Monitoring a building using deconvolution interferometry. I: Earthquake-data analysis · 2019. 8. 31. · CWP-777 Monitoring a building using deconvolution interferometry. I: Earthquake-data

CWP-777

Monitoring a building using deconvolutioninterferometry. I: Earthquake-data analysis

Nori Nakata1, Roel Snieder1, Seiichiro Kuroda2, Shunichiro Ito3,

Takao Aizawa3, & Takashi Kunimi4

1 Center for Wave Phenomena, Geophysics Department, Colorado School of Mines2 National Institute for Rural Engineering3 Suncoh Consultants Co., Ltd.4 Akebono Brake Industry

ABSTRACTFor health monitoring of a building, we need to separate the response of the

building to an earthquake from the imprint of soil-structure coupling and fromwave propagation below the base of the building. Seismic interferometry basedon deconvolution, where we deconvolve the wavefields recorded at differentfloors, is a technique to extract this building response and hence estimate veloc-ity of the wave which propagates inside the building. Deconvolution interferom-etry also allows us to estimate the damping factor of the building. Comparedwith other interferometry techniques, such as crosscorrelation and crosscoher-ence interferometry, deconvolution interferometry is the most suitable techniqueto monitor a building using earthquake records. For deconvolution interferome-try, we deconvolve the wavefields recorded at all levels with the waves recordedat a target receiver inside the building. This receiver behaves as a virtual source,and we retrieve the response of a cut-off building, a short building which is cutoff at the virtual source. Because the cut-off building is independent from thestructure below the virtual source, the technique might be useful for estimatinglocal structure and local damage. We apply deconvolution interferometry to 17earthquakes recorded during two weeks at a building in Fukushima, Japan andestimate time-lapse changes in velocity and normal-mode frequency. As shownin a previous study, the change in velocity correlates with the change in normal-mode frequency. We compute the velocities from both traveling waves and thefundamental mode using coda-wave interferometry. These velocities have a neg-ative correlation with the maximum acceleration of the observed earthquakerecords.

Key words: time-lapse monitoring, seismic interferometry, deconvolution,earthquake, building

1 INTRODUCTION

The response of a building to an earthquake has beenstudied since the early 1900s [e.g., Biot, 1933; Sezawaand Kanai, 1935; Carder, 1936]. We can estimate thefrequencies of the fundamental and higher modes ofbuildings using ambient and forced vibration experi-ments (Trifunac, 1972; Ivanovic et al., 2000; Kohleret al., 2005; Clinton et al., 2006; Michel et al., 2008).Due to the shaking caused by major earthquakes, thefrequencies of normal modes decrease (Trifunac et al.,

2001a; Kohler et al., 2005); the reduction is mostly tem-porary (a few minutes) and healing occurs with time,but some reduction is permanent (Clinton et al., 2006),who found more than 20% temporal reduction and 4%permanent reduction in the fundamental frequency ofthe motion of the Millikan Library located at the Cali-fornia Institute of Technology after the 1987 M6.1 Whit-tier Narrows earthquake. The reduction in the frequencylogarithmically correlates with the maximum accelera-tion of observed records (Clinton et al., 2006). Precip-itation, strong wind, temperature, reinforcement, and

348 N. Nakata et al.

140˚00' 140˚30' 141˚00' 141˚30' 142˚00'

36˚30'

37˚00'

37˚30'

0 km 50 km

140˚00' 140˚30' 141˚00' 141˚30' 142˚00'

36˚30'

37˚00'

37˚30'

140˚00' 140˚30' 141˚00' 141˚30' 142˚00'

36˚30'

37˚00'

37˚30'

140˚00' 140˚30' 141˚00' 141˚30' 142˚00'

36˚30'

37˚00'

37˚30'

1

2

3

4

5

6

78

9

1011

12

13

14

15

16

17

Figure 1. The building (rectangle, not to scale) and epicen-ters of earthquakes used (crosses). Numbers beside of crossescorrespond to the sequencial numbers in Table 1. The lower-right map indicates the location of the magnified area.

heavy weight loaded in a building also change the fre-quencies of normal modes (Kohler et al., 2005; Clintonet al., 2006). Because these frequencies are related toboth the building itself and the soil-structure coupling,we have to consider soil-structure interactions (Safak,1995) and nonlinearities in the response of the founda-tion soil (Trifunac et al., 2001a,b). Normal-mode fre-quencies estimated from observed records are, there-fore, not suitable for health monitoring of a building inisolation of its environment (Todorovska and Trifunac,2008b).

Snieder and Safak (2006) show that one can esti-mate an impulse response independent from the soil-structure coupling and the complicated wave propaga-tion (e.g., attenuation and scattering) below the bottomreceiver by using seismic interferometry based on decon-volution. Seismic interferometry is a technique to ex-tract the Green’s function which accounts for the wavepropagation between receivers (Lobkis and Weaver,2001; Derode et al., 2003; Snieder, 2004; Paul et al.,2005; Snieder et al., 2006b; Wapenaar and Fokkema,2006). Seismic interferometry can be based on cross-correlation, deconvolution, and crosscoherence (Sniederet al., 2009; Wapenaar et al., 2010). Deconvolution inter-ferometry is a useful technique for monitoring structuresespecially in one dimension (Nakata and Snieder, 2011,2012a,b). Because deconvolution interferometry changesthe boundary condition at the base of the building, weare able to extract the pure response of the building re-gardless of its coupling to the subsurface (Snieder andSafak, 2006; Snieder et al., 2006a).

Deconvolution interferometry has been applied toearthquake records observed in a building to retrieve the

velocity of traveling waves and attenuation of the build-ing (Oyunchimeg and Kawakami, 2003; Snieder andSafak, 2006; Kohler et al., 2007; Todorovska and Tri-funac, 2008a,b) (some studies call the method impulseresponse function or normalized input-output minimiza-tion). Todorovska and Trifunac (2008b) use 11 earth-quakes occurring over a period of 24 years to monitorthe fundamental frequency of a building after apply-ing deconvolution interferometry. The fundamental fre-quencies they estimated from the interferometry are al-ways higher than the frequencies obtained from the ob-served records because the frequencies computed fromthe observed records are affected by both the buildingitself and soil-building coupling, while the frequenciesestimated using the interferometry are only related tothe building itself. Oyunchimeg and Kawakami (2003)apply short-time moving-window seismic interferometryto an earthquake record to estimate the velocity reduc-tion of a building during an earthquake. Prieto et al.(2010) apply deconvolution interferometry to ambientvibrations after normalizing amplitudes per frequencyusing the multitaper method (Thomson, 1982) to esti-mate the traveling-wave velocity and damping factor.

In this study, we apply deconvolution interferome-try to 17 earthquakes observed at a building in Japanover a period of two weeks and monitor the changesin velocity of the building. This study is based on thework of Snieder and Safak (2006); furthermore, we ex-tend the deconvolution interferometry as proposed bySnieder and Safak (2006) to deconvolution with thewaveforms recorded at an arbitrary receiver, comparethis with crosscorrelation and crosscoherence interfer-ometry, and use interferometry for monitoring a build-ing in Japan. First, we introduce our data: geometryof receivers, locations of the building and epicenters ofearthquakes used, observed waveforms, and shapes ofthe normal modes extracted from observed records. Wealso introduce the equations of interferometry based ondeconvolution, crosscorrelation, and crosscoherence. Wefurther indicate the deconvolved waveforms obtainedfrom one earthquake and estimate a velocity as well as aquality factor (Q). Next, we apply deconvolution inter-ferometry to all observed earthquakes and monitor thechange in velocity of the building using coda-wave inter-ferometry (Snieder et al., 2002). In a companion paper,we apply the interferometry to ambient vibrations.

