# Bulletin of the Seismological Society of America

## Abstract

In this article, the causes of drift in the velocity and the displacement time history are investigated. It is found that, in addition to numerical error, drift is caused by overdeterminacy in the constants of integration. Because there are six independent prescribed at‐rest conditions (three initial and three terminal), the eigenfunctions from an eigenvalue problem, which is described by a sixth‐order ordinary differential equation satisfying the six initial and terminal at‐rest conditions, are chosen as a basis of expansion. The eigenfunctions form a dense and complete set and span a vector space.

The eigenfunctions are used to expand an accelerogram. Drift‐free consistent velocity and displacement time histories are then obtained, also in terms of the eigenfunctions, without direct integration and baseline correction. A method is also proposed to modify a real recorded accelerogram using the eigenfunctions to generate time histories compatible with the target response spectra without drift.

## Introduction

Time history analysis is widely used in seismic design, qualification, and evaluation of critical structures, such as nuclear power plants (American Society of Civil Engineers [ASCE], 2000, 2005; Canadian Standard Association [CSA], 2010; U.S. Nuclear Regulatory Commission [USNRC], 2014b). In practice, usually ground acceleration time histories rather than ground displacement time histories are recorded at a station during earthquakes. However, it has been observed that integrating ground acceleration time history often gives unrealistic and large drifts in velocity and displacement time histories.

Previous studies (Boore, 2005; Boore and Bommer, 2005) showed that drift in the displacement and velocity of real accelerograms is caused by a wide range of factors mostly due to inadequacy of the recording equipment and subsequent processing, which introduces noise. Noises cannot be avoided in real accelerograms. Noise embedded in the digitized records from analog instruments stems from characteristics of the instrument and the digitizer. So far, a considerable number of accelerograms (particularly from older earthquakes) were recorded on analog instruments, and the analog instruments still continue to be used. One of the disadvantages of analog instruments is that they need a specified threshold of acceleration to trigger and a specified threshold of acceleration to stop because they usually operate on standby to save recording medium. Thus, the accelerogram misses the pretriggering portion of the ground motion, leaving the location of the zero‐acceleration baseline unknown. Another disadvantage of analog instruments is that to use the recording in any engineering analysis it is necessary to digitize the traces, which is a primary source of noise. Accelerograms recorded on digital instruments are much better but still likely to contain noise and require processing, which prevents a true baseline from being recorded.

To resolve the problem of drift in time histories, baseline correction is used to adjust the original time histories to eliminate the drift. Various approaches of baseline correction have been proposed. Brady (1966) proposed a method of parabolic baseline correction. A cubic polynomial is used to fit the drifted velocity time history; the fitted polynomial is then used to adjust the drifted velocity time history. The derivative of the fitted polynomial is used to adjust the acceleration time history. Trifunac (1971) proposed a method for standard baseline correction. It uses multiple baseline corrections and high‐pass filtering to adjust the original accelerograms, which is largely independent of the record length. A similar method was also proposed by Graizer (1979).

Based on the characteristics of the baseline errors, Chiu (1997) proposed a three‐step algorithm to perform baseline correction. First, fit the baseline of acceleration time history using the method of least squares and remove the linear trend in the acceleration time history. Then, filter the acceleration time history to remove the remaining errors and high‐frequency random noise. Finally, subtract the initial velocity to remove the linear trend of displacement.

Converse and Brady (1992) proposed a method of high‐pass and low‐pass filtering to remove unwanted noise in the signal, and thus eliminate the drift. The original acceleration time history is padded with zeros at both ends. The padded time history is processed with Butterworth filter to obtain the adjusted acceleration time history.

The influences of using baseline correction to correct the drifted velocity and displacement time histories on structural analysis are remarkable. Boore (2001) showed that baseline correction would greatly change displacement response spectrum of the original ground motion at long periods, and displacement response spectrum is also sensitive to different methods of baseline corrections. Malhotra (2001) showed that baseline correction leads to inconsistency among acceleration, velocity, and displacement time histories. Such inconsistency imposes a significant problem on some very flexible systems, such as long‐span bridges, sloshing liquids in storage tanks, liquefiable soil deposits, tall buildings, and systems responding inelastically. Iwan and Chen (1994) also showed that velocity and displacement may be grossly underestimated by baseline correction. Baseline correction to acceleration response time history of structures may result in distorted and unrealistic responses of structures, such as deformation.

