Flow-Duration Statistics

Flow durations represent the percentage of time that a given flow is equaled or exceeded without regard to the sequence of recorded flows (Searcy, 1959). Typically, flow durations characterize the range of flow rates for the period over which data were collected. Flow durations were computed for complete water years for the entire period of record and for selected months and seasons for all 118 streamgages with 10 or more complete water years of record through water year 2022 (Ahearn and others, 2025). The streamflow data and flow statistics are based on the period of record for each streamgage, so starting and ending years vary.

Flow durations are computed by sorting the daily mean streamflows for the period of interest (such as the entire record or monthly) from largest to smallest and assigning each streamflow value a rank, starting with 1 for the largest value. The frequencies of exceedance are then computed by using the Weibull plotting-position formula (Weibull, 1939):

where For this study, flow durations were computed as follows: Flow durations for the bioperiods are computed by sorting the daily mean streamflows for the month or months (season) representing the bioperiod. For example, the Q99 in the salmonid spawning bioperiod is calculated by using all available daily mean flows for November in the period of record. Similarly, the Q99 for the overwinter bioperiod is calculated using all available daily mean flows for December, January, and February in the period of record.

Low-Flow Frequency Statistics

Low-flow frequencies typically are computed for streamgages by using annual series of selected low flows based on the lowest mean streamflow for a specified number of consecutive days (Riggs, 1972). Any combination of number of days of mean minimum flow and years of recurrence may be used to determine the low-flow frequencies. The annual series for the determination of low-flow frequencies for this study was based on a climate year. Use of a climate year rather than a water year allows for an analysis of an uninterrupted low-flow period; in Connecticut, this low-flow period typically occurs from early August through mid-October.

A given low-flow frequency statistic is the minimum consecutive D-day mean streamflow that is expected to occur once in any Y-year period, or that has a probability of 1/Y of not being exceeded in any given year. (D is the number of days, and Y is the number of years.) For this study, 7Q10 and 30Q2 low-flow frequency statistics were computed. The 7Q10 is the annual minimum mean streamflow for 7 consecutive days that has a probability of 0.10 (or 10 percent chance) of not being exceeded in a given year, and the 30Q2 is the annual minimum average streamflow for 30 consecutive days that has a probability of 0.5 (or 50 percent chance) of not being exceeded in a given year. The 7Q10 and 30Q2 are commonly used in regulating wastewater discharge to streams by many States including Connecticut and by the EPA. For the frequency analysis, the USGS Hydrologic Toolbox software was used to compute the annual minimum flows and plot the fitted log-Pearson type III probability distribution and the selected minimum flows versus recurrence intervals (Barlow and others, 2022).

In Connecticut, the 7Q10 and Q99 are often considered similar in magnitude and have been used interchangeably for water planning and permitting purposes under limited situations. In cases where streamgages do not have a long enough record to perform a frequency analysis (a minimum of 10 years is needed), the Q99 has been used as a surrogate for the 7Q10. A comparison of 7Q10 and Q99 for 40 index streamgages shows that the Q99 is slightly larger than the 7Q10 (fig. 2; table 3). In figure 2, data points plot slightly right of the one-to-one line (line of equality). For the 40 index streamgages in Connecticut and adjacent areas of nearby States, the percent difference between the Q99 and 7Q10 (table 3) ranged from a minimum of 1.7 percent (Fishkill Creek at Hopewell Junction, N.Y. [station 01372800]) to a maximum of 98.4 percent (Indian River near Clinton, Conn. [station 01195100]), with an average of 32.5 percent. The streamgages used in the calculation of percentage difference had an average record length of 45 years. In general, the magnitude of the percentage differences between two streamflow statistics (Q99 and 7Q10) are relative to the size of the drainage area. Smaller drainage areas (less than 30 mi²) typically had larger percent differences, with an average of 40.2 percent, than larger drainage areas (greater than or equal to 30 mi²), with an average of 16.2 percent.

Mean Flow, Spring Mean Flow, and Harmonic Mean Flow Statistics

Mean flow, spring mean flow, and harmonic mean flow were computed as the means of all available daily mean streamflow data for the period of record through water year 2022 (Ahearn and others, 2025) using the methods in this section.