2 BUILDING AND EARTHQUAKES

The building (rectangle in Figure 1) in which werecorded vibrations is located in the Fukushima pre-fecture, Japan. Continuous seismic vibrations wererecorded by Suncoh Consultants Co., Ltd. for two weeksusing 10 microelectromechanical-systems (MEMS) ac-celerometers, which were developed by Akebono BrakeIndustry Co., Ltd., and 17 earthquakes were observedduring the two weeks (Table 1 and Figure 1). In this

Monitoring a building by interferometry I 349

Table 1. Origin times, magnitudes, and hypocenter locations of recorded earthquakes estimated by the Japan Meteorological

Agency (JMA). The earthquakes are numbered sequentially according to their origin times. Maximum acceleration is theobserved maximum amplitude of the MEMS accelerometers at the first floor in the 120 s following the origin time of eachearthquake.

No. Origin time MJMA Latitude Longitude Depth (km) Maximum acceleration (m/s2)

1 5/31/11 16:12:20.2 3.9 37.1983 141.0900 32 0.145

2 6/1/11 01:41:19.6 4.2 37.6583 141.7617 44 0.0883 6/1/11 06:23:27.2 3.4 37.1167 140.8400 7 0.0614 6/3/11 21:44:38.7 4.2 37.2783 141.4683 36 0.0405 6/4/11 01:00:14.1 5.5 36.9900 141.2100 30 1.923

6 6/4/11 09:03:33.7 4.3 37.2683 141.4783 36 0.1477 6/4/11 10:41:10.0 4.1 37.0033 141.1933 28 0.0838 6/5/11 17:32:38.9 3.5 37.1117 140.8217 7 0.2819 6/5/11 19:46:06.3 3.9 36.9200 140.7333 14 0.175

10 6/5/11 20:16:37.0 4.4 37.0500 140.7717 11 0.84311 6/7/11 03:11:55.9 2.6 37.1233 140.8540 5 0.02012 6/9/11 19:38:32.9 5.7 36.4967 140.9700 13 0.358

13 6/11/11 00:40:55.4 3.9 36.9367 140.6833 11 0.07514 6/11/11 01:41:19.6 3.9 37.4117 141.3983 46 0.08715 6/12/11 05:08:58.4 4.5 37.2150 141.2100 21 0.15316 6/12/11 17:09:45.4 4.6 36.4117 141.0800 47 0.270

17 6/13/11 05:59:35.1 4.4 37.3350 141.3283 33 0.180

0.25

-5.10

5.85

27.05

23.55

20.25

16.55

13.05

9.45

35.00

31.10

38.95

Ele

va

tio

n (

m)

1

B

6

5

4

32

M 2 3.15

R2

R1

8

7

Flo

or

nu

mb

er

59.00 m 30.20 m

East-West North-South

Figure 2. The (left) EW and (right) NS vertical cross sec-

tions of the building and the positions of receivers (triangles).Elevations denote the height of each floor from ground level.We put receivers on stairs 0.19 m below each floor exceptfor the basement (on the floor) and the first floor (0.38 m

below). Receiver M2 is located between the first and secondfloors. Horizontal-receiver components are aligned with theEW and NS directions.

study, we focus on processing of the earthquake records,and we analyze ambient vibrations in the companionpaper. The building includes eight stories, a basement,and a penthouse (Figure 2). We installed receivers onthe stairs, located 20 m from the east side and at thecenter between the north and south sides. The samplinginterval of the records is 1 ms, and the receivers havethree components. Here, we use two horizontal compo-nents which are aligned with the east-west (EW) andnorth-south (NS) directions.

Figures 3a–3d illustrate the observed waveformsand their power spectra of earthquake No. 5, which gives

the greatest acceleration to the building. Figures 3e and3f show the spectrogram of the motion at the fourthfloor computed with the continuous-wavelet transform(Torrence and Compo, 1998). Higher frequencies quicklyattenuate and the fundamental mode is dominant forlater times in Figures 3e and 3f. The frequency of thefundamental mode in the EW component (1.17 Hz) ishigher than the frequency in the NS component (0.97Hz) because the EW side of the building is longer thanthe NS side. Both components have large amplitudesat around 0.5 Hz between 14 and 21 s. Since the 0.5-Hz component is localized in time (Figure 3e and 3f),it corresponds to a surface wave that moves the entirebuilding. However, because the frequency of the surfacewave is less than that of the fundamental mode of thebuilding, it does not excite waves that propagate withinthe building.

Figure 4 illustrates the shapes of the normal-modedisplacement computed from the real part of the Fourierspectra at different floors. We calculate displacementfrom acceleration using numerical integration (Schiffand Bogdanoff, 1967). Just as for the fundamentalmode, the frequencies of overtones in the EW com-ponent are also higher than those in the NS compo-nent. Although the displacements of both componentsin mode 1 (the fundamental mode) are almost the same,the NS-component displacement is larger than that ofthe EW component in modes 2 and 3. The amplitudeof mode 1 is much larger than the amplitudes of othermodes.


0 10 20 30 40 50 60

B1

M22345678

R1

R2East−West

a) b)

c) d)

e) f)

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

0 10 20 30 40 50 60

B1

M22345678

R1

R2North−South

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

0.1 0.2 0.3 0.5 1 1.5 2 3 4 5 6 810 20B1

M22345678

R1

R2East−West

Floo

r

Frequency (Hz)

−505101520253035

Ele

vatio

n (m

)

0.1 0.2 0.3 0.5 1 1.5 2 3 4 5 6 810 20B1

M22345678

R1

R2North−South

Floo

r

Frequency (Hz)

−505101520253035

Ele

vatio

n (m

)0 10 20 30 40 50 60

8.0

4.0

2.0

1.0

0.5

0.25

East−West

Time (s)

Freq

uenc

y (H

z)

0 10 20 30 40 50 60

8.0

4.0

2.0

1.0

0.5

0.25

North−South

Time (s)

Freq

uenc

y (H

z)

Nor

mal

ized

am

plitu

de

0

0.25

0.5

0.75

1

Figure 3. Unfiltered waveforms of earthquake No. 5 recorded at the building in (a) theEW component and (b) the NS component, and (c, d) their power spectra. (e, f) Spec-

trogram computed with continuous-wavelet transformed waveforms recorded at the fourth floor.Time 0 s represents the origin time of the earthquake. We preserve relative amplitudes of the EW and NS components.

3 DECONVOLUTION WITH ANARBITRARY RECEIVER

By deconvolving observed earthquake records, we ob-tain the impulse response of a building (Oyunchimegand Kawakami, 2003; Snieder and Safak, 2006; Kohleret al., 2007; Todorovska and Trifunac, 2008a). Whenthe height of the building is H, the recorded signal ofan earthquake in the frequency domain at an arbitrary

receiver at height z is given by Snieder and Safak (2006):

u(z) =

∞X

m=0

S(ω)Rm(ω){eik(2mH+z)e−γ|k|(2mH+z)

+ eik(2(m+1)H−z)e−γ|k|(2(m+1)H−z)}

=S(ω){eikze−γ|k|z + eik(2H−z)e−γ|k|(2H−z)}

1 − R(ω)e2ikHe−2γ|k|H , (1)


1

mode 1EW: 1.17 HzNS: 0.97 Hz

0.03


0.005


EWNS

Figure 4. Displacement of the first three horizontal normal

modes for earthquake No. 5 estimated from the real part ofthe Fourier spectra at different floors. Each mark indicatesthe displacement of a receiver. The center frequency of eachmode is shown at the top of each panel. Black horizontal

lines and the numbers on the lines show the amplitude ratioamong modes, and the box depicts the height of the building(R2 in Figure 2). The zero displacement is at the right sideof each box.

where S(ω) is the incoming waveform to the base of thebuilding, R(ω) the reflection coefficient related to thecoupling of the ground and the base of the building, kthe wavenumber, γ the attenuation coefficient, and i theimaginary unit. We use the absolute value of wavenum-bers in the damping terms because the waves attenu-ate regardless of whether the wavenumber is positive ornegative. In expression 1, we assume the wave verticallypropagates in the building (one-dimensional propaga-tion) with constant amplitude and wavenumber, andwithout internal reflections. The constant wavenum-ber implies that we assume constant velocity c becausek = ω/c. The incoming waveform S(ω) includes thesource signature of the earthquake and the effect ofpropagation such as attenuation and scattering alongthe path from the hypocenter of the earthquake to thebase of the building. The attenuation coefficient γ isdefined as

γ =1

2Q, (2)

with Q the quality factor (Aki and Richards, 2002).For m = 0 in the first line in expression 1, the first

term S(ω)eikze−γ|k|z indicates the incoming upgoingwave and the second term S(ω)eik(2H−z)e−γ|k|(2H−z)

the downgoing wave, which is reflected off the top ofthe building. The index m represents the number of re-verberations between the base and top of the building.

