2.1. Hydrologic-Market Simulation
 The simulation runs over a 12-month period, beginning on 31 December (t = 0), with the city holding some number of permanent water rights and options (NO). Initial conditions specify reservoir storage (R0) and the amount of water the city has carried over from the previous year In each of the following months, regional hydrologic conditions are simulated using data sets describing monthly reservoir inflow, outflow, and losses, with these conditions linked to both the city's water supply and the spot market price for water. This information is then combined with monthly distributions of the city's demand to make decisions regarding the purchase of leases and/or exercise of options. Multiple simulation runs for each set of initial conditions generate values for the expected annual cost of the city's portfolio, expressed as (random variables are denoted by the circumflex)
= total volume of permanent rights held by city (ac ft);
= volume of options purchased at the beginning of the year (ac ft);
= volume of exercised options (ac ft);
= volume of spot leases purchased at the end of each month (ac ft).
Within the simulation, the following constraints apply:
Non-negativity constraints also apply for all variables.
 A series of variables are used to describe regional hydrologic conditions, including it = volume of reservoir inflows for each month t; = volume of reservoir losses for each month t; and ot = volume of reservoir outflows (including spillage) for each month t.
 A water balance is maintained on the reservoir system throughout the simulation such that
 From the perspective of the individual city, total reservoir storage is less important than the volume of water available to the city itself, an amount largely determined by the city's initial supply and its share of monthly reservoir inflows Reservoir inflows available for allocation are calculated as the difference between monthly inflows and losses, multiplied by an instream loss factor (lI ∈ [0,1]), which accounts for losses incurred between the reservoir and user (which in this case is assessed prior to allocation). Inflows available for allocation to rights holders in each month (nt) are computed as
These inflows are allocated on a pro rata basis such that the fraction of monthly inflows allocated directly to the city is represented as
where R = total volume of regional water rights.
 The total volume of water available to the city in any month is assessed at the end of the preceding month, and the method of calculation changes depending on whether it is before or after the exercise month (tx). In months prior to tx, the supply available to the city in the next month (St+1) includes cumulative inflows and purchased leases, less water usage such that
where ut = city's usage in month t.
 In subsequent months, the available supply also includes exercised options, such that
The decision of whether or not to purchase leases is the last step in each month, and the decision is based on the city's available supply, specified by (9) or (10) (neither of which include consideration of leases purchased in month t). The leasing decision involves consideration of both the city's available supply and the volume of monthly inflows it expects to have allocated to it over the remainder of the year (calculated on the basis of historical records). These two values are summed to yield the city's expected water supply over the remainder of the year, such that
where = distribution of inflows allocated to the city in each month t.
 November (t = 11) inflows are considered when calculating the available supply for December, but December inflows are allocated to the following year. Therefore December's available supply and expected supply are equal (i.e., = St+1).
 Once the city's expected water supply has been calculated, the decision is made to purchase leases and/or exercise options. This is a two-part decision in which the first step involves determining whether or not to acquire water and the second involves deciding how much. Both decisions are based on the ratio of expected supply to expected demand, with the decision to acquire made by comparing this ratio against a specified threshold value (α), such that if
where t = distribution of the city's water demand during each month t.
 The question of how much to lease and/or exercise is made by comparing the ratio of expected supply to expected demand with a second specified threshold value (β). This leads to leases being purchased and/or options (NX) exercised until
In all months except tx, NX = 0 and the volume of leases purchased can be represented as
During tx, the decision process is modified such that exercising options is considered before purchasing leases. Under these conditions, the first step is to compare the exercise price (pX) with the current spot lease price If the lease price is less than the exercise price, the city will simply lease the volume defined in (14). If, however, the exercise price is less than the lease price, the city will exercise options, with the volume to be exercised expressed as follows:
In the case of the latter scenario, where options alone are insufficient to satisfy (13), the city will acquire additional water via leasing, such that
Different α and β variables can be specified for individual seasons or even individual months. In the example described later, two different parameter pairs are established, one (α1/β1) for the period running up to the month before options can be exercised (t0 tX − 1) and another (α2/β2) for the remainder of the year. Expected supply (11) is similarly partitioned, such that it is calculated relative to tX in months leading up to tX, and calculated relative to the end of the year in all subsequent months. Optimal values for α and β, those that lead to a minimum expected cost portfolio that meets reliability constraints, are determined as part of the optimization routine (see next section).