Mean flow is computed as the arithmetic mean of all daily mean flows for the data series for a designated period. For example, the mean flow for a site with 20 years of data is computed as the arithmetic mean of 7,300 daily mean values (365 daily values per year). The CT DOT drainage manual refers to this flow statistic as the “average daily” flow (CT DOT, 2023).

Spring mean flow is computed similarly to mean flow as the arithmetic mean of daily mean streamflow for March and April for a designated period. For example, the mean flow for a site with 10 years of data is computed as the arithmetic mean of 610 values (61 daily values per year). The CT DOT drainage manual refers to this flow statistic as the “average spring” flow (CT DOT, 2023).

The harmonic mean flow is defined as the reciprocal of the arithmetic mean of the reciprocal daily streamflow values. Because traditional harmonic means cannot be calculated with zeroes in the dataset—including zero-flow data results in an undefined value—an adjusted harmonic mean can be calculated that considers the proportion of zero-flow days. In this adjustment, the harmonic mean is multiplied by the proportion of zero-flow days in the record. The equation is also used by the EPA in the DFLOW model (Rossman, 1990). To estimate concentrations of toxic pollutants contained in 2 liters of water per day, which is the recommended human health criterion, Rossman (1990) recommends using the harmonic mean flow when the daily variation in the flow rate is high (EPA, 2014). Harmonic means were computed using the USGS R code Hmean, which computes the harmonic mean according to the EPA DFLOW manual (Rossman, 1990), as follows:

where

Data and Methods

Subsets of streamgages with long records (more than 30 years) were created to evaluate trends in the annual 7-day low flow during the past 30, 50, 70, and 90 climate years from 2019. All 10-year blocks within each period analyzed were required to be at least 80 percent complete so that no part of the time series would have substantial missing data. These length and completeness criteria resulted in 39 streamgages for the 30-year period from 1990 to 2019 (table 4), 28 streamgages for the 50-year period from 1970 to 2019 (table 5), 19 streamgages for the 70-year period from 1950 to 2019 (table 6), and 10 streamgages for the 90-year period from 1930 to 2019 (table 7). The number of streamgages used in the analysis decreased as the years of analysis increased because of the length of available record. The streamgages used in the trend analysis include index streamgages and regulated streamgages. Trend analysis on regulated streamgages was used to determine if the degree of regulation was detectible in the period analyzed and to help the USGS categorize streamgages affected by anthropogenic influences.

The trends were computed with methods that consider the possibility of short- and long-term persistence in the temporal data. This is an important issue that is often ignored in trend studies. Trends over time are sensitive to assumptions of whether underlying hydroclimatic data are independent, have short-term persistence, or have long-term persistence (Cohn and Lins, 2005; Koutsoyiannis and Montanari, 2007; Hamed, 2008; Khaliq and others, 2009; Kumar and others, 2009). Short- and long-term persistence may represent the occurrence of wet or dry conditions that tend to cluster from year to year (Koutsoyiannis and Montanari, 2007; Hodgkins and others, 2017). Short-term can be a couple of years, and long-term can be decades and centuries. For further discussion and references on persistence, see Hodgkins and Dudley (2011). Because the long-term time-series structure of low-flow data is not well understood, temporal trend significance with three different null hypotheses of the serial structure of the data are reported: independence, short-term persistence, and long-term persistence (Hamed and Rao, 1998; Hamed, 2008). For the serial correlation structure of data referred to as “independence,” annual 7-day flow data from year to year are independent from each other (ignoring any short or long clusters of wet and dry years).

Trends were considered statistically significant at p-value less than or equal to (≤) 0.05; this magnitude represents a 5-percent probability that a trend is due to random chance. The magnitudes of the trends were computed with the Sen slope (also known as the Kendall-Theil robust line), the median of all possible pairwise slopes in each time series (Helsel and others, 2020). The Sen slope is multiplied by the number of years of annual 7-day low flow to obtain the magnitude of the trend or total change in the annual 7-day low flow over the period analyzed. For example, a Sen slope of 9.1 cubic feet per second (ft³/s) multiplied by 90 (for the 90-year period) results in a total change in the annual 7-day low flow of 819.64 ft³/s for the Connecticut River at Thompsonville (station 01184000) streamgage (table 7).

