multiple criteria dynamic spatial optimization to manage ... - CiteSeerX

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MULTIPLE CRITERIA DYNAMIC SPATIAL OPTIMIZATION TO MANAGE WATER QUALITY ON A WATERSHED SCALE T. O. Randhir, J. G. Lee, B. Engel ABSTRACT. This article develops a dynamic spatial optimization algorithm for watershed modeling that reduces dimensionality and incorporates multiple objectives. Spatial optimization methods, which include spatially linear and nonlinear formulations, are applied to an experimental watershed and tested against a full enumeration frontier. The integrated algorithm includes biophysical simulation and economic decision-making within a geographic information system. It was observed that it is possible to achieve economic and water quality objectives in a watershed by spatially optimizing site-specific practices. It was observed that a spatially diversified watershed plan could achieve multiple goals in a watershed. The algorithm can be used to develop efficient policies towards environmental management of watersheds to address water quality issues by identifying optimal tradeoffs across objectives. Keywords. Spatial optimization, Economic and water quality modeling, Multiple criteria decision-making, Dynamic programming, Watershed, Nonpoint source pollution.

E

fforts to increase agricultural productivity can place a severe strain on land and water resources, often resulting in deteriorating water resources and ecosystems. During recent decades, unmanaged and rapid exploitation of natural resources has increased nonpoint source pollution (NPS) that has both onsite and offsite affects on human communities and ecosystems. The magnitude of this problem is reflected in the annual cost estimates on damage to water quality through agricultural sources that range from $2.2 billion (Clark et al., 1985) to $7 billion (Ribaudo et al., 1989). Given the multifarious nature of impacts of NPS pollution, many individuals, communities, and agencies are struggling with how to manage resources at watershed scales to achieve an acceptable mix of products and services (Lovejoy et al., 1997). An integrated approach to decision-making at a watershed level is necessary to combine information on spatial dynamics, multiple attributes, and processes. One such approach is to spatially optimize watershed land uses and to lessen NPS pollutants with minimal economic loss (Randhir, 1995). This approach can be used to design spatial resource policies and structural practices, to identify efficient and sitespecific production decisions, to identify efficient productservice mix in planning production of food and fiber, to plan housing and urban development, to protect

Article was submitted for publication in February 1999; reviewed and approved for publication by the Soil & Water Division of ASAE in December 1999. The authors are Timothy Randhir, Assistant Professor, Department of Natural Resources Conservation, Holdsworth Natural Resources Center, Rm. 320, University of Massachusetts, Amherst, Massachusetts; John Lee, Professor, Department of Agricultural Economics, and Bernie Engel, ASAE Member Engineer, Professor, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, Indiana. Corresponding author: T. Randhir, University of Massachusetts, Holdsworth Natural Resources Center, Rm. 320, Amherst, MA 01003, phone: 413.545.3969, fax: 413.545.3943, e-mail: .

ecosystems, and to meet industrial resource demand. This article develops such a dynamic optimization method for spatial watershed management. Typical decision-making processes in watershed management are primarily multi-objective in nature. In this process, spatial dynamics play an important role in assessing pollutant generation and transfer, site-specific economic potential, and the extent of offsite damages. Hydrology and the type of topography dictate that activities in a particular location in a watershed are not independent of those in other locations, but form a dynamic system of flows. Thus, to improve watershed planning and policy, a spatial optimization approach that integrates multiple objectives, spatial dynamics, hydrology/water quality processes, and Geographic Information Systems (GIS) is necessary. This study develops capabilities to address multiple pollutants, use of hydrologic/water quality simulation models and linkage to GIS in a watershed framework (Randhir, 1995). The general objective is to develop an integrated framework for spatial optimization in watersheds and to derive optimal practices for a case watershed. Specifically, (1) to develop a spatial search method and algorithm that can incorporate multi-criteria and biophysical simulation in a spatial optimization framework; (2) to evaluate the efficiency gain in spatial dynamic model using an experimental approach; and (3) to derive the optimal agricultural plan for an actual watershed.

