## Parameter Uncertainty Parameter Structure And Optimum Parameter Dimension

Parameter identification in a distributed-parameter system should, in principle, include the determination of both the parameter structure and its value. If zonation is used to parameterize the unknown parameters, the zonation pattern (parameter structure) is represented by the number and shape of zones. On the other hand, if the finite-element method is used for parameterization, parameter structure is represented by the number and location of nodal values of parameters.

Identifying parameter structure is much more difficult than identifying parameter values for a given parameter structure. In the past three decades, only a few studies have contributed to this topic. The question of how to determine an appropriate zonation pattern was first considered by Emsellem and de Marsily (1971), who suggested that the number of zones be gradually increased until model fit no longer improved. This approach ignores the reliability of the estimated parameters. Yeh and Yoon (1976) were the first to consider both the error in model fitting and the error associated with parameter uncertainty in determining zonation pattern; to determine if a particular zone should be subdivided into smaller zones, they used the variance of the estimation error. Sun and Yeh (1985) proposed a systematic approach that can automatically identify the optimal pattern of parameter structure and its corresponding parameter values by solving a combinatorial optimization

problem. They clearly pointed out that the identified parameter values vary with the parameter structure. As a consequence, if the parameter structure is incorrect, the identified parameter values will also be incorrect. In Carrera and Neuman (1986), the dimension of parameterization is determined by Akaike's information criteria (Akaike, 1972); these criteria can also be used to compare different zonation patterns. Bellout (1992) considered the stability of pattern identification from a mathematical analysis. Recently, Zheng and Wang (1996) used the tabu search (TS) method to find the optimal zonation structure for one-dimensional problems. Eppstein and Doupherty (1996) presented an extended Kalman filter for simultaneously estimating transmissivity values and zonation pattern. A general formulation of the inverse problem that incorporates the identification of parameter structure and its parameter values is given in Sun et al. (1998). To estimate the parameter structure, some authors have attempted to incorporate directly into the solution of the inverse problems the geological structure information obtained from well-logs and seismic measurements (Rubin et al., 1992, Sun et al., 1995; Hyndman and Gorelick, 1996; Koltermann and Gorelick, 1996).

Shah et al. (1978) showed the relationship between the optimal dimension of parameterization and observations in considerable depth. The necessity to limit the dimension of parameterization has been further studied by Yeh and Yoon (1981), Yeh, et al. (1983), and Kitanidis and Vomvoris (1983). The dimension of parameterization is directly related to the quantity and quality of data (observations). In practice, the number of observations is limited and observations are corrupted with noise. Without controlling parameter dimension, instability in the inverse solution often results (Yakowitz and Duckstein, 1980). If instability occurs, parameter values will become unreasonably small (sometimes negative, which is physically impossible) and/or large, if parameter values are not properly constrained. In the constrained minimization, instability is characterized by the fact that during the inverse solution process parameter values are bouncing back and forth between the upper and lower bounds. Reduction of parameter dimension can make the inverse solution stable. As the number of zones (in the zonation case) is increased, the modeling error (least squares) decreases while the parameter uncertainty error at some point will start to increase (Shah et al., 1978; Yeh and Yoon, 1981). A trade-off of these two types of errors can then be made, from which an optimum parameter dimension can be determined. A standard procedure is to gradually increase the parameter dimension, starting from the lowest dimension, i.e., the homogeneous case, and calculate the two types of errors for each parameterization. The error in parameter uncertainty can be represented by a norm of the covariance matrix of the estimated parameters (Yeh and Yoon, 1976; Shah et al, 1978).

All approximation of the covariance matrix of the estimated parameters in nonlinear regression can be represented by the following form (Bard, 1974; Yeh and Yoon, 1976, 1981; Shah et al, 1978; Yeh, 1986):

where J(T) = least-squares error

M = number of observations L = parameter dimension A = [J£JD]

JD = Jacobian matrix of h with respect to T

A norm of the covariance matrix has been used to represent the error in parameter uncertainty. Norms, such as trace, spectral radius (maximum eigenvalue), and determinant have been used in the literature. Equation (25) also assumes homoscedas-ticity and uncorrelated errors. This assumption is generally not satisfied and the actual covariance may be much higher than given by Eq. (25).

The covariance matrix of the estimated parameters also provides information regarding the reliability of each of the estimated parameters. A well-estimtated parameter is generally characterized by a small variance as compared to an insensitive parameter that is associated with a large variance. By definition, the correlation matrix of the estimated parameter is

where Cy's are elements of the covariance matrix of the estimated parameters. The more sensitive the parameter, the closer and quicker the parameter will converge. A correlation analysis of the estimated parameters would indicate the degree of interdependence among the parameters with respect to the objective function. Correlation of parameters is called the collinearity problem. Such problems can cause slow rate of convergence in minimization and in most cases result in nonoptimal parameter estimates. A more rigorous treatment of the collinearity problem is to use the more sophisticated statistical techniques, such as ridge regression (Cooley, 1977) and the method of principal components.

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