Trend Results in Annual 7-Day Low Flows

Results from the trend analysis for 30-, 50-, 70- and 90-year periods under the three serial correlation structures (independence, short-term persistence, and long-term persistence), magnitudes of Sen slopes, and p-values are shown in tables 4 through 7. For this study, the trend results of the annual 7-day low flow depend on the period of record analyzed and assumptions about the serial correlation structure. Decreasing trends were detected in all four periods (30, 50, 70, and 90 years) analyzed. In contrast, increasing and decreasing trends were detected in the 70- and 90-year periods. Increasing trends were found only at regulated streamgages. The trend analysis was performed using the entirety of the record. Using different parts of the streamflow record based on historical changes to streamflow in the trend analysis was outside the scope of this work. Three streamgages had trends in the three serial correlation structures: two decreasing (Pawcatuck River at Wood River Junction, R.I. [station 01117500; tables 5 and 7] and Naugatuck River at Beacon Falls, Conn. [station 01208500; table 5]), and one increasing (Connecticut River at Thompsonville, Conn. [station 01184000; table 7]).

Trend results at index streamgages (15 in the 30-year period, 11 in the 50-year period, 7 in the 70-year period, and 2 in the 90-year period) showed few statistically significant trends. Two nearby index streamgages in Rhode Island have statistically significant (decreasing) trends in two periods and in two of three of the serial correlation structures. None of the index streamgages in Connecticut have statistically significant trends in any of the periods analyzed. Trend results at regulated streamgages (24 in the 30-year period, 17 in the 50-year period, 12 in the 70-year period, and 8 in the 90-year period) showed some statistical evidence of trends in the annual 7-day low flow.

For the 30-year period (1990–2019), 39 sites (15 index streamgages and 24 streamgages with regulation) were analyzed for trends. Three of the streamgages (Quinnipiac River at Southington, Conn. [station 01195490]; Pomperaug River at Southbury, Conn. [station 01204000]; and Naugatuck River at Beacon Falls, Conn. [station 01208500]) showed a decreasing trend with short-term persistence (table 4), with a total change of 3.00, 7.29, and 35.43 ft³/s, respectively. There were no increasing trends in the 30-year period.

For the 50-year period (1970–2019), 28 sites (11 index streamgages and 17 streamgages with regulation) were analyzed for trends. Three streamgages showed a decreasing trend with independence and short-term persistence (Branch River at Forestville, R.I. [station 01111500]; Pawcatuck River at Wood River Junction, R.I. [station 01117500]; and Naugatuck River at Beacon Falls, Conn. [station 01208500]), with a total change of 9.66, 27.95, and 42.90 ft³/s, respectively (table 5). Two of these streamgages (stations 01111500 and 01117500) had decreasing trends in all three serial correlation structures. The third streamgage (station 01208500) showed decreasing trends in both the 30- and 50-year periods. There were no increasing trends in the 50-year period.

The 70- and 90-year periods included both increasing and decreasing trends (tables 6 and 7). For the 70-year period (1950–2019), 4 of the 19 streamgages analyzed showed trends with different serial correlation structures⸺three increasing (Connecticut River at Thompsonville, Conn. [station 01184000; independence, short-term persistence, and long-term persistence]; Quinnipiac River at Wallingford, Conn. [station 01196500; independence]; and Housatonic River at Stevenson, Conn. [station 01205500; independence]) and one decreasing (Pawcatuck River at Wood River Junction, R.I. [station 01117500; independence, short-term persistence, and long-term persistence]). The 01184000 streamgage (increasing trend of 1,160 ft³/s during 70 years) and 01117500 streamgage (decreasing trend of 17.74 ft³/s during 70 years) showed differing trends in the three serial correlation structures. When a trend was observed in all three serial correlation structures, there was a greater probability than not that the trend was not due to random chance. The 01196500 and 01205500 streamgages showed an increasing trend of 12.73 ft³/s and 144.83 ft³/s during 70 years, respectively.

