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.