An algorithm for the optimum distribution of a regional seismic network — II. An analysis of the accuracy of location of local earthquakes depending on the number of seismic stations

Summary Seven optimal networks consisting of 4 to 10 stations are compared for a given region, where velocity-depth profiles and the distribution of seismic intensity are known. Assuming that the standard error of arrival time is _t =0.05 s and the standard errors of the parameters of velocity-depth profiles are equal to 5% of their values, the average standard errors of the origin time and focus coordinates are estimated. The application of optimum methods to the planning of seismic networks in the Lublin Coal Basin is presented, and maps of standard errors of origin time , depth and epicenter ( _xy ) for the case of an optimum network of 6 seismic stations are given.     

Propagation of surface waves on the surface of a non-homogeneous aeolotropic cylindrical shell

Summary The object of the present paper is to investigate the propagation of surface waves on a non-homogeneous aeolotropic cylindrical shell surrounded by vacuum. The elastic constantsc _ij (i, j=1,2...) and density of the material of the shell are assumed to be of the form and respectively, where _ij, ₀ are constants andk ₁,k ₂ are any integers.     

Empirische Untersuchung zur Frage der Beziehung zwischen durchschnittlicher und kennzeichnender Wellenperiode im Seegang

Zusammenfassung Als Zusammenhang zwischen der kennzeichnenden Wellenperiode und der durchschnittlichen Periode im Seegang wird die Formel angesetzt. Mit Hilfe empirischer Unterlagen wird nachgewiesen, daßc eine Funktion des von D. E. Cartwright und M. S. Longuet-Higgins [1956] eingeführten Spektralparameters ist. Es wird eine vorläufige quantitative Beziehung zwischenc und abgeleitet.

---

Empirical investigations of the relation between the mean and the significant wave period in the sea

Summary It is supposed that the formula represents the relation between the significant wave period and the mean period in the sea. With the aid of empirical data it is demonstrated thatc is a function of the spectral parameter introduced by D. E. Cartwright and M. S. Longuet-Higgins [1956]. A preliminary quantitative relation betweenc and is derived.

---

Etudes empiriques de la relation entre la période moyenne et la période significative des vagues dans la houle

Résumé On suppose que la formule représente la relation entre la période significative des vagues et la période moyenne dans la houle. A l'aide des données empiriques on montre quec est une fonction du paramètre spectral , introduit par D. E. Cartwright et M. S. Longuet-Higgins [1956]. Une relation quantitative préliminaire entrec et est dérivée.

---

     

“Pandora box” — Matrix in the calculus of observations

Summary If the condition R(A)=k(n), whereA is the design matrix of the type n × k and k the number of parameters to be determined, is not satisfied, or if the covariance matrixH is singular, it is possible to determine the adjusted value of the unbiased estimable function of the parameters f(), its dispersion D( (x)) and ² as the unbiased estimate of the value of ² by means of an arbitrary g-inversion of the matrix . The matrix , because of its remarkable properties, is called the Pandora Box matrix. The paper gives the proofs of these properties and the manner in which they can be employed in the calculus of observations.     

Sull'approssimazione numerica di una variabile casuale normale,con applicaztone agli errori d'osservazione

Riassunto Data una variabile casuale X che segue la legge normale di probabilitl con valor medio a ed error medio y 1'A. considera un'altra variabile casuale che prende il valore intero r quando r–1/2

Summary Given a random variable X following the normal probability law, with expectation a and standard error p, the author considers another random variable , that takes the entire value r when r–1/2     

