# Bulletin of the Seismological Society of America

- Copyright © 2006 Bulletin of the Seismological Society of America

## Abstract

By means of borehole array recordings obtained at six stations from local events, we estimated coherency characteristics of body waves that propagate in a sedimentary layer–basement system in the Kanto region, Japan. The estimated coherence is characterized by an *ω*^{−2} model for *P* and *S* waves. An important result is that the values of corner frequency become nearly constant, 5–6 Hz, in travel times less than 2.5 sec for *SH* waves. This suggests that bedrock motions recorded in the pre-Tertiary basement are becoming incoherent in high frequencies. Also, the corner frequency of *P* waves is approximately twice that for *SH* waves. Hence, *P* waves propagate coherently in high frequencies as compared with *SH* waves.

## Introduction

The coherency characteristics of ground motion are affected by the scattering and attenuation in the propagating processes. Many studies on the seismic scattering were carried out to survey the statistical characteristics of solid media of the Earth by using array recordings. Among them, the coherency characteristics of seismic waves have been investigated mainly by using surface array recordings for local and regional events (McLaughlin *et al.*, 1983; Fletcher *et al.*, 1990; Vernon *et al.*, 1991, 1998) or direct surface waves such as *Lg* waves (Der *et al.*, 1984; Toksoz *et al.*, 1991). For engineering purposes, the coherency characteristics of ground motion are applied to make design indexes for pipeline systems and to construct stochastic methods for strong- motion predictions. The main purpose of these studies is to investigate so-called spatial coherency, and they found that the dependence of the spatial coherency on event (or approach azimuth of seismic waves) is not significant and the magnitude of spatial coherency decreases with increases in frequency and separation distance between observation points. The results on spatial coherency of body waves have been explained using empirical relations, in particular single- exponential coherency functions (Hindy and Novak, 1980; Loh, 1985; Luco and Wong, 1986; Menke *et al.*, 1990) or double-exponential coherency functions (Harichandran and Vanmarcke, 1986), which were empirically constructed.

On the other hand, Fletcher *et al.* (1990) and Vernon *et al.* (1991, 1998) were the only studies on the coherency characteristics of body waves along the traveling path using borehole array recordings. This is mainly due to the lack of borehole array recordings. Their studies asserted that site effects control the spatial coherency of body waves so that it decreases rapidly as compared with the coherency of body waves along the traveling path in high frequencies. However, as concluded by Vernon *et al.* (1998), the estimation of coherency by using array data from shallow boreholes whose depths are shorter than 500 m is difficult in frequencies lower than 10 Hz, because of the contamination of surface-reflected waves into incidence waves. For the engineering applications, the frequency band of 0.1–10 Hz is the most important. In the study on strong motion, in particular, a transition band between deterministic methods and stochastic methods for strong-motion predictions is 0.5–5 Hz (O’Connell, 1999). These facts require the measurement of coherency characteristics of body waves in this frequency band. This is our motivation behind the present study.

In Japan, after the Kobe earthquake of 1995, nationwide networks of strong-motion observation such as a k-net (Kinoshita, 1998) were constructed, and the data from these networks have been released through the Internet. KiK-net, one of the nationwide networks deployed by nied (National Research Institute for Earth Science and Disaster Prevention) in Japan has been producing borehole array recordings at over 500 sites, recordings that were recorded simultaneously at the bottom of boreholes and the surface at the same site by using three-component sets of negative feedback accelerometers with a frequency band of 0 to 30 Hz. In this study, we shall use borehole array recordings in the Kanto region, Japan, obtained at six borehole sites with depths of 1200–3510 m.

## Data

We used the array data recorded at six borehole stations, iwt, fch, shm, nrt, edz, and enz, deployed in the Kanto region, central Japan. The geological structures in these boreholes are illustrated in Figure 1, where the bottoms of the boreholes are 500–800 meters below the top of the pre- Tertiary basement covered by the sediment layer. We call such a geological structure the “sedimentary layer–basement system.” The locations of these six sites are given in Table 1 and shown in Figures 2a and 2b as well as the locations of the local earthquakes used for this study. The slant angle of each borehole from a vertical line is within 3°. The array recordings used in this study were obtained from the two three-component sets of negative-feedback accelerometers with a natural frequency of 450 Hz and a damping factor of 0.6–0.7, which were installed at the bottom of a borehole and at the surface at stations iwt, fch, and shm, or at the 10 m depth at stations nrt, edz, and enz. The overall response of the recording system is flat in the frequency band between 0 and 30 Hz (−3-dB point) and the sampling rate is 100 Hz or 200 Hz. The dynamic ranges of the recording system are 12 bits/sample at the iwt, fch, and shm sites (for details, see Takahashi and Hamada, 1975; Kinoshita, 1994) and 16 bits/sample at the nrt, edz, and edz sites, respectively. In this study, we used *P* and *SH* waves. The orientation of the borehole-bottom seismometers is coincident with that of the borehole-surface seismometers installed at the nrt, enz, and edz sites, that is, the north–south, east–west, and up–down directions. However, the orientation of the borehole-bottom seismometers installed at the shm, fch, and iwt sites is not coincident with that of the surface seismometers. Thus, we first compensated for the difference in the orientation of the borehole-bottom seismometers at these three sites according to the results measured by using an electric azimuth meter. After that, two horizontal components were rotated to obtain the *SH* component parallel to the directions of the maximum principal axis of the trajectory ellipse of the direct *S* phase (Kinoshita, 1994; Kinoshita and Ohike, 2002).

