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

Article in Rock Mechanics and Rock Engineering · August 2009 DOI: 10.1007/s00603-008-0009-x

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Local- and spatial joint frequency uncertainty and its application to rock mass characterisation

Steinar L. Ellefmo*1, Jo Eidsvik2

1 Department of Geology and Mineral Resources Engineering, Norwegian University of Science and Technology (NTNU), Sem Saelands vei 1, N-7491 , 2 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Alfred Getz vei 1, N-7491 Trondheim, Norway

Summary 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. Method three 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 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 five joints per meter 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 put on these kinds of analyses

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when the mine was planned. By using simulation instead of estimation, it is possible to get a clear picture of possible joint frequency values or ranges, i.e. the uncertainty.

Keywords: Joint frequency, rock mass classification, geostatistics,

*Corresponding author: Tel.: +47 73 59 48 56. Fax: +47 73 59 48 14; E-mail address: [email protected]

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1. Introduction Focus on stability is crucial in any type of underground workings like underground mining or tunneling. In mining ore grade and stability are two decisive parameters. If it is not possible to produce ore above cut-off at stable conditions, the ore is sterilized until the mining face is stabilized.

Unexpected instability incidents are expensive. They require additional expensive efforts and they are time consuming. In mining it is important to produce according to schedule to avoid loosing too much income. Tunneling projects must often be finalized within a rather strict time frame. Unexpected delays due to instabilities must therefore be avoided.

Preliminary investigations in mining and tunneling are in principle similar, but typically very different when it comes to extent. Mining projects need to perform extensive drilling campaigns to characterize and delineate the ore body. If the boreholes are logged also for geotechnical information like joint frequency and joint characteristics, they can, used along with suitable quantification techniques, provide valuable input into stability analyses.

Boreholes drilled to characterize the grade distributions in the ore, might not be suitable if one also wants to collect geotechnical data. For example, in order to assess at what rate the joint frequency decline with increasing distance from a fault, it would be best to drill perpendicular to the fault plane. Mining companies have to the authors’ knowledge acknowledged this and perform separate drilling campaigns to collect geotechnical data.

Once data is collected these must be processed and analyzed to give meaningful interpretation and decisions. Herein we present techniques based

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on geostatistics to predict the expected joint frequency and associated local- and spatial variability (uncertainty) at unsampled locations. The methods are validated by also predicting the joint frequency at sampled locations. The joint frequencies are transformed into Rock Quality Designation (RQD) values. The

RQD (Deere and Miller 1966) is a parameter used today in many mining and tunneling projects to describe the degree of jointing in a rock mass.

Geostatistics has been used in rock mass characterization. La Pointe (1980) uses geostatistics to indicate the degree of inhomogeneity in the frequencies and orientation of two distinct joint sets. Young (1987) uses indicator kriging to evaluate the local probability distribution of rock joint orientations in geological formations. Hoerger and Young (1987) use local estimates of rock mass conditions obtained through geostatistics as input into geotechnical designs. Yu and Mostyn (1993) review concepts and models used to model the spatial correlation of joint geometric parameters. Syrjänen and Lovén (2003) used geostatistics on estimated Geological Strength Index (Hoek et al. 1995).

They conclude that rock mechanical quality parameters from drill cores can be estimated using geostatistical interpolation methods. Einstein (2003) reports the use of geostatistics on RQD values. Liu and Srinivasan (2004) used multiple point statistics to simulate fracture networks. In this paper the joint frequency and its associated variability (uncertainty) is quantified using three different simulation approaches. The simulation approach enables assessment of not only expected RQD-values, but also possible RQD-values, i.e. possible RQD- ranges.

As an application, the probability that the joint frequency exceeds five joints per meter is predicted at points covering a part of the Kvannevann Iron Ore in

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Northern Norway. In effect this also estimates the probability that the RQD- value is below 90% at unsampled locations. Probability maps are generated to visualize the result.

In Chapter 2 the geology, the joint frequency geodata and the Kvannevann

Iron Ore mining operation is presented. Chapter 3 presents the objective and previous work performed on the geodata used in this paper. Chapter 4 gives a short introduction to RQD and how it is used in this study. Chapter 5 presents the three techniques applied to quantify the joint frequency and associated variability at unsampled locations. Chapter 6 presents the prediction results and discusses its implications. In Chapter 7 some concluding remarks are made.

Two of the simulation approaches are presented in more detail in the Appendix.

2. Background

2.1. Mining operation The applied mining method at the Kvannevann Iron Ore Mine is sublevel stoping. One drill drift is tunnelled in the centre of the planned stope. From this drift production holes of 3 to 4 inches in diameter, are drilled about 30 metres upwards and about 40 metres downwards in vertical fans. Another drift is tunnelled about 70 meter below the drill drift from which the production holes are drilled upwards. Typically, three or four fans are charged and blasted in one round. Blasted ore is loaded onto trucks and transported out to a crusher near the mine entrance. The stopes are about 110 metres high, 40 metres wide and

60 metres long. Figure 1 provides a mine map showing level 250, i.e. 250 meters above sea level. Crushed ore is trammed about 35 kilometres down to

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the beneficiation plant. The mine has an annual production at about 1.6 million tonnes of ore.

