The same advantages apply to the standard error of estimation of the mean SEM, which is equal to . The coefficient of variation of an estimated mean, denoted here by COVM  is the ratio of the standard error of the mean to the mean itself COVM = SEM/M. The coefficient of variation of the mean is often expressed as a percentage. When expressed as a percentage, the term "relative standard error of the mean" will be used, denoted by the symbol RSEM. Thus, RSEM = 100% is equivalent to COVM = 1.0. For comparison between analytes with different units or widely different means, the RSEM provides a standardized, unit-free non-negative numerical value for comparing the precision with which the mean concentrations of multiple analytes have been measured.

The value of RSEM depends on both the standard deviation of the population S and the number of samples N. The standard deviation is a measure of variability, and the variability in the samples depends on many factors including temporal and spatial variations in properties of the ore zone environment, well sampling methods, and laboratory measurement error. Many of these sources of variability are beyond the control of the site operator. As variability depends on site-specific characteristics, specific baseline sample sizes for a production unit cannot be determined in advance without sufficient prior data to provide at least a rough estimate of the standard deviation. Some potential constituents, particularly radionuclides and other matrix characteristics that affect the rate of extraction, may already have been measured during site exploration and development and provide a reasonable sample size for estimating the standard deviation. Other constituents may already have been monitored at nearby sites or sites with similar geochemical characteristics. Others may have no historical background databases and require a round of preliminary baseline sampling before the required sample size can be determined.
Table 7-2 contains summary statistics for the population coefficient of variation cv collected at nine sites for 35 analytes. The average coefficient of variation ranges from 4% to over 250% in this table. The upper part of this table shows the ranked statistics for analytes other than the trace metals, which are shown in the lower part of the table. Analytes other than trace metals with the highest coefficient of variation include NO2-, CO3-2, U, NH4+, and Ra-226. Of this list, only CO3-2, U, and Ra-226 have a large fraction of measurements above the detection limit. Trace metals with the highest coefficient of variation include aluminum, iron, selenium, and manganese. Among these, only iron has a large fraction of measurements over the detection limit.

Although many of the sources of variability are beyond the control of the site operator, increasing the number of baseline samples provides a way to control the RSEM for all analytes of concern to within acceptable levels. Table 7-3 shows an example of the RSEM approach. This table provides the information required to control the degree of precision obtained for mean baseline concentrations of 35 analytes at Christensen Ranch Mine Unit 6 (COGEMA 1996).
Baseline water quality sampling was conducted according to the requirements approved in the Christensen Ranch Amendment Application. The application required four rounds of sampling. The sampling was conducted over a short, 6-week period in the summer of 1996. Sampling events were spaced 2 weeks apart; 2 of the 4 samples from each baseline well were analyzed for a full suite of chemical parameters, while the remaining 2 samples were analyzed for a shorter list of constituents. The 4 rounds of analytical results from the 42 baseline wells were combined to establish baseline water quality. Table 7-2 provides a summary of the combined analytical results from the baseline wells. The list of 35 analytes in Table 7-3 includes the concentrations of major ions, trace metals, and radionuclides and other aquifer properties such as pH, alkalinity, conductivity, and total dissolved solids (TDS).
The table shows the number of samples used to compute the baseline mean and standard deviation, and the standard error and relative standard error of the mean for each analyte. At this site, up to 168 samples were used to characterize the baseline. The RSEM obtained for the analytes ranges from 0.3% to 7%. The radionuclides (U and Ra-226) are among the highest RSEM percentages; 6.8% for U and 6.6% for Ra-226. Several trace metals, including iron and aluminum, are also among the highest percentages. The high RSEM values indicate that these analytes were measured with the least relative precision.

Figure 7-1.	Ore Zone Outline and Well Locations at Christensen Ranch Mine Unit 6

The required number of baseline samples to achieve a precision of P% is calculated using:

where cv and P are expressed in percentage terms. For example, in Table 7-2, uranium has an average coefficient of variation of 105%. The required number of samples to estimate the mean uranium concentration with a precision of P = +-10% is:

The right side of Table 7-3 shows the number of samples required for each analyte to achieve a targeted RSEm of P%, for values of P ranging from 1% up to 100%. At Mine Unit 6, uranium and Ra-226 require the largest number of samples to achieve the targeted level of precision. For example, 75 samples are required to achieve a hypothetical target precision of P = 10% for uranium, and 74 samples are required for Ra-226. These sample sizes are the highest required to achieve a target precision of P = 10% for all listed analytes. If 75 or more baseline samples are collected at the mine unit, then the RSEM for all analytes is expected to be less than the target value of P = 10%. With at least 167 baseline samples for uranium and Ra-226 at Mine Unit 6, the achieved RSEM  for U and Ra-226 is considerably less than 10% for both radionuclides.

Table 7-4 shows the results of similar analyses conducted for nine ISR sites. This table summarizes results of the analyses, with emphasis on the RSEM achieved for the mean baseline uranium and Ra-226 concentrations at each site. The table also shows the size, number of baseline wells, number of wells per acre, and the average number of baseline samples per well. The right side of this table contains the actual sample size and achieved precision for uranium and Ra-226. This table also shows the sample sizes that would be required for a hypothetical targeted precision of +-10%. The highlighting in this table indicates sites meeting a hypothetical targeted precision goal of +-10%.