In many earthquake engineering applications, such as evaluation of response of inelastic structures or generation of floor response spectra, time history analysis may be required, which needs acceleration ground motions as input. Generally, there are two options for selecting proper ground motions (Villaverde, 2009) for such engineering problems: (1) select ground motions recorded at a site where the geologic, seismic, and tectonic conditions well match those of the target site, and (2) use artificial time histories, the response spectra of which well match the target response spectra of the site. In reality, it is always difficult to find earthquake ground motions that match the geologic, seismic, and tectonic conditions of the target site. Thus, generated spectrum‐compatible time histories are usually used to perform time history analysis.

Various methods have been proposed to generate spectrum‐compatible time histories. These methods can be categorized as synthetic method (Jennings *et al.*, 1968; Vanmarcke and Gasparini, 1976; Iyengar and Rao, 1979; Preumont, 1984; Khan, 1987) and modification method (Lilhanand and Tseng, 1988; Abrahamson, 1992; Mukherjee and Gupta, 2002; Suarez and Montejo, 2005; Hancock *et al.*, 2006; Cacciola, 2010).

The synthetic method was first systematically explained by Jennings *et al.* (1968). A stationary stochastic process is first generated. To simulate the transient characteristics of earthquake ground motions, the stationary stochastic process is multiplied by a user‐specified envelope shape to achieve a nonstationary random process. Based on the nonstationary random process, a synthetic time history is obtained by iteratively adjusting the nonstationary random process in frequency domain until acceptable compatibility is obtained.

Levy and Wilkinson (1976) proposed another method to generate synthetic time history for nuclear industry. A stationary time history is generated by adding a series of sine functions together. Modulating envelopes from a real earthquake ground motion are then applied to make the time history possess some characteristics of earthquake records, such as nonstationary property. The time history is modified until the difference between the computed and target response spectra is acceptable. Synthetic time histories are usually not recommended to perform nonlinear dynamic analysis of structures and facilities (Pousse *et al.*, 2006).

In the modification method, a real recorded ground motion (seed motion) is modified either in frequency domain or in time domain to make it compatible with a target response spectrum. Tsai (1972) was the first to propose a method to modify a seed motion in frequency domain to achieve a spectrum‐compatible time history. Sinusoidal functions are added or subtracted from the seed motion to achieve the final spectrum‐compatible time history. After Tsai’s work, some improvements have been proposed to modify a seed motion in frequency domain (Preumont, 1984). A seed motion is modified by repeatedly adjusting its Fourier amplitudes and simultaneously keeping its Fourier phase fixed until acceptable compatibility is obtained.

Kaul (1978) and Lilhanand and Tseng (1987, 1988) proposed methods to modify a ground motion in time domain. In these methods, one needs to repeatedly adjust the value of the ground motion locally at a time instance where response spectral value at a specific frequency occurs only until acceptable compatibility is achieved.

Based on the work by Kaul (1978) and Lilhanand and Tseng (1987, 1988), Abrahamson and coauthors (Abrahamson, 1992; Hancock *et al.*, 2006; Al Atik and Abrahamson, 2010) published three versions of wavelet methods to obtain spectrum‐compatible time histories by adding a correction function to a seed accelerogram. They use a real record as the seed motion and wavelets as the correction function. There are three versions of wavelet methods according to the choice of wavelets.

In the first version of the wavelet method, Abrahamson (1992) uses a tapered cosine wavelet to repeatedly correct a seed motion until acceptable spectrum‐compatible time history is obtained. However, the wavelets used by Abrahamson (1992) result in drift in velocity and displacement time histories. Thus, it requires baseline correction to overcome the drift.

In the second version of the wavelet method, Hancock *et al.* (2006) further proposed an improved (baseline corrected) tapered cosine wavelets that have zero initial and final displacement to modify a seed motion and obtain spectrum‐compatible time history. The resulting velocity and displacement time histories, therefore, do not require baseline correction.

Considering the numerical speed and efficiency of modifying a seed motion in previous studies, Al Atik and Abrahamson (2010) proposed an improved tapered cosine wavelet, that is, enveloped cosine wavelet, to modify the seed motion to obtain spectrum‐compatible time history, and improve the speed, stability, and efficiency of modification. Because of the envelope, integrations of the wavelet tend to be zero at both infinities but deviate from zero with a small value at end. However, the resulting velocity and displacement time histories of modified accelerogram can be still accepted as at‐rest at both ends of a finite domain.

Based on the theory of the Hilbert–Huang transform (HHT) developed by Huang *et al.* (1998), Ni *et al.* (2011, 2013) proposed an algorithm to generate spectrum‐compatible time histories using multiple real recorded ground motions, which can generate tridirectional time histories that well match target ground response spectra.

