随机偏微分方程在大地边值问题中的应用
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摘要
地球重力学不仅是地球物理学的一个重要分支,而且是大地测量学的主要支柱之一。它的基本科学任务是研究如何利用观测数据来确定地球形状、大小和外部重力场。这一科学任务用数学语言来表达就是求解一类经典位势偏微分方程的定解问题,这类问题的定解条件是在待求的地球表面上通过重力、天文和水准等测量手段得到的边值条件,人们称这类具有自由边界面的位势方程外部问题为大地测量边值问题,简称大地边值问题。作为地球重力学的研究主题,大地边值问题的研究,从地球重力学1849年由Stokes定理奠基开始,贯穿着学科发展的全过程,至今已经经历了Stokes理论、Molodensky理论、Bjerhammar理论三个里程碑,取得了举世瞩目的成果。但是,在上世纪80年代以前,这一研究基本上是属于Laplace确定性论意义的范畴。近年来,由于高新观测技术的长足进步,含重力场信息的各类高精度数据的不断获得,学科自身的发展和工程使用的需要,迫使人们去研究与当前精度相应的重力场精细建模理论与方法。一种沿非传统方向的开拓性工作,就是在上世纪90年代有Sanso等人倡议的,即从不确定性论的观点,由现代随机数学理论出发来研究地球重力学主题;将重力场看成广义随机函数一随机过程,提出所谓随机大地边值问题,并研究问题解的性质和解法。
本文研究的核心内容是:利用随机偏微分方程理论和方法研究大地测量边值问题,完成调和重力场的随机建模。
通过研究,本文给出如下主要的工作与结论,包括下面几点
首先以比较广阔的视角回顾评述了大地边值问题理论的最新进展,主要包括Stokes问题、Molodensky问题、Hotine问题及Bjerhammar问题等,探讨了现代大地边值问题某些深刻的数学内涵和问题所在,并指出了随机偏微分方程应用于大地测量学的研究目前尚未得到重视的原因。
其次推演和给出了建立随机Laplace方程的数学方法并用于求解大地测量边值问题。利用随机场理论、随机偏微分方程理论中的随机Laplace算子方程理论,并结合重力学的场论知识,来解决研究目标和研究内容中的主要关键问题;引入广义函数,结合随机数学知识,实现了Sobolev空间的随机化过程。同时在随机化了的广义函数空间上,特别是在随机化Sobolev空间中对随机边值问题的解的空间属性进行阐述。
讨论了随机偏微分方程理论用于大地边值问题的关键问题。既然广义函数在孤立点处没什么特别的意义,加之在随机Sobolev空间中求解偏微分方程,首先必须把古典导数的概念推广到弱导数(或强导数),通常还要求弱解在边界或部分边界上的前几阶导数值是给定的(经典重力梯度边值问题要求取得边界或部分边界的二阶导数),那么该广义随机函数在边界上的取值,其确切含义必须详细说明。本文将对边界Γ给出明确的解释并且对广义随机函数在边界上取值的含义予以说明,给出随机Sobolev空间的边界迹的定义。
Physical geodesy is an important branch of geodesy, its main scientific objective is the determination of the figure of the earth and the external gravity field. Accurate knowledge of the external gravity field of the earth is a prerequisite for various geodetic and geophysical investigations and applications.The determination and research of the gravity field of the earth boil down to the research of boundary value problem of geodesy. Boundary value problem be the dominating content of this subjection all the time.With the development of the modern geodetic technology in recent years, a large amount of data of different kinds on the earth's surface and in space has become available from satellite and ground measurements. This arises the over-determined problem: a boundary value problem in which more boundary conditions are given than those strictly necessary to determine the solution. Most geodetic researcher and mathematician have give attention to this problem, and moreover another problem, Stochastic Partial Differential Equation (SPDE) and its application to the Boundary Value Problems of Physical Geodesy, has drawn attention of most geodetic researcher in recent years. We can say that such research work will make a great progress from traditional study method to improve method in base domain of the gravity field. It will be based upon the modern mathematic theory, focus on main subject of earth gravity field, that is regarding gravity as generalized stochastic function. This paper puts forward the conception of stochastic earth boundary value problem, and studies the quality of solution of Stochastic Partial Differential Equation. At last, this dissertation gives the plan how to solve the equations.The mostly task in this dissertation is that using the theory and method of Stochastic Partial Differential Equation to solve the gravity boundary value problem.Through deeply investigation, a few important statement and theory has obtained in this dissertation as follow:Firstly, this dissertation discusses several classical geodetic boundary value problems, including Stokes boundary value problem, Molodenskey boundary value problem, Hotine boundary value problem, and Bjerhammar boundary value problem etc, and gives some analyses of the solutions for different boundary value problems and their relation and difference.
