Statistical modeling of daily and subdaily stream temperatures: Application to the Methow River Basin, Washington



[1] Management of water temperatures in the Columbia River Basin (Washington) is critical because water projects have substantially altered the habitat of Endangered Species Act listed species, such as salmon, throughout the basin. This is most important in tributaries to the Columbia, such as the Methow River, where the spawning and rearing life stages of these cold water fishes occurs. Climate change projections generally predict increasing air temperatures across the western United States, with less confidence regarding shifts in precipitation. As air temperatures rise, we anticipate a corresponding increase in water temperatures, which may alter the timing and availability of habitat for fish reproduction and growth. To assess the impact of future climate change in the Methow River, we couple historical climate and future climate projections with a statistical modeling framework to predict daily mean stream temperatures. A K-nearest neighbor algorithm is also employed to: (i) adjust the climate projections for biases compared to the observed record and (ii) provide a reference for performing spatiotemporal disaggregation in future hydraulic modeling of stream habitat. The statistical models indicate the primary drivers of stream temperature are maximum and minimum air temperature and stream flow and show reasonable skill in predictability. When compared to the historical reference time period of 1916–2006, we conclude that increases in stream temperature are expected to occur at each subsequent time horizon representative of the year 2020, 2040, and 2080, with an increase of 0.8 ± 1.9°C by the year 2080.

1. Introduction

[2] The health and productivity of aquatic ecosystems are highly sensitive to stream temperature [Caissie, 2006; Webb et al., 2008]. Temperature fluctuations induce changes in metabolic rates, concentrations of key water quality parameters (e.g., dissolved oxygen), biotic assemblages, and life cycle processes (e.g., spawning, migration, and mortality). Since changes in stream temperature are closely tied to meteorological and hydrological conditions [Smith and Lavis, 1975], any modification in the variability and mean state of the existing hydroclimate should have a corresponding effect on aquatic ecosystems.

[3] In response, multiple research efforts have been focused on the prediction of stream temperature under multiple future climate conditions [e.g., Webb et al., 2008; Null et al., 2012; Mantua et al., 2010]. Often, these models focus on geographical regions (e.g., Pacific Northwest) which require a hydrologic system with large spatial (sub-basin to watershed scale) and temporal (daily to weekly) resolutions [Null et al., 2012; Mantua et al., 2010; Ficklin et al., 2012]. These models typically couple hydrologic models with stream temperature models to inform broad decisions on the magnitude and general spatial extent of climate-related impacts on thermal properties of rivers and streams. Null et al. [2012] coupled the Water Evaluation and Planning System (WEAP21) hydrologic model with the Regional Equilibrium Temperature Model physical temperature model, while Ficklin et al. [2012] modified the temperature model in the Soil and Water Assessment Tool. Mantua et al. [2010] coupled output from the Variable Infiltration Capacity (VIC) model with the statistical stream temperature model of Mohseni et al. [1998] to simulate weekly stream temperature.

[4] The incorporation of the hydrologic forcings into stream temperature models has been shown to improve the predictability of models [Lowney, 2000; Webb et al., 2003; van Vliet et al., 2011]. For example, Ficklin et al. [2012] improved on an existing stream temperature model by including the contributions of multiple source water constituents (e.g., groundwater, snowmelt, surface runoff, in-stream flow) to the thermal dynamics for seven watersheds across the western United States. In addition, they adjusted the input air temperatures, precipitation, snowmelt, and groundwater to evaluate the sensitivity of stream temperature to hydroclimatic changes.

[5] The complexity of any modeling system, however, is usually directly related to the required level of quality and continuity of input data sets. There are generally two types of stream temperature models applied in assessing future climate impacts: (1) physically based models (rooted in the solution of fluid flow and heat transport equations) and, (2) empirical models (relying on the correlation strength between meteorology and stream temperature). The advantage of physically based models is the ability to estimate the spatial and temporal distribution of stream temperature at fine scales over small domains; therefore, they are typically restrictive for performing regional simulations or for predicting changes at longer time scales (e.g., years to decades) due to extensive requirements of data and computing power [Taylor, 1998; Carron and Rajaram, 2001; Brock and Caupp, 1996]. Physical models can be adapted for larger domains as described previously [e.g., Null et al., 2012]; but, usually results in a reduction in temporal resolution. Detailed, high-resolution, continuous model input is required including inputs of system geometry, meteorological forcing, and hydrological forcing. In unregulated systems, these data are often unavailable, inconsistent, or of poor quality.

[6] For these data sparse situations, stream temperatures are often modeled using statistical regression [e.g., Mohseni et al., 1998; Webb et al., 2003, 2008]. Statistical methods are appropriate at larger spatial and temporal scales, as they offer the benefit of computational efficiency [Benyahya et al., 2007]. Empirical models typically consist of regressions between stream temperature and environmental conditions, such as air temperature which is driven by the joint dependence of each on solar radiation [Benyahya et al., 2007]. Thus, these models are easily integrated with output from other modeling systems (e.g., hydrologic models forced with input from global climate models (GCMs)), which provides an opportunity to apply projected changes in hydroclimate directly, rather than making discrete adjustments (say, increasing air temperatures by 2°C) to understand model sensitivity to input changes. Statistical models are limited regarding temporal resolution, in that they are appropriate for application at daily or weekly time steps [Mohseni et al., 1998]. At shorter time steps, Mohseni et al. [1998] found that autocorrelation within the stream temperature time series makes regression increasingly difficult.

