Development of an objective-oriented groundwater model for conjunctive-use planning of surface water and groundwater

Authors


Abstract

[1] 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. The goal of conjunctive-use planning is to decrease high-nitrate concentration while maintaining groundwater levels at desired elevations and meeting water demand. We formulate a management problem that minimizes the total cost over the proper choices of the time-varying pumping and recharge rates at prespecified wells and surface ponds. To make the solution of the management problem reliable, we must have an accurate simulation model to predict groundwater level and nitrate concentration distributions under different management alternatives. The objective-oriented model construction approach seeks a representative parameter that has the simplest structure and requires the minimum data for identification but can produce reliable results for a given model application. With the data from the Warren groundwater basin, we show how to incorporate management objectives into the construction of an objective-oriented model, identify the parameter structure and its corresponding parameter values, solve the generalized inverse problem effectively by finding the worst-case parameter (WCP), evaluate the sufficiency of existing data, and find a robust experiment design when the existing data are insufficient. Results of this case study show that the presented methodology is useful in practice because (1) data sufficiency can be judged before conducting actual field experiments and (2) the identified WCP drastically reduces the computation time for constructing an objective-oriented model.

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.

2. Objective-Oriented Method for Model Construction

[10] In this section we briefly review the definitions of a parameterization representation, parameter structure, and structure error. We also give a concise derivation of the generalized inverse problem and the algorithm for solving it. Detailed discussions on the methodology of objective-oriented model construction are given by Sun and Yeh [2007a, 2007b].

2.1. Parameterization and Parameterization Representation

[11] The degree of freedom of a true distributed parameter θ(x) is infinite, and parameterization is a way to approximate a true distributed parameter by a function with a much lower degree of freedom. A general form of parameterization is

equation image

where equation image(x) is a parameterization representation (PR); the integer m is the dimension of parameterization (or the complexity level of parameter); {θj} (j = 1, 2, .., m) is a set of coefficients; and {ϕj (x, v)} is a set of basis functions with a vector set of shape parameters v. With different basis functions, equation (1) can represent continuous or discontinuous functions. The shape parameters determine the pattern of parameterization. We use a combined notation (S, equation image) to denote a PR of a distributed parameter, with S representing a parameter structure determined by m basis functions. Its complexity is determined by m and v. Vector equation image = (θ1, θ2, …, θm)T and represents the parameter values associated with the parameter structure. To represent a complex parameter structure, we can either increase the dimension of parameterization, m, or increase the number of shape parameters. When the true distributed parameter θ(x) is unknown, we prefer to increase the dimension of parameterization (m) because identifying the shape parameters is more difficult when basis functions are nonlinear with respect to the shape parameters.

[12] In the deterministic framework, a parameter structure, S, can be a simple zonal structure (Figure 1a), a complex zonal structure (Figure 1b), a continuous structure (Figure 1c) or a fracture-like structure (Figure 1d). In the stochastic framework, a parameter structure, S, can be a lognormally distributed structure (Figure 1e) or a continuous structure with a random perturbation (Figure 1f). A true distributed parameter, θ(x), may have different PRs when it is approximated by different parameter structures because θ(x) is unknown. Examples of various PRs have been shown by Sun and Yeh [2007b].

Figure 1.

Different types of parameter structures.

2.2. Generalized Inverse Problem and Structure Error

2.2.1. Generalized Inverse Problem

[13] The traditional inverse problem seeks to minimize the fitting error over the proper choices of parameter structure and its associated parameter values

equation image

subject to

equation image

where RE is the fitting residual; uDobs are observations of the state variable based on design D; (SB, equation imageB) is a PR with parameter structure SB and its associated parameter values equation imageB; uD (SB, equation imageB) are the corresponding model outputs; λ is the regularization factor; equation image0 is the prior information on parameter values; and ∥·∥D and ∥·∥P are the norms defined in the observation space and parameter space, respectively. In objective-oriented groundwater modeling, an innovation incorporates model application in the inverse solution process. Sun et al. [1998] proposed a generalized inverse problem (GIP) that adds the accuracy requirement of the model application as a constraint in parameter identification. If we use the prior information of upper and lower bounds of the parameter values as constraints to replace the regularization term in equation (2), the GIP can be formulated as

equation image

subject to

equation image
equation image

where JE is a vector of model applications, corresponding to a set of objectives E; JE(θ(x)) and JE(SB, equation imageB) are the objectives of model applications based on the true distributed parameter, θ(x) and PR (SB, equation imageB); ∥·∥E is a norm defined in the model application space; ɛ is the accuracy requirement of the model application; and equation imagemax and equation imagemin are the upper and lower bounds of parameter values.