 The choice to link decision rules to the ratio of expected supply to expected demand was based on the ability to use this value in determining both when and how much water to acquire. Alternative decision rules could have been based on the probability of shortfall, or perhaps even linked to a threshold value for the expected benefits loss that would accrue as a result of a shortfall. These types of rules may be expressed in terms more intuitive to utility personnel and/or planners (and might be explored in future work), but their use would have necessitated additional calculations to answer both the “when” and “how much” questions.
 Water is acquired just before the monthly counter changes (i.e., month t − 1 becomes month t), correspondingly St−1 St, which is then represented as
Available supply (St) is compared with a demand value (dt) obtained by either randomly sampling a monthly distribution or selecting from a monthly time series. If available supply is sufficient to meet this demand (i.e., St ≥ dt), then demand equals usage (ut = dt). If available supply is insufficient, then ut = St, leaving a shortfall of dt − St and a “failure” is recorded for that month. A distinction is made between a “failure” and a “critical failure” (St/dt ≤ 0.6) in order to recognize differences in severity and the measures that would be required to compensate for the shortfall. A running tally of both failures and critical failures is maintained throughout the simulation.
 Once available supply and demand have been compared, the process of evaluating new allocations and lease/exercise decisions repeats monthly through the end of the year. Each annual run within this probabilistic framework represents one realization of the cost and reliability of a portfolio defined by selected values for the initial conditions (R0, and decision variables (NR, NO, α1, β1, α2, β2). Multiple runs are made to determine a portfolio's expected cost (equation (2)) and expected reliability, with the latter defined as
where rf is monthly reliability against a failure and years is the number of simulated years (i.e., annual runs).
 A reasonable span of monthly reliabilities might range from 0.995 (i.e., one failure every 16.7 years) to 0.98 (one failure every 4.2 years). A similar factor (rcf) is used to measure the expected reliability relative to critical failures.
 Multiple annual runs also allow for evaluation of the probability of very high annual costs. Within the electricity and natural gas industries, a common metric used to describe the risk of high costs is the “contingent value at risk” (CVAR). Given a distribution of annual costs, the CVAR represents the mean of the annual costs falling above the 95th percentile. Something akin to the CVAR is likely to play a role in utility decisions, and this metric is used here.
 The quantity of water remaining in the city's possession at year's end is also tracked. This remaining water is not assigned any value, a shortcoming that could raise concerns that a portfolio developed within this annual framework may not bear much resemblance to the type of portfolio that would minimize costs over a longer time horizon. For instance, a portfolio that consistently left the city with very little water at the end of the year could result in very high supply costs the following year (this does not actually tend to be the case, however). While the development of long-term portfolios is beyond the scope of this work, these issues will receive some attention in the results section.
 The methodology described above involves a supply strategy that includes rights, options, and leases (strategy C); however, it is easily modified to explore alternative strategies that include permanent rights alone (strategy A) and permanent rights and options (strategy B). In the case of a city relying on strategy A, the number of rights (NR) becomes the only decision variable. With respect to strategy B, the number of decision variables increases to four (NR, NO, α2, β2) and the decision framework for acquiring water (i.e., equations (12), (13), and (15)) is similar to that described above, except that the city acquires additional water via options alone, and only in the exercise month. Strategy C involves six decision variables (α1, β1 are added) and the entire monthly decision framework described above.
2.2. Optimization Framework
 The simulation is linked to a search algorithm that identifies optimal values for the decision variables based on the following formulation (for strategy C):
Some results also incorporate an additional constraint limiting cost variability, such that
 Figure 1 illustrates a section of the optimization landscape describing expected cost as a function of the number of permanent rights and options (α1, β1, α2, β2 held constant). While the surface is relatively smooth when the volume of leases and exercised options is small (i.e., when a portfolio is mostly rights), as the volume of leases and exercised options increases so does the “noise.” This can be problematic for many gradient-based search algorithms as they can become trapped in local minima. The amplitude of the noise can be reduced by increasing the number of simulated years, but this comes at a price in terms of computational burden.
