application of the empirical mode decomposition–hilbert …€¦ · bulletin of the seismological...

1339

Bulletin of the Seismological Society of America, 91, 5, pp. 1339–1357, October 2001

Application of the Empirical Mode Decomposition–Hilbert Spectrum

Method to Identify Near-Fault Ground-Motion Characteristics

and Structural Responses

by Chin-Hsiung Loh, Tsu-Chiu Wu, and Norden E. Huang

Abstract In this article, the empirical mode decomposition method combined withthe Hilbert spectrum method (EMD � HHT) is used to analyze the free-field groundmotion and to estimate the global structural property of building and bridge structurethrough the measurement of seismic response data. The EMD � HHT method pro-vides a powerful tool for signal processing to identify nonlinear and nonstationarydata. Based on the decomposed ground-motion signal, the absolute input energy ofeach decomposed wave was studied (the fling step [pulselike wave] can be separatedfrom the recorded near-fault ground motion). Through application of the EMD �HHT method to building and bridge seismic response data, the time-varying systemnatural frequency and damping ratio can also be estimated. Damage identificationfrom seismic response data of buildings and bridges, particularly from the Chi-Chiearthquake data, is also described.

Introduction

For the purpose of damage assessment of structures dur-ing strong ground excitation, it is desirable to identify theseverity of damage based on the input–output measurement.It is believed that structural identification from input–outputdata can lead to an understanding of the deterioration mech-anism. A main disadvantage of structural identification hasbeen the lack of experimental information on an actual struc-ture. Both ambient vibration tests and forced vibration testscan provide information for structural system identificationonly from a low level of excitation. However, strong-motioninstrumentation on structures to collect earthquake responsecan also provide valuable information for the safety assess-ment of the structure.

Much work has been done on structural identificationof linear systems using both input and output data revealinga number of efficient algorithms. The recursive least-squarestime-domain identification of structural parameters providesan on-line global identification method of a linear system(Caravani et al., 1977). Identification of time-varying modelparameters is also studied through the adaptive model iden-tification method (Ljung and Soderstrom, 1983; Goodwinand Sin, 1984; Loh et al., 1996). The spectral elementmethod is also applied to identify the dynamic characteristicsof substructure from the frequency domain (Loh and Lee,1997). Damage identification based on a nonlinear dynamicmodels has not yet been widely reported. The most commonmethod to identify the nonlinear system with the prior in-formation on a nonlinear model is the application of Kalman

filter technique (Hoshiya and Sato, 1984). Modal parameteridentification methods without input information have alsobeen discussed. These methods include the two-stage least-squares method, the parametric time-frequency identifica-tion method, and the changing spectrum method. Throughthis study, the development of a nonstationary and nonlinear(earthquake) signal-processing method was discussed.

The purpose of this article is to use different identifi-cation methods to identify the fling step from near-faultground motion and to estimate a time-varying dynamic char-acteristic of building and bridge structure from seismic re-sponse data and experimental data. The empirical mode de-composition (EMD) method, developed by Huang et al.(1998), is a powerful method that can detect a nonlinear andnonstationary signal. Application of the method to the seis-mic response data of structures and free-field ground motionis carefully studied. Structural seismic response data studieswill concentrate on the signal processing of structural re-sponses using empirical mode decomposition with Hilbertspectrum analysis (called the EMD � HHT method). In orderto detect structural damage, the Hilbert marginal spectrumwas generated, from which the time-varying system naturalfrequency and damping ratio can be estimated. The identi-fication results were compared to a time domain identifica-tion method such as the system realization information ma-trix (SRIM) method. The EMD � HHT method provides apowerful method for the identification of nonstationary seis-mic response data.

1340 C.-H. Loh, T.-C. Wu, and N. E. Huang

Figure 1. Ground-motion acceleration from the Chi-Chi earthquake, collected atstations TCU068, TCU052, and TCU075. The calculated velocity, which indicateslarge amplitude and long-period waves in the velocity waveform, is also shown.

