Estimates of fracture parameters of earthquakes

A new estimate of the fracture parameters of earthquakes is provided in this paper. By theMuskhelishvili method (1953) a number of basic relations among fracture-mechanics parameters are derived. A scheme is proposed to evaluate the slip weakening parameters in terms of fault dimension, average slip, and rise time, and the new results are applied to 49 events compiled in the earthquake catalogue ofPurcaru andBerckhemer (1982). The following empirical relations are found in the paper: $$\begin{gathered} \frac{{\tau _B - \tau _f }}{{\tau _\infty - \tau _f }} = 2.339 \hfill \\ {{\omega _c } \mathord{\left/ {\vphantom {{\omega _c } {W = 0.113}}} \right. \kern-\nulldelimiterspace} {W = 0.113}} \hfill \\ \log G_c \left( {{{dyne} \mathord{\left/ {\vphantom {{dyne} {cm}}} \right. \kern-\nulldelimiterspace} {cm}}} \right) = 2 \log L (km) + 6.167 \hfill \\ \log \delta _c (cm) = 2 \log L (km) - 1.652 \hfill \\ \end{gathered} $$ whereG _c is the specific fracture energy,ω _c the size of the slip weakening zone,δ _c the slip weakening displacement,τ _B ?τ _f the drop in strength in the slip weakening zone,τ _∞?τ _f the stress drop,L the fault length, andW the fault width. The investigation of 49 shocks shows that the range of strength dropτ _B ?τ _f is from several doze to several hundred bars at depthh<400 km, but it can be more than 10³ bars ath>500 km; besides, the range of the sizeω _c of the strength degradation zone is from a few tenths of a kilometer to several dozen kilometers, and the range of the slip weakening displacementδ _c is from several to several hundred centimeters. The specific fracture energyG _c is of the order of 10⁸ to 10¹¹ erg cm^?2 when the momentM ₀ is of the order of 10²³ to 10²⁹ dyne cm.     

Constructing Constitutive Relationships for Seismic and Aseismic Fault Slip

For the purpose of modeling natural fault slip, a useful result from an experimental fault mechanics study would be a physically-based constitutive relation that well characterizes all the relevant observations. This report describes an approach for constructing such equations. Where possible the construction intends to identify or, at least, attribute physical processes and contact scale physics to the observations such that the resulting relations can be extrapolated in conditions and scale between the laboratory and the Earth. The approach is developed as an alternative but is based on Ruina (1983) and is illustrated initially by constructing a couple of relations from that study. In addition, two example constitutive relationships are constructed; these describe laboratory observations not well-modeled by Ruina’s equations: the unexpected shear-induced weakening of silica-rich rocks at high slip speed (Goldsby and Tullis, 2002) and fault strength in the brittle ductile transition zone (Shimamoto, 1986). The examples, provided as illustration, may also be useful for quantitative modeling.     

Full Moment Tensor Variations and Isotropic Characteristics of Earthquakes in the Gulf of California Transform Fault System

The full moment tensor is a mathematical expression of six independent variables; however, on a routine basis, it is a common practice to reduce them to five assuming that the isotropic component is zero. This constraint is valid in most tectonic regimes where slip occurs entirely at the fault surface (e.g. subduction zones); however, we found that full moment tensors are best represented in transform fault systems. Here we present a method to analyze source complexity of earthquakes of different sizes using a simple formulation that relates the elastic constants obtained from independent studies with the angle between the slip and the fault normal vector, referred to as angle \( \theta \) ; this angle is obtained from the full moment tensors. The angle \( \theta \) , the proportion of volume change \( \left( k \right) \) and the constant volume (shear) component \( \left( T \right) \) are numerical indicators of complexity of the source; earthquakes are more complex as \( \theta \) deviates from \( \pi /2 \) or as T and k deviate from zero as well. These parameters are obtained from the eigensolution of the full moment tensor. We analyzed earthquakes in the Gulf of California that exhibit a clear isotropic component and we observed that the constant volume parameter T is independent of scalar moments, suggesting that big and small earthquakes are equally complex. In addition, simple models of one single fault are not sufficient to describe physically all the combinations of \( \theta \) in a source type plot. We also found that the principal direction of the strike of the Transform Fault System in the Gulf of California is following the first order approximation of the normal surface of the full moment tensor solution, whereas for deviatoric moment tensors the principal direction does not coincide with the strike of the Transform Fault System. Our observations that small and large earthquakes are equally complex are in agreement with recent studies of strike-slip earthquakes.     

