Regression Model for Daily Maximum Stream Temperature David W. Neumann1; Balaji Rajagopalan2; and Edith A. Zagona3 Abstract: An empirical model is developed to predict daily maximum stream temperatures for the summer period. The model is created using a stepwise linear regression procedure to select significant predictors. The predictive model includes a prediction confidence interval to quantify the uncertainty. The methodology is applied to the Truckee River in California and Nevada. The stepwise procedure selects daily maximum air temperature and average daily flow as the variables to predict maximum daily stream temperature at Reno, Nev. The model is shown to work in a predictive mode by validation using three years of historical data. Using the uncertainty quantification, the amount of required additional flow to meet a target stream temperature with a desired level of confidence is determined. DOI: 10.1061/͑ASCE͒0733-9372͑2003͒129:7͑667͒ CE Database subject headings: Decision support systems; Regression models; California; Nevada; Streams; Water quality; Water temperature. Introduction ing decisions, the prediction must meet the following specific requirements: it must be quick, accurate, easy to use, and spatially An increasingly common problem in western U.S. river basins and temporally consistent with the operations models. To incor- and elsewhere in the world is that water storage and use for mu- porate stream temperature in the operations model, the normal nicipal, industrial, agricultural, and power production purposes operating policies are simulated and the stream temperature is leaves river biota with insufficient flow to maintain populations. predicted. Based on the prediction, decisions can be made to re- Low flows threaten biota by deteriorating habitat and/or water lease additional water, if necessary, to improve the stream tem- quality. One of the most common summer water quality problems perature. As various researchers explain ͑Beck 1987; Reckhow associated with low flows is high stream temperatures—low flows 1994; Varis et al. 1994͒, the uncertainty of any prediction should warm up more rapidly than higher flows. High stream tempera- tures reduce cold water fish populations by inhibiting growth and be quantified for decision making purposes. Thus, the temperature by killing fish at extremely high temperatures. For this reason, the prediction should also include a quantification of the uncertainty. impact of low flows and high stream temperatures on fish is an Two types of models have been developed in the past to pre- issue in many operations studies and National Environmental dict stream temperatures: empirical or regression models and Policy Act Environmental Impact Statement analyses such as physical process models. Regression models have been developed those on the Rio Grande, Colo., and Columbia basins ͑Bonneville to quantify and predict stream temperatures at various time scales. ͑ ͒ 1995; U.S. Bureau of Reclamation 1995, 2000͒. Mohseni et al. 1998 developed an S-shaped regression model to Resource managers use computer models to simulate river and predict average weekly stream temperatures at different locations reservoir operations. Computer simulations are useful to allow in the United States that account for hysteresis throughout a year. ͑ ͒ water managers to investigate the effects of varying inflows, legal Mohseni et al. 2002 also developed statistical upper boundaries policies, and operating strategies. To address the problem of for weekly stream temperatures, noting that in the upper part of warm stream temperatures, resource managers need to incorpo- the S curve, increasing air temperature results in constant stream rate stream temperature objectives in their operations models and temperatures due to back radiation and evaporation. They showed management decisions. This requires the ability to predict stream that for an arid western U.S. desert region, the maximum weekly ͑ ͒ temperature. Because the prediction will be used in daily operat- stream temperature is as high as 33°C. Hockey et al. 1982 de- veloped a daily regression model relating spot mid-day stream temperature to flow rate and daily maximum air temperature. 1Water Resource Engineer, Stetson Engineers Inc., 2171 F. Francisco Blvd., Suite 11, San Rafael, CA 94965. E-mail: davidn@ They concluded that their regression was not adequate because of ͑ ͒ stetsonengineers.com lack of data. Gu et al. 1999 produced stream temperature regres- 2Assistant Professor, Dept. of Civil, Environmental, and Architectural sion models for various weather conditions. They found that cor- Engineering, Univ. of Colorado, UCB 426, Boulder, CO 80309-0426. relation of flow to river temperature is possible and useful when E-mail: [email protected] weather parameters are decoupled from the model. 