Table 7-5 contains the summary information for uranium and Ra-226 found in the bottom two rows of Table 7-3 for nine ISR sites. The table shows the number of samples required for the RSEM for uranium and Ra-226 to be less than +-P%. The table also shows the actual number of baseline samples and the achieved precision for uranium and Ra-226 at each site. Figure 7-2 contains a scatter plot of the maximum RSEM for uranium and Ra-226 mean concentrations versus the baseline sample size at the nine production units listed in Table 7-5. A baseline sample size of approximately 80 to 100 samples is sufficient to achieve a precision of +-10% at the sites examined.

NRC's current expectations regarding baseline sampling in the ore zone are set out in their Standard Review Plan for ISR license applications (NRC 2003, p. 5-39), which specifies that one well per acre should be sampled with four independent samples taken from each well. If a typical wellfield is 25 acres, then 100 baseline samples would be required. Using the RSEM approach presented here and assuming that a DQO for the RSEM is fixed at +-10% and the ratio of the sample mean to the sample standard deviation is 1.0, then the required number of samples would be 100 or the same as current practice for a typical wellfield. This hypothetical example is supported by Figure 7-2, which shows, using actual sites, that a baseline sample size of approximately 80 to 100 samples is sufficient to achieve a precision of +-10% at the sites examined. If the DQO is reduced to +-5%, the theoretical sample size would be increased to 400. Any additional sampling could be accommodated by various combinations of additional wells, extending the sampling period, RSEM or increasing the sampling frequency. However, the operator must insure that the additional samples are independent in space and time. For example, if additional baseline sampling wells are required, they should be located in the ore zone as far as possible from other baseline sampling wells. If more samples are required from each baseline well, the additional sampling times should be roughly mid-way between current sampling times. Frequency of sampling should not exceed eight samples per year per well.

Summary

In practice, the following approach might be taken to determine the number of baseline samples:

   *    The regulator establishes the DQO for the RSEM (e.g., +-10%).
   *    The operator collects periodic samples for 1 year based on a sample density of one per acre per sampling. (From Table 7-4, it is likely that larger wellfields may require quarterly sampling, while smaller wellfields (e.g., less than 20 acres) may require monthly sampling.)
   *    The operator calculates the RSEM for each analyte by pooling sampling results.
   *    Unless the sampling results suggest otherwise, the operator determines whether either uranium or Ra-226 has an RSEM exceeding the DQO.
   *    If either uranium or Ra-226 exceeds the DQO, the operator continues sampling until the DQO is achieved.

This approach is based on the assumptions that uranium and Ra-226 are key analytes in understanding the ground water chemistry, that they will have few nondetects in the baseline sampling program, and that, based on the data reported here, they typically have high variability.

Table 7-2.	Summary Statistics for Population Coefficient of Variation (cv) of Baseline Parameters at Nine ISR Sites
                                     Rank
                                    Analyte
                                Number of Sites
                                  Minimum cv
                                  Average cv
                                  Maximum cv
                      Major Ions (mg/L) and Radiometrics
                                       1
NO2
                                       2
                                      97%
                                     121%
                                     145%
                                       2
CO3
                                       8
                                      43%
                                     110%
                                     327%
                                       3
U mg/L
                                       9
                                      42%
                                     105%
                                     213%
                                       4
NH4
                                       7
                                      11%
                                     105%
                                     433%
                                       5
Ra-226 pCi/L
                                       9
                                      61%
                                      75%
                                      87%
                                       6
NO3
                                       7
                                      6%
                                      56%
                                     222%
                                       7
K
                                       9
                                      12%
                                      47%
                                      93%
                                       8
Mg
                                       9
                                      7%
                                      38%
                                     129%
                                       9
Ca
                                       9
                                      1%
                                      21%
                                      52%
                                      10
F
                                       9
                                      6%
                                      18%
                                      35%
                                      11
Cl
                                       9
                                      7%
                                      15%
                                      36%
                                      12
HCO3
                                       9
                                      3%
                                      12%
                                      25%
                                      13
SiO2
                                       9
                                      3%
                                      12%
                                      23%
                                      14
Alk
                                       9
                                      3%
                                      9%
                                      27%
                                      15
SO4
                                       9
                                      3%
                                      8%
                                      16%
                                      16
Cond
                                       9
                                      3%
                                      7%
                                      14%
                                      17
TDS
                                       9
                                      2%
                                      6%
                                      12%
                                      18
Na
                                       9
                                      3%
                                      5%
                                      9%
                                      19
pH
                                       9
                                      2%
                                      4%
                                      7%
                              Trace Metals (mg/L)
                                       1
Al
                                       2
                                      65%
                                     262%
                                     459%
                                       2
Fe
                                       3
                                      62%
                                     253%
                                     406%
                                       3
Se
                                       7
                                      50%
                                     208%
                                     581%
                                       4
Mn
                                       4
                                      38%
                                     127%
                                     206%
                                       5
V
                                       3
                                      55%
                                     103%
                                     155%
                                       6
Cr
                                       1
                                      95%
                                      95%
                                      95%
                                       7
As
                                       6
                                      33%
                                      93%
                                     285%
                                       8
Cd
                                       2
                                      70%
                                      85%
                                     100%
                                       9
Hg
                                       1
                                      83%
                                      83%
                                      83%
                                      10
Pb
                                       1
                                      83%
                                      83%
                                      83%
                                      11
Cu
                                       3
                                      37%
                                      82%
                                     110%
                                      12
Ni
                                       2
                                      28%
                                      65%
                                     102%
                                      13
Zn
                                       5
                                      33%
                                      50%
                                      91%
                                      14
Ba
                                       1
                                      39%
                                      39%
                                      39%
                                      15
B
                                       2
                                      4%
                                      25%
                                      46%
                                      16
Mo
                                       2
                                      9%
                                      24%
                                      40%

