U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2016–1056

Prepared in cooperation with the U.S. Fish and Wildlife Service and the Bureau of Reclamation

Construction, Calibration, and Validation of the RBM10 Water 

Temperature Model for the Trinity River, Northern California
Cover: Photograph showing sunrise on the Trinity River, northern California. 

Photograph by Edward C. Jones, U.S. Geological Survey, October 29, 2008.

 
Construction, Calibration, and Validation of the 

RBM10 Water Temperature Model for the Trinity River, 

Northern California 

By Edward C. Jones, Russell W. Perry, John C. Risley, Nicholas A. Som, and Nicholas J. Hetrick

Prepared in cooperation with the U.S. Fish and Wildlife Service and the Bureau of Reclamation

Open-File Report 2016–1056

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Director

U.S. Geological Survey, Reston, Virginia: 2016

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Suggested citation:

Jones, E.C., Perry, R.W., Risley, J.C., Som, N.A., and Hetrick, N.J., 2016, Construction, calibration, and validation 

of the RBM10 water temperature model for the Trinity River, northern California: U.S. Geological Survey Open-File 

Report 2016–1056, 46 p., http://dx.doi.org/10.3133/ofr20161056. 

ISSN 2331-1258 (online)
iii

Contents

Abstract ...................................................................................................................................................................... 1

Introduction................................................................................................................................................................. 2

Study Site................................................................................................................................................................ 3

Methods...................................................................................................................................................................... 3

Structure of River Basin Model 10........................................................................................................................... 3

River Geometry ................................................................................................................................................... 5

Meteorological Data ............................................................................................................................................ 6

Boundary Conditions ........................................................................................................................................... 8

Calibration and Validation.................................................................................................................................. 14

Simulating Management Scenarios....................................................................................................................... 16

Results...................................................................................................................................................................... 17

Calibration and Validation ..................................................................................................................................... 17

Alternative Management Scenarios ...................................................................................................................... 27

Flow Augmentation Effects on Water Temperature in the Lower Klamath River ............................................... 27

Discussion ................................................................................................................................................................ 27

Acknowledgments..................................................................................................................................................... 29

References Cited...................................................................................................................................................... 30

Appendix A. River Geometry, Time Series and Water Temperatures, and Prediction Error, Trinity River, 

Northern California.................................................................................................................................................... 33

Figures

Figure 1. Map showing eight locations (circles) where observed water temperature data were measured and used 

to calibrate RBM10 at the downstream end of each meteorological reach of the Trinity River drainage basin, 

northern California ..................................................................................................................................................... 4

Figure 2. Graph showing observed shortwave solar radiation at Hoopa, California, and simulated shortwave solar 

radiation extracted from the 1 × 1 km spatial grid of Daymet, 2002–03 ..................................................................... 7

Figure 3. Graph showing extent of observed water temperature data for tributary inputs, at specified river miles 

(RMs), to the Trinity River........................................................................................................................................ 10

Figure 4. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for six 

headwater and tributary inputs between river miles 111.9 and 87.8 of the Trinity River .......................................... 12

Figure 5. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for six 

tributary inputs between river miles 79.2 and 25.0 of the Trinity River..................................................................... 13

Figure 6. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for two 

tributary inputs between river miles 19.7 and 16.4 of the Trinity River..................................................................... 14

Figure 7. Graph showing extent of observed water temperature for calibrating evaporation coefficients of RBM10 

at eight locations (RM, river mile) along the Trinity River......................................................................................... 15

Figure 8. Graphs showing simulated water temperature and residuals (simulated minus observed temperature) as 

a function of observed water temperature at five locations between river miles (RMs) 107.8 and 47.6 of the Trinity 

River. ....................................................................................................................................................................... 19
iv

Figure 9. Graphs showing simulated water temperature and residuals (simulated minus observed temperature) as 

a function of observed water temperature at four locations between river miles (RMs) 31.6 and 0.5 of the Trinity 

River ........................................................................................................................................................................ 20

Figure 10. Time series of simulated (solid line) and observed water temperature (o) at river miles 107.8, 79.3, 

58.7, and 47.6, Trinity River, 2010–2013................................................................................................................. 21

Figure 11. Time series of simulated (solid line) and observed water temperature (o) at river miles 31.6, 21.3, 

12.5, and 0.5, Trinity River, 2010–2013................................................................................................................... 22

Figure 12. Validation plots showing time series of simulated (solid line) and observed water temperature (o) at 

river miles 107.8, 79.3, 58.7, and 47.6, Trinity River, 2010–2013............................................................................ 24

Figure 13. Validation plots showing time series of simulated (solid line) and observed water temperature (o) at 

river miles 31.6, 21.3, 12.5, and 0.5, Trinity River, 2010–2013. ............................................................................... 25

Figure 14. Graphs showing 34-year historical time series of daily mean water temperatures simulated at four 

locations on the Trinity River, northern California .................................................................................................... 26

Figure 15. Simulated daily mean water temperatures at river mile 5.7 in the lower Klamath River near Klamath, 

California, during 1994, under four management scenarios that include targeted flows of 2,500, 2,800, and 3,200 

cubic feet per second, augmented by Lewiston Dam bypass, and the historical baseline “No Action” alternative 

(top). ........................................................................................................................................................................ 28

Tables 

Table 1. Source of input datasets for statistical models used to estimate boundary water temperatures for the 

Trinity River ............................................................................................................................................................... 9

Table 2. Parameter estimates of the non-linear regression model (Mohseni and others, 1998) used to estimate 

water temperature for headwater and tributary inputs to the Trinity River................................................................ 11

Table 3. Best-fit parameter estimates of the wind function shown in equation 11 and goodness-of-fit statistics 

for observed and simulated water temperatures at eight locations along the Trinity River. ..................................... 18

Table 4. Best-fit parameter estimates of the wind function shown in equation 12 and goodness-of-fit statistics

for observed and simulated water temperatures at eight locations along the Trinity River ...................................... 18

Table 5. Aggregate cross-validation estimates of prediction error for evaluating observed and simulated water 

temperatures at eight locations along the Trinity River............................................................................................ 23
v 

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Length

mile (mi) 1.609 kilometer (km)

Area

square foot (ft2) 0.09290 square meter (m2)

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Density

kilogram per cubic meter (kg/m3) 0.06242 pound per cubic foot (lb/ft3)

Energy

Watt per meter squared (W/m2) 0.0002389 kilocalories per meter squared per second 

(kcal/m2/s)

International System of Units to Inch/Pound

Multiply By To obtain

Length

meter (m) 3.281 foot (ft) 

meter (m) 1.094 yard (yd) 

Area

square meter (m2) 0.0002471 acre 

square meter (m2) 10.76 square foot (ft2) 

Wind speed

Pressure

kilopascal (kPa) 10 millibar (mb)

pascal (Pa) 0.01 millibar (mb)

millibar (mb) 0.1 kilopascal (kPa)

Wind speed

meter per second (m/s) foot per second (ft/s

Energy

kilocalories per meter squared per second 

(kcal/m2/s)

8,569 Langley per day (Ly/day)

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as °F = (1.8 × °C) + 32.
vi 

Datum

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Abbreviations

MAE mean absolute error

ME mean error

NSS Nash-Sutcliffe statistic

RBM10 River Basin Model-10

Reclamation Bureau of Reclamation

RM river mile

RMSE root mean square error

USFWS U.S. Fish and Wildlife Service

USGS U.S. Geological Survey
1 

Construction, Calibration, and Validation of the 

RBM10 Water Temperature Model for the Trinity River, 

Northern California 

By Edward C. Jones1, Russell W. Perry1, John C. Risley1, Nicholas A. Som2, and Nicholas J. Hetrick2

Abstract

We constructed a one-dimensional daily averaged water-temperature model to simulate Trinity 

River temperatures for 1980–2013. The purpose of this model is to assess effects of water-management

actions on water temperature and to provide water temperature inputs for a salmon population dynamics 

model. Simulated meteorological data, observed streamflow data, and observed water temperatures were 

used as model inputs to simulate a continuous 34-year time series of historical daily mean water 

temperature at eight locations along 112.2 river miles from Lewiston Dam near Weaverville, California, 

downstream to the Klamath River confluence. To demonstrate the utility of the model to inform 

management actions, we simulated three management alternatives to assess the effects of bypass flow 

augmentation in a drought year, 1994, and compared those results to the simulated historical baseline, 

referred to as the “No Action” alternative scenario. Augmentation flows from the Lewiston Dam bypass 

consist of temperature-controlled releases capable of cooling downstream water temperatures in hot 

times of the year, which can reduce the probability of disease outbreaks in fish populations. Outputs 

from the Trinity River water-temperature model were then used as inputs to an existing watertemperature model of the Klamath River to evaluate the effect of augmentation flow releases on water 

temperatures in the lower Klamath River.

We structured the Trinity River water-temperature model in River Basin Model-10 (RBM10), 

which uses a simple equilibrium flow model, assuming discharge in each river segment on each day is 

transmitted downstream instantaneously. The model uses a heat-budget formulation to quantify heat 

flux at the air-water interface. Inputs for the heat budget are calculated from daily mean meteorological 

data, including net shortwave solar radiation, net longwave atmospheric radiation, air temperature, wind 

speed, vapor pressure, and a psychrometric constant needed to calculate the Bowen ratio. The modeling 

domain was divided into eight reaches ranging in length from 8.8 to 20.6 miles, which were calibrated 

and validated separately with observed water temperature data collected irregularly from 1980 to 2013. 

Root mean square errors of observed and simulated water temperatures for the eight reaches ranged 

from 0.25 to 1.12 degrees Celsius (°C). Mean absolute errors ranged from 0.18 to 0.89 °C. For model 

validation, a k-fold cross-validation technique was used. Validation root mean square error and mean 

absolute error for the eight reaches ranged from 0.24 to 1.11 °C and from 0.18 to 0.89 °C, respectively.

 

1

U.S. Geological Survey.

