Factors controlling the sequence of asperity failures in a fault model

Abstract. We consider a fault with two asperities embedded in a shear zone subject to a uniform strain rate owing to tectonic loading. The static stress field generated by seismic events undergoes viscoelastic relaxation as a consequence of the rheological properties of the asthenosphere. We treat the fault as a dynamical system whose basic elements are the asperities. The system has 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 can be described in terms of one sticking mode and three slipping modes, for 5 which we provide analytical solutions. We discuss how the stress state at the beginning of the interseismic interval preceding a seismic event controls the sequence of slipping modes during the event. We focus on the events associated with the separate (consecutive) slips of the asperities and investigate how they are affected by the seismic efficiency of the fault, by the difference in frictional resistance of the asperities and by the intensity of coupling between the asperities.


Introduction
Fault dynamics can be fruitfully investigated by asperity models (Lay et al., 1982;Scholz, 2002).In this framework, it is assumed that the fault plane is characterized by the presence of one or more strong regions with a high static friction and a velocity-weakening dynamic friction.As a consequence of tectonic loading, the stress acting on the asperities is gradually increased, eventually leading to their sudden failure and to a seismic event.Thus, asperity failures account for the unstable, stick-slip sliding regime of seismogenic faults.Examples of earthquakes that can be ascribed to the failure of two asperities are the 1964 Alaska earthquake (Christensen and Beck, 1994), the 2004 Parkfield, California, earthquake (Twardzik et al., 2012), the 2007 Pisco, Peru, earthquake (Sladen et al., 2010) and the 2010 Maule, Chile, earthquake (Delouis et al., 2010).
When considering asperity models, stress accumulation on the asperities, fault slip at the asperities and stress transfer between the asperities are factors of crucial relevance.It is therefore appropriate to describe the fault as a dynamical system whose essential components are the asperities (Ruff, 1992;Turcotte, 1997).The characterization through a finite number of degrees of freedom allows the study of the long-term evolution of the system by calculating its orbit in the phase space.Fur-The aim of the present paper is to expand the model of Dragoni and Lorenzano (2015) by including elastic wave radiation and considering additional constraints on the state of the system during an interseismic phase.We solve analytically the equations of motion for each of the dynamic modes of the system.We discriminate the characteristics of a seismic event (number and sequence of slipping modes, seismic moment released, stress drops on the asperities) by identifying different subsets of states of the system.We focus on seismic events associated with the consecutive, but separate, slip of the asperities and discuss their relationship with the seismic efficiency of the fault.We retrieve additional constraints on the parameters of the system from the knowledge of the stress states originating these kinds of events.We study how many phases of alternate slips of the asperities can be involved in an earthquake and show how this feature depends on the difference in frictional resistance of the asperities and on the intensity of coupling between the asperities.