Taiwan Strong-Motion Instrumentation Program

In 1991, the Central Weather Bureau developed a pro-gram called the Taiwan Strong-Motion Instrumentation Pro-gram (TSMIP). In this program the free-field strong-motionarray as well as the structural seismic monitoring array wasdeveloped. Now over 600 free-field seismometers are de-ployed on the island and over 50 structures (including build-ings and bridges) are instrumented with a seismic monitoringsystem. This program provides valuable information on thecollection of strong ground motion data and the seismic re-sponse data of structures. For example, the Chi-Chi earth-quake with magnitude 7.3 (ML 7.3) that occurred on 21 Sep-tember 1999 directly struck the central part of Taiwan.Valuable ground-motion data with near-fault ground-motioncharacteristics were collected. Figure 1 shows the ground-motion acceleration and its calculated velocity at some sta-tion along Chelungpu fault from the Chi-Chi earthquake. Itis clear that large amplitude and long-duration pulselikewaves exist in some of the velocity wave forms. These pulse-like waves are characteristic of near-fault ground-motion.

Under the TSMIP program, many buildings and bridges

containing instrumentation also collected a lot of seismicresponse data. Among these structures, we selected the seis-mic response data of a damaged seven-story RC building, asshown in Figure 2a. In addition to the Chi-Chi earthquake,seismic response data from other earthquakes for this build-ing were also studied, as shown in Figure 2b. On thecontrary, not much seismic response data for bridges wascollected for the Chi-Chi earthquake. For example, the New-Lian River Bridge had been instrumented with a structuralarray of 30 strong-motion accelerometers along its deck, atits abutments, and at a nearby free-field location, as shownin Figure 3a. It is a continuous five-span prestress box-girderbridge located near the northwest coast of Taiwan. Besidesthe Chi-Chi earthquake, other earthquake events occurred atthe eastern coast of Taiwan, which triggered the seismic-monitoring system of this bridge (Fig. 3b shows the recordedacceleration at channel 10 from four different events). Basedon the recorded acceleration responses from the bridge struc-ture, three different levels of response ranging from weakmotion to strong motion are specified. Earthquake infor-mation (magnitude, depth, etc.) for these events is listed in

Application of the EMD–HHT Method to Identify Near-Fault Ground-Motion Characteristics and Structural Responses 1341

Figure 2. (a) and (b) side view of a seven-story building at NHCU.(c) Plot of the recorded acceleration at the roof of the building (channel 22) from fourdifferent seismic events.


Figure 3. (a) Instrumentation layout of the New-Lian River bridge. (b) Plot of themeasured acceleration at channel 10 (vertical component) from four different seismicresponse data sets of the New-Lian River bridge.

Table 1Information of Earthquake Events for New-Lian Bridge

Date Peak Acceleration (cm/sec2) Latitude Longitude

IndexExcitation

Level (yyyy/mm/dd) A3 A4 B1 B2 C4 Deg Minutes Deg Minutes MagnitudeDepth(km)

EpicentralDistance (km)

HypercentralDistance (km)

L1L2

Low1995/02/101995/04/03

7.398.08

7.789.07

5.97.78

7.668.65

1.851.25

2323

45.6556.13

121122

55.1825.9

5.165.89

24.6114.55

138.18164.59

140.36165.23

M Medium 1995/02/23 28.26 30.15 25.96 26.29 8.82 24 12.22 121 41.22 5.78 21.69 84.93 87.66H High 1995/06/25 483.24 252.47 221.61 212.12 50.79 24 36.37 121 40.11 6.51 39.88 56.95 69.53


Figure 4. (a) Instrumentation layout of the Chai-Nan River bridge (a three-spanbridge with the isolation system at the top of the bridge piers). (b) Plot of the measuredacceleration at channels 16 and 18 and channels 23 and 24 from seismic response dataof the Chai-Nan River bridge (the 22 October 1999 earthquake).

Table 1. Another example, seismic response data from theChai-Nan River Isolation Bridge, located in the southwestpart of Taiwan, was also studied. It is a three-span bridgewith base isolation at the top of a cap beam, as shown inFig. 4a. The isolators were designed to absorb the energyduring earthquake excitation. The first set of seismic re-sponse data for this isolated bridge is collected from the 22October 1999 earthquake. Figure 4b shows the recorded ac-celeration at channels 16, 18, 23, and 24. Identification was

performed on this seismic response data, so the dynamiccharacteristics of the bridge with the base isolation systemcan be identified.