The dynamics of magma mixing in a rising magma batch

The conditions under which two magmas can become mixed within a rising magma batch are investigated by scaling analyses and fluid-dynamical experiments. The results of scaling analyses show that the fluid behaviours in a squeezed conduit are determined mainly by the dimensionless number where ₁ is the viscosity of the fluid, U is the velocity, g is the acceleration due to gravity, is the density difference between the two fluids, and R is the radius of the tube. The parameter I represents a balance between the viscous effects in the uppermost magma which prevent it from being moved off the conduit walls, and the buoyancy forces which tend to keep the interface horizontal. The experiments are carried out using fluid pairs of various density and viscosity contrasts in a squeezed vinyl tube. They show that overturning of the initial density stratification and mixing occur when I>order 10^-1; the two fluids remain stratified when I 10^-3. Transitional states are observed when 10^-3<I<10^-1. These results are nearly independent of Reynolds number and viscosity ratio in the range of and Re ₁<300. Applying these results to magmas shows that silicic to intermediate magmas overlying mafic magma will be prone to mixing in a rising magma batch. This mechanism can explain some occurrences of small-volume mixed lava flows.     

Recherches complémentaires sur le bases du calculateur météorologique «Temp»

Résumé La formule de base, traduisant une propriété analytique d'une classe très générale de fonctions, est un corollaire du théorème fondamental démontré dans un mémoire précédent, d'après lequel, étant donnés une fonction continue,p(, ,t) des points (, ) d'une surface régulière fermée et du temps et le champ d'un vecteur vitesse de transfert ou d'advection tangent à et ayant des lignes de flux fermées et régulières, il existe un opérateur spatial, linéaire, non singulierA tel que la fonctionA(p+Const.) soit purement advective par rapport a (sans creusement ni comblement). Ce théorème peut être exprimé par l'équation , où est un opérateur spatial, linéaire et non singulier, fonction deA.La détermination de peut être faite, soit en comparant deux formes différentes de la solution générale de l'équation en , soit en utilisant un raisonnement a priori très simple. On arrive ainsi au résultat pour un certain scalaireu(, ).Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde l'équation résulte aussi, comme nous l'avons montré dans le mémoire précédent, de notre théorie hydrodynamique des perturbations. On montre ici que la même équation peut encore être déduite de l'équation de continuité associée à la condition d'équilibre quasi statique selon la verticale.Comme applications de la formule de base (solution générale de l'équation enM), on étudie les problèmes suivants: 1^o creusement et comblement en général; 2^o creusement et comblement des centres et des cols; 3^o mouvement des centres et des cols; 4^o instabilité d'un champ moyen; 5^o propriétés spatiales des champsp(, ,t) et des vecteurs d'advection analytiques.Après une discussion des erreurs de la prévision d'un champp(, ,t) par la formule de base, du fait des erreurs des observations et du fonctionnement du calculateur, on examine quelques particularités du transfert ou advection d'un champf ₀(, ) par le vecteur . Enfin, le dernier chapitre du mémoire donne des éclaircissements complémentaires sur la structure du calculateur électronique «Temp» (qui effectue automatiquement les opérations mathématiques de la formule de base) et expose l'état actuel de sa construction.

---

Summary The basic formula, expressing an analytical property of a very general class of functions, is a corollary of the fundamental theorem, proved in a previous paper, according to which, given a functionp(, ,t) of the points (, ) of a closed regular surface and of the time, and a transfer or advection velocity vector tangent to and having regular closed streamlines, there is a spatial, linear, non singular operatorA such thatA(p+const.) is a purely advective function in respect to (no deepening). This theorem can be expressed by the equation where is a spatial, linear, non singular operator depending onA.The determination of can be attained, either by the comparison of two different forms of the general solution of the -equation, or by a simple a priori reasonning. The conclusion is thus reached that for a certain scalaru(, ).Whenp(, ,t) is the pressure perturbation at sea level, it was shown, in the preceding paper, that the equation can also be derived from our hydrodynamical perturbation theory. We now show that for this particular case, the same equation is also a consequence of the equation of continuity together with the condition of quasi statical vertical equilibrium.The following problems are then analysed by means of the basic formula: 1^o deepening and filling in general; 2^o deepening and filling of the centres and cols; 3^o motion of the centres and cols; 4^o instability of a mean field; 5^o spatial properties of the analytical fields and advection vectors .The errors in the forecast of a field,p(, ,t) by means of the basic formula, due to the observational and computational errors, are discussed, and some peculiarities of the transfer or advection of a fieldf ₀(, ) by are examined. Finally, complementary points are disclosed on the structure of the electronic computer «Temp» which performs automatically the mathematical operations of the basic formula, and a brief report is given of the present state of its construction.

