The transient electromagnetic (TEM) method utilizes artificial transmitters and measures electromagnetic (EM) responses to reveal the resistivity information of the subsurface. The current waveform of transmitters has nonnegligible effects on induced fields. Therefore, 3-D TEM forward modeling algorithms need the capability of simulating arbitrary waveforms to obtain accurate responses. In time-stepping-based 3-D TEM forward modeling, the source term (ST) approach is frequently used, which employs the source current density to model the waveform variation during time-stepping. The ST approach, however, requires fine-time discretization to describe complex waveforms, which could significantly raise the computational cost. We present a robust convolution (Conv) approach that computes the convolution between the time derivative of the waveform and the step-off response to incorporate the waveform effects in 3-D TEM modeling. The Conv approach does not discretize the waveform using time steps. Hence, it is advantageous when modeling full-waveform cases. The developed algorithm is based on the finite-element (FE) method using unstructured grids and the implicit backward Euler approach. Both galvanic and inductive transmitters are incorporated. Ground and airborne TEM surveys are tested using an actual airborne TEM waveform, a full waveform of the 2(n) -sequence pseudorandom signal, and various synthetic waveforms. Accuracy is validated against the 1-D and 3-D solutions of published studies. The ST and Conv approaches are compared. Synthetic examples show that the latter approach simplifies the waveform incorporation in TEM modeling and substantially improves time-stepping efficiency without sacrificing accuracy.

Time-domain electromagnetic modeling in 3-D requires the solution of partial differential equations discretized on a grid. The finer grid resolution is usually required to describe rapid variations of the electromagnetic field in the near-surface, where the source and small-scale anomalies present. Since the electromagnetic field diffuses in the lossy medium, its variations become smoother with depth. The conventional finite-difference modeling approach using the staggered grid extends the fine grid resolution (needed for shallow layers) to all depths. It results in over-discretization of the problem and redundant computational costs. Here, we apply the multi-resolution (MR) grid approach to the time-domain electromagnetic modeling (TDEM). The MR grid allows us to decrease the grid resolution with depth and consequently reduce the number of degrees of freedom without compromising the accuracy of the solution. We implement a way of treating the loop source in TDEM modeling such that the definition of the source term is based on the Biot-Savart law; this allows separation of the loop source from the grid, making the source simulation more flexible.

To verify our new TDEM modeling, we perform several synthetic tests. We also apply the algorithm to model the geomagnetically induced electric field (GIE). Such modeling is an essential part of estimating hazards caused by geomagnetically induced currents (GIC). In contrast to frequency-domain modeling primarily used in previous studies, the time-domain GIE modeling allows us to consider the time variability of the source in the ionosphere in real-time.

For more realistic simulations, we use a large-scale 3-D resistivity model of Fennoscandia. An example of the MR grid GIE modeling highlights the areas of high GIE contrasts and shows that the real inhomogeneous 3-D resistivity distribution and realistic source geometry are necessary for a better estimation of GIC.

This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

3-D forward modelling and inversion techniques play an important role in data interpretation, but they are still computationally challenging tasks. Therefore, this thesis aims to improve forward modelling and inversion performance using novel multi-resolution (MR) grid approach. We implement 3-D forward modelling and data inversion for the DC Resistivity (DCR) and Time Domain Electromagnetic (TDEM) methods on the MR grid. When compared to conventional Staggered (SG) grids, the MR grid implements variable horizontal discretization (resolution) with depth, thus providing simple, but necessary flexibility in grid construction. Fine grid resolution is generally required near the surface to simulate fast variations of EM fields and to depict the shallow complex geometries and measurement configurations. Due to the lossy materials in the subsurface, the variations of the fields become smoother with depth, which is well represented by coarser grid discretization. Furthermore, this can also be viewed as decreasing sensitivity with depth, hence fine grid discretization is also less important for deep regions of the inversion model. The SG grid commonly uses a fine horizontal resolution to ensure accuracy, which is however not needed at depth and results in redundant computations. The MR grid can roughen the discretization with depth and alleviate the over-discretization. As a result, the MR grid can improve the efficiency of forward modelling while maintaining accuracy. Consequently, this improves data inversion performance while preserving the accuracy of inverse models.

