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<br /> <br />2.1 Time Series Structure <br /> <br />With the background provided in the previous section. this section <br />will proceed to further describe the mathematical structure used to <br />~~ describe the time series. The structure of the time series involved. <br />,..... in this case streamflows and stream qualities. is assumed to follow a <br />~ pattern which has been used successfully by other investigators. The <br />~J elements or components of the time series which make up the structure <br />are further described in the following subsections. <br /> <br />2.1.1 Periodic Elements <br /> <br />One of the most obvious patterns observed when daily or monthly data <br />are investigated is periodic. This is typified by the annual pattern <br />of high and low flows which commonly occur in streamflow records. <br />The pattern is usually easy to detect even by eye. A sophisticated <br />method of measuring the importance of this periodic variation is <br />used in the analysis program. It essentially fits a curve through <br />the cyclic pattern using a combination of sine and cosine functions. <br />The calculations indicate the coefficients and the importance of <br />various terms needed to describe the pattern. This analysis procedure <br />is known as a Fourier analysis and a Fourier series (composed of sine <br />and cosine terms) is used to define the pattern. <br /> <br />A less obvious annual pattern is often observed in the monthly vari- <br />ance or standard deviation of streamflow data. When data are studied <br />month by month. all the January values, February values, etc,. are <br />considered as a group. For each month a mean value (the average <br />from all the years). a standard deviation (the measure of variation <br />for the individual month). and other statistical properties can be <br />estimated. Very often these properties are found to vary from one <br />month to another and show a periodic pattern which can also be studied <br />through the Fourier analysis, <br /> <br />To look at this more mathematically, take a series, xi' of n discrete <br />values. xl' x2. x3.'.. xi'... ~, where the sample mean is calculated <br />from: <br /> <br /> n <br />x " .L L xi <br /> n i"l <br /> <br />The sample standard deviation can be calculated as: <br /> <br />6 <br /> <br />(1 ) <br />