A simple method to evaluate the seismic/infrasonic energy partitioning during an eruption using data contaminated by air-to-ground signals

A volcanic eruption transmits both seismic waves and infrasound signals. The seismo-acoustic power 7 ratio is widely used to investigate the eruption behaviors and the source dynamics. It is often the case 8 that seismic data during an eruption are signiﬁcantly contaminated or even dominated by ground 9 shaking due to infrasound (air-to-ground signals). To evaluate the contribution of infrasound-originated 10 power in the seismic data, we need a response function of the seismic station to infrasound. It is rare to 11 obtain a seismo-acoustic data-set containing only infrasound signals, though it is ideal for calculating 12 the response function. This study proposes a simple way to calculate the response function using 13 seismo-acoustic data containing infrasound and independent seismic waves. The method requires data 14 recorded at a single station and mainly uses the cross-correlation function between the infrasound data 15 and the Hilbert transform of the seismic data. It is tested with data recorded by a station at Kirishima 16 volcano, Japan, of which response function has been constrained. It is shown that the method 17 calculates a proper response function even when the seismic data contain more signiﬁcant seismic power 18 (or noise) than the air-to-ground signals. The proposed method will be useful in monitoring and 19 understanding eruption behaviors using seismo-acoustic observations. 20

[2014] distinguished air-ground coupling and ground-air coupling. In this manuscript, we call the former 31 in seismometer records as an air-to-ground signal and the latter in acoustic records as a ground-to-air 32 signal to clarify the coupling direction. 33 The generation efficiency of the air-to-ground signal is ∼ 0.1 − 10µm/s/Pa [Ichihara 2016; Novoselov 34 et al., 2020], which is much larger than that of the ground-to-air signal (∼ 0.0003 Pa/(µ m/s)) [e.g., Kim sensor and a collocated seismometer as references. They calculated the spectral power ratios between 55 them to evaluate the ground-to-air coupling and air-to-ground coupling efficiency. They used airplane 56 sound as the known acoustic signal, though it has frequency components and incident angles much higher 57 than those of volcanic signals. On the other hand, Ichihara [2016] obtained the ground response to 58 infrasound as a function of frequency at a station in Kirishima volcano, Japan (Fig. 1). This case is 59 unique in that a good infrasound source is available: Sakurajima, about 42 km away (Fig. 1c), frequently 60 transmits explosion infrasound (Fig. 1a). Because of the distance, seismic waves from the source do not 61 reach the stations at Kirishima, or if they do, they are well separated in time from the infrasound signals.

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The response function, consisting of the amplitude ratio (Fig. 1d) and the phase shift of the vertical 63 ground velocity to the pressure data (Fig. 1e), was obtained using the beginning 10 s containing the 64 strong pulse (including 3 s before the onset), and those from 15 events were stacked [Ichihara, 2016].

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In general, such an ideal infrasound source is rare in the field. Although the previous study assumed 66 the reference data, both seismic and infrasonic, include only infrasound signals, the seismic data may 67 always contain seismic waves when volcanoes are active. This paper presents a convenient method to 68 evaluate the response function and examine the contribution of the air-to-ground signals in seismometer 69 records using data containing both seismic and infrasonic signals. The method requires only an infrasound 70 sensor and a collocated seismometer without significant wind noise. If there is an additional infrasound 71 station, we can also correct for the wind noise effect.

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Basic Theory 73 We denote vertical ground velocity recorded by a seismometer as d v and pressure change recorded 74 by an infrasound sensor as d p . For simplicity, we assume the incident waves are dominated by a single 75 seismic wave (v s ) and a single infrasonic wave (p a ). We assume that d v consists of v s , an air-to-ground 76 signal (v a ) generated by p a , and wind-induced ground oscillation (v w ), while d p consists of p a and wind 77 noise (p w ). The contribution of the ground-to-air signals is assumed negligible, as mentioned above.

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These assumptions are represented by (1) A cross-correlation function between two time series, d 1 and d 2 , in a given time window [t, t + T ] is 80 represented by where τ is the time delay of d 1 to d 2 . The corresponding cross correlation coefficient where E(d; t) represents the power of a time series d in the time window [t, t + T ], namely, Hereafter, we omit t that specifies the time window.

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Among v s , v a , v w , p a , and p w in equation (1), we assume no pair except v a and p a has a correlation.

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Although v s and p a may have a correlation if their source is common, their correlation should be found 86 with a time shift much larger than that of v a to p a , considering the velocity difference between the seismic 87 waves and infrasound [Ichihara et al., 2012]. The correlation between v w and p w is small if the distance 88 between the infrasound sensor and the seismometer is larger than the correlation length of wind noise.

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For example, the distance of 5 m is enough to suppress the wind-noise correlation at frequency > 1 Hz where W p represent the power ratio of the wind noise to the waves. Under the same wind condition, 92 the infrasound data are usually more significantly affected by wind noise. Therefore, we neglect E(v w ) , and E(d p ) are calculated from the observed data. Our aim here is 95 to estimate E(v a )/E(p a ) to obtain the seismo-acoustic power ratio E(v s )/E(p a ) from the observed data.

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It is assumed that p a is propagating along the ground surface in the direction of the x-axis. The where α is the sound speed in the atmosphere, and ω is the angular frequency. When the ground is 99 a homogeneous elastic half space, and its seismic speeds are much larger than α, the vertical ground 100 velocity (v aω ) induced by p aω , is  (Fig. 1d). Therefore, it is difficult to estimate the response function theoretically. Here we 110 derive an equation relating H ps and the observed data, assuming the phase shift is −π/2. The relation 111 is tested with data in the following sections.

