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https://github.com/leouieda/phd-thesis

Latex source and figures for my PhD thesis. Slides for the thesis defense.
https://github.com/leouieda/phd-thesis

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Latex source and figures for my PhD thesis. Slides for the thesis defense.

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# PhD Thesis: Forward modeling and inversion of gravitational fields in spherical coordinates



Author: [Leonardo Uieda](http://www.leouieda.com)



Advisor: [Valéria C. F. Barbosa](http://lattes.cnpq.br/0391036221142471)



Institution: [Observatório Nacional](http://www.on.br/)



Presented: 29 April 2016



Thesis PDF (in English): [thesis.pdf](https://github.com/pinga-lab/phd-thesis/raw/master/thesis.pdf)



Thesis defense slides (in Portuguese): [presentation.pdf](https://github.com/pinga-lab/phd-thesis/raw/master/presentation/presentation.pdf)



![cover page of slides](https://github.com/pinga-lab/phd-thesis/raw/master/cover.png)



## Abstract



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.



## Contributions



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](https://github.com/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)](http://www.leouieda.com/talks/scipy2013.html).



### 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](https://github.com/pinga-lab/paper-moho-inversion-tesseroids)

for the source code and data associated with the paper.