Purpose and Scope

The USGS, in cooperation with the Wyoming Water Development Office, continues efforts to update relevant data and develop streamflow models for the StreamStats application for the State of Wyoming. StreamStats is a web application that can be used to delineate drainage areas, determine basin characteristics, and compute streamflow statistics at gaged and ungaged locations (https://streamstats.usgs.gov/ss/; Ries and others, 2017, 2024; USGS, 2019a, 2022b). The purpose of this report is to describe the methods used in developing regression equations for estimating selected streamflow statistics using current (2025) basin characteristics (table 2; Hallberg and others, 2023b) for ungaged basins in Wyoming. The regression equations were developed using newly updated streamflow statistics from Armstrong and others (2025a, b). A total of 176 streamgages were used for developing equations for mean annual maximum, mean annual, mean seasonal, and mean monthly streamflow; and annual and monthly flow-duration streamflow statistics. A subset of only 145–156 streamgages was used for developing equations for n-day statistics, which include M7D2Y, M7D10Y, M14D5Y, M30D5Y, V60D2Y, and V1D5Y. The equations presented in this report are loaded to the Wyoming StreamStats application and can be used to derive the previously mentioned streamflow statistics for ungaged locations on streams in Wyoming.

Description of Study Area

The study area includes Wyoming, an area of about 98,000 square miles, and some basins that extend into Colorado, Idaho, Utah, Montana, Nebraska, North Dakota, and South Dakota (fig. 1). The study area contains parts of four large hydrologic regions: the Great Basin, Missouri, Pacific Northwest, and Upper Colorado regions (Hallberg and others, 2023a). Wyoming has diverse ecosystems and landscapes with elevations ranging from 3,125 to 13,785 feet. These ecosystems consist of forested mountains and glaciated peaks, semiarid shrub- and grass-covered hills and plains, high-elevation plateaus, and wetlands (Chapman and others, 2004).

The diversity of Wyoming’s ecosystems is captured by the U.S. Environmental Protection Agency (EPA) level III ecoregions (EPA, 2013; Omernik and Griffith, 2014), which define spatial patterns in geology, physiography, vegetation, climate, soils, land use, wildlife, and hydrology. In the study area, seven level III ecoregions define the diverse ecology of the state. Two ecoregions, Snake River Plain, and Wasatch and Uinta Mountains, cover less area in Wyoming and are not represented as well by streamgage basins from this study. The five ecoregions that contain most of this study’s basins are the Middle Rockies, Southern Rockies, Wyoming Basin, High Plains, and Northwestern Great Plains (fig. 1). The EPA Office of Research and Development, National Health and Environmental Effects Research Laboratory (EPA, 2013) descriptions of these ecoregions are in the following paragraphs.

The Middle Rockies ecoregion is composed of steep-crested, high mountains that are largely covered by coniferous forests. These forests differ in species from the Southern Rockies, having more Pinus contorta Douglas ex Loudon (lodgepole pines). The Black Hills part of the Middle Rockies is also distinct with a more montane climate than surrounding parts of South Dakota (Chapman and others, 2004). Numerous mountain-fed, perennial streams originate in this ecoregion from snowmelt and flow into surrounding ecoregions, and annual peak streamflows are typically during the spring.

The Southern Rockies ecoregion is characterized by steep mountains, intermontane valleys, and high-elevation plateaus. Vegetation varies across this region largely by elevation: from alpine wildflowers, subalpine coniferous forests, and shrubs and grass at the lowest elevations (Chapman and others, 2004). Similar to streamflow in the Middle Rockies, the causal mechanism for streamflow is predominantly snowmelt; however, mixed rain-on-snow runoff and rainfall are more common at lower elevations.

The Wyoming Basin ecoregion is primarily a broad arid intermontane basin covered by grasslands and shrublands and flanked by the Middle Rockies to the west and east (Chapman and others, 2004). Although similar in topography and vegetation, the Wyoming Basin is drier than the Northwestern Great Plains to the northeast. Many streams are supplied by runoff from the surrounding montane ecoregions.

The Northwestern Great Plains ecoregion is defined by semiarid rolling plains in the northeastern corner of Wyoming. Cattle grazing and agriculture are common, and many streams are affected by irrigation (Chapman and others, 2004). Coal-bed methane production is also prevalent, and production water from groundwater pumping during methane extraction has effects on discharge to streams, especially during low flow. Snowmelt is often earlier in the season compared to montane ecoregions, and rainfall from summer storms also contributes to streamflow.

The High Plains ecoregion consists of undulating plains in the southeastern corner of Wyoming. The Rocky Mountains to the west cause a rain shadow, which greatly limits the precipitation in this ecoregion. Along with drought-resistant shortgrass, agricultural crops are common in this region because much of the precipitation falls during the growing season (spring–summer) (Chapman and others, 2004). Many of the basins with streamgages in this region experience some effect from irrigation.