The model
We consider a plane fault with two asperities of equal areas and different strengths, namely asperity 1 and asperity 2. The fault is enclosed between two tectonic plates moving at constant relative velocity V and embedded in a shear zone behaving like a homogeneous and isotropic Hooke solid.As a consequence of the relative motion of tectonic plates, the shear zone is subject to a uniform strain rate.We assume that coseismic stresses are relaxed with a characteristic Maxwell time Θ, as a consequence of viscoelastic relaxation in the asthenosphere following an earthquake on the fault.Following Dragoni and Lorenzano (2015), Since asperities are characterized by a much higher friction than the surrounding region of the fault, we neglect the contribution of this weaker region to seismic moment.Instead of focusing on the values of friction, slip and stress at every point on the fault, we only consider the average values of these quantities on each asperity.
We study the fault as a dynamical system with three state variables, functions of time T : the slip deficits X(T ) and Y (T ) of asperity 1 and asperity 2, respectively, and the variable Z(T ) representing the temporal variation of the difference between the slip deficits of the asperities, owing to viscoelastic relaxation in the asthenosphere.At a given instant in time, slip deficit is defined as the slip that an asperity should undergo in order to recover the relative displacement of tectonic plates that took place up to that moment.
The tangential forces on the asperities (in units of the static friction on asperity 1) are In these expressions, the terms −X and −Y represent the effect of tectonic loading, whereas the terms ± αZ correspond to the stress transfer between the asperities; finally, the terms −γ Ẋ and −γ Ẏ are forces due to radiation damping during slip, where γ is an impedance related with the seismic efficiency of the fault (Rice, 1993).The parameter α conveys the degree of coupling of the asperities.
As for friction on the asperities, we assume a simple rate-dependent law assigning a constant static friction and considering the average values of dynamic frictions during a slipping mode.This description of friction allows to replicate the typical stick-slip behaviour of fault dynamics.We assume that static friction on asperity 2 is a fraction β of that on asperity 1 and that dynamic frictions are a fraction of static frictions.
A slip event takes place over a time interval very short with respect to the typical duration of interseismic intervals.Accordingly, viscoelastic relaxation can be reasonably neglected during a slip event and the equations of motion can be solved in the limit case of purely elastic coupling between asperities.This circumstance corresponds to (Amendola and Dragoni, 2013) Accordingly, during a slip event, the equations for the slip deficits X and Y are the same as in the case of purely elastic coupling while the variable Z changes as The dynamics of the system can be characterized in terms of four dynamic modes: a sticking mode (00), corresponding to stationary asperities, and three slipping modes, corresponding to slip of asperity 1 alone (mode 10), slip of asperity 2 alone (mode 01) and simultaneous slip of both asperities (mode 11).Each of these modes is associated with a specific system of autonomous ordinary differential equations.

The sticking region
During interseismic intervals of the fault, while both asperities are stationary (mode 00) and viscoelastic relaxation of coseismic stress takes place, the orbit of the system is enclosed in a particular subset of the state space XY Z.By definition, this subset corresponds to a phase of global stick of the system: accordingly, it is defined as the sticking region of the system (Di Bernardo et al., 2008).We show how it can be identified from the conditions for the occurrence of earthquakes on the fault and a constraint on the state of stress of the fault.
During a global stick mode, the forces (1) reduce to The conditions for the onset of motion for asperity 1 and 2 are, respectively, By combination with Eq. ( 6), we get the equations defining two planes in the XY Z space, which we call Π 1 and Π 2 , respectively.
We assume a condition of no overshooting: accordingly, we require that X ≥ 0, Y ≥ 0 and that the tangential forces on the asperities are always in the same direction as the velocity of tectonic plates, that is F 1 ≤ 0, F 2 ≤ 0. Again from Eq. ( 6), it is possible to define two additional planes in the XY Z space, which we call Γ 1 and Γ 2 , where F 1 = 0 and F 2 = 0, respectively.
To sum up, the sticking region is the subset of the XY Z space enclosed by the planes X = 0, Y = 0, Π 1 , Π 2 , Γ 1 and Γ 2 : a convex hexahedron H (Fig. 1).Accordingly, the sticking region H is a subset of the sticking region defined by Dragoni and Lorenzano (2015): in fact, they did not consider any constraint on the direction of the tangential forces on the asperities, so that the global stick phase of the fault was identified by a larger set of states.The vertices of H are the origin (0, 0, 0) and the points By definition, every orbit of mode 00 is enclosed within the sticking region and eventually reaches one of the faces AECD or BCDF , belonging to the planes Π 1 and Π 2 , respectively, giving rise to a seismic event.In these cases, the system enters mode 10 or mode 01, respectively.In the particular case in which the orbit of mode 00 reaches the edge CD, the system passes to mode 11.
For later use, we introduce a point P with coordinates belonging to the edge CD and corresponding to a condition of purely elastic coupling, since Z P = Y P − X P .

Solutions of dynamic modes
We solve the equations of motion for each of the four dynamic modes of the system.We shall make use of the frequencies We consider the case of underdamping, so that γ ≤ 2: this choice is suggested by the observation that the seismic efficiency of faults is small (Kanamori, 2001) and implies that the velocity dependent terms are small with respect to dynamic frictions.Let us define the slip amplitude of asperity 1 during a one-mode event 10 in the absence of radiation (γ = 0) as Finally, we describe the effect of wave radiation by the quantity which is a decreasing function of γ, equal to 1 in the absence of radiation (γ = 0).

