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Regularization is critical for solving ill-posed geophysical inverse problems. Explicit regularization is often used, but there are opportunities to explore the implicit regularization effects that are inherent in a Neural Network structure. Researchers have discovered that the Convolutional Neural Network (CNN) architecture inherently enforces a regularization that is advantageous for addressing diverse inverse problems in computer vision, including de-noising and in-painting. In this study, we examine the applicability of this implicit regularization to geophysical inversions. The CNN maps an arbitrary vector to the model space. The predicted subsurface model is then fed into a forward numerical simulation to generate corresponding predicted measurements. Subsequently, the objective function value is computed by comparing these predicted measurements with the observed measurements. The backpropagation algorithm is employed to update the trainable parameters of the CNN during the inversion. Note that the CNN in our proposed method does not require training before the inversion, rather, the CNN weights are estimated in the inversion process, hence this is a test-time learning (TTL) approach. In this study, we choose to focus on the Direct Current (DC) resistivity inverse problem, which is representative of typical Tikhonov-style geophysical inversions (e.g. gravity, electromagnetic, etc.), to test our hypothesis. The experimental results demonstrate that the implicit regularization can be useful in some DC resistivity inversions. We also provide a discussion of the potential sources of this implicit regularization introduced from the CNN architecture and discuss some practical guides for applying the proposed method to other geophysical methods.

Electromagnetic methods are increasingly being applied in settings with steel infrastructure. These include applications such as monitoring of CO2 sequestration or even assessing the integrity of a wellbore. In this paper, we examine the impacts of the magnetic permeability of a steel-cased well on electromagnetic responses in grounded source experiments. We consider a vertical wellbore and simulate time and frequency domain data on 3D cylindrical meshes. Permeability slows the decay of surface electric fields in the time domain and contributes to a phase shift in the frequency domain. We develop our understanding of how permeability alters currents within, and external to, the casing by focussing first on the time domain response and translating insights to the frequency domain. Following others, we rewrite Maxwell's equations to separate the response into terms that describe the magnetization and induction effects. Magnetic permeability impacts the responses in two ways: (1) it enhances the inductive component of the response in the casing, and (2) it creates a magnetization current on the outer wall of the casing. The interaction of these two effects results in a poloidal current system within the casing. It also generates anomalous currents external to the casing that can alter the geometry and magnitude of currents in the surrounding geologic formation. This has the potential to be advantageous for enhancing responses in monitoring applications.

Geophysical electromagnetics (EM) plays an important role in mineral exploration and is increasingly being used to help solve other problems of relevance to society. In this article we reflect, from our perspective at University of British Columbia (UBC), on the development of EM geophysics over the years and on our attempts to enhance its understanding, and its visibility and usefulness to the community. The availability of open-source resources, and a shift within the EM community towards collaborative practices for sharing and creating software and educational resources, has been a driver of progress towards these goals. In this article we provide some background about this trajectory and how the SEG Distinguished Instructor Short Course (DISC) was a catalyst in our development of software and resources and in our broader goal of creating more collaborative connections within the EM community.

In addition to reducing anthropogenic emissions of CO$_2$, it is increasingly clear we also need to remove CO$_2$ from the atmosphere in order to avoid some of the worst case scenarios for climate change. Geologic sequestration of CO$_2$ is among the most attractive approaches because of the large global capacity and long-time scales for storage. One mechanism of geologic storage is through carbon mineralization. Some mafic and ultramafic rocks contain minerals that will react with CO$_2$ in a carbonation reaction and convert it to carbonated minerals. This is effectively a permanent CO$_2$ storage mechanism. The geologic question we are faced with is then to locate, delineate and estimate the volume of potentially reactive rocks. Using a synthetic model that emulates a prospective site for carbon mineralization in British Columbia, we simulate and invert gravity and magnetic data to delineate reactive rocks. We begin by inverting each data set independently and introduce a proxy experiment to contend with the challenging problem of choosing an appropriate physical-property threshold to estimate volumes from the recovered model. We use this proxy experiment to estimate thresholds for standard, $\ell_2$ inversion of the gravity and magnetics, as well as for inversions which use sparse and compact norms. A Petrophysically and Geologically Guided Inversion (PGI) framework is used to construct quasi-geologic models from which volumes can be estimated directly. We apply the PGI framework to the magnetics and gravity data independently. The framework is also used to jointly invert these data and produce a model that is consistent with both data sets. Cumulative volume estimates with depth are informative and can help decide whether in situ or ex situ sequestration might be appropriate. Using each of the inverted models, we estimate cumulative volume of reactive rock as a function of depth., Comment: https://fasttimesonline.co/geophysical-inversions-to-delineate-rocks-with-co-2-sequestration-potential-through-carbon-mineralization/

