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%\textbf{Supplementary Information}

\title{Numerical simulations on megathrust rupture stabilized under strong dilatancy strengthening in slow slip region}
%\date{November 7, 2012}
\author{Yajing Liu\\ Department of Earth and Planetary Sciences, McGill University \\
Department of Geology and Geophysics, Woods Hole Oceanographic Inst.}

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%\author{Yajing Liu$^{1}$}


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{\large \textbf{Supplement Information}}

{\large \textbf{Manuscript number: 2013GL055606}}
\end{center}
\listoftables
\listoffigures
%\begin{affiliations}
 %\item Department of Earth and Planetary Sciences, McGill University, Montreal, Canada
% \item Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA
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\section{Description of model parameters}

The depth distributions of friction parameters follow our previous studies of slow slip events in a Cascadia-like 2-D subduction fault model \citep{liu2010}. The temperature-dependent gabbro friction stability parameter $a-b$ \citep{he2007} show large scatter, but to the first degree can be approximated by piecewise linear segments and mapped to the thrust fault using a Cascadia geotherm \citep{peacock2002}. As shown in main text Fig. 1, the velocity-weakening (potentially seismogenic) section consists of $a-b=-0.0035$ at intermediate depths to downdip $\sim 95$ km followed by a gradual increase to neutral stability ($a-b=0$) at $\sim$ 180 km. The fault slips at velocity-strengthening condition further downdip. At shallow depth, the effective normal stress increases linearly at a lithostatic minus hydrostatic pressure gradient of 18 MPa/km, $\bar\sigma = 18z + 2.8$, where vertical depth $z$ is in kilometers and $\bar\sigma$ has the unit of MPa. A non-zero normal stress of 2.8 MPa is set at the trench to represent the pressure due to sediments. $\bar\sigma$ remains a constant of 50 MPa for $z \geq 2.62$ km ($18z + 2.8 \geq 50$), except in the slow slip event region, where $\bar\sigma$ is set to a much lower level as near-lithostatic pore pressure condition has been suggested by several lines of evidence in the ETS areas \citep{liu2007,audet2009}. The SSE $\bar\sigma$ is chosen to be in a range between 1 and 5 MPa; individual values are presented for the simulation cases in this study. In the high effective normal stress seismogenic zone, the characteristic slip distance $d_{c}$ is defined by setting a constant large ratio between the nucleation zone size $h^* = 2 \mu d_c /[\pi \nu (b-a)_{max} \bar\sigma]$ \citep{rice1993} and the discretized grid size $h$, where $(b-a)_{max}=0.0035$. $h^*/h=16$ is used in all presented cases. With parameters as shown in Table \ref{tab:parameter}, $d_c = 11$ mm in the $\bar\sigma=50$ MPa zone. In the slow slip region, $d_c$ is correspondingly smaller; a typical value is 0.2 mm for all cases with dilatancy and 0.1 mm for the reference model without dilatancy. 

\begin{table}
\begin{center}
\begin{tabular}{lll}
\hline
Notation & Definition & Value [Unit]  \\ 
\hline
$(b-a)_{max}$ & maximum velocity-weakening value & 0.0035 \\
 $f_0$ & Nominal friction & 0.6 \\
 $W$ & Total fault downdip distance & 364 [km] \\
$h$ & Model grid size &  0.1 [km] \\ 
$h^*$ & Characteristic nucleation size & 16h = 1.6 [km]   \\
$\mu$ & Shear modulus & 30 [GPa]  \\
$\nu$ & Poisson's ratio & 0.25  \\
$V_0$ & Reference slip velocity &  $10^{-6}$ [m/s] \\ 
$V_{pl}$ & Plate convergence rate & 37 [mm/yr]\\
$\delta$ & Fault dipping angle & 12$^o$ \\

\hline

\end{tabular}
\vspace{0.1in}
\end{center} \label{tab:parameter}

\caption{Typical model parameters that are constant in all simulations cases. }
\end{table}

