## Abstract

Geoscientists undertaking mineral exploration sometimes are provided with historic geological and geochemical information about a prospective mineral deposit that they would like to use in a modern mineral resource assessment. Unfortunately, these historic datasets may lack critical information describing the quality of the data, such as duplicate samples, that are required under today’s disclosure regulations. As a result, geoscientists sometimes need to generate such data quality information after the fact (*a posteriori*). This contribution describes how duplicate samples can be obtained from historic drill-core to provide an unbiased assessment of grade and measurement error, as well as ensuring equivalent geostatistical ‘support’ in all samples, while at the same time retaining at least one quarter of the drill-core in archive for all sample intervals. It also describes an alternative method that can be applied to unsampled drill-core (*a priori*) to acquire unbiased grade and measurement error estimates, ensure equal sample support, while at the same time retaining half of the drill-core in archive.

Exploration geologists commonly use drill-cores to undertake assessments of the potential resource of mineral prospects (Fig. 1A). Samples from these drill-cores are typically assayed for the elements of interest, and results are used to determine the economic viability of the mineralization during pre-feasibility and feasibility studies. Drill-cores used for this purpose are commonly divided into geologically homogeneous intervals of similar length (bounded by regular depth intervals and/or geological contacts) and split in two, either using a diamond saw (preferred) or mechanical core splitter (historic). One half of the core is then submitted for assay, whereas the other half is retained in archive to allow further (observational and/or quantitative) studies, where necessary (Fig. 1B). Ideally, no selection bias (Sinclair & Blackwell 2002) is introduced during sampling of the first half of the drill-core.

In addition to such routine resource assessment samples, drill-core is also commonly sampled in duplicate to determine the magnitude of measurement error in the assays. Although a relatively common practice in the past, such data quality assessment has only recently become mandatory because of regulatory changes invoked by National Instrument 43-101 in Canada, JORC in Australia, and analogous disclosure requirements in other countries. As a result, not all historical mineral resource databases contain such duplicate samples, and remediation of some databases is necessary before they can be used to make acceptable mineral resource estimates.

The collection of duplicate ½ drill-core samples from a proportion (say 5%) of geological intervals is advocated by many geostatisticians who make such resource estimates, and this unfortunately results in the consumption of the entire drill-core from those ‘duplicate’ intervals. As a result, many exploration geologists have been frustrated by the absence of archival drill-core samples in these geological intervals because they have been consumed by duplicate sampling (Fig. 1C).

To address this problem, some exploration geologists have used smaller parts of the drill-core (usually quarters of cores, instead of halves; Fig. 1D) for duplicate analysis in an effort to ensure retention of archival samples. Although such an approach does retain archival drill-core for future examination, the ¼ core duplicate results produce biased estimates of measurement error, as the resulting duplicate standard deviations are on average higher than those obtained from equivalent ½ core duplicates. This is because the smaller ¼ core duplicate samples have proportionally larger sampling variance (Stanley 2007). As a result, most exploration geologists have chosen to use ½ core duplicates for data quality assessment purposes, mostly at the behest of geostatisticians. Consequently, although the drill-core in duplicate intervals is consumed, measurement error estimates are consistent and unbiased because they are derived from duplicate samples with the exact same sample support (an important geostatistical property; Sinclair & Blackwell 2002) as the routine samples collected from the drill-core.

The above sampling challenges have made many exploration geologists reticent about sampling their core sufficiently in duplicate so that accurate estimates of measurement error can be determined, because they fear that potentially critical intervals of drill-core will be consumed by duplicate sampling, and this may deny future opportunities to learn important characteristics about the mineralization, or the rocks hosting it. Consequently, many mineral resource databases commonly contain relatively small numbers of duplicate samples. In some cases, the number of duplicate samples is so small that they cannot confidently be used to determine the magnitude of measurement error in resource databases. In fact, historic databases originally assembled before the advent of National Instrument 43-101, JORC, or similar requirements may contain no duplicate samples at all. Obviously, when estimates of measurement error associated with these databases are absent or based on only a small number of duplicates, the quality of the resource database is undermined because it will not meet the standards of our present regulatory regime. This reduces the economic value of the deposit under investigation because the absence of data quality information about the samples creates increased mining, and thus financial, risk.