Theoretical Background of the Empirical ModeDecomposition Method

A new signal-processing technique to detect the nonlin-ear and nonstationary signals, the EMD method and Hilbert


spectrum analysis, was introduced by Huang et al. (1998,1999). This method consists of two main parts to analyzethe time-series data. The first part is to decompose any time-series data x(t) into a set of intrinsic mode functions (IMFs)by using the EMD method. The second part is to apply theHilbert transform to each IMF. Through the application ofthe EMD with Hilbert spectrum analysis, the system naturalfrequency and damping ratio of the structures can be iden-tified. A brief description of the method is discussed.

Consider an original time-series data x(t), one may per-form the following mathematical operation

h (t) � x(t) � m (t), (1)11 11

where m11(t) is the mean value of the lower envelopes andupper envelopes for x(t). Next make the same shifting pro-cedure for h11(t), that is,

h (t) � h (t) � m (t). (2)12 11 12

Keeping the same shifting procedure until h1k(k) converges,the subchannel signal c1(t) is defined as follows:

c (t) � h (t). (3)1 1k

Thereafter, the residue signal r1(t), which is equal to x(t)minus c1(t), is treated as a new data and subjected to thesame shifting process as described previously. Then anothersubchannel c2(t) is generated. Repeating those procedures,eventually, the original data x(t) is decomposed into n em-pirical mode plus a residue term, that is,

n

x(t) � c (t) � r (t). (4)� i ni�1

A complete description of the sifting process can be foundin the article by Huang et al. (1998). The second part is toapply the Hilbert transform for each IMF component,

�1 c (s)id (t) � ds, (5)i �p �� t � s

and defined an analytic signal as

ih(t)z (t) � c (t) � d (t)j � a(t)e . (6)i i i

Then the instantaneous frequency can be calculated:

dh(t)x(t) � . (7)

dt

Both amplitude and frequency of each IMF are a function oftime and can be represented as a function of time–fre-quency–amplitude in the three-dimensional space, which isnamed the Hilbert spectrum H(x,t). Furthermore, the Hilbertmarginal spectrum h(x) is defined as

T

h(x) � H(x,t)dt. (8)�0

This marginal spectrum measures the total amplitude con-tribution from each frequency value.

The empirical mode decomposition is accomplishedthrough a sifting procedure defined as extreme sifting (ES).In ES, the location of t for x(t) � 0 is defined as the locationof extremum. The time spacing between successive extremais the extrema scale. If one examines the data more closely,then one will find that even the spacing of extrema can misssome subtle time-scale variations for there are weak oscil-lations that can cause a local change in curvature but notcreate local extremum. To account for this type of weaksignal, another sifting process, designated as curvature sift-ing (CS), was introduced. Mathematically it is equivalent tofinding the extreme value of x/(1 � x)3/2. When one obtainsthe data, one can compute the curvature according to theprevious equation. Once the curvature values are obtained,then, all the extrema of curvature values can be calculated.This combination of extrema of data and the extrema of thecurvature are treated as the total set of extrema. In general,when the signal-to-noise ratio is low, the curvature siftingprocedure could be plagued by the noise and given a falsereading.

Application of the aforementioned sifting process (ESor CS) may be difficult when the data contain intermittency,which will cause mode mixing. To overcome the mode mix-ing, a criterion based on the period length is introduced toseparate the waves of different periods into different modes.The criterion is set as the upper limit of the period that canbe included in any given IMF component. In general, the useof the intermittency criterion should be monitored carefully.Because any addition criterion introduced in the sifting pro-cess implies an intervention with a subjective condition, itcould cause bias in the final result.

Application to Damping Estimation

The EMD method is also being developed to extractdamping loss factor as a function of time and frequency (e.g.,Salvino, 2000). It is to formulate a time-dependent decayfunction for each empirical mode i (for simplicity, the modeindex i is omitted). The ith mode analytic signal in equation(6) can be written as

ih(t) ��(t)�ih(t)z(t) � a(t)e � ce . (9)

Then, a time-dependent decay function can be modeled as

˙ ˙�(t) � �a(t)/a(t). (10)

The dot in equation (10) denotes the time derivative. In thenext step, the damping ratio y(t) and the damping loss factorg(t) are formulated as


˙�2 a(t)g(t) � 2y(t) � . (11)

x (t) a(t)0

The damping loss factor g(t) can be evaluated at a giventime t and frequency x, which is modeled by the EMDmethod and Hilbert transform method:

