(Taken from Lefohn et al., 2005)
Kriging is a family of estimators used to interpolate spatial data. This family includes ordinary kriging, universal kriging, indicator kriging, co-kriging and others. The choice of which kriging to use depends on the characteristics of the data and the type of spatial model desired. The most commonly used method is ordinary kriging, which was selected for this study. A brief discussion follows on why ordinary kriging was chosen for this study rather than another method.
Indicator kriging is used when it is desired to estimate a distribution of values within an area rather than just the mean value of an area. As the purpose of the study was to estimate the mean values of the N100 and W126 exposure indices within an area rather than the distribution of values, Indicator Kriging was not selected.
Universal kriging is used to estimate spatial means when the data have a strong trend and the trend can be modeled by simple functions. Trend is scale dependent. For example Montana Tech sits on the south side of a hill high above the valley Butte , Montana . A model of the elevations around Montana Tech would show that a trend in the values exists when you look north. If you want to model the elevation to the north of Montana Tech, it can be accurately done with a simple straight line. At the scale of 1/4 mile the local data has a trend. This trend doesn’t exist for far, however. If you continue north for 60 miles, you encounter Helena , Montana . Along the way the elevation rises and falls many times as you cross mountains and valleys. On the scale of 60 miles, there is no trend in the elevation. Ozone data may display trends over small geographic areas but at the scale of the entire United States , there is no trend that can be modeled by simple functions. Because of this fact, Universal Kriging was not chosen for this study.
Co-kriging is an extension of kriging used when estimating a one variable from two variables. The two co-variables must have a strong relationship and this relationship must be defined. Use of Co-Kriging requires the spatial covariance model of each variable and the cross-covariance model of the variables. The method can be quite difficult to do because developing the cross-covariance model is quite complicated. Developing the relationship between the two variables can also be complicated. Practice in the mining industry limits Co-Kriging to the case when the variable being estimated is under sampled with respect to the second variable. If all samples have both variables, industry has found no benefit gained from the use of Co-Kriging.
Co-Kriging was not chosen for this study because the ozone indices N100 and W126 are sampled at each location. Also, there has been no study has yet been done that has identified a secondary variable from more sampling sites that is highly correlated to these exposure indices that can be used to predict these indices. Elevation may be a promising variable but there is not a sufficient number ozone monitors across a range of elevation to develop the covariance models.
Ordinary kriging was selected for this study based on how well it has performed on prior years data and because the statistical characteristics of the data in 2000 and 2003 make Ordinary Kriging the appropriate choice of estimator. The data displayed no trend at the scale of the modeling; thus universal kriging was not appropriate. The covariance models (variogram) exhibited local stationary and thus, Ordinary Kriging was the appropriate technique to use.
The authors have used ordinary kriging to make ground-level ozone models for the W126 for the years from 1982 to 2003. While the ozone values vary from year to year, the statistical character of the data remains remarkably constant from year to year. The covariance models are similar in each year and the spatial anisotropy exhibited by the co-variance models is similar in each year.
Ordinary Kriging is a spatial estimation method where the error variance is minimized. This error variance is called the kriging variance. It is based on the configuration of the data and on the variogram, hence is is homoescedastic (Yamamoto, 2005). It is not dependent on the data used to make the estimate. Recently, Yamamoto derived a error variance for ordinary kriging that is conditional to the data values. He referred to this variance as the Ordinary Interpolation variance. Yamamoto has shown that the ordinary interpolation variance is a better measure of accuracy of the kriging estimate. The ordinary kriging programs used for this study were modified to calculate the new error variance, named the Ordinary kriging interpolation variance (NKVAR) and output it along with the traditional kriging variance. The 95% error bound based on the new variance was reported also. It is believed that the new method used in this study to determine the interpolation variance is a better estimate of the error variance than the kriging variance. In particular, for skewed data, it is believed that the new variance is a much better estimate of the error variance.
Literature Cited
Lefohn, Allen S. ; Knudsen, H. Peter; and Shadwick, Douglas S. 2005. Using Ordinary Kriging to Estimate the Seasonal W126, and N100 24-h Concentrations for the Year 2000 and 2003. A.S.L. & Associates, 111 North Last Chance Gulch Suite 4A Helena , Montana 59601.
contractor_2000_2003.pdf
Yamamoto, J.K. 2005. Comparing ordinary kriging interpolation variance and indicator kriging conditional variance for assessing uncertainties at unsampled locations, In: Application of Computers and Operations Research in the Mineral Industry – Dessureault, Ganguli, Kecojevic,& Dwyer editors, Balkema.
updated: 08/21/2007