Sampling kriging

Fig. 1. Kriged and contoured cross-section of sericite / (sericite + chlorite)) 100 Index (plotted in percent) along A - A on McIntyre grid line 3500E through the HMC deposit. White diamonds indicate sample locations. Specific rock units such as C15 are labeled with abbreviations. The Cu-Au zone is indicated cross-hatching.

Fig. 4. Kriged cross-section of 5 Occ along McIntyre grid line 3500E through the HMC deposit. White dots indicate sample locations.

Semi-variogram Estimation. The first step in the kriging analysis is to estimate the semi-variogram for each site. The sample... 

At REF the estimated block means are shown for blocks whose multiplicative kriging standard deviation is less than 1.63. The blocks that are not shown were outside the area that was sampled. 

The sampled area (or the area in which kriging estimates are retained) for each site are... 

In soil science one usually has to do with two or threedimensional objects that cannot be represented by correlation constants. Here the size and the number of samples must be obtained in other ways. In this case often an intermediate step is Kriging, mapping of lines or planes of equal composition For the simplest case, punctual... 

Geostatistical methods are also termed random field sampling as opposed to random sampling [BORGMAN and QUIMBY, 1988], The spatial dependence of data and their mutual correlation can be analyzed by use of semivariograms. Statements on the anisotropy of the spatial distribution are also possible. Kriging, a geostatistical method of... 

The semivariogram as a variance function can also be used to estimate the value and the variance for new points not sampled in the investigated area. The method applied for this purpose is termed kriging. Kriging is a special regression method for interpolation of spatially or temporally correlated data with minimization of variance. The normal distribution of the data is an important condition. If the original data are not normally distributed, which is often the case for trace components in environmental compartments, the logarithm of the data or otherwise transformed data have to be applied to obtain a normal distribution of the data (see also Section 9.4). 

The advantage of kriging is that it furnishes not only an estimate of the unsampled point, but also an estimate of the variance at this location. For a sampling location the exact value is estimated with the variance being zero. If the kriging method is applied, an exact and undistorted interpolation is possible. 

If the purpose of sampling is the detailed description of the composition of an object, the character of the internal correlation has to be investigated. The methods of autocorrelation and/or semivariogram analysis, as described in Sections 6.6 and 4.4.2, may be useful for clarification of the internal spatial and/or temporal relationships existing within the parent population to be sampled. Geostatistical methods, e.g. kriging, enable undistorted estimation of the composition of unsampled locations in the area of investigation. 

Analysis of the representativeness of the samples taken by applying correlation analysis (see Sections 6.6 and 9.1.3.3) or the kriging estimation (see Sections 4.4.3 and 9.4.3.2) If the samples are not representative, secondary sampling must be repeated. Samples at additional sampling locations must be taken to obtain a closer grid. 

The number of samples which must be taken in an area which is to be investigated to record the properties of interest without loss of information is still an interesting question. In other words how many samples are necessary for representative assessment of the state of pollution A method for the determination of the minimum number of samples is suggested below. For that purpose the advantages of the kriging method are used. 

Fit all the surfaces using kriging and validate the model. Once all the variables (surfaces) have been estimated by kriging, it is important validate the metamodel, i.e. using cross validation that allows us to asses the accuracy of the model without extra sampling [2], A kriging model can be considered correct if all the errors in cross validation are inside the interval [-3,+3] standard errors. 

Figure 2. Map of arsenic distributions in groundwater from Bangladesh. The map shows a smoothed distribution determined by disjunctive kriging on 3207 samples of groundwater (<150 m depth, Gaus et al, 2001). Insets are frequency histograms of the concentrations of arsenic (pg F ) in groundwater from three selected study areas analysed in more detail...

Characterization of the variogram from actual observations permits the estimation of concentrations at points on the site which were not sampled by application of generalized least-squares type statistical regression algorithms. This type of estimation has come to be referred to as "kriging"(2). Thus, once the similarity of observations with distance has been described in terms of the variogram, contamination across the site can be estimated and... 

Journel(7-9) provides a means for generating the desired mappings in situations such as the example site which are free from any undue assumptions regarding the distributional form of concentrations within blocks. This approach simply transforms the data into the sample cumulative probability distribution. This nonparametric geostatistical approach is sometimes called "indicator kriging."... 

The concentrations in the virtual aquifer calculated in the previous example (Fig. 9.4) will be investigated by different regular arrangements of observation wells. The measured concentrations within the sampling wells are interpolated by SURFER using the Kriging-method with default parameters. The resulting interpolated concentration pattern is compared to the real concentrations of the virtual aquifer. This comparison is impossible in natural aquifers since the real concentrations are not known. 

FIGURE 2 Location map (distance units in feet) of 180 samples, histogram of lead concentration, variogram of the normal scores transform (hence the sill value of 1.0), a map of kriging estimates on a 100-ft grid and an SGS realization over the same domain. 

The Bayesian approach is beginning to come into favor since the importance of site characterization and sampling strategies has increased. This approach is similar to the geostatistical approach (kriging) in its use to develop statistically realistic descriptions of sites to aid in initial data collection and site exploration when there is a scarcity of hard data (6, 72). The Bayesian approach may provide the general framework to update parameter uncertainty when additional data are available, and to evaluate its effect on estimation or decision (13). 

Simple Kriging Consider the problem of interpolating a random function. That is, given M sample values ipi,i = 1,..., M, at each of M spatial points Xj, estimate the value of the function, 99 at the point x. 

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