2

U.S. Fish and Wildlife Service.
2 

Augmentation scenarios were based on historical hydrological and meteorological data, 

combined with prescribed flow and temperature releases from Lewiston Dam provided by the Bureau of 

Reclamation. Water releases were scheduled to achieve targeted flows of 2,500, 2,800, and 3,200 cubic 

feet per second in the lower Klamath River from mid-August through late September, coinciding with 

the upstream migration of adult fall-run Chinook salmon (Oncorhynchus tshawytscha). Water 

temperatures simulated at river mile 5.7 on the Klamath River showed a 5 °C decrease from the No 

Action historical baseline, which was near or greater than 23 °C when augmentation began in midAugust. Thereafter, an approximate 1 °C difference among augmentation scenarios emerged, with the 

decrease in water temperature commensurate to the level of augmentation. All augmentation scenarios 

simulated water temperatures equal to or less than 21 °C from mid-August through late September. 

Water temperatures equal to or greater than 23 °C are of particular interest because of a thermal 

threshold known to inhibit upstream migration of salmon. When temperatures exceed this approximate 

23 °C threshold, Chinook salmon are known to congregate in high densities in thermal refugias and 

show extended residence times, which can potentially trigger epizootic outbreaks such as of 

Ichthyophthirius multifiliis (“Ich”) and Flavobacterium columnare (“Columnaris”) that were the 

causative factors of the Klamath River fish kill in 2002. A model with the ability to simulate water 

temperatures in response to management actions at the basin scale is a valuable asset for water managers 

who must make decisions about how best to use limited water resources, which directly affect the state 

of fisheries in the Klamath Basin. 

Introduction

Water temperature influences important life-cycle parameters of fish populations such as 

survival and growth rates, which in turn dictate population dynamics. Fisheries and water managers are 

interested in understanding the effects of management actions on water temperature, and consequent 

effects on Pacific salmon (Oncorhynchus spp.) downstream of dams in the Klamath Basin. Aquatic 

research tools used to understand these relationships often include water-temperature modeling, fish 

life-cycle modeling, and mark-capture analyses (Perry and others, 2010, 2012, 2013; Plumb and others, 

2012; Benjamin and others, 2013). A water temperature model typically is developed first because it is 

used as input to fish life-cycle models. 

We present a water temperature model for the Trinity River, a major tributary to the Klamath 

River. Our model uses publically available simulated meteorological data and observed streamflow data 

to estimate daily mean water temperature. Previous water-temperature modeling studies of the Trinity 

River include Zedonis (1997), Deas (1998, 2002), and Watercourse Engineering (2007). We structured 

our model in River Basin Model-10 (RBM10; Yearsley, 2003, 2009) to be compatible with the RBM10 

model for the Klamath River developed by Perry and others (2011). Once calibrated and validated, the 

Trinity River model will be linked to the Klamath River RBM10 model to form a single model, thereby 

providing a unified framework for water temperature and salmon life-cycle models in the Klamath 

Basin. Simulated temperatures from the integrated Klamath/Trinity water temperature model will 

provide critical inputs to the Stream Salmonid Simulator (S3

; Russell Perry, U.S. Geological Survey, 

unpub.), a model being developed to simulate juvenile salmon production in the Trinity River. 

Our goal was to develop a model that accurately simulates daily mean water temperature along 

the course of the Trinity River, from Lewiston Dam (RM 112.2) to the Klamath River confluence. 

Simulated water temperatures from this model will be incorporated in a population model of Trinity 

River salmon as a basis to evaluate alternate management actions affecting salmon populations. Salmon 

population dynamics are affected by water temperature in many crucial ways—for example, egg 
3 

incubation rates, individual growth rates, migration timing, and disease susceptibility. The development 

of the RBM10 for the Trinity River is described in detail in this report. Observed water temperatures at 

eight locations along the river were used to calibrate the model, and the calibration and model validation 

results are presented. The calibrated model is then used to assess the effects of flow augmentation 

scenarios on water temperature in the lower Klamath River.

Study Site

The Trinity River, a major tributary of the Klamath River in northern California, has a drainage 

basin area of 2,970 mi2 (fig. 1). Elevation in the drainage basin ranges from 190 ft at the confluence to 

9,020 ft at its highest point, and includes parts of the Coast Ranges and Klamath Mountains. In the early 

1960s, the Bureau of Reclamation (Reclamation) completed Trinity Dam (RM 120) and Lewiston Dam 

(RM 112.2) near Weaverville, California, as part of a system of dams and canals to provide a stable 

supply of irrigation water to California’s Central Valley. The auxiliary outflow works at Trinity Dam 

provides an optional temperature-controlled release point for outflow, which bypasses the powerhouse. 

Bypass allows access to the colder hypolimnion, which is then discharged from Trinity Dam into 

Lewiston Reservoir, making cold water augmentation to the Trinity River possible at Lewiston Dam. 

The river from Lewiston Dam downstream to the Klamath River confluence, with parts of tributaries 

included, was designated as the Trinity Wild and Scenic River by the U.S. Congress in 1981. 

Methods

Structure of River Basin Model 10 

RBM10 is a one-dimensional water-temperature model that simulates daily mean water 

temperatures along the longitudinal profile of a river, based on a heat-budget formulation (Yearsley and 

others, 2001; Yearsley, 2009; Perry and others, 2011). The model structure and inputs are comprised of: 

(1) river geometry, (2) boundary conditions, and (3) meteorological data. The geometry of the Trinity 

River is defined by a series of river segments, each with a distinct set of continuity equations that 

characterize system hydraulics and geomorphic properties such as wetted channel width, depth, and 

water velocity as a function of stream discharge. Boundary conditions of the model are defined by major 

heat advection sources to the mainstem Trinity River, represented as daily inputs of discharge and water 

temperature from headwaters and tributaries. Based on river geometry and boundary conditions, 

RBM10 applies an equilibrium flow model that assumes streamflow passes instantaneously 

downstream, segment-to-segment, on a daily time-step. Heat-budget computations require daily inputs 

of meteorological data. Given these key model components, water temperatures are simulated using a 

mixed Eularian-Lagrangian numerical scheme that is computationally efficient and accurate (Yearsley, 

2009). In the Lagrangian frame of reference, the heat-budget model takes the following form: 

( air-water adv )

p

1 x

x

dT WH S

dt C A r = + (1) 

where 

 T is water temperature (degrees Celsius), 

 t is time, 

r is the density of water (kilograms per cubic meter), 

 Cp is the specific heat capacity of water (kilocalories per degree Celsius per kilogram),
4 

Figure 1. Map showing eight locations (circles) where observed water temperature data were measured and used

to calibrate RBM10 at the downstream end of each meteorological reach of the Trinity River drainage basin, 

northern California. 
5 

Ax is the cross-sectional area of the river (square meters) at location x along the river’s 

longitudinal axis, 

 Wx is the width of the river (meters) at location x, 

 Hair-water is the net heat flux at the air-water interface (kilocalories⋅per meter squared per 

second), and 

 Sadv is heat advected from tributaries (kilocalories⋅per meter squared per second).

River Geometry

To model the hydraulic properties of the Trinity River, the geometry of river segments in 

RBM10 is characterized by wetted channel width and cross-sectional area, both of which vary with river 

discharge. Continuity equations were used to quantify these relationships: 

W

W

b W aQ x = , (2) 

A

A

b A aQ x = , (3) 

where 

 Wx and Ax are wetted top width (feet) and cross-sectional area (square feet), respectively, for the 

segment beginning at river mile x, 

 Q is discharge (cubic feet per second), and

a and b are segment-specific parameters that need to be estimated. 

We formed 31 river segments for RBM10, each of which had unique parameters of the continuity 

equations. Segment breaks were defined based on location of tributaries and changes in bed slope such 

that each segment had a relatively constant average slope (mean reach length = 3.6 mi, range = 1.2–5.6 

mi; table A1). One segment (RMs 93.9–92.6) was later partitioned into two segments so that each 

segment contained no more than one input tributary. 

To estimate reach-specific parameters of the continuity equations, a HEC-RAS model (U.S. 

Army Corps of Engineers, 2010) for the Trinity River was used to simulate wetted top-width and crosssectional area for 112.2 river miles from Lewiston Dam (RM 112.2) to the Klamath River confluence 

(RM 0). For the upper 40 mi extending from the dam to the North Fork Trinity River (RM 72.2), HECRAS geometry data were available from a flood insurance study by the California Department of Water 

Resources (2014). The HEC-RAS geometry consisted of 785 cross-sectional elevation profiles over the 

40 mi of river channel and was compiled from bathymetric, lidar (light detection and ranging), and 

ground survey data. From the North Fork Trinity River confluence to the Klamath River, we used river 

geometry from a water-temperature model developed by Watercourse Engineering, Inc. (2007). 

Geometry from this model consisted of 1,168 cross-sectional elevation profiles over the 72.2 mi of river, 

which was constructed from a series of four different trapezoidal shapes to represent the channel profile. 

Geo-referenced cross-sectional profiles from both data sources were used to develop a single HEC-RAS 

model. This model was then used to simulate wetted channel top-width and cross-sectional area for each 

cross section at 15 exceedance flows, ranging from 2 to 100 percent of the 10-year exceedance flow 

(range = 801– 40,040 ft3

/s at the Klamath River confluence). To estimate parameters of the continuity 

equations ( , , , and WW A A aba b ) for each river segment in the RBM10 model, linear models of the form 

ln(y) = lna + blnQ were fit to top width and cross-sectional area for each river segment. These estimates 

defined the average hydraulic conditions for each of the original 31 river segments in the RBM10 river 

geometry (table A1). 
6 

Meteorological Data

Heat flux at the water surface is quantified by the simplified heat-budget formula of RBM10 

(Yearsley and others, 2001; Yearsley, 2003, 2009; Perry and others, 2011): 

H H HH H H H air-water sw a ar evap cond back =+- + + + ( ) (4) 

where 

 Hair-water is the net exchange of thermal energy across the air-water interface, 

 Hsw is shortwave solar radiation incident at the water surface, 

 Ha is longwave atmospheric radiation incident at the water surface, 

 Har is reflected longwave atmospheric radiation, 

 Hevap is evaporative heat flux, 

 Hcond is conductive heat flux, and 

 Hback is longwave back radiation from the water surface. 