2 2 1/2˙x(t) � [x (t) � �(t) ] . (12)0

Then, the g(t) can be further represented as a function oftime and frequency in a three-dimensional space, which isnamed the Hilbert damping spectrum g(x,t):

2 2 �1�2a(x,t) a(x,t)2 2g (x,t) � x � . (13)� � � � � �a(x,t) a(x,t)

If the frequency-dependent damping is only feature of in-terest, a root mean square time average of g(x,t) over a timeperiod from 0 to T can be written as

1/2T1 2g(x) � g (x,t)dt . (14)� � �T 0

The frequency-dependent damping was based on the resultfrom the EMD � HHT analysis. Then, the frequency-depen-dent damping loss factor can be calculated from equation (14).

Identification of a Pulselike Wave from Near-FaultGround-Motion Data

Design method is based on the premise that the energydemand during an earthquake can be predicted and that theenergy supply of a structural system can be established. Asatisfactory design implies that the energy supply should belarger than the energy demand. The energy demand duringan earthquake can be expressed as the absolute input energyEi. It was defined as (Uang and Betero, 1990):

Ei � (mm )dm , (15)t g�where t is the absolute acceleration response of a nonlinear�SDOF system, and �g is the ground displacement. An elastic-perfect plastic nonlinear model was used to evaluate thestructural response. The absolute input energy can provideinformation on the seismic demand.

Based on the ground-motion data collected near Che-lungpu fault, the EMD method was used to decompose theground acceleration into several IMFs. The peak accelerationof each IMF was identified. Because each IMF is uncorrelatedto each other then dominant frequency of each IMF can alsobe estimated. Figures 5a and 6a show the peak ground ac-celeration (PGA) of each IMF with respect to the identifieddominant frequency of that IMF. From these figures the dis-tribution of PGA of each IMF with respect to dominant fre-

quency can be observed. It is found that for data from stationTCU068 (Chi-Chi earthquake), a large PGA value was ob-served even at a very low frequency band (0.1 and 0.2 Hz).On the other hand, for data from station TCU048, no sig-nificant large PGA value was observed for frequency below0.4 Hz. The cumulative IMFs starting from the high-fre-quency signal (a1(t)) to the low-frequency signal (a14(t)) canbe used as input motion to evaluate the absolute input en-ergy. The input motion is defined as:

i�d

m (t; d) � a (t), (16)g � ii�1

where d is the cumulative number of intrinsic mode func-tions starting from highest frequency. Figure 7 shows thedecomposed IMFs and its corresponding Fourier amplitudespectrum. A total of 14 IMFs are decomposed from datacollected at station TCU068. Based on equation (16) theabsolute input energy Ei with the consideration of g(t;d) as�input is calculated, as shown in Figure 5b. For data fromstation TCU068 a significant amount of change in input en-ergy was observed in the calculated Ei value by choosingd � 8 and d � 9, respectively. This means that contributionof a9(t) IMF, which contains a large amplitude and a long-period wave (low-frequency wave), will induce a significantincrease in input energy. It means the pulselike wave wasidentified by summation of IMFs from a9(t) to a14(t). Sepa-ration of recorded acceleration into two parts can be done,one is the dynamic part ad(t) � a1(t) � . . . � a8(t), andthe other is the fling-step part ap(t) � a9(t) � . . . � a14(t).The contribution of the pulselike wave in the recordedground-motion acceleration can be obtained, as shown inFigure 5c. Figure 6b shows the Ei value from data at stationTCU049, which is different from the station TCU068 data.It is difficult to identify the low-frequency IMFs that cancause the significant change in absolute input energy. In sucha case no significant pulselike wave exists in this accelera-tion data of station TCU049. From Figure 6c one can seethat there is almost no significant contribution of pulselikewave from a low-frequency IMF. The calculation of absoluteinput energy from each IMF gives a systematic way to sepa-rate the ground motion into a dynamic part and a significantpulselike wave part.