Below, two methods (the *a priori* and *a posteriori* duplicate sampling approaches) are described that allow exploration geologists to: (1) adequately sample drill-core in duplicate in order to rigorously assess measurement error, (2) ensure that both routine and duplicate samples within the resource database all have the same sample ‘support’ (Sinclair & Blackwell 2002), and at the same time (3) allow retention of a portion of the drill-core for archival purposes.

## Methodology

### The *a priori* duplicate analysis method

The *a priori* method for collecting duplicate samples from drill-core requires the recognition, in advance of sampling, of the need to collect duplicate samples to assess the magnitude of measurement error. The *a priori* method involves the collection of routine ½ core samples from most geological intervals (Fig. 1B), and ¼ core duplicate samples (Fig. 1D) from randomly selected geological intervals. These duplicate samples are used to estimate measurement error, and are commonly sampled about 5% of the time. In large resource assessment databases, the number of these duplicate samples becomes significant, and so they can be used to provide precise estimates of measurement error in the assay database.

Once the routine ½ core and duplicate ¼ core samples have been collected, the *a priori* method requires that they be assayed, and the ¼ core duplicate means for the elements of interest be calculated for each duplicate pair using the standard formula for the mean:
(1)

where *N* equals two. Because the ¼ core duplicate samples are together volumetrically identical to the routine ½ core samples that would otherwise have been collected from the drill-core intervals, the mean assays of these duplicates are unbiased estimates of what would have been the assays of such ½ core samples. As a result, the calculated ¼ core duplicate means should be used in the resource database in lieu of the ½ core assays because they describe the grades of equivalent samples, having the same ‘sample support’ as the routine ½ core samples (Sinclair & Blackwell 2002).

The next step in this procedure involves calculating the ¼ core duplicate standard deviations using the conventional standard deviation formula: (2)

again, where *N* equals two. The average ¼ core duplicate measurement error standard deviation can then be determined from these duplicate pair standard deviations using a root mean square formula (Stanley & Lawie 2007):
(3)

where *P* is the number of duplicate pairs. The resulting ‘average’ standard deviation (*S _{RMS Total}*

_{1/4}) provides an estimate of the magnitude of total absolute measurement error in the ¼ core duplicate samples.

The average ¼ core total measurement error determined using Equation 3 includes error introduced from a number of sources, which heretofore are grouped into two parts: ‘sampling error’ (introduced during ¼ core sampling), and ‘other error’ (introduced during all other aspects of sample preparation and analysis). These errors are additive (Stanley & Lawie 2007) as variances, so: (4)

Unfortunately, although the ‘other variance’ introduced during sample preparation and analysis is likely to be identical, or almost so, for ¼ core and ½ core samples (e.g. the variance introduced during crushing and pulverizing, sub-sampling, digestion, and analysis), the ‘sampling variance’ is not, as sampling variance is inversely proportional to sample mass (Stanley 2007). This means that the sampling variance described by the ¼ core duplicates (*S ^{2}_{RMS Total 1/4}*) will be different (and thus biased) relative to the sampling variance that would be observed in ½ core samples (

*S*). This is because the masses of the ¼ core samples are smaller, making their average sampling errors, and thus their corresponding average total measurement errors, larger.

^{2}_{Total 1/2}Geostatisticians commonly use estimates of these average total measurement errors in their geostatistical calculations (they are commonly used as estimates of the ‘nugget effect’ on semivariograms). Unfortunately, they require these estimates of measurement error to be calculated from samples with sizes equal to the actual sample sizes in the resource database (e.g. Sinclair & Blackwell 2002). As a result, they need to know the average total measurement error in the routine (½ core) resource database duplicate samples, and not the average total measurement error in the ¼ core duplicate samples.

