The Spheroidal Interpolant Function

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. The spheriodal interpolant function can be used when modelling in these cases.

The spheroidal interpolant function closely resembles a spherical variogram, which has a fixed range beyond which the value is the constant sill. Similarly, the spheroidal interpolant function flattens out when the distance from X is greater than a defined distance, the range. At the range, the function value is 96% of the sill with no nugget, and beyond the range the function asymptotically approaches the sill. The chart below labels the y-axis interpolant. A high value on this axis represents a greater uncertainty relating to the known value, given its distance from X. Another way to think of this is that higher values on this axis represent a decreasing weight given to the known value.

Known values within the range are weighted proportionally to the distance from X. Known values further from X than the range will all be given approximately the same weight, and have about the same influence on the unknown value. Here, points A and B are near X and so have the greatest influence on the estimated value of point X. Points C and D, however, are outside the range, which puts them on the flat part of the spheroidal interpolant curve; they have roughly the same influence on the value of X, and both have significantly less influence than A or B:

The rest of this topic describes the parameters used to define a spheroidal interpolant. These parameters are Total Sill, Nugget, Nugget to Total Sill Ratio, Base Range, Alpha, Drift and Accuracy.

To edit the parameters for an interpolant, double-click on the interpolant in the project tree and click on the Interpolant tab. The graph on the tab shows how the interpolant function values vary with distance and is updated as you change interpolant parameters:

The yellow line indicates the Base Range. For this interpolant, the value of the interpolant is offset by the value of Nugget.

Total Sill

The Total Sill defines the upper limit of the spheroidal interpolant function, where there ceases to be any correlation between values. A spherical variogram reaches the sill at the range and stays there for increasing distances beyond the range. A spheroidal interpolant approaches the sill near the range, and approaches it asymptotically for increasing distances beyond the range. The distinction is insignificant.

Nugget

The Nugget represents a local anomaly in sampled values, one that is substantially different from what would be predicted at that point, based on the surrounding data. Increasing the value of Nugget effectively places more emphasis on the average values of surrounding samples and less on the actual data point, and can be used to reduce noise caused by inaccurately measured samples.

Nugget to Total Sill Ratio

It is the Nugget to Total Sill ratio that determines the shape of the interpolant function. Multiplying both these parameters by the same constant will result in an identical interpolant. Here, the interpolant on the left has a nugget of 3 and a sill of 10; the one on the right has a nugget of 9 and a sill of 30. Note that because the nugget and sill have been increased by the same factor, the function has the same shape.

Base Range

The Base Range is the distance at which the interpolant value is 96% of the Total Sill, with no Nugget. The Base Range should be set to a distance that is not significantly less or greater than the distance between wells, so it can reach between them. As a rule of thumb, it may be set to approximately twice the average distance between wells.

Here the effect of different range settings on the value of X is demonstrated using our trivial example of four wells:

When the range is set to 1, it is too small to describe any real effect between wells. When the range is set to 30, distant wells have more influence, increasing the spatial continuity. Also illustrated is the range set to approximately the average distance between wells (range = 4) and the range set to about twice the average distance between wells (range = 8). Of these, the range set to 8 might be the best choice.

Alpha

The Alpha constant determines how steeply the interpolant rises toward the Total Sill. A low Alpha value will produce an interpolant function that rises more steeply than a high Alpha value. A high Alpha value gives points at intermediate distances more weighting, compared to lower Alpha values. This figure charts an interpolant function for each alpha setting, using a nugget of 8, sill of 28, and range of 5000. A spherical variogram function is included for comparative purposes. The inset provides a detailed view near the intersection of the sill and range.

An alpha of 9 provides the curve that is closest in shape to a spherical variogram. In ideal situations, it would probably be the first choice; however, high alpha values require more computation and processing time, as more complex approximation calculations are required. A smaller value for alpha will result in shorter times to evaluate the interpolant.

The following demonstrates the difference between alpha = 3 and alpha = 9:

There is a measurable difference between the estimates at the point being examined, but for many purposes, using a lower alpha will result in satisfactory estimates and reduced processing time.

Drift

The Drift is a model of the value distribution away from data. It determines the behaviour a long way from sampled data.

Constant: The interpolant goes to the approximated declustered mean of the data.
Linear: The interpolant behaves linearly away from data, which may result in negative values.
None: The interpolant pulls down to zero away from data.

Here, the three Drift options for the interpolant are shown evaluated on grids:

In this example, the boundary is larger than the extent of the data to illustrate the effect of different Drift settings.

Away from the data, the value of the interpolant when Drift is Constant and Linear is not reasonable in this case, given the distance from the data. The low value when Drift is None is more realistic, given the distance from the data.

If when using the spheroidal interpolant you get a grade shell that fills the model extents, it is likely that the mean value of the data is higher than the threshold chosen for the grade shell in question. If this occurs, try setting the Drift to None.

Accuracy

Leapfrog Geothermal estimates the Accuracy from the data values by taking a fraction of the smallest difference between measured data values. Although there is the temptation to set the Accuracy as low as possible, there is little point to specifying an Accuracy significantly smaller than the errors in the measured data. For example, if values are specified to two decimal places, setting the Accuracy to 0.001 is more than adequate. Smaller values will cause the interpolation to run more slowly and will degrade the interpolation result. For example, when recording to two decimals, the range 0.035 to 0.044 will be recorded as 0.04. There is little point in setting the accuracy to plus or minus 0.000001 when intrinsically values are only accurate to plus or minus 0.005.

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