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\chapter{0-5 Hz Deterministic 3D Ground Motion Simulations for the 2014 La Habra, California, Earthquake}
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\label{chap:highf}

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We have simulated 0-5 Hz deterministic simulations for the 2014 $M_w$ 5.1 La Habra, CA, earthquake in a mesh from the Southern California Earthquake Center Community Velocity Model Version S4.26-M01 with a finite-fault source. Our simulations include statistical distributions of small-scale crustal heterogeneities (SSHs), frequency-dependent attenuation $Q(f)$, surface topography, and effects of near-surface low velocity material. Strong motion data at 259 sites within an 148 km by 140 km area are used to validate our simulations. %Amplification from the near-surface %low-velocity soil column (shear-wave %velocities $<$ 500 m/s) appears at %about 0.3 Hz, increasing to factors %of 2-3 approaching 5 Hz. 
%We find that a $Q(f)$ model with a %high-frequency exponent higher %than 0.6
%high-frequency power law $Q(f)=Q_o %V_S f^{\gamma}$ with $\gamma$ less %than 0.6 
%is inconsistent with the seismic %data. 
Accuracy of the velocity model, particularly the  near-surface low velocities and 3D structure, controls the resolution to which the anelastic attenuation can be determined. We examine parameterizations of low-frequency ($<$1 Hz) shear-wave $Q$ values ($Q_{S,0}$) as a function of shear-wave speed ($V_S$), where a faster increase of $Q_{S,0}$ for larger $V_S$  generally improves the fit to the seismic data. Power-law exponents {$\gamma$} for high-frequency ($>$1 Hz) frequency-dependent shear-wave $Q_S$ expressed as $Q_S=Q_{S,0} f^{\gamma}$ larger than about 0.6 appear inconsistent with the seismic records. 
Topography, and to a lesser extent SSHs, predominantly decrease the peak velocities and significantly increase the duration, by energy redistribution. A weathering layer with realistic near-surface low velocities is found to enhance the amplification at mountain peaks and ridges. Our results show that the effects of, and trade-offs between, near-surface low-velocity material, topography, SSHs and $Q(f)$ become increasingly important as frequencies increase.


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\section{Introduction} \label{highf:intro}
It is the ultimate goal for ground motion modelers to deliver their results to engineers and see their work used in applications beneficial for society, such as structural design. This is particularly useful in cases of infrequent observations, such as for large magnitude events at short distances from the fault, where simulations may provide a viable alternative to data. Deterministic ground motion predictions, including features such as three-dimensional (3D) velocity structure and frequency-independent anelastic attenuation are now routinely produced for frequencies up to about 1 Hz with generally satisfactory fit to recorded data \citeg{gravesSimulatingSeismicWave1996,olsen2009shakeout,roten3DSimulationsEarthquakes2012}. These simulations have proved to be useful in public earthquake emergency response and seismic hazard management \citep[e.g., the SHAKEOUT scenario; ][]{jones2008shakeout}, and complementing empirical ground motion prediction models in regions with sparse stations coverage \citeg{dayModelBasinEffects2008}.

While the results of these low-frequency simulations are promising, structural engineers need ground motions with signal content up to 5 Hz and higher for design purposes. Hybrid techniques, combining deterministic low-frequency and stochastic high-frequency signals \citeg{olsen2015sdsu,gravesKinematicGroundMotion2016} can be used to generate synthetic seismograms with frequency content up to 10 Hz and higher. However, simulating both the lower and higher frequency content using a deterministic approach has the potential to lower part of the epistemic uncertainty in the resulting ground motion estimates. In this study we investigate the feasibility of increasing the highest frequency for deterministic ground motion predictions to 5 Hz, using simulations and data for the 2014 $M_w$ 5.1 La Habra, CA, earthquake. The La Habra event was chosen due to an abundance of records available in the greater Los Angeles area, while ground motions can be considered linear due to its relatively small magnitude.


As frequencies increase above about 1 Hz, features with increasingly small length scales become important to realistically predict deterministic ground motions. For example, small-scale complexity of both the source and surrounding media, on the order of 10-100s of meters, is expected to increasingly affect ground motion predictions at higher frequencies. Frequency-independent anelastic attenuation, often chosen as proportional to the local velocity structure \citeg{gravesBroadbandGroundMotionSimulation2010} is usually a good approximation for lower frequencies \citep[up to \textasciitilde 1 Hz; e.g., ][]{liu1976velocity,fehler1992separation}. However, models of frequency-independent anelastic attenuation appear to be inconsistent with seismic records at higher frequencies where regional studies indicate that larger Q values may be more accurate \citeg{withersMemoryEfficientSimulation2015}. Finally, ground motion simulations often artificially truncate the lowest near-surface velocities due to computational limitations, which may be a reasonable approximation for lower frequencies \citeg{olsenEstimationLongPeriodSec2003}. However, stronger effects from this near-surface material emerge as frequencies increase and wavelengths decrease \citeg{pitarka2009simulating,imperatoriBroadbandNearfieldGround2013}. Here, we quantify the effects of all of these features in our 3D simulations of the La Habra event.

In southern California, two state-of-the-art 3D velocity models, namely the Community Velocity Models (CVM) versions S and H, have been developed through the Southern California Earthquake Center (SCEC). These CVMs have been validated against ground motion data in a series of studies \citeg{tabordaEvaluationSouthernCalifornia2016,savranGroundMotionSimulation2019,laiShallowBasinStructure2020}.
\citet{elyVs30derivedNearsurfaceSeismic2010} proposed a method to calibrate the near-surface material based on estimates of the time-averaged velocity in the upper 30 m ($V_{S30}$), and later improved by \citet{huCalibrationNearsurfaceSeismic2021}, specifically for sites with poor constraints for shallow rock site velocities. Here, we use the SCEC CVM-S with the update by \citet{huCalibrationNearsurfaceSeismic2021}.

The effects of irregular surface topography on ground motions also play an increasingly large role as frequencies increase \citeg{liuScatteringSeismicWaves2020}. In recent studies, theoretical and numerical methods have helped clarify the interaction between seismic waves and topography \citep[mainly scattering and trapping of waves, e.g.,][]{imperatoriRoleTopographyLateral2015,takemura2015scattering,rodgersBroadband04Hz2018}, as well as describing the characteristic effects on ground motions. Some of the most notable effects of topography found by these studies are listed in the following. (1) Amplification tends to occur at the top of relatively steep slopes for waves with comparable wavelength to the size of the topographic features; on the other hand, deamplification tends to occur at low-elevation areas \citep{trifunacAnalysisPacoimaDam1971,booreNoteEffectSimple1972,spudichDirectionalTopographicSite1996,bouchonSeismicResponseHill1996,assimakiSoilDependentTopographicEffects2005}. Amplification can range up to a factor of 10 or more between the crest and base of a topographic feature \citep{davisObservedEffectsTopography1973,geliEffectTopographyEarthquake1988,umedaHighAccelerationsProduced1987,gaffetSiteEffectStudy2000}. (2) The amplification at mountain tops is systematically larger for incident S compared with P waves, with diminishing effect when the slope decreases or the incidence angle increases \citep{bardDiffractedWavesDisplacement1982}. (3) Body and surface waves are strongly scattered by irregular topography, thus reducing ground motion amplitudes while prolonging the shaking duration \citep{sanchez-sesmaDiffractionSVRayleigh1991,leeEffectsRealisticSurface2009}. (4) Topography tends to disrupt the coherency of high-frequency ground motion and thereby distorting the S-wave radiation pattern \citep{imperatoriRoleTopographyLateral2015}. Notably, 3D models of the topography are necessary to capture the amplification effects, as noted in two-dimensional simulation results \citep{geliEffectTopographyEarthquake1988,bouchonSeismicResponseHill1996}. While geometrical characteristics, such as smoothed curvature and relative elevation, have been explored to approximate topographic effects \citeg{maufroyFrequencyScaledCurvature2015,raiEmpiricalTerrainBasedTopographic2017}, they require critical parameter constraints based on local velocity and target frequency, and are thus difficult to generalize for broadband studies.

It should be noted that previous studies discussed above included only a subset of the model features deemed to be affecting high-frequency ground motions, or omitted validation of the results. In this study, we simulate ground motions for frequencies up to 5 Hz in the widely-tested SCEC CVM-S4.26M01 including high-resolution topography, and compare to strong-motion data for the 2014 $M_w$ 5.1 La Habra, CA, earthquake, in order to constrain the relative contributions from topography, SSHs, and $Q(f)$ on the ground motions. This chapter is organized as follows. We first describe the velocity model, simulation parameters, processing of the synthetic and recorded ground motions, and the source description. Then the relative effects of model features such as topography, shallow near-surface velocities, small-scale heterogeneities, and $Q(f)$ are quantified through goodness-of-fit (GOF) measures between synthetics and data. Finally, we discuss future research directions based on our results.