The investigations of borehole geology, sonic, and density logs were conducted by nied and reported by Suzuki *et al.* (1981) and Suzuki and Omura (1999). The determinations of *P*- and *S*-wave velocity structures at the iwt, fch, and shm sites by using a down-hole method were conducted and reported by Ohta *et al.* (1980) and Yamamizu *et al.* (1981). They showed that the velocities of *P* and *S* waves propagated vertically in the pre-Tertiary basement are 5.5 km/sec and 2.5 km/sec, respectively, and the average *S*-wave velocity in the sedimentary layer is approximately 1 km/sec.

The data for investigating the coherency characteristics of direct body waves that propagate in a sedimentary layer– basement system were obtained from local events shown in Figure 2a for *P* waves and Figure 2b for *SH* waves. The data from events shown by open triangles, open circles, open squares, open reverse triangles, open rhombuses, and solid circles in Figures 2a and 2b were used for the stations shm, fch, iwt, edz, nrt, and enz, respectively. In this study, direct body waves that propagate coherently in a sedimentary layer–basement system are required for the estimation of coherence functions. Thus, we used the data from which the maximum coherence of more than 0.8 in frequencies lower than 5 Hz was estimated. In addition, the data, in which the influence of the contamination of *P* coda to *SH* waves is insignificant, are also required. The events shown in Figures 2a and 2b are selected on these conditions. The hypocenter distances are about 30 km to 150 km. The range of jma (Japan Meteorological Agency) magnitude is from 3.2 to 6.1. Two phases on deep-borehole seismograms, an incident and its surface-reflected waves, are used for the estimation of the coherence function of a body wave that propagates in a sedimentary layer–basement system. The surface- reflected body wave recorded at the bottom of a borehole is contaminated by the direct body wave when the duration of the direct body wave is longer than the two-way time at the site. The events in this study must be selected taking account of this fact. The empirical relations between direct *S* pulse (*T _{d}*) and jma magnitude (

*M*

_{jma}) obtained using data recorded at hard rock and borehole sites are as follows (Kinoshita and Ohike, 2002): and The average values of

*T*for

_{d}*M*

_{jma}6.1 are 4.2 sec and 4.4 sec for interplate and intraplate events, respectively. The estimated average values of two-way time between the surface and the bottom of borehole levels at the shm and fch sites are 4.8 sec and 4.6 sec, respectively, as will be shown later. Thus, we used data from events with a range of

*M*

_{JMA}3.2– 6.1.

## Method

The lagged coherence is defined as 1 where *P _{ij}*(

*t*,

_{d}*f*) is the cross-spectrum of body waves

*x*(

*d*,

_{i}*t*) and

*x*(

*d*,

_{j}*t*), where

*d*and

_{i}*d*indicate the depths of the recording sites

_{j}*i*and

*j*; the separation time

*t*= |

_{d}*t*−

_{i}*t*|; and

_{j}*t*are the corresponding onset times of the interested phase.

_{i}, t_{j}*P*(

_{ii}*f*) and

*P*(

_{jj}*f*) are the auto-spectra of the two seismograms recorded at depths of

*d*and

_{i}*d*, respectively. Two wave sets are used for the estimation of coherence in this study. One is to make a comparison between incidence phase and its surface-reflected one recorded at the same deep borehole. The other is to compare the direct body-wave phases recorded at the deep borehole and the surface.