Fortunately, the absolute ‘sampling error’ and ‘other error’ can be decomposed from the ‘total error’ using duplicate samples collected during sample preparation (after crushing and sieving but before any sub-sampling, yielding sample preparation duplicates). This is because these duplicates directly describe the magnitude of all of the ‘other error’ introduced during sample preparation and analysis (Stanley & Smee 2007). As a result, by subtracting the average (RMS) ‘other error’ (calculated using Equations 1 through 3 applied to these sample preparation duplicates) from the average ‘total error’, the average ‘¼ core sampling error’ can effectively be isolated and estimated (Stanley & Smee 2007): (5)

This average ‘¼ core sampling error’ can then be bias-corrected using the knowledge that sampling variance and sample size are inversely proportional (Stanley 2007): (6)

where *M _{1}* and

*M*are two different masses. In this case, the average sampling variance in ¼ core duplicates (

_{2}*S*

^{2}_{Sampling}_{1/4}) will thus be twice as large as the average sampling variances in ½ core duplicates (

*S*

^{2}_{Sampling}_{1/2}), so Equation 6 becomes: (7)

where *M* is the mass of a standard whole drill-core interval (Stanley 2007). Because of the mass-sampling variance relationship in Equation 7, the average sampling variance observed in ¼ core duplicates can now be ‘bias-corrected’ to equal the average sampling variance that would otherwise be observed in ½ core duplicates using the formula:
(8)

Once the ‘average sampling variance’ has been bias-corrected, the ‘other’ variance can be added back to re-constitute and estimate the ‘average total variance’ that would have been otherwise observed in ½ core duplicate samples: (9)

had they been collected and analyzed.

As a result, the *a priori* procedure described above will not only provide unbiased estimates of the grades of each duplicate interval with the same sample support as that in all routine drill-core intervals (½ core samples) and all intervals from which duplicate samples are collected, it will also produce unbiased estimates of the average total measurement (‘sampling’ plus ‘other’) error for use in data quality assessment and geostatistical calculations. Most importantly, this *a priori* procedure allows archival retention of one half of the drill-core in every interval within the core boxes for future examination and evaluation, because only two ¼ core duplicates (and not two ½ core duplicates) are removed from intervals sampled for data quality assessment purposes, leaving one half of the drill-core available for future investigation in all sample intervals.

### The *a posteriori* duplicate analysis method

An alternative, *a posteriori*, approach for collecting duplicate samples from drill-core may be necessary when an original drill-core assay program has already been undertaken that did not involve the collection of duplicate samples to measure errors in the assays. As a result, such resource databases commonly lack data quality assessment information describing the magnitude of measurement error. An *a posteriori* duplicate sampling program is possible only when selection bias was not introduced during the original assay program, and provided that the core has not been altered (e.g. weathered) since the original assay program.

In order to collect duplicate samples from historic drill-core, while at the same time retaining an archival sample of each interval for future examination, the archival ½ drill-core remaining in the core box from the initial drilling program can be cut into halves and one of the resulting ¼ core sub-samples can be used, with the original ½ core sample analysed during the original drill-core assay program, to make a ‘duplicate sample’. The original ½ core assays and these new ¼ core assays can then be used to obtain an unbiased estimate of grade and average measurement error in these ‘duplicate samples’ with the same sample support as the routine ½ core samples. Furthermore, because re-sampling involves the collection of only a ¼ core sub-sample, an archival ¼ core sub-sample remains in the core box in archive. A procedure for determining these unbiased total measurement errors is described below.

The *a posteriori* duplicate sampling method involves the collection and analysis of samples from drill-core in two batches, a first batch where only routine ½ core samples (Fig. 1B) are collected, and a second batch, where the archival (remaining) ½ core is itself cut in half to produce two quarters, and one of those quarters is sampled, prepared, and analysed (Fig. 1E). As a result, inter-batch error may exist and undermine this effort, and this challenge has to be addressed if valid and unbiased estimates of measurement error are to be obtained. An approach to address this challenge is described below, after presentation of the *a posteriori* duplicate sampling method.

