Showing papers in "Journal of Geodesy in 1985"

On a relativistic geodesy

Abstract: Theoretical formulas for relativistic estimation of geopotential differences are given. The relativistic geoid is defined. A technique for measuring potential differences with high precision clocks (masers or equivalent) is described. The method can operate over arbitrary terrestrial distances. Two clocks are used. The drift between the clocks is estimated by using closed loops. The clocks are used in an operational mode during the whole measuring interval. No satellite links are necessary but VLBI, GPS and ANIK-links can be used in combination with the method.

Gravity field terrain effect computations by FFT

Abstract: The widespread availability of detailed gridded topographic and bathymetric data for many areas of the earth has resulted in a need for efficient terrain effect computation techniques, especially for applications in gravity field modelling. Compared to conventional integration techniques, Fourier transform methods provide extremely efficient computations due to the speed of the Fast Fourier Transform (FFT. The Fourier techniques rely on linearization and series expansions of the basically unlinear terrain effect integrals, typically involving transformation of the heights/depths and their squares. TheFFT methods will especially be suited for terrain reduction of land gravity data and satellite altimetry geoid data. In the paper the basic formulas will be outlined, and special emphasis will be put on the practial implementation, where a special coarse/detailed grid pair formulation must be used in order to minimize the unavoidable edge effects ofFFT, and the special properties ofFFT are utilized to limit the actual number of data transformations needed. Actual results are presented for gravity and geoid terrain effects in test areas of the USA, Greenland and the North Atlantic. The results are evaluated against a conventional integration program: thus, e.g., in an area of East Greenland (with terrain corrections up to10 mgal), the accuracy ofFFT-computed terrain corrections in actual gravity stations showed anr.m.s. error of0.25 mgal, using height data from a detailed photogrammetric digital terrain model. Similarly, isostatic ocean geoid effects in the Faeroe Islands region were found to be computed withr.m.s. errors around0.03 m

Global positioning system carrier phase: Description and use

Abstract: After removing the modulation from the Global Positioning System (GPS) signal (L 1 orL 2) a pure carrier signal remains. Suppose this carrier is continuously and precisely tracked by aGPS receiver. Furthermore, suppose the phase of the carrier is periodically measured and recorded (nearly simultaneously at two or more locations) with respect to receiver oscillators having the same nominal frequency as theGPS carrier. This paper first considers alternative modeling and processing approaches to these observational data for static operations. Then an approach to dynamic relative positioning using triple differences is presented. This approach should lend itself to performing centimeter accuracy relative surveys in seconds rather than hours. An approach to fixing cycle slips, automatically, is included.

Gravity field determination from satellite gradiometry

Abstract: An orbiting gradiometer measures simultaneously several gravity quantities, ideally all six second-order derivatives of the gravitational potential. These contain information on the orbit, on the structure of the gravity field, and on the attitude of the space-craft. Due to the availability of several components simultaneously it is possible to separate orbit determination from attitude or gravity field recovery. This facilitates the analysis of the gradiometer measurements and allows the use of the principles of fast spherical harmonic analysis. The separation of gravity field recovery and orbit determination is tested numerically with a simplified gravity field (with a purely zonal spherical harmonic expansion) up to degree 300. For both the potential coefficients and for the orbit an almost exact recovery is attained after two iteration steps.

Extended free net adjustment constraints

Abstract: The contribution of geodetic measurements to the establishment of a control network can be partitioned into global and local (individual) components. The global component epitomized in a number of geometrically meaningful parameters can be estimated together with the individual point coordinates. The additional rank defect created by the extension of the parameter list is corrected by free net adjustment constraints which are extended beyond those needed for a solution of the network datum problem. Two applications of extended free net adjustment are outlined and illustrated by elementary numerical examples. A non-Cartesian (skew) reference system discussed in the Appendix provides an exotic interpretation of the estimated global and individual parameters.

The free versus fixed geodetic boundary value problem for different combinations of geodetic observables

Abstract: Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary.

Second-order design of combined linear-angular geodetic networks

Abstract: The paper deals with the solution of the weight problem for linear-angular networks on the base of criterion matrices. The observation plan of combined linear-angular networks consists of distances as well as of directions in any form. In particular the role of criterion matrices of completely isotropic structure for this type of two-dimensional network is discussed. Starting from an extreme network design, the results of a least-squares approximation are described analytically. In this case, the observation plan contains all geometrical point connections. Finally, the considerations are completed by giving as an example the second-order design of a control network for a dam of a water reservoir.

Predictability of the Earth's polar motion

Abstract: A comprehensive, experimental study of the predictability of the polar motion using a homogeneous BIH (Bureau International de l'Heure) data set is presented. Based on knowledge of the physics of the annual and the Chandler wobbles, the numerical model for the polar motion is constructed by allowing the wobble periods to vary. Using an optimum base length of 6 years for prediction, this floating-period model, equipped with a non-linear least-squares estimator, is found to yield polar motion predictions accurate from 0.012 to 0.024 inches depending on the prediction length up to one year, corresponding to a predictability of 91-83%. This represents a considerable improvement over the conventional fixed-period predictor, which does not respond to variations in the apparent wobble periods. The superiority of the floating-period predictor to other predictors based on critically different numerical models is also demonstrated.

