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1.
Summary In a combined Doppler and terrestrial net adjustment not only the known systematic discrepancies in scale and orientation between the Doppler measurements and the terrestrial results must be modelled, but also all available informations about the accuracy of these systematic differences are to be taken into account. Using the Helmert-block method for the combination procedure, no covariance matrices for the terrestrially determined coordinates must be computed, their numerical evaluation being a computational detour. The proposed procedure as applied to real nets, includes all different kinds of geometric or physical models, whereby their specific parameters are eliminated at this level. Two solutions are discussed, a three-dimensional and a two-dimensional one, but “two-dimensional” is not equivalent to “non-spatial” in this context.  相似文献   

2.
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   

3.
Summary.  GFZ Potsdam and GRGS Toulouse/Grasse jointly developed a new pair of global models of the Earth's gravity field to satisfy the requirements of the recent and future geodetic and altimeter satellite missions. A precise gravity model is a prerequisite for precise satellite orbit restitution, tracking station positioning and altimeter data reduction. According to different applications envisaged, the new model exists in two parallel versions: the first one being derived exclusively from satellite tracking data acquired on 34 satellites, the second one further incorporating satellite altimeter data over the oceans and terrestrial gravity data. The most recent “satellite-only” gravity model is labelled GRIM4-S4 and the “combined” gravity model GRIM4-C4. The models are solutions in spherical harmonics and have a resolution up to degree and order 60 plus a few resonance terms in the case of GRIM4-S4, and up to degree/order 72 in the case of GRIM4-C4, corresponding to a spatial resolution of 555 km at the Earth's surface. The gravitational coefficients were estimated in a rigorous least squares adjustment simultaneously with ocean tidal terms and tracking station position parameters, so that each gravity model is associated with a consistent ocean tide model and a terrestrial reference frame built up by over 300 optical, laser and Doppler tracking stations. Comprehensive quality tests with external data and models, and test arc computations over a wide range of satellites have demonstrated the state-of-the-art capabilities of both solutions in long-wavelength geoid representation and in precise orbit computation. Received 1 February 1996; Accepted 17 July 1996  相似文献   

4.
Doppler derived geocentric and relative geodetic positions are now widely used for detecting and controlling systematic scale and orientation errors in large classical terrestrial triangulation networks. However, the combined adjustment of terrestrial and space data raises several theoretical problems, including the choice of appropriate reference systems, the a priori weighting of the various types of observations, the modelling of systematic errors and the conditioning of the network in terms of internal and external rank deficiencies. Tests with large national networks show conclusively that, without correct modelling, systematic errors will largely be unaffected by “higher order” observations.  相似文献   

5.
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   

6.
By minimizing the global distance between the (quasi-) geoid and an ellipsoid of revolution, the best parameters of an ellipsoid and its location parameters are estimated. Input data are the absolute value of the geopotential at (quasi-) geoid level, a set of harmonic coefficients from satellite, terrestrial or combined observations, the mean rotational speed of the earth, and approximate values of the seminajor axis and the eccentricity of the ellipsoid. The output ranges from 6378137,63 m and 6378141,62 m for the best semimajor axis and from 298.259758 and 298.259651 for the reciprocal value of the best ellipsoidal flattening within WD 1 and WD 2. The best translational parameters are 6.4 cm (Greenwich-direction), 0.8 cm (orthogonal to Greenwich-direction), and 1.9 cm (parallel to rotational axis of the earth); the best rotational parameters are −0.2” (around Greenwich-direction) and 1.1” (around orthogonal to Greenwich-direction). The dependence of the datum of the (quasi-) geoid geopotential is studied in detail.  相似文献   

7.
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   

8.
This paper is intended to demonstrate the usefulness of array algebra techniques in certain multilinear least squares problems. A typical restriction of array algebra is the need for a gridded observational structure; however, the grid does not have to be uniform and in general is not limited to any particular coordinate system nor to two- or three-dimensional spaces. Another restriction comes to light when dealing with weighted multilinear least squares adjustments. The a—priori variance-covariance matrix cannot be completely arbitrary but must be expressible in terms of certain matrix products. There exist various practical ways (not discussed herein) to bridge these restrictions. The reward for using the array algebra technique when it is appropriate lies in the great computational savings. From the theoretical point of view, the backbone of most derivations are the “R-matrix multiplications” and a simple tool, demonstrated herein, called “fundamental transformation”. It follows that the least squares solution of “array observation equations” does not have to be sought by some new and complex mathematical means. The fundamental transformation allows such an adjustment problem to be rewritten in a conventional (monolinear) form; the familiar least squares solution is then written down and transformed back to the array form using the same tool. The statistical properties of the results (e.g. minimum variance) are known from the conventional approach and do not have to be rederived in the array case.  相似文献   

