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1.
Bernhard Heck 《Studia Geophysica et Geodaetica》2011,55(3):441-454
For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the
external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free
geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since
nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely,
the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known
topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed
GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among
the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect
to the classical solution of the scalar free GBVP. 相似文献
2.
A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy
in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function
for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From
the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known
to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of
the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken
into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing
potential of an ellipsoidal Earth. 相似文献
3.
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. 相似文献
4.
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
5.
The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid. 相似文献
6.
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
7.
Summary The disturbing gravitational effect of the irregularities of the shape of the coremantle boundary are studied at a point on the Earth's surface. The effect is computed for models of local and zonal distribution of the disturbing masses under the assumption that the total mass of the Earth remains constant. Formulae giving the disturbing gravitational effect as a function of the distribution of the disturbing masses and of the position of the point of observation on the Earth's surface are derived for the individual models. Numerical values of the disturbances have been calculate in all cases. 相似文献
8.
The undulation of the geoid, the gravity anomaly and the deflection of the vertical are the three basic observations describing the shape and the gravity field of the earth. The Stokes’ formula that computes the undulation of the geoid using the gravity anomaly on the geoid under spherical approximate conditions was first put forward by Stokes[1]. According to Stokes’ theory, The Vening-Meinesz formula that computes the meridian and the prime vertical components of the deflection of the ve… 相似文献
9.
Jiří Nedoma 《Pure and Applied Geophysics》1973,110(1):1899-1910
Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT
h
, which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T
h
, which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT
h
has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements. 相似文献
10.
Hasan Yildiz 《Studia Geophysica et Geodaetica》2012,56(1):171-184
Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients
from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE
gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than
those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using
Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies (Tzz), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results
show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11
E?tv?s unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE Tzz leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison
with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from
GOCE Tzz with standard deviations of 11 mGal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that
information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately
from GOCE Tzz and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However,
using covariance function obtained from GOCE Tzz has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of
predicted minus observed gravity anomalies. Therefore, GOCE Tzz should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available,
if error estimates are needed. 相似文献
11.
Richard H. Rapp 《Surveys in Geophysics》1975,2(2):193-216
The Earth's gravity field can be determined from gravity measurements made on the surface of the Earth, and through the analysis of the motion of Earth satellites. Gravity data can be used to solve the boundary value problem of gravimetric geodesy in various ways, from the classical formulation using a geoid to the concept of a reference surface interior to the masses of the Earth to a statistical method. We now have gravity information for 10 data blocks over 46% of the Earth's surface and more than several million point measurements available.Satellite observations such as range, range-rate, and optical data have been analyzed to determine potential coefficients used to describe the Earth's gravitational potential field. Coefficients, in a spherical harmonic expansion to degree 12, can be determined from satellite data alone, and to at least degree 20 when the satellite data is combined with surface gravity material. Recent solutions for potential coefficients agree well to degree 4, but with increasing disagreement at higher degrees. 相似文献
12.
Two integral transformations between the stress function, differentiation of which gives the meridian and prime vertical components of the sub-crustal stress due to mantle convection, and the satellite-to-satellite tracking (SST) data are presented in this article. In the first one, the SST data are the disturbing potential differences between twin-satellites and in the second one the line-of-sight (LOS) gravity disturbances. It is shown that the corresponding integral kernels are well-behaving and therefore suitable for inversion and recovery of the stress function from the SST data. Recovery of the stress function and the stress components is also tested in numerical experiments using simulated SST data. Numerical studies over the Himalayas show that inverting the disturbing potential differences leads to a smoother stress function than from inverting LOS gravity disturbances. Application of the presented integral formulae allows for recovery of the stress from the satellite mission GRACE and its planned successor. 相似文献
13.
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
14.
15.
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. 相似文献
16.
A detailed and accurate Earth gravity field model is important both to geophysical progress and to the precise tracking necessary for interpretation of geophysical experiments. Various satellite techniques which may be used to determine the Earth's gravity field are compared and their ability to recover the long wavelength and short wavelength features of the field are described.A high-low configuration satellite-to-satellite tracking mission is recommended for the determination of the long wavelength portion of the gravity field. Satellite altimetry and satellite gradiometry experiments are recommended for determination of the short wavelength portion of the gravity field. 相似文献
17.
