Solution of stochastic groundwater flow by infinite series,and convergence of the one-dimensional expansion

This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series solutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random field. The standard deviation of lnK is denoted . The solution method is based on a Taylor series expansion in terms of parameter , around the value =0, of the hydraulic head (H) and gradient (J). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gradient J is given by a linear first-order recursion involving a stochastic integral operator. The convergence of the -expansion solution is not guaranteed a priori. In one dimension, however, we prove convergence by solving explicitly the hierarchical sequence of equations to all orders. An infinite-order stochastic solution is obtained in the form of a -power series that converges for any finite value of . It is pointed out that other expansion methods based on K rather than lnK yield divergent series. The infinite-order solution depends on the integration method and the boundary conditions imposed on individual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningful far-field gradient conditions. The whole approach can serve as a basis for treatment of higher-dimensional problems.     

Variation of certain parameters of regional stress tensor under condition of rockburst hazard

The feasibility of gaining valuable geomechanical information derived from seismological data and its specialist interpretation for utilisation in the area of assessing hazards due to mining tremors and rockbursts has become a development of signal importance in mining seismology. Undoubtedly of particular interest is a certain knowledge of the directions of the principal stresses ₁, ₂, ₃ of the regional stress tensor. For their determination, use is made of a set of parameters from the mining tremors' regional focal mechanism solutions (angular parameters of nodal planes and axes of principal stresses in the tremor focus —P andT). Results of research conducted at the Szombierki and Wujek mines and analysis of calculated results for parameters of regional stress tensor show that there exist appreciable differences between values of these parameters and also a clear correlation with local extraction conditions that is of significance from the point of view of seismic hazard.     

Effects of rupture complexity and stress regime on scaling relations of induced microseismic events

Source parameter scaling relations are examined for microseismic events (–2.4M–0.3) occurring within higyly and moderately stressed and fractured rock masses at Strathcona mine, Sudbury, Canada. Insight into scaling is provided by waveform complexities, calculated rupture velocities, and maximum shear stresses based onin situ and numerical modelling data. The importance of normal stress on the failure process is also considered. Our results show that a strong dependence exists between stress release and seismic moment. An observed positive scaling in excess stress release (/2– _a ) is consistent with the concept of overshoot. Rupture velocities ranging from 0.2 to 0.5 and waveform complexities less than 1.5 suggested that overshoot was related to healing behind a slowly advancing rupture front. Scaling in seismic efficiency paralleled that in apparent stress, implying that seismic stress release estimates are quasi-independent of the maximum shear stress. High levels of normal stress further supported the importance of high resisting stress in the observed overshoot behaviour and its role in the failure process.     

On the Upper Bound of the Liquid Core Viscosity

By using Molodensky and Sasao (1995a) and Molodensky and Groten (1998) an approach of expansion in powers of the small parameter =(+)/ (where + and are the frequencies of nutational motion in space and in a mantle-fixed reference frame, respectively, is tidal frequency) a theory of diurnal Earth tides and nutation for a realistic model of the Earth with an inhomogeneous, viscous liquid core and an anelastic mantle is constructed. It is shown that our approach is self-consistent for semi-annual, annual, and principal nutational components (when ||1/180). By comparing the results of modern VLBI-nutational data and the results of our calculations, we have found the region of possible values of the parameters which describe the anelastic properties of the Earth's lower mantle and the viscosity of the liquid core. It is shown that modern VLBI-data are about six orders of magnitude more sensitive to the liquid core viscosity than modern seismic data or Earth free oscillations data.     

A more precise mechanical theory of askania gravity meters

Summary The measuring systems of Askania gravity meters are assumed to have 3 degrees of freedom. The axis of rotation of the beam can also be displaced in the vertical and horizontal directions. The equations of equilibrium of this system (25)–(27) were used to derive the expressions for the effect of displacing the calibration ball, for the scale equation and for the sensitivity of the gravity meter. Equations (33), (34) and (39) refer to gravity meters with a photoelectric transducer, and Eqs (46), (47) and (49) to gravity meters with a capacitive transducer.

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uum umaum au n¶rt;naam 3 mnu ¶rt;. au aa m m nam mua uuma anau. au au m um (25)–(27) ¶rt; au ¶rt; ¶rt;mu nu au aua, ¶rt; aua a u ¶rt; mummuauma. aum mmuu mm m au (33), (34), (39), ¶rt;aum m mm au (46), (47), (49).