[7] In this paper, we present the first statistical modeling effort to apply future hydroclimatological changes at subdaily time scales (i.e., hourly) with spatial resolutions at the subreach scale (i.e., meters). The proposed approach seeks to bridge the gap between daily values and the hourly requirements of high-resolution hydraulic modeling efforts. For example, this type of high-resolution data is essential for assessing the influence of mitigation efforts (e.g., channel dredging, riparian zone refurbishment) that seek to alleviate future climate impacts on stream temperature. As such, we develop a flexible statistical modeling framework that is capable of providing input at high spatial and temporal resolution to these models. We couple a generalized linear model (GLM) [McCullagh and Nelder, 1989] at the daily time scale with a K-nearest neighbor (K-nn) resampling algorithm, which provides a bias-correction function and allows disaggregation of daily values to hourly estimates of stream temperature. While we focus on the generation of a daily mean state variable (i.e., mean daily stream temperature), the modeling framework provides the opportunity to predict daily variables with a variety of distributions (i.e., continuous, logistic, discrete), such as daily maximum stream temperature, exceedance/nonexceedance of a particular thermal threshold, and number of hours exceeding that threshold, respectively. The computational efficiency of the statistical models allows the future projections of stream temperature to extend to decades, where physically based models are currently restrictive at the prescribed spatial and temporal resolution.

[8] We test the methodology in the Methow River basin in the State of Washington in a roughly 1 km reach near the confluence of the Methow and Chewuch Rivers (Figure 1). The area of interest is described in the subsequent section (section 'Study Area') along with the datasets available. The coupled GLM-VIC statistical modeling framework, K-nn methodology, and disaggregation technique are presented in section 'Methodology'. Application and results from the proposed framework including stream temperature projections for three future time periods (2020s, 2040s, and 2080s) are described in section 'Results'. Study conclusions along with a discussion of model uncertainty under the proposed framework are summarized in section 'Conclusion'.

Figure 1.

Map of the confluence of the Methow and Chewuch Rivers. Observation sites near Winthrop, WA are indicated. Inset shows the location of the study area (black square) relative to the State of Washington.

2. Study Area

[9] The Methow River in Washington (see Figure 1) offers prime spawning habitat for salmon and other cold-water fish. It is a tributary to the Columbia River system and offers a natural setting with generally unregulated flow. During the summer months, streamflow in the Methow River is typically low (less than approximately 25 cubic meters per second (cms)), resulting in both higher thermal vulnerability and cutoff side channels that limit the habitat available to these fish. Being unregulated, there is an inability to mitigate these high temperatures through upstream water releases. Under these circumstances, mitigation efforts rely heavily on stream restoration projects and riparian zone refurbishment.

[10] Future climate projections which indicate increasing air temperature in the western United States suggest the potential for increasing stream temperatures in response to changes in hydrology (i.e., snowmelt, streamflow) and additional heating [e.g., Ficklin et al., 2012; Mantua et al., 2010]. To assess the impacts of climate change in the Methow River, a statistical modeling framework is developed for generation of daily projections of stream temperature. The framework is unique in that it must provide the ability to generate subdaily meteorological forcings and hydrologic boundary conditions to be integrated with an hourly time-step, two-dimensional hydraulic model for predicting spatial variations of stream conditions and the impacts of various mitigation efforts.

[11] In this study, observations of stream temperature and environmental conditions (i.e., air temperature, precipitation, and streamflow) are used to develop statistical models of daily mean stream temperature for the Methow River near Winthrop, WA. The location is situated at the confluence of the Methow and Chewuch Rivers, which corresponds to the site of an ongoing hydraulic modeling project. Therefore, there is a need for hourly boundary condition inputs at locations upstream of the confluence, including temperature and flow projections.

2.1. Data

[12] Statistical models of stream temperature require hydrometeorological data as input. Typical hydrometeorological variables include maximum/minimum air temperature (Tax and Tan, respectively), precipitation (Pp), and streamflow (Q). The quality and quantity of the data are two primary indicators of the level of skill that can be expected from the models. For the current study, there is a limited set of water temperature data which requires a subset of data from the period 2005–2011.

[13] This research uses two types of data: observed and simulated. The observed data includes stream temperature, Ts (the independent variable for the model), and, Tax, Tan, Pp, and Q (the dependent variables). Once the models are developed using observed time series, the simulated model outputs (see section 'Snow Water Equivalent') from the VIC model can be integrated to produce future scenarios of Ts. This section describes the individual datasets by providing the metadata associated with each and quality control employed prior to use. The locations of the observed data are shown in Figure 1.

2.1.1. Stream Temperatures

[14] Near the confluence of the Methow and Chewuch Rivers, a total of seven Ts time series are available (Table 1). The sources of these data include the United States Forestry Service (USFS), US Bureau of Reclamation (USBR), Wild Fish Conservancy (WFC), and Washington State Department of Ecology (DOE). It should be noted that while the period of records and frequencies in Table 1 would indicate a wealth of data at these sites, none of the gauges provide continuous observations throughout the period of record, with most measurements occurring during the warm season. Quality control of the Ts data was performed by the respective operating agencies and, therefore, additional quality control was not performed prior to use.

Table 1. Metadata for Ts Observation Sites
SiteStreamIDEntityPeriod of RecordFrequency
Above ChewuchMethowMACUSFS30 Jun 2005 to 21 Oct 2010subdaily to 30 min
Spring CreekSpringSCUSFS/USBR/WFC2 Jul 2009 to 25 Oct 2010hourly to 30 min
Above Barkley DiversionMethowABSUSFS/USBR/WFC26 Nov 2009 to 4 Oct 2010subdaily to hourly
Chewuch MouthChewuchCMUSFS/USBR/WFC1 Jun 2005 to 5 Apr 2011subdaily to 30 min
Below ChewuchMethowMBCUSFS7 Jun 2005 to 13 Oct 2009hourly
Hwy 20 Bridge at WinthropMethow48A150DOE8 Oct 2007 to 8 Sep 2008monthly
Hwy 20 Bridge at WinthropChewuch48B070DOE8 Oct 2007 to 8 Sep 2008monthly

[15] The period of 16 July 2010 to 4 October 2010 was the only time frame which had overlapping, primarily continuous, subdaily data for more than two sites. Only four of the seven sites (i.e., Methow Above Chewuch (MAC), Spring Creek (SC), Above Barkley Silo (ABS), and Methow Below Chewuch (MBC)) are available during that time, reiterating the data availability in that portion of the Methow is highly limited and restricts the use of physically based models. Since the primary focus is on the prediction of water temperature below the confluence of the Methow and Chewuch, the subdaily values of Ts at the MBC and ABS sites in Table 1 were averaged to create a single, daily mean time series. The daily values at the downstream site were compared to the Q at the Methow above Winthrop (MAW) site (see section 'Streamflow') and meteorological data at Winthrop 1WSW (WIN; see section 'Meteorology'). This composite of daily mean Ts serves as the dependent variable in the statistical model (see Figure 3d).

