Purpose and Scope

This report (1) provides estimates of flood discharges at AEPs of 50-, 20-, 10-, 4-, 2-, 1-, 0.5- and 0.2-percent (recurrence intervals of 2-, 5-, 10-, 25-, 50-, 100-, and 500-years, respectively) for streamgages in and adjacent to Vermont and (2) describes methods, including the use of equations developed from regression analyses, to estimate flood discharges at selected AEPs for rural, ungaged, unregulated Vermont streams. Additionally, this report (3) presents a method to estimate the standard error of prediction for each estimate made using the regression equations and (4) describes methods for transferring a flood-discharge estimate for a selected AEP at a streamgage to a site upstream or downstream on the basis of drainage area.

Description of Study Area

Vermont encompasses 9,250 square miles (mi²) of land area in the northeastern United States, nearly one-eighth of the total land area of New England. The State is about 155 miles long from north to south and ranges from being about 35 miles wide (east to west) at the southern end of the State to being nearly 90 miles wide at its northern end. Vermont is bordered on the northern half of its western boundary by Lake Champlain and is bordered on its eastern border by the Connecticut River. The state of Vermont has more than 7,000 miles of rivers and streams and more than 800 lakes and ponds (Vermont Fish and Wildlife Department, 2025).

Vermont is largely forested and has rolling hills and the more mountainous terrain of the Appalachian Mountains running north-south through much of the center of the State. Land-surface elevations range from approximately 100 feet at the shoreline of Lake Champlain along the northwestern border of Vermont to more than 3,000 feet at numerous peaks. The climate of the region is temperate and humid with four distinct seasons. Precipitation is well distributed throughout the year with averages ranging from about 38 to 45 inches (in.) per year; however, the mountainous character of the State results in a great amount of variability in these ranges of precipitation. High elevation regions can receive over 70 in. of precipitation annually. Annual snowfall also varies across the State. Normal snowfall amounts range from 35 to 60 in. annually, and high elevation locations in the State receive over 100 in. of snowfall annually. The annual mean temperature ranges from 43 to 46 degrees Fahrenheit (°F) throughout the State; however, elevation and aspect can affect these ranges. The average number of days per year the minimum temperature is less than or equal to 0 °F ranges from 10 to 60 days across the State (Office of the Vermont State Climatologist, 1996).

Peak-Discharge Data Used in this Study

All available annual peak-discharge data for Vermont and adjacent physiographically similar areas in the States of New Hampshire, Massachusetts, and New York and Quebec, Canada, were considered for this study (Olson, 2025). Streamgages within 30 miles of the Vermont border that recorded annual peak-discharge data were considered adjacent. These data include records from continuously recording streamgages and crest-stage streamgages (streamgages that record only the annual peak discharge), both current (2023) and discontinued. This report includes records through water year (WY) 2023. The WY is designated by the calendar year in which it ends. It begins October 1 of the previous calendar year and ends September 30.

Of the sites considered, 156 streamgages were selected for use in this study (fig. 1; app. 1). The selection criteria required the streamgage to have a minimum of 10 years of annual peak-discharge data that were unaffected by regulation or urbanization. Regulation was assumed to have a negligible effect on peak discharges if the usable storage in the basin was less than 4.5 million cubic feet per square mile of drainage area (Benson, 1962). Peak-discharge data from sites that had greater than 4.5 million cubic feet of usable storage per square mile of drainage area were not used. None of the streamgages used for this investigation have drainage basins considered to be urbanized. The average percentage of urbanized land for basins in this study is 5.5 percent (Commission for Environmental Cooperation, 2023). The streamgages selected for this study are spatially well distributed in and adjacent to Vermont (fig. 1).

Peak-Discharge Data Stationarity

In recent decades, stationarity (the assumption that the mean and variability of data from past observations will continue unchanged in the future) of annual peak-discharge data has been challenged (Milly and others, 2008) in light of climatic and land-use changes. To determine whether trends in the annual peak-discharge data exist, a two-sided Kendall Tau trend test (Helsel and Hirsch, 1992) was done. The trend test was done with software called PeakFQ developed by the USGS to analyze peak-flow data (Flynn and others, 2006). Streamgage records with less than 30 years of annual peak-discharge data were not tested for trends because of the uncertainty in defining a trend in a period of record of this length.

