JOURNAL PAPER

Statistics or geostatistics? Sampling error or nugget effect?

I. Clark

Geostokos Ltd, Scotland

SYNOPSIS

What is a nugget effect? In the early development of geostatistics, the term 'nugget effect' was coined for the apparent discontinuity at the beginning of many semivariogram graphs. This name was chosen to reflect the large differences found between neighbouring samples in 'nuggety' mineralizations such as Wits gold reefs. Geostatistical theory assumes that the difference between a sampled value and a potential repeat sample at the same location is actually zero. Included in this 'nugget effect' would be true variation between contiguous samples due to the nature of the mineralization,micro-fracturing, nugget or crystal size, and so on. Also included in the nugget effect would be any 'random' sampling variation which might occur due to the method in which the sample was taken, the adequacy of the sample size, the assaying process, etc.
Arguments were put forward that 'sampling errors' actually exist at zero distance. Some geostatistical schools actually maintain that the 'nugget effect' is all sampling error. This would imply that 'perfect' sampling would eliminate the nugget effect entirely.
There is now a dichotomy both in the geostatistical world and in the software packages provided for geostatistical analyses. It may seem academic to argue over whether the semivariogram model should take a value of zero, a value equal to the nugget effect, or a partial value at distance zero. However, the decision can have a profound effect on both the estimated resource and in our confidence on that resource.
Whereas most geostatistical texts define the semivariogram model as taking the value of zero at zero distance, others imply that the full nugget effect should be used at zero distance. For example:
• The nugget effect refers to the nonzero intercept of the variogram and is an overall estimate of error caused by measurement inaccuracy and environmental variability occurring at fine enough scales to be unresolved by the sampling interval³
• Christensen⁴ has shown that the 'nugget effect', or nonzero variance at the origin of the sernivariogram, can be reproduced by a measurement error model
• The nugget effect is considered random noise and may represent short-scale variability, measurement error, sample rate, etc.⁵.
In many training texts and Web courses, the definition of the semivariogram is ambiguous as the formulae for semivariogram models is not actually specified at zero distance^6,7,8.

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References

1. CLARK, I. Does geostatistics work?, Proc. 16th APCOM, Thomas J O'Neil, (ed.), Society of Mining Engineers of AIME Inc, New York, 1979. pp. 213-225.         [ Links ]

2. CLARK, I. and GARNETT, R.H.T. Identification of multiple mineralisation phases by statistical methods, Trans. Inst. Min. Metall., vol. 83, 1974. pp. A43-A52.         [ Links ]

3. FORTIN, M.-J. Spatial statistics in landscape ecology, Landscape ecological analysis: issues and applications. J.H. Klopatek and R.H. Gardner (eds.). Springer Verlag, New York. 1999. pp. 253-279.         [ Links ]

4. CHRISTENSEN, R. Linear Models for Muitivariate, Time Series and Spatial Data. Springer-Verlag, New York. 1991.         [ Links ]

5. CHAMBERS, R.L., YARUS, J.M., and HIRD, K.B. Petroleum geostatistics for nongeostatisticians, Part 1, The Leading Edge, May 2000, vol. 19, no. 5, pp. 474-479;         [ Links ]

6. SINCLAIR, A.J. and BLACKWELL, H. Applied Mineral Inventory Estimation, Cambridge University Press, 2002. 381 pp.         [ Links ]

7. http://www.bioss.ac.uk/smart/unix/mvariog/slides/sl07.htm, Biomathematics & Statistics Scotland, Edinburgh.         [ Links ]

8. SAMAL, A. Basics of Variogram Analysis, Pincock Perspectives, Issue No. 84, May 2007, Pincock, Allen & Holt, Lakewood, Colorado.         [ Links ]