[cig-commits] commit: More revisions. Added some more stuff to references (not yet cited in paper).
Mercurial
hg at geodynamics.org
Wed Jan 16 13:19:58 PST 2013
changeset: 158:49532f146b9b
tag: tip
user: Brad Aagaard <baagaard at usgs.gov>
date: Wed Jan 16 13:19:49 2013 -0800
files: faultRup.tex references.bib
description:
More revisions. Added some more stuff to references (not yet cited in paper).
diff -r 4b5b01bbf44d -r 49532f146b9b faultRup.tex
--- a/faultRup.tex Tue Jan 15 14:32:56 2013 -0800
+++ b/faultRup.tex Wed Jan 16 13:19:49 2013 -0800
@@ -137,15 +137,14 @@ the static coseismic slip
the static coseismic slip
\citep{Reilinger:etal:2000,Pollitz:etal:2001,Langbein:etal:2006,Chlieh:etal:2007}.
Likewise, studies of rapid deformation associated with earthquake
-rupture propagation often approximate the loading of the crust via
-simplistic assumptions about the stress field at the beginning of a
+rupture propagation often approximate the loading of the crust at the beginning of a
rupture
\citep{Mikumo:etal:1998,Harris:Day:1999,Aagaard:etal:BSSA:2001,Peyrat:etal:2001,Oglesby:Day:2001,Dunham:Archuleta:2004}. Numerical
seismicity models that attempt to model multiple earthquake cycles,
generally simplify not only the fault loading and rupture propagation
but also the physical properties in order to make the calculations
tractable
-\citep{Ward:1992,Robinson:Benites:1995,Rundle:etal:2006,Pollitz:Schwartz:2008,Dieterich:Richards-Dinger:2010}.
+\citep{Ward:1992,Robinson:Benites:1995,Hillers:etal:2006,Rundle:etal:2006,Pollitz:Schwartz:2008,Dieterich:Richards-Dinger:2010}.
Some dynamic spontaneous rupture modeling studies have
attempted to examine a broader space-time window in order to remove
@@ -312,7 +311,8 @@ on a fault is similar to the ``traction
on a fault is similar to the ``traction at split nodes'' (TSN)
technique used in a number of finite-difference and finite-element
codes
-\citep{Andrews:1999,Bizzarri:Cocco:2005,Day:etal:2005,Duan:Oglesby:2005,Dalguer:Day:2007,Moczo:etal:2007},
+\citep{Andrews:1999,Bizzarri:Cocco:2005,Day:etal:2005,Duan:Oglesby:2005,Dalguer:Day:2007,Moczo:etal:2007}
+(ADD CITATION TO MELOSH AND RAEFSKY BSSA 1981 AS WELL, BUT CALL IT SPLIT NODES?),
but differs from imposing fault slip via double-couple point
sources. The domain decomposition approach treats the fault surface as
a frictional contact interface, and the tractions correspond directly
@@ -323,7 +323,7 @@ is often applied in dynamic spontaneous
is often applied in dynamic spontaneous rupture models with explicit
time stepping and a diagonal system Jacobian, so that the fault
tractions are explicitly computed as part of the solution of the
-uncoupled equations. In this way, the TSN technique as described by
+uncoupled equations. In this way the TSN technique as described by
\citet{Andrews:1999} could be considered an optimization of the domain
decomposition technique for the special case of dynamic spontaneous
rupture with a fault consitutitive model and explicit time stepping.
@@ -336,7 +336,7 @@ tractions are associated with the total
tractions are associated with the total strain, not the effective
plastic strain. This illustrates a key difference between this
approach and the domain decomposition approach, in which the Lagrange
-multipliers and the constrain equation directly relate the fault slip
+multipliers and the constraint equation directly relate the fault slip
to the fault tractions (Lagrange multipliers). One implication of this
difference is that when using double couple point forces, the body
forces driving slip depend on the elastic modulii and will differ
@@ -496,9 +496,12 @@ functions, we find this portion of the J
(\nabla^T + \nabla) \bm{N}_m^T \cdot
\bm{C} \cdot (\nabla + \nabla^T) \bm{N}_n \, dV.
