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\begin{flushleft}
{\Large \bf Seismic Behavior of  NPP Structures Subjected to Realistic 
3D,  Inclined  Seismic Motions, in Variable Layered Soil/Rock, on Surface or
Embedded Foundations}
\end{flushleft}

\vspace*{1cm}
\begin{flushleft}
         {B.  Jeremi{\'c}${}^{1,2}$, 
          N.  Tafazzoli${}^{1}$, 
          T.  Ancheta${}^{3}$, 
          N. Orbovi{\'c}${}^{4}$,
          A. Blahoianu${}^{4}$
          }\\
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\vspace*{1cm}
{\small ${}^{1}$~University of California, Davis, California, U.S.A.} \\ 
{\small ${}^{2}$~Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A. } \\
{\small ${}^{3}$~Risk Management Solutions, Inc., Newark, California, U.S.A.}\\
{\small ${}^{4}$~Canadian Nuclear Safety Commission, Ottawa, Canada} \\
{\small  Corresponding  author: B.  Jeremi{\'c}, phone~(530) 754-9248. fax~(530) 754-7872,
email~\texttt{jeremic@ucdavis.edu}}
\end{flushleft}

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\section{Abstract}

 
Presented here is an investigation of the seismic response of 
 a massive  NPP  structures  due to full 3D, inclined, un-correlated input motions
for  different soil and rock profiles. Of particular interest are the effects of
soil  and  rock  layering  on  the  response  and  the  changes of input motions
(frequency  characteristics)    due  to  such layering. In addition to rock/soil
layering  effects,  investigated  are  also  effects  of foundation embedment on
dynamic  response.  Significant differences were observed in dynamic response of
containment   and   internal  structure  founded  on  surface  and  on  embedded
foundations. These differences were observed for both rock and soil profiles.
%
Select  results  are used to present most interesting findings.





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\section{Introduction}
\label{Introduction}


Seismic  response  of  Nuclear  Power Plants (NPPs) has been at the forefront of
interest  from the very beginning of use of nuclear energy. Significant research
has been done over time on the subject seismic response of NPPs 
\citep{Wolf1983, Finn1986, Wagenknecht1987, Chang1988, Hanada1988, Wolf1989,     
Halbritter1998, Elaidi1998, VenancioFilho1997, Ueshima1997, Saxena2012,  Saxena2011,  Wolf1989, 
Ryu2010}. 
%


Seismic    response  of  NPPs  is  usually  quite complex, involving a number of
nonlinearities, including: 

\begin{itemize}

  \item Nonlinear response (elastic-plastic, damage, etc.) of    soil and   rock
  material adjacent to the foundation,
  
  \item  Nonlinear  response    of  contact  zone  between  soil/rock  and  the 
  foundation (gaping, slipping, elastic-plastic, damage),
  
  
  \item    Full    nonlinear  coupling between pore fluid and soil/rock skeleton
  (which strongly affects the nonlinear response of soil and rock),
  
  \item    Buoyant   effects,  where  pore fluid pressures on foundations create
  time  varying,  nonlinear forces on the foundation (which also strongly affect
  nonlinear response of the foundation -- soil/rock contact zone),
  
  \item Nonlinear response  of  the  structure  (containment  and  internal),
  
  \item Nonlinear interaction of fluids and structures/solids within the NPP.

\end{itemize}



In addition to that,
seismic  wave  field  features the following complexities:

\begin{itemize}

  \item Seismic body waves (P and S, which are fully three dimensional (3D), and
  inclined),
  
  \item  Seismic  surface  waves  (Rayleigh,    Love,  etc.,  which produce 
  three  translational   and three rotational components of motions at point
  at and near (at depth) the surface),
  
  \item  Seismic  wave  lack of correlations (or incoherence, as it is known in 
  the frequency domain)


\end{itemize}


 
 
Present  study  focuses  on  the effects soil/rock
layering  has on seismic response of an NPP subjected to realistic, 3D, inclined
seismic  wave field. Moreover, effects of foundation embedment are evaluated as
well  for  two  NPP  models,  one  on surface and the other on a $15$m
embedded foundation.

Paper  is organized in two main parts, the first describing the simulation model
development,  while  the  second  part presents a selected set of results that are
used to emphasize main findings.






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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation Model Development}
\label{SimulationModelDevelopment}

Presented  here briefly is the development of the simulation model, including
the  soil/rock profiles,  the  finite  element model for soil/rock-structure
system, input motions development and the simulation platform.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soil/Rock Profiles}