     

Ueber Herde elastischer Wellen in isotropen,homogenen Medien

Zusammenfassung 1) Es werden Multipollösungen der skalaren Wellengleichung ² f/t ² – c² div gradf=0 betrachtet. Einerseits kann man solche Lösungen direkt durch Kugelfunktionenn-ter Ordnung ausdrücken, anderseits aus der Einpollösungf=1/p F(t–p/c) durch Differentiation nachn Richtungen erhalten. Es wird der Zusammenhang zwischen den Ergebnissen der beiden Verfahren gezeigt. — 2) Für die Energiedichte und den Energiefluss durch Kugelflächen bei kleinen elastischen Verschiebungen werden Ausdrücke in Kugelkoordinaten angegeben. — 3) Für die Wellengleichung grad div –b ² rot rot werden rotationsfreie Multipollösungen angegeben und Ausdrücke für Energiedichte und Energiefluss hergeleitet. — 4) Das gleiche wird für divergenzfreie Multipollösungen durchgeführt. — 5) Es werden Multipole betrachtet, die weder rotationsfrei noch divergenzfrei sind. Als Spezialfälle werden Multipole mit zeitlich begrenzter und solche mit periodischer Erregung gezeigt, ferner Lösungen der Wellengleichung, die sowohl rotationsfrei wie divergenzfrei sind. — 6) Es wird gezeigt, wie man die elastischen Wellen, die im Sinne vonStokes von einem Herdgebiet endlicher Ausdehnung ausgehen, näherungsweise durch elastische Multipole darstellen kann. — 7) Es wird angedeutet, wie man durch Messung von Komponenten von oder u.s.w. in Punkten im Innern des Mediums die Erregung und Energie von elastischen Multipolen bestimmen kann. Ferner wird auf den Fall hingewiesen, wo ein rotationsfreier Einpol sich im Innern eines Halbraumes befindet und die Messungen an seiner Oberfläche ausgeführt werden.

---

Summary (On foci of elastic waves in isotropic homogeneous media) — 1) Multiplets as solutions of the scalar wave equation ² f/t ² – c² div gradf=0 are considered. Such solutions can be obtained either directly by aid of spherical harmonics of ordern, or by differentiating the single polef=1/p F(t–p/c) with respect ton directions. The relations between the results of those two procedures are shown. — 2) In the case of small elastic displacements , the density of energy and the flow of energy through spherical surfaces are expressed by spherical coordinates. — 3) Multiplets which satisfy the equation of motion =a ² grad div b ² curl curl and the equation curl = 0 are given, and expressions for the density and flow of energy are found. — 4) The same is done with multiplets satisfying the equation of motion and the equation div = 0. — 5) General multiplets which satisfy the equation of motion are treated. As special cases, multiplets with excitation of finite length and multiplets with periodic excitation are considered, furthermore solutions of the equation of motion and of the equations curl = 0 and div = 0 are given. — 6) It is shown how elastic waves whose origin is a region of finite extension in the sense given byStokes, can be approximated by elastic multiplets. — 7) Some indications are given on the problem of how to find the functions of excitation and the energy of an elastic multiplet by measuring components of or etc., at points in the interior of the medium. The same problem is considered in the case of the single elastic pole. = grad 1/p F (t–p/a), if the measurements are made at the surface of an elastic half space.

