We consider a plane fault with two asperities embedded in a shear zone, subject to a uniform strain rate owing to tectonic loading. After an earthquake, the static stress field is relaxed by viscoelastic deformation in the asthenosphere. We treat the fault as a discrete dynamical system with 3 degrees of freedom: the slip deficits of the asperities and the variation of their difference due to viscoelastic deformation. The evolution of the fault is described in terms of inter-seismic intervals and slip episodes, which may involve the slip of a single asperity or both. We consider the effect of stress transfers connected to earthquakes produced by neighbouring faults. The perturbation alters the slip deficits of both asperities and the stress redistribution on the fault associated with viscoelastic relaxation. The interplay between the stress perturbation and the viscoelastic relaxation significantly complicates the evolution of the fault and its seismic activity. We show that the presence of viscoelastic relaxation prevents any simple correlation between the change of Coulomb stresses on the asperities and the anticipation or delay of their failures. As an application, we study the effects of the 1999 Hector Mine, California, earthquake on the post-seismic evolution of the fault that generated the 1992 Landers, California, earthquake, which we model as a two-mode event associated with the consecutive failure of two asperities.

Asperity models have long been acknowledged as an effective means to describe many aspects of fault dynamics (Lay et al., 1982; Scholz, 2002). In such models, it is assumed that the bulk of energy release during an earthquake is due to the failure of one or more regions on the fault characterized by a high static friction and a velocity-weakening dynamic friction. The stress build-up on the asperities is governed by the relative motion of tectonic plates. Earthquakes that have been ascribed to the slip of two asperities are the 1964 Alaska earthquake (Christensen and Beck, 1994); the 1992 Landers, California, earthquake (Kanamori et al., 1992); the 2004 Parkfield, California, earthquake (Twardzik et al., 2012); and the 2010 Maule, Chile, earthquake (Delouis et al., 2010).

In the framework of asperity models, a critical role is played by stress accumulation on the asperities, fault slip at the asperities and stress transfer between the asperities. Accordingly, fault dynamics can be fruitfully investigated via discrete dynamical systems whose essential components are the asperities (Ruff, 1992; Turcotte, 1997). Such an approach reduces the number of degrees of freedom required to describe the dynamics of the system, that is, the evolution of the fault (in terms of slip and stress distribution) during the seismic cycle; also, it allows the visualization of the state of the fault and to follow its evolution via a geometrical approach, by means of orbits in the phase space. Finally, a finite number of dynamic modes can be defined, each one describing a particular phase of the evolution of the fault (e.g. tectonic loading, seismic slip, after-slip). Asperity models are capable of reproducing the essential features of the seismic source, while sparing the more complicated characterization based on continuum mechanics.

In a number of recent works, modelling of different mechanical phenomena in a two-asperity fault system has been addressed, such as stress perturbations due to surrounding faults (Dragoni and Piombo, 2015) and the radiation of seismic waves (Dragoni and Santini, 2015). In these models, the fault is treated as a discrete dynamical system with four dynamic modes: a sticking mode, corresponding to stationary asperities, and three slipping modes, associated with the separate or simultaneous failure of the asperities.

In the framework of a discrete fault model, the impact of viscoelastic relaxation was first studied by Amendola and Dragoni (2013) and then further investigated by Dragoni and Lorenzano (2015), who considered a fault with two asperities of different strengths. The authors discussed the features of the seismic events predicted by the model and showed how the shape of the associated source functions is related to the sequence of dynamic modes involved. In turn, the observation of the moment rate provides an insight into the state of the system at the beginning of the event, that is, the particular stress distribution on the fault from which the earthquake takes place.

However, no fault can be considered isolated; in fact, any fault is subject to stress perturbations associated with earthquakes on neighbouring faults (Harris, 1998; Stein, 1999; Steacy et al., 2005). Whenever a fault slips, the stress field in the surrounding medium is altered. As a result, the occurrence time and the magnitude of next earthquakes may change with respect to the unperturbed condition, which is governed by tectonic loading.