\section{Intermediate to undrained conditions}

As discussed by \citet{liu2010}, the pore pressure evolution with slip can be characterized by two dimensionless parameters: (1) a drainage parameter $U \equiv t_p/(d_c/V_{pl})$, the relative time scales for pore fluid diffusion and friction evolution, and (2) a dilatancy parameter $E \equiv  f_0(\epsilon/\beta)/(b \bar\sigma_0)$, the relative strength changes due to pore dilation and friction evolution, where $\bar\sigma_0 = \bar\sigma - p_0$ is the background effective normal stress.  When $U \gg 1$, the gouge is nearly undrained and there is no change in fluid mass. When $U \ll 1$, the gouge is nearly drained near slip rate $V_{pl}$, and there is no change in pore pressure. On a completely drained fault, $p$ instantaneously equilibrates with the ambient $p_0$ and solutions are essentially identical to the system without dilatancy (as in the Fig. 2 reference model). Thus, this study concerns intermediate drainage to nearly undrained conditions, which is investigated by using $U$ in the range of 0.01 to 100. That is, at $V_{pl}=37$ mm/yr, the characteristic pore pressure across-fault diffusion time $t_p$ is between $3\times10^{-3}$ to 30 years in the updip seismogenic zone where $d_c \approx 11$ mm. And $t_p$ is between $5.5\times 10^{-5}$ and $0.55$ years in the slow slip region where $d_c = 0.2$ mm. Note that under the quasi-dynamic approximation, the duration of modeled megathrust ruptures is typically between 400 and 600 seconds ($\sim 2 \times 10^{-5}$ years). 

As shown in Fig. \ref{fig:e1_Usz}, when the drainage parameter $U$ is varied between 0.01 and 100 independently in the updip seismogenic zone and the slow slip region, the median values of the maximum coseismic slip and downdip rupture limit are nearly independent of the choices of $U$. For each simulation case, the  modeled earthquake sequences have a median maximum coseismic slip very close to its no-dilatancy level. And the rupture stops slightly updip from the no-dilatancy termination depth. The range of variation appears to be smaller as the seismogenic zone becomes more undrained (increasing $U$). 

%\begi=n{landscape}
\begin{figure}
\begin{center}
\includegraphics[width=6.5 in]{figure/suppl/figS1.pdf}
\end{center}
\caption{Dependence of maximum coseismic slip and downdip rupture limit (top and bottom panel respectively in each subfigure) on the drainage parameter in the updip seismogenic zone ($U_{SZ}$) and SSE region ($U_{SSE}$). (a) $U_{SZ}=0.01$, (b) 0.1, (c) 1.0, (d) 10.0. $U_{SSE}$ varies from 0.01 to 10 for each fixed $U_{SZ}$. For each simulation case, thin black bar shows the variation between 10 to 90 percentile, thick black bar shows 25 to 75 percentile, red circle shows the median value. Horizontal dashed lines show the levels of the two source properties without dilatancy.} \label{fig:e1_Usz}
\end{figure}
%\end{landscape}

\section{Dilatancy effect on earthquake cycles} %to modify!
Incorporation of dilatancy affects a range of deformation modes in an earthquake cycle. Stress on the fault is perturbed which leads to large variations in the coseismic slip and interseismic period between individual earthquakes. As shown in Fig. \ref{fig:cycle}, following an earthquake (EQ1) of a similar size to that in the reference model, earthquakes EQs 2-4 have smaller coseismic slips and slower rupture speeds. In particular, EQ 4 is the only event in the sequence that does not break to the trench. In order to release the strain that is continuously accumulating but not completely released in previous cycles, the fault eventually breaks in a giant earthquake (EQ 5) with faster rupture speeds and a maximum coseismic slip of nearly 25 meters. Besides small SSEs in the low $\bar\sigma$ region, a large aseismic slip event emerges. These are unsuccessful attempts of earthquake nucleation that have slip velocities much higher than the small SSEs but not dynamic enough to reach the seismic threshold. In the cases shown in Fig. \ref{fig:cycle}, each interseismic period contains such a large aseismic event, which ``ruptures" not only the high pore pressure region but propagates further into the updip seismogenic zone and results in 1 to 2 meters of slip. 