We realize 3-D DCR forward modelling based on finite-differences discretization, which leads to solving a system of equations for electric potential. Obtained system matrices are hermitian and symmetric in both SG and MR cases. The optimal iterative solution for such systems is based on the Preconditioned Conjugate Gradient (PCG) method, which takes advantage of symmetry and has an optimal convergence rate.

The 3-D TDEM forward modelling is implemented using both the explicit scheme based on a modified version of the Du Fort-Frankel method and the implicit scheme based on a second-order backward Euler method. To implement the explicit scheme, we propose a Biot-Savart source term approach to calculate the magnetic field generated by a loop source, which makes the source calculations independent from the grid discretization and thereby improves the flexibility of the modelling setup. In the implicit scheme, the time-stepping is advanced by solving systems of equations. Similarly, the coefficient matrices are converted to be symmetric in both SG and MR grid approaches, and the equations can be efficiently solved using the PCG method as well. Since the initial guess of the solution has a substantial effect on the performance of the iterative solver, we investigate different initial guesses of the solution. Furthermore, we compare the explicit and implicit schemes of TDEM forward modelling in different resistivity scenarios to show their preferable conditions.

Based on the algorithm of explicit scheme TDEM forward modelling, we further implement modelling of Geomagnetically Induced Currents (GIC). Line currents are used to simulate the equivalent source in the ionosphere. The 3-D resistivity model of Fennoscandia is modeled with time-varying sources to investigate the inhomogeneous distribution of the induced electric fields.

Based on the explicit scheme TDEM forward modelling, we further develop the 3-D TDEM inversion algorithm. The turn-off waveform of the loop transmitter is taken into account in both forward modelling and inversion, and we highlight its importance by illustrating the result of ignoring the turn-off time. The MR grid approach is also used to discretize the inversion model and implement the pseudo modelling for sensitivity computations. We present several synthetic examples to demonstrate the improvement of inversion efficiency using the MR grid compared to the SG grid approach.

We implemented a novel multi-resolution grid approach to direct current resistivity (DCR) modeling in 3-D. The multi-resolution grid was initially developed to solve the electromagnetic forward problem and helped to improve the modeling efficiency. In the DCR forward problem, the distribution of the electric potentials in the subsurface is estimated. We consider finite-difference staggered grid discretization, which requires fine grid resolution to accurately model electric potentials around the current electrodes and complex model geometries near the surface. Since the potential variations attenuate with depth, the grid resolution can be decreased correspondingly. The conventional staggered grid fixes the horizontal grid resolution that extends to all layers. This leads to over-discretization and therefore unnecessary high computational costs (time and memory). The non-conformal multi-resolution grid allows the refinement or roughening for the grid’s horizontal resolution with depth, resulting in a substantial reduction of the degrees of freedom, and subsequently, computational requirements. In our implementation, the coefficient matrix maintains its symmetry, which is beneficial for using the iterative solvers and solving the adjoint problem in inversion. Through comparison with the staggered grid, we have found that the multi-resolution grid can significantly improve the modeling efficiency without compromising the accuracy. Therefore, the multi-resolution grid allows modeling with finer horizontal resolutions at lower computational costs, which is essential for accurate representation of the complex structures. Consequently, the inversion based on our modeling approach will be more efficient and accurate.

We discuss the implementation of multi-resolution framework to 3-D Direct Current (DC) problem. Commonly used staggered (SG) grid fixes the horizontal grid resolution for all depths. Thus, employing the fine horizontal resolution may lead to an over-discretised forward problem, subsequently affecting the performance of the inversion. We implemented a novel multi-resolution (MR) grid approach to the 3-D DC modeling and inversion problem, which allows adjustment of the horizontal resolution with depth. By using finer resolution for the near-surface regions, MR grid can ensure the modeling accuracy and describe the shallow features in the inversion model as well. The ability to use relatively coarser horizontal resolution for the deeper regions reduces the computation costs compare to the SG grid modeling. As a result, modeling and inversion can be accelerated several times by solving a smaller problem. Our grid resembles non-conformal rectangular grid, which commonly used in finite-elements modelling.