As we deconvolve a waveform recorded by a receiver

at z with a waveform observed by another receiver atza, from expression 1 we obtain

D(z, za, ω) =u(z)

u(za)

=S(ω){eikze−γ|k|z + eik(2H−z)e−γ|k|(2H−z)}

S(ω){eikzae−γ|k|za + eik(2H−za)e−γ|k|(2H−za)}

=∞

X

n=0

(−1)nn

eik(2n(H−za)+z−za)e−γ|k|(2n(H−za)+z−za)

+eik(2n(H−za)+2H−z−za)e−γ|k|(2n(H−za)+2H−z−za)o

,

(3)

where we use a Taylor expansion in the last equality. Inequation 3, the receiver at za behaves as a virtual source.Equation 3 may be unstable because of the spectral de-vision. In practice we use a regularization parameter ε(Yilmaz, 2001,Section 2.3):

D(z, za, ω) =u(z)

u(za)≈ u(z)u∗(za)

|u(za)|2 + ε〈|u(za)|2〉 , (4)

where ∗ is a complex conjugate and 〈|u(za)|2〉 the aver-age power spectrum of u(za). In this study we use ε =1%.

Note that these deconvolved waves are independentof the incoming waveform S(ω) and the ground cou-pling R(ω). When we consider substitutions S(ω) → 1,R(ω) → −1, H → H−za, and z → z−za, equation 1 re-duces to equation 3. These conditions indicate the phys-ical properties of the deconvolved waveforms: impulseresponse (S(ω) → 1), perfect reflection at the virtualsource (R(ω) → −1), and a small building (H → H−za

and z → z − za) as we discuss later.When z > za, equation 3 describes a wave that is

excited at za and reverberates between za and the topof the building. Using a normal-mode analysis (equationA4 in Appendix A), the fundamental mode of equation3 in the time domain is given by

D(z, za, t) =4πc

H − zae−γω0t sin(ω0t) cos

„

ω0H − z

c

«

,

(5)

where ω0 = πc/ {2(H − za)}. The period of the funda-mental mode is, thus,

T0 =4(H − za)

c, (6)

which corresponds to the period of the fundamentalmode of the building that is cut off at za (cut-off build-ing: Figure 5a). According to equation 3 the polar-ity change resulting from reflection at za is given by(−1)n, the reflection coefficient at the virtual source is−1. Therefore, the cut-off building is only sensitive tothe properties of the building above za, and the recon-structed wave motion in the cut-off building has thepotential to estimate local structure and local damageinstead of structure and damage for the entire building.

When z < za and n = 0 in equation 3, we


obtain two waves: an acausal upgoing wave from zto za (eik(z−za)e−γ|k|(z−za)) and a causal downgoingwave from za to z (eik(2H−z−za)e−γ|k|(2H−z−za)). Wavesfor n ≥ 1 in equation 3 account for the reverbera-tions between za and the top of the building. BecauseD(za, za, ω) = 1, the deconvolved waveforms at za is adelta function in the time domain (D(za, za, t) = δ(t));hence D(za, za, t) = 0 for t 6= 0 (clamped boundary con-dition (Snieder et al., 2006a, 2009)). The upgoing anddowngoing waves interfere destructively at za.

Although we assume, for simplicity, a constant ve-locity in equation 1, we can apply deconvolution in-terferometry to wavefields observed at a building withsmoothly varying velocities. When the velocity c varieswith height, the local wavenumber does so as well, andusing the WKBJ approximation for the phase, expres-sion 1 generalizes to

u(z) =U

1 − R(ω) exp“

2iR H

0k(z)dz

”

exp“

−2γR H

0|k(z)|dz

”

(7)

with

U =S(ω)

»

exp

„

i

Z z

0

k(z)dz

«

exp

„

−γ

Z z

0

|k(z)|dz

«

+exp

i

„

Z H

0

k(z)dz +

Z H

z

k(z)dz

«ff

× exp

−γ

„

Z H

0

|k(z)|dz +

Z H

z

|k(z)|dz

«ff–

,

and equation 3 generalizes in this case to

D(z, za, ω) =

∞X

n=0

(−1)n

×»

exp

i

„

2n

Z H

za

k(z)dz +

Z z

za

k(z)dz

«ff

× exp

−γ

„

2n

Z H

za

|k(z)|dz +

Z z

za

|k(z)|dz

«ff

+exp

i

„

(2n + 1)

Z H

za

k(z)dz +

Z H

z

k(z)dz

«ff

× exp

−γ

„

(2n + 1)

Z H

za

|k(z)|dz +

Z H

z

|k(z)|dz

«ff–

.

(8)

As for equation 3, equation 8 represents the waves in acut-off building at za, and the period of the fundamentalmode of the cut-off building depends on the slownessaveraged between za and H. Note that when z > za,D(z, za, ω) in equation 8 is only related to k(z) aboveza.

When za is at the first floor (za = 0) or at the top ofthe building (za = H), equation 3 corresponds to equa-tions 26 or 21 in Snieder and Safak (2006), respectively,which we rewrite here as

a) b)

H

H－Za

Za

Figure 5. Schematic shapes of the fundamental mode re-trieved by using seismic interferometry. (a) Fundamental

mode retrieved by deconvolving wavefields with a motionrecorded at za (equation 3). (b) Fundamental mode retrievedby deconvolving wavefields with a motion recorded at thefirst floor (equation 9).

D(z, 0, ω) =

∞X

n=0

(−1)n ×n

eik(2nH+z)e−γ|k|(2nH+z)

+eik(2(n+1)H−z)e−γ|k|(2(n+1)H−z)o

, (9)

D(z, H, ω) =1

2

n

eik(H−z)e−γ|k|(H−z) + e−ik(H−z)eγ|k|(H−z)o

.

(10)

Figure 5b illustrates a schematic shape of the funda-mental mode as given in equation 9. The period of themode in equation 9 is related to the structure of theentire building as if the building were placed on a rigidsubsurface (i.e., the reflection coefficient at the base is−1). When we put a receiver at the top floor (8 in Fig-ure 2) of a building instead of the rooftop (R1 or R2 inFigure 2), we theoretically do not obtain the responsein equation 10 because the traveling waves reflect at thetop of the building rather than at the top floor. In thiscase, the deconvolved response follows equation 3. Thisdifference may be insignificant when the wavelength ofthe traveling waves is much longer than the distancebetween the top floor and the top of the building.


4 CROSSCORRELATION AND CROSSCOHERENCE INTERFEROMETRY

In the previous section, we focused on seismic interferometry based on deconvolution. Let us consider seismic inter-ferometry based on crosscorrelation [e.g., Schuster et al., 2004] and crosscoherence [e.g., Nakata et al., 2011]; thesetwo methods are the widest-applied technique and the earliest application (Aki, 1957), respectively.