     

Estimate of earthquake hazard in the Vrancea (Romania) region

A maximum likelihood method is used to estimate the earthquake hazard parameters maximum magnitudeM _max, annual activity rate , and theb value of the Gutenberg-Richter equation in the Vrancea (Romania) region. The applied procedure permits the use of mixed catalogs with incomplete historical as well as complete instrumental parts, the consideration of variable detection thresholds, and the incorporation of earthquake magnitude uncertainty.Our imput data, comprises 105 historical earthquakes which occurred between 984 and 1934, and a complete data file containing 1067 earthquakes which occurred during the period 1935–30 August, 1986. The complete part was divided into four subcatalogs according to different thresholds of completeness. Only subcrustal events were considered, and dependent events were removed.The obtained value (=0.65) is at the lower range of the previously reported results, but it appears concurrent with conceptual and observational facts. The same concerns inferred value of _max = 7.8 and activity rate _4.0 = 5.34.     

Frequency-time analysis of geomagnetic beating-type pulsationsPc3

Summary The mechanism of beating of Pc3 type pulsations is studied. Using the method of numerical computation of a sonagram (the method of frequency-time analysis) a set of samples of pulsations from the Budkov Observatory is treated (1968–1969) mostly at K-indices equal to 2–3. By comparing f–t diagrams with the spectra of the samples an attempt has been made at interpreting the beating as a superposition of the frequency components, contained in the pulsation signal. In most observed cases it is possible to determine two close frequencies, the difference of which is on the average =5.4 mHz. The average carrier frequency of the samples was =37.6 mHz, and the average frequency of the beating =2.7 mHz. The interval of observed values of f_B amounted to 1–5 mHz. A tendency was observed for f_B to increase with increasing degree of disturbance of the geomagnetic field.     

Nonlinear Magnetoconvection and the Geostrophic Flow - Subcritical Solutions

The magnetoconvection problem under the magnetostrophic approximation is investigated as the nonlinear regime is entered. The model consists of a fluid filled sphere, internally heated, and rapidly rotating in the presence of a prescribed, axisymmetric, toroidal magnetic field. For simplicity only a dipole parity and a single azimuthal wavenumber (m = 2) is considered here. The leading order nonlinearity at small amplitude is the geostrophic flow U _g which is introduced to the previously linear model (Walker and Barenghi, 1997a, b). Walker and Barenghi (1997c) considered parameter space above critical and found that U _g acts as an equilibration mechanism for moderately supercritical solutions. However, for solutions well above critical a Taylor state is approached and the system can no longer equilibrate. More importantly though, in the context of this paper, is that subcritical solutions were found. Here subcritical solutions are considered in more detail. It was found that, at is strongly dependent on . ( is the critical value of the modified Rayleigh number is a measure of the maximum amplitude of the generated geostrophic flow while , the Elsasser number, defines the strength of the prescribed toroidal field.) R_m at proves to be the key measure in determining how far into the subcritical regime the system can advance.     

A generalized method for various data processing techniques in gravity interpretation

Summary The various data processing techniques (downward continuation, first and second derivatives and their downward continuation) used in gravity interpretation, are analogous to different types of linear filtering operations whose theoretical filter (amplitude) responses can be derived from by suitably choosingN andd, whereu andv are angular frequencies in two perpendicular directions,d the height or depth of continuation in unit of grid interval; andN denotes the order of the vertical derivative. By incorporating a mathematical smoothing function, ( being the smoothing parameter) in the theoretical filter response function, it has been possible, by selecting a suitable value of smoothing parameter, to establish an approximate equivalence of the effect of the mathematical smoothing with the inherent smoothing introduced, because of the numerical approximation (approximation error) for practically all data-processing techniques. This approximate equivalence leads to a generalized method of computing sets of weight coefficients for various data-processing techniques from filter response matching method. Several sets of weight coefficients thus have been computed with different smoothing parameter. The amplitude response curves of the various existing sets of weight coefficients have also been calculated for assessing the quality of the approximation in achieving the desired filtering operation.     

