Purpose and Scope

The purpose of this investigation was to assess the effects of impoundments on selected streamflow characteristics across the contiguous hydrologic regions in Georgia, South Carolina, and North Carolina as defined by Feaster and others (2009, 2014, 2023). Of particular interest to the South Carolina Department of Transportation are the effects that impoundments have on flood-frequency statistics for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probability (AEP) streamflows (also known as the 2-, 5-, 10-, 25-, 50-, 100-, 200-, and 500-year recurrence interval streamflows, respectively). Along with assessing changes in selected flood-frequency characteristics due to regulation from impoundments, other selected streamflow characteristics were assessed to gain a more complete understanding of impacts from regulation throughout the streamflow regime. As such, the report assesses the effects on low-streamflow characteristics, mean annual streamflow, the annual maximum streamflow, and the impacts on flow-duration statistics at selected USGS daily mean streamgages that have long-term periods of record collected during both pre- and post-regulated (before and after impoundment) streamflow conditions. The streamgages were selected based on a criterion of having at least 30 years of record in both the pre- and post-regulated periods. For an assessment of natural variability of such statistics based on period of record and hydrologic conditions captured in that record, streamflow records at selected long-term streamgages monitoring daily mean streamflows for unregulated conditions also were compared using the Indicators of Hydrologic Alteration (IHA) software developed by The Nature Conservancy (Richter and others, 1996) by analyzing the first and last half of the records. The IHA software also was used to compare selected streamflow characteristics from pre- and post-regulated long-term periods of record at USGS streamgages monitoring daily mean streamflow that had at least 30 years of record in each period. Most of the flood-frequency statistics included in this investigation were detailed in Feaster and others (2023) and are available from a USGS data release by Kolb and others (2023).

Description of Study Area

The study area includes all of Georgia, South Carolina, and North Carolina, covering an area of about 142,500 square miles (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) (Omernik, 1987; Griffith and others, 2002; U.S. Environmental Protection Agency, 2022). 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. The ecoregions were determined 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 (Griffith and others, 2002). The Fall Line separates the higher elevation 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 (Cooke, 1936; U.S. Environmental Protection Agency, 2022).

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. The Ridge and Valley ecoregion is composed of roughly parallel ridges and valleys that have a variety of widths, heights, and geologic materials. Springs and caves are relatively numerous in this ecoregion, and present-day forests cover about 50 percent of the ecoregion. The Blue Ridge ecoregion varies from narrow ridges to hilly plateaus to more massive 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 geology. The Piedmont ecoregion is composed of a transitional area between the mostly mountainous ecoregions of the Appalachian Mountains 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 featuring a mixture of cropland, pasture, woodland, and forest. The sand, silt, and clay geology of this ecoregion contrasts with the older rocks of the Piedmont ecoregion. Elevations and relief are greater than in the Southern Coastal Plain ecoregion but generally are less than in much of the Piedmont ecoregion. Streams in this area have relatively low gradient and 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 of Mexico and Atlantic coasts. This ecoregion is lower in elevation with less relief and wetter soils than the Southeastern Plains ecoregion. The Middle Atlantic Coastal Plain ecoregion consists of low-elevation flat plains and includes many swamps, marshes, and estuaries. The low terraces, marshes, dunes, barrier islands, and beaches are underlain by unconsolidated sediments. 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).

Comparison of Pre- and Post-Regulated Annual Exceedance Probability Streamflow From Long-Term Streamgages

For the 18 streamgages with both pre- and post-regulated periods (table 1 from Musser and Feaster, 2023), the percentage change in the estimated 10-, 1-, and 0.2-percent AEP streamflows, which correspond to recurrence intervals of 10, 100, and 500 years, respectively, was computed (table 3 from Musser and Feaster, 2023). In general, percentage change represents the change in a value relative to the previous value and is computed as follows:

