1 Introduction

In various mineral deposits, it is common to have drill hole samples obtained from different drilling campaigns. As each campaign has its own sampling protocol, samples from different campaigns are usually taken with different nominal lengths or supports. The construction of grade models would benefit if all available information is used for estimation purpose. However, the difference in length among samples must be considered during estimation.

One particular situation in which samples of different lengths occur is when the mineralization consists of a thin seam or vein and the drill hole samples cross the entire seam or vein. Each sample length corresponds to the seam thickness. In this case, practitioners usually work with the variables accumulation (product of grade and thickness) and thickness (Krige 1978; Bertoli et al. 2003; Marques et al. 2014). The grade estimates are obtained by dividing the accumulation estimates by the thickness estimates. This approach eliminates the vertical component (z), resulting in a two-dimensional (2D) model. The problem with this method is that a 2D block model may not be used directly for pit or stope optimization algorithms.

When a 3D block model is required, the common workflow to estimate grades using samples at different length involves the following steps: (1) compositing the samples and (2) kriging using the composites. However, compositing is not trivial when there are extreme variations among the lengths of the samples. In this case, compositing to a large length causes loss of information. In contrast, compositing to a short length breaks a large sample into pieces of equal grade, which is incorrect and at the same time artificially reduces the short-scale variability.

The use of average covariances in the kriging system to deal with samples of different support is well established in the literature (Deutsch et al. 1996; Yao and Journel 2000; Tran et al. 2001; Kyriakidis 2004; Pardo-Igúzquiza et al. 2006; Hansen and Mosegaard 2008; Liu and Journel 2009; Poggio and Gimona 2013). However, this approach has rarely been applied in estimation of grades in mineral deposits. Bassani et al. (2014) used average covariances in the kriging system to consider data obtained from large mined-out volumes. This paper aims at showing the applicability of kriging to estimate grades using drill hole samples of different lengths in a mineral deposit. A major bauxite deposit is used to illustrate the method. The grade model is validated by visual inspection, cross-validation and swath plots.

2 Kriging with Samples of Different Support

Consider the problem of estimating the average value of a continuous attribute z over the support V centred at location u, that is, zv(u). The data consist of set of n discrete grade values defined on the supports \( {v}_i\left\{ z\left({v}_i\right); i=1,\dots, n\right\} \). Support refers to the size (length, area or volume) in which the attribute z is measured. In our example, the support V(u) refers to a selective mining unit (SMU) centred at location u, and the support vi refers to the length of the ith drill hole sample. We aim to estimate the average value of an SMU. A drill hole sample represents the average grade along the sample length. In the geostatistical literature, average values over certain support (length, area or volume) are termed block values (Journel and Huijbregts 1978; Deutsch and Journel 1998; Goovaerts 1997). Equation 1 defines the ordinary kriging estimator for the block value z V (u):

$$ {z}_V^{*}(u)={\displaystyle \sum_{i=1}^n{\lambda}_i z\left({v}_i\right)} $$
(1)

where λ i is the ordinary kriging weight associated to the datum z(v i ). The kriging weights are the solution of the ordinary kriging system.

Equation 2 defines the ordinary kriging system accounting for the support of the data (Journel and Huijbregts 1978; Isaaks and Srivastava 1989; Goovaerts 1997):

$$ \left\{\begin{array}{c}\hfill {\displaystyle \sum_{j=1}^n{\lambda}_j\overline{C}\left({v}_i,{v}_j\right)+\mu =\overline{C}\left({v}_i, V(u)\right)}\hfill \\ {}\hfill {\displaystyle \sum_{j=1}^n{\lambda}_j=1}\hfill \end{array} i=1,\dots, n\right. $$
(2)

where μ is the Lagrange multiplier and \( \overline{C}\left({v}_i,{v}_j\right) \) is the covariance block to block between the block datum vi and the block datum v j . \( \overline{C}\left({v}_i,{v}_j\right) \) is calculated as the average of point covariances C(u ' i , u ' j ) defined between any discretizing point u ' i of the block datum v i and any discretizing point u ' j of the block datum v j :

$$ \overline{C}\left({v}_i,{v}_j\right) = \frac{1}{N_i{N}_j}{\displaystyle \sum_{i=1}^{N_i}{\displaystyle \sum_{j=1}^{N_j} C\left({u}_i^{\prime },\ {u}_j^{\prime}\right)}} $$
(3)

where N i is the number of discretizing points of the block datum v i and N j is the number of discretizing points of the block datum v j . The term \( \overline{C}\left({v}_i, V(u)\right) \) in Eq. 2 is the covariance block to block (Eq. 3) between the block datum v i and the block to be estimated V centred at location u.

3 Case Study

3.1 Dataset Presentation

The dataset derives from a bauxite deposit located in the northern portion of the Brazilian Amazon basin. The dataset contains 686 drill holes located on a relatively regular grid of 200 × 200 m spacing along the east (X) and north (Y) directions. The original Z coordinates were transformed into stratigraphic coordinates. The variable of interest is the percentage of the total sample mass retained at the no. 14 sieve aperture (REC14). As REC14 is an additive variable, similar to grades, the methods presented here are also suitable for grades. In 343 out of the 686 drill holes, REC14 was sampled at the nominal length of 0.5 m (white points in Fig. 1). In the remaining 343 drill holes (black points in Fig. 1), there is a single sample of REC14 whose length corresponds to the ore thickness. The sample represents the average value over the total ore thickness.