\end{equation}\end{linenomath*}
-This matches the stiffness matrix in conventional solid mechanics
-finite-element formulations. Following a similar procedure, we find
-the portion of the Jacobian associated with the constraints,
+This matches the tangent stiffness matrix in conventional solid
+mechanics finite-element formulations. In computing the residual, we
+use the expression given in equation~(\ref{eqn:residual:elasticity})
+with an implementation for infinitesimal strain and an implementation
+for small strain and rigid body motion. Following a similar procedure,
+we find the portion of the Jacobian associated with the constraints,
equation~(\ref{eqn:quasi-static:residual:fault}), is
\begin{linenomath*}\begin{equation}\label{eqn:jacobian:constraint}
\bm{L} = \int_{S_f} \bm{N}_p^T \cdot (\bm{N}_{n^+} - \bm{N}_{n^-}) \, dS.
@@ -535,18 +538,29 @@ positive side of the fault, and $p$ deno
positive side of the fault, and $p$ denotes DOF
associated with the Lagrange multipliers.
+The matrix $\bm{L}$ defined in
+equation~(\ref{eqn:jacobian:constraint}) is spectrally equivalent to
+the identity, because it just involves integration of products of the
+basis functions. This makes the traditional LBB stability criterion
+(:TODO: ADD FEM REFERENCE) trivial to satisfy by choosing the space of
+Lagrange multipliers to be exactly the space of displacements,
+restricted to the fault. In simple terms in order to specify the
+problem we need to know the distance between any pair of fault
+vertices $(v^+,v^-)$, which can be expressed as a displacement.
+
% ------------------------------------------------------------------
\subsection{Dynamic Simulations}
In dynamic simulations we include the inertial term in order to
-resolve the propagation of seismic waves. The general form of the
-system Jacobian remains the same as in quasi-static
-simulations given in equation~(\ref{eqn:saddle:point}). The integral
-equation for the fault slip constraint remains unchanged, so the
-corresponding portions of the Jacobian ($\bm{L}$) and residual
-($\bm{r}_p$) are also exactly the same as in the quasi-static
-simulations. Including the inertial term in
-equation~(\ref{eqn:quasi-static:residual:elasticity}) for time $t$
+resolve the propagation of seismic waves, with an intended focus on
+applications to earthquake physics and ground-motion simulations. The
+general form of the system Jacobian remains the same as in
+quasi-static simulations given in
+equation~(\ref{eqn:saddle:point}). The integral equation for the fault
+slip constraint remains unchanged, so the corresponding portions of
+the Jacobian ($\bm{L}$) and residual ($\bm{r}_p$) are also exactly the
+same as in the quasi-static simulations. Including the inertial term
+in equation~(\ref{eqn:quasi-static:residual:elasticity}) for time $t$
rather than $t+\Delta t$ yields
\begin{linenomath*}\begin{equation}\label{eqn:dynamic:residual:elasticity}
\begin{split}
@@ -684,12 +698,15 @@ slip remains zero, and no adjustments to
In iterating to find the fault slip and Lagrange multipliers that
satisfy the fault constitutive model, we employ the following
-procedure. We first compute the perturbation in the Lagrange
-multipliers necessary to satisfy the fault constitutive model for the
-current estimate of slip. We then compute the increment in fault slip
-corresponding to this perturbation in the Lagrange multipliers
-assuming deformation is limited to vertices on the fault. That is, we
-consider only the DOF associated with the fault
+procedure. We use this same procedure for all fault consitutive
+models, but it could be specialized to provide better performance
+depending on how the fault constitutive model depends on slip, slip
+rate, and various state variables. We first compute the perturbation
+in the Lagrange multipliers necessary to satisfy the fault
+constitutive model for the current estimate of slip. We then compute
+the increment in fault slip corresponding to this perturbation in the
+Lagrange multipliers assuming deformation is limited to vertices on
+the fault. That is, we consider only the DOF associated with the fault
interface when computing how a perturbation in the Lagrange
multipliers corresponds to a change in fault slip. In terms of the
general form of a linear system of equations ($\bm{A} \bm{u} =
@@ -775,14 +792,6 @@ slip-weakening \citep{Ida:1972}, linear
\citep{Andrews:2004}, and Dieterich-Ruina rate-state friction with an
aging law \citep{Dieterich:1979}. See the PyLith manual
\citep{PyLith:manual:1.7.1} for details.
