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\lefthead{PERFETTINI ET AL.}

\righthead{Slip on a creeping fault}

%\received{January~3,~1998}
%\revised{February~27,~1998}
%\accepted{March~31,~1998}

\paperid{}

\cpright{AGU}{1998}

\ccc{0148-0227/97/97GL-00000\$05.00}

\authoraddr{H. Perfettini, J. Schmittbuhl, Laboratoire de G\'eologie, Ecole Normale Sup\'erieure, 75231 Paris Cedex 05, France.  (e-mail: perfetti@geologie.ens.fr; schmittb@geologie.ens.fr)}
\authoraddr{J.P. Vilotte, D\'epartement de Sismologie UMR 7580 and D\'epartement de Mod\'elisation Physique et Num\'erique, Institut de Physique du Globe de Paris, 4 Place Jussieu, 75252 Paris Cedex 05, France.}


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\begin{document}


\title{SLIP CORRELATIONS ON A CREEPING FAULT}

\author{Hugo Perfettini and Jean Schmittbuhl}
\affil{Laboratoire de Geologie, Ecole Normale Superieure, Paris, France}
\author{Jean-Pierre Vilotte}
\affil{D\'epartement de Sismologie, Institut de Physique du Globe, Paris, France}
%\altaffiltext{1}{Laboratoire de Geologie, Ecole Normale Superieure, Paris}
%\altaffiltext{1}{Laboratoire de G\'eologie, Ecole Normale Sup\'erieure, Paris}
%\centerline{\it to be submitted to Geophysical Research Letters}


\begin{abstract}
Using a quasi-static three dimensional fault model which accounts for
long range elastic interactions, we examine the influence of spatial
heterogeneities of frictional strength on the slip distribution along
a creeping fault. Slip fluctuates spatially because of pinning on
local asperities. We show that three regimes of slip correlations
exist. The first regime results in a uniform slip in an homogeneous
medium. On the contrary, slip in the second regime highly fluctuates
and is controlled by heterogeneities of frictional strength.  The
third regime is intermediate and develops areas of high slip that are
much bigger than the local asperity size (self-affine properties of
the slip distribution). This particular regime illustrates the
possible misinterpretation of low frequency slip data
(e.g. interferometric and GPS data) in terms of structural or
compositional properties along the fault.

\end{abstract}

\begin{article}
%\section{Introduction}

Spatio-temporal complexity of the rupture process and the earthquake
activity have become an evidence with the increasing resolution of
near field strong ground motion records. Sources have been shown to
present large heterogeneities in the coseismic slip and the rupture
velocity \citep{Archuleta84,Brune91,Cotton95}. More recently, various
combined inversions of interferometric, GPS and strong motion data
have been used to better constrained the source tomography
\citep{Wald96,Hernandez99,Delouis2000}. Recent works
(\cite{Rubin99};\cite{Nadeau95}) have shown also the existence of
spatio-temporal correlations between micro-earthquakes occuring along
creeping faults.  The origin of this complexity is still poorly
understood and often explained as generated by a combination of a
static prestress field and fault structural or compositional
heterogeneities. Whether the observed slip and slip rate patterns may
reflect the underlying frictional or geometrical properties of the
fault remains an important question. The goal of this work is to
address, using a rather simple fault model, this hypothesis and to
study the influence of quenched strength heterogeneities on the slip
distribution of slow creeping faults.