Soil  and rock layering beneath the NPP was assumed to be horizontal. A total of
twelve soil (shear wave velocity $V_s \le 760$m/s) and rock (shear wave velocity
$V_s > 760$m/s) models (profiles) were developed.
%
Their  spatial  distribution and assumed properties (layer thicknesses and shear
wave velocities) are presented in
Table~\ref{soil_profiles_table}.    One  of  the  focuses  of  the  study was to
investigated  seismic  behavior  of  an  NPP  founded  on  a variety of geologic
conditions, from  hard rock (Case 1) all the way to soil (Case 8), with variations of uniform soil/rock
profiles  (Cases  1,  2,  4,  6,  8) and layered cases (3, 5, 7, 9, 10, 11, 12).
Soil/rock  profiles  described in Table~\ref{soil_profiles_table} extend all the
way  to a depth of $40$Km, however Table only gives the top $500$m of shear wave
velocities  (and  thus  the  soil  stiffness).  Beneath top $500$m the soil/rock
profiles  are marked as either {\bf G} (gradual increase in soil/rock
shear  wave  velocity  (stiffness)  to  values of $V_s=3200$m/s and above) or as
{\bf I}  (immediate   increase  in  soil/rock  shear  wave  velocity
(stiffness) to a value of $V_s=4000$m/s.).



%\begin{landscape}
\begin{tiny}
\begin{table}[!h]\small 
\caption{\label{soil_profiles_table} Table describing 12 soil profiles used in this
study up to a depth of $500$m. }
%\hspace*{-18.9cm}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline  
 & \multicolumn{12}{|c|}{Soil profiles with shear wave velocity [m/s]}\\
\hline
Depth [m] & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
50   &
\multirow{10}{*}{\rm   2600   }   &
\multirow{10}{*}{\rm   1500  }  &
\multirow{4}{*}{\rm  1000  } &
\multirow{10}{*}{\rm 1000 } &
\multirow{4}{*}{\rm 300   }   &
\multirow{10}{*}{\rm   300   }   &
\multirow{4}{*}{\rm  300  }  &
\multirow{10}{*}{\rm 300 } &
\multirow{2}{*}{\rm 300 } &
\multirow{2}{*}{\rm 300 } &
{\rm 300}  &
{\rm 300 }\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{12-12}\cline{13-13}
{100} &  &  &  &  &  &  &  &  &   &  & {\rm 400} & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{10-12}
{150} &  &  &  &  &  &  &  &  & \multirow{2}{*}{\rm 500 } & & \multirow{2}{*}{\rm 500 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{  200} &  &  &  &  &  &  &  &  &  &  &  & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{4-4}\cline{6-6}\cline{8-8}\cline{10-10}\cline{12-12}
{  250} &  &  & \multirow{2}{*}{\rm 1500 } &  & &  & \multirow{2}{*}{\rm 750 } &  & \multirow{2}{*}{\rm 750 } &  & \multirow{2}{*}{\rm 750 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{    300}  &    &    &    &    &    {\rm  2600}  &  &  &  &  &  {\rm 2600}  &  & {\rm 2600}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{4-4}\cline{8-8}\cline{10-10}\cline{12-12}
{  350} &  &  & \multirow{4}{*}{\rm 2000 } &  &  &  & \multirow{4}{*}{\rm 1000 } &  & \multirow{4}{*}{\rm 1000 } &  & \multirow{4}{*}{\rm 1000 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{  400} &  &  &  &  &  &  &  &  &  &  &  & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{  450} &  &  &  &  &  &  &  &  &  &  &  & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{  500} &  &  &  &  &  &  &  &  &  &  &  & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-13}
%to  $\infty$  & gradual & gradual & gradual & gradual & stiff & stiff &  stiff & gradual & stiff & stiff & stiff & stiff\\
to  $\infty$  & {\bf G} & {\bf G} & {\bf G} & {\bf G} & {\bf I} & {\bf I} &  {\bf I} & {\bf G} & {\bf I} & {\bf I} & {\bf I} & {\bf I}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
\end{tabular}
\end{table}
\end{tiny}



%\end{landscape}




It  is important to note that profiles 1 and 8 represent the two extremes, while
other soil profiles fall  in between.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Seismic Input}



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Seismic Motion Modeling}
\label{SMM}

Seismic     motions   were  developed  using  an  analytic  solution,  developed
using Green's functions \citep{Hisada1994, Hisada2003}. The
method takes into account fault slip mechanism, and
propagates the wave through elastic half space layered medium to the surface.
%
The    fault    rupture    mechanics    was  chosen  to  be  similar  to that of
Northridge  mainshock,  which generated an earthquake with a moment magnitude of
$6.7$. Fault
rupture  was  modeled  using  Boore's  source  model,  while  the  seismic  waves
propagating  from that source are modeled using Green's functions for layered
half space. As presently implemented, this model can model a minimum
period  of  $T=0.06$s (corresponding to the frequency of $f=16.67$Hz). Rupturing
fault parameters  include: fault length of $18$km, fault width of $24$km, number of 
sub-faults  along the length and width is $14 \times 14$, strike of $122^o$, and
a dip of $40^o$.  




