The Hyperbolic Method-Discussion

Evaluation of Twin and Duplicate Samples: The Hyperbolic Method: Discussion Dr. Armando Simón, R.P.Geo. Principal Geolog

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Evaluation of Twin and Duplicate Samples: The Hyperbolic Method: Discussion Dr. Armando Simón, R.P.Geo. Principal Geologist AMEC International (Chile) S.A.

The evaluation procedure of the twin and duplicate samples1 involves the preparation of Max-Min plots2, where the maximum and minimum values of the sample pairs are plotted in the y and x axis, respectively. This way, all the points are plotted above the y=x line. Precision is evaluated through the Relative Error (RE), also known as Absolute Value of the Relative Difference (AVDR). The RE is calculated as the absolute value or the difference between the original and the duplicate values, divided by the average of the two values: RE = 2*|oi – di|/(oi + di)


where oi is the original value and di is the duplicate value of the duplicate pair. Precision is treated as a qualitative parameter, inversely proportional to RE, which is a quantitative parameter: the higher the RE, the poorer the precision, and vice versa. Linear equations (y=mx; y=mx+b) are often used for evaluating the precision of duplicate data, but the typical decrease of precision near the detection limits generally leads to conciliatory, non-conventional solutions when dealing with such low values. To prevent this problem, the hyperbolic method has been developed as an alternative solution. With this method, each duplicate pair oi and di is evaluated against the hyperbolic quadratic equation: y2=m2x2+b2


for x, y≥0, where y is max [oi, di], x is min [oi, di], m is the slope of the asymptote and b the value of the intercept. The hyperbole so defined is considered as the failure line in the Max-Min plot (Figure 1). Whereas near the detection limits the hyperbolic line departs from the asymptote to allow for lower precision pairs, along the rest of the interval it tends asymptotically to a line with slope m that crosses the coordinate origin (0,0). The value of m is calculated from the RE equation (1) and the hyperbolic equation (2), considering that at the asymptote line y = m*x, and depends on the limiting RE commonly accepted for the particular type of duplicate. The m values so calculated are as follows: 1.35 for twin samples (corresponding to a 30% RE), 1.22 for coarse duplicates (corresponding to a 20% RE) and 1.11 for pulp duplicates (corresponding to a 10% RE).

1 2

Duplicate samples: generic term for twin samples, field duplicates, coarse duplicates and pulp duplicates. The Max-Min plots were developed by Scott Long (AMEC).

The value of b is conditionally established, depending on the type of material, as a multiple factor of the practical detection limit (10 to 20 times for twin samples, 6 to 10 times for coarse duplicates and 3 to 5 times for pulp duplicates). Sample pairs with relative errors exceeding the limiting values according to the equation (situated above the failure line) are considered failures and are flagged for review. A maximum failure rate of 10% is considered acceptable.

Lima, 30 September 2004

Figure 1: Failure Line Alternatives

3.0 2.7

Failure line alternatives

2.4 2.1

x=y line

Max Value

1.8 Case 1 Case 2 Case 3 45 deg

1.5 1.2

Case 1: y2=m2x2+b2 Case 2: y=mx Case 3: y=mx+b

0.9 0.6 0.3 0.0 0.0







Min Value

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