Ellipsoidal vertical deflections and ellipsoidal gravity disturbance: Case studies

First, we present three different definitions of the vertical which relate to (i) astronomical longitude and astronomical latitude as spherical coordinates in gravity space, (ii) Gauss surface normal coordinates (also called geodetic coordinates) of type ellipsoidal longitude and ellipsoidal latitude and (iii) Jacobi ellipsoidal coordinates of type spheroidal longitude and spheroidal latitude in geometry space. Up to terms of second order those vertical deflections agree to each other. Vertical deflections and gravity disturbances relate to a reference gravity potential. In order to refer the horizontal and vertical components of the disturbing gravity field to a reference gravity field, which is physically meaningful, we have chosen the Somigliana-Pizzetti gravity potential as well as its gradient. Second, we give a new closed-form representation of Somigliana-Pizzetti gravity, accurate to the sub Nano Gal level. Third, we represent the gravitational disturbing potential in terms of Jacobi ellipsoidal harmonics. As soon as we take reference to a normal potential of Somigliana-Pizzetti type, the ellipsoidal harmonics of degree/order (0,0), (1,0), (1, − 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. Fourth, we compute in all detail the gradient of the gravitational disturbing potential, in particular in orthonormal ellipsoidal vector harmonics. Proper weighting functions for orthonormality on the International Reference Ellipsoid are constructed and tabulated. In this way, we finally arrive at an ellipsoidal harmonic representation of vertical deflections and gravity disturbances. Fifth, for an ellipsoidal harmonic Gravity Earth Model (SEGEN: http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip) up to degree/order 360/360 we compute the global maps of ellipsoidal vertical deflections and ellipsoidal gravity disturbances which transfer a great amount of geophysical information in a properly chosen equiareal ellipsoidal map projection.     

To Pizzetti's Theory of the Level Ellipsoid

The topic of the Earth's reference body, which has now been established as Pizzetti's level rotational ellipsoid, is analysed. Such a body is fully determined by four parameters: a, GM, J ₂ and . At present, the largest discrepancy in determining these parameters occurs in the value of a, which may in future be replaced by the gravity potential of the mean sea level W _o, with respect to Brovar's condition.Pizzetti's four parameters of the reference body are determined by solving the Dirichlet boundary value problem. The Dirichlet problem has only a unique solution, which, however, can be expressed in infinitely many ways. It turns out that the most important part in the form of the solution is played by Lamé's conditions, which determine the type of ellipsoidal coordinates.The solutions given by Pizzetti, Molodensky and another variant are considered. The last variant leads to a simple formula for the potential of the reference ellipsoid, but the formulae for Lamé's coefficients are inconvenient. Of course, all the methods lead to identical solutions, but some of them are more convenient for the historical use of logarithms, whereas others are more appropriate for use in computers.     

Ellipsoidal Stokes Boundary-Value Problem with Ellipsoidal Corrections in the Boundary Condition

We would like to solve the Stokes boundary-value problem taking into consideration the ellipsoidal corrections in the boundary condition in ellipsoidal coordinates The original problem, i.e., the ellipsoidal Stokes boundary-value problem has been solved by Martinec and Grafarend (1997) We use the same philosophy expressed by Martinec (1998) to solve the spherical Stokes boundary-value problem with ellipsoidal corrections in the boundary condition We wish to show the magnitude of the integration kernel describing the effect of the ellipsoidal corrections in the boundary condition in a cap around the computational point.     

Far-Zone Contribution in Ellipsoidal Stokes Boundary-Value Problem

In the first attempt to solve the Stokes boundary-value problem in ellipsoidal coordinates numerically (Ardestani and Martinec, 2000), we focused on the near-zone contribution since the effect of the ellipsoidal Stokes function in the far-zone contribution is not considered. We present a method for solving the ellipsoidal Stokes integral in far-zone contribution. The numerical results of computing the magnitude of this term for an area in north of Canada are presented.     

