Application of Adjoint Approach to Oil Spill Problems

Adjoint method is applied to various oil spill problems. A three-dimensional model for describing the dispersion of a quasi-passive substance (a pollutant or a nutrient) and its adjoint model are considered in a limited sea region. Direct and adjoint estimates are used to get dual (equivalent) estimates of the mean concentration of the substance in important zones of the region. The role of dual estimates is illustrated with a few examples. They include such oil spill problems as the search of the most dangerous point of the oil tanker route, the oil dispersion with a climatic velocity, and the dependence of the oil concentration estimates on the oil spill rate. One more example is the application of optimal bioremediation strategy for cleaning a few zones polluted by oil. In this case, instead of oil, the model describes the dispersion of a nutrient released to marine environment. Balanced, unconditionally stable second-order finite-difference schemes based on the splitting method for the solution of the dispersion model and its adjoint are suggested. The main and adjoint difference schemes are compatible in the sense that at every fractional step of the splitting algorithm, the one-dimensional split operators of both schemes satisfy a discrete form of Lagrange identity. In the special unforced and non-dissipative case, each scheme has two conservation laws. Every split one-dimensional problem is solved by Thomas’ factorization method.     

Mathematical Modeling and Numerical Algorithms for Simulation of Oil Pollution

This paper deals with the mathematical modeling and algorithms for the problem of oil pollution. For solving this task, we derive the adjoint problem for the advection–diffusion equation describing the propagation of oil slick after an accident, which we call the main problem. We prove a fundamental equality between the solutions of the main and the adjoint problems. Based on this equality, we propose a novel method for the identification of the pollution source location and the accident time of oil emission. This approach is illustrated on an example for an accident in the offshore of the central part of the Vietnamese coast. Numerical simulations demonstrate the effectiveness of the proposed method. Besides, the method is verified for 1D model of substance propagation.     

Dual oil concentration estimates in ecologically sensitive zones

Propagation of the oil spilling from a damaged oil tanker is considered in a limited sea area. Direct and adjoint estimates of the average oil concentration in special zones are derived by using solutions of the 2-D main and adjoint oil transport problems, respectively. The dual estimates complement each other nicely in studying the oil spill consequences. While the direct estimates are preferable to get a comprehensive idea of the oil spill impact on the whole area, the adjoint ones are specially useful and economical when the accident site-dependence or/and the oil spill rate-dependence of the oil concentration is/are studied only in a few ecologically sensitive zones. Indeed, each adjoint estimate explicitly relates the average oil concentration in a zone to the oil spill rate using the adjoint solution values at the accident site. Being independent of the two parameters (the accident site and oil spill rate), the adjoint solution can be found for each zone regardless of a concrete accident and used repeatedly for various possible values of these parameters. Several examples explain how to decide between two estimates.Thanks to special boundary conditions, the main and adjoint problems are both well-posed according to Hadamard (1923). The dual estimates can be generalized to the three dimensions. The balanced, absolutely stable and compatibie main and adjoint 3-D numerical algorithms by Skiba (1993) can easily be adapted to the problem discussed here.     

Direct and Adjoint Oil Spill Estimates

Propagation of the oil spilling from a damaged oil tanker is considered in a limited sea area. The accident consequences are evaluated by means of direct and adjoint oil concentration estimates in ecologically sensitive zones. While the direct estimates are preferable to get a comprehensive idea of the oil spill impact on the whole area, the adjoint ones are useful and economical in studying the sensitivity of the oil concentration in some zones to variations in the accident site and oil spill rate from the tanker. Thanks to special boundary conditions set at the inflow and outflow parts of the open boundary, the main and adjoint oil transport problems are both well-posed according to Hadamard (1923). The estimates obtained in Skiba (1996a) are generalized to the three dimensions. Balanced, absolutely stable 2nd-order finite-difference schemes based on the splitting method are constructed for the two- and three-dimensional cases, both. The main and adjoint schemes are compatible in the sense that at every fractional step of the splitting algorithm, the one-dimensional split operators of both the schemes satisfy a discrete form of the Lagrange identity (Marchuk, 1995). In the special unforced and non-dissipative case, the schemes have two conservation laws each. Each split problem is solved by the factorization method.     

