1 2 3 Supplementary material 1: Basis of spectral ratio method This supplementary material describes the concepts from which the spectral ratio method for QP estimation from the long-offset WVSP data was developed and applied. 4 Figure S1 shows a schematic view of the shot–receiver configuration of the present 5 WVSP. The amplitude spectrum of the seismic record obtained by the downhole seismic 6 sensor from the shot at offset x can be expressed as 7 8 A ( f , x ) = I( f )× S ( f , x ) ×T ( f , x ) × G ( f , x ) × exp (-p ft * ( x )) , (S1) 9 where I(f), S(f,x), T(f,x), and G(x) are instrumental response, source spectrum, site 10 amplification factor, and geometrical spreading, respectively. t* is a parameter 11 describing the attenuation along the shot–receiver path, as defined in formula (2) in the 12 main text. Similarly, the amplitude spectrum observed at offset x0 is 13 A ( f , x0 ) = I( f )× S ( f , x0 ) ×T ( f , x ) × G ( f , x0 ) × exp (-p ft * ( x0 )) 14 In the WVSP, an airgun array was used as the sound source. The source waveforms 15 of the airgun shots were well controlled, and its repeatability is expected to be 16 extremely high. Therefore, we may regard that 17 18 S ( f , x ) = S ( f , x0 ) . (S2) (S3) for any combinations of the WVSP records. 19 The site amplification factor T ( f , x ) expresses the waveform distortion associated 20 with the local heterogeneities in the vicinity of the downhole sensors. If the factor is 21 independent of the incident angles of the seismic rays at the receiver, it is regarded as a 22 common factor to all the shots and can be canceled by taking the ratio of (S1) and (S2). 23 However, the factor may be dependent on the incident angle and we need to 24 confirm its behavior. In the present WVSP, the incident angle of the ray from every 25 single shot can be estimated from the arrival time differences observed by the vertical 26 array (vertical slowness). Figure S2 shows the relation between the incidental angle and 27 offset distance for the first arrivals. From the diagram, it is evident that the first arrivals 28 in the offset range of less than ~ 6 km (corresponding to R and G1 in Fig. 2) are 29 down-going waves, whereas those in the offset range beyond that (G2 and G3) are 30 up-going waves. Although the difference in incident angle is very large between the 31 down-going and up-going wave groups, the variation of the incident angles within the 32 down- and up-going wave groups is quite small. Therefore, the distance-dependent 33 variation of the spectral ratio shown in Fig. 5 depicts the attenuation effects caused by 34 increasing path lengths. 35 The remaining concern regarding the site amplification factor is the difference 36 between the factors of the down- and up-going waves. In Fig. S3, the amplitude 37 spectrum of seismic data is shown for the offset of ~ 6 km, where the incident angle 38 changes substantially from a shot to another. The observed spectra are almost similar, 39 except for a record of the shot at x = 6.35 km, and we maintain that there is no 40 systematic dependence of the fall-off rate of the high-frequency contents on incident 41 angles. 42 43 Based on these observations, we assumed that the site amplification factor is not dependent on the incident angle and that 44 T ( f , x ) = T ( f , x0 ) 45 Under the assumptions (S3) and (S4), I(f), S(f,x), and T(f,x) can be canceled by 46 taking the ratio of the amplitude spectra of the seismic records obtained at offsets x and 47 x0, 48 . A ( f , x ) G ( x, f ) = × exp éë-p f {t * ( x ) - t * ( x0 )}ùû , A ( f , x0 ) G ( x0 , f ) (S4) (S5) 49 and the parameter Dt* = t * ( x ) - t * ( x0 ) characterizing the seismic attenuation along the 50 path can be estimated. 51 When we apply the spectral ratio method to seismic waveforms traveling through 52 stratified media, the factors G(x, f) and G(x0, f) must be products of geometrical 53 spreading factors and reflection/transmission coefficients across the layer boundaries. 54 When VP is discontinuous across a boundary, the reflection/transmission coefficients are 55 independent of the frequency of seismic waves. However, the coefficients can be 56 frequency dependent if we regard a thin layer as a boundary. In this case, apparent 57 reflection/transmission coefficients can vary according to the ratio between the 58 wavelength of seismic waves and the thickness of the layer. 59 We have to consider four layer boundaries (Fig. 3) in the present QP analysis. 60 Previously obtained seismic images (Park et al. 2002; Moore et al. 2007; Bangs et al. 61 2009) in this area show that all these boundaries are distinct reflectors. Therefore, we 62 can regard them as sharp discontinuities of VP and consider that the estimated QP values 63 are not biased by the apparent frequency dependence of reflection/transmission 64 coefficients. 65 Another concern in applying the spectral ratio method to seismic signals traveling 66 through stratified media is the presence of head waves, waves traveling along layer 67 boundaries. Since waveforms of the head waves are obtained from the incident wave by 68 multiplying by a factor, 2p i 69 directly from the spectral ratio of the direct wave and head wave. In the present study, 70 all the analyzed seismic signals are not head waves (Fig. 3) and we do not have to make 71 any further corrections to take the factor into account. f (e.g. Aki and Richards, 2002), we cannot derive QP 72 73 Supplementary material 2: Data treatment 74 In this material, we discuss two issues that can affect the reliability of QP estimation 75 using the spectral ratio method: 1) window length for spectrum analysis of seismograms 76 and 2) quantitative error estimation. 