Numerical method and application of transient electromagnetic full-time responses of pseudo-random waveforms

The traditional transient electromagnetic method primarily uses square waves (trapezoidal waves), which lack high-frequency harmonic components, making it difficult to precisely characterize geological anomalies. This paper proposes emitting pseudo-random waveforms during the power supply period to improve field source resolution and enhance transient electromagnetic detection capability. Firstly, the full-time domain electromagnetic forward modeling method for pseudo-random waveforms is introduced. By using shift linear algebra operations on multiple step responses, one-dimensional pseudo-random electromagnetic responses can be calculated. For three-dimensional numerical simulations, based on the characteristics of pseudo-random waveforms, a source decoupling and Shift-and-Invert Krylov subspace technique is proposed. This method requires only one LU matrix decomposition and several dozen matrix back substitutions to solve the full-time electromagnetic field, and its accuracy is verified by comparing it with the results of the Backward Euler method. Through a combination of theoretical analysis, numerical simulation, and processing of measured data, the characteristics and detection capabilities of the secondary field excited by pseudo-random waveforms are studied in detail. The results show that compared to square waves, pseudo-random waveforms contain richer high-frequency harmonics, which can improve the detection resolution of time-domain electromagnetic methods. However, due to the relatively low energy of low-frequency harmonics, the late-stage electromagnetic field decays rapidly. The secondary field expression of pseudo-random waveforms contains both positive and negative terms, and improper zero-crossing settings can cause the late-stage secondary field to change sign. By reducing the time interval between ti and ti+1 (where i is an odd number), this phenomenon can be avoided. Moreover, comparing the secondary fields of different pulse width pseudo-random waveforms through the layered model shows that the narrow pulse width waveform has higher detection resolution. Further validation of the detection capability of pseudo-random waveforms is conducted through a three-dimensional geoelectrical model. Finally, processing field data demonstrates that the secondary field excited by pseudo-random waveforms can more finely characterize the resistivity characteristics of underground media.