%\section{Model}
We consider a simple scalar elastic model of the rupture along a fault
plane, located at $y=0$, within an unbounded homogeneous elastic solid
assuming constant normal stress.  The problem is governed by a scalar
wave equation involving a two-dimensional displacement field
$U(x,y;t)$ and a related shear traction $\sigma(x,y;t)$.  The actual
slip $u(x;t)=U(x,0^+;t)-U(x,0^-;t)$ is the slip discontinuity across
the fault plane and $\tau(x;t)$ denotes the associated perturbation of
traction.  Since we focus on creeping faults, we assume that slip
occurs quasi-statically and neglect any dynamical effects. In that
case, we can neglect the radiation damping term and the stress change
$\tau(x;t)$ due to variations of slip discontinuity along the fault is
given by (e.g. \citep{Cochard97})
\begin{equation}
\label{model2}
\tau(x;t)=\frac{\mu}{2 \pi} PV \int_{L}^{}
J(x-\xi)[u(\xi;t)-u(x;t)]
\, d\xi
\end{equation}
where integration takes place over the fault of size $L$ and $PV$
indicates the principal value.  The elastic kernel $J(x)=1/x^2$
accounts for the long range elastic interactions and $\mu$ is the
shear modulus.  For plane strain, $u=u_x$,
$\tau(x;t)=\sigma_{yx}(x,0;t)$ and $\alpha=1-\nu$ while for anti-plane
strain, $u=u_z$, $\tau(x;t)=\sigma_{yz}(x,0;t)$ and $\alpha=1$.  For
simplicity, we assume an $L$ periodic interface in the $x$ direction
such that the $1/x^2$ kernel in (\ref{model2}) transforms in $J_L
(x)=(\pi/L)^2 /\sin^2(\pi x/L)$.

To account for heterogeneous frictional properties along the
interface, we balance $\tau(x;t)$ with a random quenched friction
strength $\eta_p(x;u(x;t))$. We assume that strength fluctuations
along the fault are strongly correlated over a scale $a$, and
uncorrelated on larger scales. The scale $a$ is assumed here to be
small compared to the fault scale $L$ ($L/a \ll 1$) and defines the
scale of the fault asperities. In the remaining, the discretization
length for the fault model will be chosen as the asperity scale $a$.
Since we have assumed no spatial correlation between asperities, the
friction strength $\eta_p(x;u(x;t))$ is assumed to have a uniform
probability density over the interval $[\eta_c-\delta
\eta_c;\eta_c+\delta \eta_c]$.  For sake of simplicity, we set $a=1$
(and any length scale of the problem is given in units of $a$).

It follows that, at any time, the quasi-static evolution motion of the
 fault has to satisfy:
\begin{equation}
\label{model4}
\eta_p(x;u(x;t))\ge\tau(x;t)
\end{equation}
for all points $x$ of the interface $\Gamma$.  The evolution of the
system may be regarded as purely dissipative, i.e. all the energy
being used in frictional work \citep{Fisher98}.

We assume that the fault is creeping at an arbitrary slow displacement
rate so that only one asperity slips at a time allowing all the
configurations of the slip distribution to be explored in the spirit
of a Monte-Carlo simulation.  At each step of the calculation, only
the weakest point slips of an elementary distance $s$ which defines
the unit length in the direction of propagation.  The fault evolves at
imposed displacement and is characterized by the mean slip
$\overline{u}=(1/L)\sum_{i=1}^{L} u_i$.  Therefore the time $t$ in
equation (\ref{model4}) should be replaced by $\overline{u}$ which is
the relevant variable for describing the evolution of slip. However,
time may be introduced by noting that the average slip $\overline{u}$
of the fault obeys $\overline{u}=V_0\,t$, where $V_0$ is the tectonic
slip rate.

The weakest asperity $x_{weak}$ corresponds to the site where the yield
function $Y$ (defined as the difference between frictional strength
and shear traction) is minimum:
\begin{equation}
\label{model5}
Y(x_{weak};\overline{u})=\min_{\forall x\in \Gamma}[\eta_p(x;u(x;\overline{u}))-\tau(x;\overline{u})]
\end{equation}
The asperities are supposed to experience new frictional properties
after local slip (instantaneous healing) with
$\eta_p(x_{weak};u(x_{weak};\overline{u}))$ updated according to the
uniform probability distribution.  The asperity dislocation at
$x_{weak}$ induces, according to (\ref{model2}), a long range variation
of the shear traction $\tau(x)$ along the fault.  Then a new moving
asperity is determined and the procedure is repeated again to explore
successive equilibrium position of the fault.

The behavior of the system is controlled by the competition between
local fluctuations of the frictional strength and the effect of long
range elastic interactions.  Using (\ref{model2}) the stress drop of a
point that just slipped by an amount is $s$ is
$2\,\mu\,s/(\pi\,a)$. This allows to define a dimensionless parameter
$\gamma=\mu\,s\sqrt{12}/(\pi\,a\,\delta\,\eta_c)$ where
$2\,\delta\eta_c/\sqrt{12}$ is the rms of the frictional strength
distribution.