[16] The adjusted R2 values are computed and shown in Figure 2 to indicate the strength of correlations between the independent variables and Ts. In general, the air temperature and flow variables show strong correlations with stream temperatures. The correlation with precipitation is near zero during the summer season (Figure 2d); however, precipitation may serve as a surrogate for cloud cover and its related impacts on stream temperatures. Therefore, we include precipitation in the set of potential predictors, allowing a subset selection criterion (e.g., Bayesian Information Criteria (BIC)) to determine if the variable is applied in the final statistical models.

Figure 2.

Scatterplots of hydrometeorological variables and mean Ts for the period 16 July 2010 to 4 October 2010 with adjusted R2 given of linear fits for (a) streamflow above 15 cms, (b) maximum air temperature, (c) minimum air temperature, and (d) precipitation above 0.254 mm.

Figure 3.

Quality controlled time series of daily hydrometeorological variables for the period 1 January 2005 to 19 May 2011. Time series of (a) streamflow, (b) mean air temperature, (c) precipitation, and (d) daily mean stream temperature are shown.

2.1.2. Streamflow

[17] Streamflow data were gathered from the United States Geological Survey (USGS) for the overlapping period with the Ts data (i.e., 1 January 2001 to 19 May 2011). A total of three sites are available near the confluence of the Methow and Chewuch Rivers; however, only the site at Methow at Winthrop (USGS 12448500; MAW) was used. Unlike the Ts data, the measurements are predominantly continuous, except during transmission and equipment failures. Times with missing values and equipment issues were removed from the observational time series. As can be seen in Figure 2a, Q above 15 cms at MAW is inversely proportional and correlated (adj-R2 = 0.48) with the Ts. The excellent continuity and quality of the MAW data can be also be seen in Figure 3a. The Q at MAW will serve as one of the independent variables in the model fitting described in section 'GLMs'.

2.1.3. Meteorology

[18] Daily meteorological data were collected from the National Climatic Data Center (NCDC) for the period 1 March 1906 to 19 May 2011 for the site at Winthrop 1WSW (NCDC 459376; WIN). The daily data included multiple meteorological variables; however, only the Pp, Tax, and Tan values were considered. The daily meteorological data are fairly continuous, but did show some quality issues. For quality control, values that were unrealistic (e.g., Tan below −30°C), missing, or entered as text (e.g., “T” for trace amounts of Pp) were removed from the record. As with the stream flow at the MAW gauge, only data for the period 1 January 2005 to 19 May 2011 were used. The quality controlled daily meteorological time series for mean daily air temperature and Pp are shown in Figures 3b and 3c. The time series for Tax and Tan (not shown) showed similar continuity and quality.

[19] As seen in Figures 2b and 2c, air temperatures are well correlated (adj-R2 ≥ 0.43) with Ts in the Methow Basin; therefore, the values of Tax and Tan will serve as potential predictors in the statistical model development. Precipitation shows less correlation (adj-R2 = 0.06); but again, due to its relationship with cloud cover and the associated feedbacks on solar insolation on the stream, this variable is also included in the potential predictors (Figure 2d).

2.1.4. Snow Water Equivalent

[20] In the Methow Basin, the Natural Resources Conservation Service operates a single snowpack telemetry (SNOTEL) station at Harts Pass. The daily snow water equivalent (SWE) for the period 1 January 2005 to 19 May 2011 were downloaded and will be used as a predictor variable in the statistical model.

2.1.5. VIC Model Output

[21] The Climate Impacts Group at the University of Washington generated regional scale, future climate projections for the Pacific Northwest using 10 GCMs under different emissions scenarios (A. F. Hamlet et al., Final Project Report for the Columbia Basin Climate Change Scenarios Project, Climate Impacts Group, Seattle, Washington,, 2010). For our application, we focused on the use of the 10 climate projections based on the A1B emissions scenario (Table 2). While B1 emissions scenarios were available, the A1B scenarios serve as a proof of concept application, which the authors consider a plausible future with little greenhouse gas mitigation until the mid-21st century. The output from the downscaled GCMs was used to develop inputs to the Variable Infiltration Capacity (VIC) [Liang et al., 1994, 1996; Nijssen et al., 1997] model to assess the impacts of climate change on ecological and hydrological systems in the region.