2.2. Geology The iron ores in Valley belong to Rødingfjell Nappe Complex

(RNC) which constitutes a part of the uppermost allochthone of the

Scandinavian Caledonides. RNC contains rocks of assumed late Pre-Cambrian to Cambrian-Silurian age and consists of the Beiarn Nappe, the Slagfjell

Nappe, the Plura Nappe and the Ramnålia Nappe (Søvegjarto et. al., 1989).

Two iron-ore-bearing zones are found in Ramnålia Nappe, more specifically in the Dunderland formation. The Dunderland formation consists of dolomite- and calcite marble, mica schist, calcareous mica schist and two iron ore bearing horizons.

The economic Kvannevann Iron Ore in the Dunderland formation is a low grade meta-sedimentary iron ore with an average total iron grade around 34%.

The ore is highly banded in terms of mineral variations, grain size, grain shape and iron content. Spatially, the ore is steeply dipping.

The main minerals in the ore are and , calcite and dolomite, quartz, garnet, epidote, mica (biotite and muscovite) and amphiboles.

2.3. Joint frequency data A joint is a planar or semi-planar discontinuity in a rock formed through movement perpendicular to the fracture surface (e.g; Park 1989; Braathen and

Gabrielsen 2000).

There are three joint sets in the area (Nilsen 1979). Table 1 shows the joint strike and the joint dip of these joint sets. The strike angle is given as the

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clockwise angle from north and the dip is the angle downwards from the horizontal.

Joints represent zones of weakness in the rock mass. Thus, to assess the stability of the rock mass, the spatial joint frequency distribution is an important input. Boreholes from a 1980-drilling campaign have been logged for joint frequency by counting the number of joints pr. meter, i.e. the support in the original data is equal to one meter. Support is a geostatistical term indicating the size and shape of a sample. The logging was performed by Rana

Gruber AS personnel. The boreholes penetrating the Kvannevann Iron Ore were drilled mainly in two directions: N150oE and N330oE. This approximates a direction perpendicular to the ore strike. Consequently, the joint frequency in the boreholes is mainly influenced by joint set 1, see Table 1. For the geostatistical simulations the data were limited to those within the mineralized envelope developed in Ellefmo (2005) and regularized into 4 meter long composites or intervals. Declustered summary statistics for these 4 meter composites are given in Figure 2.

The estimation of the variogram is an important step in any geostatistical analysis. The variogram is a mathematical model that captures the spatial correlation between the joint frequency observations.

The variogram is fitted to the raw- and transformed data, see Section 5.2 and

Appendix for more information on the performed transformations. From empirical evidence an anisotropic variogram model is fitted to the empirical variogram. One model is defined along strike (N70oE), another perpendicular to strike and one down dip. The final variogram model capturing the spatial correlation in the transformed data consists of a sum of three structures. The

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first structure constitutes a nugget effect equal to 0.30. Structure two is a spherical model with sill equal to 0.55 and ranges equal to 115 meter along strike, 45 meter perpendicular to strike and 115 meter down dip. The third structure is also a spherical model with sill 0.15 and ranges equal to 400 meter along strike, 200 meter perpendicular to strike and 400 meter down dip is fitted to the data. These are validated from directional empirical plots, see Figure 3.

This model indicates the samples separated by more than 115 meter along strike are more or less uncorrelated.

The deposit is a metasedimentary iron deposit, originally deposited horizontally and subsequently folded into its final spatial arrangement

(Ellefmo, 2005). Due to these geological evidence the same variogram model has been used along strike as down dip, although the along strike variogram is relatively unstructured; see Figure 3, upper left. A similar model is fitted to the experimental variogram calculated from the untransformed data used in kriging.

3. Objective and previous work with this dataset

3.1. Objective The objective of this paper is to develop and test techniques that allow an assessment of the uncertainty in estimated joint frequency and its corresponding RQD-values at unsampled locations.

In kriging one makes a Gaussian assumption, i.e. that the underlying random field under study (here joint frequency) is Gaussian distributed. With real sample data, this assumption is almost never satisfied, especially with small counts in the data as in the joint frequency data. See also Diggle et al. (1998).

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As a consequence kriging underestimates the prediction error (the uncertainty).

For the joint frequency data presented in this paper, more sophisticated methods are used to quantify the uncertainty at unsampled locations. These methods are described in Chapter 5 and in the Appendix.

3.2. Joint frequency estimation using kriging Ellefmo (2005) used ordinary kriging to estimate the most probable joint frequency at unsampled locations. Kriging is an estimation technique, which takes the spatial correlation between the joint frequency observations into account. This spatial correlation is quantified through the variogram. See e.g.

