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1.
An integrated wavelet concept of physical geodesy 总被引:4,自引:1,他引:3
For the determination of the earth's gravity field many types of observations are nowadays available, including terrestrial
gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradio-metry, etc. The mathematical connection
between these observables on the one hand and gravity field and shape of the earth on the other is called the integrated concept
of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can
be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical
geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an
integral over an internal sphere to suitable linear combinations of observation functionals, i.e. linear functionals representing
the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential
outside and on the earth's surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of
band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding
to realistic earth's models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique
derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry and combined satellite-to-satellite tracking
and gradiometry.
Received: 28 February 1997 / Accepted: 17 November 1997 相似文献
2.
Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE 总被引:1,自引:0,他引:1
This work is dedicated to the wavelet modeling of regional and temporal variations of the Earth’s gravitational potential
observed by the GRACE (gravity recovery and climate experiment) satellite mission. In the first part, all required mathematical
tools and methods involving spherical wavelets are provided. Then, we apply our method to monthly GRACE gravity fields. A
strong seasonal signal can be identified which is restricted to areas where large-scale redistributions of continental water
mass are expected. This assumption is analyzed and verified by comparing the time-series of regionally obtained wavelet coefficients
of the gravitational signal originating from hydrology models and the gravitational potential observed by GRACE. The results
are in good agreement with previous studies and illustrate that wavelets are an appropriate tool to investigate regional effects
in the Earth’s gravitational field.
Electronic Supplementary Material Supplementary material is available for this article at 相似文献
3.
Topographic–isostatic masses represent an important source of gravity field information, especially in the high-frequency
band, even if the detailed mass-density distribution inside the topographic masses is unknown. If this information is used
within a remove-restore procedure, then the instability problems in downward continuation of gravity observations from aircraft
or satellite altitudes can be reduced. In this article, integral formulae are derived for determination of gravitational effects
of topographic–isostatic masses on the first- and second-order derivatives of the gravitational potential for three topographic–isostatic
models. The application of these formulas is useful for airborne gravimetry/gradiometry and satellite gravity gradiometry.
The formulas are presented in spherical approximation by separating the 3D integration in an analytical integration in the
radial direction and 2D integration over the mean sphere. Therefore, spherical volume elements can be considered as being
approximated by mass-lines located at the centre of the discretization compartments (the mass of the tesseroid is condensed
mathematically along its vertical axis). The errors of this approximation are investigated for the second-order derivatives
of the topographic–isostatic gravitational potential in the vicinity of the Earth’s surface. The formulas are then applied
to various scenarios of airborne gravimetry/gradiometry and satellite gradiometry. The components of the gravitational vector
at aircraft altitudes of 4 and 10 km have been determined, as well as the gravitational tensor components at a satellite altitude
of 250 km envisaged for the forthcoming GOCE (gravity field and steady-state ocean-circulation explorer) mission. The numerical
computations are based on digital elevation models with a 5-arc-minute resolution for satellite gravity gradiometry and 1-arc-minute
resolution for airborne gravity/gradiometry. 相似文献
4.
The spacetime gravitational field of a deformable body 总被引:3,自引:0,他引:3
The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation
of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation
of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body
are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental
spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents:
the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference
mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both
in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential
is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement
multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces.
The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position
of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution,
emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus
surface Love-Shida functions (h(r),l(r) versus h
′(r),l
′(r)) and Love functions k(r) versus k
′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential,
proving an excellent agreement with experimental results.
Received: December 1995 / Accepted: 1 February 1997 相似文献
5.
E. W. Grafarend 《Journal of Geodesy》2001,75(7-8):363-390
In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is
aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational
potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the
Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere
(IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical
gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry
space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of
horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections
given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions
specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the
vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators
for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space
are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity
to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward
continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced.
Received: 10 May 2000 / Accepted: 26 February 2001 相似文献
6.
A new local existence and uniqueness theorem is obtained for the scalar geodetic boundary-value problem in spherical coordinates.
The regularities H
α and H
1+α are assumed for the boundary data g (gravity) and v (gravitational potential) respectively.
Received: 27 July 1998 / Accepted: 19 April 1999 相似文献
7.
The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational
field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients
of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to
a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation
law allows a representation of the incremental mass density by the respective excitation function. A representation of an
arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing
these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In
particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments
of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely:
(1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application
of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of
inertia.
Received: 27 July 1999 / Accepted: 24 May 2000 相似文献
8.
