Water Resources Research

A grid-based approach for simulating stream temperature

Authors


Abstract

[1] Applications of grid-based systems are widespread in many areas of environmental analysis. In this study, the concept is adapted to the modeling of water temperature by integrating a macroscale hydrologic model, variable infiltration capacity (VIC), with a computationally efficient and accurate water temperature model. The hydrologic model has been applied to many river basins at scales from 0.0625° to 1.0°. The water temperature model, which uses a semi-Lagrangian numerical scheme to solve the one-dimensional, time-dependent equations for thermal energy balance in advective river systems, has been applied and tested on segmented river systems in the Pacific Northwest. The state-space structure of the water temperature model described in previous work is extended to include propagation of uncertainty. Model results focus on proof of concept by comparing statistics from a study of a test basin with results from other studies that have used either process models or statistical models to estimate water temperature. The results from this study compared favorably with those of selected case studies using data-driven statistical models. The results for deterministic process models of water temperature were generally better than the grid-based method, particularly for those models developed from site-specific, data-intensive studies. Biases in the results from the grid-based system are attributed to heterogeneity in hydraulic characteristics and the method of estimating headwater temperatures.

1. Introduction

[2] Awareness of the global scale of water-related issues has led to the development of gridded databases and environmental models applicable over regional, continental, and global domains. Solomon et al. [1968], for example, recognized that computer models that access gridded databases were effective tools for water resource planning at large scales (in their case, planning hydroelectric projects in Labrador and Newfoundland). Since then, the capability to apply gridded databases to water resources planning has advanced rapidly with the increasing sophistication of computer technology and data acquisition methods. Considerable effort has been given to the assessment of freshwater resources and their use at a global scale [Shiklomanov, 2000; World Resources Institute (WRI), 2000]. Development and application of simulation methods for assessing freshwater resources has focused primarily on macroscale hydrologic models [Alcamo et al., 2003]. Less attention has been given to developing methods for assessing water quality in spite of its importance for the health of global ecosystems. However, there is growing recognition of the need to expand the scope of ecosystems analyses to include large-scale environmental issues such as climate change [Kundzewicz et al., 2007]. Furthermore, because of the synergistic relationship between water quality and river discharge, it makes sense to develop methods that couple water quality models with gridded systems for simulating flows.

[3] This work focuses on water temperature because of its influence on the processes and functions of global ecosystems. Furthermore, potential increases in water temperature, whether the result of land cover change, water management, and/or climate change, threaten the biodiversity and integrity of aquatic ecosystems [Hester and Doyle, 2011]. Analysis of water temperature records show increasing trends in water temperature for many streams, rivers, and lakes in North American and Europe [Foreman et al., 2001; Kaushal et al., 2010; Morrison et al., 2002; Webb and Nobilis, 1995, 2007]. These studies attribute the increases in water temperature to a diversion of water from streams and rivers for irrigation and water supply, construction of impoundments, thermal discharge, and altered patterns of land use such as logging, farming, and urbanization. There is also concern that climate change will lead to further increases in water temperature [Meyer et al., 1999; Poff et al., 2002; van Vliet et al., 2010].

[4] The effects of increasing water temperature on aquatic ecosystems can be widespread and may be reflected in changes in the geographic range and productivity of aquatic species, and the resultant stress on sensitive freshwater species. These effects can be amplified when coupled with changes in hydrologic regimes [Mantua et al., 2010]. In the northwestern United States and southwestern Canada, for example, water temperature and streamflow are key factors in providing suitable habitat for important stocks of cold water fish [Elsner et al., 2010]. The Independent Scientific Group (ISG) [1996] attributes much of the risk to threatened or endangered Pacific salmon in the Columbia River system of Idaho, Oregon, and Washington to changes in water temperature and river discharge. There may also be impacts on societal needs for drinking water, recreation, irrigation, and industrial uses [Kundzewicz et al., 2007; Poff et al., 2002; van Vliet et al., 2010]. As a result, it will be necessary to develop water quality plans to mitigate or remediate impacts on the thermal regimes of aquatic environments. While it will be important to develop water quality plans at scales on the order of 10–100 m with deterministic process models of water temperature [Boyd and Kasper, 2003; Chapra et al., 2008; Theurer et al., 1984], there is also a need for water quality planning at larger scales, and in developing countries where water quality planning is limited [Kundzewicz et al., 2007].

[5] The water temperature model, RBM, described in this study has been applied to water quality issues in the Pacific Northwest [Perry et al., 2011; Yearsley, 2003, 2009] at length and timescales similar to other applications of the thermal energy budget method [Chapra et al., 2008; Cole and Wells, 2002; Sinokrot and Stefan, 1993]. In this study, however, RBM is linked to grid-based representations of surface climate, land cover, topography, and the physiography of stream channel systems. The grid-based modeling system concept developed in this study differs in a number of ways from deterministic process models such as HEATSOURCE [Boyd and Kasper, 2003], CEQUAL-W2 [Cole and Wells, 2002], and SNTEMP [Theurer et al., 1984] that are used to simulate water temperature. These deterministic models conceptualize river systems as a network of line segments or control volumes. The resulting simulations are meant to estimate a state (water temperature) in specific, idealized, natural river segments. In the grid-based approach (Figure 1), the hydrologic and water temperature models simulate the state variables essentially at nodes that aggregate stream or river properties in each cell rather than the temperatures of specific stream segments. Nodes are connected by channels in a network derived from a digital elevation model (DEM). Developing the modeling system with a grid-based approach has the advantage of providing access to the extensive gridded databases for model-forcing functions. Access to the extensive gridded databases will increase the capability for simulating water temperature and, ultimately, other water quality constituents in large river systems.

