The absolute determination of the gravitational acceleration at Sydney, Australia

An absolute measurement of the gravitational acceleration “g” has been made at the National Standards Laboratory, Chippendale, N.S.W., Australia. The determination was made by studying the free motion of a body projected vertically upwards in a vacuum and the time between its initial and final passages through two horizontal planes of known vertical separation was measured. The measured value ofg at a point 12 metres above the floor in room B. 37 of the National Standards Laboratory is 9.7967134 m/s² The corresponding value at floor level at the BMR gravity station is 9.796717 m/s² Paper presented at the meeting of the International Gravimetric Commission, Paris 7–11 September 1970.     

GPS precision as a function of session duration and reference frame using multi-point software

As an aid to survey design, we used data acquired from three European continuous GPS networks to test the precision of position estimates from static observations as a function of the length of the observing session and the number and distribution of reference stations. Our criterion was the weighted RMS of estimates over 31 days with respect to coordinates determined from 24-h sessions over a 2-year period. With a single reference station, a precision of 3 mm horizontal and 10 mm vertical could be achieved reliably only for session lengths of 3 h or longer and baselines less than 200 km. If four or more reference stations are used, these levels of precision were usually achieved with sessions as short as 2 h. With sessions 6 h or longer and four or more reference stations, the precision is typically 1–2 mm in horizontal and about 3–5 mm in vertical. Increasing the number of reference stations further provides only marginal improvement. Although there is some variation in precision in 4-station networks with the choice of reference stations, the dependence on distance and geometric distribution is weak.     

Earth rotation and network geometry optimization for very long baseline interferometers

Very Long Baseline Interferometry (VLBI) is one of the new techniques which will probably dominate geodesy and geophysics in the near future. Its main advantage lies in the fact that it brings the accuracy of direction measurements to a level previously possible only for range measurements. This closes the gap between powerful range determination techniques such as laser ranging and the much less accurate determination of directions through photographic tracking of artificial earth satellites. The technique is geometric in the sense that the relevant observations are independent of the gravity field of the earth. However, the “orbits” of the observed extragalactic radio sources with respect to an earth-fixed system are dominated and perturbed by the rotation of the earth with respect to inertial frame. This allows the determination of polar motion, precession-nutation and length-of-the-day variations, and the technique becomes also “dynamic” in this respect. The capability of determining the geometry of a network of stations within a short time interval and with a centimeter level accuracy also allows the study of the variation of network geometry with time caused by earth tides and other periodic or secular station drifts. The main objective of the present work is the exploration of the capabilities of VLBI for the recovery of earth rotation and network geometry parameters. For this purpose, a number of characteristic experimental designs based on present and candidate for the near future station locations is chosen. The results from the analysis of simulated observations for each particular design are presented in the paper. Presented at IAG International Symposium on “Optimization of Design and Computation of Control Networks”, Sopron, Hungary, 4–10 July 1977.     

Pseudo-Satellite Applications in Deformation Monitoring

In this article, three general classes of potential pseudolite applications for deformation monitoring are described. The first is GPS augmentation with pseudolite(s), which is suitable for circumstances such as urban canyons, or for monitoring in valleys and deep open-cut mines. The second is indoor applications of pseudolite deformation monitoring systems. Pseudolite arrays can, in principle, completely replace the GPS satellite constellation. This could extend the “satellite-based” deformation monitoring applications into tunnels or underground, where GPS satellite signals cannot be tracked. The last case is an inverted pseudolite-based deformation monitoring system, where a “constellation” of GPS receivers with precisely known “orbits” track a mobile pseudolite. The system consists of an array of GPS receivers, the base reference pseudolite, the mobile pseudolite, and a central processing system. However, in the case of such pseudolite-only or hybrid pseudolite-GPS deformation monitoring systems, some additional issues need to be addressed. These include multipath, atmospheric delay effects, and pseudolite location-dependent biases. To address deformation monitoring applications, some practical procedures to mitigate or eliminate their influence are suggested. Some experiments were carried out using NovAtel GPS receivers and IntegriNautics IN200CXL pseudolite instruments. The experimental results indicate that the accuracy of the height component can indeed be significantly improved – the RMS of the vertical component has been reduced by a factor of 4, to the same level as the horizontal components. Their performance will be demonstrated through case study example. ? 2002 Wiley Periodicals, Inc.     

Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation

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.     

Eye on the Ionosphere: GPS after SA

Selective Availability (SA), the intentional degradation of the accuracy of the single-frequency GPS position by the DoD, was ended on May 2, 2000. This major policy decision, promised to occur some time in the next 5 years, had an instantaneous and dramatic impact on the users of single-frequency GPS. Figure 1 shows the “before and after” of the turn-off of SA, and illustrates how significant and error source SA was. SA was turned off at 0400 UTC, the point on the graph at which the horizontal and vertical errors were markedly reduced. The magnitudes of the Standard Positioning Service (SPS) circular error probable (CEP), and sperical error probable (SEP) – the uncertainty in a fix – were reduced to less than 5 meters. ? 2001 John Wiley & Sons, