Deconvolution-Based System Identification and Finite Element Model Calibration of the UCLA Factor Building

Abstract

Analysis of wave propagation within buildings in response to earthquakes enables the tracking of changes in dynamic characteristics using impulse response functions. The velocity of traveling shear waves and the intrinsic attenuation of buildings can be retrieved, providing valuable input for system identification. The Factor Building at the University of California, Los Angeles campus (henceforth referred to as the UCLA Factor Building), an instrumented 15-story steel moment frame structure, is selected for dynamic response characterization. Shear wave travel time and attenuation are computed from wave propagation using seismic interferometry applied to recorded motions, with deconvolved waves used to compute these parameters. In this study, the natural logarithm of the envelopes of waveforms deconvolved with the basement signal provided the measure of attenuation. Additionally, the waveforms deconvolved with the basement motion, indicating the building’s fundamental mode. The frequency and time decay further constrained the shear velocity and attenuation. Shear velocity was determined using arrival times measured from deconvolved waves, resulting in an average velocity of 147.1 m/s. The observed quality factor was 10.8, with a corresponding damping ratio of 5%. The shear wave velocity and damping ratio estimates derived from deconvolved waves showed consistency with those obtained from basement deconvolved waveforms. This consistency validates wave deconvolution as an effective method for isolating building response from excitation and ground coupling. By incorporating the resonant frequencies and damping ratios derived from previous analyses into a refined element model, this study underscores the potential of wave deconvolution for extracting building dynamic characteristics, thereby enhancing our understanding of their responses to earthquakes.

1. Introduction

Numerous studies have examined the seismic responses of buildings, resulting in advancements in seismic design that aim to mitigate earthquake hazards and reduce casualties [1,2,3,4]. Wave propagation in buildings, modeled as periodic structures, involves measuring building motions, modal frequencies, shear wave velocities, and intrinsic attenuation, which has significantly enhanced the understanding of building responses [5,6,7,8,9,10]. The motion of a building primarily depends on its excitation, the building’s coupling to the ground, its size, and its mechanical properties [11,12,13]. The velocity of the shear wave and the intrinsic attenuation in the building are dominant parameters influencing the building’s response, as they control the motion and modal frequencies of the building under a given excitation. To separate the building’s response from the excitation and ground coupling, we must measure the shear wave velocity and damping ratio. Advancing the measurement of shear wave velocity and attenuation helps to accurately describe a building’s response.

Skolnik [14] studied system identification and the responses of buildings using the vibration-based approach, assuming synchronous responses at different levels. Seismic interferometry has also been proven to be a valid mathematical technique for revealing system parameters [15,16]. Wave deconvolution has been widely employed in system identification and damage detection [17,18]. Given that seismic responses are a consequence of wave propagation, a comprehensive understanding of diverse wave phenomena is essential for understanding building responses and identifying underlying structural characteristics [19,20,21,22,23,24]. This study investigates the validity of wave deconvolution in building system identification. Through rigorous response analysis, we demonstrate the method’s capability in accurately estimating building responses to seismic excitement. Our findings enhance the understanding of dynamic responses in buildings under seismic excitation and provide insight into the wave propagation pattern in buildings.

2. Methodology

2.1. Analysis Procedure and Wave Deconvolution Theory

Response characteristics can be extracted using wave deconvolution [25,26]. For buildings equipped with health monitoring systems, earthquake recordings are frequently obtained at different levels. The building’s response is determined by deconvolving wave signals recorded at floor levels with those from either the basement or the roof. The analysis procedure is demonstrated in the flowchart in Figure 1. Wave deconvolution constructs the impulse response between two signals, providing insights into the building’s response. The impulse response is used to identify the building’s normal mode and its wave propagation characteristics. The shear wave velocity and damping ratio can be determined from the identified normal modes and wave propagation. The frequency and damping ratio obtained from wave deconvolution were used to calibrate and update the building’s computer model. The updated model accurately simulates the actual recorded earthquake displacement.

