“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.     

Isotopic ratio and concentration of sulfur in the undersaturated alkaline magmas of Vulture Volcano (Italy)

Both the ³⁴ value and the total S content of products from Vulture Volcano, Italy are mainly controlled by the separation of S gases, predominantly SO₂, from high f _O2magmas containing S predominantly as SO _2- ⁴ . The addition of evaporites to such magmas appears to be a relatively uncommon and limited phenomenon. The total S content of the most primitive product of Vulture Volcano (5600 mg/kg) is very high. The high ³⁴S value of 4 indicates an origin through the partial melting of a mantle containing high S, enriched in ³⁴S of unknown origin.     

Wave propagation in a uniformly rotating fluid

Summary An unsteady flow generated by a harmonically oscillating pressure distribution of frequency acting on the paraboloidal free surface of an inviscid, incompressible fluid rotating with uniform angular velocity has been investigated. It is shown that case (i), >2 , corresponds to the usual surface waves, and case (ii), <2 , in contrast to the surface waves, corresponds to the inertial waves which are originated entirely due to rotation and have no counterpart in a non-rotating fluid motion. An explicit solution of the problem related to the above cases are obtained by the joint Laplace and Hankel transforms treatment in conjunction with asymptotic methods. The significant effects of the Coriolis force and the curvature of the free surface on the wave motions have been investigated. A comparison is made between the solutions of the problems with the horizontal and the paraboloidal free surface curvature. The analysis is concluded by exihibiting the characteristic features of the wave motions.     

Study of the velocity conditions in the earth's crust in the regions of the Bohemian massif and the carpathian system along international profiles VI and VII

u¶rt;m n uu ¶rt;u m n u ma n¶rt;aa, nu m¶rt;u u ¶rt;uau. n nuu uu ¶rt;u m n ¶rt;a¶rt; nu NoNo VI, VII. u au u n m a (x, H), ¶rt;auu an¶rt;u ¶rt;u m ¶rt; m¶rt; nu, u mua m nuu (H), aamuu an¶rt;u m a mua ¶rt;¶rt; uu ¶rt;uua u a. u a ¶rt;u m (x, H) amm ¶rt;uuu (u) a m¶rt; nu. m¶rt; u, auuu aau, om aamuam muau an¶rt;u ¶rt;u m, uu , n m u m nu aamuu a nmu, am ma a¶rt;am aa uu ¶rt;u m. aa u a¶rt;u n nu NoNo VI u VII u umuu ¶rt;a. mua m nu (H) num m ¶rt;u amu aua u anam um, ¶rt;a amu ¶rt;aua u ¶rt;- nma (nu No VII). ma ¶rt; ¶rt; aua, maa numa nm m u numa nm ma¶rt;um, a unaa nuu umnmauu a¶rt;uu ¶rt;uau amu aua.     

Origin of the fumarolic fluids of Vulcano Island,Italy and implications for volcanic surveillance

Variations in D and ¹⁸O values with H₂O contents and outlet temperatures indicate that the fumaroles of La Fossa crater have discharged mixtures of magmatic water and marine hydrothermal water, since 1979. The contribution of meteoric water was low in the period 1979–1982 and very low afterwards. The ¹⁸O values of the marine-hydrothermal component of +5 to +7.2 are due to isotopic exchange with the ¹⁸O-rich silicates of the rocks under high-temperature and low-permeability conditions. The ¹⁸O value of the magmatic end-member is generally +3.5 to +4.3, although values as high as +5.5 to +6.5 were reached in the summer of 1988, when magma degassing appears to have extended into the core of the magma body. The D values of the end-member were close to -20, typical of andesitic waters. Both the isotopic values and chemical data strongly support a dry model, consisting of a central magmatic gas column and a surrounding hydrothermal envelope, in which marine hydrothermal brines move along limited fracture zones to undergo total evaporation on approaching the conduits of magmatic fluids. The vents at the eastern and western boundaries of the fumarolic field are fed by fluids whose pressure is governed by the coexistence of vapor, liquid and halite, giving rise to a high risk of phreato magmatic explosions, should magma penetrate into these wet environments. Most La Fossa eruptions were triggered by an initial hydrothermal blast and continued with a series of phreatomagmatic explosions. The fluids discharged by the Forgia Vecchia fumaroles are mixed with meteoric water, which is largely evaporated, although subordinate loss of condensed steam may be responsible for scrubbing most of the acidic gas species. The temperatures and pressures, and the risk of a sudden pressure increase, are low. A boiling hydrothermal aquifer at 230° C is present underneath the Baia di Levante beach. This area has a minor risk of hydrothermal explosions.     

On the further generalization of trochoidal frontal waves

Summary A non-linear model of trochoidal waves is presented which represents a geometrical and kinematical generalization of Gerstner's waves and of the results of[2–4].