where

For this comparison, y2 represents the AEP streamflow estimate from the post-regulated period, and y1 represents the AEP streamflow estimate from the pre-regulated period. A negative percentage change indicates a decrease in the AEP streamflow from the pre-regulated to post-regulated period, and a positive percentage change indicates an increase. On average, at the 18 streamgages analyzed, regulation decreased the 10-, 1-, and 0.2-percent AEP streamflow estimates by about 31 percent. For the 10-percent AEP streamflows, all estimates at the 18 streamgages decreased from the pre- to post-regulated period, ranging from 2.0 to 72.0 percent (table 3 from Musser and Feaster, 2023). The median decrease was 28.9 percent with a mean decrease of 32.3 percent. For the 1-percent AEP streamflows, the percentage change ranged from −78.0 to 22.4 percent with 02089000 Neuse River near Goldsboro, N.C., and 02105500 Cape Fear River at William O. Huske Lock near Tarheel, N.C., being the only streamgages with a positive percentage change (fig. 3). It is worth noting that the peak of record at 02089000 occurred in October 2016 because of Hurricane Matthew (Weaver and others, 2016) and that the peak of record and the second largest peak of record at 02105500 occurred in September 2018 and October 2016, respectively, because of Hurricane Florence (Feaster and others, 2018) and Hurricane Matthew (Weaver and others, 2016), respectively. The median percentage change and mean percentage change for the 1-percent AEP streamflows were −32.4 and −31.6 percent, respectively. For the 0.2-percent AEP streamflows, the percentage change ranged from −83.4 to 44.7 percent with the median and mean being −31.1 and −29.8 percent, respectively. Comparisons of the full range of the flood-frequency curves from the 50- to 0.2-percent AEP streamflows are shown on figure 3.

Comparison of Annual Exceedance Probability Streamflows for Two Periods at Long-Term Streamgages on Unregulated Streams

The flood-frequency statistics computed for streamgages on unregulated streams are strongly influenced by period of record and hydrologic conditions captured in that record. This natural variability also will influence streamflow statistics computed at streamgages on regulated streams but with the added complexity of the influence from the regulation, which may enhance or even offset the natural variability in those data. Although the USGS typically designates streamgage records with 30 or more years as long term (U.S. Geological Survey, 2013), records of that length and much longer may still be too short to positively identify and quantify a real long-term change in the natural hydrologic regime (Riggs, 1985). As Lins and others (2010) noted, sometimes streamflow records of several decades may indicate a trend but, when viewed in the context of timeframes spanning many decades to centuries, may just be recognized as a short-term trend that is part of a much longer term oscillation (Stahle and Cleaveland, 1992). In their paper discussing trends in hydroclimatological data and long-term persistence, Cohn and Lins (2005, section “Discussion and Conclusions,” paragraph 27) noted that “… natural climatic excursions may be much larger than we imagine.”

To get a sense of how flood-frequency statistics have varied at unregulated USGS streamgages in Georgia, South Carolina, and North Carolina, the percentage change in the flood-frequency estimates for the 10-, 1-, and 0.2-percent AEP streamflows was computed for 18 streamgages monitoring unregulated streams by separating the record into two periods: first half of the record and last half of the record (fig. 4) (table 4 from Musser and Feaster, 2023). The record lengths analyzed ranged from 41 to 65 years for each half period with the average length being 52 years. For the 10-percent AEP streamflows, the percentage change ranged from −38.9 to 30.1 percent with a median of −9.8 percent and a mean of −8.6 percent. For the 1-percent AEP streamflows, the percentage change ranged from −48.1 to 57.0 percent with a median of −7.4 percent and a mean of −6.4 percent. For the 0.2-percent AEP streamflows, the percentage change ranged from −53.6 to 69.0 percent with a median of −9.5 percent and a mean of −5.0 percent. Comparisons of the full range of the flood-frequency curves from the 50- to 0.2-percent AEP streamflows are shown on figure 4.

