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The 2018 Gulf of Alaska earthquake (Mw 7.9) occurred in a region of the Pacific plate southwest of the Alaskan subduction zone. The earthquake was a strike-slip event, with the hypocenter located at a depth of about 25 km and a seismic moment equal to 0.96 × 1021 Nm. Two observed moment rates have been obtained by the Geoscope Observatory, France, and by the United States Geological Survey (USGS). Both of them can be interpreted as due to the failure of two asperities on the fault surface. We consider a discrete fault model, with two asperities of different areas and strengths, and show that the observed moment rates can be reproduced by appropriate values of the model parameters, as inferred from the available data. A good fit to the observed moment rates is obtained by a sequence of three dynamic modes of the system, including a phase of simultaneous slip of the asperities. The two moment rates are however characterized by different initial conditions, in terms of different initial shear stress distributions on the fault. Shear stresses on the asperities are calculated as functions of time during the event and show a similar evolution in the two cases, but with different final values. The model results show that the presence of simultaneous asperity motion can significantly increase the seismic moment of a large earthquake.

A fault containing two asperities with different strengths is considered. The fault is embedded in a viscoelastic 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 the 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 postseismic 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.

We study the dynamics of a fault containing two asperities having different strengths and subject to a constant strain rate. The fault is modeled by a discrete dynamical system with two degrees of freedom that are the slip deficits of the asperities. The radiation of elastic waves during the slipping modes is taken into account by introducing a term proportional to slip rate in the equations of motion. We give a complete analytical solution of the four dynamic modes of the system. Any seismic event can be expressed as a sequence of modes, for which the moment rate, the spectrum and the total seismic moment can be calculated. We consider the energy budget of the event and calculate the seismic efficiency. The presence of radiation may remarkably change the evolution of the system from a given state, since it moves the boundaries between the different subsets of the sticking region. In addition, the slip amplitude in a seismic event is smaller, while the slip duration is longer in the presence of radiation. The shape of the moment rate function depends on the seismic efficiency and the seismic moment decreases with increasing efficiency, at constant radiated energy. As an application of the model, we consider the 1964 Great Alaska earthquake and calculate the mode durations, the slip distribution, the moment rate function and the seismic moment of the earthquake, obtaining values that are consistent with observations.

We propose a model which may explain seismic sequences which are often observed in seismogenic regions, as for example in the Apenninic chain (Italy). In particular, we consider a normal fault and earthquakes taking place at different depths: a first shock in a shallower layer and a second in a deeper one. The normal fault is embedded in a viscoelastic half-space. As a consequence of the rheology, there are two different brittle layers, a shallower and a deeper one, where earthquakes can nucleate. Between these two layers, the rheological behavior is ductile. The thicknesses of the layers depend on the geothermal profile that is calculated taking into account the profile of the thermal and rheological parameters with depth. The fault plane, crossing layers with different rheological behavior, is heterogeneous in respect to the slip style: seismic in the brittle layers, aseismic in the ductile layer. Dislocations in the shallower layer are assumed to produce aseismic slip in the area of the fault belonging to the ductile layer. The stress concentrated, by the seismic and aseismic dislocations, on the fault plane section in the deeper brittle layer is evaluated. It is compared with the tectonic stress rate in order to calculate how much earlier the second earthquake would occur compared to if just the bare tectonic sstress was acting. It results that such an advance is comparable with typical recurrence times of earthquakes and so a mechanism of interaction between different seismic sources, mediated by aseismic slip, can be supposed. The strains and displacements at the Earth’s surface due to seismic and aseismic slip are calculated. They are large enough to be detected by present geodetic techniques.

