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

Abstract

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.

Appendix

1.1 Simulation Using Hermite Transformation and Turning Band

The idea in the turning band method is to simulate the multidimensional random field by summing contributions from a one-dimensional (1D) simulation process. The turning band method was developed by Matheron (1973). The method produces realizations z _i(x) on N lines distributed in 3D. The realizations z _i(x) are averaged to produce a realization z _s in 3D (Journel and Huijbregts Ch 1978):

$$ z_{\text{s}} = \frac{1}{{\sqrt N }}\sum\limits_{i = 1}^N {z_i \left( x \right)}. $$

(3)

The conditioning is performed through a separate kriging step (e.g. Chilès and Delfiner 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 well and produces nonconditional simulations very efficiently (Vann et al. 2002).

The turning band method 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 λ ^obs are sorted from smallest to largest and then the ith smallest data value is compared with the 100 × i/n percentile of the normal Gaussian distribution. To transform from Gaussian to data space a method based on the Gaussian anamorphosis, see e.g. Cressie (1993), using Hermite polynomials is used. Define pairs

$$ z_i^{{\text{obs}}} = G(\lambda _i^{{\text{obs}}} ),\quad \lambda _i^{{\text{obs}}} = H(z_i^{{\text{obs}}} ) $$

(4)

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

$$ H\left( z \right) = \phi _0 1 + \phi _1 z + \phi _2 \left( {z^2 - 1} \right) + \phi _3 \left( {z^3 - 3z} \right) + \cdots $$

(5)

The coefficient φ _j 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 (Fig. 11). A Q–Q plot is included in Fig. 12 to illustrate how close the transformed data is to be Gaussian. In this plot the distribution of the transformed data is compared to 50 quantiles of a Gaussian distribution with a mean and a standard deviation equal to 0.08 and 0.97, respectively.

Fig. 11

Graph illustrating the transformation to Gaussian space using Hermite polynomials

Fig. 12

Q–Q plot of the transformed data. Here 50 quantiles of N(0.08, 0.97) have been compared with the same quantiles in the transformed data distribution

Parameter estimation and prediction can be done in the Gaussian space using standard techniques for variogram fitting (Goovaerts 1997) and simulations from the turning band method. A search ellipsoid rotated N70°E with a dip equal to 70° towards north with size (200,50,200) around the new location to be predicted, i.e. longest axes along strike and down dip, has been used herein. Several realizations z ₀ are generated at unsampled locations by the turning band method. This set of values represents the range and the expected value for z ₀ at the new location can be estimated.

Afterwards the predictions z ₀ can be back-transformed into the data space by using the inverse function G in Eq. 4. The obtained results are the joint frequencies at the unsampled locations. The generated joint frequencies at the unsampled location are used to estimate the RQD with Eq. 2.

1.2 Latent Gaussian Model for the Poisson Intensity

This approach models the observed joint frequencies as Poisson variables with a spatially varying intensity for the observations λ ^obs = (λ ₁ ^obs,…, λ ^obs _n ). The log-intensity of these measurements, denoted by z = (z ₁,…,z _n ), is modelled as a latent spatially smooth process using a Gaussian distribution. Such a two-level model has been used by, 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

$$ p\left( {\lambda _i^{{\text{obs}}} \left| {z_i } \right.} \right) = {\text{Poisson}}\left( {u_i \exp \left( {z_i } \right)} \right) $$

(6)

denoting the Poisson likelihood function for joint frequency counts, where u _i = 4 m is the length of the composites of the raw data. The posterior distribution for log-intensity is then

$$ p\left( {z\left| {\lambda ^{{\text{obs}}} } \right.} \right) = \frac{{p\left( z \right)p\left( {\lambda ^{{\text{obs}}} \left| z \right.} \right)}}{{p\left( {\lambda ^{{\text{obs}}} } \right)}}. $$

(7)

The denominator is the marginal likelihood of λ ^obs that does not depend on z, while the numerator provides the trade-off between the prior imposing the 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. 7 by fitting a Gaussian approximation for z at the mode of the numerator: h(z) = p(z)p(λ ^obs|z). This is done by a Newton optimization scheme, see Breslow and Clayton (1993) and Eidsvik et al. (2008). 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. For the range parameters we use 200 m in strike and 45 m perpendicular to strike, similar to what was done in Sect. 5.1. The sill and nugget, on the other hand, are estimated by maximum marginal likelihood estimates obtained from p(λ ^obs), see Breslow and Clayton (1993). The maximum likelihood estimate for nugget is about 0.03, while the sill is about 0.1. Note that these estimates are quite different from those 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 z ₀ at a new location. This is obtained by using a search ellipsoid for kriging as above when the Gaussian posterior p(z|λ ^obs) is available. Secondly, using these realizations of intensity exp(z ₀) at the new location we generate Poisson variables from Eq. 6 to examine the expected joint frequency and associated variability. Finally, these generated joint frequencies are used to estimate RQD.