9.
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   

10.
The Euclidean spaces with their inner products are used to describe methods of least squares adjustment as orthogonal projections on finite-dimensional subspaces. A unified Euclidean space approach to the least squares adjustment methods “observation equations” and “condition equations” is suggested. Hence not only the two adjustment solutions are treated from the view-point of Euclidean space theory in a unified frame but also the existing duality relation between the methods of “observation equations” and “condition equations” is discussed in full detail. Another purpose of this paper is to contribute to the development of some familiarity with Euclidean and Hilbert space concepts. We are convinced that Euclidean and Hilbert space techniques in least squares adjustment are elegant and powerful geodetic methods.  相似文献   

11.
A new estimator for VLBI baseline length repeatability   总被引：1，自引：1，他引：0  
O. Titov 《Journal of Geodesy》2009,83(11):1041-1049
The goal of this paper is to introduce a more effective technique to approximate for the “repeatability–baseline length” relationship that is used to evaluate the quality of geodetic VLBI results. Traditionally, this relationship is approximated by a quadratic function of baseline length over all baselines. The new model incorporates the mean number of observed group delays of the reference radio sources (i.e. estimated as global parameters) used in the estimation of each baseline. It is shown that the new method provides a better approximation of the “repeatability–baseline length” relationship than the traditional model. Further development of the new approach comes down to modeling the repeatability as a function of two parameters: baseline length and baseline slewing rate. Within the framework of this new approach the station vertical and horizontal uncertainties can be treated as a function of baseline length. While the previous relationship indicated that the station vertical uncertainties are generally 4–5 times larger than the horizontal uncertainties, the vertical uncertainties as determined by the new method are only larger by a factor of 1.44 over all baseline lengths.  相似文献   

12.
Summary A datum change between two geodetic systems with points in common may be derived in three stages; slight adjustments of coordinates to make the networks of common points geometrically similar in the two systems; a scale factor to make them geometrically congruent; finally, an orthogonal transformation to swing them into coincidence. The geometrical concept is developed of a “datum screw”, not arbitrarily chosen as is the “origin” or “datum point” of a geodetic survey, but intrinsic to the geometry. The conditions under which it degenerates to a simple “datum shift” are discussed. Differential and other formulae for changes of spheroid and of datum are given, together with a set of tables of coefficients.  相似文献   

13.
The transformation of the instantaneous terrestrial coordinate system to the mean or average earth-fixed one is parameterized by the polar motion components which are continuously changing in time. Using the non-symmetricity of the connection coefficients connecting the above frames the errors (“misclosures”) are estimated which would be present if the instantaneous frame would be used as the geodetic reference frame.  相似文献   

14.
Existing position information in a network can be integrated with the densification solution in two ways: One way is to obtain a solution of the densification network followed by a merger of this and all other solutions or vice versa. Alternatively, the existing solutions (not used as weighted constraints) can be taken to be pseudo-observations in a simultaneous adjustment with the “new” observations. In both cases, all existing solutions must first be transformed to the coordinate system of the densified network and be statistically compatible with it. Simultaneous densification and integration is discussed through mathematical adjustment models in which the geometrical strength of networks is underscored. The rationale behind densifying and integrating networks either in two different steps or simultaneously is analyzed. It is concluded that the simultaneous approach should be avoided unless the various solutions turn out to be statistically compatible.  相似文献   

15.
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   

16.
This paper is to construct a “digital local, regional, region“ information framework based on the technology of “SIG“ and its significance and application to the regional sustainable development evaluation system. First, the concept of the “grid computing“ and “SIG“ is interpreted and discussed, then the relationship between the “grid computing“ and “digital region“ is analyzed, and the framework of the “digital region“ is put forward. Finally, the significance and application of “grid computing“ to the “region sustainable development evaluation system“ are discussed.  相似文献   

17.
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   

18.
The main goal of this paper is to show that the solution obtained by adjusting a free network via the inner adjustment constraint method is the minimum norm solution. The latter is a special case of the class of “minimum trace” solutions, where the trace of the variance-covariance matrix for the adjusted parameters is a minimum. The derivations are carried out in terms of pseudo-inverses, the various other forms of generalized inverses having been left out of consideration.  相似文献   

19.
Principles of North determination using suspended gyrocompasses are reviewed. Accuracy is evaluated and a procedure with two series of measurements symmetrical with respect to the zero torsion tape position is mathematically proven to be the “best” (minimum variance). Our purpose is to prove that a 20″ accuracy (1 σ) instrument was brought to a level of accuracy four times better by using multiple transit times and least squares fit. Over a total of 15 North determinations based on more than a thousand transit times, an external standard error of 4″.4 was obtained using a WildGAK-1.  相似文献   

20.
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   

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