Standard meteorological data on a daily time scale are required to compute the heat budget. 

Shortwave solar radiation incident at the water surface (Hsw ) and net longwave atmospheric radiation 

(Ha - Har) are direct inputs to RBM10, whereas the other heat-budget terms are calculated internally 

from inputs of daily mean air temperature (in degrees Celsius), daily mean wind speed (in meters per 

second), vapor pressure (in millibars), and a psychrometric constant needed to calculate the Bowen ratio 

(Yearsley and others, 2001; Yearsley, 2009). Cloud cover was used to estimate net longwave radiation. 

Meteorological inputs to RBM10 are spatially explicit, to account for the gradient in climate across the 

river’s longitudinal profile. We delineated eight meteorological reaches, each with a unique set of 

meteorological input data (tables A1 and A2). Reach definitions were influenced by the availability of 

sites with a sufficient record of water temperature measurements for model calibration of evaporation 

coefficients. Meteorological reaches ranged in length from 8.8 to 20.6 mi. 

In the absence of long time series of meteorological data from weather stations along the Trinity 

River, model inputs were constructed from simulated meteorological data. We compiled a 34-year daily 

historical record of meteorology from 1980 through 2013. Time series of daily minimum and maximum 

air temperature (in degrees Celsius), shortwave solar radiation (in Watts per meter squared), daylight (in 

seconds per day), and vapor pressure (in pascals) were extracted from the 1 × 1 km spatial grid of 

Daymet (Thornton and others, 1997, 2000, 2014; Thornton and Running, 1999) at the midpoint of each 

of the eight model reaches (see table A2 for latitude-longitude coordinates). Daily mean air temperature 

was calculated as the mean of the minimum and maximum daily air temperature. We computed cloud 

cover from daily mean air temperature following methods described in Flint and Flint (2008). Time 

series of daily mean wind speed (in meters per second) were extracted at the same eight locations (table 

A2), from a spatial grid of 1/16-degree resolution available from Livneh and others (2015).

Shortwave solar radiation, Hsw, was converted from units of watts per meter squared during 

daylight periods to kilocalories per meter squared⋅per second over the 24-hour period for input to 

RBM10. When compared to shortwave solar radiation observed at Hoopa, California (latitude, 41.0478; 

longitude, 123.6714; Western Regional Climate Center, 2015), the Daymet simulated data for that 

model reach (Reach 7) matched the observed signature reasonably well (fig. 2). 
7 

Figure 2. Graph showing observed shortwave solar radiation at Hoopa, California, and simulated shortwave solar 

radiation extracted from the 1 × 1 km spatial grid of Daymet, 2002–03. (kcal/m2)/s, kilocalories per meter 

squared⋅per second.

Longwave atmospheric radiation incident at the water surface (Ha) was calculated using the 

Stefan-Boltzmann law: 

4

a , H T eff a = e σ (5) 

where 

εeff is the effective emissivity of the atmosphere,

σ is the Stefan-Boltzmann constant, and 

 Ta is daily mean air temperature (kelvins) 

We used the model of Ångström (1918) for εeff as recommended by Flerchinger and others (2009), who 

compared estimates from different algorithms for longwave radiation to observed longwave radiation at 

21 sites across North American and China: 

0.067 0.83 0.18 10 , ae

clr e − = − × (6) 

where 

 ea is vapor pressure (kilopascals). 

Effective emissivity was then calculated by adjusting clear-sky emissivity for cloud cover using the 

model of Unsworth and Monteith (1975):

( ) 1 0.84 0.84 , eff clr e e = − c c + (7) 

where 

 c is the fraction of the sky covered by clouds. 

Reflected longwave radiation was calculated as H HA ar a ar = , where Aar is longwave reflectivity, which 

was set to 0.03. 
8 

All other components of the heat budget were calculated as described by Yearsley and others 

(2001) and Yearsley (2009), although it is useful to repeat the formulation of the evaporative heat flux 

because this term contains parameters required to calibrate RBM10: 

evap ( ) ( ), H L We e = ƒ ρ v a 0 − (8) 

where 

ρ is the density of water (kilograms per cubic meter), 

Lv is the latent heat of vaporization (kilocalories per kilogram), 

ƒ ( ) W is a function of wind speed ( ) W , and 

e0 is the saturation vapor pressure at the temperature of the water surface (millibars). 

The wind speed function often takes the general form: 

( ) 2 ƒ =+ + W bW cW a (9) 

where 

a , b , and c are empirical coefficients (Edinger and others, 1974). 

The specific form of ƒ ( ) W varies widely among studies (Edinger and others, 1974; Henderson-Sellers, 

1986). For example, Yearsley and others (2001) used the simplest form for ƒ ( ) W , assuming a = 0 and 

c = 0 ; they then estimated b through calibration. Coefficients of the wind speed function are calibration 

parameters in RBM10 (see section, “Calibration and Validation”). 

Boundary Conditions

Daily values of river discharge and water temperature must be provided at all boundaries where 

significant discharge enters the mainstem Trinity River. In addition to inputs for Lewiston Dam, we 

included the 14 tributaries as boundary conditions to RBM10 (table 1). Daily mean streamflow data for 

U.S. Geological Survey (USGS) streamgage 11525500 downstream of Lewiston Dam were used as the 

upstream boundary to the water-temperature model. All other boundary conditions were specified by 

assuming the difference in streamflow measured between streamgage 11525500 and USGS streamgage 

11530000 at Hoopa, California, represented the aggregate total inflow from all tributaries. To estimate 

the daily mean discharge of each tributary, the daily total inflow was then partitioned among tributaries 

in proportion to drainage basin area.

We assembled time series of observed daily water temperature for each boundary in our model 

(table 1); however, a complete time series was not available (fig. 3). To fill data gaps in the 34-year time 

series, we used a regression model that predicts weekly mean stream temperature as a function of 

weekly mean air temperature (Mohseni and others, 1998): 

( ) , 1 a s T T

e

γ β

a μ μ −− = +

+ (10) 
9 

Table 1. Source of input datasets for statistical models used to estimate boundary water temperatures for the 

Trinity River. 

[Source: Reclamation, Bureau of Reclamation; CDEC, California Data Exchange Center; HVT, Hoopa Valley Tribe; USFS, 

U.S. Forest Service; USFWS, U.S. Fish and Wildlife Service; USGS, U.S. Geological Survey. Data from Indian Creek were 

used as a surrogate for Reading Creek, which lacked a water temperature record. Station ID: A field identifier to distinguish 

the source data] 

Tributary Data Extent Source Station ID Latitude Longitude

Trinity River at Lewiston 1964–1983 USGS/Reclamation 11525500 40.7247 -122.8011

1987–1990 USFWS LWS -- --

1990–2015 CDEC LWS 40.6855 -122.9342

Rush Creek 2002–2004 USFWS RCTR1 40.7208 -122.8403

2011–2013 USFS rush1_H2O_Temp 40.7914 -122.8825

Grass Valley Creek 2004–2006 USGS 11525630 40.6867 -122.8600

2004–2013 CDEC GVT 40.6867 -122.8600

Indian Creek 2002–2013 USFWS ICTR1 40.7267 -122.7942

Weaver Creek 2000–2006 CDEC WVC 40.7193 -122.8036

Reading Creek -- --

Browns Creek 2000–2012 USFS Browns_H2O_temp 40.6260 -122.9591

Canyon Creek 2001–2013 USFWS CNTR1 40.7308 -123.0533

2005–2005 CDEC CJC 40.7403 -123.0492

North Fork Trinity River 2001–2013 USFWS NFTR1 40.7708 -123.1283

Big French Creek 2001–2013 USFWS BFTR1 40.7803 -123.3083

New River 2005–2005 USFS New River at Trinity River H2O temp 40.8459 -123.4799

2011–2013 USFS H2O_Temp_new1 40.9632 -123.3482

South Fork Trinity River 2001–2013 USFWS SFTR1 40.8894 -123.6028

Willow Creek 1995–2003 USFS Willow_MigrantTrap_nearmouth_H2O_temp 40.9436 -123.6324

Horse Linto Creek 1992–1993 USFS Near confluence ("HLINTO.wk3") -- --

1994–1994 USFS Near confluence ("HL94.wk3")

1997–2012 USFS HorseLinto_lowerbridge_H2O_temp 41.0052 -123.6071

2005–2006 USFS H2O_Temp_AREMP_CALIN 41.0052 -123.6062

Tish Tang Creek 2004–2013 HVT LTISHTANG -- --

where 

Ts is the weekly-mean stream temperature, 

μ is the minimum water temperature, 

α is the maximum water temperature, 

β is the air temperature at the point of inflection, 

γ represents the slope at the inflection point, and 

Ta is the weekly mean air temperature (degrees Celsius). 
10 

The parameters μaβ γ , , , and were estimated by least-squares regression using an optimization routine 

in the R statistical package (R Core Team, 2014). Weekly mean air temperatures for a tributary entering 

a given model reach were computed from the daily mean air temperatures of that reach. Following 

Mohseni and others (1998), we fit separate relationships to the rising and falling limb (that is, the part of 

the year when air temperature tends to increase or decrease; figs. 4-6). Given parameter estimates for the 

rising and falling limb of each stream (table 2), we then filled data gaps by predicting daily water 

temperature using a 7-day moving average of air temperature. No observed water temperature data were 

available for Reading Creek. Due to proximity, similarities in basin size and geomorphology, we used 

Indian Creek as a surrogate for observed water temperatures of Reading Creek.

Figure 3. Graph showing extent of observed water temperature data for tributary inputs, at specified river miles 

(RMs), to the Trinity River. Horizontal bars represent the chronological range of available data for each tributary. 