Figure 1.

Schematic of link-node network for the grid-based macroscale hydrologic/stream-temperature modeling system. Stream order, as defined here, is used in the context of the semi-Lagrangian stream-temperature model network rather than the standard Strahler definition.

[6] The water temperature model, RBM, described in earlier works [Yearsley et al., 2001; Yearsley, 2009] is formulated with state-space structure. It has been used in those and other studies [Perry et al., 2011] to obtain nominal solutions to the thermal energy budget equations, where the nominal solution is the solution for which model uncertainty is zero (deterministic model). This paper describes further development of RBM to allow for propagation of uncertainty of water temperature estimates within the grid-based system.

[7] The objective of this work is to describe an integrated grid-based hydrologic/water-temperature model system, where the water temperature model is formulated in a state-space structure. Although the ultimate goal is to develop methods for large-scale analyses of the type that have been done with hydrologic models alone, the concept is tested here on a relatively small, but data-rich, river basin (it is, however, intended for application to larger rivers). Performance standards of models derived from observations alone (statistical models) and from deterministic process models developed within the framework of the thermal energy budget are compared to the results from the test basin for purposes of evaluating model robustness.

2. Model System

[8] The modeling system integrates the water temperature model, RBM, [Yearsley, 2009] with the macroscale variable infiltration capacity (VIC) hydrologic model [Liang et al., 1994] version 4.1.2. Input and output data for the two models are managed as shown in the flow diagram of Figure 2. Input data to the VIC macroscale hydrologic model are grid-based variables that include precipitation, air temperature, downward shortwave (solar) and long-wave radiation, humidity, and land surface characteristics such as elevation, land cover type, and soil characteristics. Inputs to the VIC hydrologic model (surface meteorologic forcings) and outputs (stream discharge and surface energy budget terms) provide many, but not all, of the parameters necessary for simulating water temperature. Although parameter estimation is an issue in all model building, the scale of river basins that can be simulated with the proposed modeling system has the potential for greatly increasing the number of degrees of freedom in the structure of both the hydrologic model, VIC, and the water temperature model. Previous applications of the VIC model have relied on methods of aggregation and downscaling from other models or observations for purposes of parameter estimation [Hamlet et al., 2010; available at http://www.hydro.washington.edu/2860/report]. For the test basin in this study, there are results from field studies that can be upscaled within the context of the grid-based system for purposes of parameter estimation. More generally, data from gridded systems, as in the works of Vassolo and Döll [2005] and van Vliet et al. [2010], from satellite imagery [Durand et al., 2008], or the analysis of large data sets [Allen et al., 1994; Mohseni et al., 1998] might be used to estimate parameters other than those simulated by VIC.

Figure 2.

Flow diagram for grid-based macroscale hydrologic/water temperature modeling system.

2.1. Variable Infiltration Capacity Model (VIC)

[9] The variable infiltration capacity model (VIC) [Liang et al., 1994] is a semidistributed macroscale hydrological model that balances both the water and (optionally) surface energy budgets within a grid cell. VIC has been widely applied to study the effects of drought [Luo and Wood, 2007], floods [Hamlet and Lettenmaier, 2007], changes in snowpack [Hamlet et al., 2005], and impacts of climate change on water resources [Elsner et al., 2010] and energy supply [Hamlet et al., 2009]. The model is driven by daily or subdaily precipitation, air temperature, wind speed, downward shortwave and long-wave radiation, and vapor pressure. It simulates evapotranspiration, and surface and subsurface runoff from each grid cell at daily or subdaily time steps. Meteorological inputs can be taken either directly from climate models (e.g., when coupled to a climate model) or from gridded or in situ station observations. When using observed meteorological inputs (as in this study), the available forcings are often restricted to daily minimum and maximum air temperature, daily precipitation, and daily average wind speed. In this case, shortwave and long-wave radiation and vapor pressure can be estimated using the algorithms of Kimball et al. [1997], Thornton and Running [1999], and the Tennessee Valley Authority [1972], and the diurnal cycle can be approximated using a spline between the maximum and minimum daily air temperature. In this study, VIC was set up to run at a daily time step (hourly in the presence of snow) and to balance only the water budget.

[10] VIC is parameterized to simulate important features of river systems that include snowfall and melting and the effects of topography, vegetation, and soil types on basin hydrology. Input from a DEM is required to characterize the topographic effects on temperature and precipitation in the simulated watersheds. VIC simulates the soil temperature profile using the approximation of Liang et al. [1999].