Based on HHT, Li *et al.* (2016) further proposed two algorithms to generate spectrum‐compatible time histories that realize two approaches of generating time histories recommended by USNRC Standard Review Plan 3.7.1 (USNRC, 2014b). The first algorithm is to generate tridirectional time histories compatible with multidamping target response spectra; whereas the second algorithm is to generate tridirectional time histories compatible with single‐damping target response spectra. Time histories generated by these two algorithms satisfy requirements of USNRC Standard Review Plan 3.7.1.

For all the existing methods for generating spectrum‐compatible time histories, the following conclusions can be drawn:

All these methods except the wavelet method require baseline correction to remove drifts in velocity and displacement. However, baseline correction may undo compatibility of the generated time histories. In addition, as discussed above, different approaches of baseline correction may result in very different adjusted time histories, especially for displacement time histories.

The common prerequisite for all three versions of wavelet methods is that the seed velocity and displacement time histories obtained from the seed accelerogram should be at‐rest at both ends. However, this prerequisite is part and parcel of the wavelet method. The wavelet methods did not explain how such a velocity and displacement time histories have been obtained from a digitized accelerogram. If velocity and displacement time histories are obtained from integrating the accelerogram, they could not be made to satisfy all at‐rest conditions at both ends, as explained in the Causes of Drift section.

These existing methods have not demonstrated that the modified accelerogram, the resulting velocity time history, and the resulting displacement time history are mutually consistent.

To resolve these issues in the existing methods, the main objectives of this study are:

to generate a spectrum‐compatible time history for seismic analysis and design, which is at‐rest at the beginning and at the end;

given an accelerogram (either a real record or a synthesized time history) that is at‐rest at the beginning and at the end, to obtain from the accelerogram a velocity and displacement time histories, both of which are also at‐rest at the beginning and at the end (the at‐rest conditions have always been imposed on earthquake time histories used for seismic analysis and design);

to ensure that the accelerogram, the velocity time history, and the displacement time history are mutually consistent so that differentiation or integration of one process yields the other process. This requirement is important to the design of a multiply supported structure such as a piping system.

In this article, reasons causing drifts in velocity and displacement time histories are analyzed first. To solve the problem of drift in time histories, eigenfunctions derived from a sixth‐order ordinary differential eigenvalue problem that meet the necessary initial and terminal at‐rest conditions are employed. Velocity and displacement time histories determined from acceleration time history using the eigenfunctions do not drift. In addition, velocity and displacement time histories obtained from acceleration time history by eigenfunctions are mutually consistent. Furthermore, a method is proposed to modify a real recorded accelerogram using eigenfunctions to generate spectrum‐compatible time histories without drift.

## Drift in Velocity and Displacement Time Histories

### Causes of Drift

Velocity time history is the derivative of displacement time history *D*(*t*)=*u*(*t*), and acceleration time history is the derivative of velocity time history . Conversely, velocity is the integration of acceleration , and displacement *D*(*t*)=*u*(*t*) is the integration of velocity . If acceleration *A*(*t*), velocity *V*(*t*), and displacement *D*(*t*)=*u*(*t*) time histories satisfy these relationships, they are called consistent time histories. Furthermore, it is obvious that acceleration, velocity, and displacement time histories of earthquakes must satisfy the initial and terminal at‐rest conditions, that is, (1a)or (1b)in which *T* is the duration of the time histories.

However, there are many known problems with integrating an earthquake acceleration time history to obtain velocity and displacement time histories by numerical integration.

The velocity and displacement time histories do not satisfy all initial and terminal at‐rest conditions. The time histories will need baseline correction.

Baseline correction by incorporating polynomials into the time histories will make the adjusted displacement, velocity, and acceleration inconsistent.

The polynomials used in baseline correction will introduce unrealistic low‐frequency waves, distorting the energy content of the original earthquake time histories.

Velocity and displacement time histories are usually made to satisfy the initial at‐rest conditions when integrating the acceleration time history. Not satisfying the terminal at‐rest conditions will cause the time histories to drift indefinitely.

A realistic, nondrifting displacement process that is consistent with the accelerogram associated with it is paramount in seismic analysis of structural responses, especially responses of multiply supported systems. For example, piping is a multiply supported system. Some supports may be on different parts or different levels of a structure, whereas some may be on different structures or equipment. Each support point may undergo different displacements. Seismic qualification of a multiply supported system requires consideration of seismic anchor movements, in addition to the inertia effects. If the calculated displacement processes drift or the baselines are improperly corrected, the results of analysis from using the incorrect displacement inputs will be unrealistic and, worse yet, will lead to an erroneous conclusion about the design.