Comparison of the Distribution of the Percentage Change in Annual Exceedance Probability Streamflows for Selected Streamgages Having Long-Term Pre- and Post-Regulated Periods of Record and at Selected Unregulated Streamgages Having Two Long-Term Periods of Record

To further explore the percentage change between the 10-, 1-, and 0.2-percent AEP streamflows for the two periods from the 18 USGS streamgages with pre- and post-regulated periods of record (tables 1 and 3 from Musser and Feaster, 2023), boxplots were generated and compared with the percentage change in those same AEP streamflows for two periods of record at 18 unregulated streamgages (tables 2 and 4 from Musser and Feaster, 2023) (fig. 5). This comparison provides insight into differences that might be attributed to natural variability. Boxplots provide an informative graphical display of the distribution of a dataset (Helsel and others, 2020). The line inside the box is the median. The box height is formed from the 25th and 75th percentiles (quartiles [Q] 1 and 3, respectively) of the dataset. The difference between Q3 and Q1 is the interquartile range (IQR). The IQR provides a visual of the variation (spread) of the dataset. The location of the median line in the box indicates the skewness. For a normally distributed dataset, the median and the mean are the same. Data points that are greater than or less than 1.5 times the IQR are considered outliers. If a dataset has no outliers, the end of the whiskers extending from the box represents the minimum and maximum data points of the dataset. For datasets that have outliers, the end of the whiskers represent the minimum or maximum data point of the dataset excluding the outliers.

In general, the boxplots of the percentage change in the 10-, 1-, and 0.2-percent AEP streamflows for the two periods from the 18 streamgages monitoring unregulated streams show a variability that tends to be more balanced between positive and negative changes as compared to the pre- and post-regulated AEP streamflows (fig. 5). In comparison, the interquartile percentage change from the pre- and post-regulated periods at the 18 streamgages was all in the negative range, thereby indicating that overall, those AEP streamflows were lower in the post-regulated period than in the pre-regulated period. The median percentage change for the 10-, 1-, and 0.2-percent AEP streamflows from the pre- and post-regulated periods was 19.1, 25.0, and 21.6 percentage points lower, respectively, than the median percentage change for those same AEP streamflows of the two periods from the streamgages monitoring unregulated streams (tables 3 and 4 from Musser and Feaster, 2023). It is also interesting to note that there was at least one outlier for each of the AEP streamflows from the comparison of two unregulated periods of record but no outliers from the comparisons of the pre- and post-regulated periods of record. As noted earlier, regulation of streamflows often decreases the highest streamflows and increases the lowest streamflows, thus reducing the extremes that are typically part of natural streamflow regimes and that may tend to show up as outliers (Williams and Wolman, 1984).

Comparison of Percentage Change in Selected Pre- and Post-Regulated Daily Mean Streamflow Characteristics From Long-Term Streamgages

The IHA software (Richter and others, 1996) was used to analyze selected streamflow characteristics from the 16 USGS streamgages that had pre- and post-regulated daily mean streamflows (table 5 from Musser and Feaster, 2023). The streamflow characteristics analyzed were mean annual streamflow, 1-day maximum streamflow, 1- and 7-day minimum streamflows, low pulse count and duration, and high pulse count and duration. Other than the mean annual streamflow, the other streamflow characteristics were based on the median values for the period analyzed. The length of records available in the pre-regulated period ranged from 34 to 80 years. The length of records available in the post-regulated period ranged from 39 to 68 years. Of the 16 streamgages included in table 5 from Musser and Feaster (2023), 03513000 Tuckasegee River at Bryson City, N.C., had the lowest maximum storage index (193 acre-ft/mi²), and 02335000 Chattahoochee River near Norcross, Ga., had the highest (2,183 acre-ft/mi²). As compared to 03513000, which shows only minor differences between the pre- and post-regulated periods, the effects of regulation on the high and low streamflows at 02335000 are substantial, showing how regulation decreased the annual 1-day maximum streamflows and increased the annual 7-day minimum streamflows (figs. 6 and 7).

The percentage change in the mean annual streamflow from the pre- to post-regulated period ranged from −21.8 to 9.7 percent with a median of −0.9 percent and a mean of −1.4 percent (table 5 from Musser and Feaster, 2023). For the 1-day maximum streamflow, the percentage change ranged from −69.4 to 0.9 percent with a median of −24.4 percent and a mean of −28.2 percent. For the 1-day minimum streamflow, the percentage change ranged from −46.7 to 350 percent with a median of −4.8 percent and a mean of 29.7 percent. The percentage change for the 7-day minimum streamflow ranged from −48.2 to 166 percent with a median of 19.6 percent and a mean of 24.1 percent. These findings reflect conditions that are often found to occur from regulation; on average, the low streamflows increased, and the high streamflows decreased.