4.1 Crosscorrelation

From equation 1, the crosscorrelation of u(z) and u(za) is

C(z, za, ω) = u(z)u∗(za)

= |S(ω)|2n

eikze−γ|k|z + eik(2H−z)e−γ|k|(2H−z)o n

e−ikzae−γ|k|za + e−ik(2H−za)e−γ|k|(2H−za)o

1 − R(ω)e2ikHe−2γ|k|H − R∗(ω)e−2ikHe−2γ|k|H + |R(ω)|2e−4γ|k|H . (11)

In contrast to the deconvolution (equation 3), equation 11 depends on the incoming wave S(ω) and the groundcoupling R(ω), and does not create a clamped boundary condition (C(za, za, ω) 6= 1). Because of the presence of thereflection coefficient R(ω) and the power spectrum |S(ω)|2, it is much more complicated to estimate the properties(e.g., traveling-wave velocity and attenuation) of the building from crosscorrelation than from deconvolution.When za = 0 and za = H, equation 11 reduces to

C(z, 0, ω) = |S(ω)|2 eikze−γ|k|z + e−ikze−γ|k|(4H−z) + eik(2H−z)e−γ|k|(2H−z) + e−ik(2H−z)e−γ|k|(2H+z)

1 − R(ω)e2ikHe−2γ|k|H − R∗(ω)e−2ikHe−2γ|k|H + |R(ω)|2e−4γ|k|H , (12)

C(z, H, ω) = 2 |S(ω)|2e−2γ|k|H

n

eik(H−z)e−γ|k|(H−z) + e−ik(H−z)eγ|k|(H−z)o

1 − R(ω)e2ikHe−2γ|k|H − R∗(ω)e−2ikHe−2γ|k|H + |R(ω)|2e−4γ|k|H , (13)

respectively. If R(ω) = 0 (no reflection at the base), equation 13 is, apart from the prefactor 2|S(ω)|2e−2γ|k|H , thesame as equation 10.

4.2 Crosscoherence

Crosscoherence is defined as frequency-normalized crosscorrelation:

CH(z, za, ω) =u(z)u∗(za)

|u(z)||u(za)|

≈ u(z)u∗(za)

|u(z)||u(za)| + ε′〈|u(z)||u(za)|〉 (14)

Similar to equation 4, we use a regularization parameter ε′ in the last equality in practice. In this study, we useε′ = 0.1%. For mathematical interpretation, using Taylor expansions of

√1 + X and 1/

√1 + X for X < 1, the

crosscoherence between u(z) and u(za) is given by

CH(z, za, ω) =u(z)u∗(za)

|u(z)||u(za)| =u(z)u∗(za)

p

u(z)u∗(z)p

u(za)u∗(za)=

p

u(z)p

u∗(za)p

u∗(z)p

u(za)

=

s

{eikze−γ|k|z + eik(2H−z)e−γ|k (2H−z)} {e−ikzae−γ|k|za + e−ik(2H−za)e−γ|k|(2H−za)}{e−ikze−γ|k|z + e−ik(2H−z)e−γ|k|(2H−z)} {eikzae−γ|k|za + eik(2H−za)e−γ|k|(2H−za)}

=eik(z−za) ×

"

1 +1

2

∞X

n=1

e2ink(H−z)e−2nγ|k|(H−z)

n!An−1

ff

# "

1 +1

2

∞X

n=1

e−2ink(H−za)e−2nγ|k|(H−za)

n!An−1

ff

#

×

"

1 +

∞X

n=1

e−2ink(H−z)e−2nγ|k|(H−z)

n!An

ff

# "

1 +

∞X

n=1

e2ink(H−za)e−2nγ|k|(H−za)

n!An

ff

#

, (15)

where A0 = 1 and An = (2n − 1)!!/(−2)n. As for deconvolution interferometry, equation 15 does not depend onS(ω) and R(ω). For a complex number z = reiφ, the square root is defined as

√z =

√reiφ/2. Furthermore, the

waveforms of crosscoherence interferometry satisfies a clamped boundary condition at z = za (CH(za, za, ω) = 1,hence CH(za, za, t) = δ(t), and CH(za, za, t) = 0 for t 6= 0).For za = 0 and za = H, equation 15 simplifies to


CH(z, 0, ω) =eikz ×

"

1 +1

2

∞X

n=1


n!An−1

ff

#

×

"

1 +1

2

∞X

n=1

e−2inkHe−2nγ|k|H

n!An−1

ff

#

×

"

1 +

∞X

n=1


n!An

ff

#

×

"

1 +

∞X

n=1

e2inkHe−2nγ|k|H

n!An

ff

#

, (16)

CH(z, H, ω) =eik(z−H) ×

"

1 +1

2

∞X

n=1


n!An−1

ff

#

×

"

1 +∞

X

n=1


n!An

ff

#

,

(17)

respectively. Note that because of the complexity of crosscoherence interferometry, equations 15–17 contain pseudoevents which propagate at slower velocities than the true velocity of the building. For example, equation 16 can beexpanded into

CH(z, 0, ω)

=eikz

1 − 1

4e−4γ|k|(H−z) − 1

4e−4γ|k|H

ff

+1

4e−ikze−2γ|k|(2H−z) +

1

4e3ikze−2γ|k|(2H−z)

− 1

2e−ik(2H−3z)e−2γ|k|(H−z) − 1

2eik(2H+z)e−2γ|k|H +

1

2eik(2H−z)e−2γ|k|(H−z) +

1

2e−ik(2H−z)e−2γ|k|H

+3

8e−ik(4H−5z)e−4γ|k|(H−z) − 1

8e−ik(4H−z)e−4γ|k|H − 1

4eik(4H−z)e−2γ|k|(2H−z) − 1

4e−ik(4H−3z)e−2γ|k|(2H−z)

+3

8e−ik(4H−z)e−4γ|k|H − 1

8eik(4H−3z)e−4γ|k|(H−z) + O(e−6γ|k|(H−z)). (18)

The terms e3ikze−2γ|k|(2H−z)/4 and −e−ik(2H−3z)e−2γ|k|(H−z)/2 indicate waves that propagate at one third of thetrue velocity, and the term 3e−ik(4H−5z)e−4γ|k|(H−z)/8 at one fifth. These unphysical waves complicate the estimationof the velocity of traveling waves by applying crosscoherence interferometry to earthquake data.

4.3 Comparison of deconvolution,crosscorrelation, and crosscoherence

Each type of interferometry has different properties(e.g., amplitude or complexity). It follows from expres-sions 4, 11, and 14 that in the frequency domain thephases obtained by interferometry based on deconvolu-tion, crosscorrelation, and crosscoherence are the same.The spectral amplitude is different, though, and thisleads to different waveforms in the time domain. Welist in Table 2 the amplitudes of the first four causalwaves propagating at the true velocity of the buildingfor observed record, deconvolution, crosscorrelation, andcrosscoherence (equations 1, 9, 12, and 16, respectively).For crosscorrelation interferometry, we calculate the am-plitudes of the traveling waves in Appendix B using Tay-lor expansions. Although the amplitudes of crosscorre-lation are complicated due to the reflection coefficientR(ω) and the power spectrum |S(ω)|2, the ratios of am-plitudes for each pair of traveling waves are the sameas those for observed records. The amplitudes of thewaveforms obtained by crosscoherence are independentof incoming waveform S(ω) and reflection coefficientR(ω), but the ratio of amplitudes varies between eachpair of traveling waves. Therefore, estimating attenua-tion of the building using crosscoherence interferometry

is problematic. Since the amplitudes of the deconvolvedwaveforms are independent of S(ω) and R(ω) and de-pend exponentially on the traveled distance, deconvolu-tion interferometry can be used to estimate attenuationof the building.