It is important to understand the cause of drift in velocity and displacement time histories obtained from integrating acceleration time history. Suppose *A*(*t*) is an accelerogram, then, (2)There are four at‐rest conditions to be met, that is, *V*(0)=*V*(*T*)=*D*(0)=*D*(*T*)=0. No *C*_{0} and *C*_{1} can be found to satisfy these four independent conditions. Suppose, say, *C*_{0}=*C*_{1}=0 are selected so that *V*(0)=*D*(0)=0. Then, *V*(*T*) and *D*(*T*) may not be zero. For *t*>*T*, *V*(*t*) and *D*(*t*) will drift with *t*.

To illustrate, consider an acceleration time history (3)in which , , and *u* are a set of consistent time histories satisfying the initial and terminal at‐rest conditions (1b), and is a small constant. Obviously, acceleration *A*(*t*) given by equation (3) satisfies the initial and terminal at‐rest conditions. Integrating equation (3) yields the velocity and displacement: (4a)(4b)in which *V*(*t*) and *D*(*t*) must satisfy the last four initial and terminal at‐rest conditions in equation (1a). However, there are only two constants in the equations. As a result, the problem is overdetermined. Suppose that *V*(*t*) and *D*(*t*) are made to meet the initial at‐rest conditions.

Hence, (5a)(5b)that is, both velocity and displacement time histories do not satisfy the terminal at‐rest conditions and drift. On the other hand, if *V*(*t*) and *D*(*t*) are made to satisfy the terminal at‐rest conditions, then they will not satisfy the initial at‐rest conditions. Therefore, drift in velocity and displacement time histories from integrating acceleration time history cannot be avoided.

It is traditionally believed that numerical errors cause the problem of drift in velocity and displacement time histories obtained by integrating acceleration time history. However, it should be emphasized that, in this example, there is no numerical error involved in integrating the acceleration time history to obtain velocity and displacement time histories. This example demonstrates that it is overdeterminacy that causes drift in velocity and displacement time histories obtained by integrating acceleration time history. It is an inherent problem due to mathematical overdeterminacy in integrating acceleration time history.

Baseline correction is usually applied to remove the effect of such terms as in equation (3) to eliminate drift in velocity and displacement time histories. However, the terms removed by baseline correction are inseparable from the original acceleration time history *A*(*t*). Consequently, baseline correction generally ruins characteristics of the original acceleration time history.

### Baseline Correction

Berg and Housner (1961) and Boore (2001) showed that numerical integration of an acceleration time history often caused unrealistic drift in velocity and displacement time histories. For engineering analysis such as soil–structure interaction analysis, using drifted velocity and displacement time histories may have a negative effect on results (Yang *et al.*, 2006). Even if an artificial baseline correction is used to correct the drifted velocity and displacement time histories, it will change the characteristics of the original time history and thus yield unrealistic results.

In this subsection, a set of consistent acceleration, velocity, and displacement time histories are simulated and as shown in Figure 1. It is clearly seen that the simulated time histories do not drift. Given the simulated acceleration time history in Figure 1, its velocity and displacement time histories are calculated by numerical integration satisfying the initial at‐rest conditions:(6a)(6b)in which *t*_{m}=(*m*−1)Δ*t*, *m*=1,2,…,*N*, and Δ*t*=*T*/(*N*−1) is the time increment. Velocity and displacement time histories determined by equations (6a) and (6b) are shown in Figure 2. It is clearly seen that the displacement time history shows drift.

To eliminate drift in time histories, baseline correction is traditionally used to adjust the acceleration time histories. Two approaches of baseline correction are commonly used.

**Approach 1**: Brady (1966) proposed using a quadratic to fit the velocity time history, and then applying its derivative to adjust the original acceleration time history.

**Approach 2**: This approach, proposed by Converse and Brady (1992), has been implemented by U.S. Geological Survey to process recorded ground motions. The original acceleration time history is first padded with zeros at both ends, and then processed with Butterworth filter to obtain the adjusted acceleration time history.

The simulated acceleration time history in Figure 1 is processed by these two approaches of baseline correction and the results are shown in Figures 3 and 4, respectively.

After baseline correction, some ground‐motion parameters (such as peak ground velocity [PGV] and peak ground displacement [PGD]) of the adjusted time histories may deviate from those of the original time histories. Ground‐motion parameters for this example are listed in Table 1. Obviously, baseline correction affects PGV and PGD of the time histories.

