As an efficient geophysical exploration method, the time-domain airborne electromagnetic (AEM) data often show sign reversal in late-time channels due to induced polarization (IP) effect. The traditional imaging and inversion methods without considering the IP effect cannot recover the true electrical structure of the earth, so it is necessary to develop 3D EM forward modeling and inversion techniques with IP effect. In this paper, we propose a 3D forward modeling method for time-domain AEM with IP effect based on unstructured finite-element (FE) method. To describe the IP effect of a medium, we introduce the Cole-Cole model and transform it into fractional derivative form using frequency-time conversion. Then, we discretize it using the difference discretization format of Caputo fractional derivative. Finally, we use the vector FE method based on unstructured tetrahedral mesh and unconditionally stable second-order backward Euler’s scheme to discretize Maxwell’s equations in space and time. In this way, we can solve the 3D forward modeling problem for AEM with IP effect in time domain. We verify the accuracy of our algorithm by comparing it with the 1D semi-analytical solution for a half-space model, and then calculate EM responses for typical abnormal models and analyze the characteristics of IP effect.

The choice of data misfit measure has a great impact on the convergence of electromagnetic (EM) inversion. The conventional measure based on the $l_{2}$ -norm tends to excessively amplify the weights of a larger misfit, inadvertently neglecting data with a smaller misfit during the inversion process, thereby diminishing the resolution to a certain degree. To solve this problem, we propose a robust inversion strategy based on $l_{1}$ -norm data misfit and adaptive moment estimation (Adam). In this scheme, we use the Ekblom-type $l_{1}$ -norm to simplify the derivative computation of the absolute value function. The Adam algorithm is further applied to optimize this type of non-smooth objective function, which incorporates momentum terms and adaptive steps, allowing it to better adapt to the irregularities in gradient changes. The inversion results obtained from both synthetic models and field measurements demonstrate that the Adam method performs considerably better than the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method for optimizing the $l_{1}$ - $l_{2}$ norm of the objective function. Compared with the conventional ${l} _{2}$ -norm data misfit, the $l_{1}$ -norm data misfit can effectively avoid excessive optimization of data with large misfits and achieve high-resolution inversion results.

The flight altitude has a large effect on the airborne electromagnetic (AEM) responses. Due to the dynamic environment of the aircraft, the recorded sensor altitudes may contain errors. Research demonstrates that the AEM responses caused by a several meters altitude errors can be larger than caused by some anomalous body. Ignoring these errors will create erroneous results in AEM data interpretation. Considering that there is not yet a published 3D AEM inversion method that takes into account the flight altitude, we develop in this paper a 3D inversion algorithm for AEM with the flight height treated as an inversion parameter. For the forward modeling we use the finite element method, while for the inversion we use the Gauss-Newton optimization method. To make our inversion works for variable flight altitudes, we propose a scheme of 3D Jacobean matrix calculation for both the resistivities and flight altitudes without much increasing the computational cost. The numerical simulation result confirms that the flight altitude really has a large effect on the AEM responses. The inversions of synthetic data show that our 3D inversion method can both recover the resistivity distribution in the underground and decrease the altitude errors recorded, while the field data inversion demonstrates that our method can deliver a better inversion model with a smaller data misfit.

In a heterogeneous medium (usually called a rough medium) with fractured formations, the propagation of an electromagnetic (EM) field is a type of subdiffusion. Current mainstream geophysical EM data processing methods cannot be applied to data acquired on heterogeneous earth, as they are not governed by the classic diffusion theory. To evaluate the influence of roughness on the transient EM (TEM) signal for a complex model and contribute to data inversion, we proposed a novel three-dimensional (3D) forward modeling scheme for TEM in rough media. First, we derived the governing equation with a fractional-order time derivative for the subdiffusion of EM waves in rough media. Then, we proposed a novel time discretization using an unequal step length for the Caputo operator, which significantly reduces the total number of time steps. Finally, an implicit time-domain finite-element method using unstructured tetrahedron discretization was adopted to solve the 3D forward problem. Furthermore, an efficient time segmentation strategy combined with parallel RHS construction was proposed to accelerate modeling. The numerical results prove that the proposed method is accurate and efficient, and will be a powerful numerical method for analyzing TEM wave propagation and processing TEM data in areas with multi-scale fractures or porosity.