Goovaerts (1997) for more details on kriging and geostatistics. The horizontal section through the rock mass in Figure 4 indicates the spatial distribution of the most probable joint frequency, or the intensity. The mine map given in

Figure 1 is included for spatial reference.

The most fractured zones coincide well with stability problems experienced in the mine. The stability problems include block falls of waste rock into the western stopes and an unstable pillar in the eastern part of the mine. These areas are indicated by circles in Figure 4.

4. From joint frequency to RQD The RQD (Deere and Miller 1966) is a parameter often used to describe the degree of jointing in a rock mass. It is defined as the length of core pieces longer than 10 cm divided by the total length of core.

A rock mass can be classified according to the RQD-value (Deere and

Miller 1966, in Nilsen and Palmström 2000). See Table 2.

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Having information about joint frequency λ (joints per meter) along a scanline or borehole, the RQD can be approximated by (Priest and Hudson

1981):

RQD =100e−0.1λ (0.1λ +1) Eq. 1 Figure 5 shows the spatial distribution of RQD for the same horizontal section as in Figure 4. These RQD-values have been estimated as part of this study using the joint frequency information given in Figure 4. The RQD- classification limits given in Table 1 have been used.

All RQD-values belong to the Fair, Good and Excellent category. Despite this, the mine has experienced severe stability problems. This simply emphasizes the fact that the in-situ rock mass conditions must be seen in close relationship with planned mining activities.

5. Geostatistical simulation In this Section three methods for geostatistical simulation of joint frequencies are presented shortly. See Appendix for more details. For our purpose of rock mass characterisation these methods serve as modules for predicting the number of joints per meter and for generating realizations of possible outcomes of joint frequencies at any geographical location. This module is important since it allows us to assess the RQD in Eq. 1 and associated variability at chosen locations.

obs obs obs We denote the data of n = 1615 joint frequencies λ = (λ1 ,…, λn ). The declustered summary statistics of these observations are summarized in Figure

2. Based on the data we predict joint frequencies at new locations within the mineralised envelope.

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5.1. Simulation using Poisson-sampling Section 3.2 indicates how kriging have been used to predict the intensity λ throughout the volume of interest. By assumption, this intensity can be used in a Poisson sampling process to assess the uncertainty in joint frequency at sampled and unsampled locations and to produce probability maps. This is done as a first attempt to assess the local- and spatial uncertainty. In doing this one actually adds one layer of Poisson noise on top of the predicted result presented Ellefmo (2005). Herein, the simulation has been limited to locations within the mineralised envelope and west of x-coordinate 67.000.

5.2. Simulation using Turning Band and Hermite transformation Our second approach for prediction is constructed from a simulation technique called Turning Band Method (TBM), see Journel and Huijbregts

(1978). This technique requires Gaussian distributed data. Therefore the joint frequency data is first transformed into near-Gaussian space using Hermite polynomials. See Appendix for further details.

Parameter estimation and prediction is done in Gaussian space using standard techniques for variogram fitting, see Figure 3, and simulations from the TBM. A search ellipsoid is used to limit the data involved in the prediction at each unsampled point. Several realizations are generated by TBM and transformed into data space. This set of values represents possible joint frequencies at the unsampled location.

5.3. Latent Gaussian model for Poisson intensity The third approach models the observed joint frequencies as Poisson variables with a spatially varying intensity for the observations. The log-

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intensity of these measurements, denoted z = (z1,…,zn) is modelled as a latent spatially smooth process using a Gaussian prior distribution. The Poisson likelihood is assumed mutually independent for all observation sites at the second level. Such a two level model has been used in for instance Diggle et al.

(1998). The posterior distribution and estimates of variogram parameters can be obtained by the use of Bayes formula.

The joint frequency samples are obtained by first drawing Gaussian realizations of the log-intensity z0 at a new location. This is obtained using a search ellipsoid for kriging as in Section 5.1 when an approximate Gaussian posterior p(z|λobs) is available. See more in Appendix. Second, using these realizations of intensity exp(z0) at the new location, we generate Poisson variables to examine the expected joint frequency and associated variability.

Finally, these generated joint frequencies are used to estimate RQD.

5.4. Test locations To test the three methods presented in the previous sections five unsampled points have been chosen by the authors and five sampled points have been selected at random. Possible joint frequency realizations have been generated using the three methods. The sampled points have been excluded from the dataset when these points have been predicted.

Coordinate information of the unsampled and sampled points are given in

Table 3 and Table 4 respectively. Coordinates are given in datum NGO48, axis

IV.

The location of the points relative to the mine map from Figure 1 is given in

Figure 6.

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To apply the methods more realistically, the probability that the joint frequency is above five joints per meter has been predicted for the whole deposit. A joint frequency equal to five corresponds to a RQD close to 90%.