Figure~\ref{Map_of_Points} shows ground motion model setup.
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=14cm]{/home/jeremic/tex/works/Reports/2010/CNSC/Map_of_Points.pdf}
\includegraphics[width=12cm]{Map_of_Points.pdf}
\vspace*{-0.5cm}
\caption{\label{Map_of_Points} Analytic model  (Green's functions)   setup, with
locations of hypocenter, epicenter and the DRM model.}
\end{center}
\end{figure}
%
The  fault was  at the depth of approximately
$17$km,
while the NPP structures (the soil/rock structure model) was placed at approximately 
$17$km away from the earthquake epicenter. The main motivation for a chosen setup was 
to have a rupturing fault (hypocenter and epicenter) fairly close to the NPP, as
was    the    case    of  the  recent   earthquake at the Kashiwazaki NPP in Japan.
Moreover, close proximity (in relative terms) of the seismic source will result
in the development of
significant  3D, inclined, body and surface seismic waves. The
actual  NPP  site, was oriented in such a way to pick up
highest     horizontal    motions    in    local    X  direction,  as  shown  in
Figure~\ref{Map_of_Points}. 





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{The Domain Reduction Method.}
Seismic input is modeled using the Domain Reduction Method 
\citet{Bielak2001, Yoshimura2001}). 
%
It is a modular, two-step dynamic procedure
aimed  at  reducing the large computational domain to a more manageable size. 
%
The  DRM replaces forces from the seismic source with dynamically consistent  set
of   forces  applied  on  a  single  layer  of  finite  element  surrounding    a
soil/rock-structure NPP system. 
%
The replacement force, aka an effective force, ($P^{eff}$) and is given by :
%
%
\begin{eqnarray}
P^{eff} = \left\{\begin{array}{c} P^{eff}_i \\ P^{eff}_b \\ P^{eff}_e \end{array}\right\}
= \left\{\begin{array}{c} 0 \\ -M^{\Omega+}_{be} \ddot{u}^0_e-K^{\Omega+}_{be}u^0_e
   \\ M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b\end{array}\right\}
\label{DRMeq09}
\end{eqnarray}
%
Accelerations  ($\ddot{u}^0$)   and displacements  ($u^0$) are obtained from the
free    field  model (obtained from an analytic solution, described in section
\ref{SMM} above) and  feature realistic, 3D, inclined seismic
waves including both body (P and S) and surface (Rayleigh and Love) waves.
%
Part  of  the  model  that  is  inside  a  single layer of finite elements where
effective  forces  are  applied  can  be material nonlinear. Additional DRM modeling
details are available in \citet{Jeremic2012}.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Lack of Correlation (Incoherence) for the Original Seismic Motions}



Figure~\ref{coh_comp_x_grouped.pdf} shows the analysis
of  analytically  developed seismic motions (using Green's functions for layered
half  space)  with  incoherence  function developed from the Lotung LSST site in
Taiwan                                                                        by
\cite{Abrahamson1991,Abrahamson1992,Abrahamson2005,Abrahamson1992b}.
For each of the twelve simulations the x-components of points  1  and  2, 
were  used  to  estimate lagged coherency. To be consistent with the incoherence
function,  the  lagged coherency is calculated on the strong motion window using
an 11-point Hamming window.
%
%see
%Figure~\ref{Location_of_Points}.
%
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=12cm]{/home/jeremic/tex/works/Reports/2010/CNSC/TimAncheta/coh_comp_x_grouped.pdf}
\includegraphics[width=12cm]{coh_comp_x_grouped.pdf}
\caption{\label{coh_comp_x_grouped.pdf}      A  comparison  incoherence  of analytically
developed seismic motions (using Green's functions for layered half-space) with
incoherence  model by Abrahamson, from Lotung LSST site, for  all twelve soil/rock
profiles, grouped as rock (1-4, black markers) and soil (5-12, red markers) profiles.}
\end{center}
\end{figure}
%
Lagged  coherency  between  the pairs of the synthetic motions is  biased higher
(i.e.  more  coherent)  at  frequencies  up  to  approximately  7Hz as shown in
Figure~\ref{coh_comp_x_grouped.pdf}.  The  large spike in lagged coherency at 16
Hz  is  due  to a 16 Hz frequency limit of the synthetic motions developed using
layered half space model.




It  is  interesting  to  note  that  even  a  model with quite regular layering,
simulated over half space, is able to predict incoherence, although to a smaller
degree    than    what   the  extrapolated  model  from  Lotung LSST site shows.
These 
results  signify that a dimension of the model (in this case half-space) plays
an  important  role  in  development  of  the incoherence, since it was shown by
\cite{Walling2009},  that  a  limited  size  model (a block with sides of  $1$km)
cannot account for a majority of incoherence.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\clearpage
\subsubsection{Lack of Correlation (Incoherence) for the Abrahamson Model Motions}


In    addition   to (partially)  incoherent  motions  developed  using  layered 
half  space simulations, an additional (different) set of incoherent motions was
developed in order to test the effects incoherency has on NPP response.
%
In  order  to introduce appropriate amount of variability over the bandwidth due to
incoherence  effects,  each  point  was  used as a seed motion to generate a new
motion  across  the  grid at a separation distance of 200 meters. In other words
point  1  is modified to generate a ground motion at point 2 and vice versa. The
method   of  introducing  incoherence  effects  is  called  Frequency  Dependent
Windowing (FDW)  \citep{Ancheta2011}.
%
The method is a non-stationary simulation routine that utilizes
a   modified   short-time   Fourier  transform  (MSTF)  routine for  which  spectral
modifications  are made consistent with a selected coherency  model. The MSTF
routine  allows  preservation of the non-stationary properties of the motion and
incorporation of time-varying non-linear spectral modifications.
The routine is summarized below:
%
%short time windows of various lengths where the length of the window controls the
% frequency band over which spectral modifications are made consistent with a selected
%coherency ($\gamma$) and random Fourier amplitude model ($\sigma_{\Delta A}$).


\begin{enumerate}

 \item The seed time series is split into short time segments.
 