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¶rt;aam ¶rt; mu¶rt;a , ma m u ma u am[2–4] mu umuu u uamu mm ¶rt;u.

     

On the determination of the European gravimetric geoid

Summary The method of the automated computation of the gravimetric deflections of the vertical and of the geoidal heights for the European region is described. The work was carried out during the period 1986–1988 by the Topographic Service of the Czechoslovak Army. The computation applies to 20 sheets of the international map 1:1 000 000 (total area of =16^c, =30^c - see Fig. 1). The mean values of the free-air anomalies for each surface element =5, =7.5, approximately 9 × 9 km, were used with radius of integration of 300 km.     

Offset dependent amplitudes: Effects of wavefront and reflector curvature in homogeneous and layered media

We analyze the effect of reflector curvature on the angular dependence of reflection amplitude using ray theory. Defining the curvature effect, CE, as the ratio of reflected amplitude from a curved boundary to that from a flat boundary at the same depth, we obtain 1/CE ²=(1+Z/a ^xcos²)(1+Z/a ^y), where is the angle of incidence,Z the depth of the boundary, anda ^x anda ^y are the principal radii of curvature of the reflector in the plane, of incidence and in the perpendicular plane, respectively. At =0 this reduces to the formula given byHilterman (1975). The angular dependence ofCE involves onlya ^x, which appears to shorten at wider angles. This causes an augmentation of the dimming effect of an anticlinal geometry at far offsets. For synclinal structures, the amplitude increases with offset when |Z/a^x|<1 and decreases with offset when |Z/a^x|<1.In addition we examine the effects of wavefront curvature and of a layered overburden in modulating the curvature effect. We find that there is a significant difference in the curvature effect between plane waves and spherical waves impacting on a curved boundary. In addition results are given showing simple examples of the effect of layered overburden in distorting the curvature effect of a horizontal, but curved, reflector and, more interestingly, of a dipping, but planar, bed. These results strongly suggest that we still have but a rudimentary knowledge, in general, of the combined effects of wavefront curvature, reflector curvature and overburden layering in modulating and/or producing amplitude with offset behavior observed on seismic records.     

Hydromagnetic spin-up under conditions of the earth's core-mantle boundary

Summary The disturbances of the velocity and magnetic fields close to the Earth's core-mantle boundary, caused by sudden irregular changes in the Earth's rotation, are investigated. The problem leads to the investigation of the structure of the Ekman-Hartman hydromagnetic boundary layer, the magnetic diffusive region and the currentless region. Precise Laplacean inversions of the images of all disturbances in the Earth's core-mantle system are obtained for the limiting case of a zero magnetic Prandtl number, =0. The disturbance of the velocity in the direction of the axis of rotation (Ekman suction) in the currentless region has the nature of inertial oscillations with a frequency of 2. Additional disturbances (with respect to the case of =0) of the velocity in the azimuthal and radial directions, particularly for the EHL and MDR region, are determined for the case when 0< « 1. The disturbance in the velocity again has the character of inertial oscillations with the frequency 2, being exponentially damped in EHL asexp (–2²t) and in MDR asexp (–²t).     

The periodicity of aurorae in the years 1001–1900

Summary The sequence of aurorae, observed at latitudes up to 55° between the years 1001 and 1900 was processed by methods of spectral analysis. The same methods were applied to parts of various duration of this interval. The periods predominant in the time series under investigation were determined. In all the selected parts of the interval, these periods are always located within the same frequency band. Their position is related to the periods corresponding to mutual conjunctions of the large planets.

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¶rt; n uu, a¶rt;au a uma ¶rt; 55° nu¶rt; 1001–1900, ama nu nu m¶rt; nma aaua. a n¶rt; ¶rt; am a ¶rt;u m umaa. u n¶rt; na¶rt;au nu¶rt; u¶rt;a ¶rt;a. mu nu¶rt; ¶rt; a am umaa ¶rt;a a¶rt;m ¶rt;ua ¶rt;uaan amm. nu mum nu¶rt;au, mmmuu au u u nam.

     

Axially symmetric flow round a non-conducting obstacle in the shape of a hemisphere by a homogeneous electrical current field

m amamu n¶rt;ma au ¶rt; nmuaa mu n ma a, ¶rt;a ¶rt;¶rt; maua mu n ¶rt; nmam ¶rt;um n¶rt; nnmmu n. u m umau n aa mau a, m m nmmu ma nu ¶rt;¶rt; n naa u umuu n. maa a¶rt;aa a u um ¶rt;uam. a u nu¶rt;m um ua u au, nu u n a auu mam, n¶rt;ma [5, 6]. m um nu num m amamu au ¶rt; nmuaa mu n, n¶rt;mau u¶rt; ¶rt;a nu a¶rt;a.     