BACKGROUND Most water quality policies depend heavily on spatial information and assessment from integrated modeling. Three major components that are important for developing an integrated approach are GIS, process simulation, and economic optimization. The first component, GIS, plays a critical role in acting as a common platform to integrate various models. This also allows direct use of field parameters and databases that are site-specific. Efforts to

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update and integrate available process models into GIS are on the rise. The second component is process simulation modeling for watershed planning that includes cropgrowth, hydrology, and pollution models. A commonly used crop growth model is EPIC—Erosion Productivity Impact Calculator (Williams et al., 1989). Hydrologic models are useful to simulate transfer and fate of NPS pollutants and can be classified as distributed parametersingle event types (such as AGNPS and ANSWERS), and lumped parameter, continuous-time types (HSPF, Johanson et al., 1981; WEPP, Laflen et al., 1991; SWAT, Arnold et al., 1995). These models have excellent potential for developing a spatial decision support tool for watershed management. The third component is economic optimization, where evaluation of alternative management systems is involved. While economics of watershed management can be modeled through benefit-cost estimates within CREAMS, GLEAMS or EPIC, these models give only “lumpy” estimates (the aggregate figures are estimated without evaluation of spatial dynamics). The dynamic component involving integrated physical and economic decision making and evaluation based on the state of the system is very crude or lacking in most of these models. Several researchers (Randhir, 1995; Bouzaher et al., 1990) have used dynamic programming techniques to model the dynamics in nonpoint source pollution in an economic and water quality framework. Bouzaher et al. (1990) used a dynamic programming algorithm to generate sediment abatement cost frontiers that can be used in watershed planning. To incorporate multiple objectives into decision-making, many studies (Romero et al., 1988; Gershon and Duckstien, 1983) have used a multi-criteria decision-making approach for planning natural resource use. Romero et al. (1988) illustrated the importance and use of this technique in natural resources management. Gershon and Duckstien (1983) used a multi-objective approach in planning river basins. Less work exists in adapting these techniques for watershed planning in an integrated framework. Lack of a comprehensive spatial optimization approach can often result in less efficient management of watershed systems, conflicting objectives, unrealistic assumptions like cell-independence, and a deviation from optimal spatial plans. Furthermore, communities in watersheds often face choices involving multiple objectives rather than a single criterion.

STAGE I: MODEL DEVELOPMENT A small watershed with eight elements was used to develop a small-scale model of the spatial optimization procedure. Each cell in the watershed was assigned an arbitrary label and the aspect (flow direction) map was based on flows in the Animal Science Watershed in northcentral Indiana. This small-scale model was used to develop and test various optimal search procedures. In testing watershed practices, the trade-off frontier derived from optimizing two objectives (maximizing net returns from crop production and minimization of sediment loss from the watershed) was used as a test criterion. Primary tillage practices in the watershed include moldboard plowing, chisel plowing, and no-tillage. All possible combinations of location-specific management practices in the watershed (full enumeration set) were simulated using AGNPS model (Young et al., 1989) recursively through a FORTRAN sub-routine*. This frontier was derived from the full enumeration set containing all possible combinations of activities at each location in a watershed (fig. 1). Running AGNPS model for each change in activity in each location to estimate watershed scale outcomes generated the full enumeration set. The maximizing elements of this set of combinations form the full enumeration frontier (the boundary points identifying solutions under global maximizing). The full enumeration frontier was used as a baseline criterion to test the efficacy of various search methods. This efficacy test was also used to evaluate the extent of error in using the assumption of cell-independence. WATERSHED SEARCH METHODS Search methods aim at evaluating cost effectiveness of practices in improving surface water quality of a watershed. Search methods considered in this study vary by modeling assumptions used. Spatial Linearity. Lumped parameter models, such as EPIC, CREAMS, or USLE, have been used to generate soil erosion or nutrient runoff coefficients as inputs into farm, regional, or national mathematical programming models to evaluate the economic costs of reducing NPS pollution. This modeling procedure inherently assumes independence among locations or cells within a farm or watershed. To

METHODS The methods follow in two stages: model development (Stage I), and application (Stage II). In Stage I, a smallscale (8 cells) watershed was used to develop a full enumeration frontier determining optimal tradeoff values for economic and water quality objectives. Among various spatial optimization methods (linear, contribution ratio, and spatial dynamic programming), the method with minimal error in predicting the full enumeration frontier was identified. The identified method was then used to develop an algorithm that dynamically uses simulation and GIS information to optimize watershed practices. In Stage II, the procedure was extended to include more than two objectives, nonlinear preferences, and applied to a larger watershed to derive spatially optimal plans. 292

Figure 1–Full enumeration set and trade-off frontier in spatial optimization.