We numerically compute synthetic waveforms ex-cited at 0 m by an impulsive source (S(ω) = 1) basedon equation 1 shown in Figure 6a. In the computa-tion, we use the following parameters: H = 100 m,R(ω) = 0.5, Q = 3000, and c = 200 m/s. In ap-plying seismic interferometry, we compute deconvolu-tion ({u(z)u∗(0)} /

˘

|u(0)|2 + ε〈|u(0)|2〉¯

: Figure 6b),crosscorrelation (u(z)u∗(0): Figure 6c), and crossco-herence ({u(z)u∗(0)} / {|u(z)||u(0)| + ε′〈|u(z)||u(0)|〉}:Figure 6d), where ε = 1% and ε′ = 0.1%, using thesynthetic waveforms shown in Figure 6a. In Figures6b–6d, the virtual source is at z = 0 m. Deconvolvedwaveforms (Figure 6b) arrive at the same time as thewaves in the synthetic records, but the polarization isreversed when the wave is reflected at z = 0 m dueto the clamped boundary condition. In crosscorrelationinterferometry (Figure 6c), the causal waves arrive atthe same time as the waves in the synthetic records(Figure 6a), and the acausal waves are kinematicallyidentical to the time-reversed causal waves. Althoughfor simplicity we use S(ω) = 1 in Figure 6, the in-


0

10

20

30

40

50

60

70

80

90

100

Ele

vatio

n (m

)

Synthetic recordsa)

1 R 1

Deconvolutionb)

1 −1 1

−1 0 1

0

10

20

30

40

50

60

70

80

90

100

Time (s)

Ele

vatio

n (m

)

Crosscorrelationc)

1 R 1

−1 0 1

Crosscoherenced)

Time (s)

CH1 −1 0.5

Figure 6. (a) Synthetic waveforms based on equation 1. We numerically calculate waveforms with an impulse response (S(ω) =1) at t = 0 s at 0 m, R(ω) = 0.5, Q = 3000, H = 100 m, and c = 200 m/s. Interferometric waveforms by computing deconvolution

(panel b: equation 9), crosscorrelation (panel c: equation 12), and crosscoherence (panel d: equation 16) using waveforms shownin panel (a). The virtual source for interferometry is at the 0-m receiver. We apply a bandpass filter 0.5-1-30-40 Hz aftercomputing each waveform. The circles in each panel highlight four waves which are discussed in the main text. The numbersnear each arrow indicate the ratio of the amplitude difference between two waves highlighted by the circles apart from the

attenuation expected from the traveling distance at the correct velocity. To estimate the ratio of amplitude in panel (d), weignore wavefields which attenuate faster than e−6γ|k|(H−z), and CH1 = 2/(4 − e−4γ|k|(H−z) − e−4γ|k|H) where z = 40 m.Amplitudes in each panel are normalized by the amplitude of the first highlighted wave (at t=0.2 s). The gray line in panel (d)shows the wave which propagates at 66.67 m/s.

coming wave complicates the crosscorrelated waveformswhen we use real earthquake data, and picking the ar-rival times of the traveling waves may be difficult inthat case. Crosscoherence interferometry creates trav-eling waves which propagate at slower velocities thantrue velocity c = 200 m/s. In Figure 6d, the gray linehighlights the wave −e−ik(2H−3z)e−2γ|k|(H−z)/2, whichtravels with one third of the true wave speed (66.7 m/s).To estimate the velocity of the traveling waves, there-fore, deconvolution interferometry is useful.

We highlight the amplitudes of the waves in Fig-ure 6 with the circles. A comparison of Figures 6a and6b shows that the ratios of the amplitudes of the syn-thetic records and deconvolved waves within the first

two circles are the same, but the ratios in the secondand third circles are different. The reflection coefficientat 0 m of the synthetic records is R(ω) while the re-flection coefficient of waves obtained by deconvolutioninterferometry is −1; see the numbers next to the arrowsin Figures 6a and 6b. The difference between the reflec-tion coefficients implies that deconvolved waveforms areindependent from the ground coupling, and the decay ofamplitudes of the waves are only related to the attenua-tion of the building. The ratios of the amplitudes of thewaves highlighted by the circles in crosscorrelation inter-ferometry are the same as those in the synthetic records;see the numbers next to the arrows in Figures 6a and6c. Hence, both the building and the soil-structure cou-


Table 2. Amplitudes of traveling waves obtained from observed records and computed by seismic interferometry based on

deconvolution, crosscorrelation, and crosscoherence for za = 0 (equations 1, 9, 12, and 16). We compute the amplitudes ofcrosscorrelation interferometry in Appendix B. For crosscoherence interferometry we ignore wavefields which attenuate fasterthan e−6γ|k|(H−z).

Phase Observed record Deconvolution Crosscorrelation Crosscoherence

eikz S(ω)e−γ|k|z e−γ|k|z C1 1 − 14e−4γ|k|(H−z) − 1

4e−4γ|k|H

eik(2H−z) S(ω)e−γ|k|(2H−z) e−γ|k|(2H−z) C212e−2γ|k|(H−z)

eik(2H+z) S(ω)R(ω)e−γ|k|(2H+z) −e−γ|k|(2H+z) C3 − 12e−2γ|k|H

eik(4H−z) S(ω)R(ω)e−γ|k|(4H−z) −e−γ|k|(4H−z) C4 − 14e−2γ|k|(2H−z)

C1 = |S(ω)|2 e−γ|k|zn

1 + R(ω)e−4γ|k|Ho

( ∞X

n=0

|R(ω)|2n e−4nγ|k|H

)

C2 = |S(ω)|2 e−γ|k|(2H−z)n


( ∞X

n=0


)

C3 = |S(ω)|2 R(ω)e−γ|k|(2H+z)n


( ∞X

n=0


)

C4 = |S(ω)|2 R(ω)e−γ|k|(4H−z)n


( ∞X

n=0


)

pling influence the amplitudes of crosscorrelated wave-forms. In crosscoherence interferometry (Figure 6d), theratios of the amplitudes of the waves within the circlesare different from either synthetic records or deconvolu-tion interferometry. When we consider the amplitudesof each interferometry, deconvolution interferometry isuseful for estimating the attenuation of the building.

Based on equations 11–18, Table 2, and Figure 6,we conclude that deconvolution interferometry is suit-able for application to earthquake records to estimatethe velocity and attenuation of the building. Crosscorre-lated waveforms depend on the incoming wave S(ω) andthe ground coupling R(ω). Crosscoherence interferome-try creates pseudo events, and the decay of amplitudesof waveforms reconstructed by crosscoherence is not ex-ponentially depending on the traveled distances. There-fore, these types of interferometry are not appropriateto estimate velocity and attenuation.

Snieder et al. (2006a) show that the wavefields ob-tained from deconvolution interferometry satisfy thesame wave equation as the wavefield of the real build-ing for an external source. Using this idea, we explainwhy crosscoherence interferometry creates unphysicalevents. Following Snieder et al. (2006a), we denote thelinear differential operator that defines the wave propa-gation by L(z) (e.g., for the one-dimensional wave equa-tion L(z) = d2/dz2 + ω2/c2(z)). The operator acts onthe space variable z. For an internal source at z0, thewavefield u(z) (equation 1) satisfies L(z)u(z) = F (z0)where F is the excitation at z0. For an external source,on the other hand, u(z) satisfies L(z)u(z) = 0; this ho-mogeneous equation applies to earthquake data. Apply-ing the operator L(z) to equations 3, 11, and 14, respec-tively, gives

L(z)D(z, za, ω) = L(z)u(z)

u(za)=

1

u(za)L(z)u(z) = 0,

(19)

L(z)C(z, za, ω) = L(z)u(z)u∗(za) = {L(z)u(z)}u∗(za) = 0,(20)

L(z)CH(z, za, ω) = L(z)u(z)u∗(za)

|u(z)||u(za)| =u∗(za)

|u(za)|L(z)

„

u(z)

|u(z)|

«

6= 0,

(21)

where we used that L(z) acts on the z-coordinate only.Crosscoherence interferometry (equation 21) does notproduce a wavefield that satisfies the wave equation ofthe real building, but deconvolution and crosscorrela-tion interferometry do satisfy the wave equation. Equa-tion 21 shows that crosscoherence interferometry createsunphysical internal sources that complicate wavefieldsobtained from crosscoherence.

