**1. Introduction**

Electromagnetic technology has a wide range of applications in various fields. Nasim (2019) proposed a metamaterial platform for solving integral equations using a monochromatic electromagnetic field [1]. The application of electromagnetic technology in medical diagnosis equipment realizes the measurement of blood glucose and the detection of cancer stage [2,3]. In addition, it is necessary to consider electromagnetic fields in plasma electronics [4,5].

Here, we apply electromagnetic technology to geophysics. Grounded electrical-source airborne transient electromagnetic (GREATEM) is a useful detection geophysical method where the transmitter is set on the ground and the receiver is installed on the aircraft [6]. Combining the ground and airborne electromagnetic systems, GREATEM has the advantages of large detection depth, high resolution, wide range, and fast speed. In particular, it has unique superiority in mountains, forest-covered areas, swamps, and other special areas [7,8]. Ji (2013) used an airship as the carrier and the two-dimensional finite difference method to calculate the time-domain GREATEM response with a long wire source [9]. Ren (2017) investigated a three-dimensional time-domain airborne electromagnetic model based on the finite volume method, and then used the strategy of separating the primary field from the secondary field, which significantly saved on calculation time [10]. For airborne transient electromagnetic measurement, a three-dimensional forward simulation method which considered attitude change was proposed, and the electromagnetic response of a shallow surface was analyzed in Reference [11]. Later, the spectral element method (SE) was used to carry out three-dimensional modeling of GREATEM, effectively improving the modeling accuracy [12].

To interpret GREATEM, researchers often make the assumption that the earth is just conductive or resistive. However, there is an induced polarization (IP)effect in the earth, which is usually reflected as a reverse signal. The existence of the IP effect has a great impact on the electromagnetic response [13,14]. The IP effect is related to conductivity and frequency, and its dispersion characteristics can be expressed by the Debye model and the Cole–Cole model [15,16]. At present, numerical simulations of the polarization effect are mainly conducted in the frequency domain. Most of the early studies simulated the IP effect by converting the electromagnetic field from the frequency domain to the time domain [17,18]. Unfortunately, when broadband frequency content is considered, the methods are not so efficient [19]. In order to simulate the IP effect in the time domain, Kang (2015) proposed a method to generate a three-dimensional pseudo charge distribution using airborne time-domain electromagnetic data and effectively extracted the polarization signal [20]. Wu (2017) used the FDTD (finite difference time-domain) method to simulate the IP effect with a small loop source [21]. Commer (2017) expanded the finite difference time-domain scheme, modeled and analyzed the IP effect, and developed a simple calculation of the Cole–Cole model [22].

However, three-dimensional (3D) numerical simulations of IP effects based on FDTD require a high computation load, when including the air layer. In addition, it is difficult to calculate the source in the FDTD method; thus, the upward continuation method is often used to avoid directly loading the emission source. Mittet (2010) put forward the theory of wave field transformation and successfully applied it to marine CSEM (controllable source electromagnetic method), increasing the calculation efficiency by nearly 10-fold [23]. Mittet (2018) used the least squares fitting method to calculate the IP response of sea water in the fictitious wave field [24]. Ji (2017) improved the emission source, applied the wave field transformation method to the ground transient electromagnetic simulation, and realized a three-dimensional transient electromagnetic numerical simulation including the air layer [25]. However, it did not consider the IP effect of the earth. Therefore, here, we perform the wave field transformation in GREATEM, and we realize a three-dimensional numerical simulation in polarizable media using the Cole–Cole model. In addition, the computing efficiency is also improved.

The wave field transformation method is proposed for the grounded electrical-source airborne transient electromagnetic system in this paper. The Maxwell equations in the real diffusion field are transformed into the fictitious wave field by using the transformation relationship between the diffusion field and fictitious wave field. The fractional order Cole–Cole model is introduced into the fictitious wave field, and the Maxwell equations with the Cole–Cole model are solved using the finite difference time-domain method. Finally, an integral transformation method is used to transform the calculation results back to the real diffusion field. In the case of including polarized media, the two-field numerical simulation of the induction field and polarization field is realized. In addition, the computing efficiency is also improved.