The percentage change in the low pulse count from the pre- to post-regulated period ranged from −93.5 to 400 percent with a median of 20.0 percent and a mean of 64.6 percent (table 5 from Musser and Feaster, 2023). The percentage change in the low pulse duration ranged from −75.0 to 60.0 percent with a median of −17.5 percent and a mean of −21.6 percent. For the high pulse count, the percentage change ranged from −35.7 to 65.4 percent with a median of −5.5 percent and a mean of 6.1 percent. The percentage change in the high pulse duration ranged from −50.0 to 100 percent with a median of 12.7 percent and a mean of 13.1 percent.

Comparison of Percentage Change in Daily Mean Streamflow Characteristics From Two Periods at Long-Term Streamgages Monitoring Unregulated Streams

To assess natural variability in selected daily mean streamflow characteristics based on period of record, the IHA software (Richter and others, 1996) was used to analyze two periods (first half and last half of the records) at 17 USGS streamgages monitoring unregulated streams (table 6 from Musser and Feaster, 2023). Like the IHA analysis of the pre- and post-regulated periods at 16 USGS streamgages (table 5 from Musser and Feaster, 2023), the streamflow characteristics analyzed were mean annual streamflow, 1-day maximum streamflow, 1- and 7-day minimum streamflows, low pulse count and duration, and high pulse count and duration. The length of records available in the two periods at the 17 streamgages monitoring unregulated streams ranged from 39 to 60 years.

The percentage change in the mean annual streamflow ranged from −19.0 to 28.3 percent with a median of −3.7 percent and a mean of −2.4 percent (table 6 from Musser and Feaster, 2023). For the 1-day maximum streamflow, the percentage change ranged from −37.3 to 22.9 percent with a median of −5.8 percent and a mean of −7.3 percent. The percentage change for the 1-day minimum streamflow ranged from −74.7 to 53.9 percent with a median of −20.4 percent and a mean of −19.5 percent. Changes in the 7-day minimum streamflow were similar with a range of −71.1 to 50.0 percent and a median and mean of −24.9 and −21.4 percent, respectively. The reduction in the mean of the 1- and 7-day minimum streamflows and 1-day maximum streamflows may reflect historical drought periods that have occurred in the Southeast over the last couple of decades (South Carolina Department of Natural Resources, 2004; Weaver, 2005; Gotvald, 2016; Feaster and Guimaraes, 2017) or other influences in the basins.

The percentage change in the low pulse count from the two periods at the streamgages monitoring unregulated streams ranged from −22.2 to 28.6 percent with a median of 0.0 percent and a mean of 1.0 percent indicating a relatively normal distribution (table 6 from Musser and Feaster, 2023). For the low pulse duration, the percentage change ranged from −23.1 to 61.9 percent with a median of 28.0 percent and a mean of 25.1 percent. For the high pulse count, the percentage change ranged from −29.4 to 33.3 percent with a median of −6.3 percent and a mean of −3.1 percent. The percentage change for the high pulse duration ranged from −16.7 to 50.0 percent with a median of 0.0 percent and a mean of 2.1 percent.

Comparison of the Distribution of the Percentage Change in Daily Mean Streamflow Characteristics From the Indicators of Hydrologic Alterations Analysis at Long-Term Streamgages

Boxplots of the percentage change in daily mean streamflow characteristics for pre- and post-regulated periods at the 16 USGS streamgages (table 5 from Musser and Feaster, 2023) were compared with boxplots of the percentage change in those same daily mean streamflow characteristics for the two periods at 17 USGS streamgages monitoring unregulated streams (table 6 from Musser and Feaster, 2023) (fig. 8). The boxplots of the percentage change in mean annual streamflow for two periods at the 17 streamgages monitoring unregulated streams show a well-balanced distribution of positive and negative values with the mean and median values being similar and just slightly less than zero. The boxplots for the percentage change in mean annual streamflow for the 16 streamgages with pre- and post-regulated record also show mean and median values that are similar and just slightly less than zero but with less variability.