This has been estimated on a grid with dimensions 25 x 10 x 25 meter and probability maps have been generated.

6. Results and discussion

6.1. Results The tables presented in this section contain summary statistics for the joint frequencies obtained using the methods presented in Section 5. Summary statistics for corresponding RQD values are also included. The tabulated RQD percentiles are estimated directly from the joint frequency percentiles using Eq.

1. The 95% percentile means that 95% of all realisations are below the tabulated value, i.e. the probability that the actual value is below the tabulated value is 95%.

6.1.1. Simulation at unsampled locations Results of the joint frequencies at unsampled location using the method based on kriging with subsequent Poisson sampling and the turning band method with Hermite transformation are given in Table 5 and Table 6 respectively. Results of joint frequencies at unsampled locations using the latent Gaussian model for Poisson intensity are given in Table 7.

Comparing all three tables it can be seen that the means are comparable, but the method using the latent Gaussian model for Poisson intensity result in a larger standard deviation and consequently a wider confidence interval than the method with the two other methods. Figure 7 shows a histogram that indicate

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the most probable joint frequency value (the mode) and the associated uncertainty in point 3. This figure also shows the corresponding RQD distribution.

6.1.2. Simulation at sampled locations Results of the joint frequencies at sampled location using the method based on kriging with subsequent Poisson sampling and the turning band method with Hermite transformation are given in Table 8 and Table 9 respectively.

Results of joint frequencies at sampled locations using the latent Gaussian model for Poisson intensity are given in Table 10.

The method using the latent Gaussian model for Poisson intensity results in a larger standard deviation than the other two methods. This is the same result as with the predictions at the unsampled locations. To quantify how good the prediction is, the average squared difference between the mean values in

Tables 8, 9 and 10 and the corresponding observed values is calculated. Taking all five points into account, this indicator is smaller for the latent Gaussian method (1.8) than for the method using the Hermite transformation (4.0) and the method based on kriging with subsequent sampling (3.6). Excluding point 1 and 5, which differs most from the average joint frequency, the same indicators become 0.42, 0.36 and 0.80 for the latent Gaussian method, the Hermite transformation and the kriging based method respectively, i.e. the opposite order.

6.1.3. Probability estimation To apply the methods and assess the spatial uncertainty, the probability that the joint frequency exceeds five per meter is estimated for the deposit, limited

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to west of x-coordinate 67.000. Figure 8 shows a probability map that indicates the spatial distribution of this probability using TBM and the Hermite transformation.

Some yellow patches in the eastern and the western parts of the ore body indicate zones with a probability between 70% and 84% that the joint frequency is above five. These zones coincide well with areas where the mine has experienced stability problems, see circles in Figure 4.

The same probability as in Figure 8 has been estimated using the latent

Gaussian method. The spatial distribution of this probability is given in Figure

9.

The yellow patches clearly seen in Figure 8 cannot be seen in Figure 9.

However, the two figures show the same general trend; relatively high probability in the western part and one clearly defined patch in the eastern part.

On average, the probability that the joint frequency exceeds five is very similar for the two methods. For the Hermite transformation the average probability is

0.45 and for the latent Gaussian method it is 0.44. An impression from Figure 8 and 9 is that the latent Gaussian method provides a smoother result compared to the Hermite transformation.

Using suitable software it is a standard exercise to isolate those predictions that fall within some planned tunnel- or drift trace. Figure 10 is a horizontal section showing the predictions that fall within the trace in Figure 8. Kriging with subsequent Poisson sampling has been used to obtain this particular result.

It is clear to see how the predicted probability varies along the drift- or tunnel trace. This would constitute valuable information for the personnel who

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plans and drives the tunnel and could serve as valuable input to the estimation of required rock support.

6.2. Discussion The joint frequency predictions performed here must be considered to be a lower limit. The reason for this is that the boreholes with the joint frequency data is drilled approximately perpendicular to joint set 1 and approximately parallel to joint set 2 and 3. Thus, joint set 2 and joint set 3 are only to a limited extent present in the boreholes. That said, joint set 1 is the dominating one and taking the other two joint sets into account would probably not change the joint frequency significantly. However, having identified zones with high expected joint frequency should be taken seriously since the expected joint frequency is probably underestimated.

The methods applied herein identify zones with a relatively higher joint frequency than the surrounding rock mass. In retrospective, one can say that the stability problems in the eastern parts probably originated from the fact that too long stopes were opened. The horizontal stresses then needed to redistribute around the stope. The rock mass near the pillar is rather fractured and this rock mass would then not pick up much stress. The stresses in the pillar then exceeded the rock strength resulting in rock bursts.

The stability problems in the western part were due to the fact that the northern wall of the stope was put to far north. If the results presented herein had been used when the stope was planned, the northern wall would probably have been planned a bit further south. In doing this the mining company would loose some ore, but then the ore actually produced would have a higher grade

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and one would have fewer time consuming stability problems during development and production.