Mean annual precipitation in Wyoming for 1991–2020 ranged from less than 10 inches per year in parts of the Green River, Great Divide, Wind River, and Bighorn Basins largely defined by the Wyoming Basin ecoregion to more than 70 inches per year in mountainous areas (Middle Rockies ecoregion) in the northwestern part of the State (PRISM Climate Group, 2022). The large range of elevations and precipitation in the State results in diverse hydrology and unique mechanisms for high and low streamflows. Land uses in Wyoming include grassland pasture and range (72 percent), forest cover (12 percent), urban and special use areas (12 percent), and cropland (4 percent) (University of Wyoming, 2019). Because of the small percentage of urbanized area, streams in Wyoming are generally less affected by urbanization than streams in many other states. Physiographic regions and the orientation of basins are mainly driven by the location of mountain ranges and the Continental Divide. Much of the land cover consists of high-elevation sagebrush and grassland that receives less precipitation than the mountain ranges. Precipitation in Wyoming results from two regional climate patterns. Winter months consist of high-elevation snow accumulation and intermittent snow cover at lower elevations. Spring and early summer rainstorms and snowmelt create peak streamflow conditions on many streams and rivers during this period. Midsummer through fall is often hotter and drier than the spring and early summer, and isolated, large thunderstorms during this period can cause high-flow events on smaller streams (University of Wyoming, 2019).

Previous Studies

Previous studies in Wyoming developing streamflow regression equations focused on estimating peak-flow frequency from instantaneous streamflow records. Lowham (1988) developed regression equations for estimating peak streamflows and annual streamflow for streams in mountain, plains, and high desert physiographic regions. Miller (2003) updated peak-flow frequency regression equations for six hydrologic regions based on similar physiographic boundaries (Fenneman and Johnson, 1946) and patterns in model residuals. No previous studies in Wyoming have developed regression equations to estimate low-flow frequencies or flow-duration statistics from daily streamflow data. However, low-flow regression equations that have been developed in Montana (McCarthy and others, 2016) share basin boundaries with Wyoming. Thus, methods for estimating low-flow stream statistics in Montana were used to inform the workflow developed for this study.

Selection of Explanatory Variables

Best-subset selection was used to assess optimal explanatory variables because this technique is recommended by Helsel and others (2020). Best-subset selection requires that all possible combinations of candidate explanatory variables be tested to find the optimal regression model(s). Because this method is computationally intensive, only the most highly correlated variables were considered for each streamflow statistic group. Additionally, an arbitrary limit on the maximum number of variables, 1 per 20 streamgages, was followed to reduce the risk of model overfitting. Statsmodels version 0.14.4 (Seabold and Perktold, 2010), a Python package for statistical analysis, was used to generate weighted least squares (WLS) regression models for best-subset selection. To reduce multicollinearity between variables, only variable combinations with variance inflation factors less than 5 were considered (Helsel and others, 2020). Similarly, to avoid statistically insignificant models, combinations were excluded if any variable had a statistical probability level (p-value) greater than 0.05. The remaining combinations were ranked based on the adjusted coefficient of determination (R²) and mean squared error (MSE). The adjusted R², a metric of model fit, decreases when additional explanatory variables fail to reduce the residual sum of squares (Farmer and others, 2019). The MSE quantifies the average squared difference between predicted and observed values; a smaller MSE indicates a better model fit. After best-subset selection, final weighted left-censored regression models (described in the “Final Regression Equations” section) were developed to account for left-censored values (zero streamflow) and variation in streamgage record lengths. Because streamflow statistics estimated at streamgages with short records may have greater uncertainty (Farmer and others, 2019), streamgages were weighted by record length. Weighting was computed as the number of years of record at the streamgage divided by the mean record length across all streamgages used in the regression model (Eng and others, 2009; Lorenz, 2014).

Because of nonlinear relations between streamflow statistics and the explanatory variables, all variables were log transformed (base 10) before analysis. Basin characteristics that were computed as a percentage of the basin (percentage of forest, for example) could potentially have a value of 0; therefore, a value of 1 was added to these basin characteristics before log transformation. Transformations for respective variables are represented in the final models (app. 1, table 1.1).

A statewide (pooled) regression model for mean periodical streamflow statistics was first developed to assess candidate explanatory variables. Combining regions containing fewer streamgages can be beneficial if physiographic differences are not significant. For each streamflow statistic, WLS regression models were developed for the entire study area using basin characteristics selected by a best-subset procedure defined previously. Because spatial patterns in model residuals can indicate regional differences, residuals from pooled models were compared to those from fixed-effects models (Griffis and Stedinger, 2007), which use regional boundaries as additional explanatory variables. EPA level III ecoregions (EPA, 2013; Omernik and Griffith, 2014) and hydrologic regions from previous streamflow regression studies (Miller, 2003) were considered as explanatory variables in fixed-effects models. Ecoregions were designed to stratify the environment by the probable response to disturbance (Bryce and others, 1999), which makes them ideal candidates as variables in the fixed-effects models.

