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1.
Burkhard Schaffrin 《Journal of Geodesy》1989,63(4):395-404
The now classical collocation method in geodesy has been derived byH. Moritz (1970; 1973) within an appropriate Mixed Linear Model. According toB. Schaffrin (1985; 1986) even a generalized form of the collocation solution can be proved to represent a combined estimation/prediction
procedure of typeBLUUE (Best Linear Uniformly Unbiased Estimation) for the fixed parameters, and of type inhomBLIP (Best inhomogeneously LInear Prediction) for the random effects with not necessarily zero expectation. Moreover, “robust collocation” has been introduced by means of homBLUP (Best homogeneously Linear weakly Unbiased Prediction) for the random effects together with a suitableLUUE for the fixed parameters. Here we present anequivalence theorem which states that the robust collocation solution in theoriginal Mixed Linear Model can identically be derived as traditionalLESS (LEast Squares Solution) in amodified Mixed Linear Model without using artifacts like “pseudo-observations”. This allows us a nice interpretation of “robust collocation”
as an adjustment technique in the presence of “weak prior information”. 相似文献
2.
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
3.
Burkhard Schaffrin 《Journal of Geodesy》2008,82(2):113-121
In a linear Gauss–Markov model, the parameter estimates from BLUUE (Best Linear Uniformly Unbiased Estimate) are not robust
against possible outliers in the observations. Moreover, by giving up the unbiasedness constraint, the mean squared error
(MSE) risk may be further reduced, in particular when the problem is ill-posed. In this paper, the α-weighted S-homBLE (Best homogeneously Linear Estimate) is derived via formulas originally used for variance component estimation on
the basis of the repro-BIQUUE (reproducing Best Invariant Quadratic Uniformly Unbiased Estimate) principle in a model with
stochastic prior information. In the present model, however, such prior information is not included, which allows the comparison
of the stochastic approach (α-weighted S-homBLE) with the well-established algebraic approach of Tykhonov–Phillips regularization, also known as R-HAPS (Hybrid APproximation Solution), whenever the inverse of the “substitute matrix” S exists and is chosen as the R matrix that defines the relative impact of the regularizing term on the final result.
The delay in publishing this paper is due to a number of unfortunate complications. It was first submitted as a multi-author
paper in two parts. Due to some miscommunication among the original authors, it was reassigned to one of the J Geod special
issues, but later reassigned at this author’s request to a standard issue of J Geod. This compounded with a difficulty to
find willing reviewers to slow the process. We apologize to the author. 相似文献
4.
Parametric least squares collocation was used in order to study the detection of systematic errors of satellite gradiometer
data. For this purpose, simulated data sets with a priori known systematic errors were produced using ground gravity data
in the very smooth gravity field of the Canadian plains. Experiments carried out at different satellite altitudes showed that
the recovery of bias parameters from the gradiometer “measurements” is possible with high accuracy, especially in the case
of crossing tracks. The mean value of the differences (original minus estimated bias parameters) was relatively large compared
to the standard deviation of the corresponding second-order derivative component at the corresponding height. This mean value
almost vanished when gravity data at ground level were combined with the second-order derivative data set at satellite altitude.
In the case of simultaneous estimation of bias and tilt parameters from ∂2
T/∂z
2“measurements”, the recovery of both parameters agreed very well with the collocation error estimation.
Received: 10 October 1996 / Accepted 25 May 1998 相似文献
5.
This research deals with some theoretical and numerical problems of the downward continuation of mean Helmert gravity disturbances.
We prove that the downward continuation of the disturbing potential is much smoother, as well as two orders of magnitude smaller
than that of the gravity anomaly, and we give the expression in spectral form for calculating the disturbing potential term.
Numerical results show that for calculating truncation errors the first 180∘ of a global potential model suffice. We also discuss the theoretical convergence problem of the iterative scheme. We prove
that the 5′×5′ mean iterative scheme is convergent and the convergence speed depends on the topographic height; for Canada, to achieve an
accuracy of 0.01 mGal, at most 80 iterations are needed. The comparison of the “mean” and “point” schemes shows that the mean
scheme should give a more reasonable and reliable solution, while the point scheme brings a large error to the solution.
Received: 19 August 1996 / Accepted: 4 February 1998 相似文献
6.
The target of the spheroidal Gauss–Listing geoid determination is presented as a solution of the spheroidal fixed–free two-boundary
value problem based on a spheroidal Bruns' transformation (“spheroidal Bruns' formula”). The nonlinear spheroidal Bruns' transform
(nonlinear spheroidal Bruns' formula), the spheroidal fixed part and the spheroidal free part of the two-boundary value problem
are derived. Four different spheroidal gravity models are treated, in particular to determine whether they pass the test to
fit to the postulate of a level ellipsoidal gravity field, namely of Somigliana–Pizzetti type.