Stationary asperities (mode 00)
The variables X and Y increase steadily due to tectonic motion, while Z is governed by the Maxwell constitutive equation.
The equations of motion are where a dot indicates differentiation with respect to time T .Assuming an arbitrary initial state and initial rates the solution is with T ≥ 0. According to (21), during an interseismic interval the slip deficits of the asperities increase with time, as a result of tectonic loading, while their difference undergoes viscoelastic relaxation.
We can retrieve the time T 1 required by the orbit of mode 00 to reach the plane Π 1 by imposing the condition where we exploited Eq. ( 21).Accordingly, the slip of asperity 1 will start at where W is the Lambert function with argument Analogously, the orbit of mode 00 intersects the plane Π 2 after a time T 2 satisfying the condition Thus, the slip of asperity 2 will start at with 3.2 Slip of asperity 1 (mode 10) The equations of motion are The fault can enter mode 10 from mode 11 or from mode 00.

Case 11 → 10
After a phase of simultaneous motion, asperity 2 stops slipping and asperity 1 continues to slip alone.With initial conditions the solution is where If the orbit does not reach the plane Π 2 during the mode, asperity 1 stops slipping and the system goes back to a global stick phase; the slip duration can be calculated from the condition Ẋ(T ) = 0, yielding The final slip amplitude is then If instead the orbit reaches the plane Π 2 during the mode, the system enters again mode 11 and asperity 2 starts slipping together with asperity 1.The slip duration T 10 is then obtained by solving the equation for the unknown T .

Case 00 → 10
Due to the combined effect of tectonic loading and viscoelastic relaxation, asperity 1 fails and starts slipping alone.In this case, the initial state belongs to the plane Π 1 given by Eq. ( 8): in fact, it is defined as the set of states where the condition for the failure of asperity 1 is attained.Accordingly, and from Eq. ( 36) The solution reduces to If the orbit does not reach the plane Π 2 during the mode, asperity 1 stops slipping and the system goes back to a global stick phase; the slip duration and amplitude are, respectively, where κU is the maximum amount of slip of asperity 1 during mode 10.
If the orbit reaches the plane Π 2 before time π/ω 1 has elapsed, the system passes to mode 11 and asperity 2 starts slipping together with asperity 1.In this case, the slip duration T 10 is obtained by solving Eq. ( 39) for the unknown T with Z(T ) given by Eq. ( 44).

Slip of asperity 2 (mode 01)
The equations of motion are The fault can enter mode 01 from mode 11 or from mode 00.

Case 11 → 01
After a phase of simultaneous motion, asperity 1 stops slipping and asperity 2 continues to slip alone.With initial conditions the solution is where If the orbit does not reach the plane Π 1 during the mode, asperity 2 stops slipping and the system goes back to a global stick phase; the slip duration can be calculated from the condition Ẏ (T ) = 0, yielding The final slip amplitude is then If instead the orbit reaches the plane Π 1 during the mode, the system enters again mode 11 and asperity 1 starts slipping together with asperity 2. The slip duration T 01 is then obtained by solving the equation for the unknown T .

Case 00 → 01
As a result of the combined effect of tectonic loading and viscoelastic relaxation, asperity 2 fails and starts slipping alone.In this case, the initial state belongs to the plane Π 2 given by Eq. ( 9): in fact, it is defined as the set of states where the condition for the failure of asperity 2 is attained.Accordingly, and from Eq. ( 54) The solution reduces to If the orbit does not reach the plane Π 1 during the mode, asperity 2 stops slipping and the system goes back to a global stick phase; the slip duration and amplitude are, respectively, where βκU is the maximum amount of slip of asperity 2 during mode 01.
If the orbit reaches the plane Π 1 before time π/ω 1 has elapsed, the system passes to mode 11 and asperity 1 starts slipping together with asperity 2. In this case, the slip duration T 01 is obtained by solving Eq. ( 57) for the unknown T with Z(T ) given by Eq. ( 62).