The aim of the present paper is to discuss the combined effects of viscoelastic relaxation and stress perturbations on a two-asperity fault in the framework of a discrete fault model. In order to deal with such a problem, we base our work on the results achieved by Dragoni and Piombo (2015) and Dragoni and Lorenzano (2015). In the former work, the authors considered a two-asperity fault with purely elastic rheology and discussed the effect of stress perturbations due to earthquakes on neighbouring faults. The fault was treated as a discrete dynamical system whose state is described by two variables, the slip deficits of the asperities. In the latter work, viscoelastic relaxation on the fault was dealt with by adding a third state variable, the variation in the difference between the slip deficits of the asperities during inter-seismic intervals. In the present paper, we introduce stress perturbations as modelled by Dragoni and Piombo (2015) in the framework of the two-asperity fault considered by Dragoni and Lorenzano (2015). Accordingly, the present work represents a three-dimensional generalization of the model devised by Dragoni and Piombo (2015). Elastic wave radiation and additional constraints on the state of the fault are taken into account, as further developments with respect to previous works.

In the framework of the present model, seismic events generated by the fault are discriminated according to the number and sequence of slipping modes involved and the seismic moment released; these features are related to the particular state of the system at the beginning of the inter-seismic interval preceding the event. We discuss how stress perturbations affect the evolution of the fault in terms of changes in the state of the system and in the duration of the inter-seismic time, highlighting the complications arising from the ongoing post-seismic deformation process with respect to the purely elastic case considered by Dragoni and Piombo (2015). As an application, we consider the stress perturbation imposed by the 1999 Hector Mine, California, earthquake (Jónsson et al., 2002; Salichon et al., 2004) to the fault that caused the 1992 Landers, California, earthquake, which we model as a two-mode event due to the consecutive failure of two asperities and that was followed by remarkable viscoelastic relaxation (Kanamori et al., 1992; Freed and Lin, 2001). We propose a means to estimate the stress transfer from the knowledge of the relative positions and faulting styles of the two faults. As a further novelty with respect to the work presented by Dragoni and Lorenzano (2015), we show how the knowledge of the time interval elapsed after the 1999 earthquake can be used to constrain the admissible set of states that may have given rise to the 1992 event. We discuss the possible subsequent evolution of the Landers fault after the stress transfer from the Hector Mine earthquake, pointing out the main differences with respect to an unperturbed scenario.

We consider a plane fault containing two asperities of equal areas

In the present model we do not consider aseismic slip on the fault. It has been treated in the framework of a discrete fault model by Dragoni and Lorenzano (2017), who considered a region slipping aseismically for a finite time interval and calculated the effect on the stress distribution and the subsequent evolution of the fault. Of course, if the amplitude of aseismic slip has the same order of magnitude as that of seismic slip, the fault evolution may be affected.

In accordance with the assumptions of asperity models, we ascribe the generation of earthquakes on the fault to the failure of the sole asperities, neglecting any contribution of the surrounding weaker region to the seismic moment. Also, we do not describe friction, slip and stress at every point of the fault, but only consider their average values on each asperity.

Sketch of the model of a plane fault with two asperities. The
rectangular frame is the fault border. The state of the asperities is
described by their slip deficits

The fault is treated as a dynamical system with three state variables,
functions of time

We assume the simplest form of rate-dependent friction and associate the
asperities with constant static and dynamic frictions, the latter considered
as the average value during slip. The static friction on asperity 2 is a
fraction

During a global stick mode, the tangential forces acting on the asperities in
the slip direction are (in units of static friction on asperity 1)

An effective way to characterize fault mechanics is provided by the concept
of Coulomb stress (Stein, 1999). It is defined as the difference between the
shear stress

When considering the fault dynamics during the seismic cycle, it is possible
to identify four dynamic modes, each one described by a different system of
autonomous ODEs (ordinary differential equations): a sticking mode (00), corresponding to stationary asperities, and
three slipping modes, associated with the slip of asperity 1 alone (mode 10),
the slip of asperity 2 alone (mode 01) and the simultaneous slip of the
asperities (mode 11). A seismic event generally consists of

The sticking region of the system is defined as the set of states in which
both asperities are stationary. During a global stick phase (mode 00), the
rates

The slip of asperity 1 occurs when

We exclude overshooting during the slipping modes: accordingly, we assume

To sum up, the sticking region of the system is the subset of the

Let

The sticking region of the system, defined as the subset of the
state space

Mode 00 terminates at a point

The surface

The faces

In addition, the knowledge of the position of

As for the evolution of the variable

We now consider the perturbations of the state of the fault caused by the coseismic slip on surrounding faults. Following Dragoni and Piombo (2015), we assume that (1) the perturbations occur during an inter-seismic interval; (2) the stress transfer takes place over a time interval negligible with respect to the duration of the inter-seismic interval; and (3) at the time of the perturbation, the state of the fault is sufficiently far from the failure conditions and the stress transfer is small enough that the onset of motion of either asperity is not achieved immediately.