For the 90-year period (1930–2019), 3 of the 10 streamgages analyzed had trends in two of three serial correlation structures (independence and short-term persistence). Two streamgages (Connecticut River at Thompsonville, Conn. [station 01184000; 819.64 ft³/s during 90 years], and Quinnipiac River at Wallingford, Conn. [station 01196500; 13.72 ft³/s during 90 years]) had an increasing trend, and one streamgage (Quinebaug River at Jewett City, Conn. [station 01127000]) had a decreasing trend (64.66 ft³/s during 90 years; table 7). All three of these streamgages are at regulated sites. No trends were detected at the two index streamgages or with long-term persistence in the 90-year period. Because of the lack of strong and consistent statistical evidence of long-term trends at index streamgages in Connecticut and the adjacent areas of neighboring States, the traditional assumption of stationarity was supported for this regional regression analysis.

Regression Analysis for Estimating Selected Flow Statistics at Ungaged Stream Sites

Logarithmic (base 10) transformations were made of the flow statistics (response variable) and basin characteristics (explanatory variables) to linearize the relation between the explanatory variables and the predictor variables, stabilize the variance by obtaining equal variance about the regression line, and improve the spread of the data. OLS regression analyses using Spotfire S+ version 8.1 statistical software (TIBCO Software, Inc., 2008) were used to determine the best combinations of basin characteristics to use as explanatory variables in the multiple linear regression equations. The all-possible subsets statistical method was used for selecting explanatory variables. In all-possible subsets, all the equations created from all possible combinations of explanatory variables were examined, and the coefficient of determination (R²) was used to check for the best combination of explanatory variables. The explanatory variables were selected based on their relation to flow and correlation to other basin characteristics using Pearson’s correlation coefficient (r).

If a moderate correlation (r less than 0.6) existed between two explanatory variables, then the two variables were evaluated individually in the variable selection process. To identify the best combination of explanatory variables in the all-possible subsets method, different equations from the regression analysis were compared based on the following measurements:

The equations with a smaller standard error of estimate, Mallow’s C_p, and predicted residual sum of squares statistic and a higher adjusted-R² were preferred. In addition, the explanatory variables were selected based on statistical significance at the 95-percent confidence level, an analysis of the residuals, and how the explanatory variables might affect flows. Explanatory variables that had a 95-percent probability of effectiveness (probably a good predictor of flow and not due to chance) were classified as significant. If an explanatory variable was significant but had only a small effect on the standard error (arbitrarily chosen as less than a 2-percent change), then it was left out of the equation.

WLS and GLS regressions were used for deriving the final coefficients (Helsel and others, 2020). In OLS regression, equal weight is given to all streamgages in the analysis regardless of record length. WLS regression can account for differences in the streamgage length of record and allows more emphasis to be placed on streamgages that are considered more robust due to a longer data record. Regression coefficients in the equations for estimating flow durations and mean flows were finalized using WLS regression methods. Regression coefficients in the equations for estimating the low-flow frequency statistics (7Q10 and 30Q2) were finalized using GLS regression methods, which compensate for differences in both the variability and reliability of and correlation among the low-flow frequency statistics at the streamgages included in the analysis. Stedinger and Tasker (1985) have shown that, where streamflow record lengths vary widely and flows (and, therefore, the flow statistics) at different streamgages are highly correlated, GLS regression provides more accurate estimates of the regression coefficients, better estimates of the accuracy of the regression coefficients, and almost unbiased estimates of the model error when compared with OLS regression. GLS regression gives more weight to long-term streamgages than short-term streamgages and more weight to the streamgages where flows are the least correlated to flows at other streamgages.

WLS and GLS regression analyses were performed using the USGS Weighted-Multiple-Linear Regression (WREG) software (version 3.0; Farmer and others, 2019), written in R version 3.6.2 (R Core Team, 2019). WREG was developed in 2009 (Eng and others, 2009) and was later updated by Farmer (2021). The output of WREG provides various measures of the reliability of the regression equations including the average variance of prediction (in log units), the standard error of prediction (in percent), the standard error of estimate (in percent), the pseudocoefficient of determination (pseudo-R²), the mean squared error (in log units), the RMSE (in percent), and leverage and influence of individual observations on the regression. Equations for calculating these metrics are available in Eng and others (2009).