The response of the building is extracted by deconvolving the waves recorded at all floors, either using the signal recorded at the basement or the signal recorded at the roof [27]. For two signals ${\mathrm{u}}_1\left(\mathsf{\omega }\right)$ and ${\mathrm{u}}_2\left(\mathsf{\omega }\right)$ in the frequency domain, deconvolution is defined as the following:

$$\mathrm{D}\left(\mathsf{\omega }\right)={\mathrm{u}}_1\left(\mathsf{\omega }\right)/{\mathrm{u}}_2\left(\mathsf{\omega }\right)$$

(1)

Equation (1) is unstable when ${\mathrm{u}}_1\left(\mathsf{\omega }\right)$ approaches zero. Thus, an estimator $\mathsf{\epsilon }$ is used to stabilize Equation (1) and deconvolution is changed to

$$\mathrm{D}\left(\mathsf{\omega }\right)={\displaystyle \frac{{\mathrm{u}}_1\left(\mathsf{\omega }\right){\mathrm{u}}_2^{\mathrm{*}}\left(\mathsf{\omega }\right)}{{\left|{\mathrm{u}}_2\left(\mathsf{\omega }\right)\right|}^2+\mathsf{\epsilon }}}$$

(2)

where the asterisk denotes the complex conjugation. When $\mathsf{\epsilon }$ is zero, this expression reduces to Equation (1). In this study, the parameter ε was set to 5% of the average spectral power.

A one-dimensional shear wave propagation model is introduced with the assumption that the viscous damping ratio is independent of the frequency. The shear wave propagates from the basement to the roof and reflects back from the roof downward to the basement. The waves attenuate during propagation due to damping. With travel distance L and the constant damping ratio, this attenuation model $\mathrm{A}\left(\mathrm{L},\mathsf{\omega }\right)$ is described as

$$\mathrm{A}\left(\mathrm{L},\mathsf{\omega }\right)=\exp \left(-{\displaystyle \frac{\mathsf{\gamma }\left|\mathsf{\omega }\right|\mathrm{L}}{{\mathrm{V}}_{\mathrm{S}}}}\right)$$

(3)

In Equation (3), V_s stands for shear wave velocity. The damping ratio is $\mathsf{\gamma }={\displaystyle \frac{1}{2\mathrm{Q}}}$ where Q is the quality factor. In the frequency domain, the deconvolution of the motion at height z with the basement signal is denoted as

$$\mathrm{B}\left(\mathrm{z},\mathsf{\omega }\right)\equiv {\displaystyle \frac{\mathrm{u}\left(\mathrm{z},\mathsf{\omega }\right)}{\mathrm{u}\left(\mathrm{z}=0,\mathsf{\omega }\right)}}$$

(4)

The deconvolution with the signal at the roof is

$$\mathrm{T}\left(\mathrm{z},\mathsf{\omega }\right)\equiv {\displaystyle \frac{\mathrm{u}\left(\mathrm{z},\mathsf{\omega }\right)}{\mathrm{u}\left(\mathrm{z}=0,\mathsf{\omega }\right)}}$$

(5)

The wavenumber is defined by

$$\mathrm{k}=\mathsf{\omega }/{\mathrm{V}}_{\mathrm{s}}$$

(6)

