Seismic signals observed at the MeSO-net station during the game have several characteristic frequencies and continue from several tens of seconds to several minutes (Fig. 2). To understand the excitation of seismic waves caused by the supporters, we first checked the seismic records at Station 1145, which is located nearby supports of the steeped stadium floor for Kashiwa Reysol supporters.

Figure 3 shows the running spectra of the UD components at Station 1145. The amplitudes of the seismic signals were very high during the game and dropped at half time. They were also high before the kick-off. This time corresponds to when the players practiced in the field. Supporters started singing and jumping even before the game started. Figure 4 compares the spectra of the UD components at Stations 1145, 1149, and E.KW8M from 14:30 to 14:32. Two characteristic frequencies were observed for E.KW8M, whereas four characteristic frequencies were observed at station 1145 and 1149. The lowest characteristic frequency among these was 2.4 Hz, which was coherently observed at all stations. The other characteristic frequencies are integer multiples of the lowest characteristic frequency. Peaks were observed at 4.8, 7.2, and 9.6 Hz at 1145 and 1149, whereas only a peak at 4.8 Hz was observed at E.KW8M.

Fig. 3
figure 3

A running spectra plot of UD component of Station 1145

Fig. 4
figure 4

Spectra plots of UD components at Stations 1145, 1149, and E.KW8M between 14:30 and 14:32. A black vertical line “ × 1” represents the lowest characteristic frequency observed at the two stations, which is 2.4 Hz. Black vertical lines “ × 2”, “ × 3”, and “× 4” epresents two to four multiples of the lowest characteristic frequency (i.e., 4.8, 7.2, and 9.6 Hz, respectively)

Ellis and Ji (2004) modeled loading caused by jumping people. According to them, a single force loading on the surface owing to a constant-rate jumping individual can be written as follows:

$$Fleft(tright)=Wleft[1.0+ sum_{n=1}^{infty }{c}_{n} sinleft(frac{2npi }{T}t+{varphi }_{n}right)right],$$

(1)

where W is the weight of a person, cn is the dynamic loading factor, T is the jumping period, and ({varphi }_{n}) is the phase lag. This suggests that integer multiple characteristic frequencies should be observed in seismic records when people are jumping. Actually, we observed integer multiple characteristic frequencies in seismic records during the game as shown in Fig. 4. Therefore, the observed seismic signals are expected to be induced by jumping supporters, not other noise signals.

We tested this hypothesis by comparing the seismic records with the audio recorded in the stadium during the game. We considered that the jumping periods of supporters were controlled by the rhythm of the chants they were singing, because some supporters hit drums at the center of supporters to lead singing chants, which works as a metronome for singing and jumping supporters. Hence, we measured the rhythm of the chants from the audio records and compared it with the characteristic frequencies in the seismic records.

The rhythms of the chants were measured using the autocorrelation function of the envelope waveform of audio records. The audio records were band-passed between 100 and 200 Hz and converted to an envelope waveform, which was then low-pass filtered below 4 Hz. A bandpass filter was applied before constructing the envelope waveform to emphasize the sounds of the drums. A low-pass filter was applied to smooth the envelope waveforms. Examples of autocorrelation functions of the processed envelope functions are shown in Fig. 5. The audio record shown in Fig. 5 corresponds to the seismic record shown in Fig. 4. The calculated autocorrelation function has periodic peaks, which we define as the chant rhythm. In this case, the periodicity of the autocorrelation function was 0.421 s, which corresponds to a characteristic frequency of 2.38 Hz. This value agrees with the characteristic frequency of 2.4 Hz obtained from the seismic record shown in Fig. 4. We selected the largest peaks in the calculated autocorrelation function between lag times of 0.2 and 0.8 s (Fig. 5b), and rejected the picked frequencies if the widths of the peaks, where the correlation coefficients drop by 0.1 from the peak value, are larger than 0.2 s.

Fig. 5
figure 5

Plots of audio data for chants rhythm measurements. a 100–200 Hz band-passed waveform of sound record at 14:31:00 – 14:31:30. b Autocorrelation function of the envelope waveform converted from a. A vertical black line represents periodicity of the autocorrelation function

We made this comparison for the entire dataset used to investigate the relations between characteristic frequencies in seismic and audio records. The characteristic frequencies of the seismic records were measured using polarization analysis (Park et al. 1987; Yabe et al. 2020). The seismic waves generated by supporters are highly polarized, and polarization analysis is useful for extracting such features. Using time windows of 30 s shifted every 1 s, we calculated Fourier spectra of three-component seismograms. For every frequency, Fourier spectra covariance matrices were calculated, which were then stacked every 30 s. The stacked Fourier spectra covariance matrices are used to calculate the polarization amplitude and azimuths at every frequency and characteristic frequencies of the seismic signals. The polarization amplitude was defined as the maximum eigenvalue of the stacked covariance matrix. The polarization azimuths were calculated using a complex eigenvector for the maximum eigenvalue, which are along the direction in which the polarization dips. We also calculated singularity values defined as ({left(3tleft({S}^{2}right)-{tleft(Sright)}^{2}right)}!left/ !{2{tleft(Sright)}^{2}}right.) (Koper and Hawley 2010), where S is the stacked covariance matrix at a specific frequency and t() is the trace of a matrix. Larger singularity values mean that the polarization is more evident. The signal-to-noise ratio of every time window was calculated as well with the definition of the noise level as the log-averaged polarization amplitude between 12:00 and 13:00 h. The characteristic frequencies of seismic records were defined as frequencies at polarization amplitude peaks with signal-to-noise ratios larger than 10 and singularity values stacked for all stations larger than 0.8. The threshold of singularity values was selected so that the polarization is clearly observed in all stations. Figure 6 shows a comparison of the two characteristic frequencies read from the seismic and audio records. When the characteristic frequencies were observed, the timings were highly consistent with each other (Fig. 6a). It is also confirmed that the characteristic frequencies of the seismic records are integer multiples of the characteristic frequencies of the audio records (Fig. 6b). This observation is consistent with the model of Ellis and Ji (2004), which was explained above. Therefore, based on this result, we confirm that the seismic source of footquakes in Hitachi-Kashiwa Soccer Stadium is the earth-shaking J. LEAGUE supporters.

Fig. 6
figure 6

Scatter plots showing the comparisons of characteristic frequencies observed in seismic and sound records. a Time plot of observed characteristic frequencies. Black and red dots represent characteristic frequencies read from seismic and sound records, respectively. b Comparisons of two characteristic frequencies observed at the same timing. Gray lines represent one and two times multiple of chants rhythms

Good coincidences between characteristic frequencies in seismic and audio records provide us a good example of monitoring collective human activities through seismic waves. Figure 6a shows that characteristic frequencies varied with time, which represents supporters sang different chants during the game. It means that we can distinguish which chants supporters are singing only from seismic waves if we have a priori information on rhythms of supporters’ chants.

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