5 DECONVOLVED WAVEFORMSGENERATED FROM AN EARTHQUAKE

As an illustration of the data analysis, we first showthe application of deconvolution interferometry to therecords of earthquake No. 5. We first estimate whetherthe reflection point of the traveling wave is at R1 or R2because the building has a penthouse (Figure 2). Figure7 shows waveforms deconvolved by the motion recordedat the first floor (equation 9) for the EW and NS compo-nents. We apply a 0.4-0.5-45-50 Hz sine-squared band-pass filter to the deconvolved waveforms. Because thephysical property at the basement is different from the


B1

M22345678

R1

R2

0 1 2 3

East−Westa) b)

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

B1

M22345678

R1

R2

0 1 2 3

North−South

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

Figure 7. Deconvolved waveforms, in which the virtual source is at the first floor, of earthquake No. 5 after applying a bandpassfilter 0.4-0.5-45-50 Hz in the (a) EW and (b) NS components. Gray lines indicate the arrival time of traveling waves with thevelocity that is estimated from the least-squares fitting of the first upgoing and downgoing waves. We repeat the gray lines after

the second traveling waves based on equation 9. Solid gray lines highlight the waves in the positive polarization and dashedgray lines the waves in the negative polarization.

other floors, we do not deconvolve with the motion inthe basement in this study. During the first several hun-dred milliseconds in Figure 7, the waveforms depict atraveling wave excited at the first floor at t = 0 s. Thewave is reflected off the top of the building and prop-agates down, and then reflects again at the first floorwith the opposite polarization because the reflection co-efficient of the deconvolved waves at the first floor isequal to −1 (according to equation 9). While reverber-ating between the first floor and the top of the building,higher frequencies attenuate and the fundamental modeis dominant for later times.

To estimate the velocity of the traveling wave andthe location of the reflection point, we compute thetravel-time curve using a least-squares fitting of thepicked travel times on the first upgoing and downgo-ing waves at each floor (the first two solid gray lines inFigure 7). For picking the travel times, we seek the max-imum amplitude time in each traveling wave. We repeat-edly draw the reverberating travel-wave paths based onequation 9 using the velocity estimated from the firstupgoing and downgoing waves (in Figure 7). To avoidlarge uncertainties, we use the picked travel times be-tween floors one through five in the EW componentand between floors one through six in the NS compo-nent because at these floors the positive amplitudes ofthe upgoing and downgoing waves do not overlap. Bothtravel-time curves in the EW and NS components indi-cate that the waves reflect off the top of the penthouse(R2), and the velocity is 214 ± 9 m/s in the EW direc-tion and 158 ± 7 m/s in the NS direction, respectively,where the uncertainties are one standard deviation ofmeasurements. Because the NS side is shorter, the ve-

locity in the NS component is slower. The deconvolvedwaveforms in the NS component show large deviationfrom expected arrival times shown in the gray lines inFigure 7b, which indicates that the frequency dispersionin the NS component is larger than in the EW compo-nent. In the following, we focus on the EW-componentanalysis.

In Appendix C, we apply crosscorrelation and cross-coherence interferometry to records in the EW compo-nent. Because of the power spectrum of the incomingwave, we cannot obtain traveling waves using crosscor-relation interferometry (Figure C1c). We can estimatethe velocity of traveling waves from the waveforms cre-ated by crosscoherence interferometry, but cannot esti-mate attenuation because the fundamental mode is notreconstructed (Figure C1d).

Next, we deconvolve the wavefields with the mo-tion recorded by the receiver at the fourth floor (Figure8), where the fourth-floor receiver behaves as a virtualsource and satisfies a clamped boundary condition; thenwe apply the same bandpass filter as used in Figure 7.We can interpret waveforms in Figure 8 in two ways,which are explained using Figures 8a and 8b. We ob-tain upgoing and downgoing waves, which interfere atthe fourth floor. At the circle in Figure 8a, the upgoingwave from the bottom and downgoing wave from thetop cancel, and the deconvolved waveform at the fourthfloor vanishes for nonzero time, which is due to the factthat the waveform at the virtual source is a band-limiteddelta function.

The fourth floor also behaves as the reflection pointwith reflection coefficient −1 (equation 3), which meanswe can separate the building to two parts: above and


B1

M22345678

R1

R2

0 1 2 3

a) b)East−West

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

B1

M22345678

R1

R2

0 1 2 3

Floo

r

Time (s)

−505101520253035

Ele

vatio

n (m

)

Figure 8. (a) Deconvolved waveforms, in which the virtual source is at the fourth floor, of earthquake No. 5 after applying abandpass filter 0.4-0.5-45-50 Hz in the EW component. The gray lines indicate the travel paths expected from the velocity 195m/s and equation 3. Solid gray lines highlight the waves in the positive polarization and dashed gray lines the waves in the

negative polarization. The circle indicates the point where the positive and negative polarization waves cancel. (b) The samewaveforms shown in panel (a) but omitting deconvolved waveforms lower than the fourth floor. When we focus on the cut-offbuilding above the fourth floor, the reflection coefficient at the circle is −1.

B

1M2

2345678

R1

R2

−0.4 −0.2 0.0 0.2 0.4

East−West

Floo

r

Time (s)

−5

0

5

10

15

20

25

30

35

Ele

vatio

n (m

)

Figure 9. Deconvolved waveforms, in which the virtualsource is at the eighth floor, of earthquake No. 5 after apply-ing a bandpass filter 0.4-0.5-12-16 Hz in the EW component.The gray lines show the travel time of the waves propagat-

ing at 210 m/s. The positions of the lines are estimated fromequation 3. The thick lines have positive polarization and thedashed line negative polarization.

below the virtual source. Figure 8b shows the buildingabove the virtual source. At the circle in Figure 8b, thedowngoing wave with positive polarization from the topis perfectly reflected as the negative-polarization upgo-ing wave. Since we obtain an upgoing wave from thevirtual source and reverberations between the fourthfloor and the top of the building, this example of in-

terferometry creates the response of a cut-off buildingthat is independent from the structure below the fourthfloor (see equation 3 and Figure 8b). Similar to Figure7, the fundamental mode for the cut-off building (equa-tion 3 and Figure 5a) is dominant for later times inFigure 8b. Note the similarity between Figures 7a and8b; both figures show traveling waves and fundamentalmode. The period of the normal mode in Figure 8b isshorter than in Figure 7a as is expected from equation6. Interestingly, because the cut-off building is indepen-dent from the structure below the fourth floor, this ficti-tious building is useful for detecting local structure andlocal damage of the building.

Applying a least-squares fit of the travel times ofthe first upgoing wave at the first to fourth floors (n = 0and 0 ≤ z ≤ za in equation 3), we obtain the velocityof traveling waves to be 195 ± 25 m/s. To avoid largeuncertainties, we use the travel times at the first tofourth floors to estimate the velocity. At these floors, theupgoing waves are well separated from the downgoingwaves. For the cut-off building, by estimating the veloc-ity from the deconvolved waveforms at the floors onlybelow or above the virtual source, we can obtain thevelocity which is only related to the structure below orabove the virtual source because the virtual source sat-isfies the clamped boundary conditions. The structurebetween the first and fourth floors (below the virtualsource) contributes to the estimation of this velocity.This is the main reason why the mean velocities esti-mated from Figures 7 and 8 differ, but this discrepancyis not statistically significant.

We apply deconvolution interferometry to the mo-


0 1 2 3 4 5 6 7 8 9 10

−8

−6

−4

−2

0

2

4

Time (s)

ln(e

nvel

op)+

floor

num

ber

Figure 10. Natural logarithm of the envelopes (thin line)and linear fitting using the least-squares method (thick line).

