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1.
Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame. 相似文献
2.
In this contribution we continue our earlier research, concerning the ellipsoidal harmonic expansions of the Earth disturbing gravitational potential and its derivatives on an external reference ellipsoid confocal with respect to the normal ellipsoid and close to it. One of the results of the previous investigation is represented by a new expression for the derivative of the Jekeli’s Legendre function of the second kind, entering the ellipsoidal harmonics in the potential derivative. The derived expression depends on two Gauss hypergeometric functions which converge better than the hypergeometric functions of other authors. In the present paper we construct another expression for the derivative of the Jekeli’s Legendre function, depending on two alternative hypergeometric functions. While our earlier hypergeometric series in the expression for the derivative of this function converge better when the orders of the terms do not exceed a half of their degrees, the series constructed in the present paper converge more rapidly when the orders surpass a half of the degrees. We deduce an improved expression for the derivative of the Jekeli’s Legendre function by combining these results and then construct a corresponding new expression for the derivative of the disturbing potential. This expression is applied for constructing non-singular expressions for the components of the gradient of the potential in the local north-oriented ellipsoidal reference frame. The new expressions for these components have no these deficiencies and the expression for the potential gradient depends on very quickly convergent hypergeometric series. 相似文献
3.
Michele Caputo 《Pure and Applied Geophysics》1964,57(1):66-82
Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×1020 cm3 sec–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste. 相似文献
4.
Summary The level rotational ellipsoid, best fitting the actual Earth, rotating with the same angular velocity around a common axis of rotation, is assumed to be a mathematical model of the real Earth. The gravity potential of this body and its derivatives in the outer space are derived by means of the generalized Pizzetti method[1]. For some analyses of the structure of the Earth we need to know the gravity anomaly and thus the gravity potential and its derivatives inside the mathematical model. These values are not defined in the classical conception. In this paper, the normal potential and its derivatives in the inner space are derived up to a certain depth, which is still of significance for gravimetric research.Dedicated to RNDr Jan Pícha, CSc., on his 60th Birthday 相似文献
5.
A. Marussi H. Moritz R.H. Rapp R.O. Vicente 《Physics of the Earth and Planetary Interiors》1974,9(1):4-6
From the point of view of consistency with the Geodetic Reference System 1967, it would be desirable that the boundary surface of a Standard Earth Model is an exact equipotential ellipsoid. This is incompatible with the requirement that it be a figure of hydrostatic equilibrium. The report investigates the relation between equipotential ellipsoids and equilibrium figures. The principal conclusion is that it is possible to find an ellipsoidal model that has the same distribution of density and flattening (more precisely, of the parameter f′ as defined in the paper) as a hydrostatic model, the deviations being only of second order in the flattening. 相似文献
6.
Summary For precise geodetic computations over larger distances the reference surface of an ellipsoid of rotation should be used. However it is often replaced by a sphere of an adequate radius. The formulae are derived from figures which usually represent the conditions in a cross-section of the ellipsoid and the reference sphere through the normal plane. Equation (9) is given for the differences s of the length of the ellipse arc of the normal section and the corresponding arc of the circle with radius R. Also Eq. (19) is given for the distance d between the ellipse of the normal section and the circle (at the end point). Both equations are applied for various radii of the reference sphere. Table 1 shows the values s, Tab. 2 and Fig. 2 give the d-values for chosen lengths. It was found that especially the distance between the ellipsoid and the sphere need not always be negligible. 相似文献
7.
Mariya I. Yurkina 《Studia Geophysica et Geodaetica》1994,38(4):325-332
Summary A relation is established between coefficients of an expansion of the gravitational potential into a series of Legendre's
function of the second kind and coefficients of an expansion of gravity anomalies on the surface of the reference ellipsoid
into a series of the same functions. This connection can be useful in geodetic computations which take into account the Earth's
flattening. 相似文献
8.
Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes’ (geopotential) coefficients.
This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional
dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects,
the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to
the Earth’s surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is
harmonic in the exterior to sea level.
This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional
expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives
of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express
the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions
of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed. 相似文献
9.