Some trends in the peak discharge data from this study were found using the Kendall Tau test. The Kendall Tau statistics indicated an upward trend in annual peak-discharge data (p-value less than or equal to 0.05) for 22 of the 104 streamgages with at least 30 years of record. For an additional two streamgages, the upward trend was considered marginal (p-value less than or equal to 0.1 and greater than 0.05).

For the streamgaging data that indicate a trend, the trend can often be explained as being the result of extreme climatic anomalies near the beginning or end of the peak-discharge record. For instance, for 12 of the 22 streamgages indicating an upward trend, the trend began during the drought of 1960–69 (Hammond, 1991). The evidence of trends did not exist or was statistically insignificant when extreme events near the beginning or end of the record were eliminated. Although some streamgage data indicate trends, the annual peak-discharge data used in this study are analyzed as random, independent events that are homogeneous for a streamgage throughout the period of record. Furthermore, a consensus on best practices to adjust for with trends has not yet emerged (England and others, 2018).

Determining the Magnitude and Annual Exceedance Probabilities of Flood Discharges for Streamgages

Flood discharges with AEPs of 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent (app. 1) were computed for the 156 streamgages using methods outlined in Guidelines for Determining Flood Flow Frequency, Bulletin 17C (England and others, 2018). The Bulletin 17C guidelines incorporate the Expected Moments Algorithm, apply a Multiple Grubbs-Beck test to identify low outliers, provide procedures that accommodate interval and censored data, and compute accurate confidence intervals that account for historical data and regional skew. A summary of the historical, interval, and censored discharge data that was input is provided in appendix 2. Software developed by the USGS to analyze peak-discharge data (PeakFQ version 7.5.1) was used for these computations (U.S. Geological Survey, 2024). The annual peak-flow data that was input to the PeakFQ program were retrieved from the National Water Information System (U.S. Geological Survey, undated). Peak discharges affected by dam failure, ice jam breach, or a similar event are not included in the frequency analyses. There is evidence of mixed populations of annual peak discharges—tropical and non-tropical storm events— at streamgages in Vermont and its vicinity; however, even at streamgages with many decades of annual peak-discharge data insufficient tropical storm events were available to do mixed population analyses. Annual peak discharges through WY 2023 were included in the frequency analysis (Olson, 2025).

Magnitude and Annual Exceedance Probabilities of Flood Discharges for Streamgages

The magnitudes of flood discharges at selected annual exceedance probabilities for streamgages used in this study (Olson, 2025) are listed in appendix 3. The variance of each streamgage’s estimated discharge magnitude is reported in appendix 4. The discharges reported in appendix 3 supersede discharges reported in Olson (2014) and Smith and others (2025) because of additional available data and the updated regression equations.

Maximum Recorded Floods

The maximum recorded annual-peak discharges (app. 1) plotted in relation to drainage area for each streamgage in this study are displayed on figure 2. Annual-peak discharges that are affected by dam failure, ice jam breach, or a similar event are not included. A New England regional envelope curve of potential maximum flood discharges developed by Crippen and Bue (1977) is also shown on figure 2 along with a line developed using generalized least-squares regression analysis showing the relation between drainage area and the peak discharge with a 1-percent annual exceedance probability. Figure 2 can be used to evaluate the reasonableness of flood estimates made using techniques described in this report.

Regression Analysis and Final Regression Equations

A total of 92 basin characteristics were determined for each streamgage and tested for potential use as explanatory variables in the regression equations (app. 5). Mathematical transformations of logarithms and squares were also applied to each basin characteristic and flood-discharge statistic to obtain the most linear relations. Correlation (R Core Team, 2024a) and ordinary least squares regression (Hebbali, 2024) data computed in R (R Core Team, 2024b)—a software environment for statistical computing—were used together to evaluate which basin characteristics, transformed or untransformed, were the most significant explanatory variables while not being correlated with another explanatory variable. This initial evaluation determined that drainage area was a significant explanatory variable when discharge and drainage area were logarithmically transformed (fig. 2). This evaluation also eliminated numerous basin characteristics that were correlated with drainage area.

