The Application Of Stochastic Partial Differential Equation To The Geodetic Boundary Value Problems

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.