Te YTj4eMy22

j where the basis function (p* is chosen in such a way that it equals 1 at the particular node j and 0 at all other nodes on the element (e). Other interpolation methods for approximating the transmissivity distribution include simple polynomial approximation, cubic spline, and kriging.

Stochastic Method

In this approach, the unknown parameter is treated as a random field, characterized by its first two moments, the mean (or drift) and the covariance function. A common approach is to assume that the logarithm of the hydraulic conductivity, Y = log K, is normally distributed (Freeze, 1975; Hoeksema, 1985a). Also, the random field is represented by a constant mean and an isotropic, exponential covariance (Dagan, 1985; Hoeksema and Kitanidis, 1985b; Wagner and Gorelick, 1989):

where a\ = log hydraulic conductivity variance lY = log hydraulic conductivity correlation scale ¿¡J = distance between points xt and Xj

The hydraulic conductivity can thus be estimated by identifying the three statistical parameters pF, rrj-, and ly. In this approach, overparameterization is generally avoided, and the inverse solution obtained by the maximum-likelihood estimate and cokriging is highly stable.

In addition to the traditional approaches for parameterization mentioned above, Sun et al. (1995) suggested a geological parameterization method in which the unknown parameter (hydraulic conductivity) is directly related to the geological materials, and the geological structure of the aquifer is determined by the geostatis-tical method of kriging.

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