PhD Thesis: Forward modeling and inversion of gravitational fields in spherical coordinates
Author: Leonardo Uieda
Advisor: Valéria C. F. Barbosa
Institution: Observatório Nacional
Presented: 29 April 2016
Thesis PDF (in English): thesis.pdf
Thesis defense slides (in Portuguese): presentation.pdf
We present methodological improvements to forward modeling and regional inversion of satellite gravity data. For this purpose, we developed two open-source software projects. The first is a C language suite of command-line programs called Tesseroids. The programs calculate the gravitational potential, acceleration, and gradient tensor of a spherical prism, or tesseroid. Tesseroids implements and extends an adaptive discretization algorithm to automatically ensure the accuracy of the computations. Our numerical experiments show that, to achieve the same level of accuracy, the gravitational acceleration components require finner discretization than the potential. In turn, the gradient tensor requires finner discretization still than the acceleration. The second open-source project is Fatiando a Terra, a Python language library for inversion, forward modeling, data processing, and visualization. The library allows the user to combine the forward modeling and inversion tools to implement new inversion methods. The gravity forward modeling tools include an implementation of the algorithm used in the Tesseroids software. We combined the inversion and tesseroid forward modeling utilities of Fatiando a Terra to develop a new method for fast non-linear gravity inversion. The method estimates the depth of the crust-mantle interface (the Moho) based on observed gravity data using a spherical Earth approximation. We extended the computationally efficient Bott's method to include smoothness regularization and use tesseroids instead right rectangular prisms. The inversion is controlled by three hyper-parameters: the regularization parameter, the density-contrast between the real Earth and the reference model (the Normal Earth), and the depth of the Moho of the Normal Earth. We employ two cross-validation procedures to automatically estimate these parameters. Tests on synthetic data confirm the capability of the proposed method to estimate smoothly varying Moho depths and the three hyper-parameters. Finally, we applied the inversion method developed to produce a Moho depth model for South America. The estimated Moho depth model fits the gravity data and seismological Moho depth estimates in the oceanic areas and the central and eastern portions of the continent. We observe large misfits in the Andes region, where Moho depth is largest. In Amazon, Solimões, and Paraná Basins, the model fits the observed gravity but disagrees with seismological estimates. These discrepancies suggest the existence of density-anomalies in the crust or upper mantle, as has been suggested in the literature.
Each of the following chapters have been published or submitted for publication.
Tesseroids: forward modeling gravitational fields in spherical coordinates
We present the open-source software Tesseroids, a set of command-line programs to perform the forward modeling of gravitational fields in spherical coordinates. The software is implemented in the C programming language and uses tesseroids (spherical prisms) for the discretization of the subsurface mass distribution. The gravitational fields of tesseroids are calculated numerically using the Gauss-Legendre Quadrature (GLQ). We have improved upon an adaptive discretization algorithm to guarantee the accuracy of the GLQ integration. Our implementation of adaptive discretization uses a "stack" based algorithm instead of recursion to achieve more control over execution errors and corner cases. The algorithm is controlled by a scalar value called the distance-size ratio (D) that determines the accuracy of the integration as well as the computation time. We determined optimal values of D for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the computed tesseroids effects with those of a homogeneous spherical shell. The values required for a maximum relative error of 0.1% of the shell effects are D = 1 for the gravitational potential, D = 1.5 for the gravitational acceleration, and D = 8 for the gravity gradients. Contrary to previous assumptions, our results show that the potential and its first and second derivatives require different values of D to achieve the same accuracy. These values were incorporated as defaults in the software.
Accepted for publication in the Geophysical Software and Algorithms section of the journal Geophysics. See pinga-lab/paper-tesseroids for the source code and data associated with the paper.
Modeling the Earth with Fatiando a Terra
Geophysics is the science of using physical observations of the Earth to infer its inner structure. Generally, this is done with a variety of numerical modeling techniques and inverse problems. The development of new algorithms usually involves copy and pasting of code, which leads to errors and poor code reuse. Fatiando a Terra is a Python library that aims to automate common tasks and unify the modeling pipeline inside of the Python language. This allows users to replace the traditional shell scripting with more versatile and powerful Python scripting. The library can also be used as an API for developing stand-alone programs. Algorithms implemented in Fatiando a Terra can be combined to build upon existing functionality. This flexibility facilitates prototyping of new algorithms and quickly building interactive teaching exercises. In the future, we plan to continuously implement sample problems to help teach geophysics as well as classic and state-of-the-art algorithms.
Published in the Proceedings of the 12th Python in Science Conference (Scipy 2013).
Fast non-linear gravity inversion in spherical coordinates with application to the South American Moho
Estimating the relief of the Moho from gravity data is a computationally intensive non-linear inverse problem. What is more, the modeling must take the Earths curvature into account when the study area is of regional scale or greater. We present a regularized non-linear gravity inversion method that has a low computational footprint and employs a spherical Earth approximation. To achieve this, we combine the highly efficient Bott's method with smoothness regularization and a discretization of the anomalous Moho into tesseroids (spherical prisms). The computational efficiency of our method is attained by harnessing the fact that all matrices involved are sparse. The inversion results are controlled by three hyper-parameters: the regularization parameter, the anomalous Moho density-contrast, and the reference Moho depth. We estimate the regularization parameter using the method of hold-out cross-validation. Additionally, we estimate the density-contrast and the reference depth using knowledge of the Moho depth at certain points. We apply the proposed method to estimate the Moho depth for the South American continent using satellite gravity data and seismological data. The final Moho model is in accordance with previous gravity-derived models and seismological data. The misfit to the gravity and seismological data is worse in the Andes and best in oceanic areas, central Brazil and Patagonia, and along the Atlantic coast. Similarly to previous results, the model suggests a thinner crust of 30-35 km under the Andean foreland basins. Discrepancies with the seismological data are greatest in the Guiana shield, the central Solimões and Amazon basins, the Paraná basin, and the Borborema province. These differences suggest the existence of crustal or mantle density anomalies that were unaccounted for during gravity data processing.
Submitted for publication in the Geophysical Journal International. See pinga-lab/paper-moho-inversion-tesseroids for the source code and data associated with the paper.