Parallel Algorithms for Solving Inverse Gravimetry Problems: Application for Earth’s Crust Density Models Creation

The paper describes a method of gravity data inversion, which is based on parallel algorithms. The choice of the density model of the initial approximation and the set on which the solution is sought guarantees the stability of the algorithms. We offer a new upward and downward continuation algorithm for separating the effects of shallow and deep sources. Using separated field of layers, the density distribution is restored in a form of 3D grid. We use the iterative parallel algorithms for the downward continuation and restoration of the density values (by solving the inverse linear gravity problem). The algorithms are based on the ideas of local minimization; they do not require a nonlinear minimization; they are easier to implement and have better stability. We also suggest an optimization of the gravity field calculation, which speeds up the inversion. A practical example of interpretation is presented for the gravity data of the Urals region, Russia.

Decorrelative Mollifier Gravimetry

<p>The lecture highlights arguments that, coming from multiscale mathematics, have fostered the advancement of gravimetry, as well as those that, generated by gravimetric problems, have contributed to the enhancement in constructive approximation and numerics. Inverse problems in gravimetry are delt with multiscale mollifier decorrelation strategies. Two examples are studied in more detail: (i) Vening Meinesz multiscale surface mollifier regularization to determine locally the Earth's disturbing potential from deflections of vertical, (ii) Newton multiscale volume mollifier regularization of the inverse gravimetry problem to derive locally the density contrast distribution from functionals of the Newton integral and to detect fine particulars of geological relevance. All in all, the Vening Meinesz medal  lecture is meant as an  \lq \lq appetizer'' served to enjoy the tasty meal "Mathematical Geoscience Today'' to be shared by geoscientists and mathematicians in the field of gravimetry. It provides innovative concepts and locally relevant applications presented in a monograph to be published by Birkhäuser in the book series “Geosystems Mathematics” (2021).</p>

Stability and the inverse gravimetry problem with minimal data

The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from boundary measurements of gravitational force outside Ω. We found that about five parameters of the unknown 𝐷 can be stably determined given data noise in practical situations. An ellipse is uniquely determined by five parameters. We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse. To illustrate the technique, we use these equations in numerical examples with various location of measurements points on \partial\Omega. Similarly, a rectangular 𝐷 is considered. We consider the problem in the plane as a model for the three-dimensional problem due to simplicity.

Geodesy and Mathematics: Interactions in Multiscale Mollifier Regularization Methods

<p>The lecture highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement in Mathematics.</p><p>We particularly deal with novel applications to Geodesy in the context of multiscale approximation (MA). In fact, multiscale reconstruction and decorrelation methods are a research field originated in geophysics for, e.g., earthquake modeling some decades ago, in which today's Geodesy and Mathematics show mutual influences, especially on the subject of spectral and space data sampling.</p><p>We particularly focus the attention on inverse problems of Geodesy and multiscale mollifier regularization strategies. Two examples are studied in more detail:</p><p>(i) Vening Meinesz multiscale surface mollifier regularization to determine locally the Earth's disturbing potential from deflections of vertical,</p><p>(ii) Newton multiscale volume mollifier regularization of the inverse gravimetry problem to derive locally the density contrast distribution from functionals of the Newton integral and to detect fine particulars of geological relevance.</p><p>Neither extreme depth to explain all facets of the geodetic observational situation nor penetrative handling of mathematical obligations and technicalities can be expected. The lecture is just an \lq \lq appetizer'' served to enjoy the tasty meal "Mathematical Geodesy Today'' to be shared by geodesists and mathematicians.</p>