Received: 4 May 1999 / Accepted: 21 May 1999 相似文献
7.
The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor
structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”.
Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point
. The least-squares criterion results in a minimum-distance property implying that the vector
Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized
by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables
with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized)
method. 相似文献
8.
The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local
gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential.
The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem
of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector
(from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation
Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference
benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity
field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived
gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential
difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred
into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of
the offset of the zero point of the Iranian height datum from the geoid’s potential value W
0=62636855.8 m2/s2. According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid. 相似文献
9.
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
10.
The total optimal search criterion in solving the mixed integer linear model with GNSS carrier phase observations 总被引:3,自引:2,他引:1
Existing algorithms for GPS ambiguity determination can be classified into three categories, i.e. ambiguity resolution in
the measurement domain, the coordinate domain and the ambiguity domain. There are many techniques available for searching
the ambiguity domain, such as FARA (Frei and Beutler in Manuscr Geod 15(4):325–356, 1990), LSAST (Hatch in Proceedings of KIS’90, Banff, Canada, pp 299–308, 1990), the modified Cholesky decomposition method (Euler and Landau in Proceedings of the sixth international geodetic symposium on satellite positioning,
Columbus, Ohio, pp 650–659, 1992), LAMBDA (Teunissen in Invited lecture, section IV theory and methodology, IAG general meeting, Beijing, China, 1993), FASF (Chen and Lachapelle in J Inst Navig 42(2):371–390, 1995) and modified LLL Algorithm (Grafarend in GPS Solut 4(2):31–44, 2000; Lou and Grafarend in Zeitschrift für Vermessungswesen 3:203–210, 2003). The widely applied LAMBDA method is based on the Least Squares Ambiguity Search (LSAS) criterion and employs an effective decorrelation technique in addition. G. Xu (J Glob Position Syst 1(2):121–131,
2002) proposed also a new general criterion together with its equivalent objective function for ambiguity searching that can be
carried out in the coordinate domain, the ambiguity domain or both. Xu’s objective function differs from the LSAS function,
leading to different numerical results. The cause of this difference is identified in this contribution and corrected. After
correction, the Xu’s approach and the one implied in LAMBDA are identical. We have developed a total optimal search criterion
for the mixed integer linear model resolving integer ambiguities in both coordinate and ambiguity domain, and derived the
orthogonal decomposition of the objective function and the related minimum expressions algebraically and geometrically. This
criterion is verified with real GPS phase data. The theoretical and numerical results show that (1) the LSAS objective function
can be derived from the total optimal search criterion with the constraint on the fixed integer ambiguity parameters, and
(2) Xu’s derivation of the equivalent objective function was incorrect, leading to an incorrect search procedure. The effects
of the total optimal criterion on GPS carrier phase data processing are discussed and its practical implementation is also
proposed. 相似文献
11.
The solution of the linear Molodensky problem by analytical continuation to point level is numerically the most convenient
of all the theoretically equivalent solutions. It is obtained by successively applying the same integral operator and it does
not depend explicitly on the terrain inclination. However, its dependence on the computation point restricts somehow the computational
efficiency. The use of the Fourier transform for the evaluation of the integral operator in planar approximation lessens significantly
the burden of computations. Using this spectral approach, the problem has been reformulated and solved in the frequency domain.
Moreover, it is shown that the solution can be easily split into two steps: (a) “downward” continuation to sea level, which
is independent of the computation point, and (b) “upward” continuation from sea to point level, using the values computed
at sea level. Such a treatment not only simplifies the formulas and increases the numerical efficiency but also clarifies
the physical interpretation and the theoretical equivalence of the continuation solution with respect to the other solution
types. Numerical tests have been performed to investigate which terms in the Molodensky series are of significance for geoid
and deflection computations. The practical difficulty of differences in the grid spacings of gravity and height data has been
overcome by frequency domain interpolation.
Presented at theXIX IUGG General Assembly, Vancouver, B.C., August 9–22, 1987. 相似文献
12.
