On the formation of a plasma pressure anisotropy in the dayside magnetosheath

We present a numerical solution for the momentum equation of the magnetosheath particles that describes the distribution of the pressure anisotropy of the magnetosheath plasma in the midday meridian plane. The pressure anisotropy is a maximum near the magnetopause subsolar point (p_⊥/p_\Vert\cong10). The pressure anisotropy is caused by two factors: particles with small pitch angles (V_\Vert>V_⊥) which travel along the magnetic field lines away from the equatorial plane of the magnetosheath; and particles, after crossing the bowshock, which reach the bulk velocity component directed along the magnetic field lines again, away from the magnetosheath equatorial plane. This velocity increases with increasing distance from the subsolar point of the bowshock, and does not permit particles with large pitch angles (V_⊥>V_\Vert) to move toward the equatorial plane.     

Plasma gradient effects on double-probe measurements in the magnetosphere

The effects on double-probe electric field measurements induced by electron density and temperature gradients are investigated. We show that on some occasions such gradients may lead to marked spurious electric fields if the probes are assumed to lie at the same probe potential with respect to the plasma. The use of a proper bias current will decrease the magnitude of such an error. When the probes are near the plasma potential, the magnitude of these error signals, E, can vary as E\simT_e(n_e/n_e)+0.5T_e, where T_e is the electron temperature, n_e/n_e the relative electron density variation between the two sensors, and T_e the electron temperature difference between the two sensors. This not only implies that the error signals will increase linearly with the density variations but also that such signatures grow with T_e, i.e., such effects are 10 times larger in a 10-eV plasma than in a 1-eV plasma. This type of error is independent of the probe separation distance provided the gradient scale length is much larger than this distance. The largest errors occur when the probes are near to the plasma potential. At larger positive probe potentials with respect to the plasma potential, the error becomes smaller if the probes are biased, as is usually the case with spherical double-probe experiments in the tenuous magnetospheric plasmas. The crossing of a plasma boundary (like the plasmapause or magnetopause) yields an error signal of a single peak. During the crossing of a small structure (e.g., a double layer) the error signal appears as a bipolar signature. Our analysis shows that errors in double-probe measurements caused by plasma gradients are not significant at large scale (≫1 km) plasma boundaries, and may only be important in cases where small-scale (<1 km), internal gradient structures exist. Bias currents tailored for each plasma parameter regime (i.e., variable bias current) would o1q1improve the double-probe response to gradient effects considerably.     

Systematic retrieval of ejecta velocities and gas fluxes at Etna volcano using L-Band Doppler radar

Strombolian-type volcanic activity is characterized by a series of gas bubbles bursting at the top of a magma column and leading to the ejection of lava clots and gas emission at the surface. The quantitative analysis of physical parameters (e.g., velocity, size, and mass fluxes) controlling the emission dynamics of these volcanic products is very important for the understanding of eruption source mechanisms but remains difficult to obtain in a systematic fashion. Ground-based Doppler radar is found to be a very effective tool for measuring ejecta velocities at a high acquisition rate and close to the emission source. We present here a series of measurements carried out at Mt. Etna’s Southeast crater, using an L-band volcanological Doppler radar, during the 4 July 2001 Strombolian eruptions. Doppler radar data are supplemented by the analysis of video snapshots recorded simultaneously. We provide here a set of physical parameters systematically retrieved from 247 Strombolian explosions spanning 15 min and occurring during the paroxysm of the eruption from 21:30 to 21:45 UT. The time-average values give a maximum particle velocity of V_max^p = 94.7±24 \textm/s V_{{\max }}^p = {94}.{7}\pm {24} {\text{m/s}} , a bulk lava jet velocity of V_{\textPW - rad} = 37.6±1.9 \textm/s {V_{{{\text{PW - rad}}}}} = {37}.{6}\pm {1}.{9} {\text{m/s}} , and an initial gas velocity at the source vent of V₀^g = 118.4±36 \textm/s V_0^g = {118}.{4}\pm {36} {\text{m/s}} . The time-averaged particle diameter is found to be about D_{\textPW - rad} = 4.2±2.1 \textcm {D_{{{\text{PW - rad}}}}} = {4}.{2}\pm {2}.{1} {\text{cm}} . The volume and mass gas fluxes are estimated from time-averaged source gas velocities over the sequence duration at Q_v^g = 3 - 11 ×10³\textm³\text/s Q_v^g = {3} - {11} \times {1}{0^{{3}}}{{\text{m}}^{{3}}}{\text{/s}} and Q_m^g = 0.5 - 2 ×10³\textkg/s Q_m^g = 0.{5} - {2} \times {1}{0^{{3}}}{\text{kg/s}} , respectively.     

