Geostatistical analysis is a vital activity in developing a clear understanding of many mining projects. It can provide important information about the orebody that can heavily impact the value and economic risks associated with the project. It can be vital in determining the most effective and profitable mining methods to apply to a project. Geostatistics is used to estimate the values of properties throughout a rock or soil zone, from a population of sample values. The fundamental premise of Geostatistics is that the processes by which soil and rock materials are created and transformed in nature leads to patterns or trends in the way the properties of the material vary with direction and distance. An intuitive example is the taking of temperature samples from a beach. The temperatures sampled along the beach are likely to more closely related than samples taken closer by in an inland area where the environment is different. .
Once samples are collected from the field, they are carefully labeled with the sample location and a unique name. They will typically be analyzed by a laboratory for the attributes of interest. The three steps involved in Geostatistical analysis include: 1.
Compositing of samples over a nominated distance to ensure that each interval of material has equal representation in the sample population. 2. Histogram analysis and calculation of key statistical parameters in order to identify discrete populations, anomalous values and establish cut grades. 3. Variogram analysis to investigate the spatial trends within the sample population, including Variogram calculation, modeling and validation. 4. Use of the results of the Variogram analysis to estimate values for material throughout the orebody of zone of interest. Typically values are interpolated into block or grid models. Surpac Vision contains a comprehensive suite of geostatistical tools for the above three steps. It is important to recognize that validation of input and output data are as important as understanding geostatistical theory, the software tools and the estimation method being used. Surpac Vision stores data in a relational database. When data is loaded into the database checks are made to ensure that:
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Duplicate sample names are not present. Over lapping interval samples (in boreholes) are not present. Valid codes are entered for soil and rock descriptions. A range of user definable checks.
The following pages describe each of the four phases of geostatistical analysis.
Composting Compositing of samples ensures that each interval of material has equal representation in the sample population. Imagine that you are interested in a 5 metre zone of a drillhole. The first four metres are represented by a single sample each. The last metre, however, is represented by 5 samples at 0.2 metre intervals. Clearly the last metre would have an erroneously high level of influence on any analysis. Taking the average (a composite) of the samples over each metre would clearly result in a more valid representation. The compositing tools in Surpac are flexible and designed to composite sample data in the way that you desire. The following forms show the flexibility of the downhole compositing function.
The Geology Zones option allows the portion of the boreholes that lie within the solid model of the zone to be stored in the database and viewed as an interval on the drillhole. Surpac Vision allows you to view the data clearly in 3D.
Image: showing the orebody zone together with composited sample points.
Histogram analysis and calculation of key statistic al parameters The shape of a histogram can be used to determine if a population is bimodal. The data is examined visually to determine whether two different geology zones are present in the data and the data, indicating that the data should be separated into two populations. Histograms also help the user to determine if outliers (very high or low values) are present in the data. Surpac Vision allows you to easily modify or remove outliers from the data.
The charting tools give the user full control over the axes, number of bins and bin widths, colours and axes settings.
This graph includes the cumulative frequency curve, the x-axis maximum of 10, and a grid. Scatter plots, Q-Q and linear regression options are also available. A basic statistics function calculates a range of important parameters including data extents, percentiles, mean, variance, skewness, kurtosis etc. The user can choose from a range of reporting formats, including .csv, .pdf, .html and others.
An is otrophy It is vital to understand how data values vary with regard to distance and direction (anisotropy) from the sample points. Variogram analysis aids the geostatistician in determining the distribution of grade or quality through an orebody.
Anisotropy is the degree that values vary with direction and distance. The following image shows sections through block models with different anisotropy.
The immediate objective of geostatistical analysis is to determine the value of the properties of material throughout an orebody or orebody zone from a population of sample data acquired through drilling or surface sampling.
The first stage of the analysis is to determine the strength of continuity of data in three directions that are perpendicular to each other. These directions are called the Major, Semi Major and Minor axes and the results are expressed as the Major/semi major anisotropy ratio and the Major/minor anisotropy ratio. It is essential to be able to visualize the ellipsoid in 3D together with the data in order to be clear on the relationships. Surpac Vision has excellent tools for viewing the data together with the anisotropy ellipsoid.
Image showing the ellipsoid together with the orebody model with each viewport showing a different orientation.
Variogram analysis A Variogram is a graph which compares differences between samples against distance.
Surpac Vision supports the following variogram types:
Normal
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Logarithmic
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General Relative
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Pairwise Relative
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Normal Scores
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Variograms are used by the geostatistician to determine three characteristics of the data known as the nugget, sill and range. The nugget is a measure of the small scale intrinsic variability of the grade and is helpful in determining how the importance of samples increases as the distance to the estimation point is reduced. The term ‘nugget’ is used because the actual value of a sample will depend on whether a small or (very small) nugget of the material of interest is contained in the sample.