-
-Notice that the matrix $\bm{L}$ defined in equation~(\ref{eqn:jacobian:constraint})
-is spectrally equivalent to the identity, since it just involves a rotation
-into the local fault coordinate system. This makes the traditional LBB stability
-criterion trivial to satisfy by choosing the space of Lagrange multipliers to be
-exactly the space of displacements, restricted to the fault. In simple terms, in
-order to specify the problem we need to know the distance between any pair of
-fault vertices $(v^+,v^-)$, which can be expressed as a displacement.
% ------------------------------------------------------------------
\section{Finite-Element Mesh Processing}
@@ -880,8 +889,11 @@ classified (contains a face on the fault
classified (contains a face on the fault with this vertex). Depending
on the order of the iteration, this can produce a ``wrap around''
effect at the ends of the fault, but it does not affect the numerical
-solution because the fault slip should be zero at the edges of the
-fault.
+solution as long as the fault slip is forced be zero at the edges of
+the fault. In prescribed slip simulations this is done via the
+user-specified slip distribution, whereas in spontaneous rupture
+simulations it is done by preventing slip with artificially large
+coefficients of friction, cohesive stress, or compressive normal tractions.
% We could instead do a breadth-first classification of cells starting
@@ -978,9 +990,9 @@ PETSc preconditioners can handle the ela
PETSc preconditioners can handle the elastic portion as
discussed in the previous paragraph. In computing
$\bm{P_\mathit{fault}}$ we approximate $\bm{K}^{-1}$ with
-the inverse of the diagonal portion of $\bm{K}$. $\bm{L}$ which
+the inverse of the diagonal portion of $\bm{K}$. Because $\bm{L}$
consists of integrating the products of basis functions over the fault
-faces. Its structure depends on the quadrature scheme and the choice
+faces, its structure depends on the quadrature scheme and the choice
of basis functions. For conventional low order finite-elements and
Gauss quadrature, $\bm{L}$ contains nonzero terms coupling the
degree of freedom for each coordinate axes of a vertex with the
@@ -1233,14 +1245,16 @@ We characterize preconditioner performan
We characterize preconditioner performance in terms of the number of
iterations required for the residual to reach a given convergence
tolerance and the sensitivity of the number of iterations to the
-problem size. An ideal preconditioner would yield a small, constant
-number of iterations independent of problem size. However, for complex
-problems such as elasticity with fault slip and potentially nonuniform
-physical properties, ideal preconditioners may not exist. Hence, we
-seek a preconditioner that provides a minimal increase in the number
-of iterations as the problem size increases, so that we can
-efficiently simulate quasi-static crustal deformation related to
-faulting and post-seismic and interseismic deformation.
+problem size. Of course, we also seek a minimal overall computation
+time. We examine the computation time in the next section when
+discussing the parallel performance. An ideal preconditioner would
+yield a small, constant number of iterations independent of problem
+size. However, for complex problems such as elasticity with fault slip
+and potentially nonuniform physical properties, ideal preconditioners
+may not exist. Hence, we seek a preconditioner that provides a minimal
+increase in the number of iterations as the problem size increases, so
+that we can efficiently simulate quasi-static crustal deformation
+related to faulting and post-seismic and interseismic deformation.
For this benchmark of preconditioner performance, we examine the
number of iterations required for convergence using the PETSc additive
@@ -1277,23 +1291,23 @@ consider the sources responsible for red
consider the sources responsible for reducing the scalability and
propose possible steps for mitigation.
-The main impediment to scalability in PyLith is load imbalance during
-the solve. This imbalance is the combination of three effects: the
-inherent imbalance in partitioning an unstructured mesh, partitioning
-based on cells rather than DOF, and weighting the cohesive cells the
-same as conventional bulk cells while partitioning. In this
-performance benchmark matrix-vector multiplication (the PETSc
-\texttt{MatMult} function) has a load imbalance of up to 20\%
-on 96 processors. The cell partition balances the number of cells
-across the processes using ParMetis \citep{Karypis:etal:1999} in order
-to achieve good balance for the finite element integration. This does
-not take into account a reduction in the number of DOF associated with
-constraints from Dirichlet boundary conditions or the additional DOF
-associated with the Lagrange multiplier constraints, which can
-exacerbate any imbalance. Nevertheless, eliminating DOF associated
-with Dirichlet boundary conditions preserves the symmetry of the
-overall systems and, in many cases, results in better conditioned
-linear systems.
