Research paperFinite-element time-domain modeling of electromagnetic data in general dispersive medium using adaptive Padé series
Introduction
Time-domain electromagnetic (TEM) methods have been widely used to study subsurface conductive structures (Ward and Hohmann, 1988, Zhdanov, 2009). Compared to frequency-domain electromagnetic methods, the TEM method usually has better resolution and sensitivity to deep targets for typical transmitter-receiver configurations and broad time scales. The correct interpretation of the TEM data requires accurate forward modeling methods. There exist two major methods for solving this problem – one is based on the Fourier transform of the frequency-domain response to the time domain (e.g. Knight and Raiche, 1982, Everett and Edwards, 1993, Raiche, 1998, Mulder et al., 2007, Ralph-Uwe et al., 2008), and another exploits a direct discretization of the Maxwell's equation in both spatial and time domains (Wang and Hohmann, 1993, Commer and Newman, 2004, Maaø, 2007, Um et al., 2012, Jin, 2014, Yin et al., 2016).
Note that, the accuracy of Fourier transformation is affected significantly by the frequency sampling and the transformation methods, such as the choice of the digital filters (Li et al., 2016). The finite-difference time-domain (FDTD) methods have been used for modeling the electromagnetic response in time domain for decades (Yee, 1966).
We should note also that, in the framework of the finite-difference method, the complex geometries need to be approximated by a stair-cased model. It is well known that, these complications of finite-difference modeling, can be overcome by the finite-element approach. It has been demonstrated that the FETD method with unstructured spatial discretization can reduce the size of the problem dramatically (Um, 2011, Jin, 2014).
There are two major types of time discretization: 1) an explicit scheme, 2) an implicit scheme. The explicit scheme requires a small time step size to satisfy the Courant stability condition (Wang and Hohmann, 1993, Um, 2011, Jin, 2014), which makes this approach computationally expensive for TEM modeling with time scale from a small fraction of a second to hundreds seconds (Zaslavsky et al., 2011). The implicit approach is unconditionally stable but it requires solving a linear system of equations with the matrix depending on the time step size. This problem can be addressed by adopting modern direct solvers, since the corresponding matrix needs to be decomposed only once for a fixed time step size (Um, 2011, Jin, 2014). We adopt the FETD scheme proposed by Um (2011) for solving the TEM modeling problem. We also update the time step, in an adaptive manner, to reduce the computational cost.
The conventional modeling of TEM data usually considers a non-dispersive medium, with frequency-independent conductivity. In the presence of IP effect, the conductivity becomes frequency dependent. It was shown by Pelton et al. (1978) that the conductivity relaxation model can be well represented by the Cole-Cole model. In this paper, we consider the dispersive conductive medium with the conductivity described by the Cole-Cole model. Zhdanov (2008) introduced a more general conductivity relaxation model based on the generalized effective-medium theory of IP (so called “GEMTIP” model). It was shown by Zhdanov (2008) that the GEMTIP model reduces to the Cole-Cole model in a special case of spherical inclusions within a homogeneous background model. We could update Cole-Cole model with GEMTIP model in our FETD modeling algorithm.
Frequency-dependent dispersion models need to be represented by a convolution of the electric field in the time domain. The convolution term can be introduced into Maxwell's equation through the fractional derivative with respect to time (Zaslavsky et al., 2011, Marchant et al., 2014). Solving such equations with convolution or fractional derivative terms requires the electric field at all previous stages (Zaslavsky et al., 2011), since either the convolution or the fractional derivative correspond to a global operator. Due to this problem, the TEM data with IP effect are rarely modeled directly in time domain.
The Padé series (Baker and Graves-Morris, 1996) can be used to avoid the fractional derivative problem raised in modeling the EM field in dispersive medium (Weedon and Rappaport, 1997). The fractional differential equation can be transformed to the differential equation with integer order and further to be solved using numerical methods such as FDTD (Rekanos and Papadopoulos (2010)). Based on the work of Weedon and Rappaport (1997) and Rekanos and Papadopoulos (2010) for the FDTD method with the Padé approximation, Marchant et al. (2014) proposed a finite-volume time-domain method for simulating IP effect with the Cole-Cole model.
In all the publications cited above, in order to calculate the Padé coefficients, the Taylor series was implemented in the vicinity of one preselected center frequency. However, we will demonstrate that the accuracy of the corresponding Padé approximation depends significantly on the selected center frequency. In order to keep the same accuracy of the Padé approximation for different time moments, we propose selecting different central frequencies for early and late time moments. We call this approach the adaptive Padé series. We have implemented the FETD modeling with IP effect using this adaptive Padé approximation. Instead of using a Taylor expansion at the fixed point for calculating the Padé coefficients, we update the Padé coefficients adaptively during the FETD modeling process. This approach increases the accuracy of FETD modeling with IP effects.
Section snippets
Finite element time domain discretization of Maxwell's equation
The Maxwell's equations in time domain for the quasi-stationary EM field can be described as follows (Zhdanov, 2009, Jin, 2014):where E and H are electric and magnetic fields, Js is the current density of the source, and je is the induction current density, described by the Ohm's law:
In the last formula, σ is the electric conductivity. In a nondispersive medium, σ is time invariant.
We eliminate the magnetic field from the system of equations and obtain the diffusion
Modeling IP effects with adaptive Padé series
Previously, we assumed that the electric conductivity was time and frequency independent. However, we often encounter the frequency-dependent conductivity in geophysical exploration, and this phenomenon is manifested by the IP effect (Ward and Hohmann, 1988, Hallof and Yamashita, 1990, Luo and Zhang, 1998, Seigel et al., 2007, Zhdanov, 2009). There exist different dispersion models to describe the IP phenomenal. Pelton et al. (1978) derived the Cole-Cole relaxation model based on the equivalent
Model studies
We now demonstrate the developed algorithm using several model studies. At first, we consider a half space model with IP effect for both Debye and a general Cole-Cole conductivity relaxation. Then, we consider a model with non-dispersive half-space background and localized dispersive 3D anomaly.
Conclusions
We have developed an edge-based finite-element time-domain method for simulating electromagnetic fields in conductive and dispersive medium. We consider a total field formulation and use unstructured tetrahedral mesh to reduce the size of the problem. We also use the backward difference, which is unconditionally stable, for time domain discretization. We adopt time step doubling methods to gradually increase the step size and reduce the computational expense. The sparse system of equations is
Acknowledgement
The authors are thankful to the anonymous reviewers for their valuable suggestions. The authors acknowledge the support of the University of Utah’s Consortium for Electromagnetic Modeling and Inversion (CEMI) and TechnoImaging. The project was partially supported by the National Natural Science Foundation of China with Grant codes of 41674075, 41630317, and 41474055, the Natural Science Foundation of Guangxi Province with the code of 2016GXNSFGA380004, and the high-level innovative team and
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