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A Quantitative Approach for Estimating Coseismic Displacements

in the Near Field from Strong-Motion Accelerographs

by Emmanuel H. Javelaud,* Tatsuo Ohmachi, and Shusaku Inoue

Abstract The translational accelerations recorded by strong-motion seismometersare often contaminated by baseline offsets that prevent recovering the ground displace-ments by double integration. Detailed analysis of the K-NET95 strong-motion seis-mometer, and the improvement of the visual display of the recorded accelerations’Fourier transform by addition of long zero pads to the acceleration records, madeit possible to distinguish the origins of the long-period noise that contaminate thetranslational accelerations recorded in the near field of two studied earthquakes,the 2004 Niigata-ken Chuetsu earthquake and the 2007 Niigata-ken Chuetsu-Okiearthquake, Japan. They are the residual rotation of the instrument (residual tilt)or the 1=f digital semiconductor noise. The quantification of these two terms allowsus to discuss the low-frequency content of the records and to describe when it ispossible to obtain realistic displacement time histories. It happens when the residualtilt is removed in the time-domain or when the 1=f semiconductor noise overshadowsthe translational acceleration records up to sufficiently small frequencies. As a checkof the method, displacement time histories obtained from acceleration records arecompared with nearby 1-Hz GPS data. Good similitudes are obtained in the near fieldfor collocated instruments.

Online Material: Figures showing acceleration, velocity, and displacement timeseries, and acceleration spectra for 18 records.

Introduction

The determination of coseismic displacements fromstrong-motion accelerographs is needed for a number of im-portant purposes including tsunami prediction, estimation ofground strain, design of lifelines, and design of structuresand deep foundations, especially where structures are locatedclose to capable faults (e.g., Park et al., 2004) and the designground motions must consider both permanent and oscil-latory displacements.

Recovering displacement from acceleration records isinteresting because of the large number of accelerometersdeployed. This is, however, a challenging task: displacementtime histories can theoretically be obtained by double inte-gration of accelerograms recorded by strong-motion seis-mometers, but they usually show drift larger than expectedfor true ground displacement (Fig. 1).

The origin of the drift had been investigated, anddetailed studies of the specific components of the strong-motion seismometers had been performed. The baselineoffsets have been attributed to mechanical or electricalhysteresis in the sensor (Iwan, 1985; Shakal et al., 2001),

cross-axis effects due to misalignment of nominally orthog-onal sensors (Trifunac et al., 1973; Todorovska, 1998),analog-to-digital converter error (Boore, 2003), electronic1=f noise (Kinoshita et al., 1997; Kinoshita, 1998; Javelaudet al., 2005), or ground tilt and rotation (Graizer, 1991; Tri-funac and Todorovska, 2001; Boore, 2001; Javelaud et al.,2005, 2010; Kalkan and Graizer, 2007).

Rather, coseismic oscillatory and residual displace-ments are needed. Various processing methods have beendeveloped to remove the baseline offsets: making baselineadjustments such as simple step function, pulse followedby a step function (Iwan, 1985; Boore, 2001; Boore et al.,2002; Wang et al., 2003; Paolucci et al., 2008); parabola orother functions removed from accelerograms; filtering(Trifunac, 1971; Trifunac et al., 1973; Graizer, 1979; Grai-zer et al., 2002; Boore et al., 2002; Boore, 2003; Boore andBommer, 2005); or a combination of both (Trifunac, 1971;Boore et al., 2002; Boore and Bommer, 2005).

In this paper we focused our study on one single strong-motion seismometer, K-NET95. We investigated the perfor-mance of this instrument (theoretically and experimentally)and analyzed the contents of the acceleration time series that*Now at 703 Route de Conques, 11620 Villemoustaussou, France.

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Bulletin of the Seismological Society of America, Vol. 101, No. 3, pp. 1182–1198, June 2011, doi: 10.1785/0120100146

it delivers. Our main object was to answer the followingquestions: first, can we obtain reliable displacement timehistories from acceleration records? If yes, on which circum-stances? Second, can we obtain information about the rota-tion of the seismometer (tilt) during the earthquake, andespecially the remaining rotation of the instrument (residualtilt), from acceleration records?

The choice of the K-NET95 had been motivated by thelarge number of instruments deployed, over 1600 uniformlycovering all Japan, and by the release of its characteristics(full details in Kinoshita et al., 1997; summary in Kinoshita,1998) that enable a detailed study of the instrument.K-NET95 is used in both K-NET and KiK-net seismic net-works (Fujiwara et al., 2004). K-NET (Kyoshin Network)consists of 1000� strong-motion sensors settled on theground surface and KiK-net (Kiban-Kyoshin Network) of675 sets of boreholes and surface strong-motion seism-ometers. K-NET and KiK-net seismometers are basicallythe same. The main difference is that the data released bythe KiK-net network have a sampling frequency of 200 Hz,while the K-NET records have a sampling frequency of100 Hz (Fujiwara et al., 2004). The data used in this studywere recorded during the 2004 Niigata-ken Chuetsuearthquake, Japan (Mw 6.6, 23 October 2004, epicenter at37.29°N, 138.87°E) and the 2007 Niigata-ken Chuetsu-Okiearthquake, Japan (Mw 6.8, 16 July 2007, epicenter at 37.56°N, 136.61°E), which produced large sets of strong-motionrecords, including near-field ground motions. Both wereshallow crustal earthquakes with reverse faulting. The twoJapanese nationwide networks K-NET and KiK-net released,for each mainshock, nearly 900 digital, three components,strong-motion records. General information about these twoearthquakes, which generated crustal movements in the landarea around the source region, can be found in Asano andIwata (2009), Tabuchi et al. (2008), andMiyake et al. (2010).Unless specified otherwise, the strong-motion data comefrom these two networks K-NET and KiK-net. A fewrecords from the Japanese Meteorological Agency have alsobeen used.

