空间统计ch9variogramppt课件.ppt
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1、1,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:
2、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
3、terms variogram and semivariogram are often used interchangeably. By definition g(h) is semivariogram and the variogram is 2g(h).,2,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
4、 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): T
5、he 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
6、 for Mathematical Geology 12:115-125.,3,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
7、 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, i.e., the distance (if any) at which data are no longer correlated. Sill is the variance of the random field V(Z), disregarding the sp
8、atial structure. It is the plateau where the variogram reaches at the range, g(range).,4,Setting variogram parameters Construction of a variogram requires consideration of a few things: An appropriate lag increment for h It defines the distance at which the variogram is calculated. A tolerance for t
9、he lag increment It establishes distance bins for the lag increments to accommodate unevenly spaced observations. The number of lags over which the variogram will be calculated The number of lags in conjunction with the size of the lag increment will define the total distance over which a variogram
10、is calculated. A tolerance for angle It determines how wide the bins will span. Two practical rules: It is recommended that h is chosen as such that the number of pairs is greater than 30. 2. The distance of reliability for an experimental variogram is h D/2, where D is the maximum distance over the
11、 field of data.,5,Computing variograms An experimental variogram is calculated using the R function (in package gstat): variogrm(pH1,locgx+gy, soil87.dat) # gx: list or vector of x-coordinates # gy: list or vector of y-coordinates # pH: list or vector of a response variable,6,Covariogram and Correlo
12、gram Covariogram (analogous to covariance) and correlogram (analogous to correlation coefficient) are another two useful methods for measuring spatial correlation. They describe similarity in values between two locations. Covariogram: Its estimator: where is the sample mean. At h = 0, (0) is simply
13、the finite variance of the random field. It is straightforward to establish the relationship: The correlogram is defined as,7,Properties of the moment estimator for variogram It is unbiased: If Z(s) is ergodic, as n . This means that the moment estimator approaches the true value for the variogram a
14、s the size of the region increases. The estimator is consistent. The moment estimator converges in distribution to a normal distribution as n , i.e., it is approximately normally distributed for large samples. For Gaussian processes, the approximate variance-covariance matrix of is available (Cressi
15、e 1985). * Cressie, N. 1985. Fitting variogram models by weighted least squares. Mathematical Geology 17:563-586.,8,Properties of the moment estimator for covariance The covariance: C(h) = cov(Z(si), Z(si+h) The moment estimator: Properties: The moment estimator for the covariance is biased. The bia
16、s arises because the covariance function for the residuals, is not the same as the covariance function for the errors, For a second-order stationary random field, the moment estimator for the covariance is consistent: (h)C(h) almost surely as n . However, the convergence is slower than the varigogra
17、m. For a second-order stationary random field, the moment estimator is approximately normally distributed. Properties 1 and 2 are the reasons why the variogram is preferred over the covariance function (and correlogram) in modeling geostatistical data.,9,pHFsrf,10,Variogram models There are two reas
18、ons we need to fit a model to the empirical variogram: Spatial prediction (kriging) requires estimates of the variogram g(h) for those hs which are not available in the data. The empirical variogram cannot guarantee the variance of predicted values to be positive. A variogram model can ensure a posi
19、tive variance. Various parametric variogram models have been used in the literature. The follows are some of the most popular ones. Linear model where c0 is the nugget effect. The linear variogram has no sill, and so the variance of the process is infinite. The existence of a linear variogram sugges
20、ts a trend in the data, so you should consider fitting a trend to the data, modeling the data as a function of the coordinates (trend surface analysis).,11,Power model - where c0 is the nugget effect. The power variogram has no sill, so the variance of the process is infinite. The linear variogram i
21、s a special case of the power model. Similarly, the existence of a linear variogram suggests a trend in the data, so you should consider fitting a trend to the data, modeling the data as a function of the coordinates (trend surface analysis).,h,g(h),a 1,a 1,12,Exponential model - where c0 is the nug
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