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<br />l\) <br />~ <br />-.J <br />0) <br /> <br />z(l)i <. alz(2)i + a2z(3)i + a3z(4)i + a4z(5)i + ~ e(l) <br /> <br />The seri41 dependence (using the first-order linear autoregressive <br />model) i~ then added to form the stochastic component in which both <br />the serial and spatial dependence has been accounted: <br /> <br />Yi . rl Yi-l + P . zi <br /> <br />Note that the Y series is still (0,1) yet has the required serial <br />dependenqe as well as the proper degree of dependence on another <br />random r~sidual. Also note that this equation is only applicable <br />to the first-order linear autoregressive model. To date this is the <br />only typ~ of autoregressive model which is used in this study. The <br />subscript o~ the y values may now be better written using one sub- <br />script for the year and one for the month. The periodic mean and <br />standard 'deviations are then incorporated by: <br /> <br />Xi,j . Yi,j Sj + mj <br /> <br /> <br />(10) <br /> <br />(11) <br /> <br />(12) <br /> <br />Thi. produces the final value of the time series in month j. The monthly <br />sequence 'of x values may be grouped by 12's into years, if desired. <br /> <br />2.3 IncOrporation of Unmeasured Accretions and Depletions <br /> <br />In order ,to make a model which would produce acceptable flows, the <br />logic of;the calculations and analysis of the data had to be con- <br />structed,to reflect the proper total flow of all gaging station <br />locations as well as accretions or depletions in the reaches between <br />measurina points, This problem was handled by synthesizing the gain <br />or loss in each reach and accumulating the flows to produce the down- <br />stream total. To do this required the calculation of the intervening <br />flows by ,subtracting the upstream from the downstream historic monthly <br />value.. ;The re.ulting time series was then analyzed as the other <br />sets and its properties used as the regeneration coefficients. <br /> <br />This appjoach utilizes the analysis of flows at gaging stations around <br />the edge4 of a basin where inflows are measured on the highest tribu- <br />taries. ;However, instead of synthesizing flow at the downstream station <br />directly, the model generates the contribution for the reaches between <br />stationsiand adds it to the upper value. In this procedure, the <br />bounds of the intervening flows are specified by the modeler and can <br />be kept within reasonable bounds. <br /> <br />13 <br />