In outline, this paper begins with a practical study of theshape and geographical distribution of the baseline offsets

that contaminate the acceleration records. The example ofthe 2004 Niigata-ken Chuetsu earthquake, Japan, is detailed,and the shape of the baseline offsets estimated from thevelocity trends derived from acceleration records. Thesecond part discusses the response of the accelerometer fromtheoretical and experimental studies and the origin of theobserved baseline offsets. The nature of the different sourcesof noise, which collectively contribute to the noise in thefinal digital data, are estimated and modeled. Residualrotation of the instrument (residual tilt) and 1=f digital semi-conductor noise are the two main sources of long-periodnoise that contaminate the acceleration records deliveredby the instrument and distributed on the K-NET and KiK-netnetwork web pages. The third part of the paper describes aproposed processing scheme, based on the preceding obser-vations and analysis. A tool to retrieve the residual tilt andremove it from the acceleration records, and a method toestimate the importance of the 1=f semiconductor noise onthe records are proposed. Both are based on the use of clas-sical signal processing tools, applied consecutively to theimprovement of the Fourier spectra’s visual display by addi-tion of long zero pads. The last section compares the displac-ement time histories obtained using strong-motion records ofthe 2007 Niigata-ken Chuetsu-Oki earthquake with those of1 Hz-GPS, as a check of the proposed method.

Shape and Spatial Distribution of Baseline OffsetsEvaluated from Acceleration Records of the 2004

Niigata-ken Chuetsu Earthquake, Japan

The shapes of baseline offsets that contaminate theacceleration records are investigated from the trends of thevelocity time histories, obtained by simple integration ofthe acceleration recorded by strong-motion seismometers,after removing the pre-event mean.

Figures 2 and 3 show the spatial distribution of velocitytime histories, in the near field of the 2004Niigata-ken Chuet-su earthquake. They reveal three main features. First, thevelocity shows linear trend but not everywhere. Second,the steeper slopes are observed near the epicenter: the varia-tion of the scale used to plot the velocity can be used as a good

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Figure 1. Acceleration and displacement time series derived from a record of the 2007 Niigata-ken Chuetsu-Oki earthquake at K-NETNIG019 station, east–west component.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1183

first indicator. Also, the slope A of the linear trends, deter-mined by using the processing scheme detailed later in thispaper, is given for each record in Figures 2 and 3. It can beobserved that slope A tends to decrease when the epicentraldistance increases (Figs. 2 and 3), which is summarized inFigure 4. Third, far from the epicenter there is no more lineartrend but shapeless trends. Such shapes can also sometimes beobserved near the epicenter of the earthquake and for thevertical components of the acceleration records.

These observations show that there are at least two dif-ferent sources of baseline offsets. In the next section, the per-formance of the K-NET95 strong-motion seismometer’sresponse is investigated, with an aim at finding the originof the observed baseline offsets.

Performance of the K-NET95

Description of K-NET95 and Noise Model

The K-NET95 strong-motion seismometer consists offive building blocks as follows (Kinoshita et al., 1997;Kinoshita, 1998): a seismometer, an amplifier included inthe sensor, an analog antialias filter for the analog-to-digitalconverter, an analog-to-digital converter, and a digital anti-

alias and decimation filter. Brief descriptions of the charac-teristics of each follow.

Seismometer. The sensor (Kinoshita et al., 1997), typeV403BT, is a triaxial force-balance accelerometer with a nat-ural frequency of 450 Hz and a damping factor of 0.707(standard values). Its resolution is better than 0.1 mGal�10�3 cm=s2�. The seismometer’s sensitivity is 3.0014 V=g.

A force-balance accelerometer is basically an inertialseismometer that compensates the unknown inertial forceacting on its suspended mass with a known force (Aki andRichards, 2002; Wielandt, 2002).

The response of the seismometer, in the frequency rangeof interest, is flat to acceleration. Strictly speaking, theresponse of the accelerometer is frequency dependent (Animiand Trifunac, 1983, 1995; Aki andRichards, 2002). However,when using a damping ratio ζ � 0:707�1= ���

2p �, the amplitude

response of V403BT (Fig. 5) is nearly constant up to 150–200Hz.K-NETandKiK-net instruments release data sampledat 100 and 200 Hz, respectively. It is thus possible to retrieveinformation from their records up to 50 and 100 Hz (Nyquistfrequency) in the frequency domain. From zero to those fre-quencies, the amplitude response varies from 1 to 0.9999 and

Figure 2. Spatial distribution of velocity time histories obtained by simple integration of acceleration recorded during the 2004 Niigata-ken Chuetsu earthquake, and estimated slope A of the velocity trend (part 1 over 2: near the epicenter).

1184 E. H. Javelaud, T. Ohmachi, and S. Inoue

0.9988, respectively, and can be assumed to be constant andequal to 1. The phase varies from 0 to �9:04° and �18:3°,respectively, from 0 to 50 and 100 Hz. We do not correctfor the response of the sensor because the amplitude responseis flat to acceleration, and we neglect the phase shift.

The equation of motion (NIED, 2000; Trifunac andTodorovska, 2001; Graizer, 2005; Graizer, 2006a, 2006b)of a horizontal strong-motion transducer can be written asfollows. The right side of the equation is derived from theequation of a pendulum-like transducer.

EW∶ y001 � 2ωoζy01 � ω2oy1 � �x001 � g sinψ2 � ψ00

3r1 � x002θ1(1)

NS∶ y002 � 2ωoζy02 � ω2oy2 � �x002 � g sinψ1 � ψ00

3r2 � x001θ2:(2)

For a vertical seismometer

UD∶ y003 � 2ωoζy03 � ω2oy3 � �x003 � g�1 � cosψ1�

� ψ001r3 � x002θ3; (3)

where x00i is ground acceleration in ith direction, yi therecorded response of the seismometer, ωo � 450 Hz, andζ � 0:707, the natural frequency and the fraction of criticaldamping of the transducers.

