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We study the effects of friction heterogeneity on the dynamics of a seismogenic fault. To this aim, we consider a fault model containing two asperities with different static frictions and a rate-dependent dynamic friction. We consider the seismic events produced by the consecutive failure of the two asperities and study their properties as functions of the ratio between static frictions. In particular, we calculate the moment rate, the stress evolution during fault slip, the average stress drop, the partitioning of energy release, the seismic energy, the far-field waveforms and the spectrum of seismic waves. These quantities depend to various extent on the friction distribution on the fault. In particular, the stress distribution on the fault is always strongly heterogeneous at the beginning of the seismic event. Seismic energy and frictional heat decrease with increasing friction heterogeneity, while seismic efficiency is constant. We obtain an equation relating seismic efficiency to the parameters of the friction law, showing that the efficiency is maximum for smaller values of dynamic friction. The seismic spectrum depends on the friction distribution as to the positions and the values of the minima. However, under the model assumption that the slip durations are the same for both asperities, the corner frequency is independent of the friction distribution, but it depends on the friction law and on the coupling between asperities. The model provides a relation between the total radiated energy and the seismic moment that is consistent with the empirical relation between the two quantities. The fault model with one asperity is also considered as a particular case. The model is applied to the 1965 Rat Islands (Alaska) earthquake and shows the role of fault heterogeneity in controlling the spatial distribution of stress drop as well as the time dependence and the final amount of radiated energy.

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.

A geothermal model for the area of the Basilicata oil fields has been obtained by an analytical procedure. The model takes into account both the temperature variation due to the re-equilibrated conductive state after thrusting and frictional heating. Input parameters include heat flow density data and a series of geologically derived constraints – thrust depth, timing of thrusting, slip rate – obtained by the integration of surface and subsurface datasets. For the top 5 km of the crust, the resulting geothermal curve shows a remarkably good fit with temperatures recorded from deep oil wells. The new geotherm provides a fundamental constraint for rheological and stress accumulation modelling in the seismically active study area. Furthermore, the analytical solution provided in this study may be used as a basis to calculate the relevant geotherm for further areas and/or tectonic settings.

A fault made of two coplanar asperities subject to a constant strain rate is considered. The fault is modelled as a discrete dynamical system made of two blocks coupled by a spring and pulled at constant velocity on a rough plane. Such a system exhibits a variety of slipping modes, including the slip of single asperities and the simultaneous slip of both asperities. The associated source function can be expressed by the seismic moment rate as a function of time. The moment rate depends on the state of the system preceding the earthquake, which can be described by a single variable expressing the difference between the stresses imposed to the two asperities. We present a systematic study of the moment rate as a function of this variable and show how the moment rate changes as a function of the model parameters. The observed source function of the 2010 Maule (Chile) earthquake, that was the result of the failure of two main asperities, is interpreted in the framework of the proposed model.

A fault with two asperities is modelled as a system made of two blocks coupled by a spring and sliding on a plane under the same values of static and dynamic friction. An analytical solution is given for the simultaneous motion of the blocks and the corresponding orbits are plotted in the phase space. It is proven that, whichever the initial state is, the long-term behaviour of the system is one of an infinite number of limit cycles, characterized by a particular pattern of forces. The region where the system is located when the blocks are stationary can be divided into narrow stripes corresponding to different orbits of the points belonging to them. This implies that the system is sensitive to perturbations and has relevant implications for a fault, which is subject to stress transfers from earthquakes generated by neighbouring faults. In this case, the fault may experience a larger earthquake, with the simultaneous failure of the two asperities, which restores a stress distribution compatible with periodic behaviour. The seismic moment associated with simultaneous asperity failure is always greater than the maximum value that can be released in a limit cycle. For strongly coupled asperities, the moment can be several times larger.