Christian Hirt Sten J. Claessens Michael Kuhn Will E. Featherstone 《Studia Geophysica et Geodaetica》2012,56(4):957-975
Curtin University??s Mars Gravity Model 2011 (MGM2011) is a high-resolution composite set of gravity field functionals that uses topography-implied gravity effects at medium- and short-scales (??125 km to ??3 km) to augment the space-collected MRO110B2 gravity model. Ground-truth gravity observations that could be used for direct validation of MGM2011 are not available on Mars??s surface. To indirectly evaluate MGM2011 and its modelling principles, an as-close-as-possible replication of the MGM2011 modelling approach was performed on Earth as the planetary body with most detailed gravity field knowledge available. Comparisons among six ground-truth data sets (gravity disturbances, quasigeoid undulations and vertical deflections) and the MGM2011-replication over Europe and North America show unanimously that topography-implied gravity information improves upon space-collected gravity models over areas with rugged terrain. The improvements are ??55% and ??67% for gravity disturbances, ??12% and ??47% for quasigeoid undulations, and ??30% to ??50% for vertical deflections. Given that the correlation between space-collected gravity and topography is higher for Mars than Earth at spatial scales of a few 100 km, topography-implied gravity effects are more dominant on Mars. It is therefore reasonable to infer that the MGM2011 modelling approach is suitable, offering an improvement over space-collected Martian gravity field models. 相似文献
18.
本文采取S变换时频分析方法, 对弱震区(河南及邻近地区)形变及重力数据进行时频谱计算。 对扰动事件、 地震响应、 可能的构造运动信息、 以及固体潮瞬时频率特征参数进行对比分析, 结果表明: 重力的干扰成分主要集中在0.3～0.5 Hz、 0.001～0.1 Hz频段, 其中部分振动干扰呈条带分布; 0.1～0.18 Hz频段可能包含有构造运动的信息; 0.1～0.18 Hz信号的能量变化、 固体潮瞬时频率特征参数可作为应力场变化的表征, 特别是作为形变及重力远场变化的表征, 具有较好的前兆意义。 相似文献
19.
分析了地球自转引起的位旋转效应公式中采用近似速度的影响. 对一组GFZ的快速科学轨道、一组TUM的约化动力法轨道以及一组GFZ的事后科学轨道,计算了星历提供的速度与只有地球引力场对卫星产生作用时的卫星速度的差值,其中参考重力场模型分别采用EGM96、EIGEN2和EIGEN_CG01C. 通过比较得出：轨道数据与EIGEN2地球重力场模型的自恰性优于EGM96和EIGEN_CG01C地球重力场模型. 速度差各分量的变化具有很明显的周期性且与卫星轨道的运行周期相吻合. 当要求在卫星轨迹处获得1m2/s2精度的扰动位时,也即要求位旋转效应公式中卫星速度的近似精度小于2mm/s时,GFZ的快速科学轨道、TUM的约化动力法轨道只需要剔除那些速度精度不满足要求的卫星轨迹点;当要求在卫星轨迹处获得05m2/s2精度的扰动位时,应当重新估算上述轨道的速度信息,或采用精度更高的GFZ事后科学轨道. 相似文献
20.
Summary The paper presents comprehensive theory based on the boundary integral method for calculations of the electric potential,
electric field and corresponding magnetic field due to a pair of D.C. source electrodes near a vertical resistivity contact
in the halfspace, indlucing a 3-D disturbing body in the vicinity of the contact. Special attention is paid to the case when
the disturbing body touches the vertical contact. Results of numerical calculations are presented in the form of sounding
curves and a set of isoline graphs for potential, components of the electric and magnetic field (total and anomalous) on the
surface of the Earth. It is shown that the presence of the disturbing body at the contact is most pronounced in the electrical
characteristics. Anomalies in the magnetic field are small in comparison to the field due to the electric current in the electrode
cable and primary currents flowing from the electrodes. 相似文献