An additional criterion in selecting explanatory variables for the final regression equations was to have no more than three variables (basin characteristics). This was done to minimize overfitting the regression equation and to avoid multicollinearity among variables, which makes it difficult to evaluate the relative importance of the individual explanatory variable in the regression equation.

Hydrologic Regions

In a regional regression study, dividing a large study area into smaller, more homogeneous regions can improve the accuracy of the regression equations. Historically, regression equations to estimate flow statistics in Connecticut have been developed as a set of single statewide equations. To potentially improve the predictive accuracy and precision of the regression models in Connecticut, streamgages were grouped into different regions and exploratory linear regression analysis was performed on subsets of streamgages.

The physiographic regions of southern New England (Denny, 1982) and boundaries of the EPA northeastern coastal zone and northeastern highland level III ecoregions (Omernik, 1995; EPA, 2022) were used to subdivide the streamgages and investigate smaller hydrologic regions. The physiographic regions and level III ecoregions are based on similarities in physiography, topography, geology, hydrology, vegetation, climate, soils, land use, and ecosystems. In addition, streamgages in eastern and western Connecticut, using the Connecticut River as the dividing line, were evaluated as separate hydrologic regions. Hydrologic unit code boundaries were followed wherever possible to avoid dividing basins into multiple regions. Error metrics (mean square error and RMSE) that are commonly used for evaluating and reporting the performance of a regression model were used in assessing the models based on the physiographic and ecoregions. Results from the regional analysis of smaller subregions did not indicate improvement in the predictive accuracy and precision of the regression models. The analysis indicated that assessing Connecticut as a statewide region, as used in previous regional low-flow regression analyses, is still appropriate.

Assessment of Regression Equations

Methods of assessing the accuracy of the regression equations (quality of model fit) included both visual and numerical measures. Checking the model assumptions for collinearity (known as the variance inflation factor [VIF]), normality, and heteroscedasticity indicated no problems with the models. Collinearity was evaluated by computing VIFs for each explanatory variable. VIF values express the ratio of the actual variance of the coefficient of the explanatory variable to its variance if it were independent of the explanatory variables (Cavalieri and others, 2000). VIF values greater than 5 indicate that an explanatory variable is so highly correlated to other explanatory variables that it is an unreliable explanatory variable and should not be included in the equations because the equations may provide erroneous results. None of the explanatory variables had a VIF value greater than 5.

The models were visually assessed for patterns using residual plots and through a spatial analysis of the residuals. The residuals were plotted at the centroid of their respective drainage basins to look for geographical biases. No apparent geographical biases with either large positive or negative residuals were found. The residual plots indicated overall good performance. The equations appeared to fit the data reasonably well and adequately described the relation between the predictor and explanatory variables. The model coefficients explained the response correctly. The p-values for the regression coefficients were found to be less than or equal to 0.05, indicating the probability that the regression coefficient was significant.

Numerical measures to assess the quality of the models include adjusted-R² and RMSE (table 10). The adjusted R² identifies the percentage of variance in the response variable (flow being estimated) that is explained by the explanatory variables (basin and climatic characteristics). The RMSE will be small if the predicted responses are very close to the true responses. Conversely, the RMSE will be large if the predicted and true responses differ substantially, at least for some of the observations. A value of zero would indicate a perfect fit to the data.

The RMSE of the 47 equations developed ranged from 7.9 to 121.9 percent, with an average of 27.9 percent (fig. 4). Regression equations to estimate flows in the interquartile range (25- to 75-percent exceedances) have much smaller RMSEs than the equations to estimate extreme low flows (Q99 or 7Q10). The RMSE for the Q25 was 10.5 percent for the period of record, with an average of 16.5 percent for the six bioperiods. The RSME for the Q75 was 24.2 percent for the period of record, with an average of 21.7 percent for the bioperiods. The RMSE for the Q50 was 15.1 percent for the period of record, and ranged from 10.5 to 30.2 percent, with an average of 17.6 percent for the bioperiods. In contrast, the RMSE for the Q99 was notably larger; 105.1 percent for the period of record and ranged from 20.0 to 121.9 percent, with an average of 50.0 percent for the bioperiods. The 7Q10 is an extreme low-flow statistic and has the largest RMSE (121.9 percent) of the set of regression equations.