Next, all combinations of the remaining explanatory variables, using up to four regressors, were evaluated using ordinary least squares (Hebbali, 2024). The evaluations were based upon the R-square, adjusted R-square, root-mean-square error, Mallow’s Cp, and the Akaike information criteria of each regression model. This evaluation left 20 potential explanatory variables to be analyzed for the final regression equations.

Finally, generalized least-squares regression techniques were used to analyze and determine the final significant basin characteristics and to compute the final regression equations (Olson, 2025). The generalized least-squares regression analysis was done using the Weighted-Multiple-Linear Regression program (WREG; Farmer, 2021), a hydrologic regression package that works within the R software. The WREG package uses the generalized least-squares regression procedures outlined in Farmer and others (2019). The three basin characteristics used to develop the final regression equations—drainage area, percentage of the basin that is wetlands, and mean annual precipitation—are listed in appendix 6, by streamgage. The final regression equations (eqs. 3–10) used to estimate flood discharges on ungaged, unregulated streams in rural drainage basins in Vermont are as follows:

The procedure for estimating the flood discharge at a selected recurrence interval for an ungaged stream site using the regression equations is described in appendix 7.

Attempts were made to group streamgages with similar geographic or drainage-basin characteristics into subregions to reduce the standard error of the regression equations. This grouping would have resulted in a set of regression equations for each subregion. To evaluate whether subregions should be generated, residuals, the difference between the flood discharges estimated from the frequency analysis and the flood discharges predicted from the regression equations, were determined for each streamgage and for each regression equation. These residuals were plotted spatially at the centroid of the drainage basin of the streamgage and in relation to drainage-basin characteristics. Residuals of the 10- and 1-percent annual exceedance probability regression equations in relation to drainage-basin characteristics are shown as examples in figures 3A and 3B. No apparent trends or patterns were observed in any of the plots. Thus, the streamgages were not grouped into subregions, and the equations in this report are intended for statewide use and not for subregions.

The residual plots shown in figures 3A and 3B were used as a diagnostic tool for the regression equations. The random scatter of the points above and below the zero-reference line verifies that the model is satisfactorily meeting the assumptions of multiple-linear-regression techniques. Other diagnostics are the evaluation leverage and influence metrics of each observation—the data used to develop the regression equations. Leverage is a measure of the distance an observation is from the centroid of all the other observations giving an indication of whether the observation is unusual compared to the other observations. An influence metric, such as Cook’s D (Helsel and Hirsch, 1992), is a measure of the sensitivity of regression parameters to a particular observation (Farmer and others, 2019). The combination of high leverage and influence metrics indicate an observation may be an outlier or require additional analysis. However, after evaluating an observations’ leverage and influence metrics, the data were determined to be accurate and there was no justification for excluding any observations from the regression analysis.

Another diagnostic tool used was the variance inflation factor (VIF) (Helsel and Hirsch, 1992). The VIF may be used to evaluate collinearity of explanatory variables. VIF does not have any formal criteria, although some authors suggest that a VIF exceeding 10 may be cause for concern (Freund and Littell, 2000) because explanatory variables may then be correlated. None of the VIF values were close to this level; the greatest VIF computed for a variable used in the final regression equations was 1.2.

Limitations and Sensitivity

Basin characteristics used to develop equations 3 through 9 were determined with ArcGIS Pro (Environmental Systems Research Institute, Inc., undated) software using datasets described in “Regression Analysis and Results” section of this report. This is important because determining the basin characteristics to use in the regression equations with alternate data sources or using different computational methods than those of a GIS software may produce statistics that are different from those reported in this report and may introduce bias and yield discharge estimates that have unknown error.