Simultaneous slip of asperities (mode 11)
The equations of motion are and the solution is In the framework of a two-asperity fault model, a seismic event is generally made up of n slipping modes and can involve only one or both asperities at a time.More specifically, it is possible to distinguish three kinds of events, namely (i) events due to the slip of a single asperity, (ii) events associated with the consecutive, but separate, slips of both asperities and (iii) events involving the simultaneous slip of asperities.The present model allows to gain information on the kind of seismic event generated by the fault from a geometrical point of view, each event being originated by a particular stress state corresponding to a specific subset of the state space.In the following, we first discuss the connection between the three kinds of events discussed above with the state of the system at the beginning of the earthquake.Afterwards, we show how the number and the sequence of slipping modes in a seismic event can be univocally determined from the knowledge of the state of the system at the beginning of an interseismic interval, in the absence of stress perturbations.

Dependence on the state at the onset of the event
We showed in section 2 that the conditions for the onset of motion for asperity 1 and 2 are reached on the face AECD and BCDF of the sticking region H, respectively.Here, we discuss the different subsets in which these faces can be divided, according to the number and sequence of dynamic modes involved in a seismic event.The purpose of this analysis is to point out the relationship between the kind of seismic event generated by the fault and the state of the fault at the onset of the event itself.
Let us consider an orbit of mode 00 starting at a point P 0 inside H and reaching one of the faces AECD or BCDF at a point P k , where the earthquake begins.With reference to Fig. 2, let us first focus on the face AECD.If P k belongs to the trapezoid Q 1 , the earthquake will be a one-mode event 10; if P k belongs to the segment s 1 , the earthquake will be a two-mode event 10-01; finally, if P k belongs to the trapezoid R 1 , the earthquake will be a three-mode event 10-11-01 or 10-11-10.The specific sequence must be evaluated numerically and depends on the particular combination of the parameters α, β, γ and .The remaining portion of the face would lead to overshooting.Analogous considerations can be made for subsets Q 2 , s 2 and R 2 on the face BCDF .In the particular case in which P k belongs to the edge CD, the earthquake will be a two-mode event 11-01.
There exists a correlation between the sequence of dynamic modes associated with the subsets of the faces AECD and BCDF and the distribution of forces on the fault.Let us consider an earthquake involving n slipping modes starting with mode 10, i.e. on the face AECD.We call P i the representative point of the system at T = T i , when the system enters the i−th mode (i = 1, 2, ..., n).Finally, let d be the distance of the starting point P 1 from the edge CD.
The magnitude |F 2 | of the force acting on asperity 2 at the beginning of the event (T = T 1 ) decreases with d, as shown in Fig. 3(a), whereas the magnitude of the force F 1 acting on asperity 1 is the same everywhere (|F 1 | = 1).At T = T 2 , the force on asperity 2 is

