By Tim McLennan
Choosing an interpolant function
Leapfrog Geo uses two different base functions to form interpolants. They are the linear interpolant function and the spheroidal interpolant function. This blog covers when to use each base function, how to set the function parameters, and how to convert the parameters for a Leapfrog Mining interpolant across to Leapfrog Geo.
As explained in the Leapfrog Interpolation Basics blog article, the interpolant functions indicate how the function values are expected to vary as the distance between points increases. At small distances the values are expected to be similar and so the function values are small. At large distances the values are expected to vary considerably and so the function values are larger. The nature of this relationship means that the interpolant function is equivalent to the variogram used in geostatistical modelling.
The linear interpolant function
The base linear interpolant function is multi-scale. As a result, it is a good general purpose model. It works particularly well for lithology data which often has localised pockets of high resolution data. It is not appropriate for values with a distinct finite range of influence as it aggressively extrapolates out from the data. Unfortunately, most grade data is not well interpolated using a linear interpolant function.
However, as the linear interpolant function is multi-scale and quick to evaluate, it can be used to build an initial interpolant to visualise the data trends. This simple interpolant can then be used for determining the correct trend for the interpolant and whether compositing or a value transform may be required. After this, an appropriate interpolation function can be specified.
The spheroidal interpolant function
The extrapolation of the linear interpolant function at large distances is often unrealistic. In common cases, including when modelling most metallic ores, there is a finite range beyond which the influence of the data should fall to zero. In geostatistical modelling a finite range spherical variogram is used to model these cases. The spheroidal interpolant function was created to approximate the spherical variogram while still forming a smooth interpolant.
Setting interpolating function parameters
When editing an interpolant, the parameters for adjusting the interpolant function can be found on the Interpolant tab. In Leapfrog Geo 1.2, the shape of all the interpolant functions is determined consistently using simple terms corresponding to ones used in geostatistics. The effects of changing the parameters can be seen in real-time by looking at the embedded graph on the Interpolant tab (Figure 1). The graph shows how the interpolant function values vary with distance.
All interpolant functions pass through or close to two points defined by three common parameters. The points are (0.0, n) and (r, s), where n, r, and s are the nugget, range and sill parameters respectively. These three parameters have been annotated in grey in Figure 1.
The shape of the interpolant function in Leapfrog Geo is important as it is the shape rather than the actual value that determines how it affects the interpolant. If you scale the interpolant function by a constant value, then the interpolant and any evaluations of it will be unchanged.
It is the nugget to sill ratio that determines the shape of the interpolant function, so multiplying them both by the same number will give an identical interpolant. This can be seen in Figure 2. The function on the left has nugget of 3 and sill of 10, while the one on the right has a nugget of 9 and a sill of 30.
Spheroidal alpha parameter
The alpha parameter specifies the order of the approximating function. It can only be set to a value of 3, 5, 7 or 9.
The drift specifies the degree of the polynomial that is used to augment the interpolant. This can be thought of as specifying the degree of the expected underlying trend of the data and determines the behaviour a long way away from the data.
The accuracy determines how tightly the interpolating process adheres to the data. Ideally this value should not be set finer than the errors in the data. Using an unrealistically fine value will degrade the interpolant and cause it to process much slower. On creating an interpolant, Leapfrog Geo provides a good default estimate for the accuracy by examining the distribution of the data values.
Using the spheroidal interpolant function
The spherical variogram used in geostatistics has a fixed range beyond which the value is the constant sill. In contrast, the spheroidal interpolant function has an asymptotic sill. This means that as the distance increases, the value of the function will approach the value of the sill. However, the range parameter controls how quickly the function value approaches the sill. For Leapfrog Geo, the range is defined as the distance at which the value is equal to 96% of the sill (with no nugget). This definition is consistent with a common geostatistical method for handing asymptotic sills.
For larger values of the alpha parameter a more complex approximating function is used. This means that the spherical variogram will be approximated better, but also that it will take longer to generate and evaluate the interpolant. Figure 3 shows how the functions compare for the different alpha values with a nugget of 8, sill of 28 and range of 5000.
