Purpose and Scope

The purpose of this report is to present an updated set of regional-regression equations for estimating annual exceedance-probability discharge (AEPD; also known as peak streamflow), mean, and low streamflows for hydrologic regions of central and western Colorado. As a result, updated hydrologic regions for peak streamflow and mean and low streamflow are designated for this report. For this report, mean streamflow corresponds to mean monthly and mean annual streamflow, whereas low streamflow corresponds to flow-duration values and 7-day minimum and maximum streamflow. The PMLSRREs relate peak, mean, and low streamflows to drainage basin size, topography, hydrology, and climatology.

This report presents four sets of peak-streamflow regional-regression equations to estimate 8 AEPD statistics. These statistics are 50 percent (Q_50%) with a 2-yr interval, 20 percent (Q_20%) with a 5-yr interval, 10 percent (Q_10%) with a 10-yr interval, 4 percent (Q_4%) with a 25-yr interval, 2 percent (Q_2%) with a 50-yr interval, 1 percent (Q_1%) with a 100-yr interval, 0.5 percent (Q_0.5%) with a 200-yr interval, and 0.2 percent (Q_0.2%) with a 500-yr interval (England and others, 2019).

This report presents mean-streamflow regional-regression equations to estimate mean annual streamflow (Q_ann) and mean monthly streamflow for ungaged streams. Hereafter, in this report, these monthly statistics are denoted for each month in a water year as Q_oct, Q_nov, Q_dec, Q_jan, Q_feb, Q_mar, Q_apr, Q_may, Q_jun, Q_jul, Q_aug, and Q_sep. This report also presents the streamflow-duration values for annual exceedance probabilities of 10 percent (Q_10th), 25 percent (Q_25th), 50 percent (Q_50th), 75 percent (Q_75th), and 90 percent (Q_90th).

This report presents four sets of low-streamflow regional-regression equations to estimate three different 7-day minimum streamflow statistics with probabilities of 50 ($Q_{50}^{min})$, 10 ($Q_{10}^{min})$, and 2$Q_2^{min})$ percent, which are equivalent to annual-recurrence intervals of 2, 10, and 50 years, respectively. Hereafter, these statistics are denoted as ₇$Q_{50}^{min}$, ₇$Q_{10}^{min}$, and ₇$Q_2^{min}$, respectively. This report also presents four sets of regional-regression equations to estimate three different 7-day maximum streamflow statistics with probabilities of 50, 10, and 2 percent, which are equivalent to annual-recurrence intervals of 2, 10, and 50 years, respectively. Hereafter, these statistics are denoted as ₇$Q_{50}^{max}$, ₇$Q_{10}^{max}$, and ₇$Q_2^{max}$, respectively.

The procedure to develop regional equations for estimating peak- and 7-day maximum streamflows included generalized least-squares (GLS) multilinear regression by using base-10 logarithmic transformations of streamflow and drainage area. In contrast, the procedure to develop the regional equations for estimating mean streamflows, 7-day minimum streamflows, and streamflow-duration included ordinary least-squares (OLS) multilinear regression by using base-10 logarithmic transformations of streamflow and drainage area.

Annual peak streamflow data from streamgages with a record of at least 10 years were compiled from the USGS NWIS database (USGS, 2020a) and the Colorado Division of Water Resources (CODWR; 2020) through water year 2019. Daily mean streamflow data from streamgages with a record of at least 10 years were compiled from the USGS NWIS database (USGS, 2020b) and the CODWR (CODWR, 2020) through water year 2019. A water year is the 12-month period from October 1 through September 30, designated by the calendar year in which it ends.

The limitations and accuracy of the PMLSRREs are presented in this report. The study area was extended 50 miles outside Colorado for PMLSRRE development, because hydrology is not affected by State boundaries. However, the PMLSRREs are only applicable in Colorado. Also, the PMLSRREs presented in this report are only applicable to natural streamflow with drainage areas between 0.22 and 22,600 square miles (mi²). To clarify, the PMLSRREs are based on analysis of peak-, mean-, and low-streamflow data for streams relatively unaffected by human activities such as storage, regulation, and diversion, by return streamflows from a municipality or mining operation, or by urban development in a basin. Kircher and others (1985) provide the description of natural streamflow as streamflow from drainage basins relatively unaffected by urban development or water-management activities such as substantial reservoir storage, streamflow diversions, or return streamflows of previously diverted streamflow. Further, those authors defined natural streamflow as streamflow having less than about 10 percent of the mean annual streamflow volume at the streamgage affected by human activity. The definition by Kircher and others (1985) was used in Capesius and Stephens (2009), Kohn and others (2016), and this report.

Description of the Study Area

Colorado has a diverse landscape and climate and includes the headwaters of the major river basins of the Colorado, Green, North Platte, San Juan, South Platte, and Arkansas Rivers, and the Rio Grande. The physiographic and hydrologic differences are discussed in the next two paragraphs.

Colorado can be described by three major physiographic provinces, which trend north to south across the State (Fenneman, 1931). The Great Plains, in the eastern 40 percent of the State, consists mostly of grasslands with scattered hills, bluffs, shallow river valleys, and some cultivated areas. The southern Rocky Mountains, west of the Great Plains, includes most of central Colorado from north to south and is characterized by mountain ranges and intermountain valleys. The Colorado Plateau is in western Colorado, between the Utah State line to the west and the Rocky Mountains to the east. The landscape is distinguished by mesas, plateaus, and eroded canyon terrain including much of the western quarter of Colorado from north to south. More detailed descriptions of the major physiographic provinces can be found in Fenneman (1931) and Capesius and Stephens (2009).

Hydrologic regions of Colorado (fig. 1) were defined based on the physiographic and climatic characteristics used to develop best-fit PMLSRREs for previous studies (McCain and Jarrett, 1976; Kircher and others, 1985; Vaill, 2000; Capesius and Stephens, 2009; Kohn and others, 2016). For this report, a hydrologic region is qualitatively defined as a region of similar hydrology and climatology. The study area includes the central and western parts of Colorado located between the Colorado-Wyoming State line and the Colorado-New Mexico State line. The study area includes areas in elevation from about 5,000 feet (ft) near the Colorado-Utah State line to more than 14,000 ft in the eastern parts of the study area on the east side of the Sangre de Cristo Mountains and Colorado Front Range. The study area encompasses the headwaters of most major river basins in Colorado, where the annual peak streamflow generally is produced by snowmelt runoff. Within Colorado, the study area is defined on the east side by the 7,500-ft contour from the Wyoming State line to the Chaffee-Fremont County line, then it follows the Chaffee-Fremont County line across the Arkansas River and transitions up to the 9,000-ft contour, which is followed south to the New Mexico State line (Kohn and others, 2016). The north, west, and south extents of the study area are defined by the Wyoming, Utah, and New Mexico State lines. The Foothills and Plains hydrologic regions in Colorado as defined by Kohn and others (2016) was outside the scope of this report. For PMLSRREs development, data were included for streamgages in the Mountain, Northwest, Rio Grande, and Southwest hydrologic regions in Colorado as defined in Capesius and Stephens (2009) as well as streamgages in adjacent states (Arizona, New Mexico, Utah, and Wyoming) within 50 miles of Colorado; however, the PMLSRREs are only applicable in Colorado.

Previous Studies and Background Information

Previous studies that computed peak-streamflow regional regression equations (PSRREs) in Colorado are Patterson (1964, 1965), Patterson and Somers (1966), Matthai (1968), Hedman and others (1972), McCain and Jarrett (1976), Kircher and others (1985), Livingston and Minges (1987), Vaill (2000), Capesius and Stephens (2009), and Kohn and others (2016). Fewer studies developed mean-streamflow regional regression equations (MSRREs) and low-streamflow regional-regression equations (LSRREs) as done by Kircher and others (1985) and Capesius and Stephens (2009). Hydrologic regions in Colorado were originally defined by McCain and Jarrett (1976) and were incorporated as the regional framework in Kircher and others (1985) and Capesius and Stephens (2009). Kircher and others (1985) developed MSRREs in central and western Colorado for data collected through 1983. Capesius and Stephens (2009) published PMLSRREs for the entire State of Colorado (except for the Plains hydrologic region, where only PSRREs were published) using USGS streamflow data from the beginning of the period of record at each streamgage through water years 2006 for peak streamflow and 2007 for mean and low streamflow. Kohn and others (2015) evaluated the predictive uncertainty of the MSRREs developed by Capesius and Stephens (2009). Kohn and others (2016) published PSRREs for the Foothills and Plains hydrologic regions in Colorado using USGS streamflow data from the beginning of the period of record at each streamgage through water year 2013 and used paleoflood data at 41 streamgages to improve the uncertainty of the PSRREs.

Streamgage Selection

The streamgages selection used for this report was based on streamgages selected by Kircher and others (1985), Vaill (2000), Capesius and Stephens (2009), Kohn and others (2015), Kohn and others (2016), and the authors’ professional knowledge of hydrologic systems in Colorado. A comprehensive list of all CODWR streamgages in the Mountain, Northwest, Rio Grande, and Southwest hydrologic regions, as defined by Capesius and Stephens (2009), was compiled from the CODWR peak database (CODWR, 2020). A comprehensive list of all USGS streamgages in the study area and within 50 miles of the Colorado State line adjacent to those same hydrologic regions was compiled from the USGS (2020b) NWIS database. From the comprehensive list of candidate streamgages, those streamgages with at least 10 years of combined streamflow record through water year 2019, identified as representative of natural streamflow, were selected for this study. Following Kircher and others (1985), natural streamflow was defined as streamflow having less than about 10 percent of the mean annual streamflow volume at the streamgage affected by human activity.

Streamgages adjacent to one another and located in the same stream network were evaluated for data independence using the drainage-area ratio (DAR) and the proximity of the basin centroids, which was measured by standardized distance. Standardized distance is a measure of the normalized, or unitless distance, between the centroids of two basins, and DAR is used to determine if the size of two basins, when one basin is contained in the other, is sufficiently different such that precipitation generating the annual-maximum floods in each basin is likely to be different (Veilleux, 2009). Additional information on DAR and basin centroid proximity is found in Asquith and others (2006) and Veilleux (2009). If the DAR was less than or equal to 5.0, and the standardized distance was less than or equal to 0.5, the streamgages were determined to be redundant (Veilleux, 2009; Gotvald and others, 2012; Eash and others, 2013; Southard and Veilleux, 2014; Kohn and others, 2016). In such cases, the streamgage with the longer record was selected, and the other was removed. Excluding redundant streamgages based on relative DAR and basin-centroid location ensures the independence of the streamflow information among streamgages. This exclusion process removed redundant data or hydrologic information from the analysis. At the completion of the streamgage selection process, 418 streamgages, consisting of 15,202 years of record and a mean of approximately 36 years of record per streamgage, were used to develop the PSRREs (Kohn and others, 2026).

