水库湖泊水质分析、模拟与预测的综合数学方法及其应用
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摘要
本文在对各种水质模型与方法总结分析的基础上,针对以水库、湖泊为主的大范围、复杂地理结构区域上地表水污染的过程及规律分析、分布状态模拟、变化趋势预测给出了相应的三维数学模型和求解算法,提出了两种新的统计分析方法用于污染分布的关系分析和规律分析,并给出这些研究在密云水库水质分析中的应用实例。
1.剖析了不同类型径向基函数的性质,系统阐述了无单元配点法及其在解微分方程时的控制技巧,给出水污染质运移模型无单元配点法求解的详细过程。
2.建立了用于评价不同样本组数据分布间单调关系和协调关系的两种统计方法,给出其在湖泊水库水体、底泥、孔隙中水污染质分布分析的具体应用。
3.建立了水库湖泊BOD、DO、总氮、总磷、重金属等主要污染质迁移转化的三维水质模型,讨论了其无单元法求解技术。
4.给出了上述数值算法和统计分析工作的软件实现方法以及水污染分布的几种图形表现技术。
5.给出上述研究方法与技术在密云水库水质分析中的详细应用。获得了有关密云水库水质富营养化的成因、机理、规律、趋势及其防治措施的有指导意义的认识和结论。
The analysis of simulation and prediction on polluted state of every kind of water resources, particularly the groundwater and surface water, is an important mission of environment science. In the water environment problem research area, the mathematic model of water quality is the main technical means on the study of water environment, water quality analysis and prediction. Water quality model techniques have already stepped into the age of multi-types and multi-projects through its development more than a half century. But the mathematic determinant model of variety of equations as representative is the most efficient all the time for description in the internal process of migrating and translating of water contamination. Therefore it is still the preferred and main tools for analysis, simulation and prediction of water quality. Because the equation models are usually complex, they are solved numerically. The finite element method and finite difference method are the main numerical methods for differential equations. But it is very difficult to solve the 3D water quality model of reservoir and lake with large area and complex geographic structure by finite element method or finite difference method, so mostly study of 3D problem are actually solved by predigesting the 3D model to 2D or ID model. Although some works have done to solve the 3D model directly, these works had to deal with the different depth of bottom and boundary by geometric approximations. The difficult still remains in solving 3D water quality model by traditional numerical methods directly.
Recently, radial basis functions method (Element-Free Method), as a new function interpolation method or a new numerical method for differential equations, has been paid much more attention. The method can translate the multi-dimension problems to one dimension problems in order to solve them easily. The method doesn't need grid dissection, has the predominance especially for discrete points data model. At the same time, data and information digging technology based on statistic methods is widely applicable foreground. Water quality analysis methods based on statistic technology can overcome the lack of systematic parameters by qualitative analysis correction and complement the conclusions from quantitative analysis.Given the booming development of computer technology, the calculation cost decreases dramatically, accompanied by the advances of capability and efficiency, the calculation methodology and implement projects improve constantly. These progresses provide new sustentions for various researches on water environment, but it brings forward higher desires.The present paper offers corresponding 3D mathematic models and the solution methods which aims at the process, principle analysis, distribution simulation and transformation trend forecast of surface water pollution, primary in the reservoirs and lakes, which are large in area and have complex geographic geometry. We develop two new statistic analysis methods used for distribution relation and principle analysis of contamination, combined with the research works of National major project of basic research development "The mechanism and regulation principle of air, water and land pollution in Beijing and around (Item No. G1999045700)", gives some application examples of these theories.The concrete working project is as follows:1. By in-depth anatomy of radial basis functions and correlated theories, starting with types choice and parameters option of radial basis functions, boundary conditions manage techniques and equations discretization methods, gives detailed constructive expatiation and discussion for some basic problems such as function interpolation and numerical solution for
differential equations by radial basis methods, thereby offers the numerical solution of surface water quality 3D models by radial basis functions.2. Creates appropriate statistic methods, and makes qualitative analysis and estimation for distribution law, characteristics and changing trend of water quality.3. By taking the full advantage of high-powered calculating capacity and excellent graph display technology of modern computers, devises a suit of lifelike intuitionistic simulation projects and develops corresponding specialized software. The software not only implements radial basis function methods 3D solutions of water quality models and statistic analysis of water quality distribution, but also implements computer 3D visualized static and dynamic water quality simulation of reservoirs and lakes.The innovated points are mainly listed as what follows:1. Anatomizes properties of various types of radio basis functions. Discusses various situations and managed methods (such as types selection, evaluating methods of radio basis functions coefficients, resolvents of different boundary conditions) referring to the solution of water quality models by radio basis functions methods. All these are difficult problems which have not been solved ideally.2. The complexity of solving 3D problems are simplified by directly solving 3D water quality models using radio basis functions methods, and increasing the adapted ability for problems on complex areas and irregular discrete data points.3. Two new statistical methods are presented — the measurements of concentration distributing monotonous correlativity of the same contamination and co-correlativity of distribution of one contamination between layers or distributions of two different contaminations, to dig latent information from practical data resource sufficiently.4. Water quality models are built which include the biologic effect as a source in equations, and can more detailedly and more all-sided depict the migration and translation laws of water contaminations.