[11] As VIC simulates the runoff and base flow in grid cells one at a time and independent of one another, a separate routing model must be used to simulate streamflow. The runoff and base flow from each cell simulated by VIC are routed as surface flow in stream “channels” representing all of the stream channels in each grid cell. The routing model used in this study is based on the work of Lohmann et al. [1996]. The simulated runoff in each cell is convolved with a unit hydrograph to produce a hydrograph of streamflow from the point of origin to the channel network. The streamflow from each cell is then routed through a channel network derived from a DEM with linearized St. Venant equations. In addition, the water temperature model requires stream depth and an effective reach speed to solve the one-dimensional, time-dependent thermal energy budget equation. As described below, these variables are estimated with empirical methods that relate hydraulic properties of rivers and river basin characteristics.

2.2. Water Temperature Model

[12] The development of the thermal energy budget model with state-space structure is derived from the model of nonlinear discrete-time dynamic systems [Schweppe, 1973]:

display math
display math

where

C() is a vector of state variables,

z() is a vector of observations of the state variables,

F() is a nonlinear vector function of the state variables,

G() is a matrix for the uncertain processes, w(),

B() is a matrix for the known inputs, u(),

u() is a vector of known inputs,

H() is a vector mapping state variables to a measurement device,

w, v are vectors of uncertain processes,

n are the number of time steps, n = 0, 1, 2, … , N, and

Δ is the computational time step.

[13] The methods for propagating uncertainty and Bayesian estimation (smoothing, filtering, and predicting) can be developed from (1) and (2) if the system model is linearized about the nominal solution to the thermal energy budget equation [Beck, 1987; Schweppe, 1973]

display math

[14] For the linearized system, the Taylor series expansion of (1) about the nominal solution gives the following result for propagating uncertainty, Γ:

display math

where F(1) is a matrix of the first-order terms in the Taylor series expansion of (3),

display math

and

display math

The derivation of (4) assumes that the uncertain processes, w, have Gaussian distributions with zero mean, and the differences between the nominal solutions, Cnom, and the true states, C, are small enough that terms higher than first-order in the Taylor series expansion of F() in (4) can be neglected.

[15] The discrete-time thermal energy budget in a Lagrangian frame of reference [Yearsley, 2009] having the form of (1) is given by

display math

where

T() is the water temperature, °C,

ρ is the density of water, Kg m−3,

Cp is the specific heat capacity of water, J kg−1 °C−1,

FT are the thermal energy budget components that are a function of the water temperature, T,

hT() is the vector transpose of the known inputs to the thermal energy balance that is not a function of the water temperature, T(),

D(nΔ, xj) is the channel depth at time, nΔ, and distance, xj, m,

Φ() are the advected sources of thermal energy from tributaries, groundwater, and hyporheic flow, °C s−1,

wT(nΔ) is the uncertain process for the thermal energy budget at time, nΔ,°C,

Δ(xj) is the time increment for the Lagrangian parcel to travel from starting point, inline image, to the boundary of the j-th grid cell at xj.

[16] The time, Δ(xj), to traverse the j-th grid cell is

display math

where

U( j) the river speed in the j-th grid cell, m s−1,

inline image is the upstream location of a Lagrangian parcel, m,

xj is the downstream location of a Lagrangian parcel after a time of travel, Δ(xj), m, and

ΣΔ(xj) = Δ, the computational time step, s.

[17] The water temperature model, RBM, that solves the thermal energy budget equation (7) with a semi-Lagrangian numerical method [Yearsley, 2009], is used to simulate nominal water temperatures, Tnom, where for the nominal solution, wT(nΔ) is zero. RBM has been used to simulate nominal daily water temperatures in the Columbia River and its major tributaries in Washington and Idaho [Yearsley, 2009] and the Klamath River of southern Oregon and northern California [Perry et al., 2011].

[18] In the semi-Lagrangian framework, parcels always end on a grid cell boundary. The starting point, determined in this model by reverse particle tracking for the duration of the computational time step, Δ, may or may not be a cell boundary. When the starting point is not a cell boundary, the starting temperature is determined by interpolating between upstream and downstream boundaries using second-order Lagrangian polynomials.

[19] The results for the nominal solution to the thermal energy budget method developed by Yearsley [2009] can be extended by treating parameters in the function, hT() as augmented state variables in the state vector, C(),

display math

where

qsw (nΔ, xj) is shortwave radiation, W m−2 s−1,

qlw(nΔ, xj′) is long-wave radiation, W m−2 s−1,

Uwind (nΔ, xj′) is the wind speed, m s−1,

Tair(nΔ, xj′) is the air temperature, °C,

evapor(nΔ, xj′) is the vapor pressure of air above water surface, mb, and

D(nΔ, xj′) is the water depth, m.

[20] Equation (4) can be applied to propagate the uncertainty of water temperature estimates given uncertainty in the components of the thermal energy budget. Specific details are given in the appendix. The nominal solution to the time-dependent equations of the one-dimensional thermal energy budget, the basis for the development of (3) and (4) in the grid-based system, is described more fully by Yearsley [2009].

[21] Nominal state estimates for river systems idealized as line segments rather than as a grid-based network produced results in the Columbia and Klamath River Basins that are the basis for the assumption that differences between the nominal solution and the true state are sufficiently small [Yearsley, 2009; Perry et al., 2011]. In those examples, the water temperature observations were assumed to represent the true state. As part of the proof of concept for the grid-based modeling system, additional tests of the nominal solution and propagated variances of water temperature estimates for the grid-based approach are described below.

