The transient electromagnetic method (TEM) has been widely used as a geophysical exploration method in recent years. When Maxwell's equations are discretized in the time domain by the direct solution method, the initial field is used as a substitute for the source, so the electromagnetic response of a shallow three-dimensional anomalous body cannot be calculated. Maxwell's equations in the frequency domain are simple in form, and the current source can be directly added without calculating the initial field. However, large linear equations must be calculated. The coefficient matrix is large, and the calculation speed is slow, which limits their application. Based on Yee grids, this paper combines octree grids with the finitedifference frequency-domain (FDFD) method. Ensuring a sufficiently large computational area, octree grids are used to refine the area of anomalous bodies, while coarse grids are still used to reduce the total number of grids and improve the efficiency. In the numerical simulation, the vacancy positions are set to zero to solve the data storage problem of the coarse grids and fine grids. The binary paraboloid interpolation method is used to solve the electromagnetic field component transfer problem at the intersection of the coarse grid and fine grid. Finally, the electromagnetic response curve in the time domain is obtained by a frequency-time transformation. By comparing the calculation results of typical models, the correctness of the FDFD method based on octree grids is verified. By comparing the computational time of complex anomalous bodies for Yee grids and octree grids, it can be concluded that the efficiency of the FDFD method based on octree grids is improved to a certain extent. INDEX TERMS Finite-difference frequency-domain (FDFD), frequency-time transformation, numerical simulation, octree grids, transient electromagnetic method (TEM).
An improved finite-difference time-domain (FDTD) solution is developed for three-dimensional transient electromagnetic modeling in source-free media. Based on Guptasarma's linear filtering method and the fast Hankel transform, the improved method selects the electric fields (E) at the first two time-steps, and the time derivative of the magnetic induction (∂B/∂t) at the second time-step as initial conditions. This selection can avoid approximating ∂E/∂t at initial times and improve initial accuracies. During the iterations, the electric fields and ∂B/∂t are both defined at the same integer time indices and the deriving equations are modified based on the FDTD method. Consequently, it can avoid the approximation of the electric fields at intermediate time indices and magnetic fields at integer time indices, and improve the iteration precisions. The convolutional perfectly matched layer (C-PML) boundary condition is employed to reduce the reflection errors and the effects are verified. At last the proposed approach is checked via homogeneous half-space models, layered models and 3D anomalous models with complex backgrounds. The results show that the improved method can reduce simulation errors twice or thrice and deepen effective exploration depth by 20%, which has verified the effectiveness of the improved method for three-dimensional transient electromagnetic modeling, especially for long-time and high-resistance modeling.
Summary Superparamagnetic (SPM) effects lead to 1/t power-law delay approximately during the middle and late periods of transient electromagnetic responses. Ignoring SPM effects can produce large errors and cause incorrect data interpretation. Furthermore, because of the nonlinearity of the Chikazumi magnetic susceptibility model, it is difficult to discretize the model in the time domain. Therefore, developing an algorithm for three-dimensional (3D) modeling of SPM effects in the time domain directly is difficult. In view of the above problems, we propose a 3D modeling method of SPM effects for airborne transient electromagnetic (ATEM) data. We also introduce the widely used Chikazumi magnetic susceptibility model into the finite-difference frequency-domain (FDFD) method. According to the mapping relations between the magnetic field intensity and the magnetic permeability on the grid after discretization, we re-establish the scientific matrix of the magnetic permeability. And we adopt arithmetic averaging of the two adjacent cubes’ magnetic permeability as the magnetic permeability in boundary, which is a different setting method with the electrical conductivity. The 3D modeling of SPM effects is successfully realized, and its maximum relative error is less than 7%. We subsequently analyze the intrinsic properties of the abnormal magnetic medium, as well as the effect of the SPM unit's parameters, such as the depth, thickness, electrical conductivity, zero-frequency magnetic susceptibility, and the height of the receiver, on the response. Then, we calculate the SPM responses of 3D magnetic bodies and analyze the influence of magnetic susceptibility and electrical conductivity on magnetite responses. The present study can provide theoretical guidance for ATEM detection in magnetic media and lay a foundation for 3D inversion of SPM effects.
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