Purpose and Scope

This report documents surrogate regression-model development at six springs in southern Gooding County, south-central Idaho, using linear regression models. The six springs in upstream-to-downstream order are: (1) Niagara Springs at Diversion Number 2 near Buhl (U.S. Geological Survey [USGS] site 13093692), (2) Briggs Springs near Buhl (USGS site 13095175), (3) Banbury Springs near Buhl (USGS site 424120114491901), (4) Box Canyon Springs near Wendell (USGS site 13095500), (5) Box Canyon Springs below aqueduct diversion near Wendell (USGS site 4242271144904), and (6) Hatchery Springs near Hagerman (USGS site 424547114513101) (fig. 1; U.S. Geological Survey, 2021). The springs are referred to with abbreviated site names throughout the report, as defined in table 1. The statistical regression models relate specific conductance to nitrate concentrations from 2018 to 2022 in spring water to determine if specific conductance can be used as a surrogate for nitrate concentration changes, providing the USFWS with less expensive method to determine nitrate concentrations compared to measuring nitrate concentrations directly.

Specific conductance, streamflow, and nitrate concentrations fluctuate throughout the year in the springs, typically increasing from summer to late autumn and then decreasing until the following summer. The day of the year is evaluated as an explanatory variable in the surrogate regression models to account for the fluctuation of specific conductance and nitrate concentrations throughout the year. Two springs, Briggs Spring and Upper Box Canyon Spring, have existing USGS streamgages near the continuous water-quality measurement location, and hence streamflow also is evaluated as an explanatory variable in the surrogate regression models. Specific-conductance measurements at the six springs started during May 2018–November 2019, depending on the spring, and continued through December 2022. The discrete water-quality data used to build the surrogate regression models were measured monthly. Two springs, Briggs Spring and Niagara Springs, also have continuous specific-conductance, streamflow, and nitrate concentration data (monitored at 15- or 20-minute intervals) that were used to build the surrogate models for at least part of the period (Skinner, 2023).

Description of Study Area

The six springs are in canyons along the north side of the Snake River in southern Gooding County. The spring-monitoring locations are upstream from the endangered BSL and threatened BRS populations where possible. The springs’ water source is discharge from the ESRP aquifer. Groundwater in the ESRP aquifer moves from the northeast to the southwest (Rupert, 1997), discharging to the springs. The ESRP aquifer is a regional basalt aquifer that is composed primarily of a series of vesicular and fractured olivine basalt flows (Quaternary age) of the Snake River Group (Whitehead, 1992). These basalt flows average from 20 to 25 feet (ft) thick with a regional basalt aquifer estimated maximum thickness of about 6,300 ft (Twining and Bartholomay, 2011; Rupert and others, 2014). The top of the basalt generally is less than 100 ft below land surface. The layered basalt flows in the ESRP aquifer yield exceptionally large volumes of water to wells and springs.

Regional groundwater in the ESRP aquifer originates as good-quality, mountain-front recharge along the north margin of the ESRP or as recharge through undeveloped rangeland (Skinner and Rupert, 2012). Most of the land near the Snake River and overlying the downgradient parts of the ESRP aquifer is agricultural. Recharge through these agricultural lands acquires large amounts of nutrients from fertilizers and cattle manure, ultimately mixing with the regional groundwater before discharging to springs (Skinner and Rupert, 2012).

Continuous Water-Quality Monitoring

The six spring sites were instrumented to measure specific conductance at 15-minute intervals and two of the springs, Briggs and Niagara Springs, also had continuous nitrate sensors installed measuring nitrate concentrations every 20 minutes. Briggs and Upper Box Canyon Springs had the continuous water-quality sensors installed next to existing streamgages. Sondes at these two springs also continuously measured water temperature, dissolved oxygen, and pH at 15-minute intervals. The other four springs without streamgages had measurements of water depth concurrently with water temperature. Continuous sensor deployment, calibration, maintenance, and data reporting followed the methods described in Pellerin and others (2013) and Wagner and others (2006). These guidelines and procedures include site selection, routine sensor calibration, data corrections for drift and fouling, and evaluating data representation (that is, cross-section data corrections). Springs have very low sediment content, a major source of potential water-quality sensor fouling; hence, minimal fouling occurred. The continuous nitrate sensors measure nitrate plus nitrite and, therefore, are reported as nitrate plus nitrite as nitrogen concentrations (Pellerin and others, 2013). However, nitrite typically is not detected in spring water at these six sites so the nitrate plus nitrite as nitrogen concentrations represent nitrate concentrations, and hereinafter are referred to as just nitrate.