 Implicit filtering is a finite difference search method in which the difference increment (i.e., the size of the finite difference stencil) is varied as the optimization progresses [Kelley, 1999]. In this way, local minima which are artifacts of low-amplitude noise do not trap the iteration, and the noise is “implicitly filtered” out. This is in contrast to methods which explicitly try to filter out high-frequency components of the objective function [Moré and Wu, 1997; Kostrowicki and Piela, 1991]; such methods are designed for problems with high-amplitude high-frequency terms and should be thought of as global optimization algorithms. Implicit filtering is not a global optimization method, and is designed to efficiently solve problems, such as those presented in this paper, which have noisy but not violently oscillatory optimization landscapes (see Figure 1). Methods such as steepest descent, which are based on gradients, can be trapped in the small-scale local minima that noisy surfaces exhibit, and may fail if this results in an optimization surface that is not differentiable. In this problem, as in many others, the noise results from using an expected value (cost) as the objective function. The frequency and amplitude of the noise increases with greater use of leases and exercised options (probabilistic variables) and decreases with the number of simulated years used to generate an expected cost estimate of each portfolio. While an infinite number of simulations for each portfolio would generate a perfectly smooth optimization surface (which could be optimized using some form of steepest descent approach), implicit filtering allows for efficient optimization of the problem by allowing the search to progress while reducing the number of simulated years required to generate expected cost values during each iteration.
 Implicit filtering uses the finite difference gradient (as described by the difference between points on the finite difference stencil) to compute a search direction for descent. Unlike the classical steepest descent method, in which the negative gradient (or an approximation of the negative gradient) is used, implicit filtering uses a quasi-Newton model of the Hessian to scale the gradient, thereby accelerating convergence in the terminal phase of the iteration. The theory for implicit filtering [Kelley, 1999; Stoneking et al., 1992] and related algorithms [Audet and Dennis, 2003; Torczon, 1997; Kelley, 1999] explains how such methods overcome low-amplitude noise and also gives insight into the limitations of these methods. In particular, there is no guarantee that a global minimum will be found. While implicit filtering cannot ensure convergence to a global minimum (this can only be proven for methods that undertake exhaustive efforts to asymptotically sample a dense subset of the design space), there is a rich literature describing the convergence of this class of methods, generally distinguished by the “polling” of stencil points throughout an iteration [Audet and Dennis, 2003; Kelley, 1999; Torczon, 1997]. This body of work demonstrates that for problems involving a smooth objective function and inequality constraints, any limit point of an iteration satisfies the first-order necessary conditions for optimality, which is the typical conclusion in convergence theorems for iterative methods for optimization. These results have also been generalized to both nonsmooth [Audet and Dennis, 2003; Finkel and Kelley, 2004] and noisy problems [Choi and Kelley, 2000; Stoneking et al., 1992].
 In this application, the implementation code, implicit filtering for constrained optimization (IFFCO), uses the difference gradient stencil for more than computation of the gradient [Choi et al., 1999]. The gradient-based optimization is augmented with a coordinate search using the stencil points. If the result of the coordinate search is better than the result from the descent method, IFFCO accepts the coordinate search result. The coordinate search is also used in one of the termination tests for optimization (for details, see Choi et al.  and Kelley ). IFFCO handles constraints in two ways. Simple bound constraints on variables (e.g., NO ≥ 0) are enforced at each iteration by setting variables that exceed the bounds to the value of the nearest bound. Indirect constraints (e.g., reliability) are handled by assigning slightly higher values to the objective function of points where the constraint is violated. These failed points are always at the edges of the stencil, and they act to steer the search away from the infeasible region. IFFCO's combination of stencil-based sampling and gradient-based optimization is most effective when the function to be minimized is a smooth surface with low-amplitude perturbations. Such problems are common in a number of applications, and while implicit filtering has not been applied to water resource management problems, it has been successfully employed in some related settings, including the design of groundwater remediation systems [Battermann et al., 2002; Fowler et al., 2004].