Empirical ground-motion relations using moderate earthquakes recorded by Medellín–Aburrá Valley (Colombia) strong-motion networks

We tested attenuation relations obtained for different regions of the world to verify their suitability to predict strong-motion data recorded by Medellín and Aburrá Valley Accelerographic Networks. We used as comparison criteria, the average of the difference between the observed and the predicted data as a function of epicenter distance and its standard deviation. We also used the approach developed by Sherbaum et al. (Bull Seism Soc Am 94:2164–2185, 2004) that provides a method to evaluate the overall goodness-of-fit of ground-motion prediction equations. The predictive models selected use a generic focal depth. We found that this parameter has an important influence in the ground-motion predictions and must be taken into account as an independent variable. We also found important to characterize the local soil amplification to improve the attenuation relations. We found empirical relations for peak horizontal acceleration PGA and velocity PGV based on the Kamiyama and Yanagisawa (Soils Found 26:16–32, 1986) approach. $$\begin{aligned} \log _{10} (PGA)=0.5886M_L -1.0902\log _{10}(R)-0.0035H+C_{st}\pm 0.\text{29} \end{aligned}$$ $$\begin{aligned} \log _{10} (PGV)=0.7255M_L -1.8812\log _{10}(R)-0.0016H+C_{st}\pm 0.36 \end{aligned}$$ where PGA is measured in cm/s $^{2}$ and PGV in cm/s, $M_{L}$ is local magnitude in the range 2.8–6.5, $R$ is epicentral distance up to 290 km, $H$ is focal depth in km and $C_{st}$ is a coefficient that accounts for the site response due to soil conditions of each recording station. The introduction of focal depth and local site conditions as independent variables, minimize the residuals and the dispersion of the predicted data. We conclude that $H$ and $C_{st}$ are sensitive parameters, having a strong influence on the strong-motion predictions. Using the same functional form, we also propose an empirical relation for the root mean square acceleration a $_\mathrm{rms}$ : $$\begin{aligned} \log _{10} \left( {a_{rms} } \right)=0.4797M_L -1.1665\log _{10} (R)-0.00201H+C_{st}\pm 0.40 \end{aligned}$$ where a $_\mathrm{rms}$ is measured in cm/s $^{2}$ , from the S-wave arrival and using a window length equal to the rupture duration. The other variables are the same as those for PGA and PGV. The site correction coefficients $C_{st}$ found for PGA, PGV and a $_\mathrm{rms}$ show a similar trend indicating a good correlation with the soil conditions of the recording sites.     