Non-Gaussian probability distributions of solar wind fluctuations

The probability distributions of field differences x()=x(t+)-x(t), where the variable x(t) may denote any solar wind scalar field or vector field component at time t, have been calculated from time series of Helios data obtained in 1976 at heliocentric distances near 0.3 AU. It is found that for comparatively long time lag , ranging from a few hours to 1 day, the differences are normally distributed according to a Gaussian. For shorter time lags, of less than ten minutes, significant changes in shape are observed. The distributions are often spikier and narrower than the equivalent Gaussian distribution with the same standard deviation, and they are enhanced for large, reduced for intermediate and enhanced for very small values of x. This result is in accordance with fluid observations and numerical simulations. Hence statistical properties are dominated at small scale by large fluctuation amplitudes that are sparsely distributed, which is direct evidence for spatial intermittency of the fluctuations. This is in agreement with results from earlier analyses of the structure functions of x. The non-Gaussian features are differently developed for the various types of fluctuations. The relevance of these observations to the interpretation and understanding of the nature of solar wind magnetohydrodynamic (MHD) turbulence is pointed out, and contact is made with existing theoretical concepts of intermittency in fluid turbulence.     

A scaling law of internal run-up duration

Wave tank experiments with long internal waves of elevation, of different initial length l, moving in a two-fluid system, interacting with a weak slope of 0.045 rad, show an onshore flow of the dense water, at the undisturbed pycnocline-slope intersection, of duration $11.3\sqrt{l/g'}Wave tank experiments with long internal waves of elevation, of different initial length l, moving in a two-fluid system, interacting with a weak slope of 0.045 rad, show an onshore flow of the dense water, at the undisturbed pycnocline-slope intersection, of duration 11.3?{l/g￠}11.3\sqrt{l/g'} (g′ reduced gravity). This period corresponds to that of a strong bottom current event measured in the stratified ocean at the Ormen Lange gas field, at 850 m depth, lasting for 24 hrs, corresponding to 11.2?{l/g￠}11.2\sqrt{l/g'}, using the width l = 300 km of the Norwegian Atlantic Current (NAC) at the site as length scale, suggesting a lateral sloshing motion of the NAC causing the event. The onshore velocity of the dense fluid has a maximal velocity of 0.4?{g￠h₂}0.4\sqrt{g'h_2} in laboratory and 0.5 ms^-1=0.3?{g￠h₂}^{-1}=0.3\sqrt{g'h_2} in the field (h ₂ mixed upper layer thickness). Run-up of the dense fluid, beyond the undisturbed pycnocline-slope intersection, has initially a front velocity of 0.35?{g￠h₂}0.35\sqrt{g'h_2}, corresponding to the velocity of the head of a density current on a flat bottom. Due to disintegration, an initially depressed pycnocline results in comparatively smaller run-up and velocity. While moving past the turning point, a dispersive wave train is formed in the back part of the depression wave, developing by breaking into a sequence of up to eight boluses moving by the undisturbed pycnocline-slope intersection.     