Table 2. List of the 10 GCM Used in the A1B Scenarios
ModelSponsor, Country
hadcm3Hadley Centre for Climate Prediction and Research/Met Office, UK
cnrm_cmMeteo-France/Centre National de Recherches Meteorologiques, France
ccsm2National Center for Atmospheric Research, USA
echam5Max Planck Institute for Meteorology, Germany
echo_gMeteorological Institute of the University of Bonn, Meteorological Research Institute of the Korea Meteorological Administration and Model Data Group, Germany/Korea
cgcm3.1_t47Canadian Centre for Climate Modelling and Analysis, Canada
pcm1National Center for Atmospheric Research, USA
miroc_3.2Center for Climate System Research (University of Tokyo), National Institute for Environmental Studies, and Frontier Global Change Center for Global Change, Japan
ipsl_cm4Institut Pierre Simon Laplace, France
hadgem1Hadley Centre for Climate Prediction and Research/Met Office, UK

[22] The VIC model is a spatially distributed hydrologic model that solves the water and energy balance at each model grid cell. The model initially was designed as a land-surface model to be incorporated in a GCM so that land-surface processes can be more accurately simulated; however, it is often run in standalone mode. For climate change impact studies, VIC is run in what is termed the water balance mode that is less computationally demanding than an alternative energy balance mode, in which a surface temperature that closes both the water and energy balances is solved for iteratively. Using the University of Washington VIC applications, the water balance mode is driven by daily weather forcings of precipitation, maximum and minimum air temperature, and wind speed. Additional model forcings that drive the water balance, such as solar (shortwave) and longwave radiation, relative humidity, vapor pressure, and vapor pressure deficit, are calculated within the model. The VIC outputs are configurable but typically include grid cell moisture and energy states through time (i.e., soil moisture, snow water content, snowpack cold content) and water leaving the basin either as evapotranspiration, base flow, sublimation, or runoff, where the latter represents the combination of faster-response surface runoff and slower-response base flow. Additional details on the VIC setup and development are well documented in Wood and Lettenmaier [2006], Wood et al. [2004], and Maurer et al. [2002]. An overview of the Columbia Basin Climate Change Scenarios Project is currently in press, which provides additional details on the integration of the GCM output and VIC model, including implementation and calibration [Hamlet et al., 2013]. The reader is, therefore, referred to these references for further details.

[23] For the current study, meteorological variables (e.g., Tax, Tan, and Pp) and bias-adjusted flow (Q) were extracted from 10 different VIC simulations for three future climate horizons: 2010–2039 (representative of year 2020); 2030–2059 (representative of year 2040); and 2070–2099 (representative of year 2080). Two 1/16th degree latitude-longitude grid cells were used for the extraction. The first location was centered at 48.46875°N, 120.15625°W, which is in close proximity to the MAW gauge. This site serves as the source for the predictor variables of meteorology and flow, with the assumption that the weather is consistent over the grid box and that the flow is accumulative over the entire watershed. The second grid location was centered at 48.71875°N, 120.65625°W and contains the SNOTEL site at Harts Pass. This site serves as the source the SWE predictor variable; however, comparison with the observed gauge data requires an adjustment factor of 0.65 be applied to the Harts Pass gauge for direct comparison (not shown). These differences arise from point to area reduction and universal assignment of vegetative class at the grid scale in the VIC model which fails to allow for point-specific estimates of SWE. In addition, the same data were available from a single historical VIC model simulation. The VIC output provides historical projections at daily time steps for the period 1916–2006 for a total of 91 years of daily data, which serves as a reference period. The VIC output is also provided as a time series of 91 years of data representative of the three future climate horizons of 2020, 2040, and 2080.

3. Methodology

[24] In this study, mean daily Ts is modeled as a function of Tax, Tan, and Pp at the WIN meteorological site; daily mean Q at the MAW gauge, and daily SWE at the Harts Pass gauge. This section will describe the methodology for developing the GLM and the K-nn technique used to perform bias adjustment of the VIC scenarios prior to application in the statistical modeling framework, including disaggregation (Figure 4).

Figure 4.

Flow chart of the statistical framework.

3.1. GLMs

[25] A flexible statistical framework has often been applied in the assessment and prediction of many water quality variables [Neumann et al., 2003, 2006; Towler et al., 2010a, 2010b; Caldwell and Rajagopalan. 2011]. Neumann et al. [2003, 2006] used regression to model the daily maximum Ts, while Towler et al. [2010a, 2010b] incorporated the GLM into a seasonal risk analysis of meeting turbidity requirements for a water supplier in the Pacific Northwest. Caldwell and Rajagopalan. [2011] advanced the work of Neumann et al. [2003, 2006] by predicting multiple Ts characteristics (e.g., daily Ts range and exceedance/nonexceedance) using a GLM coupled with a stochastic weather generator to assess the seasonal risk to salmon fisheries in the Sacramento River. The current application builds on the work of Caldwell and Rajagopalan. [2011] by integrating the output from the VIC hydrologic model to generate future Ts scenarios to assess the impact of climate change on fisheries in the Methow River.

[26] In a GLM, the response or the dependent variable Y can be assumed to be a realization from any distribution in the exponential family with a set of parameters [McCullagh and Nelder, 1989]. A smooth and invertible link function transforms the conditional expectation of Y to a set of predictors (equation (1)).

display math(1)

[27] G(.) is the link function, X is the set of predictors or independent variables, E(Y) is the expected value of the response variable and ɛ is the error assumed to be normally distributed with variance (σɛ). In a linear model (the standard linear regression), the function G(.) is identity and Y is assumed to be normally distributed. Depending on the assumed distribution of Y, there exist appropriate link functions [McCullagh and Nelder, 1989]. The model parameters, β, are estimated using an iterated weighted least squares method that maximizes the likelihood function as opposed to an ordinary least squares method in linear modeling. The GLM can be used to model a variety of response variables––for skewed variables with a lower bound of 0 such as daily mean water temperature, the Gamma distribution assumption of Y and its associated link function is appropriate. We refer the readers to McCullagh and Nelder [1989] for information about a variety of distributions, link functions, and parameter estimation.

[28] To obtain the best set of predictors for the model there are objective criteria such as the Akaike Information Criteria (AIC) or BIC—both of which penalize the likelihood function based on the number of parameters [Venables and Ripley, 2002]. Models are fit using all possible subsets of predictors and also link functions; for each, the AIC and BIC are computed and the model with lowest AIC or BIC is selected as the “best model.” We used BIC in this study as it tends to be slightly more parsimonious compared to AIC. This subset selection procedure accounts for correlation between independent variables. Models can also be tested for significance against a null model or an appropriate subset model. For our case, we compare the GLM model to a simple linear regression model, typical in statistical models of stream temperature [Benyahya et al., 2007], where the daily mean stream temperature is function only of daily mean air temperature.