The motion at height z in the time domain is defined as

$$\begin{array}{ll}\mathrm{u}\left(\mathrm{z},\mathrm{t}\right)=\mathrm{A}\left(\mathrm{z},\mathrm{t}\right)\times & \mathrm{s}\left(\mathrm{t}-{\displaystyle \frac{\mathrm{z}}{{\mathrm{V}}_{\mathrm{s}}}}\right)+\mathrm{A}\left(2\mathrm{H}-\mathrm{z},\mathrm{t}\right)\times \mathrm{s}\left(\mathrm{t}-{\displaystyle \frac{2\mathrm{H}-\mathrm{z}}{{\mathrm{V}}_{\mathrm{s}}}}\right)+\mathrm{r}\left(\mathrm{t}\right)\times \mathrm{A}\left(2\mathrm{H}+\mathrm{z},\mathrm{t}\right)\times \mathrm{s}\left(\mathrm{t}-{\displaystyle \frac{2\mathrm{H}+\mathrm{z}}{{\mathrm{V}}_{\mathrm{s}}}}\right)\\ & +\mathrm{r}\left(\mathrm{t}\right)\times \mathrm{A}\left(4\mathrm{H}-\mathrm{z},\mathrm{t}\right)\times \mathrm{s}\left(\mathrm{t}-{\displaystyle \frac{4\mathrm{H}-\mathrm{z}}{{\mathrm{V}}_{\mathrm{s}}}}\right)+\dots \end{array}$$

(7)

where H is the building height, s stands for the input ground motion at the basement, and r is the reflection coefficient of the basement. With the sum of the upward- and downward-traveling waves and the attenuation assumption, the motion at height z in the frequency domain is expressed as

$$\mathrm{u}\left(\mathrm{z},\mathsf{\omega }\right)=\sum _{\mathrm{n}=0}^{\mathrm{\infty }}\mathrm{S}\left(\mathsf{\omega }\right){\mathrm{R}}^{\mathrm{n}}\left(\mathsf{\omega }\right)\left\{{\mathrm{e}}^{\mathrm{i}\mathrm{k}\left(2\mathrm{n}\mathrm{H}+\mathrm{z}\right)}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|\left(2\mathrm{n}\mathrm{H}+\mathrm{z}\right)}+{\mathrm{e}}^{\mathrm{i}\mathrm{k}\left(2\left(\mathrm{n}+1\right)\mathrm{H}-\mathrm{z}\right)}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|\left(2\left(\mathrm{n}+1\right)\mathrm{H}-\mathrm{z}\right)}\right\}$$

(8)

In Equation (8), $\mathrm{S}\left(\mathsf{\omega }\right)$ denotes the input ground motion at the basement, and $\mathrm{R}\left(\mathsf{\omega }\right)$ is the reflection coefficient of the basement. They are transformed from parameters s and r in the time domain. The reflection coefficient $\mathrm{R}\left(\mathsf{\omega }\right)$ is the function of the frequency. Equation (8) suggests that the amplitudes and phases are altered when the upward- and downward-going waves cross from one floor to another due to reflections at the interface. These alterations are characterized by the reflection coefficient $\mathrm{R}\left(\mathsf{\omega }\right)$, and are different for upward- and downward-going waves. With Equation (8), $\mathrm{B}\left(\mathrm{z},\mathsf{\omega }\right)$ and $\mathrm{T}\left(\mathrm{z},\mathsf{\omega }\right)$ are converted as

$$\mathrm{B}\left(\mathrm{z},\mathsf{\omega }\right)=\sum _{\mathrm{n}=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{n}}\left\{{\mathrm{e}}^{\mathrm{i}\mathrm{k}(2\mathrm{n}\mathrm{H}+\mathrm{z})}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|(2\mathrm{n}\mathrm{H}+\mathrm{z})}+{\mathrm{e}}^{\mathrm{i}\mathrm{k}(2\left(\mathrm{n}+1\right)\mathrm{H}-\mathrm{z})}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|(2\left(\mathrm{n}+1\right)\mathrm{H}-\mathrm{z})}\right\}$$

(9)

$$\mathrm{T}\left(\mathrm{z},\mathsf{\omega }\right)={\displaystyle \frac{1}{2}}\left\{{\mathrm{e}}^{\mathrm{i}\mathrm{k}(\mathrm{z}-\mathrm{H})}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|(\mathrm{z}-\mathrm{H})}+{\mathrm{e}}^{\mathrm{i}\mathrm{k}(\mathrm{H}-\mathrm{z})}{\mathrm{e}}^{-\mathsf{\gamma }\left|\mathrm{k}\right|(\mathrm{H}-\mathrm{z})}\right\}$$