Discrete Water-Quality Data Collection

Water-quality sampling occurred monthly at the six spring sites and followed the protocols in the USGS National Field Manual for the Collection of Water-Quality Data (U.S. Geological Survey, [variously dated]). All discrete water-quality samples were analyzed by the USGS National Water Quality Laboratory (Denver, Colorado). Water-quality samples were collected as grab samples at the continuous measurement locations to optimize comparability. During each spring visit, field parameters were measured (water temperature, specific conductance, dissolved oxygen, and pH) and a nutrient sample was collected. The nutrient sample included analyses for ammonia (NH₃ + NH₄⁺) as nitrogen, nitrate plus nitrite as nitrogen, nitrite as nitrogen, and orthophosphate as phosphorus. Nitrate plus nitrite as nitrogen concentrations were used in the surrogate regression models and as verification data for the continuous nitrate concentration measurements. Nitrite concentrations are less than laboratory detection levels of 0.0010 milligrams per liter (mg/L) in 237 of the 273 (87 percent) samples from all six springs. The nitrate samples have an average nitrite concentration of 0.0024 mg/L and a maximum nitrite concentration of 0.0061 mg/L in the 36 samples with a nitrite detection. Considering the low nitrite concentrations (most are less than the detection limit), the nitrate plus nitrite concentrations effectively represent just nitrate concentration.

Quality Assurance and Quality Control

Quality assurance (QA) and quality control (QC) sequential replicate nutrient samples were collected to identify, quantify, and document variability in the nitrate concentration data resulting from collecting, processing, handling, and analyzing samples (Mueller and others, 2015). The QA/QC samples were collected following protocols described in the USGS National Field Manual for the Collection of Water-Quality Data (U.S. Geological Survey, [variously dated]). Five sequential replicate samples were collected at each of the six springs for a total of 30 nitrate concentration replicate samples. The relative percent difference (RPD) of the nitrate concentration replicates ranged from 0.01 to 4.43 percent, with a mean 1.11 percent RPD.

Continuous water-quality measurement followed the methods described in Pellerin and others (2013) and Wagner and others (2006), which prescribe criteria for water-quality meter calibration and data corrections. Continuous water-quality data corrections only are done when the cause(s) of data error(s) can be validated or explained as a data collection or monitor error such as fouling, calibration drift, and other errors explained by field personnel, field notes, comparison with other water-quality parameters at the spring, or information from nearby sites. For example, sondes at a couple of spring sites were slow to adjust to ambient water conditions after being pulled from the spring for calibration and maintenance; this recovery period was deleted. Data corrections are needed when a data error exceeds certain criteria. For continuously measured specific conductance, errors greater than the criteria of plus or minus (±) 5 microsiemens per centimeter or ± 3 percent of the measured value, whichever is greater, must be corrected (Wagner and others, 2006). However, the criteria can be reduced based on study needs, and because of the small range of continuously measured water-quality parameters at the six springs, data corrections were applied at a smaller criterion such as specific conductance errors being corrected at 2 percent of the measured value or less.

Nitrate concentrations from monthly grab samples and laboratory analysis as well as from continuous field measurements were included in the same surrogate regression models for Briggs Spring and Niagara Springs. Both measurement methods of nitrate concentration produce the same type of results (nitrate + nitrite as nitrogen), allowing laboratory nitrate concentrations to be used as bias corrections, such as biological fouling to the continuously measured nitrate concentrations as noted by Pellerin and others (2013). In the presence of both nitrate-measurement methods, the corrected continuous nitrate concentrations were used in the surrogate regression-model development. The two methods of nitrate concentration measurement were compared for the 52 coincident measurements at both springs, with an average difference of 0.01 mg/L, a minimum difference of 0.00 mg/L, and a maximum difference of 0.21 mg/L (fig. 2).