Application of possibility-probability distribution in assessing water resource risk in Yiwu city

The possibility-probability risk calculated using the interior-outer set model is referred to as fuzzy risk. A fuzzy expected value of the possibility-probability distribution is a set with E _α(x) and [`(E)]<sub>a</sub>\bar E_\alpha (x) as its boundaries. The fuzzy expected values E <sub>α</sub>(x) and [`(E)]_a\bar E_\alpha (x) of a possibility-probability distribution represent the fuzzy risk values being calculated. Under such an α level, three risk values can be calculated: conservative risk value, venture risk value and maximum probability risk value. As α adopts all values throughout the set [0, 1], it is possible to obtain a series of risk values. Therefore, the fuzzy risk can be a multi-valued risk or set-valued risk. Calculation of the fuzzy expected value of Yiwu city’s water resource risk has been performed based on the interior-outer set model. We can get a conservative risk value (R _C ) of 800 mm for Yiwu city’s water resource risk, a venture risk value (R _V ) of 1020 mm, and a maximum probability risk value (R _M ) of 988 mm for the α = 0.1 level cut set.     

Coda wave attenuation in the Parecis Basin,Amazon Craton,Brazil: sensitivity to basement depth

Small local earthquakes from two aftershock sequences in Porto dos Gaúchos, Amazon craton—Brazil, were used to estimate the coda wave attenuation in the frequency band of 1 to 24 Hz. The time-domain coda-decay method of a single backscattering model is employed to estimate frequency dependence of the quality factor (Q _c) of coda waves modeled using Q_c = Q₀ f^hQ_{\rm c} =Q_{\rm 0} f^\eta , where Q ₀ is the coda quality factor at frequency of 1 Hz and η is the frequency parameter. We also used the independent frequency model approach (Morozov, Geophys J Int, 175:239–252, 2008), based in the temporal attenuation coefficient, χ(f) instead of Q(f), given by the equation c(f)=g+\fracpfQ_e \chi (f)\!=\!\gamma \!+\!\frac{\pi f}{Q_{\rm e} }, for the calculation of the geometrical attenuation (γ) and effective attenuation (Q_e^-1 )(Q_{\rm e}^{-1} ). Q _c values have been computed at central frequencies (and band) of 1.5 (1–2), 3.0 (2–4), 6.0 (4–8), 9.0 (6–12), 12 (8–16), and 18 (12–24) Hz for five different datasets selected according to the geotectonic environment as well as the ability to sample shallow or deeper structures, particularly the sediments of the Parecis basin and the crystalline basement of the Amazon craton. For the Parecis basin Q_c = (98±12)f^(1.14±0.08)Q_{\rm c} =(98\pm 12)f^{(1.14\pm 0.08)}, for the surrounding shield Q_c = (167±46)f^(1.03±0.04)Q_{\rm c} =(167\pm 46)f^{(1.03\pm 0.04)}, and for the whole region of Porto dos Gaúchos Q_c = (99±19)f^(1.17±0.02)Q_{\rm c} =(99\pm 19)f^{(1.17\pm 0.02)}. Using the independent frequency model, we found: for the cratonic zone, γ = 0.014 s^− 1, Q_e^-1 = 0.0001Q_{\rm e}^{-1} =0.0001, ν ≈ 1.12; for the basin zone with sediments of ~500 m, γ = 0.031 s^− 1, Q_e^-1 = 0.0003Q_{\rm e}^{-1} =0.0003, ν ≈ 1.27; and for the Parecis basin with sediments of ~1,000 m, γ = 0.047 s^− 1, Q_e^-1 = 0.0005Q_{\rm e}^{-1} =0.0005, ν ≈ 1.42. Analysis of the attenuation factor (Q _c) for different values of the geometrical spreading parameter (ν) indicated that an increase of ν generally causes an increase in Q _c, both in the basin as well as in the craton. But the differences in the attenuation between different geological environments are maintained for different models of geometrical spreading. It was shown that the energy of coda waves is attenuated more strongly in the sediments, Q_c = (78±23)f^(1.17±0.14)Q_{\rm c} =(78\pm 23)f^{(1.17\pm 0.14)} (in the deepest part of the basin), than in the basement, Q_c = (167±46)f^(1.03±0.04)Q_{\rm c} =(167\pm 46)f^{(1.03\pm 0.04)} (in the craton). Thus, the coda wave analysis can contribute to studies of geological structures in the upper crust, as the average coda quality factor is dependent on the thickness of sedimentary layer.     