We show envelopes at only the middle-second to eighth floorsbecause the first floor is a virtual source and the basementfloor has a different physical condition.

tion recorded by the receiver at the eighth floor, whichis the highest receiver in the building (Figure 9). Sniederand Safak (2006) found that this procedure gives onlyone pair of upgoing and downgoing waves; however,since the eighth-floor receiver is about 12 m below thetop of the building (R2), the deconvolved waveformsin Figure 9 satisfy equation 3 instead of equation 10. InFigure 9, we apply a 0.4-0.5-12-16 Hz sine-squared band-pass filter to deconvolved waveforms. Because the qual-ity of the data is not enough to accurately pick traveltimes, we cannot estimate the wave velocity from Figure9. The gray lines in Figure 9 indicate the arrival time ofthe traveling waves at 210 m/s as inferred from Figure7a.

From the resonant waves in Figures 7 and 8, we canestimate Q following the method proposed by Sniederand Safak (2006). Figure 10 shows the logarithmic en-velopes of the deconvolved waveforms in Figure 7a ateach floor except the basement and the first floor (thinlines), and their least-squares linear fits (thick lines).Because we use the waveforms deconvolved with thefirst floor, the estimated Q is for the entire building.We assume Q is constant in the entire building becausethe wavelength in the frequency range used is muchlonger than the height of the building (the resonant fre-quency is 1.17 Hz). In Figure 10, the average slope ofthe fitting lines indicates that Q−1 = 0.098 based onthe fundamenta-mode frequency 1.17 Hz.

6 MONITORING A BUILDING USING 17EARTHQUAKES

Using the 17 earthquakes recorded in the two weeks (Ta-ble 1 and Figure 2), we monitor the change in the shear-wave velocity of the building. Figure 11 illustrates thewaveforms which are deconvolved by the wave recordedat the first floor of each earthquake (left panel) andthe power spectra of the deconvolved waveforms (rightpanel). The virtual source is at the first floor (similarto Figure 7a). The frequency component around 1.5 Hzshows the fundamental mode and around 5 Hz the firstovertone. From the bottom to top traces for each earth-quake, the traces are aligned from the first to eighthfloors, and the waves propagating between the bottomand the top are visible. Comparing the fundamental-mode waves for later times among earthquakes, we canroughly estimate changes in velocity from a visual in-spection, e.g., the velocities in earthquakes 5, 8, 10, and12 are slower. The earthquakes, which show slower ve-locity, indicate lower normal-mode frequencies as shownby Todorovska and Trifunac (2008b). The ratio of the re-ductions in velocity and frequency are almost the same.

The amplitude of each resonant wave provides anestimate of attenuation. For example, the attenuationis strong for earthquake No. 5 because the amplitudeof the fundamental mode fades away at around 2.5 s.For some earthquakes, although the fundamental modeis dominant at later times, deconvolved waveforms stillshow upgoing and downgoing waves (e.g., at 2.5 s ofearthquake No. 15), which implies either that the at-tenuation at higher frequency is relatively weak at thetime these earthquakes occurred, or that the overtonesare strongly excited. We estimate the velocity of trav-eling waves using the method that is the same as forFigure 7a (the black symbols in Figure 12). The blackmarks in Figure 12b illustrate a negative correlation be-tween the velocities and the maximum acceleration ofobserved records.

To estimate velocities, we also apply coda-wave in-terferometry as developed by Snieder et al. (2002) todeconvolved waveforms. Coda-wave interferometry al-lows us to estimate a relative velocity change fromtwo waveforms by computing crosscorrelation. Coda-wave interferometry has been applied to multiplets [e.g.,Poupinet et al., 1984; Snieder and Vrijlandt, 2005] andto waveforms which are obtained by seismic interferome-try [e.g., Sens-Schonfelder and Wegler, 2006]. By usingcoda-wave interferometry, we estimate velocities fromthe deconvolved waves between 1-3 s in Figure 11. Thewaves in the time interval are mostly the fundamen-tal mode. We choose earthquake No. 5 as a referenceand estimate the relative velocity for each earthquakefrom the reference earthquake. In coda-wave interfer-ometry, we stretch and interpolate one waveform andcompute a correlation coefficient (CC) with a referencewaveform (uref ) in the time domain (Figure 13) (Lobkis


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

0 1 2

Ear

thqu

ake

num

ber

Time (s)0 4 8Frequency (Hz)

Figure 11. The waveforms of each earthquake in the EW component after deconvolution with the waves recorded on thefirst floor in the time domain (left panel), and the power spectra of the waveforms (right panel). We apply a bandpass filter

0.4-0.5-45-50 Hz. We show the traces from the first floor to the eighth floor in each earthquake.

and Weaver, 2003; Hadziioannou et al., 2009; Weaveret al., 2011):

CC(α) =

R t2t1

u(t(1 − α))uref (t)dtq

R t2t1

u2(t(1 − α))dtR t2

t1u2

ref (t)dt, (22)

where t1 and t2 denote the time window, and in thisstudy we use 1-3 s. At the maximum of CC(α),

α =v − vref

vref, (23)

where v and vref are the velocities at each earthquakeand the reference earthquake, respectively. For comput-


1 4 5 6 7 8 9 10 12 14 15 16 17200

220

240

260

280

300

Earthquake number

Vel

ocity

(m/s

)

a) b)traveling wavestretching

10−2 10−1 100200

220

240

260

280

300

Acceleration (m/s2)

Vel

ocity

(m/s

)

traveling wavestretching

Figure 12. (a) Velocities estimated from traveling waves (black) and by coda-wave interferometry using the stretching method(gray) of each earthquake. The error bars of the velocities estimated from traveling waves (black) are one standard deviation

of individual arrival times, and the bars in the stretching method are calculated byq

σ25 + σ2

δv . We illustrate only velocities

which have smaller than 10% velocity uncertainty (for traveling waves) or are estimated from more than three traces whichhave a correlation coefficient greater than 0.9 (for the stretching method). (b) Crossplot of estimated velocities with maximumacceleration observed at the first floor.

ing CC, we apply the same bandpass filter as for Figure11, and the waves are mostly the fundamental mode.

Note that even though we use the waves between1-3 s for applying coda-wave interferometry, the origintime for stretching is at time 0 s. The gray symbols inFigure 12 are the velocities estimated by coda-wave in-terferometry using the stretching method. Because weuse earthquake No. 5 as a reference (uref ), the esti-mated velocities of each earthquake using the stretch-ing method (the gray symbols in Figure 12) are the rel-ative velocities with respect to the velocity of earth-quake No. 5. Therefore, the velocities estimated fromtraveling waves and by the stretching method in earth-quake No. 5 are, by definition, the same. The standarddeviation of the velocity change (the gray bars in Fig-ure 12) for each earthquake is estimated by

p

σ25 + σ2

δv,where σ5 is the standard deviation of the velocity mea-surements estimated from traveling waves at differentfloors in earthquake No. 5, and σδv is the standard de-viation of the relative velocity measurements betweeneach earthquake and earthquake No. 5 estimated by thestretching method at different floors. The gray symbolsin Figure 12b indicate that the velocities obtained by thestretching method also have a negative correlation withthe acceleration, but the slope is steeper than that forthe traveling waves. Since the waves in 1-3 s are mostlyfundamental mode and the main difference between thetraveling waves and the fundamental mode is the fre-quency (the traveling waves contain higher frequenciesthan the fundamental mode), the difference in slopesindicates dispersion. The steeper slope of the gray sym-

bols in Figure 12b indicates that the imprint of accel-eration is stronger for lower frequencies than for higherfrequencies.

