots of 100, 40, 20, and 10 curves, respectively. For the ten 100‑year curves, the estimates of the 100‑year recurrence interval flow ranged from about 6,500 to 9,200 ft³/s, reflecting the variation in such estimates just based on sampling from different time periods (time-sampling error).

Updating the regional skew also will affect the flood‑frequency estimates at a streamgage. The regional skew from the previous rural flood‑frequency study for Georgia, South Carolina, and North Carolina by Feaster and others (2009), Gotvald and others (2009), and Weaver and others (2009), respectively, used a regional skew of −0.019 with a mean square error (MSE) of 0.143. The updated regional skew was 0.048 with an MSE of 0.092, which also is noted as the average variance of prediction in appendix 1. In addition, the previous study was based on techniques from Bulletin 17B (Interagency Advisory Committee on Water Data, 1982) whereas, the current study is based on updated techniques in Bulletin 17C (England and others, 2019).

The 10‑ and 1‑percent AEP streamflow-regression lines developed from this study were compared with the regression lines from the previous rural flood‑frequency study for Georgia, South Carolina, and North Carolina (figs. 15 and 16; Feaster and others, 2009; Gotvald and others, 2009, and Weaver and others, 2009). For the comparisons, the simplified equations that are only a function of drainage area were used (table 3). The 10‑ and 1‑percent AEP streamflows were computed using the simplified equations from the current and previous studies for streamgages in Georgia, South Carolina, and North Carolina that drain 100 percent from a single hydrologic region and were included in the regression analyses for this study (table 1 from Kolb and others, 2023). The percentage change ([current − previous]/previous) of the 10‑ and 1‑percent AEP streamflows for the streamgages meeting this criterion were computed and the mean and median values for each hydrologic region are shown in table 6.

The percentage differences in hydrologic regions 1 and 2 were considered reasonable when comparing regression equations generated from two sets of data for two different time periods (table 6). The largest percentage differences were for the streamgages in Sand Hills hydrologic region (HR3). In the past decade, there have been several historical flood events that impacted the Sand Hills and Coastal Plain regions (Feaster and others, 2015, Feaster, Weaver, and others, 2018; Weaver and others, 2016), which would influence the updated regression equations. In the previous study, the regression lines for the Blue Ridge and Sand Hills regions exhibited a different slope than the other regions. In the current study, the Blue Ridge regression line still has a different slope, but the Sand Hills does not (fig. 13A). As such, the divergence of the regression lines for the Sand Hills region below 100 mi² is accounting for much of the difference between the current and previous regression estimates (figs. 15 and 16), which also is likely related to the historical flooding previously noted. It should also be noted that HR3 has a smaller number of streamgages than the other hydrologic regions except for HR5, which makes the regression lines more sensitive to changes in the at-site statistics used in the regression analysis. The historical flooding also is likely a strong influence in the increase in regression estimates for the Coastal Plain hydrologic region (HR4).

The mean and median percentage changes in the 10‑ and 1‑percent AEP regression estimates for streamgages in the Lower Tifton Upland region (HR5) were the second largest of the five hydrologic regions in this study (table 6). In the spring of 2009, historical flooding occurred in southern Georgia in the area that includes HR5 (Gotvald, 2010), with many streamgages having peak flows exceeding the 1‑ and 0.2‑AEP streamflows. As such, and like the Sand Hills and Coastal Plain regions, that historical flooding is likely a strong influence in the increase in the regression estimates for HR5.

Maximum Floods

In a flood‑frequency analysis, it is inherent that for floods with a specified probability of recurrence (such as the 1‑percent AEP streamflows), the regression line is a best-fit line for linear regression (or a best-fit plane for multiple regression) through a series of statistically specified P-percent AEP streamflows from some number of gaged locations in a given region. The regression line is fit so that the variance about the line is minimized; therefore, approximately half of the data points will plot above the regression line and half will plot below the regression line. For many engineering design projects, this level of uncertainty is acceptable and often is compensated for by including a factor of safety in the design process. In certain projects that include concerns about high risk, the assessment of maximum measured floods is another tool that can be used to evaluate the reasonableness of a flood‑frequency estimate.

Crippen and Bue (1977) developed envelope curves, which are curves encompassing the maximum values in a dataset, from maximum flood data for 17 regions in the conterminous United States (fig. 17). Crippen (1982) later provided equations that described the envelope curves. The curves were not associated with specific probabilities or frequencies but were provided as a tool to help assess the maximum flood that might be expected in the regions for a given drainage-area size. Costa (1987) developed an envelope curve of the maximum rainfall-runoff floods in the United States, but his dataset did not include any streamgages in the southeastern United States.

Maximum peak flows were plotted with drainage area using data from the streamgages included in table 1 from Kolb and others (2023) (figs. 18 and 19). In addition, a few streamgages that currently are regulated but had appreciable large floods recorded prior to regulation also were included (table 10 from Kolb and others, 2023). The streamgages were grouped based on having at least 75 percent of the basin located either above or below the Fall Line (fig. 1), which allows for comparisons with the Crippen (1982) curves. An envelope curve was then drawn for both regions that encompasses the largest floods. The streamgages above the Fall Line include data from the Ridge and Valley, Blue Ridge, and Piedmont ecoregions, and the streamgages below the Fall Line include data from the Southeastern Plains, Middle Atlantic Coastal Plain, and Southern Coastal Plain ecoregions (fig. 1). Similar to the Crippen (1982) curves, the envelope curves generated from the maximum flow data included in this study indicate maximum flood-streamflow potential for a range of drainage areas for the two regions, which are (1) the area above the Fall Line and (2) the area below the Fall Line (table 7).