 The simulation-optimization procedure includes 10,000 annual simulation runs for each set of decision variables, generating values for expected costs, reliability, critical reliability, and the CVAR which are generally reproducible to three significant figures. These parameters, as well as the α and β values, are passed to IFFCO which then guides the search of the optimization landscape. A search duration of 50 calls to the function (i.e., simulation) per decision variable was generally found to provide a resolution with respect to the expected cost and portfolio composition that corresponded to less than 1% and 200 ac ft, respectively. In some cases, 50 calls were insufficient to reach this resolution, and in these instances the solution from the first 50 calls (or a close approximation) was used as a starting point and the process repeated until changes in the solution were within these tolerances.
2.3. Study Region
 The U.S. side of the Lower Rio Grande Valley (LRGV) derives its water supply almost entirely from the Rio Grande, with flows managed via the Falcon and Amistad reservoirs (Figure 2). The two reservoirs have a combined storage capacity of approximately 5.8 million ac ft (MAF), with an additional 2.1 MAF of capacity set aside for flood protection (dead storage is roughly 30,000 ac ft). The storage in these reservoirs is strictly divided between the United States and Mexico according to the treaty of 1944 [Schoolmaster, 1991], with each countries' share of storage, inflows, outflows, and losses calculated as single system-wide values (Table 1). Since the two reservoir came on line in 1968, combined U.S. storage in these structures has varied from a low of approximately 0.7 MAF to a high of 4.0 MAF. The hydrologic data record extends from 1970 to 2002, and while there have been subtle shifts in the purpose of the diversions over that period (municipal use increased from 7% to 13% of regional total), average annual usage and monthly usage patterns have remained largely unchanged. The U.S. share of reservoir inflows is allocated to the LRGV's nearly 1600 water rights holders by the Rio Grande Watermaster's Office, which also administers transfers between rights holders.
Table 1. Simulation Data Summarya
|Reservoir inflows (it), × 1000 ac ft||Mean||89.6||88.5||91.0||100.2||142.2||159.0||159.8||195.8||246.8||203.2||106.3||87.5|
|Reservoir outflow (ot), × 1000 ac ft||Mean||82.3||70.0||94.5||143.2||159.6||152.5||124.7||132.8||97.4||100.6||61.7||59.5|
|Reservoir losses (lt), × 1000 ac ft ||Mean||17.4||21.5||34.3||41.9||46.0||52.6||57.7||55.0||40.8||33.0||22.4||16.9|
|Spot lease prices (pLta),b $/ac ft||Mean||17.0||17.4||16.8||14.6||16.2||16.7||15.2||12.7||15.8||13.8||14.4||16.3|
|Spot lease prices (pLtb),b $/ac ft||Mean||27.9||28.5||27.6||26.2||28.0||25.3||23.4||23.5||26.7||25.0||24.9||24.4|
|Demand (dt), ac ft||Mean||1569||1457||1681||1714||1919||1957||2073||2075||1692||1639||1547||1572|
 Ideally, the simulation described would be developed using long time series data sets that cover the same period for each hydrologic parameter (e.g., inflows, outflows), such that serial correlation in and between the data could be preserved. In cases where serial correlation is strong, expected supply and expected demand values would be estimated using conditional probability distributions based on current conditions (or those in the immediate past). In this case, however, the hydrologic data set is relatively limited (32 years) and use of only the sequential record would have reduced the analysis to a fairly narrow set of conditions. Attempts to expand consideration to a wider range of conditions by fitting existing hydrologic data to standard population models (e.g., lognormal, log-Pearson type III) using chi-square tests yielded very poor fits. The level of serial correlation in data sets and potential relationships between data sets were also explored to determine what other methods of hydrologic input could be used within the simulation.