—A model is proposed to study the modification of the stress field at a transcurrent plate boundary due to frictional heat production at depth. Two cases are considered a stable and a stretched lithosphere. The model is applied to those weak faults where the dynamic friction is small compared to a static one; if the deformation along the brittle portion of the fault is entirely accommodated by a series of seismic ruptures in a quasi-static state where the fault has been moving for millions of years, the long-term thermal field perturbation due to these ruptures results in only a few degrees and can be neglected. The boundary zone is considered as a viscoelastic body subject to a constant strain rate. The lower section of the boundary is assumed to slip aseismically along a vertical transcurrent fault and to completely accommodate the plate motion, while the upper section is locked. The slipping zone is divided into a semi-brittle zone, placed between the isothermal surfaces of 300°C and 450°C, and a ductile zone beneath. The frictional heat is calculated by assuming a linearly decreasing friction in the semi-brittle and a constant friction in the ductile zones. The heat modifies the temperature field, producing an upward movement of the semi-brittle and ductile fault sections. As a consequence, the thickness of the brittle fault section is reduced and friction at the base of this section is less. The stress field in the boundary zone is calculated as a function of time for different friction profiles and slip rates on the fault. Owing to heat production, a greater stress concentration is produced on the brittle fault section, while shear stress is lowered in regions occupied by the uplifted semi-brittle layer. These effects are found to be remarkable only in the case of a stable zone, with a standard unperturbed geotherm, while they are irrelevant in a stretched zone with a high geothermal gradient. In any case, the role of the semi-brittle layer appears to be more prominent in the case of boundaries with higher slip rates, due to the presence of higher stress values.

We consider a fault surface which differs slightly from a plane due to a jog. The fault is placed in an elastic space and is subject to a uniform stress field. The orientation of the fault is such that the normal traction is greater on the jog determining a higher static friction. A considerable increase in friction and the formation of a strong asperity can occur due to repeated slip episodes on the fault. Slip produces an elastic deformation of the fault faces in correspondence of the asperity, causing an increase in normal traction and hence in friction. This process can be described as a tensile Somigliana dislocation, accompanied by partial fracturing of the fault face material, which can produce fault gouge.

The problem of post-seismic stress evolution at the lower dislocation edge of great shallow earthquakes is investigated by a model reproducing a transcurrent boundary zone subject to a constant strain rate. Dislocation models with frictional stress threshold for slip arrest must be considered in order to study post-seismic stress evolution in the proximity of dislocation edges. It is found that the boundary zone is vertically divided in two regions: an upper region, where stress increases with time and a lower region, where stress decreases. Dislocations which are nucleated in the seismogenic region can propagate downward into the aseismic region: in this case earthquakes are followed by relaxation of the stress concentration at the lower dislocation edge. This may have effects on the space-time distribution of aftershocks.

Tectonic boundary zones are characterized by a shallow, brittle region (seismogenic region) where frictional slip occurs on fault planes, and by a lower, ductile and aseismic region. The nucleation depth and the extent of slip planes in great interplate earthquakes are determined by the relationships between frictional resistance, applied stress and stress drop. The post-seismic stress evolution on a fault plane is investigated by a model which reproduces a transcurrent boundary zone subject to a constant strain rate. Dislocation models with frictional stress threshold for slip arrest must be considered in order to study post-seismic stress evolution in the proximity of dislocation edges. It is found that dislocations which are nucleated in the seismogenic region can propagate downward into the aseismic region: in this case earthquakes are followed by relaxation of the stress concentration at the lower dislocation edge. This has effects on the space-time distribution of aftershocks.

SUMMARY During 1997 several slow earthquakes have been recorded by a geodetic interferometer located beneath Gran Sasso, central Italy. The strain rise times of the events range from tens to thousands of seconds and strain amplitudes are of the order of 10−9. Amplitudes scale with the square root of the rise time and this suggests a diffusive behaviour of the slip propagation along the fault. In this work, we develop a model in which slip diffusion is the result of the presence of a gouge layer between fault faces, with a viscoplastic rheology. The fluid velocity field in the gouge layer diffuses in the directions of fault length and fault thickness, with different characteristic times. This model reproduces the relation between amplitude and rise time of measured strain signals. Synthetic straingrams, obtained for a horizontally layered, flat Earth and a source located a few kilometres from the instrument, are in agreement with observed signals.

