Local and Spatial Joint Frequency Uncertainty and its Application to Rock Mass Characterisation

  • Steinar L. EllefmoEmail author
  • Jo Eidsvik
Original Paper


Stability is a key issue in any mining or tunnelling activity. Joint frequency constitutes an important input into stability analyses. Three techniques are used herein to quantify the local and spatial joint frequency uncertainty, or possible joint frequencies given joint frequency data, at unsampled locations. Rock quality designation is estimated from the predicted joint frequencies. The first method is based on kriging with subsequent Poisson sampling. The second method transforms the data to near-Gaussian variables and uses the turning band method to generate a range of possible joint frequencies. The third method assumes that the data are Poisson distributed and models the log-intensity of these data with a spatially smooth Gaussian prior distribution. Intensities are obtained and Poisson variables are generated to examine the expected joint frequency and associated variability. The joint frequency data is from an iron ore in the northern part of Norway. The methods are tested at unsampled locations and validated at sampled locations. All three methods perform quite well when predicting sampled points. The probability that the joint frequency exceeds 5 joints per metre is also estimated to illustrate a more realistic utilisation. The obtained probability map highlights zones in the ore where stability problems have occurred. It is therefore concluded that the methods work and that more emphasis should have been placed on these kinds of analyses when the mine was planned. By using simulation instead of estimation, it is possible to obtain a clear picture of possible joint frequency values or ranges, i.e. the uncertainty.


Joint frequency Rock mass classification Geostatistics Iron ore 



The authors would like to acknowledge Rana Gruber AS for the permission to use their joint frequency data.