     

A remark on the hydrostatic approximation

Summary For an autobarotropic atmosphere the exact linearized equations of motion can be solved for the pressure oscillation when =0 and =2 (=frequency of oscillation, =angular velocity of the earth's rotation). The pressure computed in this way is shown to be in close agreement to the pressure computed by using the hydrostatic approximation.     

Anisotropic magnetic susceptibility of rocks under mechanical stresses

Summary The magnetic susceptibility of a rock under a uniaxial compression () decreases along the axis of compression and increases along the direction perpendicular to the axis, with an increase of . Thus, the magnetic susceptibility of a compressed rock becomes anisotropic.The decrease of longitudinal susceptibility,K (), and the increase of transverse susceptibility,K (), are theoretically derived from a model of rock which assumes the uniaxial anisotropy and the isotropic magnetostriction of magnetic minerals in rocks and a random orientation of the minerals. Results show thatK () decreases toward zero whereasK () increases and approaches a finite asymptotic value with an increase of , and –(/)K () is twice as large as /K () for small values of . These results are in good agreement with experimental data.

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Zusammenfassung Die magnetische Suszeptibilität eines Steines unter zunehmender uniachsigen Druckspannung () nimmt ab längs der Achse der Druckspannung und nimmt zu längs der Richtung senkrecht der Achse. Somit wird die magnetische Suszeptibilität des gedrückten Steines anisotrop.Die Abnahme der longitudinalen Suszeptibilität,K (), und die Zunahme der transversalen Suszeptibilität,K (), werden theoretisch von einem Modell eines Steines hergeleitet, das die uniachsige Anisotropie, die isotrope Magnetostriktion, und eine nichtbevorzugte Orientierung der magnetischen Minerals im Stein annimmt. Die Ergebnisse zeigen, dass mit einer Zunahme des ,K () gegen Null abnimmt, währendK () zunimmt und sich einem begrenzten asymtotitschen Wert nähert und, dass für kleine Werte des , –(/)K () zweimal so gross wie /K () ist. Diese Ergebnisse stimmen gut mit den Versuchangaben überein.

     

A note on the stress-dilatancy relation for simulated fault gouge

Theoretical constraints on the stress-dilation relation for a deforming Coulomb material requirev ifC=0 andv sin^-1( _m / _m ) always, wherev is the dilation angle, is the friction angle,C is cohesion, _m is the maximum shear stress, and _m is the mean effective stress. Recent laboratory measurements of friction and dilatancy of simulated fault gouge show that small amplitude shear-load cycling causes compaction and consolidation. Comparison of the data with theory indicates that such load cycling produces: (1) increased coefficient of friction (or friction angle), (2) increased cohesion, and (3) increased dilatancy rate (or dilation angle). Under certain conditions of load cycling without significant plastic shear strain accumulation ( ^p <0.005) we find thatv exceeds both and, in contrast to theory, sin^-1( _m / _m ). This result is interpreted in terms of enhanced cohesion and overconsolidation, which lead to residual stresses within the gouge. An analogy is drawn between these special loading conditions and those extant on natural faults. In particular, our results imply that jostling and minor stress variations associated with microearthquakes may produce strengthening of fault gouge and changes in the fault zone's stress-dilatancy relation. Hence, compaction associated with microseismicity may lead to subsequent dilation of fault gouge, even for faults with large displacement rates and large net offsets (e.g., San Andreas). In regions where such dilation persists over sufficient displacements (on the order of the critical slip distance for seismic faulting) it may tend to inhibit unstable slip.     

Analysis of State of Stress During The 1997 Earthquake Swarm In Western Bohemia

Focal mechanisms of 70 events of the January 1997 earthquake swarm were analysed using the Gephart and Forsyth method to determine the state of stress in the West Bohemia/Vogtland region and to reveal possible stress variations during the swarm activity. The method was applied to the whole set of 70 focal mechanisms and to several subsets distinguishing magnitudes of events and the spatial and temporal distribution of the swarm. The three representative stress models A ( ₁ ,A: 0/45 (az/dip), ₂ A: 134/35, ₃ A: 243/25), B ( ₁ ,B: 162/38, ₂ B: 263/14, ₃ B: 10/49), and C ( ₁ C: 135/30, ₂ C: 1/50, ₃ C: 239/23) which could characterise the state of stress in the region were found. Model C can be considered to be the most probable stress model because of its consistency with the European stress field. The results of the stress analysis applied to the individual subsets of focal mechanisms indicate that the state of stress was more uniform during the first phase of the swarm. To distinguish between the fault plane and the auxiliary plane of fault plane solutions a statistical approach was used. The results showed that the fault planes striking NW-SE and dipping 45° to NE were activated during the swarm.     