Figure 3. Spatial distribution of velocity time histories obtained by simple integration of acceleration recorded during the 2004 Niigata-ken Chuetsu earthquake, and estimated slope A of the velocity trend (part 2 over 2: the seismometers are more distant from the epicenter).

Figure 4. Slope of the velocity time histories (A in Gal) versusepicentral distance, 2004 Niigata-ken Chuetsu earthquake.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1185

The expressions g sinψi and g�1 � cosψ1� are, respec-tively, the horizontal and vertical contribution to the responsefrom tilt, where g is the gravitational acceleration and ψi arotation of the ground surface about xi axis. When the groundis tilted by an amount of ψ, the response of the system isexactly identical with a horizontal acceleration of magnitudeg sinψ or a vertical acceleration of magnitude g�1 � cosψ�.The difference of sensitivity to tilt between the horizontaland vertical seismometers is sometimes used (Kalkan andGraizer, 2007; Graizer and Kalkan, 2008) to identify the cor-ner frequency at which the long-period contamination due totilt occurs in the horizontal components.

The expression x00i θj is the contribution to the responsefrom cross-axis sensitivity. Cross-axis sensitivity is thesensitivity of a transducer to motion perpendicular to its prin-cipal axis. For K-NET95 we are not aware of any measure-ments of cross-axis sensitivity. However, Kinoshita et al.(1997) measured the overall cross talk of the instrumentby using a shaking table. These measured values of cross talkprovide a global quantification (mechanical and electrical) ofsignal leakage from one channel to another channel. At fre-quencies ranging from 1 to 10 Hz, the cross talk is less than1%. It is smaller at lower frequencies. We do not correct forthe cross-axis sensitivity, because it is much smaller than thetranslational acceleration, especially at low frequencies.

The expression ψ00krj is the inertial force term due to

angular rotation (torsion of the instrument housing), whereψ00k is the angle of pendulum rotation, and rj the length of

the pendulum arm. For strong-motion seismometers, rj isusually small and equal to zero in some cases. In the descrip-tion of the K-NET95 (Kinoshita, 1998), there is no mentionthat the acceleration records are dependent on this term.However, it is possible to estimate an upper bound of theangular rotation’s effect from the work of Lee and Trifunac(2009), considering that the length of the pendulum arm isidentical or smaller than the one of older instruments. Forexample, r ≈0; 008 m for the SMA-1 strong-motion seism-ometer (Trifunac and Todorovska, 2001). The estimation ofthe inertial force due to the angular rotation term shows that itis always much smaller than the translational acceleration

at low frequencies. It may, however, not be neglected forapplications requiring detailed information about the veryhigh frequency content of the acceleration records.

Analog Antialias Filter. This analog high-cut filter consists(Kinoshita, 1998) of a two-stage resistor-capacity filter withtime constants of 1=12 600 s and 1=62 600 s. We simulate thefilter’s response. It is flat to 1000 Hz, and the phase shifts are0 to 100 Hz: from 0 to 50 Hz (K-NET) and 100 Hz (KiK-net), the amplitude response varies from 1 to 0.9997 and0.9987, respectively, and the phase varies from 0 to �1:72°and �3:43°, respectively. Therefore, both the amplitude andphase can be assumed to be constant.

Analog-to-Digital Converter. The analog-to-digital converter (ADC) consists (Kinoshita et al., 1997; Kinoshita,1998) of a 1-bit sigma-delta modulator (Mitra and Kaiser,1993) and a digital decimation filter. It can be approximatedto a 24-bit type converter.

The effect of the ADC noise on the final displacement isestimated. Basically the ADC converts the output of theamplifier (3.0014 V=g) into integers. The maximum measur-able acceleration of K-NET95 is 2000 Gal. This means that ateach sampling time, for each analog input ranging from�6 V to �6 V corresponds an integer value among the224 � 16777216 possible ones. The smallest input voltagechange that causes the output value of the ADC to increaseor decrease by one unit, known as the least significant bitvalue or Q, is (Scherbaum, 2001)

Q � Full Scale Voltage2n

; (4)

for an n-bit ADC.Assuming that the conversion induces a uniformly dis-

tributed white noise (Scherbaum, 2001), the standard devia-tion of the noise is

σa � Q=������12

p: (5)

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Figure 5. Variation of the amplitude response with damping and frequency. V403BT response is the simulation where ζ � 0:707.

1186 E. H. Javelaud, T. Ohmachi, and S. Inoue

Double integration of this white noise is a random variablewith zero mean and nonzero standard deviation. The standarddeviation of the final displacement (σdend ; Boore, 2003) is

σdend ��T3Δt

3

�12

σa; (6)

where T is the duration of the time series and Δt is thesampling interval.

K-NET95 has aQ value of 7:15 × 10�7 V=count, whichis 2:34 × 10�4 Gal=count. With this value, equations (5) and(6) give σdend of 0.020 cm and 0.014 cm for the K-NETand KiK-net instruments, respectively, after integration of300 s. Thus, this model of the analog-to-digital convertershows that the ADC noise can be neglected.

Digital Antialias Filter. The digital high-cut filter is athree-pole Butterworth filter with a corner frequency of30 Hz. This filter was designed by applying the bilineartransform to an analog Butterworth filter (Kinoshita, 1998).We used the same procedure to simulate the filter’s response,whose amplitude is shown in Figure 6.

The effect of the digital antialias filter on the displace-ment is investigated. A simple model (Fig. 7) that simulatesthe amplitude response of a high-cut filter (Fig. 7a) is used. Itseparates its amplitude response into an all-pass filter(Fig. 7b) and a boxlike function (Fig. 7c). The boxlike func-tion has an amplitude of 0 up to 20 Hz, then it linearlyincreases up to 80 Hz (KiK-net) where it reaches a valueof 1. Above 80 Hz, the amplitude is 1. This function is sub-tracted from the all-pass filter.

