## Ordinary kriging

Kriging refers to a family of least-square linear regression algorithms that attempt to predict values of a variable at locations where data are not available based on the spatial pattern of the available data. The description of kriging theory and its application are given in detail by [5]. A semivariogram, y(h) represents the spatial variability in the data and is defined as Eq. (1):

1 N(h) 2

Y(h) = E [Z(xj + h) — Z(x )]2, (1)

2N(h) 7=1

where N(h) is the number of pairs of points separated by lag distance h, Z(x) and Z(x+h) are random values at locations x and x+h.

In this study, the exponential model have been used to fit the sample semivariograms, this model parameterizes the semivariogram in the following way:

Y(h) = C0 + C1 [1 — exp(-h / a)], (2)

where C0, Cj, a are called nugget, sill, and range respectively.

The objective of ordinary kriging procedure is to estimate data values at unsampled locations x0 using information available elsewhere in the domain (x1, x2,……………………………………………………………………………………………… , xn) . This can be carried out

by expressing Z(x0)as a linear combination of the data Z(x1), Z(x2),…………………… Z(xn) :

Z(x0) = EX7Z (x0).

The optimal weights “ A7 ” are calculated assuming that the estimation Z(x0) by Z(x0) is

unbiased, that is, the expected value of the estimates is the same as that of the known data. The

n

condition needed for unbiased estimator is 2 A = 1.

7=1 7