Lateral Interpolation vs Kriging

The lateral interpolation method for enhancing subsurface geophysical datasets proved to be quite effective. In order to test its validity against more traditional methods, a three- dimensional reference model was calculated using a method of ordinary kriging. Kriging is a statistical approach to solving scarce sampling issues through interpolation using the

variability of known data points in the vicinity (Samsonov, 2007). Ordinary kriging is a method of geostatistical interpolation that does not rely on knowing the mean of the data

and thus it is the simplest application to this problem (Chilès and Delfiner, 2012). The

kriging approach fits a variogram to nearby points to assess the variability and

interpolates a model that fits the statistical trend of the variogram as well as the known

data points (Journel, 1974). For comparison to the lateral interpolation model, two-

dimensional interpolation using ordinary kriging was carried out on each of the layers of

the x-y plane. While three-dimensional kriging is possible, this two-dimensional method

provides a simple, computationally inexpensive way to directly compare the results of the two techniques.

A Gaussian variogram was chosen because it had the best correlation with the data

(Figure 3.24). Kriging was completed layer-by-layer in Matlab, using code adapted from Schwanghart (2010). A sample cross-section of the resulting three-dimensional reference

model is displayed in figure 3.25. Even before this reference model is used in a

geophysical inversion, it displays some interesting characteristics that the laterally interpolated model did not. It is clear that the kriging method is over-smoothing the data points across the full extent of the mesh, creating a layer of high susceptibility

approximately 400 metres thick. While it seems that the kriging method has smoothed

out the high susceptibility region over entire x-y planes at shallow depths, some of the

structure seen in the lateral interpolation model is still prevalent. The issue with

smoothing could be the result of the minimization of the estimation variance which often

causes the kriging process to smooth true spatial fluctuations (Journel, 1974). The

cut the anomalous region. Since the deposit is subhorizontal, the variance for values is going to be relatively small, resulting in the overestimation of densities outside of the boundaries of the deposit due to smoothing. Another interesting point to consider is the

sensitivity of the ordinary kriging process to the specification of the variogram (Chilès

and Delfiner, 2012). An inappropriately chosen variogram fit can result in kriging estimates that differ from the ideal estimates. While the Gaussian variogram fit the data reasonably well, there will be some error contributed from this part of the process.

Figure 3.24: A sample variogram showing the fit of the data to a Gaussian trend. Created with code modified from Schwanghart (2010).

Figure 3.25: Sample cross-section from the three-dimensional reference model for core density data created using ordinary kriging methods (northing line 6885890 [UTM], facing north). The reference model shows evidence of lateral over-

smoothing. Units on the right side represent density in g/cm3 _{above or below 2.67}

g/cm3_{. Scale on the left is elevation ASL in metres.}

The kriging model was then used as a reference model and its variances were used to calculate the maximum and minimum boundary constraints for an inversion of gravity

data. Figure 3.26a shows a cross-section from the resulting inversion. The shallow

regions of this model display some of the features of the subhorizontal high density layer seen in the lateral interpolation model as well as some of the layering seen at moderate depths. However, the over-smoothing overprints a large portion of the shallow structure

and results in an overall smooth model. Figure 3.26b is a cross-section of the previous

laterally interpolated model with the same density scaling as the kriging model for comparison. The lateral interpolation technique proposed earlier appears to do a better

job of resolving a reliable reference model for the given drillhole geometry and subsurface structure. A modified, more statistically robust version of the lateral

interpolation method described in the previous sections was also tested for its validity. Rather than assign ranges for the maximum and minimum boundary models based solely on the uncertainty value for a given cell, the weighted standard deviation plays a role in this method. The weighted standard deviation was based on the deviations of each of the values binned into each cell and combined with the uncertainty value to assign error tolerances. This method did not appear to enhance the resulting inversions; rather it appeared to hinder them due to not allowing a large enough range for the cells. This method could be investigated further by increasing the role that the uncertainty value plays in the range, although it also might be due to the fact that individual cells generally do not have large enough populations of data to calculate a reliable standard deviation.

Figure 3.26: Comparison cross-sections of constrained gravity inversions for the Thor Lake study area (northing line 6885270, facing north). In the top figure, a reference model obtained via ordinary kriging was used while in the bottom figure, the laterally interpolated model has been rescaled for comparison. Units on the right side represent density in g/cm3 _{above or below 2.67 g/cm}3_{. Scale on the left is}

elevation ASL in metres.

a)