Theoretical Analysis and Validation of Multiple-Mode Doppler Curves

Acoustic-seismic coupling is a prominent phenomenon in seismology. Seismologists can track airborne traffic events by analyzing the Doppler curves in acoustic-seismic coupled signals. We conducted a theoretical analysis to explain the phenomenon of multiple-mode Doppler curves seen in the actual data and point out that the multiple-mode Doppler curves in seismic signals originate from the periodic signals generated during the flight of flying objects. Numerical simulations further confirm this theoretical viewpoint: when the signals generated during aircraft flight can be decomposed into multiple single-frequency periodic signals, we can observe a corresponding number of Doppler curves in seismic signals. Conversely, if the airplane signals cannot be decomposed into periodic single-frequency signals, we cannot observe the presence of Doppler curves in the seismic record. If the frequency of the signals generated by the aircraft varies linearly with time, i.e., exhibits the characteristics of linear frequency modulation, a regular frequency shift curve may still appear in the time-frequency spectrum of the seismic record. In such cases, the Doppler curves need to be corrected to be used for accurate tracking of the aircraft. These research findings provide a solid theoretical foundation for seismologists to utilize Doppler curves in seismic data to track airborne events.