(10)

When $\mathrm{t}>{\displaystyle \frac{2\mathrm{H}-\mathrm{z}}{{\mathrm{V}}_{\mathrm{s}}}}$, the deconvolved waveforms are changed into the sum of the damped modes:

$$\mathrm{B}\left(\mathrm{z},\mathrm{t}\right)={\displaystyle \frac{4\mathsf{\pi }{\mathrm{V}}_{\mathrm{s}}}{\mathrm{H}}}\sum _{\mathrm{m}=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}(-\mathsf{\gamma }{\mathsf{\omega }}_{\mathrm{m}}\mathrm{t})\mathrm{c}\mathrm{o}\mathrm{s}({\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{m}}\left(\mathrm{H}-\mathrm{z}\right)}{{\mathrm{V}}_{\mathrm{s}}}})\mathrm{s}\mathrm{i}\mathrm{n}({\mathsf{\omega }}_{\mathrm{m}}\mathrm{t})$$

(11)

with

$${\mathsf{\omega }}_{\mathrm{m}}={\displaystyle \frac{(\mathrm{m}+\frac{1}{2})\mathsf{\pi }{\mathrm{V}}_{\mathrm{s}}}{\mathrm{H}}},\hspace{1em}\hspace{1em}\mathrm{m}=0,1,2,\dots$$

(12)

In Equation (12), the sum is exponentially damping. The fundamental mode with m = 0 has the smallest damping. That means for larger time ($\mathrm{t}\gg 2\mathrm{H}/\mathsf{\pi }{\mathrm{V}}_{\mathrm{s}}$), the fundamental mode dominates. Hence,

$$\mathrm{B}\left(\mathrm{z},\mathrm{t}\right)\approx {\displaystyle \frac{4\mathsf{\pi }{\mathrm{V}}_{\mathrm{s}}}{\mathrm{H}}}\mathrm{e}\mathrm{x}\mathrm{p}(-\mathsf{\gamma }{\mathsf{\omega }}_0\mathrm{t})\mathrm{c}\mathrm{o}\mathrm{s}({\displaystyle \frac{{\mathsf{\omega }}_0\left(\mathrm{H}-\mathrm{z}\right)}{{\mathrm{V}}_{\mathrm{s}}}})\mathrm{s}\mathrm{i}\mathrm{n}({\mathsf{\omega }}_0\mathrm{t})$$

(13)

and the resonance frequency becomes

$${\mathsf{\omega }}_0={\displaystyle \frac{\mathsf{\pi }{\mathrm{V}}_{\mathrm{s}}}{2\mathrm{H}}}$$

(14)

The period corresponding to the angular frequency is given by

$${\mathrm{T}}_0={\displaystyle \frac{4\mathrm{H}}{{\mathrm{V}}_{\mathrm{s}}}}$$

(15)

From the Equations (6) and (10), if there is no attenuation, the deconvolved waves with the roof recording in the time domain are given by a superposition of upward- and downward-propagating delta functions, as in the following:

$$\mathrm{T}\left(\mathrm{z},\mathrm{t}\right)=\mathsf{\pi }\left\{\mathsf{\delta }\left(\mathrm{t}-{\displaystyle \frac{\mathrm{z}-\mathrm{H}}{{\mathrm{V}}_{\mathrm{s}}}}\right)+\mathsf{\delta }(\mathrm{t}+{\displaystyle \frac{\mathrm{z}-\mathrm{H}}{{\mathrm{V}}_{\mathrm{s}}}})\right\}$$

(16)

The aforementioned form the fundamental theory of wave deconvolution [2], frequently applied for seismic interferometry analysis. Based on the theory, dynamic response characteristics are analyzed for the UCLA Factor Building.

