Multifractal measures,especially for the geophysicist

This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach inMandelbrot (1974). The generalization from fractalsets to multifractalmeasures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function (), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity is called Hölder exponent. In terms of the alternative functionf() used in the approach of Frisch-Parisi and of Halseyet al., one has ()=f()–E for measures supported by the Euclidean space of dimensionE. Whenf()0,f() is a fractal dimension. However, one may havef()<0, in which case is called latent. One may even have <0, in which case is called virtual. These anomalies' implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantityD _q, which is shown forq>1 to be a critical dimension for the cuts. An enhanced multifractal diagram is drawn, includingf(), a function called (q) andD _q.This text incorporatesand supersedes Mandelbrot (1988). A more detailed treatment, in preparation, will incorporateMandelbrot (1989).     

Détermination des paramètres d'Ångström par des observations actionmétriques courantes

Summary By comparing the equalities expressing the solar flux within a wide spectral region on the assumption of an extintion within the atmospheric aerosol given by ^– and that of an extintion represented by ₁^–1, a relationship is established by means of which: 1) The error is discussed which is done when the air opacity is expressed by ₁, in the case when 1; 2) A simple and quick method is worked out for determining the parameters and from actinometric observations, carried out within the spectral regions <525 m and 525 m<<625 m.     

L'ordre du coefficient de récombinaison dans la coucheE

Résumé On propose une méthode de détermination de l'ordre du coefficient équivalent de récombinaison en n'utilisant que les données ionosphériques. A cette fin on évalue l'apport de la vitesse des changements passagersdN _m E/dt dans la densité électroniqueN _m E. L'évaluation se fait en présence d'une série de valeurs choisies de , on observe laquelle des courbes définis _m (t) dans cette étude coïncidera de plus près avec _m (t) mesurée. Les changements de montrés sur la figure 4 sont obtenus de cette manière. La valeur diurne de varie de (0.5÷1) 10^–7 cm³ sec^–1; peu après le lever et peu avant le coucher du soleil devient >10^–7 cm³ sec^–1. Lorsque cos>0, d'après la règle décrcit et devient 10^–8 cm³ sec^–1. Les explications des changements de obtenus de cette manière sont données en rendant compte des changements par rapport aux ions atomiques et moléculaires, à la disparition rapide des ions positifs pendant le coucher du soleil en présence d'un coefficient de récombinaison plus grand et des changements de température eventuels. La symétrie ou l'asymétrie des fréquences critiquesf ₀ E quand cos sont égaux permettent dans la marche diurne de juger de l'ordre de . Toutes ces évaluations indiquent également des valeurs de environ 10^–7 cm³ sec^–1. En précisant des mesures def ₀ E il est possible de définir non seulement l'ordre, mais aussi la valeur de elle-même.

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Summary A method for the determination of the order of the equivalent recombination coefficient is suggested, by using ionospheric data only. The increase in the speed of temporary changes indN _m /dt in the electronic densityN _m E is estimated for this purpose. The estimation is done with series of selected for the sake of expediency values of , following at the same time which curves _m (t), determined in the course of work, will coincide most closely with the _m (t) measured. The changes in , shown in figure 4, have been obtained in this way. The diurnal value of is in the range of (0.5 to 1) 10^–7 cm³ sec, being >10^–7 cm³ sec a little after sunrise and a little prior to sunset. At cos<0, by rule decreases and becomes 10^–8 cm³ sec. Explanations of the changes thus obtained in are indicated, taking into consideration the relation of atomic and molecular ions, the rapid disappearance of positive ions at sunset with a higher recombination coefficient and eventual temperature changes. From the symmetry or asymmetry of the critical frequencies off ₀ E at equal cos in the course of the day it can also be judged for the order of . All those estimates show values of in the range of 10^–7 cm³ sec. In the case of precise measurements off ₀ E, not only the determination of the order but also the real value of is possible.