3.2. K-Nearest Neighbor Resampling

[29] Since the output from the VIC model spans a much longer period of record than observed, a K-nn resampling algorithm [Rajagopalan and Lall, 1999; Buishand and Brandsma, 2001] is employed to select days from the historical record that are representative of the hydrometeorological conditions in the model output. The K-nn algorithm is a method for classifying objects based on closest training examples in a feature space. To account for the shifting hydrology and meteorology throughout the year, the training examples (or observed weather and flow vectors) are selected from a 30 day window on either side of the current Julian date. In our case, there are typically 6 years of data (2005–2011) with a total of 61 potential neighbors from each year. The length of this window of time can be adjusted by the user, but should maintain the general statistics of the hydroclimate for that time of year. Here, we selected 30 days to account for potential shifts in the seasonality of snow accumulation and melt. For example, if the current Julian day is 31 (i.e., 31 January), first the weather-flow vector is extracted from the current VIC run. The potential neighbors are then 1 January to 1 or 2 March (depending on leap year) from each year 2005 through 2011. The Mahalanobis distances between the daily VIC and the potential neighbor's weather-flow vectors from the observed record (i.e., 1 January 2005 to 19 May 2011) are computed [e.g., Yates et al., 2003]. After ordering the distances from closest to farthest, each neighbor is prescribed a weight based on the cumulative sum of 1/K with the closest neighbor receiving the highest weight. A neighbor is then selected at random based on weight; the index of that neighbor is used to extract the associated weather-flow vector from the observational record. The selected weather-flow vector (i.e., neighbor) from the historical record is then used to replace the VIC model output prior to use as predictors in the statistical modeling framework. An additional benefit of the K-nn algorithm is selection of particular dates, which can be used to disaggregate the daily simulations to hourly.

3.3. Spatiotemporal Disaggregation

[30] Hourly hydrometeorological inputs are required for the two-dimensional hydraulic model being developed at the Bureau of Reclamation. Since the GLM models predict daily Ts at the MAW site, spatial disaggregation is required to distribute the daily hydrometeorological variables to two synthetic upstream boundary condition locations: upstream Methow (MUS) and upstream Chewuch (CUS). CUS is assumed to represent a location near the site of the CM Ts measurements; MUS is assumed to represent a location near the site of MAC Ts measurements. In addition, the downstream location (MDS) is assumed to represent the location of the MBC and ABS Ts observations. Henceforth, we will use the CUS, MUS, and MDS abbreviations for clarity. The daily values then must be disaggregated temporally to generate inputs for the hydraulic modeling system. Since the quality and availability of continuous, daily and subdaily measurements in the Methow River Basin were severely limited, we perform the spatial and temporal disaggregation by developing proportion vectors at the daily and hourly time scale, using the methods employed in Nowak et al. [2010]. Nowak et al. [2010] applied a nonparametric stochastic approach to disaggregation of annual flow values via K-nearest neighbor resampling of daily proportion vectors which captured the observed statistics quite well. We further this approach by using both hourly and daily proportion vectors for both streamflow and stream temperature.

[31] Daily mean Ts values are computed for three locations (MUS, MDS, and CUS) using the subdaily and daily measurements at four sites (i.e., ABS and MBC for MDS; MAC for MUS; and, CM for CUS) from the period 1 January 2005 to 31 December 2011. If all three sites do not have a daily mean, then the period mean is used at each of the sites. The proportion of each upstream, gauge relative to the downstream gauge is calculated to yield the daily temperature proportion vectors (DTPV). Only 329, of a possible 2555, DTPVs are unique (Figure 5). The mean values for CUS (1.05) and MUS (0.93) approach one but signify that the Chewuch is generally warmer than the Methow inflows due to differences in riparian cover and flow.

Figure 5.

Histogram of the daily temperature proportion vectors for the (a) Chewuch Upstream and (b) Methow Upstream sites, shown as a fraction of the daily mean stream temperature downstream.

[32] The hourly Ts proportion vectors (HTPVs) use the available hourly Ts measurements at the same sites. For each day of the calendar year (1–365), the hourly mean Ts is calculated. At least one of the sites must be reporting for a given hour; otherwise, the monthly mean for that hour is imposed. Separate HTPVs are calculated as the fractional contribution of each hour to the daily sum for days 1–365 at the two upstream and one downstream site to account for the impact of flow differences between the tributaries and the cumulative flow downstream. The variety of diurnal fluctuations inherent in the HTPVs at each site can be seen in Figure 6. Using the corresponding date (or its Julian day of the year) picked during the K-nn resampling, the appropriate proportion vectors for both spatial and temporal disaggregation can be selected for distributing the predicted value from the GLM to both upstream points at hourly time steps. Validation statistics for the spatiotemporal disaggregation technique are presented in section 'Validation of Spatiotemporal Disaggregation'.

Figure 6.

Hourly stream temperature proportion vectors for the Methow Upstream (MUS), Chewuch Upstream (CUS), and Methow Downstream (MDS) sites, shown as the fractional value of the daily accumulative stream temperature.