An experimental study of the seismic ringing of the moon

Summary The seismic records obtained from the seismic station ALSEP (Apollo Lunar Surface Experiments Package) on the moon, due to the impact of the LM (Lunar Module) of the Apollo XII and the S-IVB rocket state of the Apollo XIII moon missions show the seismic signals decaying in amplitude slowly with time. The observed ringing phenomenon on these records along with the fact that the disturbances lasted for a long period were unexpected. Laboratory experiments on homogeneous spheres have produced this ringing. The physical explanation of this may be complicated, and it is quite possible that this observed ringing is due to the combined effect of the filtering, reflections and scattering of energy due to the curvature of the homogeneous (in the seismic sense) sphere. The explanation of the ringing of the moon may very well lie in such a study.     

The role of the tropopause in stratospheric-tropospheric exchange processes

Summary In a previous studyReiter andMahlman [37]³) have estimated the amount of stratospheric air intruding into the stable layer of the jet stream front in a case of cyclogenesis not accompanied by surface radioactive fallout. In the present report the same case is examined on a more general basis. Outflow from, as well as inflow into, the stratosphere is estimated over the entire thickness of the tropopause gap.The research reported in this paper was carried out under contract AT(11-1)-1340 with the U.S. Atomic Energy Commission.     

On the heat flux related to stretching in the NW-Mediterranean continental margins

Summary The surface thermal flux of the continental margins of the northwestern Mediterranean Sea is interpreted on the basis of a 1-D instantaneous pure shear stretching model of the lithosphere in terms of three components: the background heat flowing out from the asthenosphere (38 mW m^–2), the transient contribution depending on the rift age and extension amount (35 mW m^–2 at the most), and the contribution due to the radiogenic elements of the lithosphere. The radiogenic component is estimated at the continental margins of the Ligurian-Provençal basin and Valencia trough, and in the surrounding mainland areas by means of available data of surface heat generation from Variscan Corsica, Maures-Estérel and the Central Massif along with a geophysical-petrological relationship between heat production and seismic velocity. The lithosphere radiogenic heat contribution q_l decreases with the thinning factor according to the exponential law: q_l() = a exp(-b), in which factor b is greater for that part of the lithosphere below the uppermost 10 km. Considering also the heat generated by radioactive isotopes in sediments, the stable Variscan lithosphere produces an average thermal flux of 30 mW m^–2 which decreases by about one half where the lithosphere is thinned by one third. Although the surface heat generation is 2·1 – 3·3 µW m^–3 in the Maures-Estérel massif — excepting small outcrops of dioritic rocks with lower heat production — and 1·8 µW m^–3 for most of Corsica, the radiogenic heating within the lithosphere for such areas is nearly the same and does not explain the higher heat flux of the Corsica margin. This asymmetric thermal pattern with surface heat flux which is 10 – 15 mW m^–2 higher than predictions is probably of upper mantle origin, or can be ascribed to penetrative magmatism.     

EISCAT measurements of the solar wind

EISCAT observations of interplanetary scintillation have been used to measure the velocity of the solar wind at distances between 15 and 130R (solar radii) from the Sun. The results show that the solar wind consists of two distinct components, a fast stream with a velocity of 800 km s^–1 and a slow stream at 400 kms^–1. The fast stream appears to reach its final velocity much closer to the Sun than expected. The results presented here suggest that this is also true for the slow solar wind. Away from interaction regions the flow vector of the solar wind is purely radial to the Sun. Observations have been made of fast wind/slow wind interactions which show enhanced levels of scintillation in compression regions.     

Polarization characteristics of ulf electromagnetic waves in an inhomogeneous anisotropic ionosphere of the Earth

Summary Drawing on [6], the height profiles of local complex polarization and local polarization characteristics of electromagnetic waves in several models of the Earth's ionosphere are analysed. The profiles were obtained with the aid of a computer program for modelling propagating [5]. The analysis was carried out a) for a flxed given configuration of the external magnetic field of the EarthB ₀ at a number of discrete frequencies f<5 Hz, b) for chosen model of the ionosphere at a fixed given frequency f=1 Hz and in connection with a change of the dip of the lines of force and of the magnitude of the external (homogeneous) magnetic field|B ₀|, c) for various models at f=1 Hz and a varying configuration of the external magnetic field, reflecting the change in geomagnetic latitude. The results of the analysis will serve as an aid to the interpretation of the results of solving the problem of wave propagation through the ionosphere.     