Seismic moment tensor representations and radiation patterns in unbounded media with elipsoidal cavities driven by low frequency pressure

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Die Matrix der Fische XV. Über das Verhalten von Elritzen,Phoxinus laevis Agass., gegenüber Licht verschiedener Wellenlänge

Summary 30 minnows (Phoxinus laevis Agass.) have been simultaneously exposed to nearly monochromatically filtered light with the wave lenghts of λ-640 mμ (red), λ=580 mμ (yellow), λ=520 mμ (green), λ=491 mμ (blue) and λ=444 mμ (violet). The resulting disturbances of the fish were measured with seismic methods. We observed a considerable reaction in red and violet light; however, the fish remained relatively undisturbed by yellow, green and blue radiation. These reactions occurred, although the difference between the wave lenghts for maximum transmittance of the red and yellow filter was 60 mμ, and of the blue and violet filter 47 mμ only.

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14. Mitteilung siehe diese Zeitschrift13, 31–35 (1951).     

椭球坐标系下多震相地震射线追踪

地震射线追踪方法技术在地震学领域有着较为广泛的应用,然而大多数算法建立在直角坐标系或球坐标系下,实际地球并非完美的球体,而是两极略扁的椭球体,因此,球坐标系下计算结果与真实情况存在一定误差.传统的做法一般是在球坐标系下进行计算,而后进行椭球校正.本文提出了一种直接在椭球体模型中采用分区多步最短路径算法进行多震相地震射线追踪的方法技术,实现了椭球坐标系下多震相地震波射线路径追踪和走时计算.与解析解的对比表明:该算法具有较高的计算精度,适用于任意形状的椭球体,且不需要进行额外的走时校正.数值模拟结果表明,计算所得P波和PcP反射波的走时与AK135走时表的误差小于0.1 s.当震中距较大时,使用球对称模型和椭球体模型计算所得的走时差异显著,说明采用椭球坐标系的必要性.     

Geoid Determination through Ellipsoidal Stokes Boundary-Value Problem

Martinec and Grafarend (1997) have shown how the construction of Green's function in the Stokes boundary-value problem with gravity data distributed on an ellipsoid of revolution is approached in the O(e ₀ ² )-approximation. They have also expressed the ellipsoidal Stokes function describing the effect of ellipticity of the boundary as a finite sum of elementary functions. We present an effective method of avoiding the singularity of spherical and the ellipsoidal Stokes functions, and also an analytical expression for the ellipsoidal Stokes integral around the computational point suitable for numerical solution. We give the numerical results of solving the ellipsoidal Stokes boundary-value problem and their difference with respect to the spherical Stoke boundary-value problem.     

Multifractal objective analysis: conditioning and interpolation

 We investigate various ways of statistically estimating multifractal fields from sparse data. First, the problem is set in the general framework of conditional expectations, and the effect of (multi) fractal sampling on the statistics of the measured process is investigated, showing how analytical expressions describing the statistical properties of the phenomenon should be modified by the sampling. Then, several techniques are introduced, our goal being to estimate the intensity of a field at resolution λ, given samples of the process collected by networks at higher resolutions Λ>λ. The general strategy underlying all the estimating techniques presented is to approximate the unknown field values at resolution λ by means of most likely estimates conditional to the available information at resolution Λ>λ. Finally, the procedures are tested on simulated lognormal multifractal fields sampled by means of fractal networks, and the propagation of the errors in a scaling framework is also discussed. These techniques are necessary for estimating geophysical processes in regions where no monitoring stations are present, a scenario often encountered in practice, and may also be of great help in studying natural hazards and risk assessment.     

Solution to the Stokes Boundary-Value Problem on an Ellipsoid of Revolution

We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e₀ of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.     

