A complete suite of closed analytical expressions is presented for the surface displacements, strains, and tilts due to inclined shear and tensile faults in a half-space for both point and finite rectangular sources. These expressions are particularly compact and free from field singular points which are inherent in the previously stated expressions of certain cases. The expressions derived here represent powerful tools not only for the analysis of static field changes associated with earthquake occurrence but also for the modeling of deformation fields arising from fluid-driven crack sources.

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Surface deformation due to shear and tensile faults in a half-space

This is a study of the formulation, some basic solutions, and applications of the Biot linearized quasistatic elasticity theory of fluid-infiltrated porous materials. Whereas most previously solved problems are based on idealizing the fluid and solid constituents as separately incompressible, full account is taken here of constituent compressibility. Previous studies are reviewed and the Biot constitutive equations relating strain and fluid mass content to stress and pore pressure are recast in terms of new material parameters, more directly open to physical interpretation as the Poisson ratio and induced pore pressure coefficient in undrained deformation. Different formulations of the coupled deformation/diffusion field equations and their analogues in coupled thermoelasticity are discussed, and a new formulation with stress and pore pressure as basic variables is presented that leads, for plane problems, to a convenient complex variable representation of solutions. The problems solved include those of the suddenly introduced edge dislocation and concentrated line force and of the suddenly pressurized cylindrical and spherical cavity. The dislocation solution is employed to represent that for quasi-static motions along a shear fault, and a discussion is given, based on fracture mechanics models for fault propagation, of phenomena involving coupled behavior between the rupturing solid and its pore fluid, which could serve to stabilize a fault against rapid spreading. Also, the solution for a pressurized cylindrical cavity leads to a time-dependent stress field near the cavity wall, and its relevance to time effects in the inception of hydraulic fractures from boreholes, or from drilled holes in laboratory specimens, is discussed. Various limiting cases are identified, and numerical values of the controlling porous media elastic parameters are given for several rocks.

Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents

Recently, Bouchon (1979a) reinterpreted strong motion seismograms obtained during the Parkfield earthquake of 1966 using a new method applicable to a finite propagating dislocation source in a layered medium. His results and other pertinent data, interpreted in terms of the barrier model of Das and Aki (1977), suggest that the rupture may be stopped by a barrier with the specific fracture energy of about 109 erg cm−2. Using the formulas of Ida (1973a), we estimated parameters of the barrier as follows: breaking slip of about 20 cm, cohesive stress of about 100 bars, and length of end zone (nonelastic zone) of a few hundred meters. The barrier parameters for the great 1857 earthquake were also obtained from the description of surface fault breaks by Wallace (1968). The result led to the estimation of maximum acceleration of about 1.5g near the fault, under the assumption that the end zone length is proportional to the diameter of individual crack of the barrier model. Barriers for other earthquakes are discussed, and they are classified into geometrical barriers such as fault bend and corner and inhomogeneous barriers such as the high-velocity anomaly straddling the San Andreas fault near San Juan Bautista. The barriers act not only as a stopper of rupture but also as an initiator of rupture, as well as a stress concentrator, causing twin earthquakes and migration or progression of major earthquakes along the plate boundary.

Characterization of barriers on an earthquake fault

Shear cracks with finite cohesive forces can propagate by skipping past barriers. The barriers left behind may remain unbroken or may eventually break because of subsequent increase in dynamic stress depending on the ratio of barrier strength to tectonic stress. This model can explain a variety of observations on rupture in the earth, including (1) segmentation of the fault or ruptured zone in earthquakes and rock bursts, (2) ripples in seismograms which cannot be explained by path effect, and (3) departure of the scaling law of the seismic spectrum from that based upon the similarity assumption. The model also explains why the simple uniform dislocation model sometimes works better than the crack model without barriers. It also predicts, contrary to common belief, that an earthquake with low average stress drop may generate relatively greater amounts of high-frequency waves than an earthquake with high average stress drop. One important consequence of our barrier model is the possibility of predicting the occurrence of aftershocks by analyzing the source spectrum of the main shock.

