Comparing Variogram Methods for Improved Ordinary Kriging Interpolation in Earth Science

Interpolation is a common technique used in geoscience to estimate values at unsampled locations based on a set of known sample points. A popular interpolation method is ordinary kriging, which uses a spatial model based on the spatial autocorrelation of the data to generate estimates. The accuracy of ordinary kriging depends heavily on the choice of variogram model used to estimate the spatial autocorrelation. In this article, we will explore different variogram methods for ordinary kriging and discuss their advantages and disadvantages.

What is a variogram?

A variogram is a mathematical function that describes the spatial autocorrelation of a data set. It is used in kriging to model the spatial dependence between data points. The variogram is computed by measuring the variance of the differences between pairs of data points at different distances. The resulting curve shows how the variance of the differences changes with distance. The curve typically has a characteristic shape, with low variance at small distances (because nearby points are highly correlated) and high variance at large distances (because distant points are uncorrelated).

There are several different types of variogram, including the classical variogram, the robust variogram, and the indicator kriging variogram. Each type has its own strengths and weaknesses, and the choice of variogram depends on the specific characteristics of the data being analyzed.

The classic variogram

The classical variogram is the most commonly used variogram model in ordinary kriging. It assumes that the data are normally distributed and that the spatial autocorrelation is isotropic (i.e., the correlation between points is the same in all directions). The classical variogram is defined as

γ(h) = 0.5 * VarZ(x) – Z(x + h)

where h is the lag distance (i.e., the distance between pairs of points), Z(x) is the value of the variable at location x, and Var is the variance of the differences. The classical variogram can be estimated using a method of moments or a maximum likelihood estimator.

A disadvantage of the classical variogram is that it assumes isotropy, which may not always be the case in practice. If the data exhibit anisotropy (i.e., the correlation between points varies with direction), the classical variogram may produce inaccurate results.

The Robust Variogram

The robust variogram is a modification of the classical variogram that is more resistant to outliers and non-normal data. It is defined as

γ(h) = Median|Z(x) – Z(x + h)|

where Median is the median of the absolute differences. The robust variogram is less sensitive to extreme values than the classical variogram and can produce more accurate results when the data contains outliers or is non-normal.
A disadvantage of the robust variogram is that it is more computationally intensive than the classical variogram because it involves sorting and calculating medians. It may also be less accurate than the classical variogram when the data is normally distributed and free of outliers.

The indicator kriging variogram

The indicator kriging variogram is a type of variogram used in indicator kriging, which is a variation of ordinary kriging used to interpolate categorical data (i.e., data that can take only a finite number of values). The indicator kriging variogram models the spatial dependence between categories rather than the values themselves. It is defined as

γ(h) = PZ(x) ≠ Z(x + h)

where P is the probability that the difference between Z(x) and Z(x + h) is not zero. The indicator kriging variogram can be used to estimate the probability of a given category occurring at unsampled locations.

A disadvantage of the indicator kriging variogram is that it is only applicable to categorical data. It cannot be used to interpolate continuous data.

Inference

In summary, the choice of variogram method for ordinary kriging depends on the nature of the data being analyzed. The classical variogram is the most commonly used method and is appropriate for normally distributed data with isotropic spatial autocorrelation. The robust variogram is more resistant to outliers and non-normal data, but is more computationally expensive. The indicator kriging variogram is only applicable to categorical data. When choosing a variogram method, it is important to carefully consider the characteristics of the data and choose the method that is most appropriate for the specific analysis.

It is also worth noting that there are many other types of variograms, including the spherical, exponential, and Gaussian variograms. Each of these models has its own specific form and assumptions, and may be appropriate for certain types of data. In practice, it is often best to try several different variogram models and compare their performance using cross-validation or other statistical measures.
In any case, the choice of variogram is only one aspect of the ordinary kriging interpolation process. Other important considerations include the choice of interpolation algorithm, the selection of the appropriate neighborhood size, and the assessment of the accuracy of the resulting estimates. By carefully considering all of these factors, it is possible to produce accurate and reliable estimates of values at unsampled locations, which can be extremely useful in a number of geoscience applications.

FAQs

What is ordinary kriging?

Ordinary kriging is a common interpolation method used in Earth science to estimate values at unsampled locations based on a set of known sample points. It uses a spatial model based on the spatial autocorrelation of the data to produce estimates.

What is a variogram?

A variogram is a mathematical function that describes the spatial autocorrelation of a set of data. It is used in kriging to model the spatial dependence between data points.

What is the classical variogram?

The classical variogram is the most commonly used variogram model in ordinary kriging. It assumes that the data is normally distributed and that the spatial autocorrelation is isotropic (i.e., the correlation between points is the same in all directions).

What is the robust variogram?

The robust variogram is a modification of the classical variogram that is more resistant to outliers and non-normal data. It is less sensitive to extreme values than the classical variogram and can produce more accurate results when the data contains outliers or is non-normal.

What is the indicator kriging variogram?

The indicator kriging variogram is a type of variogram used in indicator kriging, which is a variation of ordinary kriging used to interpolate categorical data. It models the spatial dependence between the categories rather than the values themselves.

What are the advantages of the classical variogram?

The classical variogram is the most widely used method and is generally appropriate for normally distributed data with isotropic spatial autocorrelation. It is easy to compute and can provide accurate results when the data meets its assumptions.

What are the disadvantages of the robust variogram?

The robust variogram is more computationally expensive to calculate than the classical variogram since it involves sorting and calculating medians. It may also be less accurate than the classical variogram when the data is normally distributed and free of outliers.