+The main impediment to scalability in PyLith is load imbalance in
+solving the linear system of equations. This imbalance is the
+combination of three effects: the inherent imbalance in partitioning
+an unstructured mesh, partitioning based on cells rather than DOF, and
+weighting the cohesive cells the same as conventional bulk cells while
+partitioning. In this performance benchmark matrix-vector
+multiplication (the PETSc \texttt{MatMult} function) has a load
+imbalance of up to 20\% on 96 processors. The cell partition balances
+the number of cells across the processes using ParMetis
+\citep{Karypis:etal:1999} in order to achieve good balance for the
+finite element integration. This does not take into account a
+reduction in the number of DOF associated with constraints from
+Dirichlet boundary conditions or the additional DOF associated with
+the Lagrange multiplier constraints, which can exacerbate any
+imbalance. Nevertheless, eliminating DOF associated with Dirichlet
+boundary conditions preserves the symmetry of the overall systems and,
+in many cases, results in better conditioned linear systems.
We evaluate the parallel performance via a weak scaling
criterion. That is, we run simulations on various numbers of
@@ -1465,7 +1479,10 @@ time step size of five years. This time
time step size of five years. This time step corresponds to one tenth
of the viscoelastic relaxation time; hence it tests the accuracy of
the viscoelastic solution for moderately large time steps relative to
-the relaxation time.
+the relaxation time. Recall that the quasti-static formulation does
+not include inertial terms and time stepping is done via a series of
+static problems so that the accuracy depends only on the temporal
+variation of the boundary conditions and constitutive models.
Figure~\ref{fig:savage:prescott:profiles} compares the numerical
results extracted on the ground surface along the center of the model
@@ -1636,15 +1653,15 @@ rupture propagation.
% ----------------------------------------------------------------------
% Notation -- End each entry with a period.
\begin{notation}
- $\bm{A}$ & Matrix associated with Jacobian operator for the entire system of equations.\\
- $\bm{C}$ & Four order tensor of elastic constants.\\
+ $\bm{A}$ & matrix associated with Jacobian operator for the entire system of equations.\\
+ $\bm{C}$ & fourth order tensor of elastic constants.\\
$\bm{d}$ & fault slip vector.\\
$\bm{f}$ & body force vector.\\
$\bm{l}$ & Lagrange multiplier vector corresponding to the fault traction vector.\\
- $\bm{L}$ & Matrix associated with Jacobian operator for constraint equation.\\
- $\bm{K}$ & Matrix associated with Jacobian operator for
+ $\bm{L}$ & matrix associated with Jacobian operator for constraint equation.\\
+ $\bm{K}$ & matrix associated with Jacobian operator for
elasticity equation.\\
- $\mu_f$ & coefficient of friction.\\
+ $\bm{N}_m$ & matrix for $m$ basis functions.\\
$\bm{n}$ & normal vector.\\
$\bm{P}$ & preconditioning matrix.\\
$\bm{P}_\mathit{elastic}$ & preconditioning matrix associated with elasticity.\\
@@ -1661,8 +1678,10 @@ rupture propagation.
$V$ & spatial domain of model.\\
$V_p$ & dilatational wave speed. \\
$V_s$ & shear wave speed.\\
- $\Delta t$ & Time step.\\
+ $\Delta t$ & time step.\\
+ $\eta^{*}$ & nondimensional viscosity used for numerical damping.\\
$\pmb{\phi}$ & weighting function.\\
+ $\mu_f$ & coefficient of friction.\\
$\rho$ & mass density.\\
$\bm{\sigma}$ & Cauchy stress tensor.
\end{notation}
@@ -1670,10 +1689,10 @@ rupture propagation.
% ------------------------------------------------------------------
\begin{acknowledgments}
- We thank Ruth Harris and Fred Pollitz for their careful reviews of
- the manuscript. Development of PyLith has been supported by the
- Earthquake Hazards Program of the U.S. Geological Survey, the
- Computational Infrastructure for Geodynamics (NSF grant
+ We thank Sylvain Barbot, Ruth Harris, and Fred Pollitz for their
+ careful reviews of the manuscript. Development of PyLith has been
+ supported by the Earthquake Hazards Program of the U.S. Geological
+ Survey, the Computational Infrastructure for Geodynamics (NSF grant
EAR-0949446), GNS Science, and the Southern California Earthquake
Center. SCEC is funded by NSF Cooperative Agreement EAR-0529922 and
USGS Cooperative Agreement 07HQAG0008. PyLith development has also
@@ -1766,14 +1785,17 @@ rupture propagation.