A probabilistic model of seismicity: Kamchatka earthquakes

The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω, $ \tilde F $ \tilde F P}. Each earthquake is treated as an outcome ω _i in the space of elementary events Ω whose cardinality for the period under consideration is given by the number of events. In turn, ω _i is characterized by a system of random variables, viz., energy class k_i, latitude φ _i , longitude λ _i , and depth h _i . The time of an outcome has been eliminated from this system in this study. The random variables make up subsets in the set $ \tilde F $ \tilde F and are defined by multivariate distributions, either by the distribution function $ \tilde F $ \tilde F (φ, λ, h, k) or by the probability density f(φ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequency interpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law for probability density f(k), where the initial value of the distribution function f(k ₀) is the basic data [Bogdanov, 2006] rather than the seismic activity A ₀, we proceed to show that for different intervals of coordinates and time the distribution f _elim(k) of an earthquake catalog with the aftershocks eliminated is identical to the distribution f _full(k), which corresponds to the full catalog. It follows from our calculations that f ₀(k) takes on nearly identical numeral values for different initial values of energy class k ₀ (8 ≤ k ₀ ≤ 12) f(k ₀). The difference decreases with an increasing number of events. We put forward the hypothesis that the values of f(k ₀) tend to cluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesis that statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a distribution function with the initial value f(k ₀) = 2/3. We discuss statistical distributions of earthquake hypocenters over depth and the epicenters over various areas for several periods     

Geoid Determination Through Ellipsoidal Stokes Boundary-Value Problem by Splitting Its Solution to the Low-Degree and the High-Degree Parts

The ellipsoidal Stokes boundary-value problem is used to compute the geoidal heights. The low degree part of the geoidal heights can be represented more accurately by Global Geopotential Models (GGM). So the disturbing potential is splitted into a low-degree reference potential and a higher-degree potential. To compute the low-degree part, the global geopotential model is used, and for the high-degree part, the solution of the ellipsoidal Stokes boundary-value problem in the form of the surface integral is used. We present an effective method to remove the singularity of the high-degree of the spherical and ellipsoidal Stokes functions around the computational point. Finally, the numerical results of solving the ellipsoidal Stokes boundary-value problem and the difference between the high-degree part of the solution of the ellipsoidal Stokes boundary-value problem and that of the spherical Stokes boundary-value problem is presented.     

Higher degree moment tensor inversion of Mani earthquake using far-field broadband recording

Breakthrough point source model, extended earthquake source model is used to calculate more seismic source parameters in this paper. We express seismic source using higher degree moment tensors, to reduce a large number terms originally presenting in higher degree moment tensor representation, Haskell rupture model is used. We inverted the source parameters of Mani earthquake in Tibet using broad-band body wave of 32 stations of Global Seismograph Network (GSN), the results show that it is a strike-slip fault, rupture direction is 75° , rupture duration is 19 s, the fault plan is f =77° , d =88° , l =0° , the auxiliary plane is f =347° , d =90° , l =178° , and the fault dimension is 47 km′ 28 km. These results will give new quantitative data for earth dynamics and have practical meaning for seismic source tomography research.     

Investigation of the broadband ELF turbulence by observations of the FAST satellite

Scaling properties of variable electric fields in the topside ionosphere have been investigated on scales s from ∼30 m to 2 km by FAST electric field observations with sample rate of 512 s⁻¹, in sixteen events of the broadband ELF turbulence. It is shown that down to scales of a few hundred meters, the power of turbulent electric fluctuations is a power law, ∼s ^α. Scaling index α derived from the slope of logarithmic diagrams (LD) constructed by the discrete wavelet transform of data can be estimated as α = 2.2 ± 0.3, which is close to α estimate earlier reported for scales 1–30 km by electric field observations of the Dynamics Explorer 2 satellite. The behavior of α index is analyzed near the scale of the order of electron inertial length λ_e = c/ω₀ (ω₀ being the electron plasma frequency). At altitudes considered (700–2500 km), λ_e makes 100–900 m. We demonstrate that at scales ≤λ_e, a decrease of LD slope and deviation from the power law are typically observed. As pointed out in the discussion, this feature cannot be identified as a transition to the diffusion range, where dissipation of the turbulence occurs.     