Many streamgages only collect annual peak streamflow data and do not collect daily streamflow, whereas other streamgages are only operated seasonally, so daily streamflow is collected during only a part of the year. As a result, only 323 of the 418 streamgages could be used to develop the mean- and low-streamflow regional regression equations (MLSRREs). Of those 418 streamgages, 379 were operated by USGS; 13 were operated by CODWR; and 26 were operated by both USGS and CODWR during the period of record. A map showing the location of the streamgages is in figure 2; each of the 418 streamgages and ancillary information are listed in appendix 1 (table 1.1); and each of the 418 streamgages, ancillary information, and basin and climatic characteristics are available in Kohn and others (2026). USGS streamgage names and other identifying characteristics also can be accessed through the USGS NWIS database (USGS, 2020b).

Peak Streamflow Analysis

The series of annual peak-streamflow data at 418 continuous-record, seasonal, and crest-stage streamgages were used to estimate AEPDs, such as the 100-year flood. The AEPDs from streamgage data provide the basis for developing the current peak-streamflow regression equations and were computed by using the USGS software program PeakFQ version 7.4 (Veilleux and others, 2014) with inputs of systematic data from the beginning of the record for a given streamgage through water year 2019. The peak-streamflow equations in this report express flood-frequency estimates in terms of annual exceedance probabilities (AEP), which are the reciprocals of the recurrence intervals. AEP can also be represented in percent, and a particular flood-frequency estimate is then termed the “P-percent chance streamflow,” where P is the probability (in percent) that the streamflow will be equaled or exceeded in any year. For example, a 10-year flood is the same as having a 0.10 AEP; this streamflow also is described as a 10-percent flood or Q_10% (Southard and Veilleux, 2014).

For this report, the log-Pearson Type III frequency distribution was fit to the logarithms of the annual peak streamflows to determine flood frequency estimates by following the guidelines established by the Interagency Advisory Committee on Water Data (IACWD; 1982) and by England and others (2019). The mean and standard deviation of the annual peak streamflow, and streamgage skew coefficient at each streamgage were used to fit the distribution to describe the midpoint, slope, and curvature of the flood-frequency curve, respectively (Gotvald and others, 2012). Estimates of the P-percent AEPDs for each streamgage are computed by inserting the three statistics of the frequency distribution into equation 1:

The streamgage-skew coefficient is a measure of the asymmetry of the frequency distribution and is greatly affected by the presence of high or low outliers (annual peak streamflows substantially higher or lower than the trend of the data). Large positive streamgage skews typically are the result of high outliers, and large negative streamgage skews typically are the result of low outliers (Southard and Veilleux, 2014).

Mean Streamflow Analysis

Mean-annual streamflow values for the 323 selected streamgages were computed from daily mean streamflow data in the USGS NWIS database (USGS, 2020b) and in the CODWR database (CODWR, 2020) from the beginning of the record for a given streamgage through water year 2019. For each period of record representative of natural streamflow conditions, values of mean-annual streamflow were computed as the arithmetic mean of the daily mean streamflow in each water year. Streamgages without a minimum of 10 years of data were excluded from the computations of mean annual streamflow. Streamgages operated seasonally were also omitted from this computation not to bias the statistics. As a result, mean annual streamflows greater than zero were computed at 323 streamgages.

Similar to calculating mean-annual streamflow values, mean-monthly streamflow data from the 323 selected streamgages were computed from daily mean streamflow data in the USGS NWIS database (USGS, 2020b) and in the CODWR database (CODWR, 2020) through water year 2019. Mean monthly streamflows were computed as the arithmetic mean of each daily mean streamflow, in each month of the period of record representative of natural-streamflow conditions, by the USGS software program SWToolbox version 1.0.5 (Kiang and others, 2018). Streamgages without a minimum of 10 years of daily mean streamflow in a month were excluded from the mean-annual streamflow computations for that month. As a result, mean monthly streamflows were computed at streamgages ranging from a total of 325 for December to 347 for August with only three of those streamgages having mean monthly streamflows of zero for four to seven months of the year. The mean monthly streamflow data used and generated by SWToolbox for all streamgages are available in Kohn and others (2026).