The boxplot for the percentage change in the 1-day maximum streamflows for the pre- and post-regulated periods was mostly in the negative range with the median and mean values being −24.4 and −28.2 percent, respectively (fig. 8), showing that the 1-day maximum streamflows tended to be lower for the post-regulated periods of record (table 5 from Musser and Feaster, 2023). As previously noted, this is consistent with the observation that regulation tends to reduce high streamflows. The boxplot of the percentage difference in the 1-day maximum streamflows for the two unregulated periods of record tends to be more balanced between both positive and negative percentages with the median and mean values being −5.8 and −7.3 percent, respectively (table 6 from Musser and Feaster, 2023).

Comparisons of the boxplots of the percentage change in the 1- and 7-day minimum streamflows for the pre- and post-regulated streamflows and the two periods of unregulated streamflows show that the percentage change of the pre- and post-regulated streamflows had a larger variability with the mean values both being positive, thereby suggesting (as previously noted) that, on average, the low streamflows for the post-regulated period tended to be higher than those for the unregulated periods (fig. 8).

For the low and high pulse counts and durations, boxplots were generated for the actual values instead of the percentage change (figs. 9A, B and 10A, B). The low pulse count from the pre-regulated period of record was somewhat like the two periods from the streamgages on unregulated streams, which show no substantial differences between the two periods, but with less variability (fig. 9A). The post-regulated period had a higher mean and median low pulse count and a much greater variability than the pre-regulated period and the two unregulated periods. Although not substantially different, the post-regulated low pulse duration had greater variability and a lower mean and median duration than the pre-regulated period (fig. 9B). The low pulse duration for the first half and last half of the streamgages monitoring unregulated streams had greater variability and higher means and medians than the pre- and post-regulated low pulse durations.

For the pre- and post-regulated periods at the 16 streamgages, the median values for the high pulse count were similar, but the post-regulated distribution of high pulse count had much greater variability (fig. 10A). Like the low pulse count (fig. 9A), the high pulse count for the first and last half of the record at the 17 streamgages monitoring unregulated streams was quite similar (fig. 10A). The pre- and post-regulated high pulse durations also were similar, with the post-regulated period having only slightly more variability than the pre-regulated period (fig. 10B). Like the high pulse count (fig. 10A), the distribution of the high pulse duration for the first and last half of the record at the 17 streamgages monitoring unregulated streams was similar (fig. 10B).

Comparison of Flow-Duration Curves

Flow-duration curves are cumulative frequency curves that show the percentage of time during which specified streamflows were equaled or exceeded during the period of record analyzed (Searcy, 1959). Flow-duration curves can be useful for understanding the broad range of streamflow characteristics of a gaged stream and can be useful for comparisons with other streams. Flow-duration curve data from the IHA analysis for the 17 USGS streamgages with pre- and post-regulated daily mean streamflow were used to generate flow-duration curves (table 1 from Musser and Feaster, 2023) (fig. 11). The flow-duration curves provide a graphical comparison of the effects of regulation throughout the full range of daily mean streamflows. For some of the streamgages, the extremes tend to show the greatest differences between the pre- and post-regulated periods of record, which as previously noted is a common occurrence for certain types of regulation.

To gain an understanding of how flow-duration curves may differ based on period of record, a comparison of two periods at the 17 streamgages monitoring unregulated streams also was made (table 2 from Musser and Feaster, 2023) (fig. 12). For the streamgages monitoring unregulated streams, the flow-duration curves for the two periods tend to have similar shapes. The streamgages in the Edisto and Flint River Basins tend to show lower streamflows throughout much of the flow-duration curve for the last half of the records. This tendency could be due to the historical droughts captured in those records or other influences in the basins.