On the seismic response of under-designed caisson foundations

The seismic behaviour of caisson foundations supporting typical bridge piers is analysed with 3D finite elements, with due consideration to soil and interface nonlinearities. Single-degree-of freedom oscillators of varying mass and height, simulating heavily and lightly loaded bridge piers, founded on similar caissons are studied. Four different combinations of the static ( $\text{ FS }_\mathrm{V}$ FS V ) and seismic ( $\text{ FS }_\mathrm{E}$ FS E ) factors of safety are examined: (1) a lightly loaded ( $\text{ FS }_\mathrm{V}= 5$ FS V = 5 ) seismically under-designed ( $\text{ FS }_\mathrm{E} < 1$ FS E < 1 ) caisson, (2) a lightly loaded seismically over-designed ( $\text{ FS }_\mathrm{E} >1$ FS E > 1 ) caisson, (3) a heavily loaded ( $\text{ FS }_\mathrm{V} = 2.5$ FS V = 2.5 ) seismically under-designed ( $\text{ FS }_\mathrm{E} < 1$ FS E < 1 ) caisson and (4) a heavily loaded seismically over-designed caisson. The analysis is performed with use of seismic records appropriately modified so that the effective response periods (due to soil-structure-interaction effects) of the studied systems correspond to the same spectral acceleration, thus allowing their inelastic seismic performance to be compared on a fair basis. Key performance measures of the systems are then contrasted, such as: accelerations, displacements, rotations and settlements. It is shown that the performance of the lightly loaded seismically under-designed caisson is advantageous: not only does it reduce significantly the seismic load to the superstructure, but it also produces minimal residual displacements of the foundation. For heavily loaded foundations, however ( $\text{ FS }_{V} = 2.5$ FS V = 2.5 ), the performance of the two systems (over and under designed) is similar.     

Mantle Transition Zone Structure Beneath Southeastern China and its Implications for Stagnant Slab and Water Transportation in the Mantle

We determined depth variation of the 410- and 660-km discontinuities beneath southeastern China by common-converted-point stacking of \(\rm P\) -wave receiver functions of 121 permanent Chinese seismic stations. We then combined the results with seismic velocity variation to estimate temperature and water content variations in the mantle transition zone of the region. Previous tomographic studies have shown a stagnant slab in the mantle transition zone in eastern Asia that is connected to subduction of the western Pacific. Temperature variations obtained clearly outline the shape of the stagnant slab, with its western edge at 113.5 \(^\circ\) E and the southern edge at 28.5 \(^\circ\) N. The correlation between the location of the stagnant slab and surface tectonics suggests that the Cenozoic extension in eastern China is closely associated with the subduction of the western Pacific and its eastward migration. The water content of the stagnant slab is lower than in surrounding slabs, suggesting that the water has already been released from the subducting slab into the upper mantle.     

Compatible ground-motion prediction equations for damping scaling factors and vertical-to-horizontal spectral amplitude ratios for the broader Europe region

In a companion article Akkar et al. (Bull Earthq Eng, doi:10.1007/s10518-013-9461-4, 2013a; Bull Earthq Eng, doi:10.1007/s10518-013-9508-6, 2013b) present a new ground-motion prediction equation (GMPE) for estimating 5 %-damped horizontal pseudo-acceleration spectral (PSA) ordinates for shallow active crustal regions in Europe and the Middle East. This study provides a supplementary viscous damping model to modify 5 %-damped horizontal spectral ordinates of Akkar et al. (Bull Earthq Eng, doi:10.1007/s10518-013-9461-4 2013a; Bull Earthq Eng, doi:10.1007/s10518-013-9508-6, 2013b) for damping ratios ranging from 1 to 50 %. The paper also presents another damping model for scaling 5 %-damped vertical spectral ordinates that can be estimated from the vertical-to-horizontal (V/H) spectral ratio GMPE that is also developed within the context of this study. For consistency in engineering applications, the horizontal and vertical damping models cover the same damping ratios as noted above. The article concludes by introducing period-dependent correlation coefficients to compute horizontal and vertical conditional mean spectra (Baker in J Struct Eng 137:322–331, 2011). The applicability range of the presented models is the same as of the horizontal GMPE proposed by Akkar et al. (Bull Earthq Eng, doi:10.1007/s10518-013-9461-4 2013a; Bull Earthq Eng, doi:10.1007/s10518-013-9508-6, 2013b): as for spectral periods $0.01 \hbox { s}\le \,\hbox {T}\le \,4\hbox { s}$ as well as PGA and PGV for V/H model; and in terms of seismological estimator parameters $4\le \hbox {M}_\mathrm{w} \le 8, \hbox { R} \le 200 \hbox { km}, 150\hbox { m/s}\le \hbox { V}_\mathrm{S30}\le $ 1,200 m/s, for reverse, normal and strike-slip faults. The source-to-site distance measures that can be used in the computations are epicentral $(\hbox {R}_\mathrm{epi})$ , hypocentral $(\hbox {R}_\mathrm{hyp})$ and Joyner–Boore $(\hbox {R}_\mathrm{JB})$ distances. The implementation of the proposed GMPEs will facilitate site-specific adjustments of the spectral amplitudes predicted from probabilistic seismic hazard assessment in Europe and the Middle East region. They can also help expressing the site-specific design ground motion in several formats. The consistency of the proposed models together with the Akkar et al. (Bull Earthq Eng, doi:10.1007/s10518-013-9461-4 2013a; Bull Earthq Eng, doi:10.1007/s10518-013-9508-6, 2013b) GMPE may be advantageous for future modifications in the ground-motion definition in Eurocode 8 (CEN in Eurocode 8, Design of structures for earthquake resistance—part 1: general rules, seismic actions and rules for buildings. European Standard NF EN 1998-1, Brussels, 2004).     