Flood‑frequency Estimation at a Streamgage

Bulletin 17C (England and others, 2019) recommends that better flood‑frequency estimates for a streamgage can be obtained by combining (weighting) streamgage flow estimates determined from the log-Pearson Type III analysis of the annual peaks with flow estimates obtained for the streamgage from regression equations. Optimal weighted flow estimates can be obtained if the variance of prediction for each of the two estimates is known or can be estimated accurately and precisely. The variance of prediction can be thought of as a measure of the uncertainty in either the streamgage estimate or the regional regression results. If the two estimates can be assumed independent and are weighted in inverse proportion to the associated variances, the variance of the weighted estimate will be less than the variance of either of the independent estimates.

The variance of prediction corresponding to the streamgage flow estimate from the LPIII analysis is computed using the asymptotic formula given in Cohn and others (2001) with the addition of the mean-squared error of generalized skew (Griffis and others, 2004). This variance varies as a function of the length of record, the fitted LPIII distribution parameters (mean, standard deviation, and weighted skew), and the accuracy of the method used to determine the generalized skew component of the weighted skew. The variance of prediction for the streamgage estimate generally decreases with length of record and the quality of the LPIII distribution fit. The variance of prediction values for the streamgage flow estimates for the 807 streamgages in Georgia, South Carolina, and North Carolina are listed in table 3 from Kolb and others (2023), which includes both the redundant streamgages and those included in the regression analyses.

The variance of prediction from the regional regression equations is a function of the regression equations and the values of the independent variables used to develop the flow estimate from the regression equations. This variance generally increases as the values of the independent variables move further from the mean values of the independent variables. For the streamgages included in the regression analysis, the variance of prediction is provided as part of the WREG output (table 3 from Kolb and others, 2023). The average variance of prediction values for the regional regression equations used in this study are listed in table 4 and can be used for weighting of streamflow estimates (eq. 15) for streamgages not included in the regression analysis.

Once the variances have been computed, the two independent flow estimates can be weighted using the following equation:

The weighted (best) flow estimates were computed using equation 15 along with the variance of prediction values for the regression equations (table 4) and the variance from the at-site EMA analyses for the 807 streamgages in Georgia, South Carolina, and North Carolina (table 3 from Kolb and others, 2023). When the variance of prediction corresponding to one of the estimates is high, the uncertainty is also high and so the weight for that estimate is relatively small. Conversely, when the variance of prediction is low, the uncertainty is also low and so the weight is correspondingly large. The variance of prediction associated with the weighted estimate, V_w(s), is computed using the following equation:

Confidence intervals for the weighted estimate also can be computed. The upper and lower 95‑percent confidence intervals (95%CI) on the weighted AEP estimate can be computed as

An example of the application of the procedure described above is the following steps for computation of the weighted 1‑percent AEP streamflow for streamgage 02116500, Yadkin River at Yadkin College, N.C. (site 256 on fig. 2; tables 1 and 3 from Kolb and others, 2023):

Flood‑frequency Estimation for an Ungaged Site Near a Streamgage

Sauer (1974) presented the following method to improve flood‑frequency estimates for an ungaged site near a streamgage, on the same stream, with 10 or more years of peak-flow record. To obtain a weighted peak-flow estimate (Q_w(u)) for P-percent AEP at the ungaged site, the weighted flow estimate for an upstream or downstream streamgage (Q_w(s)) must first be determined by using equation 15 provided in the previous section. The weighted estimate for the ungaged site (Q_w(u)) is then computed using the following equation:

Use of equation 18 gives full weight to the regression equation estimates when the drainage area for the ungaged site is equal to 0.5 or 1.5 times the drainage area for the streamgage and increasing weight to the streamgage estimates as the drainage-area ratio approaches 1. The weighting procedure should not be applied when the drainage-area ratio for the ungaged site and streamgage is less than 0.5 or greater than 1.5.

An example application of this procedure is the computation of the weighted 1‑percent AEP streamflow for a hypothetical ungaged site on the Yadkin River located above the USGS streamgage 02116500, Yadkin River at Yadkin College, N.C. (site 256 on fig. 2; table 1 from Kolb and others, 2023), discussed in the previous section. The regulated streamgage 02115360, Yadkin River at Enon, N.C. (site 248 on fig. 9; table 8 from Kolb and others, 2023), is used as a hypothetical unregulated and ungaged site for purposes of this example:

For an ungaged site that is located between two streamgages on the same stream, two flow estimates can be made using the methods and criteria outlined in this section. In addition to evaluating the differences in hydrologic regions of the two streamgages compared to the hydrologic region(s) for the ungaged site, additional hydrologic expertise and judgment may be necessary to determine which of the two estimates (or some interpolation thereof) is most appropriate. Other factors that might be considered when evaluating the two estimates include differences in the length of record for the two streamgages and the hydrologic conditions present during the data-measurement period for each streamgage (that is, whether the time series represents a climatic period that was predominately wet or dry).