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\section{Model Features and Computational Aspects}\label{approach}

In this section we describe the simulation method and model setup, and summarize the features included in our model. In addition, we outline the processing parameters for simulations and data, and introduce our goodness-of-fit (GOF) measures to validate our simulations.

\subsection{Numerical method for simulating ground motions}
We use the staggered-grid finite-difference (FD) code AWP-ODC \citep[Anelastic Wave Propagation, Olsen-Day-Cui, from the authors of the code, hereafter denoted by AWP;][]{cuiScalableEarthquakeSimulation2010}, which is $4^{\text{th}}$-order accurate in space and $2^{\text{nd}}$-order accurate in time, to generate ground motion predictions for the La Habra event. AWP has been adapted to GPU accelerators for kinematic sources \citep{cui2013physics}, and provides support for frequency-dependent viscoelastic attenuation \citep{withersMemoryEfficientSimulation2015} and topography using a curvilinear grid \citep{oreillyHighorderFiniteDifference2021}. 

The accuracy of AWP has been thoroughly verified. For example, large-scale earthquake simulations in realistic 3D earth models with strong heterogeneities and complex finite-fault source descriptions \citep{bielakShakeOutEarthquakeScenario2010,bielak2016verification}, revealed good agreement between AWP, another staggered-grid FD code and a finite-element code. 
%The DM method implemented in the scalable GPU version of AWP was verified against uniform mesh solutions for the $M_w$ 5.1 La Habra earthquake \citep{rotenHighfrequencyNonlinearEarthquake2018}. 
The implementation of frequency-dependent anelastic attenuation was tested by \citet{withersMemoryEfficientSimulation2015} against a frequency-wavenumber solution, and the accuracy of the curvilinear topography implementation \citep{oreillyHighorderFiniteDifference2021} was verified against SPECFEM3D \citep{komatitschSpectralelementSimulationsGlobal2002}.


\subsection{Velocity Model}
We used a model domain of lateral dimensions 148 km by 140 km, rotated 39.9$^{\circ}$ clockwise with a depth extent of 60 km (see \cref{fig:highf-1}). The mesh was extracted from the SCEC CVM-S4.26-M01, an updated version of the original CVM-S4 model \citep{magistraleSCECSouthernCalifornia2000,kohlerMantleHeterogeneitiesSCEC2003} with iterative 3D tomography inversions in Southern California \citep{leeRapidFullwaveCentroid2011}. The SCEC Uniform Community Velocity Model software framework \citep[V19.4; ][]{smallSCECUnifiedCommunity2017} was used for the extraction of seismic P-wave velocity ($V_P$), $V_S$ and the material density. The choice of CVM-S4.26-M01 (hereafter abbreviated with CVM-S) for this study was based on the results by \citet{tabordaEvaluationSouthernCalifornia2016} who concluded from a comprehensive validation of four velocity models with 30 earthquakes in the greater Los Angeles region that this model consistently yielded the best fit to ground motion data using a variety of metrics. The model includes the near-surface $V_S$ tapering method proposed by \citet{huCalibrationNearsurfaceSeismic2021}.

\subsection{Small-scale Heterogeneities}
Small-scale crustal heterogeneities (on the order of tens to hundreds of meters) are known to exist in nature \citeg{savranModelSmallscaleCrustal2016} but are insufficiently resolved in state-of-the-art velocity models. Instead, small-scale heterogeneities are commonly included in numerical simulations via statistical models of property fluctuations \citeg{imperatoriBroadbandNearfieldGround2013,savranGroundMotionSimulation2019}. Here, we superimpose a statistical model of velocity and density perturbations onto CVM-S, defined via a von K\'arm\'an shape function \citep{frankelFiniteDifferenceSimulations1986}:

\begin{equation}\label{eq:highf-1}
  \Phi_{v, a}(r)=\sigma^{2} \dfrac{2^{1-v}}{\Gamma(v)}\left(\dfrac{r}{a}\right)^{v} K_{v}\left(\dfrac{r}{a}\right)
\end{equation}

\noindent which has Fourier transform:

\begin{equation}\label{eq:highf-2}
  P(k)=\dfrac{\sigma^{2}(2 \sqrt{\pi} a)^{E} \Gamma(v+E / 2)^{v+E / 2}}{\Gamma(v)\left(1+k^{2} a^{2}\right)}
\end{equation}
\noindent in which $k$ is the wave number and $E$ is the Euclidean dimension, $\Gamma$ denotes the Gamma function, and $K$ stands for the modified Bessel function of the second kind with order $\nu$. The parameters of the von K\'arm\'an autocorrelation function include correlation length $a$, standard deviation $\sigma$ and Hurst number $\nu$. This approach generates a random field with zero mean, and the desired standard deviation is guaranteed by scaling the random variable at each computational node. We used a fixed Hurst number of 0.05 and introduced elliptical anisotropy with a ratio of horizontal-vertical correlation lengths of 5, and tested correlation lengths between 100-5000 m, and standard deviation of 5\% and 10\%, based on previous studies in Southern California \citeg{nakata2015stochastic,savranModelSmallscaleCrustal2016}. In our model, the random perturbations extend to a depth of 7.5 km, and then linearly tapered to a standard deviation of 0 at 10 km depth \citep{olsen2018constraints}. \Cref{fig:highf-2} shows an example realization of small-scale heterogeneities, compared to the original CVM-S in terms of $V_S$ at the surface.

\subsection{Topography}
While the basins of the greater Los Angeles region, including near the epicentral area of the La Habra event, are characterized by relatively flat topographic relief, the San Gabriel and Santa Ana Mountains bound the area to the north and east, respectively (see \cref{fig:highf-1}). To quantify the effects of topography on ground motions from the La Habra event, we use the curvilinear grid approach by \citet{oreillyHighorderFiniteDifference2021}. In this version of AWP, surface topography is incorporated by stretching the computational grid in the vertical direction, while keeping the horizontal grid spacing unchanged, so that the surface grid locations conform to the shape of the topography. We include surface topography into our model domain via the $\dfrac{1}{3}$ arc-second resolution Digital Elevation Model in southern California from the U.S. Geological Survey \citep{USGS3DEP}.

\subsection{Anelastic Attenuation}
Anelastic attenuation is needed for accurate simulation of seismic wave propagation through earth models at distances further than the dominant wavelength to account for the loss of intrinsic energy. Frequency-independent attenuation, resulting in identical seismic energy loss per cycle across a frequency bandwidth, has successfully been used to validate ground motion recordings for frequencies up to about 1 Hz \citeg{olsenEstimationLongPeriodSec2003,graves2004observed}. However, as frequencies increase above about 1 Hz, data often supports frequency-dependent $Q$ \citeg{raoof1999attenuation,eberhart2014imaging,wangUsingDirectCoda2017}. To address these findings, \citet{withersMemoryEfficientSimulation2015} developed an efficient coarse-grained memory variable approach to model frequency-dependent attenuation using a power law formulation

\begin{equation}\label{eq:highf-3}
  Q(f)=Q_{0} *\left(\dfrac{f}{f_{0}}\right)^{\gamma}, \quad f \geq f_{0},
\end{equation}
\noindent where $Q_0$ is a frequency-independent $Q$ value applied for $f<f_{0}$.  %\citet{withersMemoryEfficientSimulat%ion2015} and %\citet{savranGroundMotionSimulation2%019} found that $\gamma$ values of %to 0.6-0.8 produced the best fit to %seismic records of the 2008 Chino %Hills, CA, earthquake.

A widely-used parameterization of $Q_0$ is proportional to local seismic velocity, with separate values $Q_{P,0}$ and $Q_{S,0}$ for $V_P$ and $V_S$ quality factors, respectively, producing an expected stronger attenuation for lower velocity material, as pointed out by \citet{haukssonAttenuationModelsThree2006}. \citet{taborda2014ground} revised the formula expressed by \citet{brocher2008compressional} and applied a $6^{\text{th}}$-order polynomial function for $Q_{S,0}$ from $V_S$, and $Q_{P,0}=\dfrac{3}{4}\left(V_P/V_S\right)^2Q_{S,0}$. We test a variety of these parameterizations of $Q_0$ for the La Habra event (see \cref{fig:highf-A1}).

\subsection{Near-surface Geotechnical Layer}
CVM-S includes geotechnical data which integrates geology and geophysics data from surficial and deep boreholes, oil wells, gravity observations, seismic refraction surveys and empirical rules calibrated based on ages and depth estimates for geological horizons in southern California \citep{magistraleGeologybased3DVelocity1996,magistraleSCECSouthernCalifornia2000}. While recent validation studies, such as \citet{tabordaEvaluationSouthernCalifornia2016}, have shown that the basin structure included in CVM-S is reasonably accurate, unrealistically large surface rock site velocities (see \cref{fig:highf-2}) motivated the method by \citet{elyVs30derivedNearsurfaceSeismic2010} to reduce the $V_S$ in the top 350 m based on available $V_{S30}$ values. Recently, \citet{huCalibrationNearsurfaceSeismic2021} proposed a method to further adjust the near-surface structure to a depth of 1000 m, resulting in an improved fit between simulated and recorded Fourier spectra below 1 Hz for the La Habra earthquake, which will be used in the simulations in this study.