Dependence on the state at the beginning of the interseismic interval
We now discuss how the location of the initial point P 0 of any orbit of mode 00 affects the number and the sequence of slipping modes in the seismic event.Our aim is to illustrate how the kind of seismic event generated by the fault depends on the state of the fault at the beginning of the interseismic interval preceding the event itself.Dragoni and Lorenzano (2015) showed the existence of a transcendental surface Σ which allows to discriminate the first slipping mode in a seismic event.In fact, this surface divides the sticking region H in two subsets H 1 and H 2 .Given any initial state P 0 ∈ H, the seismic event starts with mode 10 if P 0 ∈ H 1 or with mode 01 if P 0 ∈ H 2 ; in the particular case in which P 0 ∈ Σ, the seismic event starts with mode 11.The surface Σ does not depend on the parameter γ; thus, it is not affected by seismic efficiency.
We now describe an additional surface inside each of the subsets H 1 and H 2 , allowing to distinguish the number of slipping modes in a seismic event.
Let P 1 be the point where the orbit of mode 00 starting at P 0 ∈ H 1 reaches the face AECD.In order that P 1 belongs to the segment s 1 , its coordinates must satisfy Eq. (B4).Introducing the solutions (21) of mode 00 in Eq. ( B4) and replacing T 1 with its expression (23), we obtain the equation of a transcendental surface Σ 1 where W is the Lambert function with argument γ 1 defined in Eq. ( 24).The surface Σ 1 is shown in Fig. 5.It lies beneath the surface Σ, so that the subset H 1 is divided into two sections H − 1 and H + 1 , respectively below and above Σ 1 .If P 0 ∈ H − 1 , then P 1 ∈ Q 1 and the earthquake will be a one-mode event, whereas if P 0 ∈ H + 1 , then P 1 ∈ R 1 and the earthquake will be a three-mode event, as discussed in the previous section.By definition, the segment s 1 belongs to Σ 1 and no orbit can cross Σ 1 : accordingly, if P 0 ∈ Σ 1 , its orbit remains on Σ 1 and reaches the segment s 1 , giving rise to a two-mode event.We now repeat the analysis for the subset H 2 .Let P 2 be the point where the orbit of mode 00 starting at P 0 ∈ H 2 reaches the face BCDF .In order that P 2 belongs to the segment s 2 , its coordinates must satisfy Eq. (B11).Introducing the solutions (21) of mode 00 in Eq. ( B11) and replacing T 2 with its expression (26), we obtain the equation of a transcendental surface Σ 2 where the argument γ 2 has been defined in Eq. ( 27).The surface Σ 2 is shown in Fig. 6.It lies above the surface Σ, so that the subset H 2 is divided into two sections H − 2 and H + 2 , respectively below and above Σ 2 .If P 0 ∈ H − 2 , then P 2 ∈ R 2 and the earthquake will be a three-mode event, whereas if P 0 ∈ H + 2 , then P 2 ∈ Q 2 and the earthquake will be a one-mode event.By definition, the segment s 2 belongs to Σ 2 and no orbit can cross Σ 2 : accordingly, if P 0 ∈ Σ 2 , its orbit remains on Σ 2 and reaches the segment s 2 , giving rise to a two-mode event.
In the purely elastic case, the surfaces Σ 1 and Σ 2 reduce to two lines in the XY plane that were defined by Dragoni and Santini (2015).It is clear from their definitions ( 71) and ( 72) that both Σ 1 and Σ 2 depend on the maximum amount of slip allowed to asperity 1 in a one-mode event 10.Therefore, their position inside the sticking region changes as a function of γ.
For larger values of γ, they are both closer to Σ, so that the subsets H + 1 and H − 2 are smaller.This feature shows that higher values of γ reduce the possibility of simultaneous slip of the asperities, in agreement with the results obtained by Dragoni and Santini (2015).

Seismic moment and stress drops on the asperities
The seismic moment released during an earthquake involving n slipping modes can be retrieved from the knowledge of the total slip amplitudes of the asperities.
During the i−th mode, starting at time T = T i when the state of the system is (X i , Y i , Z i ), the slips of asperity 1 and 2 are, respectively, with i = 1, 2, ..., n.The final slip amplitudes of asperity 1 and 2 are, respectively, Accordingly, the final seismic moment is given by where M 1 is the seismic moment associated with a one-mode event 10 in the absence of wave radiation (γ = 0).The slip rates of the asperities in an n-mode event are where H(T ) is the Heaviside function.The moment rate of an n-mode event is then Figures ( 7) and ( 8) show the evolution of the slip amplitude and the moment rate function associated with one-mode events 10 and 01, respectively, for a given choice of the parameters of the system.
The knowledge of the slip amplitudes of the asperities and the stress transferred from one asperity to the other allows to evaluate the static force drops on the asperities associated with the n-mode event.At the end of the earthquake, the static force drop on asperity 1 is where we used the definitions of F 1 and U 1 given in Eq. ( 6) and Eq. ( 74), respectively.Analogously, the static force drop on asperity 2 is where we used the definitions of F 2 and U 2 given in Eq. ( 6) and Eq. ( 74), respectively.
The values of M 0 , ∆F 1 and ∆F 2 can be discriminated according to the position of the point P 1 where the seismic event starts, as summarized in Table 1.For events involving the slip of a single asperity, the force drop on the stationary asperity is negative, since stress is accumulated on it.The static stress drop on the asperities can be straightforwardly obtained dividing the static force drops by the area of the asperities.
6 Events due to the consecutive slip of the asperities We focus on seismic events associated with the consecutive, but separate, slip of the asperities.First, we consider two-mode events 10-01 and 01-10 and discuss how they are affected by the seismic efficiency of the fault.Afterwards, we exploit the knowledge of the stress states giving rise to such events in order to obtain additional constraints on the parameters of the system.Finally, we study how many phases of alternate slips of the asperities can be involved in an earthquake and how these particular sequences of dynamic modes are related to the parameters of the system.

