Water Temperature Regime
3.1 Introduction
3.2.1.1 Linear regression analysis
The development of a linear relationship between air and water temperatures is based on the assumption that the rate of change in heat storage in a river can be related to air temperature change (Stefan and Preud'homme 1993). This approach relies on the (generally) linear relationship between air and water temperatures. Inherent in this approach is that the data conform to the basic assumptions of linear regression (McConway et al. 1999), viz.that the:
• Mean of the response variate (Yi) is linear; or in other words that the mean is stable;
• Variation about the mean (s.) has a normal distribution;
• Variance (c;) is the same for all values of the explanatory variable (i.e.
constant variance);
• Ciare independent of each other.
Air temperature is widely measured, which makes it convenient to use as a basis for predicting water temperatures (Webb and Nobilis 1997). Typically, such studies are undertaken using large (1-100km) areas and coarse time scales (5 days to months). Stefan and Preud'homme (1993)·used weather stations which were an average of 70 kilometres from the rivers; correlations decreased when the distance exceeded 160 kilometres. Smith (1981) found significant simple linear relationships between monthly air and water temperatures for rivers in Great Britain. Crisp and Howson (1982) found a simple linear relationship between air and water temperatures to be adequate to predict 5- or 7-day mean water temperatures for streams in the north Pennines and English Lake District. Webb and Nobilis (1997) examined the relationship between monthly mean air and water temperatures for a small catchment in Austria that had data for 90 years, and found a significant relationship between monthly airand water temperatures.
However, a simple linear relationship between air and water temperatures may not always be adequate in describing water temperatures completely. Relationships may vary between catchments because of factors such as slope and aspect (Webb and Nobilis 1997), which
makes generally applicable models elusive. Furthermore, Webb and Nobilis (1997) found between-month differences in regression slopes, suggesting that the relationship between air and water temperatures is influenced by other seasonally-dependant variables, such as flow volume and flow rate. As flow volumes drop, so do water velocities, giving more time for a body of water to approach thermal equilibrium with the air above it (Essig, 1998). The instability of the relationship between air and water temperatures during the year implies that it is not one of true cause and effect. Water temperature depends on a number of meteorological and hydrological parameters, such as relative humidity, water depth, flow and sediment load (Stefan and Preud'homme 1993). Other factors, such as adjusting temperatures for flow, need to be considered (Webb 1996), and these can be included into more complex water temperature models using multiple linear regressions, with the basic form of the model shown in Equation 3.1. However, "multiple regression analysis has suggested that the effects of rainfall and river flow on water temperature are small compared with those of air temperature" (Webb and Nobilis 1997).
n
y = ''LJ3;x;
+r
;=0
[3.1]
wherey is the response variable, Xi is the ith explanatory variable, Pi is the ith coefficient, and y is a random error term.
Crisp and Howson (1982) found that a simple linear relationship was not adequate for predicting water temperatures below O°C. According to Mohseni and Stefan (1999), interpolations of water temperatures outside the range 0-20°C are less accurate because the essentially linear relationship becomes logistic. High air temperatures often coincide with low flow periods, and in the northern hemisphere at low temperatures, snowmelt complicates water temperature prediction (Mohseni and Stefan 1999). An S-shaped logistic function derived from daily air temperatures plus two stream temperature variables (Equation 3.2) was developed by Mohseni et al. (1998) to compensate for these non- linearities, which also captured the stochastic nature of stream temperature.
a-fJ- Ts= fJ-+-1--y-( p'---r a-)
+e
[3.2]
where Tsis the estimated stream temperature; Tais the air temperature measured at or near the stream gauging site; a is the estimated maximum stream temperature;f1 is the estimated minimum stream temperature; yis a measure of the steepest slope of the logistic function;
and
p
represents the air temperature at the inflection point (or curve midpoint).3.2.1.2 Time series analysis
A set of observations collected sequentially over time, and represented graphically, constitutes a time series. Due to the earth's annual revolution around the sun, and the twenty-four hour rotation on its axis, meteorological data exhibit seasonal patterns,ranging upwards from cycles with a 24-hour periodicity (Abraham and Ledolter 1983). A river's thermal regime is subject to the same series of periodicity that other meteorological data are. The traditional approach to modelling seasonal data is to decompose it into the following components (Chatfield 1980; Abraham and Ledolter 1983):
• Seasonal effect;
• Other cyclic changes, such as daily variation;
• Trend, for example long term climatictrends;
• Randomlstochastic fluctuations (residuals).
These components are agglomerated into what is known as an "additive" model (Equation 3.3). Both the trend and seasonal components can be fitted using least-squares methods, with the trend component being modeled using low-order polynomials (such as linear regression), while seasonal components are modeled using trigonometric functions (such as in spectral analysis) (Abraham and Ledolter 1983). Time series analyses involve the decomposition of a time series into these various components. (Stochastic) time series models are concerned with modelling the errors (et) or residuals, once the seasonal and trend components are removed. The errors, and therefore the observations, are assumed to be uncorrelated, described by a normal distribution probability curve, free of trend (stationary; Le. constant mean and variance), and free of seasonal influences (cf. Section 3.2.1). In practice, the residuals are rarely uncorrelated, and there is usually some degree of serial correlation,especially if the data are collected in sequence (Abraham and Ledolter
1983). Time senes models should thus incorporate the correlation structure of the residuals.