[14] Because θ(x) is unknown, JE (θ(x)) cannot be calculated directly in equation (3b). But we can calculate the distance between any two known PRs, (SA, equation imageA) and (SB, equation imageB), in the model application space. If SA is close to the structure of the true parameter, from the conservation point of view we can use the following condition to replace the constraint (3b)

equation image

subject to

equation image

where SE(SA, SB) is called the structure error (or model application error in this paper), measured in the model application space when using parameter structure SB to replace parameter structure SA. From the definition, the following simple propositions of the structure error can be concluded [see Sun and Yeh, 2007a]: (1) SE(SA, SB) ≠ SE(SB, SA). (2) When SB is a simplification of SA, then SE(SB, SA) = 0. (3) The homogenization error SE(SA, S1), i.e., the structure error of using the homogeneous structure S1 to replace SA, is the upper bound of all structure errors {SE(SA, SB)}, where SB is any structure used to replace SA.

2.2.2. Worst Case Parameter

[15] The maximum-minimum problem (equation (4)) is extremely difficult to solve. To simplify the solution, we apply the concept of the worst-case parameter (WCP) proposed by Sun [2005] to convert the maximum-minimum problem to a minimization problem. The WCP is defined as

equation image

subject to

equation image

A PR (SB, equation imageA,B) is called a projection of (SA, equation imageA) onto the structure SB when

equation image

where Θ(equation imageA) is a set consisting of 2m vertices (if the dimension of SA is m) of Θ(SA). The SE is obtained together with the WCP in equation (5) as the maximum value of the objective function. On the basis of the theorem presented by Sun [2005], the WCP must be a vertex of the admissible region Θ(SA), and can be obtained by searching 2m vertices rather than the entire admissible region. If the WCP, equation imageA, can be efficiently found, the calculation of SE can be reduced to the solution of the following minimization problem:

equation image

A detailed explanation of formulating the SE and constructing the WCP are given by Sun [2005] and Sun and Yeh [2007a].

2.3. Algorithm for Solving GIP

[16] Once we have an efficient method to find the WCP, we use the stepwise regression method presented by Sun et al. [1998] to solve the GIP. A sequence of parameter structures is formed as

equation image

In equation (8), S1 is a homogeneous structure. The sequence of the identified parameter structures starts from a homogeneous structure, and then increases the parameter structure dimension one by one until a convergence criterion is satisfied. This downscaling procedure avoids overparameterization.

[17] Two optimization problems are involved in solving the GIP. At each parameter structure complexity level, besides calculating the structure error, SEm, the fitting residual, REm, also is calculated. After solving the REm (equation (3a)) and the SEm (equation (4)) for each parameter structure dimension (m), the following four cases emerge: (1) If REm > 2η and SEm > ɛ, increase the parameter structure dimension from m to m + 1. (2) If REm < 2η and SEm < ɛ, stop and use m as the identified model. (3) If REm > 2η and SEm < ɛ, stop and use m as the identified model. (4) If REm < 2η and SEm > ɛ, additional data must be collected.

[18] The symbol η is the upper bound of the observation error as a norm measured in the observation space, and ɛ is the accuracy requirement of the model application as a norm measured in the objective space. Case 1 implies that the existing data still have the potential of providing more information, but the structure error is not satisfied. Hence, we increase the parameter structure dimension. When structure complexity increases, the values of both SEm and REm will decrease. In cases 2 and 3, the identified model already satisfies the accuracy requirement of the given model application; we can stop the GIP solution and use PR (Sm, equation imagem) as the identified parameter. In case 4, the information content in the existing data is insufficient to identify a reliable model, and thus we must collect additional data. Figure 2 shows the overall steps for constructing an objective-oriented groundwater model.

Figure 2.

Flowchart for constructing an objective-oriented groundwater model.

3. An Optimal Conjunctive-Use Model for the Warren Groundwater Basin

3.1. Background of the Warren Groundwater Basin

[19] The Warren subbasin is located 100 miles east of Los Angeles in the southwestern part of the Mojave Desert. The areal extent of the subbasin is 19 square miles, bounded by both the San Bernardino Mountains and Pinto Mountain Fault on the northwest and the Little San Bernardino Mountains on the southwest. Its water-bearing deposits form the Warren groundwater basin, which has an area of 5.5 square miles (Figure 3). Faults separate the groundwater basin into five hydrogeologic units: the west, midwest, mideast, east and northeast. Using information obtained from lithologic and downhole geophysical logs, the alluvial deposits are divided into four aquifers: the upper, middle, and lower alluvial aquifers, and the deep aquifer that consists of essentially sedimentary deposits [Nishikawa et al., 2003].

Figure 3.

Model grid with pumping wells (triangles) and recharge ponds (squares) for the groundwater flow and transport model of the Warren subbasin, California. Groundwater level constraints locations (crosses, H1–H3) and nitrate constraints locations (circles, C1–C4).