The effect of the boxlike function on the velocity iscalculated. Let B�f� be the boxlike function in the frequencydomain and b�t� its inverse Fourier transform in the timedomain. Thus,

B�f� �Z �∞�∞

b�t�e�j2πftdt; (7)

when f � 0, B�0� � R�∞�∞ b�t�dt, where

R�∞�∞ b�t�dt≈R tend

0 b�t�dt is the residual velocity obtained at the end of therecord tend by single integration of b�t�. Moreover, B�0� � 0

by definition, as can be seen in Figure 7. Thus, B�0� �R tend0 b�t�dt � 0, and adding or subtracting such a boxlikefunction has no effect on the residual velocity and thenon the residual displacement. This simple model showsthat the presence of the digital antialias filter has no effectwhen double integrating the acceleration to obtain thedisplacement.

Other Sources of Noise. As all seismographs using semi-conductor circuits, the K-NET95 output acceleration data arecontaminated at low frequencies by 1=f noise (Kinoshitaet al., 1997; Kinoshita, 1998).

What Are We Really Recording?

According to the previous explanation, the K-NET95acceleration time series delivered on the network web pagecontains, for each component, three kinds of information: thetranslational acceleration itself x00, 1=f noise at low frequen-cies, and information about tilt.

In this section, we first investigate the presence of 1=fnoise in the record and describe how to determine its level foreach accelerometer. We then experimentally confirm the tilteffect on the acceleration record and discuss it in terms oftransient and residual tilt.

Characterization of the 1=f Noise at Low Frequency. Thepresence of 1=f semiconductor noise within modern strong-motion seismometers is described by Kinoshita et al. (1997)and Kinoshita (1998). After checking its presence in theKinemetrics K2 Altus series, an instrument similar in thatrespect to the K-NET95 and available for test in the labora-tory, we then describe a practical method to determine the1=f noise level of each accelerometer used in the K-NETand KiK-net networks.

Figure 6. Simulation of the amplitude response of the digitalantialias filter.

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Figure 7. Simple model used to investigate the influence of thedigital antialias filter on displacement.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1187

Experimental Response of Kinemetrics Inc. K2 Acceler-ometer. The performance of the K2 strong-motion seis-mometer is evaluated by comparing its theoretical responseto the observed one.

Kinemetrics Inc. (see Data and Resources) determinedan empirical model of its K2 sensor: two pairs of conjugatepoles were found to represent well the instrument’s transferfunction. The amplitude of the instrument’s overall responseis simulated according to the Kinemetrics’ model and shownin Figure 8.

Each accelerograph’s response can be tested by usingambient vibration noise records. Figure 9a shows a 240-sacceleration time history recorded in the laboratory. Figure 9b

is its Fourier acceleration spectrum. The comparison of thisspectrum with the Fourier amplitude spectrum of the micro-tremors recorded simultaneously by a sensitive velocimetershows that the K2 instrument recorded the natural responseof the building from 1 to 20 Hz. Above 20 Hz, the noiserapidly decreases in amplitude: the ground acceleration(input of the instrument) is high-cut filtered by the instru-ment’s response (Fig. 8). At low frequencies, up to 0.5 Hz,1=f digital semiconductor noise predominates (slope �1 in alog–log plot).

The real amplitude response of the accelerometer istherefore the sum of its theoretical response and of 1=f noiseat low frequency (Fig. 10).

From our experience, the 1=f noise level varies accord-ing to each sensor and with time. It must therefore bedetermined for each accelerometer’s component beforeprocessing the records.

Practical Determination of the 1=f Noise. The re-sponse of each accelerometer and the 1=f noise level canbe evaluated by using ambient vibration noise records. Theserecords arevery similar to the pre-event part of themainshocks(same amplitude), but of longer duration: we usually use 100 slong records. Such records can be found among the large num-ber of aftershocks recorded by the K-NET and the KiK-netnetworks. Figure 11 shows the acceleration power spectrumof the ambient vibration noise recorded at the bottom of anobservation borehole at the KiK-net NIGH04 station. Thespectrum shows the characteristics of electrical instruments

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Figure 9. (a) Kinemetrics Inc. K2 accelerometer ambient vibration noise record and (b) its Fourier transform (below).

1188 E. H. Javelaud, T. Ohmachi, and S. Inoue

(Smith, 1999). At low frequencies, up to 0.1 Hz, digital 1=fsemiconductor (Kinoshita, 1998) noise predominates.Between 0.1 and 20 Hz, the signal consists of white noise.Above 20Hz, the noise rapidly decreases in amplitude (digitalantialias filter). At very high frequencies, the analog/digitalconverter noise predominates. It is therefore possible toprecisely estimate the 1=f noise level.

Effect of Tilting on the Seismometer’s Output. The re-sponse of the seismometer to tilt is first experimentally con-firmed. Then the effect of tilt (transient and residual) on thedisplacement time histories derived from acceleration isevaluated.

Experimental Effect of Tilting on the SeismometerOutput. Theoretically, when the ground is tilted by anamount of ψ, the system’s response is exactly identical witha horizontal acceleration of magnitude g sinψ or a verticalacceleration of magnitude g�1 � cosψ�.

We experimentally rotated the Altus K2 strong-motionseismometer around its X axis. The angle of rotation was1.10°. Figure 12 shows the acceleration recorded beforeand after the tilting of the instrument, the latter lasting fromthe sixteenth to the twenty-second second of the record.According to equation (2), the acceleration offset causedby the rotation and recorded by the Y axis should be980 sin��1:10°� � �18:81 Gal, which is identical, at theexperimental rotation error, to what can be observed in

Figure 12, Y component. Inversely, an angular rotation of�1:10° would give an output of �18:81 Gal.