5. By the combination of qualitative analysis with quantitative analysis, the combination of equation model with statistic estimation and the combination of Microcosmic (shortdated) analysis with microscopical (long-term) analysis, the models are applied to depict and calculate the microcosmic (shortdated) partial quantitative water quality distribution and change. Statistic methods are applied to obtain the microscopical (long-term) holistic qualitative trend and the analysis of water quality evolution. Equation models, based on internal law of water contamination migration and translation, can overcome many difficulties and localizations such as the lack of sample data, very long distance of different sampling time, very short count of sample groups, etc;Statistic estimation analysis, starting with only the data on-the-spot survey and depending on internal correlative information, can overcome efficiently many disadvantages such as much system parameters, difficulty to determine values of parameters, difficulty to reflect and to control secondary infection factors, to be incapable of shield random disturbances in equation models. The combined application of these two methods supply for each other, inaugurating new approach for water analysis, simulation and forecast.6. Several kinds of visual representation technology are developed for contaminations concentration distribution analysis: Building continuous spectrum of a graph of contaminations concentration distribution or free optioned contour plots of a graph of contaminations concentration distribution through rendered images;Displaying the concentration of every points when it is pointed by mouse;Establishing color-geographical digital text of concentration distribution table.Conclusions such as distribution, changing laws, effecting mechanisms, forming causes, fathering project and controlling measures, etc of various contaminations in MiYun reservoir are obtained by applying these technologies to calculate and analyze the observed data in continuous tow yeas in the reservoir.The methods how to solve 3D water quality models by element-free method discussed by
this paper don't need mesh-dissection, so their applications are even more convenient, agile and easily to program for calculation. Generally 3D finite element method mainly adopts mesh of pole shape, so the geometric error is commonly large for vary depth complex area. Compare with finite element method, element-free method can depict complex boundary shape exactly, so it has evident predominance for dealing with the water quality analysis problems in complex three dimensional geometric areas. The arithmetic presented in this paper is the same with two dimensional or one dimensional problem. It is needed only to delete the redundant coordinate variables and corresponding coefficients, then the discrete solving process by element-free method of two dimensional or one dimensional problem is obtained.
1.剖析了不同类型径向基函数的性质,系统阐述了无单元配点法及其在解微分方程时的控制技巧,给出水污染质运移模型无单元配点法求解的详细过程。
2.建立了用于评价不同样本组数据分布间单调关系和协调关系的两种统计方法,给出其在湖泊水库水体、底泥、孔隙中水污染质分布分析的具体应用。
3.建立了水库湖泊BOD、DO、总氮、总磷、重金属等主要污染质迁移转化的三维水质模型,讨论了其无单元法求解技术。
4.给出了上述数值算法和统计分析工作的软件实现方法以及水污染分布的几种图形表现技术。
5.给出上述研究方法与技术在密云水库水质分析中的详细应用。获得了有关密云水库水质富营养化的成因、机理、规律、趋势及其防治措施的有指导意义的认识和结论。
The analysis of simulation and prediction on polluted state of every kind of water resources, particularly the groundwater and surface water, is an important mission of environment science. In the water environment problem research area, the mathematic model of water quality is the main technical means on the study of water environment, water quality analysis and prediction. Water quality model techniques have already stepped into the age of multi-types and multi-projects through its development more than a half century. But the mathematic determinant model of variety of equations as representative is the most efficient all the time for description in the internal process of migrating and translating of water contamination. Therefore it is still the preferred and main tools for analysis, simulation and prediction of water quality. Because the equation models are usually complex, they are solved numerically. The finite element method and finite difference method are the main numerical methods for differential equations. But it is very difficult to solve the 3D water quality model of reservoir and lake with large area and complex geographic structure by finite element method or finite difference method, so mostly study of 3D problem are actually solved by predigesting the 3D model to 2D or ID model. Although some works have done to solve the 3D model directly, these works had to deal with the different depth of bottom and boundary by geometric approximations. The difficult still remains in solving 3D water quality model by traditional numerical methods directly.