Another correction typically applied to continuous water-quality measurements is a cross-section correction based on the variability of a water-quality parameter horizontally and vertically across the spring (Pellerin and others, 2013). Because the specific-conductance and nitrate-concentration measurements and nitrate samples are all collected from the same point location, cross-sectional variability does not affect data quality as related to the surrogate regression models. However, the incorporation of streamflow in two surrogate regression models (Briggs and Upper Box Canyon Springs) requires cross-sectional variability to be assessed and corrected if needed because streamflow is measured using the full cross-sectional area of a stream and not at a single location like the water-quality parameters. Five separate cross-sectional profiles of specific conductance were measured at Briggs Spring, and RPDs between these profiles and the corresponding point measurements ranged from 0.22 to 3.83 percent, with anveragee RPD of 1.36. These low RPD values reflect a good relation between specific-conductance point measurements and cross-sectional streamflow. Upper Box Canyon Springs cannot have cross sections measured because of extreme safety concerns; however, the water quality and streamgage location is just downstream from a waterfall, which likely causes good mixing resulting in a uniform distribution of water quality.

Regression Model Development

The surrogate regression models were developed using the Surrogate Analysis and Index Developer tool (SAID) (Domanski and others, 2015). The SAID tool relates surrogate explanatory variables to a response variable using ordinary least squares linear regression models. SAID provides several overall model summary statistics and observational diagnostic statistics using multiple plots, tabular data views, and output reports. To use ordinary least squares linear regression to develop surrogate models, the explanatory variables need to be matched with the response variables by time. SAID performs the time matching between the variables, allowing for a user-specified maximum time offset. A 5-minute maximum matching time difference was allowed to match the explanatory and response variables.

Ordinary least squares linear regression models require several assumptions, depending on the purpose of the regression (Helsel and others, 2020). Linear regression assumptions for the surrogate regression models are as follows:

SAID provides summary statistics, such as the probability plot correlation coefficient, and related plots to identify linearity amongst the surrogate and response variables and residual variance and normality, as well as data transformation tools to correct for these issues. Autocorrelation, or serial correlation, can occur with continuous data if data close in time are strongly correlated with each other. To avoid autocorrelation, the continuous datasets (including specific conductance, nitrate concentrations, and streamflow collected at Briggs and Niagara Springs) were reduced to single monthly means from the 15- and 20-minute measured data.

All surrogate model variables at the six springs have a periodic increase and decrease of specific conductance, streamflow, and nitrate concentrations throughout the year. The large fluctuations of these variables each year are related to upgradient agriculture (Skinner and Rupert, 2012). This is exemplified well at Briggs Spring because it has the longest measurement duration. Specific conductance and nitrate concentrations are highest from October to early November and lowest from April to July depending on the spring. Streamflow at Briggs and Upper Box Canyon Springs also peaks from October to early November and is lowest from April to June of each year (fig. 3). To account for the periodic nature of the variables, the day of the year was evaluated as an explanatory variable, using both the periodic functions sine and cosine of the day of the year.

The surrogate regression models were evaluated based on overall model-fit statistics, the coefficient of determination (R² and adjusted R² if more than one explanatory variable was used) and the F-statistic compared to constant model and corresponding probability value (p-value). The explanatory variables’ individual p-values also were evaluated to confirm the significance of each explanatory variable within the surrogate regression model and the Prediction Error Sum of Squares (PRESS) statistic was used to compare alternate regression models at the same spring using different explanatory variables (Helsel and others, 2020). The surrogate regression model with the lowest PRESS statistic produces the least error when using the surrogate regression model to make predictions.

Observational diagnostic statistics provided by SAID were used to identify outliers and individual data values with a high influence on the surrogate regression model. Observation diagnostics include residual plots, leverage, and the influence statistics: Cook’s D and DFFITS (difference in fit) (Helsel and others, 2020). Removal of outliers identified by residual, leverage, and influence statistics alone may overestimate the certainty of the surrogate regression models (Helsel and others, 2020, Stone and Klager, 2022); therefore, it is important to consider whether each outlier represents an error or a rare but real event.

