# The Importance of Corner Frequency in Site-Based Stochastic Ground Motion
Models

Maijia Su  Mayssa Dabaghi  Marco Broccardo

###### Abstract

Synthetic ground motions (GMs) play a fundamental role in both deterministic
and probabilistic seismic engineering assessments. This paper shows that the
family of filtered and modulated white noise stochastic GM models overlooks a
key parameter—the high-pass filter’s corner frequency, $f_{c}$. In the
simulated motions, this causes significant distortions in the long-period
range of the linear-response spectra and in the linear-response spectral
correlations. To address this, we incorporate $f_{c}$ as an explicitly fitted
parameter in a site-based stochastic model. We optimize $f_{c}$ by
individually matching the long-period linear-response spectrum (i.e., $Sa(T)$
for $T\geq 1$s) of synthetic GMs with that of each recorded GM. We show that
by fitting $f_{c}$ the resulting stochastically simulated GMs can precisely
capture the spectral amplitudes, variability (i.e., variances of
$\log(Sa(T))$), and the correlation structure (i.e., correlation of
$\log(Sa(T))$ between distinct periods $T_{1}$ and $T_{2}$) of recorded GMs.
To quantify the impact of $f_{c}$, a sensitivity analysis is conducted through
linear regression. This regression relates the logarithmic linear-response
spectrum ($\log(Sa(T))$) to seven GM parameters, including the optimized
$f_{c}$. The results indicate that the variance of $f_{c}$ observed in natural
GMs, along with its correlation with the other GM parameters, accounts for 26%
of the spectral variability in long periods. Neglecting either the $f_{c}$
variance or $f_{c}$ correlation typically results in an important
overestimation of the linear-response spectral correlation.

###### keywords:

Synthetic Ground Motion , Stochastic Model , High-Pass Filter , Corner
Frequency

††journal: EESD

[inst1]organization=Department of Civil, Environmental and Mechanical
Engineering, University of Trento, Trento,country=Italy
[inst2]organization=Department of Civil and Environmental Engineering,
American University of Beirut, Beirut,country=Lebanon

##  1 Introduction

A proper selection or generation of ground motion (GM) time series is vital
for designing earthquake-resistant buildings and conducting seismic risk
assessments. Efforts to synthesize GMs include site-based stochastic models,
which have evolved since the 1950s, progressing from stationary white-noise
models to fully temporal- and spectral-nonstationary models. It is critical
that synthetic GMs capture the essential characteristics and natural
variability of recorded GMs. In recent years, site-based stochastic GMs [7, 4,
10] have gained widespread attention due to their computational efficiency,
straightforward formulations, and reliance on input that consists of readily
available information about earthquake source, path, and site characteristics
(e.g., the earthquake magnitude, source-to-site distance, and shear-wave
velocity of the site).

Site-based GM models are directly fitted to a recorded GM catalog of interest,
which includes GMs selected from a specified range of earthquake scenarios
defined by their source, path and site characteristics. Specifically, GMs in
the catalog are individually fitted using a prescribed parametric GM model.
This study focuses on stochastic GM models based on simulating modulated and
filtered white noises. The random process of the filtered white-noise can be
simulated in two ways: in the time domain by solving a linear filter excited
by Gaussian white noise [6] or in the frequency domain by using spectral
representations based on Priestley’s evolutionary theory [1]. Typically, these
filtered processes are subsequently refined using a high-pass filter, ensuring
(1) that artificially generated GMs achieve zero residual velocity and
displacement at the end of the process and, more importantly, (2) that their
low-frequency contents are adjusted to match those of recorded GMs.

