## 1 Introduction  
In the field of geophysical exploration, electromagnetic methods serve as pivotal detection techniques, extensively applied in oil and gas exploration, mineral resource surveys, and groundwater research. These methods infer subsurface structures and their physical parameters by measuring the response of underground media to electromagnetic waves,with resistivity being a key parameter. Traditional electromagnetic approaches typically deploy transmitters and receivers on the surface to capture the electromagnetic (EM) responses of subsurface media. However, due to the diffusive nature of electromagnetic fields, surface-based EM methods often face limitations in detecting deep targets [1]. The borehole-to-surface electromagnetic method emerges as a superior technique for deep exploration, offering advantages unattainable by conventional surface EM methods. This approach involves placing the transmitter within a borehole, in close proximity to the target, emitting electromagnetic signals at various frequencies and measuring the EM response components at the surface to delineate subsurface electrical structures. The reduced distance between the transmitter and the target enhances the clarity of the surface-received EM responses, thereby improving observational accuracy and efficiency [2; 3]. In the early development of the borehole-to-surface electromagnetic method, pioneering work was carried out by researchers from the former Soviet Union and Japan. In 1958, based on the methodological framework of downhole seismic exploration, research teams from both countries systematically introduced the concept of joint borehole-surface observations into electromagnetic exploration. This marked the formal proposal of the theoretical concept of \"borehole-to-surface electromagnetics\" and was accompanied by a landmark experimental validation [4; 5]. This innovative, cross-disciplinary technology transfer not only broke through the vertical resolution limitations of traditional surface electromagnetic exploration but also established a solid methodological foundation for the high-precision detection of deep geological structures through the synergistic use of downhole transmitters and surface array receivers. In 2013, [PERSON] et al. [6] employed computer simulations to study CO\({}_{2}\) monitoring at the Kevin Dome sequestration site in Montana, USA, validating the feasibility of the borehole-to-surface electromagnetic method for deep reservoir CO\({}_{2}\) monitoring. Their results demonstrated that the method is not only efficient but also capable of clearly delineating the distribution of CO\({}_{2}\) within subsurface strata [6]. In 2015, [PERSON] et al. [7] investigated the applicability of borehole-to-surface electromagnetic technology for monitoring fluid movements in German oil fields using a three-dimensional forward modeling approach. Their simulations indicated that the method is highly sensitive to resistivity variations deep within reservoirs and that augmenting the system with downhole acquisition near the reservoir significantly enhances resolution [7]. In 2024, [PERSON] et al. [8] developed a three-dimensional forward modeling code for the borehole-to-surface electromagnetic method using C++ and validated its accuracy and precision by comparing it with semi-analytical solutions. This study further explored the effectiveness and sensitivity of the method in mineral and oil and gas reservoir exploration [8]. Due to computational resource limitations, the current numerical simulation methods for borehole-to-surface electromagnetic techniques generally assume isotropic conditions, which impose certain limitations when describing subsurface media with anisotropic characteristics [9; 10]. In reality, factors such as thin-layer deposition or particle alignment in sediments can induce significant variations in the measured data responses owing to the electrical anisotropy present in the Earth's crust [11; 12].