Fragility curves for risk-targeted seismic design maps

Seismic design using maps based on “risk-targeting” would lead to an annual probability of attaining or exceeding a certain damage state that is uniform over an entire territory. These maps are based on convolving seismic hazard curves from a standard probabilistic analysis with the derivative of fragility curves expressing the chance for a code-designed structure to attain or exceed a certain damage state given a level of input motion, e.g. peak ground acceleration (PGA). There are few published fragility curves for structures respecting the Eurocodes (ECs, principally EC8 for seismic design) that can be used for the development of risk-targeted design maps for Europe. In this article a set of fragility curves for a regular three-storey reinforced-concrete building designed using EC2 and EC8 for medium ductility and increasing levels of design acceleration \((\hbox {a}_\mathrm{g})\) is developed. These curves show that structures designed using EC8 against PGAs up to about 1 m/s \(^{2}\) have similar fragilities to those that respect only EC2 (although this conclusion may not hold for irregular buildings, other geometries or materials). From these curves, the probability of yielding for a structure subjected to a PGA equal to \(\hbox {a}_\mathrm{g}\) varies between 0.14 ( \(\hbox {a}_\mathrm{g}=0.7\) m/s \(^{2})\) and 0.85 ( \(\hbox {a}_\mathrm{g}=3\) m/s \(^{2})\) whereas the probability of collapse for a structure subjected to a PGA equal to \(\hbox {a}_\mathrm{g}\) varies between 1.7 \(\times 10^{-7}\) ( \(\hbox {a}_\mathrm{g}=0.7\) m/s \(^{2})\) and 1.0 \(\times 10^{-5}\) ( \(\hbox {a}_\mathrm{g}=3\) m/s \(^{2})\) .     

Application of the Pi theorem to the wear rate of gouge formation in frictional sliding of rocks

A simple law of wear rate is examined for the process of gouge generation during the frictional sliding of simulated faults in rocks, by use of the Pi theorem method (dimensional analysis) and existing experimental data. The relationship between wear rate (t/d) and the applied stress can be expressed by the power-law relations $$\frac{t}{d} = C_\sigma \sigma ^{m\sigma } ,\frac{t}{d} = C_\tau \tau ^{m\tau }$$ wheret is the thickness of the gouge generated on the frictional surfaces,d is the fault displacement, σ and τ are normal stress and shear stress, respectively, andC _σ,C _τ,m _σ andm _τ are constants. These results indicate that the exponent coefficientsm _σ andm _τ and the coefficientsC _σ andC _τ depend on the material hardness of the frictional surfaces. By using the wear rates of natural faults, these power-law relationships may prove to be an acceptable palaeopiezometer of natural faults and the lithosphere.     