The power spectral densities of the time history before and after baseline correction are shown in Figure 5; it shows that baseline correction significantly changes power spectral density of the original time history in low‐frequency range. This may affect responses of flexible structures, such as long‐span bridges and pipelines. In addition, different approaches of baseline correction usually yield different processed time histories.

This example demonstrates that

for a set of consistent time histories satisfying initial and terminal at‐rest conditions, velocity and displacement time histories may be determined by integrating the acceleration time history drift;

blindly applying baseline correction to acceleration time histories may ruin characteristics of the original time histories.

In this example, drifts in velocity and displacement time histories still appear under the condition that no external noise is embedded in the simulated acceleration time history. Therefore, noise is not the only source leading to drift in velocity and displacement time histories; overdeterminacy and numerical error aforementioned also cause the drift in velocity and displacement time histories. Because the simulated acceleration time history contains no external noises, baseline correction unreasonably removes significant low‐frequency components from the original acceleration time history and yields unrealistic adjusted acceleration time history.

To resolve the problem of drift in generating velocity and displacement time histories from acceleration time history, eigenfunctions of a sixth‐order ordinary differential eigenvalue problem are used to expand the acceleration time history.

## Expansion Using Eigenfunctions

### Sixth‐Order Eigenvalue Problem

To obtain consistent acceleration, velocity, and displacement time histories, it is critical to find consistent basis functions satisfying all initial and terminal at‐rest conditions. Because there are six initial and terminal at‐rest conditions, six constants in the solution function are needed to satisfy these conditions. Hence, consider the following sixth‐order ordinary differential eigenvalue problem (7)satisfying six boundary conditions, that is, the six initial and terminal at‐rest conditions: (8)The general solution of equation (7) is (9)Differentiating with respect to *t* gives (10)(11)Substituting equations (9)–(11) into the six initial and terminal at‐rest conditions (8) yields a system of six homogeneous linear algebraic equations for the coefficients *C*_{1},*C*_{2},…,*C*_{6}: (12)in which For *C*_{1},*C*_{2},…,*C*_{6} to have nontrivial solutions, the determinant of the coefficient matrix must be 0, resulting in the eigenequation (13)The eigenequation (13) is a transcendental equation and has infinitely many roots or eigenvalues. It can be shown that there are two sets of roots:

the first set of roots is given exactly by :

*ν*=2*kπ*,*k*=1,2,…;the second set of roots is 9.4270555709, 15.7079533785, 21.9911486180, 28.2743338821,…, which are given approximately by :

*ν*=(2*k*+1)*π*,*k*=1,2,…. It seems impossible to find analytically exact solutions; but for*ν*>30, the approximation is extremely good with relative error less than 10^{−12}.

In summary, the eigenvalues can be written as *ν*_{n}=*λ*_{n}*T*=(*n*+1)*π*, *n*=1,2,…, in which the results are exact when *n* is odd and approximate when *n* is even.

For each eigenvalue *ν*_{n}, the corresponding eigenvector **C**_{n}={*C*_{n1},*C*_{n2},…,*C*_{n6}}^{T}, *n*=1,2,…, can be determined from equation (12) and the eigenfunction is obtained from equation (9). Analytical expressions of eigenfunctions can be easily obtained using a symbolic computation software such as Maple.

As an example, for *T*=25, *ν*_{19}=*λ*_{19}*T*=20*π*, , , and are shown in Figure 6. It is clearly seen that, unlike sine and cosine functions used in Fourier series, , , and satisfy the initial and terminal at‐rest conditions.

It is easy to show that the eigenfunctions possess the following orthogonality properties: (14)The eigenfunctions could be used as basis functions to expand a time history *u*(*t*) of duration *T* as (15)in which *a*_{n} are constant coefficients. If the eigenfunctions are normalized such that , then Parseval’s inequality becomes (16)The expansion is an Euler–Fourier expansion. The convergence is uniform and pointwise. Therefore, integration or differentiation can be done term by term. In addition, the order of operation, that is, integration (or differentiation) and summation, can interchange.