Previous models have treated the corner frequency, $f_{c}$, of the high-pass
filter as a constrained parameter and have used various methods to estimate
its value, including using a constant value of 0.1Hz [3, 7] or 0.2Hz [10],
values from the flat-file of the PEER-NGA database [6], and values predicted
using equations based on earthquake magnitude [4, 8] or earthquake motion
duration [5, 9]. This results in synthetic GMs with distorted long-period
linear-response spectra and incorrect linear-response spectral correlation
structures, as illustrated in Section 2. In this context, we show that $f_{c}$
plays a key role and needs to be a free parameter of the parametric stochastic
GM model. Consequently, Section 3 introduces a novel fitting procedure to
optimize $f_{c}$. Specifically, the procedure requires matching the long-
period content of the simulated linear-response spectra with those computed
from recorded motions. The resulting synthetic GMs with optimized $f_{c}$ also
have their linear-response spectral correlation structure consistent with the
recorded motions. In Section 4, a comprehensive sensitivity analysis is
conducted through linear regression to gain fundamental insights into the
impact of the corner frequency on linear-response spectra and spectral
correlation structures.

##  2 Effect of $f_{c}$ on the Linear-Response Spectra of Simulated Motions

This paper builds upon the findings of a previous study by Broccardo and
Dabaghi [2], which compared statistical quantities of the 5%-damped elastic
pseudo-acceleration response spectrum ($S_{a}(T)$) between a recorded GM
catalog111The recorded catalog has 71 GMs which are downloaded from the PEER
NGA-West2 database, see details in [7, 2]. The search conditions use reverse
fault type, source-to-site distances $10\sim 90$ km, magnitudes $6\sim 7.6$,
and soil shear-wave velocity $V_{S30}\geq 600$ m/s. and its corresponding
synthetic catalogs. The study found that the synthetic catalog generated using
the temporal-based GM model proposed by Rezaeian and Der Kiureghian [7]
overestimates spectral correlations (i.e., Pearson’s correlation coefficients
of $\log(S_{a}(T))$ at periods $T_{1}$ and $T_{2}$). In contrast, the
frequency-domain equivalent model [1] of the GM model [7] exhibits improved
correlations. This paper reveals that the choice of GM simulation method in
the time or frequency domain is not critical. The improved correlation pattern
arises from a different approach in estimating $f_{c}$ in [1]. Moreover, the
improper selection of $f_{c}$ not only skews the spectral correlation but also
affects the spectral marginal (i.e., the marginal probability distribution of
$S_{a}(T)$) at periods $T\geq 1$s.

To illustrate our findings, we compare the $S_{a}(T)$ statistics of synthetic
GM catalogs to those of the corresponding recorded GM catalog studied in [7,
2]. These synthetic catalogs are obtained using both the temporal- and
spectral-based model formulations [7, 1], along with various methods of
estimating $f_{c}$. Figure 1 and Figure 2 respectively summarize the improved
spectral marginal and correlation when using the optimized $f_{c}$ method
presented in Section 3 in comparison to four other traditional $f_{c}$
methods. These traditional methods involve using a constant value
$f_{c}=0.1$Hz [7], an earthquake-magnitude-based prediction equation [4], a
GM-duration-based prediction equation [5] and the values provided in NGA flat-
file [6]. The comparisons show no significant difference between using the
temporal-based and the spectral-based GM model. The differences arise from the
choice of the $f_{c}$ computation method. Specifically, Figure 1 compares four
key features of the spectral marginal: the 5% quantile, 50% quantile (i.e.,
median), 95% quantile, and standard deviation (s.t.d) of $\log(S_{a}(T))$. The
comparison reveals that all $f_{c}$ estimation methods accurately capture the
response spectrum at short periods ($T\leq 1$ s); however, only the optimized
method consistently provides a good fit at long periods ($T>1$ s). Traditional
$f_{c}$ estimation methods tend to overestimate the quantiles of the spectral
responses but underestimate the variance of the spectral marginal. Figure 2
compares the spectral correlations between periods $T_{1}\in[0.05,10]$s and
four fixed periods $T_{2}=0.1,0.5,1,4$s. Overall, the optimization-based
method exhibits the closest fit, while the fixed $f_{c}$ results in large
over-estimations of the correlation between short- and long-period ranges.