77 We applied Hanning windowing to reduce estimation error of the spectrum. Since 78 Hanning windowing is a smoothing process, resolution in the frequency domain must 79 be fine enough to obtain the correct spectrogram shape, such as the slope measured in 80 this study. Therefore, longer data length is preferable. On the other hand, we must be 81 cautious about contamination of later arrivals when selecting the data length in our case. 82 As the window becomes longer, more energy of later arrivals could be included in the 83 obtained spectrum, and those later arrivals may have different frequency content from 84 the target (mostly the first arrival) signals. We computed spectral ratios with three 85 different window lengths: 0.256 s (128 samples), 0.512 s (256), and 1.024 s (512). 86 Figure S4 shows examples of estimated spectral ratios in the offset range where the 87 arrival times of later phases are close to the first arrivals, judging from the record 88 section shown in Fig. 2. The difference in overall shape of the spectra is small in the 89 frequency range for the slope estimation. However, high frequency levels tend to be a 90 little higher in the spectral ratios with the window length of 1.024 s than in the others, 91 and contamination of the later arrivals is suspected. We regarded the difference between 92 the spectra ratios with 0.256 s and 0.512 s window lengths to be small enough and 93 chose 0.512 s as the optimum length of spectral analysis in this study because the longer 94 length may be better in terms of frequency resolution. 95 The errors of the QP values were derived from the following calculations: 96 estimating errors of spectral ratio, estimating error of t*, and evaluation of QP errors. 97 The spectrum was obtained after smoothing using a Hanning window with order m, and 98 the coefficient of variation (C.V., standard deviation normalized to mean value) is 99 defined as C.V. = 100 1 . 2m +1 (S6) 101 In this study, we set m = 8, which makes C.V. ~ 0.24. The errors of spectral ratio can be 102 obtained by the law of error propagation. Using the estimated errors of spectral ratios, 103 t* was estimated with its error by using a standard weighted least square fitting. Figure 104 6 shows t* and the errors thus obtained. The estimated errors of t* were taken into 105 account in the least square estimation of QP-1 values, and their errors are shown in Table 106 1. 107 108 Supplementary material 3: Reliability of the QP model 109 The QP model was derived solely through an analysis of the spectral ratio of the 110 observed seismograms in this study, but it is worthwhile to confirm whether the QP 111 model is consistent with the observed amplitude in time domain. We can obtain not only 112 t* but also log ê 113 of the seismograms. Here, we attempt to verify whether the estimate values are 114 consistent with 115 We calculated G(x) by using an asymptotic ray theory (Cerveny et al. 1977). In Fig. 116 S5, the 117 shown. We set the reference offset (x0) at 3.1 km, where we took the reference trace in 118 calculating the spectral ratios (Fig. 4). Most of the estimated geometrical spreading 119 factors are well explained by the assumed Vp, reinforcing the consistency of the 120 obtained t* with the observed data. The underestimation of 121 could be due to distortion of the spectral ratio due to contamination by later arrivals. We 122 further attempted to calculate the signal amplitude as a function of offset distance by 123 equation (S5) using the 124 structure model obtained by this study. Figure S6 compares the calculated and observed 125 normalized amplitudes é G ( x) ù ú in equation (1) by the least square fitting to the spectral ratios êë G ( x0 ) úû G ( x) , which can be calculated from the assumed VP structure model. G ( x0 ) G ( x) estimated from the spectral ratio and calculated from the VP model are G ( x0 ) G ( x) at x = 9.1 km G ( x0 ) G ( x) calculated from the assumed Vp model and the QP G ( x0 ) A ( f , x) . In the calculation, we assumed f = 20 Hz. For the A ( f , x0 ) 126 observed amplitudes, we took the maxima of the absolute amplitude in the time window 127 for the spectrum analysis. The diagram shows that the combination of the assumed Vp 128 model and our best-fit QP model explain the observed variation of amplitude quite well, 129 whereas the amplitudes expected from the attenuation model with QP = 100 for the 130 basement layer (L3) are substantially smaller than the observed ones. 131 Finally, we verified whether our seismic waveform data were more consistent with 132 a frequency independent Q model than a frequency dependent Q model. Frequency 133 dependence is often expressed as Q ( f ) = Q0 f a . Here, we assume = 0.66, moderate 134 frequency dependence, which is obtained from the results of Yoshimoto et al. (1998). In 135 Fig. S7, we compare the observed spectral ratio with the calculated ones, assuming = 136 0 (frequency independent Q) and = 0.66 (frequency dependent Q). Inconsistency 137 increases when the frequency dependent Q is introduced; therefore, we prefer the 138 frequency independent model. 139 140 References 141 Aki K and Richards PG (2002) Quantitative Seismology, second edition, University 142 Science Books, Sausalito, pp. 700 143 Cerveny VI, Molokov A, Psencík I (1977) Ray Method in Seismology, University 144 Karova, Prague, pp. 214 145