\noindent\includegraphics{figs/solvertest_scaling}
\caption{Plot of parallel scaling for the performance benchmark with
the algebraic multigrid preconditioner and fault block custom
- preconditioner. The finite-element integrations for the Jacobian
- and residual exhibit good weak scaling with minimal sensitivity to
- the problem size. The linear solve (solid lines in the top panel)
- does not scale as well, which we attribute to the poor scaling of
- the algebraic multigrid setup and application as well as limited
- memory and interconnect bandwidth. We attribute fluctuations in
- the relative performance to variations in the machine load
- from other jobs on the cluster.}
+ preconditioner. The stages shown include the numerical integration
+ of the residual ({\tt Reform Residual}) and Jacobian ({\tt Reform
+ Jacobian}) and setting up the preconditioner and solving the
+ linear system of equations ({\tt Solve}). The finite-element
+ integrations for the Jacobian and residual exhibit good weak
+ scaling with minimal sensitivity to the problem size. The linear
+ solve (solid lines in the top panel) does not scale as well, which
+ we attribute to the poor scaling of the algebraic multigrid setup
+ and application as well as limited memory and interconnect
+ bandwidth. We attribute fluctuations in the relative performance
+ to variations in the machine load from other jobs on the cluster.}
\label{fig:solvertest:scaling}
\end{figure}
diff -r 4b5b01bbf44d -r 49532f146b9b references.bib
--- a/references.bib Tue Jan 15 14:32:56 2013 -0800
+++ b/references.bib Wed Jan 16 13:19:49 2013 -0800
@@ -1563,9 +1563,8 @@
rupture simulation},
journal = JGR-SE,
volume = {112},
- number = {82},
+ number = {B02302},
year = {2007},
- pages = {art. no.--B02302},
doi = {10.1029/2006JB004467},
abstract = {},
}
@@ -1582,6 +1581,246 @@
doi = {10.4401/ag-3201},
abstract = {},
}
+
+ at article{Hillers:etal:2006,
+ author = {Hillers, G. and Ben-Zion, Y. and Mai, P.~M.},
+ title = {Seismicity on a fault with rate- and state-dependent
+ friction and spatial variations of the critical slip
+ distance},
+ journal = JGR-SE,
+ volume = {111},
+ number = {B01403},
+ year = {2006},
+ doi = {10.1029/2005JB003859},
+ abstract = {We perform systematic simulations of slip using a
+ quasi-dynamic continuum model of a two-dimensional
+ (2-D) strike-slip fault governed by rate- and
+ state-dependent friction. The depth dependence of
+ the a − b and L frictional parameters are treated in
+ an innovative way that is consistent with available
+ laboratory data and multidisciplinary field
+ observations. Various realizations of heterogeneous
+ L distributions are used to study effects of
+ structural variations of fault zones on
+ spatiotemporal evolution of slip. We demonstrate
+ that such realizations can produce within the
+ continuum class of models realistic features of
+ seismicity and slip distributions on a fault. We
+ explore effects of three types of variable L
+ distributions: (1) a depth-dependent L profile
+ accounting for the variable width of fault zones
+ with depth, (2) uncorrelated 2-D random
+ distributions of L with different degrees of
+ heterogeneity, and (3) a hybrid distribution
+ combining the depth-dependent L profile with the 2-D
+ random L distributions. The first type of L
+ distribution, with relatively small L over the depth
+ range corresponding to the seismogenic zone and
+ larger L elsewhere, generates stick-slip events in
+ the seismogenic zone and ongoing creep above and
+ below that region. The 2-D heterogeneous
+ parameterizations generate frequency-size statistics
+ with event sizes spanning 4 orders of magnitude. Our
+ results indicate that different degrees of
+ heterogeneity of L distributions control (1) the
+ number of simulated events and (2) the overall
+ stress level and fluctuations. Other observable
+ trends are (3) the dependency of hypocenter location
+ on L and (4) different nucleation phases for small
+ and large events in heterogeneous distributions.},
+}
+
+ at article {Ampuero:Rubin:2008,
+ author = {Ampuero, J.~P. and Rubin, A.~M.},
+ title = {Earthquake nucleation on rate and state faults – {Aging} and slip laws},
