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.     

Dynamic behaviour of a phase boundary under non-uniform surface loads

Summary. A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long time-expansions of the solution can be obtained from series expansion and integration of the Laplace transform along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions.
For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t ^1/2) as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface).
For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber k _L of the surface excitation is such that κ k ²_Lτ≤ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k ²_Lτ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ⁻¹ k ⁻²_L.
When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km.     

Oxygen consumption in Macrotrachela musculosa and Trichotria truncata (Rotatoria) from the High Arctic

Oxygen consumption by rotifers Macrotrachela musculosa and Trichotria truncata from Spitsbergen tundra (77°N) was measured using the method of Cartesian divers. The metabolic rate of M. musculosa was: 0.205 10⁻³mm³ 0₂ per g 10⁻⁶ per hour at 2°C, 0.201 10⁻⁶mm³ at 6°C and 0.616 10⁻³mm³ 0₂ per g 10⁻⁶ per hour at 10°C. The metabolic rate of Trichotria truncata at 6° was 0.103 10⁻³mm³ per g 10⁻⁶ per hour. The relation between body weight and oxygen consumption by M. musculosa at 2°C is expressed with the equation R = 0.18W^0.67, with R – oxygen consumption in mm³10⁻³ per individual per hour and W – wet weight of an animal in g 10⁻⁶.     

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.     

Coseismic and other short-term strain changes recorded with Sacks-Evertson strainmeters in a deep mine, South Africa

Summary. During 1977 March and April, three Sacks-Evertson borehole dilatometers were installed at the ends of boreholes drilled into the sidewall of an experimental tunnel at a depth of 3.1 km in the ERPM gold mine near Johannesburg. In the following year coseismic strain changes ranging from 5 ± 10⁻¹⁰ to values exceeding 5 ± 10⁻⁶ were recorded for hundreds of mine tremors in the magnitude range -1 to 3.7 and at hypocentral distances of 50 m to about 2 km. Hypocentral coordinates and magnitudes were determined from seismograms recorded from an underground array of geophones. Amplitudes and polarities of the coseismic strain steps are generally in excellent agreement with theoretical expectations based on point-source dislocation theory; specifically, the strain steps are proportional to the seismic moment divided by the cube of hypocentral distance. At a strain level of 5 ± 10⁻⁹ or greater the tremors do not appear to be preceded by any short-term indications of instability even for tremors producing coseismic steps greater than 5 ± 10⁻⁶ and for which the strainmeters were within a source radius of the hypocentre. Continuous strain changes observed at the times when the mine excavation, at a distance of about 100 m, is extended are in good agreement with calculated changes based on the theory of elasticity. A similar calculation is consistent with post-seismic strain changes observed to follow some of the closer tremors. These post-seismic strains show a logarithmic dependence on time following the tremor and appear to be due to the interaction of a tremor with the adjacent mine excavation rather than to deformation within the actual seismic source region.     

A near-regional explosion source model for tuff

Summary. A variety of near-regional (300 km) data, including spectral amplitudes of Pg , surface-wave forms, and close-in (5–10 km) accelerograms have been used to build an elastic seismic source model for a 1-Mton explosion in tuff at near-regional distances. The model consists of: (1) a pressure pulse which injects 3 × 10¹² cm³ of volume into the medium, (2) a vertical, upward force impulse that imparts 10¹⁸ dyn-s of momentum to the medium, each source component having a time duration of 0.6 s and a depth of 1.3 km. The force impulse appears to be required by two considerations: (a) the striking similarity, apart from sign, of explosion surface waves with those of their cavity collapses, (b) the observation of considerable SV energy leaving the source of the 1-Mton explosions JORUM and HANDLEY . Scaling curves have been constructed which fit the proposed source model. These scaling curves employ: very slow decrease, as (yield)^−0.10 of the primary corner frequency; decay as (frequency)⁴ or (frequency)³ to high frequency. While these scaling curves are unconventional, they appear to be the only ones which can satisfy the near-regional data. The slow scaling with yield of the spectral carner frequency suggests that it is caused by something other than the equivalent elastic radius, e.g. the time duration of motion at the source. The results, at odds with similar studies at teleseismic distances, suggest that significantly different equivalent elastic sources are required at near-regional (as compared with teleseismic) distances; therefore, the effect of the upward impulse might not be seen at teleseismic distances. Consequently, these results probably do not pertain to the seismic discrimination problem at teleseismic distances.     

