A fault containing two asperities with different strengths is considered. The fault is embedded in a shear zone subject to a constant strain rate by the motions of adjacent tectonic plates. The fault is modelled as a discrete dynamical system where the average values of stress, friction and slip on each asperity are considered. The state of the fault is described by three variables: the slip deficits of the asperities and the viscoelastic deformation. The system has four dynamic modes, for which analytical solutions are calculated. The relationship between the state of the fault before a seismic event and the sequence of slipping modes in the event is enlightened. Since the moment rate depends on the number and sequence of slipping modes, the knowledge of the source function of an earthquake constrains the orbit of the system in the phase space. If the source functions of a larger number of consecutive earthquakes were known, the orbit could be constrained more and more and its evolution could be predicted with a smaller uncertainty. The model is applied to the 1964 Alaska earthquake, which was the effect of the failure of two asperities and for which a remarkable post-seismic relaxation has been observed in the subsequent decades. The evolution of the system after the 1964 event depends on the state from which the event was originated, that is constrained by the observed moment rate. The possible durations of the interseismic interval and the possible moment rates of the next earthquake are calculated as functions of the initial state.

Many aspects of fault dynamics can be reproduced by asperity models (Lay et al., 1982; Scholz, 1990), assuming that one or more regions of the fault have a much higher friction than the adjacent regions. Several large and medium-size earthquakes that occurred in the last decades were the result of the failure of two distinct asperities, such as the 1964 Alaska earthquake (Christensen and Beck, 1994), the 1995 Kobe earthquake (Kikuchi and Kanamori, 1996), the 2004 Parkfield earthquake (Johanson et al., 2006) and the 2010 Maule, Chile, earthquake (Delouis et al., 2010).

In the framework of an asperity model, the evolution of asperities in terms of stress accumulation, seismic slip and mutual stress transfer plays a key role. Therefore, the dynamical behaviour of a fault can be fruitfully investigated by means of discrete models describing the state of asperities (e.g. Ruff, 1992; Rice, 1993; Turcotte, 1997). An advantage associated with a finite number of degrees of freedom is that we can predict the evolution of the system at long term by calculating its orbit in the phase space.

A discrete fault model with two asperities was originally proposed by Nussbaum and Ruina (1987) and further investigated by Huang and Turcotte (1990, 1992), McCloskey and Bean (1992) and others. Dragoni and Santini (2012, 2014) solved analytically the equations of motion in the case of a two-asperity fault with different strengths in an elastic medium.

In the long-term evolution of a fault, the rheological properties of Earth's lithosphere play an important role. Lithospheric rocks are not perfectly elastic but have a certain degree of anelasticity (Carter, 1976; Kirby and Kronenberg, 1987; Ranalli, 1995; Nishimura and Thatcher, 2003; Bürgmann and Dresen, 2008). As a consequence, the static stress fields produced by fault dislocations undergo a certain amount of relaxation during the interseismic intervals, which alters the stress distribution on faults and modifies the occurrence times of seismic events (Kusznir, 1991; Kenner and Segall, 2000; Smith and Sandwell, 2006; Piombo et al., 2007; Ding and Lin, 2014).

A preliminary study of the effects of viscoelastic relaxation on a fault containing two asperities was made by Amendola and Dragoni (2013), in the case of asperities with the same frictional strength. It was shown that the stresses on the asperities increase non-linearly during the interseismic intervals, although the tectonic loading takes place at constant rate. As a consequence, earthquakes are anticipated or delayed with respect to the case of an elastic medium. In addition, the stress rate is different for the two asperities, so that the stress distribution changes during loading and the asperity subject to the greater stress at a given instant in time is not necessarily the first one to fail in the next earthquake.

The present paper generalizes Amendola and Dragoni (2013) in that it considers two asperities with different strengths and a larger set of possible states for the fault in the interseismic intervals. We investigate which subsets of states drive the system to the failure of one asperity or both. Whether the failure starts at one asperity or the other has consequences on the position of the earthquake focus as well as on its source function and seismic moment.

The model is applied to the 1964 Alaska earthquake, for which a sufficiently long time interval has elapsed to allow for observation of a remarkable post-seismic relaxation (Zweck et al., 2002). The moment rate of the earthquake was modelled by Dragoni and Santini (2012), showing that it can be approximately represented as a two-mode event with the consecutive failure of the two asperities. We study the subsequent evolution of the system in the presence of viscoelastic relaxation and calculate the duration of the interseismic interval and the possible source functions of the next earthquake.

We consider a plane fault with two asperities of equal areas, that we name
asperity 1 and 2, respectively (Fig. 1). Following Amendola and Dragoni (2013),
all quantities are expressed in nondimensional form. We assume that
the fault is embedded in a homogeneous and isotropic shear zone, subject to a
uniform strain rate by the motion of two tectonic plates at relative velocity

The fault model. The state of the fault is described by the slip
deficits

We do not determine stress, friction and slip at every point of the fault
but, instead, the average values of these quantities on each asperity. We
define the slip deficit of an asperity at a certain instant

The state of the fault is described by three variables:

The forces

Fault slip is governed by friction, which is best described by the
rate-and-state friction laws (Ruina, 1983; Dieterich, 1994). According to the
premise, we use a simpler law assuming that the asperities are characterized
by constant static frictions and consider the average values of dynamic
frictions during fault slip. We assume that the static friction of asperity 2
is a fraction

If we call

The dynamics of the system has four different modes: a sticking mode, corresponding to stationary asperities (mode 00), and three slipping modes, corresponding to the failure of asperity 1 (mode 10), the failure of asperity 2 (mode 01), and the simultaneous failure of both asperities (mode 11). Each mode is described by a different system of differential equations.