Equatorial ionospheric response to isolated auroral substorms over a solar cycle (1980-85): evidence of longitudinal anomaly

The equatorial ionospheric response to 228 isolated, rapid-onset auroral substorms (as defined from the auroral electrojet index AE) was found from enhancements of the virtual (minimum) height of the F-region (h′F) in the declining phase of a solar cycle (1980-85). The responses, found for three longitudinal sectors at the equator: Africa (Ouagadougou and Dakar), Asia (Manila) and America (Huancayo), were compared with the response close to the auroral source region at Yakutsk (northern Siberia). The auroral substorm onsets were centered at 17 and 15 UT at sunspot maximum (1980-82) and minimum (1983-85), preceding by 3–5 h the period of post-sunset height rise in the African sector whereas other sectors were in the early afternoon (Huancayo) and morning (Manila). The African response, particularly at Ouagadougou, was distinctly different from other sectors. In the sunspot maximum years (1980-81) the auroral surges were followed after about 3 h by a sharp depression (h′F<0) in the post-sunset height rise, with a period of little or no response (h′F=0) in 1982. A response polarity reversal (h′F>0) was noted in this sector for sunspot minimum (1983-85) when large h′F enhancements were observed at the sunset region. The responses in the Asian and American sector were positive except for a case in Huancayo when response was negative, following an auroral surge before the sunset at this station. It appears that the aurorally generated large-scale travelling ionospheric disturbances (LSTIDs), which first cause positive height enhancements in a sub-auroral location (Yakutsk), subsequently affect the unstable post-sunset ionosphere in the equatorial Africa.     

Revision of the amplitude-distance relation for local magnitude estimation in Italy

Maximum amplitude observations of local and regional seismic phases(S _g and L _g, with 800 km) recorded by stations of the I.N.G.National Seismological Network in the period 1994–1999 have beenused to investigate the attenuation of such phases in Italy. The propagationmodel adopted herein accounts for both geometrical spreading, with a termdepending on a given power of distance, and inelastic absorption of themedium, with a term depending exponentially on the distance. Fitting theparameters of the model with experimental data, we considered two cases:assuming a quality factor Q proportional to the frequency f orindependent of f. We compared our results with the values tabled byRichter in 1935, which are still used for the local magnitude estimate in theI.N.G. bulletins. The values found in this study are different than thosegiven by Richter. This suggests that these new values should be used insteadto compute the local magnitude in Italy.     

Geometrical spreading factor of head waves in cased and open boreholes

Factors (coefficients) of geometrical spreading of compressional and shear head waves are calculated for an impulse multipole source of elastic oscillations in boreholes. It is shown that the length of the logging tool (i.e., the distance between the source and the nearest receiver) used for sonic measurements and the velocities of elastic waves in the medium both contribute to the factor of geometrical spreading. For a high-velocity formation (the shear wave velocity in the rock is higher than the compressional wave velocity in the fluid that fills the borehole) and a sufficiently long sonic tool with a monopole source, the coefficient of geometrical spreading is approximated by asymptotic formula 1/Z [Roever et al., 1974; Krauklis and Krauklis, 1976], where Z is the length of the tool; i.e., the amplitude of the compressional head wave decreases proportionally to the distance between the source and the receiver. In acoustically soft formations, this approximation is inapplicable even for long tools with length Z > 4 m. Waveforms in cased boreholes have a significant frequency dispersion even in case of good-quality cementing, and the factor of geometrical spreading there depends considerably on the length of the tool and the elastic properties of the rocks.     