Tidal and rotational perturbations of Earth's gravity gradients

Summary. In view of present attempts to build orbiting gravity gradiometers with an accuracy of about 10⁻⁴ E, perturbing gravitational gradients arising from tide-generating bodies and from rotational deformations induced by polar motion are evaluated.     

The Influence of Surface Inhomogeneities On Deep Electromagnetic Soundings of the Earth

Summary. Asymptotic expressions for components of the electromagnetic field of a grounded electric dipole are considered for the model consisting of a thin surface-layer overlapping a stratified medium with a highly resistive screen on the roof. It is shown that the method of spatial derivatives makes it possible to obtain proper estimates of the impedance at distances of r ≥|λ₀| from the nearest edge of the surface anomaly (|λ₀| being the effective depth of the field penetration in the underlying section). the magnetotelluric methods allow one to obtain the true values of impedance, provided r ≥ max {|λ₀|, |/( S ⁻¹+ Z ⁻₀|^1/2} where S is the integrated conductivity of the surface layer, is the transverse resistance of the screen, and Z ⁻₀ is the Tikhonov—Cagniard impedance for the medium underlying the surface layer.     

High-frequency radiation from crack (stress drop) models of earthquake faulting

We present a theory for the radiation of high-frequency waves by earthquake faults. We model the fault as a planar region in which the stress drops to the kinematic friction during slip. This model is entirely equivalent to a shear crack. For two-dimensional fault models we show that the high frequencies originate from the stress and slip velocity concentrations in the vicinity of the fault's edges. These stress concentrations radiate when the crack expands with accelerated motion. The most efficient generation of high-frequency waves occurs when the rupture velocity changes abruptly. In this case, the displacement spectrum has an ω^-2 behaviour at high frequencies. The excitation is proportional to the intensity of the stress concentration near the crack tips and to the change in the focusing factor due to rupture velocity. We extend these two-dimensional results to more general three-dimensional fault models in the case when the rupture velocity changes simultaneously on the rupture front. Results are similar to those described for two-dimensional faults. We apply the theory to the case of a circular fault that grows at constant velocity and stops suddenly. The present theory is in excellent agreement with a numerical solution of the same problem.
Our results provide upper bounds to the high-frequency radiation from more realistic models in which rupture velocity does not change suddenly. The ω^-2 is the minimum possible decay at high frequencies for any crack model of the source.     

Explicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems

We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax = b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A ⁻ is found using one-step backprojections on the Penrose condition AA ⁻≈ I , from which we calculate the statistical properties of the solution. The computation of A ⁻ can easily be parallelized, each column being constructed independently.
The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A ⁻ is not accurate enough to actually compute the solution x , the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 10⁶ non-zero elements in A . Computation time is proportional to the number of non-zero elements in A . If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns x _i , a further gain in efficiency can be obtained by neglecting the small elements in A , but a more accurate estimation of the correlation requires a full treatment of even the smaller A _ij . We finally develop a formalism to compute a damped version of A ⁻.     

Modelling of short-interval silent slip events in deeper subduction interfaces considering the frictional properties at the unstable–stable transition regime

Recent high-resolution observations of crustal movements have revealed silent slip events (SSEs) with propagation velocities of around 10–15 km d⁻¹ and with intervals of 3–14 months along the deeper parts of the Cascadia and Nankai subduction zones. This study develops 2-D and 3-D models of these short-interval SSEs considering the frictional behaviour that was confirmed experimentally by Shimamoto for the unstable–stable transition regime. To represent this frictional behaviour, a small cut-off velocity to an evolution effect is introduced in a rate- and state-dependent friction law. When the cut-off velocity to the evolution effect is significantly smaller than that to a direct effect, steady-state friction exhibits velocity weakening at low slip velocities and velocity strengthening at high slip velocities. At the deeper Cascadia and Nankai subduction interfaces, the pore pressure is inferred to be high because of the dehydration of materials in the descending plate. Under conditions where the pore-fluid pressure is nearly equal to the lithostatic pressure and the critical weakening displacement is very small, short-interval SSEs with propagation velocities and slip velocities of 4–8 km d⁻¹ and  2 − 4 × 10⁻⁷  m s⁻¹, respectively, can be reproduced. The propagation velocity of short-interval SSEs is in proportion to the slip velocity. The results also show that during the nucleation process of large earthquakes, the occurrence of short-interval SSEs becomes irregular because of the accelerated slips that occur at the bottom of the seismogenic zone. Our results suggest that monitoring of short-interval SSEs might be useful for forecasting the main earthquakes.     

