Geostatistical modelling of offshore diamond deposits

Transcription

1 Geostatistical modelling of offshore diamond deposits JEF CAERS AND LUC ROMBOUTS STANFORD UNIVERSITY, Department of Petroleum Engineering, Stanford, CA , USA TERRACONSULT, Oosterveldlaan, 273 Mortsel-Antwerp, Belgium Abstract Offshore diamond deposits are extremely challenging in terms of geostatistical ore reserve evaluation. The variogram of the number of stones per sample usually has a high nugget due to the extreme grade variability and the global histogram of grades is discrete and extremely skew. Yet the geological information available testifies of a highly structured deposit. Offshore deposit of the Namibian coast reveal the existence of potholes of diamonds, only a few meters in diameter, yet containing possibly a high number of stones. The potholes are clustered in areas of typically a few hundred meters across. Based on data of a typical Namibian offshore deposit, a large unconditional simulation is constructed that honors the histogram and variogram of the number of stones per sample and that depicts the conceptual geological model for the deposit. This unconditional model is constructed using a doubly stochastic Neyman-Scott point process (diamonds are but points distributed in space) reflecting the most important geological features. This unconditional simulation is used as a so-called training image from which important mining information can be extracted such as a volume-variance relationship or other change-ofsupport variables. It is shown that the volume variance relation extracted from the training image is close to the volume-variance from the already mined part of the current deposit (the study was done before any mining started, hence the significance of this result). To obtain conditional simulations for assessing distributions of local grade variability, a novel conditional simulation technique is presented. Introduction Diamond deposits are one of the world s most important mineral resources in terms of economically feasible exploration and mining, particularly in an era of downturn within the industry. Nevertheless, in terms of resource evaluation, diamond deposits pose a serious challenge. Secondary diamond deposits (alluvial, coastal and marine) are characterized by a highly structured clustering of stones, due to the nature of deposition and due to variations of roughness in the bedrock. This leads to sample statistics of the counts of stones per sample that are geostatistically very unfavorable: highly nuggeted variograms and skew histograms. Furthermore, many of the traditional geostatistical kriging and simulation techniques cannot be applied to secondary diamond deposits. Traditional Gaussian simulation calls for a transformation to a continuous Gaussian space which is not possible due to the inherent discrete nature

2 of the variable (number of stones per sample). Indicator-type methods such as indicator simulation are not practical because it is impossible to obtain interpretable experimental indicator variograms from the data. The geostatistical modelling of diamond deposits therefore has to proceed in a non-classical way. Since the geo-structure of these deposits are difficult to obtain from the variogram, we propose to build a geological model of the diamond deposit. This geological model need not be conditioned to any sample data, yet it should reveal the most important structures as interpreted by the geologist, for example from geophysical data. Furthermore, the geological model should honor, or at least not be contradictory or inconsistent with the sample statistics obtained from the actual deposits. The sample statistics consist of the histogram and variogram of counts. In geostatistics the geological model is termed a training image or training data set. Since the training image is exhaustive, it is possible to extract from it socalled multi-point statistics, i.e. correlation between multiple location at the same time, rather than the two-point correlation modelled through the variogram. In this paper we show how these multi-point statistics can be used to obtain volume-variance relationships from the training image. Next we show, a novel conditional simulation method that renders realizations, conditioned to local sample data, and that reproduces the spatial patterns of the training image. The namibian off-shore diamond deposits The offshore diamond deposits under study are located along the Namibian coast. The diamond-bearing gravels occur in water depths of 40 to 90 metres at a distance of 3 to 12 km from the coastline. Diamonds tend to enrich in gullies less than 100m wide, formed into hard rock, as well as in deeper channels, about 300m wide, which could represent older river valleys. Favorable locations for diamond enrichment are found by using a variety of offshore exploration techniques such as side scan sonar seafloor imagery, high resolution CHIRP sub-bottom profiling, continuous bathymetric profiling and checks with vibrocore drilling. Regional survey lines are spaced at 1 km in a north-south direction. Areas of interest are covered by a denser orthogonal grid of 150 by 300m lines. Samples for diamond recovery are usually between 4 to 10 m. The fraction between 1.5 to 16mm is retained for diamond recovery. The regional sampling ratio is less than 1:15,000, but resources definition requires better sampling ratios. Measured Resources require a sampling ratio better than 1/1,500, while indicated resources require sampling ratios better than 1:15,000. Total sediment thickness in most areas is less than 5 metres. The sediments are mainly gravels covered by a thin layer of sand. The areas of interest have most often an average sediment thickness of 2 metres or less. From geology to random function model Based on the geological observation that diamonds cluster in space and that these clusters exhibit themselves a distinct spatial variation, it is imperative to build a model that reflects this geological reality. The model we propose is not new and has been proposed in Caers et al. (1996) for alluvial diamond deposits. We will first review the model and then show how its parameters can be fitted to sample data. The model is algorithmically defined as follows (see Figure 1): 1. Using Voronoi tesselation we generate regions (polygons) over a given domain of constant cluster density. Voronoi tesselation are generated through the simulation of a Poisson point process (Matern, 1960). The intensity of that Poisson process, denoted as, defines the size distribution of the polygons. The size distribution of the polygons depends itself on the range of the variogram

