## 1. Introduction

[2] Over the last 4 decades, physically based simulation models have been used to analyze groundwater flow, contaminant transport, seawater intrusion, and biodegradation in aquifers. Newly developed computational techniques have allowed hydrogeologists to simulate more and more complex physical, chemical, and biological phenomena occurring in groundwater. However, the lithology of an aquifer is usually very complex and unknown, and the available geological data for model calibration very limited in both quantity and quality. As a result, the reliability of model predictions is often questionable because of the uncertainties in identified parameter structure and parameter values.

[3] It may be possible to fit the model output well with existing data by increasing model complexity; however, this may lead to an incorrect parameter structure. In fact, a small fitting residual does not promise the reliability of model prediction, especially when the aquifer is heterogeneous. If model complexity has exceeded the capability of the data to support the identification, overparameterization occurs [*Yeh and Yoon*, 1981; *Kitanidis and Vomvoris*, 1983; *Yeh*, 1986; *Sun*, 1994]. To avoid overparameterization *Yeh and Yoon* [1981] used a norm of the covariance matrix of the estimated parameter to represent parameter uncertainty [*Yeh and Yoon*, 1976]. Model selection criteria based on statistics, such as the AIC [*Akaike*, 1974], BIC [*Schwarz*, 1978], HIC [*Hannan*, 1980], and KIC [*Kashyap*, 1982] also can be used to determine model complexity. However, the reliability of model application remains uncertain depending on data sufficiency. Although the simultaneous identification of the parameter dimension, parameter pattern, and the corresponding parameter values has been reported to reduce model uncertainties by a number of researchers [*Sun and Yeh*, 1985; *Eppstein and Dougherty*, 1996; *Zheng and Wang*, 1996; *Heredia et al.*, 2000; *Tsai et al.*, 2003a, 2003b], the identified parameter pattern and its corresponding parameter values from fitting observations alone may not be reliable for model application.

[4] In the stochastic framework, parameter identification is considered an information transformation procedure. In particular, it is treated as the shift of information from the measured data to the unknown parameter with the help of a model and prior information. Markov Chain Monte Carlo (MCMC) random sampling methods may be used to find the posterior distribution numerically for nonlinear models [*Mosegaard and Sambridge*, 2002; *Lu and Zhang*, 2003; *Qian et al.*, 2003]. However, the multivariate probability distributions of relevant parameters are needed before implementing the Monte Carlo simulation. In order to cover the entire statistical ensemble, a very large number of realizations is necessary. Additionally, finer discretizations in space/time also are required for accuracy. Even though this approach is computationally demanding, the reliability of the estimated parameters for model application is not guaranteed.

[5] Prediction from a single conceptual model may lead to statistical bias and underestimation of model uncertainty. To address this, *Beven and Binley* [1992] proposed the generalized likelihood uncertainty estimation (GLUE) method for the identification of several alternative parameter structures and the postulation of a prior probabilistic model of parameter uncertainty. The approach relies on Monte Carlo simulation to generate the posterior probability of hydrologic behaviors. Alternative parameter structures are discarded on the basis of a subjectively defined rejection criterion. *Gaganis and Smith* [2001] proposed a Bayesian approach to quantify the effect of model error on model predictions. This approach, similar to the GLUE method, depends on Monte Carlo simulation without model calibration or subjective model selection criteria. *Hoeting et al.* [1999] provided a tutorial on Bayesian Model Averaging (BMA). *Neuman* [2003] proposed a maximum likelihood Bayesian model averaging (MLBMA) approach to assess the uncertainty of model prediction. *Ye et al.* [2008] used the model averaging approach, which considered several covariance structures, for a real case study and concluded that the overall model-averaged performance of KIC is better than that of other criteria.

[6] Without knowing the real structure of an aquifer, the assessment of data sufficiency becomes extremely difficult. To identify a parameter structure, there must be sufficient information contained in the observation data. A more complex parameter structure requires more data for identification. An optimal experimental design problem usually combines a simulation model with a decision-making model in its solution procedure. The traditional experimental design approach assumes that the parameter structure is known [*Hsu and Yeh*, 1989; *Loaiciga*, 1989; *Nishikawa and Yeh*, 1989; *Mayer et al.*, 2002]. Although some authors have considered the reliability of the design solution when there is uncertainty in the simulation model by conducting a Monte Carlo analysis [*Meyer et al.*, 1994; *Wagner*, 1995; *Mayer et al.*, 2002; *Chang et al.*, 2005], the method can only handle a very limited number of realizations. When the structure of the identified parameter is unknown, it is not possible to judge data sufficiency from a design because identifying a more complex structure requires more data. Hence, the optimal experimental design becomes meaningless, while the robustness of the design cannot be assured.

[7] From the above discussion, it is evident that the traditional approaches to model calibration and parameter identification cannot provide adequate answers to the following questions: (1) How do we determine an appropriate structure for the identified parameter without knowing its true structure? (2) How do we design a field experiment to collect sufficient data for the identification without knowing the complexity of the identified parameter? (3) How do we estimate the reliability of the identified parameter without knowing its structure error? In other words, the classical theory of inverse problems, the classical statistical methods for uncertainty estimation, and traditional experimental design concepts cannot be used for cases where significant model errors exist.

[8] As pointed out by *Box and Luceno* [1997], “all models are wrong, but some are useful.” On the basis of this premise, *Sun* [2005] and *Sun and Yeh* [2007a, 2007b] proposed an objective-oriented modeling approach, which seeks to identify a representative parameter with the simplest structure and assure its reliability for predetermined model applications. When the reliability of model applications is incorporated into the parameter identification procedure, the problem of determining the complexity of parameter structure and the problem of judging the sufficiency of data become solvable. Employing the concept of structure identifiability defined by *Sun* [2005], a simpler parameter structure can be used to replace a more complex parameter structure provided that the information contained in the observation data can overcome the impact of observation and structure errors. For a given structure, the worst-case parameter (WCP) is the one that produces the maximum deviation in model applications when its structure is simplified. *Sun* [2005] proved that if the WCP is identifiable then all other parameters with the same structures or simplified structures also can be identified. As a result, if an experimental design is sufficient for identifying the WCP, then the design must be robust. The data sufficiency problem thus becomes meaningful for the given objectives of model application.

[9] In this paper, we construct an objective-oriented model for conjunctive-use planning of surface water and groundwater for the Warren groundwater basin in southern California. Section 2 briefly reviews the definition and derivation of the generalized inverse problem (GIP). Section 3 introduces a general background of the groundwater basin and formulates the model application objective of an optimal conjunctive-use model. Section 4 formulates the model fitting residual and structure error for the groundwater basin, solves the GIP, and analyzes the inverse solutions. In section 5, we solve the GIP to judge the sufficiency of the existing data. When the existing data are insufficient for constructing a reliable model for the specified model application objectives, we design a robust and cost-effective field experiment for collecting the necessary data to make the calibrated model reliable for the stipulated model application objectives. Section 6 draws conclusions.