An analytical method for the evaluation of the in-plan irregularity of non-regularly asymmetric buildings

This paper describes a new method for the evaluation of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ of uncoupled torsional to lateral frequencies in real multi-storey buildings. The above-mentioned parameters greatly affect the lateral-to-torsional coupling of the response of asymmetric systems and thus are of paramount importance in the assessment of the in-plan irregularity of buildings. The proposed method, which is a generalization of that suggested by Calderoni et al. (Earthq Spectra 18(2):219–231, 2002), allows the calculation of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ from the structural response to arbitrary distributions of forces and torsional couples. The effectiveness of the method is validated on some regularly and non-regularly asymmetric buildings characterised by different in-plan irregularity. The analyses demonstrate that the results of the method are rigorous in the case of regularly asymmetric systems and only slightly depend upon the heightwise distribution of the forces in the case of non-regularly asymmetric systems. Finally, the values of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ resulting from the proposed method are compared to those obtained by means of the procedure suggested by Makarios and Anastassiadis in (Struct Des Tall Spec Build 7(1):33–55, 1998a; Struct Des Tall Spec Build 7(1):57–71, 1998b) .     

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form $$\dot \varepsilon = A'\left[ {sinh (\alpha \sigma )} \right]^n \exp \left[ {{{ - (E * + PV * )} \mathord{\left/ {\vphantom {{ - (E * + PV * )} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right]$$ where \(\dot \varepsilon \) This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both.     

Empirical ground-motion models for point- and extended-source crustal earthquake scenarios in Europe and the Middle East

This article presents the latest generation of ground-motion models for the prediction of elastic response (pseudo-) spectral accelerations, as well as peak ground acceleration and velocity, derived using pan-European databases. The models present a number of novelties with respect to previous generations of models (Ambraseys et al. in Earthq Eng Struct Dyn 25:371–400, 1996, Bull Earthq Eng 3:1–53, 2005; Bommer et al. in Bull Earthq Eng 1:171–203, 2003; Akkar and Bommer in Seismol Res Lett 81:195–206, 2010), namely: inclusion of a nonlinear site amplification function that is a function of $\text{ V }_\mathrm{S30}$ and reference peak ground acceleration on rock; extension of the magnitude range of applicability of the model down to $\text{ M }_\mathrm{w}$ 4; extension of the distance range of applicability out to 200 km; extension to shorter and longer periods (down to 0.01 s and up to 4 s); and consistent models for both point-source (epicentral, $\text{ R }_\mathrm{epi}$ , and hypocentral distance, $\text{ R }_\mathrm{hyp}$ ) and finite-fault (distance to the surface projection of the rupture, $\text{ R }_\mathrm{JB}$ ) distance metrics. In addition, data from more than 1.5 times as many earthquakes, compared to previous pan-European models, have been used, leading to regressions based on approximately twice as many records in total. The metadata of these records have been carefully compiled and reappraised in recent European projects. These improvements lead to more robust ground-motion prediction equations than have previously been published for shallow (focal depths less than 30 km) crustal earthquakes in Europe and the Middle East. We conclude with suggestions for the application of the equations to seismic hazard assessments in Europe and the Middle East within a logic-tree framework to capture epistemic uncertainty.     

Scaling Transition in Earthquake Sources: A Possible Link Between Seismic and Laboratory Measurements

