Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a nonlinear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.
We propose a novel dictionary learning add-on for the Inverse Problem Matching Pursuit (IPMP) algorithms for approximating spherical inverse problems such as the downward continuation of the gravitational potential. With the add-on, we aim to automatize the choice of dictionary and simultaneously reduce the computational costs. The IPMP algorithms iteratively minimize the Tikhonov–Phillips functional in order to construct a weighted linear combination of so-called dictionary elements as a regularized approximation. A dictionary is an intentionally redundant set of trial functions such as spherical harmonics (SHs), Slepian functions (SLs) as well as radial basis functions (RBFs) and wavelets (RBWs). In previous works, this dictionary was chosen manually which resulted in high runtimes and storage demand. Moreover, a possible bias could also not be ruled out. The additional learning technique we present here allows us to work with infinitely many trial functions while reducing the computational costs. This approach may enable a quantification of a possible bias in future research. We explain the general mechanism and provide numerical results that prove its applicability and efficiency.
<p>A fundamental problem in the geosciences is the downward continuation of the gravitational potential. It enables us to learn more about the system Earth and, in particular, the climate change.</p><p>Mathematically, we can model a (downward continued) signal in a 'best basis' consisting of local and global trial functions from a dictionary. In practice, our dictionaries include spherical harmonics, Slepian functions and radial basis functions. The expansion in dictionary elements is obtained by one of the Inverse Problem Matching Pursuit (IPMP) algorithms.</p><p>However, it remains to discuss the choice of the dictionary. For this, we further developed the IPMP algorithms by introducing a learning technique. With this approach, they automatically select a finite number of optimized dictionary elements from infinitely many possible ones. We present the details of our method and give numerical examples.</p><p>See also: V. Michel and N. Schneider, <em>A first approach to learning a best basis for gravitational field modelling,</em> arxiv: 1901.04222v2</p>
<p align="justify">We attempt the reconstruction of the solid earth&#8217;s interior three-dimensional structure using seismic wave observations. The interior structure of the mantle deviates moderately from spherically symmetrical reference models and therefore seismological observables also vary moderately from spherically symmetrical predictions. Hence we consider here the linearized inverse problem of seismic traveltime tomography.</p> <p align="justify">Usually, the solution is approximated in a fixed basis system: either global (e.g. polynomials) or local (e.g. finite elements) basis functions. Here we use a dictionary-based approximation approach, called the Learning Regularized Functional Matching Pursuit (LRFMP). A dictionary is an intentionally redundant set of diverse trial functions from which iteratively an approximation in a best basis is built. The next best basis element is chosen such that the Tikhonov functional is minimized.</p> <p align="justify">The methods have been used for a variety of spherical as well as tomographic tasks from the geosciences as well as medical imaging. Here we apply them to seismic traveltime tomography for the first time. We discuss relevant developments and challenges in the process of tailoring the methods to the problem and show first promising results.</p>
<p>The three-dimensional structure of the Earth's interior shapes its geomagnetic and gravity fields, and can thus be constrained by observing these fields. 3-D Earth structure also causes seismological observables to deviate from those predicted for approximated, spherically symmetrical reference models. Travel time tomography is the inverse problem that uses these observed differences to constrain the 3-D structure of the interior.<br>On the planetary scale, i.e. in a spherical geometry, this linearized inverse problem has been parameterized with a variety of basis systems, either global (e.g. spherical harmonics) or local (e.g. finite elements). The Geomathematics Group Siegen has developed alternative approximation methods for certain applications from the geosciences: the Inverse Problem Matching Pursuits (IPMPs). These methods combine different basis systems by calculating an approximation in a so-called best basis, which is chosen iteratively from a so-called dictionary, an intentionally overcomplete set of diverse trial functions. In each iteration, the choice of the next best basis element reduces the Tikhonov functional. A particular numerical expertise has been gained for applications on spheres or balls. Hence, the methods were successfully applied to, for instance, the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging.<br>Our aim is to remodel the IPMPs for travel time tomography. This includes developing the data-dependent operator, deciding for specific trial functions and applying the operator to them. We also have to define termination criteria and develop the regularization in theory and practice. We introduce the IPMPs and show results from our remodelling.</p>
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.