\subsection{Ground Motion Simulations}
\Cref{tab:highf-1} lists the parameters used in our simulations. All simulations have the same duration of 120 s and resolve wave propagation up to $f_{max}=5$ Hz by at least 5 points per minimum S-wavelength. We use AWP-topo that supports a uniform regular, curvilinear mesh to model wave propagation in composite models including topography and other features, with the minimum $V_S$ clamped at 500 m/s to reduce computational cost. $V_P$ in the low-velocity material is determined by the $V_P/V_S$ ratio from (the un-clamped) CVM-S. We use a kinematic source generated following \citet{gravesKinematicGroundMotion2016}, which creates finite-fault rupture scenarios with stochastic characteristics optimized for California events. The focal mechanism was taken from the U.S. Geological Survey \citep[strike=233$^\circ$, dip=77$^\circ$, rake=49$^\circ$;][]{usgsEarthquakeEventsFocal2014} with a moment magnitude 5.1.


\subsection{Data Processing}
259 strong-motion seismic stations were used to validate the simulations. The strong motion recordings (velocity time series) are obtained from SCEC (F. Silva, Personal Communication, 07/2020), with hypocentral distance up to 90 km and signal-to-noise ratio above 3 dB. The processing procedure included the following steps: (1) low-pass filtering of the time series below 10 Hz using a zero-phase filter; (2) interpolating the time series linearly to a uniform time step; (3) tapering of at the last 2 seconds using the positive half of a Hanning window; (4) zero padding the last 5 seconds; (5) filtering the seismograms to the desired frequency, and (6) converting velocities to accelerations by a time derivative. Except for the initial 10 Hz low-pass filter, all filters used a low-cut frequency of 0.15 Hz to avoid noise interference (in the data). $4^{\text{th}}$-order Butterworth filters were used in all cases. Finally, our horizontal synthetic seismograms were rotated 39.9$^\circ$ counter-clockwise.

\subsection{Intensity Measures}
We use 7 different intensity measures to characterize the performance of our ground motion models for the La Habra earthquake, namely peak ground velocity (PGV), peak ground acceleration (PGA), energy duration (DUR), cumulative energy (ENER), response spectral acceleration averaged between 0.1 and 10 s (RS), smoothed Fourier amplitude spectrum (FAS), as well as Arias intensity (AI). We computed the SA at frequencies linearly spaced from 0.2 to 5 Hz. Cumulative energy is calculated as $E N E R=\int v(t)^{2} d t$. Both ENER and DUR are defined on the interval between the arrival of 5 and 95 percent of the total energy. Arias intensity describes the cumulative energy per unit mass \citep{arias1970measure}, and is defined as $AI=\dfrac{\pi}{2 g} \int a(t)^{2} d t$, where $a(t)$ is the acceleration time series, and $g$ is the gravitational acceleration. 

\section{Results}
\subsection{Source Models}
Due to the stochastic characteristics of the kinematic source generator by \citet{gravesKinematicGroundMotion2016}, a series of 40 source realizations with different random seeds are evaluated based on comparisons between spectral accelerations with records at stations with epicentral distance of 31 km or less (R. Graves, Personal Communication, 03/04/2020). The 40 source models, using a fault area of 2.5 km x 2.5 km, were rated based on the average absolute bias between synthetics and data up to 5 Hz for the median pseudo-spectral acceleration rotated over all azimuths (rotD50), from which we selected the three best performing source descriptions with hypocentral depths at 5, 5.5 and 6 km (see \cref{fig:highf-A2}). The rupture duration of the source descriptions is less than 2 s and sampled at an interval of 0.0006 s, identical to the time step used in our simulations. The three sources tend to generate overall similar patterns of PGV within the same bandwidth of low (< 2.5 Hz) or high (> 2.5 Hz) frequencies (\cref{fig:highf-A3}). Based on this result, we carry out our analysis with Source 1 (\cref{fig:highf-3}) only in order to limit the computational requirements.

\subsection{Minimum \textbf{$V_S$}}
Southern California features several low-velocity basins where the lowest $V_S$ in CVM-S can be much lower than the minimum value of 500 m/s that we imposed in our models (see \cref{tab:highf-1}). As previous studies have pointed out, soft soils, characterized by lower $V_S$, have been found to generate significant amplification of ground motions \citeg{anderson1984model}. Reducing the minimum $V_S$ will, however, increase the computational cost for the series of 3D numerical simulations needed in our analysis beyond the available resources. For example, clamping the minimum $V_S$ at 200 m/s instead of 500 m/s requires about 40 times more node hours for a single simulation. 

For this reason, we use a computationally much less expensive 1D method, which models vertically-incident SH waves in a horizontally-layered halfspace \citeg{dayRMSResponseOnedimensional1996,thompsonTaxonomySiteResponse2012}, to account for effects of the material with $V_S$ less than 500 m/s. At each site, we calculated the response from two 1D models, one using the velocity profile from our models, and the other using the same profile but with minimum $V_S$ clamped at 500 m/s. In this way, the ratio of the two 1D results characterizes the effects of the material with $V_S$ less than 500 m/s, which is then convolved with our 3D simulations. Because the SH1D method considers SH waves only, we will apply this calibration to horizontal components only. 
%This procedure is encouraged by the %results of 3D simulations with %minimum $V_S$ of 200 m/s and 500 m/s %using the domain in %\cref{fig:highf-1}, showing %negligible differences on the %vertical component %(\cref{fig:highf-A4}).

\Cref{fig:highf-4} illustrates the results of applying the SD1D method low-velocity correction for an example site. The two profiles show similar SH1D responses below about 0.3 Hz, above which the SH1D response ratio slowly trends upward with frequency, depicting the amplification from the material with $V_S$ less than 500 m/s. The PGV of the horizontal synthetic with the correction is increased by 32\% relative to that with $V_S$ clamped at 500 m/s (\Cref{fig:highf-4}c). The correction leaves the shape of the waveform almost unchanged. The smoothed Fourier spectra (\Cref{fig:highf-4}d) further suggests that clamping $V_S$ at 500 m/s may be reasonable for frequencies up to 0.8-1 Hz. Similar effects are observed for all profiles with near-surface velocity lower than 500 m/s, and we therefore apply this technique to all our 3D simulations with minimum $V_S$ clamped at 500 m/s.



\subsection{Topography}
In this section we investigate the effects of topography, which are often ignored in numerical simulations \citeg{graves2004observed,olsenStrongShakingAngeles2006,savranGroundMotionSimulation2019}. Our analysis of topographic effects uses a reference model with topography removed.
\Cref{fig:highf-5} shows the percent difference between models with and without topography for PGV, DUR and AI for bandwidths of 0.15-1 Hz, 1-2.5 and 2.5-5 Hz. It is clear that topography complicates the wavefield pattern significantly, even at frequencies below 1 Hz in terms of DUR and thus AI. Consistent with previous studies \citeg{hartzell1994initial, leeEffectsRealisticSurface2009}, we observe a weak deamplification of PGV below 1 Hz in basin areas, while that mountain peaks and ridges may amplify PGV by less than 50\%. In addition, we find that PGV is reduced by about 30\% in the Chino Basin and northwest of San Gabriel Mountains. These results are in agreement with \citet{maEffectsLargeScaleSurface2007} who found that the San Gabriel Mountains scatter surface waves from a northern rupture on the San Andreas Fault and reduce the PGVs in the LAB by up to 50\%. We interpret these results as shielding and focusing effects on the front and back sides of the mountains, respectively, which become more significant at higher frequencies, in agreement with \citet{liuScatteringSeismicWaves2020}. In addition, at frequencies increase above 2.5 Hz, we observe a clear pattern of “amplification-deamplification-amplification” along the N-S (short) axis of the San Gabriel Mountains, which is predicted in numerical experiments by \citet{liuScatteringSeismicWaves2020}.

It is particularly noticeable that DUR within 10 km of the source is strongly amplified for both low and high frequencies, mostly to the north (northwest end of the Santa Ana Mountains). Here, topography seems to act as a significant source of scattering that increases the wave duration on the sides of the mountain facing the incoming wavefields, while DUR is reduced on the ``back'' sides of the mountain seen from the source location. At further distance from the source, our results show a clear negative correlation between the effects on PGV amplification and DUR lengthening, suggesting that topography redistributes seismic energy from the large-amplitude first arrivals to the adjacent coda waves. These results are in agreement with \citet{leeEffectsRealisticSurface2009} who noticed that the effects from topography can interfere with those from path and directivity.