Connectivity in priority area selection for conservation

The spatial relations of sites within networks of priority areas for conservation is critical to the long-term maintenance of key genetic, population and ecosystem processes. However, these relations have received relatively little attention in the development of mathematical methods for objectively identifying such networks. Here we present a novel heuristic for incorporating connectivity explicitly as part of the model constraints, provide an integer linear programming formulation for the same problem, describe an integer cutting procedure which defines a sequence of non-decreasing lower bounds on the optimal solution and report the results of some computational experiments using these algorithms.     

Air Quality Assessment And Control Of Emission Rates

Mathematical methods based on the adjoint model approach are given for the air-pollution estimation and control in an urban region. A simple advection–diffusion-reaction model and its adjoint are used to illustrate the application of the methods. Dual pollution concentration estimates in ecologically important zones are derived and used to develop two non-optimal strategies and one optimal strategy for controlling the emission rates of enterprises. A linear convex combination of these strategies represents a new sufficient strategy. A method for detecting the enterprises, which violate the emission rates prescribed by a control, is given. A method for determining an optimal position for a new enterprise in the region is also described.     

Air Pollution Estimates in Guadalajara City

A method for estimating the impact of industrial emissions is suggested and applied to the Guadalajara City Metropolitan Area (GCMA). The method is based on solutions to the pollution transport model and its adjoint. Two equivalent direct and adjoint mean pollution concentration estimates are considered for ecologically important zones of the GCMA. The dependence of these estimates on the number, positions and emission rates of industrial plants, as well as on the wind and initial pollution distribution in the GCMA is qualitatively and quantitatively examined. It is shown that the adjoint model solutions serve as the influence functions providing valuable information on the role of each of the industrial plants in polluting different zones within the GCMA. These solutions have been calculated with a balanced and absolutely stable second-order finite-difference scheme based on the splitting method. A method for an optimal allocation of a new industrial plant is considered.     

Industrial pollution transport. Part 1. Formulation of the problem and air pollution estimates

A pollution transport problem is formulated in a limited area. As the pollution sources we take emissions from industrial plants. Physically and mathematically suitable conditions are prescribed on the open boundaries. We show that the problem (as well as its adjoint) is well posed in the sense that a weak solution exists, is unique and depends continuously on its data. Direct and adjoint estimates of the average pollution concentration in an ecologically important zone are given, and the sensitivity of these estimates to perturbations in model parameters is analyzed.     

Industrial pollution transport. Part 2. Control of industrial emissions

Solutions of the pollution transport problem and its adjoint are used to monitor mean pollution concentration in an ecologically important zone. Four strategies of control over pollutants released into the atmosphere by industrial plants are suggested. They differ by the restrictions imposed on the emission rate of each plant. All the strategies use solutions of the adjoint transport problem and assure the fulfillment of the sanitary norm in the zone. A linear interpolation of these strategies also brings pollution level in the zone down to the sanitary norm. A method of detecting the plants violating the prescribed emission rates is also given. A simple example is given to illustrate the strategies suggested.     

A Three-Dimensional, Integrated Seasonal Separate Advection–Diffusion Model (ISSADM) to Predict Water Quality Patterns in the Chahnimeh Reservoir

Seasonal rivers are the main sources of discharge for many lakes and reservoirs. These rivers can deliver pollutants into these water bodies, especially during large events. The fate and distribution of these pollutants within lakes is difficult to predict. Here, a three-dimensional, finite-volume model for predicting lake water quality is used to account for internal advection and diffusion, including the impacts of the inflowing rivers on the velocity field. We used parsimonious sub-models for the source/sink terms for temperature, dissolved oxygen, ammonia nitrogen, phosphorous, phytoplankton, and zooplankton concentrations and tested the model predictions against field measurements from the Chahnimeh Reservoir in Iran. The modeled water quality parameters were in good agreement with the measured values. Results were notably poorer when the three-dimensionality of the model was removed. This study suggests that properly simulating three-dimensional advection is important to properly predict the distribution of pollutants within some lakes and reservoirs and that this model may be directly applicable to systems similar to Chahnimeh Reservoir.     