[20] Groundwater is the sole source of water supply for the Town of Yucca Valley. Because of overpumping, water level in the groundwater basin declined more than 300 feet from the late 1940s to 1994. In order to prevent further groundwater level decline, the local water district, the Hi-Desert Water District (HDWD), executed an artificial recharge program in February 1995 that used imported surface water to replenish the groundwater by surface recharge ponds. Because of the lack of storage facilities in the groundwater basin, it was also desirable to store the imported water in the groundwater basin for future use. The artificial recharge program resulted in groundwater level recoveries of as much as 250 ft. However, nitrate (NO3) concentrations in certain parts of the basin, particularly in the midwest and mideast hydrogeologic units, increased and exceeded the U.S. Environmental Protection Agency (USEPA) maximum contaminant level (MCL) of 44mg/L (background concentration is 10 mg/L) during operation of the artificial recharge program. According to Nishikawa et al. [2003], the primary source of the high-nitrate concentrations was septage, the waste stored in septic tanks, discharged through the septic system, and the mechanism responsible for the rapid increase of nitrate concentration in the groundwater was the rising of the groundwater levels as a result of the artificial recharge program.

3.2. Objective

[21] After validating the mechanism of high-nitrate concentrations, in 1996, HDWD proposed a conjunctive-use project that includes surface recharge ponds and pumping wells near the ponds. The purpose of the project is to decrease the high-nitrate concentration while maintaining the groundwater levels at desired elevations as well as meet the water demand. Surface recharge ponds are used to infiltrate clean water imported from the California State Water Project (SWP) in order to recover groundwater levels, and pumping wells are used to control the rates at which the groundwater rises while removing high-nitrate concentrations. Hence the management objective is to find an optimal pump and recharge strategy by controlling the rates of surface recharge and pumping.

3.3. Optimal Conjunctive-Use Model

[22] The management goal is to design a conjunctive-use program to decrease the high-nitrate concentrations while maintaining groundwater levels at desired elevations and meeting water demand. Accordingly, a suitable management model optimizes the pump and recharge strategy, where the objective is to minimize the total cost over the proper choices of the time-varying pumping and recharge rates at prespecified wells and surface ponds. The management model is formulated as JE = min J, where

equation image

subject to

equation image
equation image
equation image
equation image
equation image
equation image

where J is the objective function; Qp,t is the pumping rate (ft3/d) during the tth stress period at pumping well p; and ap,t is the corresponding cost coefficient, which includes the cost for pumped and treated water during the tth stress period; Rj,t is the recharge rate (ft3/d) during the tth stress period for surface recharge pond j with the corresponding cost coefficient, aj,t; Np is the number of pumping wells; NJ is the number of recharge ponds; T is the operation horizon of the management problem; himax is the maximum allowable water level at location i; hi,t is the simulated groundwater level at time t and location i; I is the set of locations for the groundwater level constraints; MCL is the maximum contaminant level; K is the set of locations for the concentration constraints; Ck,t is the simulated concentration at time t and location k; Dt is the water demand during the tth stress period; At is the total available imported water for recharge during the tth stress period; Qpmax is the pumping capacity of well p; and Rjmax is the recharge capacity of pond j. The decision variables are Qp,t and Qj,t. The total cost includes pumping, which is lumped with treated water costs, and recharge.

[23] After consulting with USGS's staff and acquiring the data needed for this management model from HDWD, the proposed conjunctive-use project includes three five-acre recharge ponds (NJ = 3, site 3, 6, and 7) and four pumping wells (Np = 4, 6W, 9E, 18E, and HDWD MON2). The recharge pond, site 3, is located in the west hydrogeologic unit; site 6 is located in the midwest hydrogeologic unit; and site 7 is located in the mideast hydrogeologic unit. The four pumping wells are distributed in the west (6W), midwest (9E), mideast (18E), and east (HDWD MON2) hydrogeologic units (Figure 3).

[24] To demonstrate the proposed methodology, the planning horizon (T) is assumed to be 3 years and the stress periods for the decision variables, Qp,t and Rj,t, to be six months. Hence, there are a total of 42 decision variables for three recharge ponds and four pumping wells [6(3 + 4)]. HDWD provides the water demand for each period (Dt), the total available imported water for recharge for each period (At), and the capacities for the pumping well (Qpmax) and recharge pond (Rjmax). The groundwater levels at three locations (H1–H3) in the west, midwest, and mideast hydrogeologic units are constrained, all of them near the recharge ponds (Figure 3). HDWD requires that groundwater levels be maintained at least 150 ft below the land surface. This determines the maximum allowable groundwater levels at locations H1, H2 and H3. The nitrate concentrations at four locations (C1–C4) distributed in the west, midwest, mideast, and east hydrogeologic units (Figure 3) are constrained. HDWD requires that nitrate concentrations be less than the MCL of 44 mg/L after 3 years of operation. To maintain feasibility, the MCLs are set at 60, 55, 53, 49, 45, and 42 mg/L at the end of each of the six stress periods. In the model, ap,t = 0.003(1 + r)t, where r is the discount rate and equals 0.025 (= 0.05/2); and 0.003 is the unit price (dollars/ft3) for pumping groundwater and treated contaminant water. In addition, aj,t = 0.006(1 + r)t, where 0.006 is the unit price (dollars/ft3) for recharge water. HDWD provides the unit prices, and the discount rate is assumed to be five percent per annum. Because of the nonlinearity of groundwater levels and constrained nitrate concentration levels, we use MINOS [Murtagh and Saunders, 1995], a nonlinear optimization solver, to solve this management problem.