Table 7-3.	Baseline Statistics and Number of Samples Required at Christensen Mine Unit 6 for the Relative Standard Error of the Baseline Mean to Be Less Than P% for 35 Analytes with Summary Statistics for U and Ra-226
                                    Analyte
                              Number of Samples*
                                     Mean*
                                       M
                              Standard Deviation*
                                       S
                          Standard Error of the Mean
                                      SEM
                      Relative Standard Error of the Mean
                                     RSEM
Number of Samples Required for Relative Standard Error of Baseline Mean Less Than P%

                                     Total
                                       N
                                  >D.L.[a]

                                     P=1%
                                      2%
                                      3%
                                      4%
                                      5%
                                      6%
                                      7%
                                      8%
                                      9%
                                      10%
                                      15%
                                      20%
                                      25%
                                      30%
                                      40%
                                      50%
                                      75%
                                     100%
MAJOR IONS  -  mg/L

Ca
                                                                            164
                                                                            164
                                                                         26.238
                                                                          4.947
                                                                          0.386
                                     1.5%
                                                                            356
                                                                             89
                                                                             40
                                                                             23
                                                                             15
                                                                             10
                                                                              8
                                                                              6
                                                                              5
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Mg
                                                                            166
                                                                            162
                                                                          4.455
                                                                          1.417
                                                                          0.110
                                     2.5%
                                                                           1012
                                                                            253
                                                                            113
                                                                             64
                                                                             41
                                                                             29
                                                                             21
                                                                             16
                                                                             13
                                                                             11
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Na
                                                                            168
                                                                            168
                                                                        239.762
                                                                         10.091
                                                                          0.779
                                     0.3%
                                                                             18
                                                                              5
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
K
                                                                            162
                                                                            162
                                                                           6.61
                                                                          2.163
                                                                          0.170
                                     2.6%
                                                                           1071
                                                                            268
                                                                            119
                                                                             67
                                                                             43
                                                                             30
                                                                             22
                                                                             17
                                                                             14
                                                                             11
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
CO3
                                                                             85
                                                                             78
                                                                           5.49
                                                                          2.968
                                                                          0.322
                                     5.9%
                                                                           2923
                                                                            731
                                                                            325
                                                                            183
                                                                            117
                                                                             82
                                                                             60
                                                                             46
                                                                             37
                                                                             30
                                                                             13
                                                                              8
                                                                              5
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
HCO3
                                                                             85
                                                                             83
                                                                          74.48
                                                                         11.834
                                                                          1.284
                                     1.7%
                                                                            253
                                                                             64
                                                                             29
                                                                             16
                                                                             11
                                                                              8
                                                                              6
                                                                              4
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
SO4
                                                                            168
                                                                            168
                                                                        530.643
                                                                         40.372
                                                                          3.115
                                     0.6%
                                                                             58
                                                                             15
                                                                              7
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Cl
                                                                            168
                                                                            168
                                                                          4.539
                                                                          1.091
                                                                          0.084
                                     1.9%
                                                                            578
                                                                            145
                                                                             65
                                                                             37
                                                                             24
                                                                             17
                                                                             12
                                                                             10
                                                                              8
                                                                              6
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
NH4
                                                                             86
                                                                             19
                                                                          0.111
                                                                          0.031
                                                                         0.0033
                                     3.0%
                                                                            780
                                                                            195
                                                                             87
                                                                             49
                                                                             32
                                                                             22
                                                                             16
                                                                             13
                                                                             10
                                                                              8
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
NO2
                                                                             86
                                                                              0

