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

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aM pT(t) = QT(t) • (13) Using instantaneous measurements introduces four new errors: U4 = log[Q(t)] — log[QT(t)] = error in instanta- neous discharge measured with TFM, U5 = log[Q(t)] — log[QT(t)] = error in instanta- neous discharge measured with a portable flowmeter, U6 = log [a(t)] — log [aT(t)] = error in instantaneous power meter reading, U7 = log[pT(t)] — log(PT) = error in instantaneous PCC, or the difference between true instan- taneous PCC, and true PCC for the period. Therefore, when the PCC is estimated using equa- tion 12, the PCC error U3 may be broken down into three components, U3 = U6 — U5 + U7 , (14) which again is shown using the definitions of the various errors. Combining equations 11 and 14 gives the final expression for the error in total pumpage as estimated by the PCC approach, log (V) — log (VT) = U2 — U6 + Us — U7 - (15) The difference in log- transformed instantaneous discharge (diffQ) as measured by a portable flowmeter and a TFM may be expressed as the difference of two errors, diffQ = log[Q(t)] — log[Q(t)] = U5 — U4 . (16) Similarly, the difference between log- transformed pumpage computed by the PCC approach and TFM approach (diffP) may be computed by subtracting the TFM error (U1) from both sides of equation 15 to yield diffP = log(V) — log(V) (17) U2— U6 +U5— U7—U1. The expression for diffP in equation 17 has one additional component, namely U1 , that is not contained in the actual error for the PCC approach (that is, the error relative to true total pumpage) given by equation 15. That is, TFM errors in an actual appli- cation of the PCC approach would not be observed. The differing signs in these expressions indicate that some of the errors can be compensating. A posi- tive error in one term may cancel a negative error in another, giving a smaller overall error. While such cancellation may hold for certain pairs of terms, other pairs of errors may be independent of each other. For example, U5 , the error in instantaneous discharge measured with a portable flowmeter, would not be expected to be related to U6, the error in instanta- neous power meter reading. For variables that are uncorrelated, the signs make no difference in the contribution to total variance, because the variance of a difference of two uncorrelated random variables is the same as the variance of the sum. The errors U4 (error in instantaneous discharge measured with a TFM) and U5 (error in instantaneous discharge measured with a portable flowmeter) repre- sent deviations of instantaneous discharge from true discharge. Because true discharge is unknown, there is no estimate of size of these component errors. Data are available only for diffQ, which, in equation 16, is the difference between these two individual errors. The values in table 4 indicate bounds on the variance of U5 under different conditions. If consistency between two (or more) portable flowmeter methods is an indi- cation that the methods are both accurate, in the sense of being a good estimate of the true instantaneous discharge, then the error variance (0.000539) from table 4 would be a good estimate of the variance of U5 . In this case, the site- and date - variance compo- nents in table 4 would be mostly attributable to error in the TFM, U4. However, an upper bound for the vari- ance of U5 would be the sum of variance components in table 4, or 0.002639. Use of this value as an esti- mate of the variance of U5 would assume that the TFM is error -free. The error U7 (error in instantaneous PCC) in equation 17 represents deviation of the instantaneous PCC from some long -term true value, which is assumed to be constant. Thus, the average magnitude of U7 depends on how much temporal variability exists in the time series { pT(t) }. An in -depth study of temporal variability of the PCC, including trends, seasonality, and magnitude of serial correlation, would 32 Comparison of Two Approaches for Determining Ground -Water Discharge and Pumpage in the Lower Arkansas River Basin, Colorado, 1997 -98