3 Figure 1: Steps in construction of the model, (top-left) construction of Voronoi polygon, (top-right) Seeding of cluster centers, (bottom-left) Spreading the stones in cluster, (bottom-right) final result.

4 (Matern, 1960). Within each polygon we assign a constant cluster density that is drawn from a continuous distribution. We consider to be a two-parameter Gamma distribution. 2. Cluster centers are generated in each polygon according to Poisson process with intensity. Each cluster center is assigned independently a number of stones, which is a discrete random variable drawn from a distribution. We take to be a one parameter Sichel distribution (Sichel, 1973). 3. Stones are uniformily spread inside a cluster, the cluster itself is a circle with constant radius. We denote the uniform spatial distribution as where is the vector of spatial coordinates of a single stone in the cluster. In order to construct a simulation of this model, we need to determine the parameters, and of the distributions,, the parameter and the value, all this based on the sample data. The sample data consists of the number of stones counted with fixed sample size, spread on a regular grid. We denote the set of sample counts as, the coordinates of the sample. The sample size is denoted as. The parameters, and can be determined from the sample histogram if (Caers et al., 1996) The size of the clusters is known from geology (geophysical data). The size of the samples and clusters is much smaller than the size of the polygons. This is in reality always the case. In this particular case, one can calculate, given,,, and the model histogram that will be generated when the above algorithm is applied (see Caers, 1996). We denote that model histogram as Evidently the model histogram depends on the sample size. Given the sample data one construct the sample histogram and applies an iterative technique (Press et a., 1990, p. 292) for fitting the parameters by minimzing a -squared type objective function (1) In Caers (1996), a computationally efficient recurrence relationship between the probabilities is derived that makes the iterative procedure fast. Similarly, Caers et al. (1996) derived the covariance model for the model shown in Figure 1. In short, it is shown that the covariance between and can be calculated as with, (2) and where equals the probability that two samples at distance belong to the same polygon. For Voronoi polygons this can be calculated as follows (Matern, 1960)