Therefore, any rotation of the strong-motion instrumentis recorded simultaneously to the translational groundacceleration. The acceleration time series delivered by the in-strument includes the record of the instrument’s rotation dur-ing the transient part of the shaking (transient tilt), but it alsoincludes any residual rotation (residual tilt) that will be re-corded as a constant value lasting until the end of the record.

Effect of Transient Tilting on the AccelerationRecord. Published comparisons of oscillatory displace-ments from 1-Hz GPS instruments with those derived fromnearby K-NET and KiK-net accelerograms, after accountingfor a baseline offset using the pre-event signal, show verygood similitudes (figure 2 in Miyazaki et al., 2004; chapter 4in Clinton, 2004). The results shown later in this paper (espe-cially at the Ojiya stations where the instruments are only600 m apart) confirm the observations. We therefore assumethat the effect of the transient ground tilt on the displacementcan be neglected.

Effect of Residual Tilting on the Acceleration Re-cord. Any residual tilt is recorded by strong-motion seism-ometers as a constant value lasting until the end of the record.To get insights on the effect of this addition of low amplitudebut long duration information, it is convenient to model theresidual tilt. The simplest method to simulate the residual tiltis to consider a step function of amplitude A and duration T,ending at the end of the record. When processing the data,finite length records are used. The step function becomes aboxlike function lasting until the end of the record. The Four-ier transform of that boxlike function has an amplitude of

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Figure 10. The observed response of the accelerometer is thesum of its theoretical response and of 1/f low-frequency semicon-ductor noise.

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Figure 11. Power spectrum calculated from a noise record atSekikawa NIGH04 station, underground seismometer, recordedsoon after the 2004 Niigata-ken Chuetsu earthquake, Japan. Thepower spectrum representation allows comparison with Kinoshita’soriginal papers.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1189

F�f� � AT sin�πfT�πfT . It shows three main features. First, when

the frequency f is equal to or is a multiple of 1T, F�f� � 0.

Then, F�f� → AT as f → 0. Furthermore, the envelope ofthe signal has a slope of 1=f.

Summary. The analysis of the K-NET95 performanceshows that its output contains four different elements ofinformation. They are the translational acceleration and thetilt of the ground, both of them being modified during therecording process by the response of the seismometer (the-oretical response, and the addition of electronic 1=f noise atlow frequencies). Therefore, the content of the accelerationtime series must be discussed in terms of the four differentelements’ relative importance.

The low-frequency content of the acceleration record isalways dominated by either 1=f semiconductor noise or byresidual tilt. Two main cases can be distinguished accordingto the relative position, in the frequency domain, of the trans-lational acceleration, 1=f semiconductor noise and resi-dual tilt.

The first case happens when 1=f semiconductor noisedominates the low-frequency content of the record, oversha-dowing both the translational acceleration and any residualtilt. An example of this situation is shown in Figure 13. Inthis case, it is usually not possible to recover the displace-ment time histories, unless the frequency at which the 1=fnoise stops to dominate the record is so small that the 1=fnoise would not affect the displacement time histories fordurations of interest.

The second situation happens when the residual tiltdominates the low-frequency content of the record, oversha-dowing both the translational acceleration and the 1=f semi-conductor noise.

Based on this analysis of the K-NET95’s performance,the next section proposes a processing scheme for both situa-

tions. It first starts with the selection of the most suitablesignal processing tools to carry out the Fourier transformof the record and to improve the visual display of the Fourierspectra at low frequencies.

Processing Scheme

Zero Order Correction

We apply the usual condition that the acceleration is zerobefore the earthquake shaking starts: the pre-event mean ofthe acceleration record is subtracted from the whole record.

Fourier Transform and Zero Padding

The long-period (low frequency) information containedin the acceleration records is investigated. So far, we studied

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Figure 12. Experimental rotation of a K2 Altus strong-motion seismometer around its X axis, and effect on the seismometer output.

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Figure 13. Power spectrum calculated from a record atSekikawa NIGH04 site, underground seismometer, during the 2004Niigata-ken Chuetsu earthquake. The power spectrum representa-tion allows comparison with Kinoshita’s original papers.

1190 E. H. Javelaud, T. Ohmachi, and S. Inoue

the response of the instrument. We now consider the mod-ifications that the change from continuous-time signal (inputof the instrument) to discrete-time signal (output of the in-strument) is implying on the Fourier transform and discusshow to best apply the Fourier transform on the records.

During the recording process, the infinite continuous-time signal, input of the instrument, is transformed to a dis-crete-time signal. Finite length records of seismic events arestored. In terms of the Fourier spectra, this corresponds to atransformation from a continuous-time Fourier transform to adiscrete-time Fourier transform (DTFT). The best Fouriertransform that one can expect to obtain from the data releasedis the DTFT.

However, computers perform the discrete Fourier trans-form (DFT) of finite length records, such as the accelerationrecords distributed by the networks. The DFT computes theFourier amplitudes only at a finite number of frequencies, fk(Oppenheim and Schafer, 2010). It implies a loss of informa-tion. Indeed, if the sampling spacing between the frequenciesfk is not small enough, important information of the DTFTspectrum located between the frequencies fk does not appearin the computed Fourier spectrum. It is therefore necessary toreduce the interval between the frequencies fk as much aspossible, when performing the DFT, to calculate the Fourieramplitudes of the DTFT at frequencies sufficiently dense sothe computed Fourier spectra is indistinguishable from theDTFT itself.

In signal processing, the frequency spacing of the dis-crete Fourier transform is Δf � 1

N:Δt, where Δt is the sam-pling interval and N is the number of data. WhenN increases,Δf decreases; it is therefore possible to compute the record’sFourier amplitudes at closer frequencies by addition of zeropads before taking the DFT (Oppenheim and Schafer, 2010).The addition of a high degree of time-domain zero is a usualstep in classical signal processing. It does not add informa-tion nor improve the ability to resolve close frequencies(which depends, for example, on the sampling frequency).Adding zeroes before and after the acceleration time seriesresults in a smaller frequency spacing of the computedFourier transform, allows the user to recover the record’sDTFT, and simply allows him to make the most use of theavailable data.