Magnetic anisotropy of polycrystalline haematite induced by A D. C. magnetic field

Summary The changes in the anisotropy of initial susceptibility of haematite ores, induced by a D.C. magnetic field are studied. After applying stronger magnetic fields, the changes in the anisotropy parameters are considerable and depend on the manner in which the basal planes of the haematite grains are arranged in the samples. The experimental results are interpreted on the basis of the single-domain theory. The proposed simple model of induced anisotropy is based on the relation between the directional susceptibility of the sample and the preferred orientation of the magnetic moments of the grains in polycrystalline haematite.

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am uu aumnuu aa nuuumuamum ¶rt;, a nm aum n. nuu u aum n uu aumnuu aam u aum m na n¶rt;u nmamum aa. numa. ¶rt;a umnmua unau ¶rt;¶rt; muu. ¶rt;a nma ¶rt; a aumnuu aa a auumu u nuuumu m umauu aum m nuumauamum.

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Presented at the 5th Scientific Assembly of IAGA in Prague 1985.     

Approximative ray tracing and the accuracy of earthquake localization in media approximated by block structures

Summary Localization errors made by approximating a given medium by a homogeneous rectangular block model and due to the approximative solution of two-point ray tracing are estimated for four different structures. The separate influence of the individual characteristics of the medium is discussed. The iterative Geiger method supplemented by a generalized inversion of the damped least-squares and Marquardt algorithm is used for localizating five hypothetical hypocentres using 9 stations. Travel times in the original structures, used as input data for localization, are calculated using a special ray tracing system.

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am uu auauu a annuau a ¶rt; ¶rt; ma u ma ¶rt;¶rt; u annuau m a ¶rt; munau nuuum .

     

Stresses and displacements due to a stationary point source of heat in an elastic halfspace

am amu ¶rt; ¶rt; mnu annu u u, a maua m umu mna n nnmam. a auu uu naa, m uau anu m anuu amua muna u am am u n ummamuu anu uu nmu nnmama. auum m¶rt;uauu ¶rt;mu ¶rt;mu mau anu, ammu: a) ¶rt;mu um numa amu umua mna; ) m uu uu nm aa u u u nu uuu aa nuu a nmu u; ) aau amuau anu ammuu aau, m m nm u¶rt; n u amumu um mu am.     

Single- and inter-station transfer functions for non-synchronous geomagnetic arrays — I. Theory of methods used in data analysis

Summary The theory of the multivariate coherence analysis(spectral domain approach) is developed for calculating single- and inter-station transfer functions and corresponding vector induction characteristics from time variations of the geomagnetic field components. An alternative approach of calculating similar induction characteristics using a time domain algorithm is shown.

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aam mu m aaua(nma n¶rt;¶rt;) nuuu ama ¶rt;-u -mau n¶rt;am u u mmmu m aamumu u¶rt;uu n anua auau mau aum n u. u¶rt;um ma m¶rt; u aau aamumu u¶rt;uu nm ama amu un mua um. nuam aum na auu am u aamum ¶rt; n ama naam.

     

Reaction of linear gravimetric measuring systems to disturbances of equilibrium state

Summary The reaction is studied of measuring systems of quartz gravity meters to external disturbances. The actual elastic system is approximated by a model of a linear oscillator with one degree of freedom. The appropriate differential equation of motion is solved for some of the typical shapes of determinate and random disturbance input signals. The result represents expressions which describe the time behaviour of the output signals.

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¶rt;m au uum um aaum m u u. mua na uma annuum ¶rt; u uma ¶rt; mn ¶rt;. mmm au ¶rt;uu a ¶rt; m munuu u¶rt; n¶rt; a au ua a ¶rt;. mam m au, nuau n¶rt;u ua a ¶rt; auumu m u.

     

Transformations of spherical harmonics and applications to geodesy and satellite theory

am ¶rt; mauau uu u u aamuam u a. m num u m a¶rt;a muu nmuaa u ¶rt;uau nmu¶rt;uu.

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Invited Paper, 9^th Brazilian Congress in Cartography, Curitiba, Paraná, Brazil, February 4–9, 1979.     