\subsection{Anelastic Attenuation}\label{highf:qf}
\Cref{fig:highf-6} shows a comparison between PGVs extracted from records and synthetics for a simulation with Model 1, and
\Cref{fig:highf-7} shows the horizontal- and vertical-component FAS and FAS bias from Models 1, 3, 4 and 5 (see \cref{tab:highf-2}). As expected, the frequency-dependent and constant $Q$ models diverge above 1 Hz. Among the four attenuation models, $Q_S=0.075V_Sf^{0.4}$ (Model 4) fits the vertical component the best, $Q_S=0.1V_S$ (Model 3) and $Q_S=0.05V_Sf^{0.6}$ (Model 5) provide the best fit to the FAS of the data for horizontal components, though Model 3 (Model 5) underpredicts the vertical component above 3 Hz (below 3 Hz). 

As for FAS, the spatial distribution of peak ground motions, e.g. PGV, varies significantly between various $Q(f)$ models (\cref{fig:highf-8}). The $Q_S=0.1V_Sf^{0.6}$ model strongly overpredicts the high-frequency (>2.5 Hz) PGVs in the basins toward the south and west, while providing a fairly good match at distances greater than 40 km. The more attenuated $Q_S=0.075V_Sf^{0.4}$ and $Q_S=0.1V_S$ models, on the other hand, generate moderate overprediction in the near-source regions while underpredicting the PGVs at farther distances. 

The consistent overprediction within 40 km for all $Q$ models may be caused by the omission of the near-surface low-velocity material ($<$500 m/s) when inverting for the source description. Another potential cause of the near-source overprediction could be a very low $Q$ in the shallow sediments (more abundant in the near-source area as compared to larger epicentral distances), as proposed by \citet{houghAttenuationAnzaCalifornia1988} and examined in the numerical simulations by \citet{withersGroundMotionIntraevent2019}. The presence of such thin, near-surface layer with very low Q would also alleviate at least part of the overprediction of the duration (DUR), as obtained for most models. On the other hand, it is intriguing that the PGVs predicted by all tested $Q$ models appear to decay faster than data beyond 40 km, roughly at the boundaries of the LAB. This observation may suggest unrealistically large contrasts in the shear impedance near the basin boundaries in the CVM, causing excessive entrapment of waves (see \cref{fig:highf-2}). Another explanation for the underprediction of ground motions beyond epicentral distances of 40 km could be higher $Q$ in the surrounding mountain areas compared to those in the sedimentary basins. We recommend further research into these discrepancies.

The presence of shallow low velocities is crucial in determining the best-fitting $Q$ models. \Cref{fig:highf-9} shows the comparison of PGV and DUR in 0.15-2.5 Hz and 2.5-5 Hz for models with and without shallow velocity taper. In addition to the amplification on PGV due to increasing $Q$, the 1000 m taper lowers the shallow velocity and increase the PGVs beyond 30-40 km, where most rock sites locate. Furthermore, the desired larger ground motion amplitudes at farther distance cannot be achieved by even larger $Q$, which will generate substantial overprediction of DUR (\cref{fig:highf-9}); while velocity taper has minor influence on DUR and thus is preferred.

\subsection{Small-scale heterogeneities}
\Cref{fig:highf-10} shows the effects of adding a von K\'arm\'an distribution of small-scale heterogeneities (SSHs) with $\sigma = 5\%$ and horizontal correlation length of 5000 m compared to a reference model without SSHs on PGV and DUR in the frequency bands 0.15-2.5 Hz and 2.5-5 Hz. \citet{savranGroundMotionSimulation2019} studied a smaller region up to 2.5 Hz, roughly in the center of our simulated domain, found that SSH presents itself as a second-order source of misfit that yields relatively small influence on ground motions. Our results generally agree with their findings, and further show that SSHs have stronger effects at farther distance and in higher frequencies.

\citet{przybillaEstimationCrustalScattering2009} performed analysis using elastic radiative transfer theory and showed that the direction dependence of scattering can be identified by $ak$, where $a$ is the correlation length and $k$ is the wave number. For $ak \approx 1$, waves interact with heterogeneous medium most intensively because the wavelength and correlation length are in the same order. When $ak\gg 1$ waves are predominantly scattered in the forward direction, which generates geometric focusing in the early arrivals and leads to larger peak amplitudes, and vice versa for $ak \ll 1$. In the case with $a=5000$ m, our model generally confines to the forward scattering regime and PGV is amplified, which is shown quantitatively in \Cref{fig:highf-11}. As the frequency increases, $k$ and thus $ak$ tends to increase as well, causing generally weaker scattering effects as deviating from the most intensive scattering regime, thus the median difference closer to 0 for both PGV and DUR. However, higher frequencies wavefields are capable of resolving finer structure, and may induce larger spatial variability in scattering effects, represented by larger standard deviations. More SSH realizations with various standard deviations and correlation lengths of the random field are examined (see \Cref{fig:highf-A5}), which shows a general trend that larger standard deviation and correlation length yield stronger SSH effects.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion and Conclusions}
In this study we have explored the effects of a series of different model features on the resulting ground motions for the 2014 M5.1 La Habra, CA, earthquake. Clearly, trade-offs between the parameters complicates or even inhibits determining a unique set of model parameters creating a best fit to the data. 
In order to quantitatively rate the performance of the different model features, we used a modified subset of the goodness-of-fit (GOF) metrics proposed by the methods of \citet{andersonQuantitativeMeasureGoodnessOfFit2004} and \citet{olsenGoodnessoffitCriteriaBroadband2010} developed for the comparison of broadband seismic traces (0 to 10+ Hz).
The GOF score for each metric is defined as
\begin{equation}\label{qe:highf-4}
  G_{\text {metric }}=10 \operatorname{erfc}\left(\dfrac{2|x-y|}{x+y}\right).
\end{equation}
\noindent where $x$ and $y$ are two positive scalars from the selected metrics. $G_{\text {metric}}$ is computed for each metric and combined into a weighted average using all 3 components. We used weights of 0.5, 0.5, 1, 0.5, 0.5, 1, and 1 for for PGV, PGA, DUR, AI, ENER, RS, and FAS, where reduced weights are chosen due to correlation between metrics \citep{olsenGoodnessoffitCriteriaBroadband2010}. The GOF score for the entire simulation is calculated as the average of $G_{\text {station }}$ across all 259 stations. GOF values between two signals above 4.5 and 6.5 are considered fair and very good fit, respectively.

The GOF scores for the models (see \cref{tab:highf-2}) explored in this study are shown in \Cref{fig:highf-A7}, see also \Cref{fig:highf-12} and \Crefrange{fig:highf-A11}{fig:highf-A32} in \Cref{highf:app} for results for individual models. 
Note that the model ID does not represent any ranking of their goodness of fit against seismic records. The models generally achieve GOF in the range of 5.1 to 5.4 for the low frequencies (< 2.5 Hz), with the lowest values for models 15, 16 and 20, and the highest value for models 4, 6, and 11. The GOF scores for the high frequencies (> 2.5 Hz) are typically about an increment of one smaller than those for the low frequencies, caused by a combination of increased uncertainty in the source description and model parameters of the former. Among the investigated models, Models 3, 4, 6, and 21 achieve the highest GOF values for 2.5-5 Hz, while Models 9, 15, and 19 reflect the lowest values. Note that Model 9, characterized by a 350 m taper of the near-surface velocity modification, yields fair GOF values for frequencies below 2.5 Hz while sacrificing the fit for high-frequency waves, consistent with the recommendations of a 1000 m taper by \citet{huCalibrationNearsurfaceSeismic2021}. The smallest GOF value is obtained for model 15, indicating that SSH with standard deviation of 10\% may be too heterogeneous. The consistently lower GOF above 2.5 Hz for Models 17-22 indicates the necessity of including mountain topography in high-frequency simulations.


%\subsection{Topography and Shallow Velocity Taper}\label{highf:topoandtaper}

\citet{huCalibrationNearsurfaceSeismic2021} showed that reducing shallow velocities in CVM-S at poorly constrained sites in the greater Los Angeles area by a generic taper function based on $V_{S30}$ was able to improve the fit of ground motion synthetics to data for the La Habra earthquake, particularly in regions constrained by limited geological information.
However, since their tests were restricted to frequencies below 1 Hz and models with a flat free surface. Here, we examine the efficacy of the method for frequencies up to 5 Hz, while adding topography to the models from \citet{huCalibrationNearsurfaceSeismic2021}.
As in their study we divided the 259 strong motion recording sites into two groups: type A sites representing sites with good geological constraints, and type B sites with poor geological constraints, characterized by unrealistically large surface $V_S$ in the top 500-1000 m in CVM-S. \Cref{fig:highf-13} shows the median FAS for both types of sites from various models (see \cref{tab:highf-2} for a list of model features). As observed by \citet{huCalibrationNearsurfaceSeismic2021}, type A sites are largely unaltered by the shallow velocity tapering methodology. The original CVM-S (Model 11) significantly underpredicts the FAS at type B sites up to 5 Hz. 