### Decompose an Acceleration Time History by Eigenfunctions

An earthquake acceleration time history *A*(*t*) of duration *T* can be decomposed using a set of *N* eigenfunctions: (17)Multiplying both sides of equation (17) by , *m*=1,2,…,*N*, and integrating from 0 to *T* yields (18)If *A*(*t*) is sampled at *N* discrete time instances *t*_{i}=(*i*−1)Δ*t*, *i*=1,2,…,*N*, Δ*t*=*T*/(*N*−1), then (19)Equation (18) becomes (20)Equation (20) gives a system of *N* linear algebraic equations for *N* unknown coefficients *a*_{n}, *n*=1,2,…,*N*, which can be readily solved. The velocity and displacement time histories can then be determined as (21)In equations (17) and (21), , , and , *n*=1,2,…,*N*, are the *n*th eigenfunction and its derivatives are given by equations (9)–(11).

Because , , and are consistent and satisfy all initial and terminal at‐rest conditions, the acceleration *A*(*t*), velocity *V*(*t*), and displacement *D*(*t*) time histories given by equations (17) and (21) are guaranteed to be consistent and satisfy all initial and terminal at‐rest conditions, hence no drift.

It should be emphasized that the proposed eigenfunctions can also be used to expand any time series provided the time series is continuous and is equal to zero at the beginning and at the end. Convergence of the expansion is pointwise.

It is necessary to mention that eigenfunctions cannot be used to determine permanent ground displacement, because eigenfunctions satisfy initial and terminal at‐rest conditions. Actually, it is quite difficult to determine permanent ground displacement; advanced techniques such as Global Positioning System are usually applied to determine permanent ground displacement. However, it is beyond the scope of this study.

## Generate Time Histories Using Eigenfunctions

Existing methods of modifying a ground motion need baseline correction to remove drifts in velocity and displacement time histories. In this section, a method of modifying a ground motion using eigenfunctions to make it compatible with a target response spectrum is proposed. Spectrum‐compatible time histories generated by this method do not need baseline correction, because the eigenfunctions satisfy initial and terminal at‐rest conditions.

### Optimization

Optimization plays an important role in the analysis of physical systems. To set up an optimization problem, an objective function and variables or unknowns affecting the objective are required. Because the variables are usually constrained, the goal of optimization is to search suitable values of the variables within the constrained region to maximize or minimize the objective function. A general form of optimization with nonlinear constraints is given by: (22)in which **x** is the vector of variables, *f*(**x**) is the objective function, *g*_{j}(**x**) is the nonlinear equality or inequality constraint function, *m*_{e} is the number of nonlinear equality constraint functions, *m* is the total number of constraint functions, and **x**_{L} and **x**_{U} represent the lower and upper bounds of **x**, respectively.

### Procedure to Generate Time Histories Using Eigenfunctions

The flow chart for a procedure to modify a real recorded ground motion (seed motion) using eigenfunctions to obtain a spectrum‐compatible time history is shown in Figure 7. Details of the procedure are explained as follows:

select a seed motion

*X*_{S}(*t*) of duration*T*(the spectral shape of the seed motion should be similar to that of the target response spectrum to some extent);decompose the seed motion using eigenfunctions by equations (17)–(20), to obtain constant coefficients

*a*_{n};based on frequencies over which the target response spectrum is defined, say ,

*i*=1,2,…,*N*_{s}, in which*N*_{s}is the number of frequencies, extract*N*_{s}eigenfunctions from step 2 for which concentrated frequencies are closest to the frequencies of target response spectrum. The concentrated frequency of eigenfunctions is determined by (23)These extracted eigenfunctions are expressed as ,*i*=1,2,…,*N*_{s}, called matching eigenfunctions. Matching eigenfunctions will be used to modify the seed motion to make it compatible with the target response spectrum by linearly scaling their amplitudes.The modified time history

*X*(*t*,**s**) is expressed as: (24)in which**s**={*s*_{1},*s*_{2},…,*s*_{Ns}} is the vector of scaling factors, which are initially set as zero.The following optimization model is used to determine the scaling factors: (25)(26)in which

*η*_{i}is the weighting factor, which is used to emphasize important frequency ranges and mitigate effects of less important frequency ranges of the target response spectrum. and , both of which are functions of**s**, are the time instances when Arias intensity of the time history reaches 5% and 75%, respectively.*ρ*is the correlation coefficient between the generated time history and other spectrum‐compatible time histories. S_{T}(*f*_{i},*ζ*) is acceleration spectral value of the target response spectrum, and ⌊⌋ denotes the floor function. Response spectrum of the modified time history is calculated by (27)in which*ω*=2*πf*and*ζ*are the circular natural frequency and damping ratio, respectively, and * denotes the convolution integration.In the optimization model, the difference between the computed and target response spectra is set as the objective function (25), and the compatibility requirements according to USNRC SRP 3.7.1 (USNRC, 2014b) are set as constraint functions (26).