Fault plane with barriers: A versatile earthquake model

Rupture in individual earthquakes apparently is limited to regions between bends in faults. This is illustrated for eight events that have occurred since 1966. A model based on geometric concepts describes why this is so and clarifies earlier ideas of "asperities" and "barriers" used to explain earthquake initiation and termination processes. Because of their importance in the rupture process, bend zones should be monitored for precursory effects.

http://science.sciencemag.org/content/sci/228/4702/984.full.pdf

Role of fault bends in the initiation and termination of earthquake rupture.

Summary The arrest of a semi-infinite longitudinal shear crack is caused by either (1) the finiteness of available strain energy, or (2) an increase in fracture energy along the trajectory of the running crack. In the former case the following relationship may be used to evaluate the fracture energy: yo = RAa2/2ptn where yo is the fracture energy per unit length along the crack edge per unit extension of the cracktip (erg cm-2), R is the characteristic radius of the fault (cm), ACJ is the stress drop (dyne cm-2) and p is the rigidity (dyne cm-2). This leads to the following relationship: log Aa = -3 log R +f log (2ptny0) or from the Keylis-Borok relationship (1959): log Mo = 5/2 log R +* log (2ptny0)+0*64 where M, is the seismic moment in dyne cm-'. These two relationships are statistically acceptable for Southern California faults and the TongaKermadec Arc earthquakes. The fracture energy is found to vary from lo3 to lo9 erg cm-' with fresh fracture being associated with 10'-lQ9 erg cm-2 while frictional rupture with 103-107 erg cm-2. These values are in good agreement with other independent estimates.

The Fracture Energy of Earthquakes

Summary This paper is the first of a series that will examine the effect of earth structure on earthquake displacement, strain and tilt fields at the Earth's surface. Its purpose is to develop the numerical techniques to be applied in the papers that foliow. A general computational procedure for the evaluation of the integral expressions for the surface displacements due to an arbitrary point dislocation source in a layered medium is described. It is shown to be rapid and inexpensive to use, and its accuracy appears to be entirely adequate for practical purposes.

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Elastic Dislocations in a Layered Half-Space—I. Basic Theory and Numerical Methods

Summary The static displacement and strain fields caused by the introduction of a shear dislocation point source into a layered elastic half space have been evaluated using a Thomson-Haskell matrix method (Singh). The point source was found to be an adequate representation of a fault at distances greater than four fault lengths. Gross earth structure (oceanic, shield and tectonic models) cause fields that differ little from those of a uniform half space. However, significant departures from the uniform half-space fields are found to be caused by low rigidity layers, both at the surface (representing sediment cover) and at depth (representing possible zones of partial melt). Both of these features cause complexities in the strain fields that depend on the source orientation and source-receiver distance, and these may result in amplification or attenuation of the uniform halfspace fields by factors of up to 10.

Elastic Dislocations in a Layered Half‐Space–II The Point Source

The displacement fields due to a very long vertical strike-slip fault are calculated for earth models consisting of two layers over a half-space. It is shown that if zones of low rigidity are present beneath the Earth9s surface, they will result in an amplification of the displacement field observed at the Earth9s surface. The amount of amplification depends both on the structure and the distance from the fault. If thin soft layers exist in the upper mantle, it is shown that they will have a very strong effect on the observed displacements. It may eventually be possible to use static observations at the surface to detect such layers.

Effect of earth layering on earthquake displacement fields

Summary

An inversion method is developed for estimating the source parameters of seismic and aseismic events from static strain data. The inversion method consists of a non-linear least squares algorithm of Marquardt combined with an incremental procedure. The inversion method is quite stable if the strain fields are modelled by a point source in a homogeneous halfspace. The seismic moment, direction of strike, slip direction, and dip of the San Fernando earthquake of 1971 February 9, and the Gifu, Japan earthquake of 1969 September 9, have been estimated from the static strain data. The estimated parameters from the static data are in good agreement with the seismically estimated source parameters. The success with which the static strain steps may be inverted to obtain the source parameters of two seismic events suggests that static strain steps are indeed a real manifestation of the static fields associated with displacement on a fault.

Dushan B. Jovanovich

Papers

The Fracture Energy of Earthquakes

Elastic Dislocations in a Layered Half-Space—I. Basic Theory and Numerical Methods

Elastic Dislocations in a Layered Half‐Space–II The Point Source

Effect of earth layering on earthquake displacement fields

An Inversion Method for Estimating the Source Parameters of Seismic and Aseismic Events from Static Strain Data