4. Results

4.1. Statistical Model Development

[33] Since the statistical model is predicting the mean daily Ts at the downstream location, only the sites at MBC and ABS were used to create a composite daily Ts time series for the period 1 January 2005 to 19 May 2011. A total of 738 days were available during the period with observed temperatures above 0°C; and, the highest number of days were available by month during the warm season (Figure 7). For the days with available Ts data, the GLM was fit by month using the hydrometeorological predictors from WIN (e.g., Tax, Tan, and Pp), the MAW (e.g., Q) site, and the Harts Pass location (e.g., SWE). Therefore, there are a total of twelve statistical models, one for each month to forecast mean daily stream temperature. Using the BIC for subset selection, the preferred predictors for Ts include Tax, Tan, and Q. Only 2 months (February and March) include Pp in the best subset from BIC, primarily during the rainy, cool season. The SWE variable is selected during the cool season as well during the months of January, March, November, and December. The results of the BIC subset selection can be found in Table 3.

Figure 7.

Counts of the number of days with mean daily Ts available by month during the period 1 January 2005 to 19 May 2011.

Table 3. Subset of Predictors Selected by the GLM Using BIC
JanTax, Tan, Pp, Q, SWE
FebTax, Tan, Pp, Q
MarTax, Tan, Pp, SWE
AprTax, Tan, Q
MayTax, Tan, Q
JunTax, Tan, Q
JulTax, Tan, Q
AugTax, Tan, Q
SepTax, Tan, Q
OctTax, Tan
NovTax, Tan, Q, SWE
DecTax, Tan, P, Q, SWE

4.2. GLM Verification

[34] To test the skill of the GLM, cross validation is performed by randomly dropping 10 percent of the points over 100 iterations and computing the root mean square error (RMSE). Scatterplots of the observed versus predicted values (Figure 8) and RMSE boxplots (Figure 9) indicate excellent skill throughout the year. To highlight the seasonal performance of the models, only relationships for January, April, July, and October are shown, though the relationships are similar in other months. Correlations are high in all months (with adj-R2 ≥ 0.60); and, all are statistically significant based on a t test for the significance of the correlation coefficient at significance level, α = 0.05, such that t is defined in equation (2) as:

display math(2)

where t is the test statistic, r is the coefficient of correlation, and n is the sample size. The warm season months generally exhibit highest RMSE values, with mean RMSE approaching 0.8°C. The largest variability in RMSE is during the months of April and July large variability in Ts occurs due to snowmelt and high air temperatures, respectively. Lowest RMSE values in the winter are related to daily Ts that remain near 0°C as the Methow River freezes.

Figure 8.

Scatterplots of observed versus predicted values for January, April, July, and October from the GLM model fitting. The one-to-one line is overlaid for reference.

Figure 9.

Boxplots of RMSE for January, April, July, and October from the cross-validation. Errors generally average less than 1°C, except during the summer when mean errors approach 1.2°C.

4.3. Integration of GLM and Climate Change Scenarios

[35] Outputs from the historical climate model simulations are combined to generate a large predictor matrix of daily Tax, Tan, and Pp (January 1915 to December 2006); Q (October 1915 to December 2006); and SWE (January 1915 to December 2006) for use in the GLM framework. The historical predictor matrix is the overlapping subset that covers 91 complete calendar years for the period from January 1916 to December 2006. Using the twelve monthly GLM models, the historical predictor matrix is used to predict daily mean Ts at the MAW site. Boxplots and probability distribution functions (PDFs) of daily mean Ts for the 4 months representing each season (i.e., January, April, July, and October) indicate that the variability is increased in the historical simulations compared to observations (PrV in Figures 10 and 11) when using the GLM-VIC model output without adjustment. Similar results are observed for other months (not shown). To address the issue, we implemented a K-nn resampling technique as described in section 'K-Nearest Neighbor Resampling' to perform bias adjustment on the VIC model output. Difference in variance and means between the observations and historical runs, both with VIC and adjusted using K-nn, were tested using an F-test and the appropriate t test, respectively. Results indicate that while the variance is similar at the 95% confidence level for the VIC runs, there is a significant deviation in the mean. On the other hand, the variance is reduced in the K-nn simulations (a known limitation to K-nn); however, the simulated mean daily stream temperatures are approximately 50% closer to the observed values than in the VIC alternative. Both the K-nn and the VIC simulations of daily mean stream temperature underestimate the mean. The center location and shape of the resulting boxplots and PDFs related to the K-nn simulations match much more closely with the observed distributions (see PrK in Figures 10 and 11). As a result, the bias-adjusted VIC output then serves as input to the monthly GLM models.

Figure 10.

Boxplots of observed versus predicted values of mean stream temperature for January, April, July, and October when using the historical simulation as input to the GLM, using the historical VIC (PrV) simulations compared to the bias-adjusted, K-nn simulations (PrK).

Figure 11.

PDFs of daily mean stream temperature for observed (black) versus predicted with historical VIC (PrV; red) and predicted with bias-adjusted K-nn VIC output (PrK; green).

[36] The future climate projection scenarios are processed in the same manner. First, climate change predictor matrices are developed for each of the 30 coupled VIC scenarios. Then, the K-nn resampling is applied to each predictor matrix to generate new bias-adjusted matrices, which are used to predict daily mean Ts through the GLM (not shown).

4.4. Validation of the Integrated VIC-GLM Model

[37] To assess the performance of the coupled VIC GLM, the mean daily stream temperature from the historical climate simulations were compared to daily values at each site using the following statistics: Nash-Sutcliffe (NS) coefficient [Nash and Sutcliffe, 1970]; ratio of the root mean square error to the standard deviation of the observed data (RSR); percent bias (PBIAS), the ratio of the sum of residual errors between the simulated and observed data and the sum of the observed data; the mean error (ME); the RMSE; and the adjusted R2 (Adj R2) or the squared correlation between the observed and predicted values adjusted for sample size and degrees of freedom. Guidelines established by Moriasi et al. [2007] and applied in Ficklin et al. [2012] where NS > 0.50, RSR < 0.70, and PBIAS between ± 25% are used to qualify the validation results as satisfactory. Comparison to the null model described in section 'GLMs', indicates the GLM model exhibits increased values of NS, decreased values of RSR, equivalent absolute PBIAS and ME, and lower RMSE (Table 4), indicating enhanced skill over simple linear regression. In addition, the adjusted R2 values using the GLM for the months shown in Figure 8 were equal to or higher than those from the null model, particularly in April and July when correlation was more than double (Table 5). The robustness of a model to small changes in input parameters provides an estimate of the sensitivity of the model to individual parameters. For example, in Ficklin et al. [2012], model sensitivity was analyzed using a normalized, dimensionless sensitivity index (I) shown in equation (3), where

display math(3)

where y2 is the perturbed predicted values at x2 = x0 + Δx and y1 is the perturbed predicted values at x1 = x0 Δx, y0 is the original predicted values, and Δx = x2 − x1.