Electrical and electromagnetic (EM) methods can be diagnostic geophysical imaging tools for monitoring applications, such as carbon capture and storage or hydraulic fracturing. In these settings, it is common that steel-cased wells and other steel infrastructure are present. Grounded source methods, which use electrodes to inject current into the earth are of interest for casing integrity and monitoring applications. Electrostatic, or direct current (DC) resistivity, experiments form the basis of our understanding of the physics of grounded source experiments in terms of charges, currents, and electric fields. Steel-cased wells are highly conductive and although their presence makes numerical modelling more challenging, they can help targets of interest be detected because they channel charges and currents to depth. Time-domain EM experiments use a time-varying transmitter current. Understanding the EM response requires that we consider both galvanic, or DC currents, as well as image currents that are induced in the subsurface. As compared to DC experiments, the physics of EM is more complex because of the multiple current systems, as well as the need to consider magnetic permeability of steel-cased wells. However, EM experiments have the advantage that they can provide a large data set that is sensitive to a target of interest. Ultimately this will increase the potential for being able to extract information about the target.

The standard smooth electrical resistivity tomography inversion produces an estimate of subsurface conductivity that has blurred boundaries, damped magnitudes, and often contains inversion artifacts. In many problems the expected conductivity structure is well constrained in some parts of the subsurface, but incorporating prior information in the inversion is not a trivial task. In this study we developed an electrical resistivity tomography inversion algorithm that combines parametric and smooth inversion strategies. In regions where the subsurface is well constrained, the model was parameterized with only a few variables, while the rest of the subsurface was parameterized with voxels. We tested this hybrid inversion strategy on two synthetic models that contained a well constrained highly resistive or conductive near-surface horizontal layer and a target beneath. In each testing scenario, the hybrid inversion improved resolution of feature boundaries and magnitudes and had fewer inversion artifacts than the standard smooth inversion. A sensitivity analysis showed that the hybrid inversion successfully recovered subsurface features when a range of regularization parameters, initial models, and data noise levels were tested. The hybrid inversion strategy can potentially be expanded to a range of applications including marine surveys, permafrost/frozen ground studies, urban geophysics, or anywhere that prior information allows part of the model to be constrained with simple geometric shapes.

In a previous paper, we introduced a framework for carrying out petrophysically and geologically guided geophysical inversions. In that framework, petrophysical and geological information is modelled with a Gaussian Mixture Model (GMM). In the inversion, the GMM serves as a prior for the geophysical model. The formulation was confined to problems in which a single physical property model was sought, with a single geophysical dataset. In this paper, we extend that framework to jointly invert multiple geophysical datasets that depend on multiple physical properties. The petrophysical and geological information is used to couple geophysical surveys that, otherwise, rely on independent physics. This requires advancements in two areas. First, an extension from a univariate to a multivariate analysis of the petrophysical data, and their inclusion within the inverse problem, is necessary. Second, we address the practical issues of simultaneously inverting data from multiple surveys and finding a solution that acceptably reproduces each one, along with the petrophysical and geological information. To illustrate the efficacy of our approach and the advantages of carrying out multi-physics inversions, we invert synthetic gravity and magnetic data associated with a kimberlite deposit. The kimberlite pipe contains two distinct facies embedded in a host rock. Inverting the datasets individually leads to a binary geological model: background or kimberlite. A multi-physics inversion, with petrophysical information, differentiates between the two main kimberlite facies of the pipe. Through this example, we also highlight the capabilities of our framework to work with interpretive geologic assumptions when minimal quantitative information is available. In those cases, the dynamic updates of the Gaussian Mixture Model allow us to perform multi-physics inversions by learning a petrophysical model., Comment: 47 pages in one column format; 14 figures; to be published in Geophysical Journal International

The potential for extracting and interpreting induced polarization (IP) data from airborne surveys is now broadly recognized. There is, however, still considerable discussion about the conditions under which the technique can provide knowledge about the subsurface and thus, its practical applications. Foremost among these is whether, or under what conditions, airborne IP can detect chargeable bodies at depth. To investigate, we focus on data obtained from a coincident-loop time-domain system. Our analysis is expedited by using a stretched exponential rather than a Cole-Cole model to represent the IP phenomenon. Our paper begins with an example that illuminates the physical understanding about how negative transients (the typical signature of an IP signal in airborne data) can be generated. The effects of the background conductivity are investigated; this study shows that a moderately conductive and chargeable target in a resistive host is an ideal scenario for generating strong IP signals. We then examine the important topic of estimating the maximum depth of the chargeable target that can generate negative transients. Lastly, some common chargeable earth-materials are discussed and their typical IP time-domain features are analyzed. The results presented in this paper can be reproduced and further explored by accessing the provided Jupyter notebooks., Comment: 39 pages, 14 figures