Topography causes generally small effects in terms of FAS (\Cref{fig:highf-13}, Model 1 versus Model 17), except for the increase on the vertical component for type B sites for frequencies larger than 1.5 Hz. %It is interesting that we %observed equally strong %effects in vertical components %compared to horizontal %components, which is somewhat %different from previous %experiments \citep[][and the %references %therein]{massaOverviewTopograp%hicEffects2014}. Moreover, the %models with $\gamma=0.6$ %(Model 1, 10, 16) show a trend %%of growing overprediction %above 2-3 Hz, which can be %improved by reducing $\gamma$ %to 0.0 (Model 3, %frequency-dependent $Q$). %However, given the tendency of %slight overprediction of the %SH1D correction above 2.5 Hz, %this zero $\gamma$ value is %supposed to be aggressive. %Additionally, the vertical %component, which is not %affected by the SH1D %correction, suggests that %$\gamma=0$ may underpredict %the FAS between 3-5 Hz, and %thus an intermediate $\gamma$ %may be appropriate.
Previous studies attempting to capture the effects of topography on ground motions and establish proxies to characterize such effects typically have used simple homogeneous models of earth material, e.g. \citet{maufroyFrequencyScaledCurvature2015} and \citet{raiEmpiricalTerrainBasedTopographic2017}.  These studies found that topographic curvature is a good proxy characterizing irregular surface in evaluating topographic effects. However, assessment of topographic effects on ground motions are complicated by amplification due to the presence of shallow weathering layer of low velocities, typically present in mountain regions, omitted in these studies. Here, we re-assess these findings including the amplification effects from the modification of near-surface material at type B sites proposed by \citet{huCalibrationNearsurfaceSeismic2021}. We calculated the smoothed curvature of topography with a smoothing window of 640 m.
%(a much shorter smoothing %window of 120 m is also tested %considering our high frequency %bound, as shown in %\Cref{fig:highf-A3}, which %yields similar results).
Steeper relief is characterized by larger curvature values, while flatter regions are of curvature close to zero. \Cref{fig:highf-14} shows the percent difference in PGV caused by including topography for varying curvature in the simulated region. The response of two models are shown, one with 1000 m $V_S$ taper \citep{huCalibrationNearsurfaceSeismic2021} and one without (original CVM-S). The model without the near-surface low velocities introduced by the $V_S$ taper method tends to reduce the PGVs by about up to 40\% for most curvatures below 2.5 Hz, with a broader spectrum of de-amplification (up to 75\%) and amplification (up to 40-100\%, most pronounced for the largest curvatures) at frequencies between 2.5 and 5 Hz. These trends, however, becomes much more notable when the shallow low velocities are present, where steep topography (e.g., mountain summits and local steep hills) increases PGVs. This result may partly explain why in previous studies, omitting the near-surface model complexity, topographic effects on ground motions tend to underestimate the amplification at mountain tops compared to observations \citeg{pischiuttaTopographicEffectsHill2010,lovati2011estimation}.


The accuracy of the source description is critical for obtaining reliable estimates of the parameters controlling model features such as $Q(f)$. The overprediction for near-source epicentral distances and distance decay faster than that for data at further distances of the PGVs (see \cref{fig:highf-8,fig:highf-9,fig:highf-12}) may at least partly be explained by uncertainty in the source moment and/or fault area. While the moment is likely relatively well constrained, a somewhat larger fault area  would decrease near-source PGVs and facilitate propagation of additional seismic energy to further distances. We recommend using simulations with additional (ensembles of) sources, to further examine the model features in this work.


In addition to the source description, inaccuracies in the velocity structure further complicates estimation of the optimal model parameters, in particular for the $Q$ model, as pointed out by \citeg{savranGroundMotionSimulation2019,laiShallowBasinStructure2020}. Withers et al. (2015) estimated higher $\gamma$ values (near 0.8) from their modeling of the Chino Hill earthquake. However, this estimate may be biased by the use of a CVM-S with near-surface rock velocities biased high, as well as smaller source-station distances and lower maximum frequency. Savran and Olsen (2019) used $\gamma$ equal to 0.6, but with limited high-frequency resolution up to 2.5 Hz. 
Nevertheless, future improvement in community velocity models, wider access to computational resources, more efficient numerical codes and guidance from this study are bound to further constrain the ground motion models, leading to more accurate seismic hazard analysis.


\section*{Data and Resources}
The UCVM program used to extract velocity meshes can be obtained from SCEC on \url{https://github.com/SCECcode/UCVMC} (last accessed 12/2020). The simulations were performed on Summit at the Oak Ridge Leadership Computing Facility in Tennessee. Most of the data-processing work was done using Python and the Generic Mapping Tools package (\url{https://www.generic-mapping-tools.org}, last accessed 04/2021).


\section*{Acknowledgements}
\addcontentsline{toc}{section}{\protect\numberline{}Acknowledgements}

This research was supported through the U.S. Geological Survey External Program (award \#G19AS00021), as well as the Southern California Earthquake Center (SCEC; Contribution Number xx). SCEC is funded by the National Science Foundation (NSF) Cooperative Agreement EAR-1600087 and the U.S. Geological Survey (USGS) Cooperative Agreement G17AC00047. We thank Robert W. Graves for providing the source models and Fabio Silva for providing the station records of the 2014 La Habra earthquake.

\Cref{chap:highf}, in full, is a reformatted version of a paper currently in preparation for submission for publication: Hu, Z., Olsen, K.B. and Day, S.M. (2021), 0-5 Hz Deterministic 3D Ground Motion Simulations for the 2014 La Habra, California, Earthquake. The dissertation author was the primary investigator and author of this paper.


\newpage
\section*{Tables and Figures}
\addcontentsline{toc}{section}{\protect\numberline{}Tables and Figures}%

%% For very long table
% \clearpage
% \begin{sidewaystable}[!ht]
% \caption{Coregionalization matrix $\mathbf{P}^\mathbf{3}$}
% \begin{adjustbox}{width=\textwidth,center}
% \begin{tabular}{|c|cccccccccccccccccccccccccccccccc|c|}
% \end{tabular}
% \label{tab:5-S3}
% \end{adjustbox}
% \end{sidewaystable}

\begin{table}[!ht]
  \vrule depth 2pt width 0pt
  \caption{Simulation parameters used for the deterministic ground motion simulations of the 2014 La Habra earthquake.}
  \label{tab:highf-1}
  \resizebox{0.7\textwidth}{!}{%
    \renewcommand{\arraystretch}{1.2}%
    \begin{tabular}{@{}lc@{}}
      \toprule
      \textbf{Model}               &  \\ \midrule
      Topography                    & Yes\\
      Length                        & 147.840 km\\
      Width                         & 140.400 km\\
      Depth                         & 58.000 km\\
      Northwest corner              & \multicolumn{1}{l}{-118.0154409, 34.8921683}\\
      Southwest corner              & \multicolumn{1}{l}{-118.9774168, 33.9093124}\\
      Southeast corner              & \multicolumn{1}{l}{-117.7401908, 33.0695780}\\
      Northeast corner              & \multicolumn{1}{l}{-116.7729754, 34.0429241}\\[1.5mm]%  \midrule
      \textbf{Spatial resolution}   & \multicolumn{1}{l}{}\\[1mm]%  \midrule
      Maximum frequency             & 5 Hz\\
      Minimum $V_S$                 & 500 m/s\\
      Points per minimum wavelength & 5\\
      Grid discretization           & 20 m\\
      Number of cells               & 150,486,336,000\\
      Number of GPU processors      & 1,512\\
      Wall-clock time               & 5 hr\\[1.5mm]% \midrule
      \textbf{Temporal resolution}  & \multicolumn{1}{l}{}\\[1mm]% \midrule
      Time discretization           & 0.0006 s\\
      Simulation time               & 120 s\\
      Number of timesteps           & 200,000\\ \bottomrule
    \end{tabular}%
  }
\end{table}
\clearpage