+ journal = JGR-SE,
+ volume = {113},
+ number = {B1},
+ doi = {10.1029/2007JB005082},
+ year = 2008,
+ abstract = {We compare 2-D, quasi-static earthquake nucleation on
+ rate-and-state faults under both “aging†and “slipâ€
+ versions of the state evolution law. For both
+ versions mature nucleation zones exhibit 2 primary
+ regimes of growth: Well above and slightly above
+ steady state, corresponding respectively to larger
+ and smaller fault weakening rates. Well above steady
+ state, aging-law nucleation takes the form of
+ accelerating slip on a patch of fixed length. This
+ length is proportional to b−1 and independent of a,
+ where a and b are the constitutive parameters
+ relating changes in slip speed and state to
+ frictional strength. Under the slip law the
+ nucleation zone is smaller and continually shrinks
+ as slip accelerates. The nucleation zone is
+ guaranteed to remain well above steady state only
+ for values of a/b that are low by laboratory
+ standards. Near steady state, for both laws the
+ nucleation zone expands. The propagating front
+ remains well above steady state, giving rise to a
+ simple expression for its effective fracture energy
+ Gc. This fracture energy controls the propagation
+ style. For the aging law Gc increases approximately
+ as the square of the logarithm of the velocity
+ jump. This causes the nucleation zone to undergo
+ quasi-static crack-like expansion, to a size
+ asymptotically proportional to b/(b−a)2. For the
+ slip law Gc increases only as the logarithm of the
+ velocity jump, and crack-like expansion is not an
+ option. Instead, the nucleation zone grows as an
+ accelerating unidirectional slip pulse. Under both
+ laws the nucleation front propagates at a velocity
+ larger than the slip speed by roughly μ′/bσ divided
+ by the logarithm of the velocity jump, where μ′ is
+ the effective elastic shear modulus. For this
+ prediction to be consistent with observed
+ propagation speeds of slow slip events in subduction
+ zones appears to require effective normal stresses
+ as low as 1 MPa.},
+}
+
+ at article {Matsuzawa:etal:2010,
+ author = {Matsuzawa, T. and Hirose, H. and Shibazaki, B. and Obara, K.},
+ title = {Modeling short- and long-term slow slip events in the seismic cycles of large subduction earthquakes},
+ journal = JGR-SE,
+ volume = {115},
+ number = {B12},
+ doi = {10.1029/2010JB007566},
+ year = 2010,
+ abstract = {Slow slip events (SSEs) occur in the deeper extents of
+ areas where large interplate earthquakes are
+ expected in subduction zones, such as the Nankai
+ region of Japan and the Cascadia region of North
+ America. In the Nankai region, SSEs are divided into
+ long- and short-term SSEs, depending on their
+ duration and recurrence interval. We modeled and
+ examined the occurrence of long- and short-term SSEs
+ and changes in their behavior during the seismic
+ cycles of large interplate earthquakes. In these
+ numerical simulations we adopted a rate- and
+ state-dependent friction law with cutoff velocities
+ and assumed that the distribution of pore fluid
+ controls the recurrence interval of both long- and
+ short-term SSEs. The recurrence intervals of
+ reproduced short-term SSEs decrease during a
+ long-term SSE, as observed in western Shikoku, in
+ the Nankai region. The recurrence intervals of both
+ types of SSEs become shorter in the later stages of
+ interseismic periods. Large interplate earthquakes
+ nucleate between the region where SSEs occur and the
+ locked region of the large earthquakes, as suggested
+ from observations of the 1944 Tonankai
+ earthquake. Our numerical results suggest that the
+ stress buildup process in a seismic cycle affects
+ the recurrence behavior of SSEs.},
+}
+
+ at incollection{Lapusta:Barbot:2012,
+ author = {Lapusta, N. and Barbot, S.},
+ title = {Models of earthquakes and aseismic slip based on
+ laboratory-derived rate-and-state friction laws},
+ booktitle = {The Mechanics of Faulting: From Laboratory to Real Earthquakes},
+ publisher = {Research Signpost},
+ year = 2012,
+ pages = {153--207},
+}
+
+ at article {Barbot:etal:2012,