Low Streamflow Analysis

Daily mean streamflow data at the 418 streamgages were compiled from the USGS NWIS database (USGS, 2020b) and the CODWR database (CODWR, 2020) from the beginning of the record for a given streamgage through water year 2019 to compute the 7-day minimum and maximum streamflows for return periods of 2-, 10, and 50-years (AEPs of 0.50, 0.10, and 0.02, respectively). The USGS software program SWToolbox version 1.05 (Kiang and others, 2018) was used to compute the 7-day minimum and maximum streamflows from the daily mean streamflow data. SWToolbox assumes the log-Pearson Type III distribution is suitable for low- and high-flow frequency analysis in Colorado. Daily mean streamflow values were computed as the arithmetic mean of the 15-minute streamflow recordings at a streamgage for each day in the entire period of record representative of natural-streamflow conditions. The SWToolbox computed the 7-day streamflow values by determining the product moments of mean, standard deviation, and streamgage skew of the logarithms of the streamflow values in a sliding window from April through March for minimum streamflows and from October through September for maximum streamflows. Finally, a log-Pearson Type III distribution was fit to the product moments, and the quantiles for the 2-year, 10-year, and 50-year recurrence intervals were extracted. Streamgages without 10 years of non-zero daily mean streamflow, in a month, were excluded from the 7-day minimum and maximum streamflow computations. Streamgages operated seasonally were omitted from this computation not to bias the statistics. As a result, the 7-day minimum and maximum streamflows were computed at streamgages ranging from 284 for the 7-day minimum streamflows to 319 for the 7-day maximum streamflows, with between 4 and 44 streamgages having 7-day minimum streamflows of zero, depending on the return period, and with 4 streamgages having 7-day maximum streamflows of zero (for all return periods). The 7-day minimum and maximum streamflow data used and generated by SWToolbox for all streamgages are available in Kohn and others (2026).

The streamflow-duration values were computed by ranking the daily mean streamflow data for the period of record representative of natural-streamflow conditions and linearly interpolating to the five exceedance percentiles of interest (10, 25, 50, 75, and 90) using an R script (R Core Team, 2022). As a result, the streamflow-duration values were computed for 323 of the 418 streamgages selected for the study. At those 323 streamgages, the streamflow-duration values were zero at 2–27 streamgages, depending on the exceedance probability, with two streamgages having streamflow-duration values of zero for all return periods. The streamflow-duration values used and generated by the R script (R Core Team, 2022) for all streamgages are available in Kohn and others (2026).

Basin and Climatic Characteristics

Based on previous studies conducted in Colorado and neighboring States (Kircher and others, 1985; Soenksen and others, 1999; Vaill, 2000; Rasmussen and Perry, 2000; Miller, 2003; Waltemeyer, 2008; Capesius and Stephens, 2009; Lewis, 2010; and Kohn and others, 2016) and on the availability of data, 97 characteristics (55 basin and 42 climatic characteristics) were evaluated as candidate explanatory variables in the regression analysis (table 1.2). The 97 characteristics consist of physical properties of the basin, precipitation amount, snowpack data, temperature, types of land cover, surficial lithology, and soil characteristics.

Basin and climatic characteristics were determined for the PSRRE analysis for each of the 418 streamgage drainage basins using Esri ArcMap 10.6.1 techniques, tools, and algorithms (Esri, 2017). The 1/3 arc-second 3-dimensional elevation data (3DEP) were processed through the ArcMap Spatial Analyst tool, as well as through the StreamStats online application (USGS, 2017, 202250) to delineate basins for the 418 streamgages found in this report. Additionally, 3DEP data (USGS, 2017) were analyzed to produce the characteristics of outlet elevation; basin elevation minimum and maximum values; mean, minimum, and maximum slope values; and the percentage of basin at an elevation above 7,500 ft, as listed in table 1.2. The basin perimeter, drainage area, and centroid latitude and longitude characteristic calculations were done on basin and pour-point geometry using the ArcMap Geometry Calculator tool. The percentage of basin with slopes greater than 30 percent characteristic was computed using StreamStats (USGS, 2022). The mean precipitation, precipitation amount, frequency, mean air temperature, soil, surficial lithology, snow data, and land cover data were processed with basin outlines using the National Water Quality Assessment Area-Characterization Toolset (Price and others, 2010) to produce the elevation-related characteristics for each basin presented in table 1.2, which also includes the data source.

Definition of Peak-, Mean-, and Low-Streamflow Regions

McCain and Jarrett (1976) originally defined five unique hydrologic regions in Colorado based on physiographic and climatic characteristics and peak streamflow. These same five hydrologic regions were used for all such Colorado streamflow studies with minor modifications (Kircher and others, 1985; Vaill, 2000; Capesius and Stephens, 2009) until Kohn and others (2016) subdivided the Plains hydrologic region into two unique hydrologic regions: the Plains and the Foothills hydrologic regions, which resulted in a total of six unique hydrologic regions in Colorado. The scope of this report was to update the PMLSRREs in central and western Colorado, which is defined as the Mountain, Northwest, Rio Grande, and Southwest hydrologic regions in Capesius and Stephens (2009). The study area was extended 50 miles into adjacent States to include more streamgages and improve statistical robustness for development of PMLSRREs.

After analyzing the study area for potential regional subdivisions based on river basin, elevation, latitude and longitude, and previous studies (Kircher and others, 1985; Vaill, 2000; Capesius and Stephens, 2009), the study area was divided into four updated hydrologic regions based on mean basin elevation in feet above North American Vertical Datum of 1988 (less than 8,014 feet, between 8,015 and 9,492 feet, between 9,493 and 10,490 feet, and greater than 10,490 feet) for the PSRREs and based on river basin (Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores) for the MLSRREs, which resulted in the smallest standard model error (SME, in percent) and standard error of prediction (SEP, in percent) and largest pseudocoefficient of determination (pseudoR², dimensionless).