Feasibility of grain-size analysis methods for determination of vertical hydraulic conductivity of streambeds

Accurate estimation of streambed vertical hydraulic conductivity (K_v) is of great importance in the analysis of water quantity exchange and solute transfer between a stream and its sediments. The paper analyzed the inaccuracy of hydraulic conductivity values of sediments derived from grain-size distribution (K_g), which were determined from eight empirical grain-size methods to represent streambed K_v. In this study, the values of K_v for a streambed were derived using falling-head standpipe permeameter tests conducted at eight study sites in the Elkhorn River, Nebraska, and the tested streambed columns were then collected for grain-size analysis by sieving. These empirical methods were used to calculate the K_g values of the streambed from grain-size distribution data of sediments. Unlike many other studies, this study verifies K_g from grain-size distribution with K_v from permeameter tests on the basis of the same samples of streambed sediments. The K_g values derived from the eight empirical methods were larger than the K_v from permeameter tests; there are five methods that give K_g values of about 3–6 times larger than these K_v. The K_g values from the Kozeny formula followed by the Hazen formula give the largest overestimation error if they are used to represent the K_v of the streambed. The USBR and Shepherd formulas generated K_g values close to K_v, but these K_g values are still larger in general than the K_v values. Moreover, the new values of coefficient C for the empirical formulas were revised so that they can be used to calculate the approximate K_v of a streambed. Among the eight methods, the ratios of the original C values to the average new C range from 1.3 to 5.9. It can be hypothesized that smaller C values must be used in the estimation of K_v for general soil samples if these empirical formulas are used to calculate K_v.     

Mesopause temperatures calculated from the O2(a 1<Delta> g ) twilight airglow emission recorded at Maynooth (53.2°N, 6.4°W)

Spectra of the O₂(a¹_g) airglow emission band at 1.27 µm have been recorded during twilight at Maynooth (53.2°N, 6.4°W) using a Fourier transform spectrometer. Synthetic spectra have been generated for comparison with the recorded data by assuming a particular temperature at the emitting altitude, and modelling the absorption of each line in the band as it propagates downward through the atmosphere. The temperature used in generating the synthetic spectra was varied until an optimum fit was obtained between the recorded and synthetic data; this temperature was then attributed to the altitude of the emitting layer. Temperatures derived using this technique for 91 twilight periods over an 18-month period exhibit a strong seasonal behaviour with a maximum in winter and minimum in summer. Results from this study are compared with temperatures calculated from the OH(3, 1) Meinel band recorded simultaneously. In winter OH temperatures exceed O₂ values by about 10 K, whereas the opposite situation pertains in summer; this result is interpreted in terms of a possible change in the altitude of the mesopause as a function of season. Estimates of the twilight O₂(0, 0) total band intensity indicate that its intensity is lower and that its decay is more rapid in summer than in winter, in agreement with earlier observations.     

Edge Detection and Depth Estimation Using a Tilt Angle Map from Gravity Gradient Data of the Kozaklı-Central Anatolia Region,Turkey

In this paper the application of an edge detection technique to gravity data is described. The technique is based on the tilt angle map (TAM) obtained from the first vertical gradient of a gravity anomaly. The zero contours of the tilt angle correspond to the boundaries of geologic discontinuities and are used to detect the linear features in gravity data. I also present that the distance between zero and ±p\mathord

Experimental investigation of conditions of lateral shear reinforcements in RC columns accompanied by buckling of longitudinal bars

This study presents a cyclic load test of four RC columns to obtain data on stresses and strains on lateral shear‐reinforcing bars that contact buckled longitudinal reinforcing bars. The specimens include columns laterally jacketed with all‐elastic fibre‐reinforced polymer (FRP) sheets. The buckling lengths and modes in the longitudinal bars of the four column specimens stabilized at a buckling deflection (= lateral deformation of buckled longitudinal bar) from 0.3 to 0.6 mm. The yield portion ratio r_by of shear reinforcement around the buckled longitudinal bars was introduced as an index of the development of buckling conditions. Here, the yield portion ratio r_by was defined as the ratio of the length of the region where the shear reinforcements yield l_by, to the buckling length l_b. The r_by values of the tested columns ranged from 0.45 to 0.76. The test results show that the buckling stress and the specific compressive stress of the longitudinal bars in the columns were smaller than those of the bare bars. Copyright © 2007 John Wiley & Sons, Ltd.     