(a) Temporal-based model

(b) Spectral-based model

Figure 1: Comparing statistics ( 5% quantile, 50% quantile, 95% quantile and
standard deviation) of 5%-damped elastic response spectra between a real GM
catalog and 5 corresponding synthetic catalogs using different $f_{c}$
estimation methods: (a) Synthetic catalogs simulated using the temporal-based
model [7]; (b) Synthetic catalogs simulated using the spectral-based modal [1]

(a) Temporal-based model

(b) Spectral-based model

Figure 2:  Comparing the spectral correlations between a real GM catalog and 5
corresponding synthetic catalogs using different $f_{c}$ estimation methods:
(a) Synthetic catalogs simulated using the temporal-based model [7]; (b)
Synthetic catalogs simulated using the spectral-based modal [1]

##  3 Optimized Corner Frequency Fitting

This section proposes a method to optimize the fitting of parameter $f_{c}$
for stochastic GM models (e.g., [7, 1, 10] ) that require a high-pass filter
to eliminate the unrealistic low-frequency contents. To make this paper self-
contained, this section briefly introduces the temporal-based [6] and
spectral-based [1] stochastic GM models; readers can refer to their original
papers for more details.

We integrate the temporal-based [6] and spectral-based [1] stochastic GM
models into a consistent framework. Both models start by applying a time-
varying linear filter to a Gaussian white-noise excitation. The resulting
colored noise has spectral nonstationarity due to the time-varying filter.
Next, the signal is adjusted with a time-modulating function to achieve
temporal nonstationarity. Finally, a high-pass filter is applied to adjust the
low-frequency contents of the simulations to match a specific recorded GM and
assure zero velocity and displacement at the end of the motion. The
distinction between the two models lies in the definition of the time-varying
linear filter: temporal-based models utilize an evolutionary impulse response
function $h_{X}(t-\tau|\boldsymbol{\theta}_{F}(\tau))$, while spectral-based
models use an evolutionary frequency-response function
$H_{XX}(\omega|\boldsymbol{\theta}_{F}(t))$. The parameter vector
$\boldsymbol{\theta}_{F}(t)$ determines the transient properties of the linear
filter. In summary, both stochastic GM models follow a four-step procedure,
including:

  * 1.

Step 1: Simulate a filtered white noise process $X_{1}(t)$

    * (a)

either in the time domain:
$X_{1}(t)=\int_{0}^{t}h_{X}(t-\tau|\boldsymbol{\theta}_{F}(\tau))dW(\tau)$,

    * (b)

or in the frequency domain:
$X_{1}(t)=\int_{-\infty}^{+\infty}H_{XX}(\omega|\boldsymbol{\theta}_{F}(t))\exp(-i\omega
t)d\hat{W}(\omega),$
where $W(\tau)$ is a real Gaussian white-noise process, and $\hat{W}(\omega)$
is a zero-mean complex valued process defined as the Fourier spectrum of
$W(\tau)$.

  * 2.

Step 2: Normalize $X_{1}(t)$ to a unit-variance process, denoted as
$X_{2}(t)=X_{1}(t)/\sigma_{X_{1}}(t)$, where $\sigma_{X_{1}}(t)$ is the
process variance of $X_{1}(t)$.

  * 3.

Step 3: Envelop $X_{2}(t)$ with a time-modulating function
$q(t|\boldsymbol{\theta}_{T})$ and obtain
$X_{3}(t)=q(t|\boldsymbol{\theta}_{T})X_{2}(t)$, where
$\boldsymbol{\theta}_{T}$ is the vector of parameters that control the
temporal energy distribution.

  * 4.

Step 4: Convolve ${X}_{3}(t)$ with a high-pass filter $h_{f}(t|f_{c})$ and the
filter outputs the simulated GM.