Table 4. Comparison of Daily Validation Statistics Between the Null and GLM Models
Table 5. Comparison of Adjusted R2 Values Between the Null and GLM Models
Adj R2JanAprJulOct

[38] As in Ficklin et al. [2012], we apply Δx variations of ± 10 percent to each of the predictor variables (i.e., Tx, Tn, Pp, Q, and SWE). The index I then represents the change in output variable (mean and variance) resulting from a change in the inputs to the model. The absolute value of I can be ranked into four classes based on level of sensitivity as given in Ficklin et al. [2012] from Lenhart et al. [2002], such that higher values indicate increasing sensitivity. We calculate the sensitivity index I for both the mean and variance of the predicted variable Ts (Table 6). For the prediction of the mean, Tx was the most sensitive parameter with Tn being of medium sensitivity, and the other variables found to be insensitive. Using the variance as an indicator, the variable Tax was of medium sensitivity with all other variables insensitive, suggesting that the primary drivers for the mean state of Tax and its variance are air temperature related, with lesser influence from precipitation, flow, or snow water equivalent.

Table 6. Sensitivity Indices for Each of the Predictor Variablesa
  1. a

    High (H) and medium (M) sensitivity denoted.


4.5. Validation of Spatiotemporal Disaggregation

[39] Simulations of daily mean stream temperature using the historical VIC inputs were spatially disaggregated to both upstream points and temporal disaggregation applied at all three locations. To test the validity of the results, the same statistics were computed using the hourly data as for the daily values, including: NS, RSR, PBIAS, ME, and RMSE (Table 7). While marginal results were evident for the MUS location (NS = 0.50, RSR = 0.70, PBIAS = −5.6), both the CUS and MDS locations were considered acceptable with NS>0.50, RSR<0.70, and PBIAS between ± 25%. Scatterplots of the observed versus predicted hourly values at each site show modest correlation with adjusted R2 values exceeding 0.73 at CUS and MDS and 0.57 at MUS (Figure 12).

Table 7. Comparison of Hourly Validation Statistics at the Three Disaggregated Sites
Figure 12.

Scatterplots of observed versus predicted values of hourly stream temperature for (a) Chewuch Upstream, (b) Methow Upstream, and (c) Methow Downstream. The one-to-one line is overlaid for reference in red.

4.6. Implications of Climate Change on Stream Temperatures

[40] Mean daily values of Ts from each of the VIC simulations were concatenated by month and assembled to identify the mean and standard deviations (hence, confidence intervals) (Table 8), as well as to show the variability in predictions. Figure 13 shows the boxplots for the months of June through September, the primary season of evident shifts. The ensemble means were computed for each of the future time horizons of 2020, 2040, and 2080, by averaging the results by day of each of the 10 climate modeling scenarios. PDFs of each time horizon were plotted relative to the historical simulation. Mean daily Ts increases by ∼2°C during the warm season by the year 2080, with lesser magnitude positive shifts in the distribution at the 2020 and 2040 time horizons (Figure 14). Shifts during the cool season (see Table 8) were less pronounced. These results suggest that a warming climate will likely be associated with warming Ts in the Methow River Basin. Figure 15 indicates that the greatest increase in mean daily Ts will occur during the summer months. Given the anticipated shift to warmer conditions in future climate, a one-tailed, unpaired t test indicates the simulated daily mean stream temperature time series from the historical run is significantly different from the simulated values at each future period at the α = 0.05 level. Maximum differences between the historical run and the 2080 time horizon are 2.8 ± 4.7°C (in July) and 0.8 ± 1.9°C (annual average), respectively. The values of maximum differences between the historical run and the 2020 and 2040 time horizons range from 1.4 ± 3.5°C (2020) to 2.4 ± 4.4°C (2040). Mean differences are 0.4 ± 1.6°C and 0.7 ± 1.8°C, respectively, for the 2020 and 2040 time horizons.

Table 8. Mean Monthly Daily Mean Stream Temperature (°C) With 90% Confidence Interval for Differences Between Historical and Each Future Period in Parentheses
Jan2.22.2 (±1.3)2.3 (±1.3)2.3 (±1.4)
Feb3.03.1 (±1.3)3.2 (±1.3)3.2 (±1.4)
Mar5.55.8 (±1.8)6.0 (±1.8)6.1 (±1.9)
Apr6.96.9 (±0.9)7.0 (±1.0)7.0 (±1.1)
May7.47.5 (±0.9)7.5 (±1.0)7.5 (±1.0)
Jun8.89.4 (±2.1)10.0 (±2.8)10.4 (±3.3)
Jul12.213.6 (±3.5)14.6 (±4.4)15.0 (±4.7)
Aug14.114.9 (±1.5)15.4 (±1.7)15.5 (±1.7)
Sep11.712.1 (±1.4)12.5 (±1.5)12.6 (±1.5)
Oct8.08.1 (±1.2)8.3 (±1.3)8.4 (±1.2)
Nov4.64.8 (±2.1)5.0 (±2.1)5.1 (±2.1)
Dec3.23.5 (±1.6)3.8 (±1.8)3.8 (±1.9)
Figure 13.

