点源重力场模拟的Dirichlet积分双通道配置核方法
A Dirichlet-integral based dual-access collocation-kernel approach to point-source gravity-field modleing
摘要：
在$\mathbb{R}^3$中的问题是解决相关的矢量场的标量位满足拉普拉斯方程在一些无界的外部区域，是近似的未知（点）源包含在互补的子区域。两个特定的字段的几何形状被认为是：$\mathbb{R}^3$半空间和外部的$\mathbb{R}^3$球，这是两个标准的地球物理和地球勘探重力问题的设置。对于这些几何形状，它表明，存在一种新类型的核空间，这是标记的狄利克雷积分双访问配置核空间（DIDACKS），这是非常适合于许多应用程序。研究的DIDACKS例子与再生核希尔伯特空间有关，它们具有一个复制核（与再生核相反），该复制核具有场点和相应源点之间距离的倒数的普遍存在的形式。这种方法的基础是三个普遍感兴趣的基本数学关系。这些关系中的两个--对应于两种几何形状--产生精确的封闭形式的内积，从而产生用于各种类型的对应点源强度的精确线性方程组（即，质点、点偶极和/或点四极集合）。给定的场不仅在点配置意义上被重构，而且在（加权）场能量误差最小化意义上被重构。
英文摘要：
Problems in $\mathbb{R}^3$ are addressed where the scalar potential of an associated vector field satisfies Laplace's equation in some unbounded external region and is to be approximated by unknown (point) sources contained in the complimentary subregion. Two specific field geometries are considered: $\mathbb{R}^3$ half-space and the exterior of an $\mathbb{R}^3$ sphere, which are the two standard settings for geophysical and geoexploration gravitational problems. For these geometries it is shown that a new type of kernel space exists, which is labeled a Dirichlet-integral dual-access collocation-kernel space (DIDACKS) and that is well suited for many applications. The DIDACKS examples studied are related to reproducing kernel Hilbert spaces and they have a replicating kernel (as opposed to a reproducing kernel) that has the ubiquitous form of the inverse of the distance between a field point and a corresponding source point. Underpinning this approach are three basic mathematical relationships of general interest. Two of these relationships--corresponding to the two geometries--yield exact closed-form inner products and thus exact linear equation sets for the corresponding point source strengths of various types (i.e., point mass, point dipole and/or point quadrupole sets) at specified source locations. The given field is reconstructed not only in a point collocation sense, but also in a (weighted) field-energy error-minimization sense.
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