Shear flow instabilities in the Earth’s magnetotail

Shear flow instability is studied in the Earth’s magnetotail by treating plasma as compressible. A dispersion relation is derived from the linearized MHD equations using the oscillating boundary conditions at the inner central plasma sheet/outer central plasma sheet (OCPS) interface and OCPS/plasma-sheet boundary layer (PSBL) interface, whereas the surface-mode boundary condition is used at the PSBL/lobe interface. The growth rates and the real frequencies are obtained numerically for near-Earth (\midX\mid\sim10-15 R_E) and far-Earth (\midX\mid\sim100 R_E) magnetotail parameters. The periods and wavelengths of excited modes depend sensitively on the value of plasma-sheet half thickness, L, which is taken as L=5 R_E for quiet time and L=1 R_E for disturbed time. The plasma-sheet region is found to be stable for constant plasma flows unless M_A3>1.25, where M_A3 is the Alfvén Mach number in PSBL. For near-Earth magnetotail, the excited oscillations have periods of 2–20 min (quiet time) and 0.5-4 min (disturbed time) with typical transverse wavelengths of 2–30 R_E and 0.5-6.5 R_E, respectively; whereas for distant magnetotail, the analysis predicts the oscillation periods of \sim8-80 min for quiet periods and 2–16 min for disturbed periods.     

Horizontal hydraulic conductivity of shallow streambed sediments and comparison with the grain‐size analysis results

Streambed horizontal hydraulic conductivity (K_h) has a substantial role in controlling exchanges between stream water and groundwater. We propose a new approach for determining K_h of the shallow streambed sediments. Undisturbed sediment samples were collected using tubes that were horizontally driven into streambeds. The sediment columns were analysed using a permeameter test (PT) on site. This new test approach minimizes uncertainties due to vertical flow in the vicinity of test tube and stream stage fluctuations in the computation of the K_h values. Ninety‐eight PTs using the new approach were conducted at eight sites in four tributaries of the Platte River, east‐central Nebraska, USA. The K_h values were compared with the nondirectional hydraulic conductivity values (K_g) determined from 12 empirical grain‐size analysis methods. The grain‐size analysis methods used the same sediment samples as K_h tests. Only two methods, the Terzaghi and Shepherd methods, yielded K_g values close to the K_h values. Although the Sauerbrei method produced a value relatively closer to K_h than other nine grain‐size analysis methods, the values from this method were not as reliable as the methods of Terzaghi and Shepherd due to the inconsistent fluctuation of the average estimates at each of the test sites. The Zunker, Zamarin, Hazen, Beyer, and Kozeny methods overestimated K_h, while the Slichter, US Bureau of Reclamation (USBR), Harleman, and Alyamani and Sen methods underestimated K_h. Any of these specific grain‐size methods might yield good estimates of streambed K_h at some sites, but give poor estimates at other sites, indicating that the relationship between K_g and K_h is significantly site dependent in our study. Copyright © 2011 John Wiley & Sons, Ltd.     