The optimization is performed until an optimal vector of suitable scaling factors

**s**^{*}is obtained.

To generate a set of tridirectional time histories using the above procedure, the following three steps are followed:

Generate the first horizontal time history compatible with the horizontal target response spectrum.

Generate the second horizontal time history compatible with the horizontal target response spectrum. The correlation coefficient between this time history and the first horizontal time history needs to be considered.

Generate the vertical time history compatible with the vertical target response spectrum. The correlation coefficients between this time history and each of the two horizontal time histories generated in steps 1 and 2 need to be considered.

### Applications

The eigenfunctions could be used to determine realistic velocity and displacement time histories from a given acceleration time history. Furthermore, they can be applied to generate spectrum‐compatible time histories. In this section, two examples of determining velocity and displacement time histories and two examples of generating spectrum‐compatible time histories are presented.

#### Example 1: Determining Velocity and Displacement Time Histories

##### El Centro Earthquake

The El Centro earthquake ground motion (east–west component, including consistent acceleration, velocity, and displacement time histories) from the Pacific Earthquake Engineering Research (PEER) strong‐motion database (see Data and Resources) is used to demonstrate the capacity of eigenfunctions to determine velocity and displacement time histories. The acceleration time history is first decomposed by eigenfunctions using equations (17)–(20), obtaining the constant coefficients *a*_{n}. Using these constant coefficients, the acceleration time history is reconstructed, and the velocity and displacement time histories are determined by equation (21). The reconstructed acceleration time history, and determined velocity and displacement time histories are presented in Figure 8. It is clearly seen that the velocity and displacement time histories determined by eigenfunctions exactly match (overlap) the time histories from the PEER strong‐motion database.

##### Coyote Lake Earthquake

One horizontal ground motion recorded at the San Ysidro School station during the Coyote Lake earthquake (United States) in 1979 (see Data and Resources) is selected from the Center for Engineering Strong Motion Data (CESMD). This ground motion has been postprocessed. The velocity and displacement time histories obtained by integrating the recorded acceleration time history are shown in Figure 9. It can be clearly seen that the displacement time history drifts, which is due to the inherent problem of mathematical overdeterminacy and numerical error.

Because initial and terminal values of the ground motion from CESMD are not zero, zero values are padded at both ends so that it is suitable to use eigenfunctions to determine velocity and displacement time histories. The zero padding will not affect critical characteristics of the original ground motion. The velocity and displacement time histories determined using eigenfunctions are presented in Figure 10. It is clearly seen that the velocity and displacement time histories satisfy initial and terminal at‐rest conditions; that is, no drift. This clearly demonstrates how well the eigenfunction method can work to eliminate drift in velocity and displacement in time histories due to numerical error and overdeterminacy.

#### Example 2: Generating Spectrum‐Compatible Time History

##### NUREG R.G. 1.60

The design response spectra from NUREG R.G. 1.60 (USNRC, 2014a) are taken as the target response spectra. Tridirectional ground motions recorded at Norcia‐Zona Industriale station during the Umbria Marche earthquake (Italy) in 1997 (see Data and Resources) are selected as seed motions. These seed motions satisfy the initial and terminal at‐rest conditions. The generated tridirectional time histories are presented in Figures 11–13. It is clear that the generated time histories well match the target response spectra and do not drift.

##### CENA UHS

The central and eastern North American (CENA) uniform hazard spectrum (UHS; Atkinson and Elgohary, 2007) is taken as the target response spectrum. Seed motions are selected from different earthquakes to generate time histories compatible with CENA UHS. One ground motion observed at the Long Valley Dam station during the Mammoth Lakes earthquake (United States) in 1980 (see Data and Resources) is selected as the seed motion to generate the first horizontal time history, one ground motion observed at the WAHO station during the Loma Prieta earthquake (United States) in 1989 (see Data and Resources) is selected as the seed motion to generate the second horizontal time history, and one ground motion observed at the Calexico Fire station during the Imperial Valley earthquake (United States) in 1979 (see Data and Resources) is selected as the seed motion to generate the vertical time history. Because different seed motions have different durations, these three seed motions are truncated or padded zero at the ends so they possess the same duration.

The generated tridirectional time histories are presented in Figures 14–16. It is seen that the generated time histories well match the target response spectra except at some high frequencies, and do not drift. The reason for the relatively poor match at some high frequencies is that CENA UHS has extraordinarily high amplitudes at high frequencies, which is quite unusual for a real earthquake ground motion.