We estimate the corner frequencies of 20 crustal seismic events from mainshock–aftershock sequences in different tectonic environments (mainshocks 5.7 < M _W < 7.6) using the well-established seismic coda ratio technique (Mayeda et al. in Geophys Res Lett 34:L11303, 2007; Mayeda and Malagnini in Geophys Res Lett, 2010), which provides optimal stability and does not require path or site corrections. For each sequence, we assumed the Brune source model and estimated all the events’ corner frequencies and associated apparent stresses following the MDAC spectral formulation of Walter and Taylor (A revised magnitude and distance amplitude correction (MDAC2) procedure for regional seismic discriminants, 2001), which allows for the possibility of non-self-similar source scaling. Within each sequence, we observe a systematic deviation from the self-similar \( M_{0} \propto \mathop f\nolimits_{\text{c}}^{ - 3} \) line, all data being rather compatible with \( M_{0} \propto \mathop f\nolimits_{\text{c}}^{ - (3 + \varepsilon )} \) , where ε > 0 (Kanamori and Rivera in Bull Seismol Soc Am 94:314–319, 2004). The deviation from a strict self-similar behavior within each earthquake sequence of our collection is indicated by a systematic increase in the estimated average static stress drop and apparent stress with increasing seismic moment (moment magnitude). Our favored physical interpretation for the increased apparent stress with earthquake size is a progressive frictional weakening for increasing seismic slip, in agreement with recent results obtained in laboratory experiments performed on state-of-the-art apparatuses at slip rates of the order of 1 m/s or larger. At smaller magnitudes (M _W < 5.5), the overall data set is characterized by a variability in apparent stress of almost three orders of magnitude, mostly from the scatter observed in strike-slip sequences. Larger events (M _W > 5.5) show much less variability: about one order of magnitude. It appears that the apparent stress (and static stress drop) does not grow indefinitely at larger magnitudes: for example, in the case of the Chi–Chi sequence (the best sampled sequence between M _W 5 and 6.5), some roughly constant stress parameters characterize earthquakes larger than M _W ~ 5.5. A representative fault slip for M _W 5.5 is a few tens of centimeters (e.g., Ide and Takeo in J Geophys Res 102:27379–27391, 1997), which corresponds to the slip amount at which effective lubrication is observed, according to recent laboratory friction experiments performed at seismic slip velocities (V ~ 1 m/s) and normal stresses representative of crustal depths (Di Toro et al. in Nature in press, 2011, and references therein). If the observed deviation from self-similar scaling is explained in terms of an asymptotic increase in apparent stress (Malagnini et al. in Pure Appl Geophys, 2014, this volume), which is directly related to dynamic stress drop on the fault, one interpretation is that for a seismic slip of a few tens of centimeters (M _W ~ 5.5) or larger, a fully lubricated frictional state may be asymptotically approached.     

Near real-time automatic moment magnitude estimation

In this paper we describe a stable automatic method to estimate in real time the seismic moment, moment magnitude and corner frequency of events recorded by a network comprising broad-band and accelerometer sensors. The procedure produces reliable results even for small-magnitude events $\hbox {M}_{\mathrm{W}}\approx 3$ . The real-time data arise from both the Transfrontier network at the Alps-Dinarides junction and from the Italian National Accelerometric Network (RAN). The data is pre-processed and the S-wave train identified through the application of an automatic method, which estimates the arrival times based on the hypocenter location, recording site and regional velocity model. The transverse component of motion is used to minimize conversion effects. The source spectrum is obtained by correcting the signals for geometrical spreading and intrinsic attenuation. Source spectra for both velocity and displacement are computed and, following Andrews (1986), the seismic moment and the first estimate of the corner frequency, $f_{0}$ , derived. The procedure is validated using the recordings of some recent moderate earthquakes (Carnia 2002; Bovec 2004; Parma 2008; Aquila 2009; Macerata 2009; Emilia 2012) and the recordings of some minor events in the SE Alps area for which independent seismic moment and moment magnitude estimates are available. The results obtained with a dataset of 843 events recorded by the Transfrontier and RAN networks show that the procedure is reliable and robust for events with $\hbox {M}_{\mathrm{W}}\ge 3$ . The estimates of $f_{0}$ are less reliable. The results show a scatter, principally for small events with $\hbox {M}_{\mathrm{W}}\le 3$ , probably due to site effects and inaccurate locations.     