SUMMARY An effusive volcanic eruption results from a sequence of different processes, such as the pressurization of a magma chamber, the propagation of a dyke and the flow of lava at the Earth' surface. The aim of this paper is to establish relationships between the different quantities describing such processes. We consider a spherical magma chamber filled with a low-viscosity magma and included in a homogeneous and isotropic elastic half-space. We assume that, as a result of the inflow of fresh magma or a phase transition, the pressure in the chamber increases slowly during a finite time interval. Assuming that the pressure increase is linear in time, we calculate the stress field generated in the surrounding medium considering the chamber as a centre of dilation. We assume that a vertical tensile fracture originates at the top of the magma chamber after the rock strength is exceeded. The fracture is assumed to propagate quasi-statically along a vertical plane, driven by the stress distribution: both the cases of positive and negative buoyancy force are considered. The problem is solved in two dimensions by considering the fracture as a tensile Somigliana dislocation and expanding the associated stress release into Chebyshev polynomials. The fracture may reach the Earth's surface or not, depending on the depth and radius of the magma chamber, the rate and duration of pressure increase, the rock and magma densities and the rock strength. When the fracture reaches the Earth's surface, we assume that it becomes a vertical conduit. Magma pours out from the vent, driven by the pressure gradient in the conduit. Under the assumption of laminar flow of a Newtonian fluid, we evaluate the initial effusion rate as a function of the relevant model parameters. The flow rate is found to be a non-linear function of the density contrast. We also establish a relationship between the flow rate in the conduit and the initial thickness of the ensuing lava flow, in the case of effusion on a steep slope.

We present here the results from dynamical and thermal models that describe a channeled lava flow as it cools by radiation. In particular, the effects of power-law rheology and of the presence of bends in the flow are considered, as well as the formation of surface crust and lava tubes. On the basis of the thermal models, we analyze the assumptions implicit in the currently used formulae for evaluation of lava flow rates from satellite thermal imagery. Assuming a steady flow down an inclined rectangular channel, we solve numerically the equation of motion by the finite-volume method and a classical iterative solution. Our results show that the use of power-law rheology results in relevant differences in the average velocity and volume flow rate with respect to Newtonian rheology. Crust formation is strongly influenced by power-law rheology; in particular, the growth rate and the velocity profile inside the channel are strongly modified. In addition, channel curvature affects the flow dynamics and surface morphology. The size and shape of surface solid plates are controlled by competition between the shear stress and the crust yield strength: the degree of crust cover of the channel is studied as a function of the curvature. Simple formulae are currently used to relate the lava flow rate to the energy radiated by the lava flow as inferred from satellite thermal imagery. Such formulae are based on a specific model, and consequently, their validity is subject to the model assumptions. An analysis of these assumptions reveals that the current use of such formulae is not consistent with the model.

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.

A power-law, viscoelastic constitutive equation for lithospheric rocks, is considered. The equation is a nonlinear generalization of the Maxwell constitutive equation, in which the viscous deformation depends on the n-th power of deviatoric stress, and describes a medium which is elastic with respect to normal stress, but relaxes deviatoric stress. Power-law exponents equal to 2 and 3, which are most often found in laboratory experiments, are considered. The equation is solved by a perturbative method for a viscoelastic layer subjected to a constant, extensional or compressional, strain rate and yields stress as a function of time, temperature and rock composition. The solution is applied to an ideal extensional boundary zone and shows that the base of the crustal seismogenic layer may be deeper than predicted by a linear rheology.

A model is proposed describing the mechanical evolution of a shear zone along compressional and extensional plate boundaries, subject to constant strain rate. The shear zones are assumed as viscoelastic with Maxwell rheology and with elastic and rheological parameters depending on temperature and petrology. Stress and strain are computed as functions of time and depth. For both kinds of boundaries the model reproduces the existence of a shallow seismogenic zone, characterized by a stress concentration. The thickness of the seismogenic layer is evaluated considering the variations of shear stress and frictional strength on faults embedded in the shear zone. Assuming that a fault dislocation takes place, the brittle-ductile transition is assumed to occur at the depth at which the time derivative of total shear stress changes from positive to negative values. The effects of different strain rates and geothermal gradients on the depth of the brittle-ductile transition are studied. The model predictions are consistent with values inferred from seismicity data of different boundary zones. © 1993 Birkhäuser Verlag.