Acceleration time histories are first padded with zeroes.Figure 14 shows the Fourier transform of the accelerationrecorded at station NIG019 (east–west component, 2007Niigata-ken Chuetsu-Oki earthquake) after adding a variablenumber of zeroes. It can be seen there that the Fourieramplitude spectrum without zero pads is very jagged atlow frequencies due to the small number of sampling defin-ing the curve there. This situation is improved by increasingthe number of zero pads in the acceleration record beforetaking the discrete Fourier transform. Adding zeroes to makethe record 223 samples long results in a smaller frequencyspacing of the Fourier transform. It provides more detailsat low frequency for the purpose of this method, withouttaking much computing time (from 1 s to 2 s with current

computers). This tool reveals the characteristics of the long-period information contained in acceleration record.

Case 1: At Low Frequencies, the 1=f SemiconductorNoise Overshadows the Translational Accelerationand the Residual Tilt

During the 2004 Niigata-ken Chuetsu earthquake,NIGH04 station was 111 km from the epicenter of the main-shock. The acceleration power spectrum of the mainshockrecorded at station NIGH04 is shown in Figure 13, aswell as the 1=f noise model determined from the noiserecord (Fig. 11).

At low frequencies, the 1=f digital noise predominates.Let Aacc�f� be the sum in the frequency domain of Ag�f� andAnoise�f�, where Ag�f� is the true ground translational accel-eration and Anoise�f� is the sum of the digital 1=f and whitenoises. The same relation holds for the velocity V�f�. Thevelocity V�f� � Vg�f� � Vnoise�f�, where Vg�f� is the trueground velocity and Vnoise�f� is the sum of a digital 1=f2 and1=f noise. By definition of the Fourier transform, for a 300 srecord,

v�t�⇔V�f� �Z �∞�∞

v�t�e�j2πftdt

�Z

T�300 sec

0

v�t�e�j2πftdt; (8)

when f � 0,

V�0� �Z

T�300 s

0

v�t�e�j2π0tdt �Z

T�300 s

0

x0�t�dt

� �x�t��T�300 s0 � displacement�t � 300 s�: (9)

Thus, the residual displacement is equal to the Fourieramplitude of the velocity at zero frequency. BecauseV�0� � Vg�0� � Vnoise�0�, where Vg�f� ≪ Vnoise�f�, instru-mental noise completely overshadows the Fourier amplitudeof the ground acceleration at low frequencies. It is thereforenot possible to recover the exact residual ground displace-ment in such cases.

However, if the frequency at which the 1=f noise stopsto dominate the record is small (i.e., Vg�f� ≪ Vnoise�f� istrue only at very low frequencies), it is possible to obtainstable displacement time histories over the duration ofengineering interest. As an example shown in Figure 15, thedouble integration of the acceleration recorded by the under-ground KiK-net NIGH12 station during the 2004 Niigata-ken Chuetsu earthquake leads to a stable residual displace-ment of 14 cm. The frequency at which the 1=f noise stops todominate the record (0.004 Hz determined from the Fourieracceleration spectra) is so small that the 1=f noise doesnot affect much the displacement time histories.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1191

Case 2: At Low Frequencies, the Residual TiltOvershadows the Translational Acceleration andthe 1=f Semiconductor Noise: Example of theNIG019 Station, East–West Component

After adding zero pads, we find that in this case thelow-frequency content of the acceleration time history(Figs. 14, 16) is similar to the Fourier spectra of a step func-tion (Fig. 16a,b).

The amplitude A and duration T used to define a stepfunction in the time domain (Fig. 16a) can be extracted inthe frequency domain from the Fourier transform of the stepfunction (Fig. 16b): by definition, the Fourier amplitude atzero frequency is A:T, and the frequency when the Fourieramplitude is zero for the first time is 1=T. Therefore, Aand T of a step function can be uniquely derived from itsFourier transform. It is important to note that A and T arenot handpicked at random, but really extracted from theFourier transform.

In Figure 16, we compare the acceleration record and itsFourier transform (Fig. 16c,d), with a simple step function andits Fourier transform (Fig. 16a,b). It can be seen that in thefrequency domain, the low-frequency part of the record(Fig. 16d, in the box) is identical to a step function (Fig. 16b).We therefore observe that the acceleration record (Fig. 16c) iscontaminated by a step function, whose amplitude and start-ing time can be determined uniquely from the Fourier spectra

of the acceleration record. In this example, we obtain the fol-lowing values from the Fourier transform of the accelerationrecord (Fig. 16d):A:T � 11:7177 Gal s (Fourier amplitude atzero frequency) and 1=T � 0:0036467 1=s (frequency whenthe Fourier amplitude is zero for the first time). Thus,T � 1=�1=T� � 274:2186 s, A � A:T=T � 0:042731 Galand the starting time of the step function �ts� �duration of the record �T � 300 s �T � 25:7814 s. InFigure 17, the superposition in the frequency domain ofthe acceleration record and the step function of characteristicsA and t[s] shows extremely good match at low frequencies.

Subsequent processing is applied in the time domain. Astep function of amplitude A � 0:042731 Gal and of startingtime ts � 25:7814 s is subtracted from the accelerationrecord (Fig. 17a,b). Note that we corrected the accelerationtime series from a step function starting at t[s] (until the endof the record). While zero pads are added after the accelera-tion record, it is strictly speaking a rectangular box ofamplitude A, starting at ts, until the end of the record (300 sin K-NET case).

Double integration of the corrected acceleration timehistory shows no more drift: the displacement is perfectlystable after the shaking (Fig. 17c,d). The residual displace-ment is estimated at �3:2 cm.