Recently, radial basis functions method (Element-Free Method), as a new function interpolation method or a new numerical method for differential equations, has been paid much more attention. The method can translate the multi-dimension problems to one dimension problems in order to solve them easily. The method doesn't need grid dissection, has the predominance especially for discrete points data model. At the same time, data and information digging technology based on statistic methods is widely applicable foreground. Water quality analysis methods based on statistic technology can overcome the lack of systematic parameters by qualitative analysis correction and complement the conclusions from quantitative analysis.Given the booming development of computer technology, the calculation cost decreases dramatically, accompanied by the advances of capability and efficiency, the calculation methodology and implement projects improve constantly. These progresses provide new sustentions for various researches on water environment, but it brings forward higher desires.The present paper offers corresponding 3D mathematic models and the solution methods which aims at the process, principle analysis, distribution simulation and transformation trend forecast of surface water pollution, primary in the reservoirs and lakes, which are large in area and have complex geographic geometry. We develop two new statistic analysis methods used for distribution relation and principle analysis of contamination, combined with the research works of National major project of basic research development "The mechanism and regulation principle of air, water and land pollution in Beijing and around (Item No. G1999045700)", gives some application examples of these theories.The concrete working project is as follows:1. By in-depth anatomy of radial basis functions and correlated theories, starting with types choice and parameters option of radial basis functions, boundary conditions manage techniques and equations discretization methods, gives detailed constructive expatiation and discussion for some basic problems such as function interpolation and numerical solution for
differential equations by radial basis methods, thereby offers the numerical solution of surface water quality 3D models by radial basis functions.2. Creates appropriate statistic methods, and makes qualitative analysis and estimation for distribution law, characteristics and changing trend of water quality.3. By taking the full advantage of high-powered calculating capacity and excellent graph display technology of modern computers, devises a suit of lifelike intuitionistic simulation projects and develops corresponding specialized software. The software not only implements radial basis function methods 3D solutions of water quality models and statistic analysis of water quality distribution, but also implements computer 3D visualized static and dynamic water quality simulation of reservoirs and lakes.The innovated points are mainly listed as what follows:1. Anatomizes properties of various types of radio basis functions. Discusses various situations and managed methods (such as types selection, evaluating methods of radio basis functions coefficients, resolvents of different boundary conditions) referring to the solution of water quality models by radio basis functions methods. All these are difficult problems which have not been solved ideally.2. The complexity of solving 3D problems are simplified by directly solving 3D water quality models using radio basis functions methods, and increasing the adapted ability for problems on complex areas and irregular discrete data points.3. Two new statistical methods are presented — the measurements of concentration distributing monotonous correlativity of the same contamination and co-correlativity of distribution of one contamination between layers or distributions of two different contaminations, to dig latent information from practical data resource sufficiently.4. Water quality models are built which include the biologic effect as a source in equations, and can more detailedly and more all-sided depict the migration and translation laws of water contaminations.
5. By the combination of qualitative analysis with quantitative analysis, the combination of equation model with statistic estimation and the combination of Microcosmic (shortdated) analysis with microscopical (long-term) analysis, the models are applied to depict and calculate the microcosmic (shortdated) partial quantitative water quality distribution and change. Statistic methods are applied to obtain the microscopical (long-term) holistic qualitative trend and the analysis of water quality evolution. Equation models, based on internal law of water contamination migration and translation, can overcome many difficulties and localizations such as the lack of sample data, very long distance of different sampling time, very short count of sample groups, etc;Statistic estimation analysis, starting with only the data on-the-spot survey and depending on internal correlative information, can overcome efficiently many disadvantages such as much system parameters, difficulty to determine values of parameters, difficulty to reflect and to control secondary infection factors, to be incapable of shield random disturbances in equation models. The combined application of these two methods supply for each other, inaugurating new approach for water analysis, simulation and forecast.6. Several kinds of visual representation technology are developed for contaminations concentration distribution analysis: Building continuous spectrum of a graph of contaminations concentration distribution or free optioned contour plots of a graph of contaminations concentration distribution through rendered images;Displaying the concentration of every points when it is pointed by mouse;Establishing color-geographical digital text of concentration distribution table.Conclusions such as distribution, changing laws, effecting mechanisms, forming causes, fathering project and controlling measures, etc of various contaminations in MiYun reservoir are obtained by applying these technologies to calculate and analyze the observed data in continuous tow yeas in the reservoir.The methods how to solve 3D water quality models by element-free method discussed by
this paper don't need mesh-dissection, so their applications are even more convenient, agile and easily to program for calculation. Generally 3D finite element method mainly adopts mesh of pole shape, so the geometric error is commonly large for vary depth complex area. Compare with finite element method, element-free method can depict complex boundary shape exactly, so it has evident predominance for dealing with the water quality analysis problems in complex three dimensional geometric areas. The arithmetic presented in this paper is the same with two dimensional or one dimensional problem. It is needed only to delete the redundant coordinate variables and corresponding coefficients, then the discrete solving process by element-free method of two dimensional or one dimensional problem is obtained.
引文
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