Ray-centred coordinate systems in anisotropic media

Whereas the ray-centred coordinates for isotropic media by Popov and Pšenčík are uniquely defined by the selection of the basis vectors at one point along the ray, there is considerable freedom in selecting the ray-centred coordinates for anisotropic media. We describe the properties common to all ray-centred coordinate systems for anisotropic media and general conditions, which may be imposed on the basis vectors. We then discuss six different particular choices of ray-centred coordinates in an anisotropic medium. This overview may be useful in choosing the ray-centred coordinates best suited for a particular application. The equations are derived for a general homogeneous Hamiltonian of an arbitrary degree and are thus applicable both to the anisotropic-ray-theory rays and anisotropic common S-wave rays.     

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

Source parameters of bohai earthquake, July 18, 1969 and yongshan earthquake, May 11, 1974 determined by synthetic seismograms of teleseismic P waves

The source parameters of the Bohai Sea earthquake, July 18, 1969 and Yongshan, Yunnan earthquake, May 11, 1974 were determined by full — wave theory synthetic seismograms of teleseismic P waves. P+pP+sP wereform were calculated with WKBJ approximation and real integral paths. One — dimensional unilateral, finite propagation source was also considered. By trail — and — error in comparing the theoretical seismograms with the observational ones of WWSSN stations, the source parameters were obtained as follow: for Bohai earthquake, φ=195°, δ=85°, λ=65°,M _o=0.9×10¹⁹Nm,L=59.9km.V _R=3.5km/s, ∧ _R =160°; for Yongshan earthquake, φ=240°, δ=80°, ∧=150°,M _o=1.3×10¹⁸Nm,L=48.8km,V _R=3km/s, ∧ _R =−10°, where φ is strike, δ dip angle, λ slip angle,M _o seismic moment,L rupture length,V _R rupture propagation speed. As III type fractures the faulting propagated along the fault planes, and ∧ _R is the angle from the strike to the propagation direction. Yongshan earthquake showed complexity in its focal process, having four sub—ruptures during the first 60 seconds. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,13, 1–8, 1991.     

Dynamics of solar activity and anomalous weather in summer 2010: 1. Sector boundaries: Anticyclone formation and destruction

A close synoptic relationship has been found, on the one hand, between the sector structure of the solar and interplanetary magnetic fields and, on the other hand, the structure of tropospheric parameters (the near-Earth pressure and temperature) in Central Russia during an anomalously hot anticyclonic weather in June–August 2010. It has been stated that the Earth crossed the magnetic sector boundaries in full agreement with the boundaries of meteorological parameters, structuring the formation, stabilization, and decay dynamics of anticyclones according to observations performed at IZMIRAN (Troitsk, φ = 55°, λ = 37°).     

New Early Cretaceous paleomagnetic results from Qilian orogenic belt and its tectonic implications

Lower Cretaceous red sedimentary rocks from the depositional basin of East Qilian fold belt have been collected for a paleomagnetic study. Stepwise thermal demagnetization reveals two or three components of magnetization from dark red sandstones. Low-temperature magnetic component is consistent with the present Earth Field direction in geographic coordinates. High-temperature magnetic components are mainly carried by hematite. The mean pole of 19 sites for high-temperature magnetic components after tilt-correction is λ=62.2°N, φ=193.4°E, A95=3.2°, and it passes fold tests at 99% confidence level and reversal tests at 95% confidence level. The paleopole is insignificantly different from that of Halim et al. (1998) from the same sampling area at the 95% confidence level. Compared with paleomagnetic results for North China, South China, and Eurasia, our results suggest that no significant relative latitudinal displacement has taken place between Lanzhou region and these blocks since Cretaceous time. Remarkably, the pole of Lanzhou shows a 20° clockwise rotation with respect to those of North China, South China, and Eurasia. Geological information indicates that the crustal shortening in the western part of Qilian is greater than that in eastern part. In this case, the clockwise rotation of sampling area was related to India/Eurasia collision, and this collision resulted in a left-lateral strike-slip motion of the Altun fault in north Tibetan Plateau after the Cretaceous.