The study of surface deformation due to seismic activity is often made using dislocations with uniform slip and simple geometries. A better modeling of coseismic and postseismic surface displacements can be obtained by using dislocations with variable slip and nonregular shapes. This is consistent with the asperity model of fault surfaces, assuming a friction distribution on faults made of locked zones with much higher friction than surrounding zones. In this paper we consider the 1997–1998 Colfiorito seismic sequence. The coseismic surface displacements in the Colfiorito zone are used in order to infer the slip distribution on the fault surface at different stages of the sequence. The displacement field has been modeled varying the slip distribution on the fault, and comparing the deformation observed by SAR and GPS techniques with model results. The slip distribution is calculated by Monte Carlo simulations on a normal fault with the dip angle equal to 40°. A good approximation is obtained by using square asperity units of 1.5 times 1.5 km2. In the first stage, we employed a simplified model with uniform slip, in which each asperity unit is allowed to slip a constant amount or not to slip at all, and in the second stage, we evaluate the slip distribution in the dislocation area determined by the Monte Carlo inversion: in this case we allow unit cells to undergo different values of slip in order to refine the initial dislocation model. The results show that the 1997 seismic events of the sequence can be modeled by irregular dislocations, obtaining a good fit to the DInSAR and GPS observations. The model also confirms the results of previous studies by a different methodology, defining the distribution of asperities on the fault plane using the fault geometry, the geodetic data and the seismic moment of the 1997–1998 Colfiorito seismic sequence. Furthermore, the analysis of 1997 aftershocks in the seismogenic region shows a strong correlation between most events and the asperity distribution, which can be considered as an independent test of the validity of the model.

The 1964 Alaska earthquake was the second largest seismic events in the 20th century. The aim of this work is the use of surface deformation data to determine asperity and slip distributions on the fault plane of the Alaska earthquake: these distributions are calculated by a Monte Carlo method. To this aim, we decompose the fault plane in a large number of small square asperity units with a side of 25 km; this allows us to obtain plane surfaces with an irregular shape. In the first stage, each asperity unit is allowed to slip a constant amount or not to slip at all, providing the geometry of the dislocation surface that best reproduces the observed displacements. To this purpose, a large number of slip distributions have been tried by the use of the Monte Carlo method. The slip amplitude is the same for all the asperities and is equal to the average fault slip inferred from the seismic moment. In the second stage, we evaluate the slip distribution in the dislocation area determined by the Monte Carlo inversion: in this case, we allow unit cells to undergo different values of slip in order to refine the initial dislocation model. The results confirm the previous finding that the slip distribution of the great Alaska earthquake was essentially made of two dislocation areas with a higher slip, the Prince William Sound and the Kodiak asperities. Analysis of the post-1964 seismicity in the rupture region shows a strong correlation between the larger earthquakes (Mw≥6) and the distribution of locked asperities following the 1964 event, which can be considered as an independent test of the validity of the model. We do not find slip values higher than 25 m for any of the patches, and we determine two separate high-slip zones: one correspondent to the Prince William Sound asperity, and one (∼18 m slip) to the Kodiak asperity. The slip distribution connected with the 1964 shock appears to be consistent with the following seismicity in the region.