In order to solve the' problem about geodetic boundary value problem, this paper deduces and gives the method to establish the stochastic Laplace equation at the first time interiorly. Using the theory of stochastic field and stochastic partial differential equation, and moreover the theory of stochastic Laplace operator equation, combining theory of fields, this paper redefines the generalized function in L2 space. With the modern mathematics theory, the paper gives the Stochastic Sobolev Space, then analyses the space quality of the solution for Stochastic Partial Differential Equation.The paper also discusses the key problem for applying the theory of SPDE to geodetic boundary value problem. Since it is unreasonable to think over the value of generalized function at a isolated point, and for obtain the solution of SPDE from the given Stochastic Sobolev Space, the dissertation generalizes the conception of classical derivative to weak derivative style.It is necessary to define the boundary value of week solution on boundary region or on partial boundary, such as classical gravity grads boundary value problem needs to know seconds rank derivative,- we must explain the particular situation of the value for the generalized function on boundary.And moreover this paper gives the conception of trace for stochastic Sobolev space, defines the boundary r of domain and boundary value in Stochastic Sobolev Space, and moreover the trace definition is provided at last.After the definition of trace, we can establish the model of random field function, and present stochastic gravity boundary value problem. The general solution of stochastic Laplace equation also been provided.In order to solve the boundary value problem with chaos or complicated boundary dates, such as singularity etc, it is necessary to establish the model of stochastic partial differential equation for the gravity field, this paper give the stochastic Poisson integral as a generalized stochastic functional, We also compare the relationships of stochastic model and determined model of Poisson integral, indicate that determined model be just the one of special situation of stochastic Poisson integral.With the development of GPS and other geodetic technology, it became possible to get large amount of observation data and therefore the earth's figure can be determined. In this way, the formulation of the principal problem of gravimetry will be changed, so that the original of boundary value problems (GBVPs), i.e. the free GBVP, has reduced to the fixed GBVP. The purpose of this paper is to try and develop the theory of advanced GBVPs.
First of all, we give a stochastic model for processing the GBVP with continuous observation data, and secondly, we establish the continuous observation equation, and we also find out the fact that the error is dependent upon its range variance only.
本文研究的核心内容是:利用随机偏微分方程理论和方法研究大地测量边值问题,完成调和重力场的随机建模。
通过研究,本文给出如下主要的工作与结论,包括下面几点
首先以比较广阔的视角回顾评述了大地边值问题理论的最新进展,主要包括Stokes问题、Molodensky问题、Hotine问题及Bjerhammar问题等,探讨了现代大地边值问题某些深刻的数学内涵和问题所在,并指出了随机偏微分方程应用于大地测量学的研究目前尚未得到重视的原因。
其次推演和给出了建立随机Laplace方程的数学方法并用于求解大地测量边值问题。利用随机场理论、随机偏微分方程理论中的随机Laplace算子方程理论,并结合重力学的场论知识,来解决研究目标和研究内容中的主要关键问题;引入广义函数,结合随机数学知识,实现了Sobolev空间的随机化过程。同时在随机化了的广义函数空间上,特别是在随机化Sobolev空间中对随机边值问题的解的空间属性进行阐述。
讨论了随机偏微分方程理论用于大地边值问题的关键问题。既然广义函数在孤立点处没什么特别的意义,加之在随机Sobolev空间中求解偏微分方程,首先必须把古典导数的概念推广到弱导数(或强导数),通常还要求弱解在边界或部分边界上的前几阶导数值是给定的(经典重力梯度边值问题要求取得边界或部分边界的二阶导数),那么该广义随机函数在边界上的取值,其确切含义必须详细说明。本文将对边界Γ给出明确的解释并且对广义随机函数在边界上取值的含义予以说明,给出随机Sobolev空间的边界迹的定义。
Physical geodesy is an important branch of geodesy, its main scientific objective is the determination of the figure of the earth and the external gravity field. Accurate knowledge of the external gravity field of the earth is a prerequisite for various geodetic and geophysical investigations and applications.The determination and research of the gravity field of the earth boil down to the research of boundary value problem of geodesy. Boundary value problem be the dominating content of this subjection all the time.With the development of the modern geodetic technology in recent years, a large amount of data of different kinds on the earth's surface and in space has become available from satellite and ground measurements. This arises the over-determined problem: a boundary value problem in which more boundary conditions are given than those strictly necessary to determine