Residuals from pooled models were examined spatially to assess geographical patterns of bias. Heteroskedasticity, unequal variance across residuals, was minor with some regional differences in residuals (app. 1); however, many models demonstrated large residuals for streamgages in the High Plains ecoregion as compared to streamgages in other regions. Because of the lack of representation of hydrologic conditions in this region, streamgages within the High Plains ecoregion were ultimately excluded from the final regression models.

Regionalization was further assessed using fixed-effects models. Fixed-effects models were used to test for differences between model intercepts and slope coefficients for basins within a given ecoregion as compared to a common model. To test the difference between intercepts of each region, fixed-effects models using indicator variables, defined as the percentage of drainage area of the respective streamgage basin within a selected ecoregion, were assessed (Griffis and Stedinger, 2007). Using a percentage allows for a smooth transition for ungaged locations on streams that flow through multiple ecoregions (Feaster and others, 2023). Differences in slope can be assessed by including interaction terms in the fixed-effects model, which are the cross products of the pooled model explanatory variables and the percentage of the drainage areas in the respective ecoregions (Griffis and Stedinger, 2007). The Middle Rockies ecoregion was assigned as the base ecoregion, and thus all indicator variables and interaction terms corresponded to other ecoregions. For example, all indicator variables and interaction terms equal 0 when estimating streamflow for a basin entirely in the Middle Rockies ecoregion. Although physiographic differences may exist among ecoregions, these differences may be insignificant in the coefficients of the modeled slope or intercept, implying the base ecoregion is representative of all ecoregions evaluated. The statistical significance of explanatory variables was assessed at the 10-percent level (p-value greater than 0.1) for a chi-square distribution. A consistent slope between regions may indicate that, across ecoregion boundaries, the time of concentration for streamflow has a consistent linear relation to the explanatory variables. For example, precipitation may relate differently to peak streamflow statistics in basin or plains regions, where floods are often caused by localized rainstorms, compared to mountain regions, where floods typically result from rain on snow and snowmelt (Miller, 2003). However, for low streamflows, snowpack is a dominant factor statewide, and the model slope does not vary across regions, which is the case for the low-flow n-day regression models in this study. A difference in intercept alone may indicate differences in the volume of runoff in ecoregions because of differences in soil, land cover, storage, and slope (Griffis and Stedinger, 2007). In this study, interaction terms were often more statistically significant for evaluated models than indicator variables, indicating streamflow and model explanatory variables have different linear relations across ecoregions. The p-values for all explanatory variables used in the fixed-effects models are listed in table 1.1 to highlight the dominant patterns in the significance of regional explanatory variables and the potentially unnecessary (insignificant) regional variables for selected statistics.

Log-transformed drainage area and mean April 1 snow-water equivalent were ranked high in best-subset selection for mean periodical and flow-duration statistics groups. Because these groups contain a range of streamflow statistics calculated over various temporal periods, fixed-effects models were first assessed for individual flow statistics to identify regional patterns; flow statistics were then regrouped based on model similarity. For example, monthly flow-duration statistics generally indicated similar statistical significance for ecoregion interaction terms for most exceedance intervals, and thus, the same explanatory variables were used for the final models. Seasonal and monthly mean statistics demonstrated slightly different statistically significant ecoregion interaction terms and thus have unique explanatory variables. For example, regional patterns are similar for most streamflow statistics associated with early spring months (April–May), which often do not have statistically significant interaction terms from the Southern Rockies ecoregion. This similarity of regional patterns could be due to spring snowmelt contributing to runoff in both the Southern Rockies ecoregion and Middle Rockies base ecoregion. For many streamflow statistics during the summer months (July–September), the model interaction terms for snow-water equivalent for the Wyoming Basin ecoregion are generally not statistically significant, which may reflect streamflow response to the runoff from the Middle Rockies ecoregion that supplies and sustains many streams in the Wyoming Basin. Most annual flow-duration statistics have statistically significant interaction terms for all ecoregions, although for low-percentage durations, few terms were statistically significant, which may emphasize the significance of mountain snowpack in high streamflow statewide.

Low-flow and high-flow n-day statistics produced different best-subset results for pooled models; thus, regionalization was evaluated separately for each statistics group using fixed-effects models. For low-flow n-day statistics models, drainage area interaction terms were statistically significant across all ecoregions. In contrast, the log-transformed mean March precipitation (table 2) interaction term was generally not statistically significant in the Southern Rockies ecoregion and Wyoming Basin relative to the Middle Rockies base ecoregion, which may indicate that low streamflows in these regions respond similarly to late-winter precipitation.