We show the results of application of Differential SAR Interferometry to the MW 7.4, August 17, 1999, Izmit earthquake, Western Turkey. The differential interferogram is obtained using an interferometric ERS2 ascending pair with a time interval of 35 days (August 13th - September 17th). The fringe pattern clearly defines the coseismic displacement field extended in an area of about 100 km N-S and 120 km E-W. The analysis of the interferogram shows the right-lateral strike-slip movement on the activated section of the North Anatolian fault system. The maximum SAR-detected displacement ranges between 117.6 cm and 134.4 cm in the proximity of Gölcük. We invert SAR data for uniform dislocation on a single fault plane using a Montecarlo procedure, with the aim of testing a large set of a priori possible asperity distributions on the fault. We then use a forward modeling approach to evaluate the slip variability for the dislocation using additional constraints as surface offsets and seismicity distribution: in this case we allow unit cells to undergo different values of slip in order to refine the initial dislocation model. Misfits between SAR data and modeled slant range displacements are generally low for all our models (~ 12 cm). Our results indicate that slip is concentrated in the central-western part of the fault, in the upper 10-15 km, tapering to the fault tips. For the Izmit case, we note that a well constrained fault model can be obtained only integrating DInSAR data with additional observations. This is mainly due to an undersampling of the displacement field by DInSAR, caused by decorrelation and lack of image data.

The process of earthquake nucleation is studied assuming that faults are rupture surfaces on which sliding is controlled by friction. Earthquakes are assumed to arise through an instability of frictional sliding. Empirical slip laws indicate that, under constant ambient conditions, friction depends on time, slip rate and slip history. Regular stick-slip behaviour is induced by velocity weakening, a negative dependence of friction on slip rate. Velocity weakening is introduced into a model for a propagating Somigliana dislocation under slowly increasing ambient shear stress. The instability occurs when the rate at which friction decreases becomes greater than the rate at which the applied stress must increase to produce an advance of fault slip. The possibility that this condition is fulfilled depends on the velocity dependence and on the spatial distribution of friction on the fault. A critical nucleation width of the dislocation is associated with the instability and is controlled by the friction distribution, which determines the size of the initial slipping patch. Depending on the stress drop and the characteristic slip distance, the critical nucleation width may be greater for small earthquakes than for large earthquakes, with respect to the initial slipping patch.

A 2-D model is presented for the propagation of a Somigliana dislocation along a fault with nonuniform friction. Fault slip is driven by a uniform ambient shear stress, slowly increasing with time. The dislocation is nucleated in the lowest-friction region of the fault plane and is confined by the surrounding higher-friction regions (asperities). The case studied involves asperities which have smooth borders, characterized by a constant friction gradient. For values of ambient shear stress near to the weak-zone friction, the propagation of dislocation is slowed down by the presence of asperities. Only when it goes beyond the border does the dislocation front move at increasing velocity. The model shows that, at a given value of ambient shear stress, the slip amplitude is larger in the case of finite and constant friction gradient than in the case of asperities with sharp borders. Unlike the propagation velocity, slip rate is ever increasing during the dislocation process. The model shows to what extent a dislocation is influenced by the distribution of friction on the fault. A detailed knowledge of slip rate and slip history is needed to understand the mechanism of frictional instability on faults.

A model is proposed for studying the mechanical behaviour of faults during their interseismic periods. The model considers a plane fault surface in an elastic medium, subject to a uniform shear stress which increases slowly with time. A1-D friction distribution is assumed on the fault, characterized by asperities and a weaker zone. The traction vector on the fault plane has an arbitrary orientation: in particular, it can be nonperpendicular to the asperity borders. Aseismic fault slip takes place when the applied stress exceeds the frictional resistance: slip starts in weak zones and is confined by asperities, where it propagates at increasing velocity. Propagation into asperities is characterized by a dislocation front, advancing perpendicularly to the asperity border. Fault slip does not take prate in the direction of traction, except when traction is perpendicular or parallel to the asperity border. The propagation of such aseismic dislocations produces a stress redistribution along the fault and can play a key role in determining the conditions which give rise to earthquakes.

