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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.
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Figure S1 shows a schematic view of the shot–receiver configuration of the present
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WVSP. The amplitude spectrum of the seismic record obtained by the downhole seismic
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sensor from the shot at offset x can be expressed as
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A ( f , x ) = I( f )× S ( f , x ) ×T ( f , x ) × G ( f , x ) × exp (-p ft * ( x )) ,
(S1)
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where I(f), S(f,x), T(f,x), and G(x) are instrumental response, source spectrum, site
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amplification factor, and geometrical spreading, respectively. t* is a parameter
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describing the attenuation along the shot–receiver path, as defined in formula (2) in the
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main text. Similarly, the amplitude spectrum observed at offset x0 is
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A ( f , x0 ) = I( f )× S ( f , x0 ) ×T ( f , x ) × G ( f , x0 ) × exp (-p ft * ( x0 ))
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In the WVSP, an airgun array was used as the sound source. The source waveforms
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of the airgun shots were well controlled, and its repeatability is expected to be
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extremely high. Therefore, we may regard that
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S ( f , x ) = S ( f , x0 )
.
(S2)
(S3)
for any combinations of the WVSP records.
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The site amplification factor T ( f , x ) expresses the waveform distortion associated
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with the local heterogeneities in the vicinity of the downhole sensors. If the factor is
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independent of the incident angles of the seismic rays at the receiver, it is regarded as a
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common factor to all the shots and can be canceled by taking the ratio of (S1) and (S2).
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However, the factor may be dependent on the incident angle and we need to
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confirm its behavior. In the present WVSP, the incident angle of the ray from every
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single shot can be estimated from the arrival time differences observed by the vertical
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array (vertical slowness). Figure S2 shows the relation between the incidental angle and
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offset distance for the first arrivals. From the diagram, it is evident that the first arrivals
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in the offset range of less than ~ 6 km (corresponding to R and G1 in Fig. 2) are
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down-going waves, whereas those in the offset range beyond that (G2 and G3) are
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up-going waves. Although the difference in incident angle is very large between the
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down-going and up-going wave groups, the variation of the incident angles within the
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down- and up-going wave groups is quite small. Therefore, the distance-dependent
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variation of the spectral ratio shown in Fig. 5 depicts the attenuation effects caused by
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increasing path lengths.
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The remaining concern regarding the site amplification factor is the difference
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between the factors of the down- and up-going waves. In Fig. S3, the amplitude
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spectrum of seismic data is shown for the offset of ~ 6 km, where the incident angle
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changes substantially from a shot to another. The observed spectra are almost similar,
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except for a record of the shot at x = 6.35 km, and we maintain that there is no
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systematic dependence of the fall-off rate of the high-frequency contents on incident
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angles.
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Based on these observations, we assumed that the site amplification factor is not
dependent on the incident angle and that
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T ( f , x ) = T ( f , x0 )
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Under the assumptions (S3) and (S4), I(f), S(f,x), and T(f,x) can be canceled by
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taking the ratio of the amplitude spectra of the seismic records obtained at offsets x and
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x0,
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.
A ( f , x ) G ( x, f )
=
× exp éë-p f {t * ( x ) - t * ( x0 )}ùû ,
A ( f , x0 ) G ( x0 , f )
(S4)
(S5)
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and the parameter Dt* = t * ( x ) - t * ( x0 ) characterizing the seismic attenuation along the
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path can be estimated.
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When we apply the spectral ratio method to seismic waveforms traveling through
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stratified media, the factors G(x, f) and G(x0, f) must be products of geometrical
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spreading factors and reflection/transmission coefficients across the layer boundaries.
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When VP is discontinuous across a boundary, the reflection/transmission coefficients are
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independent of the frequency of seismic waves. However, the coefficients can be
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frequency dependent if we regard a thin layer as a boundary. In this case, apparent
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reflection/transmission coefficients can vary according to the ratio between the
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wavelength of seismic waves and the thickness of the layer.
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We have to consider four layer boundaries (Fig. 3) in the present QP analysis.
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Previously obtained seismic images (Park et al. 2002; Moore et al. 2007; Bangs et al.
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2009) in this area show that all these boundaries are distinct reflectors. Therefore, we
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can regard them as sharp discontinuities of VP and consider that the estimated QP values
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are not biased by the apparent frequency dependence of reflection/transmission
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coefficients.