Data exploration typically begins with determining the correlation between data points at time t and time t+1 (the autocorrelation coefficient at lag k),using the raw data. Plotting the autocorrelation coefficient (rk) against time lag k in a correlogram provides an indication of the degree of seasonality of the time series, as well as trend. If the time series contains seasonal oscillations, the correlogram will also exhibit seasonal fluctuations, which show as a sine wave on the correlogram. Trend (non-stationarity) is apparent where values of rk are significantly different from zero for more than three lag periods, or in other words
r«
remains significantly large until k gets big enough so that random error components dominate (Makridakis et al. 1983). Stationarity (removal of trend) can be achieved through nth order differencing techniques. Usually, first-order differencing is adequate to remove trend (z',= ZI+1- z.). Variances may be smoothed using appropriate data transformations, such as logarithmic transformations. The effects of seasonality may be removed using data smoothing techniques, such as moving averages. For data that are stationary and free of seasonal effects, the autocorrelation coefficients should be non- significantly different from zero, if the pattern is completely random. In cases where the correlation coefficients are significantly different to zero, this may be attributed to short- term correlations, such as when a cold day is followed by one or two successive cold days. A time series model will be able to be developed if the data are stationary and non- seasonal, and if the autocorrelations drop to values not significantly different to zero within two to three time lags.Simple time series models may be developed using deseasonalized data that has been
"smoothed" by calculating moving averages. Autoregressive techniques provide a more sophisticated method for modelling time series. Such models (Equation 3.4) are similar to multiple linear regression models, although in this case the water temperature, x, at time t, is regressed on past values, to yield models of order p. A more sophisticated approach involves the use of ARIMA (Autoregressive Integrated Moving Average) models, which integrate autoregressive and moving averages models. ARIMA models may be used for seasonal and non-seasonal data. Non-seasonal ARIMA models are of the form ARIMA (P,d,q) where p, d, and q refer to the order of autoregressive, differencing and moving average components respectively. Seasonal ARIMA models are of the form ARIMA
(P,d,q)(P,D,Q)", where nrefers to the seasonal periodicity of the seasonal component of the model, and P, D and
Q
refer to the autoregressive, differencing and moving average components of the seasonal model respectively (Makridakis et al. 1983).[3.3]
where ZI is the time senes, TI is the trend component for time t; SI IS the seasonal component, andClis the stochastic process.
P
XI =La;xl_ ; +ZI
;=\
[3.4]
whereXI is water temperature at time t,XI-i is water temperature at time intervalipreceding t, ads a coefficient, andZIis a random term.
There may be a chance that a river's temperature changes over time as a result of anthropogenic influences such as impoundments and pollution. Time series analyses make use of historical records to predict future temperatures (Webb 2000). For example, Hostetler (1991) used time series analysis to examine the impacts of timber logging on stream temperatures. The study examined twenty years of data; it was known when logging began. By fitting a time-series model to the data, it was possible to separate the temperature time series into various components. Effects accounted for by climate and flow could be excluded, and any remaining trend components could be attributed to logging. However, trends in water temperatures are generally difficult to show because of a general lack of long-term water temperature records (Webb 2000).
3.2.1.3 Spectral analysis
Spectral analysis, which is also called Fourier or Harmonic analysis (Makradakis et al.
1983) decomposes a time series with a seasonal component, into a set of sine waves, and is fitted using least-squares methods, i.e. a derivativeof a regression model. Such analyses involve the discovery of hidden periodicities in a given time series (Chatfield 1980). In the case of water temperatures, spectral analysis makes use of harmonic curves to approximate the annual cycle of water temperature (Webb 2000). Harmonic functions will produce
smooth water temperature curves that account for the seasonal component of a time series, and therefore trend and variability are not inherently included in such models. Stefan and Preud'homme (1993) broke water temperature down into a deterministic component (a sinusoidal function of the time of year) and a stochastic component (a function of air temperature on the same day). The inclusion of a random term brings a stochastic element into the harmonic curves (Webb 2000). First-order harmonic curves, which take the general form of Equation 3.5, have been found to approximate the seasonal variability of water temperatures, when the variability is symmetric over seasons (Smith 1981; Ward
1985; Hostetler 1991; Stefan and Preud'homme 1993; Caissie et al. 2001). Such models are appropriate for time series with seasonal patterns where the amplitude and phases are fixed (Abraham and Ledolter 1983). Seasonality adds an asymmetric element, which can be simulated using higher order harmonic curves (Long 1976; Ward 1985; Webb 2000), where the amplitude and phase shift are able to change dynamically, based on previous values (Abraham and Ledolter 1983).
[3.5]
where T, is daily water temperature; Tis the mean annual air temperature; A is the amplitude of the sine curve; N is the number of observations within a cycle; t, is the itlz observation within the cycleN, }'is a random term. Two optional additional terms may be included in the sine function,