When $\gamma$ is much greater than 1, the elastic interactions screen
the effect of the fluctuations of the frictional strengths.  The
heterogeneities are not strong enough to pin the front and slip can
never be arrested.  In our model for imposed displacement, the slip
runaway is controlled, in contrast with a stress controlled boundary
condition.  For this pulse-like propagative regime, heterogeneities
play a minor role and therefore this regime will be refered as the
{\it weakly heterogeneous} (WH) regime.


When $\gamma$ is much smaller than one, we observe an extreme regime
where recurrent slip activity can occur at the same point since the
stress drop due to elastic interactions is not big enough (compared to
$\delta \eta_c$) to unload the point. This regime will be refered as
the {\it strongly heterogeneous} regime (SH). In this case, the slip
field is expected to have the same statistical distribution than the
fluctuations of the frictional strength, {\it e.g.} a white noise in
the present study.

For the intermediate regime, the magnitude of the elastic interactions
are of the order of the amplitude of the frictional strength
variations. This is the {\it pining regime} (P) where interactions
between frictional heterogeneities and elastic stress transfers lead
to non trivial spatio-temporal correlations of slip in particular a
self-affine scale invariance. Such a scale invariance is also observed for a crack front propagation \citep{Schmittbuhl97}.

\begin{figure}
\figbox*{}{}{\epsfig{file=FIGURES/slip_distrib_fn_g.eps, height=6.5cm}}
\caption{\protect\small\rm Examples of the slip pattern in the
stationary regime for $L=2048$ and $\gamma=10^{-4}$, 0.1 and $10^{3}$.
Slip is normalized by its maximum value $u_{max}$ over the fault.  In
the case $\gamma=0.1$ no characteristic length is observed. Slip
distribution exhibits a the self-affine scale invariance.
\label{profil}}
\end{figure}

The simulations start with a zero displacement field.  Figure
\ref{profil} shows various slip distributions in steady state related
to $\gamma=10^{-4}$, $10^{-1}$ and $10^{3}$ illustrating the three
regimes mentioned above. 


%\section{Results}
In the {P} regime, the slip distribution reaches a stationary state
which is relevant for mature creeping faults ({\it i.e.} faults which
have undergone a sufficient amount of slip). Figure \ref{profil} for
$\gamma=0.1$ shows a typical slip distribution in the steady state
regime: No characteristic length scale appears. The self-affine
invariance of the slip fluctuations along the fault is characterized
by the Hurst exponent $\zeta$ which can be estimated using the
auto-correlation function 
$g(x,x';\overline{u})=<[(u(x;\overline{u})-u(x';\overline{u})]^2>_x$ 
(where $<...>_x$ means average over the whole fault).  For a
self-affine signal, the auto-correlation function behaves like a power
law i.e. $g(x,x';\overline{u})\propto |x-x'|^{2\zeta}$ and implies
${\delta u}\propto \delta x^{\zeta}$ where ${\delta u}=<u(x+\delta
x)-u(x)>_x$ represents the mean fluctuation of slip on the fault over
a length $\delta x$.  Many methods (variable bandwidth, return
probability, Fourier spectrum, wavelet analysis, etc..., as discussed
for instance in \cite{Schmittbuhl95b}) may be used to measure $\zeta$.
For example the power spectrum $P(k)$ (i.e. Fourier transform of the
auto-correlation function mentioned above) of a self affine signal
with Hurst exponent $\zeta$ shows a power law behavior $P(k)\propto
k^{-1-2\zeta}$.  Using this last method, we found $\zeta\simeq 0.35$ a
value consistent with previous estimates of $\zeta$ in similar models
(\citep{Schmittbuhl95a,Schmittbuhl99}).

\begin{figure}
\figbox*{}{}{\epsfig{file=FIGURES/furu_fn_dt_g_0.1_new_theta.eps, height=6.5cm}}
\caption{\protect\small\rm Power spectrum in the steady state regime
of the incremental slip when $\gamma=0.1$ for different increment
displacements $\Delta \overline{u}$ and a system size of $L=4096$.
Starting from short range correlations, the system develops
correlations at every length scales as $\Delta\overline{ u}$
increases.
\label{temporal}}
\end{figure}