5 Variogram of 73 sample data Distance in m 2-D Reference Data East East Figure 2: (top-left) Variogram from sample data, (top-right) model with individual stones, (bottom-left) pixelgrid of counted stones per sample (bottom right) pixelgrid of indicator variable and with the surface area of a circle with radius and the surface area of the intersection of such two circles. Recall that is the intensity of the Poisson process that generates the polygons. Given an experimental variogram, we can fit the model (2). Note that the model (2) depends on the (known) sample size, the (unknown) parameter, the (assumed) distribution of stones within a cluster, the expected value and variance of the number of stones per cluster (which is known from the fitted histogram model (1)) and the variance of the number of clusters per polygon (which is also known from the fitted histogram model (1)). Through analysis of a large number of secondary deposits around the world, the authors found that usually the size of clusters, measured by, is larger than the sample size, yet the cluster size is smaller than the sample spacing. In this way, the small scale variation or distribution of diamonds within clusters cannot be detected on the experimental variogram because the smallest lag-distance is larger than the cluster size, i.e. the clusters appear as a nugget effect. Because the number of stones per cluster varies considerably from cluster to cluster, that nugget effect is extremely high (70-90 % in most cases) and the resulting sample histogram extremely skew with a high percentage of samples containing no stones at all (40-70 % in most cases).

6 Building the model Offshore diamond deposits are particularly challenging, because of the low sample density (due to high sample cost). In this paper, we present a particular section of the offshore Namibian deposits from which approximately 75 samples were retained on an area of a couple of square kilometers. For confidentiality reasons we cannot show Detailed sample locations, sample size. Histograms or any other univariate statistics. Maps or simulations revealing the shape of the deposit. Instead simulations are shown on a square grid. The histogram of the sample data is modelled using the above outlined procedure, giving the following parameter values: Figure 2 shows the omni-directional variogram of the sample data (the variogram is determined to be isotropic), from which we deduct a range of 500m. This entails that the value of, the Poisson intensity of the polygons equals per unit area. Given these numbers, one can construct a large unconditional simulation of the point process model using the above described algorithm. Figure 2 (topright) shows a part of the unconditional simulation. Next, a regular grid of pixels having the same sample size as the actual sample size is used to count the number of stones. Figure 2(bottom left) shows the counting result. The large unconditional simulation is then upscaled (Figure 3) to various target block-sizes (used in actual mining). The global mean and dispersion variance of the upscaled block image can be calculated to obtain a volume-variance or a volume-coefficient-of-variance relationship as shown in Figure 3. For this particular deposit, the calculated volume-coefficient-of-variance differed only 1 % from the actual mined deposit. This is a significant result because this simulation study was performed before actual mining started. Conditional simulation A training image Conditional simulations need to be constructed for mine planning, assessing local grade variability and uncertainty on block estimates. For the above mentioned confidentiality reasons we will only show a part of the deposit (25 samples) on a square grid. A direct construction of a simulation as proposed in Figure 1 that is conditional to the data is not straightforward. Data from the actual deposit consists only of counts of the number of stones in small samples. Other information, needed for the above described simulation methodology, such as the local size of the polygons, the local intensity of the clusters per polygon are simply not observed. Such information is only known globally, i.e. over the entire deposit. An alternative approach is therefore proposed that uses the unconditional simulation method indirectly. For that purpose, a large unconditional simulation, such as constructed in Figure 2d can be utilized as a so-called training image. A training image contains important information about the structure of the deposit, yet need not be conditional to any data. Since a training image is exhaustive, important statistics can be extracted/borrowed that describe the spatial variation of the phenomenon. These statistics can

7 D Reference Data D Reference Data East East coeff. var block volume times xx Figure 3: (top-left) pixelgrid for blocksize A, (top-right) pixelgrid for blocksize B, (bottom) Coefficient of variation versus blocksize (not shown) for block size A (*) and blocksize B (o). be histograms, exhaustive two-point statistics, i.e. complete covariance tables, or even multiple point statistics, i.e. statistics that describe the correlation between multiple spatial locations. The task of conditional simulation is to anchor these statistics to the available local data. In the next section, we describe what statistics are actually borrowed from the training image and how they are anchored to the local sample data. For simplicity, we will only simulate the indicator variable, which equals 1 if there is a stone at and equals zero if not. The training image under consideration is that of Figure 2 (bottom-right). The local data is denoted as,. Doubly stochastic conditional simulation Since the training image is constructed as a doubly stochastic point process, we propose to borrow statistics from the training image in a two-step approach as follows. A second training image is constructed, see Figure 4a, that is obtained through locally averaging (convolving) the original training image with a square window, i.e. each pixel in the second training image is calculated using where are the various lag distances between the center location and nodes in the window. The random variable is termed the locally varying mean, because it describes, locally, the mean of the discrete variable. The idea is to simulate for each realization the locally varying mean,, on the actual deposit, using the full-covariance table obtained from the second training image, then perform an indicator simulation of the variable with that simulated locally varying mean. Next, these two steps are explained in greater detail.