Optimization and Decision Heuristics for Chesapeake Bay Nutrient Reduction Strategies

Regional policies to achieve water quality goals assign a unique pollution control technology to every pollution source in a watershed, thereby defining a watershed strategy. For watersheds with even a modest number of pollution sources and control alternatives, the decision problem has combinatorial complexity. The perception of complexity—manifested in innumerable feasible watershed strategies—commonly induces the use of simplifying decision heuristics and ad hoc decision rules that reduce decision complexity by limiting the choice set to a “manageable” number of alternatives. In problems with large complex choice sets, these decision heuristics simplify decision making by excluding the vast majority of feasible alternatives a priori. In contrast, watershed-scale optimization enables decision makers to consider all feasible alternatives implicitly, exploiting rather than restricting the complexity of the feasible choice set. This contrast is illustrated using mixed-integer linear programming to identify interstate watershed strategies that achieve Chesapeake Bay nutrient reduction goals for the Potomac River Basin. The use of optimization in collaborative decision making helped refine and capture decision makers’ underlying values and preferences in policy-relevant constraints reflecting equity and political feasibility. Optimization formulations incorporating these constraints identified more effective and desirable management alternatives that would not otherwise have been considered using familiar decision heuristics and traditional comparisons among a limited number of ad hoc scenarios. Incorporating optimization in collaborative decision making generated superior watershed strategies and eased the cognitive limitations on decision making by substituting the computational burden of solving mixed-integer linear programs for decision makers’ cognitive burden of enumerating alternatives and scenarios for environmental systems with combinatorial complexity.     

广佛珠江河网区石油类污染数值模拟研究

基于EFDC模型,构建了广佛珠江感潮河网区三维水动力模型;同时,通过耦合石油类污染物水力输运与悬沙吸附一沉降过程,构建石油类水质数学模型,从而更全面客观地模拟水体石油类污染物的迁移转化过程。通过2001年1月实例模拟及验证,表明模型能较好地模拟再现珠江水体石油类浓度变化过程,为日后珠江水环境质量监控与水质改善、城市河涌整治成果评价及预测等提供技术支持。     

Migration of pollutants in groundwater. II. adsorbable pollutants and numerical dispersion reduction

We discuss here the partial differential equations governing the migration of a decomposing pollutant adsorbing according to a Langmuir isotherm and undergoing 2-dimensional flow in a saturated aquifer. The equation governing the mass transfer of the pollutant to the surfaces within the aquifer are solved in closed form, permitting the use of larger values of the time increment t in the numerical integration of the dispersion-advection equation governing the behavior of the dissolved pollutant. In this numerical integration transverse numerical dispersion is eliminated by using conformal coordinates (velocity potential and stream function), and longitudinal numerical dispersion is very substantially reduced by use of an asymmetrical 4-point formula to represent the advection term. Some representative results are given as contour maps. The mass transfer rate coefficient is estimated as the least positive eigenvalue of a diffusion problem.     