In addition, although wetland area and annual precipitation values were used in the equations to estimate flood discharges in Vermont in Olson (2014) and in this investigation, the source of these basin characteristics is not the same. Basin characteristics used in the equations published in 2014 cannot be used in these new set of equations, nor vice versa. For instance, the annual precipitation values used in the equations published in 2014—which were 30-year precipitation normals from 1981–2010—should not be applied to the equations developed for this study—which use the 30-year precipitation normals from 1991–2020. Use of basin characteristics from the wrong source will produce unreliable results with unknown error.

The regression equations are applicable to sites on ungaged, unregulated streams in rural basins that are within the region encompassed by the drainage basins used for this investigation. Use of these equations is appropriate for sites with drainage-basin characteristics that are within the range of drainage-basin characteristics used to develop the equations. In general, the uncertainty in the regression equation results will be greater when the explanatory variables used in the equations are further from their mean. The mean and ranges of drainage-basin characteristics used for the analysis are shown in table 2 and figure 4. If the independent variables used in the regression equations are outside of these ranges, the results of the equations are considered extrapolations, and the accuracy of the predictions is unknown. For sites that have drainage-basin characteristics outside the acceptable ranges, a simplified equation that uses drainage area as the only explanatory variable is used for this study and is provided in this report in the section titled “Drainage-Area-Only Regression Equations.” For sites that are considered urban, Moglen and Shivers (2006) describe techniques that transform rural flood-discharge frequency estimates to estimates for urban watersheds.

The sensitivity of each regression equation to changes in the magnitude of the independent variables was tested to evaluate the amount of error that can be introduced if basin characteristics are incorrectly computed. The sensitivity analysis was done by adjusting basin characteristic values by plus or minus 10 percent while holding the other basin characteristics constant at their respective mean magnitudes (Olson, 2025). The results of the sensitivity analysis listed in table 3 indicate that the regression equations are most sensitive to changes in drainage area and changes in the basin-wide mean of the average annual precipitation. However, a 10-percent change in the basin-wide mean of the average annual precipitation value accounts for a greater change in the computed flood discharge than does a 10-percent change in drainage area.

Accuracy of the Regression Equations

The accuracy of regression equations can be measured in several ways. The coefficient of determination, R², indicates the proportion of variability of the dependent variable and is explained by combining the explanatory variable in the regression model. The pseudo coefficient of determination, R²_pseudo, can be described in the same way but accounts for the effect of the time-sampling error (Farmer and others, 2019). The closer the R² or R²_pseudo is to 1, the better the regression explains the variation in the dependent variables. The R² and R²_pseudo for each regression equation (Olson, 2025) is in table 4.

One of the most common measures of accuracy is the root mean square error (table 4); it measures how much the regression results deviate from the observed data. The root mean square error represents the standard deviation of the residuals.

Another measure of accuracy is the average standard error of prediction (table 4). The average standard error of prediction has a model error component—the error resulting from the model—and a sampling error component—the error that results from model parameters that are developed from samples of the population. Thus, the average standard error of prediction is a measure of the expected accuracy of a regression model when applied at an ungaged location with basin characteristics like those used to develop the regression equation. This measure of accuracy is needed because regression equations are typically used for ungaged locations. About two-thirds of the estimates made using a regression equation for ungaged locations will have errors less than the average standard error of prediction for that equation.

Accuracy Analysis of Individual Estimates from the Regression Equations

Because the coefficient of determination, the pseudo coefficient of determination, the root mean square error, and the average standard error of prediction (table 4) are computed using available streamgage data, they are approximations of the overall accuracy of the regression equations for ungaged sites. Techniques for computing the accuracy of individual regression equation estimates for ungaged sites are available and are discussed in this section. Individual estimates are measured for accuracy by means of the standard error of prediction and prediction intervals.

