Approximate expression for the hilbert transform of a certain class of functions and their application to the ray theory of seismic waves

Summary Approximate expressions for the Hilbert transform of the functionf(t)=exp(- ₀ ² t ²/²) cos( ₀ t+v) are determined. This function, given a suitable choice of the three parameters ₀, and v, approximates a wide class of seismic signals very well. The approximate expressions for the Hilbert transform enable very simple formulae to be given for the elementary seismograms of the individual seismic body waves (in the zero approximation of the ray theory). This accelerates the computation of ray theoretical seismograms considerably.     

Explanation of the internal air pressure effect on the worden gravity meter dial constant

Summary A model, explaining the effect of air pressure changes in the space of the measuring system on the fine dial counter constant of the Worden and similar gravity meters, is presented. It is based on the changes of the bellows volume with compensating the gravity changes. It is shown that the resetting the gravity meter in the whole range has practically no influence on the fine dial counter constant.

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u¶rt;a ¶rt;, na m uu ¶rt;au ¶rt;a nmam uum um a nm a umaauma ¶rt; u nuaum. ¶rt; aa a uu a ua umu uma nu nauu uu u mmu. aa, m nma ¶rt;uanaa um namuu um a uu nm a uma.

     

Three-dimensional inverse problem for inhomogeneous transversely isotropic media

Summary The basic formula used in the presented paper gives the relation between the P wave travel-time perturbation and the perturbation of an inhomogeneous transversely isotropic medium, expressed by four perturbations of elastic parameters and by two angles of orientation of the axis of symmetry of transverse isotropy in space. The travel time perturbation is computed along the ray in the unperturbed inhomogeneous isotropic medium. Four elastic parameters and two angles are parametrized in the model under study and a system of equations for many rays is constructed. The equations are linear in the sought elastic parameters and nonlinear in the sought angles, and the iterative Levenberg-Marquardt algorithm is thus used to solve them. The theoretical 3-D inverse problem was solved in the presented numerical example. The data, simulating teleseismic data, were computed in the direct problem and then inverted. The results indicate the applicability and limitation of the presented algorithm in real problems.

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a a, unaa n¶rt;aa am, ¶rt;am mu ¶rt; uu u na u uu ¶rt;¶rt; nn umn ¶rt;, a m nuu naamau u ¶rt; au umauu u umuu nn umnuu nmam. u u na um ¶rt; a aa ¶rt;¶rt; umn ¶rt;. nu naam u ¶rt;a a naamuua ¶rt;u u nma uma au ¶rt; u . au u n um nu naama u u n um a umauu umuu, nm un m umamu aum a-aa¶rt;ma ¶rt; u u. am nu¶rt; ¶rt; m u nu. nu muu ¶rt;a aaa a na a¶rt;aa u am ¶rt;a a. mam naam auu u mu nuu nu¶rt;uma a a.

     

Seismic response analysis of two-dimensional absorbing structures

Summary A computational method (based on finite differences) is proposed for the seismic response analysis of the Earth's structures. The method is applied to the earth fill and the sedimentary basin. It was found that the lateral inhomogeneity of non-absorbing structures caused remarkable effects on synthetic seismograms, viz. strong horizontal reflections at later arrivals. In absorbing structures these effects are much less pronounced and the response is determined primarily by the variable thickness of the structures. The responses of structures with non-causal or causal absorption differ only slightly, but they are strongly influenced by the magnitude of the absorption itself.     

Effect of the additional low-sensitivity channel on the amplitude response of the electromagnetic seismograph

u¶rt;m ¶rt; anum¶rt;-amm aamumu maum a aa nu mummu (). ma¶rt;am aamumu uu mau auma uu am auu a uu aaa aa.     

Changes in the shape of the circumpolar vortex due to heating from above

Summary Based on model considerations it is shown that, under certain assumptions, zonalization of tropospheric circulation may be expected in the region of the auroral oval as a result of heat released at the time energetic electrons penetrate from the Sun into the lower stratosphere.

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a auu ¶rt; a¶rt;u naa, m nu m n¶rt;nu u¶rt;am auau mn uuu amu aa aa mam ¶rt;u mna nuu mumu m u a u mam.