Relations entre le creusement et le comblement des perturbations et leur déplacement

Résumé On commence par définir le creusement et le comblement d'une fonctionp(, t) du tempst et des points (, ) d'une surface régulière fermée en se donnant, sur cette surface, un vecteur vitesse d'advection ou de transfert tangent à . Le creusement (ou le comblement) est la variation dep sur les particules fictives se déplaçant constamment et partout à la vitesse , A chaque vecteur et pour un mêmep(, ,t) correspond naturellement une fonction creusementC (, ,t) admissible a priori; mais une condition analytique très générale (l'intégrale du creusement sur toute la surface fermée du champ est nulle à chaque instant), à laquelle satisfont les fonctions de perturbation sur les surfaces géopotentielles, permet de restreindre beaucoup la généralité des vecteurs d'advection admissibles a priori et conduit à des vecteurs de la forme: , oùT est un scalaire régulier, () une fonction régulière de la latitude , le vecteur unitaire des verticales ascendantes etR/2 une constante. Ces vecteurs sont donc une généralisation naturelle des vitesses géostrophiques attachées à tout scalaire régulier. Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde, le vecteur d'advection par rapport auquel on doit définir le creusement est précisément une vitesse géostrophique: on a alors ()=sin etT un certain champ bien défini de température moyenne.On déduit ensuite une formule générale de géométrie et de cinématique différentielles reliant la vitesse de déplacement d'un centre ou d'un col d'un champp(, ,t) à son champ de creusementC (, ,t) et au vecteur d'advection correspondant. Cette formule peut être transformée et prend la forme d'une relation générale entre le creusement (ou le comblement) d'un centre ou d'un col et la vitesse de son déplacement, sans que le vecteur d'advection intervienne explicitement. On analyse alors les conséquences de ces formules dans les cas suivants: 1^o) perturbations circulaires dans le voisinage du centre; 2^o) perturbations ayant, dans le voisinage du centre, un axe de symétrie normal ou tangent à la vitesse du centre; 3^o) évolution normale des cyclones tropicaux.Finalement, on examine les relations qui existent entre le creusement ou le comblement d'un champ, le vecteur d'advection et la configuration des iso-lignes du champ dans le voisinage d'un centre.Ces considérations permettent d'expliquer plusieurs propriétés bien connues du comportement des perturbations dans différentes régions.

---

Summary The deepening and filling (development) of a functionp(, ,t) of the timet and the points (, ) of a regular closed surface is first of all defined, in respect to a given advection or transfer velocity field tangent to , as the variation ofp on any fictitious particle moving constantly and everywhere with the velocity . For a givenp(, ,t) and to any there corresponds a well defined development fieldC (, ,t). All theseC fields are a priori admissible, but a very general analytical condition of the perturbation fields in synoptic meteorology (the integral of the development fieldC (, ,t) on any geopotential surface vanishes at any moment), leads to an important restriction to advection vectors of the form: , whereT is any regular scalar, () any regular function of latitude, the unit vector of the ascending verticals andR/2 a constant. These vectors are a natural generalisation of the geostrophic velocities attached to any regular scalar. Whenp(, ,t) is the pressure perturbation at sea level, its development must be defined in respect to a geostrophic advection vector belonging to the above defined class of vectors with ()=sin andT a well defined mean temperature field.A general formula of the differential geometry and kinematics ofp(, ,t) is then derived, giving the velocity of any centre and col of ap(, ,t) as a function of the advection vector and the corresponding development fieldC (, ,t). This formula can be transformed and takes the form of a general relation between the deepening (and filling) of a centre (or a col) of ap(, ,t) and its displament velocity, the advection vector appearing no more explicitly. A detailed analysis of the consequences of these formulae is then given for the following cases: 1^o) circular perturbations in the vicinity of a centre; 2^o) perturbations having, in the vicinity of a centre, an axis of symmetry normal or tangent to the velocity of the centre; 3^o) normal evolution of the tropical cyclones.Finally, the relations between the developmentC (, ,t) of a fieldp(, ,t), the advection velocity vector and the configuration of the iso-lines in the vicinity of a centre are analysed.These theoretical results give a rational explanation of several well known properties of the behaviour of the perturbations in different geographical regions.

