Estimation Methods
Estimation Methods
Spatial interpolation methods are used to estimate measured data to the nodes in grids that do not coincide with measured points. The spatial interpolation methods differ in their assumptions, methodologies, complexity, and deterministic or stochastic nature.
Inverse distance weighted averaging (IDWA) is a deterministic estimation method where values at grid nodes are determined by a linear combination of values at known sampled points. IDWA makes the assumption that values closer to the grid nodes are more representative of the value to be estimated than samples further away. Weights change according to the linear distance of the samples from the grid nodes. The spatial arrangement of the samples does not affect the weights. IDWA has seen extensive implementation in the mining industry due to its ease of use. IDWA has also been shown to work well with noisy data. The choice of power parameter in IDWA can significantly affect the interpolation results. As the power parameter increases, IDWA approaches the nearest neighbor interpolation method where the interpolated value simply takes on the value of the closest sample point. Optimal inverse distance weighting is a form of IDWA where the power parameter is chosen on the basis of minimum mean absolute error.
Splining is a deterministic technique to represent two-dimensional curves on three-dimensional surfaces. Splining may be thought of as the mathematical equivalent of fitting a long flexible ruler to a series of data points. Like its physical counterpart, the mathematical spline function is constrained at defined points. Splines assume smoothness of variation. Splines have the advantage of creating curves and contour lines that are visually appealing. Some of splining's disadvantages are that no estimates of error are given and that splining may mask uncertainty present in the data. Splines are typically used for creating contour lines from dense regularly spaced data. Splining may, however, be used for interpolation of irregularly spaced data.
Geostatistical Methods (Kriging)
Kriging is a stochastic technique similar to inverse distance weighted averaging in that it uses a linear combination of weights at known points to estimate the value at the grid nodes. Kriging is named after D.L. Krige, who used kriging's underlying theory to estimate ore content. Kriging uses a variogram (a.k.a. semivariogram) which is a representation of the spatial and data differences between some or all possible "pairs" of points in the measured data set. The variogram then describes the weighting factors that will be applied for the interpolation. Unlike other estimation procedures investigated, kriging provides a measure of the error and associated confidence in the estimates. Cokriging is similar to kriging except it uses two correlated measured values. The more intensely sampled data is used to assist in predicting the less sampled data. Cokriging is most effective when the covariates are highly correlated. Both kriging and cokriging assume homogeneity of first differences. While kriging is considered the best linear unbiased spatial predictor (BLUP), there are problems of nonstationarity in real-world data sets.
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