\begin{figure}\begin{center}
\includegraphics[width=3.25 in]{figure/suppl/figS2.pdf}
\end{center}
\caption{(a) Slip history and (b) slip budget, with low dilatancy $\epsilon/\beta=0.1$ MPa at all depths. The five earthquakes, EQs 1-5, have maximum coseismic slips of 14, 10.8, 10.3, 4.8 and 23.9 meters and intervals from the respective preceding earthquake are 323, 281, 279, 262 and 454 years. The  slip budget is calculated for a cycle that consists of interseismic slip prior to EQ1, coseismic and postseismic slip of EQ1 in (a).} \label{fig:cycle}
\end{figure}


\section{Extremely small and large dilatancy effect cases on Figure 4 phase diagram}

Figs. \ref{fig:e0025e005} and \ref{fig:e0175e05} show the evolution of slip and slip rate for the modeled 3000-year history, for two cases representing extremely small ($\epsilon/\beta$ of 0.025 MPa in the updip seismogenic zone and 0.05 MPa in the downdip slow slip region) and large ($\epsilon/\beta$ of 0.175 MPa and 0.5 MPa, respectively, in the two regions) dilatancy effects. The two cases are also shown by white crosses in Fig. 4c phase diagram in the main text. 

In Fig. \ref{fig:e0025e005}, except for the first two megathrust events, small and large earthquakes occurred in pairs. The small earthquakes ruptured only a small portion of the fault; approximately between downdip 80 and 200 km for EQ 1, and between 100 and 185 km for EQs 3 and 5. The maximum coseismic slip is small as well, reaching 3.4, 2.2 and 1.5 meters, respectively, during EQs 1, 3 and 5. Results from these partial fault rupturing events are not included in the variational range and median value calculations in main text Fig. 4. The large earthquakes of the pairs, on the other hand, have maximum coseismic slips much greater than the no-dilatancy level, reaching 23.9, 22.0 and 18.7 meters, respectively, during EQs 2, 4 and 6. Updip rupture propagates all the way to the trench, and the downdip rupture limit is also slightly deeper than the no-dilatancy value. Such exceptionally large coseismic slip and spatial rupture extent thus contribute to Regime I in Figs. 4a and 4c phase diagrams where the two source characteristics go above the no-dilatancy level. 

In Fig. \ref{fig:e0175e05}, due to the strong dilatancy-strengthening effect, the coseismic rupture process usually breaks into several sub-events, each separated by several months of average slip rates 100 to 1000 times faster than the plate loading rate. Take the final rupture sequence in Fig. \ref{fig:e0175e05} for example, the first sub-event EQ 1 ruptured downdip approximately 60 to 110 km, accumulating $\sim 1.9$ meters of coseismic slip; EQ 2 nucleated where EQ 1 updip rupture stopped and propagated bilaterally updip to the trench and downdip to the SSE region with a maximum slip of 8.5 meters, the largest among this sequence; nucleation of EQ 3 is slightly further updip, due to the stress transfer, and rupture took place at the same spatial extent as EQ 2 but with a smaller amount of slip 6.2 meters. The intervals between the subsequent events are 104 days and 186 days. Modest slip accumulated during the inter-event periods due to the higher than $V_{pl}$ slip rate; for example, more than 4 meters of slip is released aseismically at the trench between EQ 2 and EQ 3. 

For the calculation of the percentile range and median value of the two earthquake characteristics in the main text and Supplement, only megathrust events with a maximum coseismic slip greater than 4 meters are included. The 4 m threshold was optimized choice such that (1) it is large enough to exclude earthquakes which only rupture a small portion of the megathrust fault, and (2) it is small enough to include small earthquakes produced at high $(\epsilon/\beta)_{SZ}$, such as 0.175 MPa as shown in Fig. S3. Those events do rupture the entire VW zone, but with a lot of preseismic and postseismic slip accumulation. 