---

---

Communication à la 2ème Assemblée de la «Società Italiana di Geofisica e Meteorologia» (Gênes, 23–25 Avril 1954).     

Anisotropy of Cs137 gamma-ray absorption in rocks

Summary The anisotropy of the Cs¹³⁷ gamma-ray absorption coefficient has been investigated in layered models and several rock samples. It was found that in contrast to other petrophysical parameters the gamma-ray absorption coefficient is characterized by a relatively small coefficient of anisotropy (3–5%). Therefore, the possible effect of the anisotropy can be neglected in the measurements of for rocks as it is usually below current observational errors.     

Radiance and color of the sky at twilight: Perturbations caused by stratospheric haze

Layers of stratospheric aerosol with optical thicknesses as small as 10^–4 cause noticeable perturbations in the monochromatic logarithmic brightness gradient,G, and the color ratio,C, of the twilight sky. Modeling of the twilight's radiant properties shows that definite single-valued relationships exist between maxima inC or minima inG and optical thickness, , physical thickness h, and mean altitude, , of stratospheric layers. It is therefore possible to determine , h and and monitor their variations by performing either single wavelength measurements ofG or two-wavelength spectrophotometric measurements ofC. The presence of haze in the lower troposphere and the occurrence of multiple scattering both have relatively minor influences on the recovery of the stratospheric dust properties, provided that 10< <30 km.Formal mathematical inversions of the single-scattering twilight equations are possible in principle, but difficult in practice because of non-linearities. Inversions incorporating an iterative linearization process with constrained smoothing, successfully recovered the features of the haze layer, but tended to oversmooth the vertical profile and underestimate the mean altitude of the haze layer.     

Molecular refraction in the Earth's mantle

The Drude law (molecular refraction) for the temperature radiation in a monoatomic model of the Earth's mantle is derived. The considerations are based on the Lorentz electron theory of solids. The characteristic frequency (or eigenfrequency) of independent electron oscillators (in energy units, ) is identified with the band gapE _G of a solid. The only assumption is that solid material related to the Earth's mantle has the mean atomic weight A21 g/mole, and its energy gap (E _G) is about 9 eV. In this case the value of molecular refraction (in cm³/g) is (n ²–1)/=0.5160.52, where andn are the density and the refractive index at wavelength _D=0.5893 m (sodium light), respectively. The average molecular refraction of important silicate and oxide minerals with A21, obtained byAnderson andSchreiber (1965) from laboratory data, is , where denotes the mean arithmetic value calculated from three principal refractive indices of crystal. For the rock-forming minerals with 19A<24 g/mole the new relation was found byAnderson (1975).     

Hydrated cluster ion formation rates and winter anomaly

Expressions for the effective cluster formation rates are written in terms of neutral densities and recent laboratory data on reaction rate coefficients. At D-region heights these parameters show seasonal variations due to mesospheric temperature changes. Reduction in cluster formation rates for winter months would result in a decrease in the concentration of hydrated ions and hence reduced effective recombination coefficient for electrons. The results show latitude and seasonal variations which are similar to that of a normal winter anomaly in radio wave absorption.     

Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional (l) stationary. Fermat's functional (l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral (l) = (x ^k,x ^k ')du, taken along curve l, where (x ^k,x ^k ') is the relevant Lagrangian, x ^k are coordinates, u is a parameter used to specify the position of points along l, and x ^k ' = dx ^k÷du. If Lagrangian (x ^k,x ^k ') is a homogeneous function of the first degree in x ^k ', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k '). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k ') to derive the relevant Hamiltonian ⁽¹⁾(x ^k,p _k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ⁽¹⁾(x ^k,x ^k '). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ⁽²⁾. Parameter u along the curves is taken to correspond to travel time , and the second-degree Lagrangian ⁽²⁾(x ^k, ^k ) is then introduced by the relation ⁽²⁾(x ^k, ^k ) = [⁽¹⁾(x ^k, ^k )]², with ^k = dx ^k÷d. The second-degree Lagrangian ⁽²⁾(x ^k, ^k ) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ⁽²⁾(x ^k,p _k) from Lagrangian ⁽²⁾(x ^k, ^k ), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ⁽²⁾(x ^k, ^k ) and ⁽²⁾(x ^k,p _k) can be expressed in terms of the wave propagation metric tensor g _ij(x ^k, ^k ), which depends not only on position x ^k, but also on the direction of vector ^k . It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ⁽²⁾(x ^k, ^k ), not to the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). It is also shown that relations ⁽²⁾(x ^k, ^k ) = ; and ⁽²⁾(x ^k,p _k) = are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x ⁱ. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.     

Das Spektrum der internen Bewegungsvorgänge der westlichen Ostsee im Periodenbereich von 0,3 bis 60 Minuten

Zusammenfassung Für ein Beobachtungsbeispiel (Stromgeschwindigkeit, Dichte) aus der Kieler Bucht wird eine Deutung ausgeprägter Maxima der zugehörigen Spektren zwischen 0,3 und 60 Minuten durch interne Wellen gegeben. Mit einer beobachteten Periode von zwei Minuten durchgeführte Rechnungen ergeben eine starke Abhängigkeit der Wellenlänge von der mittleren Strömung . Im Falle erhält man bei 23,5 m Wassertiefe eine Wellenlänge von etwa 70 m, im Falle von etwa 85 m. Die berechneten Schwingungen stellen uneigentliche interne Wellen dar (W. Krauß [1966]). Die Interpretation durch eine Grenzflächenwelle führt auf eine Wellenlänge von 86 m, die nur geringfügig von denen der internen Wellen 1. Ordnung in stetig sich ändernder Strömung abweicht.In einer theoretischen Untersuchung werden kleinräumige Anfangsstörungen (z. B. momentane Druckänderungen an der Meeresoberfläche) als mögliche Ursache für die Entstehung kurzperiodischer interner Wellen erkannt. Es zeigt sich, daß kurzzeitig wirksame Anfangsbeschleunigungen in ihrem Einwirkungsbereich stehende, allmählich abklingende interne Wellen erzeugen, während in der Umgebung gleichzeitig fortschreitende Wellen entstehen, deren Amplituden mit wachsender Entfernung vom Erregungsgebiet abnehmen. Die Perioden der Schwingungen haben größere Werte als die zu einer exponentiellen Schichtung gehörige Väisäläperiode und verändern sich in Abhängigkeit von der Größe des Anregungsgebietes wie die zellularer Stabilitätsschwingungen.

---

The powerspectrum of internal motions in the western baltic between the periods 0.3 and 60 minutes. Part 1: Interpretation of the wavelike component of the internal unrest in the sea

Summary The powerspectra of the internal unrest in the sea show marked peaks in the range of periods between 0.3 and 60 minutes. An interpretation of these phenomena is given in terms of internal waves for a specific example obtained from short periodic current and density variations in the Kieler Bucht. The numerical calculations for an observed period of two minutes show an important influence of the vertical distribution of the current on the wavelength. In the case of the wavelength amounts to about 70 m, where as in the case of the length is about 85 m, assuming a depth of the sea of 23.5 m. The computed oscillations represent improper internal waves (W. Krauß [1966]). Interpretation by internal boundary waves yields a wavelength of 86 m, which is slightly different only from those of the first mode of internal waves in the case of continuously varying .By a theoretical investigation it is shown that short periodic internal waves may be caused by local initial perturbations (for instance by sudden variations of pressure at the surface). The solution of the problem describes slowly decreasing standing internal waves, which are generated within the area upon which short-dated initial accelerations have acted. At the same time a train of progressive waves is developed in the environment travelling away from the centre of the excitation. The amplitudes of these waves diminish with increasing distance from the origin. The periods of the computed oscillations yield higher values than the Väisäläperiod belonging to an exponential stratification. The variability in these periods is caused by variations in depth, by variations in stability, and by changes in the horizontal dimensions of the area of initial perturbation. This dependence is similar to that of cellular oscillations of stability.