[22] The Lagrangian coordinate, xj, the location of a parcel at time, (nΔ), is a function of the stream speed, U, and is not explicit in the state-space model of (7). As a result, any uncertainty associated with stream speed is not included in (4). In this study, uncertainty in state estimates due to uncertainty in estimating the Lagrangian coordinate from the stream speed, U, is treated separately from (4).

3. Model Tests

3.1. Study Area

[23] The Salmon River Basin, Idaho (Figure 3) was chosen as the prototype for testing the combined hydrologic and water temperature modeling system. Streamflow and water temperature were simulated at a spatial resolution of 1/16th degree. The Salmon River is a major tributary of the Snake River and drains an area of ∼36,300 km2 in Central Idaho. The mean annual streamflow of the Salmon River at the US Geological Survey (USGS) streamflow gaging station (USGS gage 1331700) near Whitebird, Idaho, is 316 m3 s−1 for the period of record. Because of its importance for salmon habitat, the Salmon River Basin is being studied by a number of natural resource agencies. These studies include a network of water-temperature monitoring sites. This network is well-suited for testing the concept described here because of the length and quality of the record. Monitoring sites were selected from NOAA (available at https://www.webapps.nwfsc.noaa.gov/WaterQuality/) and USGS monitoring programs (available at http://waterdata.usgs.gov/id/nwis/qw) based on length of record. Table 1 and Figure 3 show the sites used in this study.

Figure 3.

Location map for Salmon River Basin.

Table 1. NOAA and USGS Monitoring Sites Used to Compare Simulated and Observed Water Temperatures in the Salmon River Basin
Site Name (Gage)StreamMeasurement TypeRiver (km)Drainage Area (km2)Latitude/Longitude
Valley Creek (13295000)Valley CreekTemperature0.4038144.223°N
  Flow  114.930°W
Yankee Fork (13296500)Salmon RiverTemperature590.5207744.268°N
  Flow  114.733°W
Salmon River at Salmon (13302500)Salmon RiverFlow416.6967945.18°N
     113.90°W
SF Salmon near Krassel (13310700)South Fork Salmon RiverFlow63.985545.99°N
     115.73°W
Little Salmon at Riggins (13316500)Little Salmon RiverFlow0.8149245.41°N
     116.33°W
Whitebird (13317000)Salmon RiverTemperature86.436,30045.751°N
  Flow  116.324°W
SawtoothSalmon RiverTemperature617.277744.147°N
Hatchery    114.882°W
Marsh CreekMiddle Fork Salmon RiverTemperature179.818944.400°N
     115.167°W
Knox BridgeSouth Fork Salmon RiverTemperature111.525944.658°N
     115.703°W
Thomas CreekSouth Fork Salmon RiverTemperature97.551844.717°N
     115.017°W
Secesh RiverSecesh RiverTemperature26.151045.215°N
     115.810°W
Profile CreekaProfile CreekTemperature44.96°N
     115.43°W
Vanity CreekaVanity CreekTemperature44.55°N
     115.06°W
Morse CreekaMorseTemperature44.61°N
 Creek   113.82°W

3.2. Hydrologic Simulations

[24] Simulations using the VIC macroscale hydrologic model require an extensive data collection and parameter estimation process. Gridded data sets prepared for previous applications of VIC to streams and rivers in the Columbia River Basin by Elsner et al. [2010] were used for this study. Gridded daily values of precipitation, maximum and minimum air temperature, and wind speed are the primary forcing functions for VIC simulations. Other forcings (downward short- and long-wave radiation, and vapor pressure) are derived from the daily temperature and temperature range using methods described by Maurer et al. [2002]. The forcing data set spans the period 1915–2006, and has a spatial resolution of 1/16th degree latitude by longitude.

[25] VIC simulations require four parameter files that include a soil parameter file, a vegetation parameter file, a vegetation library, and a snowbands parameter file. All four parameter files used for this study are essentially as described by Elsner et al. [2010]. Runoff and base flows were simulated with VIC for the period 1950–2006. The resulting runoff and base flows were routed through the 1/16th degree grid system with a version of the Lohman et al. [1996] routing model modified for the structure of the integrated modeling system (Figure 2). Simulated and observed mean daily flows at selected USGS gaging stations (Table 1) in the Salmon River Basin are shown in Figure 4.

Figure 4.

Daily average simulated and observed flows at selected USGS gage locations in the Salmon River Basin.

[26] Metrics used in this study to assess the simulations results include the Nash-Sutcliffe efficiency index (NSC) [Nash and Sutcliffe, 1970], the linear correlation coefficient, R2, center of timing of flow (CT), mean annual flow (MA), and mean summer flow (MS). CT is defined as the date at which half of the annual flow has been exceeded (i.e., the median) and MS was calculated as the mean flow for the period beginning on the first day after 1 June when flows fall below the mean annual flow and ending 30 September, regardless of the starting date [Wenger et al., 2010]. The metrics for selected gaging stations are shown in Table 2.

