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The KRIGE2D Procedure |

As an illustration, consider a hypothetical example in which an organic solvent leaks from an industrial site and spreads over a large area. Assume the solvent is absorbed and immobilized into the subsoil above any ground-water level, so you can ignore any time dependence.

For you to find the areal extent and the concentration values of the solvent, measurements are required. Although the problem is inherently three-dimensional, if you measure total concentration in a column of soil or take a depth-averaged concentration, it can be handled reasonably well with two-dimensional techniques.

You usually assume that measured concentrations are higher closer to the source and decrease at larger distances from the source. On top of this smooth variation, there are small-scale variations in the measured concentrations, due perhaps to the inherent variability of soil properties.

You also tend to suspect that measurements made close together yield similar concentration values, while measurements made far apart can have very different values.

These physically reasonable qualitative statements have no explicit probabilistic content, and there are a number of numerical smoothing techniques, such as inverse distance weighting and splines, that make use of large-scale variations and "close distance-close value" characteristics of spatial data to interpolate the measured concentrations for contouring purposes.

While characteristics (i) and (iii) are handled by such smoothing methods, characteristic (ii), the small-scale residual variation in the concentration field, is not accounted for.

There may be situations, due to the use of the prediction map or due to the relative magnitude of the irregular fluctuations, where you cannot ignore these small-scale irregular fluctuations. In other words, the smoothed or estimated values of the concentration field alone are not a sufficient characterization; you also need the possible spread around these contoured values.

The key assumption in applying the SRF formalism is that the measurements come from a single realization of the SRF. However, in most geostatistical applications, the focus is on a single, unique realization. This is unlike most other situations in stochastic modeling in which there will be future experiments or observational activities (at least conceptually) under similar circumstances. This renders many traditional ideas of statistical inference ambiguous and somewhat counterintuitive.

There are additional logical and methodological problems in applying a stochastic model to a unique but partly unknown natural process; refer to the introduction in Matheron (1971) and Cressie (1993, section 2.3). These difficulties have resulted in attempts to frame the estimation problem in a completely deterministic way (Isaaks and Srivastava 1988; Journel 1985).

Additional problems with kriging, and with spatial estimation methods in general, are related to the necessary assumption of ergodicity of the spatial process. This assumption is required to estimate the covariance or semivariogram from sample data. Details are provided in Cressie (1993, pp. 52 -58).

Despite these difficulties, ordinary kriging remains a popular and widely used tool in modeling spatial data, especially in generating surface plots and contour maps. An abbreviated derivation of the OK estimator for point estimation and the associated standard error is discussed in the following section. Full details are given in Journel and Huijbregts (1978), Christakos (1992), and Cressie (1993).

Here, is the fixed, unknown mean of the process, and is a zero mean SRF representing the variation around the mean.

In most practical applications, an additional assumption is
required in order to estimate the covariance *C*_{z} of
the *Z*(*r*) process. This assumption is
second-order stationarity:

This requirement can be relaxed slightly when you are using the semivariogram instead of the covariance. In this case, second-order stationarity is required of the differences rather than :

By performing local kriging, the spatial processes
represented by the previous equation for Z(*r*) are more general than
they appear. In local kriging, at an unsampled location
*r _{0}*, a separate model is fit using only data
in a neighborhood of

Given the *N* measurements *Z*(*r _{1}*), ... ,

Linearity requires the following form for :

Applying the unbiasedness condition to the preceding equation yields

Finally, the third condition requires a constrained
linear optimization involving and
a Lagrange parameter 2*m*. This constrained
linear optimization can be expressed in terms of the
function given by

Define the *N* ×1 column vector by

The optimization is performed by solving

The resulting matrix equation can be expressed in
terms of either the covariance *C*_{z}(*r*) or
semivariogram .In terms of the covariance, the preceding equation results
in the following
matrix equation:

The solution to the previous matrix equation is

Using this solution for and *m*, the
ordinary kriging estimate at *r _{0}* is

with associated prediction error

where ** c_{0}** is

These formulas are used in the best linear unbiased prediction (BLUP) of random variables (Robinson 1991). Further details are provided in Cressie (1993, pp. 119 -123).

Because of possible numeric problems when solving
the previous matrix equation,
Duetsch and Journel (1992)
suggest replacing the last
row and column of 1s in the preceding matrix
** C** by

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