3Director, Center for Advanced Decision Support for Water and In contrast to regression based models, many physical process Environmental Systems ͑CADSWES͒, Univ. of Colorado, UCB 421, models have been developed. Physical process models attempt to Boulder, CO 80309-0421. E-mail: [email protected] model the underlying processes that affect stream temperature Note. Associate Editor: Robert G. Arnold. Discussion open until De- such as channel geometry, conduction, radiation, advection, and cember 1, 2003. Separate discussions must be submitted for individual dispersion. Among various work, Taylor ͑1998͒; Carron and Ra- papers. To extend the closing date by one month, a written request must ͑ ͒ ͑ ͒ be filed with the ASCE Managing Editor. The manuscript for this paper jaram 2001 ; and Brock and Caupp 1996 developed stream was submitted for review and possible publication on February 26, 2002; temperature models using mechanistic one- or two-dimensional approved on August 13, 2002. This paper is part of the Journal of En- heat advection/dispersion transport equations. Although a mecha- vironmental Engineering, Vol. 129, No. 7, July 1, 2003. ©ASCE, ISSN nistic temperature model could, in theory, give very accurate re- 0733-9372/2003/7-667–674/$18.00. sults, this type of model requires numerous detailed input data, is JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / JULY 2003 / 667 quality of the Truckee River, particularly in the lower reaches where the river flattens out in the desert between Reno and Pyra- mid Lake. The water acquired by the WQSA will be stored in upstream reservoirs and released as necessary to mitigate down- stream water quality problems. In particular, this WQSA water will be released on a daily basis to meet a target daily maximum stream temperature. The stream temperature of the Truckee River between the confluence with the Little Truckee River and Reno is influenced mainly by natural warming, that includes solar radia- tion and conduction. Downstream of Reno, wastewater effluent and irrigation return flows enter the river, making accurate tem- perature predictions much more complex and uncertain. As a first step to improve Truckee River water quality, this paper investi- gates the temperature at Reno. A diagram of the study section is shown in Fig. 1. Fig. 1. Diagram of study section Stream Temperature Model computationally intensive and is, therefore, difficult to incorpo- rate in a river and reservoir operations model. Empirical, regres- The goal of regression models is to fit a set of data with an sion based models can be computationally less intensive, there- equation, the simplest being a linear equation. The linear regres- fore quick to implement and easy to validate. With regression sion model takes the form models it is possible to easily quantify the uncertainty. Tˆ ϭa ϩa x ϩa x ϩ...ϩa x (1) In this paper, we develop a regression model to predict low 0 1 1 2 2 n n ˆ ϭ ϭ flow summer stream temperatures on the Truckee River at Reno. where T stream temperature; a0 ,a1 ,a2 ,...,an coefficients; ϭ The model is developed using a stepwise linear regression proce- and x1 ,x2 ,...,xn independent predictors. dure that selects the significant predictors. The regression model The available data are summarized in Table 1 with the loca- provides uncertainty estimates using standard linear regression tions of the gaging sites shown in Fig. 1. Most of the temperature theory. We develop a strategy to use the uncertainty information data were collected after 1993. Since 1993 and 1994 were dry to determine the additional flow required to meet a temperature years with low flows and high river temperatures, the same con- target with a given confidence level. ditions that the prediction will be used, these are the most appro- This paper is organized as follows. We present the water qual- priate years to use in the empirical relationships. In addition, only ity issues on the Truckee River. Next, we describe the develop- data from June, July, and August will be used. We did not include ment of the regression model and present statistical model diag- September because the river cools in the latter half of the month. nostics. We validate the model using historical data and present It is likely that the model developed will be applicable to the first strategies to use the uncertainty of the prediction. Finally, we half of September. We chose to look at data for which the flow at discuss the results and summarize the findings. Farad was less than 14.2 m3/s ͑500 cfs͒ because at flows above this threshold, there is rarely a temperature problem in the study reach. Also, 14.2 m3/s ͑500 cfs͒ is a logical cutoff because, ac- Truckee River Background cording to U.S. Bureau of Reclamation water managers ͑Scott, personal communication, 2001͒, additional water to mitigate tem- The methodology developed is applied to the Truckee River in perature problems will not be released when the flow at Farad is California and Nevada. The Truckee River, like other basins in above the legal flow target of 14.2 m3/s ͑500 cfs͒. the western U.S., does not have the water resources to meet ag- Candidate predictors for the stream temperature prediction at ricultural, municipal, and industrial purposes and still provide ad- Reno include: equate habitat for fish.