*

Timothy W. Pritchard of Ohio State University developed this FORTRAN sub-routine while at the Department of Agricultural Economics, Purdue University. TRANSACTIONS OF THE ASAE

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represent this method, let S(Xij) be the sediment deposition rate in the ith element† with jth management practice of the watershed, R(Xij) be the economic returns from the ith element using jth management practice, and Xij is the optimal practice j in the element i. The spatially linear model can be represented as maximizing a linear objective function (eq. 1), subject to equation 2.

∑∑

Maximize: Xij

i

Subject to:

(1)

X ij ≤ L

(2)

Xij

Subject to:

X sij =

j

∑ SC

∑∑R

∑∑S

X ij

(3)

X ij × θ i ≤ L

(4)

j

j

where θi is the contribution ratio (relative contribution of element i to the total pollutant loading_ at the outlet cell) associated with the ith element, and L is total sediment leaving the watershed. The contributions ratio for each cell was evaluated using results from the AGNPS model. This approach attempts to capture the dynamic relationship by linking each element of the watershed with the outlet element through the use of contribution ratios. A problem with this procedure is that it does not trace the path and element-to-element interactions in identifying and updating Xij, but calculates only the net impact at the watershed outlet. The contribution ratio can be misleading because of predetermined contribution levels are used instead of recursive calculation. Spatial Dynamic Programming. This dynamic model is a mathematical representation that represents cell-to-cell dynamics in a watershed. The net watershed return is maximized (eq. 5), subject to a dynamic equation 6 (state of pollutant stock) and a watershed outlet condition (eq. 7).

The terms element, cell, and location are used interchangeably to represent a spatial unit.

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(5)

∑ S P X sij j

X sij . . . ∀ s,i ∈ WS

(6)

j

∑S

X Tij ≤ L

(7)

j

In this model, s represents the spatial stage of the flow, and WS represents the set of all elements in the watershed. The indexes C and P refer to the current element and elements of downstream stages, respectively. This methodology is based on the idea that each stage is a subwatershed, consisting of all the elements in the current stage and elements of all downstream stages determined by aspect-flow direction map (backward recursion). The total terminal sediment output (eq. 7) can be parameterized to create a dynamically efficient frontier for identifying optimal management practices. Evaluating this system of equations requires extensive computing requirement for large watersheds and for smaller cell resolution, which could lead to the “curse of dimensionality” (exponential increase in the number of combinations to optimize). However, with the use of Bellman’s equation (Bellman, 1957) applied over geographic space, dimensionality of the problem can be substantially reduced. Bellman’s approach starts at a terminal stage and iteratively solves for each stage backwards (backward recursion). The recursion continues for each stage and optimizes using the values from earlier stages. The model can be parameterized for sediment loading or other hydrologic/water quality parameters at the terminal cell. All cells in the watershed are linked to each other through the state equation 6, which relates cell-to-cell dynamic movement of sediment (stock variable in dynamic programming). Thus, spatial decision making through dynamic programming involves maximizing an objective function subject to the spatial state equation and a terminal condition (Randhir, 1995). A multi-objective method (eqs. 8 and 9) is used to test efficiency gains from multiple criteria, as opposed to a single objective in dynamic modeling. Because watershed planning involves tradeoff in multiple objectives, this is an important component for effective policies. Minimize: L∞ = d

(8)

Xij

Z* O – Subject to: φ

O

∑ ∑ ZO X ij i

*

Z O–

j

∑ ∑ Zl O i

†

X sij

j

j

i

i

∑S

Subject to:

and

In this formulation, it is assumed that interdependence among cells is absent. The model identifies the elementmanagement combination that maximizes total watershed returns,_constrained by the total sediment deposition at the outlet (L). By parameterizing the total allowable sediment level, trade-off frontiers can be generated. Contribution ratio. Several studies have adopted sediment-delivery ratios in conjunction with estimated soil erosion rates as a proxy for evaluating NPS sediment yields watersheds. This approach is predominant in water quality studies to evaluate offsite effects and to derive policies regarding least cost means of reducing NPS pollution. The problem representing the contribution ratio approach, presented in equation 3 and equation 4. Maximize:

i

j

∑∑S i

Xij

+ R X ij

∑s ∑ ∑ R

Maximize:

≤d

(9)

j

where L∞ is the Lp metric at p = ∞, i.e., the maximum deviation from among the objectives (d) is minimized. Z*o is the best possible value of the oth objective function and 293