Comparisons between high-degree models of the Earth’s topographic and gravitational potential may give insight into the quality and resolution of the source data sets, provide feedback on the modelling techniques and help to better understand the gravity field composition. Degree correlations (cross-correlation coefficients) or reduction rates (quantifying the amount of topographic signal contained in the gravitational potential) are indicators used in a number of contemporary studies. However, depending on the modelling techniques and underlying levels of approximation, the correlation at high degrees may vary significantly, as do the conclusions drawn. The present paper addresses this problem by attempting to provide a guide on global correlation measures with particular emphasis on approximation effects and variants of topographic potential modelling. We investigate and discuss the impact of different effects (e.g., truncation of series expansions of the topographic potential, mass compression, ellipsoidal versus spherical approximation, ellipsoidal harmonic coefficient versus spherical harmonic coefficient (SHC) representation) on correlation measures. Our study demonstrates that the correlation coefficients are realistic only when the model’s harmonic coefficients of a given degree are largely independent of the coefficients of other degrees, permitting degree-wise evaluations. This is the case, e.g., when both models are represented in terms of SHCs and spherical approximation (i.e. spherical arrangement of field-generating masses). Alternatively, a representation in ellipsoidal harmonics can be combined with ellipsoidal approximation. The usual ellipsoidal approximation level (i.e. ellipsoidal mass arrangement) is shown to bias correlation coefficients when SHCs are used. Importantly, gravity models from the International Centre for Global Earth Models (ICGEM) are inherently based on this approximation level. A transformation is presented that enables a transformation of ICGEM geopotential models from ellipsoidal to spherical approximation. The transformation is applied to generate a spherical transform of EGM2008 (sphEGM2008) that can meaningfully be correlated degree-wise with the topographic potential. We exploit this new technique and compare a number of models of topographic potential constituents (e.g., potential implied by land topography, ocean water masses) based on the Earth2014 global relief model and a mass-layer forward modelling technique with sphEGM2008. Different to previous findings, our results show very significant short-scale correlation between Earth’s gravitational potential and the potential generated by Earth’s land topography (correlation +0.92, and 60% of EGM2008 signals are delivered through the forward modelling). Our tests reveal that the potential generated by Earth’s oceans water masses is largely unrelated to the geopotential at short scales, suggesting that altimetry-derived gravity and/or bathymetric data sets are significantly underpowered at 5 arc-min scales. We further decompose the topographic potential into the Bouguer shell and terrain correction and show that they are responsible for about 20 and 25% of EGM2008 short-scale signals, respectively. As a general conclusion, the paper shows the importance of using compatible models in topographic/gravitational potential comparisons and recommends the use of SHCs together with spherical approximation or EHCs with ellipsoidal approximation in order to avoid biases in the correlation measures. 相似文献
9.
Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high
degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of
orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from
a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics.
This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals
of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and
obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests
with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes
and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series
also offers a computational savings of at least one third. 相似文献
10.
A general scheme is given for the solution in a least-squares sense of the geodetic boundary value problem in a spherical,
constant-radius approximation, both uniquely and overdetermined, for a large class of observations. The only conditions are
that the relation of the observations to the disturbing potential is such that a diagonalization in the spectrum can be found
and that the error-covariance function of the observations is isotropic and homogeneous. Most types of observations used in
physical geodesy can be adjusted to fit into this approach. Examples are gravity anomalies, deflections of the vertical and
the second derivatives of the gravity potential.
Received: 3 November 1999 / Accepted: 25 September 2000 相似文献
11.
A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential 总被引:6,自引:0,他引:6
J. Li 《Journal of Geodesy》2002,76(4):226-231
A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential
is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified
Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted
to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the
corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation.
Received: 27 December 2000 / Accepted: 3 September 2001 相似文献
12.
Simplified techniques for high-degree spherical harmonic synthesis are extended to include gravitational potential second
derivatives with respect to latitude.
Received: 23 July 2001 / Accepted: 12 April 2002
Acknowledgement. The authors would like to thank Christian Tscherning for recommending Laplace's equation as an accuracy test. Our use of
Legendre's differential equation, as the most direct means for extending our simplified synthesis methods to second-order
derivatives, was a direct result of this suggestion.
Correspondence to: S. A. Holmes 相似文献
13.
When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with
two problems: (1) the absence of O(|x|−2) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation
of the disturbing potential, is of the order O(|x|−1) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator
fails; (2) the N×N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN)
s
), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed.
(1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension
3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the
boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier.
The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility
of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate
sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any
modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system
of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel
in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast
algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear
system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent
of the number of unknowns, which indicates good conditioning of the system matrix.
Received: 16 October 1999 / Accepted: 28 February 2001 相似文献
14.
The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical
harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression
of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of
the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing
similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means
of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a
preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three
coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated
for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for
the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various
problems of satellite geodesy and celestial mechanics. 相似文献
15.
