空间域位场延拓新方法研究

在空间域进行位场延拓,需要数值求解第一类Fredholm积分方程,由于所得方程组系数矩阵不是稀疏矩阵,求解该方程组需要的计算机内存大,计算量大,导致延拓算法在一般计算机上难以实现,阻碍了对空间域位场延拓方法的研究.在分析系数矩阵结构特征的基础上,本文证明了方程组系数矩阵是对称的分块Toeplitz型矩阵.利用系数矩阵的对称性和分块Toeplitz型矩阵与向量相乘的快速算法,解决了系数矩阵的存储和计算问题,使得空间域位场延拓成为可能,为研究新的位场延拓方法和分析延拓误差提供了一条新的途径.利用模型数据和实测资料,对空间域位场向上延拓、空间域积分迭代法向下延拓进行了检验,结果证实了空间域位场延拓的可行性和正确性.

Research on spatial domain continuation method for potential field

To apply continuation of potential field in spatial domain,Fredholm integral equation of the first kind has to be solved numerically.Because the coefficient matrix that comes from discretizing integral equation is not sparse and has very large scale,it requires large computer memory to save the matrix,and operation on it consumes very long time.The continuation method in spatial domain is not developed in practice.In this paper,based on the analysis of structure of coefficient matrix,we find an important fact that the coefficient matrix is a symmetric,block Toeplitz type matrix.A mathematical proof is given in the paper.By using fast algorithm that already exists to solve block Toeplitz system,we realize spatial domain upward continuation and downward continuation.Model data and aeromagnetic data test shows the feasibility and correctness of spatial continuation of potential field.