Scattering of waves by a quasi-thin body of arbitrary shape

Summary. The problem of scattering of a wave with a front of arbitrary shape by a curved quasi-thin body bounded by contour L of arbitrary shape is presented. The problem is solved using the Kirchhoff approximation (KA). The reflecting (scattering) properties of the body are given in terms of the reflection (transmission) coefficients. The routine technique of computation in the KA is complemented by significant improvements. The procedure of smooth continuation of the body surface S beyond its boundary contour L is suggested. The invariant presentation of the expressions for the scattered field u _s obtained in the paper is extremely important.
The leading term in the formula for the field u _s is obtained as a product u _s= u _ray W , where u _ray is the reflected wavefield computed, according to the formulae of ray theory, as if the body surface is boundless; W is the so-called weakening function, which takes account of the modification of the ray field caused by diffraction at the edges (the contour L ).
The arguments of the weakening function are dimensionless quantities depending on the relation between the body size and the radii (half-axes) r _f^{( i )} ( i = 1, 2) of the Fresnel zones on the surface S. The values of r _f^{( i )} are obtained in the paper. The method of computation of the function W turns out to depend essentially on the reciprocal position of the Fresnel zone and body contour. When the contour L is of a complex form, the surface S is divided into simple surfaces S_i , each of which is limited by a contour L , which permits the carrying out of a single valued (in the sense of the leading part of the field) smooth continuation of S_i .     

Near-field waveforms from an arbitrarily expanding, transparent spherical cavity in a prestressed medium

Summary. A simple, approximate ('transparent') solution is derived for the near-field radiation emitted by a spherical cavity expanding in an initial pure shear prestress field. Near-field terms, their propagation and decay are discussed for a variety of growth histories, and are shown to be rather insensitive to the detailed variations of rupture velocity. The transparency approximation is shown to be adequate in the near field as well as in the far field; the main effect is a slight narrowing of far-field pulses. Time domain moment estimators at close range are more reliable for the S wave than for the P wave since transverse pulses are not as strongly contaminated by near-field effects.     

3-D seismic velocities calculated from lattice-preferred orientation and reflectivity of a lower crustal section: examples of the Val Sesia section (Ivrea zone, northern Italy)

To quantify the seismic properties of lower crustal rocks and to better constrain the origin of the lower crustal seismic reflectivity, we determined the complete 3-D seismic properties of a lower crustal section. Eight representative samples of the main lithologic and structural units outcropping in the Val Sesia (Ivrea zone) were studied in detail. The seismic velocities were calculated using the single crystal stiffness coefficients and the lattice preferred orientation (LPO) of each mineral in all samples. The 21 stiffness coefficients characterizing the elastic behaviour of each rock are determined. Mafic and ultramafic rocks such as pyroxenite and pyroxene-bearing gabbros display complex shear wave properties. These rocks are weakly birefringent (maximum 0.1 kms⁻¹) and it is difficult to find consistent relationships between the seismic properties and the rock structure. On the other hand, seismic properties of deformed felsic rocks are essentially controlled by mica. They display strong S -wave birefringence (0.3 km s⁻¹) and relatively high V _p anisotropy (7.6 per cent). Amphibole also strongly influences the rock birefringence patterns. For both kind of rocks, the foliation is highly birefringent and the fast polarized shear wave is systematically oriented parallel to the foliation. We show that the number of mineral phases in the rock strongly controls the anisotropy. The seismic anisotropy has a complex role in the P -wave reflectivity. Compared to the isotropic case, anisotropy enhances the reflection coefficient for about 60 per cent of the possible lithological interfaces. For 40 per cent of the interfaces, the reflection coefficient is much lower when one considers the medium as anisotropic.     

The influence of water on the stress supported by experimentally faulted Westerly granite

Summary. Fault zones in wet Westerly granite deformed at temperatures of 300° and 400°C require markedly lower shear stresses for sliding than when dry, even when the effective confining pressure is held constant between the wet and dry tests, provided that the strain rate is lower than 10⁻⁷s⁻¹. The rate of strength reduction is enhanced by increasing the pore water pressure. The deformation rate is a power function of the applied stress where the stress exponent is approximately 7 for pore water pressure of 100 MPa and 21 for pore water pressure of 20 MPa.
The experimental results are extrapolated to conditions believed to occur at depths of 10 km along the San Andreas Fault Zone. It is suggested that for slow tectonic deformation at strain rates of 10⁻¹¹ and 10⁻¹⁴s⁻¹ the shear stress for sliding on faults in granite would be approximately 60 and 20 MPa, respectively, at pore water pressures equal to the hydrostatic head. Fluid overpressures of c. 0.8 lithostatic pressure are required to lower the shear stress for sliding into the 10 MPa range at the slower strain rate.     