Final slip amplitudes

Let

The components of

The vector

Since the stress perturbation does not alter the friction coefficients of
rocks, it is reasonable to assume that the ratio

The stress transfer resulting from earthquakes on neighbouring faults alters
several parameters of the model. A first remarkable change concerns the
strength of the asperities. After the perturbation, we can define a new ratio

The variations in static frictions entail different conditions for the onset
of motion of the asperities. Taking Eq. (

Following the changes in static frictions, the surface

As a consequence of the changes in dynamic frictions, the amount of slip that
asperities undergo during a seismic event is modified. In turn, the
perturbation alters the seismic moment associated with an earthquake. The
variations in the final slip amplitudes

The variations in tangential stresses and static frictions discussed so far
entail a change in the Coulomb forces assigned to the asperities. Combining
Eq. (

On the whole, the effect of the stress perturbation can be discussed in terms
of the quantity

Changes in the final slip amplitudes

As already stated, stress perturbations can anticipate or delay the
occurrence of an earthquake produced by a certain asperity. We now quantify
this effect in terms of the variation in the duration of the inter-seismic
interval. Generally speaking, the perturbation vector

Let us first focus on the case in which the unperturbed state

If instead

According to the model, rock rheology plays a critical role in the response
to stress perturbations. In the case of purely elastic coupling between the
asperities, Dragoni and Piombo (2015) showed that the changes in the
duration of the inter-seismic interval prior to the failure of asperity 1 and
2 are, respectively,

Conversely, in the viscoelastic case there is no straightforward connection
between the sign of

We study the effects of the 16 October 1999

The 1992 Landers earthquake was a right-lateral strike-slip event that can be
approximated as the result of the slip of two coplanar asperities (Kanamori
et al., 1992): a northern one (asperity 1) and a southern one (asperity 2),
with average slips

The 1992 event was followed by remarkable post-seismic deformation, which can
be interpreted as the result of several processes. For the sake of the
present application, we assume viscoelastic relaxation as the most
significant mechanism. We assign a viscosity

Geometry of the Landers (LAN) and Hector Mine (HM), California, faults that generated the 1992 and 1999 earthquakes, respectively. The stars indicate the hypocentres of the seismic events. The labels 1 and 2 identify the asperities on the Landers fault.

We model the 1992 earthquake as a two-mode event 01-10 starting from mode 00.
Accordingly, the orbit of the system during mode 00 lies inside the subset

Dragoni and Tallarico (2016) studied the 1992 Landers earthquake under the
hypothesis of purely elastic coupling between the asperities. Following the
authors, we take

Every state

The 1999 Hector Mine earthquake was generated by right-lateral strike-slip
faulting located at 34.59

The stress transferred to the asperities at Landers can be evaluated
employing the model of Appendix

We now introduce the effect of the perturbation in the framework of the
discrete model. The changes in the tangential forces (

In order to characterize the effect of the perturbation, let us consider the
difference

In order to improve our knowledge on the state that gave rise to the 1992
Landers earthquake and on the possible future events generated by that fault,
we exploit the seismic history between 1999 and the present date. After the
perturbation caused by the Hector Mine earthquake, the inter-seismic time

As a consequence, we can constrain the admissible states on the segment

These conclusions would have to be reconsidered if new stress perturbations
from neighbouring faults were to affect the post-seismic evolution of the
Landers fault in the future. In addition, if no earthquakes were to be
observed for some time on the Landers fault, the refining procedure discussed
above could be repeated and the admissible subsets of segments

Finally, we discuss the features of the next seismic event generated by the 1992 Landers fault, highlighting the changes due to the Hector Mine earthquake.

Every state

The number and the sequence of dynamic modes in the earthquake depend on the
sub-interval of

Another significant result of the stress perturbation concerns the variation
in the inter-seismic time before the next seismic event. As in Sect.

Future earthquakes generated by the 1992 Landers, California, fault,
as functions of the variable

Change in the seismic moment released during the next event on the
1992 Landers, California, fault, as a result of the stress perturbation due
to the 1999 Hector Mine, California, earthquake. On the horizontal axis, the
variable

Some peculiar features stand out. First, we notice that, for all states

Change in the inter-seismic time before the next event on the 1992
Landers, California, fault, as a result of the stress perturbation due to the
1999 Hector Mine, California, earthquake. On the horizontal axis, the
variable

At the occurrence of the next earthquake produced by the Landers fault, the number and sequence of dynamic modes involved and the energy released will reveal more about the state of the system, thus allowing a further refinement of the specific conditions that gave rise to the 1992 event.