In mode 00, the velocities

A seismic event takes place when the orbit of the system reaches one of the
faces

The equations of motions of the four dynamic modes and the corresponding
solutions are given below. Viscoelastic relaxation is negligible during the
slipping modes; therefore, the equations for

The sticking region

The variables

The equations of motion are

Case 11

the solution is

where

The slip duration, calculated from the condition

and the final slip amplitude is

Case 00

and from Eq. (

The solution reduces to

If the orbit does not reach the face

If the orbit reaches

where

The equations of motion are

Case 11

the solution is

where

The slip duration, calculated from the condition

and the final slip amplitude is

Case 00

and from Eq. (

The solution reduces to

If the orbit does not reach the face

If the orbit reaches

where

The equations of motion are

Case 10

and the constants are

Case 01

The constants

Case 00

The constants

In general, a seismic event will involve

The surface

During the interseismic intervals, the fault is subject to continuous
tectonic loading due to the motion of adjacent plates and to the effect of
viscoelastic relaxation of the stress accumulated by previous seismic
activity. Given any state

Any curve (Eq.

The faces

After an orbit reaches one of the faces

Therefore, events involving the simultaneous failure of asperities can take place only from particular subsets of states of the system. In general, a three-mode event can result from four different sequences of modes: 10–11–10, 10–11–01, 01–11–10, and 01–11–01. In particular cases, sequences like 10–01–10 or 01–10–01 are also possible.

The reasons for the different sequences of modes involved in the earthquake
are clear if we consider the forces acting on the asperities in the different
states. If we consider the face

This analysis enlightens the relationship between the state of the fault before an earthquake and the sequence of modes in the seismic event. It also suggests that the knowledge of the source function of an earthquake may allow us to constrain the orbit of the system in the phase space.

The number and the sequence of slipping modes involved in a seismic event
determine the moment rate of the earthquake. Let

This allows us to constrain the evolution of the system to a certain subset of the phase space and, when the next earthquake occurs, the knowledge of its moment rate will allow us to further constrain this subset. Hence, if we knew the source functions of a sufficiently large number of consecutive earthquakes, we could constrain more and more the orbit of the system and its evolution could be predicted with a smaller uncertainty.

The 1964 Alaska earthquake was one of the largest earthquakes in the last
century, with a seismic moment

For the Alaska earthquake there is clear evidence that post-seismic deformation
occurred in the decades following the event (Zweck et al., 2002; Suito and
Freymueller, 2009). Part of the deformation has been ascribed to aseismic
slip of the fault and part to viscoelastic relaxation. The latter shows a
characteristic time

According to the present model, the seismic event was a sequence of modes 10–01
starting from mode 00. Since the first mode was 10, the orbit of the
system in mode 00 was in the subset

From Eq. (

Then, according to Eqs. (

On the basis of a purely elastic model, Dragoni and Santini (2012) predicted
that the next large earthquake involving the 1964 fault would take place
about 350 years later and would be due to the failure of asperity 2 alone. If
we introduce viscoelastic relaxation, a wider range of possibilities appears.
Since the segment

Examples of possible moment rates

According to the present model, the duration of the interseismic interval
between 1964 and the next earthquake is

In general, the next event will be an

Hence, the interval [

The corresponding values of the seismic moment

Examples of moment rates

We considered a fault with two asperities of different strengths placed in a shear zone subject to a constant strain rate by the motion of adjacent tectonic plates. The equations of motion were written under the hypothesis that the asperities have the same area: this is a reasonable approximation for many earthquakes. The system has been represented by a discrete model described by three variables: the slip deficits of the asperities and the viscoelastic deformation. The system dynamics has one sticking mode and three slipping modes, for which we solved analytically the equations of motion.

If the state of the fault at a given instant in time is known in terms of the system variables, we can calculate the orbit of the system in the phase space and hence predict its evolution. The state of a fault is not directly measurable, but the model shows that the knowledge of the earthquake source functions allows us to constrain the orbit of the system.

The study of the sticking region of the phase space shows how the state of the system before a seismic event controls the sequence of slipping modes in the event. Since the moment rate depends on the number and the sequence of slipping modes, the knowledge of the source function of an earthquake constrains the possible states of the system, hence its orbit in the phase space. Then, if we knew the source functions of a sufficiently large number of consecutive earthquakes, we could constrain the orbit more and more and predict its evolution with a smaller uncertainty.

As an example, we considered the fault that originated the 1964 Alaska earthquake. This earthquake was due to the failure of two distinct asperities; being a large-size event, it was followed by remarkable post-seismic deformation; in addition, more than 50 years have elapsed since the earthquake, allowing such a deformation to be observed over a sufficiently long period of time. The knowledge of the source function of this earthquake allows us to determine the subset of phase space in which the system was before 1964 and the subset to which it came afterwards. This constrains the evolution of the system to a certain bundle of orbits in the phase space but still allows for a wide range of possible occurrence times and source functions for the next earthquake. However, when the next earthquake occurs, the knowledge of its moment rate will allow us to further constrain the orbit, and so on.

The present model is of course a simplification of a real fault, but it suggests how the accumulation of knowledge on the seismic activity of a fault may allow us to constrain the state of the fault and to predict its future activity.

The authors are grateful to the editor Ilya Zaliapin, to J. Freymueller and to an anonymous referee for useful comments on the first version of the paper. Edited by: I. Zaliapin Reviewed by: J. Freymueller and one anonymous referee