+ author = {Barbot, S. and Lapusta, N. and Avouac, J.-P.},
+ title = {Under the hood of the earthquake machine: {Toward}
+ predictive modeling of the seismic cycle},
+ journal = {Science},
+ volume = {336},
+ number = {6082},
+ pages = {707--710},
+ month = may # {11},
+ doi = {10.1126/science.1218796},
+ year = 2012,
+ abstract = {Advances in observational, laboratory, and modeling
+ techniques open the way to the development of
+ physical models of the seismic cycle with
+ potentially predictive power. To explore that
+ possibility, we developed an integrative and fully
+ dynamic model of the Parkfield segment of the San
+ Andreas Fault. The model succeeds in reproducing a
+ realistic earthquake sequence of irregular moment
+ magnitude (Mw) 6.0 main shocks—including events
+ similar to the ones in 1966 and 2004—and provides an
+ excellent match for the detailed interseismic,
+ coseismic, and postseismic observations collected
+ along this fault during the most recent earthquake
+ cycle. Such calibrated physical models provide new
+ ways to assess seismic hazards and forecast
+ seismicity response to perturbations of natural or
+ anthropogenic origins.},
+}
+
+ at article {Igarashi:etal:2003,
+ author = {Igarashi, T. and Matsuzawa, T. and Hasegawa, A.},
+ title = {Repeating earthquakes and interplate aseismic slip in the northeastern {Japan} subduction zone},
+ journal = JGR-SE,
+ year = 2003,
+ volume = {108},
+ number = {B5},
+ doi = {10.1029/2002JB001920},
+ abstract = {On the basis of a waveform similarity analysis, we
+ detected 321 earthquake clusters with very similar
+ (cross-correlation coefficient >0.95) waveforms on
+ the plate boundary in the northeastern Japan
+ subduction zone. Most of them were not found within
+ the subducting Pacific plate with a few
+ exceptions. Moreover, even on the plate boundary,
+ they were not located in the large moment release
+ areas of large interplate earthquakes that occurred
+ recently or in the areas where the plates are
+ inferred to be strongly coupled from GPS data
+ analyses. These observations suggest that these
+ similar earthquakes are caused by repeating slips of
+ small asperities with a dimension of around 0.1 to 1
+ km surrounded by stable sliding areas on the plate
+ boundary. If the aseismic slip portion in these
+ small asperities is negligible, we can estimate the
+ cumulative amount of aseismic slip in the area
+ surrounding each asperity. In other words, repeating
+ earthquake data potentially can be used to estimate
+ the spatiotemporal aseismic slip distribution on the
+ plate boundary. We estimated the spatial
+ distribution of slip rate on the plate boundary from
+ repeating earthquake data. The scaling relation
+ between seismic moment and seismic slip by Nadeau
+ and Johnson [1998] is used for the estimation of the
+ slip amount by each repeating earthquake
+ cluster. Obtained spatial distribution is consistent
+ with that estimated from GPS data on land.},
+}
+
+ at article {Ito:etal:2007,
+ author = {Ito, Y. and Obara, K. and Shiomi, K. and Sekine, S. and
+ Hirose, H.},
+ title = {Slow earthquakes coincident with episodic tremors and slow slip events},
+ journal = {Science},
+ volume = {315},
+ number = {315},
+ pages = {503--506},
+ month = jan # {26},
+ doi = {10.1126/science.1134454},
+ year = 2007,
+ abstract = {We report on the very-low-frequency earthquakes
+ occurring in the transition zone of the subducting
+ plate interface along the Nankai subduction zone in
+ southwest Japan. Seismic waves generated by
+ very-low-frequency earthquakes with seismic moment
+ magnitudes of 3.1 to 3.5 predominantly show a long
+ period of about 20 seconds. The seismicity of
+ very-low-frequency earthquakes accompanies and
+ migrates with the activity of deep low-frequency
+ tremors and slow slip events. The coincidence of
+ these three phenomena improves the detection and
+ characterization of slow earthquakes, which are
+ thought to increase the stress on updip megathrust
+ earthquake rupture zones.},
+}
+
@book{Aki:Richards:2002,
author = {Aki, K. and Richards, P.~G.},
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