 \item A discrete Fourier transform (DFT) is applied to the segments.
 
 \item 
 Fourier  amplitudes  and  phase  angles  at  each  frequency  within  a desired
 frequency  range  (dependent  on  segment  length) are modified consistent to a
 coherency function for each segment \citep{Ancheta2010}.
 
 \item 
 The  new  set  of  Fourier phase angles is combined with the new set of Fourier
 amplitudes  and  transformed  into  the  time  domain  with  an inverse Fourier
 transform (IFT).


 
 \item  The  modified short time segments are recombined to form a modified time
 series.
 
 \item Steps 1-5 are performed multiple times for multiple segment lengths, with
 each  segment length having a specified frequency range over which phase angles
 are modified. Hence, multiple modified time series are created. Segment lengths
 and corresponding frequency limits used are shown in Table \ref{AnchetaTable1}.
 
 \item  The  multiple  modified  time series are match-filtered (i.e., band-pass
 filtered   with   the  limits  of  the  pass-band  matching  the  band  of  the
 modification) to combine the modified frequency bands in the frequency domain.
 
 \item  The  non-overlapping  frequency bands are inverted with an IFT to create
 the final broadband modified time series.

\end{enumerate}

\begin{table}[!htb]
\begin{center}
\caption{\label{AnchetaTable1}  Segment duration (L) and frequency bands (b) used in
the FDW routine. $T$ is the duration of the seed series.}
%\hspace*{-18.9cm}
\begin{tabular}{|c|c|}
\hline
Segment duration, $L_i$ [sec] & Frequency limits , $b_i$ [Hz] \\
\hline
1.28      &  2-Nyquist \\ 
2.56      &  1-2 \\ 
5.12      &  0.5-1 \\ 
10.24     &  0.25-0.5 \\ 
20.48     &  0.12-0.25 \\ 
$T$      &  0-0.12 \\  \hline
\end{tabular}
\end{center}
\end{table}


%A description of the incoherence models used is presented next.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Description of Selected Incoherence Models.}
Synthetic  models  created  by  analytic  solution  for layered half space, with
different  profiles but similar source were used as seed motions for incoherence
motion      development.      Soil/Rock     profiles     as     described     in
Table~\ref{soil_profiles_table} were used
and  grouped  as  rock  cases (1-4) with $V_s \ge 760$m/s  and soil cases (5-12)
with $V_s < 760$m/s.


Separate coherency and random Fourier amplitude models are selected for
rock (1-4) and soil (5-12) profiles separately. For soil cases 1-4, a plane wave
coherency ($\gamma_{pw}$) and $\sigma_{\Delta A}$
developed by \cite{Abrahamson2007} from the Pinyon Flat array recordings located
in        California        was    selected.   These   models   are   shown   in
equations~\ref{Ancheta01} and \ref{Ancheta02} and Table~\ref{Ancheta03}
%
\begin{eqnarray}
\gamma_{pw} (f, \xi) =
\left[
1+
\left(
\frac{f       \tanh(      a_3      \xi)}{a_1 f_c(\xi)}
\right)^{n1(\xi)}
\right]^{-1/2}
%
\left[
1+
\left(
\frac{f       \tanh(      a_3      \xi)}{a_2}
\right)^{n2}
\right]^{-1/2}
\label{Ancheta01}
\end{eqnarray}
%
%
\begin{eqnarray}
\sigma_{\Delta A} (f, \xi)
=
0.79 (1-e^{-0.45 - 0.0017 \xi)f})
\label{Ancheta02}
\end{eqnarray}



\begin{table}[!htb]
\begin{center}
\caption{\label{Ancheta03}  Plane-wave  coherency  model  coefficients  for  the
horizontal component.}
%\hspace*{-18.9cm}
\begin{tabular}{|l|l|}
\hline
Coefficient name  & Horizontal coefficient value\\
\hline
$a_1$ & $1.0$ \\ \hline
$a_2$ & $40$ \\ \hline
$a_3$ & $0.4$ \\ \hline
$n_1(\xi)$ & $3.80-0.040 \ln{(\xi+1)}+0.0105 \left[\ln{(\xi+1)}-3.6\right]^2$ \\ \hline
$n_2$ & $16.4$ \\ \hline
$f_c(\xi)$ & $27.9-4.82 \ln{(\xi+1)}+1.24 \left[\ln{(\xi+1)}-3.6\right]^2$\\ \hline
\end{tabular}
\end{center}
\end{table}




