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15 percent, and the year -to -year variance was about <br />nine times the date- within -year variance. This indi- <br />cates that year -to -year variability of the PCC may <br />make a significant contribution to errors in the PCC <br />approach for estimating pumpage. The conclusions <br />based on this analysis are based on an assumption that <br />the State - approved PCC's made from 1994 -97 are of <br />the same quality as the 1998 study PCC's. <br />Details of Analysis and Results <br />To determine total pumpage for some specified <br />period of time, such as one pumping season, at a given <br />well, the following terms are used: <br />VT = true total pumpage volume for the monitoring <br />period, <br />V = pumpage volume as measured by a TFM, <br />V = pumpage volume as estimated by the PCC <br />approach, <br />AT = true electrical power consumption for the <br />period, <br />A = total electrical power consumption as <br />measured by a meter, <br />PT = true PCC for the period, and <br />p = estimated PCC <br />True values in these definitions cannot be measured <br />directly, but are still assumed to exist. Total pumpage <br />estimated by the PCC approach ( V ) is computed using <br />metered power consumption ( A) and the estimated <br />PCC (p) from the equation <br />V = A (9) <br />P <br />(The conversion factor to account for different units of <br />measure will for simplicity be taken to be unity in this <br />section.) <br />At this point, no assumption is made about how <br />p is obtained. The true PCC ( PT) is the value that, by <br />definition, yields a correct value for total pumpage <br />when divided into true power consumption, or <br />however, the true values AT and VT are generally <br />unknown, so PT also is unknown. <br />Again, logarithmic transformations are used to <br />express all errors; that is, an error is the difference <br />between a log- transformed quantity that is measured <br />or estimated and the log transform of the corre- <br />sponding true value. Hence the errors are defined as: <br />U1 = log(V) — log(VT) = TFM pumpage error, <br />U2 = log (A) — log (AT) =electrical power meter <br />error, <br />U3 = log (p) — log (pT) = error in the estimated <br />PCC. <br />One relation of interest is the error in the PCC <br />approach, given by <br />log (V) —log (VT) = U2 —U3 (11) <br />the difference between power meter error and PCC <br />error. This relation is derived using the definitions and <br />equations 9 and 10. <br />If the estimated PCC is obtained using a <br />measured instantaneous discharge, as in the data set <br />analyzed in the section "Comparison of Ground -Water <br />Pumpage Estimates ", and if measurements are made at <br />some single time, t, the following can be defined: <br />QT(t) = true instantaneous discharge at time t, <br />Q(t) =instantaneous discharge as measured by a <br />TFM, <br />Q(t) =instantaneous discharge as measured by a <br />portable flowmeter, <br />aT(t) = true instantaneous electrical power <br />consumption at time t, <br />a(t) = instantaneous electrical power consumption <br />determined from a power meter, <br />PT(t) = true instantaneous PCC, and <br />p = PCC estimated with one instantaneous <br />discharge measurement. <br />Thus, p is calculated by <br />P = a(t) <br />Q(t) <br />(12) <br />A and total pumpage estimated by the PCC approach is <br />T ' (10) equation 9 with p used for p. The true instantaneous <br />PT = VT PCC is defined to be <br />SOURCES OF DISCREPANCY BETWEEN PUMPAGE ESTIMATES 31 <br />