4. Construction of an Objective-Oriented Groundwater Model

[25] Once we determine the model application objective for the Warren groundwater basin, we can implement the procedure of constructing an objective-oriented model. The GIP seeks to identify a representative parameter with the simplest structure that satisfies the accuracy requirement of a given model application. This approach is different from the traditional data-driven model calibration in that, in addition to residual error minimization and regularization, an additional model application objective is included as a constraint in the inverse solution.

4.1. Numerical Model

[26] The USGS developed and manually calibrated a groundwater flow model (MODFLOW2000 [Harbaugh et al., 2000]) as well as a mass transport model (MT3DMS [Zheng, 2005]) by trial and error [Nishikawa et al., 2003]. Aquifer properties were assumed to be uniform for each layer. The spatial discretization of the model consisted of three horizontal layers, each one constituted by a 25 × 75 grid. Layer 1 represented the upper and middle aquifers (unconfined aquifers), layer 2 represented the lower aquifer (convertible aquifer), and layer 3 represented the deep aquifer (confined aquifer). The period from 1956 to 2001 was simulated and divided into two parts: one simulation from 1956 to 1994 and the other from 1995 to 2001. The fitting residual (RE) calculation was conducted on the basis of the second part of the simulation (1995–2001). The first part of the simulation was used as a validation process for the results of the fitting residual.

[27] Model boundaries for the groundwater flow model were simulated as no-flow boundaries, except the eastern boundary, which is a general head boundary (GHB). Specified flux conditions were used to simulate natural and artificial recharge (septage, irrigation return flow, and the HDWD artificial recharge operations). As for the mass transport model, model boundaries were associated with groundwater flow boundaries. Concentration values were specified at the GHB and specified flux conditions for any inflowing water. For the flow and mass transport models, initial model layer properties were set at the trial-and-error values obtained by Nishikawa et al. [2003], where a uniform property was assumed for each layer. Figure 3 shows the model grid for the groundwater flow and transport model of the groundwater basin, the locations of faults (F1–F7), and hydrogeologic units.

4.2. Generalized Inverse Problem

[28] In the Warren groundwater basin, the high-nitrate concentrations occur predominately in the first layer, where most observation data were taken. We therefore choose to increase the structure complexity of the horizontal hydraulic conductivity for the first layer (K1). We also conduct a sensitivity analysis of the rest of the parameters, i.e., the hydraulic conductivity for the second and third layer (K2 and K3), specific yield (Sy), specific storage (Ss), hydraulic characteristic (F1–7), and general head boundary conductance (GHB). The results indicate that the model is insensitive to changes of these parameters with respect to the management objective, and are not chosen for downscaling. Therefore, the PR in this case study is denoted as (S, K1).

[29] Two optimization problems REm and SEm are involved in solving the GIP. The fitting residual REm for this case study is formulated as

equation image

where hnhobs and Cncobs are the observed groundwater levels and nitrate concentrations; hnh (Sm, K1m) and Cnc (Sm, K1m) are the corresponding simulation outputs; Nh is the number of point groundwater level observations; Nc is the number of point nitrate concentration observations; and σh2 and σc2 are the variances of the groundwater level and concentration observation errors, respectively.

[30] We formulate the structure error for this case study as

equation image

where JE is the minimal total cost in the optimal conjunctive-use model described in section 3.3 (model application), and JE (SA, K1A) and JE (SB, K1B) are the costs obtained with the corresponding PRs (SA, K1A) and (SB, K1B).

[31] In this study we use a global-local optimization scheme [Tsai et al., 2003a, 2003b; Mahinthakumar and Sayneed, 2005] that includes a genetic algorithm (GA) [Goldberg, 1989; Michalewicz, 1994] and a modified Gauss-Newton method to calculate the REm and SEm. First we use GA to find an approximate global optimal solution, then use a modified Gauss-Newton method to improve the solution obtained from the GA. We modify the GA program developed by D. L. Carroll (FORTRAN genetic algorithm driver, version 1.7a, 2001, available at http://cuaerospace.com/carroll/ga.html) to serve our purpose, and implement the modified Gauss-Newton method according to Yoon and Yeh [1976] and Yeh and Yoon [1981]. In the modified Gauss-Newton method part of the optimization scheme, the gradient of JE with respect to the parameter, K1B (if the dimension of K1B is m) must be calculated, so we use a finite difference method to approximate the derivative. That is,

equation image

We determined the parameter perturbation value (δ) to be 10% by trial and error.