7 CONCLUSION

We obtain impulse responses of the building and theirchanges in velocity by applying deconvolution interfer-ometry to 17 earthquake records. We estimate the reflec-tion point of the traveling wave, which is at the top ofthe penthouse, from the deconvolved waveforms. Sincethe shape of the ground plan of the building is rect-angular, the velocities of the traveling wave in two or-thogonal horizontal components are different. Accord-ing to the properties of deconvolution, the responsesare independent from the soil-structure coupling andthe effect of wave propagation below the bottom re-ceiver. Because the cut-off building is independent ofthe structure below the virtual source, one might beable to use the cut-off building to investigate localstructure and local damage. Crosscorrelation interfer-ometry cannot separate the building response from thesoil-building coupling and the wave propagation belowthe virtual source. Crosscoherence interferometry pro-duces unphysical wavefields propagating at slower ve-locity than the true wave speed of the real building, andthe attenuation of the waveforms obtained from crossco-herence do not correspond to the travel distance of thewaves. Hence, in contrast to deconvolution interferom-etry, these types of interferometry are not appropriate


−0.5 −0.25 0.0 0.25 0.5−0.2

0

0.2

0.4

0.6

0.8

1C

orre

latio

n co

effic

ient

α

0.21

0.93a) b)

0 1 2 3−1

−0.5

0

0.5

1

Time (s)

Nor

mal

ized

Am

plitu

de

Eq. 5Eq. 9Eq. 9 with streching

Figure 13. (a) Correlation coefficient (CC) as a function of α (equation 22) between deconvolved waveforms computed from

earthquakes No. 5 and No. 9 at the eighth floor. Dashed arrows point to the maximum CC value and its value of α. Forcomputing CC, we use only the waveforms from 1.0 s to 3.0 s. (b) Deconvolved waveforms at the eighth floor of earthquakesNo. 5, No. 9, and No. 9 with stretching for α = 0.21 (see panel (a)).

for applying to earthquake records for estimating veloci-ties and attenuation of buildings. We estimate velocitiesfrom both traveling waves and the fundamental mode ofdeconvolved waveforms. The velocities estimated fromeach earthquake and maximum acceleration have a neg-ative correlation.

ACKNOWLEDGMENTS

We thank sponsor companies of the Consortium Projecton Seismic Inverse Methods for Complex Structures. Wethank JMA for providing the earthquake catalog. Weare grateful to Diane Witters for her help in preparingthis manuscript.

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APPENDIX A: NORMAL-MODE ANALYSIS OF DECONVOLUTION INTERFEROMETRY

In equation 3, we analyze the deconvolution interferometry based on superposition of traveling waves using a Taylorexpansion. Here, we analyze equation 3 based on summation of normal-mode waves while using contour integration asfollowing the procedure proposed by Snieder and Safak (2006). Applying the inverse Fourier transform to D(z, za, ω)and using the relationship k = ω/c, the deconvolved response in the time domain is given by

D(z, za, t) =

Z ∞

−∞

e−iω(t− zc )e−γ|ω| z

c + e−iω(t− 2H−zc )e−γ|ω| 2H−z

c

eiω zac e−γ|ω| za

c + eiω 2H−zac e−γ|ω| 2H−za

c

dω. (A1)

For t > (2H − z)/c, the locations of poles (ω†) of the integrand in equation A1 are

ω† = ωl(±1 − iγ), (l = 0, 1, 2, · · · ) (A2)

where the normal-mode frequencies are given by

ωl =

`

l + 12

´

πc

H − za(A3)

Using the residue theorem, equation A1 can be written as the summation of normal-mode wavefields:

D(z, za, t) =4πc

H − za

∞X

l=0

(−1)le−γωlt sin(ωlt) cos

„

ωlH − z

c

«

. (A4)

APPENDIX B: AMPLITUDE OF CROSSCORRELATION INTERFEROMETRY

In this appendix, we compute the amplitude of the crosscorrelated waveforms in equation 12. Using Taylor expansions,we rewrite the equation 12 as

C(z, 0, ω) = |S(ω)|2n

eikze−γ|k|z + e−ikze−γ|k|(4H−z) + eik(2H−z)e−γ|k|(2H−z) + e−ik(2H−z)e−γ|k|(2H+z)o

×

(

∞X

n=0

(R(ω))n e2inkHe−2nγ|k|H

) (

∞X

m=0

(R∗(ω))m

e−2imkHe−2mγ|k|H

)

. (B1)

Here, we only focus on the waves with phases eikz, eik(2H−z), eik(2H+z), and eik(4H−z), and define their amplitudesas C1, C2, C3, and C4, respectively. From equation B1, C1 is obtained by

C1eikz = |S(ω)|2

"

eikze−γ|k|z∞

X

n=0

n

(R(ω))n e2inkHe−2nγ|k|H (R∗(ω))n

e−2inkHe−2nγ|k|Ho

+e−ik(2H−z)e−γ|k|(2H+z)∞

X

n=0

n

(R(ω))n+1 e2i(n+1)kHe−2(n+1)γ|k|H (R∗(ω))n

e−2inkHe−2nγ|k|Ho

#

C1 = |S(ω)|2 e−γ|k|zn


(

∞X

n=0


)

. (B2)

Similar to expression B2, we obtain C2, C3, and C4 as


B

1M2

2345678

R1

R2 a)

Floo

r

Time (s)

b)

Time (s)

−5

0

5

10

15

20

25

30

35

Ele

vatio

n (m

)

B

1M2

2345678

R1

R2

−0.5 0 0.5 1

c)

Floo

r

Time (s)−0.5 0 0.5 1

d)

Time (s)

−5

0

5

10

15

20

25

30

35

Ele

vatio

n (m

)

Figure C1. Waveforms obtained by applying (a,c) crosscorrelation and (b,d) crosscoherence interferometry, in which the virtualsource is at the first floor, to the records of earthquake No. 5 in the EW component. These waveforms are applied bandpass

filters (a,b) 0.4-0.5-45-50 Hz, (c) 1.3-1.6-45-50 Hz, and (d) 1.4-2-45-50 Hz. Note that the bandpass filter for panels (a) and (b)is the same filter as for Figure 7. The amplitude scales in each panel are different. The solid gray lines in panel (c) and (d)are the positive-polarity traveling waves with the velocity estimated from Figure 7a based on equations 12 and 16. The dashedgray line in panel (d) indicates the wave that propagates at one third of the velocity of Figure 7a.

C2 = |S(ω)|2 e−γ|k|(2H−z)n


(

∞X

n=0


)

, (B3)

C3 = |S(ω)|2 R(ω)e−γ|k|(2H+z)n


(

∞X

n=0


)

, (B4)

C4 = |S(ω)|2 R(ω)e−γ|k|(4H−z)n


(

∞X

n=0


)

. (B5)

Appendix C: Applying crosscorrelation andcrosscoherence interferometry to real data

In Figure C1, we apply crosscorrelation and crosscoher-ence interferometry to real data to draw similar figuresas Figure 7a. When we use the same bandpass filter as

for Figure 7, crosscorrelation interferometry enhancesthe fundamental-mode frequency, and only the funda-mental mode is visible (Figure C1a). This is caused bythe power spectrum of the incoming wave (equation 12).When we cut the fundamental-mode frequency, the first-


higher mode is dominant in the crosscorrelated wave-forms (Figure C1c). Therefore, we cannot estimate thevelocity of traveling waves.

Figure C1b illustrates upgoing and downgoingwaves from t = 0 s, and the waves propagate at the samevelocity as the traveling waves reconstructed by de-convolution interferometry (Figure 7a). However, thesepropagating waves may be affected by the wave whichpropagates at slower velocities. The fundamental mode

is not clear in Figure C1b. This is due to the spectral ra-tio used in crosscoherence interferometry, as a result theamplitude in later times is much smaller than the ampli-tude of deconvolution interferometry (Table 2). Becauseof the frequency we used, the wave propagating withslower velocity is not clear (the dashed gray line in Fig-ure C1d). The negative amplitudes around the dashedline might be related to the wave with slower velocity.

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