Source mechanism of the deep Colombian earthquake of 1970 July 31 from the free oscillation data

Summary. The method proposed by Mendiguren to determine the source parameters from free oscillation data is applied to the 1970 July 31 deep Colombian earthquake. The results indicate a source propagating horizontally for about 150 km along the lithosphere and cutting across its width. The slab behaves as a guide for source propagation. The horizontal propagation velocity is determined as 3.8 km/s. The intensity of the source grew proportionately to the second power of the propagation distance. This rate of source intensity growth may be interpreted either by a fan-shaped fault model or by a cone-shaped volume source. The average slip and stress drop are estimated as 360 cm and 300 bar for the fault model. For the volume source model the transformational shear strain and stress are estimated as 11 × 10⁻⁵ and 160 bar. There is no evidence of a double couple radiation preceding the P origin time. It is shown that the isotropic and deviatoric components of the moment tensor cannot be uniquely resolved when only observations of a single mode are available. It is observed that, from a statistical basis, the available ₀ S_n data for Colombian shock can be equally well explained by a pure deviatoric source model or by a source model including an isotropic component. Numerical experiments indicate that the inclusion of higher mode data does not change this situation. But, on the other hand, numerical experiments show that the available data and the scheme used for the inversion would not result in a solution including an artificial implosive component if the actual source were pure deviatoric. If the departure from a pure deviatoric source is produced by noise, it has to be non-random, as it could be produced by lateral heterogeneities not included in the inversion scheme.     

Evaluation of stress and displacement fields due to an elliptical plane shear crack

Summary. Following the classic work of Eshelby, the slip and stress distributions due to an elliptical plane shear crack are evaluated. The relation between average (or maximum) slip on the crack and the (constant) static stress drop, for faults of different aspect ratios, is found. The slip vector is not parallel to the applied stress but makes a small angle to it, except when the stress is applied along the major or minor axis of the ellipse. The stress -distribution around the crack shows that in addition to the expected stress concentration along the crack edge, there are broad regions of stress increase off the crack plane for circular and elliptical cracks, similar to those known to exist for in-plane but not for antiplane shear cracks. Whether the stress- intensity factor at the end of one axis is greater or less than that at the end of the other axis ( k_a ≶ k_b ), depends on the condition: √ b/a ≶ (1 − v ) where a and b are the semi-axes of the ellipse, k_a and k_b are the stress-intensity factors at the end of the a- and b -axes and v is Poisson's ratio. The total stress-intensity factor varies smoothly along the edge of the ellipse from one axis to the other and it is found that this variation is rather small.     

Thrust faulting and crust—upper mantle structure in East Australia

Summary. The mid-crustal earthquake of 1973 March 9 (m_b= 5.5, h ≤ 20 km) located 60 km south-west of Sydney, Australia, provides unambiguous evidence of contemporary thrust faulting in South-eastern Australia — a region of high heat flow and Cenozoic basaltic volcanism. Aftershock locations suggest a steeply dipping fault in the depth range from 8 to 24 km with a lateral extent of about 8 km. The mechanism solution is consistent with a tectonic stress field that is dominated by east—west horizontal compression. A seismic moment of 5.7 ± 10²³± 20 per cent dyne-cm was computed from surface-wave amplitudes. Minimum values of slip and stress drop, 2 cm and 1 bar respectively, were estimated from the moment and a fault size taken' from aftershock locations.
Refinement modelling by a controlled Monte Carlo technique was used to provide unbiased models directly from multimode group velocities. The dispersion of fundamental and higher mode surface waves recorded at the digital high-gain station at Charters Towers, Queensland, and the WWSSN station at Adelaide, South Australia, is satisfied by crust- and upper-mantle models which have neither pronounced S-wave low-velocity zones nor thick high-velocity lids within 140 km of the Earth's surface. These models have subcrustal shear velocities of 4.20–4.32 km/s which are 0.4–0.5 km/s slower than Canadian shield shear velocities (CANSD).     

