  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.