For soil cases (5-12), a lagged coherency ($|\gamma|$) and $\sigma_{\Delta A}$
 developed by \cite{Ancheta2010} from the
Lotung LSST (Large Scale Seismic Test) and BVDA (Borrego Valley Differential Array)
recordings located in Taiwan and California, respectively, was selected. The models are
shown in equations~\ref{Ancheta04} and \ref{Ancheta05}.
%
\begin{eqnarray}
\tanh^{-1}{|\gamma(f,\xi)|}
=
\left(3.79-0.499(\ln{\xi})\right)
\exp{\left((-0.115 - 0.0084 \xi)f\right)}
+
\frac{1}{3}(\xi) f^{-0.878(\xi)}
+
0.35
\nonumber \\
\label{Ancheta04}
\end{eqnarray}
%
\begin{eqnarray}
\sigma_{\Delta A} (f, \xi)
=
(1-e^{-0.1005 - 0.0025 \xi)f})
\nonumber
\\
\label{Ancheta05}
\end{eqnarray}


Plane wave coherency is typically lower than lagged coherency as it includes random
variations in the wave passage effect. The plane wave coherency model was selected
instead of a lagged coherency model as there are no available lagged coherency models
available for rock sites comparable to eastern North America rock sites.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Range of Incoherence Model Applicability}

Based on a report by \cite{Abrahamson2007} the incoherence models
developed  from  the  Pinyon  Flat  array  appears  to be applicable  to separation
distances  of  $5$ to $150$~meters and a frequency range of from $5$ to $40$~Hz.
Since the model is developed from
Pinyon Flat recordings alone. it is unknown how well this model fits similar rock site
array recordings or possible future array recordings located in the eastern North America
region.

The incoherence models developed by \cite{Ancheta2010} are a slight modification of
incoherence models developed by \cite{Abrahamson1992} and are applicable to separation
distances of $6$ to $160$~meters and frequencies greater than $1$~Hz.

 Both  models  have been extrapolated to $200$~meters for the process of including
 incoherence  using the FDW routine. In a comparison of coherency from the SMART
 1  and  LSST  array  in  Taiwan, \cite{Abrahamson1992} concluded that the decay in
 coherency was similar up to $200$m.
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Finite Element Model}



Full  3D  finite  element  models were developed for both surface foundation and
embedded    foundation    NPPs.  Structural  models    (containment and internal
structures) consisted of modal equivalent stick model.

\paragraph{Containment Structure Modal Equivalent Stick Model} was modeled using
twelve  (12)  3D  elastic  beam  elements,  where  each stick has its properties
calculated  from  the  cylindrical  and/or  half  sphere part of the containment
structure.  Three  moments of inertia were used (two bending and one torsional),
while lumped masses included both translational and mass moments of inertia.



\paragraph{Internal  Structure Modal Equivalent Stick Model} consists of sixteen
nodes,  connected with elastic beam elements as well as rigid connections (where
center of mass is not on the beam axes). Stiffness properties were characterized
with  bending  moments  of inertia (two in 3D), coordinates of stiffness center,
moment  of  inertia in torsion and shear area. Since for the internal structure,
centers    of  mass  do  not  coincide  with  centers  of  the  stiffness, rigid
connection elements were used to place lumped  mass elements away from stiffness
centers.

\paragraph{Foundation  Slab}  was  placed  at the bottom of both containment and
internal  structures  (connection  at  the same node). Since foundation slab was
modeled  using brick (solid) elements, a rectangular/cross beam structure was placed on top
of   the  foundation  slab  to  provide  for  stability  of  above  beam  models
(containment  and  internal  stick  models).  In  addition  to providing for the
stability  of containment and internal stick models, this rectangular/cross beam
structure  also  provided  additional stiffness for the foundation slab that was
missing    since    containment    structure  (in  reality  a  thick  shell with
certain  horizontal  spatial extent) is replaced with a stick model. Addition of
rectangular/cross  beam  structure  was  done  for  both  surface  and  embedded
foundation models.
%
The foundation slab has dimensions of $100$m $\times$ $150$m
and  a  thickness  of  $3.5$m.  For  the  surface foundation, the lower level of
concrete is at the depth of $5$m while for the embedded it is at the depth of $15$m.
%