---

Spectre des oscillations internes de la mer Baltique Ouest pour des périodes comprises entre 0,3 et 60 minutes. 1ère Partie: Interprétation des éléments ondulatoires de mouvement

Résumé Pour un cas d'observation (vitesse de courant, densité) en baie de Kiel, des maximums bien marqués des spectres correspondants entre 0,3 et 60 minutes s'expliquent par des ondes internes. Des calculs effectués avec une période de deux minutes montrent que la longueur d'onde dépend beaucoup du courant moyen, . Pour , par 23,5 m de profondeur, on obtient une longueur d'onde environ 70 m; pour , une longueur d'environ 85 m. Les oscillations calculées représentent des ondes internes qui ne sont pas des ondes propres. L'interprétation par une onde de surface limite conduit à une longueur d'onde de 86 m très peu différente de celles des ondes internes du premier ordre dans un courant constamment variable.Une étude théorique montre que des perturbations initiales peu étendues (par exemple variations momentanées de la pression à la surface de la mer) peuvent être à l'origine d'ondes internes à courte période. Il apparaît que des accélérations initiales, agissant brièvement, font naître dans leur zone d'action des ondes internes stationnaires qui s'amortissent peu à peu, tandis qu'en même temps aux alentours se produisent des ondes progressives dont l'amplitude décroît à mesure qu'elles s'éloignent de la région où elles ont pris naissance. Les périodes des oscillations ont des valeurs plus grandes que celle de la période de Väisälä rapportée à une stratification exponentielle, et elles varient suivant la grandeur de la zone où elles ont pris naissance comme les oscillations de stabilité cellulaire.

     

Predicting earthquakes by analyzing accelerating precursory seismic activity

During 11 sequences of earthquakes that in retrospect can be classed as foreshocks, the accelerating rate at which seismic moment is released follows, at least in part, a simple equation. This equation (1) is ,where is the cumulative sum until time,t, of the square roots of seismic moments of individual foreshocks computed from reported magnitudes;C andn are constants; andt _fis a limiting time at which the rate of seismic moment accumulation becomes infinite. The possible time of a major foreshock or main shock,t _f,is found by the best fit of equation (1), or its integral, to step-like plots of versus time using successive estimates oft _fin linearized regressions until the maximum coefficient of determination,r ²,is obtained. Analyzed examples include sequences preceding earthquakes at Cremasta, Greece, 2/5/66; Haicheng, China 2/4/75; Oaxaca, Mexico, 11/29/78; Petatlan, Mexico, 3/14/79; and Central Chile, 3/3/85. In 29 estimates of main-shock time, made as the sequences developed, the errors in 20 were less than one-half and in 9 less than one tenth the time remaining between the time of the last data used and the main shock. Some precursory sequences, or parts of them, yield no solution. Two sequences appear to include in their first parts the aftershocks of a previous event; plots using the integral of equation (1) show that the sequences are easily separable into aftershock and foreshock segments. Synthetic seismic sequences of shocks at equal time intervals were constructed to follow equation (1), using four values ofn. In each series the resulting distributions of magnitudes closely follow the linear Gutenberg-Richter relation logN=a–bM, and the productn timesb for each series is the same constant. In various forms and for decades, equation (1) has been used successfully to predict failure times of stressed metals and ceramics, landslides in soil and rock slopes, and volcanic eruptions. Results of more recent experiments and theoretical studies on crack propagation, fault mechanics, and acoustic emission can be closely reproduced by equation (1). Rate-process theory and continuum damage mechanics offer leads toward understanding the physical processes.