     

On the possibility of long-range forecasts of geomagnetic activity

Summary The relation between geomagnetic activity and solar activity is studied in long intervals and in individual 11-year cycles, characterized by certain basic parameters, in connection with investigating the prognostic significance of long-term fluctuations of geomagnetic activity [1, 2]. A number of properties as well as sufficiently close relations with a prognostic significance have been found, such as the secular variation of geomagnetic activity, the 22-year cycle of change in the form of the behaviour of geomagnetic activity in 11-year cycles, long-term concurrence of solar and geomagnetic activity and its changes within the 11-year cycle, relations between solar and geomagnetic parameters characterizing the 11-year cycle.

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a auu uu u ¶rt; uaum amum ¶rt;u nma u u m¶rt; 11-mu ua am mu ¶rt; ¶rt;m nuauaum amumu.

     

Non-linear dynamo waves in an incompressible medium when the turbulence dissipative coefficients depend on temperature

Non-linear - dynamo waves existing in an incompressible medium with the turbulence dissipative coefficients depending on temperature are studied in this paper. We investigate of - solar non-linear dynamo waves when only the first harmonics of magnetic induction components are included. If we ignore the second harmonics in the non-linear equation, the turbulent magnetic diffusion coefficient increases together with the temperature, the coefficient of turbulent viscosity decreases, and for an interval of time the value of dynamo number is greater than 1. In these conditions a stationary solution of the non-linear equation for the dynamo waves amplitude exists; meaning that the magnetic field is sufficiently excited. The amplitude of the dynamo waves oscillates and becomes stationary. Using these results we can explain the existence of Maunders minimum.     

On some sources of uncertainty in the Lyman-α ionization-rate calculation

Summary The effect of the uncertainty and variability in the absorption cross-section, ionization cross-section and the Lyman- line shape on the Lyman- ionization-rate calculation is studied. The effect of the variability of the Lyman- line shape seems to be negligible. The effect of the ionization cross-section is rather small. The greatest and very significant effect, particularly at lower altitudes, is due to the discrepancy in the absorption cross-section data. Some ionospheric and atmospheric consequences are briefly discussed.     

Actinometric determination of turbidity parameters in India

Summary A study of the wavelength exponent of aerosol scattering in the Ångström relation for extinction by aerosol has been made from the ground-based measurments of direct solar radiation using Ångström pyrheliometer with and without Schott filters. It has been observed that in India, mainly for the middle part of the year this exponent is zero or even negative which means that the aerosol scattering is nearly neutral which is in marked contrast with the condition prevailing in middle latitudes. It is evident from the -values that the aerosol size distribution in India is far different from that prevailing in middle latitudes. At four representative stations in India, the values of the wavelength exponent and the atmospheric turbidity coefficient have been determined using the method introduced byÅngström [1,2]) and are discussed here.     

Quasi-static thermal deformations of a sphere by radiation from a point source

Summary In this paper the quasi-static temperature and stress distributions set up in an elastic sphere by radiation from a point source at a finite distance from the centre of the sphere and out-side it, have been discussed. The temperature boundary condition has been taken in the general form involving an arbitrary function of time. The final solutions have been obtained in terms of series involving Legendre polynomials. Numerical calculations have been done on IBM 1620 Computer and a desk calculator. The results have been represented in graphs.Notation the del operator - u the displacement vector - T the excess of temperature over that at state of zero stress and strain - , Lamé's constants - /2(+) Poisson's ratio - coefficient of linear expansion - 2(1+) - a radius of the sphere - d distance of the point source from the centre of the sphere - d _o a/d - K coefficient of thermal conductivity - h heat transfer coefficient of the surface     

The role of various ionization sources in the A3 lf radio-wave absorption in middle latitudes

Summary The 272 kHz ionospheric absorption has been measured at the Prhonice Observatory by the A3 method (oblique incidence) for a long period. The results of the comparison of these absorption data with indices of solar, geomagnetic and cosmic ray activity are presented for large solar zenith angles. The day-time absorption (x 75° – 85°) is described well by these indices and appears to be affected particularly by direct solar Lyman- radiation. The night-time absorption (x>100°), however, is not described sufficiently by the indices used.     