Influence of the seismic efficiency
We illustrate how two-mode events 10-01 and 01-10 are affected by the radiation of elastic waves.To this aim, we study the effect of a variation of the parameter γ in the interval [0, 2].In the following, we shall use a superscript 0 when referring to quantities defined in the absence of wave radiation (γ = 0).
The lengths l 1 and l 2 of segments s 1 and s 2 , respectively, as well as their distances d 1 and d 2 from the edge CD are provided in Appendix B. In the limit case γ = 0, the maximum amount of slip κU of asperity 1 that is present in their expressions must be replaced by U defined in Eq. ( 16), where U ≥ κU .In Fig. 9 we plot the ratios l 1 /l 0 1 and l 2 /l 0 2 as functions of γ.The trends clearly point out that an increase in γ entails a lengthening of both segments s 1 and s 2 .As a matter of fact, the lengths of these segments depend on the coordinates of their end points, which are in turn constrained by the no overshooting conditions.Since wave radiation reduces the maximum amount of slip allowed to the asperities, the number of states satisfying the no overshooting conditions is increased and more states are included in the segments s 1 and s 2 .As γ grows, the probability that the system gives rise to a two-mode event 10-01 or 01-10 is thus enlarged.
According to Eq. (B15), the ratio d i /d 0 i is the same for both segments s 1 and s 2 .It is shown in Fig. 10 as a function of γ.
Evidently, an increase in γ takes both segments s 1 and s 2 closer to the edge CD of the sticking region.This can be explained if one considers the already discussed correlation between the different subsets of the faces AECD and BCDF and the forces acting on the asperities (section 4.1).Taking into account that wave radiation lowers the slip of the asperities, the stress transferred by one asperity to the other during a slip event is reduced as well.Thus, the segment s 1 must be closer to the edge CD, so that the value of F 2 at the beginning of mode 10 is large enough for the stress transferred by asperity 1 to asperity 2 to trigger mode 01.Analogous considerations can be made for the segment s 2 on the face BCDF .
A direct consequence of the smaller distance between segments s 1 and s 2 and the edge CD is that the areas A Q i of the subsets Q 1 and Q 2 are enlarged, while the areas A R i of the subsets R 1 and R 2 are reduced.This is shown in Fig. 11, where we plot the ratios This feature provides an additional proof that higher seismic efficiency progressively reduces the possibility of simultaneous slip of the asperities.

Additional constraints on the parameters of the system
We introduced in section 2 the constraint F 1 ≤ 0, F 2 ≤ 0, requiring that the tangential forces on the asperities are always in the same direction as the velocity of tectonic plates.Accordingly, the ratio F 1 /F 2 must always be a positive quantity.We now exploit the knowledge of the particular stress states yielding to two-mode events 10-01 and 01-10 to establish additional constraints on the parameters of the system.Let us first consider a two-mode event 10-01 taking place on the segment s 1 on the face AECD of the sticking region.
Introducing the coordinates of any of the end points (B3)-(B5) of s 1 in the expressions (6) of the forces acting on the asperities, we find that the stress state at the onset of the event is such that Imposing the condition F 1 /F 2 ≥ 0 , we find Let us now focus on a two-mode event 01-10 taking place on the segment s 2 on the face BCDF of the sticking region.
Introducing the coordinates of any of the end points (B10)-(B12) of s 2 in the expressions (6) of the forces acting on the asperities, we find that the stress state at the onset of the event is such that Imposing the condition To sum up, the parameters of the system are subject to the condition Although these constraints have been obtained considering two particular seismic events, they represent a general feature of the present model.