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 two hydrologic regions above the Fall Line (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 AEP streamflow normally is curvilinear; however, the relation between the logarithms of drainage area and the logarithms of the AEP streamflows for selected probabilities (P-percent; for example, 50, 20, or 10 percent) 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 are another assumption for OLS regression. Transformation of the AEP streamflow and the explanatory variables to logarithms often enhances the homoscedasticity of the data about the regression line. 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 geographic distribution in the uncertainty of the predictions, which provides information on potential subregions that might reduce the uncertainty. This assessment of independent variables for percentage of drainage area from hydrologic regions 1 and 2 indicated that there was not a statistically significant difference between the two regions, and therefore, the regression analysis was done using a single region above the Fall Line.

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 (VIF). A VIF greater than 10 indicates highly correlated explanatory variables and warrants additional investigation (Montgomery and others, 2012), and a VIF less than 5 is preferred (Farmer and others, 2019). The VIF for the final variables included in the regression analysis was close to 1.

As part of the update of rural flood-frequency regional regression equations for Georgia, South Carolina, and North Carolina, Feaster and others (2023) tested 26 basin characteristics as potential explanatory variables. In that analysis, drainage area and percentage of hydrologic regions were included in the final regional regression equations. Based on that analysis, drainage area and percentage of hydrologic regions 1 and 2 were tested with a series of other characteristics related to impoundments in this investigation. The additional characteristics tested were

The maximum storage and surface area were obtained from the U.S. Army Corps of Engineers (2020) NID database. The distance to the first and last upstream reservoirs was computed using Esri ArcMap (Esri, 2021). The ArcMap Utility Network Analyst tool “Find Path” was used to select segments of streamline from the National Hydrography Dataset (U.S. Geological Survey, 2021a) High Resolution layer NHDFlowline between a streamgage and a reservoir. The attribute values of LengthKM from the NHDFlowline layer were summed for these segments and converted to miles. The remaining distance along the selected streamline that extended downstream from the streamgage was then determined using the “Measure” tool and subtracted from the summed total distance.

All-possible-subsets regression methods were tested using the candidate explanatory variables related to impoundments along with drainage area, percentage of hydrologic regions 1 and 2, and a cross product of drainage area and percentage of hydrologic region 2, which was statistically significant in the rural regression equations from Feaster and others (2023). The final explanatory variables for the exploratory regression analysis were selected based on several factors, including (1) standard error of the estimate, (2) Mallow’s Cp statistic (Helsel and others, 2020), (3) statistical significance of the explanatory variables, (4) coefficient of determination (R²), and (5) ease of measurement of explanatory variables. Based on the OLS assessments of various groupings of explanatory variables and the criteria noted previously, a regression model including drainage area and maximum storage index was selected using only streamgages for which the basin drained 75 percent or more from above the Fall Line.

Final 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 streamflow regression equations. The analysis was performed using R statistical software (R Core Team, 2020) with the R package WREG (a weighted least squares regression for streamflow frequency statistics program), version 3.0 (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-streamflow records at streamgages are of different and widely varying lengths and when concurrent streamflows at different streamgages are correlated. GLS regression techniques give less weight to streamgages that have shorter periods of record than to streamgages with longer periods of record. Less weight also is given to streamgages where concurrent peak streamflows 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 geographic distribution, regression diagnostics also were reviewed to assess high leverage and high influence. The leverage metric measures how unusual 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 transcription errors in reported values of some independent variables may produce high leverage or influence metrics. Streamgages with high influence or leverage were given additional review to determine if such errors had been made or if the streamgage should be excluded for other reasons. From those reviews, nothing was found to warrant removing any streamgages from the regression analyses. For the final regression analyses, 39 streamgages were included with the distribution by State shown in table 1 and figure 2.

The final regression equations for estimating peak streamflows at the selected AEPs are listed in table 2. The equations allow for the computation of AEP streamflows for regulated rural basins that drain 75 percent or more from above the Fall Line. Figure 13 shows plots of the observed and predicted 10-, 1-, and 0.2-percent AEP streamflows and provides a visual of this uncertainty in the regression estimates, which is discussed in detail in the next section. Figure 13 also includes the observed and predicted estimates from the 24 streamgages that were not included in the regression analysis because the basin was found to be redundant but still had 75 percent or more of the basin drainage coming from above the Fall Line (table 7 from Musser and Feaster, 2023). The estimates from these 24 streamgages provide a cross validation of the regression model and show that, although they were not included in the regression analysis, the plot of the observed and predicted estimates from these 24 streamgages are well within the mix of the streamgages that were included in the regression analysis.

