## 1 Introduction  
The electrical source TEM, as an important geophysical exploration technique [1], is widely used in fields such as metal mineral exploration [2, 3], coalfield geological hazard investigation [4], hydro-geophysical prospecting [5], and tunnel advance prediction [6, 7] due to its high efficiency and deep detection capabilities [8, 9]. However, traditional electrical source TEM primarily uses square waves (or trapezoidal waves), with fieldwork also employing triangular waves, semi-sine waves, and other waveforms [10]. Overall,these waveforms generally lack high-frequency harmonic components, making them inadequate for the fine characterization of poor geological bodies [11]. Moreover, the volume effects in TEM diffusion fields are prominent, and the resolution capability for electrical interfaces is limited when applying diffusion field-based migration imaging and inversion methods [12]. Therefore, exploring new TEM detection techniques and imaging methods is of significant value for improving the interpretation of TEM data. This study investigates a multi-resolution pre-stack migration imaging method for electrical source TEM virtual wavefields. It is well known that traditional field sources use square waves with a fixed fundamental frequency to excite TEM fields, with harmonic energy primarily dominated by low-frequency harmonics, which limits the detection resolution [13]. To enhance the detection capabilities of the TEM, some scholars have focused on the transmitted waveform during ON-Time as a research entry point [14,15]. [PERSON] et al. investigated the effect of different transmitted waveforms on the response of airborne electromagnetic systems, and provided the relationship between the optimal pulse width of commonly used semi-sine waves, square waves, triangular waves, and trapezoidal waves and detection time [16]. [PERSON] et al. derived the electromagnetic field expressions for a loop source in free space for these four waveforms, studied their impact on electromagnetic detection, and pointed out that square and trapezoidal waves are suitable for ground-based electromagnetic methods, while semi-sine and triangular waves are more suitable for airborne electromagnetic methods [17]. [PERSON] et al. conducted a detailed study on the detection capability of semi-sine and trapezoidal waves for geological targets and suggested that, based on actual exploration requirements, optimal airborne electromagnetic parameters can be designed to improve operational efficiency [18]. According to recent research, the traditional square wave lacks high-frequency harmonic components, making it difficult to fundamentally improve the detection resolution in time-domain electromagnetic methods [19]. [PERSON] et al. highlighted advancements in urban underground space imaging methods, noting that by transmitting differential pulses with varying pulse widths during the ON-Time for scanning detection, a multi-resolution TEM diffusion field can be generated, thereby enhancing the resolution of TEM detection [13]. Therefore, this paper employs differential pulse excitation to generate electromagnetic fields, and through multi-pulse scanning, enables multi-resolution detection of geological anomalies at varying depths and scales. To achieve high-resolution imaging of TEM data, the kinematic characteristics of the wavefield can be extracted from the diffusion field based on the reverse transformation theory of TEM wavefields [20,21,22,23,24]. However, the reverse transformation equation for the virtual wavefield is a first-kind Fredholm integral equation, which is inherently ill-posed [25]. When the reverse transformation equation is linearized and discretized, the resulting coefficient matrix is severely ill-conditioned, making it difficult to accurately recover the virtual wavefield using conventional numerical methods [26]. To mitigate the ill-conditioning of the coefficient matrix, [PERSON] et al.