The above 4-step procedure includes three parametric models:
$h_{X}(t-\tau|\boldsymbol{\theta}_{F}(\tau))$ (or
$H_{XX}(\omega|\boldsymbol{\theta}_{F}(t))$), $q(t|\boldsymbol{\theta}_{T})$,
and $h_{f}(t|f_{c})$. This study chooses $h_{f}(t|f_{c})$ as the impulse
response function of a critically damped oscillator, represented by
$h_{f}(t|f_{c})=t\exp(-2{\pi}f_{c})$. The parameter estimations for the first
two models are thoroughly studied in [6, 1] and they are assumed to be already
fitted in this study. Next, we focus on fitting the final parameter, the
corner frequency $f_{c}$, by proposing an optimization criterion.

The impact of applying a high-pass filter is well-known: it slightly alters
the simulated linear-response spectrum at short periods ($T<1$s) but
significantly affects long-period responses. For example, Figure 4 presents
the spectrum $S_{a}(T)$ of a GM recording from the Chi-Chi, Taiwan earthquake
(NGA#1517 downloaded from the PEER-west2 database). The figure also shows the
mean logarithmic spectrum $\mathbb{E}(\log(S_{a}(T)))$ of the corresponding
simulated ground motions, generated using fitted model parameters
$\hat{\boldsymbol{\theta}}_{F}(t)$ and $\hat{\boldsymbol{\theta}}_{T}$, with
various $f_{c}$ values ranging from $0$ to $2$ Hz222When $f_{c}=0$Hz, the
high-pass filter is not used.. In Figure 4, it is apparent that setting
$f_{c}=0.5$ Hz provides the closest match to the recorded spectrum. Further
detailed results with $f_{c}=0.5$ Hz are shown in Figure 4. The figure
illustrates 10 samples from the fitted GM model together with the mean and
$\pm$2$\sigma$-confidence bounds of 100 GM samples. These observations in
Figure 4 and 4 inspire us to optimize $f_{c}$ such that the recorded spectrum
best falls within the confidence bounds of the simulated spectra.

Figure 3: Comparison of $\log(Sa(T))$ of a recorded GM and
$\mathbb{E}(\log(S_{a}(T)))$ of synthetic GMs when setting the corner
frequency $f_{c}$ to different values.

Figure 4: Comparison of the logarithmic spectrum response $\log(Sa(T))$ of a
recorded GM and synthetic GMs generated from the fitted stochastic GM model
using $f_{c}=0.5$Hz.

The mismatch between the recorded and simulated spectra is quantified as their
relative bias accumulated over the long-period region (i..e, $T\in[1,10]$).
Let $S_{a}^{\mathrm{real}}(T)$ and $S_{a}^{\mathrm{sim}}(T|f_{c})$ denote the
as-recorded and corresponding simulated spectra, respectively. The simulated
spectrum $S_{a}^{\mathrm{sim}}(T|f_{c})$ is conditional on $f_{c}$ and is a
random variable due to the white-noise excitation. The optimization objective
is to minimize the summation of the relative bias, denoted as:

$\epsilon(f_{c})=\left|\int_{1}^{10}\frac{\log(S_{a}^{\mathrm{real}}(T))-\mu[\log(S_{a}^{\mathrm{sim}}(T|f_{c}))]}{\sigma[\log(S_{a}^{\mathrm{sim}}(T|f_{c}))]}d\log(T)\right|,$
(1)

where $\mu[\cdot]$ and $\sigma[\cdot]$ are operators that respectively yield
the mean and standard deviation of the input random variables. The bias is
measured on the logarithmic scale and normalized by the standard deviation. In
practice, we solve the logarithmic integrals in Eq.(1) by summing the biases
on 30 logarithmically spaced points within the integral region. Additionally,
the statistical quantities in Eq.(1) are estimated through Monte Carlo
Simulation by drawing 100 samples from the fitted GM model. Finally, we
determine the optimized value of $f_{c}$ that minimizes Eq.(1) within a grid
ranging from $[0,2]$ Hz, with an interval of $\Delta f_{c}=0.01$ Hz. For the
recorded GM used in Figure 4, we obtain $f_{c}=0.54$ Hz.