Long-period geomagnetic variations and mantle conductivity: an inversion using Bailey's method

Summary. We have implemented an algorithm which is based on Bailey's solution of the inverse problem of electromagnetic induction in the Earth. The study was motivated by recent determinations of very long period data and also benefited from recent redeterminations of high frequency data. The algorithm has been successfully tested to provide reliable estimates of conductivity down to a depth of 2000 km, using synthetic data in the period range from 4 days to 11 years. Smooth data sets, which are required for the inversion, were constructed from various sources. At a given depth, the range of inverted models is less than one order of magnitude. Due to the lack of high frequency data, the conductivity of the upper 600 km of the mantle, which is found to be of the order of 10⁻¹Ω⁻¹ m⁻¹, may be overestimated. The algorithm performs well in the middle mantle, where conductivity rises steadily from 1 to 50 Ω⁻¹ m⁻¹. The lack of very low frequency data and limitations of the algorithm prevent one from obtaining meaningful estimates in the lower mantle. However, the study of the propagation of the late 1960s secular variation acceleration provides an estimate of the mean conductivity of the whole mantle. Thus, a complete mantle profile can be constructed. It is found that deep mantle conductivity probability does not exceed a few hundred Ω⁻¹ m⁻¹.     

Numerical computation of the three-dimensional normal ellipsoidal field with a 10−2 m and 10−2 mgal accuracy

Summary. A new computation of the relevant geometrical and mechanical quantities pertaining to the three-dimensional normal ellipsoidal field (Bocchio) has been worked out with reference to a spatial grid ranging from 0 to 9000 m in dynamic height and from 0 to 90° in latitude, with a mesh size of 500 m and 1° respectively giving a better accuracy. An estimate of the latter at the upper end of the vertical interval gives 10⁻² m for the radii of curvature of the equipotential surfaces and 10⁻²mgal for normal gravity. Twenty-six functions of interest have been numerically computed including the metric tensor, Christoffel's symbols and the tensor of gravity gradients giving about 45 000 tabulated data.     

Numerical simulation of the propagation of P waves in fractured media

We study the propagation of P waves through media containing open fractures by performing numerical simulations. The important parameter in such problems is the ratio between crack length and incident wavelength. When the wavelength of the incident wavefield is close to or shorter than the crack length, the scattered waves are efficiently excited and the attenuation of the primary waves can be observed on synthetic seismograms. On the other hand, when the incident wavelength is greater than the crack length, we can simulate the anisotropic behaviour of fractured media resulting from the scattering of seismic waves by the cracks through the time delay of the arrival of the transmitted wave. The method of calculation used is a boundary element method in which the Green's functions are computed by the discrete wavenumber method. For simplicity, the 2-D elastodynamic diffraction problem is considered. The rock matrix is supposed to be elastic, isotropic and homogeneous, while the cracks are all empty and have the same length and strike direction. An iterative method of calculation of the diffracted wavefield is developed in the case where a large number of cracks are present in order to reduce the computation time. The attenuation factor Q ⁻¹ of the direct waves passing through a fractured zone is measured in several frequency bands. We observe that the attenuation factor Q ⁻¹ of the direct P wave peaks around kd = 2, where k is the incident wavenumber and d the crack length, and decreases proportionally to ( kd ) ⁻¹ in the high-wavenumber range. In the long-wavelength domain, the velocity of the direct P wave measured for two different crack realizations is very close to the value predicted by Hudson's theory on the overall elastic properties of fractured materials.     

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.     

Seismic sources with observable glut moments of spatial degree two

Let ζ_Λ and r _Λ. be the hypocentral position and time of an extended indigenous seismic source. Backus showed that the force moment tensors of the source, Γ^{( m +1, n )}(ζ_Λ, r _Λ), determine and are determined by the motion which the source produces. For small m + n , only the long-period motion is relevant. The glut moment tensor Λ^{( m,n )} (ζ_Λ, r _Λ.) can be calculated uniquely from γ^{( m +1, n )}(ζ_Λ r _Λ) only if m = 0 or m = 1. The tensor G =Λ^(2,0)(ζ_Λ) gives the spatial variance tensor W_Λ of the source, and W_Λ. roughly describes the size, shape and orientation of the source region. Therefore the failure of the observed F =Γ^(3,0)(ζ_Λ) to determine G uniquely is of seismological interest. In the present paper we show that F determines G uniquely if we assume the source to be a simple straight line source (SSLS) or an ideal fault in an isotropic medium with isotropic prestress (IFIMIP). We give tests on F which determine whether it can come from a SSLS, from an IFIMIP or from a simple plane surface source (SPSS). If we assume the source to be a SPSS then knowing F and the fault plane determines G to within an unknown scalar multiple of a certain tensor tangent to the fault plane. Moreover F determines the fault plane uniquely unless F can come from a SSLS. If it can, then F determines this virtual source line uniquely, and F permits the fault plane to be any plane containing the virtual source line.     