By using simulation instead of estimation it is possible to get a clear picture of not only the expected joint frequency and the expected RQD, but also a picture of possible values or ranges.

The case herein is from a meta-sedimentary iron ore. Although this is a specific case with certain geologic conditions, the methods used have a general validity. As long as enough joint frequency data is collected through surface mapping or borehole logging, the methods can be used.

The method in Section 5.2 would require more data than the method in 5.3 since 1) Simulation by Hermite polynomial requires an empirical calculation of the variogram and 2) Fitting the polynomials and kriging the transformed variable would need a sufficient amount of data to be conditionally unbiased.

The method in Section 5.2 has the advantage of CPU time. It is fast to fit the

Hermite polynomials and this is done only once. A user would have to select a threshold for the number of terms in the polynomial.

The method in Section 5.3 has the advantage of modelling a two level model that has received much attention in statistics and several domains of applications. The method is also consistent from the modelling assumption perspective and in the prediction of Poisson data for new geographical locations. Finding the posterior approximation takes some more time than in the standard kriging based method of Section 5.2 since fitting the posterior for latent log intensity requires optimization. For variogram parameter estimation a matrix of size n x n must be inverted with the latent Gaussian model. This becomes very CPU demanding when the data size is large. Empirical

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variogram fitting which is used for the Hermite polynomial method is still fast to do.

When the method presented in Section 5.1 is used, one assumes that the estimated intensity is without uncertainty. Although the estimated intensity is the most probable one, the kriging variance obtained through the kriging exercise indicates that this in fact is not valid. One solution might be to view the realisations from the TBM (Section 5.2) as intensities and perform a subsequent Poisson sampling, but this is questionable since no assumptions of

Poisson counts are used in the original Hermite transformation of the data.

The results show that the kriging based methods (Section 5.1 and Section

5.2) give a smaller standard deviation than the method using the latent

Gaussian model for Poisson intensity. The mean values however, are comparable.

7. Conclusions The applied methods can be used to predict the joint frequency and the associated variability. A clear picture of the spatial distribution can be obtained. Geostatistical methods and simulation is used here to produce spatial maps for RQD. Such maps can be important for the planning of mining or tunnelling activities and should have been used to a greater extent in the presented case. Three methods are presented and compared. On average the methods produce results that are similar, but the methods based on kriging seem to produce a smaller standard deviation than the latent Gaussian method.

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

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9. Appendix

9.1. Simulation using Hermite transformation and Turning Band The idea in the turning band method (TBM) is to simulate the multidimensional random field by summing contributions from a one- dimensional simulation process. TBM was developed by Matheron (1973). The method produces realizations zi(x) on N lines distributed in 3D. The realizations zi(x) are averaged to produce a realization zs in three dimensions

(Journel and Huijbregts 1978):

1 N z = z ()x s ∑ i Eq. 2 N i=1

The conditioning is performed through a separate kriging step (e.g. Chilès and Definer 1999).

The N lines are distributed regularly (Lantuéjoul 2002) or independently and uniformly (Journel 1974) or according to sequences with weak discrepancy on the unit sphere (Bouleau 1986).

The method replicates the variogram especially good and it produces non- conditional simulations very efficient (Vann et al. 2002).

The TBM requires Gaussian distributed data. Therefore the joint frequency data is first transformed into Gaussian space. This transformation produces a new dataset by processing the raw joint frequencies.

To transform from data- to Gaussian space the data values λobs are sorted from smallest to largest and then the i-th smallest data value is compared with the 100 x i/n percentile of the normal Gaussian distribution. To transform from

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Gaussian- to data space a method based on the Gaussian anamorphosis, see e.g.

Cressie (1993), using Hermite polynomials is used. Define pairs

zobs = G(λ obs), λ obs = H(z obs) i i i Eq. 3 for all data locations i = 1,…,n, where H denotes the set of Hermite polynomials and G are inverse transformations. The Hermite polynomials are orthogonal and represents the natural polynomials in a Taylor expansion of the

Gaussian given by

Hz()=++φφφ113 z z23 −+ φ z −+ z 01 2( ) 3( ) K Eq. 4

The coefficient φj in front of polynomial j is fitted from the range of raw data and standard Gaussian percentiles covering the support of the standard

Gaussian, say (-3.5,3.5). From a finite set of data, a perfect match is not possible. Moreover, the polynomials typically fluctuate in the far tails of the

Gaussian, see Figure 10.

Parameter estimation and prediction can be done in the Gaussian space using standard techniques for variogram fitting (Goovaerts, 1997) and simulations from the TBM. A search ellipsoid rotated N70oE with a dip equal to 70o towards north with size (200,50,200) around the new location to be predicted, i.e longest axes along strike and down dip have been used herein.

