## 1. Introduction

[2] Conjunctive use of surface water and groundwater resources for irrigated agriculture is beneficial because it can buffer the natural variability of supplies obtained from surface water by relying to a greater extent on groundwater, which is typically more expensive. In many arid and semiarid regions, surface water reservoirs have been built to protect against the natural variability of runoff. However, reservoirs constructed for drought mitigation usually perform multiple tasks that conflict with water supply for irrigation, such as flood protection [*Yeh*, 1985; *Labadie*, 2004]. Groundwater provides additional insurance against reductions in agricultural production during droughts [*Bredehoeft and Young*, 1983; *Tsur*, 1990]. A drawback is that increased groundwater pumping during extended droughts may lead to severe aquifer head drawdowns, resulting in high pumping costs and seawater intrusion in coastal aquifers [*Willis and Finney*, 1988; *Reichard and Johnson*, 2005; *Schoups et al.*, 2006]. When developing conjunctive water management strategies, all these issues should be taken into account. In addition, given the large uncertainty in surface water supply, the proposed strategies should be reliable over a wide range of streamflow scenarios. Therefore a two-pronged approach is needed. First, a tool is needed to quantify streamflow uncertainty, which is commonly achieved by means of a stochastic streamflow model. Second, optimal water management decisions need to be identified that account for streamflow uncertainty, which results in a stochastic optimization problem.

[3] The literature on stochastic time series analysis applied to streamflow modeling is vast and many methods have been developed for a wide range of problems [*Salas*, 1993]. In this paper, we are concerned with the concurrent generation of monthly streamflow at multiple reservoirs such that observed autocorrelations and cross correlations are preserved at both monthly and multiannual timescales. Particularly, in terms of sustainable water management it is essential that the observed drought characteristics are preserved in the generated time series [*Loaiciga*, 2005]. Either parametric or nonparametric methods may be used for this type of problem. Parametric models make assumptions about the form of the distributions (e.g., Gaussian) and typically require a large number of parameters to be estimated when both short-term and long-term correlations need to be preserved [*Rasmussen et al.*, 1996]. Parametric disaggregation methods [*Koutsoyiannis*, 2001] are available to deal with this problem by e.g., first generating annual streamflows, followed by disaggregation to monthly flows. Alternatively, nonparametric models such as bootstrapping methods [*Vogel and Shallcross*, 1996] and kernel-based methods [*Tarboton et al.*, 1998] have been developed that do not make any prior assumptions about the shapes of the distributions. More recently, hybrid methods have been introduced that combine the strength of both parametric and nonparametric approaches [*Srinivas and Srinivasan*, 2005].

[4] The next step is to solve the problem of optimal water management under uncertain water supply. Two main approaches can be distinguished, namely implicit and explicit stochastic optimization [*Labadie*, 2004]. Implicit stochastic optimization (ISO) relies on deterministic optimization methods to find management strategies that are optimal over a long historical record [*Lund and Ferreira*, 1996] or over a large number of shorter synthetic streamflow realizations [*Young*, 1967; *Bhaskar and Whitlatch*, 1980]. Postoptimization regression analysis [*Hiew et al.*, 1989] or neural network analysis [*Raman and Chandramouli*, 1996] of the optimization results then yields general operating rules that, for example, specify reservoir releases as a function of current storage. Explicit stochastic optimization (ESO) on the other hand works directly with streamflow probabilities, which can either be included in the objective function, as in stochastic linear programming [*Jacobs et al.*, 1995; *Seifi and Hipel*, 2001] and stochastic dynamic programming [*Loaiciga and Marino*, 1986; *Tejada-Guibert et al.*, 1995; *Faber and Stedinger*, 2001], or in the constraints through chance-constrained programming [e.g., *Loucks and Dorfman*, 1975], or in both the objective function and the constraints [*Reichard*, 1995]. The ESO approach explicitly accounts for the lack of perfect knowledge of future events, but it can lead to computationally intractable optimization problems of multireservoir systems.

[5] The current paper focuses on conjunctive use in one of the most important agricultural regions in Mexico, the 6800 km^{2} Yaqui Valley near the Sea of Cortez in the state of Sonora. The objective here is to derive conjunctive surface water and groundwater operating rules for a wide range of streamflow scenarios. In particular, the goal is to find a balance between unnecessary reservoir spills during wet periods and sustained irrigated agriculture during drought conditions, while avoiding excessive aquifer head drawdowns and seawater intrusion.

[6] We build on a deterministic spatially distributed numerical simulation-optimization model for the Yaqui Valley developed by *Schoups et al.* [2006], and present here for the first time an approach that involves a combination of methods enabling us to consider a more complex, multiannual, regional water management problem under uncertainty. Methodologically, *Schoups et al.* [2006] studied elements of water management with a deterministic model of the hydrologic systems (surface water reservoirs and alluvial coastal aquifer), and a model that simulates crop production. Using these model components, a series of annual profit maximization problems were solved with large-scale constrained gradient-based optimization. Unlike that prior work, here we consider a substantially different problem. We study multiobjective trade-offs in surface water reservoir operation that arise from uncertainty in streamflow. In addition, our water management model is multiannual and integrates a variety of modern methodological advances.

[7] There are three primary contributions of the current work. First, water management operating rules are identified in a hierarchical framework by means of interannual optimization of agricultural sustainability and spill minimization using a multiobjective global optimization algorithm [*Vrugt et al.*, 2003], with a nested series of annual profit-maximizing models solved with nonlinear gradient-based optimization [*Gill et al.*, 2002]. As a starting point, our hierarchical optimization framework was based on the approach of *Cai et al.* [2001], who used a genetic algorithm for multiannual optimization and linear programming for annual optimization. Ours is a nonlinear optimization problem and therefore the approach is a generalization of the one presented by Cai et al. However, more importantly, our problem is a multiobjective one. Cai et al. considered only one objective, namely agricultural sustainability. Our approach on the other hand, explicitly quantifies trade-offs between two objectives: sustaining irrigated agriculture during droughts and minimizing unnecessary reservoir losses during wet periods. Our study also builds upon the valuable approach of *Lund and Ferreira* [1996]. As in their work, we assume perfect knowledge of a long historical record of monthly streamflows. However, our approach differs from that study in that optimal management is parameterized by a linear operating rule. Our method is similar to the piecewise linear operating rules of *Oliveira and Loucks* [1997]. The slope and intercept of the linear rule are used as decision variables in the multiannual optimization. A second contribution of this paper that was not within the scope of *Schoups et al.* [2006] is that uncertainty in surface water supply is quantitatively addressed. To accomplish this, we use a large number of equally probable streamflow realizations generated by the stochastic streamflow generation method of *Srinivas and Srinivasan* [2005]. Our third contribution involves performance evaluation. Optimal operating rules are evaluated by means of a postoptimization Monte Carlo analysis using a large set of generated streamflow records. In that sense, our approach is related to the implicit stochastic optimization (ISO) method discussed earlier.

[8] The paper is organized as follows. First, we give some background on the Yaqui Valley study area. This is followed by an outline of the hierarchical optimization approach, and a brief discussion of the stochastic streamflow model. Results are then presented for each part of the analysis and finally our findings are summarized in the conclusions.