This proposed processing scheme is applied in thenext section to acceleration records of the 2007 Niigata-kenChuetsu-Oki earthquake, Japan. The displacement time his-

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Figure 14. Effect of padding the acceleration history recorded at NIG019 station, east–west component, during the 2007 Niigata-kenChuetsu-Oki earthquake.

1192 E. H. Javelaud, T. Ohmachi, and S. Inoue

tories obtained from acceleration records are compared with1-Hz GPS data, when collocated, as a check of the method.

Example of the 2007 Niigata-kenChuetsu-Oki Earthquake

The K-NET and KiK-net acceleration time historiesrecorded in the near field during the 2007 Niigata-ken Chuet-su-Oki earthquake were processed according to the scheme

described in the preceding section. The records of twoJapanese Meteorological Agency (JMA) instruments are alsoadded by extension of the method.

In the present section, we first detail the processing stepsapplied to accelerograms recorded near the epicenter of theearthquake (map in Fig. 18), then compare the displacementtime histories obtained from acceleration records with 1-HzGPS data, when collocated. Finally, we draw a map of theresidual displacements obtained by the two methods.

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Figure 15. Acceleration, Fourier amplitude of the acceleration, acceleration derived velocity and displacement time histories of the 2004Niigata-ken Chuetsu earthquake, NIGH12 downhole accelerometer, north–south component.

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Figure 16. Comparison of (a), (b) step function and (c), (d) acceleration record in both time and Fourier domains. Example from theNIG019 station, east–west component.

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1193

Summary of Processing Steps Applied

The accelerograms recorded at the stations shown inFigure 18 have been processed. Each accelerogram, its Four-ier transform, velocity, and displacement time histories aswellas the processing steps applied are given inⒺFigures S1–S18in the electronic supplement to this paper. The processingdetails of the acceleration times series are summarized inTables 1 and 2.

In Tables 1 and 2, “no correction applied” does not meanthat the residual tilt is equal to zero, but that it is so small thatother sources of noise, especially the 1=f semiconductornoise, predominates at low frequencies.

Stable displacement time histories are obtained in pro-cessing case 1 (4 records) and case 2 (8 records). However,for intermediate situations (5 records), stable residual dis-placements are usually not obtained, and so an estimationof the displacement is given.

Note that at two stations, the processing scheme couldnot be applied. A retaining wall collapse had been reported atstation NIG018 by NIED (2007), altering the accelerationrecord. Also, one record, JMA Kashiwasaki, north–southcomponent, shows more than one slope in the velocity timehistory. An additional pulse can be observed in the accelera-tion time series at about 40 s.

Comparison of Displacements Obtained fromAcceleration Records with 1-Hz GPS Data

The processing scheme’s efficiency is tested: the dis-placement time series obtained by processing strong-motion

records are compared with nearby 1-Hz GPS data as a checkof the method.

During the earthquake, the K-NET NIG019 OJIYAstation and the 1-Hz GPS OJIYA station recorded the event.Both stations are 600 m distance apart. Displacement timehistories calculated from processed acceleration records werecompared with the 1-Hz GPS data (Fig. 19). For the east–west and the north–south components, the seismic displace-ment and the 1-Hz GPS displacement time histories showextremely good similitude for both the residual and theoscillatory parts of the displacements.

In Kashiwasaki, two strong-motion accelerograms,NIG018 and JMA Kashiwasaki, as well as two 1-Hz GPSstations, K1 and K2, recorded the event. The area is in thevery near field of the earthquake and was subjected to verylarge displacements. It is known for having experienced soilrelated problems including liquefaction: at GPS station K2,tilt of the antenna pillar (Tabuchi et al., 2008) had beenreported; also, at NIG018 station, collapse of a retaining wallaround the station was reported (NIED, 2007). The data ofthese two instruments were removed from the set of recordsused. The records remaining at hand, that is, the displace-ment time histories obtained from the JMA accelerometerand the K1 1-Hz GPS station (situated 3.5 km apart), werecompared. For the east–west component, we found a goodagreement for the residual displacement at about 14 cm.From the JMA seismometer, the amplitude of the oscillatorydisplacement reached almost 80 cm. There is a good agree-ment between the acceleration-derived displacement and the1-Hz GPS displacement. The differences can be due to thequickly varying field of displacement in the area and to

Figure 17. (a) Superposition of the acceleration record and the step function of amplitude A (not on scale) and starting time t[s] used tocorrect the acceleration record. (b) The superposition in the frequency domain of the acceleration record and a step function of amplitude Aand starting time t[s], determined in (a), shows extremely good match at low frequency. (c), (d) Uncorrected and corrected velocity anddisplacement time histories. The corrected displacement history is obtained by subtracting the step function from the acceleration record, thendouble integrating the corrected acceleration time history. The displacement is stable after the earthquake.

1194 E. H. Javelaud, T. Ohmachi, and S. Inoue

possible misorientation of the instruments during their instal-lation. Regarding the north–south component of the accelera-tion record, a pulse at 40 s made the processing schemeimpossible to apply.

At Takayanagi, the JMA accelerometer and the 1-Hz GPSstation are 1.2 km apart. They show similar residual dis-placement and good similitude during the oscillatory partof the displacement, considering the distance between thestations.

For the other two stations, the displacements obtained bythe two methods show very good similitude considering thatthe distance between the two sensors is several kilometers.

Comparison of Coseismic Residual Displacement

In the near field of the 2007 Niigata-ken Chuetsu-Okiearthquake, residual displacements calculated as suggestedpreviously were compared with displacements measured at

Figure 18. Comparison of coseismic residual displacement obtained from strong-motion seismometers and surrounding GPS stationsduring the 2007 Niigata-ken Chuetsu-Oki earthquake. Two JMA records were also processed by extension of the method.