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Zlo is the worst possible value in attaining the oth objective derived from optimization of that objective. This method, called compromise programming in the optimization literature, is capable of combining multiple objectives. The parameter φo is the weight assigned to each objective, with n o Σ φ = 1. The model minimizes the maximum distance from o=1 the target and worst possible cases, thus balancing on the efficient tradeoff frontier. Spatial Optimization. To apply these models at a watershed level, it is necessary to integrate the efficient model formulation into a spatial optimization algorithm. This algorithm can be used in developing a complete model (SMDP-Spatial Multi-attribute Dynamic Programming). The algorithm (fig. 2) and its sub-components represent an integrated procedure. Let the stages (s) of each cell be indexed from 1 to T. The complete algorithm searches the watershed for the outlet cell based on the fact that it represents the only outflow point in a watershed. Once this cell is identified, it is labeled as stage T. Tracing through the aspect map, all the cells that empty into stage T are identified as stage T–1. This procedure is repeated to complete all locations of the watershed, terminating with the elements in the boundary (Stage 1). The algorithm performs the optimization procedure on each stage, starting at the last stage (indexed T). The AGNPS model simulated the levels of sedimentation and nutrient runoff, while crop productivity estimates are derived using the Erosion Productivity Impact Calculator (EPIC). The cost under each management scenario was calculated for each element using prices in the region. The multi-criteria approach was utilized to optimize the management practices at each stage, and the optimized choices are identified for this stage. After these calculations, the algorithm then shifts to the next stage (T–1). The simulations and economic performances are again optimized for each of the cells in this stage (T–1), and optimized with a constraint describing the transition equation that links stages (T–1) and T. Because the coefficients for the optimization are calculated for the entire sub-watershed starting at (T–1) through the final stage (T), all the dynamic interactions and paths between these stages are fully accounted for. The optimal choices are again fixed for this stage (T–1) and the downstream stage (T), before moving to the next stage (T–2). This recursive iteration continues until optimal management practices were identified for the entire watershed. The next phase in the integrated model is to regroup cellblocks with similar practices and clumping, using GIS. This will enable

Figure 2–Spatial multi-attribute dynamic programming algorithm. 294

formation of contiguous cellblocks that are easier implementation of cropland management practices. This is needed in watersheds were the optimal outcome need to account for local agricultural practice. For example, tillage operations can be difficult to change for each cell. Identical blocks can be delineated for tillage practice by clumping areas with similar practices and by using expansion functions available in GIS. STAGE II: WATERSHED APPLICATION In implementing the SMDP algorithm at a larger scale, the system of hydrologic, biophysical simulation models, optimization routines and GIS modules were integrated into a model called WISDOM or Watershed Integrated Spatial Dynamic Optimization Model (Randhir, 1995). In this model, a location (cell or element) was defined as a typical analytical land unit following GIS standards (like discrete raster cells or polygons) and referenced by geographic coordinates. The structure of the WISDOM model and its components is presented in figure 3. Two biophysical simulation models, AGNPS (Young et al., 1989) and EPIC (Williams et al., 1989) were used to model spatial hydrology and crop growth processes. The GIS data were linked using the GRASS-AGNPS interface developed by Srinivasan and Engel (1994) to manage and manipulate input and output data from AGNPS. The information on spatial flow and growth processes simulated from these models was stored as appropriate layers of information in a multi-layered database. The AGNPS model was used to generate pollutant loading and spatial transfers for various cropping systems at each stage (identified in staging module) of the dynamic system. Pollution information generated included nitrogen and phosphorus pollution in runoff water, nutrients attached to sediment, and direct sediment loading. Crop productivity and soil erosion estimates under each management practice