Mixed Integer-Real Valued Adjustment (IRA) Problems: GPS Initial Cycle Ambiguity Resolution by Means of the LLL Algorithm 总被引:4,自引:0,他引:4
Erik W. Grafarend 《GPS Solutions》2000,4(2):31-44
In order to achieve to GPS solutions of first-order accuracy and integrity, carrier phase observations as well as pseudorange
observations have to be adjusted with respect to a linear/linearized model. Here the problem of mixed integer-real valued
parameter adjustment (IRA) is met. Indeed, integer cycle ambiguity unknowns have to be estimated and tested. At first we review
the three concepts to deal with IRA: (i) DDD or triple difference observations are produced by a properly chosen difference
operator and choice of basis, namely being free of integer-valued unknowns (ii) The real-valued unknown parameters are eliminated
by a Gauss elimination step while the remaining integer-valued unknown parameters (initial cycle ambiguities) are determined
by Quadratic Programming and (iii) a RA substitute model is firstly implemented (real-valued estimates of initial cycle ambiguities)
and secondly a minimum distance map is designed which operates on the real-valued approximation of integers with respect to
the integer data in a lattice. This is the place where the integer Gram-Schmidt orthogonalization by means of the LLL algorithm (modified LLL algorithm) is applied being illustrated by four examples. In particular, we prove
that in general it is impossible to transform an oblique base of a lattice to an orthogonal base by Gram-Schmidt orthogonalization where its matrix enties are integer. The volume preserving Gram-Schmidt orthogonalization operator constraint to integer entries produces “almost orthogonal” bases which, in turn, can be used to produce the integer-valued
unknown parameters (initial cycle ambiguities) from the LLL algorithm (modified LLL algorithm). Systematic errors generated
by “almost orthogonal” lattice bases are quantified by A. K. Lenstra et al. (1982) as well as M. Pohst (1987). The solution point of Integer Least Squares generated by the LLL algorithm is = (L')−1[L'◯] ∈ ℤ
m
where L is the lower triangular Gram-Schmidt matrix rounded to nearest integers, [L], and = [L'◯] are the nearest integers of L'◯, ◯ being the real valued approximation of z ∈ ℤ
m
, the m-dimensional lattice space Λ. Indeed due to “almost orthogonality” of the integer Gram-Schmidt procedure, the solution point is only suboptimal, only close to “least squares.” ? 2000 John Wiley & Sons, Inc. 相似文献
13.
C. C. Tscherning 《Journal of Geodesy》1978,52(1):85-92
The term “entity” covers, when used in the field of electronic data processing, the meaning of words like “thing”, “being”,
“event”, or “concept”. Each entity is characterized by a set of properties.
An information element is a triple consisting of an entity, a property and the value of a property. Geodetic information is
sets of information elements with entities being related to geodesy. This information may be stored in the form ofdata and is called ageodetic data base provided (1) it contains or may contain all data necessary for the operations of a particular geodetic organization, (2)
the data is stored in a form suited for many different applications and (3) that unnecessary duplications of data have been
avoided.
The first step to be taken when establishing a geodetic data base is described, namely the definition of the basic entities
of the data base (such as trigonometric stations, astronomical stations, gravity stations, geodetic reference-system parameters,
etc...).
Presented at the “International Symposium on Optimization of Design and Computation of Control Networks”, Sopron, Hungary,
July 1977. 相似文献
14.
Global plate tectonics and the secular motion of the Pole 总被引:1,自引:0,他引:1
Astronomical data compiled during the last 70 years by the international organizations (ILS/IPMS, BIH) providing the coordinates
of the instantaneous pole, clearly shows a continuous drift of the “mean pole” (≡barycenter of the wobble cycle with respect
to the Conventional International Origin (CIO).
This study was undertaken to investigate the possibility of an actual secular motion of the barycenter (approximated by the
earth's maximum principal moment of inertia axis or axis of figure) due to differential mass displacements from lithospheric
plate rotations. The method assumes the earth's crust modeled as a mosaic of 1°×1° blocks, each one moving independently with
their corresponding absolute plate velocities. The differential contributions to the earth's second-order tensor of inertia
were computed, resulting in no significant displacement of the earth's axis of figure.
In view of the above, the possibleapparent displacement of the “mean pole” as a consequence of station drifting due to absolute plate motions was also analyzed, again
without great success. As a further step the old speculation of the whole crust possibly sliding over the upper mantle is
revived and the usefulness of the CIO is questioned.
Presented at the IAU Symposium No. 78, “Nutation and the Earth Rotation”, Kiev, 22–29, May, 1977. 相似文献
15.