  1. Barton N, Lien R, Lunde J (1974) Engineering classification of rock masses for the design of rock support. Rock Mech 6:189–236CrossRefGoogle Scholar
  2. Bieniawski ZT (1973) Engineering classification of jointed rock masses. Trans S African Inst Civ Eng 15(12):335–344Google Scholar
  3. Bieniawski ZT (1989) Engineering rock mass classifications. Wiley, New York, 251pGoogle Scholar
  4. Bouleau N (1986) Probabilités de l’Ingéniur. Herman, ParisGoogle Scholar
  5. Breslow NE, Clayton DG (1993) Approximate inference in generalized linear mixed models. J Am Stat Assoc 88:9–25CrossRefGoogle Scholar
  6. Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York, 635pGoogle Scholar
  7. Cressie N (1993) Statistics for spatial data, Wiley, New YorkGoogle Scholar
  8. Deere D, Miller RD (1966) Engineering classification and index properties for intact rock. University of Illinois, Tech report no. AFWL-TR-65-116Google Scholar
  9. Diggle P, Tawn JA, Moyeed RA (1998) Model-based geostatistics. J R Stat Soc Ser C 47:299–350CrossRefGoogle Scholar
  10. Eidsvik J, Martino S, Rue H (2008) Approximate Bayesian inference for spatial generalised linear mixed models. Scand J Stat (to appear)Google Scholar
  11. Einstein HH (2003) Uncertainty in rock mechanics and rock engineering—then and now, pp 281–293. ISRM 2003—Technology roadmap for rock mechanics, South African Institute of Mining and MetallurgyGoogle Scholar
  12. Ellefmo S (2005) From deposit to product. A probabilistic approach to the value chain of iron ore mining. Doctoral thesis. Norwegian University of Science and Technology, 271pGoogle Scholar
  13. Gabrielsen RH, Braathen A, Dehls J, Roberts D (2002) Tectonic lineaments of Norway. Norsk Geologisk Tidsskrift 82:153–174Google Scholar
  14. Goovaerts P (1997) Geostatistics for natural resources evaluation. Applied geostatistics series. Oxford University Press, Oxford, 442pGoogle Scholar
  15. Green P, Hjort NL, Richardson S (2003) Highly structured stochastic systems, Oxford University Press, OxfordGoogle Scholar
  16. Hoek E, Kaiser PK, Bafden WF (1995) Support of underground excavations in hard rock. A.A. Balkema, RotterdamGoogle Scholar
  17. Hoerger, Steven H, Young, Dae S (1987) Predicting local rock mass behavior using geostatistics. 28th US symposium on rock mechanics, Tuscon 29 June–1 July 1987, pp 99–106Google Scholar
  18. Journel AG (1974) Geostatistics for conditional simulation of ore bodies. Econ Geol 69(5):673–687CrossRefGoogle Scholar
  19. Journel AG, Huijbregts Ch J (1978) Mining geostatistics. Academic Press, New York, 580pGoogle Scholar
  20. La Pointe PR (1980) Analysis of the spatial variations in rock mass properties through geostatistics. In: Proceedings of the 21st symposium on rock mechanics, pp 570–580Google Scholar
  21. Lantuéjoul C (2002) Geostatistical simulation. Models and algorithms. Springer, Heidelberg, 239pGoogle Scholar
  22. Liu X, Srinivasan S (2004) Merging outcrop data and geomechanical information in stochastic models of fracture reservoirs. SPE 90643. SPE International petroleum conference in Puebla, Mexico, 8–9 November 2004Google Scholar
  23. Matheron G (1973) The intrinsic random functions and their application. Adv Appl Prob 5:439–468CrossRefGoogle Scholar
  24. Nilsen B (1979) Stabilitet av høye fjellskjæringer. Report 11, Geol. Inst., NTH, Trondheim, 271pGoogle Scholar
  25. Nilsen B, Palmstrøm A (2000) Engineering geology and rock engineering. Handbook 2. Norwegian Group for Rock Mechanics (NBG), 201pGoogle Scholar
  26. Nilsen B, Shresta KK, Panthi KH, Holmøy K (2003) RMR vs Q vs RMiS. Tunnels Tunneling Int 35(5):45–48Google Scholar
  27. Palmström A (1995) RMi—a rock mass characterization system for rock engineering purposes. Ph.D. thesis, University of Oslo, Norway. 400pGoogle Scholar
  28. Park RG (1989) Foundations of structural geology, 2nd edn. Blackie Academic & Professionals. 140pGoogle Scholar
  29. Priest SD, Hudson JA (1981) Estimation of discontinuity spacing and trace length using scanline surveys. Int J Rock Mech Min Sci Geomech Abstr 18:183–197CrossRefGoogle Scholar
  30. Stavropoulou M, Exadaktylos G, Saratsis G (2007) A combined three-dimensional geological–geostatistical–numerical model of underground excavations in rock. Rock Mech Rock Eng 40(3):213–243CrossRefGoogle Scholar
  31. Syrjänen P, Lovén P (2003) 3-D modeling of rock mass quality, pp 1175–1178. ISRM 2003−Technology roadmap for rock mechanics, South African Institute of Mining and Metallurgy, 2003Google Scholar
  32. Søvegjarto U, Marker M, Gjelle S (1989) STORFORSHEI 2027 IV, berggrunnskart, M = 1:50.000 Norges geologiske undersøkelseGoogle Scholar
  33. Vann J, Bertoli O, Jackson S (2002) An overview of geostatistical simulation for quantifying risk, pp 13–29. In: Searston SM, Warner RJ (eds) Proceedings from quantifying risk and error symposium. Geostatistical Association of AustraliaGoogle Scholar
  34. Young Dae S (1987) Indicator Kriging for unit vectors: rock joint orientations. Math Geol 19(6):481–501CrossRefGoogle Scholar
  35. Yu YF, Mostyn GR (1993) Spatial correlation of rock joints, pp 241–255. In: Li KS, Lo S-CR (eds) Probabilistic methods in geotechnical engineering. Balkema, Rotterdam, 331pGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Geology and Mineral Resources EngineeringNorwegian University of Science and Technology (NTNU)TrondheimNorway
  2. 2.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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