NO3
                                                                             86
                                                                              5
                                                                          0.076
                                                                          0.015
                                                                         0.0016
                                     2.1%
                                                                            390
                                                                             98
                                                                             44
                                                                             25
                                                                             16
                                                                             11
                                                                              8
                                                                              7
                                                                              5
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
F
                                                                             85
                                                                             85
                                                                          0.142
                                                                          0.049
                                                                         0.0053
                                     3.7%
                                                                           1191
                                                                            298
                                                                            133
                                                                             75
                                                                             48
                                                                             34
                                                                             25
                                                                             19
                                                                             15
                                                                             12
                                                                              6
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
SiO2
                                                                             86
                                                                             86
                                                                          4.058
                                                                          0.927
                                                                          0.100
                                     2.5%
                                                                            522
                                                                            131
                                                                             58
                                                                             33
                                                                             21
                                                                             15
                                                                             11
                                                                              9
                                                                              7
                                                                              6
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
TDS
                                                                            167
                                                                            167
                                                                         861.09
                                                                         50.746
                                                                          3.927
                                     0.5%
                                                                             35
                                                                              9
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Cond μmho/cm
                                                                            167
                                                                            167
                                                                        1261.26
                                                                          66.99
                                                                          5.184
                                     0.4%
                                                                             29
                                                                              8
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Alk CaCO3
                                                                            166
                                                                            166
                                                                         70.753
                                                                          6.804
                                                                          0.528
                                     0.7%
                                                                             93
                                                                             24
                                                                             11
                                                                              6
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
pH
                                                                            164
                                                                            164
                                                                           8.83
                                                                          0.369
                                                                          0.029
                                     0.3%
                                                                             18
                                                                              5
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
TRACE METALS  -  mg/L
Al
                                                                             86
                                                                             20
                                                                          0.127
                                                                          0.082
                                                                          0.009
                                     7.0%
                                                                           4169
                                                                           1043
                                                                            464
                                                                            261
                                                                            167
                                                                            116
                                                                             86
                                                                             66
                                                                             52
                                                                             42
                                                                             19
                                                                             11
                                                                              7
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
As
                                                                            167
                                                                            118
                                                                          0.003
                                                                          0.001
                                                                        0.00008
                                     2.6%
                                                                           1112
                                                                            278
                                                                            124
                                                                             70
                                                                             45
                                                                             31
                                                                             23
                                                                             18
                                                                             14
                                                                             12
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Ba
                                                                             86
                                                                              0

B
                                                                             86
                                                                              0

Cd
                                                                             86
                                                                              0

Cr
                                                                             86
                                                                              1
                                                                           0.05
                                                                              0

                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Cu
                                                                             86
                                                                              0

Fe
                                                                             86
                                                                              3
                                                                          0.107
                                                                          0.066
                                                                         0.0071
                                     6.7%
                                                                           3805
                                                                            952
                                                                            423
                                                                            238
                                                                            153
                                                                            106
                                                                             78
                                                                             60
                                                                             47
                                                                             39
                                                                             17
                                                                             10
                                                                              7
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
Pb
                                                                             86
                                                                              0

Mn
                                                                             86
                                                                              0

Hg
                                                                             86
                                                                              3
                                                                         0.0003
                                                                              0

                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Mo
                                                                             86
                                                                              7
                                                                          0.057
                                                                          0.005
                                                                        0.00054
                                     0.9%
                                                                             77
                                                                             20
                                                                              9
                                                                              5
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Ni
                                                                             86
                                                                              4
                                                                          0.068
                                                                          0.019
                                                                         0.0020
                                     3.0%
                                                                            781
                                                                            196
                                                                             87
                                                                             49
                                                                             32
                                                                             22
                                                                             16
                                                                             13
                                                                             10
                                                                              8
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Se
                                                                            168
                                                                             13
                                                                         0.0020
                                                                          0.001
                                                                        0.00008
                                     3.9%
                                                                           2500
                                                                            625
                                                                            278
                                                                            157
                                                                            100
                                                                             70
                                                                             52
                                                                             40
                                                                             31
                                                                             25
                                                                             12
                                                                              7
                                                                              4
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                                                              2
V
                                                                             86
                                                                              3
                                                                          0.107
                                                                          0.059
                                                                         0.0064
                                     5.9%
                                                                           3041
                                                                            761
                                                                            338
                                                                            191
                                                                            122
                                                                             85
                                                                             63
                                                                             48
                                                                             38
                                                                             31
                                                                             14
                                                                              8
                                                                              5
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
Zn
                                                                             86
                                                                             21
                                                                          0.015
                                                                          0.008
                                                                        0.00086
                                     5.8%
                                                                           2845
                                                                            712
                                                                            317
                                                                            178
                                                                            114
                                                                             80
                                                                             59
                                                                             45
                                                                             36
                                                                             29
                                                                             13
                                                                              8
                                                                              5
                                                                              4
                                                                              2
                                                                              2
                                                                              2
                                                                              2
RADIOMETRICS

U mg/L
                                                                            164
                                                                            148
                                                                         0.0126
                                                                         0.0109
                                                                        0.00085
                                     6.8%
                                                                           7484
                                                                           1871
                                                                            832
                                                                            468
                                                                            300
                                                                            208
                                                                            153
                                                                            117
                                                                             93
                                                                             75
                                                                             34
                                                                             19
                                                                             12
                                                                              9
                                                                              5
                                                                              3
                                                                              2
                                                                              2
Ra-226 pCi/L
                                                                            168
                                                                            167
                                                                        105.858
                                                                         90.928
                                                                          7.015
                                     6.6%
                                                                           7379
                                                                           1845
                                                                            820
                                                                            462
                                                                            296
                                                                            205
                                                                            151
                                                                            116
                                                                             92
                                                                             74
                                                                             33
                                                                             19
                                                                             12
                                                                              9
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                  Maximum for
                                   U, Ra-226
                                                                            168
                                                                            167
                                                                              -
                                                                              -
                                                                              -
                                     6.8%
                                                                           7484
                                                                           1871
                                                                            832
                                                                            468
                                                                            300
                                                                            208
                                                                            153
                                                                            117
                                                                             93
                                                                             75
                                                                             34
                                                                             19
                                                                             12
                                                                              9
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                       P