2.2. Building Instrumentation and Monitoring Data

The UCLA Factor Building (Figure 2a), a 15-story high instrumented steel moment-resisting frame structure with a basement, serves as the case study to conduct wave deconvolution analysis and finite element model calibration. The 234-foot-tall building comprises 15 above-ground stories, one full basement, and one partial basement. Typical floor plans for stories 1–9 and floors 1–15 are presented in Figure 2b,c, respectively. Floors 1–9 feature a rectangular shape with protrusions in all four corners, and a wing extending outwards on the west side. The rectangular area measures 126 ft 4 in × 74 ft 4 in, while the wing on the east side measures approximately 35 ft 6 in × 24 ft 5 in. From the 10th floor to the roof, the main rectangular area extends outwards by 4 ft 2 into the east and 13 ft 10 into the west. The building is supported by 35 columns from floors 1–9. Floors 10 up to the roof are supported by 44 columns. Eight steel tube sections are used to reinforce the protrusions at the building’s corners. All of the beams are assumed to be connected using moment-resisting connections. Except for steel tube sections, all of the steel sections used in this building are typical W-sections. The building’s foundations are supported by spread footings.

The seismic array installed in the Factor Building comprises 72 force balance accelerometers (Kinemetrics FBA-11) distributed across every floor. The force balance accelerometer sensors have a natural frequency of 50 Hz, normal damping at 70% of the critical damping ratio, a dynamic range of 135 dB from 0.01 to 50 Hz, and an output range scale of ±4 g. The locations of the accelerometers are shown in Figure 3a. Typically, each floor is equipped with four uniaxial sensors oriented north, south, east, and west. The accelerometers installed in the north–south directions are parallel to each other and so, similarly, are the sensors in the east–west direction. In both basements, two uniaxial sensors are installed orthogonally to measure horizontal motions, while two uniaxial sensors are aligned vertically to record vertical motions. This arrangement allows for the computation of the building’s rocking motion. In 2003, the U.S. Geological Survey’s National Strong Motion Project upgraded from 18-bit data loggers to 24-bit IP-based data loggers with fiber-optic data transmission to enhance real-time seismic monitoring. In 2005, a 100 m-deep borehole and overhead surface accelerometers were installed in the nearby botanical gardens, approximately 25 m away from the building. This setup includes a three-component shallow 1 g accelerometer at 100 m depth and a 2 g accelerometer at the surface. The accelerometer network in the Factor Building continuously records data, which are archived for public use at the IRIS Data Management Center. The recorded motions during the 2004 M6.0 (moment magnitude) Parkfield earthquake were used for this study. The earthquake’s epicentral distance was approximately 260 km away from the building. Figure 3b,c present the recorded waveforms in the east—west and north—south directions, along with inferred torsion from the first basement to the 15th floor. The peak acceleration at the roof reached approximately 0.25% g in the east–west direction and about 0.20% g in the north–south direction. The building’s response was essentially linear elastic due to the low intensity of the earthquake. For this event, the P waves arrived at approximately 10 s, and the strongest S waves arrived at around 35 s. Amplified long-period surface waves, attributed to the basin structure in Los Angeles [28], arrive at t = 75 s and induce a resonance with an amplitude that increases with the floor level. The recordings were analyzed using wave deconvolution to obtain the dynamic characteristic parameters.