Energy regimes in the Earth's magnetosphere

Summary The dynamics of the main processes of energy accumulation and dissipation in the Earth's magnetosphere at various geomagnetic disturbance levels is examined. The results of the relevant calculations are tabulated. The relationships between the energy parameters of the solar wind and the Earth's magnetosphere are analyzed. Some conclusions concerning the field-aligned currents in polar caps, the Joule dissipation of energy and the energy injection into the ring current, the energy releases in the upper ionosphere, etc., are drawn.

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ama ¶rt;uaua n anu u ¶rt;uunauu uu aum u nu au aum mu. mam uu n¶rt;ma maua. auum au ¶rt; muuu aamumuau ma u aum u. u¶rt;m m au n¶rt; ma n an, uu ¶rt; ¶rt;uunauu, uu uuu m, uu, ¶rt; u, u m. ¶rt;.

     

Effects of a uniaxial compression on remanent magnetizations of igneous rocks

Summary Characteristics of remanent magnetization of an igneous rock under a uniaxial compression are divided into those of the soft component and those of the hard component. The soft component of remanent magnetization irreversibly decreases with an increase of compression () regardless whether the axis of is parallel or parpendicular to the direction of magnetization. The hard component changes reversibly with : the magnetic intensity of hard component increases with when the axis of is perpendicular to the magnetization, whereas it decreases with when the axis of is parallel to the magnetization.Theoretically, the irreversible change of the soft component is attributable to the irreversible movement of 90° domain walls which results in the pressure demagnetization effect, while the reversible change of the hard component is due to the reversible rotation of spontaneous magnetization within those domains which are so tightly fixed by large effective anisotropy energy that the induced magnetoelastic energy cannot drive their 90° walls.In natural remanent magnetization of igneous rocks, the soft component corresponds to either one or all of isothermal remanent magnetization, viscous remanent magnetization and piezo-remanent magnetization, whereas the hard component is mostly due to thermo-remanent magnetization.     

On the structure of the interplanetary magnetic field under non-stationary boundary conditions

Summary The structure of the interplanetary magnetic field (IMF) is investigated theoretically in a kinematic approximation under frozen-in conditions in the solar wind and non-stationary boundary conditions. As an example, a time-dependent model of the IMF, created in the case of a change of the general magnetic field of the Sun represented by the dipole term, is analyzed. Very simple assumptions as to the field of velocities in the solar wind are made. The results show the formation of zero IMF points of two types(O, X). Points of theO-type are formed and move radially in the equatorial plane. They are surrounded by magnetic clouds with loops of lines of force. Points of theX-type are formed and move radially above the poles.

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muu u¶rt;aa mma nam aum n() uamu nuuuu nu uu mu m u mauaau uu. am nua ama ¶rt; , ua nu uuu aum n a, n¶rt;ma ¶rt;un , auu m u. mum n m m ¶rt;a nmu n¶rt;nu. mam naam aau m ¶rt; mun(O, X). uO-muna uam u ¶rt;um a¶rt;ua amua nmu. u aumu aau nmu u uu. uX-muna uam u ¶rt;um a¶rt;ua a¶rt; nau.

     

Spectral properties of clouds

Summary In the shortwave region the variations of the spectral properties of clouds are due only to the variations of optical characteristics of water. It results in small variations in the scattering pattern. In the longwave region scattering and absorption are of essentially diffraction character. Results of computation for a typical cloud drop are given; the accuracy of «the layer of precipitated water» method is analysed; thicknesses of cloud layers are investigated when the flux incoming into a cloud gradually deforms into the blackbody radiation flux.

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. , , .

     

On the diffusion of the perturbing toroidal magnetic field from the core to the mantle

Zusammenfassung Es werden die Wurzeln einer charakteristischen Gleichung für die Randbedingung der Diffusion des toroidalen magnetischen Störungsfeldes berechnet, das in dünner Schicht unterhalb der Grenze Kern — Mantel, als eine Folge von Störung der Winkelgeschwindigkeit dieser Schicht in Anwesenheit des magnetischen Dipolfeldes induziert wird. Die Wurzeln werden für verschiedene Werte des Verhältnisses der Leitfähigkeit des Erdmantels und Erdkerns ausgerechnet. Es wird gezeigt, dass das aus dem Milieu mit der elektrischen Leitfähigkeit in das Milieu mit elektrischer Leitfähigkeit _m diffundierende toroidale magnetische Störungsfeld vom Werte _m/<10^–2 an von dem Verhältnis elektrischer Leitfähigkeiten beider Milieus unabhänging bleibt.