The habitat-forming (March–April) bioperiod has the smallest RMSE, ranging from 8.2 to 20.0 percent for the Q25 to Q99. Typically, flows are substantially higher during the habitat-forming bioperiod than other bioperiods (months or seasons). In contrast, the rearing and growth (July–October) bioperiod, which has the lowest flow conditions of all bioperiods, has the largest RMSE, ranging from 24.3 to 121.9 percent for the Q25 to Q99. The adjusted coefficient of determination (adjusted-R²) of the 47 equations ranged from 73.4 to 99.5 percent, with an average of 95.1 percent, which indicates that overall about 95.1 percent of the observed variation can be explained by the model’s inputs.

Diagnostic checks on the regression equations included evaluating outliers and influential observations. The presence of outliers is a subtle form of nonnormality, and influential observations are data that substantially change the fit of the regression line. The influence of an individual observation on the regressions is measured with Cook’s D statistic (Helsel and others, 2020). Cook’s D statistic is a measure of the change in the parameter estimates when an observation is deleted from the regression analysis. No influential observations that appreciably altered the slope of the regression line were found with Cook’s D statistic.

Scatterplots of the predicted flow from the regression model and observed flow from the streamgage were used to visualize the performance of the regression models. Scatterplots of the select streamflow statistics are shown in figures 5 and 6. The x-axis represents the observed streamflow (flow statistic from the streamgage record), and the y-axis represents the predicted streamflow (flow statistic from the regression model). When the points are close to the one-to-one (1:1) line (line of equality), then the predicted data are close to the observed data. Ideally, if the predictions are perfect, then all the points will lie on the 1:1 line. Scatterplots of the higher flows—mean (fig. 5A) and spring mean (fig. 5B)—show a narrow spread between the observed and predicted values. Scatterplots of low and medium flows—harmonic mean (fig. 5C) and 30Q2 (fig. 5D)⸺show a moderate spread between the observed and predicted data. Scatterplots of extremely low flows⸺7Q10 (fig. 5E) and Q99 (fig. 5F)⸺show a large spread between the observed and predicted data. Scatterplots of the Q50 for five of the six bioperiods (fig. 6A, B, C, D, E) show a narrow spread between the observed and predicted data. The Q50 scatterplot for the remaining bioperiod, the period with the lowest flow—rearing and growth (fig. 6F)—shows a moderate spread.

A wide range between the observed and predicted data is generally found in regression equations for estimating extreme low flows, also indicated by the RMSE in this study; 121.9 percent for 7Q10 and 105.1 percent for Q99. Predicting extreme low flows using regression equations remains challenging, as can be seen from the adjusted-R² values (73.4 and 78.1 percent for the 7Q10 and Q99, respectively), suggesting that about 25 percent of variation in the streamflow is not explained by the explanatory variables (basin characteristics) in the equations. This may be improved with a more comprehensive look at the individual streamgages and effects of local geology and climatic characteristics.

In this study, bias correction factors were not used because they generally are very small in Connecticut. The transformation of the base-10 log-transformed regression equations to unlogged (original) units to calculate specific streamflow statistics at an ungaged site can introduce bias in the streamflow estimate. Bias correction factors were used in some studies in Massachusetts to remove the bias from the estimates from regression equations (Ries, 1994a, b; Ries and Friesz, 2000; Archfield and others, 2010); however, the bias correction factors were generally very small. Archfield and others (2010) shows that the bias correction factor resulted in a 0.3-percent increase in the 1-percent annual exceedance probability streamflow and 2.4-percent increase in the 99-percent annual exceedance probability streamflow for regression equations to estimate streamflow quantiles in Massachusetts. In a Rhode Island study (Bent and others, 2014), bias correction factors were not used because if they had been, then the streamflows estimated from the regression equations would not have an equal chance of being higher or lower than their actual values (Julie Kiang, USGS, oral commun., 2011). In studies by Risley (1994) in Massachusetts, Stuckey (2006) in Pennsylvania, Armstrong and others (2008) in Massachusetts; and Ahearn (2010) in Connecticut, bias correction factors were not used, likely because they were generally very small.