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. Uncertainty, or error, 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 time-sampling error. Model error measures the ability of a set of explanatory variables to estimate the values of peak-streamflow characteristics calculated from the streamgage records used to develop the equation. The model error depends on the number and predictive power of the explanatory variables in a regression equation. Time-sampling error measures the ability of a finite number of streamgages with a finite number of recorded annual peak streamflows to describe the true characteristics of the entire peak-streamflow record for a streamgage. The time-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 time-sampling error variance and is computed using the following equation:

where

Assuming that the explanatory variables for the streamgages in a regression analysis are representative of all streamgages in the 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:

where the remaining variables are as previously defined in equation 4.

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 by using AVP, in log units, and the following transformation formula:

where

The S_p,ave is a measure of the average uncertainty 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 pseudo R² described by Griffis and Stedinger (2007b). 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 formula:

where When pseudo R² is closer to 1, the regression equation better explains the variation in the response variable. 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 3.

Users of the regression models may be interested in a measure of uncertainty at a particular 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. The prediction interval is the range that likely contains the streamflow characteristic for a new observation not included in the development of the regression equations. Tasker and Driver (1988) determined that a 100(1–α) prediction interval for the true value of a streamflow characteristic for an ungaged site from the regression equation can be computed as follows:

where where where The values for γ² and U are presented in table 4.

The following limitations should be recognized when using the final regional regression equations for regulated rural basins in Georgia, South Carolina, and North Carolina:

The maximum storage for the individual reservoirs upstream from the streamgages included in the regulated rural regression analysis ranged from 185 acre-feet (acre-ft) to 3,820,000 acre-ft (U.S. Army Corps of Engineers, 2020). Of those reservoirs, 90 percent had a maximum storage greater than or equal to 5,220 acre-ft with the mean maximum storage being 328,000 acre-ft and the median being 40,600 acre-ft.

Comparison of Regulated Rural Flood-Frequency Estimates With Unregulated Rural Flood-Frequency Estimates

For the 10-, 1-, and 0.2-percent AEP streamflows, the at-site regulated flood-frequency estimates were compared with the regulated regression estimates by using the equations in table 2 and with the unregulated rural flood-frequency estimates from the regional regression equations from Feaster and others (2023). The comparisons were done using boxplots of the percentage change from the at-site regulated flood-frequency estimates for the 39 streamgages included in the regulated regression analysis to the regulated regression estimates and to the unregulated regression estimates for the same group of 39 streamgages (fig. 14). In general, the boxplots show that the regulated regression estimates better match the at-site regulated flood-frequency estimates than do the unregulated flood-frequency estimates. The median and mean percentage change shows that the unregulated flood-frequency estimates tend to overestimate the at-site regulated flood-frequency estimates and have greater variability than do the percentage change from the regulated regression estimates.

The percentage change between the at-site regulated estimates and regulated regression estimates for the 10-percent AEP streamflows ranged from −62 to 126 percent with a median and mean of −2.9 and 10 percent, respectively (fig. 14). The percentage change between the at-site regulated estimates and the unregulated regression estimates for the 10-percent AEP streamflow predictions ranged from −48 to 332 percent with a median and mean of 23 and 52 percent, respectively.

For the 1-percent AEP streamflows, the percentage change between the at-site regulated estimates and the regulated regression estimates ranged from −67 to 182 percent with a median and mean of 6.6 and 12 percent, respectively (fig. 14). The percentage change between the at-site regulated estimates and the unregulated regression estimates for the 1-percent AEP streamflows ranged from −53 to 442 percent with the median and mean being 28 and 62 percent, respectively.

