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

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there is a 95- percent probability that the difference in aggregated pumpage between the TFM and PCC approach would be between about —3.41 and 1.59 percent. The analysis indicates that the difference in aggregated pumpage would be expected to be smaller as the number of wells becomes larger. Assuming the distribution of total TFM pumpage is the same for 1998 data set, there is a 95- percent proba- bility that the difference in aggregated pumpage between the TFM and the PCC approach for any given year for a network of 1,000 wells would be between —1.71 and —0.11 percent. This assumes that the large differences in pumpage are confined to wells with smaller pumpage. It also is important to emphasize that only 1998 pumpage data were used for this anal- ysis, so the effect of temporal variations (over a period greater than 1 year) of PCC's on total network pumpage is not known. Details of Analysis and Results The difference in total pumpage between the PCC and TFM approaches for n wells, Dn , is denoted as, n n n Dn = Vi — Vi = (Vi — Vi) (18) where V i denotes the PCC - estimated total pumpage at well i (i =1, 2,..., n), and Vi is the corresponding value of TFM- measured total pumpage at well i. To determine the difference in total pumpage for n wells, D,, it may be assumed that Dn is approxi- mately normally distributed. Once the mean of Dn and the standard deviation of Dn are defined, proba- bility statements may be made on the likely magnitude of network differences from year to year. It is assumed that TFM- measured pumpage values Vi are fixed (non- random), and the mean of Dn and the standard deviation of Dn are expressed relative to total network TFM- measured pumpage. Complications arise in computing the mean and standard deviation of Dn primarily because of the nonnormality of the individual well differences, diffP, and the fact that these differences appear to have a tendency to vary in magnitude depending on how large ground -water pumpage, V, is. This variation necessitates using a stratification scheme. The effect of using the logarithmic transformation also must be considered. Specifically, analysis of how the errors (differences) at individual wells is propagated to total network errors (differences) requires that three rele- vant issues be considered in some detail: the effect of the logarithmic transformation, the effect of changes of the distribution of differences depending on volume pumped at a well, and the effect of nonnormality of the distribution of differences between (logarithmically transformed) TFM and PCC pumpage volumes.The effect of the logarithmic transformation becomes an issue because, when computing total network pumpage for a number of wells, it is the untrans- formed values that need to summed. Thus, results from analyses using logarithmically transformed data first need to be back transformed. This back - transformation results in a so- called transformation bias. If the differences between the log- transformed pumpage volumes were identically and normally distributed, then estimating the magnitude of this bias would be straightforward. However, as shown in figure 9A, there is indication of a tendency of the distribution of differences to change depending on total pumpage and of nonnormality. Therefore, stratifi- cation is used to account for changes in the distribu- tion of differences, and a parameter- estimation procedure that does not rely on an assumption of normality is used. Much of the problem is associated with the rela- tively few number of paired measurements that have a much larger difference in pumpage than most of the data (fig. 9A). The rank transformation that was used in the analysis of variance down - weighted the effect of these differences and, therefore, produced results that are representative of the central tendency of the data. However, when summing volumes over all wells in a network, the small number of data that have large differences will be included; therefore, the potential effect of these data cannot be ignored. The data associ- ated with the large differences were examined, and a valid reason for deleting them from the analysis was not found. In addition, the nature of the data did not lend itself to fitting a common probability distribution or to description of the exact pattern of the non- uniform variations in the distribution with respect to pumpage. Thus, the approach taken below is essen- tially nonparametric and should be viewed as an attempt to explore the sensitivity of total network pumpage to these large errors (differences). 34 Comparison of Two Approaches for Determining Ground -Water Discharge and Pumpage in the Lower Arkansas River Basin, Colorado, 1997 -98