 The Pearson test for serial independence was applied to the inflow time series, yielding evidence of weak autocorrelation in the monthly inflow data using both a 1- and 2-month lag (R2 of 0.15 and 0.05, respectively). The relatively low level of serial dependence is likely a function of the longer time step (i.e., monthly), as well as the arid nature of the watershed and its lack of features that might enhance the system's hydrologic “memory” (e.g., snowpack/snowmelt). Autocorrelation in monthly data is therefore unlikely to play a significant role in simulating regional supply conditions, particularly given that the Valley's regional reservoir capacity is approximately 4 times average annual inflow. This capacity is sufficiently large that ignoring the weak autocorrelation in the data is unlikely to significantly affect simulated reservoir levels, and while interannual correlation of inflows could be an issue in multiyear simulations, it is not a factor in the single annual cycles evaluated in this work. A similar evaluation of the 10-year record of monthly municipal usage (normalized by population) yielded a statistically significant, but weak serial correlation using a 1- and 2-month lag.
 With respect to relationships between variables, little evidence of correlation was observed between reservoir inflows and municipal water usage (R2 = 0.12, as measured by the Spearman test for trend), a situation that is likely due to climatic differences between central Mexico, where the majority of inflows originate, and the Valley, which is hundreds of miles away on the Gulf Coast. Correlations were also weak between reservoir outflow and municipal usage (R2 = 0.18), as outflow is dominated by irrigation releases, which in the Valley's semiarid climate are largely dependent on a fixed schedule and tend to obscure the relatively small amount directed to municipal use. These analyses suggest that assuming independence in monthly values for inflow, outflow, and municipal usage could provide a reasonable basis for simulating regional conditions. As a result, values for these variables are randomly selected from the appropriate monthly data list within the simulation. Values for expected supply and expected demand are also computed directly from these monthly distributions (as opposed to conditional probability distributions predicated on current conditions).
 Allocations to regional rights holders (equation (7)) are calculated using an instream loss factor (lI) of 0.175, and distributed pro rata across the region's 1.9 million ac ft of water rights (R). As the number of regional rights substantially outstrips the annual average volume of available reservoir inflows, each acre foot of rights is allocated around 0.725 ac ft of water in an average year. December (initial) reservoir storage levels (R0) are varied across historical December levels ranging from 0.8 to 2.2 MAF. The city's share of this storage at the beginning of the year is specified as a fraction of the total rights that the city holds such that = While it might seem logical to assume that high/low levels of R0 and would coincide, this is not necessarily the case. A substantial percentage of annual inflows occur in the fall, so even when year-end storage is below average, fall allocations can result in a city beginning the year with a significant volume of carryover water. Three values are chosen to represent low, normal, and high values for both (0.1, 0.3, 0.5) and R0 (0.8, 1.5, 2.2 MAF), and paired combinations of these values represent initial conditions for each simulation. The city's water demand is based on usage records for Brownsville, Texas, a town of 120,000 using an average of approximately 21,000 ac ft per year (Table 1).
 The vast majority (85%) of regional water use is agricultural, much of it directed toward relatively low valued irrigation activities (e.g., cotton), and a growing municipal population (expected to double by 2050) provides substantial economic incentives for agricultural to urban water transfers. While economic incentives alone do not always translate to an increased volume of trades [DeMouche et al., 2003], this does appear to be the primary driver in the Valley [Chang and Griffin, 1992]. The regional water market is relatively efficient and has presided over the steady transfer of permanent rights from irrigators and urban users in recent years [Griffin, 1998]. Permanent transfers are almost always approved but must navigate a regulatory process that can take over a year to complete. Leases tend to raise fewer concerns over third-party impacts and are subject to a simplified approval process that is often concluded in a few days [Griffin and Characklis, 2002]. Lease transactions require only that the buyer and seller deliver a one-page document to the watermaster detailing their respective account numbers and the volume of water to be transferred (price information is optional). The ease of completing these transactions contributes to the high level of market activity, with an average of nearly 70,000 ac ft of water transferred via leases each year [Watermaster's Office, 2004]. The structure of the market leads to the assumption that spot market transaction costs are essentially negligible. While this assumption is reasonable within the Valley, it may not be so in many other regions, a factor which may bias this analysis in favor of spot market leases.