Time-Domain System Identification FormInput/Output Data

Different from using the high performance signal pro-cessing technique (EMD � HHD method) directly on themeasured response data, system identification can also beperformed from the input/output seismic response data ofbuilding and bridge structures. One group of techniques usesa nonparametric approach to determine the input–outputmap using the least-squares method. The input–output mapis characterized by a system model that may not have anyphysical meaning explicitly. A system realization algorithm


Figure 5. (a) Plot of PGA with respect to the dominant frequency for each IMF (top,station TCU052; bottom, station TCU068; 01:47). (b) Calculated absolute input energyby using the cumulative IMF as input (station TCU068, 01:47, T � 1.0 sec, dampingratio � 5%, E–W direction). (c) Separation of original ground acceleration (fromstation TCU068, E–W direction) into the dynamic part ad � a1 � . . . � a8 and thepulselike wave part ap � a9 � . . . � a13.


Figure 6. (a) Plot of PGA with respect to the dominant frequency for each IMF (top,station TCU049; bottom, station TCU082; 01:47). (b) Calculated absolute input energyby using the cumulative IMF as input (station TCU049, 01:47, T � 1.0 sec, E–Wdirection). (c) Separation of original ground acceleration (from station TCU049, E–Wdirection) into the dynamic part ad � a1 � . . . � a10 and the pulselike wave partap � a9 � . . . � a13.


Figure 7. Application of the EMDmethod to ground-motion data from sta-tions (a) TCU068 and (b) TCH049 (01:47,E–W direction). Fourteen IMF values andtheir Fourier amplitude spectra were ob-tained.


with information matrix was used to study the building re-sponse data. Based on the mathematical framework the sys-tem realization using the information matrix (SRIM) is es-tablished (Junag, 1997). A brief description of the methodis discussed. Consider a state space linear discrete time sys-tem model

x(k � 1) � Ax(k) � Bu(k), (17)

y(k) � Cx(k) � Du(k), (18)

where x is system state, y is system output, and u is input.The SRIM is derived using the state space discrete time linearequation to form a data correlation matrix for system iden-tification. Based on the discrete-time ARX model the fol-lowing matrix equality can be developed:

� Y (k) � bU (k), (19)p p

where

� � [� � . . . � ],0 1 p�1

(19a)b � [b b . . . b ],0 1 p�1

and Yp(k) and Up(k) are expressed in the form shown inequation (19a) and (19b).

y(k) y(k�1) L y(k�N � 1)y(k�1) y(k�2) L y(k�N)

Y (k) � ,p M M O M� �y(k�p�1) y(k�p) L y(k�p�N�2)

(19b)

u(k) u(k�1) L u(k�N�1)u(k�1) u(k�2) L u(k � N)

U (k) � .p M M O M� �u(k�p�1) u(k�p) L u(k�p�N�2)

(19c)

Postmultiplying equation (19) by , one can obtain theTU (k)p

correlation matrix:

�1 T�[R � R R R ] � 0, (20)yy yu uu uy

or

�R � 0 , (21)hh mxpm

where

�1 TR � R � R R R .hh yy yu uu uy

One may take a singular value decomposition to factor thematrix Rhh, then the state matrix A can be obtained. Withthe estimated system matrix A, the dynamic characteristics

(system natural frequency and damping ratio) of the struc-ture can be estimated.

Seismic Response Identification of aSeven-Story Building

The seven-story reinforced concrete (RC) building struc-ture (Fig. 2 shows the side view) located at National ChungHsing University (NCHU), Taichung. From the reconnais-sance report of the 1999 Chi-Chi earthquake, this buildingwas identified as having moderate damage. In this sectionboth the EMD � HHD signal-processing technique and sys-tem realization identification algorithm are applied to esti-mate the dynamic characteristics of this seven-story buildingfrom its seismic response data. In order to compare thechange of dynamic characteristics of this building, seismicresponse data collected from this building under differentearthquake loading was also selected for identification.Three low-level seismic response data sets (i.e., data fromthe 23 February 1995 earthquake, the 25 June 1995 earth-quake, and the 5 March 1996 earthquake, respectively), ofthis building were selected for this study. Figure 2c showsthe comparison on the recorded acceleration at the roof ofthe building in the east–west direction from four differentearthquakes.