Variations of theD-region aeronomical parameters under non-equilibrium conditions

Summary In this paper an attempt is made to define the time variations of theD-region main aeronomical parameters under non-equilibrium conditions by using experimental data by the A3 method, phase-height method and satellite data for the short-wave X-radiation. The SID effect of day-time and time variations of 21 August 1969 electron concentrationN(t) and recombination coefficient (t) are investigated. It is shown that during SID effect (t) increases.

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Résumé Dans le présent travail on a procédé à une expérience pour déterminer les variations temporelles des principaux paramètres aéronomiques en utilisant des données expérimentales par la méthode A3 de mesure de l'absorption ionosphérique, la méthode phase-altitude et des données satellites aux rayons X durs. On a examiné les variations diurnes et les variations temporelles lors de SID effet durant le 21 Août 1969, de la densité électroniqueN(t) et du coefficient de récombinaison (t). On a montré que lors de SID effet (t) s'accroît.

     

Cyclic organochlorines in North Sea sediments,relation with size and organic matter

Summary The geographical distribution of organic matter and particles <20 m in North Sea sediments as well as the correlation between organic content, <20 m fraction and PCB (as the sum of content of 24 individual components), p, p-DDE, HCB, -HCH and Lindane (-HCH) is reported. High amounts of organic matter and fine particles have been found in Fladen Ground, Kattegat/Skagerrak/Norwegian Channel, and the inner German Bight. Organic matter settling in the Kattegat/Skagerrak/Norwegian Channel is probably degraded by benthic organisms, including mineralization by bacteria. The smelter in south Norway and the rivers Ems and Elbe are point sources for HCB, the mud-area south-east of Helgoland is a point source for HCB and PCBs. No point source, however, was found for p,p-DDE. Sediment content of PCB, p,p-DDE and HCB show a good correlation to the amount of organic matter and to the <20 m fraction. -HCH and Lindane are only poorly adsorbed onto sediments.

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Zyklische organische Chlorverbindungen in Nordseesedimenten: Verteilung von organischer Substanz und feinpartikulärem Material

Zusammenfassung Es wird über die Verteilung von organischer Substanz und feinpartikulärem Material <20 m in Nordseesedimenten sowie über die Korrelation der Sedimentparameter mit den Konzentrationen von polychlorierten Biphenylen (24 Einzelisomere), p,p-DDE, HCB, -HCH und Lindan (-HCH) berichtet. Hohe Gehalte an organischer Substanz und der <20 m Fraktion wurden im Fladengrund, Kattegat/ Skagerrak/Norwegische Rinne sowie in der inneren Deutschen Bucht gefunden. Die Ergebnisse legen den Schluß nahe, daß ein Großteil des sedimentierten organischen Materials vor der norwegischen Küste von Benthosorganismen und Bakterien verstoffwechselt wird. Durch die mit Søderbergelektroden arbeitenden Metallschmelzen Südnorwegens sowie durch die Flüsse Elbe und Ems werden die Sedimente punktuell mit HCB belastet. Das Schlammgebiet südöstlich von Helgoland tritt als Emittent für PCBs in Erscheinung. Für p,p-DDE konnte keine Punktquelle nachgewiesen werden. Im Gegensatz zu -HCH und Lindan korrelieren die Sedimentkonzentrationen an PCBs, HCB und p,p-DDE gut mit den Gehalten an organischer Substanz bzw. feinpartikulärem Material.