Based upon a data set of 25 points of the Baltic Sea Level Project, second campaign 1993.4, which are close to mareographic
stations, described by (1) GPS derived Cartesian coordinates in the World Geodetic Reference System 1984 and (2) orthometric
heights in the Finnish Height Datum N60, epoch 1993.4, we have computed the primary geodetic parameter W
0(1993.4) for the epoch 1993.4 according to the following model. The Cartesian coordinates of the GPS stations have been converted
into spheroidal coordinates. The gravity potential as the additive decomposition of the gravitational potential and the centrifugal
potential has been computed for any GPS station in spheroidal coordinates, namely for a global spheroidal model of the gravitational
potential field. For a global set of spheroidal harmonic coefficients a transformation of spherical harmonic coefficients
into spheroidal harmonic coefficients has been implemented and applied to the global spherical model OSU 91A up to degree/order
360/360. The gravity potential with respect to a global spheroidal model of degree/order 360/360 has been finally transformed
by means of the orthometric heights of the GPS stations with respect to the Finnish Height Datum N60, epoch 1993.4, in terms
of the spheroidal “free-air” potential reduction in order to produce the spheroidal W
0(1993.4) value. As a mean of those 25 W
0(1993.4) data as well as a root mean square error estimation we computed W
0(1993.4)=(6 263 685.58 ± 0.36) kgal × m. Finally a comparison of different W
0 data with respect to a spherical harmonic global model and spheroidal harmonic global model of Somigliana-Pizetti type (level
ellipsoid as a reference, degree/order 2/0) according to The Geodesist's Handbook 1992 has been made.
Received: 7 November 1996 / Accepted: 27 March 1997 相似文献
16.
The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse
Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation”
of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to
applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local
tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of
the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson
integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and
the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error
budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×106km.
Received: 18 March 1997 / Accepted: 19 January 1998 相似文献
17.
The passive satellite GFZ-1 has been orbiting the Earth since April 1995. The purpose of this mission is to improve the current
knowledge of the Earth's gravity field by analysing gravitational orbit perturbations observed at unique low altitudes, below
400 km. GFZ-1 is one target of the international satellite laser ranging ground network. An evaluation of the first 30 months
of GFZ-1 laser tracking data led to a new version of the global GRIM4-S4 satellite-only gravity field model: GRIM4-S4G. Information
was obtained from GFZ-1 data for spherical harmonic coefficients up to degree 100, which was not possible in any earlier satellite-only
gravity field solution. GFZ-1's contribution to a global 5 × 5° geoid and gravity field representations is moderate but visible
with a 1 cm and 0.1 mGal gain in accuracy on a level of 75 cm and 5 mGal, respectively.
Received: 10 November 1998 / Accepted: 19 April 1999 相似文献
18.
Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography 总被引:1,自引:1,他引:0
The determination of the gravimetric geoid is based on the magnitude of gravity observed at the surface of the Earth or at
airborne altitude. To apply the Stokes’s or Hotine’s formulae at the geoid, the potential outside the geoid must be harmonic
and the observed gravity must be reduced to the geoid. For this reason, the topographic (and atmospheric) masses outside the
geoid must be “condensed” or “shifted” inside the geoid so that the disturbing gravity potential T fulfills Laplace’s equation everywhere outside the geoid. The gravitational effects of the topographic-compensation masses
can also be used to subtract these high-frequent gravity signals from the airborne observations and to simplify the downward
continuation procedures. The effects of the topographic-compensation masses can be calculated by numerical integration based
on a digital terrain model or by representing the topographic masses by a spherical harmonic expansion. To reduce the computation
time in the former case, the integration over the Earth can be divided into two parts: a spherical cap around the computation
point, called the near zone, and the rest of the world, called the far zone. The latter one can be also represented by a global
spherical harmonic expansion. This can be performed by a Molodenskii-type spectral approach. This article extends the original
approach derived in Novák et al. (J Geod 75(9–10):491–504, 2001), which is restricted to determine the far-zone effects for
Helmert’s second method of condensation for ground gravimetry. Here formulae for the far-zone effects of the global topography
on gravity and geoidal heights for Helmert’s first method of condensation as well as for the Airy-Heiskanen model are presented
and some improvements given. Furthermore, this approach is generalized for determining the far-zone effects at aeroplane altitudes.
Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the size and distributions of these
effects. 相似文献
19.
Lars E. Sj?berg 《Journal of Geodesy》1988,62(2):93-101
The spherical harmonic coefficients of the Earth’s gravitational potential are conveniently determined by integration of gravity
data or potential data (derived from satellite altimetry) over a sphere. The major problem of such a method is that the data,
given on the non-spherical surface of the Earth, must be reduced to the sphere.
A new integral formula over the surface of the Earth is derived. With this formula improved first order topographic corrections
to the spherical formulas are obtained. 相似文献
20.
G. Ramillien 《Journal of Geodesy》2002,76(3):139-149
A fast spherical harmonic approach enables the computation of gravitational or magnetic potential created by a non-uniform
shell of material bounded by uneven topographies. The resulting field can be evaluated outside or inside the sphere, assuming
that density of the shell varies with latitude, longitude, and radial distance. To simplify, the density (or magnetization)
source inside the sphere is assumed to be the product of a surface function and a power series expansion of the radial distance.
This formalism is applied to compute the gravity signal of a steady, dry atmosphere. It provides geoid/gravity maps at sea
level as well as satellite altitude. Results of this application agree closely with those of earlier studies, where the atmosphere
contribution to the Earth's gravity field was determined using more time-consuming methods.
Received: 14 August 2000 / Accepted: 19 March 2001 相似文献