In the second sampling and analysis campaign, ¼ core duplicate samples are collected from approximately 5% of the originally sampled intervals so that they, along with the corresponding original ½ core samples, also become significant in number and can provide acceptable estimates of measurement error. Unfortunately, unlike in the *a priori* method, above, the combined mass of these ½ and ¼ core duplicates is not physically identical to the mass of the original routine ½ core samples. This is because their collective mass is ¾ of the original core interval (not ½, as in the *a priori* method). As a result, the average grades (or weighted average grades) of these ‘duplicate samples’ should not be used in lieu of routine ½ core samples in the resource database because they will have different sample supports (Sinclair & Blackwell 2002). Rather, the first duplicate, consisting of the original ½ core sample, should be used in the resource database, because its estimate of grade has exactly the same sample support as routine ½ core samples (which it is), and temporal errors caused by drift between the original and re-sampling batches of analyses will not exist.

Then, in contrast to the *a priori* method described above, the weighted averages and standard deviations of these ½ core-¼ core duplicates must be determined in order to obtain an average unbiased estimate of measurement error. These weighted statistics can be calculated using the following formulae, which employ weights equal to the proportion of core that each ‘duplicate’ represents (½ and ¼):
(10)

and: (11)

respectively, where *N* = 2, *W _{1}* = ½ and

*W*= ¼.

_{2}Now, as before, the average standard deviation of these ½ core-¼ core duplicate standard deviations can be determined using a root mean square approach (Equation 3), and the average ½ core-¼ core sampling variance for this *a posteriori* method is thus:
(12)

derived in a manner analogous to that in the *a priori* method (Equation 5).

These ½ core-¼ core duplicate samples have a sampling variance determined using a different sample support (mass) than the routine ½ core samples and the ¼ core duplicates used in the *a priori* method, as their total mass equals 3/4 of the drill-core interval mass (not ½). This causes the sampling variance estimate to be biased. Fortunately, this sampling variance can be bias-corrected in a manner similar to that used in the *a priori* method, using the same strategy employed in the *a priori* method (Stanley 2007). In this case, the average mass of these ½ core-¼ core duplicates equals *3/8 M* instead of *1/4 M*. As a result, a different bias-correction factor is required for *a posteriori* duplicate variance calculations. The bias-correction factor to employ derives from application of the above average masses into Equation 6, yielding:
(13)

which reduces to: (14)

As a result, the total absolute measurement error for ½ core samples, derived using this *a posteriori* duplicate method, is:
(15)

As noted above, the *a posteriori* method of duplicate drill-core analysis involves grade measurement in two separate analytical batches. As a result, for the calculations associated with the *a posteriori* method to be valid, each of the analytical batches from which these duplicate analyses derive must exhibit the same levels of accuracy and precision; otherwise, an additional source of variation is introduced into the analysis (inter-batch error). To ensure that accuracy and precision are commensurate in each batch, reference materials analysed in the first (historical) batch of ½ core samples must also be analysed in the second batch (with the ¼ core duplicates). Then, if both the means and standard deviations of the reference materials in each batch are not significantly different, based on results of appropriate inference tests, the assumption of ‘no inter-batch bias’ is valid, and use of the *a posteriori* technique will also be valid.

Unfortunately, if the first batch of analyses was analysed before National Instrument 43-101 or JORC requirements came into effect, it is possible (or even likely) that this first batch of analyses lacks not only duplicate samples, but also reference materials. If this is the case, archived powders of routine samples from the original batch (exhibiting a representative range in grade) can be re-analysed, along with the ¼ core duplicates in the second batch. These powder duplicates can then be used in lieu of reference materials to illustrate, again using inference tests, that there is no bias between the two analytical batches.

## Applications

To illustrate how the calculations associated with both *a priori* and *a posteriori* methods can be undertaken to: (1) obtain accurate estimates of measurement error; (2) provide unbiased concentration estimates with consistent sample support; and (3) retain archival drill-core for future examination, several anonymous data quality assessment datasets associated with two exploration programs on a West African gold project are evaluated below. These were produced during historical and recent rounds of drilling on this mesothermal gold property.