Electromagnetics has an important role to play in solving the next generation of geoscience problems. These problems are multidisciplinary, complex, and require collaboration. This is especially true at the base scientific level where the underlying physical equations need to be solved, and data, associated with physical experiments, need to be inverted. In this paper, we present arguments for adopting an open-source methodology for geophysics and provide some background about open-source software for electromagnetics. Immediate benefits are the reduced time required to carry out research, being able to collaborate, having reproducible results, and being able to disseminate results quickly. To illustrate the use of an open-source methodology in electromagnetics, we present two challenges. The first is to simulate data from a time domain airborne system over a conductive plate buried in a more resistive earth. The second is to jointly invert airborne TDEM and FDEM data with ground TDEM. SimPEG, Simulation and Parameter Estimation in Geophysics, (https://simpeg.xyz) is used for the open-source software. The figures in this paper can be reproduced by downloading the Jupyter Notebooks we provide with this paper (https://github.com/simpeg-research/oldenburg-2018-AEM). Access to the source code allows the researcher to explore the simulations and inversions by changing model and inversion parameters, plot fields and fluxes to gain further insight about the EM phenomena, and solve a new research problem by using open-source software as a base. By providing results in a manner that allows others to reproduce, further explore, and even extend them, we hope to demonstrate that an open-source paradigm has the potential to enable more rapid progress of the geophysics community as a whole.

Inversions of airborne EM data are often an iterative process, not only requiring that the researcher be able to explore the impact of changing components such as the choice of regularization functional or model parameterization, but also often requiring that forward simulations be run and fields and fluxes visualized in order to build an understanding of the physical processes governing what we observe in the data. In the hope of facilitating this exploration and promoting reproducibility of geophysical simulations and inversions, we have developed the open source software package, SimPEG. The software has been designed to be modular and extensible with the goal of allowing researchers to interrogate all of the components and to facilitate the exploration of new inversion strategies. We present an overview of the software in its application to airborne EM and demonstrate its use for visualizing fields and fluxes in a forward simulation as well as its flexibility in formulating and solving the inverse problem. We invert a line of airborne TDEM data over a conductive vertical plate using a 1D voxel-inversion, a 2D voxel inversion and a parametric inversion, where all of the forward modelling is done on a 3D grid. The results in this paper can be reproduced by using the provided Jupyter notebooks. The Python software can also be modified to allow users to experiment with parameters and explore the physics of the electromagnetics and intricacies of inversion.

The work in this paper is motivated by the increasing use of electrical and electromagnetic methods in geoscience problems where steel-cased wells are present. Applications of interest include monitoring carbon capture and storage and hydraulic fracturing operations, as well as detecting flaws or breaks in degrading steel-casings -- such wells pose serious environmental hazards. The general principles of electrical methods with steel-cased wells are understood, and several authors have demonstrated that the presence of steel-cased wells can be beneficial for detecting signal due to targets at depth. However, the success of a DC resistivity survey lies in the details. Secondary signals might only be a few percent of the primary signal. In designing a survey, the geometry of the source and receivers, and whether the source is at the top of the casing, inside of it, or beneath the casing will impact measured responses. Also the physical properties and geometry of the background geology, target, and casing will have a large impact on the measured data. Because of the small values of the diagnostic signals, it is important to understand the detailed physics of the problem and also to be able to carry out accurate simulations. This latter task is computationally challenging because of the extreme geometry of the wells, which extend kilometers in depth but have millimeter variations in the radial direction, and the extreme variation in the electrical conductivity (typically 5-7 orders of magnitude between the casing and the background geology)., Comment: Geophysical Journal International