     

The optimal shape design method applied to modelling thermal thinning of the oceanic lithosphere near hot spots

Summary Using the optimal shape design method, which is generally described, and von Herzen's et al. measurements of the heat flow, the shape of the lithosphere and its thermal field is computed in the vertical plane parallel to the hot spot source versus the plate velocity at a distance of about 250 km from the axis of the Hawaiian Island chain. The results are compared with the computations based on Crough's idea of thermal rejuvenation of the oceanic lithosphere above a hot spot source. If we assume that the lateral cross-section of the lithospheric bottom is described by the Gaussian curve h=h₀ exp (–y²/2²), we obtain h₀35 km and 130 km, where h is the value of lithospheric thinning and y the lateral coordinate. We thus obtain the lower limit of the lateral dimension of the Hawaiian anomaly.

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u m¶rt; nmua nua amu, m u ma nuam, u ¶rt;a mn nm, ua a um u mn n mua nmu, naa mu um mum umua mu u ¶rt;a nuuum 250 m uuuaa aunaa. mam a uuu, au a u¶rt;u aa (Crough), aauu mn mu au um a¶rt; umu mu. u n¶rt;num, m ama nn u umu ¶rt;a nuam u aa h=h₀ exp (–y²/2²), m num h₀ 35 u 130 ,¶rt; h—umu mu u —amaa ¶rt;uama. ¶rt;am, =130 m u n¶rt; ama aaaa aauu.

     

Analytical One-Way Plane-Wave Solution in the 1-D Anisotropic “Simplified Twisted Crystal” Model

Analytical expressions for the exact 2 × 2 one-way propagator matrix of a plane S wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, are presented. The analytical equations are useful in testing the applicability and accuracy of various approximate wavefield modelling methods, especially of the coupling ray theory and of its various quasi-isotropic approximations and various numerical implementations.In addition to the exact analytical solution of the elastodynamic equation in the simplified twisted crystal model, the analytical solutions of the equations of the four ray methods are given. The ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. These four approximate solutions of the elastodynamic equation are roughly compared with the exact solution. Both the exact analytical solution and the analytical ray-theory solutions in the simplified twisted crystal model are also helpful in debugging computer codes for various approximate wavefield modelling methods, especially for the coupling ray theory.     

Closed formulae for transformation of the cartesian coordinate system into a system of geodetic coordinates

Summary Relations for the direct transformation of the Cartesian coordinate system into a system of geodetic coordinates.

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u ¶rt;u , , z ¶rt;uu , , h.

     

On the estimation of the heat flow at the earth's crust lower boundary

Summary The paper deals with a comparison of two models used in determining the heat flow q_M at the lower boundary of two-dimensional geothermal models of the Earth's crust. Method I is based on a linear relationship between the component of the surface heat flow, which is not generated by heat sources inside the model, and the heat flow q_M. This method uses a regularization process, in which the variation of the sought function q_M is limited from above. Method II is the most frequently used iterative method, in which the (i + 1)-th approximation of q_M is determined from the surface heat flow, corresponding to the i-th approximation of q_M used as a boundary condition in solving the direct problem. The comparison of both methods has revealed that the solution obtained by method I satisfies the supposed reality better than the solution obtained by method II. Method II is attractive especially for its simplicity. To eliminate the local variations of the estimated heat flow q_M, which are due to automatic transmission from the surface heat flow, a combination of method II with some smoothing procedure could be applied.

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¶rt;am ¶rt;a m¶rt;a uu mn nma q_M a uau ¶rt; muu ¶rt; . m¶rt; unm u ¶rt; ma nm nma, ma uaa umuau mna mu ¶rt;u, u nm q_M. m m¶rt; nu n uauu, auuau auau u uu q_M . m m¶rt;, mu nm ¶rt; u n a , umau: (i + 1)-a annuau nma q_M um n auma nm nma, ma i- annuauu q_M. au ¶rt;m, m nm q_M, a¶rt; n m¶rt;, mam n¶rt;naa ¶rt;mummu , u, n m m¶rt;. m m¶rt; nuam nmm. ¶rt; mau a auau a¶rt; nma q_M, uu ¶rt;mu amamu na u auau nm nma, ¶rt;a uuam au-u¶rt; m¶rt; auau.

     

Die Lösung der Strahlungs-Übertragung in der Erdatmosphäre anhand der Monte-Carlo-Methode

ua bmam umumu ¶rt;uauu mu ¶rt;u , n no mo¶rt; me a [3] u no mo¶rt; [5] ¶rt; u a mauu uu amum. na¶rt;u ¶rt; m a h₀=6°.     