Modeling of Hydrodynamics and Cohesive Sediment Processes in an Estuarine System: Study Case in Danshui River

The Danshui River estuarine system is the largest estuarine system in northern Taiwan and is formed by the confluence of Tahan Stream, Hsintien Stream, and Keelung River. A comprehensive one-dimensional (1-D) model was used to model the hydrodynamics and cohesive sediment transport in this branched river estuarine system. The applied unsteady model uses advection/dispersion equation to model the cohesive sediment transport. The erosion and deposition processes are modeled as source/sink terms. The equations are solved numerically using an implicit finite difference scheme. Water surface elevation and longitudinal velocity time series were used to calibrate and verify the hydrodynamics of the system. To calibrate and verify the mixing process, the salinity time series was used and the dispersion coefficient of the advection/dispersion equation was determined. The cohesive sediment module was calibrated by comparing the simulated and field measured sediment concentration data and the erosion coefficient of the system was determined. A minimum mean absolute error of 4.22 mg/L was obtained and the snapshots of model results and field measurements showed a reasonable agreement. Our modeling showed that a 1-D model is capable of simulating the hydrodynamics and sediment processes in this estuary and the sediment concentration has a local maximum at the limit of salinity intrusion. Furthermore, it was indicated that for Q ₅₀ (the flow which is equaled or exceeded 50% times), the turbidity maximum location during neap tide is about 1 km closer to the mouth compared to that during spring tide. It was found that deposition is the dominant sediment transport process in the river during spring–neap periods. It was shown that, while sediment concentration at the upstream depends on the river discharge, the concentration in the downstream is not a function of river discharge.     

Monitoring nutrient transport in large rivers

The problem of estimating nutrient transport in large rivers and the uncertainty of such load estimates was studied both empirically and theoretically. In the empirical part of the study, time series of data from the Rhine, Meuse, Vistula and Oder Rivers were examined. The results of this data analysis justify the use of linear interpolation to estimate concentrations prevailing between sampling occasions. A special study of the spatial variation of concentrations within different cross-sections of the Vistula river showed that such variation can contribute substantially to the uncertainty of load estimates. In general, however, sampling at one point in the cross-section did not result in biased load estimates. In the theoretical part of the study, simple ARMA (autoregressive-moving average) models were used to derive generally applicable formulas for the expected mean square error of load estimates based on serially dependent concentration data. These formulas were then used to estimate the uncertainty of calculated nutrient loads in the Rhine and the Vistula, respectively.     

Natural background concentrations of nutrients in the German Bight area (North Sea)

Natural background concentrations of nutrients are needed for the assessments of eutrophication processes and their status. Natural background concentrations of total nitrogen (TN) and total phosphorus (TP) were modelled for the rivers discharging into the German Bight and the Rhine considering individual catchment sizes, freshwater flows and soil types. These data were validated by comparison with data from unpolluted rivers. The consistency of modelled and some compiled nutrient concentrations was confirmed by their area-specific load dependency on freshwater discharges. Pristine inorganic nutrient concentrations were deduced from modelled relations to TN and TP in unpolluted rivers. Pristine nutrient gradients between rivers and offshore waters were estimated by linear mixing until a salinity of 32, continued by hyperbolic fits towards recent mean offshore values (salinity 34.5?C35). Based on these gradients and recent mean salinities, maps of pristine surface gradients were plotted for the whole German Bight. Variability was transferred from recent conditions as percentage of standard deviation. Reported historical nutrient data and concentrations from unpolluted rivers, coastal and offshore North Sea waters are discussed concerning their relations to natural background conditions.     

The importance of small urbanized watersheds to pollutant loading in a large oligotrophic subalpine lake of the western USA