For the 0.2-percent AEP streamflows, the percentage change between the at-site regulated estimates and the regulated regression estimates ranged from −82 to 229 percent with a median and mean of 5.8 and 15 percent, respectively (fig. 14). The percentage change between the at-site regulated estimates and the unregulated regression estimates for the 0.2-percent AEP streamflows ranged from −68 to 495 percent with a median and mean of 30 and 67 percent, respectively.

Figure 15 provides a visual comparison of the regulated regression flood-frequency curves and the unregulated regression flood-frequency curves for the 10-, 1-, and 0.2-percent AEP flows. The unregulated flood-frequency curves are from Feaster and others (2023) and assume that basins drain fully from hydrologic regions 1 and 2, which are the two hydrologic regions above the Fall Line (fig. 2). Because the regulated flood-frequency equations are functions of drainage area and maximum storage index (table 2), two curves are shown (fig. 15), with one holding the maximum storage index constant at the minimum value and the other holding the maximum storage index constant at the maximum value from the streamgages included in the regulated regression analysis (table 5). As can be seen, as the maximum storage index increases from the minimum to the maximum value included in the regression analysis, the flood-frequency estimates decrease (fig. 15). For the minimum value of the maximum storage index, the flood-frequency curve is more in the range of the unregulated curves. It should be noted that the regulated flood-frequency analysis includes 39 streamgages and the unregulated flood-frequency analysis for hydrologic regions 1 and 2 includes 352 and 113 streamgages, respectively, that drain 75 percent or more within one hydrologic region (table 1 from Kolb and others, 2023). Also, holding the maximum storage index at the minimum and maximum values is useful to provide a one-dimensional visual comparison; however, with the regulated regression being a function of two independent variables, the regression analysis is fitting a plane and not a linear curve within a range of maximum storage index values between the minimum and maximum values (table 5). As such, differences in unregulated regression curves and the regulated regression curve for the minimum value of the maximum storage index would be expected.

Estimation at a Streamgage

B17C (England and others, 2018) recommends that improved flood-frequency estimates for a streamgage can be obtained by combining (weighting) streamgage streamflow estimates determined from the LPIII analysis of the annual peak streamflows with streamflow estimates obtained for the streamgage from regression equations. Optimal weighted streamflow estimates can be obtained if the variance of prediction for each of the two estimates is known or can be estimated accurately. 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 to be 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 from the EMA analyses at a streamgage is provided for each AEP estimate as part of the output in PeakFQ (table 13 from Musser and Feaster, 2023; U.S. Geological Survey, 2022). The variance from the streamgage analysis is related to the years of record, with long records tending to have a lower variance than short records. 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 streamflow 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. The average variance of prediction values for the regional regression equations used in this investigation are listed in table 3.

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

where

For the 63 streamgages in Georgia, South Carolina, and North Carolina that drained 75 percent or more from above the Fall Line (table 13 from Musser and Feaster, 2023), the weighted streamflow estimates were computed using equation 11 with the variance from the at-site EMA analysis along with one of the following:

Of those 63 streamgages, 39 were included in the regression analyses, and 24 were redundant and had basins that drained 75 percent or more from above the Fall Line. 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 at the streamgage, V_w(s), is computed using the following equation: where the remaining variables are as previously defined in equation 11.

Confidence intervals for the weighted estimate also can be computed (England and others, 2018). The upper and lower 95-percent confidence intervals (95%CI) on the weighted AEP estimate can be computed as

where the remaining variables are as previously defined in equations 11 and 12.

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, that has 10 or more years of peak-streamflow record. To obtain a weighted peak-streamflow estimate for P-percent AEP at the ungaged site, Q_w(u), the weighted streamflow estimates for an upstream or downstream streamgage, Q_w(s), must first be determined by using equation 11. The weighted AEP estimate for the ungaged site, Q_w(u), is then computed using the following equation:

where

Use of equation 14 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 gives 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. For streamgages with large drainage areas, hydrologic judgement should be applied as to whether shifting the weighted estimate to the full limits of the drainage area ratio (0.5 or 1.5) is appropriate. In addition, all other limitations noted in the report should be adhered to as well.