                                                                           P=1%
                                                                             2%
                                                                             3%
                                                                             4%
                                                                             5%
                                                                             6%
                                                                             7%
                                                                             8%
                                                                             9%
                                                                            10%
                                                                            15%
                                                                            20%
                                                                            25%
                                                                            30%
                                                                            40%
                                                                            50%
                                                                            75%
                                                                           100%
a - >D.L. is greater than detection limit
* Source: COGEMA 1996, Table 6, mean is arithmetic mean for all detectable samples
Table 7-4.	Wellfield Characteristics and Comparison of Actual and Target Baseline Sample Size at Nine ISR Production Units

                                     Site

                                     Acres

                           Number of Baseline Wells

                       Number of Baseline Wells per Acre

                      Average Number of Samples per Well

                      Target Number of Samples per Well*

                 Maximum Target Number of Samples for RSEM=10%
                                       U
                                    Ra-226

                                   Actual N
                                 Achieved RSEM
                                   Target N
                                  Target RSEM
                                   Actual N
                                 Achieved RSEM
                                   Target N
                                  Target RSEM
Crow Butte MU1
                                                                            9.3
                                                                             12
                                                                            1.3
                                                                            8.4
                                                                              8
                                                                             94
                                                                            101
                                                                           9.6%
                                                                             94
                                                                            10%
                                                                            101
                                                                           7.7%
                                                                             60
                                                                            10%
Highland A
                                                                            3.0
                                                                              5
                                                                            1.7
                                                                            1.0
                                                                              8
                                                                             38
                                                                              5
                                                                          18.6%
                                                                             18
                                                                            10%
                                                                              5
                                                                          27.4%
                                                                             38
                                                                            10%
Highland B
                                                                           13.8
                                                                             20
                                                                            1.4
                                                                              -
                                                                             24
                                                                            445
                                                                              -
                                                                              -
                                                                            445
                                                                            10%
                                                                              -
                                                                              -
                                                                             47
                                                                            10%
Irigaray MU1 to MU9
                                                                             30
                                                                             46
                                                                            1.5
                                                                            4.3
                                                                              4
                                                                            162
                                                                            200
                                                                           9.0%
                                                                            162
                                                                            10%
                                                                            200
                                                                           7.0%
                                                                             76
                                                                            10%
Christensen MU2 North
                                                                            5.7
                                                                              8
                                                                            1.4
                                                                            4.0
                                                                              9
                                                                             69
                                                                             32
                                                                          14.7%
                                                                             69
                                                                            10%
                                                                             32
                                                                          10.8%
                                                                             37
                                                                            10%
Christensen MU3
                                                                           20.4
                                                                             22
                                                                            1.1
                                                                            4.0
                                                                              6
                                                                            132
                                                                             64
                                                                          14.3%
                                                                            132
                                                                            10%
                                                                             64
                                                                           8.3%
                                                                             44
                                                                            10%
Christensen MU4
                                                                             12
                                                                             12
                                                                              1
                                                                            4.0
                                                                             14
                                                                            164
                                                                             48
                                                                          18.5%
                                                                            164
                                                                            10%
                                                                             49
                                                                          11.9%
                                                                             70
                                                                            10%
Christensen MU5
                                                                             25
                                                                             25
                                                                              1
                                                                            4.0
                                                                              3
                                                                             69
                                                                             99
                                                                           5.8%
                                                                             34
                                                                            10%
                                                                             99
                                                                           8.3%
                                                                             69
                                                                            10%
Christensen MU6
                                                                             42
                                                                             42
                                                                              1
                                                                            3.9
                                                                              2
                                                                             75
                                                                            164
                                                                           6.8%
                                                                             75
                                                                            10%
                                                                            168
                                                                           6.6%
                                                                             74
                                                                            10%

                                                                            Key

Sites meeting hypothetical targeted precision goal of 10%.

                                      [*]
An alternative is to increase the number of wells.