The updated hydrologic regions for the PSRREs will be designated as (1) Alpine (mean basin elevation greater than 10,490 feet), (2) Sub-Alpine (mean basin elevation between 9,493 and 10,490 feet), (3) Mid-Elevation (mean basin elevation between 8,015 and 9,492 feet), and (4) Plateau (mean basin elevation less than 8,014 feet) hydrologic regions (fig. 3). Figure 3 shows the streamgages used in the development of the PSRREs color coded by hydrologic region; because the hydrologic regions are based on mean basin elevation, geographic region boundaries cannot be drawn. The distribution of the streamgages by color in figure 3 provides spatial context. Each hydrologic region included 104–105 streamgages, and figure 3 shows the hydrologic region boundaries previously defined by Capesius and Stephens (2009) and Kohn and others (2016). The new hydrologic regions for the MLSRREs will be designated as (1) Colorado-East Slope Headwaters, (2) Green River, (3) Rio Grande, and (4) San Juan-Dolores (fig. 4).

Figure 4 includes each streamgage used in the development of the MLSRREs color coded by hydrologic region; the updated hydrologic regions based on major river basin are also included. Figure 4 provides spatial context of the study area by color coding each streamgage: 170 in the Colorado-East Slope Headwaters hydrologic region, 57 in the Green River hydrologic region, 42 in the Rio Grande hydrologic region, and 54 San Juan-Dolores hydrologic region. The eastern edge of the study area was established by Vaill (2000), and this report follows that boundary as described in the “Description of the Study Area” section of this report. The hydrologic regions outside the study area in eastern Colorado remain unchanged and designated as the Foothills and Plains hydrologic regions as defined by Kohn and others (2016).

The subdivision of central and western Colorado into four hydrologic regions, for both PSRREs and MLSRREs, was determined through the OLS regression procedures of maximizing the coefficient of determination (R², dimensionless) and the adjusted coefficient of determination (adjR², dimensionless) and by minimizing the standard error (Mallow’s C_p) and the predicted residual sum of squares (PRESS) statistic. By using GLS regression procedures, it was confirmed that the updated regions reduced uncertainty in the PSRREs compared to the unsubdivided regions used in previous studies (Kircher and others, 1985; Vaill, 2000; Capesius and Stephens, 2009). Prior to finalizing the updated regions, maps of streamgage residuals were created and various other subdivisions were analyzed based on latitude, longitude, basin outlet elevation, mean basin elevation, maximum basin elevation, and drainage area.

Development of the Peak-Streamflow Regional-Regression Equations

The PSRREs were developed for use in estimating peak streamflow at selected AEPs, from 50 to 0.2 percent, at gaged and ungaged locations for basins in the Alpine, Sub-Alpine, Mid-Elevation, and Plateau hydrologic regions of central and western Colorado. Regression analyses using OLS were performed to select the basin and climatic characteristics for use as independent variables. A linear relation between the dependent and independent variables is needed for OLS regression. To satisfy this criterion, variables often are transformed, and in hydrologic analyses, typically the log-transformation is used (Southard and Veilleux, 2014; Farmer and others, 2019). The dependent response variable is the AEPDs, and the independent explanatory variables are the basin and climatic characteristics that describe the AEPDs. For this study, the dependent variable AEPDs were transformed to base 10 logarithms, but the drainage area was the only independent variable transformed to base 10 logarithms to increase linearity prior to the development of the PSRREs.

Homoscedasticity, a constant variance in the dependent variable for the range of the independent variables about the regression line, and normality of residuals are also criteria for OLS regression. Transformation of the AEPDs and certain other variables to base 10 logarithms can enhance the homoscedasticity of the data about the regression line (Southard and Veilleux, 2014). Linearity, homoscedasticity, and normality of residuals were examined in residual plots.

The hydrologic model used in the regression analysis in this report is of the form (eq. 1):

If the dependent variable Q_P and the independent variable A are the only logarithmically transformed variables, then the hydrologic model has the following linear form (eq. 4):

The basin and climatic characteristics (table 1.2) with the strongest correlation to peak streamflow were identified, checked for any substantial cross correlation with other variables in the group using the variance inflation factor (VIF) statistic, and were selected as the potential explanatory variables for the PSRREs. The final AEPD values for the 418 streamgages used in the regional-regression analysis are available in Kohn and others (2026).

Generalized-Least Squares Regression

Multilinear regression using GLS, as described by Stedinger and Tasker (1985), Tasker and Stedinger (1989), Griffis and Stedinger (2007), and Farmer and others (2019), is a method of weighting streamgage AEPD data in the regression analysis according to differences in streamflow reliability (record lengths) and variability (record variance) and according to spatial cross correlations of concurrent streamflow among streamgages. Comparison of OLS, weighted-least squares (WLS), and GLS in a study by Stedinger and Tasker (1985) indicated that the weighted methods (WLS and GLS) produced results more accurate than the OLS regression method. When streamflow records of varying lengths or with correlated concurrent streamflows occurred in the dataset, the weighting technique used in GLS produces equations that are both improved in estimates of streamflow statistics and of the predictive accuracy of the statistics (Stedinger and Tasker, 1985).