Quasi-Static Strain and Stress Fields due to a Moment Tensor in Elastic–Viscoelastic Layered Half-Space

We derived explicit expressions in the time domain for 3-D quasi-static strain and stress fields, due to a point moment tensor source in an elastic surface layer overlying viscoelastic half-space under gravity. The expressions of strain in the elastic surface layer were directly obtained from the expressions of displacement in our previous paper. The conversion of strain into stress is easy, because the stress–strain relation of elastic material is linear. In the viscoelastic substratum, the expressions of strain were obtained by applying the correspondence principle of linear viscoelasticity to the associated elastic solution. The strain–stress conversion is not straightforward, as the stress–strain relation of viscoelastic material is usually given in a differential form. To convert strain into stress, we used an integral form of the stress–strain relation instead of the usual differential form. The expressions give the responses of elastic half-space at \( t = 0 \) , and the responses of an elastic plate floating on non-viscous liquid at \( t = \infty \) . The moment tensor is rationally decomposed into the three independent force systems, corresponding to isotropic expansion, shear faulting and crack opening, and so the expressions include the strain and stress fields for these force systems as special cases. As the first numerical example, we computed the temporal changes in strain and stress fields after the sudden opening of an infinitely long vertical crack cutting the elastic surface layer. Here, we observe that the stress changes caused by the sudden crack opening gradually decay with time and vanish at \( t = \infty \) everywhere. After the completion of stress relaxation, a characteristic pattern of shear strain remains in the viscoelastic substratum. Since the strain and stress fields at \( t = \infty \) can be read as the strain- and stress-rate fields caused by steady crack opening, respectively, this numerical example demonstrates the realization of a steady stress state supported by steady viscous flow in the asthenosphere, associated with steady seafloor spreading at mid-ocean ridges. For the second numerical example, we computed the temporal changes in strain and stress fields after the 2011 Tohoku-oki mega-thrust earthquake, which occurred at the North American-Pacific plate interface. In this numerical example, the stress changes caused by coseismic fault slip vanish at \( t = \infty \) in the viscoelastic substratum, but remain in the elastic surface layer. The coseismic stress changes (and also strain changes) in the elastic surface layer diffuse away from the source region with time, due to gradual stress relaxation in the viscoelastic substratum.     

Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff

Earth and environmental variables are commonly taken to have multivariate Gaussian or heavy-tailed distributions in space and/or time. This is based on the observation that univariate frequency distributions of corresponding samples appear to be Gaussian or heavy-tailed. Of particular interest to us is the well-documented but heretofore little noticed and unexplained phenomenon that whereas the frequency distribution of log permeability data often seems to be Gaussian, that of corresponding increments tends to exhibit heavy tails. The tails decay as powers of ? $ \alpha $ where 1 <  $ \alpha $  < 2 is either constant or grows monotonically toward an asymptote with increasing separation distance or lag. We illustrate the latter phenomenon on 1-m scale log air permeabilities from pneumatic tests in 6 vertical and inclined boreholes completed in unsaturated fractured tuff near Superior, Arizona. We then show theoretically and demonstrate numerically, on synthetically generated signals, that whereas the case of constant $ \alpha $ is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy noise, a random field (or process) subordinated to truncated fractional Gaussian noise, the case of variable $ \alpha $ is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy motion (tfLm), a random field subordinated to truncated fractional Brownian motion. Whereas the first type of signal is relatively regular and characterized by Lévy index $ \alpha $ , the second is highly irregular (punctuated by spurious spikes) and characterized by the asymptote of $ \alpha $ values associated with its increments. We describe a procedure to estimate the parameters of univariate distributions characterizing such signals and apply it to our log air permeability data. The latter are found to be consistent with a collection of samples from tfLm with $ \alpha $ slightly smaller than 2, which is easily confused with a Gaussian field (characterized by constant $ \alpha $  = 2). The irregular (spiky) nature of this signal is typical of observed fractured rock properties. We propose that distributions of earth and environmental variable be inferred jointly from measured values and their increments in a way that insures consistency between these two sets of data.     