Final Regression Equations

Final regression equations are listed in table 10, along with the number of stations used in the regression analysis and several performance metrics. The explanatory variable names in the final equations are the StreamStats labels for the variables (USGS, 2024a). Nine basin characteristics—drainage area (DRNAREA); percentage of area with coarse-grained, stratified deposits (CRSDFT); stream density (total length of streams divided by the drainage area; STRDEN); mean basin slope (BSLDEM10M); mean basin elevation (ELEV); percentage of area with hydrologic soil group A (SSURGOA); mean monthly precipitation in November (NOVAVPRE10); mean precipitation in the winter (average of December, January, and February; PRCWINTER10); and mean annual temperature (TEMP)—are used as explanatory variables in the equations. The performance metrics used to report the quality of the final regression equations include the adjusted-R² (in percent) and the RMSE (in percent).

Drainage area (DRNAREA) is an explanatory variable in all the equations and is considered a primary cause of streamflow variation between sites. Streamflow could logically be expected to increase in proportion to the size of the drainage area. The second most common explanatory variable is the percentage of the area with coarse-grained, stratified deposits in the basin. In general, the physical processes controlling streamflow during late summer or early fall in Connecticut are related to geologic and soil characteristics. Studies by Wandle and Randall (1994) and Cervione and others (1982) found drainage basins underlain with a large percent of coarse-grained, stratified deposits have larger base flows than drainage basins underlain with glacial till and bedrock. The differences in streamflow between basins may be increased further by variations in the infiltration capacity and delayed subsurface runoff indicated by the percentage of soil type in the basin. A basin’s soil index (percentage of the area with hydrologic soil type A) represents the potential maximum infiltration and average moisture conditions, which affect streamflow. For this study, the percentage of area with coarse-grained, stratified deposits or hydrologic soil type A were important explanatory variables for the Q25 to Q99 and low-flow frequencies.

Three climatic characteristics—monthly mean precipitation for November, seasonal mean precipitation for December through February, and annual mean temperature—were statistically significant explanatory variables in the salmonid spawning, overwinter, and habitat forming bioperiods that span November through April. Streamflow commonly reflects the regional precipitation and temperature. The western and eastern uplands have cooler temperatures than the other areas in the State and typically receive precipitation in the form of snow in winter, whereas the lowlands more often experience rain events, which could explain the importance of these climatic variables in these three bioperiods. The climatic variables have positive coefficients, thereby causing an increase in the estimated flow value when the climatic variable increases.

For this study, three of nine explanatory variables have negative coefficients, causing a decrease in the estimated flow value. Stream density, which is the total length of streams divided by the drainage area, was a significant explanatory variable in the flow duration, low-flow frequency, and mean flow regression equations. A greater stream density in a basin compared to a basin with a smaller stream density allows base flow to be routed out of the basin earlier in the runoff recession (causing less base flow to be available during lower flows) through its larger network of flow paths intercepting the water table. The decreasing streamflow is represented by the negative coefficient (Bent and others, 2014). In several other studies, stream density had a negative coefficient in equations for estimating low flows in Pennsylvania (Stuckey, 2006) and for estimating the probability of streams flowing perennially in Massachusetts (Bent and Archfield, 2002).

Two other significant physiographic characteristics with negative coefficients are mean basin values for elevation and slope. Elevation was included in the overwinter bioperiod for Q25 to Q95 and slope was an important predictor for mean flow. Although elevation may not directly cause streamflow variations, it may serve as an index for other factors (such as temperature or vegetation) that cause streamflow variation. Generally higher elevations and steeper slopes are associated with different geologic characteristics and subsequently different base flows. The geomorphic and climatic indices in the regressions can be measured conveniently using GIS technology and generally are perceived to estimate low and mean streamflow with reasonable accuracy.