The 3D finite element models for a soil/rock block  have dimensions of 
$200$m  (lenght)  $\times$  $140$m (width) $\times$ $60$m (depth). 
They  consist  of  eight  node  brick  elements  with  linear  interpolation  of
displacements and dimensions of $5{\rm  m}  \times  5{\rm  m} \times 5{\rm m}$.
With  chosen  dimension  of  the  finite  element,  different types of soil/rock
material  models  (stiffness)  will  be  able  to  accurately  model different
frequencies of motions:
  \begin{itemize}
    \item  Rock with shear wave velocity of $v_s=2600$m/s, and with the element
    size  of  $h=5$m is able to model (with small error, below 10\%) frequencies
    of  up  to  $f_{max}=65$Hz, while any frequencies above that will be modeled
    with increased error,
    \item  Similar  to  the  above  case  for  rock  with shear wave velocity of
    $v_s=1500$m/s,   and  element  dimension  of  $h=5$m,  accurate  modeling  of
    frequency  of  $f_{max}=37$Hz  is  expected,  while anything above that will
    introduce larger error,
    \item Rock with $v_s=1000$m/s, and element size  $h=5$m, will accurate model $f_{max}=25$Hz;
    \item  While soil with shear wave velocity of $v_s=300$m/s, and element size
    of      $h=5$m,  will  accurately  model  $f_{max}=7$Hz,  while  for  higher
    frequencies, the error will be larger.
  \end{itemize}


For all the finite element models, Newmark numerical time stepping algorithm was
used  (\cite{Newmark1959}),  with  time  step increment of $\delta t = 0.015$s. A
small  amount  of  numerical  damping  was introduced through Newmark algorithms
constants  $\gamma  =  0.6000$  $\beta=  0.3025$  in  order  to  damp  out  high
frequencies  that  are  present due to the finite element discretization process
\cite{local-86}, \cite{Argyris91}.




Figures~\ref{DRMSurfaceModel05} and \ref{DRMModel04}   show  a  3D  finite element model for a case of
surface and embedded foundation, respectively.
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=10cm]{/home/jeremic/tex/works/Reports/2010/CNSC/ModelPlots/VisIt_Mesh_Surface/NPP_surface01.jpeg}
\includegraphics[width=10cm]{NPP_surface01.jpeg}
\caption{\label{DRMSurfaceModel05}  Surface  foundation    model with underlying
rock/soil  and  the  equivalent  elastic stiffness and mass superstructure model
representing NPP.}
\end{center}
\end{figure}
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=10cm]{/home/jeremic/tex/works/Reports/2010/CNSC/ModelPlots/VisIt_Mesh_Excavated/NPP_embedded02.jpeg}
\includegraphics[width=10cm]{NPP_embedded02.jpeg}
\caption{\label{DRMModel04} Embedded foundation  model with underlying rock/soil
and  the equivalent elastic stiffness and mass superstructure model representing
NPP.}
\end{center}
\end{figure}
%
In    addition    to  the  modal  equivalent  stick models for a containment and
internal structures, model includes a   concrete  slab, the DRM layer, the soil/rock inside the DRM
layer  and  the  material  outside  the  DRM  layer.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{Simulation Platform}


Finite element program used in this study was developed using a
number    of    numerical    libraries,  algorithms,  material models and finite
elements  available through one of open source licenses.
%
%
The finite  element  domain  was  managed  using  Modified OpenSees Services (MOSS)
library 
\citep{Jeremic2004d,  Jeremic2007d,  Jeremic2008a,   McKenna97}.
%
On      a   lower functional  level,   a   set   of   NewTemplate3Dep   numerical   libraries
\citep{Jeremic2000f}
was used  for  constitutive  level modeling, while nDarray numerical libraries
\citep{Jeremic97d} are used to handle vector, matrix
and  tensor  manipulations, and  element  libraries from UCD
CompGeoMech  FEMtools    \citep{Jeremic2004d}    are  used to supply other
necessary libraries and components.
%
%
%
The solution    of  system of equations was  provided by  UMFPACK solver
\citep{Davis97b, Davis97, Davis1999, Davis2004, Davis2004b}.
%


Numerical  simulation  system  used in this study is a precursor to the NRC ESSI
Simulator  modeling  and  simulation  system that is currently undergoing a very
active development (expansion, documentation, verification and validation, etc.) \citep{Jeremic2011a}.










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\section{Select Results}
\label{ComparissonResults}


A  large  number  of test cases were analyzed and a wealth of results developed.
Presented  here  is  a  select number of results that are used to emphasize main
findings    related    to  the influence of variable layered soil/rock under the
surface and embedded  foundation slab.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{Comparison    of    Responses  Between  NPP Structures on a Variable
Stiffness Single Layer Base Soil/Rock}


Presented  here  are  results  for  behavior  of the NPP on surface and embedded
foundation with variable thickness of a uniform soil/rock layer.
%
%
In particular, four rock/soil profiles (all with uniform rock/soil up to the
depth of $500$m) are used to emphasize differences.
%
The following profiles are used:
%
\begin{itemize}
\item Case 1 ($v_s=2600$m/s to a depth of $500$m),
\item Case 2 ($v_s=1500$m/s to a depth of $500$m),
\item Case 4 ($v_s=1000$m/s to a depth of $500$m),  
\item Case 8  ($v_s=300$m/s  to  a depth of $500$m)
\end{itemize}
%
where a more detailed description of profiles is given in
Table~\ref{soil_profiles_table}.