4.3. Identification of the WCP

[32] The WCP and the homogenization error are sensitive to available prior information. With more prior information, the shape of the WCP would be simpler and the homogenization error smaller. Otherwise, the shape of the WCP would be more complex and the homogenization error larger. In this study, we use the stratigraphic unit and geophysical data for the Warren groundwater basin from Nishikawa et al. [2003] as the prior information for WCP and the structure pattern. We use the geophysical data, which identify the locations of faults (F2, F3, F4, and F5), to generate the zonal structures, S1S5. Then we use the geophysical data and the stratigraphic unit data to generate zonal structures S8S15. After compiling all prior information related to K1, we consider the following series of zonal structures: S1, S2, S3, S4, S5, S8, S12 and S15. Zonal structures S3 to S15 are shown in Figures 4a4f. Zonal structure S1 is homogeneous and S2, S3, S4, and S5 are determined from the geophysical data, F2–F5. Zonal structures S8, S12 and S15 are formed using the geophysical data and stratigraphic unit data. The estimated transmissivities of the middle aquifer based on specific capacity data from wells perforated in the upper and middle aquifer are from about 920 ft2/d to 6450 ft2/d [Nishikawa et al., 2003]. The thickness of the middle aquifer is from about 120 ft to 250 ft. Hence, the upper and lower bounds of K1 are assumed to be 50 (ft/d) and 5 (ft/d).

Figure 4.

WCP obtained for a series of structures, i.e., (a) 3-zone, (b) 4-zone, (c) 5-zone, (d) 8-zone, (e) 12-zone, and (f) 15-zone structure.

[33] To determine the WCP complexity, we calculate the homogenization error, SE(SA, S1), which is the structure error between a structure, SA, and a homogeneous structure. We determine SA from all available prior information. In this study, we consider a series of homogenization errors: SE(Sm, S1), m = 1,2,3,4,5,8,12,15. In each complexity level, we only need to search 21, 22, 23, 24, 25, 28, 212, or 215 vertices instead of the entire continuous admissible region, thus drastically reducing computational efforts. The results appear in Table 1. From Table 1 we find that the structure error increases with increasing parameter structure dimension because a more complex structure is more difficult to homogenize. The homogenization error SE(Sm, S1) increases from 2.29 × 105 to 9.01 × 105 when m is increased from two to 15. The corresponding WCPs of SA appear in Figures 4a4f. Table 1 shows that the homogenization error approaches an asymptotic value at SE(S12, S1). Hence, we conclude that using a 12-zone structure to represent the complex heterogeneous aquifer is sufficient. We therefore proceed to solving the GIP and conducting an experimental design.

Table 1. Parameter Structure Dimension Versus Homogenization Error
Parameter Structure DimensionSE (×105) (dollars)
S10.00
S22.29
S34.16
S45.83
S57.25
S88.20
S128.87
S159.01

4.4. Results of GIP

[34] We use 398 groundwater level observations and 221 nitrate observations to calculate REm. The σh and σc values are assumed to be 1.0 (ft) and 1 (mg/L), respectively. The manually calibrated results from the USGS [Nishikawa et al., 2003] indicate that the fitting residual for the groundwater level is around 40 ft; hence, we set the stopping criterion for the fitting residual for the groundwater level, REm, at 40 ft (2η1). Because of the lack of information on concentration, we only consider REm for the groundwater level in our analysis. The accuracy requirement of model application (ɛ1) is assumed to be 5.0 × 105. This is based on our discussions with the USGS and HDWD staff. Following the procedure described in section 2.3, the 12-zone structure first is replaced by a homogeneous structure S1. By solving the inverse problem with only one unknown parameter, K11, the optimized results show that the minimized residual, RE1, is 46.57 ft when K11 = 27.6 ft/d. The corresponding structure error, SE1, is 8.87 × 105. Since RE1 is larger than 2η1 = 40 ft, we increase the parameter structure dimension from one zone to two zones. During the global-local optimization procedure for the two-zone structure, the fitting residual, RE2, decreases to 42.53 ft when K121 = 26.1 ft/d and K122 = 30.7 ft/d, and the corresponding structure error, SE2, decreases to 5.38 × 105. Although the reduction in fitting residual is apparent, the model application requirement, SE2, is still larger than ɛ1 = 5.0 × 105. Hence, we continue to increase the parameter structure dimension by one. Repeating the optimization procedure for the three-zone structure, the RE3 is reduced to 38.81 ft when K131 = 28.5 ft/d, K132 = 36.1 ft/d, and K133 = 12.1 ft/d; SE3 is reduced to 3.66 × 105. Now RE3 is smaller than 2η1 = 40 ft, but SE3 already satisfies the model application requirement. Consequently, we stop the stepwise regression procedure and conclude that the identified three-zone structure is acceptable. The fitting residual and structure error results appear in Table 2, and the structure patterns appear in Figures 5a5c. From the GIP results, we conclude that with the existing observation data, the identified 3-zone structure is sufficient to replace the 12-zone structure WCP as the true system and can produce reliable results to meet the model application requirement.