3. Results Analysis

3.1. Wave Deconvolution Analysis

The motions in the east–west direction were deconvolved using motions recorded at different floor levels. Figure 4a shows the deconvolved waveforms for the building’s height relative to the recorded signal at the basement. Since deconvolving a signal with itself yields a delta function, the deconvolved wave at level B1 manifests as a single spike. This upward-propagating impulse signal serves as a theoretical input used to excite the building. The deconvolved waves observed on each floor are triggered by the input impulse, indicating that no waves are present at any level before the impulse appears. The initial appearance of the deconvolved waves begins with an upward-propagating wave. It was observed that the time delay between successive peaks increased with floor level. The wave travels upward to the roof and is reflected down. The roof reflection creates a second peak in the deconvolved waves. The reflected wave then propagates downward to level B1, generating a subsequent upward-propagating impulsive wave. The process of upward and downward wave propagation continues cyclically over time. As indicated by Equations (9) and (11), the deconvolved waves comprise an infinite number of cycles. During the early periods (0–3 s), the deconvolved waves are characterized by the superposition of upward- and downward-propagating waves, making it challenging to distinguish individual waves due to their interference. The later part (>3 s) standing waves result from the building’s resonance excited by the input impulse signal. This resonance increases in amplitude with floor level and is relatively monochromatic.

Earthquake waves preferentially transmit low-frequency components through the building, while attenuating high frequencies, resulting in minimal reflection coefficients across the floors. However, the reflection coefficient for upward-traveling waves on the free surface approaches unity [29], which significantly exceeds individual floor-level reflection coefficients. Consequently, the roof is the dominant reflecting boundary, with no reflection occurring on the floors. Equations (10) and (16) confirm that the roof-deconvolved waveforms comprise a superposition of a single upward-propagating wave and a single downward-traveling wave. The waveforms obtained from deconvolving the roof signal are significantly simpler than those obtained with the basement signal, as illustrated in Figure 4b. The simplicity of the deconvolved waveforms suggests that wave propagation is predominantly one-dimensional at the frequencies used. The reflection coefficient of elastic waves by a floor depends on the product of the frequency and the mass of the floor. The absence of reflection from floors is due to the relatively low frequencies used for wave deconvolution analysis. Additionally, the dominant wavelength of the shear waves extends across multiple floors, which further diminishes reflections from individual floors. At higher frequencies, the wavelength becomes smaller relative to the height of the floors. The deconvolved wave may become more complex due to the reflections from individual floors as the wave travels through the building. In cylindrical structures, wave propagation becomes extremely complex due to multiple reflections and refractions, as well as possible structural inhomogeneity caused by reinforcement. In particular, the presence of Pochhammer—Chree type waves allows longitudinal waves, torsional waves, longitudinal bending waves, and rare spiral waves to propagate within them, greatly increasing the difficulty and accuracy requirements for analyzing wave propagation behavior [30].

The deconvolved waveforms in the north–south and torsional directions show similar characteristics to those in the east–west direction. However, the size difference between the east–west and north–south directions results in varying stiffnesses. Therefore, the shear waves in the east–west and north–south directions travel at various velocities.

The normal modes in the deconvolved response, as indicated by Equation (11), are determined solely by the building’s characteristics. Given that the fundamental mode is dominant, Equations (13)–(15) are used to extract the response of the Factor Building using the deconvolved waveforms in two directions. Equation (12) describes the deconvolved waves with the superposition of an upward-propagating and downward-propagating wave. By analyzing the deconvolved waveforms with the signals at the basement and roof, the shear wave velocity and damping ratio are determined to check the consistency of wave deconvolution.

The cross-power spectra of the deconvolved waves (Figure 4a) reveal pronounced peaks at 0.468 Hz and 1.526 Hz in the east–west direction (Figure 5a). The frequencies were estimated to be 0.505 Hz and 1.624 Hz in the north–south direction, and 0.521 Hz and 1.644 Hz for torsions. These frequencies represent the first two-order modal frequencies for the Factor Building. The dominant frequencies obtained from wave deconvolution are listed in Column 2 of

Table 1. The building’s fundamental frequencies, as determined by deconvolution and system identification of the 2004 M6.0 Parkfield earthquake data and those obtained from the OpenSEES model.