A model is proposed which may explain the multiple occurrence of earthquakes which is often observed in the same seismogenic region. Such earthquake sequences or swarms may consist of events several tens of kilometres apart in space and a few days to months apart in time. The model considers a long, vertical strike-slip fault embedded in an elastic half-space. The fault is heterogeneous with respect to strength and to slip style (seismic or aseismic) and is subject to a slowly increasing ambient shear stress. We assume that conditions for stable aseismic slip are present in a part of the fault plane together with unstable asperities. The failure of an asperity, producing one earthquake of the sequence, is assumed to produce the onset or the acceleration of a seismic slip on a distant area of the fault, in proximity to another asperity. The conditions under which the latter asperity may fail, producing another earthquake of similar magnitude, are studied as functions of ambient stress rate, asperity strength, amount of aseismic slip, and aseismic slip area. It is found that the effect of aseismic slip can be remarkable in anticipating the occurrence of the second earthquake.

SUMMARY A 2-D model which represents a slipping fault with non-uniform Coulomb friction is studied. The fault plane is subject to a uniform ambient shear stress, slowly increasing with time. Aseismic fault creep is assumed to start in a weak zone, when the ambient stress reaches a strength threshold. The solution for the resulting dislocation is worked out analytically using a technique based on Chebyshev polynomials. It is found that the dislocation partially propagates into the adjacent asperities, concentrating stress onto them and preparing the conditions which will produce the asperity failure and the accompanying earthquake. Propagation is not self-similar and occurs at increasing velocity. A non-linear slip hardening effect is reproduced. The nearness to earthquake instability is measured by a dimensionless parameter which depends on Coulomb friction and ambient shear stress and decreases to zero with time. An upper boundary to the critical value of this parameter, at which instability may occur, is estimated and is found to depend on the ratio between the sizes of the asperity and the weak zone.

A crack model in antiplane shear configuration is shown representing creep processes interpreted in terms of ‘viscous’ deformation of a narrow plastic layer, characterized by inhomogeneous rheological properties, embedded within a homogeneous elastic medium. The evolution in time of slip and stress over the crack plane is studied through a truncated expansion in Chebyshev polynomials, and convergence is proved to be fast in the simple examples considered. Finite‐stress solutions are found which are compatible with constitutive relations of elasto‐plastic materials and furthermore these allow us to simulate creep propagation and stress transfer between locked and unlocked fault segments. This model provides a simple interpretation of the shallow depth of the seismogenic layer observed in several areas of the world and lends itself to modelling creep processes during either post‐seismic rebound or pre‐seismic stress buildup. Stress transfer is accomplished mostly by the slow extension of the creeping section. During a seismic cycle it is envisaged that different regimes dominate over deep, intermediate and shallow sections of faults: (i) slow pre‐seismic stress build‐up accompanied by creep and stress migration toward intermediate depths; (ii) brittle fracture over shallow and intermediate sections of faults; (iii) post‐seismic rebound over intermediate and deep sections of faults. The present crack model, while providing finite‐stress solutions, allows a better understanding of how stress may accommodate at different depths over a fault plane during a seismic cycle.

Summary. A strike-slip plate boundary is considered with non-uniform strength. Depth-dependent slip on the fault surface is assumed to occur aseismically in the lower fault section, in response to a basal shear stress, and seismically in the brittle upper section. The upper fault section acts as a barrier, preventing aseismic slip from reaching shallower depths. If, however, friction decreases in the proximity of the Earth's surface, brittle fracture may occur beyond the barrier. It is found that, when friction in the uppermost fault section exceeds the applied shear stress, fault slip occurs there with a smaller amplitude than at larger depth. This may explain the discrepancies, which have been sometimes reported, between fault offsets observed at the Earth's surface and fault offsets inferred from seismograms by using simpler dislocation models.

Summary. We consider a long strike-slip fault in a lithosphere modelled as an elastic slab. To the base of the slab a shear stress distribution is applied which simulates the viscous drag exerted by the asthenosphere. The resulant stress on the fault plane may directly fracture the lithosphere in its brittle upper portion; alternatively it may give rise at first to a stable aseismic sliding in the lower portion. In the latter case, stress concentration due to the deep aseismic slip is the relevant feature of the pre-seismic stress acting on the upper section of the lithosphere. The two cases are examined by use of dislocation theory and their observable effects compared. Different depths of the aseismic slip zone and the presence or absence of a uniform friction on the seismic fault are allowed for. If the model is applied to the San Andreas fault region, where a steady sliding condition actually seems to be present at shallow depth, it turns out that the slip amplitudes commonly associated with large earthquakes are consistent with average basal stress values which can be substantially lower than a few bars, a value often quoted as the steady state basal stress due to a velocity gradient in the upper asthenosphere.

