pyGIMLi: An open-source library for modelling and inversion in geophysics

This review gives an overview of the development of machine learning in geoscience. A thorough analysis of the codevelopments of machine learning applications throughout the last 70 years relates the recent enthusiasm for machine learning to developments in geoscience. I explore the shift of kriging toward a mainstream machine learning method and the historic application of neural networks in geoscience, following the general trend of machine learning enthusiasm through the decades. Furthermore, this chapter explores the shift from mathematical fundamentals and knowledge in software development toward skills in model validation, applied statistics, and integrated subject matter expertise. The review is interspersed with code examples to complement the theoretical foundations and illustrate model validation and machine learning explainability for science. The scope of this review includes various shallow machine learning methods, e.g., decision trees, random forests, support-vector machines, and Gaussian processes, as well as, deep neural networks, including feed-forward neural networks, convolutional neural networks, recurrent neural networks, and generative adversarial networks. Regarding geoscience, the review has a bias toward geophysics but aims to strike a balance with geochemistry, geostatistics, and geology, however, excludes remote sensing, as this would exceed the scope. In general, I aim to provide context for the recent enthusiasm surrounding deep learning with respect to research, hardware, and software developments that enable successful application of shallow and deep machine learning in all disciplines of Earth science.

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70 years of machine learning in geoscience in review

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 Botts 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 Guyana Shield, the central Solimoes and Amazonas Basins, the Parana Basin, and the Borborema province. These differences suggest the existence of crustal or mantle density anomalies that were unaccounted for during gravity data processing.

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Fast nonlinear gravity inversion in spherical coordinates with application to the South American Moho

We present an updated oceanic crustal age grid and a set of complementary grids including spreading rate, asymmetry, direction, and obliquity. Our data set is based on a selected set of magnetic anomaly identifications and the plate tectonic model of Muller et al. (2019, 10.1029/2018TC005462 ). We find the mean age of oceanic crust is 64.2 Myr, slightly older than previous estimates, mainly due to the inclusion of pockets of Mesozoic aged crust in the Atlantic and Mediterranean and improvements to the Jurassic Pacific triangle. This older crust is partly compensated by additional Cenozoic‐aged back‐arc basin crust not included in previous models. The distribution of spreading modes based on area of preserved crust is relatively equal between slow (20–55 mm/yr) and fast (75–180 mm/yr) spreading systems at 33% and 39%, respectively. Crust transitional between fast and slow, or intermediate systems (55–75 mm/yr), cover 20% of the preserved ocean floor with much smaller proportions of crust formed at ultraslow (5%) and super‐fast (3%) spreading systems. Slow and intermediate spreading systems exhibit the most stable behavior in terms of spreading asymmetry and obliquity, with the widest distribution of obliquities occurring at ultraslow spreading systems, consistent with present‐day observations. Our confidence grid provides a complementary resource for nonexperts to identify those parts of the age grid that are least well constrained. Our grids in 6, 2, and 1 arc min resolution as well as our python workflow, isopolate, used to compute our data sets are freely available in online repositories and on the GPlates data portal.

A Global Data Set of Present-Day Oceanic Crustal Age and Seafloor Spreading Parameters

We present a case study of applying 3D inversion of gravity gradiometry data to iron ore exploration in Minas Gerais, Brazil. The ore bodies have a distinctly high-density contrast and produce well-defined anomalies in airborne gravity gradiometry data. We have carried out a study to apply 3D inversion to a 20  km2 subarea of data from a larger survey to demonstrate the utility of such data and associated inversion algorithm in characterizing the deposit. We examine multiple density contrast models obtained by first inverting Tzz; then Txz, Tyz, and Tzz jointly; and finally all five independent components to understand the information content in different data components. The commonly discussed Tzz component is sufficient to produce geologically reasonable and interpretable results, while including additional components involving horizontal derivatives increases the resolution of the recovered density model and improves the ore delineation. We show that gravity gradiometry data can be used to deli...

3D inversion of airborne gravity gradiometry data in mineral exploration: A case study in the Quadrilátero Ferrífero, Brazil

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2019GC008515

The Generic Mapping Tools Version 6

We have developed the open-source software Tesseroids, a set of command-line programs to perform 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 have determined optimal values of D for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the comp...

Tesseroids: Forward-modeling gravitational fields in spherical coordinates

We have developed a new gravity gradient inversion method for estimating a 3D density-contrast distribution defined on a grid of rectangular prisms. Our method consists of an iterative algorithm that does not require the solution of an equation system. Instead, the solution grows systematically around userspecified prismatic elements, called “seeds,” with given density contrasts. Each seed can be assigned a different density-contrast value, allowing the interpretation of multiple sources with different density contrasts and that produce interfering signals. In real world scenarios, some sources might not be targeted for the interpretation. Thus, we developed a robust procedure that neither requires the isolation of the signal of the targeted sources prior to the inversion nor requires substantial prior information about the nontargeted sources. In our iterative algorithm, the estimated sources grow by the accretion of prisms in the periphery of the current estimate. In addition, only the columns of the sensitivity matrix corresponding to the prisms in the periphery of the current estimate are needed for the computations. Therefore, the individual columns of the sensitivity matrix can be calculated on demand and deleted after an accretion takes place, greatly reducing the demand for computer memory and processing time. Tests on synthetic data show the ability of our method to correctly recover the geometry of the targeted sources, even when interfering signals produced by nontargeted sources are present. Inverting the data from an airborne gravity gradiometry survey flown over the iron ore province of Quadrilatero Ferrifero, southeastern Brazil, we estimated a compact iron ore body that is in agreement with geologic information and previous interpretations.

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Robust 3D gravity gradient inversion by planting anomalous densities

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.

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Leonardo Uieda

Papers

The Generic Mapping Tools Version 6

Tesseroids: Forward-modeling gravitational fields in spherical coordinates

Fast nonlinear gravity inversion in spherical coordinates with application to the South American Moho

Robust 3D gravity gradient inversion by planting anomalous densities

Modeling the Earth with Fatiando a Terra