Several realizations z0 are generated at unsampled locations by the TBM. This set of values represents the range and the expected value for z0 at the new location can be estimated.

Afterwards the predictions z0 can be back-transformed into data space by using the inverse function G in Eq. 3. The obtained results are the joint

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frequencies at the unsampled locations. The generated joint frequencies at the unsampled location are used to estimate the RQD with Eq. 1.

9.2. Latent Gaussian model for Poisson intensity This approach models the observed joint frequencies as Poisson variables

obs obs obs with a spatially varying intensity for the observations λ =( λ1 ,…, λn ). The log-intensity of these measurements, denoted z = (z1,…,zn) is modelled as a latent spatially smooth process using a Gaussian distribution. Such a two level model has been used in for instance Diggle et al. (1998). The main objective is then to predict this log-intensity with uncertainty for any geographical location, and then simulate joint frequencies from the Poisson distribution with these sampled intensities

Let p(z) be the Gaussian prior probability density for log-intensity. This is modelled as above with a spherical variogram defined from sill, nugget and two correlation ranges in the strike and perpendicular-to-strike directions. Let further observations be mutually independent given the log intensity, with

p(λobs z )= Poisson(u exp(z )) i i i i Eq. 5 denoting the Poisson likelihood function for joint frequency counts, where

u4i = m is the length of the composites of raw data. The posterior distribution for log intensity is then

p(z)p(λobs z) p()z λobs = p()λobs Eq. 6

The denominator is the marginal likelihood of λobs that does not depend on z, while the numerator provides the trade off between the prior imposing the

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assumed spatial smoothness in the latent log-intensity and the Poisson likelihood enforcing a reasonable match to the observed joint frequency counts.

We approximate the posterior in Eq. 6 by fitting a Gaussian approximation for z at the mode of the numerator: h(z)=p(z)p(λobs|z). This is done by a

Newton optimization scheme, see Breslow and Clayton (1993) and Eidsvik et al. (2006). The covariance of the Gaussian is fitted from the curvature using second derivatives of log[h(z)] at the mode of h(z).

The parameters in the prior consist of smoothness values in a variogram for the log-intensity z that are fitted for the spherical model defined by Eq. 6. For the range parameters we use 200 m in strike and 45 m perpendicular to strike, similar to what was done in Chapter 5.1. The sill and nugget, on the other hand, are estimated by maximum marginal likelihood estimates obtained from p(λobs), see Breslow and Clayton (1993). The maximum likelihood estimate for nugget is 0.03, while the sill is 0.1. Note that these estimates are quite different from the ones obtained with the Hermite transformation. This is natural since the data are modelled differently.

The joint frequency predictions are obtained by first drawing Gaussian realizations of the log-intensity z0 at a new location. This is obtained by using a search ellipsoid for kriging as above when the Gaussian posterior p(z|λobs) is available. Secondly, using these realizations of intensity exp(z0) at the new location we generate Poisson variables from Eq. 5 to examine the expected joint frequency and associated variability. Finally, these generated joint frequencies are used to estimate RQD.

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References

Bouleau, N. 1986. Probabilités de l'Ingéniur, Herman, Paris. Breslow, N. E. and Clayton, D. G., 1993, Approximate inference in generalized linear mixed models, Journal of the American statistical association, 88, 9- 25. Chilès, J.P., Delfiner, P. 1999. Geostatistics: modeling Spatial Uncertainty. John Wiley & Sons, Inc. 635p. Cressie, N., 1993, Statistics for spatial data, Wiley. Deere, D. and Miller, R.D. 1966. Engineering classification and index properties for intact rock. Univ. of Illinois, Tech Rept. No. AFWL-TR-65- 116. Diggle, P., Tawn, J. A., and Moyeed, R. A., 1998, Model-based geostatistics, Journal of the royal statistical society, Series C, 47, 299-350. Einstein, H.H. 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 Metallurgy. Eidsvik, J., Martino, S., and Rue, H., 2006, Approximate Bayesian inference for spatial generalised linear mixed models, Technical report 2, 2006, Dept of Mathematical Sciences, NTNU. 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. 271p. Gabrielsen, R.H, Braathen, A., Dehls, J. and Roberts, D. 2002. Tectonic lineaments of Norway. Norsk Geologisk Tidsskrift, vol. 82, pp. 153-174. Hoek, E, Kaiser, P.K., Bafden, W.F. 1995. Support of Underground Excavations in Hard Rock, Rotterdam: A.A. Balkema. Hoerger, Steven H. and 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 – 106. Journel, A. G. 1974. Geostatistics for conditional simulation of ore bodies. Econ. Geology, vol. 69, no. 5, p. 673 - 687. Journel, A. G., Huijbregts, Ch. J. 1978. Mining geostatistics. Academic Press. 580p. La Pointe, P. R. 1980. Analysis of the spatial variations in rock mass properties through geostatistics. Proceedings of the 21st Symposium on Rock Mechanics, 570-580. Lantuéjoul, C. 2002. Geostatistical simulation. Models and algorithms. Springer. 239p. Liu, X., Srinivasan, S. 2004. Merging Outcrop Data and Geomechanical Information in Stochastic Models of Fracture Reservoirs. SPE 90643. SPE