Table 7-5.	Number of Samples Required at Nine Production Units for Relative Standard Error of U and Ra-226 Mean Baseline Concentrations to Be Less Than +-P%
                                     Site
                                Wells per Acre
                               Samples per Well
Number of Samples (N) Required for RSEM of U and Ra-226 Baseline Means to Be Less Than +-P%

                                     P=1%
                                      2%
                                      3%
                                      4%
                                      5%
                                      6%
                                      7%
                                      8%
                                      9%
                                      10%
                                      15%
                                      20%
                                      25%
                                      30%
                                      40%
                                      50%
                                      75%
                                     100%
Crow Butte MU1
                                     1.29
                                     8.40
                                                                          9,359
                                                                          2,340
                                                                          1,040
                                                                            585
                                                                            375
                                                                            260
                                                                            191
                                                                            147
                                                                            116
                                                                             94
                                                                             42
                                                                             24
                                                                             15
                                                                             11
                                                                              6
                                                                              4
                                                                              2
                                                                              2
  Actual P and N for (U, Ra)

9.6% (101,101)

Highland A
                                     1.67
                                     1.00
                                                                          3,741
                                                                            936
                                                                            416
                                                                            234
                                                                            150
                                                                            104
                                                                             77
                                                                             59
                                                                             47
                                                                             38
                                                                             17
                                                                             10
                                                                              6
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
  Actual P and N for (U, Ra)

                                                                    27.4% (5,5)

Highland B
                                     1.38
                                       -
                                                                         45,374
                                                                         11,344
                                                                          5,042
                                                                          2,836
                                                                          1,815
                                                                          1,261
                                                                            926
                                                                            709
                                                                            561
                                                                            454
                                                                            202
                                                                            114
                                                                             73
                                                                             51
                                                                             29
                                                                             19
                                                                              9
                                                                              5
  Actual P and N for (U, Ra)

Irigaray MU1-9
                                     1.53
                                     4.35
                                                                         16,127
                                                                          4,032
                                                                          1,792
                                                                          1,008
                                                                            646
                                                                            448
                                                                            330
                                                                            252
                                                                            200
                                                                            162
                                                                             72
                                                                             41
                                                                             26
                                                                             18
                                                                             11
                                                                              7
                                                                              3
                                                                              2
  Actual P and N for (U, Ra)

9% (200,200)

Christensen MU2 North
                                     1.40
                                     4.00
                                                                          6,877
                                                                          1,720
                                                                            765
                                                                            430
                                                                            276
                                                                            192
                                                                            141
                                                                            108
                                                                             85
                                                                             69
                                                                             31
                                                                             18
                                                                             12
                                                                              8
                                                                              5
                                                                              3
                                                                              2
                                                                              2
  Actual P and N for (U, Ra)

                                 14.7% (32,32)

Christensen MU3
                                     1.08
                                     4.00
                                                                         13,142
                                                                          3,286
                                                                          1,461
                                                                            822
                                                                            526
                                                                            366
                                                                            269
                                                                            206
                                                                            163
                                                                            132
                                                                             59
                                                                             33
                                                                             22
                                                                             15
                                                                              9
                                                                              6
                                                                              3
                                                                              2
  Actual P and N for (U, Ra)

14.3% (64,64)

Christensen MU4
                                     1.00
                                     4.00
                                                                         16,352
                                                                          4,088
                                                                          1,817
                                                                          1,022
                                                                            655
                                                                            455
                                                                            334
                                                                            256
                                                                            202
                                                                            164
                                                                             73
                                                                             41
                                                                             27
                                                                             19
                                                                             11
                                                                              7
                                                                              3
                                                                              2
  Actual P and N for (U, Ra)

                                                                  18.5% (48,49)

Christensen MU5
                                     1.00
                                     3.96
                                                                          6,838
                                                                          1,710
                                                                            760
                                                                            428
                                                                            274
                                                                            190
                                                                            140
                                                                            107
                                                                             85
                                                                             69
                                                                             31
                                                                             18
                                                                             11
                                                                              8
                                                                              5
                                                                              3
                                                                              2
                                                                              2
  Actual P and N for (U, Ra)

                                                                   8.3% (99,99)

Christensen MU6
                                     1.00
                                     3.90
                                                                          7,484
                                                                          1,871
                                                                            832
                                                                            468
                                                                            300
                                                                            208
                                                                            153
                                                                            117
                                                                             93
                                                                             75
                                                                             34
                                                                             19
                                                                             12
                                                                              9
                                                                              5
                                                                              3
                                                                              2
                                                                              2
  Actual P and N for (U, Ra)

                                                                 6.8% (164,168)

                      Summary Statistics Across All Sites
                                Wells per Acre
                               Samples per Well
Number of Samples (N) Required for Coefficients of Variation of U and Ra-226 Baseline Means to Be Less Than +-P%

                                     P=1%
                                      2%
                                      3%
                                      4%
                                      5%
                                      6%
                                      7%
                                      8%
                                      9%
                                      10%
                                      15%
                                      20%
                                      25%
                                      30%
                                      40%
                                      50%
                                      75%
                                     100%
                                    Average
                                     1.24
                                     4.18
                                                                         13,922
                                                                          3,481
                                                                          1,547
                                                                            870
                                                                            557
                                                                            387
                                                                            285
                                                                            218
                                                                            172
                                                                            140
                                                                             62
                                                                             35
                                                                             23
                                                                             16
                                                                              9
                                                                              6
                                                                              3
                                                                              2
                                    Minimum
                                     1.00
                                     1.00
                                                                          3,741
                                                                            936
                                                                            416
                                                                            234
                                                                            150
                                                                            104
                                                                             77
                                                                             59
                                                                             47
                                                                             38
                                                                             17
                                                                             10
                                                                              6
                                                                              5
                                                                              3
                                                                              2
                                                                              2
                                                                              2
                                    Maximum
                                     1.67
                                     8.40
                                                                         45,374
                                                                         11,344
                                                                          5,042
                                                                          2,836
                                                                          1,815
                                                                          1,261
                                                                            926
                                                                            709
                                                                            561
                                                                            454
                                                                            202
                                                                            114
                                                                             73
                                                                             51
                                                                             29
                                                                             19
                                                                              9
                                                                              5

