Stochastic Modeling of Water Temperatures in Running Waters
Water temperature is a very important parameter not only in water quality studies but also in biological studies. For instance, salmonids can be adversely affected by natural high stream water temperatures or by those resulting from anthropogenic sources such as deforestation. To predict stream water temperatures, two different approaches have been used; the deterministic and stochastic approaches. The deterministic approach consists of a physical model based on the energy budget (solar radiation, convection, etc.) and the physical characteristics of the stream (water depth, stream cover, etc.). The stochastic modelling approach consists of studying the structure (autocorrelation) of the stream water temperature time series and its dependence on air temperatures (cross-correlation). The purpose of this study is to develop and test the performanoe of a stream water temperature model using a stochastic approach to predict water temperatures in rivers under natural conditions. The performance of such an approach was tested using data from Catamaran Brook, a small stream in New Brunswick (Canada). It differs from previous studies in that most others were on larger river systems. This stochastic model incorporates the Box and Jenkins method (1976) which relates the time series data to both water and air temperature residuals. To calculate the residuals of both air and water temperatures, a seasonal component was first estimated using Fourier series analysis. This seasonal component better represents the long-term trend in air and water temperatures for the studied period or season (i.e. increasing water temperatures at first, then reaching a maximum during the early part of August and decreasing again later in the season). The Fourier series with one harmonic was chosen for the analysis as it has been shown in previous studies that the first harmonic represents most of the variation within the stream water temperature variable. The model was calibrated using the Box and Jenkins method and Catamaran Brook data from 1990. This analysis consist of determining a transfer function relating present water temperature residuals to past water and air temperature residuals including present air temperature residuals and a random component. The random component (also called « noise series ») of the model is a normally distributed variable with a standard deviation calculated using the calibration period. After the calibration period, subsequent years or post-calibration years were analyzed to predicted stream water temperatures with the model using air temperature data only. A study of residuals between predicted and measured stream water temperatures showed very good results during the calibration year (1990) with a calculated root-mean square error of 0.75°C. The predicted temperatures during post calibration years (i.e. 1991 and 1992) were good and the root-mean-square errors were similar to previous studies (e.g. Marceau et al. 1986) with values of 1.45°C and 2.10°C respectively. The measured stream water temperatures during the post-calibration years were only used to estimate the relative performance of the model as opposed to a forecasting model which utilizes actual measured temperatures. At Catamaran Brook is has been observed that natural variation in air temperatures can have an influence on the performanoe of the model. When air temperatures were recorded higher or lower than the long term values (normal temperatures) calculated by the Fourier series analysis, the predicted water temperatures was not as good. For instance it was observed that during September of 1992, during which time the air temperature was higher that normal, the performance of the model was not as good with a root me an squared error of 2.96°C. However, during July 1992, below normal air temperatures were also recorded and a very good prediction of stream water temperatures in Catamaran Brook was achieved with a root me an square error of 0.98°C. In general, satisfactory prediction of stream water temperatures was achieved using the Box and Jenkins stochastic modelling approach.