空间统计ch9variogram.pptVIP

* Chapter 9 – (Semi)Variogram models Given a geostatistical model, Z(s), its variogram g(h) is formally defined as where f(s, u) is the joint probability density function of Z(s) and Z(u). For an intrinsic random field, the variogram can be estimated using the method of moments estimator, as follows: where h is the distance separating sample locations si and si+h, N(h) is the number of distinct data pairs. In some circumstances, it may be desirable to consider direction in addition to distance. In isotropic case, h should be written as a scalar h, representing magnitude. Note: In literature the terms variogram and semivariogram are often used interchangeably. By definition g(h) is semivariogram and the variogram is 2g(h). Robust variogram estimator Variogram provides an important tool for describing how the spatial data are related with distance. As we have seen it is defined in terms of dissimilarity in data values between two locations separated by a distance h. It is noted that the moment estimator given in the previous page is sensitive to outliers in the data. Thus, sometimes robust estimators are used. The widely used robust estimator is given by Cressie and Hawkins (1980): The motivation behind this estimator is that for a Gaussian process, we have Based on the Box-Cox transformation, it is found that the fourth-root of ?12 is more normally distributed. * Cressie, N. and Hawkins, D. M. 1980. Robust estimation of the variogram, I. Journal of the International Association for Mathematical Geology 12:115-125. Variogram parameters The main goal of a variogram analysis is to construct a variogram that best estimates the autocorrelation structure of the underlying stochastic process. A typical variogram can be described using three parameters: Nugget effect – represents micro-scale variation or measurement error. It is estimated from the empirical variogram at h = 0. Range – is the distance at which the variogram reaches the plateau,