Christoph Schwarzbach

A framework for the upscaling of the electrical conductivity in the quasi-static Maxwells equations

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.

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.

3D Cooperative Parametric Inversion of Frequency and Time-domain Airborne Electromagnetic Data

We developed a 3D frequency and time-domain parametric level-set AEM inversion code for multiple bodies with independent resistivities and applied a cooperative approach for spatially overlapping data sets. We tested the method on both synthetic and field data, and the synthetic results showed that the cooperative parametric approach recovered the most accurate inversion model. We then applied the method to overlapping AEM data sets at the Committee Bay greenstone belt in Nunavut, Canada to image linear thin dipping conductors where the dip information is crucial for the exploration program.

Finite Element Based Inversion For Electromagnetic Problems Using Stochastic Optimization

SUMMARY In this paper we discuss the solution of large scale electromagnetic inverse problems that arise in hydrocarbon exploration of the ocean floor. Such problems require handling domains of considerable extension, bathymetry and a large number of sources and receivers. We show that by combining finite element discretization and stochastic optimization it is possible to efficiently deal with such large scale problems.

A Multiscale Finite Volume Method with Oversampling for Geophysical Electromagnetic Simulations

on, we develop a Multiscale finite volume (MSFV) method with oversampling for the quasistatic Maxwell’s equations in the frequency domain. Our method begins by assuming a coarse mesh nested into a fine mesh, which accurately discretizes the setting. For each coarse cell, we solve independently a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. To solve the local Maxwell’s system, we use the fine mesh contained in the extended domain and the Mimetic Finite Volume method. Afterwards, these local solutions, called basis functions, together with a weak continuity condition are used to construct a coarse-scale version of the global problem that is much cheaper to solve. The basis functions can be used to obtain the fine-scale details from the solution to the coarse-scale problem. Our approach leads to a significant reduction in the size of the final system of equations to be solved and in the amount of computational time of the simulation, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using a heterogeneous 3D mineral deposit model.

3D Multiple Body Parametric Inversion of Time-domain Airborne EM Data

We developed a 3D parametric inversion for time-domain airborne EM data using a skewed ellipsoid representation for multiple conductive or resistive anomalies. The approach aims to simplify the task of imaging thin, potentially highly conductive, anomalies with 3D EM inversion. The algorithm finds the optimal location, shape, size and resistivity of the anomalies in a homogeneous or heterogeneous background by employing a Gauss-Newton style optimization. Our parametric method is tested on a synthetic and field data set. The synthetic model is composed of two narrow dipping conductive anomalies in a resistive background along with a vertical narrow conductor. The survey layout and resistivity structure is based off field data from a greenstone setting. The parametric inversion accurately recovers the spatial extent and dips of the three synthetic anomalies, although the depth extent of the anomalies is exaggerated. In the greenstone field example, the inversion defines the spatial location, extent and dips of three conductive anomalies to provide a new conductivity interpretation of an area where little information is known regarding the true nature of the conductors.