Results of four different uniform profiles (from hard rock, to
 soil,    Cases 1, 2, 4 and 8) are used to show that softer foundation soil/rock
 does  filter    out   higher  frequencies  as  well  as de-amplify motions for higher
 frequencies  at  the top of both containment and internal  structures. Behavior
 is quite the opposite  at lower frequencies, where softer
 soil/rock  amplify  motions  and  contributes  to  (in  some  cases) significant
 amplification of motions (particularly for frequencies below $1$Hz).


Figures
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
and 
\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf} 
show response for surface and embedded foundation models, respectively.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
%\subsubsection{Containment  and  Internal Structure on Embedded Foundation,
%Variable Stiffness Single Layer Base Soil/Rock}


%\clearpage


\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_internal_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Base_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
Comparison  of acceleration FFTs for cases  1, 2, 4, 8 for horizontal direction  at the 
(upper) {\bf top of containment structure},
(middle) {\bf top of  internal structure}, and 
(bottom)  {\bf the base of containment and internal structures} on a {\bf surface foundation}.}
\end{center}
\end{figure}
%



%\clearpage

\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Base_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
Comparison  of acceleration FFTs for cases 1, 2, 4, 8 in horizontal direction at the
(upper) {\bf top of  containment structure}, 
(middle) {\bf top of  internal  structure}, 
(bottom) {\bf   base  of  internal  and  containment  structures}  on  an  {\bf embedded
foundation}.}
\end{center}
\end{figure}
%



There are a number of observations that can be made:

\begin{itemize}

\item  There is a slight increase in base motions for all four cases for surface
foundation  (when compared to embedded foundations) for lower frequencies, below
$4$Hz,    while    there    is  a  larger decrease (for surface foundations when
compared to embedded foundations) for frequencies above $4$Hz for all
cases,  with  particularly  large  decrease  for  soil  (Case 8) (compare bottom
in Figures
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
and
\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}).
%
%These  differences can be explained by the fact that surface waves (in this case
%the  Rayleigh  waves)  do  influence  motions at depth but decay fast, so that a
%single  wave  length  (in  depth)  is  where these influences usually disappear.
%Hence,  for  the embedded foundation, less high frequency surface motions affect
%the NPP.


\item For structural response on embedded foundation
(compare middle and upper in figures 
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf})
and
Figures~\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf})
the internal  and containment structure react quite differently at frequencies above
$2$Hz, for different soil/rock stiffness beneath the foundation.
%
It  is  particularly  important  to  note  that the embedded foundation seems to
reduce  motions  of both containment and internal structures for frequency range
of   $1$Hz  to  $5$Hz (significantly) while motions remain almost the same for a
frequency range above $5$Hz.  This  can  be explained by the fact that for the embedded
foundation,  the  energy  of surface waves (Rayleigh waves in this case) is much
smaller  for  embedded  foundation  and  with that, the rotational components of
motions are not exciting structure (containment and internal) as much as for the
surface  foundation case. This conclusion, however cannot be drawn in general as
there    might  be  cases  where  the  lack  of  surface  waves (at depth) might
not be beneficial to the dynamic response.



\item  The soil subsurface condition (Case 8)  does consistently show the lower
response for higher frequencies, and higher response for lower frequencies,
This  is, of course to be expected, and will only be amplified if real soil/rock
non-linearities  are taken into account (only linear elastic material properties
were used in this study).



\item  Response of the structures on surface foundations
is  fairly similar for containment structure
with significant divergence of response only  after about $4$Hz (again the soil case does
diverge  the most). 
%
On  the  other  hand,  the  internal  structure has a very different response on
different  soil/rock  layers. 
%
Soil  profile  shows again the most divergence for
both containment and internal structure
reduction in response only for frequencies higher than $6$Hz.


\end{itemize}






















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{Comparison      of    Responses   Between NPP Structures on Variable
Thickness Soft Soil Layer}




In addition to the above uniform profiles, analyzed were profiles with a varying
thickness        of    a    soil  layer ($v_s=300$m/s)  (cases 6, 5,  10 and 12)
where the thickness was varying:

\begin{itemize}

 \item Case 6, top $500$m of soil with  $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
 
 \item Case 5, top $200$m of soil with  $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
 
 \item Case 10, top $100$m of soil with  $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
 
 \item Case 12, top $50$m of soil with  $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s).

\end{itemize}





Figure~\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf},
shows    seismic  response  at  the  top  and  base  of containment and internal
structures
for the surface foundation, while 
Figure~\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
show response of the top and base of containment and internal structures on embedded foundation.

\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_internal_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Base_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
Comparison  of acceleration FFTs for cases 5, 6, 10, 12 for horizontal direction  
at the  
(upper) {\bf  top  of  a  containment  structure}, 
(middle) {\bf  top  of  an  internal  structure}, and 
(bottom) {\bf  base of containment and internal structures} on a {\bf
surface foundation}.}
\end{center}
\end{figure}
%


\begin{figure}[!htb]
\vspace*{-5.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Base_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
Comparison  of acceleration FFTs for cases 5, 6, 10, 12 for horizontal direction  at
the 
(upper) {\bf top of a containment structure},
(middle) {\bf top of a internal structure}, and
(bottom) {\bf  base  of  containment  and  internal  structures} on an {\bf embedded
foundation}.}
\end{center}
\end{figure}
%



%
A number of observation can be made:
%
\begin{itemize}

\item  A  significant  differences  are  observed  in responses between  surface and embedded
foundation structures (both for containment and internal structures). 