## Conclusion

In this article, the cause of the drift of velocity and displacement time histories resulting from integrating acceleration time history is investigated. It is discovered that, in addition to numerical error, the drift is caused by mathematical overdeterminacy. To overcome the problem, eigenfunctions of a sixth‐order ordinary differential eigenvalue problem are employed to satisfy the six initial and terminal at‐rest conditions. The orthogonality and completeness of the proposed eigenfunctions are discussed. According to Parseval’s theorem, the eigenfunctions form a complete set.

Given an acceleration time history satisfying initial and terminal at‐rest conditions, a procedure for decomposing an acceleration time history using eigenfunctions is presented. Using the constant coefficients from decomposing the acceleration time history, realistic velocity and displacement time histories satisfying initial and terminal at‐rest conditions are determined. Examples are presented to demonstrate the advantages of using eigenfunctions to determine velocity and displacement time histories from an acceleration time history. Special attention is drawn to Figures 8–10, which clearly demonstrate how well the eigenfunction method can work.

The velocity *V*(*t*) obtained from numerical integration of acceleration *A*(*t*) and the displacement *D*(*t*) obtained from numerical integration of *V*(*t*) will drift due to numerical errors and overdeterminacy in the constants of integration. Although the numerical errors can be kept to a minimum via various techniques such as using a superior integration scheme and filtering of noises, the drifts due to overdeterminacy are intrinsic and will always be present in the results of integration. A case in point is the theoretical example, in which the integration is exact yet the resulting processes drift. In contrast, the velocity *V*(*t*) and the displacement *D*(*t*) produced by the eigenfunction method are drift free.

The existing practices in baseline correction resort to adding polynomials to the time series. There are three drawbacks to such a measure. First, adding polynomials has no mathematical basis. Second, the polynomials distort the frequency content of the processes. Third, the polynomials render the processes no longer mutually consistent. For example, differentiation of the corrected displacement time history does not yield the corrected velocity time history, and differentiation of the corrected velocity time history does not yield the corrected acceleration time history. The same goes for the integrations.

Spectrum‐compatible time histories are required for time history analysis. A method for modifying a seed motion using eigenfunctions is proposed to make it spectrum‐compatible. Because the eigenfunctions satisfy initial and terminal at‐rest conditions, spectrum‐compatible time histories generated are consistent and do not drift.

It is noted that the proposed eigenfunctions can also be used to expand any time series provided the time series is continuous and is equal to zero at the beginning and at the end. Convergence of the expansion is pointwise.

The method of eigenfunctions in this study has some limitations. It is not to be applied to a situation where permanent ground displacement is involved, because the eigenfunctions have zero displacement at the end of the duration. The method of eigenfunctions is not yet available in any existing commercial program. The method requires appreciable computational efforts. For wide applications of the method, a user‐friendly computer program is needed.

## Data and Resources

The ground motions used in this study were obtained from Pacific Earthquake Engineering Research (PEER) strong‐motion database and Center for Engineering Strong Motion Data (CESMD). These data are publicly available on the following websites: El Centro earthquake ground motion was obtained from the PEER strong‐motion database at http://ngawest2.berkeley.edu/spectras/18927/searches/17832/edit (last accessed June 2016, login required); the ground motion observed at the San Ysidro School station during the Coyote Lake earthquake in 1979 was obtained from the CESMD at http://www.strongmotioncenter.org (last accessed June 2016); ground motions observed at Norcia‐Zona Industriale station during the Umbria Marche earthquake in 1997 were obtained from the PEER strong‐motion database at http://ngawest2.berkeley.edu/spectras/18928/searches/17833/edit (last accessed June 2016, login required); ground motion observed at the Long Valley Dam station during the Mammoth Lakes earthquake in 1980 was obtained from the PEER strong‐motion database at http://ngawest2.berkeley.edu/spectras/18928/searches/17833/edit (last accessed June 2016, login required); ground motion observed at the WAHO station during the Loma Prieta earthquake in 1989 was obtained from the PEER strong‐motion database at http://ngawest2.berkeley.edu/spectras/18928/searches/17833/edit (last accessed June 2016, login required); ground motion observed at the Calexico Fire station during the Imperial Valley earthquake in 1979 was obtained from the PEER strong‐motion database at http://ngawest2.berkeley.edu/spectras/18928/searches/17833/edit (last accessed June 2016, login required).

## Acknowledgments

The research for this article was supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by University Network of Excellence in Nuclear Engineering (UNENE).

- Manuscript received 27 June 2016.