\begin{table}[!ht]
  \begin{threeparttable}
    \vrule depth 2pt width 0pt
    \centering
    \caption{Summary and main features of the models used in this study.}
    \label{tab:highf-2}
    \resizebox{\textwidth}{!}{%
      \renewcommand{\arraystretch}{1.4}%
      \begin{tabular}{@{}ccccc@{}}
        \toprule
        Model ID & Topography & $Q(f)$\tnote{\textsuperscript{*}} & SSH\tnote{\textdagger} & $V_S$ Taper Depth (m)\\ \midrule
        1        & Yes        & $Q_S=0.1V_Sf^{0.6}$               & No                     & 1000  \\
        2        & Yes        & $Q_S=0.1V_Sf^{0.3}$               & No                     & 1000  \\
        3        & Yes        & $Q_S=0.1V_S$                      & No                     & 1000  \\
        4        & Yes        & $Q_S=0.075V_Sf^{0.4}$             & No                     & 1000  \\
        5        & Yes        & $Q_S=0.05V_Sf^{0.6}$              & No                     & 1000  \\
        6        & Yes        & $Q_S=Q_{seg}(V_S)f^{0.4}$\tnote{\textsuperscript{**}}            & No                  & 1000  \\
        7        & Yes        & $Q_S=Q_{poly}(V_S)f^{0.4}$\tnote{\textsuperscript{***}}           & No                  & 1000  \\
        8        & Yes        & $Q_S=0.1V_Sf^{0.6}$               & No                     & 700\\
        9        & Yes        & $Q_S=0.1V_Sf^{0.6}$               & No                     & 350\\
        10        & Yes        & $Q_S=0.1V_Sf^{0.6}$               & No                     & 0  \\
        11        & Yes        & $Q_S=0.1V_S$               & No                     & 0  \\
        12        & Yes        & $Q_S=Q_{seg}(V_S)f^{0.4}$\tnote{\textsuperscript{**}}        & $\sigma=5\%$, $a=100m$ & 1000\\
        13        & Yes        & $Q_S=Q_{seg}(V_S)f^{0.4}$\tnote{\textsuperscript{**}}        & $\sigma=5\%$, $a=500m$ & 1000\\
        14       & Yes        & $Q_S=Q_{seg}(V_S)f^{0.4}$\tnote{\textsuperscript{**}}        & $\sigma=10\%$, $a=100m$ & 1000\\
        15       & Yes        & $Q_S=Q_{seg}(V_S)f^{0.4}$\tnote{\textsuperscript{**}}        & $\sigma=10\%$, $a=500m$ & 1000\\
        16        & Yes        & $Q_S=0.1V_Sf^{0.3}$              & $\sigma=5\%$, $a=5000m$ & 1000\\
        17       & No         & $Q_S=0.1V_Sf^{0.6}$               & No                     & 1000  \\
        18       & No         & $Q_S=0.1V_Sf^{0.6}$               & No                     & 700\\
        19       & No         & $Q_S=0.1V_Sf^{0.6}$               & No                     & 350\\
        20       & No         & $Q_S=0.1V_Sf^{0.6}$               & No                     & 0  \\
        21       & No         & $Q_S=0.1V_S$                      & No                     & 1000  \\
        22       & No         & $Q_S=0.1V_S$                      & No                     & 0  \\\bottomrule
        \\[-5mm]
      \end{tabular}%
    }
    \begin{tablenotes}
      \item[\textsuperscript{*}] \footnotesize $Q_P=2Q_S$\\[-6pt]
      \item[\textsuperscript{**}] \footnotesize $Q_{seg}(V_S)=0.075V_S$, for $V_S <= 1000$ m/s; $Q_S=0.2V_S - 125$, for $V_S > 1000$ m/s\\[-3pt]
      \item[\textsuperscript{***}] \footnotesize $Q_{poly}(V_S)=10.5 - 16V_S + 153V_S^2 - 103V_S^3 + 34.7V_S^4 - 5.29V_S^5 + 0.31V_S^6$, for $V_S$ in km/s, see \citet{taborda2014ground}\\[-4pt]
      \item[\textdagger] \footnotesize When included, hurst number = 0.05 and orizontal to vertical correlation length ratio = 5
    \end{tablenotes}
  \end{threeparttable}
\end{table}
\clearpage

% %%%%%%%%%%%%% figures 

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth]{figures/figure_highf_1.png}
  \caption{Simulation domain for the La Habra earthquake (purple solid rectangle) and locations of 259 strong motion stations (black triangles). The star denotes the epicenter. The geographical coordinates of the corners of the simulated domain is listed in \cref{tab:highf-1}, which is used in subsequent map views.}
  \label{fig:highf-1}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.85\textheight,keepaspectratio]{figures/figure_highf_2.png}
  \caption{Illustration of the imprint of small-scale heterogeneities at the surface. (a) $V_S$ extracted from the CVM-S. (b) Same as (a) but superimposed with a statistical model of heterogeneities with a correlation length of 100 m, anisotropy factor of 5, Hurst number of 0.05 and standard deviation of 5\%. Topography is removed in (b) for clarity. The epicenter for the La Habra earthquake is depicted with a star.}
  \label{fig:highf-2}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.85\textheight,keepaspectratio]{figures/figure_highf_3.pdf}
  \caption{Description of the selected source model used in this study. (a) Slip distribution (shading), with contours representing rupture time at a 0.4 s interval starting from 0. (b) and (c) represent the sum of the moment rates for all subfaults and the Fourier amplitude spectrum, respectively. A Brune-type  $\omega^{-2}$ decay source \citep{brune1970tectonic} that fits the source spectrum is plotted for reference.}
  \label{fig:highf-3}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_4.pdf}
  \caption{Illustration of the SH1D method used to include the effects of material with $V_S$ less than 500 m/s in our 3D simulations for an example site. (a) $V_S$ profile extracted from CVM-S (red dashed curve) and clamped at 500 m/s (blue). (b) SH1D response ratio between the profiles without clamping and with clamping of $V_S =500$ m/s. (c) Synthetics from a 3D simulation using $V_S=500$ m/s, with and without the SH1D response ratio. (d) Fourier amplitude spectra corresponding to the waveforms in (c).}
  \label{fig:highf-4}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_5.png}
  \caption{Percent difference of PGV (the first row) and DUR (the second row) at the surface determined by the model with topography and the model without topography for (left) 0.15-1 Hz, (center) 1-2.5 Hz, and (right) 2.5-5 Hz. Positive (negative) values colored in red (blue) indicate amplification (deamplification). The star denotes the epicenter.
  }
  \label{fig:highf-5}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_6.png}
  \caption{Comparison of interpolated PGVs measured at 259 stations, depicted by triangles, for (a) data and (b) synthetics using Model 1 (including topography, 1000 m shallow velocity refinement and frequency-dependent attenuation $Q_S=0.1V_Sf^{0.6}$, $Q_P=2Q_S$; see \cref{tab:highf-2}). The star denotes the epicenter. (c) PGV against $R_{hypo}$ for data and synthetics. The left and right columns show band-limited results for 0.15-2.5 Hz, and 2.5-5 Hz, respectively.
  }
  \label{fig:highf-6}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_7.pdf}
  \caption{FAS computed from records and models with various attenuation models (blue: Model 1, violet: Model 3, red: Model 4, green: Model 5). The left (right) column shows results for the horizontal (vertical) components. The top row shows the FAS amplitudes and the bottom show shows the FAS bias between models and records, calculated as the 10-based log between simulations and data. The solid lines depict the median FAS over all 259 stations. The shading shows the 95\% confidence interval (CI) and the dashed lines denote one standard deviation centered at the median.
  }
  \label{fig:highf-7}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_8.png}
  \caption{(a-c) Spatial distribution of the three-component bias for PGV, band pass filtered between 2.5 and 5 Hz. The bias values are computed as the base 10 logarithm of the ratio between simulations and records at each strong motion site. Positive (negative) values represent overprediction (underprediction). (d) Moving average of the bias of PGV using a 20-point window from the three $Q$ models (red: Model 1, green: Model 3, blue: Model 4; see \cref{tab:highf-2}) shown in (a-c) versus hypocentral distance.
  }
  \label{fig:highf-8}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_9.pdf}
  \caption{Bias of (a-b) PGV and (c-d) DUR for passbands (left) 0.15-2.5 Hz and (right) 2.5-5 Hz at all 259 stations. The bias is calculated in the same way as for \Cref{fig:highf-7}. The solid lines depict the moving average of the bias using a 20-point window for each of the $Q$ models (blue: Model 11, red: Model 10, green: Model 3, orange: Model 1; see \cref{tab:highf-2}) versus hypocentral distance.
  }
  \label{fig:highf-9}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_10.png}
  \caption{Difference in (top row) PGV and (bottom) DUR (bottom row) from Model 16, including SSH with $\sigma = 5\%$ and $a = 5000$ m, versus Model 1 (no SSHs). Left (right) columns show results for bandwidths 0.15-2.5 Hz (2.5-5 Hz). The star depicts the epicenter.
  }
  \label{fig:highf-10}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_11.pdf}
  \caption{Probability density histogram of the difference between Model 16, including SSH with $\sigma = 5\%$ and $a = 5000$ m, and Model 1 (no SSHs). The definition of percent difference (x-axis) is the same as in \Cref{fig:highf-5}.
  }
  \label{fig:highf-11}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_12.pdf}
  \caption{Bias of (top row) PGV and (center row) DUR and (bottom row) GOF for bandwidths (left column) 0.15-2.5 Hz and (right column) 2.5-5 Hz at all 259 stations for Model 6 (see \cref{tab:highf-2} for a list of model features). The bias is calculated in the same way as for \Cref{fig:highf-9}. The solid line depicts the moving average using a 20-point window. The shading denotes the standard deviation centered at the mean.}
  \label{fig:highf-12}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_13.pdf}
  \caption{Bias of FAS on the (a) east-west, (b) north-south and (c) vertical components, calculated from models labeled by their IDs. A positive (negative) value depicts overprediction (underprediction). The left and right columns shows type A and B sites, respectively. The solid lines depict the median of FAS, where the narrow band is the 95\% confidence interval of the median, and the dashed lines depict the standard deviation centered at the median.
  }
  \label{fig:highf-13}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_14.pdf}
  \caption{
 Density of PGV change for models with topography relative to models without topography for bandwidths of (left column) 0.15-2.5 Hz and (right column) 2.5-5 Hz, and models with (top row) and without (bottom row) modified shallow velocities. The y-axis depicts topographic curvature smoothed using a 2-D window of dimensions 640 m $\times$ 640 m. Values toward the top right (bottom left) denote strong amplification at steep topography (deamplification at flat topography). Note that density intervals do not correspond to constant bin sizes.}
  \label{fig:highf-14}
\end{figure}
\clearpage