Figure 7-2.	Scatter Plot of the Maximum RSEM for U and Ra-226 Mean Concentrations versus Baseline Sample Size at Nine Production Units

Determining the Number of Monitoring Wells Required to Detect Noncompliance

This section presents a statistical approach to determining the number of post-operational monitoring wells. The number of wells to monitor in the post-operational period will depend on the size of the field and the connectivity (i.e., the degree to which the wells are hydraulically connected) of the wells. One approach to this question involves the theory of hypergeometric sampling. The monitoring wells discussed in this section are wells within the ore zone and not the perimeter monitoring wells used to detect excursions discussed in Section 3.3.1.

If the wells in the production unit have high connectivity, then fewer monitoring wells would be required to achieve adequate coverage. A modified approach for wellfields with some measure of connectivity is discussed in Section 7.2.2. In practice, it may prove difficult to quantify the qualitative measures of connectivity discussed in that section. If that proves to be the case, the hypergeometric sampling approach presented in the following section is most appropriate.

Determining the Number of Monitoring Wells based on Hypergeometric Sampling

Experience has shown that it is unlikely that every monitored well in a production unit will be in  compliance. Hypothesis tests are used to determine if the monitored wells have returned to baseline conditions and have achieved steady state. Tests for comparing the data collected at the site after restoration with baseline conditions are discussed in Section 7.3. Tests for trends are discussed in Section 7.7. We assume that the test employed is adequate for the task of determining which of the monitored wells are in compliance and which are not. Every test has a "gray region" where the test may have relatively high decision error rates, but these tests perform with almost 100% accuracy when the monitored wells are not near the gray region. Although the tests are performed only on the monitored wells, we are interested in extrapolating the test results obtained for the monitored wells to other wells in the production unit, assuming they would have a similar pattern of test outcomes if all wells were monitored.

If the test outcomes indicate that all monitored wells demonstrate compliance, or if only a few do not, then these results also provide information about the wells which were not monitored. Suppose the question is phrased in this manner:

      If all wells were monitored, what percentage of the wells would be expected to not demonstrate compliance?

If all wells were monitored, this percentage would be known. When fewer than all wells are monitored, the percentage can be known only with uncertainty. The statistical approach involves confidence levels for the unknown parameter Q defined as the percentage of wells that would not demonstrate compliance if all wells were monitored. The compliance determinations for the monitored wells are based on statistical tests that will have Type 1 and Type 2 errors. Hence, a test indicating that a monitored well is in compliance does not mean that the well truly is in compliance, and a test indicating that a monitored well is not in compliance does not mean that the well truly is not in compliance. This limitation also is applicable for Q, which is a measure of the proportion of wells with test results that are not in compliance.

In the following discussion, the set of all wells in the production unit form the "population" from which a "sample" of monitored wells is selected. In many sampling problems, it is assumed that the population to be sampled is much larger than the intended size of the sample. Essentially, the population size is assumed to be of infinite size. When a population of finite size is sampled, the required size of the sample is less than the corresponding sample size for an infinite population. In a population with a relatively small size (in the hundreds), the savings can be significant. With a population of finite size, it is theoretically possible to determine exactly any population parameter of interest by monitoring all of the wells. When only a sample of wells is monitored, then population parameter estimates will have uncertainty and a statistical approach is used for estimating these parameters.

There are several statistical approaches for estimating the population parameter Q. Two parametric statistical approaches are the traditional frequentist statistical approach, such as maximum likelihood estimation and the Bayesian approach.

In maximum likelihood estimation, the goal is to find the "best" point estimate for Q, defined as the value of Q which maximizes the likelihood function l(Q, X) for the observed set of sample values X=(x1, ..., xn). This estimate is a "statistic," meaning that it depends on the sample values X. The most commonly used maximum likelihood estimate is the sample mean. Since this estimate is a function of the sample values, the estimate would vary if the sampling process was repeated. Confidence intervals for the "true" value of Q are formed by calculating the frequency of values of Q that would occur when the sampling process is repeated.

In the Bayesian approach, it is assumed that the value of the population parameter Q is always uncertain, both before and after the sampling process. It is also assumed that this uncertainty can be expressed in the form of a probability distribution for the unknown parameter Q. This probability distribution expresses the uncertainty in the estimate of Q. If Q is relatively well known, then the probability distribution is a narrow distribution, but if the value of Q is very uncertain, then the probability distribution is wide.