Overall Model Summary Results

Specific conductance, day of the year, and streamflow, where available, were able to model nitrate concentrations in various combinations for four of the six springs. The two springs with streamgages (Briggs and Upper Box Canyon Springs) included streamflow as an additional explanatory variable. The surrogate regression models and their explanatory variables and ranges are listed in table 2, and the model statistic results are listed in table 3. The springs farthest to the east have the most upgradient agricultural input, and springs to the west have the least upgradient agricultural input. This relation is indicated by decreasing specific conductance and nitrate concentrations from the eastern to western springs (table 2). The adjusted R² (R² if only one explanatory variable) surrogate regression summary statistics for the models range from 0.15 to 0.94. Surrogate regression models performed well at four of the six springs considering all of the diagnostic statistical results.

The two springs with streamflow available as an explanatory variable in the surrogate regression models (Briggs and Upper Box Canyon Springs) resulted in the best surrogate regression models with adjusted R² surrogate regression summary statistic values of 0.94 and 0.89, and low PRESS values of 0.51 and 0.48 respectively. The root mean squared error or standard error of the regression (table 3) provides the average model error in units of the response variable (nitrate concentration) and ranges from 0.05 to 0.23 mg/L for all the surrogate regression models. The surrogate regression models for the six springs all have a probability plot correlation coefficient greater than or equal to 0.84 indicating normality of residuals, which is one of the required assumptions necessary for linear regression. Along with the probability plot correlation coefficients, the probability plots themselves were visually evaluated to confirm normality or residuals. All of the surrogate regression models except for one of the Banbury Springs surrogate regression models have p-values of less than 0.01 for the overall surrogate regression model F-tests, indicating that the regression relations are statistically significant and not due to chance.

Observational Diagnostic Model Results

Individual observation diagnostic statistics support the assumptions required for linear regression models. The observational diagnostic statistics were evaluated for all of the surrogate regression models and are available in the related data release (Skinner, 2023). Some of the surrogate regression model assumptions were partially or completely invalid and required changes to the models. These linear regression assumption issues are discussed and exemplified in the paragraphs that follow.

Plots of the explanatory variables and measured nitrate concentrations help us evaluate if the required linear relation exists between them. Upper and Lower Box Canyon Springs are good examples in that the linear relation between specific conductance and nitrate concentrations change through time. Both Upper and Lower Box Canyon Springs have a flat or nonlinear relation between specific conductance and nitrate concentrations between specific conductance values of 403 to 410 microsiemens per centimeter (µS/cm) at 25 °C. From 410 µS/cm to the maximum specific conductance value of 431 µS/cm at 25 °C, as specific conductance values increase, nitrate concentration also increases signifying the required linear relation (fig. 4).

Plots of surrogate regression model residuals with time are used to evaluate serial correlation and determine if the model varies with time. Serial correlation is identified when nearby samples (nearby in time for this study) are similar and correlated with each other. This is identified in plots of surrogate regression model residuals with time whereby points tend to follow each other and form a pattern. The Briggs Springs residual compared to time plot from the surrogate regression model (using the continuously measured explanatory variables) shows the presence of serial correlation and that nearby points in time have similar residuals (fig. 5A). However, the surrogate regression model using the monthly mean explanatory variables shows that the serial correlation has been minimized or removed (fig. 5B).

The Banbury Springs residual compared to time plot (fig. 6) indicates a possible residual trend over time or a step change, starting in autumn 2021, that shows more negative residuals. A surrogate regression model with trending residuals invalidates an assumption for linear regression models and weakens the model’s ability to predict values.

Plots of predicted nitrate concentrations and their residuals (difference between the observed and predicted nitrate concentrations) in the surrogate regression model were used to identify the presence of heteroscedasticity and curvature (Helsel and others, 2020). All of the surrogate regression models show fairly uniform variance or homoscedasticity, as exemplified by Lower Box Canyon Springs in figure 7.

Individual Surrogate Regression Model Results