## 1 Introduction  
Transient electromagnetic method (TEM) is a classic geophysical electromagnetic (EM) exploration technique and has been widely used in mineral and petroleum exploration, engineering and environmental monitoring, groundwater and geothermal detection, as well as geological disaster prediction, etc. TEM utilizes artificial sources and normally adopts a periodic step-off source waveform to emit EM signals. During the switch-off period, the underground medium modifies the induced EM fields that decay over time. By measuring and analyzing the transient response, the information of the underground anomalous bodies can be obtained, which helps to infer the spatial distribution of the target body. TEM is characterized by lightweight instruments, high sensitivity, and strong resolution. Additionally, TEM observes the pure secondary EM fields with fewer requirements on the transmitter-receiver offset compared to the frequency-domain EM methods, and therefore, TEM has attracted more attention in recent years. [PERSON], [PERSON], [PERSON], and [PERSON] were among the first to use simple spherical models to study the impact of magnetic susceptibility on electromagnetic response [1; 2; 3; 4]. [PERSON] included susceptibility effects in a study using 1-D layered models [5]. [PERSON] et al. investigated the effect of the magnetic susceptibility of underground media on the EM field induced by a ring current source [6]. [PERSON] and [PERSON] demonstrated that the effect of magnetic susceptibility is observable for geological models with a high-magnetic-susceptibility thin layer [7; 8]. [PERSON] and [PERSON], using a 1-D layered model, with known resistivities, obtained the 1-D vertical distribution of magnetic susceptibility from the inversion of frequency-domain EM data [9]. [PERSON] and [PERSON] experimentally demonstrated that, under certain conditions, assuming constant magnetic susceptibility in the inversion could not determine the location of the anomalous body [10]. However, when the magnetic susceptibility was treated as the inversion parameters, the location, resistivity, and magnetic susceptibility values of the anomalous body could be accurately determined. [PERSON] et al. developed a 1-D inversion algorithm for magnetotelluric (MT) data that simultaneously inverts both conductivity and magnetic susceptibility, which was later applied to the interpretation of 3-D data [11]. [PERSON] et al. published a 1-D inversion algorithm for TEM using a loop source, which simultaneously inverts both conductivity and magnetic susceptibility [12]. Currently, research on EM forward modeling and inversion considering the non-homogeneity of magnetic susceptibility is mainly focused on the MT method. In TEM forward and inverse modeling, it is typically assumed that the magnetic permeability or the magnetic susceptibility of the entire medium is homogeneous. The magnetic permeability of a vacuum is normally adopted, with calculations and analyses focusing on the non-homogeneity of resistivity in the subsurface medium. However, in areas with high magnetic susceptibility, such as regions with igneous rocks of high magnetic content or areas rich in ferromagnetic minerals [13], neglecting the effect of magnetic susceptibility may lead to incorrect data-interpretation results. Therefore, it is important to develop 3-D forward modeling and inversion algorithms for TEM data that take into account the influence of magnetic susceptibility. Based on the current state of research, this paper develops a 3-D forward and inverse modeling algorithm for ground-based loop-source TEM data, which can simultaneously handle the inhomogeneous condition for both resistivity and magnetic susceptibility models. Several synthetic examples are presented to demonstrate the accuracy of the forward modeling algorithm and the inversion results for the resistivity and magnetic susceptibility models.