\item  For example,
internal structure (middle figures in Figures 
\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf} 
and
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf})    
show    significant    differences,  and  in  general the embedded
foundation  internal  structure has amplified motions when compared with the surface
foundation case.


\item On the other hand, containment structure (upper figures in Figures
\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf} 
and
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf})   
 show  amplification   for embedded foundation case only below about $1.5$Hz and
 above  $4$Hz,  while  in the range $1.5-4$Hz, the surface foundation has larger
 amplifications.


\item  The thickness of the soft soil layer (it varies from $50$m
to  $500$m)  does not affect much the response of either containment or internal
structures at higher frequencies.  Presented  Fourier  Amplitudes  suggest  that
a  small  variation of response is to be expected for variable thickness of soft
soil  layer,  particularly for frequencies above approximately $2$Hz. Below such
frequency,  response  does differ, in some cases more significant that in other.
This  behavior  is  somewhat  expected,  as  the  soft soil layer "isolates" the
structures  from  higher frequencies, while the lower frequencies do get certain
amplification,  depending  on  the  layer thickness, stiffness and mass. This is
most  obvious  for  the  frequencies  of  about  $0.8$Hz where the amplification
factors seems to differ by an order of magnitude between various cases.

\item  Independent  of  the  soft soil layer thickness, significant reduction of
higher frequencies is noticeable, which is  apparent in  Figure
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}.

\end{itemize}

%OVDE

% Time Ancheta Comments
%  When comparing the relative amplification of the base motion to the:
%  
%  1.  I  see the same pattern of amplification/de-amplification in the uniform Vs
%  profiles,  and  the  variable thickness cases. It seems however, that the input
%  motion  at  the  base for the uniform Vs profile has a relatively flat spectrum
%  between  0.1-10  Hz (except case 8 in Figure 6/7) while the input motion at the
%  base for the variable thickness profiles has more spectral peaks and troughs.
%  
%  2.  Are  the relative spectral peaks and troughs dependent on a property of the
%  profiles?
% 







%\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}


Presented    in    this  paper was an investigation of the seismic response of a
massive  NPP soil/rock-structure system (both containment and internal structures, with the foundation slab
(surface  and/or  embedded) and with a significant soil/rock volume) due to full
3D,  inclined,  un-corellated  body  and surface seismic motions for a different
soil/rock  profiles. Presented was an investigation of the importance of soil/rock
layering on response of containment and internal structures to realistic seismic
motions.  In  addition  to  that,  an  investigation  of  effects  of foundation
embedment was also performed for all soil/rock layering cases.

A  number  of  specific conclusions can be made for each of the soil/rock layer 
cases,  and  such  conclusions  were  made  in the above results section. It is,
however,  very  important  to  note  that presented results do suggest that each
soil/rock  layering  system  features  behavior  specific  for  that  particular
layering, for particular structural system (containment and/or internal) and for
a  particular  (realistic) earthquake motion, landing to the conclusion that the
interaction  of  all  three  components,  the  earthquake, the soil/rock and the
structure play crucial role in dynamic response of an NPP.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgment}
%
Work  presented  here  was  funded  by  a grant from the Canadian Nuclear Safety
Commission (SNSC) and such support is greatly appreciated.
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage


\newpage

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\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
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  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi

\bibitem[Abrahamson(1992{\natexlab{a}})]{Abrahamson1992}
N.~Abrahamson.
\newblock Spatial variation of earthquake ground motion for application to
  soil-structure interaction.
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\bibitem[Abrahamson(1992{\natexlab{b}})]{Abrahamson1992b}
N.~Abrahamson.
\newblock Generation of spatially incoherent strong motion time histories.
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\newblock ISBN 90 5410 060 5.

\bibitem[Abrahamson(2005)]{Abrahamson2005}
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\newblock Updated coherency model.
\newblock \emph{Report Prepared for Bechtel Corporation, April}, 2005.

\bibitem[Abrahamson(2007)]{Abrahamson2007}
N.~A. Abrahamson.
\newblock Hard rock coherency functions based on {P}inyon {F}lat data.
\newblock unpublished data report, 2007.

\bibitem[Abrahamson et~al.(1991)Abrahamson, Schneider, and
  Stepp]{Abrahamson1991}
N.~A. Abrahamson, J.~F. Schneider, and J.~C. Stepp.
\newblock Empirical spatial coherency functions for applications to
  soil--structure interaction analysis.
\newblock \emph{Earthquake Spectra}, 7\penalty0 (1):\penalty0 1--27, 1991.

\bibitem[Ancheta et~al.(2012)Ancheta, Stewart, and Abrahamson]{Ancheta2011}
T.~Ancheta, J.~Stewart, and N.~Abrahamson.
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\end{thebibliography}


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