Figure 3–Structure and operation of the WISDOM model. TRANSACTIONS OF THE ASAE

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and soil type were simulated using EPIC in the multilayered structure. The multi-layered structure also used separate layers for different processes and practices that are spatially variable. Information on the relative preference to each of the economic and environmental attributes (relative weights) and the nature of the individual utility function defining preferences under each attribute (form of utility function) was used to develop a multi-attribute utility function within the Multi-Attribute Utility module. This function was designed using the Multi-Attribute Utility Theory (MAUT) with preference weights, a nonlinear objective function, and limits (extreme values) on certain attributes. This form of utility formulation enabled a non-linear ranking of different attributes, taking into account relative weights associated with attribute values. This module also used various attribute levels from the multi-layered structure. A separate staging module indexes each cell into various stages in the dynamic system of the watershed. This module used GIS analytical functions to delineate the watershed locations into different stages based on an aspect mapping (direction of flow). The SMDP algorithm used information from three modules: multi-attribute utility module, multi-layered database, and the search module. In the search module, an interlayer search was performed to locate targeted values in the watershed. The search within a layer was to identify transition from one geographic state of flow to another. Iteratively, the optimal values of each location were derived using the SMDP module. These optimal values were used to create new layers of information in the multilayered structure. Economic and environmental attribute values generated by the optimal plan were evaluated through comparison to a baseline, which represents the current cropping patterns and practices within a watershed. The WISDOM model was applied to a 332 ha Animal Sciences watershed located in northcentral Indiana, a typical watershed in the Midwest. Extensive spatial watershed information on hydrology, topography, land ownership, and land use is available. An initial step in preparing the watershed for using the WISDOM model is the definition of locations. A grid of 200 m × 200 m (4 ha each) was used. A total of 83 cells cover the entire watershed. For reference, each location is labeled for use in the spatial optimization model. The aspect map depicting the flow direction in the Animal Sciences watershed is presented in figure 4. The EPIC model was used to estimate crop yield distributions for various combinations of cropping systems, input levels, tillage practices, and weather conditions (Randhir, 1995). The model conditions simulated crop yields using 25 years of weather data to generate yield estimates under each cropping system. The simulations were also repeated for alternative placement of crops in a sequence. The revenue stream from each cropping system was calculated using historical prices in the region and deflated for inflationary effect with “All Crop Price Index” with 1980 as a base. The cropping system considered included continuous corn (CC) and corn-soybean (CS); chisel tillage (CT), moldboard tillage (MT), and no till (NT); and a moderate level of fertilizer (MF), and high level of fertilizer (HF). The C-values (cropping factor in AGNPS) for alternative cropping systems were calculated VOL. 43(2): 291-299

Figure 4–Aspect/Flow direction map of the animal sciences watershed.

using Revised Universal Soil Loss Equation (RUSLE). Because AGNPS is an event-based model, a design storm of 50.8 mm of rainfall with an Erosivity Index (EI) of 32.3 was used in the watershed. This storm event produces erosion comparable to average annual soil erosion from the watershed and was constructed with expert opinion from agricultural engineers familiar with the region.

RESULTS AND DISCUSSION SPATIAL DYNAMICS EXPERIMENT FOR THE EIGHT-CELL WATERSHED A sub-watershed of 6.5 ha was identified in the Animal Sciences watershed and used to conduct the spatial dynamics experiment. The experimental watershed had three stages. All the three models (linear model, contribution-ratio model, and spatial dynamic programming model) were implemented in the experimental watershed to generate trade-off frontiers. The tradeoff values from all the three search methods are presented in table 1. The tradeoff between income and sediment in the case of element-independence model indicates that income was $1,968 for unrestricted sediment loss, and declined to $195 for the 98% reduction of sediment loss. The trade-off frontier that defines the returns-percent reduction is piecewise linear, with a weakening tradeoff. In the linear model with element independence, the dynamic component was absent. In the next two models, the dynamic component was incorporated into the decisionmaking. The first is an approximated dynamic model that uses a constant ratio (contribution ratio), to capture inherent dynamics in the watershed system. The parameterization results of the model with contribution ratios are also presented in table 1. The trade-off frontier obtained using the model with contribution ratios was concave (fig. 5). The trade-off values were also different from those that were obtained from the spatially linear model with independence assumptions. However, the tradeoff frontiers do not lie on the efficiency frontier (obtained from full enumeration) and do not capture the true element-to-element interaction. The results of the dynamic programming model are presented in table 1. The trade-off frontier obtained from the dynamic model was derived at various levels of 295

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Table 1. Trade-off results for spatial search methods Sediment at the Outlet (tons)

Returns ($ )

Sediment Reduction (%)

Trade-off with Element Independence 0.1 0.3 0.6 1.7 3.7 4.4 5.7 6.3

195 486 973 1744 1840 1872 1936 1968

98 95 90 73 41 31 10 0

Results with Contribution Ratio 0.3 0.6 1.0 1.8 3.7 5.0 5.7 6.3

399 959 1565 1744 1840 1904 1936 1968

96 90 84 72 41 21 10 0

Spatial Dynamic Programming Results 0.1 0.3 0.6 1.2 1.8 2.1 2.5 3.2

466 1210 1712 1808 1872 1904 1936 1968

96 90 83 62 45 34 20 0

Figure 5–Overlay of frontiers generated by search methods.