Based on the exploratory results from OLS regression and explanatory variable multicollinearity analysis, the USGS R package WREG (Eng and others, 2009; Farmer, 2021) was used to compute GLS PSRREs from the most correlated candidate variables. Thirteen basin and climatic characteristics were used as potential explanatory variables to develop the PSRREs with Q_1% AEPD streamflows based on OLS exploration. The general form of the equation was defined by equation 4. Final GLS regression models for PSRREs were selected based on minimizing values of the SME and the SEP and maximizing values of the pseudoR² derived from OLS exploration.

In Capesius and Stephens (2009) and Kohn and others (2016), the error associated with the PMLSRREs was characterized using the SEP, pseudoR²SME, and adjR². The SEP describes the sum of the model error and sampling error. The SEP is the square root of the mean GLS variance of prediction (Tasker and Stedinger, 1989; Eng and others, 2009). The pseudoR² value is a measure of the percentage of the variation explained by the basin or climatic characteristics (explanatory variables) included in the model and is calculated based on the degrees of freedom in the regression (Eash and others, 2013). The adjR² is a measure of the percentage of the variation explained by the basin or climatic characteristics (explanatory variables) included in the model, but it also compensates for multiple explanatory variables such that adding additional explanatory variables increases the value of adjR² only when the predictive capability of the model increases (Lee and others, 2012). Griffis and Stedinger (2007) describe how the pseudoR² is more appropriate than the R² or adjR² in measuring the true variation explained by the explanatory variables in the GLS model. The SME measures only the error of the model and does not include sampling error regression (Eash and others, 2013). The SME is the square root of the regression model error variance (Tasker and Stedinger, 1989).

Models using GLS use a correlation model and weighting matrix. The correlation smoothing function relates the correlation between annual peak streamflows at two streamgages to the geographic distance between the streamgages for every paired combination of the streamgages with a given number of years of concurrent streamflow. The correlation smoothing function was defined in equation 18 in Eng and others (2009), with an alpha of 0.001 and theta of 0.99. The correlation smoothing function was used by WREG to compute a weighting matrix for the four hydrologic regions of streamgages based on the mean elevation included in the development of the GLS PSRREs. Figure 5 displays the predicted 100-year peak streamflow using the regression equation compared to the streamgaged (observed) peak streamflow at each station in the four hydrologic regions. Figure 5 indicates a reasonable scatter distribution around the line of equality across the range of streamflows, particularly for the Alpine and Sub-Alpine hydrologic regions. A separate model was selected for each of the four hydrologic regions. Within each hydrologic region, the same explanatory variables were applied for each of the eight PSRREs in that region for consistency amongst different exceedance probabilities. All explanatory variables were statistically significant at a p-value of less than or equal to (≤) 0.05, and the VIF for explanatory variables was less than 3.

Development of the Mean-Streamflow Regional-Regression Equations

The MSRREs were developed for use in estimating the mean-annual and the 12 mean-monthly streamflows at gaged and ungaged locations for basins in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions of central and western Colorado. Regression techniques using OLS were performed to select the basin and climatic characteristics for use as independent variables following the same method as used for the PSRREs. The dependent response variables are the mean annual and mean monthly streamflows, and the independent explanatory variables are the basin and climatic characteristics. Logarithmic transformation of the values for mean-annual and mean-monthly streamflows and certain other variables can enhance the homoscedasticity of the data about the regression line (Southard and Veilleux, 2014). Therefore, to enhance homoscedasticity and to increase linearity prior to the development of the equations, values for mean-annual and mean-monthly streamflows were transformed to base 10 logarithms and the drainage areas (as the only dependent variable) were transformed to base 10 logarithms. Linearity, homoscedasticity, and normality of residuals were examined in residual plots.

The basin and climatic characteristics (table 1.2) with the strongest correlation to mean streamflow were identified, checked for any substantial cross correlation with other variables in the group, and were selected as the potential explanatory variables for the MSRREs. The final mean annual and the 12 mean monthly streamflows for the 323 streamgages used in the regional-regression analysis are available in Kohn and others (2026).

Ordinary Least-Squares Regression

The methods of Farmer and others (2019) were used to develop the MSRREs. The MSRREs were developed using OLS in R (R Core Team, 2022) with the allReg function in the smwrStats R package (Lorenz, 2015). Because the mean annual and mean monthly streamflows were computed without using the log-Pearson Type III frequency distribution, GLS regression cannot be used for the MSRREs. The potential explanatory variables were assessed for linear correlation with the mean annual and the 12 mean monthly streamflows using plots, VIF statistic, and the statistical significance (p-values) of each explanatory variable when regressed with peak streamflow to determine the strongest predictors of mean annual and the 12 mean monthly streamflows.

The most correlated OLS regression models (with one-, two-, three-, four- and five-explanatory variables using the best of all subsets routine) for prediction of the mean annual and the 12 mean monthly streamflows were identified. Following the PSRREs methods, multicollinearity (high correlation among the explanatory variables) can make results, which are based on a multiple linear-regression model, misleading or erroneous. This generally disqualifies the use of both variables within a single final MSRRE. Early in the variable selection process, the VIF statistic and plots comparing explanatory variables were used to make preliminary assessments of potential multicollinearity. Final OLS regression models for mean annual and the 12 mean-monthly streamflows were selected based on minimizing values of SEP and maximizing values of adjR².