_{j}As an example, Figure 3 exhibits the transverse components of velocity seismograms that were converted from the original acceleration seismograms recorded at site shm for the earthquake of 16 January 1988. The top and bottom of Figure 3 are recordings at free surface and 2,300 m depth, respectively. It may be easy to identify direct *S* phases on both borehole and surface recordings and surface-reflected phases on the borehole seismogram, phases that are traveling in a sedimentary layer–basement system. We first determine the start time *t _{i}* of the direct

*SH*phase, visually, as marked by “A” in Figure 3. By using a time window with length shorter than the two-way travel time estimated according to the

*S*-wave velocity structure at the site (Yamamizu

*et al.*, 1981), the cross-correlation function between the direct

*SH*phase and its surface-reflected one is calculated as shown in the top right of Figure 4a. The lag time at which the estimated cross-correlation has its maximum is the separation time

*t*, and it is the two-way travel time of

_{d}*S*waves throughout the sedimentary layer–basement system at site shm. The surface-reflected phase whose start time is given by

*t*=

_{j}*t*+

_{i}*t*, as marked by “B” in Figure 3, is thus determined. By using the two

_{d}*SH*phases whose start points are marked by “A” and “B,” having the same window length as the one given by the portion whose start and end point are marked by “B” and “BB,” respectively, coherence is estimated by applying Welch’s periodogram-averaging method (Proakis and Manolakis, 1996) as shown in the bottom right of Figure 4a. Auto-spectral densities of the direct

*SH*phase and its surface-reflected one are given by the top left and bottom left of Figure 4a, respectively.

As mentioned previously, two direct incident phases recorded at both the deep borehole and surface are also used to estimate the coherency characteristics of body waves that propagate in a sedimentary layer–basement system. As shown in Figure 3 (top), for example, the onset of the *SH* phase is marked by “C,” also determined by using the cross- correlation method. In this case, the two *SH* phases used for the estimation of coherence are portions with the start points marked by “A” and “C,” having the same window length between “C” and “CC,” respectively. The corresponding spectral density, one-way travel time, and coherence are shown in Figure 4b.

The coherence shown in Figure 4a is estimated using the window length of 4.03 sec, which is shorter than the two-way time of 4.63 sec. According to Vernon *et al.* (1991), the estimation of coherence is robust to window length. They tested two window lengths, 0.5 sec and 2 sec, and found no significant difference on the resultant coherence for body waves from local events. Similarly, Kinoshita (2003) showed that the estimated results of coherence using different windows with length between one-way travel time and two-way travel time did not show significant difference. On the contrary, window lengths that were longer than the two- way time produced significantly different coherence. This may be due to the contamination of surface-reflected waves to incident body waves that appeared on deep-borehole seismograms. Thus, we use data windows with length between the one-way time and two-way time in this study.

## Results

### Coherence of *P* Waves

To study the coherence of *P* waves, we used vertical component seismograms only.

Coherence characteristics estimated from the comparison between the direct

*P*phase and its surface-reflected one on borehole seismograms: Figures 5a and 5b are results obtained by using the data from events shown in Figure 2a at stations shm and fch for 11 and 18 events, respectively. Solid circles are mean coherence and dotted lines are the 95% confidence levels of regression coefficients. Applying a nonlinear least-squares fitting method, the solid line in Figures 5a and 5b can be best fitted by the following*ω*^{−2}model: 2 The model parameters for station shm are = 0.83 ± 0.02 and*f*= 9.21 ± 0.36 Hz, where the error values are the 95% confidence levels of regression coefficients. The mean value with standard error of two-way time estimated at the shm site is 1.73 ± 0.05 sec. Assuming_{c}*n*is a free parameter, we obtain that*n*= 2.28 ± 0.22, = 0.80 ± 0.02, and*f*= 9.46 ± 0.36 Hz, where the error values are the 95% confidence levels of regression coefficients. The corresponding parameters at the fch site are = 0.83 ± 0.02 and_{c}*f*= 9.86 ± 0.37 Hz in the_{c}*ω*^{−2}model, and the two-way travel time is 2.00 ± 0.05 sec. Assuming that*n*is a free parameter, the best- fitted parameters are as follows:*n*= 2.34 ± 0.22, = 0.80 ± 0.02, and*f*= 10.13 ± 0.39 Hz, where the error values are the 95% confidence levels of the regression coefficients. Data length used for the estimation of coherence is the length of two-way time − 0.3 sec for each event. In this case, two-way time means the lag time at which the cross-correlation between the direct phase and its surface-reflected one is its maximum._{c}Coherence characteristics estimated from the comparison between direct phases recorded at the bottom of a deep borehole and surface: Figures 6a and 6b are results obtained by using the data from events shown in Figure 2a at the shm and fch sites, respectively. Solid circles are mean coherence and dotted lines are the 95% confidence levels of the regression coefficients. Total numbers of events used for the estimation of coherence are 21 and 22 for stations shm and fch, respectively. Again, the estimated coherence can be best interpreted by an