Data for Indian Creek were used as a surrogate for Reading Creek, which lacked a water temperature record. Data 

prior to 1980 were not used because of a lack of corresponding meteorological data.
11 

Table 2. Parameter estimates of the non-linear regression model (Mohseni and others, 1998) used to estimate 

water temperature for headwater and tributary inputs to the Trinity River. 

Tributary Meteorological 

reach Limb

First 

week of 

limb

Minimum 

water 

temperature

μ 

Maximum 

water 

temperature

α

Air 

temperature 

at the point 

of inflection

β

Slope at 

inflection 

point

γ

Trinity River at Lewiston 1 rising 51 8.7 19.3 -0.2 0.156

falling 30 4.8 23.2 14.1 0.032

Rush Creek 1 rising 51 -7.1 35.1 29.0 0.092

falling 30 4.4 23.6 11.4 0.153

Grass Valley Creek 1 rising 51 -14.0 42.0 38.9 0.056

falling 30 -0.2 28.2 17.9 0.072

Indian Creek 1 rising 51 1.3 26.7 16.4 0.131

falling 30 2.4 25.6 13.1 0.133

Weaver Creek 2 rising 51 -7.0 35.0 23.5 0.099

falling 30 -3.9 31.9 19.1 0.105

Reading Creek 2 rising 51 1.4 26.6 16.0 0.136

falling 30 2.3 25.7 13.1 0.135

Browns Creek 2 rising 51 4.6 23.4 14.0 0.207

falling 30 3.8 24.2 14.7 0.170

Canyon Creek 3 rising 51 -22.0 50.0 35.4 0.082

falling 30 1.6 26.4 12.0 0.167

North Fork Trinity River 3 rising 51 -14.0 42.0 28.7 0.095

falling 30 0.3 27.7 14.2 0.154

Big French Creek 4 rising 51 -0.5 28.5 17.4 0.114

falling 30 2.4 25.6 12.7 0.145

New River 5 rising 51 2.4 25.6 14.0 0.181

falling 30 -7.8 35.8 21.3 0.126

South Fork Trinity River 6 rising 51 -2.2 30.2 15.0 0.178

falling 30 -1.0 29.0 13.4 0.185

Willow Creek 6 rising 51 -3.9 31.9 20.7 0.101

falling 30 -1.0 29.0 16.1 0.097

Horse Linto Creek 7 rising 51 -6.5 34.5 24.3 0.093

falling 30 3.3 24.7 12.0 0.157

Tish Tang Creek 7 rising 51 1.7 26.3 15.7 0.132

falling 30 3.1 24.9 11.9 0.161
12 

Figure 4. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for six 

headwater and tributary inputs between river miles 111.9 and 87.8 of the Trinity River. Two lines in each panel 

represent the fitted non-linear regression model (Mohseni and others, 1998) used to predict water temperatures for 

rising and falling limbs of air temperature.
13 

Figure 5. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for six 

tributary inputs between river miles 79.2 and 25.0 of the Trinity River. Two lines in each panel represent the fitted 

non-linear regression model (Mohseni and others, 1998) used to predict water temperatures for rising and falling 

limbs of air temperature.
14 

Figure 6. Scatter plots of observed weekly mean water temperature and weekly mean air temperature for two 

tributary inputs between river miles 19.7 and 16.4 of the Trinity River. Two lines in each panel represent the fitted 

non-linear regression model (Mohseni and others, 1998) used to predict water temperatures for rising and falling 

limbs of air temperature.

Calibration and Validation

Calibration parameters of RBM10 include coefficients that influence how wind speed affects 

evaporative heat flux. Heat lost through evaporation is proportional to the wind speed function, ƒ ( ) W , 

and to the vapor pressure deficit, a e e 0 − (eq. 8). We assessed the fit of the water-temperature model to 

observed data using two forms of the wind function. First, we used a common form of the wind speed 

function, 

( ) , ƒ W a bW r r rr = + (11) 

where 

r is reach 1,..., 8 corresponding to each unique set of meteorological data. 

Observed temperatures at eight locations along the river were close to the end points for each reach (fig. 

7, tables A1 and A2), providing a direct correspondence among observed water temperature data, 

meteorological data, and wind-function parameters. To calibrate the model, we used an optimization 

routine to find parameter values of r a and r b that minimized the sum of squared deviations between 

simulated and observed temperatures. We calibrated one reach at a time, from upstream to downstream, 

because water temperatures at the end of reach r form the boundary conditions at the start of reach r+1. 

For example, observed water temperatures at the end of reach 1 were used to estimate 1 1 a b and .Next, 

1 1 a b and were set to their best-fit values and observed water temperatures at the end of reach 2 were 

used to estimate 2 2 a b and , and so on. 
15 

Figure 7. Graph showing extent of observed water temperature for calibrating evaporation coefficients of RBM10 

at eight locations (RM, river mile) along the Trinity River. Horizontal bars represent the chronological range of 

available data for each tributary.

We also calibrated the water-temperature model using a wind speed function with seasonal 

evaporation coefficients similar to the approach of Deas and Orlob (1999): 

( ) ,1 ,1

,2 ,2

 if April 1 November 1

 if November 1 < April 1

r rr

r

r rr

a bW d

W a bW d ƒ  + ≤ ≤ =  + < 

 (12) 

where 

 d is the day of year and

a bj rj rj , , and , 1, 2 ( ) = { } are coefficients that apply to either the period April 1 to November 1 

(“summer”) or November 1 to April 1 (“winter”).

Based on initial calibration results, we incorporated an additional calibration parameter that 

scaled the heat flux at the air-water interface. Initial calibrations indicated that the best-fit parameters of 

the evaporation function were near zero, suggesting that the calibration was minimizing evaporative 

cooling in an effort to increase simulated water temperatures. Despite minimal evaporative cooling, 

simulated water temperatures were negatively biased. The model appeared to capture the timing of 

fluctuations in temperature well, suggesting that the advection of heat was being simulated accurately. 

Therefore, we introduced an additional calibration parameter, θr , that scaled the effect of water depth 

on heat exchange at the water surface in the water-temperature model (see eq. 1): 

1 air-water adv

p rx x

dT H S

dt C D A r θ

  =   +

 

 (13) 

where 

Dx is the average depth of the river channel at location x. 
16 

The parameter, θr , was also reach-specific and estimated simultaneously with the evaporation 

parameters for each reach. 

All available data were used for calibration so that parameters were estimated across a wide 

range of annual variation in weather and hydrology. We assembled an extensive dataset comprised of 

23,565 measurements of daily mean water temperatures with 7–13 years of data at each site (fig. 7). 

Observed water temperature data used in calibration were acquired from U.S. Fish and Wildlife Service 

(USFWS), Arcata, California, except for data from streamgage 11530000 at Hoopa, California, which 

was sourced from the U.S. Geological Survey National Water Information System. 

We used statistical and graphical methods to determine model fit and to compare the two models 

with different wind functions (eqs. 11 and 12). Goodness-of-fit statistics included the root mean square 

error (RMSE), mean error (ME), mean absolute error (MAE), and the Nash-Sutcliffe statistic (NSS). To 

assess systematic departures of the model, we plotted observed and simulated water temperatures and 

residuals versus observed water temperatures. We compared model fit between wind functions and then 

selected the wind function that yielded the minimum bias and error for use in all subsequent water 

temperature simulations.

To evaluate model simulations with data not used to fit the model, we used k-fold cross 

validation (Davison and Hinkley, 1997). This technique involves (1) splitting the calibration dataset into 

k = 1,..., K subsets; (2) leaving out the kth subset (the “assessment set”) while the remaining data (the 

“training set”) are used to estimate the model parameters; and then (3) using the parameter estimates 

from the training set to simulate water temperatures of the assessment set. This process is repeated for 

all K subsets, goodness-of-fit statistics for each assessment set are calculated, and then an aggregate 

measure of fit is calculated as the mean of a particular statistic over all subsets. For our cross-validation, 

we split the calibration dataset into subsets by reach and year and conducted a cross-validation for each 

reach. The best-fit parameters estimated during the calibration process were used as initial values. 

Following the three steps described above, we then set aside observed data from year k in reach r, used 

the remaining data to estimate , , and rk rk b −− a (where -k indicates exclusion of the kth year), and then 

simulated water temperatures for year k in reach r using estimates of , , and . rk rk b −− a Because the K

subsets were unequal in size due to missing data in some years, we used a weighted average of the 

goodness-of-fit statistics with weights equal to the fraction of the observations in the kth year. 

Simulating Management Scenarios

We used RBM10 to simulate water temperatures under three flow augmentation scenarios to 

demonstrate how the model can be used to assess potential water-management decisions. The goal of 

these scenarios was to evaluate how cold water released at Lewiston Dam would influence water 

temperatures in the lower Klamath River during the severe drought of 2015, under the planned 

operational schedule at Iron Gate Dam. Because the constructed model inputs limited our simulations to 

1980–2013, we selected 1994 as a surrogate year with drought conditions similar to those of 2015 in 

terms of meteorology and river flows. 

To simulate water temperatures in the lower Klamath River, we used the Trinity River RBM10 

model and the Klamath River RBM10 model of Perry and others (2011). Discharge and water 

temperature values simulated at RM 0.5 on the Trinity River were input as a boundary condition of the 

Klamath River RBM10 model. Tributary inputs that define boundary conditions were otherwise 

unchanged, and the historical meteorological data were used. The river geometry of the Klamath River 

model was truncated to establish Iron Gate Dam as a headwater boundary. Using the historical and 

prescribed dam releases, we were then able to assess the cold water Trinity River augmentation effects 
17 

on lower Klamath River water temperature by comparing the simulated water temperatures at Klamath, 

California. 