J. Feltens 《Journal of Geodesy》2011,85(4):239-254
The vector-based algorithms for biaxial and triaxial ellipsoidal coordinates presented by Feltens (J Geod 82:493–504, 2008; 83:129–137, 2009) have been extended to hyperboloids of one sheet. For the backward transformation from Cartesian to hyperboloidal coordinates,
of two iterative process candidates one was identified to be well suited. It turned out that a careful selection of the center
of curvature is essential for the establishment of a stable and reliable iteration process. In addition, for zero hyperboloidal
heights a closed solution is presented. The hyperboloid algorithms are again based on simple formulae and have been successfully
tested for various theoretical hyperboloids. The paper concludes with a practical application example on a cooling tower construction. 相似文献
16.
E. W. Grafarend 《Journal of Geodesy》1981,55(4):286-299
WhenH. Moritz (1967, 1971) studied “kinematical geodesy” for the purpose of separation of gravitation and inertia, especially within combined
accelerometer-gradiometer systems, it was hard to believe that within five years time inertial survey systems would be available,
exactly operating according to his theoretical design. Here, we attempt to give a geodetic introduction into the fundamental
equation of inertial positioning materialized by inertial survey systems with emphasis on a careful error model, including
36 parameters of type time interval, initial positions, initial gravity, varying acceleration, varying gravity gradients,
accelerometer bias, accelerometer random uncertainty, accelerometer non-orthogonality, initial misalignment angles, accelerometer
scale factor uncertainty. The notion of “multipoint” boundary value problem and initial value problem of inertial positioning
is reviwed. So-called “post-mission” adjustment techniques for inertial surveys are discussed.
Presented at the 2nd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981. 相似文献
Sommaire QuandH. Moritz (1967, 1971) a étudié la géodésie cinématique dans le but de séparer la gravitation et l’inertie, spécialement en combinant accéléromètres et gradiomètres, il était difficile de croire qu’en cinq ans les systèmes d’arpentage inertiels seraient disponibles et fonctionneraient exactement selon ses prévisions théoriques. Ici, nous allons tenter de donner une introduction géodésique à l’équation fondamentale du positionnement inertiel, matérialisée par un système d’arpentage inertiel en soulignant l’importance d’un modèle d’erreur incluant 36 paramètres du genre intervalle de temps, positions initiales, gravité initiale, accélération variable, gradient de la gravité variable, déviation des accéléromètres, incertitude aléatoire des accéléromètres, non-orthogonalité des accéléromètres, angles initiaux des défauts d’alignement, incertitude du facteur d’échelle des accéléromètres. La notion de problème “multipoint” aux limites et du problème de la valeur initiale du positionnement inertiel y est revue. Les techniques de compensation “après la mission” y sont discutées.
Presented at the 2nd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981. 相似文献
17.
Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation 总被引:5,自引:2,他引:3
Frank Neitzel 《Journal of Geodesy》2010,84(12):751-762
In this contribution it is shown that the so-called “total least-squares estimate” (TLS) within an errors-in-variables (EIV)
model can be identified as a special case of the method of least-squares within the nonlinear Gauss–Helmert model. In contrast
to the EIV-model, the nonlinear GH-model does not impose any restrictions on the form of functional relationship between the
quantities involved in the model. Even more complex EIV-models, which require specific approaches like “generalized total
least-squares” (GTLS) or “structured total least-squares” (STLS), can be treated as nonlinear GH-models without any serious
problems. The example of a similarity transformation of planar coordinates shows that the “total least-squares solution” can
be obtained easily from a rigorous evaluation of the Gauss–Helmert model. In contrast to weighted TLS, weights can then be
introduced without further limitations. Using two numerical examples taken from the literature, these solutions are compared
with those obtained from certain specialized TLS approaches. 相似文献
18.
A strict formula for geoid-to-quasigeoid separation 总被引:3,自引:2,他引:1
Lars E. Sjöberg 《Journal of Geodesy》2010,84(11):699-702
The paper presented by Flury and Rummel (J Geod 83:829–847, 2009) discusses an important topographic correction to the traditional
formula for the quasigeoid-to-geoid separation. Nevertheless, as their formula is approximate, the reader may ask for its
relation to the strict one (defined as the one consistent with Bruns’s formula and the boundary condition of physical geodesy),
which is now derived. Although the result formally differs from that of Flury and Rummel, we show that the two formulas agree
to the centimetre level all over the Earth. We also discuss the practical computation of the topographic correction. 相似文献
19.
Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude 总被引:2,自引:0,他引:2
A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude λ and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values
of the TM coordinates are given by conveniently truncated Gauss–Krüger series expansions, we use the maximum norm of the absolute
error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals
0.21 mm, whereas for the region of the ellipsoid bounded by the meridians ±20° the accuracy is equal to 0.3 mm. Our approach
is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43 mm. The small parameter of the power series expansion is the square of the eccentricity
of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincaré method of the small parameter.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
20.
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