Based on the EMD method, the ES, CS, and intermit-tence sifting processes were used to analyze the seismic re-sponse data, and the Hilbert spectrum was used to analyzeeach IMF. The average Hilbert spectrum from the signal ofeach channel was calculated. Figure 8a shows the compari-son of frequency-time-amplitude of the response signal atchannel 22 between the 25 June 1995 event and the 21 Sep-tember 1999 event. From observation of Figure 8a, it wasfound that the frequency-time-amplitude distribution fromstrong-motion excitation (21 September 1999 event) showedmore low-frequency signals than the results from seismicresponse of weak ground motion excitations. From the Hil-bert marginal spectrum, as shown in Figure 8b, the identifieddominant frequency from Chi-Chi earthquake shifted sig-nificantly to lower frequency as compared to the results ofthe 25 June 1995 earthquake. The average damping loss fac-tor g(x) from data of channel 22 is shown in Figure 8c. InFigure 8c, the identified damping loss factor of this buildingfrom the data of the 21 September 1999 earthquake is greaterthan the results from other weak seismic event data, particu-larly in the low-frequency range. It is clear that the dampingloss factor of this building was increased during the Chi-Chiearthquake (especially at the low-frequency range). Com-parison of the amplitude distributions of the average Hilbertmarginal spectrum at different floor levels for differentevents is also plotted in Figure 9. The x axis of Figure 9shows the sensor location along the height of this building,the y axis shows the frequency value, and the h(x) is dis-played with color scale. Darker red color indicates the lo-cation of dominant frequency. In Figure 9, the identifiednatural frequency of this building is concentrated at 2.0 Hz


Figure 8. (a) Comparison of the average Hilbert spectrum with building responsedata (channel 22 of the Civil Engineering building at NCHU, seventh floor) by usingdata from the 25 June 1995 earthquake and the 21 September 1999 earthquake.(b) Comparison of the Hilbert marginal spectrum with four different seismic responsedata sets (channel 22 of NCHU). (continued)

from the analysis of 21 September 1999 Chi-Chi earthquake.On the contrary the identified results from the weak excita-tion events are concentrated at 3.0 Hz. It is obvious that thenatural frequency of this building significantly changed un-der the excitation of the Chi-Chi earthquake (damage of thebuilding was clearly observed).

The aforementioned SRIM method was also applied toidentify the system’s natural frequency and damping ratio ofthis seven-story RC building in National Chung Hsing Uni-versity. The identified natural frequencies and damping ra-tios are listed at Table 2. It is clear that the identified naturalfrequency from two different methods were very consistent,


Figure 8. (Continued) (c) Estimated frequency-dependent damping loss factor(channel 22) from different seismic response data.

but the identified damping ratios from the EMD differ withthe results of SRIM. Note that the estimated damping lossfactor g(x) from the EMD method was frequency dependent,but the identified damping ratio from the SRIM method re-sulted from an equivalent linear system.

Application of EMD � HHT to Seismic ResponseData of Bridges

Case 1 Study: Identification of New-LianRiver Bridge

In the case 1 study, the application of the EMD � HHTmethod was used to study the seismic response data of New-Lian River bridge (as shown in Fig. 3) from both weak andstrong ground excitation, as listed in Table 1. Based on theEMD � HHT method, the ES, CS, and intermittence siftingprocedure was used to analyze the seismic response data.Hilbert transform was used to analyze each IMF, and thenthe Hilbert spectrum and damping loss factor for each chan-nel were calculated. Fig. 10a shows the frequency-time-amplitude of the average Hilbert spectrum (results using ei-ther the ES, CS, or intermittence sifting procedure) at channel

10 from two seismic events. From the Hilbert marginal spec-trum (also shown in Fig. 10b), it is observed that for datafrom both the 25 June 1995 and the 10 February 1995 earth-quakes, high-frequency signals were identified at channel10. The average damping loss factor is also plotted in Figure10c. The Hilbert marginal spectrum and damping loss factorestimation from the data of two events at channel 10 looksalmost the same. The amplitude distribution of the averageHilbert marginal spectrum of both vertical and transversemotions along the bridge deck is also examined, as shownin Figures 11. The x axis shows spatial sensor location alongthe bridge deck, the y axis shows frequency value, and am-plitude of h(x) is displayed in red-scale, with the darker redcolor indicating a larger amplitude value. Figure 11a showsthe amplitude distribution of the average Hilbert marginalspectrum of vertical vibration from channels 10 and 12 ofthe bridge. In Figure 11a, the identified frequency from the25 June 1995 earthquake shows that high-frequency signalswere observed in vertical vibration. The amplitude distri-bution of the vibration in transverse direction of the bridgeis also plotted in Figure 11b. From observation of Figure11b, it is found that the natural frequency at the central partof the bridge was lower than the results at the two sides ofthe bridge, and the maximum amplitude of Hilbert marginal


Figure 9. Plot of the amplitude distribution fromthe average Hilbert marginal spectrum at channels 27,15, and 22 (first, fourth, and seventh floors, respec-tively) from the seismic response data for a seven-story building at NCHU: (a) 25 June 1995, (b) 21 Sep-tember 1999.