A full-scale study of Darcy-Weisbach friction factor for channels vegetated by riparian species

In this article, an open channel flow resistance equation, deduced applying dimensional analysis and incomplete self-similarity condition for the flow velocity distribution, was tested using measurements carried out in a full-scale channel equipped with three types of riparian plants (Salix alba L., Salix caprea L. and Alnus glutinosa L.). In the experimental channel, having banks lined with boulders, the vegetation branches were anchored in a concrete bottom. For each species, the measurements were carried out with plants having different amounts of leaves, different plant density and plant area index. The relationship between the scale factor Γ of the velocity profile and the Froude number was separately calibrated by measurements carried out without and with vegetation. The component of Darcy-Weisbach friction factor corresponding to the riparian vegetation f_v was calculated as the difference between the measured friction factor value (channel grain roughness + vegetation) and that calculated for the channel without vegetation in the same hydraulic conditions. Using these f_v values, the relationship between the scale factor Γ and the Froude number was calibrated. In this last relationship, a scaling coefficient a varying with the investigated vegetation type was introduced. This coefficient, as expected, gives the highest friction factor values for vegetation having branches with leaves. The theoretical flow resistance law, coupled with the relationship for estimating the Γ function having a scaling coefficient different for each investigated vegetation type, allowed an accurate estimate of the Darcy-Weisbach friction factor (errors less than or equal to 20% for 82.6% of the investigated cases). Finally, for the investigated vegetation species that are characterized by a condition with few leaves or leafless, the scaling coefficient a resulted strongly related to the bending stiffness. This analysis demonstrated that the highest Darcy-Weisbach friction factors correspond to vegetation species characterized by the highest values of bending stiffness. The friction factor values calculated for this last condition are characterized by errors that were less than or equal to ±20% for 90.6% of cases.     

Finite element analysis of the seismic response of ancient columns

This paper presents a numerical study of the response to earthquake actions of eight ancient Greek multidrum columns, which were chosen as representative of a broad spectrum of the ancient columns preserved to date. The study was conducted using the FE software Abaqus, in three steps. At first, the software efficiency to predict the rocking response of dry-stone structures was verified. Afterwards, various numerical models of one typical ancient multidrum column were seismically excited. Records of four real earthquakes differing in frequency content were used for the excitation of the models. Each column model was different from the others at least in one geometric or modeling parameter. Although the examined parameters affected the numerical results, their variation did not alter significantly the overall behavior of the column. In the last step of the study, numerical models of eight columns were seismically excited using four seismic records. In these analyses, among the simulation approaches tested in the second step, the simpler one was adopted. On the basis of the numerical results, conclusions were drawn regarding the seismic resistance of the columns and its correlation with the dynamic characteristics of the columns (size, slenderness, and frequency parameter) and with the basic intensity measures and frequency content indicator of the seismic motions (PHA, PHV, PHD, and T_g). Thus, criteria of seismic collapse for the multidrum columns are proposed, which can be used for the approximate assessment of the seismic vulnerability of free-standing ancient columns, provided that the columns are in good preservation state.     

A probabilistic model of seismicity: Kamchatka earthquakes

The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω, $ \tilde F $ \tilde F P}. Each earthquake is treated as an outcome ω _i in the space of elementary events Ω whose cardinality for the period under consideration is given by the number of events. In turn, ω _i is characterized by a system of random variables, viz., energy class k_i, latitude φ _i , longitude λ _i , and depth h _i . The time of an outcome has been eliminated from this system in this study. The random variables make up subsets in the set $ \tilde F $ \tilde F and are defined by multivariate distributions, either by the distribution function $ \tilde F $ \tilde F (φ, λ, h, k) or by the probability density f(φ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequency interpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law for probability density f(k), where the initial value of the distribution function f(k ₀) is the basic data [Bogdanov, 2006] rather than the seismic activity A ₀, we proceed to show that for different intervals of coordinates and time the distribution f _elim(k) of an earthquake catalog with the aftershocks eliminated is identical to the distribution f _full(k), which corresponds to the full catalog. It follows from our calculations that f ₀(k) takes on nearly identical numeral values for different initial values of energy class k ₀ (8 ≤ k ₀ ≤ 12) f(k ₀). The difference decreases with an increasing number of events. We put forward the hypothesis that the values of f(k ₀) tend to cluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesis that statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a distribution function with the initial value f(k ₀) = 2/3. We discuss statistical distributions of earthquake hypocenters over depth and the epicenters over various areas for several periods