Density-gravity covariance functions produced by overlapping rectangular blocks of constant density

Abstract

A set of N rectangular blocks of unit density form the basis of a linear vector space. This space will be a Hilbert space with reproducing kernel, when equipped with the L2 norm (and inner product). The kernel, K ( P , Q), will be the product sum of the (maximally N) orthonormalized base functions evaluated at two points P, Q, in space. Multiplied by an appropriate scaling constant, this function will be a covariance function for a random density (anomaly) function. Using covariance propagation, auto-and cross-covariance functions for and with quantities related to the (anomalous) gravity field created by the random density function may be computed.When blocks of varying volume are used, the density variance will vary within the structure described by the blocks. For disjoint blocks, the variance is inversely proportional to the volume. The use of overlapping blocks generally creates correlations between density values in different blocks, a possibility which does not exist when using disjoint blocks.Computational examples are presented which show that it is possible to create realistic covariance models for the gravity anomaly, and that the cross-correlation between surface gravity and surface density may obtain values in the interval from 0.4 to 0.7. These values are more realistic than values obtained in earlier studies using quasi-harmonic density models.

Low Temperature Fourier Transform Infrared Spectroscopic Studies of Hydrogenated Amorphous Silicon Nitride Films

“…They range from purely local approaches, like a block ansatz, to completely global tools like a representation in orthogonal polynomials. In the case of a block ansatz, the Earth, or a part of it, is subdivided into predefined (overlapping or non-overlapping) blocks with a very simple density structure, such as a constant density, per block, see, for instance [24,26,30,56]. Also within this category of purely local methods are those approaches which use point masses, see, for example, [9,53].…”

“…This is demonstrated in [38] for the inverse gravimetric problem and in [2] for the seismic travel time tomography problem. In the context of the inverse gravimetric problem, such approaches have also been employed as collocation methods with a more statistical background (see [24,45,47,56,58,60]) and as spline methods based on the theory of reproducing kernel Hilbert spaces (see [18,38] and the historical references in [1] where the use of similar methods for different applications is discussed). It should be noted that since the model space is of finite dimensions the ill-posedness is circumvented.…”

Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals

“…More elaborate ways of choosing this added matrix could be based on covariance matrices, see, e.g. Tscherning 1991Tscherning , 1995Tscherning , 1996.…”

Non-stationary covariance function modelling in 2D least-squares collocation

“…Also, non-stationary covariance functions provide a more flexible tool to study density-gravity covariance functions (Tscherning 1991), which leads to the solution of the inverse gravimetric problem in geophysical exploration because a non-stationary mass density distribution is more realistic than a stationary one. …”

“…-Other examples: Tscherning (1991) proposed an approach based on mass-density anomaly considerations, but it was not pursued due to the limited capability of computers in the early 1990s. Tscherning (1999) suggested another possibility to use Riesz representers, but which was not practically applicable because of the lack of a sufficient numerical algorithm.…”