It is important to note that the proposed optimized $f_{c}$ fitting method is
versatile and can be applied to other high-pass filters, such as the
Butterworth filter. To adapt it for different filters, adjustments can be made
to the grid range and the discretization step size $\Delta f_{c}$.

##  4 Sensitivity of Response Spectrum to $f_{c}$

In this section, we conduct a sensitivity analysis to quantify the impact of
$f_{c}$ variability on the variability of the linear-response spectrum. This
analysis utilizes a linear regression model, where seven GM parameters are
used as input variables and $\log(Sa(T))$ as the output. Among these
parameters, one is $f_{c}$, and the remaining six are parameters from the
stochastic GM model proposed by Rezaeian and Der Kiureghian [7]. These six GM
parameters include the logarithmic Arias intensity ($\log(AI)$), effective
duration ($D_{5-95}$), time taken to reach the middle of the strong-shaking
phase ($t_{mid}$), and three key GM parameters at $t_{mid}$: the predominant
frequency $\omega(t_{mid})$, the rate of change of the filter frequency
$\omega^{\prime}(t_{mid})$, and the filter bandwidth $\zeta(t_{mid})$.

The linear regression model is represented as follows:

$Y=\beta_{0}+\sum\limits_{n=1}^{7}\beta_{n}\theta_{n}+\epsilon=\beta_{0}+\boldsymbol{\beta}^{\mathrm{T}}\boldsymbol{\theta}+\epsilon,$
(2)

where the input random vector refers to
$\boldsymbol{\theta}=[\log(AI),D_{5-95},t_{mid},\omega(t_{mid}),\omega^{\prime}(t_{mid}),\zeta(t_{mid}),f_{c}]^{\mathrm{T}}$.
The input vector covariance matrix is $\Sigma_{\boldsymbol{\theta\theta}}$,
with the $n$-th diagonal entry representing the variance
$\sigma^{2}_{\theta_{n}}$. The regression error $\epsilon$ is a zero-mean
Gaussian random variable, incorporating two error sources: truncation error
due to linearization and unaccounted GM information excluded from the
parameters $\boldsymbol{\theta}$. The regression is repeated for multiple
spectral ordinates by setting $Y=\log(Sa(T))$ for $T\in[0.05,10]$s. The linear
regression in Eq.(2) facilitates the decomposition of the spectral variance
$\operatorname{Var}(Y(T))$ into two terms and the spectral covariance
$\operatorname{Cov}(Y(T_{1}),Y(T_{2}))$ into four terms:

$\operatorname{Var}(Y(T))=\boldsymbol{\beta}^{\mathrm{T}}(T)\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}(T)+\operatorname{Var}(\epsilon(T)),$
(3) $\begin{aligned}
\operatorname{Cov}(Y(T_{1}),Y(T_{2}))&=\boldsymbol{\beta}^{\mathrm{T}}(T_{1})\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}(T_{2})+\boldsymbol{\beta}^{\mathrm{T}}(T_{1})\operatorname{Cov}(\boldsymbol{\theta},\epsilon(T_{2}))\\\
&+\boldsymbol{\beta}^{\mathrm{T}}(T_{2})\operatorname{Cov}(\boldsymbol{\theta},\epsilon(T_{1}))+\operatorname{Cov}(\epsilon(T_{1}),\epsilon(T_{2}))\end{aligned}.$
(4)

In this study, the regression parameters $\boldsymbol{\beta}$ are fitted by
the ordinary least-squares method. The variance
$\operatorname{Var}(\epsilon(T))$ and covariances
($\Sigma_{\boldsymbol{\theta\theta}}$,
$\operatorname{Cov}(\epsilon(T_{1}),\epsilon(T_{2}))$ and
$\operatorname{Cov}(\boldsymbol{\theta},\epsilon(T))$) are computed from the
GM parameters and regression errors using the corresponding empirical
(co)variance formula.