The 2003 Mw 7.2 Fiordland subduction earthquake, New Zealand: aftershock distribution, main shock fault plane and static stress changes on the overlying Alpine Fault

The 2003 August 21 Fiordland earthquake ( M _L7.0, M _W7.2) was the largest earthquake to occur in New Zealand for 35 yr and the fifth of M6+ associated with shallow subduction in Fiordland in the last 15 yr. The aftershocks are diffuse and do not distinguish between the two possible main shock fault planes implied by the Harvard CMT solution, one corresponding to subduction interface thrusting and the other corresponding to steeply seaward dipping thrusting. The distinction is important for calculating the induced stress changes on the overlying Alpine Fault which has a history of very large earthquakes, the last possibly in 1717. We have relocated the aftershocks, using data from temporary seismographs in the epicentral region and the double difference technique. We then use the correlation between aftershock hypocentres and regions of positive changes in Coulomb Failure Stress (CFS) due to various candidate main shock fault planes to argue for concentrated slip on the shallow landward dipping subduction interface. Average changes in CFS on the offshore segments of the Alpine Fault are then negative, retarding any future large events. In our models the change in CFS is evaluated on faults of optimal orientation in the regional stress field as determined by inversion of P -wave polarities.     

Dynamic stress field of a kinematic earthquake source model with k-squared slip distribution

Source models such as the k -squared stochastic source model with k -dependent rise time are able to reproduce source complexity commonly observed in earthquake slip inversions. An analysis of the dynamic stress field associated with the slip history prescribed in these kinematic models can indicate possible inconsistencies with physics of faulting. The static stress drop, the strength excess, the breakdown stress drop and critical slip weakening distance D _c distributions are determined in this study for the kinematic k -squared source model with k -dependent rise time. Several studied k -squared models are found to be consistent with the slip weakening friction law along a substantial part of the fault. A new quantity, the stress delay, is introduced to map areas where the yielding criterion of the slip weakening friction is violated. Hisada's slip velocity function is found to be more consistent with the source dynamics than Boxcar, Brune's and Dirac's slip velocity functions. Constant rupture velocities close to the Rayleigh velocity are inconsistent with the k -squared model, because they break the yielding criterion of the slip weakening friction law. The bimodal character of D _c / D _tot frequency–magnitude distribution was found. D _c approaches the final slip D _tot near the edge of both the fault and asperity. We emphasize that both filtering and smoothing routinely applied in slip inversions may have a strong effect on the space–time pattern of the inferred stress field, leading potentially to an oversimplified view of earthquake source dynamics.     

Focal mechanisms of recent earthquakes in the Southern Korean Peninsula

We evaluate the stress field in and around the southern Korean Peninsula with focal mechanism solutions, using the data collected from 71 earthquakes ( M_L = 1.9–5.2) between 1999 and 2004. For this, the hypocentres were relocated and well-constrained fault plane solutions were obtained from the data set of 1270 clear P -wave polarities and 46 SH / P amplitude ratios. The focal mechanism solutions indicate that the prevailing faulting types in South Korea are strike-slip-dominant-oblique-slip faultings with minor reverse-slip component. The maximum principal stresses (σ₁) estimated from fault-slip inversion analysis of the focal mechanism solutions show a similar orientation with E–W trend (269°–275°) and low-angle plunge (10°–25°) for all tectonic provinces in South Korea, consistent with the E–W trending maximum horizontal stress (σ_Hmax) of the Amurian microplate reported from in situ stress measurements and earthquake focal mechanisms. The directions of the intermediate (σ₂) and minimum (σ₃) principal stresses of the Gyeongsang Basin are, however, about 90 deg off from those of the other tectonic provinces on a common σ₂–σ₃ plane, suggesting a permutation of σ₂ and σ₃. Our results incorporated with those from the kinematic studies of the Quaternary faults imply that NNW- to NE-striking faults (dextral strike-slip or oblique-slip with a reverse-slip component) are highly likely to generate earthquakes in South Korea.     