During the initial, historical drilling program, the original samples consisted of ½ drill-core, but no duplicate samples and only a limited number of reference materials were collected and analysed. As a result, in order to obtain the data quality assessment information necessary to bring this drill-core dataset to National Instrument 43-101 or JORC standards, ¼ core re-sampling was undertaken on 5% of the original samples to obtain a subset of duplicate samples that could be used to document the measurement error in the analyses (producing *a posteriori* dataset # 1). This re-sampling was undertaken using a quasi-random (random-stratified) method where representative numbers of samples exhibiting concentrations from different grade ranges were collected at random to ensure that all grade ranges were appropriately represented in the subsequent dataset. Furthermore, only samples with original assays exceeding 0.1 g/t were sampled and analysed in duplicate because samples with lower concentrations were not considered of import for resource calculation purposes.

In addition, because the three reference materials analysed in the first round of sampling were no longer available during this second round of sampling, powders from a different set of samples with Au assays > 0.1 g/t were collected using a similar random-stratified approach and assayed to create duplicates from which inter-batch errors could be quantified (producing *a posteriori* dataset # 2). The calculations applied to these two datasets are used below to illustrate the *a posteriori* method of duplicate analysis.

At the same time that the above re-sampling and analysis of the original samples was being undertaken, a second round of drilling was underway on the property. Because the importance of data quality assessment samples was now fully appreciated before this second round of drilling took place, duplicate ¼ core samples were randomly selected, collected and analysed to document sampling error (producing *a priori* dataset # 1), and a different set of randomly collected powder duplicates from these samples were also assembled and analysed to document analytical error (producing *a priori* dataset # 2). These two datasets include samples with gold grades below 0.1 g/t obtained from samples collected at random, and are used below to illustrate the calculations associated with the *a priori* method of duplicate analysis.

In all cases, drill-core samples were prepared by crushing and pulverizing the entire sample, and 30-g splits of the powders were submitted for analysis by fire assay pre-concentration with ICP-MS finish. Results were reported to the nearest 0.001 g/t, and the instrumental detection limit reported by the laboratory was 0.001 g/t. Concentrations less than detection were recoded to half the detection limit (0.0005 g/t).

In addition, because the duplicate samples and duplicate powders in each of these drilling programs come from different samples (Fig. 2A), these duplicates are not part of a balanced, stratified duplicate sampling ‘tree’ (Fig. 2B). As a result, the mean concentrations of the duplicate samples and duplicate powders cannot be assumed to be equal. Because the formulae describing the additive character of the absolute variances, above, are invalid if the samples do not have equal means, a condition guaranteed if the duplicate samples derive from such a stratified sampling tree (Fig. 2B; also known as a staggered, unbalanced ANOVA design; Bainbridge 1963), these absolute variance formulae were not used. Instead, analogous formulae involving relative variances were used to estimate the sample and powder relative errors (Garrett 1983), as these accommodate the differences in duplicate means with no additional loss of generality. These relative error formulae are presented in Appendix 1.

### An *a priori* method example

The data quality assessment dataset of ¼ core sample duplicates obtained in the second round of diamond drilling on the West African gold property consists of 89 duplicate Au assays of HQ and NQ drill-core (5% of 1769 original samples) collected and analysed at the same time as the routine ½ core samples (*a priori* dataset # 1). Duplicate sample lengths range from 20–100 cm. Duplicate sample rock densities were measured using the ‘buoyancy’ method on ½ core, and results range from 2.65–2.97 g/ml. Half core sample masses were calculated from these data, and range from 0.472–4.671 kg.

A second data quality assessment dataset consists of analytical duplicates derived from 30-g powders from the same number (89) of, but different, quasi-randomly selected samples (*a priori* dataset # 2). These were used to calculate the ‘other’ error present in the samples. Both of these datasets are presented in the spreadsheet file ‘A Priori & Posteriori Datasets.xls’ available on-line at the author’s website: http://www.acadiau.ca/~cstanley/datasets.html.

The total observed relative error (one standard deviation divided by the mean; CV, in %) for the ¼ core sample duplicates in *a priori* dataset # 1, calculated using the mean and standard deviation formulae of Equations 1 and 2, is 55.88% (Fig. 3). Unfortunately, this result is determined assuming that all of the ¼ core samples are of the same mass (i.e. they have the same sample support), which is not true because the drill-core intervals have different lengths and densities. As a result, corrections for the varying sample support in this database must be made to ensure an unbiased result. These corrections should only be undertaken on the sampling error because that is all that these sample support variations affect (Stanley 2007).