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Another concern in applying the spectral ratio method to seismic signals traveling
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through stratified media is the presence of head waves, waves traveling along layer
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boundaries. Since waveforms of the head waves are obtained from the incident wave by
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multiplying by a factor, 2p i
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directly from the spectral ratio of the direct wave and head wave. In the present study,
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all the analyzed seismic signals are not head waves (Fig. 3) and we do not have to make
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any further corrections to take the factor into account.
f
(e.g. Aki and Richards, 2002), we cannot derive QP
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Supplementary material 2: Data treatment
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In this material, we discuss two issues that can affect the reliability of QP estimation
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using the spectral ratio method: 1) window length for spectrum analysis of seismograms
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and 2) quantitative error estimation.
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We applied Hanning windowing to reduce estimation error of the spectrum. Since
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Hanning windowing is a smoothing process, resolution in the frequency domain must
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be fine enough to obtain the correct spectrogram shape, such as the slope measured in
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this study. Therefore, longer data length is preferable. On the other hand, we must be
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cautious about contamination of later arrivals when selecting the data length in our case.
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As the window becomes longer, more energy of later arrivals could be included in the
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obtained spectrum, and those later arrivals may have different frequency content from
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the target (mostly the first arrival) signals. We computed spectral ratios with three
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different window lengths: 0.256 s (128 samples), 0.512 s (256), and 1.024 s (512).
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Figure S4 shows examples of estimated spectral ratios in the offset range where the
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arrival times of later phases are close to the first arrivals, judging from the record
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section shown in Fig. 2. The difference in overall shape of the spectra is small in the
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frequency range for the slope estimation. However, high frequency levels tend to be a
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little higher in the spectral ratios with the window length of 1.024 s than in the others,
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and contamination of the later arrivals is suspected. We regarded the difference between
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the spectra ratios with 0.256 s and 0.512 s window lengths to be small enough and
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chose 0.512 s as the optimum length of spectral analysis in this study because the longer
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length may be better in terms of frequency resolution.
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The errors of the QP values were derived from the following calculations:
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estimating errors of spectral ratio, estimating error of t*, and evaluation of QP errors.
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The spectrum was obtained after smoothing using a Hanning window with order m, and
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the coefficient of variation (C.V., standard deviation normalized to mean value) is
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defined as
C.V. =
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1
.
2m +1
(S6)
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In this study, we set m = 8, which makes C.V. ~ 0.24. The errors of spectral ratio can be
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obtained by the law of error propagation. Using the estimated errors of spectral ratios,
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t* was estimated with its error by using a standard weighted least square fitting. Figure
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6 shows t* and the errors thus obtained. The estimated errors of t* were taken into
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account in the least square estimation of QP-1 values, and their errors are shown in Table
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1.
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Supplementary material 3: Reliability of the QP model
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The QP model was derived solely through an analysis of the spectral ratio of the
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observed seismograms in this study, but it is worthwhile to confirm whether the QP
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model is consistent with the observed amplitude in time domain. We can obtain not only
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t* but also log ê
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of the seismograms. Here, we attempt to verify whether the estimate values are
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consistent with
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We calculated G(x) by using an asymptotic ray theory (Cerveny et al. 1977). In Fig.
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S5, the
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shown. We set the reference offset (x0) at 3.1 km, where we took the reference trace in
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calculating the spectral ratios (Fig. 4). Most of the estimated geometrical spreading
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factors are well explained by the assumed Vp, reinforcing the consistency of the
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obtained t* with the observed data. The underestimation of
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could be due to distortion of the spectral ratio due to contamination by later arrivals. We
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further attempted to calculate the signal amplitude as a function of offset distance by
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equation (S5) using the
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structure model obtained by this study. Figure S6 compares the calculated and observed
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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 )
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observed amplitudes, we took the maxima of the absolute amplitude in the time window
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for the spectrum analysis. The diagram shows that the combination of the assumed Vp
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model and our best-fit QP model explain the observed variation of amplitude quite well,
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whereas the amplitudes expected from the attenuation model with QP = 100 for the
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basement layer (L3) are substantially smaller than the observed ones.
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Finally, we verified whether our seismic waveform data were more consistent with
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a frequency independent Q model than a frequency dependent Q model. Frequency
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dependence is often expressed as Q ( f ) = Q0 f a . Here, we assume  = 0.66, moderate
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frequency dependence, which is obtained from the results of Yoshimoto et al. (1998). In
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Fig. S7, we compare the observed spectral ratio with the calculated ones, assuming  =
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0 (frequency independent Q) and  = 0.66 (frequency dependent Q). Inconsistency
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increases when the frequency dependent Q is introduced; therefore, we prefer the
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frequency independent model.
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References
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Aki K and Richards PG (2002) Quantitative Seismology, second edition, University
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Science Books, Sausalito, pp. 700
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Cerveny VI, Molokov A, Psencík I (1977) Ray Method in Seismology, University
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Karova, Prague, pp. 214
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