The number of observations is finite and limited, whereas the spatial domain is continuous. For an inhomogeneous aquifer, the dimension of, for example, the transmissivity is theoretically infinite. In practice, the infinite parameter dimension must be reduced to a finite dimensional form. The reduction of the number of parameters from the infinite dimension to a finite dimensional form is called parameterization (Yeh and Yoon, 1976, 1981; Yeh, 1986; Sun, 1994a). Parameterization can be achieved by either a deterministic method or by a stochastic model. In general, parameterization can be achieved by the following methods.
In this approach, the flow region of the aquifer is divided into a number of zones, and a constant parameter value is used to characterize the aquifer property in each zone. The unknown transmissivity function is then represented by a number of constants, which is equal to the number of zones. Here, we mention the work of Coats et al. (1970), Emsellem and de Marsily (1971), Yeh and Yoon (1976), and Cooley (1977, 1979).
In principle, the zonation pattern and its corresponding parameter values should be determined simultaneously (Sun and Yeh, 1985; Sun et al., 1998).
If, for example, finite elements are used as the interpolation method, the unknown parameter distribution in the flow region is discretized into a number of elements connected by a number of nodes. Each node is associated with a chosen local basis function. The unknown transmissivity distribution is then approximated by a linear combination of the basis functions, where the parameter dimension is equal to the number of unknown nodal transmissivity values (DiStefano and Rath, 1975; Yoon and Yeh, 1976; Yeh and Yoon, 1981):
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