## 6 Conclusions  
This paper proposes the multi-resolution pre-stack migration imaging method of the virtual wave field in TEM exploration with electrical sources. By exciting the electromagnetic field with differential pulses of varying pulse widths, the detection resolution of TEM can be effectively improved. To capture the electrical interface characteristics of the subsurface layers, this study employs the preconditioned precise integration scanning wave field inverse transformation method to extract the virtual wave field from the diffusion field. The finite-difference method is then used for migration imaging, and the migration results from different differential pulses are stacked to obtain a high-resolution imaging profile from shallow to deep regions. The following conclusions are drawn:  
* When square waves are used as the excitation waveform, the dominance of low-frequency harmonics hinders the achievement of high-resolution detection. In contrast, differential pulses effectively enhance high-frequency harmonics, thereby mitigating the impact of low-frequency harmonics on detection performance. * Based on the buried thrust fault model, square waves and differential pulses with different pulse widths were used as excitation sources. From the relative anomaly maps of \(\mathbf{b}_{\mathrm{z}}\), it is observed that the anomaly response increased by 53.7%, indicating that differential pulses provide higher detection resolution. * The TEM diffusion field exhibits a stronger volumetric effect. Converting it into a virtual wavefield can effectively mitigate the impact of the volumetric effect. However, the virtual wavefield cannot accurately represent the subsurface features, and it is necessary to apply migration imaging methods to obtain the geometric shapes of the stratigraphic interfaces. * Square waves with different pulse widths can effectively reflect the geometric characteristics of the shallow subsurface layers. As the pulse width of the square wave increases, although the resolution of the deep subsurface layers improves to some extent, the enhancement remains limited. * Imaging results from the buried thrust fault model and the complex geological model demonstrate that differential pulses enable multi-resolution detection across different depth ranges, which can subsequently be used to obtain high-resolution imaging through correlation stacking. * This paper investigates the use of differential pulses with varying pulse widths to excite transient electromagnetic fields, and employs migration imaging methods to image the interfaces of geological anomalies. Future research will focus on developing resistivity imaging methods using differential pulses with different pulse widths. * This study primarily addresses the fundamental theoretical research on transient electromagnetic multi-resolution detection. The corresponding equipment is currently in the final stage of debugging. We plan to conduct the first experimental validation once the equipment is completed, with the experimental validation expected to begin in August 2025. At that time, a systematic evaluation of the proposed method will be conducted. In summary, this research provides guidance for time-domain electromagnetic exploration and data interpretation. Future studies should explore how to design the optimal combination of differential pulse waveforms tailored to specific geological structures. Since this study primarily focuses on the foundational theoretical work and the development of the proposed method, the next step will be to apply the presented data processing techniques to field data to further validate the feasibility of the method. Writing--original draft, validation, methodology, and formal analysis, K.L. ; writing--review and editing, funding acquisition, and conceptualization: K.L., X.L. and J.Y. ; reviewand editing, analysis and interpretation: Y.F., Q.Y. and X.T. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China (Grant No.