Boxplots of daily mean stream temperature for the months of June through September for the historical and each future climate horizon of 2020, 2040, and 2080, indicating the range of values predicted and uncertainty associated with each month.

Figure 14.

PDFs of ensemble mean values from the GLM when coupled with the VIC output using K-nn for the historical (black) and each future period of 2020 (orange), 2040 (red), and 2080 (dark red). Abscissa line at daily mean stream temperature of 13.9°C indicates upper thermal threshold used to designate properly functioning fish habitat.

Figure 15.

Monthly ensemble means of Ts from the GLM when coupled with the VIC output using K-nn for each future period 2020 (orange), 2040 (red), and 2080 (dark red). Historical from K-nn shown in black.

[41] The National Marine Fisheries Service and U.S. Fish and Wildlife Service prescribe general guidelines for Ts to assess properly functioning habitat for salmonids [Bjornn and Reiser, 1991; U.S. Bureau of Reclamation, 2008, Appendix I, Table I-4]. The upper limit of 13.9°C is used to discriminate Ts that are beyond an acceptable limit, that thermal threshold at which initial effects begin to occur on salmonids, according to these two studies. Several other studies use a slightly higher value around 15°C. The flexibility of the modeling system allows this to be a user defined parameter as the value may differ based on species, life cycle stage of the fish, and river system. We highlight the probabilities of exceeding these values in Figure 14. The potential to exceed the threshold value in future climates is evident as the area under the curves is larger than historical, particularly in months June to September (Figure 15), with increasing impacts at each successive 2020, 2040, and 2080 time horizon, respectively. Table 9 shows the corresponding increases in the probability of mean daily Ts above 13.9°C during the summer season at each time horizon. These threshold exceedances are more than 40 percent more likely during the months of July and August by the year 2080.

Table 9. Probability of Ts > 13.9°C for the Ensembles at Each Time Horizon for the Months of June Through September

5. Conclusion

[42] The current study develops a methodology for using available daily Ts data and hydrometeorological inputs for generating statistical models of daily Ts for each month. In addition, the GLM provides a mechanism for ingesting the output from climate change scenarios to evaluate future impacts on Ts in the Methow River Basin. Despite the sparse data in the region, the statistical model fits were excellent with small values of RMSE across all months. Predicted values from the historical climate scenarios indicated that there was some limitation to skill during the winter months when data was most scarce; however, the use of a K-nn resampling approach improved the skill relative to using the raw output from the coupled VIC modeling framework. As such, the K-nn resampling technique was applied, as well, to perform bias adjustment on the future climate projections at each time horizon of 2020, 2040, and 2080.

[43] The ensemble mean plots of future Ts indicated a mean annual warming of 0.8 ± 1.9°C by the year 2080 with increases relative to historical at each of the future climate horizons, all significant at the α = 0.05 level. Despite the large range in the 90 percent confidence intervals, the projections remain skewed toward positive changes in stream temperature under future climate scenarios, with the primary focus during the summer months. Using a threshold Ts as a metric of viable habitat for fish, we note the increased probability of days with temperatures exceeding the threshold. The threshold of 13.9°C is conservative due to the use of daily mean instead of instantaneous values, which suggests that the impact to viable habitat may be more profound.

[44] Though the proposed framework provides a flexible approach to incorporate projected climate change into stream temperature modeling, there are several key uncertainties that will need to be kept in perspective while applying the methodology. There are three primary sources of uncertainty in the current study: (i) the K-nn resampling algorithm; (ii) the coupled VIC modeling system; and (iii) the statistical regression. The K-nn resampling approach restricts sampling only from observed data, thereby possibly providing a low estimate of future climate-driven stream temperature given the assumption of warming; however, testing of the range of future values compared to historical values showed > 2% of the daily values exceeded those from the observed record. The small fraction of projected future days that fall outside the bounds of historic observations is recognized as a potential suppression of extreme events for use in impacts analysis. In addition, future days that fall close to the bounds of the historical observations may be resampled disproportionately, resulting in reduced variability. Changes in the hydrologic character of the system (e.g., riparian vegetation) may also be partially neglected through the limitations of the resampling approach. Since extreme events are of primary interest to impact modelers, this limitation should be considered before direct application of the approach outlined in the current study. The coupled VIC modeling system itself has several uncertainties including the process of statistical downscaling from large-scale GCM grids to finer-scale grids needed for hydrologic modeling. Also, the quality of the calibrated VIC model will govern the quality of the flow outputs. Third, developing the regression models from limited data, and not being able to include variables such as the groundwater components of runoff could impact projected stream temperature results.

[45] Subsequent modeling efforts will investigate the impact of these future scenarios on the spatial and temporal distribution of stream temperature within the confluence region of the Methow and Chewuch Rivers by using a two-dimensional hydraulic model. Therefore, we also present an efficient method for disaggregating the daily mean projections to hourly increments. While skill scores indicated satisfactory results at the CUS and MDS sites, additional calibration is required for the MUS site. This is potentially due to the limited data at that location. In an unregulated system like the Methow River, the mitigation efforts to reduce the impacts of climate change will focus on restoration projects that provide additional habitat, where suitable water temperatures can be maintained over the longest duration. Restoration efforts will focus on identifying these areas by analyzing outputs from the two-dimensional hydraulic model. We expect that these efforts will be essential in supporting healthy habitat in the Methow River Basin under climate change, particularly for cold-water fishes, such as salmon.


[46] The authors would like to acknowledge the National Aeronautics and Science Administration's Earth-Sun Science Applied Sciences Program, Grant NNX08AK72G and the associated authors of that grant for providing funding to complete this work and for encouraging the continual advance of scientific endeavors through remarkable contributions to the research community. In addition, we would like to thank the Bureau of Reclamation, Research and Development Office for providing support for application in the Methow River Basin through the Science and Technology Program (X6507).