Figure 6 shows the predicted mean annual streamflow using the OLS regression equation compared to the streamgaged (observed) mean annual streamflow at each station in the four hydrologic regions. Figure 6 indicates a reasonable scatter distribution around the line of equality across the range of streamflows. Separate models were fit for each mean annual and mean monthly streamflow statistics for each of the hydrologic regions. All explanatory variables were statistically significant at a p-value of ≤0.05, and the VIF for explanatory variables was less than 3.

Development of the Low-Streamflow Regional-Regression Equations

The LSRREs were developed for use in estimating the 7-day minimum and maximum streamflows for return periods of 2-, 10-, and 50-years (AEP of 0.50, 0.10, and 0.02) and streamflow-duration values for five exceedance percentiles of interest (10, 25, 50, 75, and 90) at gaged and ungaged locations for basins in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions of central and western Colorado. Regression techniques using OLS were performed to select the basin and climatic characteristics for use as independent variables following the same method as used for the PSRREs. The dependent response variable is the 7-day minimum and maximum streamflows and streamflow-duration values, and the independent explanatory variables are the basin and climatic characteristics that describe the 7-day minimum and maximum streamflows and streamflow-duration values. For this study, the dependent variable 7-day minimum and maximum streamflows and streamflow-duration values were transformed to base 10 logarithms, but the drainage area was the only independent variable that was transformed to base 10 logarithms to increase linearity prior to the development of the LSRREs. For the 7-day minimum streamflows, all zero streamflow values were substituted with the smallest streamflow in each statistic (0.01 ft³/s for ₇$Q_{2\%}^{min}$, 0.001 ft³/s for ₇$Q_{10\%}^{min}$, and 0.005 ft³/s for ₇$Q_{50\%}^{min}$) prior to model fitting so the dependent variable could be logarithmically transformed. In addition, for the 7-day minimum streamflows, a censored regression technique using the lowest non-zero value as the censoring level was analyzed. This technique yielded models and coefficients similar to the OLS, so it was determined that an OLS would be the preferred approach for the 7-day minimum streamflows. For the streamflow-duration values, all zero streamflow values were substituted with a value of 0.005 ft³/s prior to model fitting so the dependent variable could be logarithmically transformed. For the 7-day maximum streamflows, no zero streamflows were present, and no substitutions were made.

Homoscedasticity about the regression line and normality of residuals also is a criterion for OLS regression. Transformation of the 7-day minimum and maximum streamflows and streamflow-duration values and certain other variables to base 10 logarithms can enhance the homoscedasticity of the data about the regression line (Southard and Veilleux, 2014). Linearity, homoscedasticity, and normality of residuals were examined in residual plots.

The basin and climatic characteristics (table 1.2) with the strongest correlation to peak streamflow were identified, checked for any substantial cross correlation with other variables in the group, and were selected as the potential explanatory variables for the LSRREs. The final 7-day minimum and maximum streamflows and streamflow-duration values for the 323 streamgages used in the regional-regression analysis are available in Kohn and others (2026).

Ordinary Least-Squares Regression

The methods of Farmer and others (2019) were used to develop the LSRREs. A previous study (Hortness, 2006) determined that using GLS regression techniques for 7-day minimum streamflows is not preferred over OLS or WLS because of the extreme cross correlations present between streamgages. As a result, GLS regression was only used for the 7-day maximum streamflows. The WLS regression was analyzed for the 7-day minimum streamflows and streamflow-duration values using weights of record length and the variance of the logarithms of the observed values. After reviewing the uncertainty metrics for each model, it was determined that WLS regression did not improve the uncertainty of the models, and so the OLS regression methods were used for the 7-day minimum streamflows and streamflow-duration values.

The potential explanatory variables were assessed for linear correlation with the 7-day minimum streamflows and streamflow-duration values using plots, VIF statistic, and the statistical significance of each explanatory variable when regressed with peak streamflow (p-values), to determine the strongest predictors of 7-day minimum streamflows and streamflow-duration values. Final OLS regression models for 7-day minimum streamflows and streamflow-duration values were selected based on minimizing values of SEP and maximizing values of adjR².

Figures 7 and 8 display the predicted 7-day minimum streamflows and streamflow-duration values, respectively, using the OLS regression equation compared to the streamgaged (observed) 7-day minimum streamflows and streamflow-duration values at each station in the four hydrologic regions. Figures 7 and 8 indicate a reasonable scatter distribution around the line of equality across the range of streamflows with larger uncertainty present in all equations for lower streamflows. Figures 7 and 8 display the substituted zero streamflow values as a vertical line above the 1:1 line, indicating the equations overpredict the statistic when streamflows are very low or zero, which is more common in the San Juan-Dolores and Green River hydrologic regions. The plots are in log units, and this phenomenon occurs when almost all the predicted streamflows would be less than 1 ft³/s, so caution may be beneficial when applying these equations when streamflows are very low (≤ 1 ft³/s). Separate equations were fit for each 7-day minimum streamflow and streamflow-duration value statistic for each hydrologic region. For the 7-day minimum streamflow, within each hydrologic region, the same explanatory variables based on the ₇$Q_{10\%}^{min}$ statistic were applied for each of the three flow-duration statistics (₇$Q_{2\%}^{min}$, ₇$Q_{10\%}^{min}$, ₇$Q_{50\%}^{min}$) in that region for consistency amongst different exceedance probabilities. For the streamflow-duration values, within each hydrologic region, the same explanatory variables based on the Q_50th statistic were applied for each of the five flow-duration statistics (Q_10th, Q_25th, Q_50th, Q_75th, Q_90th) in that region for consistency amongst different exceedance probabilities. All explanatory variables were statistically significant at a p-value of ≤0.05, and the VIF for explanatory variables was less than 3.