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Les Organochlorines cycliques dans les Sédiments de la Mer du Nord, relation avec la taille et les matières organiques

Résumé La distribution géographique de matière organique et des particules de taille inférieure à 20 m dans les sédiments de la Mer du Nord ainsi que la corrélation entre le contenu organique, (la fraction <20 m) et PCB (somme des contenus de 24 composants individuels), p, p-DDE, HCB, -HCH et Lindane (-HCH) est décrite. D'importantes quantités de matières organiques et de particules fines ont été trouvées à Fladen Ground, Kattegat/Skagerrak/Norwegian Channel et dans le baie intérieur allemende. Les composants organiques qui se trouvent dans le Kattegat/ Skagerrak/Norwegian Channel sont probablement soumis à la dégradation par des organismes (benthic), et aussi à une minéralisation par bactéries. Les usines de métallurgie du Sud de la Norvège et la rivière l'Ems et l'Elbe sont des points sources pour HCB, les zones de boue au Sud Est de Helgoland sont des points sources pour HCB et PCB. Aucun point source ne fût trouvé cependant pour p, p-DDE. Le contenu en sédiment de PCB, p, p-DDE et HCB montre une bonne corrélation avec la quantité de matière organique et la fraction <20 m. -HCH et Lindane ne sont que peu absorbés dans les sédiments.

     

Analytical study of the energy rate balance equation for the magnetospheric storm-ring current

We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, E_y, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of E_y, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, _m, are found to be very sensitive to the adopted efficiency factor, , decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of and the resulting values for the decay time parameter, _r, do not vary greatly as varies. As a consequence, the relative values of between the main and the recovery phase, _m/_r, can be greater or smaller than one as varies from 0.1 to 1.     

Friction constitutive law with rate and state dependences

A general formula for the Dieterich-Ruina friction constitutive law with rate and state (n-state variables,n=1, 2,...) dependences has been obtained and discussed under the assumption that the slip acceleration a varies ion a linearly with the slip displacement , namelya = a ₀ + (-₀). Wherea ₀, ₀ are initial constants, is the acceleration rate and constant.a ₀ and may be arbitrary constants (positive, negative or zero).The extreme value of frictional resistance and the existence condition of the extreme value, which are very important and govern to some degree the motion process of a frictionally slipping mechanical system, have been analyzed. A critical value _c which is the measure of the velocity weakening and velocity strengthening of the mechanical system, and its properties and the relationship to the extreme problem have been studied. Again, according to the critical value _c, the concepts of light or strong velocity weakening (or strengthening) are introduced.A possibly new phenomenon that frictional resistance may vary in some kind of decayed oscillation is found. Finally, the condition for the smallest frictional resistance for a slipping mechanical system with nonuniform acceleration has been obtained.     

Single- and inter-station transfer functions for non-synchronous geomagnetic arrays-II. the results of the field data analysis

Summary The theory of methods of computing single- and inter-station transfer functions in both the spectral and time domains was developed in paper[1]. Both approaches are applied to the variation data recorded at field stations along two non-simultaneous profiles traversing the eastern margin of the Bohemian Massif, where a zone of anomalous induction seems to mark an important geological boundary of formations with different histories of development. The results of both analyses are found to coincide within reasonable bounds of 20–30% in the principal induction characteristics.

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u m¶rt; ama ¶rt;-u -mau n¶rt;am u nma u am a ua am[1]. am nua m am a n¶rt;¶rt;a nu ¶rt; aaua ¶rt;aaum auau aumua a n mau ¶rt; u nu, nau m au aua,¶rt; aa a aa u¶rt;uu. a, u¶rt;u, mamau a¶rt;a ¶rt; ¶rt; ¶rt;uuau au umuu aumu. mam aau nma u am auam a 20–30% ¶rt; u¶rt;u naam.

     

Linear stability of a non-constantly stratified horizontal layer penetrated by an azimuthal magnetic field

Summary The convection in a rapidly rotating, electrically conducting, horizontal fluid layer, non-constantly stratified and penetrated by an inhomogeneous magnetic field, is studied. The convection is investigated for various ratios of the thickness of the stable and unstable stratified part of the layer. The thermal model of the layer, as well as the analysis of the results have been treated with regard to the physical conditions in the liquid core of the Earth.