Consequently, the ‘other’ error introduced during sample preparation and analysis must be first subtracted from this observed total error to isolate the sampling error (Equation A5). Because some relative variances for duplicate pairs equal zero, we cannot generally subtract the ‘other’ relative variance from the observed relative variance for each sample because some of the resulting variances will be negative, a situation that is impossible. As a result, these corrections must be undertaken using the average total and average ‘other’ relative variances in order to obtain the average sampling relative variance. Given that the average ‘other relative error’ reported by powder duplicates in *a priori* dataset # 2 is 26.55% (Fig. 4), the average sampling relative error can be calculated. This average ¼ core sampling relative error is thus 49.17%, calculated using Equation A5.

Two corrections for sample support then need to be made, first because the duplicate samples have different diameters, lengths and densities, and second because the duplicate samples are ¼ core instead of ½ core. Both corrections employ Equation A8 to modify the sampling variance to what would be expected with a constant ‘ideal’ sample support, here assumed to be 100 cm long, NQ ½ core duplicates with nominal densities of 2.65 g/ml. An appropriate average sample support factor first needs to be determined and used to obtain an unbiased estimate of the average relative sampling error for ¼ core duplicates with constant diameter, length, and density.

Because random errors are additive as variances, the appropriate ‘average’ sample support factor should be calculated using a root-mean-square approach. The sample support factor for each sample will be the ratio of the mass of a standard sample support (the ‘ideal’ 100 cm long, ½ NQ core sample represented by most routine samples in the dataset, with an assumed density of 2.65 g/ml and calculated mass of 2.358 kg) divided by the actual sample support of each pair of duplicate samples (which is also ½ core), calculated from the interval diameter, length and density of that duplicate sample (derived from Equation A6). The ‘average’ sample support factor can then be calculated using the root-mean-square value of all of the resulting sample support factors in this dataset, and is 1.233.

Multiplying this ‘average’ sample support factor by the square of the average relative sampling error (Equation A8), and then taking the square root, yields the sample support-corrected average relative error for ¼ core duplicates, equal to 54.60%.

Once the effects of the different sample supports on ¼ core sampling error have been addressed, the sampling error must be further corrected to estimate the average sampling error that would be observed in ½ core duplicate samples. This approach differs from the sample support correction, above, only because the correction factor (the ratio of the mass of a ½ core sample to ¼ core sample = 0.5; from Equation A8) is the same for every duplicate sample. Consequently, the ‘average’ (RMS) correction factor automatically equals ½, and the sample support-corrected relative error for ½ core duplicates can be calculated simply by multiplying the average sample support corrected relative error for ¼ core duplicates by the square root of ½ (from Equation A8). This results in an average sample support corrected relative sampling error for ½ core duplicates of 38.61%. Adding back in the average ‘other’ relative error of 26.55% using Equation A9, we obtain a total, unbiased relative error estimate for ½ core duplicate samples in *a priori* dataset # 1 of 46.86% (Fig. 3).

This calculated total relative error is what should be used to describe the magnitude of measurement error in ½ core DDH samples from the second drilling program. Note that a total relative error of 46.86% is not extraordinarily high for a gold deposit (cf. Stanley & Smee 2007), particularly a mesothermal gold deposit containing visible gold, and thus likely indicates that these DDH assays are ‘fit for purpose’. Consequently, these data can be confidently used in subsequent resource estimates of the deposit. Interestingly, because the average ‘other’ error for the samples collected in this second round of drilling is 26.55% and the corresponding sample support corrected relative sampling error for ½ core duplicates is 38.61%, relative sampling error contributes approximately (68%) of the variation to these data, while sample preparation and analysis contributes the balance (32%). This ‘other’ relative error is proportionally large compared to many Au deposits (cf. Stanley & Smee 2007). Although this may indicate that improvements to sample preparation and analysis procedures are required, sample preparation involved the pulverization of entire samples, so no sub-sampling was undertaken during sample preparation. As a result, because the analysis for gold does not typically introduce large amounts of error to assays (cf. Stanley & Smee 2007), the ‘other’ error is more likely due to a ‘nugget effect’ created during sub-sampling of the powders. This would indicate that larger masses of samples (> 30 g) of this material would need to be analysed in order to reduce this anomalous sub-sampling error (nugget effect).