Figure 5.

The identified horizontal conductivity structures, i.e., (a) one-zone, (b) two-zone, and (c) three-zone structure.

Table 2. GIP Results
Parameter Structure DimensionREm Head (ft)/NO3 (mg/L)SEm (×105) (dollars)K1 (ft/d)
S146.57/28.78.8718.3
S242.53/27.45.3839.1/11.7
S338.81/26.73.6645.6/8.8/15.6

[35] The results may be obvious for the following two reasons: (1) observation data are lacking in the far east and northeast hydrogeologic units and (2) the locations of these units are quite remote from the pumping and recharge actions; therefore, it is not necessary to increase the parameter structure dimension by adding F4 or F5 in the identification procedure. The GIP results are reasonable and justified in that the east part (K133) of the identified structure can be a homogeneous structure, which is the simplest structure, yet the identified model is reliable for management applications. We also calculated the structure error (SE) of using the model with the trial-and-error parameter values [Nishikawa et al., 2003] to replace a 12-zone structure (Figure 4e), and the SE is equal to 9.45 × 105. From the results, we see that different parameter structures may fit the existing data equally well (RE3 = 38.81 ft versus RE = 40.03 ft with the trial-and-error parameter values), but they produce significantly different application results (SE3 = 3.66 × 105 versus SE = 9.45 × 105 with trial-and-error parameter values). This demonstrates that the identified parameter pattern and its corresponding parameter values from fitting observations alone may not be sufficient for model application.

[36] We conduct a sensitivity analysis to determine the impact of dispersivity on the solution of the GIP. We use the identified three-zone structure obtained from the GIP procedure as the base model, then change the longitudinal dispersivity value (the horizontal and vertical transverse dispersivities are changed according to the specified proportionality) and calculate the SE. The longitudinal scale of this problem is about 33,000 ft. For this scale, the longitudinal dispersivity ranges from 17 to 20,000 ft [Gelhar et al., 1992]. Hence, two different longitudinal dispersivity values, 7500 and 75 ft, are used in the sensitivity analysis, which produce the corresponding SE values of 3.70 × 105 and 3.64 × 105. From this simple sensitivity analysis, we conclude that the SE is insensitive to the dispersivity.

4.5. Comparison of Computation Time

[37] One of the major advantages of applying the WCP concept in calculating the structure error is reduction in computation time. Because the WCP must be a vertex of the admissible region [Sun, 2005], the global optimal solution can be found only by searching 2m vertices (m is the parameter structure dimension) instead of the entire continuous domain of the admissible region. Tables 3a and 3b shows the comparison of computation time between solving the structure error with and without applying the WCP. We first used a homogeneous structure to replace a two-zone structure, as in our first test case. The results show that the optimization scheme applying the WCP required 1.8 h of CPU time on a PC with Pentium IV 2.26 GHz and 512 MB RAM (four evaluations) to find the global optimal solution, but the scheme without the WCP required 125.8 h of CPU time (304 evaluations) to approximate the optimal solution (Tables 3a and 3b). Our second test case used a homogeneous structure to replace a three-zone structure. The optimization scheme with the WCP required 3.3 h of CPU time (eight evaluations) to obtain the optimal solution, but the scheme without the WCP could not converge to the optimal solution even after 335.1 h of CPU time (800 evaluations) (Tables 3a and 3b). The advantage of using the WCP clearly is demonstrated by this simple example.

Table 3a. Comparison of Computation Time Between Using the WCP and Without Using the WCP for Case One
ParametersUsing the WCPWithout Using the WCP
One ZoneTwo ZonesOne ZoneTwo Zones
K11 (ft)22.250.022.049.6
K12 (ft)5.05.0
SE (×105) (dollars)2.292.292.252.25
CPU time (h)1.8 125.8 
Table 3b. Comparison of Computation Time Between Using the WCP and Without Using the WCP for Case Two
ParametersUsing the WCPWithout Using the WCP
One ZoneThree ZonesOne ZoneThree Zones
K11 (ft)39.250.029.023.0
K12 (ft)5.016.3
K13 (ft)50.045.9
SE (×105) (dollars)4.164.162.582.58
CPU time (h)3.3 335.1 

