Comparison of Two Approaces for Determining Ground-Water Discharge and Pumpage in the Lower Arkansas River Basin Colorado 1997-98

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K Var(DJ _ Tk6k (26) k =1 where Tk is the sum of the V? for all wells in the kth stratum and 6k = Var(ed`ffPi) for each well i in the kth stratum. If the number of strata K =1, that is, if the assumption of identical distribution holds, equation 25 gives ED = E(ediffP —1) (27) n I Vi i 1 The important implication in this equation is that, if the differences have the same distribution, bias in the difference in total network pumpage relative to the magnitude of total (TFM) network pumpage is the same magnitude as relative bias for an individual well given in equation 22. For example, a 5- percent bias per well translates into a 5- percent bias for the total network. If K is greater than 1, then according to equation 25, network relative bias is a pumpage - weighted average of the individual stratum biases X1,11 �t2, ..., ItK . Likewise, if K =1, the standard deviation of total network error as a fraction of total network pumpage is given by FSD(ediffP) SD(Dn) = (28) n n Vi Vi i =1 i =1 In this equation, the ratio involving Vi on the right - hand side tends to decrease as the number of wells (n) increases. The rate of decrease is in proportion to I/ Fn . Thus, the random component of difference in total network pumpage tends to decrease and become less important compared to the bias compo- nent, represented in equation 25, which does not diminish with number of wells, n. If K >l, it may be shown that the standard deviation of Dn , computed from equation 26, relative to total network pumpage, will still tend to grow smaller as number of wells (n) increases, again roughly in proportion to I/ Fn . Use of equations 25 and 26 requires estimates of the parameters N and ak . Let nk be the number of measurements from the kth stratum, and denote these measurements by diffpk1, diffpk2, ..., diffpknk . If it can be assumed that these observations are normally distributed, there are special widely used techniques based on this assumption that can be used to estimate the parameters. Because the normal assumption is not a good one, however, the parameters are estimated by and nk Itk — nk I (ediffPkj — 1) (29) . J =1 nk 6k = n (edlffPk�— 1 —!kk)2 . (30) Wk � =1 Equations 29 and 30 are the ordinary sample mean and sample standard deviation of the ediffP —1 values in the kth stratum. These estimates are essentially the "smearing estimates" for nonparametric retransforma- tion discussed by Duan (1983) in the context of regression. The K strata for this analysis are formed by dividing the range of log (Vi) values for the 553 paired - pumpage measurements for 1998 into K equal intervals. The number of wells was n = 103. The diffPkj used for estimating the mean and the standard deviation in equations 29 and 30 consist of the differ- ences log( V i) — log (Vi) for all the log (Vi) in the kth stratum. The correct or most appropriate value of K is not known, so computations in equations 25 and 26 were done using parameter estimates from equa- tions 29 and 30, for K ranging from 1 to 50. As K increases, the outcome of this analysis is essentially equivalent to randomly selecting a PCC- estimated value at each well for computing pumpage at that site. Results are shown in figure 11. When K = 1, the tendency for error magnitude to diminish for larger pumpage is ignored, so that large pumpage could conceivably have errors as large as 239 percent (the 36 Comparison of Two Approaches for Determining Ground -Water Discharge and Pumpage in the Lower Arkansas River Basin, Colorado, 1997 -98