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First, the phase of d v is shifted by π/2, using the Hilbert transform, where Imag[·] takes the imaginary part. The phase-shifted time series are denoted with a subscript h, 114 namely, The last approximation holds with negligible wind noise. We use equations (12) and (5) to obtain .
This equation estimates H ps using only the observed data.

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When the wind noise is significant, we need another infrasound station close by to evaluate the relation 127 It is noted that this equation is applicable on the condition that an infrasound singal p a exists. Equation

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(14) indicates that we should refer to H 2 ps for interpreting the observed power ratio, seismic power E(v s ) increases the power ratio above H 2 ps . Equation (14) is particularly useful when 131 the air-to-ground signal is comparable with the seismic power. We can also identify the condition of cannot provide the meaningful seismic-acoustic energy 133 partitioning.

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As the infrasound signal is larger than the background wind noise (Fig. 1a), we assume W p = 0 in 144 the following analysis. On the other hand, there is noticeable oscillation before the arrival of infrasound 145 in the seismograph (Fig. 1b), indicating that the background seismic signal is comparable with the air-146 to-ground signal. Specifying the origin of the background signal is out of the scope of this study. Here 147 we treat all signals other than v a in d v as v s . 148 We use the four frequency bands: 1-3.5 Hz, 3.5-7 Hz, 7-12 Hz, and 12-18 Hz. For the selection 149 of the frequency bands, we refer to the feature of the known response function shown in Fig. 1d and   150 1e. The phase shift of d v to d p is constant around −π/2 at 1-7 Hz, while the amplitude ratio H ps is 151 relatively stable in 7-18 Hz. In general cases without known response functions, we may arbitrarily select 152 the frequency bands. 153 We apply a zero-phase-shift band-pass filter to d p and d v and calculate the cross-correlation coefficient, the period in all the frequency bands. 160 We use the band-pass-filtered data in 120 s after time zero as d  that the phase shift of −π/2 was successfully corrected.

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The data used in this analysis are ideal to evaluate H ps because d p and d v mainly include p a and 173 v a , respectively. We do not always have such data sets. We evaluate the effect of v s with the assumption 174 that v s is independent of p a . We artificially add v s made in the following way. We generate a random 175 function that is centered at zero and has the same length as d v . We apply the same band-pass filter to 176 this function and call it as d bg . We normalize d bg to have the same energy as d v , by multiplying it by . Then, we amplify it by an arbitrary factor Γ to make a hypothetical v s . Namely,

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The actual √ E(v s )/E(v a ) should be larger than Γ because d v contains some background signal other 180 than v a . We may see it by comparing the amplitudes before and after the infrasound arrival in d v and 181 d p (Fig. 2). It is also supported by Fig. 3b that R max < 1.

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Including v s into d v , we follow the same procedure as the previous paragraph to calculate H ps . For 183 each frequency and each Γ, we tested with 100 random functions. The mean and the standard deviation 184 of H ps , τ max , and R max are presented as functions of Fig. 4a, 4c, and 4d, respectively. 185 We also show the relation of the individual H ps and R max in Fig. 4b. As the background signal power, 186 E(v s ), increases, the cross-correlation coefficient decreases (Fig. 4d). On the other hand, H ps stays 187 around the expected value ( Fig. 4a and 4b).
The average values are almost the same as those without 188 artificial noise, H 0 ps (the closed circles on the vertical axis in Fig. 4a).

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The same test is performed using actual seismic data recording volcanic tremors. Shinmoe-dake, an infrasound and can average a long time window or many windows, we may estimate H ps more accurately.

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We developed equation (13) that relates the ground response to infrasound (H ps ) and the observed 205 seismo-acoustic data (d v and d p ). We have confirmed that it provides a proper value of H ps , even 206 when the observed seismic data contain significant seismic signals as well as the air-to-ground signals.

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Constraining H ps is essential for interpreting the observed seismic-acoustic power ratio to the energy 208 partitioning between seismic waves and infrasound at the source.

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Availability of data and materials 210 The datasets used and/or analysed during the current study are available from the corresponding author 211 on reasonable request.

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Competing interests 213 The authors declare that they have no competing interests.   infrasound at the SMN station. The red dots represent the response function at 1-7 Hz used by Ichihara [2016]. All the plots are reproduced from Ichihara [2016], but the phase delay in (e) has been converted from the range of [−π, π] to [−2π, 0].

Figure 2
The cross-correlation analyses of the data in Fig. 1 at (a) 1-3.5 Hz, (b) 3.5-7 Hz, (c) 7-12 Hz, and (d) 12-18 Hz. In each panel, the top and middle panels show the band-passed infrasonic data (dp) and seismic data (dv), respectively. The bottom panel shows the cross-correlation coe cient between the seismic data, dv, and infrasonic data, dp, with the vertical axis representing the time delay, τ , of dv to dp. The horizontal dashed line indicates τ = 0.  . 1d and 1e). (b) The corresponding cross-correlation coe cient between the phase-shifted seismic data, dhv 316 , and the infrasonic 317 data, dp. The colors indicate the frequency bands as in (a).

Figure 4
The effect of background seismic signals, vs, in the seismic data, dv, on the estimation of Hps. The colors indicate the frequency bands: 1-3.5 Hz (red), 3.5-7 Hz (green), 7-12 Hz (blue), and 12-18 Hz (purple). A hundred random functions are band-pass ltered and used as vs. The horizontal axis is the square-root of power ratio of vs to the original dv that is approximated as va. It is noted that the actual power ratio