Effect of viscosity structure on fault potential and stress orientations in eastern Canada

Previous investigations of the causal relationship between postglacial rebound and earthquakes in eastern Canada have focused on the mode of failure and the observed timing of the pulse of earthquake/faulting activity following deglaciation. In this study, the observational database has been extended to include observed orientations of the contemporary stress field and the rotation of stress since deglacial times. It is shown that many of these observations can be explained by a realistic ice history and a viscoelastic earth with a uniform 10²¹ Pa s mantle.
The effects of viscosity structure on the above predictions are also examined. It is shown that, since most of the above observations are found within the ice margin, they are not very sensitive to lithospheric thickness. Also, the inclusion of a 25 or 50 km ductile layer within the lithosphere will not decouple the seismogenic upper crust. High viscosity (10²² Pa s) in the lower mantle is rejected by the stress orientation and rotation observations. A low-viscosity (6 times 10²⁰Pa s) upper mantle with 1.6 times 10²¹ Pa s in the upper part of the lower mantle and 3 times 10²¹ Pa s in the lower part of the lower mantle below 1200 km depth has been found to give predictions that are in general agreement with the observations.     

The elastodynamic near field

Summary. The elastodynamic fields of point forces and shear dislocations of finite source duration are analysed with the aim of establishing the frequency and time-domain characteristics of the field in the near-source region. Criteria are obtained for amplitude dominance in regions where the source–sensor distance is much smaller than the wavelength.
It is shown that in the frequency domain , the Green's tensor (and hence the displacement field of a single point force) attenuates like R ⁻¹ in the near-source region and there exists no region in which the 'near-field' term becomes dominant such that the 'far-field' term can be neglected. Hence, there is no real 'near-field' term for the elastodynamic Green's tensor. The near-field terms of the displacements, velocities and accelerations excited by a shear dislocation attenuate like R ⁻², since the R ⁻³ and R ⁻⁴ terms tend to be eliminated due to mutual cancellation of P and S motions in the near-source region.
In the time domain , the corresponding near field of the displacement field is defined for the steady amplitude interval (away from transients) R /β < t < R /α+ T by the condition R ≤βT where β is the shear velocity and T is the source's duration. The relative strengths of all other arrivals will depend on the particular time window under consideration.
The particle motion patterns due to a single force in the near-source region are shown to be similar to rotating hyperbolas with an axis along the force direction, which are quite different from the 'smoke ring' motion patterns of the so-called 'near-field' term itself.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

Stress accumulation between volcanoes: an explanation for intra-arc earthquakes in Nicaragua?

Destructive upper crustal earthquakes in Central America are often located between active volcanic centres—a geometric relationship that we study using finite element Coulomb failure stress (CFS) models that incorporate the rheologically heterogeneous nature of the volcanic arc. Volcanoes are simulated as mechanically weak zones within a stronger crust. We find that deformation of the volcanic centres within a regional stress field dominated by dextral shear causes stress increases in surrounding crust, with a maximum CFS change between neighbouring volcanoes. This increase in CFS enhances the probability of fault slip on arc-normal faults that are located between volcanic centres; for example, the Tiscapa fault, which ruptured during the 1972 December 13,   M _s   6.2 Managua earthquake. The amount of stress increase due to long-term (100 yr) volcano shearing is on the order of 0.1–0.6 bars, similar to values estimated for subduction zone earthquakes.     

Surface joint systems, tectonic stresses and geomorphology: a reconciliation of conflicting observations

This note seeks to reconcile the widespread small-scale fractographic observations indicating that surface joints are extension fractures with the frequently observed occurrence of such joints as what appear to be nearly orthogonal conjugate sets, their strikes fitting the shear lines of the neotectonic stress field. It is suggested that the local appearance of joints as extension fractures may have nothing to do with their orientation in the large-scale neotectonic stress field, inasmuch as extrapolations from a local scale to plate-tectonic dimensions are quite speculative. Thus, the local extension-characteristics of the joint surfaces may be acquired at the latest stage of their genesis as a result of the corresponding rock faces becoming exposed, but the geometrically orientational attributes may be conditioned by the shear in the surrounding large-scale neotectonic stress field; a new possible mechanism for reconciling the conflicting local and large-scale observations is suggested.     

The solution of the inverse gravity potential problem for cylindrical bodies

Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution.
(1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ₁( z )ρ₂( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ₁( R , ø), the length, form and orientation of the cylinder if we know the function ρ₁( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem.
(2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ₁(τ)ρ₂(θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ₁(θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ₁(τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.     