Table 2. Mean Annual Flow, Mean Summer Flow, Center of Flow Timing, Nash-Sutcliffe, and Sample Correlation Coefficients for Simulated and Observed Stream Flows at Selected USGS Gages in the Salmon River Basin
GageMean Annual Flow (MA) (m3 s−1)Mean Summer Flow (MS) (m3 s−1)Center of Flow Timing (CT) Day of YearNash-Sutcliffe CoefficientCorrelation Coefficient, R2
GageVICGageVICGageVIC
132950005.94.023.261.681601420.380.71
1329650028.521.017.18.621631440.470.73
1330250055.752.034.528.81651480.400.72
1331070014.811.05.913.411511540.830.93
1331650021.815.08.525.431471540.750.90
13317000316.0295.0164.0128.01571560.860.92

[27] MA and MS show a negative bias for all six selected gaging stations. Lower relative bias in MA at the two gaging stations with the greatest drainage area (Salmon River at Salmon and Whitebird) is consistent with the hypothesis that VIC simulations show less bias at larger scales [Wenger et al., 2010]. Timing of low flows, however, is particularly important and simulated flows at the three gaging stations in the eastern region of the basin (Valley Creek, Salmon River below Yankee Fork, and Salmon River at Salmon) have CT's that are more than 2½ weeks earlier than the observed. Differences in the CT's of the simulated and observed peak flows at these gaging stations results in much poorer performance in terms of efficiency (NSC) and linear correlation, R2. These results for the Salmon River Basin are similar to the findings of Wenger et al. [2010] in their comparison of VIC simulations of daily flows with observed hydrographs at 55 US Geological Survey (USGS) gaging stations in the Columbia River system with drainage areas of 27 km2–2318 km2. They found a median value of NSC of 0.43 for the 55 hydrographs in their study and values of NSC that varied between −10.05 and 0.81. They were also unable to predict low flows accurately at daily time steps at the scale of their study and attributed these issues to a number of factors that are also present in hydrologic simulations of streamflows in the Salmon River Basin.

[28] These include negative bias associated with snowmelt regions, unmodeled within-cell variability, limitations of calibration, or errors in interpolating weather station data. In the absence of biases in timing and the volume of low flows, simulated streamflows and a knowledge of stream hydraulic geometry can be used to estimate stream speed, U(j), in (8), that are essential for solutions to (7). A method for estimating both width and depth, based on geographic information system (GIS)-derived characteristics from Ames et al. [2009], was used in this study. In their study of 98 unregulated and undiverted streams in Idaho, they found the following relationships for estimating stream depth, D, and width, W as a function of basin drainage area, Ad,

display math
display math

where D is the stream depth, m, at the 2-yr flood, W is the stream width, m, at the 2-yr flood, and Ad is the drainage area, km2. Equations (10) and (11) can be used to estimate a cross-sectional area in each of the line segments that link nodes. Stream speed, U(j), needed to estimate the time, Δ(xj), in (8), which can then be calculated by dividing the simulated flow exiting a cell by the cross-sectional area.

[29] Because of the nature of the biases in simulated flows during the summer, estimates of U(j) from simulated flows often result in unrealistically low values. Furthermore, (10) and (11) are constant in time and would give rise to unrealistically low stream speeds, even in the absence of negative bias in the simulated flows. As a way of accounting for bias at low flow, a lower limit was imposed on the minimum stream speeds estimated from the simulated stream flows with (10) and (11). Limits were based on an analysis of 20 USGS gaging stations in the Salmon River Basin where cross-sectional characteristics are routinely measured (available at http://waterdata.usgs.gov/id/nwis/measurements). Minimum stream speeds for these 20 gaging stations ranged from 0.14 m s−1 to 0.42 m s−1. Based on these data, two thresholds for stream speed were evaluated spanning the range of minimum values. In case I, stream speeds were not allowed to go below 0.15 m s−1 and in case II they were not allowed to go below 0.45 m s−1.

3.3. Stream Temperature Simulations

[30] The stream temperature model, RBM, assumes that an exchange of energy across the air-water interface and advected energy from tributaries and point sources capture the important features of temperature change in rivers dominated by advection. The results from the hydrologic simulations described above provide estimates of hydraulic properties and stream speed. The disaggregated meteorological forcing from the VIC hydrologic simulations provides the data required to estimate the flux of thermal energy at the air-water interface for each grid cell. Although the flux of thermal energy from the stream bottom is also a surface-area related phenomenon, it is generally small [Sinokrot and Stefan, 1994] and assumed to be negligible in this study. Shade from topography and riparian vegetation can significantly affect the thermal energy budget of streams and rivers [Sinokrot and Stefan, 1994; Boyd and Kasper, 2003]. However, since the objective of this work is essentially development and demonstration of the concept, the thermal energy budget was kept simple; the same components as those in the earlier development of the nominal solution [Yearsley, 2009] were kept in this solution.

[31] Initial conditions for water temperature as a function of time for the headwater nodes of all stream reaches are also required input. In this study, estimates of headwater temperature were based on the work of Mohseni et al. [1998], a widely applied statistical method to establish relationships between air and water temperatures. Their approach uses local observations of water and air temperatures to develop a nonlinear regression model of the form:

display math

where

Thead is the daily headwaters temperature, °C,

Tsmooth is the smoothed air temperature, °C, and

α, β, γ, μ are the regression parameters.