We investigate space-time (where time should be understood as mean
slip) in the stationary regime by studying the increment evolution
between two states of the system separated by a slip $\Delta
\overline{u}$.  The evolution can be analyzed computing the power
spectrum of the slip increment $\Delta u(x;\Delta\overline{ u})=
<u(x;\overline{u}+\Delta\overline{
u})-u(x;\overline{u})>_{\overline{u}}$ where $<...>_{\overline{u}}$
means averaging over all the configurations $\overline{u}$.  The power
spectrum $P_{\Delta u}(k)$ of $\Delta u(x;\Delta \overline{u})$ is
plotted in figure \ref{temporal} for a system of size $L=4096$ and
various values of $\Delta\overline{ u}$.  For small increments
$\Delta\overline{ u}$, the spectrum is essentially flat showing that
the spatial correlations between successive slip distributions occur
at very short wavelengths.  As $\Delta\overline{ u}$ increases, the
spectrum reaches a power law trend characteristic of self-affinity
where correlations exist at all length scales. Self-affinity develops
only over scales lower than a characteristic spatial scale
$\xi(\Delta\overline{u})$ which defines an activity zone that spreads
with the evolution of the slip increment $\Delta \overline{u}$ as
(e.g. \citep{Tanguy98})
\begin{equation}
\label{result1}
\xi\propto \Delta \overline{u}^{1/z}
\end{equation}
where $z$ is a dynamic exponent. In the Fourier domain, the power
spectrum should exhibit a power law trend for frequencies higher than
$cste/\xi(\Delta\overline{ u})$ and be flat for lower frequencies.
This behavior is illustrated in figure \ref{temporal}.  The spreading
of the activity continues until the activity zone $\xi$ reaches the
system size $L$ defining a characteristic slip increment
$\Delta\overline{ u}_L$. At that stage, the power spectrum will not
evolve for slip greater than $\Delta\overline{ u}_L$ since
correlations have developed over the whole system size.  The slope of
the linear trend of the power spectra in figure \ref{temporal} is
-1.7, a value consistent with the self-affine scaling of
$\overline{u}$ ($\xi\simeq 0.35$).

The spatial correlations of slip can be related to the temporal ones
since $z=\zeta+1$ (e.g. \citep{Tanguy98}).  After a slip increment
$\Delta \overline{u}$, the activity is localized within a region of
extension $\xi\propto \Delta\overline{ u}^{1/z}$.  The number of
asperities that have slipped during that move is proportional to the
slip area {\it i.e.} $\xi \Delta \overline{u}\propto \xi^{1+\zeta}$,
where the relation $\Delta \overline{u}\propto \xi^{\zeta}$ has been
used.  The number of asperities is also proportional to the slip
increment $\Delta \overline{u}$ so that $\xi^{1+\zeta}\propto \xi^{z}$
giving the expected relation.

\begin{figure}
\figbox*{}{}{\epsfig{file=FIGURES/activity_map_fn_g_zoom.eps, height=6.5cm}}
\caption{\protect\small\rm Activity maps for $L=2048$ and different
values of $\gamma$.  (top) The regime {WH} ($\gamma=10^{3}$) is
dominated by the elastic interactions which results in a pulse-like
propagation of slip. (middle) The intermediate regime {P}
($\gamma=10^{-1}$) shows the existence of spatio-temporal correlations
and the clustering of the activity.  (bottom) regime {SH}
($\gamma=10^{-4}$) is controlled by the fluctuations of the frictional
strength and a random diffusion of the activity all through the fault.
\label{activity_map}}
\end{figure}


The spreading of the activity with increasing displacement in the { P}
regime results from two competing effects: long range elastic
interactions and frictional strength heterogeneities.  The former
tends to strongly correlate slip laterally while the latter tends to
pin the slip at one site.  Figure \ref{activity_map} shows the
activity maps or locations of active sites as slip $\overline{u}$ is
evolving for various values of the parameter $\gamma$
($\gamma=10^{-4}$, $10^{-1}$ and $10^{3}$).