For the augmentation scenarios, Reclamation provided three release schedules with specific 

release-point temperatures for Lewiston Dam and the planned 2015 operational release schedule for Iron 

Gate Dam on the Klamath River. Water releases from Lewiston Dam were scheduled to achieve flow 

targets of 2,500, 2,800, and 3,200 ft3

/s in the lower mainstem Klamath River (RM 5.7) near Klamath, 

California, when combined with tributary flows and the planned 2015 release schedule for Iron Gate 

Dam. Water temperatures downstream of Iron Gate Dam in 2014 were used as the uppermost boundary 

condition of the Klamath River model, with the selection based on similarities to 2015 with regard to 

reservoir management, Iron Gate Dam release schedules, and drought-condition meteorology. The 

release schedules covered the period of July 20–November 30, with flow targets timed to coincide with 

the upstream migration of adult fall-run Chinook salmon. We also simulated a baseline referred to as the 

“No Action” alternative scenario that was based on historical tributary flows, meteorology, and dam 

releases observed in 1994. Although conditions in 2015 were not exactly the same as in 1994, differing 

in historical tributary flows, temperatures, and meteorology, alternative scenarios based on 1994 were 

exactly the same except for dam outflows and temperatures. Thus, temperature difference between 

targeted flow scenarios and the No Action baseline allowed us to directly assess the effect of each 

scenario on water temperature in the lower Klamath River in 1994, a year with conditions similar to 

those of 2015. 

Results

Calibration and Validation

Based on measures of precision and bias, we selected the wind function without a seasonal 

component (eq. 11) for simulating water temperatures of the Trinity River. For every reach, measures of 

precision (RMSE and MAE) were slightly less for the model with constant evaporation coefficients 

(table 3) compared to the model with seasonal evaporation coefficients (table 4). Furthermore, bias (as 

measured by the mean error) was smaller for the model with constant evaporation coefficients (table 3) 

than for the model with seasonal evaporation coefficients (table 4). For the model with constant 

evaporation coefficients, the residuals as a function of water temperature indicated little bias for reaches 

1–4, a slight negative bias at low water temperature for reaches 4–8, and a slight positive bias at high 

water temperature for reaches 7 and 8 (figs. 8 and 9). We used the wind function with constant 

evaporation coefficients for all subsequent water temperature simulations (table 3).

Overall, RBM10 performed well in simulating water temperatures of the Trinity River. 

Simulated water temperature tracked observed temperature closely, capturing both seasonal and shortterm fluctuations in observed water temperatures (figs. 10 and 11). Therefore, precision for each model 

reach was favorable. The RMSE ranged from 0.25 to 1.12 °C and MAE ranged from 0.18 to 0.89 °C 

among locations (table 3, figs. 8 and 9). 
18 

Table 3. Best-fit parameter estimates of the wind function shown in equation 11 and goodness-of-fit statistics for observed and simulated water 

temperatures at eight locations along the Trinity River. 

[Statistics are defined as follows: n, sample size; NSS, Nash-Sutcliffe statistic; RMSE, root mean square error; ME, mean error (mean of simulated-measured); 

MAE, mean absolute error]

Meteorological 

reach

Location 

(river mile) a�r b�r θ�r n NSS RMSE ME MAE 

1 107.7 2.59E-09 2.19E-09 12.213 3,464 0.974 0.250 -0.013 0.178

2 79.2 1.68E-10 8.07E-14 0.408 3,205 0.963 0.704 -0.172 0.550

3 58.6 3.58E-13 5.21E-12 1.570 3,270 0.972 0.790 -0.160 0.608

4 47.5 5.11E-10 4.58E-14 0.781 3,276 0.965 0.951 -0.289 0.742

5 31.5 6.35E-10 2.06E-11 0.581 3,036 0.955 1.119 -0.511 0.887

6 21.2 2.16E-11 4.61E-11 1.747 1,877 0.965 0.954 -0.160 0.701

7 12.5 5.51E-11 3.31E-10 0.999 2,225 0.972 0.914 -0.310 0.729

8 0.5 1.33E-10 2.02E-10 2.508 3,212 0.967 1.025 -0.437 0.821

Table 4. Best-fit parameter estimates of the wind function shown in equation 12 and goodness-of-fit statistics for observed and simulated water 

temperatures at eight locations along the Trinity River. 

[Statistics are defined as follows: n, number of days with observed water-temperature data; NSS, Nash-Sutcliffe statistic; RMSE, root mean square error; ME, 

mean error (mean of simulated-observed); MAE, mean absolute error]

Meteorological 

reach

Location 

(river mile) a�r,1 b�r,1 a�r,2 b�r,2 θ�r n NSS RMSE ME MAE 

1 107.7 1.70E-08 4.68E-10 7.40E-09 1.78E-09 12.177 3,464 0.974 0.251 -0.028 0.176

2 79.2 3.57E-12 4.48E-13 1.01E-09 3.18E-13 0.398 3,205 0.961 0.721 -0.177 0.567

3 58.6 7.67E-11 1.71E-12 8.16E-10 5.45E-11 1.782 3,270 0.971 0.804 -0.225 0.618

4 47.5 1.56E-09 1.99E-11 3.04E-13 8.49E-12 0.645 3,276 0.960 1.013 -0.405 0.793

5 31.5 4.00E-10 4.19E-12 7.00E-10 9.15E-11 0.559 3,036 0.952 1.146 -0.520 0.908

6 21.2 9.53E-11 2.25E-11 4.19E-09 8.28E-10 4.430 1,877 0.963 0.973 -0.256 0.713

7 12.5 2.72E-11 7.46E-12 8.30E-09 6.80E-09 3.729 2,225 0.967 0.995 -0.531 0.801

8 0.5 3.97E-10 2.23E-10 2.81E-11 6.08E-11 0.890 3,212 0.964 1.074 -0.472 0.864
19 

Figure 8. Graphs showing simulated water temperature and residuals (simulated minus observed temperature) as 

a function of observed water temperature at five locations between river miles (RMs) 107.8 and 47.6 of the Trinity 

River. Red lines in the four graphs in the right column show a LOWESS smoother (locally weighted polynomial 

regression) to indicate trend in residuals.
20 

Figure 9. Graphs showing simulated water temperature and residuals (simulated minus observed temperature) as 

a function of observed water temperature at four locations between river miles (RMs) 31.6 and 0.5 of the Trinity 

River. Red lines show a LOWESS smoother (locally weighted polynomial regression) to indicate trend in residuals.
21 

Figure 10. Time series of simulated (solid line) and observed water temperature (o) at river miles 107.8, 79.3, 

58.7, and 47.6, Trinity River, 2010–2013. Water temperature was simulated using best-fit coefficients from model 

calibration. Complete 34-year time series is shown in figures A1–A4.
22 

Figure 11. Time series of simulated (solid line) and observed water temperature (o) at river miles 31.6, 21.3, 12.5, 

and 0.5, Trinity River, 2010–2013. Water temperature was simulated using best-fit coefficients from model 

calibration. Complete 34-year time series is shown in figures A5–A8. 
23 

Although the calibrated model fit the observed data well, it is important to consider performance 

when the model is used to simulate water temperatures that are outside the range of data used to 

calibrate the model. Our cross-validation analysis indicated that the model performed well when 

compared with observed data excluded from the calibration process (table 5, figs. 12 and 13). Aggregate 

estimates of prediction error from the cross-validation were nearly equal (table 5) to goodness-of-fit 

statistics from the calibration (table 3). If the parameter estimates had depended strongly on a particular 

year of observed data, the prediction error may have been considerably larger than indicated by the 

calibration statistics. We attribute the favorable prediction error to the structure of RBM10 and the 

extensive set of observed water temperature data used to inform the model. With such a dataset, 

excluding any given year from the calibration has a minor effect on the parameter estimates and 

subsequent simulations (table A3).

On average across the 34-year historical time series, the mean of daily mean water temperature 

ranged from about 7 °C in early January to 11 °C in late July at Lewiston Dam and from 5 to 23 °C at 

Hoopa, California (fig. 14). The 2.5th and 97.5th percentiles of daily mean temperature for the historical 

time series yields a 3 °C interval about the mean at Lewiston Dam and a 6 °C interval at Hoopa, 

California (fig. 14). The complete time series at each of the calibration sites is shown in figures A1–A8.

Table 5. Aggregate cross-validation estimates of prediction error for evaluating observed and simulated water 

temperatures at eight locations along the Trinity River. 

[The aggregate cross-validation estimates are the weighted averages of the yearly estimates shown in table A3. Statistics are 

defined as follows: n, number of days with observed water-temperature data; NSS, Nash-Sutcliffe statistic; RMSE, root mean 

square error; ME, mean error; MAE, mean absolute error]

Reach River mile Number

of years n NSS RMSE ME MAE 

1 107.8 11 3,464 0.971 0.242 -0.013 0.178

2 79.3 13 3,205 0.952 0.695 -0.172 0.550

3 58.7 13 3,270 0.960 0.783 -0.160 0.608

4 47.6 13 3,276 0.948 0.948 -0.290 0.742

5 31.6 12 3,036 0.938 1.113 -0.511 0.887

6 21.3 11 1,877 0.945 0.931 -0.160 0.701

7 12.5 7 2,225 0.969 0.912 -0.310 0.729

8 0.5 12 3,212 0.957 1.018 -0.437 0.821
24 

Figure 12. Validation plots showing time series of simulated (solid line) and observed water temperature (o) at 

river miles 107.8, 79.3, 58.7, and 47.6, Trinity River, 2010–2013. For each year k, water temperature was 

simulated by estimating model coefficients with kth year of observed data excluded, and then using those 

coefficients to simulate water temperature in year k (see table A3).
25 

Figure 13. Validation plots showing time series of simulated (solid line) and observed water temperature (o) at 

river miles 31.6, 21.3, 12.5, and 0.5, Trinity River, 2010–2013. For each year k, water temperature was simulated

by estimating model coefficients with kth year of observed data excluded, and then using those coefficients to 

simulate water temperature in year k (see table A3).
26 

Figure 14. Graphs showing 34-year historical time series of daily mean water temperatures simulated at four 

locations on the Trinity River, northern California. Thick line shows mean temperature on a given day, shaded area 

shows the interquartile range (25th–75th percentiles), and thin lines bound 2.5th–97.5th percentiles. 
27 