The estimation of the Earth’s gravitational potential energy E was obtained for different density distributions and rests
on the expression E = − (Wmin + ΔW) derived from the conventional relationship for E. The first component Wmin expresses minimum amount of the work W and the second component ΔW represents a deviation from Wmin interpreted in terms of Dirichlet’s integral applied on the internal potential. Relationships between the internal potential
and E were developed for continuous and piecewise continuous density distributions. The global 3D density model inside an
ellipsoid of revolution was chosen as a combined solution of the 3D continuous distribution and the reference PREM radial
piecewise continuous profile. All the estimates of E were obtained for the spherical Earth since the estimated (from error
propagation rule) accuracy σE of the energy E is at least two orders greater than the ellipsoidal reduction and the contribution of lateral density inhomogeneities
of the 3D global density model. The energy E contained in the 2nd degree Stokes coefficients was determined. A good agreement
between E = EGauss derived from Gaussian distribution and other E, in particular for E = EPREM based on the PREM piecewise continuous density model and E-estimates derived from simplest Legendre-Laplace, Roche, Bullard
and Gauss models separated into core and mantle only, suggests the Gaussian distribution as a basic radial model when information
about density jumps is absent or incomplete. 相似文献
10.
11.
Summary The present work aims at investigating the effects of anisotropy on the apparent resistivity measurements over the surface of a three layer Earth. An appropriate model with the anisotropic layer embedded between two isotropic media is chosen for study. The point current source is assumed to be present at the surface of such a model. After writing the values of the potentials in each layer, expressions for apparent resistivities for Wenner and Schlumberger configurations are derived. Also, the expressions for potential for the limiting cases in which the bottom layer is infinitely resistive or infinitely conducting are derived. It is shown that these expressions can be recast into simpler forms corresponding to isotropic layer (s) on an isotropic half-space. 相似文献
12.
Vladimir Yu Semenov 《Acta Geophysica》2015,63(1):103-124
The first decade of 21st century is characterized by the appearance of new approaches to deep induction soundings. The theory of magnetovariation and magnetotelluric soundings was generalised or corrected. Spatial derivatives of response functions (induction arrows) were obtained for the ultra-long periods. New phenomena have been detected by this method: secular variations of the Earth’s apparent resistivity and the rapid changes of induction arrows over the last 50 years. The first one can be correlated with the number of earthquakes, and the second one–with geomagnetic jerks in Central Europe. The extensive studies of geoelectrical structure of the crust and mantle were realized in the frame of a series of international projects. New information about geoelectrical structures of the crust in Northern Europe and Ukraine was obtained by deep electromagnetic soundings involving controlled powerful sources. An influence of the crust magnetic permeability on the deep sounding results was confirmed. 相似文献
13.
Tanju Akar 《水文科学杂志》2020,65(6):984-994
ABSTRACTSediment accumulation in a river reservoir is studied by stochastic time series models and analytical approach. The first-order moving average process is found the best for the suspended sediment discharge time series of the Juniata River at Newport, Pennsylvania, USA. Synthetic suspended sediment discharges are first generated with the chosen model after which analytical expressions are derived for the expected value and variance of sediment accumulation in the reservoir. The expected value and variance of the volume of sediment accumulation in the reservoir are calculated from a thousand synthetic time series each 38 years long and compared to the analytical approach. Stochastic and analytical approaches perfectly trace the observation in terms of the expected value and variability. Therefore, it is concluded that the expected value and variance of sediment accumulation in a reservoir could be estimated by analytical expressions without the cost of synthetic data generation mechanisms. 相似文献
14.
Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation 总被引：1，自引：0，他引：1
Moritz Rexer Christian Hirt Sten Claessens Robert Tenzer 《Surveys in Geophysics》2016,37(6):1035-1074
Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution. 相似文献
15.
In order to perform resistivity imaging, seismic waveform tomography or sensitivity analysis of geophysical data, the Fréchet derivatives, and even the second derivatives of the data with respect to the model parameters, may be required. We develop a practical method to compute the relevant derivatives for 2.5D resistivity and 2.5D frequency-domain acoustic velocity inversion. Both geophysical inversions entail the solution of a 2.5D Helmholtz equation. First, using differential calculus and the Green's functions of the 2.5D Helmholtz equation, we strictly formulate the explicit expressions for the Fréchet and second derivatives, then apply the finite-element method to approximate the Green's functions of an arbitrary medium. Finally, we calculate the derivatives using the expressions and the numerical solutions of the Green's functions. Two model parametrization approaches, constant-point and constant-block, are suggested and the computational efficiencies are compared. Numerical examples of the derivatives for various electrode arrays in cross-hole resistivity imaging and for cross-hole seismic surveying are demonstrated. Two synthetic experiments of resistivity and acoustic velocity imaging are used to illustrate the method. 相似文献
16.