Table 2The Identified Natural Frequencies and Damping Ratios

for NCHU Based on the SRIM Method

23 February 1995 1st mode 2nd mode

x direction Frequency (Hz) 3.25 3.80Damping ratio (%) 1.85 2.21

y direction Frequency (Hz) 3.20Damping ratio (%) 2.27

25 June 1995 1st mode 2nd mode

x direction Frequency (Hz) 3.17Damping ratio (%) 2.37


5 March 1995 1st mode 2nd mode



21 September 1999 1st mode 2nd mode


y direction Frequency (Hz) 2.07 2.29Damping ratio (%) 4.41 1.63

spectrum appears at the central part of the bridge (located atthe center of span).

Case 2 Study: Identification of Chai-Nan RiverIsolation Bridge

Identification of dynamic characteristics from the seis-mic response of bridge structure with isolation (isolator wasdesigned and install at the top of two piers) was conducted,as shown in Figure 4a. Seismic response data of Chai-NanRiver bridge from the 22 October 1999 earthquake was col-lected. The identification will concentrate on the seismic re-sponse data between input and output of the isolator, and theconstruction joint at the left side of the bridge. By adaptingthe EMD � HHT method the average Hilbert spectrum fromdata recorded at channels 16 and 18 (i.e. longitudinal direc-tion at the top of pier and at the center span of bridge girder,respectively) was calculated first, as shown in Figure 12a.

From the estimated frequency-time Hilbert spectrum atchannels 16 and 18, one can find that most of the high-frequency signals were filtered from the data of channel 18,from which the effect of the isolation system can be clearlyobserved. It also shows that the estimated damping loss fac-tor at channel 18 (at the bridge deck) is much higher than atchannel 16 (at the top of the bridge pier and underneath ofisolator). Also, from data at channels 24 and 23 (i.e., at theleft side of the bridge where construction joint is located),the average Hilbert spectrum from each recorded data wasalso calculated, as shown in Figure 13a. It was found thatthe Hilbert spectrum calculated from channel 23 (transversedirection) preserved more high-frequency signal than thespectrum from channel 24 (longitudinal direction). The en-ergy distribution of channel 24 was concentrated at 1.0 Hz,but some high-frequency signals were identified from chan-nel 23. This situation is due to the fact that impact phenom-enon occurred between the shear key and the bridge girder(at left abutment) in the transverse direction. The frequency-dependent amplitude distribution of the average Hilbert mar-ginal spectrum and the frequency-dependent damping lossfactor g(x) are also plotted in Figure 12b for channels 16and 18 and in Figure 13b for channels 23 and 24. The am-plitude distribution of average Hilbert marginal spectrum atlocation of accelerometer along the bridge deck for the Chai-Nan Bridge is also shown in Figure 14. The x axis showsthe spatial sensor location along the bridge deck, and the yaxis shows the frequency value. Figure 14a is obtained bycomputing the Hilbert marginal spectrum for response datafrom channels 23, 14, 15, 17, 20, and 22 along the bridgegirder in the transverse direction; and Figure 14b is obtained


Figure 10. Comparison of the New-Lian River bridge seismic response data (atchannel 10) from two seismic events (the 10 February 1995 and 25 June 1995 earth-quakes): (a) frequency-time-amplitude distribution, (b) Hilbert marginal spectrum, and(c) frequency-dependent damping loss factor.


Figure 11. Plot of the amplitude distribution ofthe average Hilbert marginal spectrum for vibrationin the (a) vertical direction (channels 10 and 12) and(b) transverse direction along the deck of the New-Lian bridge.

from channels 24, 16, 18, and 21 in the longitudinal direc-tion. Because of the impact between the shear key and bridgegirder, from the observation of Figure 14, it is clear thatsignificant high-frequency energy was identified in the trans-verse direction than in the longitudinal direction particularlyat the abutment of the bridge. The isolation system at the topof the pier can reduce the high-frequency signals transmittedfrom the pier response in longitudinal direction.