Mode No.	Frequency (Hz)			Mode Shape
	Deconvolution	Parkfield Data	OpenSEES
1	0.468	0.468	0.468	Bending in X direction
2	0.505	0.504	0.477	Bending in Y direction
3	0.521	0.680	0.654	Torsion
4	1.521	1.439	1.249	Bending in X direction
5	1.624	1.615	1.352	Bending in Y direction
6	1.644	1.826	1.892	Torsion
7	N/A	2.506	2.013	Bending in X direction
8	N/A	2.686	2.327	Bending in Y direction
9	N/A	3.787	2.935	Torsion

. These values reflect the monochromatic nature of the resonance in both the horizontal and vertical directions. Using the frequencies and the total building height of 74.8 m, measured to the second basement, the shear wave velocities were computed as 140.0 m/s and 151.1 m/s in the east–west and north–south directions, respectively, using Equation (15).

The arrival times of the upward- and downward-propagating waves were determined by identifying the peak amplitudes of these waves, as marked by circles in Figure 4b for the east–west direction. For the upward-propagating wave, travel distances relative to the roof were assigned a negative value, while distances for the downward-propagating wave were assigned positive values. The measurements in the east–west direction, shown in Figure 5b, display a near-linear relationship between travel time and distance, suggesting constant shear wave velocities in this direction. By fitting a straight line to the data, the slope of 0.0068 was determined for the east–west direction, yielding an estimated shear wave velocity of 147.1 m/s. A similar linear fit in the north–south direction results in a slope of 0.0065, yielding an estimated shear wave velocity of 153.8 m/s. Shear wave velocities predicted by the wave propagating and the normal modes are practically identical. The differences observed were 5.1% in the east–west direction and 1.8% in the north–south direction. These discrepancies were expected, as the wave deconvolution method assumed a homogeneous medium with uniform mass and stiffness properties.

Considering the arrival times at different floors, the floor heights and the shear wave velocities in the east–west and north–south directions were calculated. Shear wave velocity profiles in the two directions were identified and are demonstrated in Figure 6. An identical trend was observed in both directions. Due to the outward extension’s in-plane size at the 10th floor and above, the shear wave velocities changed at an identical level. This profile effectively demonstrates the variation in stiffness throughout the building.

Expression (13) indicates the attenuation of the waveforms deconvolved with the signal at the base. For the deconvolved waveforms in the east–west direction in Figure 4a, envelopes are applied from floor 9 to the roof, and the natural logarithm of the envelopes is shown in Figure 7a. Between 1 s and 13 s, the logarithm decays linearly with time. Therefore, straight lines were fitted for the top nine floors using the solid thick lines to determine the attenuation. The floor numbers were added to each curve for clarity due to the dependency of the slopes on attenuation. Offsetting the data does not influence the result, as the slope does not change with the offset. The deconvolved waveforms in the north–south direction were analyzed similarly, using the time from 1 s to 10 s in the logarithm of the envelopes. The slopes for both the east–west and north–south directions are listed in

Table 2. The slopes of the fitted lines for the natural logarithm of the envelopes from Floor 9 to the roof.

Floor No.	Slope in EW	Slope in NS
9	−0.1384	−0.1589
10	−0.1398	−0.1592
11	0.1406	−0.1583
12	−0.1420	−0.1599
13	−0.1306	−0.1505
14	−0.1286	−0.1506
15	−0.1343	−0.1560
Roof Average	−0.1351 −0.1362	−0.1562 −0.1562

. The average slopes for the east–west and north–south directions were used to compute the damping ratios, using the following relationship: the slope is equal to (−ω0/2Q). Given a resonance frequency of 0.468 Hz in the east–west direction and 0.505 Hz in the north–south direction, the estimated quality factor in both directions was 10.8, corresponding to an estimated damping ratio of 5% in the Factor Building.