Alternative Management Scenarios

Flow Augmentation Effects on Water Temperature in the Lower Klamath River

We selected the drought year of 1994 to model a historical baseline “No Action” alternative to 

evaluate the effects of targeted cold water releases at Lewiston Dam on water temperatures in the lower 

Klamath River. With No Action, the simulated historical daily mean water temperature at RM 5.7 on the 

lower Klamath River was greater than or near 23 °C for most of August (fig. 15). In mid-August, when 

flow augmentation through the bypass was ramped up, water temperatures were simulated to decrease to 

less than the historical baseline under all targeted flow augmentation scenarios (2,500, 2,800, and 3,200 

ft3/s in the lower Klamath River). The three augmentation scenarios uniformly produced an initial 5 °C 

decrease in daily mean water temperature in response to a short-duration tribal “Boat Dance” pulse flow 

scheduled as part of the anticipated 2015 operations by Reclamation that was common to all 

augmentation alternatives. After the initial temperature decrease triggered by the Boat Dance release, an 

approximate 1 °C difference in water temperature emerged between each of the three augmentation 

scenarios, with higher targeted flows associated with greater reductions in water temperature. Under 

each targeted flow scenario, daily mean water temperatures were projected to be less than 21 °C for the 

duration of the augmentation period. In late September with No Action, lower Klamath River water 

temperatures decreased to about 21 °C, and a steady decrease in daily mean water temperature 

commenced as the calendar shifted towards winter. Augmentation was curtailed at that time, and the 

projected water temperatures for all scenarios converged. 

Discussion

Water temperature is a critical water-quality parameter influencing fish populations, as well as 

the structure of the food webs on which they depend. Understanding how water-management actions in 

the Trinity River drainage basin affect water temperatures not just in the Trinity River, but also farther 

downstream in the lower Klamath River, is therefore a critical element in forecasting how such actions 

will influence salmonid populations of the Klamath Basin. Here, we developed a water-temperature 

model for the Trinity River capable of simulating daily mean water temperature along the longitudinal 

gradient of the river from RM 112.2 (Lewiston Dam) to the Klamath River confluence. We calibrated 

and validated the model to an extensive set of observed water temperatures, allowing us to recreate a 34-

year historical record of daily mean water temperature. The model proved useful in comparing the 

effects of bypass flow augmentation scenarios that targeted flows of 2,500, 2,800, and 3,200 ft3

/s in the 

lower Klamath River, which were intended to lower water temperatures and decrease risk of disease in 

adult salmon returns in a drought year. Key findings from these simulations indicated that (1) bypass 

augmentation can reduce water temperatures in the lower Klamath River by as much as 5 °C, (2) 

reduction in water temperature is proportional to the augmentation rate, and (3) a flow target as low as 

2,500 ft3

/s was likely to keep daily mean water temperatures equal to or less than 21 °C from midAugust to late September, assuming tributary hydrology and metrology in 2015 is consistent with the 

hydrology and metrology in 1994 used in the simulations. Given that model accuracies are about ±1 °C 

for both Trinity and Klamath RBM10 models (table 4, figs. 8–9; Perry and others, 2011) and 

uncertainties and potential differences in tributary inflows and metrology between 1994 and 2015, a 

flow target of 2,800 ft3

/s would more conservatively ensure that temperatures would remain less than 

23 °C.
28 

Figure 15. Simulated daily mean water temperatures at river mile 5.7 in the lower Klamath River near Klamath, 

California, during 1994, under four management scenarios that include targeted flows of 2,500, 2,800, and 3,200 

cubic feet per second, augmented by Lewiston Dam bypass, and the historical baseline “No Action” alternative 

(top). Difference in temperature (targeted flow scenario minus No Action alternative) shows relative effects of 

bypass augmentation (bottom).
29 

In evaluating the simulated temperature effects of the various management alternatives, 

temperatures near or exceeding 23 °C are of particular interest, as this has been identified as a threshold 

for impairing the upstream migration of adult salmon (Strange, 2010a)—a threshold regarded as a 

“thermal migration barrier.” Under the No Action scenario, water temperatures in the lower Klamath 

River were simulated to approach or exceed 23 °C during most of August (fig. 15). Adult fall-run 

Chinook salmon (Oncorhynchus tshawytscha) typically begin entering the Klamath River in late

August, and a thermal migration barrier can cause fish to congregate at high densities in the lower river, 

particularly in years with below average streamflow. Stream conditions of this type can lead to epizootic 

outbreaks of Ichthyophthirius multifiliis (“Ich”) and Flavobacterium columnare (“Columnaris”), 

diseases that were associated with the massive Klamath River fish kill of 2002 (Guillen, 2003a, 2003b; 

Turek and others, 2004). Although water temperature is an important factor of epizootic outbreaks, 

other factors such as fish density and streamflow must also be considered (Bodensteiner and others, 

2000; Strange, 2010b; U.S. Fish and Wildlife Service and National Oceanographic and Atmospheric 

Administration, 2013; Belchick, 2015). Nonetheless, the augmentation scenarios we present in this 

study provide a working example of how the Trinity and Klamath RBM10 models can be used to inform 

management decisions as needed to protect salmon populations in the Klamath Basin given limited 

water resources.

As meteorological and streamflow data become available, the Trinity River water-temperature 

model can easily be updated in the future. The next step is to make available an integrated 

Trinity/Klamath RBM10 water-temperature model that can be used to directly inform real-time resource 

management decisions such as the example presented in this report, as well as function as a critical sub 

model for salmon life-cycle modeling efforts currently underway as an additional decision-support tool 

for managers.

Temperatures near or greater than 23 o

C may not be common in the lower Trinity River at the 

time when adult fall-run Chinook salmon begin their annual migration (fig. 14). However, climate 

change projections suggest higher temperatures will become increasingly common in the future (Perry 

and others, 2011; Flint and Flint, 2012), which will place additional stressors on adult salmon runs. The 

Trinity/Klamath RBM10 water-temperature model likely will prove beneficial to further our 

understanding of the relationship between water temperature and salmon populations of the Klamath 

Basin, as it will improve ability of managers to make informed water-management decisions during 

drought-like conditions assumed to increase in frequency as a result of climate change. 

Acknowledgments

Construction of this water-temperature model was a significant undertaking and would not be 

possible without the help of many. Funding was provided by Bureau of Reclamation (Agreement 

R14PG00145ed) and U.S. Fish and Wildlife Service (Agreement 4500054074). We thank California 

Department of Water Resources and Watercourse Engineering, Inc., for the river geometry data. Scott 

Evans of the U.S. Geological Survey processed the river geometry with the Hydrologic Engineering 

Centers River Analysis System to output the data fundamental to characterizing the hydraulics of the 

Trinity River. Andrea McBroom with the U.S. Forest Service, Paul Zedonis with the Bureau of 

Reclamation, Edward Oldenberg with the Hoopa Valley Tribe, and Aaron Martin with the Yurok Tribe 

all helped gather the extensive water-temperature dataset required to construct and calibrate the model. 

Ben Livneh provided the gridded wind data. The map of the watershed was artfully created by Mark 

Hereford of the U.S. Geological Survey. Finally, we must thank Paul Zedonis, again, and Danielle 

Hereford of the U.S. Geological Survey for their constructive reviews of this report. To the unsung 

heroes who collected the field data, and others we did not mention, please know we are grateful. 
30 

References Cited

Ångström, A., 1918, A study of the radiation of the atmosphere: Smithsonian Institute Miscellaneous 

Collection, v. 65, p. 159–161. 

Belchik, M., 2015, An Outbreak of Ichthyophthirius multifiliis in the Klamath and Trinity Rivers in 

2014 (draft report): Yurok Tribal Fisheries Program Data Series Report, 56 p. 

Benjamin, J.R., Connolly, P.J., Romine, J.G., and Perry, R.W., 2013, Potential effects of changes in 

temperature and food resources on life history trajectories of juvenile Oncorhynchus mykiss: 

Transactions of the American Fisheries Society, v. 142, no. 1, p. 208–220. 

Bodensteiner, L.R., Sheehan, R.J., and Wills, P.S., 2000, Flowing water—An effective treatment for 

Ichthyophthiriasis: Journal of Aquatic Animal Health, v. 12, no. 3, p. 209–219. 

California Department of Water Resources, 2014, Trinity River flood insurance study hydraulic 

analysis—Technical summary, March 2014: California Department of Water Resources, Northern 

Region Office, Cooperating Technical Partners, grant nos. EMF-2008-GR-0808 and EMF-2011-GR1109, 12 p. 

Davison, A.C., and Hinkley, D.V., 1997, Bootstrap methods and their application—Cambridge series in 

statistical and probabilistic mathematics (1st ed.): Cambridge, United Kingdom, Cambridge 

University Press, 592 p. 

Deas, M.L., 1998, Trinity Reservoir water temperature simulation model: Prepared for Trinity County 

Planning Department, Natural Resources Division, August 1998, 27 p. 

Deas, M.L. 2002, Trinity Reservoir inflow temperature monitoring study: Prepared for Trinity County 

Planning Department, June 2002. 

Deas, M.L., and Orlob, G.T., 1999, Klamath River modeling project: Davis, University of California, 

report no. 99-04, 235 p. + appendixes. 

Edinger, J.E., Brady, D.K., and Geyer, J.C., 1974, Heat exchange and transport in the environment: 

Electric Power Research Institute, Cooling Water Discharge Research Project, RP-49, 125 p. 

Flerchinger, G.N., Xaio, W., Marks, D., Sauer, T.J., and Qiang, Y., 2009, Comparison of algorithms for 

incoming atmospheric long-wave radiation: Water Resources Research, v. 45, no. 3, 13 p. 

Flint, L.E., and Flint, A.L., 2008, A basin-scale approach to estimating stream temperatures of 

tributaries to the Lower Klamath River, California: Journal of Environmental Quality, v. 37, 68 p. 