sediment and income. This parameterization can be replicated for more intermediate levels of these trade-offs. The dynamic model generated a concave trade-off frontier that represents substitution between income and sediment reduction. The trade-off frontiers from various methods are overlaid over the full enumeration frontier to evaluate the extent of deviation from efficient points (full enumeration frontier). The full enumeration set (fig. 1) represents a global set to be optimized. Along this frontier, income decreased for each additional increase in sediment restriction. The decline in income was marginally higher for higher sediment restriction levels. An overlay-analysis of the frontiers generated by each model is presented in figure 5. The optimal solutions are represented by trade-off frontiers generated by each model. It was observed that the element-independence model had the highest deviation from the full enumeration frontier compared to all other models. The deviation in estimates increased for each addition in sediment restriction. An 296

important observation was that at unrestricted levels, where no constraints are placed on sediment yields in the model, all models resulted in the same estimate. However, for higher levels of restriction in sediment, deviation from the true estimate value increased. The element-independence and contribution-ratio methods overestimated the cost (foregone profit) of obtaining a given level of sediment reduction compared to the dynamic programming approach. The difference in profit estimates is insignificant for below 70% restriction in sediment in the eight-cell experimental watershed with three stages. This deviation can be substantial in applications to watershed modeling with higher resolution, more stages of flows, and higher heterogeneity. An alternate interpretation is that for a given expectation of profit, there could exist a wide disparity in assessment of sediment reductions from various approaches. For example in figure 5, for an expectation of $1,700, full enumeration and spatial dynamic model estimate 62% reduction in sediment, while linear and contribution-ration models predict 46%. This difference could have policy implications. It is clear from figure 5 that in all models, higher deviation among methods was observed at increasing levels of restriction in sediment loss. The shape of the frontier generated by the dynamic model coincided with the full enumeration frontier. This indicates that the dynamic component can be critical in developing reliable estimates of trade-offs for use in decision-making and policy. The multi-criteria evaluation in the dynamic model was accomplished in the small-scale experiment using a compromise-programming framework with an income to sediment reduction preference ratio of 7:3. The plan generated a return of $1,712 with 0.6 tons of sediment, which coincides with the frontier generated in the dynamic model. Weights used represent a function (a straight line in the case of two objectives) that is similar to the indifference curves. Because unique optima were obtained, compromise programming can achieve reliable results. In cases of more than two objectives and nonlinear preferences, this may not always be unique. Changes in the weights assigned to objectives in a compromise programming formulation reflect a tangential movement of the preference line along the efficiency frontier. This clearly demonstrates the applicability and scope of multicriteria methods in watershed modeling and opens vistas for more realistic incorporation of multiple objectives with appropriate nonlinear preference models. WATERSHED-SCALE APPLICATION The entire Animal Sciences watershed was discretized into 83 grid cells for applying the WISDOM model. The application of the staging procedure in the Animal Sciences watershed resulted in 12 dynamic stages. Each stage represents a particular state in the SMDP algorithm. A listing of the cells and stages is presented in table 2. The final outlet cell (cell number 80), forms the last stage (T) of the dynamic process (by backward recursion in the staging process). This cell forms the initial stage in the spatial optimization process. All cells (73, 74, and 81) that flow directly into cell number 80 were included in the next stage (stage 2). A similar procedure was repeated for all

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Table 2. Dynamic stages in the animal sciences watershed for subdivision into 83 cells Stage

Cell Label

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage 9 Stage10 Stage 11 Stage 12

80 73, 74, 81 63, 64, 82, 75, 76 53, 54, 65, 66, 67, 77, 83 52, 40, 55, 56, 57, 58, 68, 78 39, 26, 27, 41, 42, 43, 44, 59, 69, 79 38, 25, 13, 14, 28, 29, 30, 45, 46, 60, 70 12, 1, 2, 15, 16, 17, 18, 31, 32, 33, 47, 61, 71 3, 4, 5, 19, 20, 21, 34, 48, 62, 72 6, 7, 8, 22, 35, 49 9, 23, 36, 50 12, 10, 11, 24, 37, 51