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International Petroleum Conference in Puebla, Mexico, 8-9 November 2004. Matheron, G. 1973. The intrinsic random functions and their application. Advances in Applied Probability, vol. 5, pp. 439-468. Nilsen, B.1979. Stabilitet av høye fjellskjæringer. Report 11, Geol. Inst., NTH, Trondheim. 271p. Nilsen, B, Palmstrøm A. 2000. Engineering geology and rock engineering. Handbook 2. Norwegian Group for Rock Mechanics (NBG). 201p. Park, R.G. 1989. Foundations of structural geology. 2nd ed. Blackie Academic & Professionals. 140p. Priest, S.D., Hudson, J.A. 1981. Estimation of discontinuity Spacing and Trace Length Using Scanline Surveys. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., vol. 18, pp 183-197. Syrjänen, P. and 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, 2003. Søvegjarto, U., Marker, M., Gjelle, S. 1989. 2027 IV, berggrunnskart, M=1:50.000 Norges geologiske undersøkelse. Young, Dae S. 1987. Indicator Kriging for Unit Vectors: Rock Joint Orientations. Mathematical Geology. Vol. 19, no. 6. pp 481-501. Yu, Y.F., Mostyn, G.R. 1993. Spatial correlation of rock joints, pp. 241-255. In: Li, K.S. and Lo S-C.R. (eds). Probabilistic Methods in Geotechnical Engineering. Balkema, Rotterdam. 331p.

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Figure captions

Figure 1 Mine map at level 250. Coordinates given in datum NGO48, axis

IV.

Figure 2 Joint frequency, 4 meter long composites. Histogram and summary statistics

Figure 3 Directional empirical variogram plots. Upper left: N70oE, upper right: N160oE, lower left: down dip.

Figure 4 Horizontal section through the rock mass indicating the spatial distribution of the most probable joint frequency.

Figure 5 Horizontal section through the rock mass indicating the spatial distribution of the RQD. Seen in relation to Table 1, all values are from Fair to

Excellent.

Figure 6 Left: unsampled locations. Right: Sampled locations. Both relative to the same mine map as in Figure 1.

Figure 7 Histogram showing the joint frequency- and the RQD distribution for point 3, unsampled locations.

Figure 8 Horizontal section showing the probability that joint frequency is above five joints per meter using the Hermite transformation. Some yellow patches can be seen in the eastern and the western part of the deposit indicating a probability between 70 and 84% that the joint frequency is above five. Two black lines indicate a hypothetical tunnel- or drift trace further used in Figure

10.

Figure 9 Horizontal section showing the probability that joint frequency is above five joints per meter using the latent Gaussian method. The yellow

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patches in Figure 8 can not be seen, however the general trend is similar; relatively high probability in the western and one patch in the eastern part.

Figure 10 Horizontal section showing the probability that joint frequency is above five joints per meter along a hypothetical tunnel- or drift trace. The probability has been predicted using kriging based method.

Figure 11 Graph illustrating the transformation to Gaussian space using

Hermite Polynomials.

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Figure 1

JointFreq 0 10 20

Nb Samples: 1615 Minimum: 0 0.3 Maximum: 23.2492 0.3 Mean: 5.03993 Std. Dev.: 2.24287 Frequencies

0.2 0.2 Frequencies

0.1 0.1

0.0 0.0 0 10 20 JointFreq Isatis Kvannevann/CompBreak4mLinesFile(InsideMinEnvWestOf67000) root - Variable #1 : JointFreq Weight Variable: WeightJointFreq4mCompInsideMinEnvWestOf67000 Aug 07 2007 12:08:09 Kvannevann_Break Monovariate Statistics : - Nb. samples : 1615 - Minimum : 0 - Maximum : 23.2492 - Mean : 5.03993 - Std. Dev. : 2.24287 - Coef of Var. : 0.445

Figure 2

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Figure 3

Figure 4

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Figure 5

Figure 6

900 900

800 800

700 700

600 600

500 500

400 400 Frequency Frequency

300 300

200 200

100 100

0 0 0 2 4 6 8 10 12 14 60 66 72 78 84 90 96 Joint frequency RQD

Figure 7

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Figure 8

Figure 9

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Figure 10

Gaussian values -5 0 5

20 20 JointFreq

10 10 JointFreq

0 0

-5 0 5 Gaussian values Isatis Kvannevann/CompBreak4mLinesFile(InsideMinEnvWestOf67000) root Raw Variable : JointFreq Gaussian Variable : GaussJointFreq4mCompInsideMinEnvWestOf67000 Aug 07 2007 11:17:55 Weight Variable : WeightJointFreq4mCompInsideMinEnvWestOf67000 Kvannevann_Break Nb Sample : 3122

Figure 11

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Table captions

Table 1 Joint set characteristics.