Flint, L.E., and Flint, A.L., 2012, Estimation of stream temperature in support of fish production 

modeling under future climates in the Klamath River Basin: U.S. Geological Survey Scientific 

Investigations Report 2011–5171, 31 p. 

Guillen, G., 2003a, Klamath River fish die-off September 2002, Causative factors of mortality: U.S. 

Fish and Wildlife Service, Arcata Fish and Wildlife Office, Arcata, California, report no. AFWO-F02-03, 115 p. 

Guillen, G., 2003b, Klamath River fish die-off September 2002, Report on estimate of mortality: U.S. 

Fish and Wildlife Service, Arcata Fish and Wildlife Office, Arcata, California, report no. AFWO-01-

03, 28 p. 

Henderson-Sellers, B., 1986, Calculating the surface energy balance for lake and reservoir modeling—A 

review: Reviews of Geophysics, v. 24, no. 3, p. 625–649. 

Livneh B., Pierce, D.S., Bohn, T.J., Munoz-Ariola, F., Nijssen, B., Cayan, D., Vose, R., and Brekki, 

L.D., 2015, A spatially comprehensive, hydrometeorological data set for Mexico, the U.S., and 

Southern Canada 1950–2013: Nature Scientific Data, article no. 150042, 12 p. 

Mohseni, Omid, Stefan, H.G., and Erickson, T.R., 1998, A nonlinear regression model for weekly 

stream temperatures: Water Resources Research, v. 34, no. 10, p. 2,685–2,692. 
31 

Perry, R.W., Plum, J.M., Fielding, S.D., Adams, N.S., and Rondorf, D.W., 2013, Comparing effects of 

transmitters within and among populations—Application to swimming performance of juvenile 

Chinook salmon: Transactions of the American Fisheries Society, v. 142, no. 4, p. 901–911. 

Perry, R.W., Risley, J.C., Brewer, S.J., Jones, E.C., and Rondorf, D.W., 2011, Simulating daily water 

temperatures of the Klamath River under dam removal and climate change scenarios: U.S. Geological 

Survey Open-File Report 2011–1243, 78 p. 

Perry, R.W., Romine, J.G., Adams, N.S., Blake, A.R., Burau, J.R., Johnston, S.V., and Liedtke, T.L., 

2012, Using a non-physical behavioural barrier to alter migration routing of juvenile Chinook salmon 

in the Sacramento-San Joaquin River Delta: River Research and Applications, v. 30, no. 2, p. 192–

203. 

Perry, R.W., Skalski, J.R., Brandes, P.L., Sandstrom, P.T., Klimley, A.P., Ammann, A., and 

MacFarlane. B., 2010, Estimating survival and migration route probabilities of juvenile Chinook 

salmon in the Sacramento-San Joaquin River Delta: North American Journal of Fisheries 

Management, v. 30, no. 1, p. 142–156. 

Plumb, J.M., Connor, W.P., Tiffan, K.F., Moffitt, C.M., Perry, R.W., and Adams, N.S., 2012, 

Estimating and predicting collection probability of fish at dams using multistate modeling: 

Transactions of the American Fisheries Society, v. 141, no. 5, p. 1,364–1,373. 

R Core Team, 2014, R: A language and environment for statistical computing: Vienna, Austria, R 

Foundation for Statistical Computing, http://www.R-project.org/.

Strange, J.S., 2010a, Upper thermal limits to migration in adult Chinook salmon—Evidence from the 

Klamath River Basin: Transactions of the American Fisheries Society, v. 139, no. 4, p. 1,091–1,108. 

Strange, J.S., 2010b, Summary of scientific evidence to guide special flow releases to reduce the risk of 

adult fall Chinook salmon mass disease mortality in the lower Klamath River: Yurok Tribal Fisheries 

Program, accessed February 24, 2016, at www.trrp.net. 

Thornton, P.E., Hasenauer, H., and White, M.A., 2000, Simultaneous estimation of daily solar radiation 

and humidity from observed temperature and precipitation—An application over complex terrain in 

Austria: Agricultural and Forest Meteorology, v. 104, p. 255–271, http://dx.doi.org/10.1016/S0168-

1923(00)00170-2. 

Thornton, P.E., and Running, S.W., 1999, An improved algorithm for estimating incident daily solar 

radiation from measurements of temperature, humidity, and precipitation: Agriculture and Forest 

Meteorology, v. 93, p. 211–228, http://dx.doi.org/10.1016/S0168-1923(98)00126-9. 

Thornton, P.E., Running, S.W., and White, M.A., 1997, Generating surfaces of daily meteorological 

variables over large regions of complex terrain: Journal of Hydrology, v. 190, p. 214–251, 

http://dx.doi.org/10.1016/S0022-1694(96)03128-9. 

Thornton, P.E., Thornton, M.M., Mayer, B.W., Wilhelmi, N., Wei, Y., Devarakonda, R., and Cook, 

R.B., 2014, Daymet—Daily surface weather data on a 1-km grid for North America, version 2 data 

set, spatial range: N=41.14, S=40.65, E=-122.86, W=-123.69, temporal range—1980/01/01–

2014/12/31: Oak Ridge, Tennessee, Oak Ridge National Laboratory Distributed Active Archive 

Center, accessed July 29, 2015, at http://daac.ornl.gov. 

Turek, S., Rode, M., Cox, B., Heise, G., Sinnen, W., Reese, C., Borok, S., Hampton, M., and Chun, C., 

2004, September 2002 Klamath River fish-kill—Final analysis of contributing factors and impacts: 

California Department of Fish and Game, 183 p. 

Unsworth, M.H., and Monteith, J.L., 1975, Long-wave radiation at the ground—I, Angular distribution 

of the incoming radiation: Quarterly Journal of the Royal Meteorological Society, v. 101, p. 13–24. 
32 

U.S. Army Corps of Engineers, 2010, HEC-RAS—River analysis system, user's manual version 4.1, 

January 2010, CPD-68: U.S. Army Corps of Engineers, Institute for Water Resources, Hydrological 

Engineering Center, 790 p., accessed Sept. 13, 2011, at www.hec.usace.army.mil. 

U.S. Fish and Wildlife Service and National Oceanic and Atmospheric Administration, 2013, 2013 Fall 

flow release recommendation: U.S. Fish and Wildlife Service and National Oceanic and Atmospheric 

Administration, Memorandum from Irma Lagomarsino and Nicholas Hetrick to Brian Person, 

Reclamation Northern California Area Manager, 46 p. 

Watercourse Engineering, Inc., 2007, Trinity River flow and temperature modeling project: Prepared for 

Trinity County Planning Department Natural Resources Division and the Trinity River Restoration 

Program, 172 p. 

Western Regional Climate Center, 2015, RAWS USA Climate Archive: Reno, Nevada, Western 

Regional Climate Center Web site, accessed May 21, 2015, at http://www.raws.dri.edu.

Yearsley, J., 2003, Developing a temperature total maximum daily load for the Columbia and Snake 

Rivers, simulation methods: U. S. Environmental Protection Agency, Region 10, Seattle, Washington, 

Report 901-R-03-003, 30 p. 

Yearsley, J., 2009, A semi-Lagrangian water temperature model for advection-dominated river systems: 

Water Resources Research, v. 45, 19 p. 

Yearsley, J., Karna, D., Peene, S., and Watson, B., 2001, Application of a 1-D heat budget model to the 

Columbia River system: U.S. Environmental Protection Agency, Region 10, Seattle, Washington, 

Report 901-R-01-001, 65 p. 

Zedonis, Paul, 1997, A water temperature model of the Trinity River: U. S. Fish and Wildlife Service, 

Arcata, California, 97 p. 
33 

Appendix A. River Geometry, Time Series and Water Temperatures, and Prediction Error, Trinity River, 

Northern California 

Table A1. River geometry used in RBM10 for the Trinity River including the estimated a and b parameters of the continuity equations used to 

describe the hydraulic properties of the river.

Segment description Start 

mile

End 

mile

Meteorological 

reach aA bA aW bW Tributary name River mile

Lewiston Dam to Rush Cr. 112.2 107.7 1 5.366 0.624 16.859 0.322 -- --

Rush Cr. to Grass Valley Cr. 107.7 104.0 1 2.902 0.683 8.666 0.402 Rush Cr. 107.6

Grass Valley Cr. to Steel Bridge 104.0 99.0 1 5.958 0.592 19.464 0.276 Grass Valley Cr. 103.9

Steel Bridge to Indian Cr. 99.0 95.4 1 3.530 0.670 11.382 0.346 -- --

Indian Cr. to Weaver Cr. 95.4 93.9 1 2.360 0.707 18.440 0.291 Indian Cr. 95.2

Weaver Cr. to Douglas City 93.9 92.8 2 3.911 0.631 10.993 0.324 Weaver Cr. 93.8

Douglas City to Reading Cr. 92.8 92.6 2 3.911 0.631 10.993 0.324

Reading Cr.