Cells (no.) 1 3 5 7 8 10 11 13 10 6 4 5

stages, terminating in a cell (boundary) that does not possess inflows. At this point, it is essential to check that the cells in each stage cannot flow into each other, but can have flows into the cells in the next stage. If flows occur between elements in a particular stage, that indicates an error in the assignment of the dynamic flows. SPATIAL OPTIMIZATION RESULTS The optimal plan derived by applying the WISDOM model to the Animal Sciences watershed is presented in figure 6. It was observed that spatial diversification of cropping systems (cultivation of several cropping systems that are distributed throughout the watershed) is necessary to accomplish multiple objectives. A corn-soybean rotation with moldboard plowing and a medium level of fertilizer was found optimal for 84 ha. This allocation was distributed in five contiguous blocks (see fig. 6) spread throughout the watershed. A cropping system of corn-soybeans under notill and medium level of fertilizer application was optimal in four blocks forming a total area of 112 ha. The cornsoybean rotation under chisel plowing and medium level of fertilizer use was optimal in two cells (8 ha). An optimal crop plan of corn-soybeans under no-till and high fertilizer use was generated in six blocks accounting to an area of 64 ha in the watershed. Cultivating continuous corn using chisel plow and applying a medium level of fertilizer was optimal for the 16 ha block. Continuous corn under no-till and high level of fertilizer was optimal for the block of 16 ha. The optimal cropping system in a block of 32 ha was continuous corn under no-till and medium level of fertilizer use. A general result of this optimization is the need to match site-specific characteristics in a watershed to cropping systems using spatial hydrologic dynamics (fig. 6). Such a spatial optimization procedure can be used to achieve optimal values of multiple attributes. Bouzaher et al. (1990) and Braden et al. (1989) obtained similar results. No-till cultivation is required in areas adjacent to channels in the watershed. In areas of higher erodibility, no-till cultivation was also found to be preferable. The spatial distribution of the optimal plan is an indication of the need for micro-level targeting that incorporates site-specific information on the release and deposition rates of NPS pollutants. This highly diversified mosaic of agricultural cropping activities is not surprising because the reduction in NPS pollution needs to account for the relative position of location and physical flows in the watershed. VOL. 43(2): 291-299

Figure 6–Optimal spatial plan for animal sciences watershed. Note: A block represents a collection of cells with the same cropping system and the cells of a block are contiguous to each other. Such reclassification is useful in implementing the optimal plan.

Furthermore, locations with higher vulnerability to chemical pollution need minimal field operations, while other locations may have different agricultural activities. It is observable from the optimal plans that by varying the type of cropping system at specific locations in a watershed, overall loading of NPS pollutants is reduced. Less fertilizer application to vulnerable areas is necessary to reduce pollutant loading from the overall watershed. Under the optimal plan, the nitrogen attached to sediment was reduced by 0.5 kg/ha from the pre-optimization (PO) level. The nitrogen concentration in runoff water dropped by 1.66 ppm from the pre-optimization level. Phosphorus in the sediment is dropped by 0.25 kg/ha from the PO level, while phosphorus loading in the runoff dropped by 1.82 kg/ha (0.15 ppm) from the PO level. The reductions can be substantial for the watershed area. Clay content in the sediment declined to 71.9 tons from a baseline level of 85.5 tons. Out of the total sediment loading of 728.8 tons delivered to the outlet, 100.4 tons was in the form of silt, 491 tons as small aggregates, 50 tons as large aggregates, and 15.4 tons in the form of sand. It was observed that at the pre-optimization (baseline) level, the discounted aggregate income stream in the Animal Sciences watershed was $0.12 million with a variance in income of $3 million. The aggregate variance is calculated as an additive sum of income from different cells without taking into account the covariance among cells. This corresponded to a coefficient of variation (CV) of 14%. The income in the optimal plan was lower ($0.11 million) than the baseline with a risk of $4.4 million 297

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Table 3. Spatial distribution of income and variance in the optimal plan Area (Ha) 88 108 68 8 12 4 16 28

Cell Nos. 1, 2, 3, 12, 15, 24, 34-37, 49-51, 61, 65, 68, 71-74, 78, 81 4-8, 10, 13, 14, 16, 18, 27- 29, 31, 33, 41, 44, 46-48, 54, 55, 59, 60, 66, 70, 79 9, 19, 21-23, 26, 32, 38-40, 45, 52, 57, 62, 67, 69, 77 11, 53 17, 65, 78 30 20, 76, 82, 83 42, 43, 56, 63, 64, 75, 80