8 Simulating the locally varying mean The second training image is exhaustive, hence one can calculate the full-covariance table of the image. This table describes the covariance between any point and any other point in space. Figure 4b shows the covariance table of the second training image. Next, we perform a conditional sequential Gaussian simulation of the locally varying mean using the full covariance table (see Yao, 1998). The conditioning data for simulating are constructed from the actual local data, as follows. Using the pair of indicator training image (Figure 2d) and locally varying mean training image (Figure 4a), one estimates the following probabilities and describe the distribution of given the indicator data at any. Thus, for each local datum, a local datum is constructed by randomly drawing from the appropriate distribution. The set of local data,, is then used to perform conditional simulation. Figure 4c,d shows the set of 25 indicator and a single realization of the locally varying mean data. Figure 4e shows a conditional simulation of the local mean. Note that one has to construct as many realizations of ) as are needed to simulate in the next step. Simulating the indicator variable The next step consists of constructing an indicator simulation of the variable, given the local data and given the locally varying mean. For this purpose we use a sequential indicator simulation with locally varying mean, i.e. at each node in the sequential path we need to determine, using indicator kriging, the weights in where is the set of orginal data and previously simulated nodes. The indicator covariance used is the residual covariance that can be obtained by calculating the covariance of an image that is the pixelwise substraction of the second training image from the first training image. Although not shown, that variogram is almost pure nugget. Figure 4f shows a single realization of and if compared with the training image shows that the structure of the training image is well reproduced. Conclusions In this paper, we present a novel geostatistical approach for modelling secondary diamond deposits. Diamond deposits often have extremely skew histogram/ high-nugget variogran of the number of stones per sample because of the geological phenomenon of clustering of stones on a rough bedrock. To take into account this geological reality, we construct a so-called training image, that reflects the believed geological variation and is conditioned to simple sample statistics such as histogram and variogram. From such image one can extract information on the statistics of support related variables such as dispersion variances or coefficients of variation. However, such analog/synthetic deposit can be used to extract spatial statistics that need to be reproduced in a conditional simulation. In this paper we propose to extract and reproduce information in two stages: the mean of an indicator variable, then conditional to that mean, the indicator variable itself. This allowed to mimick the actual geological variation that also comes in two stages: a variation of the mean number of clusters, and a variation within the clusters themselves.

9 (a) second training image: local mean (b) covariance table East East (c) map of indicator data (1) (d) single realization mean "data" (e) single realization local mean (f) single realization indicator East East Figure 4: (a) Second training image obtained through smoothing the first training image (Figure 2 (bottom-right)), (b) Covariance table of second training image, (c) local indicator data, (d) single realization of the local mean data, (e) single realization of local mean variable conditioned to data in (d), (f) sibgle realization of indicator variable.

10 References Caers, J., Gelders, J., Vervoort, A. Rombouts, L. (1996). Non-conditional and conditional simulation of a spatial point process : applications to diamond deposits. International Geostatistics Congress (Eds. E. Baafi and N. Schofield), Wollongong, Australia, Kluw.Ac. Publ., Caers, J.(1996). A general family of counting distributions suitable for modelling cluster phenomena. Mathematical Geology, 28, 5, Yao, T. (1998). Automatic covariance modelling and conditional spectral simulation with fast Fourier transform. Unpublished Ph.D. thesis, Stanford University, Stanford, USA.