Approximate solutions of the surface mass loading of a spherical Earth model

Summary Simple linear representation of the components of an approximate plane solution of point mass loading of the Earth's surface in a conveniently chosen coordinate system leads to selection of a 2nd-degree curve which is the best fit of the spherical solution for the given Earth model. The new approximate solution, which, analogously to the plane solution, can be called a parabolic solution, enables the simple input parameters of the plane solution to be used also for substantially larger angular distances. The comparison with the spherical solution is carried out by computing the effects of the M₂-wave of ocean tides. The results of the computations for the tidal station Brussels prove the two solutions to be in sufficient agreement for global problems as well.

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m nu uau aa nuuum nm u m m au nmu u n¶rt;¶rt; um ¶rt;uam ¶rt;m ma u m mnu, ma auu a n¶rt;¶rt;um u u ¶rt; ¶rt;a ¶rt;u u. nuuum u, m n aauu nm u aam naauu u, nm unam nm ¶rt; ¶rt;a nm u ¶rt;a ¶rt; m u amu. au uu u m uu uu ₂ u nuu. mam uu ¶rt; nuu mauu ¶rt;aam ¶rt;mum au u u ¶rt;a ¶rt;a a¶rt;a.

     

The gravitational effect of bodies with variable density

Summary Determination of the gravitational effect of some bodies, the density of which is supposed to be variable in the vertical direction, possibly in the horizontal direction, too.

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n¶rt;uaumau ma m m, nmm m um mua uu muma anau.

     

Iterative geophysical tomography

Summary The algorithm of iterative geophysical tomography is presented. The medium is approximated smoothly by means of B-splines. The tww-point problem of ray computation is solved with the aid of paraxial approximation. The parameters of the medium are obtained from the iterative algorithm of minimizing the quadratic form. Two numerical 2-D examples are given.

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u¶rt; au umamuuu mauu. ¶rt;a annuuaa n nu nu -na. ma na aa a nu nu naaua annuauu. aam ¶rt; n a umamu aua uuauauu a¶rt;amu . am nu¶rt; ¶rt;a 2-D u nua.

     

On the non-tidal secular acceleration of the Earth's rotation

Summary The secular positive acceleration of the Earth's rotation has been computed on the basis of the observed secular decrease of the second zonal harmonic[5]. It corresponds to the observed secular deceleration of the Earth's rotation which should be greater because of oceanic tides.

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¶rt; u a¶rt;a u m aauu [5], u ¶rt;a num u au. m mmmu a¶rt;a u mu au u, m.. n¶rt; n m uu , ¶rt; m n auu nuua.

     

Constraints to the three-dimensional non-hydrostatic density distribution in the earth

Summary The paper is concerned with the properties of a density distribution within the Earth. A system of density parameter constraints involving Stokes' coefficients of the gravity field and the parameters describing the Earth's figure is derived. A density model, whose parameters fit these constraints, accounts for the fine structure of the gravity field and Earth's figure. Additional condition imposed on the average spherical density model are derived; they guarantee that the average spherical model is compatible with the 3-D density model.

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aamuam ma an¶rt;u nmmu mu u. ¶rt;a uma u, auuau au naam nmm ¶rt;u; uma am m naamaumau n u naam nuau u u. mma ¶rt;, naam m ¶rt;mm mu auuau u, nm n¶rt;um uu umnmau m mmaumau n u u u. ¶rt; ma ¶rt;num u, aa¶rt;au a ¶rt; u ¶rt; nmmu, m nuam ¶rt;uu ¶rt; u ¶rt;u m nmm ¶rt;.

     

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.     

Curvature of a group of difeomorphisms and the instability of Kármán's vortex street

a auu uu nmama uau u¶rt;a u¶rt;mu (nn ¶rt;uu au m a) ua na ááa m a muau¶rt;¶rt;uaua uma. mumu mmmm uu aa nu¶rt;umu n au uu ma a u ma mu.     

Time patterns of polarization currents generated in measuring the electrical conductivity of rocks

amamuu aau m¶rt;a uu mn¶rt;mu n¶rt; nu u mnama, m num , naa, m uau nuau nmu um a mam uu.     

Regularization of downward heat flow continuation

Summary Some regularization approaches to downward heat flow continuation from the Earth's surface to the depth of interest are presented, provided the heat transfer is described by the homogeneous steady-state equation. The Fourier transform of horizontal coordinates is used and the relation between the regularization of the heat flow spectrum and the regularization of the heat flow in the spatial domain are discussed. The utilization of an a priori information about the subsurface heat flow is mentioned.