Generalized-Least Squares Regression

For the 7-day maximum streamflows, regression techniques using GLS were performed to select the basin and climatic characteristics for use as independent variables following the same method as used for the PSRREs. Thirteen basin and climatic characteristics were used as potential explanatory variables to develop the 7-day maximum streamflows, and separate equations were created for each dependent variable in the four hydrologic regions. For the 7-day maximum streamflow, within each hydrologic region, the same explanatory variables based on the ₇$Q_{10\%}^{min}$ statistic were applied for each of the three flow-duration statistics (₇$Q_{2\%}^{min}$, ₇$Q_{10\%}^{min}$, ₇$Q_{50\%}^{min}$) in that region to maintain consistency between different exceedance probabilities.

The correlation smoothing function was defined in equation 18 in Eng and others (2009) with an alpha of 0.001 and theta of 0.99. The correlation smoothing function was used by WREG to compute a weighting matrix for the four hydrologic regions of streamgages based on the mean elevation included in the development of the GLS PSRREs. Final GLS regression models for 7-day maximum streamflows were selected based on minimizing values of SME and SEP and maximizing values of pseudoR².

Figure 9 displays the predicted 7-day maximum streamflow using the regression equation compared to the streamgaged (observed) 7-day maximum streamflow at each station for the four hydrologic regions. Figure 9 indicates a reasonable scatter distribution around the line of equality across the range of streamflows, particularly for the Rio Grande and San Juan-Dolores hydrologic regions. All explanatory variables were statistically significant at a p-value of ≤0.05, and the VIF for explanatory variables was less than 3.

Final Peak-, Mean-, and Low-Streamflow Regional-Regression Equations

The PSRREs in the Alpine, Sub-Alpine, Mid-Elevation, and Plateau hydrologic regions were developed using a total of 105, 104, 104, and 105 streamgages, respectively, and no streamgages were used in more than one region. The selection of the final basin and climatic characteristics and the evaluation of the accuracy of the PSRREs were based on the Q_1% AEPD for each hydrologic region. Maintaining consistency between explanatory variables minimizes the possibility of predictive inconsistencies between estimates of different probabilities, so that streamflow estimates will increase as the streamflow probability decreases. The final peak-streamflow regional-regression equations from GLS regression with pseudoR², SEP, SME, and the model coefficients are listed for each hydrologic region in table 1. The performance metrics pseudoR² and SME describe how well the PSRREs perform on the streamgages used in the regression analyses, and the SEP measures how well the GLS regression models can predict AEPDs at ungaged sites (Eash and others, 2013).

The MSRREs in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions were developed using a total of 170, 57, 42, and 54 streamgages, respectively, and no streamgages were used in more than one region. The final mean annual and the 12 mean monthly streamflow regional-regression equations from OLS regression with adjR², SEP, and the model coefficients are listed for each hydrologic region in table 2.

The 7-day minimum streamflow regional-regression equations in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions were developed using a total of 156, 44, 40, and 44 streamgages, respectively, and no streamgages were used in more than one region. The $Q_{10\%}^{min}$selection of the final basin and climatic characteristics and the evaluation of the accuracy of the 7-day minimum streamflow equations were based on the ₇$Q_{10\%}^{min}$ statistic for each hydrologic region. Maintaining consistency between explanatory variables minimizes the possibility of predictive inconsistencies between estimates of different probabilities, so predicted streamflow will increase as the streamflow probability decreases. The final 7-day 2-, 10-, and 50-year minimum streamflow regional-regression equations from OLS regression with adjR², SEP, and the model coefficients are listed for each hydrologic region in table 3.

The streamflow duration values and regional-regression equations in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions were developed using a total of 170, 57, 42, and 54 streamgages, respectively, and no streamgages were used in more than one region. The selection of the final basin and climatic characteristic and the evaluation of the accuracy of the streamflow-duration equations were based on the Q_50th statistic for each hydrologic region. The final streamflow regional-regression equations for streamflow-duration values for annual exceedance probabilities of 10, 25, 50, 75, and 90 percent from OLS regression with adjR², SEP, and the model coefficients are listed for each hydrologic region in table 4.

The 7-day maximum streamflow regional-regression equations in the Colorado-East Slope Headwaters, Green River, Rio Grande, and San Juan-Dolores hydrologic regions were developed using a total of 167, 52, 35, and 52 streamgages, respectively, and no streamgages were used in more than one region. The $Q_{10\%}^{max}$selection of the final basin and climatic characteristics and the evaluation of the accuracy of the 7-day maximum streamflow equations were based on the ₇$Q_{10\%}^{max}$ statistic for each hydrologic region. The final 7-day 2-, 10-, and 50-year maximum streamflow regional-regression equations from GLS regression with pseudoR², SEP, SME, and the model coefficients are listed for each hydrologic region in table 5. The final regression models for each equation and hydrologic region are available in Kohn and others (2026).