### An *a posteriori* method example

*A posteriori* dataset # 1 used to illustrate the numerical procedures involved in implementing the *a posteriori* duplicate analysis method contains 201 duplicate samples (5% of 4026 original assays) derived from NQ and HQ drill-cores. These samples have variable interval lengths between 39 and 100 cm, and densities that range between 2.65 and 2.98 g/cc. Calculated sample masses thus range from 0.920 to 4.687 kg. Note that geological intervals with fire assay Au grades greater than 0.1 g/t in the first (½ core) duplicate were, as above, sampled in duplicate using ¼ core samples. Consequently, some statistics calculated from these data may be biased because of the selective, random-stratified sampling undertaken when collecting this subsequent round of ¼ core duplicates. Fortunately, the relative error statistic is not likely to be one of them, so this dataset suffices to illustrate the calculations and important factors that need to be considered when assessing measurement error using the *a posteriori* method.

Because these ½-core-¼-core duplicate samples were analysed in two batches separated over a significant time span, potential inter-batch error could exist. As a result, 100 randomly selected powders were re-analysed with the ¼ core duplicate samples to allow comparison of the analytical qualities of the two batches of analyses (Fig. 5). Although the average relative error for these inter-batch duplicate powders is high (Fig. 6; 25.87%; likely due to the nugget effect, see above), it is similar to the average relative error obtained from duplicate powders analysed in the same batch (26.55%; documenting the magnitude of the ‘other error’ in the *a priori* duplicate analysis example, above; *a priori* dataset # 2; Fig. 4). Nevertheless, an inference test should probably be used to rigorously determine whether significant inter-batch analytical bias exists between these duplicate pairs. This can be undertaken by calculating the mean (
) and standard error (
) of the differences between the duplicate pairs, and evaluating whether
includes zero (α = 0.05; Evans & Rosenthal 2010). Because it does (
and
), the null hypothesis that these duplicates are equal cannot be rejected, indicating that they are (on average) not significantly different, there is no inter-batch analytical bias between the first and second analyses, and ½ core-¼ core duplicate samples can be used to assess sampling error in the dataset containing samples from the first round of drilling.

Because these powder duplicates were derived from both analytical batches, just like the ½ core-¼ core sample duplicates, they fortuitously represent the correct material to use to determine the magnitude of the ‘other’ error present in these samples. Consequently, these two datasets can be used to illustrate the *a posteriori* duplicate sampling method, and are also presented in the spreadsheet file ‘A Priori & Posteriori Datasets.xls’ (http://www.acadiau.ca/~cstanley/datasets.html).

Because the sample duplicates used in this analysis have different masses (½ core and ¼ core), calculation of the total average relative error (one standard deviation divided by the mean; CV, in %) for the 201 ½ core-¼ core sample duplicates was made using the ‘weighted’ mean and standard deviation formulae of Equations 10 and 11. The resulting total average relative error is 44.23% (Fig. 7). Unfortunately, as above, this result is determined assuming that all of the ½ core and ¼ core samples have the same mass (the same sample support), an assumption that is not true.

As a result, corrections for these varying sample supports analogous to those used in the above example must be made to ensure an unbiased result. Subtracting the ‘other’ relative error (25.87%) estimated by the 100 duplicate samples analysed in two batches, above (using a RMS approach: Equation A5; Fig. 6) from the total average relative error of 44.23%, yields an estimated ½ core-¼ core sampling average relative error of 35.88%. Then multiplying the square of the appropriate ‘RMS calculated’ sample support correction factor of 1.035 by the ½ core-¼ core average relative sampling variance and taking the square root yields a sample support-corrected relative sampling error for ½ core-¼ core duplicates of 35.60%. Then, using the formula in Equation A14 to convert this sample support-corrected, ¼ core-½ core relative sampling error to ½ core-½ core relative sampling error gives 31.61%. Finally, adding back the ‘other’ relative error (25.87%; Equation A15) to determine the total average ½ core-½ core relative error in the data yields 40.85% (Fig. 7).