For high-flow n-day statistics, the best-subset models incorporated additional physiographic variables beyond drainage area and precipitation. The explanatory variables in the pooled model included drainage area, relief, mean November temperature, and mean April 1 snow-water equivalent. In fixed-effects models, only the Northwestern Great Plains had statistically significant interaction terms for both weather variables; however, for V1D5Y, the interaction term for snow-water equivalent is not statistically significant, indicating regional differences are mostly captured by drainage area.

Final Regression Equations

Regression equations were developed for all streamflow statistics using weighted left-censored regression models (Helsel and others, 2020). Weights were based on the number of years of record available at each streamgage (Armstrong and others, 2025b). Because many streamgages in this study had values of 0 for some streamflow statistics, common regression techniques such as WLS or generalized least squares could create additional bias (Helsel and others, 2020). Thus, zero-flow values were treated as left-censored data using weighted left-censored regressions (Kroll and Stedinger, 1996; Kroll and Vogel, 2002).

To develop the final regression equations, weighted left-censored regression was implemented using the smwrQW R package developed by the USGS (Lorenz, 2014). When flow statistics do not have censored data, weighted left-censored regressions produce results equivalent to WLS (Helsel and others, 2020). Censored data in this study were defined as streamflows below a perceptible threshold of 0.1 cubic foot per second; therefore, all streamflow statistics with observed values less than 0.1 cubic foot per second were treated as left censored (Kroll and Stedinger, 1996; Kroll and Vogel, 2002). For statistics in which fewer than 20 percent of the streamgages are censored, weighted left-censored regression was considered appropriate (Helsel and others, 2020). When 20 percent or more of the streamgages had censored data, an alternative modeling approach, such as logistic regression, would be required. Such statistics were uncommon (table 1.1) and were removed from this study. The generalized least squares regression method (Tasker and Stedinger, 1989), which is often used for frequency statistics, such as n-day statistics, was not suitable because many statistics had censored data (Eng and others, 2009).

The mean standard error of estimates, in percent, for the regression equations is given in table 1.1. The mean standard error of estimates is a measure of the uncertainty of predictions made for all streamgages used in the model; however, the standard error of prediction should be calculated for individual estimates at the ungaged site of interest and subsequently used to determine the prediction intervals for the selected streamflow statistic. The following equation can be used to calculate the standard error of prediction for an ungaged site:

The SEP can be used to calculate a confidence interval for the same estimate using equation 2, and the prediction interval can be estimated using equation 3.

Limitations of Regression Equations

The regression equations were developed to estimate selected streamflow statistics for ungaged streams in Wyoming that are unaffected or minimally affected by streamflow alteration. Alteration indices defining the degree of alteration from dams or diversion are available for streamgage basins within Wyoming (Armstrong and other, 2025b) but do not comprehensively extend to basins partially contained in neighboring states. Also, the alteration threshold defined by these indices may not be applicable to all basins in Wyoming because water-use data may be incomplete. Streamflow statistics used to develop these regression equations were selected from streamgages (table 1) classified as unaltered for the period of record analyzed. Therefore, the final regression equations (table 1.1) can be used for predicting streamflows at similarly unaltered basins in Wyoming.

Multivariate models, such as these regression equations, may have unreliable results when estimating streamflow statistics for basins with explanatory variables outside the range of values used to develop the equations. The range of explanatory variables for respective regression equations is listed in table 1.1.

Streamgages with high leverage and influence were flagged in the final weighted left-censored regression models using thresholds defined in this section. Leverage is a function of the distance from a dependent variable value to the regression model mean. Leverage identifies streamgages with explanatory variables far from the mean of the joint distribution of all explanatory variables in the model. The leverage value for a prediction should not exceed the largest leverage from the modeled dataset (Helsel and others, 2020).

Streamgages with high influence can have a disproportionate effect on the estimated regression parameters in equations. Influence is measured by Cook’s D (Cook, 1977), an estimate of the change in model fit if a particular observation is omitted. Equations for determining high leverage and influence are described in Helsel and others (2020). High influence is defined as Cook’s D greater than F(p+1, n−p) at α=0.1, where F(p+1, n−p) is the value of the F-distribution at 10 percent for p+1 and n−p degrees of freedom. All streamgages also were checked for high leverage, defined as leverage greater than 3p/n, where p is the number of coefficients in the model and n is the number of observations. For most models, few streamgages demonstrated high leverage and influence; for the three streamgages that did (USGS streamgages Muscrat Creek near Shoshoni, Wyoming [06239000]; East Fork Nowater Creek near Colter, Wyo. [06267400]; and Fifteen Mile Creek near Worland, Wyo. [06268500]), records did not indicate significant alteration or other reasons to remove them from the final models.