[32] The parameters, α = 16.1, β = 12.1, γ = 0.12, and μ = 0.1, were estimated from data collected and reported by Donato [2002] during the summer of 2000. These data are limited in duration, spanning only the period 15 July 2000 to 10 September 2000, but are from locations in second-order streams at relatively undisturbed sites. The selected sites were chosen for their low maximum daily average temperatures and their proximity to their respective headwaters. The four parameters were estimated using least squares minimization. The smoothed air temperatures, Tsmooth, were obtained from daily air temperatures, Tair, output by the VIC hydrologic model and smoothed in the following way:

display math

[33] In addition, the estimates from (12) were adjusted for elevation using a value of −0.4°C per 1000 m based on the work of Donato [2002] and Isaak et al. [2010].

[34] The data described above were organized following the flow diagram shown in Figure 2. Simulations of streamflow and water temperature were performed on daily time steps for the 1057 1/16th degree grid cells composed of the Salmon River Basin for the period 1950–2006. Simulated estimates of the average daily nominal water temperature for the two scenarios of stream speeds (cases I and II) are compared with observed values at four of the sites listed in Table 1 and shown in Figure 5. The sites were selected to represent a range of drainage basin areas to show how results vary with stream size.

Figure 5.

Simulated and observed daily-average water temperatures at selected sites in the Salmon River Basin.

[35] Statistics that assess the quality of the fit for the differences between the nominal state estimates and the observed include the Nash-Sutcliffe coefficient (NSC) to assess the efficiency of the fit, the linear correlation coefficient, R2, the root-mean-square deviation (RMSD) between observed (O) and simulated (P),

display math

and bias (BIAS),

display math

[36] One of the objectives of this work is to evaluate the results from the grid-based approach in light of other approaches for estimating stream temperatures. A number of studies from the literature were chosen for purposes of comparison. The studies represent a variety of methods, including deterministic process models [Foreman et al., 1997; Sinokrot and Stefan, 1993; Sullivan and Rounds, 2004], multiple regression models [Isaak et al., 2010; Pilgrim et al., 1998; van Vliet et al., 2010], time series models [Caissie et al., 1998], and neural network models [Risley et al., 2002]. The number and type of metrics used to assess model performance vary from study to study, but all report some measure of the second moment of the difference between simulated (predicted) and observed, similar to the RMSD of (14). They are variously described as root-mean-square error (RMSE), root-mean-square (RMS), standard error of estimate (SEE), and standard error of prediction (SEP). Measures of the second moment of the difference between simulated and observed from these studies are shown in Table 3. Measures of RMSD for the simulated results from the grid-based approach, as well as the other statistical measures described above, are shown in Table 4 for selected monitoring sites in the Salmon River Basin.

Table 3. Performance Statistics Found in Other Studies Using Statistical (S) or Process (P) Models to Estimate Water Temperaturea
River SystemStatistical SeasureModelSimulation Period (Timescale)Statistic Value (°C)
  • a

    In the Isaak et al. [2010] studies (T) indicates the results for training data and (V) indicates results for “validation” data.

Five Minnesota riversSSEMNSTRM (P)Days0.16–1.04
  [Sinokrot and Stefan, 1993]Hours 
Santiam River (Oregon)RMSECE-QUAL-W2 (P)Months0.50–1.16
  [Sullivan and Rounds, 2002]Hours 
Fraser River (British Columbia)RMSFJQHW97 (P) [Foreman et al., 1997]Months Daily0.72–1.50
Columbia River (Idaho and Washington)RMSERBM (P)Years0.61–0.97
  [Yearsley, 2009]Daily 
Klamath River (California and Oregon)RMSERBM (P)Years0.82–1.5
  [Perry et al., 2011]Daily 
Catamaran BrookRMSEStochastic (S)Years0.59–1.68
  [Caissie et al., 1998]Daily 
39 Minnesota riversSEPRegression (S)Years 
  [Pilgrim et al., 1998]Daily3.53
   Weekly2.58
   Monthly1.91
   Yearly1.25
Miramichi RiverRMSEEquilibrium (P)Years0.95–1.95
  [Caissie et al., 2005]Daily 
13 Rivers worldwideRMSEMohseni regression (S) [Van Vliet et al., 2010]Years Daily1.7–3.5
Boise River Basin, IdahoRMSERegressionYears2.51–2.85(T)
  Nonspatial (S)Weekly2.78–3.63(V)
  [Isaak et al., 2010]  
Boise River Basin, IdahoRMSERegressionYears1.41–1.60(T)
  Spatial (P/S) [Isaak et al., 2010]Weekly2.51–2.85(V)
Table 4. Bias (BIAS), Root-Mean-Square Deviation (RMSD), Nash-Sutcliffe Efficiency (NSC), and Linear Correlation Coefficient, R2, for Simulated Water Temperature Using the Mohseni Method to Estimate Headwaters Temperaturesa
Sampling SiteBIAS (°C)RMSD (°C)NSCR2
Case ICase IICase ICase IICase ICase IICase ICase II
  • a

    Minimum stream speed for case I = 0.15 m s−1 and for case II = 0.45 m s−1.