Multiple consecutive slips
In the following, we investigate the conditions under which the system can generate a n-mode event involving the consecutive, but separate, slip of the asperities, with n > 2. To this aim, we recall that the slip deficit of asperity 1 is reduced by an amount κU each time it slips alone; analogously, the slip deficit of asperity 2 is reduced by an amount βκU each time it slips alone.
6.3.1 Three-mode events 10-01-10 At the end of a two-mode event 10-01, starting at a point P 1 = (X 1 , Y 1 , Z 1 ) on the segment s 1 on the face AECD of the sticking region, the system is at a point P 2 with coordinates The event will then continue with a third mode 10 if P 2 ∈ Π 1 : thus, introducing the coordinates of P 2 in Eq. ( 8) and bearing in mind that we get the following condition: As 0 < β < 1, this result is unacceptable, since α is defined as positive.We conclude that, if we consider seismic events involving the alternate slips of the asperities, starting with the slip of asperity 1, the system can only generate a two-mode event 10-01.Any additional slip phase is prevented by the stronger frictional resistance of asperity 1 with respect to asperity 2.
6.3.2Three-mode events 01-10-01 At the end of a two-mode event 01-10, starting at a point P 1 = (X 1 , Y 1 , Z 1 ) on the segment s 2 on the face BCDF of the sticking region, the system is at a point P 2 with the same coordinates as given in Eq. ( 86).The event will then continue with a third mode 01 if P 2 ∈ Π 2 : thus, introducing the coordinates of P 2 in Eq. ( 9) and bearing in mind that we get the following condition: Since 0 < β < 1, the constraint α * ≥ 0 is always satisfied.Accordingly, under the particular condition α = α * , the system can give rise to three-mode events 01-10-01.
6.3.3Four-mode events 01-10-01-10 At the end of a three-mode event 01-10-01, the system is at a point P 3 with coordinates The event will then continue with a fourth mode 10 if P 3 ∈ Π 1 : thus, introducing the coordinates of P 3 in Eq. ( 8), bearing Eq. (B11) in mind and taking into account that α = α * , we end up with which is unacceptable, since β is defined as positive.We conclude that, if we consider seismic events involving the alternate slips of the asperities, starting with the slip of asperity 2, the system can only generate two-mode events 01-10 and, under particular conditions related with the geometry of the fault and the coupling between the asperities, three-mode events 01-10-01.
To sum up, according to our analysis, the present model predicts n−mode events with n ≤ 3; specifically, the sole seismic event involving three slipping modes (i.e., n = 3) is associated with the particular sequence 01-10-01, which can only take place under the condition (90).The existence of events involving more than three slipping modes in the framework of the present model may be object of future works.
We now consider the face BCDF .The vertices of the trapezoid Q 2 are the point F given in Eq. ( 13) and the points The segment s 2 lies on the line and its end points are the points I 2 and The vertices of the trapezoid R 2 are the end points of s 2 and the points J 1 and J 2 .
The lengths of segments s 1 and s 2 are, respectively, The distances of segments s 1 and s 2 from the edge CD are, respectively, Competing interests.The authors declare that they have no conflict of interest.Fig. 9 The lengths l 1 /l 0 1 and l 2 /l 0 2 of segments s 1 and s 2 as functions of γ (α = 1, β = 0.5, = 0.7).Larger values of the ratios l i /l 0 i entail a higher probability of a two-mode event associated with the separate slip of both asperities Fig. 10 The distance d/d 0 of segments s 1 and s 2 from the edge CD as a function of γ (α = 1, = 0.7).The smaller the distance, the more homogeneous the stress distribution on the fault at the beginning of a two-mode event associated with the separate slip of both asperities Fig. 11 The areas

List of
R 1 and R 2 as functions of γ (α = 1, β = 0.5, = 0.7).As the ratios A Q i /A 0 Q i increase, the possibility of simultaneous slip of asperities is reduced.The converse holds for the ratios Tables Table 1.Final seismic moment M0 and static force drops ∆F1, ∆F2 on asperity 1 and 2 following an earthquake involving n slipping modes, as a function of the state P1 where the event started.The entry e.n. is the abbreviation for evaluated numerically.
where the constants A, B, C, D, E 1 , E 2 and E 3 depend on initial conditions and are listed in Appendix A. The duration T 11 of mode 11 must be evaluated numerically: letting T x and T y be the smallest positive solutions of the equations Ẋ(T ) = 0 and Ẏ (T ) = 0, respectively, we have T 11 = min(T x , T y ).11 Solid Earth Discuss., https://doi.org/10.5194/se-2018-31Manuscript under review for journal Solid Earth Discussion started: 7 May 2018 c Author(s) 2018.CC BY 4.0 License.