Simulating direct current resistivity, frequency domain electromagnetics and time domain electromagnetics in settings where steel cased boreholes are present is of interest across a range of applications including well-logging, monitoring subsurface injections such as hydraulic fracturing or carbon capture and storage. In some surveys, well-casings have been used as "extended electrodes" for near surface environmental or geotechnical applications. Wells are often cased with steel, which has both a high conductivity and a significant magnetic permeability. The large physical property contrasts as well as the large disparity in length-scales, which are introduced when a steel-cased well is in a modeling domain, makes performing an electromagnetic forward simulation challenging. Using this setting as motivation, we present a finite volume approach for modeling electromagnetic problems on cylindrically symmetric and 3D cylindrical meshes which include an azimuthal discretization. The associated software implementation includes modeling capabilities for direct current resistivity, time domain electromagnetics, and frequency domain electromagnetics for models that include variable electrical conductivity and magnetic permeability. Electric and magnetic fields, fluxes, and charges are readily accessible in any simulation so that they can be visualized and interrogated. We demonstrate the value of being able to explore the behaviour of electromagnetic fields and fluxes through examples which revisit a number of foundational papers on direct current resistivity and electromagnetics in steel-cased wells. The software implementation is open source and included as a part of the SimPEG software ecosystem for simulation and parameter estimation in geophysics., Comment: Submitted to Computers & Geosciences. Code to run the examples is available at: https://github.com/simpeg-research/heagy_2018_emcyl and archived at https://doi.org/10.5281/zenodo.1220427

Fluid flow in the vadose zone is governed by Richards equation; it is parameterized by hydraulic conductivity, which is a nonlinear function of pressure head. Investigations in the vadose zone typically require characterizing distributed hydraulic properties. Saturation or pressure head data may include direct measurements made from boreholes. Increasingly, proxy measurements from hydrogeophysics are being used to supply more spatially and temporally dense data sets. Inferring hydraulic parameters from such datasets requires the ability to efficiently solve and deterministically optimize the nonlinear time domain Richards equation. This is particularly important as the number of parameters to be estimated in a vadose zone inversion continues to grow. In this paper, we describe an efficient technique to invert for distributed hydraulic properties in 1D, 2D, and 3D. Our algorithm does not store the Jacobian, but rather computes the product with a vector, which allows the size of the inversion problem to become much larger than methods such as finite difference or automatic differentiation; which are constrained by computation and memory, respectively. We show our algorithm in practice for a 3D inversion of saturated hydraulic conductivity using saturation data through time. The code to run our examples is open source and the algorithm presented allows this inversion process to run on modest computational resources., Comment: Computers and Geosciences (2018)

Electromagnetic simulations of complex geologic settings are computationally expensive. One reason for this is the fact that a fine mesh is required to accurately discretize the electrical conductivity model of a given setting. This conductivity model may vary over several orders of magnitude and these variations can occur over a large range of length scales. Using a very fine mesh for the discretization of this setting leads to the necessity to solve a large system of equations that is often difficult to deal with. To keep the simulations computationally tractable, coarse meshes are often employed for the discretization of the model. Such coarse meshes typically fail to capture the fine-scale variations in the conductivity model resulting in inaccuracies in the predicted data. In this work, we introduce a framework for constructing a coarse-mesh or upscaled conductivity model based on a prescribed fine-mesh model. Rather than using analytical expressions, we opt to pose upscaling as a parameter estimation problem. By solving an optimization problem, we obtain a coarse-mesh conductivity model. The optimization criterion can be tailored to the survey setting in order to produce coarse models that accurately reproduce the predicted data generated on the fine mesh. This allows us to upscale arbitrary conductivity structures, as well as to better understand the meaning of the upscaled quantity. We use 1D and 3D examples to demonstrate that the proposed framework is able to emulate the behavior of the heterogeneity in the fine-mesh conductivity model, and to produce an accurate description of the desired predicted data obtained by using a coarse mesh in the simulation process., Comment: 27 pages, 13 figures

Simulations and inversions of electromagnetic geophysical data are paramount for discerning meaningful information about the subsurface from these data. Depending on the nature of the source electromagnetic experiments may be classified as time-domain or frequency-domain. Multiple heterogeneous and sometimes anisotropic physical properties, including electrical conductivity and magnetic permeability, may need be considered in a simulation. Depending on what one wants to accomplish in an inversion, the parameters which one inverts for may be a voxel-based description of the earth or some parametric representation that must be mapped onto a simulation mesh. Each of these permutations of the electromagnetic problem has implications in a numerical implementation of the forward simulation as well as in the computation of the sensitivities, which are required when considering gradient-based inversions. This paper proposes a framework for organizing and implementing electromagnetic simulations and gradient-based inversions in a modular, extensible fashion. We take an object-oriented approach for defining and organizing each of the necessary elements in an electromagnetic simulation, including: the physical properties, sources, formulation of the discrete problem to be solved, the resulting fields and fluxes, and receivers used to sample to the electromagnetic responses. A corresponding implementation is provided as part of the open source simulation and parameter estimation project SimPEG (http://simpeg.xyz). The application of the framework is demonstrated through two synthetic examples and one field example.