5. Robust Experimental Design

[38] The GIP results obtained from section 4 allow us to judge data sufficiency. If the existing data are insufficient to identify the parameter structure, an experimental design for data collection is needed. Sun [2005] and Sun and Yeh [2007b] proposed a new approach to experimental design for constructing objective-oriented groundwater models. In the approach the minimal cost and maximal information content are used as the objectives, while the data sufficiency and feasible designs are treated as constraints. The optimal experimental design problem is formulated as

equation image

where D* is the optimal design, f(D) is the cost of experimental design D, and {FRD} is a feasible and robust set for parameter structure identification. A feasible design may be constrained by the maximum number of observation wells, the maximum pumping rate and the maximum available cost. Sun [2005] proved that if a design D is sufficient for identifying the WCP(SA), then the design is robust and sufficient for identifying all parameters with structure SA or with a structure that is a simplification of SA. Thus the design obtained through equation (13) is cost-effective, optimal, robust and reliable. Moreover, data sufficiency can be determined before actually conducting the field experiments.

5.1. Heuristic Algorithm for Solving Robust Experimental Design

[39] An experimental design generally consists of two parts, the excitation part and the observation part [Sun, 1994]. In this study, we only consider the observation part. The first step is to compile all available prior information, including the objectives of model application and their accuracy requirement. On the basis of prior information, we estimate a structure SA as the true structure and find its WCP, equation imageA. The complexity of WCP (the dimension of structure SA) depends on the available prior information. More prior information may produce a simpler WCP, and as a result less information will be needed from the designed experiment. If less prior information is available, the degree of freedom should be large enough to represent the true system, and more information will be needed from the designed experiment. After determining the WCP(SA), we find a feasible solution set, i.e., potential locations for installing the observation wells. Afterwards, we select the observation wells from the potential well locations and use these wells to generate observations by running the simulation model with WCP(SA). Finally, we generate a series of parameter structures one by one and solve the GIP problem.

[40] We use a norm of the following information matrix to select the potential observation well locations:

equation image

where Jh = ∂h/∂equation image1 is the Jacobian matrix of the groundwater level, equation image1 is the estimated horizontal hydraulic conductivity for the first layer, and σ2 is the estimated variance of the groundwater level. We use a maximal information criterion that maximizes the trace of the information matrix, Tr(I), to evaluate the competing designs [Cleveland and Yeh, 1990].

[41] The overall heuristic algorithm of finding a robust and optimal experimental design for parameter structure identification is summarized in the following steps. (1) Compile all available prior information to estimate the structure SA as the true structure and find its WCP, equation imageA. (2) Determine potential observation well locations. (3) Calculate the trace of the information matrix for all potential observation well locations and rank them. (4) Select one observation well with the highest rank and generate observations by running the simulation model, WCP(SA). (5) Generate a series of parameter structures one by one, calculate the optimal structure pattern and parameter value (Sm, K1m) by calculating REm, and check the structure error, SEm. (6) Evaluate data sufficiency. If data are sufficient, stop. Otherwise, go to step 7. (7) If the data are insufficient (REm < 2η and SEm > ɛ), go to step 4. Otherwise, go to step 8. (8) If the model application requirement is not satisfied (REm > 2η and SEm > ɛ), increase the parameter structure dimension by one and go to step 5 directly.

5.2. Experimental Design

[42] To test the applicability of the proposed robust experimental design, we tighten the model application requirement to ɛ2 = 1.0 × 105 and keep the stopping criterion for REm the same; thus the identified three-zone structure is no longer adequate to replace the WCP (SE3 = 3.66 × 105 > ɛ2). However, the minimized RE3 of 38.81 ft is less than 2η1. Hence we conclude that the existing data are insufficient to identify the WCP and satisfy the model application requirement. We therefore proceed with the experimental design to collect additional data for parameter structure identification.

[43] The purpose of experimental design is to select a set of locations and observation frequencies for observation wells such that the cost of design is minimized while the information is maximized. For simplification, we make the following assumptions: (1) observation wells are selected from the existing wells only, (2) the sampling cost for each observation well is the same, and (3) the selected observation wells are used to generate observations for the entire simulation horizon, i.e., the observation frequency is not considered in the design. With these assumptions, the optimization problem is reduced to a search for the minimum number of observation wells. We also assume each observation well is independent and observation wells are selected one at a time.

[44] From the USGS study [Nishikawa et al., 2003], there are nine potential observation well locations (BSGC17, 7W, 11W, 17E, 7E, 14E, HDWD MON1, 5E, and 28N1), shown in Figure 6. We run the simulation model with a 12-zone structure WCP (Figure 4e), obtained by calculating the homogenization error reported previously, to generate a set of observation data from these potential observation well locations. The generated observations are corrupted with Gaussian noise with zero mean and 1 ft standard deviation (η2 = 1). We calculate and rank the trace of the information matrix, Tr(I), for each potential well location for the selection process.

Figure 6.

Experimental design results (a four-zone structure). Open diamonds are the unselected potential well locations; solid diamonds are the selected potential well locations.