We illustrate this analysis for the same GM catalog as used in Figure 1 and 2.
The GM parameters $\log(AI)$, $D_{5-95}$ and $t_{mid}$ can be directly
computed by their definitions. The parameters
$\omega(t_{mid}),\omega^{\prime}(t_{mid}),\zeta(t_{mid})$ are extracted from
the short-time Thomson’s multiple-window spectrum of GMs (see details in [3,
1]). The final parameter $f_{c}$ is optimized using the method proposed in
Section 3. The statistical observations of these extracted parameters, along
with their corresponding fitted probabilistic models, are provided in A.

The fitting outcomes over the spectrum ordinate $T\in[0.05,10]$s are depicted
in Figure 6 and 6. Figure 6 illustrates the coefficient of determination
($R^{2}$) and compares two cases of $f_{c}$: the optimized values versus a
constant value of 0.1Hz. A constant $f_{c}$ implies neglecting the $f_{c}$
variance (i.e., $\sigma^{2}_{\theta_{7}}=0$) and its covariance with the other
six parameters, i.e.,
$[{\Sigma}_{\boldsymbol{\theta\theta}}]_{7,n}=[{\Sigma}_{\boldsymbol{\theta\theta}}]_{n,7}=0$
for $n\in[1,2,...,6]$. The coefficient $R^{2}$ serves as an evaluation metric
for the linear models, indicating the proportion of the output variance that
the linear regression can predict, i.e.,
$R^{2}=\boldsymbol{\beta}^{\mathrm{T}}\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}/\operatorname{Var}(Y)$.
The figure indicates a significant enhancement in accounting for the total
variance for periods $T\geq 1$s when using the optimized $f_{c}$. This
improvement becomes more substantial with the increase of $T$, reaching up to
26% at $T=10$s. Moreover, by using the optimized $f_{c}$, a linear
relationship among the input-output pairs already explains $81\%\sim 95\%$ of
spectral variances; the remaining variance due to the regression error is
small333B additionally shows that the linear-regression covariance
$\boldsymbol{\beta}^{\mathrm{T}}(T_{1})\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}(T_{2})$
in Eq.(4) explains $70\%\sim 130\%$ of the total covariance
$\operatorname{Cov}(Y(T_{1}),Y(T_{2}))$.. Figure 6 compares the weighted
regression coefficients ${\beta}_{n}\times\sigma_{\theta_{n}}$ of the seven GM
parameters. The sum of squares of these coefficients equals the regression
variances (i.e.,
$\boldsymbol{\beta}^{\mathrm{T}}\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}$)
when the input variables are uncorrelated. The figure again indicates that the
coefficient related to $f_{c}$ becomes pronounced after $T=1$s and gains more
significance with increasing $T$. At long period $T>5$s, the $f_{c}$
coefficient outweighs the coefficients of the other 6 GM parameters.

Figure 5: Coefficient of determination $R^{2}$

Figure 6: Weighted regression coefficients
${\beta}_{n}\times\sigma_{\theta_{n}}$

Next, we investigate the impact of the $f_{c}$ variability in the regression
model by considering two scenarios: (1) neglecting both the $f_{c}$ variance
and covariance by treating $f_{c}$ as a constant, e.g., $f_{c}=0.1$Hz, and (2)
neglecting only the $f_{c}$ covariance. These two modifications change the
spectral variance in Eq.(3) and the spectral covariance in Eq.(4). The
comparisons before and after the modifications are shown in Figure 7 and 8.
Figure 7 suggests that neglecting either the $f_{c}$ variance or covariance
leads to an underestimation of the spectral variance in the long-period range.
The shaded areas in Figure 7 illustrate that the $f_{c}$ variance contributes
more significantly than the $f_{c}$ covariance. Specifically, the $f_{c}$
variance can contribute up to a maximum of 28% s.t.d of $\log(Sa(T))$ to the
total at $T=10$s, while the $f_{c}$ covariance contributes up to a maximum of
11% at around $T=5.4$s. Figure 8 demonstrates the impact of the $f_{c}$
variability on the spectral correlation, i.e.,
$\rho(T_{1},T_{2})=\operatorname{Cov}(Y(T_{1}),Y(T_{2}))/\allowbreak\sqrt{\operatorname{Var}(Y(T_{1}))\operatorname{Var}(Y(T_{2}))}$.
It is evident that neglecting either the $f_{c}$ variance or covariance tends
to overestimate the spectral correlation, especially between the short- and
long-period ranges. This outcome aligns with the finding in Figure 2, where
improper $f_{c}$ values in the stochastic GM model result in overestimated
correlations. The detailed analysis in Figure 8 reveals that the biased
spectral correlation could be more influenced by the $f_{c}$ covariance than
the $f_{c}$ variance.