. 92.7

Douglas City to Browns Cr. 92.6 87.8 2 3.483 0.654 11.568 0.318 -- --

Browns Cr. to Soldier Cr. 87.8 83.8 2 3.376 0.664 8.139 0.361 Browns Cr. 87.7

Soldier Cr. to Canyon Cr. 83.8 79.2 2 2.264 0.711 4.325 0.468 -- --

Canyon Cr. to Coopers Bar 79.2 75.0 3 1.814 0.733 4.673 0.460 Canyon Cr. 79.1

Coopers Bar to N. Fork Trinity R. 75.0 72.6 3 2.158 0.718 9.334 0.357 -- --

NF Trinity to Eagle Cr. 72.6 69.6 3 3.247 0.677 19.454 0.247 N. Fork Trinity R. 72.5

Eagle Cr. to Price Cr. 69.6 64.0 3 3.457 0.659 43.641 0.161 -- --

Price Cr. to Big French Cr. 64.0 58.6 3 4.206 0.647 29.855 0.195 -- --

Big French Cr. to Italian Cr. 58.6 53.8 4 3.460 0.696 25.038 0.227 Big French Cr. 58.5

Italian Cr. to Burnt Ranch 53.8 48.7 4 5.374 0.633 34.309 0.184 -- --

Burnt Ranch to Transfer Stn. 48.7 47.5 4 3.614 0.688 33.147 0.194 -- --

Transfer Stn. to Burnt Ranch Falls 47.5 44.9 5 1.803 0.686 15.753 0.216 -- --

Burnt Ranch Falls to New R. 44.9 43.7 5 2.339 0.642 63.372 0.069 -- --

New R. to Icebox Cr. 43.7 39.2 5 3.702 0.631 44.572 0.128 New R. 43.6

Icebox Cr. to Quinby Cr. 39.2 34.7 5 6.976 0.644 32.334 0.208 -- --

Quinby Cr. to S. Fork Trinity R. 34.7 31.5 5 3.777 0.690 20.900 0.237 -- --

S. Fork Trinity R. to China Cr. 31.5 28.2 6 11.355 0.630 34.812 0.225 S. Fork Trinity R. 31.3

China Cr. to Willow Cr. 28.2 25.0 6 8.964 0.609 33.569 0.210 -- --

Willow Cr. to W.C. Screw Trap 25.0 21.2 6 2.988 0.695 40.771 0.185 Willow Cr. 24.9
34 

Segment description Start 

mile

End 

mile

Meteorological 

reach aA bA aW bW Tributary name River mile

W.C. Screw Trap to Horse Linto Cr. 21.2 19.7 7 10.203 0.599 41.079 0.182 -- --

Horse Linto Cr. to Tish Tang Cr. 19.7 16.4 7 3.314 0.666 40.864 0.166 Horse Linto Cr. 19.6

Tish Tang Cr. to Hoopa 16.4 12.4 7 3.698 0.676 29.703 0.213 Tish Tang Cr. 16.3

Hoopa to Mill Cr. 12.4 8.6 8 3.278 0.714 61.678 0.160 -- --

Mill Cr. to Norton Cr. 8.6 3.9 8 14.600 0.584 54.780 0.158 -- --

Norton Cr. to Klamath R. 3.9 0.0 8 4.879 0.636 69.348 0.109 -- --

Table A2. Midpoint coordinates of the eight meteorological reaches used to extract gridded data.

Reach Latitude Longitude Reach length

(miles)

1 40.6939 -122.8605 18.3

2 40.6453 -122.9567 14.7

3 40.7702 -123.1276 20.6

4 40.7889 -123.4389 11.1

5 40.8743 -123.5300 16.0

6 40.9444 -123.6317 10.3

7 41.0256 -123.6407 8.8

8 41.1386 -123.6857 12.4
35 

Figure A1. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 107.8 near Rush Creek.
36 

Figure A2. Time series of simulated (solid line) and observed water temperature (o) at river mile 79.3 near 

Canyon Creek.
37 

Figure A3. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 58.7 near Big French Creek.
38 

Figure A4. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 47.6 near the Burnt Ranch Transfer Station.
39 

Figure A5. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 31.6 near South Fork Trinity River.
40 

Figure A6. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 21.3 near Willow Creek.
41 

Figure A7. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 12.5 at Hoopa.
42 

Figure A8. Time series of simulated (solid line) and observed water temperature (o) of the Trinity River at river 

mile 0.5 near the Klamath River confluence.
43 

Table A3. Year-specific prediction error from the k-fold cross-validation. 

[Prediction error is estimated by calibrating to observed data without year k and then simulating water temperature in year k. 

Statistics are defined as follows: n, number of days with observed water-temperature data, SS, sum of squares; NSS, NashSutcliffe statistic; RMSE, root mean square error; ME, mean error; MAE, mean absolute error]

Reach

(r) Year (k) n SS NSS RMSE ME MAE 

1 2002 240 13.95 0.966 0.241 -0.052 0.176

2003 281 11.18 0.977 0.199 -0.089 0.155

2005 256 14.28 0.974 0.236 0.069 0.169

2006 365 14.45 0.975 0.199 0.051 0.153

2007 276 16.30 0.982 0.243 -0.095 0.202

2008 289 10.10 0.974 0.187 0.017 0.138

2009 365 13.91 0.988 0.195 -0.002 0.130

2010 365 11.12 0.987 0.175 -0.054 0.140

2011 365 29.17 0.964 0.283 -0.093 0.188

2012 366 52.11 0.939 0.377 0.076 0.260

2013 296 29.74 0.961 0.317 0.017 0.252

2 2001 16 10.35 0.212 0.804 -0.401 0.713

2002 183 45.48 0.962 0.499 -0.032 0.405

2003 99 55.35 0.884 0.748 -0.174 0.562

2004 139 27.52 0.964 0.445 -0.055 0.337

2005 177 58.93 0.952 0.577 -0.150 0.460

2006 269 108.59 0.964 0.635 -0.239 0.493

2007 276 211.73 0.959 0.876 -0.394 0.719

2008 289 148.95 0.939 0.718 0.094 0.566

2009 365 191.85 0.970 0.725 -0.078 0.588

2010 365 241.95 0.939 0.814 -0.425 0.653

2011 365 223.85 0.951 0.783 -0.374 0.593

2012 366 138.81 0.967 0.616 -0.184 0.482

2013 296 126.39 0.972 0.653 0.151 0.516

3 2001 16 28.84 -0.486 1.343 -0.408 0.841

2002 239 179.30 0.965 0.866 0.000 0.680

2003 281 227.39 0.953 0.900 -0.372 0.703

2004 170 95.44 0.937 0.749 -0.339 0.534

2005 177 65.18 0.970 0.607 -0.162 0.449

2006 269 103.52 0.980 0.620 -0.148 0.474

2007 276 192.18 0.978 0.834 -0.279 0.633

2008 85 44.74 0.927 0.725 0.077 0.637

2009 365 250.35 0.976 0.828 0.072 0.666

2010 365 177.37 0.972 0.697 -0.342 0.553

2011 365 228.15 0.968 0.791 -0.302 0.600

2012 366 194.42 0.972 0.729 -0.247 0.573

2013 296 255.63 0.967 0.929 0.285 0.725
44 

Reach

(r) Year (k) n SS NSS RMSE ME MAE 

4 2001 13 26.86 -0.543 1.437 -0.604 1.002

2002 183 123.15 0.953 0.820 0.207 0.673

2003 99 138.78 0.820 1.184 -0.494 0.877

2004 170 152.90 0.914 0.948 -0.525 0.719

2005 256 245.03 0.955 0.978 -0.521 0.779

2006 365 292.19 0.960 0.895 -0.520 0.692

2007 276 311.84 0.968 1.063 -0.481 0.839

2008 289 281.47 0.947 0.987 0.053 0.759

2009 296 251.94 0.973 0.923 0.053 0.759

2010 365 279.06 0.961 0.874 -0.550 0.702

2011 365 322.24 0.960 0.940 -0.457 0.736

2012 332 283.29 0.956 0.924 -0.210 0.717

2013 267 253.13 0.970 0.974 0.082 0.736

5 2001 13 24.70 -1.199 1.378 -0.523 0.925

2002 215 273.71 0.958 1.128 -0.265 0.879

2003 281 459.58 0.929 1.279 -0.833 1.024

2005 178 106.09 0.960 0.772 -0.379 0.601

2006 190 176.25 0.931 0.963 -0.640 0.746

2007 266 281.13 0.974 1.028 -0.412 0.814

2008 279 372.37 0.939 1.155 -0.166 0.896

2009 299 357.23 0.944 1.093 -0.535 0.850

2010 365 483.35 0.940 1.151 -0.856 0.954

2011 365 540.05 0.942 1.216 -0.767 0.965

2012 350 453.59 0.948 1.138 -0.704 0.948

2013 235 276.23 0.968 1.084 0.390 0.866

6 1999 140 274.70 0.908 1.401 0.360 1.052

2001 213 213.69 0.934 1.002 -0.302 0.759

2002 41 23.25 0.437 0.753 -0.557 0.667

2003 323 364.88 0.953 1.063 -0.517 0.811

2005 229 266.96 0.957 1.080 -0.569 0.849

2007 160 75.61 0.981 0.687 -0.042 0.468

2008 173 143.96 0.958 0.912 0.390 0.689

2009 141 97.54 0.963 0.832 0.205 0.607

2010 151 118.47 0.971 0.886 0.063 0.668

2011 196 100.80 0.979 0.717 -0.170 0.492

2012 110 29.70 0.969 0.520 -0.286 0.418

7 2007 91 100.71 0.889 1.052 -0.804 0.842

2008 344 286.34 0.970 0.912 -0.236 0.755

2009 365 248.01 0.982 0.824 -0.256 0.663

2010 365 278.45 0.969 0.873 -0.466 0.707

2011 360 293.43 0.972 0.903 -0.502 0.696

2012 366 299.84 0.970 0.905 -0.530 0.702

2013 334 353.20 0.970 1.028 0.307 0.830
45 

Reach

(r) Year (k) n SS NSS RMSE ME MAE 

8 2002 178 155.80 0.941 0.936 0.077 0.762

2003 43 48.35 0.832 1.060 -0.906 0.911

2004 189 347.25 0.861 1.355 -0.780 1.049

2005 186 177.62 0.964 0.977 -0.599 0.786

2006 341 322.12 0.966 0.972 -0.560 0.748

2007 365 312.13 0.977 0.925 -0.321 0.744

2008 284 288.88 0.966 1.009 -0.268 0.791

2009 211 227.48 0.972 1.038 -0.204 0.881

2010 365 312.01 0.966 0.925 -0.614 0.790

2011 365 298.53 0.972 0.904 -0.610 0.725

2012 366 488.51 0.952 1.155 -0.749 0.943

2013 319 396.80 0.965 1.115 0.088 0.887
46

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http://wfrc.usgs.gov/
Jones and others—Construction, Calibration, and Validation of the RBM10 Water Temperature Model, Trinity River, California—Open-File Report 2016–1056

ISSN 2331-1258 (online)

http://dx.doi.org/10.3133/ofr20161056