Comparing the *a priori* and *a posteriori* results of these bias-corrected, standardized sample support, ½ core, average total relative errors from the two rounds of drilling indicates that the two datasets exhibit similar levels of error (46.86% v. 40.85%; note that these total relative errors are not significantly different (just), based on an F-test of equality of (relative) variances that uses a test statistic of 1.316, degrees of freedom of 88 and 200, respectively, α = 0.05, and thus a critical value of 1.335). As a result, there is no reason to indicate that samples from each round of drilling cannot be combined into a global dataset for use in calculating an acceptable resource estimate for this deposit. Collectively, these drill-core data exhibit a weighted average relative error of 42.78%, and this can be used in subsequent ore resource and reserve calculations as a fair and un-biased estimate of total relative error in these gold analyses.

Finally, corrections to the *a priori* and *a posteriori* initial average relative measurement errors for sample support and core size caused an overall reduction in the relative measurement error (by 16 and 8%, respectively). This is because the initial estimates were made using sample masses that were on average smaller than the routine samples collected during diamond drilling, and so over-estimated error. Although the presence of HQ drill-core in these databases caused an under-estimation of total measurement error, drill-core intervals less that 1 m and the smaller size of ¼ core-½ core and ¼ core-¼ core duplicates counter-acted this under-estimation, resulting in a net over-estimation that, via the above corrections, was accommodated to produce unbiased estimated of total measurement error. Consequently, failing to make core diameter, density, and interval length corrections, as well as the required ½ core-¼ core and ¼ core-¼ core bias corrections, can drastically change the measurement error estimates made during data quality assessment of drill-core samples, and thus can very substantially undermine perceptions about the quality of the assay data.

## Conclusions

The *a priori* and *a posteriori* methods of duplicate drill-core analysis produce unbiased and equivalent estimates of measurement error, and ensure that: (1) samples in resource databases exhibit constant sample support; (2) total measurement errors determined by ¼ core-¼ core or ½ core-¼ core duplicates are unbiased; and (3) at least a quarter of drill-core is retained for archival purposes. As a result, these two duplicate analysis techniques satisfy the needs of both exploration geologists and geostatisticians, ensuring that each can achieve their objectives without negatively impacting the ability of the other to achieve their goals.

Because the total measurement errors calculated from *a priori* or *a posteriori* duplicates using the methods described above are unbiased, they can be used as input in any error propagation exercise applied to mineral deposit resource databases (Stanley & Smee 2007; Stanley & Lawie 2007). Such exercises are typically undertaken to assess the grade dependence of a mineral deposit during pre-feasibility or feasibility studies assessing the economic viability of the mineral deposit.

## Acknowledgments

This paper benefitted from comments made by Dr. R.G. (Bob) Garrett and those of an anonymous reviewer.

## Appendix 1

In this paper, various types of errors have been described in absolute terms, as standard deviations. The approaches advocated herein can also be undertaken in a relative context, as relative standard deviations, also known as coefficients of variation, which can be calculated using means (Equation 1) and standard deviations (Equation 2):

Below are presented the corresponding equations that can be used to undertake both *a priori* and *a posteriori* duplicate analysis using relative errors instead of absolute errors. These are the very equations used in the two examples presented in this paper. All of these relative error equations are linked to their corresponding absolute error equations by equation number.

*A priori* duplicate analysis relative error equations

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

*A posteriori* duplicate analysis relative error equations

In this case the coefficient of variation is calculated using the weighted mean (Equation 10) and weighted standard deviation (Equation 11):

Again, these relative error equations are linked to their corresponding absolute error equations by equation number.

(A12)

(A13)

(A14)

(A15)

- © 2014 AAG/The Geological Society of London