Table 2 Rock mass classification based on RQD.

Table 3 Unsampled locations

Table 4 Sampled locations

Table 5 Results of joint frequencies at unsampled locations kriging and subsequent Poisson sampling. Joint frequency percentiles in the table are used for RQD estimations.

Table 6 Results of joint frequencies at unsampled locations using TBM and

Hermite polynoms. Joint frequency percentiles in the table are used for RQD estimations.

Table 7 Results of joint frequencies at unsampled locations using the latent

Gaussian model for Poisson intensity.

Table 8 Results of joint frequencies at sampled locations using kriging and subsequent Poisson sampling. RQD is estimated from the joint frequency percentiles and the observed value (joint frequency) is included for comparison with the prediction.

Table 9 Results of joint frequencies at sampled locations using TBM and hermite polynoms. RQD is estimated from the joint frequency percentiles and the observed value (joint frequency) is included for comparison with the prediction.

Table 10 Results of joint frequencies at sampled locations using the latent

Gaussian model for Poisson intensity. Observed value included for comparison with the prediction.

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Table 1 Joint name Strike Dip Joint set Foliation joints N65oE – N85oE 60oN - 90o 1 Steep traverse joints N150oE – N210oE 70oØ - 90o / 70oV - 90o 2 Flat traverse joints N150oE – N210oE <25oE / <25oW 3

Table 2 Term RQD Very poor < 25 Poor 25 – 50 Fair 50 – 75 Good 75 – 90 Excellent 90 – 100

Table 3 Point number 1234 5 East (X) 66058 66310 66805 66500 66356 North (Y) 939198 939307 939442 939444 939446 Z 250 300 250 150 200

Table 4 Point number 1234 5 East (X) 66485.75 66295.45 66696.49 66860.29 66086.2 North (Y) 939334.6 939293 939425.3 939501.9 939212.6 Z 332.6 95.8 416.11 350.97 304.54 Observed value 2.32 4.75 6.25 4.49 3.50

Table 5 Point number 1 2 3 4 5 Mean 6.5 4.7 4.7 4.7 6.8 Stdv 2.6 2.2 2.2 2.2 2.6 95% percentile 11 8 9 9 11 5% percentile 3 2 1 1 3 95% percentile RQD 69.9 80.7 77.2 77.2 69.9 5% percentile RQD 96.3 98.2 99.5 99.5 96.3

Table 6 Pointnumber 12345 Mean 6.6 5.6 4.5 4.8 6.3 Stdv 2.3 2.2 1.9 2.2 2.8 95% percentile 11.0 9.4 8.2 8.9 12.2 5% percentile 3.8 2.7 2.0 2.8 2.9 95% percentile RQD 70.0 75.9 80.1 77.5 65.5 5% percentile RQD 94.5 96.9 98.2 96.8 96.5

Table 7 Point number 1 2 3 4 5 Mean 6.6 4.8 4.9 5.2 6.3 Stdv 3.3 2.6 2.6 2.8 3.4 95% percentile 13 10 10 10 13 5% percentile 2 1 1 1 2

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95% percentile RQD 62.7 73.6 73.6 73.6 62.7 5% percentile RQD 98.2 99.5 99.5 99.5 99.5

Table 8 Point number 1 2 3 4 5 Mean 4.2 4.1 5.4 5.6 7.0 Stdv 2.0 2.0 2.4 2.4 2.7 95% percentile 8 8 10 10 12 5% percentile 1 1 2 2 3 Observed value 2.32 4.75 6.25 4.49 3.50 5% percentile RQD 80.2 80.9 73.6 73.6 66.3 95% percentile RQD 99.5 99.5 98.2 98.2 96.3

Table 9 Point number 1 2 3 4 5 Mean 4.3 4.4 5.7 5.3 7.4 Stdv 1.3 1.2 1.7 1.4 2.3 95% percentile 6.6 6.4 9.1 8.0 12.0 5% percentile 2.6 2.8 3.7 3.5 4.4 Observed value 2.32 4.75 6.25 4.49 3.5 5% percentile RQD 85.8 86.5 76.9 80.9 66.3 95% percentile RQD 97.2 96.7 94.6 95.1 92.7

Table 10 Point number 1 2 3 4 5 Mean 3.1 5.2 6.1 5.5 6.2 Stdv 2.1 2.8 2.9 2.8 3.3 95% percentile 8 10 11 11 13 5% percentile 1 1 2 2 2 Observed value 2.32 4.75 6.25 4.49 3.50 5% percentile RQD 80.1 73.5 70.0 70.0 62.7 95% percentile RQD 99.5 99.5 98.2 98.2 98.2

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