\begin{figure}\begin{center}
\includegraphics[width=6 in]{figure/suppl/figS3.pdf}
\end{center}
\caption{Extremely small dilatancy effect case: $(\epsilon/\beta)_{SZ}=0.025$ MPa in the updip seismogenic zone, $(\epsilon/\beta)_{SSE}=0.05$ MPa in the slow slip event region between downdip 140 and 215 km. Drainage parameter is $U=1.0$ uniformly along the entire fault. (a) Slip history on the fault. Black lines are interseismic slip every 50 years. Red lines are coseismic slip every 20 seconds.  A median coseismic slip of 22 meters is calculated for earthquakes with slip larger than 4 m (here, 24.8, 15.5, 23.9, 22.0, 18.7 meters respectively), and the median of their rupture downdip limits is 238.65 km. This is further downdip than the no-dilatancy case, due to larger coseismic slip in general. (b) Maximum slip rate on the fault in $\log_{10}(V_{max}/V_{pl})$. Vertical dashed line represents the cutoff threshold for ``coseismic" slip. (c) Maximum slip rate for a 50-yr SSE periods.}\label{fig:e0025e005}
\end{figure}

\begin{figure}\begin{center}
\includegraphics[width=6 in]{figure/suppl/figS4.pdf}
\end{center}
\caption{Extremely high dilatancy effect case: $(\epsilon/\beta)_{SZ}=0.175$ MPa in the updip seismogenic zone, $(\epsilon/\beta)_{SSE}=0.5$ MPa in the SSE region. $U=1.0$. (a) Slip history on the fault. Black lines are interseismic slip every 50 years. Red lines are coseismic slip every 20 seconds. A median coseismic slip of 7.7 meters is calculated for modeled earthquakes with slip larger than 4 meters (here, 7.5, 7.5, 9.1, 6.2, 8.6, 8.0, 8.5, 6.2 meters respectively). The median of their rupture downdip limits is 142.4 km. (b) Maximum slip rate on the fault in $\log_{10}(V_{max}/V_{pl})$. Vertical dashed line represents the cutoff threshold for ``coseismic" slip. (c) Maximum slip rate during a 3-yr period containing the final rupture sequence EQ 1 to EQ 3. 
}\label{fig:e0175e05}
\end{figure}
\clearpage
\section{Long-term and short-term SSEs}
As discussed in the main text, the transition from shot-term to long-term SSEs behavior as dilatancy effect increases (Fig. 4) provides insights to constructing a model that can produce both types of SSEs simultaneously with properties similar to those observed in Tokai and Bungo Channel, SW Japan \citep{hirose2005,miyazaki2006,sekine2010}. Fig. \ref{fig:ltst} shows such an example. The SSE zone is defined from downdip 110 to 215 km, under a low $\bar\sigma=1$ MPa and $d_c=0.05$ mm. Its updip portion (110-145 km) is under strong dilatancy $\epsilon/\beta=0.5$ MPa, and downdip portion (145-215 km) is under low dilatancy $\epsilon/\beta=0.01$ MPa, and the rest of the fault has $\epsilon/\beta=0.1$ MPa. Drainage parameter $U=1$ at all depths. As shown in Fig. \ref{fig:ltst}a, short-term SSEs repeat roughly every 0.3 year, with an average duration of 1-2 weeks, and slip is mostly limited to 145-200 km. Long-term SSEs repeat roughly every 2.5 years, with an average duration of 0.3 years, and slip takes place 110-200 km. More interestingly, short-term and long-term SSEs may occur simultaneously on the fault. The relative spatiotemporal distribution of the two types of SSEs are qualitatively similar to those observed in the Bungo Channel, Nankai Trough, although extensive parameter space search is needed in order to quantitatively reproduce the frequency and slip amplitude observations \citep{hirose2005}.

\begin{figure}
\begin{center}
\includegraphics[width=4 in]{figure/suppl/figS5.pdf}
\end{center}
\caption{An example of spatiotemporal relative distribution of short-term and long-term SSEs under heterogenous dilatancy conditions.}\label{fig:ltst}
\end{figure}


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% \multicolumn5l{(a) Algorithms from Numerical Recipes}\\
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