%% SH1D correction effects. Discarded 071721 fro conciseness.
% \begin{figure}[!ht]
%   \centering
%   \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_15.pdf}
%   \caption{\purple{Bias of (top row) PGV and (center row) DUR and (bottom row) GOF for bandwidths (left column) 0.15-2.5 Hz and (right column) 2.5-5 Hz at all 259 stations for model 6, with (blue) and without (red) the 1D correction the (see \cref{tab:highf-2} for a list of model features). The bias is calculated in the same way as for \Cref{fig:highf-9}. The solid line depicts the moving average using a 20-point window.}}
%   \label{fig:highf-15}
% \end{figure}
% \clearpage

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\section*{Appendix}\label{highf:app}
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%% Left out on 071721
% \subsection{Correction for Near-surface Material with $V_S$ $<$ 500 m/s}\label{highf:vs200} 
% In this study, we constrained our 3D simulations to using $V_S$ $>$ 500 m/s due to limitations in computational resources, and added 1D corrections for the effects of the material with $V_S$ $<$ 500 m/s. In order to assess the effect of these 1D corrections we used a 3D simulation using CVM-S with minimum $V_S$ of 200 m/s with a discontinuous mesh version of AWP
% \citet{nieFourthOrderStaggered2017}. The wavefields are exchanged within an overlap zone between two media partitions with a factor-of-three change in grid spacing, which significantly reduces the number of grid points needed in high-velocity regions and thereby improves the efficiency. The improved efficiency of the DM approach allows us to lower the minimum $V_S$ to 200 m/s. The simulated domain (\Cref{tab:highf-1}) is discretized into three partitions: 1) dx=8 m from the surface to 1,472 m, 2) dx=24 m between 1,472 m and 10,336 m, and 3) dx=72 m at deeper levels. The reason for not using the more efficient DM code for other simulations in this study is that the code is limited to a flat free surface condition.
% %limit comparison to flat %sites?
% % Kim, you mean filtering sites by elevation lower than some threshold, or low velocity (soil sites), or by curvature ? I tried using sites with elevation <= 1000m, or sites with surface Vs <= 1000m/s, which both have limited influence.

% \Cref{fig:highf-A4} shows the FAS bias of two 3D simulations with minimum $V_S$ of 200 m/s and 500 m/s, both using a flat free surface boundary condition. We notice  that the lower velocity material has very little effect on the vertical component, justifying only applying the correction to the horizontal components. The SH1D correction is applied to the horizontal components of the simulation with minimum $V_S$ of 500 m/s. For the horizontal components, the corrected simulation matches the minimum $V_S$ of 200 m/s simulation fairly well below about 2.5 Hz. The moderate overprediction for higher frequencies likely is caused by vertical resonance effects primarily in the 1D model.


%%%sub-resolution sampling %of the low-velocity %material for the %simulation with minimum %$V_S$ of 200 m/s (5 %points per minimum %$V_S$).



\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.85\textheight,keepaspectratio]{figures/figure_highf_S1.pdf}
  \caption{
  Shear-wave quality factor ($Q_S$) plotted against $V_S$ (m/s) for several attenuation models widely used in the literature \citeg{olsenEstimationLongPeriodSec2003,taborda2014ground,savranGroundMotionSimulation2019,withersGroundMotionIntraevent2019} and investigated here. The inset figure in the upper left corner zooms into $V_S <= 1600$ m/s, denoted by the dashed black box. Note that these $Q_S$ relations are valid for constant $Q$ models, or frequency-dependent $Q$ models for frequencies below 1 Hz.}
  \label{fig:highf-A1}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S2.pdf}
  \caption{Description of three candidate source models used in this study. (top) Slip distribution (shading) for sources 1, 2 and 3 (left to right), characterized by their hypocentral depths at 5, 5.5 and 6 km, respectively. Contours depict rupture time at a 0.4 s interval starting from 0. (bottom) (left) sum of the moment rates for all subfaults and (right) Fourier amplitude spectrum, respectively.  Sources 1, 2 and 3 (from left to right in the first row) are characterized by their hypocentral depths at 5, 5.5 and 6 km, respectively. The contours represent rupture time at a 0.4 s interval starting from 0. Source 1 is the default source model used elsewhere in this chapter.}
  \label{fig:highf-A2}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S3.png}
  \caption{PGVs for sources 1, 2 and 3 (from left to right; see \cref{fig:highf-3}). The top and bottom rows represent the band-pass filtered results for 0.15-2.5 Hz and 2.5-5 Hz, respectively. The star denotes the epicenter.}
  \label{fig:highf-A3}
\end{figure}
\clearpage

%% Discarded on 071721
% \begin{figure}[!ht]
%   \centering
%   \includegraphics[width=0.9\textwidth,height=0.85\textheight,keepaspectratio]{figures/figure_highf_S4.pdf}
%   \caption{
% FAS bias between data and synthetics with minimum $V_S$ clamped at 200 m/s (blue), 500 m/s (green), and 500 m/s with 1D correction (red) for (a) E-W, (b) N-S, and (c) vertical component. A positive (negative) bias depicts overprediction (underprediction). The solid lines show the median FAS bias over all 259 stations, shading depicts the 95\% confidence interval (CI) and the dashed lines denote one standard deviation centered at the median.}
%   \label{fig:highf-A4}
% \end{figure}
% \clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S5.pdf}
  \caption{Probability density histogram of the PGV difference caused by SSH effects, between Models 12-14 with Model 6 (blue, green and red) , and Model 16 with Model 2 (cyan). The definition of percent difference (x-axis) is the same as in \Cref{fig:highf-11}.
  }
  \label{fig:highf-A5}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S6.pdf}
  \caption{
  Density of PGV change for models with topography relative to models without topography for bandwidths of (left column) 0.15-2.5 Hz and (right column) 2.5-5 Hz, and models with (top row) and (bottom row) without modified shallow velocities. The y-axis depicts topographic curvature smoothed using a 2-D window of 120 m $\times$ 120 m. Values toward the top right (bottom left) denote strong amplification at steep areas (deamplification at flat areas). Note that density intervals do not correspond to constant bin sizes.
  }
  \label{fig:highf-A6}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/GOF_mean_SH1D_tabels.pdf}
  \caption{
  GOF scores for a subset of the metrics used in this study, for frequency bands 0.15-1 Hz, 1-2.5 Hz, and 2.5-5 Hz. Model IDs are listed in \Cref{tab:highf-2}.} 
 
  \label{fig:highf-A7}
\end{figure}
\clearpage

% \begin{figure}[!ht]
%   \centering
%   \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_16.png}
%   \caption{Similar with \Cref{fig:highf-8}, but with seismic moment reduced from $M_w$ of 5.14 ($M_0 = 5.79 \times 10^{16}$ Nm) to $M_w$4.56e16 Nm.
%   }
%   \label{fig:highf-A8}
% \end{figure}
% \clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S11.pdf}
  \caption{Bias of (top row) PGV and (middle row) DUR and GOF (bottom row) for bandwidths of (left column) 0.15-2.5 Hz and (right column) 2.5-5 Hz at all 259 stations for Model 1 (see \cref{tab:highf-2} for model features). The bias is calculated in the same way as \Cref{fig:highf-9}. The solid line depicts the moving average of the bias of PGV using a 20-point window versus hypocentral distance. The shading denotes the standard deviation centered at the mean.
  }
  \label{fig:highf-A11}
\end{figure}
\clearpage

\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S12.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 2.
  }
\label{fig:highf-A12}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S13.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 3.
  }
\label{fig:highf-A13}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S14.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 4.
  }
\label{fig:highf-A14}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S15.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 5.
  }
\label{fig:highf-A15}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S17.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 7.
  }
\label{fig:highf-A17}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S18.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 8.
  }
\label{fig:highf-A18}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S19.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 9.
  }
\label{fig:highf-A19}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S20.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 10.
  }
\label{fig:highf-A20}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S21.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 11.
  }
\label{fig:highf-A21}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S22.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 12.
  }
\label{fig:highf-A22}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S23.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 13.
  }
\label{fig:highf-A23}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S24.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 14.
  }
\label{fig:highf-A24}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S25.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 15.
  }
\label{fig:highf-A25}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S26.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 16.
  }
\label{fig:highf-A26}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S27.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 17.
  }
\label{fig:highf-A27}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S28.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 18.
  }
\label{fig:highf-A28}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S29.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 19.
  }
\label{fig:highf-A29}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S30.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 20.
  }
\label{fig:highf-A30}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S31.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 21.
  }
\label{fig:highf-A31}
\end{figure}
\clearpage


\begin{figure}[!ht]
  \centering
  \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_S32.pdf}
  \caption{Same as \Cref{fig:highf-A11}, but for Model 22.
  }
\label{fig:highf-A32}
\end{figure}
\clearpage




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%\endrefsection




%% Discarded

% This finding is in agreement with the inversion results in Southern California by \citet{linFrequencyDependentAttenuationWaves2018}, who estimated an optimal power-law exponent of about 0.4.