*ω*^{−2}model, empirically, as shown by the solid line. The best- fitted parameters are as follows: = 0.86 ± 0.03 and*f*= 11.32 ± 0.91 Hz at the shm site, and the one-way travel time is 0.86 ± 0.03 sec. The result obtained at the fch site shown in Figure 6b yields = 0.91 ± 0.03 and_{c}*f*= 10.07 ± 0.74 Hz in the_{c}*ω*^{−2}model. The one- way travel time is 0.97 ± 0.05 sec. Data length used for the estimation of coherence is the length of twice the one- way time − 0.2 sec for each event. In this case, one-way time means the lag time at which the estimated cross- correlation between direct phases recorded at the bottom of a borehole and surface is its maximum.

### Coherence of *S* Waves

For *S* waves, we used the *SH* component only. The station and event locations are shown in Figure 2b.

Coherence characteristics estimated from the comparison between the direct

*S*phase and its surface reflection recorded at the deep borehole: Figures 7a and 7b are results obtained at the shm and fch sites, respectively. Solid circles are mean coherence and dotted lines show the error ranges of the 95% confidence levels of the regression coefficients. Total numbers of events used for the estimation of coherence are 30 and 26 for stations shm and fch, respectively. The solid line is the nonlinear fitting to relation (2). The best-fitted parameters are = 0.79 ± 0.02 and*f*= 3.92 ± 0.17 Hz at the shm site, and two-way travel time is 4.81 ± 0.10 sec. Assuming_{c}*n*is a free parameter, we get the following best-fitted parameters:*n*= 2.09 ± 0.17, = 0.78 ± 0.03, and*f*= 4.00 ± 0.24 Hz. The result obtained at the fch site shown in Figure 7b yields = 0.73 ± 0.02 and_{c}*f*= 4.42 ± 0.18 Hz in the_{c}*ω*^{−2}model. The two-way travel time is 4.70 ± 0.14 sec. Assuming that*n*is a free parameter, we get the following best-fitted parameters:*n*= 1.80 ± 0.13, = 0.76 ± 0.03, and*f*= 4.20 ± 0.26 Hz. Data length used for the estimation of coherence is the length of two-way time − 0.6 sec for each event. In this case, two-way travel time means the lag time at which the cross-correlation between direct phase and its surface-reflected one is its maximum._{c}Coherence characteristics estimated from the comparison between direct phases recorded at the bottom of a deep borehole and surface: Figures 8a, 8b, and 8c are the coherence characteristics estimated for stations shm, fch, and iwt, respectively. The coherence characteristics estimated at the nrt, enz, and edz sites are also shown in Figures 8d, 8e and 8f, respectively. Solid circles are the mean values of coherence and dotted lines are the 95% confidence error levels of regression coefficients. The total numbers of events used for the estimation of coherence are 22, 16, 15, 25, 10, and 13 for stations shm, fch, iwt, nrt, enz, and edz, respectively. The best-fitted parameters to relation (2) are as follows: = 0.94 ± 0.05 and

*f*= 6.00 ± 0.42Hz, = 0.97 ± 0.06 and_{c}*f*= 5.89 ± 0.55Hz, = 0.98 ± 0.08 and_{c}*f*= 4.09 ± 0.45 Hz, = 0.92 ± 0.08 and_{c}*f*= 5.83 ± 0.08 Hz, = 0.95 ± 0.11 and_{c}*f*= 5.95 ± 1.39 Hz, and = 0.99 ± 0.08 and_{c}*f*= 5.39 ± 0.51 Hz for sites shm, fch, iwt, nrt, enz, and edz, respectively. Also, estimated one-way times are 2.40 ± 0.04 sec, 2.30 ± 0.08 sec, 3.03 ± 0.13 sec, 1.72 ± 0.04 sec, 0.55 ± 0.04 sec, and 1.48 ± 0.04 sec at the shm, fch, iwt, nrt, enz, and edz sites, respectively. Data lengths used for the estimation of coherence are 2.82 sec, 2.82 sec, 2.82 sec, 2.56 sec, 0.95 sec, and 2.56 sec at the shm, fch, iwt, nrt, enz, and edz sites, respectively._{c}