Salmon River at Whitebird0.51−1.082.121.690.830.850.950.95
Marsh Creek2.521.722.922.240.230.590.950.94
Salmon River below Yankee Fork2.060.832.251.410.710.920.970.97
Valley Creek1.450.502.241.560.770.910.960.97
Knox Bridge2.451.922.372.080.570.700.960.95
Thomas Creek1.020.432.001.560.760.870.960.96
Salmon River at Sawtooth Hatchery1.09−0.113.382.130.480.810.930.94
Secesh River2.742.102.742.330.470.750.900.91

3.4. Uncertainty Analysis

[37] Uncertainty in the simulations of water temperature arises from several sources including uncertainty in initial (headwaters) water temperature, meteorologic forcings downscaled from large-scale analyses, river hydraulic properties estimated with empirical coefficients (10) and (11), and flows simulated by the macroscale hydrologic model, VIC. Uncertainty in simulations because of uncertainty in initial conditions and meteorologic forcings are propagated as the time-dependent variances in (4). The state-space formulation of (4) for the water temperature model uses the equations given in the appendix. Equation (4) is an approximate solution, given the stated assumption of the Gaussian errors and the small differences between nominal and true state estimates. The elements of the covariance matrix, Q(), describing the uncertainty, w(), in (6), are measures of the uncertainty associated with downscaling the meteorologic forcing functions and estimating the depth with (10). For this study, variances for the meteorologic forcing variables were based on levels of uncertainty reported by Maurer et al. [2002]. The variance of the depth as estimated by (10) is based on the results of Ames et al. [2009]. Specific values of the variance used to propagate uncertainty are shown in Table 5. Q(1,1), the variance of wT(nΔ) in (7), was assumed to be zero such that all of the propagated variance for the water temperature was a result of uncertainty in the meteorologic forcings and water depth. The initial value of the variance of water temperature, Γ11(0), in (4) was assigned a value of 4.0°C2 to represent the uncertainty in estimating the headwaters temperatures.

Table 5. Values of Variance Used to Define the Parameter, Q(), in (6)
Water Temperature (°C2)Long-wave Radiation (J/s/m2)Shortwave Radiation (J/s/m2)Wind Speed (m/s)2Air Temperature (°C)2Vapor Pressure (mb)2Depth (m2)
0.0625.0625.00.094.00.250.36

[38] After specifying the required parameters, (A1)(A4) can be solved to provide insights into the way uncertainty in the estimates of water temperature is propagated due to uncertainty in specifying headwaters temperatures and meteorologic forcings. The nature of the thermal energy budget for streams is such that uncertainty in initial conditions will decay with time depending on water depth and the rate at which heat is lost due to back radiation and sensible and latent heat transfer [Edinger et al., 1968]. In this regard, propagated variances due only to uncertainty in headwaters temperatures along a longitudinal section of the main stem Salmon River on 1 August 2000 are shown in Figure 6 for purposes of illustrating the concept as well as providing a measure of the length scale of headwaters effects. The results from the two scenarios limiting stream speeds (cases I and II) are shown to illustrate the effect of stream speed on the longitudinal distribution of uncertainty. In this example, the main stem Salmon River, the influence of the headwaters temperatures have nearly disappeared within ∼150 km for case I (lower minimum stream speed) and within ∼200 km for case II. Uncertainty in the simulation results downstream from these locations will, therefore, depend on the uncertainty in estimates of the thermal energy budget.

Figure 6.

Propagation of downstream standard deviation due to uncertainty in estimates of headwaters temperatures for 1 August 2000.

[39] Structural uncertainty due to uncertainty in meteorologic forcing is a function of the nominal estimates of the state variables and the magnitude of the parameters, Q(), in (6). This uncertainty persists after uncertainty in the headwaters conditions has decayed. Of interest, is the variance, Γ11(nΔ), of the water temperature model estimates. The time history of Γ11(nΔ) at each node includes whatever residual remains of the uncertainty in the headwaters estimates, Γ11(0), plus the structural uncertainty estimated with (4). One standard deviation, estimated as the square root Γ11(), is plotted for selected monitoring sites as error bars in Figure 7. For the structure of the integrated hydrologic/stream-temperature modeling system developed in this study, the method of propagating variance captures much of the structural uncertainty in water temperature estimates due to uncertainty in the surface thermal energy budget.

Figure 7.

Simulated (RBM) water temperatures plus one standard deviation compared with observations in the Salmon River Basin. Umin = 0.45 m s−1. Simulation year 2000.

4. Discussion

[40] For the Salmon River Basin, the test case used for this study, measures of the mean square difference between simulated and observed values using the grid-based approach for simulating water temperature are comparable to the results from selected studies that used statistical models (S) (Tables 3 and 4). The best results, in terms of this metric, are where site-specific, data-intensive studies were used to develop parameters for deterministic process models (P) [Sinokrot and Stefan, 1993; Sullivan and Rounds, 2004]. The resulting simulations using deterministic process models have generally lower standard deviations than the statistical models and the grid-based model developed here. However, the objectives of the statistical models of Isaak et al. [2010], Pilgrim et al. [1998], and van Vliet et al. [2010] are more similar to the goals of the grid-based approach represented by the integrated hydrologic/stream-temperature model of this study. These studies are for using models to extend the use of data to broader scales of analysis, while the objective of the process models has been to address water quality issues in specific stream segments. It is generally the case that more site-specific data can improve model results, as shown by the improvement in almost all of the statistical measures for the Salmon River Basin (Table 4) when basin-specific data for stream speeds are used in the simulations.