Using higher-order ray theory, we derived exact elastodynamic Green functions for three simple types of homogeneous anisotropy. The first type displays an orthorhombic symmetry, the other two types display transverse isotropy. In all cases, the slowness surfaces of waves are either ellipsoids, spheroids or spheres. All three Green functions are expressed by a ray series with a finite number of terms. The Green functions can be written in explicit and elementary form similar to the Stokes solution for isotropy. In two Green functions, the higher-order ray approximations form a near-singularity term, which is significant near a kiss singularity. In the third Green function, the higher-order ray approximations also form a near-field term, which is significant near the point source. No effect connected with the line singularity was observed. 相似文献
17.
18.
N. A. Chujkova L. P. Nasonova T. G. Maximova 《Izvestiya Physics of the Solid Earth》2014,50(3):427-443
The formulas that allow, within the quadratic approximation, for the contribution of the anomalous masses, distributed along the height relative to the reference ellipsoid, in the Stokes parameters are derived. It is shown that the contribution of the quadratic terms is largest and commensurate, by the order of magnitude, with the linear contribution if the anomalous masses have a dipole distribution along the height. The quadratic contribution is particularly significant for Mars, where the span of relative variations in the surface topography is by an order of magnitude larger than in the Earth. The problem is solved and the method is developed for finding the depths of compensation for the topographical harmonics of different order and degree. The most probable levels of compensation for topographic irregularities are determined by the analysis of the distribution histograms for the depths of compensation. The maps of lateral distributions of the compensating masses at the selected levels are calculated. It is shown that the observed anomalous structures generate the anomalies in the internal gravity field, which may serve as a cause for the convective motion in the mantle and core of the planet. Besides, the probable nonisostatic vertical stresses in the crust and mantle of the Earth and Mars are calculated. 相似文献
19.
A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:
- - Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.
- - Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.
- - Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.
- - Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.
- - Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.
- - The removed effects at the application points are restored on the surface of reference ellipsoid.
- - Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.
- - Computation of the geoid of Iran has successfully tested this new methodology.
Keywords: Geoid computations; Ellipsoidal approximation; Ellipsoidal boundary value problem; Ellipsoidal Bruns’ formula; Satellite altimetry; Astronomical observations 相似文献
20.
Milan Burša 《Studia Geophysica et Geodaetica》1966,10(4):401-410
Summary Orbital problems of satellite geodesy are based on the assumption that the geocentric orbital elements, giving the geocentric
position of the satellite being observed, are known for the moment of observation. The determination of such elements, however,
is possible only if the observed topocentric quantities were with sufficient accuracy reduced to geocentric ones. The reduction
assumes, however, that the position of the satellite stations with respect to the centre of mass of the Earth is known. This,
however, is not known and the determination of the geocentric positions is substantially the main problem of present-day geodesy.
In the case of natural celestial bodies, relatively far from the basic body, inaccuracies in correcting for the parallax practically
do not influence the solution. But in the case of near artificial Earth satellites these inaccuracies are not negligible.
The present paper analyzes from this aspect the case when the orbital elements are calculated from one position (three coordinates)
and three velocity components. Relations are derived (14, 15, 19, 22, 32, 38) giving the exactly geocentric elements. The
way in which they differ from the well-known relations of classical celestial mechanics (10, 11, 17, 20, 24–26, 36), giving
only certain approximate elements (we call them quasi-geocentric) consists in considering parameters defining the position
of the origin and direction of the axes of the used reference system in the Earth's body. In the derivation we neglected expressions
equal in order to the squares of the coordinates of the centre of the reference ellipsoid with respect to the centre of mass
of the Earth and to the squares of the angles between the axis of rotation of the ellipsoid and the mean rotation axis of
the Earth's inertia as well between the planes of the initial (Greenwich) astronomic and geodetic meridians.
Адрес: Politickych vězňů 12, Praha 1-Nové Město. 相似文献
Адрес: Politickych vězňů 12, Praha 1-Nové Město. 相似文献