On conditions for the unique inversion of seismic travel-time curves

It is shown that when the travel-time curve of a refracted wave from a surface source is known and at least one of the following conditions is satisfied, i.e. when

1. the travel-time curve of a wave reflected from a horizontal interface lying below the deepest low velocity layer is known, or
2. the travel-time curve of a wave from a deep source situated below the deepest low velocity layer is known, or
3. the measureH(u)=mes {z∶z≥0,v ^?1 (z)≥u} is analytical in some segment [c, d], where \(0< c< d< \infty , c< a_n , H(a_n ) = \bar z_n ,\bar z_n\) is the depth of the lower end of the deepest low velocity layer and in the interval [c, ∞) an analytical functionH(u)) exists which providesH(u)≡H(u)) ifu∈[c, d], then (1) velocityv(z) outside the low velocity layers and (2) the measureH _k (u)=mes {z∶z∈L _k,v ^?1 (z)≥u} for each low velocity layerL _k,k=1, 2, ..., n, are defined unambiguously.

     

Approximating the Distribution of Pareto Sums

Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human-induced processes, and the sums of heavy tailed rvs represent a particularly important construction in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 – x^– for 1/2 < < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sums distribution. Two of the proposed approximations are based on the Generalized Central Limit Theorem, which establishes the general limit for the sums of independent identically distributed rvs in terms of stable distributions; these approximations work well for large numbers of summands. Another approximation, which replaces the sum with its maximal summand, has less than 10% relative error for the upper quantiles when < 1. A more elaborate approach considers the two largest observations separately from the rest of the observations, and yields a relative error under 1% for the upper quantiles and less than 5% for the median. The last approximation is specially tailored for the lower quantiles, and involves reducing the non-Gaussian problem to its Gaussian equivalent; it too yields errors less than 1%. Approximation of the observed cumulative seismic moment in California illustrates developed methods.     

A method of searching for radial velocity anomalies in the lower mantle using interfering signals

Radial velocity anomalies in the lower mantle that give rise to triplications in the travel-time curve for short-periodP waves will produce arrivals havingdT/d values that differe by roughly 0.2–0.5 s/deg. The first two arrivals associated with such triplications will be separated by less than one second over a distance range of 4°–10° they may not, therefore, be separable visually on single seismograms, so that their presence can only be inferred from some measurable property that depends on their mutual interference. If there are lateral variations in the regions of anomalous velocity gradients, the interfering signals will also have different azimuths of arrival. Using two synthetic wavelets we have investigated the effect of interference on bothdT/d and azimuth measurements at the Yellowknife Array. We found that if the interfering pulses have a dominant frequencyv, there is a range of time separations (0.30/v0.55/v) over which the measureddT/d and azimuth values may fluctuate by much more than the differences indT/d and azimuth between the interfering signals. We have evaluated the following empirically defined functions for three different primary signals, and for three different relative amplitudes of the interfering signals:f (t), the drift function, which expresses how the measured slownesses,p, and azimuths, , differ from the slownesses and azimuths of the primary wavelets; f(), the range function, which describes the behaviour of the upper and lower bounds ofp and as a function of the difference in arrival times of the signals, andf , studied the properties of these functions, and have outlined how these properties provide criteria based on the numerical and statistical characteristics of the arrival vectors, and on the waveform of the signal that will enable small radial velocity anomalies to be more clearly delineated.Contribution No. 863 from the Earth Physics Branch.     

Source characteristics of the 1992 Nicaragua tsunami earthquake inferred from teleseismic body waves

We analyzed the broadband body waves of the 1992 Nicaragua earthquake to determine the nature of rupture. The rupture propagation was represented by the distribution of point sources with moment-rate functions at 9 grid points with uniform spacing of 20 km along the fault strike. The moment-rate functions were then parameterized, and the parameters were determined with the least squares method with some constraints. The centroid times of the individual moment-rate functions indicate slow and smooth rupture propagation at a velocity of 1.5 km/s toward NW and 1.0 km/s toward SE. Including a small initial break which precedes the main rupture by about 10 s, we obtained a total source duration of 110 s. The total seismic moment isM _o =3.4×10²⁰ Nm, which is consistent with the value determined from long-period surface waves,M _o =3.7×10²⁰ Nm. The average rise time of dislocation is determined to be 10 s. The major moment release occurred along a fault length of 160 km. With the assumption of a fault widthW=50 km, we obtained the dislocationD=1.3 m. From andD the dislocation velocity isD=D/0.1 m/s, significantly smaller than the typical value for ordinary earthquakes. The stress drop =1.1 MPa is also less than the typical value for subduction zone earthquakes by a factor of 2–3. On the other hand, the apparent stress defined by 2E _s /M _o , where andE _s are respectively the rigidity and the seismic wave energy, is 0.037 MPa, more than an order of magnitude smaller than . The Nicaragua tsunami earthquake is characterized by the following three properties: 1) slow rupture propagation; 2) smooth rupture; 3) slow dislocation motion.