Über die komplexe Natur des Seeganges

Zusammenfassung Unter der Annahme, daß der Seegang nacheinander drei Hauptentwicklungsstufen durchläuft, bis er voll ausgereift ist, lassen sich Beziehungen zwischen den charakteristischen Wellengrößen, der Windgeschwindigkeit, der Wellenlaufstrecke und der Zeitdauer der Windeinwirkung ableiten. Diese Beziehungen sind in graphischer Form für die praktische Seegangsvorhersage dargestellt. Sie gestatten u. a. die Vorausbestimmung der mittleren Periode, d. h. des Mittelwertes der Zeitintervalle zwischen aufeinanderfolgenden Wellenkämmen an einem festen Ort, der häufigsten Periode, der Breite des Periodenspektrums sowie der mittleren und maximalen Wellenhöhe.

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On the complexity of wind waves — 2^nd part: The growth of waves under the wind's influence

Summary Proceeding from the supposition that wind generated waves have to pass through three principal stages before they will be fully developed, one may establish relations between wind velocity, fetch, duration of wind action, and the characteristic wave dimensions. Graphical representations of these relations are given for the practice of wave forecasting. Among others, they allow to predict the mean period, i. e. the mean value of the time intervals between the passage of successive wave crests at a given point, the most frequent period and the width of the spectrum of periods as well as the mean and maximum wave heights.

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La nature complexe des vagues de vent- 2^ième partie: Accroissement des vagues sous l'effect du vent

Résumé En supposant que les vagues de vent doivent parcourir trois phases principales pour atteindre les maxima de leurs dimensions et de leur force on peut établir des relations entre la vitesse du vent, le fetsch, la durée de l'action du vent et les dimensions caractéristiques des vagues. Pour les besoins de la prévision pratique des vagues de vent on a établi des représentations graphiques de ces relations qui permettent entre autres de prévoir la période moyenne, c'est à dire de prévoir la moyenne des intervalles de temps entre le passage par un point donné des crêtes des vagues consécutives, la période la plus fréquence, la largeur du spectre de la période et les moyennes et maxima des hauteurs des vagues.

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Aus den Arbeiten des Department of Meteorology and Oceanography an der New York University für das Office of Naval Research, Washington, D. C., unter Contract No. Nonr — 285(05).     

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, evanescent or inhomogeneous waves can be studied and modelled using the notion of complex rays. The wavefront or eikonal equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates. In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays. Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismograms. For anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.     

Radioactivity and geochemistry of granulites and some related rocks from the saxonian granulite complex

Summary The distribution of radioactive(Th, U, K), major and selected trace(Rb, Sr, Ba, Y, Zr, V, Cr, Ni) elements of granulites from the Saxonian Granulite Complex was studied. Similarly to the South Bohemian granulites, the Saxonian granulites can be divided according to the contents of their major and trace elements into two main groups, groupA containing mostly acid and subacid granulites (K ₂ O>2.5%, SiO ₂ >68%), and groupB containing mostly intermediate and basic granulites (K ₂ O<2.5%, SiO ₂ <68%). Statistically significant differences between groupsA andB were found for all major oxides and several trace elements(Rb, V, Cr, Ni). The Saxonian granulites follow the same calc-alkaline trend as the South Bohemian, granulitesA being placed mostly in the rhyolite field and granulitesB mostly in the dacite, andesite and basalt fields of this trend. The investigated granulites are characterized by a considerable scatter ofTh andU contents accompanied by very variableTh/U ratios; theTh andU concentrations of granulitesA are substantially lower than is usual for rocks of corresponding acidity.

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¶rt;a an¶rt;u a¶rt;uamu(Th, U, K) u ua ¶rt;u(Rb, Sr, Ba, Y, Zr, V, Cr, Ni) m aum n¶rt;a aaum na. naa, m u¶rt;aum n uu aam n aaum u ¶rt;u am aua, u u uu. aum n u uu ma a¶rt;um ¶rt; ¶rt;nn; nnA nua¶rt;ama a au¶rt; u au¶rt;aum (K ₂ O>2,5%, Si O ₂ >68%), nnB ¶rt;u u aum (K ₂ O<2,5%, SiO ₂ <68%). ¶rt; muunnau mm mamumuu m au ¶rt; a u u ¶rt; m ¶rt;u m(Rb, V, Cr, Ni). auaum n¶rt;¶rt;m um- m¶rt; a u -uaum;aumA a¶rt;ma a uum n, uaumB a a ¶rt;aum, a¶rt;um u aam n m m¶rt;a. ¶rt;aum — u unnA — aamum uu ¶rt;au da¶rt;uamu mTh uU.

