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Producing probability maps to assess risk of exceeding critical threshold value of soil EC using geostatistical approach  Suresh Tripathi Technical Director, Geosun Pty Ltd P O Box 207, Boronia, VIC, 3155, Australia Email: s.tripathi@geosun.net Assumptions and Geostatistical Variogram Model: Geostatistical models are handled with the help of a probabilistic framework which is traditionally presented in terms of continuous random processes. In this paper, I consider a point s in d-dimensional Euclidean space Rd and suppose that a random variable {z(s):sÎD} at spatial location s is a realization of a stochastic process {Z(s):sÎD}. Geostatistical approach requires the stationary assumptions as described below. The process is said to be stationary in the mean if the following condition met as where h (lag) is a vector in Rd . If the covariance cov (Z(s), Z(s+h)) is finite and depends upon only h, then the function C(h), called the covariogram, can be defined by If C(h) is a function of only ||h|| then C(h) is called isotropic. If the variance of the difference between Z(s) and Z(s+h) is finite, and depends only on h then the variogram 2g(h) can be defined as The process {Z(s)} is said to be intrinsic stationary or, alternatively, to satisfy the intrinsic hypothesis if it satisfies equations (1.1) and (1.3). In this case the variogram can be written as the expected squared difference between random variables Z(s) and Z(s+h) as below. 
Note that g(0) = 0, g³0 and g(-h) = g(h). If var [Z(s)] = C(0) (finite), and second-order stationarity is satisfied, then the variogram can be written in terms of the covariogram as 
which implies that Z(s) is also intrinsically stationary. In this case the variogram is bounded and C(0) is known as the sill of the variogram. Indicator approach requires data to be converted into binary based on the certain threshold value as Zk below. 
Experimental indicator variogram can be calculated from the data as given below.  where 2N(h) is total number of pairs. Spherical varogram model was used to fit experimental indicator variogram values obtained from the data and is given below. 
where C(0), Cs and a are sill, nugget variance and range (spatially correlated distance) respectively. Geostatistical Model: In the stationarity framework, the regionalized variable Z(s) can be modelled as the sum of a deterministic part , a mean function called the drift, and a zero-mean stochastic process d(s) called the residual random part. The mean function can be thought of as the large scale variation representing the variable's global trend over D. The zero-mean stochastic process can be thought of as the small scale variation representing the spatial dependence after the trend is removed. This type of model is known as a stochastic model or a probabilistic model (Cressie 1991) and can be expressed as equation (1.6) 
If m(s) varies slightly over a region, or the expected value of the drift is not constant but varies over the region D then the drift may possibly be expressed as a linear combination of suitable base functions and we can write 
where s is a location index,bj-1 , j = 1,...,p+1 are called the unknown drift coefficients, and fj-1 , j=1,...,p+1 are known base functions at s. The model expressed in equation (1.7) is known as the universal kriging model in geostatistics. A special case of the universal kriging model is the ordinary kriging model, when m (s) is unknown but constant and the simple kriging model when the mean function is known. Indicator Kriging At any unsample location S0 , the ordinary kriging estimator for threshold value used to interpolate at unknown location is defined as below.  Where the weight  are obtained by solving the indicator kriging system of (n+1) equations (Goovaerts, 1995).
whereY (Zk) is a Lagrange parameter.
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