the solution. Most geodetic researcher and mathematician have give attention to this problem, and moreover another problem, Stochastic Partial Differential Equation (SPDE) and its application to the Boundary Value Problems of Physical Geodesy, has drawn attention of most geodetic researcher in recent years. We can say that such research work will make a great progress from traditional study method to improve method in base domain of the gravity field. It will be based upon the modern mathematic theory, focus on main subject of earth gravity field, that is regarding gravity as generalized stochastic function. This paper puts forward the conception of stochastic earth boundary value problem, and studies the quality of solution of Stochastic Partial Differential Equation. At last, this dissertation gives the plan how to solve the equations.The mostly task in this dissertation is that using the theory and method of Stochastic Partial Differential Equation to solve the gravity boundary value problem.Through deeply investigation, a few important statement and theory has obtained in this dissertation as follow:Firstly, this dissertation discusses several classical geodetic boundary value problems, including Stokes boundary value problem, Molodenskey boundary value problem, Hotine boundary value problem, and Bjerhammar boundary value problem etc, and gives some analyses of the solutions for different boundary value problems and their relation and difference.
In order to solve the' problem about geodetic boundary value problem, this paper deduces and gives the method to establish the stochastic Laplace equation at the first time interiorly. Using the theory of stochastic field and stochastic partial differential equation, and moreover the theory of stochastic Laplace operator equation, combining theory of fields, this paper redefines the generalized function in L2 space. With the modern mathematics theory, the paper gives the Stochastic Sobolev Space, then analyses the space quality of the solution for Stochastic Partial Differential Equation.The paper also discusses the key problem for applying the theory of SPDE to geodetic boundary value problem. Since it is unreasonable to think over the value of generalized function at a isolated point, and for obtain the solution of SPDE from the given Stochastic Sobolev Space, the dissertation generalizes the conception of classical derivative to weak derivative style.It is necessary to define the boundary value of week solution on boundary region or on partial boundary, such as classical gravity grads boundary value problem needs to know seconds rank derivative,- we must explain the particular situation of the value for the generalized function on boundary.And moreover this paper gives the conception of trace for stochastic Sobolev space, defines the boundary r of domain and boundary value in Stochastic Sobolev Space, and moreover the trace definition is provided at last.After the definition of trace, we can establish the model of random field function, and present stochastic gravity boundary value problem. The general solution of stochastic Laplace equation also been provided.In order to solve the boundary value problem with chaos or complicated boundary dates, such as singularity etc, it is necessary to establish the model of stochastic partial differential equation for the gravity field, this paper give the stochastic Poisson integral as a generalized stochastic functional, We also compare the relationships of stochastic model and determined model of Poisson integral, indicate that determined model be just the one of special situation of stochastic Poisson integral.With the development of GPS and other geodetic technology, it became possible to get large amount of observation data and therefore the earth's figure can be determined. In this way, the formulation of the principal problem of gravimetry will be changed, so that the original of boundary value problems (GBVPs), i.e. the free GBVP, has reduced to the fixed GBVP. The purpose of this paper is to try and develop the theory of advanced GBVPs.
First of all, we give a stochastic model for processing the GBVP with continuous observation data, and secondly, we establish the continuous observation equation, and we also find out the fact that the error is dependent upon its range variance only.
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