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¶rt; m zuau n¶rt;¶rt; n¶rt;u u nmz mnz nma n¶rt;nuu, m n mna nuam ¶rt;¶rt; maua au (au anaa). na naau zuma ¶rt;uam u ama ¶rt; zuau mnz nma u uau z nma. nma m nuu anu u n¶rt;nmz mnz nma.

     

Optimum determination of Euler Angles

a m m n¶rt; uua ¶rt; au anauu a nmu n¶rt;nuu, m u ¶rt;a a ¶rt;u -um, ma u amu um ¶rt;uam. a namu unm ¶rt;a anau, nu a¶rt;u m m aua n-a. nuam n¶rt; n¶rt;u au — D-nmuamu — naa uu anau ¶rt; n¶rt;u a.     

Secular changes of the geoid in the Fennoscandian land uplift region

Summary Two alternative methods are presented of determining the secular changes of the geoid in the Fennoscandian uplift region. Both are based on the Airy-Heiskanen hypothesis of isostatic compensation. The first alternative is the usual Stokes integration of changes of free-air anomalies in the given region. Another alternative was derived, which is based on calculating the gravitational potential of the surface uplift layer and of the same layer at the depth of compensation. The calculated changes of the shape of the geoid were then compared with the results of Bjerhammar's geophysical interpretation of the expansion of the Earth's gravity potential (regional depression of the geoid in the Fennoscandian uplift region). Regression analysis indicates a strong correlation between the two quantities (r=0.88). The time required to level this depression of the geoid is estimated. A relation was derived for improving the accuracy of the observed values of Fennoscandian uplifts, which takes into account the elastic reaction of the Earth's body to the change of gravitational potential and perturbations, caused by the secular decrease of the depth of the northern part of the Baltic Sea.

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¶rt;ma ¶rt;a auama m¶rt;a n¶rt;u uuu¶rt;a amu n¶rt;a a¶rt;uu. a u¶rt;m uunm uamu nauu u-aa. auam — umuau ma uu aau ¶rt; ¶rt; ¶rt;a amu. ¶rt; m auam, a a uuuaumau nmuaa nm n¶rt;a, u a au nauu. am u uu u¶rt;a auam mamau aaa nuu umnmauu au nmuaa u mmu u (ua ¶rt;nuuu¶rt;a amu a¶rt;uauu). u aau aam a u au u uu (r=0,88). uam , ¶rt;u ¶rt; auau a ¶rt;nuuu¶rt;a. ¶rt; mu ¶rt; mu u au n¶rt; a¶rt;uauu, a n au u a uuaumau nmuaa u u, mau u uu amu amu .

     

Precession-nutation torque in terms of the stokes constants

um nu-mau m nuuau u auumu m nm ma u. m mu u¶rt;uu¶rt;a m ma nuua u uu.     

The Earth's structure in the region of the crimea according to the Rayleigh waves observations

Summary The Earth's crust and mantle structure in the region Crimea are investigated on the surface waves observations. For this investigation the long-period seismographs have been installed for the triangle of the seismic stations Simferopol, Alushta, Theodosiya. The phase velocities dispersion has been determined on the base of surface waves for the 8 earthquakes. The observed dispersion curves agree with the theoretical ones for the Earth's model with the crust's thickness of 36–40 km and with (in the mantle) the low velocity layer (V _s =4.3 km/sec at the depth of about 120 km). . , , . 8 . 36–40 . (V _s=4.3 /) 120 .Presented to the IASPEI General Assembly, Madrid, 1969.     

The four primary geodetic parameters

Summary The four primary geodetic parameters defining the geodetic reference system are discussed from the point of view of their physical meaning and current estimation of their actual accuracy. The geopotential scale factor has been treated as the primary geodetic parameter defining the Earth's dimensions.

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¶rt;am m nu¶rt;uu naama, n¶rt;u¶rt;u um mumu, mu u uu a u mmu. ama amnmuaa ¶rt;am am nu¶rt;u naama, n¶rt; a u.