% \subsection{Anelastic Attenuation and $\kappa$}
% The shape and amplitude of the FAS for strong ground motions reveal fundamental information about source rupture and wave propagation through the earth's crust. At high frequencies, above the corner frequency up to the Nyquist frequency, data shows that the FAS decays exponentially \citep{anderson1984model}, which can be described by the parameter $\kappa$. Both \citet{hanksFmax1982} and \citet{anderson1984model} agree that the behavior of $\kappa$ is primarily caused by the subsurface geological structure close to the recording site. Anderson and Hough found $\kappa$ values consistently in the range of 0.04-0.07 s for southern California, with a small increase with distance. 


% We calculated $\kappa$ from a least-squares fit to the spectra from synthetics and data, using the average of the two horizontal components for frequencies between 2 and 5 Hz in frequency-log (FAS) space. The spectra were then stacked, either for the whole set of stations or stations grouped into bins. Our calculated $\kappa$ from data at the entire set of stations is 0.061, which is consistent with that measured by \citet{anderson1984model}. 
% \Cref{fig:highf-15} shows the stacked FAS for a series of models. As compared to the $\kappa$ values from the data, the model with frequency-independent attenuation ($Q_S=0.1V_S$) appears to decay too rapidly and, generating $\kappa$ values that underpredict those from data above 1-1.5 Hz. On the other hand, frequency-dependent attenuation models produce a slower FAS decay as frequencies increase, and generally provide $\kappa$ values closer to those from the data. We find that topography and $V_S$ taper depth cause 2nd-order effects on the stacked FAS and are not shown.

%  especially in the distance range with dense site distribution, e.g., 20-45 km. We observed that $\kappa$ first increases slowly with distance, as reported by \citet{anderson1984model}, but then decreases to a low level rapidly at some distance bins beyond 45 km. This is likely due to the relatively fewer site numbers, or the fact that $\kappa$ is a strong localized parameter with spatial variability determined by site effects, which is evidence in \citet{anderson1984model} as well. Nonetheless, The smaller $\kappa$ at distance greater than 45 km may indicate stronger attenuation in these regions compared to near-source regions which are mostly sedimentary basins.
% The results show, as expected, that the
% critical parameter controlling $\kappa$ is the power-law exponent $\gamma$ of Q(f). However, determination of an optimal $\gamma$ value from $\kappa$ complicated by spatial variability, the maximum frequency of 5 Hz, and number of records.
% Within 10-20 km away from the source, which is generally deemed to be dominated by source effects, velocity heterogeneities and topography can induce ground motion variability. SSHs are found to result more PGV amplification than topography, which almost has no influence; while topography increases the duration remarkably over the full bandwidth we studied. Considering the relatively moderate relief near the source, topographic effects should be underlined in ground motion prediction even in less mountainous reginos.

% Based on accessibility of more accurate surface elevation information, mathematical descriptions of small-scale heterogeneities and frequency-dependent attenuation and community efforts in assessing shallow material structure, we explored various features and the complicated trade-offs between them. We have shown that physics-based deterministic numerical simulations are capable of capturing the most important characteristics of the ground motions incited by the $M_w$ 5.1 2014 La habra earthquake. Our results show that such high-frequency simulations can be a valuable complement to the stochastic models that evaluate seismic hazard up to 25 Hz. Because of our great advance in pushing up the frequency range of deterministic simulations (from 1-2 Hz in most previous simulations to 5 Hz), the uncertainties in stochastic methods generating higher-frequency synthetics can be reduced significantly.
% The fact that the GOFs at lower frequencies tend to be higher than at high frequencies is attributed to lack of enough high-resolution geological structure. The best geologically-constrained fine structure is surface topography, which increases peak ground motions at mountain tops and yields stronger effects at higher frequency, especially decreasing the ground shaking amplitude and prolonging the duration at low reliefs. The inclusion of more realistic low velocities is of relative minor importance, due to its sparsity in the model. However, stations underlain by very soft soils, with $V_S$ as low as 200 m/s, can still experience at most 80\% overprediction in PGV that is not captured when velocities are clamped at a higher value, e.g. 500 m/s. Small-scale heterogeneities (SSHs), generally reduce the peak ground motions and increase the durations, in a way similar to that of topography. More importantly, the models with SSHs always generate a better fit in duration compared to the model without SSHs. The frequency-dependent attenuation mainly amplifies the ground motions, and is critical in fitting the spectra decay according to our examination of the high-frequency decay ($\kappa$) as a function of distance.

% Generally speaking, the measurements related to peak values, response spectra and duration are almost always best predicted; while the fits of the Arias intensity, energy and Fourier amplitude spectra are more difficult to improve. Furthermore, among the poor measures, the large standard deviations of Arias and energy across various models indicate the even greater difficulty in fitting, in contrary to Anderson’s (\citeyear{andersonQuantitativeMeasureGoodnessOfFit2004}) speculation; while the Fourier spectra shows a very low standard deviation, which implies some systematic misfit. By examining the velocity profiles and FAS at rock sites, we found the unrealistic near-surface velocities in the CVM-S likely explains this misfit. Furthermore, introducing near-surface low velocities, e.g. via a $V_{S30}$-based method, can greatly improve the underprediction. [[Brief description of the refinement model up to 5 Hz]]
% With the effects and importance of the feature summarized, a subset of our well-performed models with these features included, achieve a fair improvement in terms of the proposed GOF criterion. In contrast, the simplest models without incorporating any additional features yield consistently lower GOF scores with the increase of frequency. Our results suggest that the major source of misfits is the poor constraints in the underlying velocity model at the high resolution required for our high-frequency simulations.  The composite models that generated the best fit suggests that incorporating topography, frequency-dependent attenuations (e.g. $Q_S=0.05V_Sf^{0.6}$ or $Q_S=0.1V_Sf^{0.6}$) and SSHs is a valid supplement to the velocity model in numerical simulations.

%% Discarded on 071721
% We estimated the effects of the near-surface low velocities by comparing the PGV, DUR and GOF with (Model 3) and without (Model 6) the 1D correction for material with $V_S$ $<$ 500 m/s, see \Cref{fig:highf-15}. As expected from \cref{fig:highf-A4} the correction generates small effects for frequencies below 2.5 Hz, while it controls the $Q$ model for the high frequencies. The near-surface low velocities tend to amplify the PGVs with little influence on DUR. Including the low velocities improves the fit at distances larger than 40 km, but tends to deteriorate in the near-source regions due to overamplification above 2.5 Hz.



%Previous studies attempting to %constrain $Q$ models in basin %areas Why do we find different %optimal Q models from previously %published models (Withers, Olsen, %Taborda, Graves)? Because omission %of model features (topo, LVL, SSH, %max frequency, source models) vary %inall those studies and bias the %results. [ELABORATE]
%\section{Conclusions}
% The fractal structure of topography, causing interaction of seismic waves at different wavelengths, greatly complicates interpretation of its effects on the ground motion at frequencies, as noted by \citet{booreNoteEffectSimple1972} and \citet{panzeraEvidenceTopographicEffects2011}. We thus expect stronger ground motion variability caused by complex topography, which will generate stronger regional concentration of amplification and deamplification, especially at higher frequencies.

%% Discarded
% \begin{figure}[!ht]
%   \centering
%   \includegraphics[width=0.9\textwidth,height=0.9\textheight,keepaspectratio]{figures/figure_highf_15.pdf}
%   \caption{$\kappa$ values calculated for data and different $Q(f)$ models from the FAS between 2 and 5 Hz. (a) FAS staked over all stations for data and for 5 models (labels denote model ID, see \cref{tab:highf-2}, as well as the attenuation model for $Q_s$. All models use $Q_p$ = $2Q_s$. We find an average $\kappa$ value of 0.061 for the data. (b) Average $\kappa$ values calculated at stations binned in 5 km intervals (count) for data and models from (a).
%   }
%   \label{fig:highf-15}
% \end{figure}
% \clearpage