Prediction Intervals

Flow estimates obtained from regression equations have a related degree of uncertainty that can be described by prediction intervals. Prediction intervals indicate the probability that the true flow for a site is within the given bounds of flow. For example, the 90-percent prediction interval for a flow estimate at a site indicates that there is a 90-percent confidence that the true flow for the site is between the given flow values obtained from the regression equation. The lower and upper boundaries of the 90-percent prediction intervals can be computed by:

where where where

The values of t_(α/2,n−p) and U needed for equations 4 and 5 for the 47 regression equations are listed in table 11. The values of γ² (model error variance, in log units) needed in equation 5 can be calculated by squaring the value of the mean square error (log₁₀) in table 11. Example computations of prediction intervals can be found in Ahearn and Hodgkins (2020) and Bent and others (2014).

Limitations of Regression Equations

Use of the regression equations presented in this report in determining selected streamflow statistics is limited by the range of the basin characteristics data used to develop the equations and by the accuracy of the estimates. The regression equations developed in this study are not intended to be used at ungaged sites in which the basin characteristics are outside of the range of those used to create the regression equations. The ranges of the basin characteristics data used as explanatory variables to develop the regression equations are listed in table 12; the corresponding accuracies of the estimates calculated by these equations are listed in table 11. The use of these regression equations requires that the physical and climatic basin characteristics be determined using the same datasets (Ahearn and others, 2025) that were used to develop the equations described in this report.

The equations, which are based on data from streams with little to no flow alteration, will give useable estimates only for the natural flows for selected sites. They will not give estimates for sites where the flow is altered by structures and artificial processes, such as dams, surface-water withdrawals, groundwater withdrawals (pumping wells), diversions, and wastewater discharges. The equations are not applicable at sites where the groundwater-contributing areas and the surface-water drainage areas to stream sites are substantially different in size. In these areas, groundwater can flow from one surface-water drainage area into another. Therefore, in basins whose groundwater-contributing areas are larger than their surface-water drainage areas, the equations would likely underestimate streamflows. Conversely, for areas whose groundwater-contributing areas are smaller than their surface-water drainage areas, the equation would likely overestimate streamflows. Additionally, the equations are not applicable to streams with losing stream reaches. Losing streams are defined as streams or stream reaches that lose water to the groundwater system (Winter and others, 1998, p. 9–10 and 16–17). Generally, a stream reach is losing where the groundwater table does not intersect the streambed in the channel (the water table is below the streambed) during low-flow periods. Losing stream reaches commonly begin where a stream flows from an area of the basin underlain by till or bedrock onto an area underlain by stratified deposits (such as where hillsides meet river valleys). At such junctures, a stream can lose a substantial amount of water through its streambed.

The accuracy of the regression equations is a function of the quality of the data used to develop the equations. These data include the streamflow data used to estimate the statistics, information about possible unknown flow alterations to the stream above a site, and the measured basin characteristics. Basin characteristics used in the development of the regression equations are limited by the accuracy of the digital data layers available and used at the time of this study [2024].

StreamStats Application

The streamflow statistics for the 118 streamgages and 47 regression equations for estimating flow statistics developed are expected to be included as part of this study in the USGS National StreamStats web application (USGS, 2024a) that provides analytical tools for water-resources planning and management and for engineering and design purposes. For nearly a quarter century, the StreamStats web application has been a source of information used by Federal, State, and local governments, the private sector, and other organizations to make decisions about how streamflow may affect the safety and well-being of the public (Ries and others, 2024). StreamStats (USGS, 2024a; Ries and others, 2017) provides users with the ability to obtain streamflow statistics such as the 1-percent annual exceedance probability, a peak flow, and the 7Q10, a low flow, from USGS streamgage sites, as well as estimates of streamflow statistics from regression equations for user-selected ungaged sites. An interactive geoprocessing map-based interface allows user to click on the centerline for any stream site to calculate selected streamflow statistics and the associated 90-percent prediction intervals from the regression equations. The basin-characteristic values for a user-selected stream site used as input for the regression equations are determined from digital map data from ArcGIS (Esri, 2024). StreamStats outputs include a map of the drainage basin boundary for the stream site, the values of the GIS-measured basin characteristics, the estimated streamflow statistics, and prediction intervals for the estimates.