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Even though it is not assumed that the differ- <br />ences at individual sites, Vi — Vi , have any particular <br />distribution, the network difference (Dn) , which is <br />the sum of a number of independent random variables, <br />will, under some general conditions, be approximately <br />normally distributed. This follows from central limit <br />theory, and, because of the stratification that is applied <br />below, central limit results for random variables that <br />are not identically distributed need to be used. Experi- <br />ments at randomly selecting values from the stratified <br />population of total well pumpage to estimate network <br />pumpage indicate that normality is a good approxima- <br />tion for total network pumpage. Given that Dn has an <br />approximately normal distribution, only the mean and <br />variance (or standard deviation) need evaluation. The <br />main purpose of the analysis that follows is to obtain <br />expressions for the mean and standard deviation. <br />Assume that the Vi are a set of fixed <br />(nonrandom) values, and that the deviation of Vi <br />from Vi is described by a random error. The differ- <br />ence between log- transformed PCC- estimated <br />pumpage and log- transformed TFM- measured <br />pumpage at well i is <br />diffPi = log (Vi) —log (Vi) (19) <br />These errors are all assumed to be associated with <br />different wells, so they will be assumed throughout <br />to be independent. <br />Exponentiating both sides of equation 19 gives <br />the relation <br />Vi = Vie dtiffPi <br />(20) <br />between the untransformed variables. The additive <br />error on the log- transformed variables becomes a <br />multiplicative error on the untransformed variables. <br />The mean difference between the PCC and the TFM <br />pumpage volume for well i is <br />E(Vi — Vi) = VYEed`ffPi — Vi <br />= V,E(ediffPi — 0. <br />E(Vi - Vi) = E(ediffli -1) (22) <br />Vi <br />If EdiffPi = 0 , then it may be shown that <br />Eed�ffPi > 1 , or E(ediffPi —1) > 0. Thus, even if <br />the errors in the log- transformed variables have mean <br />zero, there is a positive bias when looking at untrans- <br />formed variables. This is important because, when <br />looking at network -wide aggregates, the untrans- <br />formed variables need to be summed, so the absence <br />of bias in the log- transformed variables does not auto- <br />matically translate into a lack of bias for network -wide <br />aggregates. Bias in the present situation, however, is <br />not limited to bias caused by the logarithmic transfor- <br />mation. Additional bias is introduced by large positive <br />errors that reflect nonnormality of diffPi (fig. 9A), <br />and the variance of these errors changes with Vi , <br />which motivates the need for the stratification that <br />follows. <br />The mean, or expected, difference (also referred <br />to as bias) is given by <br />n <br />EDn = V iE(ed`ffPi —1) , (23) <br />Z = I <br />and the variance is given by <br />n <br />Var(Dj _ V?Var(ed`ffPi) . (24) <br />i =1 <br />To deal with the error distribution dependence <br />on total pumpage, the population of wells is stratified <br />with respect to the magnitude of total pumpage at a <br />well, Vi' and it is assumed that the errors within <br />each stratum are identically distributed. Equation 23 <br />leads to <br />K <br />EDn = I Bk K <br />(21) k = I <br />where E denotes mathematical expectation, or mean, <br />and Vi is assumed to be fixed. If the mean deviation is <br />expressed as a fraction of TFM pumpage Vi , it is <br />(25) <br />where K is the number of strata, Bk is the sum <br />of the Vi for all wells in the kth stratum, and <br />µk = E(ecl ' — 0 for each well i in the kth stratum. <br />Likewise, equation 24 yields <br />ESTIMATION OF TOTAL NETWORK PUMPAGE 35 <br />