Use of Regression Equations for Streamgages

An estimate of flood discharge at a selected annual exceedance probability made at a streamgage can be improved by combining regression equation results with the frequency curve computed from the streamgage record. The procedure recommended by Cohn and others (2012) is to compute a flood discharge using the regression equation estimate and the result of the frequency analysis of the streamgage record for a given annual exceedance probability weighted by the inverse of the variance of each of the discharge estimates. The procedure was applied to all the streamgages used in this study, and the weighted flood-discharge results are in appendix 3. Generally, the weighted flood discharge estimate provides better estimates of the true discharges than those determined from either the flood-frequency analysis or the regression analysis alone. The weighted discharges were computed using the following equation:

Prediction intervals for the weighted discharge (Q_w) can be computed with equations 15 and 16; however, the standard error of prediction for the weighted discharge (SE_w) is to be substituted for the standard error of prediction for the regression estimate (SE_pred). The standard error of prediction for the weighted discharge can be computed with the following formula:

Use of Regression Equations for Areas near Streamgages

Estimates of flood-discharge magnitude at selected annual exceedance probabilities for ungaged sites that are not at, but are near, a streamgage and are on the same unregulated stream can be improved by combined use of the regression equations and the nearby streamgage data. A method to adjust the weighted discharge for a streamgage (Q_w) developed from equation 17 to a site of interest upstream or downstream from the streamgage is given in this report. This method increases the weight of a discharge estimate derived from a regression-equation over the weight of a discharge estimate derived from streamgage data the farther upstream or downstream the site of interest is from the streamgage. This method is modified from a technique described in Ries (2007) that adjusts the weight linearly in logarithmic units, satisfying the logarithmic relation between flood discharge and drainage area. The logarithmic slope (c) of a line that goes through the weighted estimate of the discharge at the streamgage and converges on a location upstream or downstream where full weight will be given to the regression equations at a selected annual exceedance probability is computed as follows:

The value of a determines where, as a percentage of the gaged drainage area, full weight will be given to the regression equation. Hence, when the site of interest is upstream from the streamgage, a=0.5, the site is required to have a drainage area no smaller than 50 percent of the streamgage drainage area. When the site of interest is downstream from the gage, a=1.5, the site is required to have a drainage area no larger than 150 percent of the streamgage drainage area.

The final step is to compute the weighted flood-frequency estimate for the ungaged site, Q_u, using

An example that uses this technique is in appendix 7. As with any technique used to compute a weighted flood-discharge estimate, unexpected results could happen if there is a substantial difference between the discharges being weighted. If the difference in discharges being weighted is substantial, c could become negative, indicating discharge and drainage area are inversely related and making the procedure invalid

Drainage-Area-Only Regression Equations

For some ungaged sites, the percentage of the basin covered by wetlands or the basin-wide mean of the average annual precipitation may be outside the acceptable ranges required for the full regression equations (eqs. 3–10). The acceptable basin characteristic ranges are described in this report in the section titled “Limitations and Sensitivity.” In addition, some users of the equations may not have access to basin characteristics beyond drainage area. Because of this, a set of simplified regression equations (Olson, 2025) that incorporate drainage area as the only independent variable was developed. Generalized least-squares regression techniques were used to compute the coefficients in the equations. The simplified regression equations (eqs. 21–28) used to estimate flood discharges from ungaged, unregulated streams in rural drainage basins in Vermont are as follows:

The same 156 streamgages used to develop the previous regression equations were used to develop the simplified regression equations. The simplified regression equations are still applicable to sites with drainage areas ranging from 0.47 to 851 mi². Because the equations have only one explanatory variable, they are less accurate than the full regression equations in this report. The coefficient of determination, the pseudo coefficient of determination, the root mean square error, and the average standard error of prediction of the simplified equations are in table 6.

Although drainage-area-only regression equations are less accurate than the multivariable equations, these simplified equations are still valuable. The exponent in each of the drainage-area-only regression equations is the slope of the average linear logarithmic relation between drainage area and flood discharge for a selected AEP. Hence, the exponent can be used as an alternate method to adjust flood-frequency data from a streamgage to flood-frequency data from streamgages upstream and downstream. This use of the method is to be limited to streamgages within 50- to 150-percent of the streamgage drainage area (Wandle, 1983). To use this method, one would use equation 19 with the exponent from the simplified regression equation at a selected recurrence interval substituted for c.