Urban land use has been implicated as a major contributor of nonpoint source pollution in aquatic systems. Through increased nonpoint delivery of pollutants, including constituents found in stormwater, Lake Tahoe is undergoing a marked decline in its transparency, primarily due to increasing production of algae from enhanced nutrient loading and delivery of fine particles to the lake from the watershed. In response to these findings, a regional restoration effort is underway to improve basin watersheds and the water quality in Lake Tahoe. In this study, stormwater autosamplers were used to collect flow-weighted composite samples that characterized event mean concentrations for event and nonevent conditions within a small, urbanized watershed in the Tahoe basin. An event-specified constant-concentration water quality model was then applied to the event mean concentration and continuous streamflow data to estimate pollutant loads for nitrate, nitrite, ammonia, orthophosphate, and suspended sediment. These data were compared with previously reported load estimates from 10 primary monitored streams in larger watersheds of the Tahoe basin. Results from a linear regression analysis demonstrate strong and significant relationships between watershed impervious area and pollutant loadings from Lake Tahoe watersheds. These small, urbanized watersheds and intervening zones, which only comprise 10 % of the total Lake Tahoe drainage area, include a significant portion of the total Lake Tahoe impervious area. The findings of this study suggest that small, urbanized watersheds and intervening zones are disproportionately important contributors of nonpoint source pollution, including nutrients and suspended particles.     

Modelling the dispersion of treated wastewater in a shallow coastal wind-driven environment,Geographe Bay,Western Australia: implications for environmental management

Numerical models are useful for predicting the transport and fate of contaminants in dynamic marine environments, and are increasingly a practical solution to environmental impact assessments. In this study, a three-dimensional hydrodynamic model and field data were used to validate a far-field dispersion model that, in turn, was used to determine the fate of treated wastewater (TWW) discharged to the ocean via a submarine ocean outfall under hypothetical TWW flows. The models were validated with respect to bottom and surface water current speed and direction, and in situ measurements of total nitrogen and faecal coliforms. Variations in surface and bottom currents were accurately predicted by the model as were nutrient and coliform concentrations. Results indicated that the ocean circulation was predominately wind driven, evidenced by relatively small oscillations in the current speeds along the time-scale of the tide, and that dilution mixing zones were orientated in a predominantly north-eastern direction from the outfall and parallel to the coastline. Outputs of the model were used to determine the ‘footprint’ of the TWW plume under a differing discharge scenario and, particularly, whether the resultant changes in TWW contaminants, total nitrogen and faecal coliforms would meet local environmental quality objectives (EQO) for ecosystem integrity, shellfish harvesting and primary recreation. Modelling provided a practical solution for predicting the dilution of contaminants under a hypothetical discharge scenario and a means for determining the aerial extent of exclusion zones, where the EQOs for shellfish harvesting and primary recreation may not always be met. Results of this study add to the understanding of regional discharge conditions and provide a practical case study for managing impacts to marine environments under a differing TWW discharge scenario, in comparison to an existing scenario.     

Interval-parameter Fuzzy-stochastic Semi-infinite Mixed-integer Linear Programming for Waste Management under Uncertainty

An interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming (IFSSIP) method is developed for waste management under uncertainties. The IFSSIP method integrates the fuzzy programming, chance-constrained programming, integer programming and interval semi-infinite programming within a general optimization framework. The model is applied to a waste management system with three disposal facilities, three municipalities, and three periods. Compared with the previous methods, IFSSIP have two major advantages. One is that it can help generate solutions for the stable ranges of the decision variables and objective function value under fuzzy satisfaction degree and different levels of probability of violating constraints, which are informative and flexible for solution users to interpret/justify. The other is that IFSSIP can not only handle uncertainties through constructing fuzzy and random parameter, but also reflect dynamic features of the system conditions through interval function of time over the planning horizon. By comparing IFSSIP with interval-parameter mixed-integer linear semi-infinite programming and parametric programming, the IFSSIP method is more reasonable than others.     

Migration of pollutants in groundwater. III. Numerical dispersion reduction with cartesian coordinates

The partial differential equation governing the movement of a decomposing pollutant undergoing 2-dimensional flow in a saturated aquifer is examined. The analytical solution of the equation is usually not possible, and use of mesh numerical integration techniques causes excessive numerical dispersion to arise from the advection term. We apply two asymmetrical upwind formulas to approximate the advection term. These markedly reduce numerical dispersion without requiring the use of coordinate systems obtained by conformal mapping. Flow within a right angle and flow toward a sink in an otherwise uniform field are analyzed to illustrate the methods.