Income ($)

Variance

605

6,919

378 381 460 605 405 477 519

12,385 11,568 13,070 2,800 14,774 2,242 1,928

(a coefficient of variation of 19%). The optimal plan generated a distribution of income and risk that vary over space (table 3). The spatial variation in economic outcomes can be attributed to soils, cropping systems, and temporal changes in crop yield and practices. The blocks of 76 ha generated an average profit of $605/ha, with a variance of $6,919 in income (CV is 0.14). The average income was $378/ha in 100 ha with a variance of $12,385 in income. A CV of 29% indicates that the income can vary within the range of ± 29% from the mean level. The block of 60 ha generated an average income of $381/ha with a variance of $11,568 in income (CV is 0.28). In two cells, the income was $460/ha with a variance of $13,070 in income, corresponding to a CV of 0.25. The blocks formed by 12 ha had an income level of $605/ha and a variance of $2,800 in income (CV is 0.09). A 16 ha block generated an average income level of $477/ha with a variance of $2,242 in income (CV is 0.10). An income of $380/ha was generated by the optimal plan in a 68 ha block, and with a variance of $11,568 in income (CV is 0.28). A 28 ha block generated an income of $519/ha with a variance of $1,928 in income (CV is 0.08). The average income per hectare, from the entire watershed, with optimal spatial plan was $336, while the variance in income as $5,473 (a CV of 22%). A general conclusion is that by using the spatial optimization method, one can make scientific decisions on watershed use, especially when multiple objectives are involved.

CONCLUSION Resource managers and policy administrators face tough decisions when trying to decrease agricultural NPS pollutants with minimal economic loss. An integrated multi-criteria optimization approach that incorporates spatial dynamics of pollutant generation and movement and the spatial distribution of economic activities can be used to model decision-making in watersheds. This study presents the development of a spatial optimization procedure using spatial dynamic programming. Using an experimental approach, it was demonstrated that the model with the assumption of element-independence could result in inferior solution in identifying optimal watershed practices compared to other models that incorporate spatial dynamics. The use of contribution ratios to capture spatial dynamics can also result in sub-optimal watershed planning. By using Bellman’s dynamic programming approach applied over geographic space, a spatial dynamic optimization model was developed for watershed planning. It was found that the spatial optimization technique could be used to identify optimal land use plans in achieving multiple criteria decisions in watersheds. The efficiency is 298

tested as a deviation from the frontier generated using a full enumeration experiment. The role of multiple criteria was identified in this study using a compromise-programming framework to identify tradeoff among multiple objectives. The spatial optimization technique, along with the multi-criteria procedure, was used in developing a spatial multi-attribute dynamic programming algorithm. This algorithm incorporates multi-objective evaluation to identify sitespecific management decisions in a watershed. The spatial optimization algorithm is used in developing an integrated framework (WISDOM) consisting of GIS, surface water quality and crop growth simulation models, and their interfaces. By applying the integrated model to a watershed in northcentral Indiana, it was observed that it is possible to achieve economic and water quality objectives in the watershed by spatially optimizing site-specific practices. Similar to results of previous studies, spatial diversification of cropping activities was required for optimality. Most pollutants were reduced from baseline levels with minimal economic loss. The model and solution can be superior to methods that target a single pollutant and resulting in higher levels of other pollutants. It was observed that location-specific changes were needed in the watershed to achieve optimal land use in watersheds. Because the spatial optimization procedure incorporates spatial dynamics in evaluating each stage of the dynamic system, it is an important policy tool for resource managers. The study has developed a comprehensive framework for studying multicriteria and spatial issues arising from managing watersheds to achieve economic and water quality objectives.

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USDA Soil Conservation Service. 1991. State Soil Geographic Database (STATSGO) Data Users Guide. U.S. Dept of Agriculture Misc. Pub. No. 1492. Washington, D.C.: USDA/SCS. Williams, J. R., C. A. Jones, J. R. Kiniry, and D. A. Spanel. 1989. The EPIC crop growth model. Transactions of the ASAE 32(2): 497-511. Young, R. A., C. A. Onstad, D. D. Bosch, and W. P. Anderson. 1989. AGNPS: A nonpoint source pollution model for evaluating agricultural watersheds. J. Soil & Water Conserv. 44(2): 168-173.

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