Solid
Earth Discuss., https://doi.org/10.5194/se-2018-31Manuscript under review for journal Solid Earth Discussion started: 7 May 2018 c Author(s) 2018.CC BY 4.0 License.owing to the stress transfer from asperity 1.If the magnitude of F 2 (T 1 ) is large enough that |F 2 (T 2 )| = β, the slip of asperity 1 triggers the slip of asperity 2, so that mode 10 is followed by mode 01 or 11.This condition is verified by states P 1 ∈ s 1 and P 1 ∈ R 1 , respectively, as shown in Fig.3(b); conversely, |F 2 (T 2 )| < β for states P 1 ∈ Q 1 and mode 10 is followed by mode 00.Similar considerations hold for the face BCDF , with |F 2 | = β everywhere.This is shown in Fig.4.The boundaries of the subsets of the faces AECD and BCDF can be identified taking into account the no overshooting conditions and the constraint on the orientation of the tangential forces acting on the asperities discussed in section 2. The details are provided in Appendix B.

)
Author contributions.E. L. developed the model, produced the figures and wrote a preliminary version of the paper; M. D. checked the equations and revised the text.Both authors discussed extensively the results.

Fig. 1 Fig. 3 Fig. 4 Fig. 5 Fig. 6
Fig.1The sticking region of the system: a convex hexahedron H (α = 1, β = 1).The point P , corresponding to purely elastic coupling between the asperities, is shown.Seismic events take place on the faces AECD and BCDF Fig.2The faces AECD and BCDF of the sticking region and their subsets, which determine the number and the sequence of dynamic modes during a seismic event (α = 1, β = 1, = 0.7).The events taking place on the face AECD (BCDF ) start with mode 10 (01) Fig. 3 Force F 2 on asperity 2 during an earthquake involving n slipping modes and starting with mode 10, as a function of the distance d of the initial state P 1 , measured on the face AECD from the edge CD of the sticking region H (α = 1, β = 0.5, γ = 1, = 0.7) : (a) magnitude of F 2 at the onset of the event (T = T 1 ); (b) magnitude of F 2 after the initial slip of asperity 1 (T = T 2 ).The labels indicate the subsets of the face AECD corresponding to different intervals of d.The dashed line indicates the condition for the slip of asperity 2 (|F 2 | = β), which is reached only for states P 1 ∈ s 1 and P 1 ∈ R 1 Fig. 4 Force F 1 on asperity 1 during an earthquake involving n slipping modes and starting with mode 01, as a function of the distance d of the initial state P 1 , measured on the face BCDF from the edge CD of the sticking region H (α = 1, β = 0.5, γ = 1, = 0.7) : (a) magnitude of F 1 at the onset of the event (T = T 1 ); (b) magnitude of F 1 after the initial slip of asperity 2 (T = T 2 ).The labels indicate the subsets of the face BCDF corresponding to different intervals of d.The dashed line indicates the condition for the slip of asperity 1 (|F 1 | = 1), which is reached only for states P 1 ∈ s 2 and P 1 ∈ R 2Fig.5The surface Σ 1 in the subset H 1 of the sticking region, discriminating the number of slipping modes in a seismic event starting when the orbit of the system reaches the face AECD (α = 1, β = 1, γ = 1, = 0.7, V Θ = 1) Fig.6The surface Σ 2 in the subset H 2 of the sticking region, discriminating the number of slipping modes in a seismic event starting when the orbit of the system reaches the face BCDF (α = 1, β = 1, γ = 1, = 0.7, V Θ = 1) Fig. 7 (a) Slip amplitude and (b) moment rate function associated with a one-mode event 10 (α = 1, γ = 1, = 0.7) Fig. 8 (a) Slip amplitude and (b) moment rate function associated with a one-mode event 01 (α = 1, β = 0.5, γ = 1, = 0.7)