[45] Following the heuristic algorithm described in section 5.1, the experimental design procedure starts from a homogeneous structure and gradually increases the parameter structure dimension until the model application requirement is satisfied. For a homogeneous structure, the first selected observation well is 14E, which has the highest information ranking. The optimized results are RE1 = 0.62 ft (<2η2 = 2) and SE1 = 8.31 × 105(>ɛ2 = 1.0 × 105). From the results, we see that the data are insufficient and new observation data are needed. Hence, we add the observation data obtained from the second well, 17E, into the experimental design. The fitting residual for two observation wells, RE1, is equal to 3.85 ft (>2η2) and the corresponding structure error, SE1, is equal to 7.75 × 105(>ɛ2). The model application requirement still is not satisfied, but the data contain more information. Consequently, we increase the parameter structure dimension from one to two. For a 2-zone structure, two observation wells, 14E and 17E, still are required for identifying the 12-zone structure WCP. When the parameter structure dimension increases from one to two, the fitting residual, RE2, decreases to 1.08 ft (<2η2) and the structure error, SE2, decreases to 4.94 × 105(>ɛ2). The results show that the data are insufficient and additional observation wells are needed. Therefore, we select the third well, HDWD MON1, and repeat the optimization procedure. The results show that a 2-zone structure with data from three observation wells for identifying the 12-zone structure WCP still cannot satisfy the model application requirement (SE2 = 4.19 × 105 > ɛ2). This suggests that the parameter structure dimension should be increased from two to three (RE2 = 3.22 > 2η2). On the basis of the rank of the information criterion, we sequentially select the fourth well, 11W, and the fifth well, 7W, for the three-zone structure, and repeat the procedure for each additional well until the model application requirement is satisfied. The results show that the data obtained from five observation wells with a three-zone structure meet the residual error minimization criterion (RE3 = 2.55 > 2η2), but the model application requirement remains unsatisfied SE3 = 1.96 × 105. This again suggests that the parameter structure dimension should be further increased from three to four. We repeat the experimental design procedure, and, finally, the optimized 4-zone structure with six observation wells is sufficient to identify the 12-zone structure WCP. The optimized results are RE4 = 1.74 ft (<2η2) and SE4 = 0.86 × 105(<ɛ2), and the six observation wells are 14E, 17E, HDWD MON1, 11W, 7W, and 7E. At the mean time, this design is robust and optimal, and should be sufficient for identifying all parameters with the same structure or with a structure that is a simplification of the 12-zone WCP. The summary of the experimental design results and the values of REm and SEm appear in Table 4. Figure 6 shows locations of the selected observation wells and the identified four-zone structure.

Table 4. Results Obtained From Robust Experimental Design
Parameter Structure DimensionNumber of WellsRE (ft)SE (×105) (dollars)ConclusionDecision
S110.628.31InsufficientIncrease observation wells
 23.857.75More information availableIncrease structure complexity
S221.084.94InsufficientIncrease observation wells
 33.224.19More information availableIncrease structure complexity
S331.352.82InsufficientIncrease observation wells
 41.682.35InsufficientIncrease observation wells
 52.551.96More information availableIncrease structure complexity
S450.751.20InsufficientIncrease observation wells
 61.740.86SufficientStop

6. Conclusion

[46] In this study, we constructed an objective-oriented model for conjunctive-use planning of surface water and groundwater for the Warren groundwater basin in southern California. We successfully implemented the GIP concept. We incorporated a management objective for conjunctive-use planning of surface water and groundwater into the model calibration process. The management model optimized the time-varying pumping and recharge rates at specified locations. We successfully demonstrated the concept of WCP using data collected from the Warren groundwater basin. We developed a global-local optimization scheme with GA and modified the Gauss-Newton method for solving the fitting residual and structure error for this real case study.

[47] On the basis of the results, a simple model with 12 zones was sufficient to represent a real and complex heterogeneous aquifer. From the GIP results, the Warren groundwater basin can be represented by a three-zone structure that satisfies the specified requirement of model application, ɛ1 = 5.0 × 105. When a more stringent criterion, ɛ2 = 1.0 × 105, was chosen for model application, the existing data were insufficient to identify the 12-zone structure WCP and a robust experimental design was undertaken. We used the maximization of the trace of the information matrix as the criterion for selecting the observation well locations. The results show that the observation network design with the optimized well locations can provide sufficient information for solving the GIP when the 12-zone structure WCP is used as the true parameter. This design is robust in that it is also sufficient for identifying all other parameters with the same or simplified structures. The key advantage of experimental design is that the design was obtained using simulation and optimization prior to conducting actual field experiments; thus, it achieved a tremendous saving in cost.

Acknowledgments

[48] This material is based on work supported by NSF under award EAR-0336952 and USGS under grant 05HQGR0161. We would like to thank three anonymous reviewers and the Associate Editor for their in-depth and constructive reviews.