Variography merges the science of data comparison, the intuition of geology and the art of modelling to communicate a geologist’s understanding of grade continuity. Understanding how sample values relate to each other in space can help to infer possible similarities between known samples and points that have not been sampled. This information can then be used to generate an estimate (e.g. using ordinary kriging) of the grade at unsampled locations.
The four main aspects of variography are:
The accuracy of the continuity directions and variogram modelling parameters depends on the quality of the resource analyst’s understanding of the controls on mineralisation. It is therefore imperative that he/she has a sound understanding of the orebody including structural, weathering and lithological controls on mineralisation.
Clear domaining almost always leads to simpler and clearer variography (refer to our previous article on data analysis). Conversely, poor domaining results in confusing variograms and incompatible variogram models for estimation.
Consider plotting the differences between the sample values against the separation distance between samples. Intuitively, samples further apart are more different from each other than samples close together. A variogram is created by plotting the average variability for all sample pairs at a certain distance apart against that separation distance, termed the lag or “h” (e.g. 10 m). When calculating a directional variogram, “h” refers to the distance and direction vector (e.g. 10 m north-south).
The gamma symbol (γ) is the standard symbol for variability in a variogram. On the variogram we plot γ (h) being the average variability (or variogram value) of all sample pairs separated by vector “h”. The average variability is calculated for a series of lags and plotted against lag distance to create a variogram plot.
Superimposing the pair count on the variogram can be used as a guide to selecting the optimum lag size.
In order to use a variogram for estimation, variograms need to be modelled in the three orthogonal directions to define a three dimensional ellipsoid. This allows variogram values to be calculated by the estimation software for any separation distance in any orientation. The three directions are called the major (direction 1), semi-major (direction 2) and minor (direction 3) axes.
Geological knowledge is the best guide to defining your directions of continuity and all directions should be assessed to ensure they make geological sense. Variogram maps or fans are a useful tool to confirm these directions or defining them if there is insufficient geological information.
Once the three orthogonal directions have been determined, then the variograms need to be interpreted and a best fit mathematical model fitted to each direction. There are a number of variogram model shapes you can choose from including:
- Hole-effect models
The most common variogram models are the spherical and exponential models. There is no requirement to choose any one model over another. The most important consideration is whether the model shape reflects the calculated variogram based on the sample data. Standard model shapes can be accumulated to create more detailed variogram model shapes and these are commonly referred to as nested variogram models. When using nested models remember to keep it simple and use as few structures as possible.
Features of a variogram model are:
- The nugget effect (C0) describes the expected difference between samples when the separation distance is almost zero and comprises two parts – the inherent small scale (geological) variability and any errors in the data (e.g. sampling errors and database errors).
- The total sill (C0 + C1) represents the total variability inherent in the data. The sill is equivalent to the total data variance that would be calculated using the variance statistic, although most variograms are standardised to a total sill of 1 for simplicity.
- The range (a1) of continuity is the lag or separation distance at which the variability reaches the sill. When samples are separated by distances beyond the range of continuity they have no spatial correlation.
When interpreting variograms, there are some rules which should be followed. Because one three dimensional model of the grade continuity is being defined, it is important that the models in the three orthogonal directions all have the same nugget effect and the same sill values. This also means that there must be the same number of nested structures and same model types used. Supervisor links the nugget effect and sill values between the three directions, along with the model type and number of structures, to ensure that these rules are followed. The ranges however, can be different for each of the directions.
The nugget effect is an important part of the variogram model and typically has the biggest impact on the sample weights (and hence the resultant estimates). Ideally, the closest spaced data, which is typically in the downhole direction, should be used to model the nugget value. A true downhole variogram can be generated in Supervisor using the hole number as a key field, which provides the best variogram to model the nugget effect.
Once the nugget effect has been modelled, it’s time to model the three directions of continuity. Using up to three nested structures, the variogram models can be adjusted using interactive sliders until a reasonable fit is achieved for all directions (directions 1, 2 and 3).
When dealing with skewed distributions (e.g. gold grades in most gold deposits), high grade outliers can have a significant impact on the variogram calculation and obscure the underlying continuity and directions. If this is the case, then it is advisable to transform the data prior to calculating the variograms, using either a normal scores transform or an indicator transform. Both of these transforms, along with the required back transform of the normal scores variogram model sill values, can be performed easily in Supervisor.
Snowden’s Supervisor software provides all the tools necessary to make modelling variograms easy, fast and intuitive. Multiple windows allow you to view and model variograms for all directions simultaneously and in 3D. Auto-fitting of variogram models and interactive adjustment of lags saves time and tests the stability of your model.