The sill is the difference between the maximum variability of the samples and the nugget. It is also the distance at which the range is attained.
The range (sometimes abbreviated as the letter “A”) represents the maximum distance which sample pairs can be said to have some relationship to their separation distance. Beyond the range, there is no relationship.
Along the x-axis of a variogram is a series of bins. Each bin represents a distance between sample points (lag). For example the first bin might contain sample points that are from .5 to 1.5 metre apart. Each bin has a width, and in this example the width is 1 metre. Finding the best possible variograms is vital to determining the way that properties vary throughout an orebody. Surpac Vision contains excellent tools to aid the user in varying the directions and properties of the variograms to gain the best possible understanding the properties of the material. Let us look more closely at the variogram graph and modeling tools.
The above variogram is captured from Surpac Vision’s geostatistics module. One of the strengths of Surpac Vision is the ability for the user to vary the width of the lag bins using the sliding bar control, dynamically viewing the effects on the variogram graph and map. This creates a powerful mechanism for exploring the subtle effects of the bin width – essential for analysis of data which displays valid variograms across only a narrow range of parameters. Variogram map A variogram map presents the variogram graphs in each direction around the 360 degree circle simultaneously.
Selecting the next and previous model button shifts the variogram clockwise to the next or previous direction. The combination of the ability to dynamically adjust the lag distance together with the variogram map and ability to step through the directions helps the user to find the best possible variograms. The user is able to explore the variograms in two planes known as the primary and secondary variogram maps. It is important to orientate the primary variogram map in the appropriate direction in terms of dip and dip direction in order for the resulting anisotropy ellipsoid to be corrected orientated. The major axis of the anisotropy ellipsoid will lie on the primary variogram map plane. The direction of maximum continuity (the maximum range) on the primary variogram map will be the major axis of the anisotropy ellipsoid. The secondary variogram map is perpendicular to the primary variogram map and also perpendicular to the major axis. Surpac Vision allows you to save each of the two variogram maps to a DTM which can be viewed in 3D together with any other data or models.
This image shows the primary and secondary variogram maps in the left pane, with the orebody model and anisotropy ellipse in the right viewport. The shape of the secondary variogram map in the above image is due to the narrow shape of the orebody, which means there are no samples pairs in certain orientations. Once we have determined the directions of the major, semi major and minor axes, we then calculate the anisotropy ratios from the value of the ranges of the variograms for each of those directions. For example, if the range of the variogram in the major axis direction is 200 and the range of the variogram map in the semi major axis direction is 100, then we have a 2:1 ratio for the major to semi major axes. In formal terms we have a ‘major to semi-major anisotropy ratio of 2:1’. What this means is that in the direction of the major axes, sample values will be related to each other over a longer distance than in the semi-major direction by a ratio of 2:1.
Block model estimation >> More information on the Surpac Vision block model The purpose of the geostatistical analysis is to determine how the data varies with direction and distance so that those rules can be applied when generating values throughout the orebody using a block or grid model. The Block Model estimation options available are shown in the Block Model estimation menu:
As the menu shows, Surpac Vision interfaces with GSLIB (installed during Surpac Vision installation) and provides the options for conditional simulation as well as traditional kriging techniques using the GSLIB libraries. The parameters generated by the geostatistical analysis are entered during the block model estimation process.
The first form (above) requires you to nominate the block model attribute that will contain the estimated values.
In the next form you are able to nominate attributes for storing information that aids is investigating the levels of data support for each of the estimated block values.
The search parameters form is where we enter the parameters obtained from the geostatistical analysis. Ie the orientation of the anisotropy ellipsoid and the axes ratios.
If you wish, you can view the anisotropy ellipsoid. The function allows you to use any of the rotation conventions for defining the ellipse that you may be familiar with, as shown in the list on the form.
Finally the variogram file name, structure and discretisation values are entered. The estimation function allows the user to limit the estimation to the blocks contained in the zone of interest. >> more information on the block model constraints engine. Applying the block model estimation function estimates the values of the blocks throughout the model. Surpac Vision is very flexible in the ways of viewing and reporting on the block model. Viewing is especially important as it aids the use to visually validate the results of the modeling operations.
The left-hand viewport shows all of the blocks on an orebody colored by grade. Top right is a long section of a portion of the model, while the bottom right viewport shows only the blocks that satisfy a constraint. Reporting Surpac Vision has very flexible block model reporting tools. The following form creates a block model report of tonnes and grade of the material broken down by bench broken down into grades ranges within each bench. The report takes only moments to produce and can be created in any of a range of formats including .csv, .pdf or .html.