Reference coordinate systems and frames: Concepts and realization

Abstract: Geodynamics has become the subject of intensive international research during the last decade, involving plate tectonics, both on the intra-plate and inter-plate scale, i.e., the study of crustal movements, and the study of earth rotation and of other dynamic phenomena such as the tides. Interrelated are efforts improving our knowledge of the gravity and magnetic fields of the earth. A common requirement for all these investigations is the necessity for a well-defined reference coordinate system (or systems) to which all relevant observations can be referred and in which theories or models for the dynamic behavior of the earth can be formulated. In view of the unprecedented progress in the ability of geodetic observational systems to measure crustal movements and the rotation of the earth, as well as in theory and model development, there is a great need for the theoretical definition, practical realization, and international acceptance of suitable coordinate system(s) to facilitate such work. This article deals with certain aspects of the establishment and maintenance of such a coordinate system.

The role of frame definitions in the geodetic determination of crustal deformation parameters

Abstract: A study of the role of coordinate frame definitions in the determination of crustal deformation parameters is first carried out for the theoretical case where displacement information between two discrete time epochs is continuously available for all area points. The obtained results are next applied to the realistic case where the required continuous information is derived by means of an interpolation of the known coordinate variations at the points of a horizontal geodetic network. The problem is different from the usual one of frame-invariant interpolation, since not only the domain of definition, but also the interpolated quantities, depend on independent choices of coordinate frames. Specific necessary and sufficient conditions for the invariance of derived crustal deformation parameters are given for linear type of interpolations of either the coordinates at the second epoch or of the displacements. With the help of the above conditions the invariance characteristics of two commonly used types of linear interpolations are finally derived, in order to illustrate the practical significance of the results.

A note on observation decorrelation, variances of residuals, and redundancy numbers

Abstract: A technique for processing correlated observations suitable for large, sparse, least-squares adjustments is reviewed Correlated coordinate differences derived from the Global Positioning System are used as illustrative examples However, the methods examined are suitable for all types of correlated observations The computation of variances of residuals, redundancy numbers, and marginally detectable errors is considered for sparse systems

A robust approach to global problems in physical geodesy

Abstract: A technique for renormalization of integral equations is used for obtaining very robust solutions. The number of multiplications used for inverting the integral equations can be reduced dramatically and mostly only weighted means will be needed. Theoretical gain in computer time might be up to 100 000 times for the most favourable cases when using 1000 unknowns. (Practical gains will be considerably less.) Solutions have been obtained with increased accuracy compared to the classical technique over integral equations. Surface elements might be of arbitrary size but the method is optimal for a global approach with equal area elements. The solutions were found strictly invariant with respect to the depth to the internal sphere, when using simpler models.

A unified approach to post-mission processing of intertial data

Abstract: The outpout of inertial survey systems is available to the user in two basic forms: as Kalman filtered information at updates or as integrated velocity and position information at regular time intervals. In case of the second data type, the post-mission processing starts with the approximation of the velocity error curve. This approximation is either based on a system error model as in Kalman filtering or uses curve fitting techniques. From there on, smoothing or adjustment procedures are used as further steps in the post-mission treatment of both data types.

Evaluation of deflections of the vertical using topographic-isostatic and astrogeodetic data for the area of Greece

Abstract: The evaluation of deflections of the vertical for the area of Greece is attempted using a combination of topographic and astrogeodetic data. Tests carried out in the area bounded by 35°≤ϕ≤42°, 19°≤λ≤27° indicate that an accuracy of ±3″.3 can be obtained in this area for the meridian and prime vertical deflection components when high resolution topographic data in the immediate vicinity of computation points are used, combined with high degree spherical harmonic expansions of the geopotential and isostatic reduction potential. This accuracy is about 25% better than the corresponding topographic-Moho deflection components which are evaluated using topographic and Moho data up to 120 km around each station, without any combination with the spherical harmonic expansion of the geopotential or isostatic reduction potential. The accuracy in both cases is increased to about 2″.6 when the astrogeodetic data available in the area mentioned above are used for the prediction of remaining values. Furthermore the estimation of datum-shift parameters is attempted using least squares collocation.

On the meaning of geodetic orientation

Abstract: After deriving models for changes of coordinates and azimuths due to rotations, the investigation considers methods for modeling terrestrial orientation in adjustments of geodetic networks. If a misorientation of a geodetic network exists, this can be due to systematic errors in astronomic longitude or in astronomic azimuth, or in both. A separation of these two effects is not possible in practice. The initial azimuth at the datum origin contributes to the orientation only as much as any other azimuth of the same weight.