     

Theoretical pv-magnitude calibrating curves for deep foci

Summary The theoretical medium period PV-magnitude calibrating curves were computed for the Earth model PREM and a wide range of focal depths. The calculated set of curves reflects the distribution of model parameters, the influence of source radiation was not taken into account. The widely used Gutenberg's empirical calibrating curves were compared with the theoretical ones. Pronounced deviations in the shape and differences in the level of isolines were found. Thus, if model PREM is considered to be representative of the Earth's structure, the empirical calibrating curves (D, h) for magnitude determination currently used in seismological practice, have to be verified.

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auma mmuu au u u ¶rt; n¶rt;u aum¶rt; - n ¶rt;nu¶rt; annaam ¶rt; ¶rt;u u RE u ¶rt;uanau aa. uu maam an¶rt;u naam ¶rt;u ma uu u umua. u nu nuuu au u ma u a mmuuu uu. u a¶rt; mu auu n u u. au a, u num ¶rt; RE n¶rt;maum ¶rt; mu u, m ¶rt;u num nuuu au u (D, h) ¶rt; n¶rt;u aum¶rt;, m am unm u namu.

     

Influence of the IMF sector boundaries on cosmic rays and tropospheric vorticity

Summary The effect of the IMF sector boundary crossing (IMF SBC) in the vorticity area index (VAI) — the well-known dip in the VAI after IMF SBC — is found to be independent of the IMF SBC effect in the cosmic ray flux. This finding refutes a recent suggestion by Lundstedt [1] that the IMF SBC effect in VAI is caused by a decrease in cosmic ray flux, but supports the concept of the IMF SBC effects in the ionosphere and atmosphere developed by Latovika [2–4]. Cosmic rays seem to affect the troposphere in another way.

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¶rt;mu nu mau nam aum n ( ) a u¶rt; na¶rt;u aumu () — um uu n — a¶rt; auu m ma nm uu . mm mam nam ¶rt;a n¶rt;u ¶rt;m¶rt;a [1], m m a nuu nma uu , n¶rt;¶rt;uam nu m u u am, aum amu [2–4]. am m uu u m um a mn ¶rt;u a.

     

On one possible type of upper cyclogenesis

Summary Based on model considerations it is shown that, under certain assumptions, upper cyclogenesis may be expected in the region of the auroral oval as a result of heat released at the time energetic particles of extraterrestrial origin penetrate into the lower stratosphere or even troposphere.

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a auu ¶rt; a¶rt;u naa, m nu m n¶rt;nu u¶rt;am u amu aa aa mam ¶rt;u mna nuu muu amu nu¶rt;u u mam uu mn.

     

A three-dimensional numerical model of advection and diffusion of urban source pollutants

ama mam maum n¶rt; n¶rt;um ma u ¶rt;, ¶rt;u ¶rt;unu nu nu au muu u. ¶rt;aa ¶rt; aa a uu au mm ¶rt;uuu n m¶rt;a am. u ¶rt;a n¶rt;aam m ¶rt; muu m nau am u u a um anu u m umu au, m ¶rt;aa ¶rt; a nm n¶rt;um nmam an¶rt;u mauu nu .     

Seismic velocity-ratio in the crust and uppermost mantle in southeast Wellington province,New Zealand

The regional variation of the seismic velocity-ratio () over a 200 km long traverse has been studied by means of microearthquake surveys. The Wadati-plot method is used with a minimum of four P and S arrivals for each of 49 earthquakes. The area as a whole is found to be characterized by a value of 1.74–1.76 for earthquakes of depth 12–40 km, except in a 50 km long section near Wellington, where is low at 1.60. This low has been attributed to the fault zones in the region. A small change of is observed between the upper crust (5 km) and lower crust (12 km), but there is no change of between the lower crust and uppermost mantle.     

High heat flow measured in Ostrava-Karviná coal basin

aamuam mam uu mn nma ma-au a. a¶rt; uua¶rt;uma mnam, uuma mnn¶rt;mu u mn nma m ¶rt;uamau au uu uma, n¶rt; m n¶rt; mn nm nu¶rt; a. 2. a u au Q=1,99 u 2,06×10^–6 /² am ¶rt;au mu n ua amu, ma mu (2,5×10^–6 a/² ) aum nam mm ¶rt;uana. um mm am na m¶rt;, ¶rt;¶rt; nu¶rt;m am mam.