From Figure 3 you can see that the spheroidal for alpha=9 is the best approximation to the spherical variogram. Between zero and the range, the value for the alpha=3 spheroidal is greater than the alpha=9 spheroidal. At distances above the range this behaviour is reversed, as seen in the inset graph. Naturally, the alpha=5 and 7 spheroidals are between the other two spheroidals.
Converting from Leapfrog Mining
The interpolation parameters for a Leapfrog Mining interpolant can be accessed on the right hand side of the Variogram tab. The spheroidal interpolant function uses the same parameters in both Leapfrog products, but the range parameter is defined differently. Additionally, Leapfrog Geo does not support an alpha value of 1, because with that value the spheroidal is a very poor approximation to the spherical variogram.
In Leapfrog Mining the range is defined using a weighted integral of the function values. In practical terms this means the Leapfrog Mining spheroidal interpolant functions are very close to the spherical variogram near the origin, but do not approximate it so well beyond the range. It also makes it more difficult to describe the interpolant in simple terms that can be compared with more traditional geostatistical methods. Figure 4 shows how the functions compare for the different alpha values with the same parameters as the previous image.
By design, the alpha=9 spheroidal is identical in both Leapfrog products. Again, it is the best approximation to the spherical variogram.
For an alpha value of 3, 5 or 7, identical results to a Leapfrog Mining spheroidal interpolant function can be obtained in Leapfrog Geo by scaling the range parameter appropriately. The equation linking the ranges is given by:
rangeGeo = rangeMining * scale factor
where the scale factor depends on the value of alpha as in the table below:
|Scale factor||1.39||1.11||1.03||1.0 (unchanged)|
Figure 5 shows the variogram tab from Leapfrog Mining for a grade interpolant.
The range in Leapfrog Mining is 300, alpha is 3 and so the scale factor is 1.39 and
rangeGeo = 300 * 1.39 = 417
The calculation can be verified using the interpolant diagram in Leapfrog Mining. The interpolant value at the Leapfrog Geo range should be 96% of the sill. This is shown in Figure 6.
Using the linear interpolant function
In Leapfrog Geo, the linear interpolant function uses the same key parameters as the spheroidal interpolant function. This makes it easy to switch between the two. Although, unlike the spheroidal interpolant function, the linear interpolant function does not have a range in a traditional geostatistics manner. Furthermore, if there is no nugget value, then the function is multi-scale. This means that the interpolant is not sensitive to the scale of the input data. If you scale the locations of the input data by a factor, then the final interpolant will be spatially scaled by the same factor. Figure 7 shows an annotated screenshot of the function graph for a linear interpolant function.
Converting from Leapfrog Mining
Leapfrog Mining uses alternative parameters for specifying a linear interpolant function, slope and nugget. These are based on the standard mathematical equation of a line, y = m * x + c; where c is the y-intercept (nugget), and m is the slope.
The slope is determined as rise over run, so the graph can be used to determine the relationship between the parameters in the two Leapfrog products.
slope = rise / run = (sill – nugget) / range
The above equation can be rearranged to give:
sill = slope * range + nugget
This equation can be used to determine the sill from the slope for any desired range. To convert the Leapfrog Mining parameters, choose a range and use the equation. Usually it makes sense to use the estimated range provided when the interpolant in Leapfrog Geo is created.
If the nugget is zero, then the interpolant function is multi-scale, and the interpolant is not sensitive to the other parameters. The results will be the same in Leapfrog Geo no matter what values are used for the range and sill.
As the interpolant function shape in Leapfrog Geo is determined purely by the nugget to sill ratio, the initial values for the sill and nugget can be scaled if necessary.
Suppose the Leapfrog Mining parameters are slope = 0.2 and nugget = 5.0. Choose a range of 80.0 and calculate sill = 0.2 * 80.0 + 5.0 = 21.0.
Again, you can verify the calculation by using the interpolant diagram in Leapfrog Mining. In this case the value at the range is the sill as can be seen in Figure 8.