 All water markets exhibit idiosyncrasies. In the case of the LRGV, the most noteworthy is that current rules allow for permanent rights to be transferred between agricultural and urban users, but only allow lease transactions between similar user types (e.g., urban to urban), giving rise to two spot lease markets [see Characklis et al., 1999]. The municipal market has fewer transactions, as cities tend to hold volumes of permanent rights well in excess of average usage, while the agricultural lease market is much more active (1514 transactions over the period 1994–2003; average price $22.60 per ac ft). Efforts to eliminate this prohibition on intersectoral leasing are currently being undertaken [South Central Texas Regional Water Planning Group, 2000], and when this occurs it seems likely that the lower marginal value of irrigation water will lead to regional lease prices similar to those observed in the agricultural market. These simulations assume this is the case and that lease prices from the agricultural market are representative of what would be observed in agricultural to urban transactions.
 It should also be noted that while there is some evidence of serial correlation (again using the Pearson test) in the spot price data set as a whole, once the data are separated into these two subsets the effects of serial correlation becomes quite weak (1-month lag typically has an R2 < 0.10). In effect, it appears that when reservoir storage drops below (rises above) the threshold level, the mean monthly price increases (decreases), but subsequent price variation about the mean is essentially random. This randomness in spot market prices is likely due, in large part, to the decentralized nature of the market. While the prices of the most recent lease transactions can be obtained from the watermaster's office, it seems clear that most transactions are completed with only a general knowledge of the current level of water scarcity (i.e., reservoir level is low or its not). This leads to a spread in prices, even those observed in the same month with similar reservoir levels. Such behavior might suggest that a high-volume buyer, motivated by large potential savings, could find a lower price by increasing the amount of time and effort spent looking for a seller. However, correlations between spot market prices and the volume purchased yielded no evidence of a statistically significant relationship. Finally, consideration was also given to adjusting the spot price data to reflect real prices over the period 1993–2002. Both the producer price index for all farm products (which rises from 106.3 to 111.5 over this period) and the Texas index of prices received for farm products (which falls from 98.0 to 93.0 over the same period) seem likely to be strong indicators of variation in the marginal benefits of irrigation water over time, but the mixed directions and small changes in these indices led to the decision to use unadjusted (nominal) lease prices. Chi-square tests yielded little evidence that monthly lease data fit any standard distribution type, so lease prices are represented as monthly data lists. The simulation is set up to randomly sample spot prices from one of monthly lists, with the decision as to which made according to the current storage level are sampled monthly from the appropriate list.
 Option contracts have been discussed in the LRGV but are not yet actively traded. Their introduction into the market, however, would appear to be a logical step with few bureaucratic hurdles. Within the simulation, a single type of European call contract is offered, with the option purchased on 31 December (t = 0) and exercised on 31 May (t = 5). The date 31 May falls just before the peak usage months in both the municipal and agricultural sectors and therefore seems to provide a logical point for users to assess their current supplies and make choices. There are, of course, a host of other call dates that might be suitable as well, and consideration might even be given to developing option contracts with multiple exercise dates, but such considerations are left for future work. Given an initial reservoir storage (R0), the conditional probability of May storage (R5) being above or below 1.43 MAF can be computed, and it is assumed that the market incorporates this information into option pricing. As a result, equation (1) is modified such that the option price is conditional on R0, with
The exercise price (pX) is set at $15 per ac ft, a level in line with the mean spot lease price when reservoir level is above the threshold level and one that is therefore assumed to be sufficient to attract enough irrigators to create an options market. Using this exercise price, the resulting option prices are $13.26, $11.36, and $2.18 per ac ft when initial storage levels are 0.8, 1.5, and 2.2 MAF, respectively. The annualized price of permanent rights (pR) is $22.60 per ac ft, but considering that only about 0.7 ac ft are allocated to these rights in an average year, the effective annualized cost of water obtained via these rights is $31.17 per ac ft. There is considerable anecdotal evidence that the price of permanent rights has been growing relatively rapidly in recent years (C. Rubinstein, Rio Grande watermaster, personal communication, 2003), but this is difficult to quantify directly since the sales prices for permanent rights are not recorded by the watermaster. The annualized cost of rights corresponds to a $1000 per ac ft purchase price amortized over 40 years at a 6% discount rate, and assumes that the real value of the right increases at around 4% per year over that period.