We consider a vertical strike-slip fault which is embedded in an elastic layer overlying an elastic half-space. The layer and the half-space have different rigidities. In such a model, dislocations on the fault are usually taken to occur with uniform slip, due to the difficulty of solving analytically the general problem of variable-slip (Somigliana) dislocations in this case. We adopt the Somigliana dislocation representing a crack in a homogeneous half-space and examine under which conditions this can be considered a good approximation to a crack in the layered medium. Better approximations can be obtained by super-positions of Chebyshev polynomials of the first kind and odd degree.

Summary The Somigliana dislocation theory is applied to the study of strike-slip faulting in an isotropic, homogeneous, elastic half space in the presence of localized distributions of strain nuclei. Vanishing shear tractions on the dislocation surface are assumed and the slip amplitude as a function of depth is derived, together with the stress drop pattern and the energy release. The results show indications that the condition of continuous stable sliding prevailing at depth along the locked sections of faults may strongly affect the slip, the stress drop and the energy release of a seismic dislocation event. Moreover, fault widths derived from dislocation models driven from underneath are greater than those evaluated from models with a uniform initial stress field and thus tend to explain in simple terms the common finding in some regions that earthquake foci are deeper than evaluated widths of dislocation surfaces.

SUMMARY We propose a model which describes the formation of a strong asperity on a fault. We consider a fault surface which differs slightly from a plane due to a jog-like topographic variation. The fault is placed in an elastic space and is subject to a uniform stress field. The orientation of the fault is such that the normal traction (which is compressive) is greater on the topographic variation, determining a higher static friction and hence an asperity. The value of friction on this asperity depends on the magnitude of shear stress. For times of seismological interest, the increase in shear stress, at rates typical of tectonic processes, does not produce a sensible increase in friction with respect to the adjacent fault segments. A considerable increase in friction and the formation of a strong asperity (or even a barrier) can occur due to repeated seismic-slip episodes on the fault. Slip results in an elastic medium deformation, causing an increase in normal traction on the asperity and hence in friction. This process is described with the aid of a tensile Somigliana dislocation. Regions with high friction undergo partial fracturing of the fault-face material, which can produce fault gouge. The tensile dislocation introduces a small non-double-couple component in the seismic moment of the seismic event, the magnitude of this component depending mainly on the relative size of the asperity.

Summary. Two‐dimensional crack problems in elastic homogeneous isotropic media are considered which describe rupture over a fault surface characterized by non‐uniform stress drop. Solutions can be found in which the stress field is finite at the crack tips and the rupture surface is not assigned a priori, but is part of the solution. These crack models are found to be consistent with the frictional stress threshold criterion for slip arrest over pre‐existing fault surfaces. A crack is found to stop when its contribution to the stress field is opposite to the stress drop at the crack tips. The quasi‐static propagation of a crack up to the arrest configuration is studied in terms of the minimum energy principle. The crack spontaneously propagates in such a way as to make the value of the stress intensity factor at one tip equal to the value at the other tip. Furthermore a tip propagating in a region with higher friction is found to move more slowly than the other tip propagating in a region with lower friction. Simple criteria for fracture arrest are derived, in terms of a properly averaged stress drop. Piecewise constant stress drop profiles are explicitly considered yielding a variety of solutions which can be applied to modelling asperities or barriers over a fault plane. The evaluation of the amount of the energy released during the quasi‐static crack propagation shows that stopping phases cannot be efficiently radiated if the crack comes to rest in a low friction region.