All estimates of coherence obtained for the vertical component of *P* waves and *SH* component of *S* waves show that it is empirically reasonable to use the *ω*^{−2} model for the explanation of coherence characteristics of body waves traveling in a sedimentary layer–basement system. Thus we use this model hereafter to explain coherence characteristics. Estimated coherence shown in Figures 5a, 5b, 7a, and 7b is flat in low frequencies. This is due to the all-pass characteristics of transfer functions As demonstrated by Kinoshita (1999), the transfer function for a sedimentary layer– basement system, with a direct wave as input and its surface reflection as output, has maximum-phase characteristics, and thus all-pass characteristics. However, a transfer function whose input and output waves are direct phases on deep- borehole and surface seismograms, respectively, is a minimum-phase system having strong feedback (Kinoshita, 1999). This means that the conventional cross-spectrum method cannot apply to the estimation of coherence (Akaike, 1966), if the input wave is contaminated by feedback signals that are mainly due to surface-reflected waves, as in the case of shallow-borehole data. The use of deep-borehole data can relax this constraint so that a remarkable dip structure in estimated coherence does not appear in low-frequency bands.

## Discussion

The main point is the characteristics of the empirical *ω*^{−2} model of coherence that is proposed in the present study. The relationship between travel time and *f _{c}*/2 for

*SH*waves is shown in Figures 9a for

*n*= 2 and 9b for

*n*as free parameter in relation (2). The corresponding zero-frequency magnitude of coherence, versus travel time, is also shown in Figures 10a (

*n*= 2) and 10b (

*n*= free parameter). There is a tendency that the values of corner frequency become nearly constant (5–6 Hz) for travel time less than 2.5 sec. Then, the values of corner frequency decrease with increasing travel time. However, since we have only one data point for travel time 3–4.5 sec, to find the precise relationship between corner frequency and travel time in this time range we need more data. Figures 9a, 9b, 10a, and 10b indicate that the coherence of

*SH*waves for travel time less than 2.5 sec is well explained by a characteristic

*ω*

^{−2}model with a nearly constant corner frequency and > 0.9. The advent of constant corner frequency suggests that the bedrock motion in the pre-Tertiary basement is incoherent in high frequencies. Assuming the

*ω*

^{−2}model of coherence with ≅ 1, we get

*γ*

^{2}(

*f*/3) ≅ 0.9,

_{c}*γ*

^{2}(

*f*/2) ≅ 0.8, and

_{c}*γ*

^{2}(

*f*) = 0.5. According to this

_{c}*ω*

^{−2}model, we can evaluate quantitatively the transition band explained briefly in the introduction. For example, if we define the transition band by using

*γ*

^{2}= 0.8, then the frequency range of

*f*/2, 2.5–3 Hz, is the transition band. Of course, it is more natural to define the transition band by using the magnitude range of coherence, for example,

_{c}*γ*

^{2}= 0.8–0.9. Then, the transition band is approximately from 1.5 to 3 Hz for

*SH*waves.

Similarly, Figures 5a, 5b, 6a, and 6b indicate that the coherence of *P* waves is explained by the *ω*^{−2} model. The ratios of the estimated corner frequencies of the *ω*^{−2}-model obtained for *P* waves to the corresponding corner frequencies for *SH* waves shown in Figures 7a, 7b, 8a, and 8b are 2.3, 2.2, 1.9, and 1.7, respectively. Thus, the corner frequency of *P* waves is approximately twice that of *SH* waves Hence, *P* waves propagate coherently in high frequencies as compared with *SH* waves in a sedimentary layer–basement system.

## Conclusion

A fundamental issue in strong-motion seismology and earthquake engineering is to determine the frequency band in which body waves propagate coherently in a sedimentary layer–basement system. To respond to this issue, we investigated down-hole array recordings including deep-borehole seismograms obtained in the Kanto region, Japan, and obtained the following results:

Coherence characteristics of body waves from local events and traveling in a sedimentary layer–basement system are empirically modeled by , where

*n*is approximately 2.The corner frequency

*f*in the_{c}*ω*^{−2}model for*SH*waves intends to increase with a decrease in travel time. However, the values of corner frequency become nearly constant, independent of travel time, for travel time less than 2.5 sec. In the case of*SH*waves the values of corner frequency are about 5–6 Hz, and the values of are larger than 0.9.The corner frequency in the

*ω*^{−2}model for*P*waves is approximately twice that for*SH*waves. Hence,*P*waves propagate coherently in high frequencies as compared with*SH*waves in a sedimentary layer–basement system.

## Acknowledgments

The authors deeply appreciate Dr. Anshu Jin for her constructive suggestions and critical review of this manuscript. This study was supported in part by grant-in-aid for scientific research No. 15560408 in Japan and by the jnes (Japan Nuclear Energy Safety Organization) open application project for enhancing the basis of nuclear safety.

- Manuscript received 20 April 2005.