Implementation of the SRIM method for the data col-lected from the Chai-Nan River bridge was also studied. Theidentified dominant frequency of the bridge is shown in Ta-ble 3. By comparison on the identified results between theEMD � HHT method and SRIM method, it is clear that the

identified natural frequency from two different methods wasconsistent, but the identified damping ratios from the EMDdiffer with the results of SRIM. The EMD � HHT methodprovides a technique to identify the time- and frequency-dependent signals.

Conclusions

This article applied the EMD method and the absoluteinput energy of SDOF system to estimate the pulselike wavein the near-fault ground motion, particularly from the Chi-Chi earthquake data. Application of the EMD � HHTmethod to estimate the dynamic characteristics of buildingand bridge structures from their seismic response data is alsoconducted. Through the analysis, the following conclusionsare made:

1. The empirical mode decomposition method provides avery powerful tool for the analysis of frequency–time do-main signals. The nonlinear and nonstationary character-istics of signals can be observed through this analysis.The application of EMD method with Hilbert transformto the seismic response data of building structure can de-tect the time-varying system natural frequency and damp-ing ratio.

2. By considering the IMFs, decomposed by the EMDmethod, one can evaluate the absolute input energy of aSDOF nonlinear system. Through the judgment of abruptchange of input energy from each IMF, the pulselike waveof near-fault ground motion can be identified. Thesepulselike waves exist in a low-frequency component ofnear-fault ground motion.

3. By comparing the identified results between the EMD �HHT method and the SRIM method, the identified naturalfrequency from two different methods was consistent, butthe identified damping ratios from the EMD � HHTmethod can have the frequency-dependent characteristicsof damping value, which differ from the results of SRIM.The damping loss factor g(x) was frequency dependent,

Table 3Identified Dominant Frequency and Damping Ratioof the Chai-Nan River Bridge (from the 22 October

1999 Earthquake)

Output, Channel 18/Input, Channels 16 and 21 1st mode


Output, Channel 18/Input, Channels 7–12 1st mode


Output, Channel 18, Input, Channels 8 and 11 1st mode



Figure 12. Comparison of the (a) estimated frequency-time-amplitude Hilbert spec-tra and (b) estimated Hilbert marginal spectrum and the estimated frequency-dependentdamping loss factor of channels 16 and 18 of the Chai-Nan River Bridge with theisolation system (in longitudinal direction of the 22 October 1999 earthquake).


Figure 13. Comparison of the (a) frequency-time-amplitude Hilbert spectra and (b)estimated Hilbert marginal spectrum and the estimated frequency-dependent dampingloss factor of channels 24 and 23 of the Chai-Nan River bridge with the isolation system(22 October 1999 earthquake).


Figure 14. The amplitude distribution of averageHilbert marginal spectrum at the location of the ac-celerometer along the bridge deck for the Chai-NanRiver bridge: (a) transverse distribution, (b) longitu-dinal distribution.

but the identified damping ratio from SRIM was a valueusing an equivalent linear system.

4. From the identification of a seven-story building atNCHU, the EMD � HHD method can also provide a fre-quency-time-amplitude spectrum from which the damageof the structural system can be observed. Through com-parisons between the Chi-Chi earthquake data and othersmall event data the building damage condition can beobserved.

5. In the analysis of seismic response data of Chai-NanRiver bridge, the characteristics of the isolation system

and the impact at the abutment during earthquake loadingcan be identified. With the Hilbert damping spectrummethod, the damping loss factors are estimated as a func-tion of time and frequency.

Acknowledgments

The authors wish to express their thanks to the National ScienceCouncil for the support on this research under Grant No. NSC89-2211-E-002-019. The seismic response data provided by Central Weather Bureauis also acknowledged.

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National Taiwan UniversityCivil Engineering DepartmentTaipei, Taiwan

(C.-H.L.)

National Center for Research on Earthquake EngineeringTaipei, Taiwan

(C.-H.L.)

National Science and Technology Program for Hazard Mitigation OfficeNational Taiwan UniversityTaipei, Taiwan

(T.-C.W.)

NASA Goddard Space Flight CenterOcean and Ice BranchGreenbelt, Maryland 20711

(N.E.H.)

Manuscript received 31 July 2000.

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