Micromonas pusilla (Prasinophyceae) as part of pico-and nanoplankton communities of the Barents Sea

Micromonas pusilla (Butcher) Manton & Parke appears to be a prominent member of the Barents Sea picoplankton community as revealed by the serial dilution culture method. Cell numbers frequently exceeded 10⁷ cells 1⁻¹, though they usually varied between 10³and 10⁶ cells l⁻¹. A number of other identified and unidentified taxa were recorded and quantified. Distribution relative to the marginal ice zone is reported.     

Dissolved and total organic and inorganic carbon in some British rivers

Rivers transport both organic and inorganic carbon from their sources to the sea. Results of ~800 organic and inorganic analyses from various British rivers of contrasting size and land use are presented here: (1) the headwater River Tern, a rural river of 852 km² catchment; (2) the Ouseburn, a small urban 55 km² catchment; (3) the River Tyne, a larger river system of ~3000 km² catchment; (4) a spatial survey from 205 sample sites on ~60 rivers from SW England. We found that, with the exception of peat-rich headwaters, DIC concentration is always greater than DOC. DIC is primarily in the form HCO₃ ⁻ , with DIC concentrations highest in highly urbanised catchments, typically greater than those observed in catchments with carbonate bedrock, demonstrating a significant and previously unrecognised anthropogenic inorganic carbon input to urban rivers.     

Source tomography by simulated annealing using broad-band surface waves and geodetic data: application to the Mw= 8.1 Chile 1995 event

We invert surface-wave and geodetic data for the spatio-temporal complexity of slip during the M _w =8.1 Chile 1995 event by simulated annealing. This quasi-global inversion method allows for a wide exploration of model space, and retains the non-linearity of the source tomography problem. Complex source spectra are obtained from 5 to 45 mHz from first- and second-orbit fundamental-mode Rayleigh waves using an empirical Green's function cross-correlation technique. Coseismic displacement vectors were measured at 10 GPS sites near Antofagasta. They are part of a French-Chilean experiment which monitors the Northern Chile seismic gap. The spectra, together with the geodetic data, are inverted for the moment distribution on a 2-D dipping fault, under the physical constraints of slip positivity and causality. Marginal a posteriori distributions of the model parameters are obtained from several independently inverted solutions. In general, features of the slip model are well resolved. Data are well fitted by a purely unilateral southward rupture with a nearly uniform velocity around 2.5–3.0 km s⁻¹, and a total duration of 65 s. Several regions of moment release were imaged, one near the hypocentre, a major one 80 km south of it and a minor one 160 km south of it. The major patch of moment release seemed to have propagated to relatively shallow depths near the trench, 100 km SSW of the epicentre. The region of major slip is located updip of the 1987, M _w =7.5 earthquake, suggesting a causal relationship. Most of the slip occurred updip of the hypocentre (36 km), but the entire coupled plate interface (20–40 km) ruptured during the Chile 1995 event.     

Magnetic and viscous coupling at the core–mantle boundary: inferences from observations of the Earth's nutations

Dissipative core–mantle coupling is evident in observations of the Earth's nutations, although the source of this coupling is uncertain. Magnetic coupling occurs when conducting materials on either side of the boundary move through a magnetic field. In order to explain the nutation observations with magnetic coupling, we must assume a high (metallic) conductivity on the mantle side of the boundary and a rms radial field of 0.69 mT. Much of this field occurs at short wavelengths, which cannot be observed directly at the surface. High levels of short-wavelength field impose demands on the power needed to regenerate the field through dynamo action in the core. We use a numerical dynamo model from the study of Christensen & Aubert (2006) to assess whether the required short-wavelength field is physically plausible. By scaling the numerical solution to a model with sufficient short-wavelength field, we obtain a total ohmic dissipation of 0.7–1 TW, which is within current uncertainties. Viscous coupling is another possible explanation for the nutation observations, although the effective viscosity required for this is 0.03 m² s⁻¹ or higher. Such high viscosities are commonly interpreted as an eddy viscosity. However, physical considerations and laboratory experiments limit the eddy viscosity to 10⁻⁴ m² s⁻¹, which suggests that viscous coupling can only explain a few percent of the dissipative torque between the core and the mantle.