Table 1East–West Component

A (Gal) Residual Displacement (cm) Processing Type

NIG017 0.0048 ≈� 2:0 IntermediateNIG018 – Problem of wall stability –NIG019 0.04273 �3:2 Case 2NIG020 No correction applied �2 Case 1NIG021 0.00963 �2:3 Case 2NIG023 No correction applied �0:2 Case 1NIG028 No correction applied �2:1 Case 1NIGH12 0.0048 ≈� 1:5 IntermediateJMA Takayanagi 0.2366 �1:8 Case 2JMA Kashiwasaki �1:0365 �13:8 Case 2

Approach for Estimating Coseismic Displacements in the Near Field from Strong-Motion Accelerographs 1195

surrounding GPS stations. When GPS and seismometer sta-tions are collocated (a few hundred meters apart), the residualdisplacements obtained by the two methods are almostidentical. At other locations where accelerometers and GPSstations are a few kilometers distance apart, residual dis-placements obtained by processing seismometer recordsare consistent with displacements measured by GPS stations.

Discussion and Conclusions

This paper proposes a simple method based on theaddition of long zero pads to investigate the low-frequencycontent of acceleration time histories recorded by the strong-motion seismometer K-NET95 and delivered on the Japa-nese K-NET and KiK-net networks’ web pages. It points

out that the translational acceleration is contaminated atlow frequency by 1=f electronic noise and by the residualtilt. The shapes of the baselines offsets that contaminate theacceleration records (observed, for example, in the sectionShape and Spatial Distribution of Baseline Offsets Evaluatedfrom Acceleration Records of the 2004 Niigata-ken ChuetsuEarthquake, Japan for a typical earthquake) can be related tothe predominance of residual tilt (linear trend in the velocitytime history) or 1=f noise (shapeless trend in the velocitytime history) at low frequencies.

We find that in some circumstances, it is possible toobtain reliable estimates of the displacement time historiesfrom acceleration records. This happens when the low-frequency content of the acceleration record is dominated by1=f electronic noise up to very small frequencies (case 1 in

Table 2North–South Component

A (Gal) Residual Displacement (cm) Processing Type

NIG017 0.0092 2.1 Case 2NIG018 – Problem of wall stability –NIG019 �0:0472 2.3 Case 2NIG020 �0:0023 1.8 Case 2NIG021 �0:0095 ≈3:0 IntermediateNIG023 0.00068 0.6 IntermediateNIG028 No correction applied 0.8 IntermediateNIGH12 No correction applied 1.2 Case 1JMA Takayanagi �0:1449 6.8 Case 2JMA Kashiwasaki 2.7449 See text Double slope

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OJIYA (600m)GPS n°950240 & K−NET NIG019

MASHIMA (8 km)GPS n°970807 & K−NET NIG017

TAKAYANAGI (1.2km)GPS n°020962 & JMA n°65056

MATSUNOYAMA (10km)GPS n°950244 & K−NET NIG023

KASHIWASAKI (3.5 km)GPS n°940051 & JMA n°65025

Accelerometer, uncorrected

GPSAccelerometer, corrected

Figure 19. Accelerometer displacement time series vs. nearby 1-Hz GPS data recorded during the 2007 Niigata-ken Chuetsu-Oki earth-quake, Japan.

1196 E. H. Javelaud, T. Ohmachi, and S. Inoue

the Processing Scheme section) or by constant residual tilt(case 2 in the Processing Scheme section).

Other situations, including large 1=f noise contaminat-ing the translation acceleration, make it impossible to retrieveaccurately the displacement time histories (processingscheme, intermediate situation in Example of the 2007Niigata-ken Chuetsu-Oki Earthquake). This study thereforeshows the importance of the deployment of instruments withas low as possible 1=f electronic noise. The recent replace-ment of K-NET95 by K-NET02 seismometers, instrumentswith lower electronic noise, is providing great improvementin the determination of displacement time histories.

Another limitation of the method appears when acceler-ograms contain motions from more than one event or fromlate pulse such as the JMA Kashiwasaki, north–south com-ponent record, as this may create multiple baseline offsets.Also, the method requires a minimum original data’s length.K-NET and KiK-net usually release 300 s long signals re-corded at a sampling frequency of 100 Hz or 200 Hz, whichgive good results when processed according to the proposedmethod.

Regarding the tilt, we observed that the transient tiltusually has no significant effect on the derivation of the dis-placement time histories, whereas any residual tilt of lowamplitude but long duration seriously alters the velocitiesand displacements. We propose a systematic method toestimate the residual tilt (case 2 in the Processing Schemesection) from acceleration records.

Data and Resources

Seismograms were provided by the National ResearchInstitute for Earth Science and Disaster Prevention (Tsukuba,Japan) available through the K-NET at www.k‑net.bosai.go.jp (last accessed March 2011) and KiK-net at www.kik.bosai.go.jp (last accessed March 2011) networks, and bythe Japanese Meteorological Agency, while GPS measure-ments were provided by the Geographical Survey Institute,Ministry of Land, Infrastructure and Transport, Ibaraki,Japan. The K2 user’s manual (K2 and Makalu User’sManual, Document 302200) is available from KinemetricsInc. at www.kinemetrics.com (last accessed May 2010).

Acknowledgments

The first author wishes to acknowledge the Japanese government forfunding his research term through the Monbukagakusho scholarshipprogram. He also thanks Y. Murakami for his contribution to this workand L. Tabiso for his encouragement. The authors are grateful to ArthurMcGarr and two anonymous reviewers for their valuable comments andsuggestions.

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Tokyo Institute of TechnologyDepartment of Built Environment4259, Nagatsuta, Midori-kuYokohama 226–8502, Japan

(E.H.J.)

Tokyo Institute of TechnologyDepartment of Built Environment4259, Nagatsuta, Midori-kuYokohama 226-8502, Japan

(T.O., S.I.)

Manuscript received 28 May 2010

1198 E. H. Javelaud, T. Ohmachi, and S. Inoue