Before the sampling is done, the prior state of knowledge about the parameter Q is expressed in a prior distribution written as p(Q). After the sampling is completed, the data are used to update the prior state of knowledge about the parameter Q (using Bayes' theorem) to a new, narrower distribution called the posterior distribution p(Q|X). The posterior distribution is proportional to the product of the prior distribution times the likelihood function: p(Q|X) ≈ p(Q) l(Q, X)

In this application, a non-informative prior distribution is used for Q. The only prior knowledge assumed is that Q must be between 0 and 1, and no value of Q in between these limits is more likely that any other value. These assumptions imply that the prior distribution for Q is a uniform distribution between 0 and 1, i.e., p(Q)=1 for all Q. Note that p(Q|X) ≈ l(Q, X) with this choice of a prior distribution. The value of Q, which maximizes the likelihood function, is the same value that maximizes the Bayesian posterior distribution for Q. Hence, the maximum likelihood estimate is equal to the Bayesian highest posterior density estimate using a non-informative prior distribution.

In addition to the point estimate generated using the Bayesian approach, both 1-sided and 2-sided intervals of highest posterior density (HPD) may be constructed containing 90%, 95% or any desired level of posterior probability. The shape of the posterior distribution is, in general, not symmetric around the highest posterior density point estimate. This means that the HPD intervals for Q constructed using the Bayesian approach are in general asymmetric. The degree of asymmetry increases at values of Q near 0 or 1. The HPD intervals are approximately symmetric when Q is near (1/2).

In this section, it is assumed that a sufficient number of samples have been collected from each monitored well to determine if the well demonstrates compliance. (Sections 7.6 and 7.8 discuss the procedures for determining compliance of an individual well.) After the sampling is completed, some of the monitored wells will have demonstrated compliance and others will not. These results are used to determine the maximum likelihood (and Bayesian) point estimate of Q equal to the percentage of monitored wells not demonstrating compliance:

This point estimate of Q will have statistical uncertainty. The required number of monitoring wells is determined by placing an upper bound on the posterior distribution for Q. If there are M wells in a production unit, then a sufficient number N are designated as monitoring wells, such that the Bayesian 1-sided highest posterior density interval for Q is within the specified bounds.
If all monitored wells are found to demonstrate compliance, then the point estimate for Q is 0%. However, this estimate also has uncertainty, as there still may be unmonitored wells that would be expected not to demonstrate compliance. If a well production unit has only 5 wells, then monitoring of all wells may be required to achieve the desired level of confidence in the estimate of Q. If there are 200 wells in a production unit, it will be necessary to monitor more than 5 wells, but not to monitor all the wells. However, if the number of noncomplying wells in the unit is small relative to M, it is very difficult to detect that there are wells out of compliance without very extensive monitoring.

A Bayesian analysis of the hypergeometric distribution is used to determine the number of monitoring wells required to demonstrate, with a relatively high degree of assurance, that no more than q percent of the wells in the field would not demonstrate compliance if all wells were monitored. Let X denote the number of monitored wells that do not demonstrate compliance in the set of N monitoring wells. Let Y denote the unknown number of wells in the field that would not show compliance if all wells were tested. Then the random variable X has a hypergeometric distribution with parameters N, M and Y. A realization of the random variable X is a nonnegative integer denoted by the lower case symbol x. Table 7-6 shows the parameters of the hypergeometric distribution and the range of values for each parameter.

Table 7-6.	Parameter Definitions for the Hypergeometric Distribution
                                   Parameter
                         All Wells in Production Unit
                                 (Population)
                      Monitored Wells in Production Unit
                                   (Sample)
                                Number of Wells
                                       M
                                    M>0
                                       N
                                   1<=N<=M
                    Number of Wells not Showing Compliance
                                       Y
                                  x <=y<= M
                                       X
                       max(0,N-M+y) <= x <= min(N, y)

A "sample" of N monitoring wells should be selected from the total number of wells. The monitoring wells should be representative of the "population" of all M wells in the production unit. Using the hypergeometric distribution, the probability that x of the N monitored wells do not show compliance, given that Y=y of all M wells in the production unit do not show compliance, is:

                                       .

Here the combinatorial coefficient  for a >= b, and the symbol a! denotes the factorial of the integer a = (1)(2)(3)...(a). By convention, 0! = 1. Note that the term (M-y) is the number of wells in the production unit that would demonstrate compliance if all wells were tested, and (N-x) is the number of monitored wells that demonstrate compliance.

For ease of discussion, we denote a test outcome that demonstrates compliance as a "Pass;" otherwise, the test outcome is a "Fail." The posterior distribution for the (unknown) number of wells Y in the population with outcome "Fail," given that x wells had the outcome "Fail" in the sample of N wells, is:

                                    with .

Here Y is a random variable with integer values y = x, ... , M - (N - x), the posterior distribution gives the probability of each possible value of Y, and k is a normalizing constant such that these probabilities add to one. The parameter Q is the unknown percentage of wells in the population that would have an outcome of "Fail" if all wells had been monitored. For each field size M there are M+1 discrete values possible for Q, one for each possible value of Y: . The posterior distribution for Q is derived from the posterior distribution for Y by linear transformation.