We considered a plane fault embedded in a shear zone, subject to a uniform strain rate owing to tectonic loading. The fault is characterized by the presence of two asperities with equal areas and different frictional resistance. The coseismic static stress field due to earthquakes produced by the fault is relaxed by viscoelastic deformation in the asthenosphere.

The fault was treated as a discrete dynamical system with three degrees of freedom: the slip deficits of the asperities and the variation of their difference due to viscoelastic deformation. The dynamics of the system was described in terms of one sticking mode and three slipping modes. In the sticking mode, the orbit of the system lies in a convex hexahedron in the space of the state variables, while the number and the sequence of slipping modes during a seismic event are determined by the particular state of the system at the beginning of the inter-seismic interval preceding the event. The amount of slip of the asperities and the energy released during an earthquake generated by the fault can be predicted accordingly.

The effect of stress transfer due to earthquakes on neighbouring faults was studied in terms of a perturbation vector yielding changes to the state of the system, its sticking region and the energy released during a subsequent seismic event. The specific effect on the evolution of the fault is related with the orientation of this vector in the state space.

We investigated the interplay between the ongoing viscoelastic relaxation on the fault and a stress perturbation imposed during an inter-seismic interval. Following a stress perturbation due to earthquakes on neighbouring faults, an increase in the Coulomb stress associated with a given asperity directly yields the anticipation of the slip of that asperity, and vice versa, if a purely elastic rheology is assumed for the receiving fault (Dragoni and Piombo, 2015). According to the present model, this property no longer holds if the change in Coulomb stress occurs while viscoelastic relaxation is taking place on the receiving fault. In fact, even if the change in the inter-seismic intervals of the asperities can still be evaluated from a theoretical point of view, the specific effect of the stress perturbation could be univocally inferred only if the particular states of the fault at the time of the stress perturbation and right after it were known. The information on the change in Coulomb stress on the fault do not suffice any more.

We applied the model to the stress perturbation imposed by the 1999 Hector Mine, California, earthquake to the fault that caused the 1992 Landers, California, earthquake, which was due to the failure of two asperities and was followed by significant viscoelastic relaxation. We modelled the 1992 Landers earthquake as a two-mode event associated with the separate slip of the asperities and showed how the event is compatible with a number of possible initial states of the fault, which can be screened on the basis of the seismic history to date. The details of the stress transfer associated with the 1999 Hector Mine earthquake were calculated using the relative positions and faulting styles of the two faults as a starting point. We discussed the effect of the stress perturbation, pointing out the complexity of its influence on the possible future events generated by the 1992 Landers fault in terms of the associated energy release, the sequence of dynamic modes involved and the duration of the inter-seismic interval. Specifically, we showed that the consequences of the 1999 Hector Mine earthquake on the post-seismic evolution of the 1992 Landers fault depend on the specific state of the Landers fault at the time of the 1999 earthquake and immediately after it, even if the variations in the Coulomb stress on the asperities at Landers are known. On the whole, the application allowed the exemplification of the critical unpredictability of the effect of a stress perturbation occurring while viscoelastic relaxation is taking place.

Another source of complication may be represented by the interaction between viscoelastic relaxation and stable creep on the fault. This problem is beyond the scope of the present work, but it may be object of future research by combining elements of the present model with the one of Dragoni and Lorenzano (2017).

All data and results supporting this work were gathered
from the papers listed in the References and are freely available to the
public. Specifically, data on the 1999 Hector Mine, California, earthquake
were collected from the Finite-Source Rupture Model Database (SRCMOD)
available at

We consider two plane faults, namely fault 1 and fault 2, embedded in an
infinite, homogeneous and isotropic Poisson medium of rigidity

We define a coordinate system

placing the origin at the centre of fault 1:

clockwise rotation about the

counterclockwise rotation about the

Geometry of the model employed to study the stress transfer between
neighbouring faults. Fault 1 is the perturbing fault, while fault 2 is the
receiving fault. The coordinates

The perturbing fault is treated as a point-like dislocation source (a double
couple of forces) located at the origin. This is a good approximation for
non-overlapping regions (Dragoni and Lorenzano, 2016). Let

EL developed the model, produced the figures and wrote a preliminary version of the paper; MD checked the equations and revised the text. Both authors discussed the results extensively.

The authors declare that they have no conflict of interest.

The authors are thankful to the editor Richard Gloaguen, to Sylvain Barbot and to an anonymous referee for their useful comments on the first version of the paper. Edited by: Richard Gloaguen Reviewed by: one anonymous referee