On the use of multiple comparisons test for the analysis of levelling discrepancies

Abstract: In a number of papers dating from 1955,A.M. Wassef and his co-worker,F. Z. A. Messih, have demonstrated the application of the analysis of variance technique to the study of levelling discrepancies. The usual method of testing for the significance of means by the Fisher or F-test was used by these authors.

Bayesian inference for the derivation of less sensitive hypothesis tests

Abstract: The matrixt distribution is derived and its properties are shown which are applied for the Bayesian inference in the multivariate linear model. By this approach hypothesis tests for the multivariate model are derived which are less sensitive than the tests of the sampling theory. Examples of their application in the analysis of data for the detection of deformations are given.

Polar motion for an elastic earth model using a canonical theory

Abstract: The purpose of this paper is to develop a canonical formulation of the rotational motion for an elastic Earth model. We have obtained the canonical equations for the precession and nutation motion in an inertial frame, and from this we have deduced the equations in an Earth-fixed frame. The linearized equations deduced for polar motion are equivalent to those obtained using Liouville's equations.

A note on the computation of Meissl’s smoothing coefficients βℓ

Abstract: This note gives a new, short and non-recursive formula for Meissl’s smoothing coefficients βl.

Estimability of astronomical longitude and latitude only from theodolite observations within three-dimensional networks of terrestrial type

Abstract: From a two-dimensional network adjustment it is well understood that the one orientation unknown of a theodolite frame is estimable, once the orientation datum parameter, e.g., one azimuth, is fixed. In three-dimensional networks the problem of estimability of three orientation unknowns inherent in a theodolite frame is more complex. Here we prove that not only the classical horizontal orientation unknown is estimable (up to the datum degrees of freedom), but also astronomical longitude and astronomical latitude which can be considered as two additional orientation unknowns of the theodolite frame moving with respect to an earth-fixed equatorial frame of reference. Thus the theodolite instrument can be considered—at least theoretically—a gradiometer measuring the variation of the directional parameters of the gravity vector from one point to another. Or up to the datum degrees of freedom astronomical longitude and astronomical latitude can be determined from only theodolite observations between exclusively terrestrial points. Monicke (1982), has shown that despite the refraction problem the method works sufficiently well in practice.

On the ergodicity of the covariance function in the case of a plane

Abstract: The Bessel functions can be represented as a limit of spherical harmonics This fact serves as a basis for the transition from the covariance function of the gravity field on the sphere to the covariance function in the plane It is proved that for the limit R→∞ the variance of the empirical covariance function becomes zero so that the famous nonergodicity proof of Lauritzen holds for the sphere but not for the plane

New method for reduction of astrolabe observations using rectangular coordinates on the celestial sphere

Abstract: A new reduction method usingx, y, z-coordinates is derived for astrolabe observations. By this method, the latitude and the longitude of the station are computed without the need of a priori knowledge of the station position. This method is a significant development in data reduction of astrolabe and other almucantar observations due to its mathematical exactness, simplicity, and the ease of handling the associated statistics.

Accuracy of gravimetric deflections of the vertical and optimization of gravity networks

Abstract: In order to solve the problems of determining the shape of a part of the earth of national or continental extent, that is, of rigorous constituting and computing of the astrogeodetic network, it is required to determine gravimetric deflections of the vertical with an accuracy of, say, 0″.3. For this it is adequate to carry out additional gravity surveys in the neighborhoods of computation points, in addition to a given uniform gravity survey (normal density gravity survey).

Sur l'histoire de la notion du potentiel

Abstract: Huygens undoubtedly used the notion of the potential in his well-known work “Discours de la cause de la pesanteur” (Leyden, 1690), investigating the Earth's figure by means of Newton's famous canals. Maupertuis used this notion in a similar work published in London in 1733. The notion of potential was used by Clairaut in the “Theorie de la figure de la Terre” (Paris, 1743) and by D'Alembert in many articles (1752, 1761, 1768, 1773, 1780). Euler used it also in many of his works since 1736. These examples of application of the potential were not noticed because of peculiarities of terminology.

Horizontal pendulum with electrical recording

Abstract: This paper discusses new aspects in the design of horizontal pendulums and gives a description of the construction of a novel horizontal pendulum with capacitive transducer. Functions and characteristics describing the behaviour of the pendulum are also presented.

Indian geoid on GRS 67 from geopotential coefficients

Abstract: The geoid on GRS 67 for the Indian subcontinent has been determined using the geopotential coefficients of degree and order 52 and 180. A broad comparison has also been made between these geoids and the previously determined generalized gravimetric and detailed astrogeodetic-gravimetric geoids. It has been found that still higher degree and order geopotential coefficient solutions from adequate reliable terrestrial data are required to achieve precision of geodetic standards.

Notes on the use of complex number notation

Abstract: Views are expressed on the use of the complex number notation with applications to well known geodetic transformations and relations.