Fig. 1
figure 1

Location map of the drill hole collars

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Figure 2a shows the histogram of REC14 weighted by the length of the samples and summary statistics. The distribution is fairly symmetric around the mean with a low coefficient of variation. As the drill holes are approximately regularly spaced, these statistics are representative of the study area. Figure 2b shows the QQ plot between the drill hole data (length weighted) and the declustered data (obtained with a nearest neighbour estimate). The points in Fig. 2b are close to the line y = x, showing that the two distributions are similar, as expected. Figure 2c shows the histogram of the length of the samples. The length of the samples varies from 0.25 to 7.88 m. Since roughly 80 % of the samples are short, whose lengths are between 0.25 and 0.75 m, the geomodeler may feel tempted to retain only these short samples for estimation. However, keeping only the short samples results in excessive loss of information.

Fig. 2
figure 2

Histogram of REC14 (a), QQ plot between drill hole and declustered data (b), histogram of the length of the samples (c)

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3.2 Variogram Analysis and Modelling

Equation 4 describes the variogram model of REC14:

$$ \gamma (h)=0.15+0.50\cdot \mathrm{S}\mathrm{p}\mathrm{h}\left(\frac{\mathrm{NS}}{250 m},\frac{\mathrm{EW}}{250 m},\frac{\mathrm{vert}}{4.10 m}\right)+0.35\mathrm{S}\mathrm{p}\mathrm{h}\left(\frac{\mathrm{NS}}{4500 m},\frac{\mathrm{EW}}{4500 m},\frac{\mathrm{vert}}{4.20 m}\right) $$
(4)

Only the samples with length between 0.25 and 0.5 m were used to calculate the experimental variogram. As kriging with samples of different support needs a variogram at a point scale to calculate the average covariances, long samples were not used to calculate the experimental variogram.

3.3 Estimation

Ordinary kriging considering the different support in the data was used to estimate REC14. The drill hole samples were discretized along the main direction of the sample. The spacing of the discretization points corresponds to the length of the small-scale data used to calculate the experimental variogram. These discretization points of the samples were used to calculate the covariances block to block (Eq. 3) between the samples. The estimation was performed at a block model with block size of 50 × 50 × 0.5 m along X, Y and Z, respectively. The block discretization was set to 5 × 5 × 1. The estimates were constrained to the blocks inside the interpreted geological model.

3.4 Model Validation

The grade model was checked with the following techniques: (1) visual inspection, (2) swath plot and (3) cross-validation.

Visual inspection consists in comparing visually the grade model with the samples. The grade model must be consistent with the data.

Swath plot consists in first defining a series of swaths or slices along the X, Y and Z directions. Then, the average grade of the model and the declustered average grade of the samples within the slices are compared. The samples were declustered with a nearest neighbour estimate.

In the cross-validation, first one sample at a particular location is removed. Second, the value is estimated at that location using the remaining samples. The same estimation parameters used in the estimation of the block model were used in the cross-validation. The estimation error (difference between the estimated and true value) was calculated. Furthermore, the estimated and true values were compared using a scatter plot.

4 Results and Discussion

Figure 3 shows a plan view of the block model together with the samples. The high-grade areas of the block model are close to the high-grade samples, as expected.

Fig. 3
figure 3

Block model and samples. The lines represent the blocks while the points represent the samples

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The swath plots show that the block model reproduced the trend in the data along the X, Y and Z directions (see Fig. 4a–c). The local mean of the block model is similar to the local declustered mean for the three directions. In addition, the estimates neither overestimate nor underestimate systematically the local mean (see Fig. 4a–c; the local mean of the block model is not always either above or below the local declustered mean).

Fig. 4
figure 4

Swath plots along the X (a), Y (b) and Z (c) directions

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Figure 5a shows the histogram of estimation errors weighted by the length. The grade model is globally accurate, as the mean error is 0.14 %, which is close to zero. It represents a relative difference of 0.21 % in comparison to the mean of the data, which is 65.52 % of REC14 (see summary statistics in Fig. 2a). The scatter plot between the estimated and true values (Fig. 5b) shows that the regression line y = ax + b (red line in Fig. 5b) is fairly close to the first bisector line. This diagram indicates that the grade model does not suffer from a substantial conditional bias.

Fig. 5
figure 5

Histogram of estimation errors (a) and scatter plot between estimated and true values (b)

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The results highlight the strength of kriging to deal with samples of different support. Practitioners often overlook the fact that the kriging system handles data of different support (Journel 1986). In this paper, we used kriging considering the support of the data to estimate grades with samples of different lengths in a mineral deposit.

5 Conclusions

The study shows that kriging considering the support of the data is suitable to estimate grades in a mineral deposit using drill hole samples of different lengths. The methodology was applied to a bauxite deposit. The resultant grade model is visually consistent with the data and reproduced the trend of the data. In addition, the grade model is accurate and does not have significant conditional bias.