In the propagative regime { WH} (top panel), the slip propagates
laterally as observed in the zoom window.  The size of the activity is
expected to scale with displacement as $\xi\propto \overline{u}$
($z=1$) since the slip of an asperity triggers only the motion of its
immediate neighbor.

In the strongly heterogeneous regime {SH} (middle panel of
fig. \ref{activity_map}), the activity is analogous to a brownian
motion. The activity is spread all over the fault since the elastic
coupling is too weak to overcome the frictional strength fluctuations.
In this case, $\xi\propto \overline{u}^{1/2}$ ($z=2$, $\zeta=-1/2$) as
expected for a classical diffusive process. In the pining regime {P}
(bottom panel), we observe a clustering of slip as shown within the
zoom window of the activity map (fig. \ref{activity_map}).  We found
$z\simeq 1.35$ as in \cite{Schmittbuhl95a}, a value bracketed by the
$z=1$ of the WH regime and the $z=2$ of the SH regime.

\begin{figure}
\figbox*{}{}{\epsfig{file=FIGURES/spec_energy_cum.eps, height=6.5cm}}
\caption{\protect\small\rm Power spectrum of the large scale
frictional work $W$ for a system size $L=4096$.  The best fit has a
slope $-2$ ($W(\omega) \propto \omega^{-2}$) leading to a scaling
$W(\overline{u}) \propto \overline{u}^{1/2}$.
\label{energy}}
\end{figure}

We have studied so far spatial correlations of the asperity slips.  At
the scale of the fault, one can introduce a friction force
$F_c(\overline{u})$ defined as $Y(x_{weak};\overline{u})$ given in
(\ref{model5}).  $F_c(\overline{u})$ is the yield force above which
the fault is moving.  The energy (frictional work) dissipated by the
fault after slip $\overline{u}$ is
$W(\overline{u})=\int_{0}^{\overline{u}} F_c(\lambda)\,d\lambda$.  For
the pinning regime P, the energy power spectrum $P_W(\omega)$ exhibits
in figure~\ref{energy} a power law scaling $W(\omega)\propto
\omega^{-2}$. Such $\omega$-square decay could be related to
seismological observations of radiated spectrum for the far-field
displacement \citep{Aki67,Houston86}.


%\section{Discussion}
We have studied a simple model of a creeping fault moving under a slow
imposed displacement with long range elastic interactions and
frictional heterogeneities.  Three different regimes of slip
fluctuations have been observed according to a non dimensional
parameter $\gamma$ which compares elastic stress coupling and
frictional strength fluctuations: a first regime ($\gamma \gg 1$) with
a pulse-like propagation of the slip activity; an intermediate regime
($\gamma \simeq 1$) where the slip activity remains clustered over a
characteristic length $\xi$, which results from the competition
between long range elastic interactions and frictional strength
fluctuations, and that evolves with the imposed displacement; a third
regime ($\gamma \ll 1$) where activity spreads as a brownian motion
and where slip fluctuations map frictional strength fluctuations.

The analysis may have interesting implications for interpretation of
low frequency slip inversions using strong motion, GPS and SAR
measurements. The slip fluctuations observed in our model should have
a strong signature in GPS or SAR signals due to the power law decay of
the slip power spectrum. Slip clusters of size $\xi$, as observed in
the pinning regime (P), will appear at low frequencies as areas of
uniform slip. In the light of the present analysis, slip patches
inferred from GPS or SAR measurements may be quite misleading when
mapped onto fault structural heterogeneities . Indeed, in the pinning
regime the size of a slip activity zone is independent of the asperity
size.
 

%Spatial correlations between frictional asperities have been studied
%by \cite{Schmittbuhl99} using a similar model and considering
%short and long-range Gaussian correlations.
%The% latter affect the value of the $\zeta$ and $z$ exponents
%in the {\bf WP} regime, leave regime {\bf P} unchanged,
%and slip correlations in regime {\bf SP} is expected to be identical
%to spatial fluctuations of the frictional strength.





\acknowledgments
We are grateful to R. Madariaga, A. Cochard, and J. P. Ampuero
for their generous help.


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\bibliography{slip_correl}

\end{article}
\end{document}