[41] Model bias is also an important measure of model performance. For the results of this study, bias in simulated temperatures is due in great part to the magnitude of the stream speeds during low flow periods and estimates of headwaters temperatures. The effect of stream speed on bias at low flows is apparent in the results of Table 4. The effect is greater in the upper reaches of the stream segments, although the extent of the headwaters influence is also a function of stream speed. This is illustrated in Figure 6 showing the propagated variance due to uncertainty in the headwaters estimate for the two cases of minimum stream speed. The variance decays with time due to the same dynamics that cause stream temperatures to tend toward equilibrium. The time constant associated with this decay is a function of surface thermal energy loss terms and water depth [Edinger et al., 1968] at any monitoring site. The contribution of headwaters uncertainty to variance will, as a result, be greater at a specific site for higher speeds because the travel time from the headwaters to the site is less.

[42] As the effect of headwaters uncertainty decays in the downstream direction, the role of stream speed becomes more important. When stream temperatures are initially lower than the equilibrium temperature, as is almost always the case for high-altitude streams like those in the Salmon River Basin, travel time will be an important factor in determining the bias. In the case of the downstream effects due to differences in stream speed, the surface thermal energy budget is the control rather than the headwaters temperatures. When stream speeds are low there will be more time to reach equilibrium at a given location and, if the equilibrium temperature is higher than the initial temperature, simulated temperatures will also be higher.

[43] The findings with respect to stream speed and headwaters temperatures suggest that more site-specific data can lead to improved results for the grid-based approach. Nevertheless, it is encouraging that based on the statistical measure used here for assessing model performance, the integrated modeling system performs as well or better than the statistical models and within the range of some site-specific applications of process models.

5. Conclusions

[44] The structure of the VIC/RBM hydrology and stream-temperature modeling system (Figure 1) is designed to access the growing inventory of information available in gridded databases. However, the number of parameters that must be specified to run the modeling system is very large. Rarely, if ever, are there sufficient data for estimating all model parameters. As a result, it is necessary to estimate the parameters by aggregation (land surface characteristics), by upscaling results from localized studies (the Mohseni method for estimating initial headwaters temperatures), and by downscaling results from large-scale studies (meteorologic forcing). The robustness of the modeling system under these conditions can be measured by how model performance compares to results from other studies, and by the way model response is affected by changes in important model parameters. In this study, differences between observed and simulated water temperatures have root-mean-square deviations comparable to those of other studies of water temperature (Tables 3 and 4). The model response of the integrated model system is affected by changes in hydrology and initial conditions, although the resulting metrics are still comparable to those of other studies.

[45] The VIC/RBM modeling system described in this study provides a basis for linking a grid-based hydrological model to a stream-temperature prediction method applicable to intermediate (thousands of km2) and larger river basins. The results also suggest ways of improving performance in this specific case and can provide guidance for other grid-based applications. Although the Salmon River Basin is near the lower end of this range, application of the methods to large river basins such as the Columbia or the Mississippi is limited only by the availability of gridded data sets for model parameters.

Appendix A:: Propagation of Uncertainty for State Estimates

[46] The nonlinear form of the state-space water temperature model for the portion of the stream segment, which has no advected sources (Φ[nΔ, xj] = 0), is

display math

where, wT(nΔ) is assumed to be zero and FT[] and hT() are obtained from the elements of net transfer of thermal energy across the air-water interface: net transfer equals long-wave radiation plus shortwave radiation minus back radiation plus latent heat transfer plus sensible heat transfer or:

display math

where

σ is the Stefan-Boltzman constant, 5.682 × 10−8 J °K−1,

ρ is the density of water, 1000 Kg m−3,

λ is the latent heat of vaporization, 2.4995 × 106 J kg−1,

N is the empirical constant, 1.59 × 10−9 s−1 m−1 mb−1,

rb is the Bowen ratio = 0.64,

Tair is the air temperature, °C,

e0 is the vapor pressure in the air, mb, and

eW is the saturation vapor pressure at water temperature

equal to inline image, mb.

[47] Replacing FT[] and hT() in (A1) with the specific terms in (A2) and rearranging gives:

display math

where the back radiation from the water surface has been linearized over the temperature range 0°C–25°C as

display math

The state space structure of the remaining state variables in (9) is of the form

display math

The uncertainty in (4) is propagated by keeping only the first-order terms in a Taylor series expansion as in (5). The Taylor series expansion of the state variables in (A4) is straightforward, while that of the thermal energy budget (A3) is

display math
display math
display math
display math
display math
display math
display math
display math

[48] The uncertainty of advected sources is propagated with the first-order methods of Lettenmaier and Richey [1979]

display math

where Γds refers to the resulting uncertainty when main stem (main) sources are combined with tributary (trib) sources.

Acknowledgments

[49] Data and information for this study were provided by the Columbia Basin Climate Change Scenarios Project (PI: Alan F. Hamlet), University of Washington, (available at http://www.hydro.washington.edu/2860/). GIS support was provided by Robert Norheim, Climate Impacts Group, University of Washington.

Ancillary