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am u¶rt;m u m aa mn¶rt; u¶rt;uma nm mamuuau, nua ¶rt;¶rt; aum n. u u¶rt;m ¶rt; a mu m u mu u mu mamuuuao amu . ua ¶rt; , a u aau mam, n¶rt;a anm uuu u u¶rt; ¶rt; u.

     

Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust

Permeability exerts significant control over the development of pore pressure excess in the crust, and it is a physical quantity sensitively dependent on the pore structure and stress state. In many applications, the relation between permeability and effective mean stress is assumed to be exponential and that between permeability and porosity is assumed to be a power law, so that the pressure sensitivity of permeability is characterized by the coefficient and the porosity sensitivity by the exponent . In this study, we investigate experimentally the dependence of permeability on pressure and porosity in five sandstones with porosities ranging from 14% to 35% and we review published experimental data on intact rocks, unconsolidated materials and rock fractures. The laboratory data show that the pressure and porosity sensitivities differ significantly for different compaction mechanisms, but for a given compaction mechanism, the data can often be approximated by the empirical relations. The permeabilities of tight rocks and rock joints show relatively high pressure sensitivity and low porosity sensitivity. A wide range of values for and have been observed in relation to the mechanical compaction of porous rocks, sand and fault gouge, whereas the porosity sensitivity for chemical compaction processes is often observed to be given by 3. We show that since the ratio / corresponds to the pore compressibility, the different dependences of permeability on porosity and pressure are related to the pore structure and its compressibility. Guided by the laboratory data, we conduct numerical simulations on the development of pore pressure in crustal tectonic settings according to the models ofWalder andNur (1984) andRice (1992). Laboratory data suggest that the pressure sensitivity of fault gouge is relatively low, and to maintain pore pressure at close to the lithostatic value in the Rice model, a relatively high influx of fluid from below the seismogenic layer is necessary. The fluid may be injected as vertically propagating pressure pulses into the seismogenic system, andRice's (1992) critical condition for the existence of solitary wave is shown to be equivalent to >1, which is satisfied by most geologic materials in the laboratory. Laboratory data suggest that the porosity sensitivity is relatively high when the permeability is reduced by a coupled mechanical and chemical compaction process. This implies that in a crustal layer, pore pressure may be generated more efficiently than cases studied byWalder andNur (1984) who assumed a relatively low porosity sensitivity of =2.     

Intensity of vlf and elf phenomena in the outer ionosphere

Summary Spatial and temporal variability of natural ELF-VLF phenomena field intensity has been studied using data from Interkosmos satellites with apogees below 2000 km. The results show a distinct dependence of the diurnal variation and latitudinal intensity distribution on the magnetic activity. The L-value at which the maximum of plasmaspheric noise occurs, e.g., correlates better with the Dst-index than with other indicies.

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mama u a auau anmu n mm - u a ua a uu ¶rt;a uu m an u 2000 . mam naam m auum m ¶rt;a u um an¶rt;u uu m aum amumu. uua L-naama, a m nm au naa uu, um Dst-u¶rt; ¶rt;uu u¶rt;auaum amumu.

     

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.

     

On the general theory of equivalent projection

Summary The paper deals with the general theory of equivalent projection. By specifying the results derived, special cases may be treated. Examples of methods are described which enable free functions to be found in a general solution. A detailed procedure of solving the problem with a numerical example is given, taking into account the optimization according to the global criterton, established for extreme angular distortion.

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aamuam a mu auu nu; ¶rt;am m am au. nu m¶rt;, nu amu ¶rt; uu uu. umam nmuauau n aamumu au, ma ¶rt; aua uau , u nuam n¶rt; m¶rt; u u nu.

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Dedicated to Professor Frantiek Fiala on the Centenary of his Birthday     

Analysis of the magnetometric vibration method

n¶rt;m mu ¶rt; mu uau aumma. mu aum u mu u m u mu auauu m¶rt;a. u¶rt;m u naam nm nmmuna uau aumma.