In summary, the proposed linear regression accounts for most of the spectral
variance and spectral covariance. Specifically, it shown that a fixed value of
$f_{c}$ results in a distorted linear-response spectrum for long periods.
Moreover, neglecting the covariance between $f_{c}$ and the other six GM
parameters severely underestimates spectral variance and significantly
overestimates spectral correlation.

Figure 7: Effect of neglecting either the $f_{c}$ variance or the $f_{c}$
covariance on spectral variance. Figure 8: Effect of neglecting either the
$f_{c}$ variance or the $f_{c}$ covariance on spectral correlation.

##  5 Conclusions

This study emphasizes the importance of fitting the corner frequency ($f_{c}$)
parameter in site-based stochastic GM models and proposes a method to optimize
its fitting. Compared to previously used $f_{c}$ estimation methods, GM models
using the optimized $f_{c}$ show markedly enhanced accuracy in capturing the
linear-response spectrum statistics of recorded GMs. A sensitivity analysis of
the linear-response spectrum is conducted via linear regression. The
regression incorporates seven GM parameters including $f_{c}$ as input
variables. The analysis reveals that the value and variance of $f_{c}$ and its
correlation with the other six parameters contribute significantly to the
linear-response spectral ordinates, variability, and correlation.

##  Appendix A Statistics of extracted parameters from the 71-GM catalog

Figure 9 presents the statistical analysis of the parameters extracted from
the 71-GM catalog [7, 2]. In the diagonal positions, histograms of individual
parameters are displayed alongside their fitted probability density functions.
These marginal models for the GM parameters except for $f_{c}$ use the same
distribution types as in the Rezaeian and Der Kiureghian model [7]; $f_{c}$ is
fitted by an exponential distribution. The upper triangle exhibits Pearson’s
correlation coefficients for each pair of parameters. The lower triangle
illustrates the bi-variate distribution incorporating Gaussian dependency,
including their observed data points.

Figure 9: Statistics of seven GM parameters extracted from the 71-GM catalog.

##  Appendix B Percentages of spectral covariance decomposed by the regression
model

Figure 10 shows the percentages of the total spectral covariance explained by
the four individual terms in Eq.(4). The figure indicates that the linear-
regression covariance,
$\boldsymbol{\beta}^{\mathrm{T}}(T_{1})\Sigma_{\boldsymbol{\theta\theta}}\boldsymbol{\beta}(T_{2})$,
accounts for $70\%\sim 130\%$ of the total covariance. Contributions from the
second and third terms, i.e.,
$\boldsymbol{\beta}^{\mathrm{T}}(T_{1})\operatorname{Cov}(\boldsymbol{\theta},\epsilon(T_{2}))$
and
$\boldsymbol{\beta}^{\mathrm{T}}(T_{2})\operatorname{Cov}(\boldsymbol{\theta},\epsilon(T_{1}))$,
are negligible. The percentages of the last term, i.e., covariance between
regression errors at distinct periods, compensate for the linear-regression
covariance almost to 100%.

Figure 10: Percentage contributions of the four terms in Eq. (4) to the total
spectral covariance.

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