Equation (10) indicates that the propagating waves decay while traveling through the building, as evident in roof-deconvolved waveforms. Figure 4b demonstrates that the amplitudes of downward-propagating waves are consistently smaller than the amplitudes of upward-propagating waves. The propagating wave decay, modeled by the expression exp(−πfz/QV_s), is quantified by computing the ratios of the amplitudes of the downward-propagating waves to the upward-propagating waves, with the two-way distance measured relative to the roof. The relationship between the natural logarithm of the ratio and the two-way distance is fitted by a straight line in both the east–west and north–south directions, as shown in Figure 7b. Deconvolved waves showed dominant frequencies of approximately 2.75 Hz in the east–west direction and 3.25 Hz in the north–south direction. Considering these frequencies, along with the calculated shear wave velocities and quality factors in the two directions, the fitted lines suggest similar attenuation rates with the distance of the Factor Building.

3.2. Results and Discussion

The fundamental frequencies of the Factor Building were also estimated directly from the power spectra of the building response during the Parkfield earthquake at various floors, as reported in Column 3 in Table 1. A three-dimensional finite element model of the building was created in OpenSEES [31]. The model comprised 962 nodes, 1442 beams, and 714 columns. The material parameters were defined using the uniaxial material steel, with an elasticity modulus of 141,590 kg/m², a yield strength of 293 kg/m², a shear modulus of 54,458 kg/m², and a hardening ratio of 0.02. The resonant frequency identified from wave deconvolution in the east–west and north–south directions, along with the first nine frequencies identified using power spectra, were used to calibrate the computer model. The first nine modes, shown in Figure 8a, correspond to the EW direction, NS direction, and torsion. Frequency calibration was used for model updating. Mass and stiffness distributions were adjusted to align with the fundamental frequencies of those observed in the recordings, while mode shapes were simultaneously refined to eliminate differences. The updated frequencies are listed in Column 4 of Table 1. The comparison of the first and second modes obtained from the earthquake data and simulated is presented in Figure 8b.

After updating the model, Rayleigh damping was applied by using a damping ratio of 5% estimated from deconvolution, as indicated in Figure 9a. The red dots in the figure represent the modal damping ratios (%) corresponding to the natural frequencies of each mode.Rayleigh damping was applied, resulting in calculated mass and stiffness coefficients of 0.2537 and 0.0047, respectively. Dynamic analysis was carried out for the updated model using these parameters. The east–west and north–south recordings at basement one for the Parkfield earthquake were used as input ground motions. Relative displacements were compared between the recorded data and the computed data, as shown in Figure 9b. The displacement calibration between recorded and simulated data demonstrated consistency, validating the model’s performance within a linear state.

4. Conclusions

Wave deconvolution was applied to extract the normal mode and the wave propagation in horizontal directions for the Factor Building. The deconvolved waves were used to estimate the shear wave velocity and attenuation. Waves deconvolved from the roof’s motion signal indicated the presence of upward- and downward-propagating waves, while waveforms deconvolved from the basement’s motion signal revealed the building’s fundamental mode. The frequency and decay over time provided constraints on the shear velocity and attenuation. The shear velocity was determined from the arrival times of the propagating waves and computed directly using the expression 4 Hf, as indicated by the normal mode. The damping ratio was constrained by analyzing the natural logarithm of the envelopes of the deconvolved waves across the top nine floors. The consistent shear wave velocity estimates computed from both roof and basement deconvolved wave signals validate the reliability of the wave deconvolution method. This alignment confirms wave deconvolution’s ability to isolate building response from excitation and ground coupling influences. With the information acquired from the deconvolved waves, the model of the building was calibrated and updated in detail. The analysis validates the effectiveness of wave deconvolution in building system identification, enabling precise prediction of building response. Wave deconvolution enables the interpretation of wave propagation patterns in buildings, enhancing the understanding of building dynamic response to seismic excitation. This study expands the application of building-health monitoring data by providing novel insights into structural dynamics. The proposed method surpasses traditional seismic interferometry techniques due to higher sample rates, resulting in enhanced accuracy. The strict control of sampling rates is essential to ensure optimal results.