     

The intensity of the main phase of geomagnetic storms in relation to the parameters of the solar wind

Summary On the basis of investigating 10 storms (1965–1967) good correlation was found between the density of the solar wind energy (₂=1/2mNv²) and the intensity of the main phase of the geomagnetic storms, expressed in terms of the maximum decrease of the horizontal intensity (B=H/cos). The relation between ₂, or Nv², and B could then be used to determine the quantities and ₀ ( is the factor expressing the increase in energy density in the magnetosphere, ₀ is the energy density of the particles in a quiet magnetosphere). A comparison with the directly observed distribution of the energy density of the particles in the magnetosphere indicates that the computed value of ₀ seems to be realistic. The magnitude of the factor will have to be checked again.     

On the possibilities of horizontal propagation ofPc1 pulsations in the ionosphere

Summary The vertical distribution of the contribution of the energy flux density due to the Alfvén(ordinary) wave, guided by the geomagnetic field(and propagating through the ionosphere to the Earth's surface) in the horizontal direction is demonstrated in the mechanism of the horizontal propagation of the Pc1 signal. The distribution with height is shown of the variations of the polarization characteristics of the propagating wave(e.g. the rotation of the polarization plane, changes in ellipticity, attenuation, etc.), which are the result of coupling in the denser layers of the low ionosphere in which also suitable isotropic(extraordinary) modes are generated. The results obtained using the method described in[4, 13] are demonstrated on a model of the daytime ionosphere under incidence of ordinaryL-modes, frequency f=0.3 Hz, and various meridional angles at the ionosphere.

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auauma anmau uaa Pc1 naa m an¶rt;u ¶rt;u nmmu ma uu uma anauu maum n n¶rt; , anma u nmu. naa m an¶rt;u uu aamumu nuauu anma (nauau nmu nuauu, uu unmumu, amau u m.¶rt;.), m m ¶rt;mu au¶rt;mu na uu u . ¶rt; mum n¶rt;¶rt;u umn() ¶rt;. mam num m¶rt; [4, 13] ¶rt;mua ¶rt;u ¶rt; u nu na¶rt;uu a u L-¶rt; amm f=0,3 n¶rt; au u¶rt;uau au.

     

Deep structure of the lithosphere beneath the territory of Bulgaria

Summary Teleseismic P residuals calculated for waves arriving from various azimuths and angles of incidence, and a 3-D inversion of the residuals provided the basis for characterizing the uppermost mantle structure beneath Bulgaria. The Moesian Platform and the Rhodopean Massif are two different blocks characterized by a lithosphere thickness of about 130–140 km with a zone of lithosphere thinning along their contact. Both units have opposite patterns of the directional dependence of relatively high and low P velocities. This directional dependence is interpreted by dipping anisotropic structures in the subcrustal lithosphere, which probably represent remnants of paleosubductions of an old oceanic lithosphere.

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auma u nu¶rt;u n¶rt; au aumau u au. a uu mu ¶rt;am ¶rt; aamuauu mm amuu n¶rt; au. uua nama u ¶rt;nu au, ¶rt;a au a mu um nuuum 130–140 ¶rt; uma m ¶rt; u mama. a a aamua nmun ana auum mum u u mu . ma anaa auum umnmuaa a nu aumn mm amu. mu mm n¶rt;maum a mamu na¶rt;u ma au um.

     

Sunspot indices characterizing the 11-year cycle as a whole and their relationship to the analogous indices of geomagnetic activity

Summary Sunspot indices characterizing the 11-year cycle as a whole were determined on the basis of Wolf's sunspot numbers and their total area as well as on the basis of the primary physically indices of the spot-generating activity of the Sun.Their interrelationship is shown and their relationship to the indices of geomagnetic activity aa and C_i is examined. It is shown that under high solar activity a certain saturation of the Earth's magnetosphere occurs and that the variations of geomagnetic activity correlate with the variations of the global sunspot indices (Wolf number and total area) but correlate but slightly with the primary physically sunspot indices.

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u u n¶rt;;u nm u uuu nu u¶rt;; nmaoam ¶rt;;mmu q, n¶rt;; u¶rt;;u nm, aamuu -mu u . aaa u aua u u u mu u¶rt;;a ¶rt;;a¶rt;;um amumu aa u _i. aa, m nu amumu mynam am aum u m auauu ¶rt;;a¶rt;;umo amumu um auauau ¶rt;; u¶rt;; nm (u a, aa na¶rt;), a um uuu nuu u¶rt;u nm.

     

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.

     

On the problem of sudden commencements of geomagnetic storms as manifestations of shock waves

nma nu SS u nmau u nau a¶rt;a ¶rt;u nma u u a au 11-mu ua. u ¶rt; nm ¶rt; n. a¶rt;am nuu nu SS u.