Kriging estimation

Figure 6. Isoarea map of kriging estimates of lead concentrations (pg/g) in soil (5).

The projection theory indicates that a projection estimator of the unknown I(2c z), such as a kriging estimator, is also a projection estimator (kriging) of its conditional expectation Fx(z (N)), (7) Consider then the indicator kriging estimator ... 

The kriging process (11) is repeated as many times as there are different cut-offs (z ) retained to discretize the interval of variability of the concentration P(x). The different kriging estimates i(x zit) are then pieced together to provide an estimate of the conditional cdf Fx(z (N)) ... 

Note that if the kriging weights Xi(z) are data values-independent, the indicator estimates (11) are not hence, the final estimate of the conditional cdf Fx(z (N)) is data values-dependent. For each cut-off Z], the kriging estimator (11) requires a different indicator covariance model ... 

Model Fitting When experimental variograms reveal the structure and distribution pattern in an ore body then for any further estimation, it is necessary to fit a mathematical model to experimental variograms, which are called theoretical variograms. These mathematical models will be used in Kriging estimation. Several predefined models (Linear, Spherical, Gaussian, etc.). 

The kriging system of linear equations is derived so that their solution gives kriging weights such that the kriging estimator is a "best linear unbiased estimator." The estimator is linear because the estimator is a weighted sum. It is unbiased because the system... 

It is assumed that the intrinsic hypothesis holds within the limited neighborhood that is used in calculating the kriging estimates. That is, eight measurements will be used by the kriging estimator and within the limited neighborhood (with a radius of approximately 1000 feet) that these measurements occur it is assumed that there is no drift or systematic trend. 

Kriging Estimates and Standard Deviations, performed on the natural logarithms of the... 

The kriging estimates of the mean concentration (ppm lead) over a 250 foot by 250 foot block and the kriging standard deviation for each block are shown in Figures 9 through 14. At RSR and DMC the estimated block means are shown for blocks whose multiplicative kriging standard deviation was less than 2. (Since the measurements are transformed using the natural logarithm, the standard deviations... 

Figure 9. Kriging estimates of the mean lead concentration for 250 ft by 250 ft blocks at RSR.

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

Fig. 4-6. Kriging estimation of nickel near a metallurgical factory [EINAX and SOLDT, 1995], (a) Representation of the isolines, (b) three-dimensional plot...

The mean of these quotients Az2rei should be situated near zero and their standard deviation near unity, i.e. it should be a standardized normal distribution if the kriging estimation is distortion-free. 

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. 

Fig. 9-22 a and b). They can be found in the immediate vicinity of the emission source. In a small area (see Tab. 9-10) the concentrations of cadmium exceed the index value of the category B of the Dutch hst [1988] (Tab. 9-11) which implies a need for more detailed investigation. By means of the kriging estimation the critical polluted area can be well assessed. The locations of the concentration maxima enable conclusions to be drawn about the source of emission. The blast furnace works in the north of the Maxhiitte proves to be the main source of iron and cadmium. This also corresponds to the impact of particulate emissions in the territory investigated (see also Fig. 9-19). 

Fig. 9-22. Kriging estimation for some selected elements (isolines in ig g ) (a) iron, (b) cadmium, (c) lead, (d) calcium, (e) potassium...

The indicator kriging estimator, I (x z), appears as a linear combination of the indicator data i(xa z), in the neighborhood, n," surrounding the point for which a prediction is to be made. This can be symbolized as ... 

Kiiging is an exact estimator, that is, the kriging estimator at a data location will be the data value. The minimized error variance or kriging variance can be calculated for all estimated locations ... 

Kriging estimates are smooth. The kriging variance is a quantitative measme of the smoothness of the kriging estimates. There is no smoothing when kriging at a data location, a =0. There is complete smoothness when kriging with data far from the location being estimated the es-... 

Draw a random residual (u) that follows a normal distribution with mean of 0.0 and a variance of (u). Add the kriging estimate and residual to get a simulated value. The independent residual (u) is drawn with classical Monte Carlo simnlatioa... 

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. 

Ordinary kriging is a best linear unbiased estimate of the parameter. It is linear because its estimates are weighted linear combinations of the available data, unbiased because it tries to have the mean error equal to zero, and best because it aims at minimizing the variance of the error (7). The kriging estimator, Z is described as. 

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