Characteristics of rain electricity in Nigeria II — Experimental and theoretical relations

Analysis of data, covering four rainy seasons, of rain current, point-discharge current and potential gradient reveal novel relations in the form (i) $$Q_{r + } /Q_{r - } = k_1 (T_{r + } /T_{r - } )^{1.1} $$ for rain charge and duration ratios; and (ii) $$Q_{p - } /Q_{p + } = k_2 (T_{p - } /T_{p + } )^{1.1} $$ for point charge and duration ratios, where thek's are constants; and (iii) $$i_r = - \alpha (i_p - c)$$ for rain and point-discharge current densities, where α has the same value for all types of rain andc is a constant controlled by the rainfall intensityR. For rain not associated with point discharge the relation takes the familiar form $$i_r = - AR(E - \bar E)$$ Theoretical values are obtained for \ga andA on the basis of the Wilson ion-capture theory as worked out in detail by Whipple and Chalmers.     

Flow velocity profiles in the Lower Scheldt estuary

Recent acoustic Doppler current profiler (ADCP)-measurements in the Scheldt estuary near Antwerp, Belgium, revealed anomalous, i.e. anti-clockwise circulations in a left bend during the major part of the flood period; these circulations were established shortly after the turn of the tide. During ebb, anti-clockwise circulations persisted, as predicted by classical theory. These data were analysed with a 3D and a 1DV-model. The 3D simulations reveal that the anomalous circulations are found when salinity is included in the computations—without salinity “normal” circulations were found. From analytical and 1DV simulations, it is concluded that a longitudinal salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ may induce a near-bed maximum in flow velocity reversing the direction of the secondary currents. The 1DV-model was then used to assess the contribution of various processes one by one. It was found that because of a reduction in vertical mixing, the vertical velocity profile is not at equilibrium during the first phase of accelerating tide, further enhancing the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ . A small vertical salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ appeared to have a very large effect as the crosscurrents of the secondary circulations induced by ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ became an order of larger magnitude. However, at the site under consideration, the effects of transverse salinity gradients, generated by differential advection in the river bend, were dominant: adverse directions of the secondary circulations were found even when the vertical velocity profile became more regular with a more or less logarithmic shape, i.e. when the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ and ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ did not play a dominant role anymore. It is argued that data on the secondary velocity structure, which can be measured easily owing to today’s developments in ADCP equipment, may serve as an indicator for the accuracy at which the salinity field is computed with 3D numerical models. Moreover, the large effect of the salinity structure on the velocity field must have a large impact on the morphological development of estuaries, which should therefore be accounted for in morphological modelling studies.     

A Crustal Model for the Eastern Alps Region and a New Moho Map in Southeastern Europe

In the last two decades, south-central Europe and the Eastern Alps have been widely explored by many seismic refraction experiments (e.g., CELEBRATION 2000, ALP 2002, SUDETES 2003). Although quite detailed images are available along linear profiles, a comprehensive, three-dimensional crustal model of the region is still missing. This limitation makes this region a weak spot in continental-wide comprehensive representations of crustal structure. To improve on this situation, we select and collect 37 published active-source seismic lines in this region. After geo-referencing each line, we sample them along vertical profiles—every 50?km or less along the line—and derive P-wave velocities in a stack of homogeneous layers (separated by discontinuities: depth of crystalline basement, top of lower crust, and Moho). We finally merge the information using geostatistical methods, and infer S-wave velocity and density using empirical scaling relations. We present here the resulting crustal model for a region encompassing the Eastern Alps, Dinarides, Pannonian basin, Western Carpathians and Bohemian Massif, covering the region within $45^{\circ}\text{--}51^{\circ}\hbox{N}$ and $11^{\circ} \text{--} 22^{\circ}\hbox{E}$ with a resolution of $0.2^{\circ} \times 0.2^{\circ}.$ We are also able to extend and update the map of Moho depth in a wider region within $35^{\circ}\text{--}51^{\circ}\hbox{N}$ and $12^{\circ}\text{--}45^{\circ}\hbox{E},$ gathering Moho values from the collected seismic lines, other published dataset and using the European plate reference EPcrust as a background. All the digitized profiles and the resulting model are available online.