Regional mid-ocean stress and a proposed focal mechanism of 'stress-discordant' earthquakes

Summary. Rock stress measurements in Iceland show maximum horizontal compression perpendicular to the trend of Reykjanes Ridge crest and of its extension, the active volcanic zone of Iceland. Fault-plane solutions of dormant stage earthquakes are consistent with the measured stress orientations, but strike—slip earthquakes associated with volcanic surges and some earthquake swarms in active geothermal areas exhibit apparent reversals of mechanism and are here defined as 'stress-discordant' in the sense that they yield deduced stress orientations 90° from the regional stress field as determined by hydrofracturing and strain relief methods. It is proposed, supported by comparison with the pore-pressure induced Denver earthquakes, that the 'stress-discordant' volcanic earthquakes are triggered by increased pore pressure and probably involve stick-slip motion similar to that reported for some laboratory tests of the pore pressure effect, characterized by gradual onset and sudden stopping of each slip episode. The question is raised as to whether stress-discordant earthquakes are dominated by a stopping phase or terminal shock with consequent reversal of the deduced shear couple. A possible stopping mechanism is suggested: the dilatant stiffening of fault gouge during shear.
It is proposed that direct measurements of stress orientation be made by hydrofracturing tests at other places along the mid-ocean ridge crest and on the margins of the Red Sea and East African rifts. The Icelandic stress data indicate the need for sceptical re-examination of some fundamentals of plate tectonics theory.     

Focal mechanisms of small earthquakes in the southeastern Brazilian shield: a test of stress models of the South American plate

Focal mechanisms of small earthquakes with magnitudes of about 3 in the SE Brazilian shield are calculated using S / P amplitude ratios. Low attenuation ( Q _p from 400 to 800) in the shield upper-crustal layers allowed sharp S arrivals to be recorded up to distances of 100 km. Besides P -wave polarities, SH -wave first motions were also used to constrain the nodal-plane orientations. Normal and reverse faulting mechanisms with strike-slip components were found. The inversion of four mechanisms to estimate the stress tensor indicated a strike-slip stress regime with roughly E–W-orientated σ ₁ and N–S σ ₃. Both the orientations and the shape factor ( φ =0.7) of the inverted stress are in excellent agreement with theoretical predictions for that part of Brazil from the driving-force model of Coblentz & Richardson (1996) . Good agreement with the nature of the stress, as well as its orientation, was also found for the model of Meijer (1995) . Both of these theoretical models include spreading stresses along the continent/ocean lithospheric transition. Because the earthquakes are more than 300 km from the continental shelf they should not be affected by the local flexural forces caused by sediment load in the marginal basins. The agreement between observed and theoretical stresses then confirms the importance of continental spreading forces in modelling intraplate stresses.     

Fast transform eigenfunction methods for boundary value problems

An extension of the classic eigenfunction expansion method is presented for solving boundary value problems on boundaries of essentially arbitrary shape in R ³. the algorithm yields approximate solutions in terms of simple eigenfunctions of the underlying differential operator. the coefficients of these eigenfunctions are calculated with two-dimensional FFTs. the algorithm has many applications in geophysics, such as potential theory, scattering theory, earthquake source dynamics, and wave propagation, where non-separable boundaries are common.     

2001 August earthquake swarm at Shadwan Island, Gulf of Suez, Egypt

The earthquake swarm that struck Shadwan Island at the entrance of the Gulf of Suez in 2001 August included 408 events. Almost all of these events (94 per cent) were microearthquakes and only 6 per cent had small measurable magnitudes  (5.0 > M _L≥ 3.0)  . Most of the earthquakes were weak and followed each other so closely in time that they could not be identified at more distant stations. The fault plane solutions of the strongest events of the swarm show almost identical focal mechanisms, predominately normal faulting with a significant sinistral strike-slip component for nodal planes trending NW–SE. A comparison with the mechanisms of the 1969 and 1972 events which took place 20 km north of the swarm epicentral region shows similarities in faulting type and orientation of nodal planes. The azimuths of T -axes determined from focal mechanisms in this study are oriented in the NNE–SSW direction. This direction is consistent with the present-day stress field derived from borehole breakouts in the southern Gulf of Suez and the last phase of stress field changes in the Late Pleistocene, as well as with recent GPS results.
The source parameters of the largest  ( M _L≥ 3.0)  events of the 2001 August Shadwan swarm have been estimated from the P -wave spectra of the Egyptian National Seismograph Network (ENSN). Averaging of the values obtained at different stations shows relatively similar source parameters, including a fault length of  0.65 ≤ L ≤ 2 km  , a seismic moment of  7.1 × 10¹²≤ Mo ≤ 3.0 × 10¹⁴ N m  and a stress drop of  0.4 ≤Δσ≤ 10  bar.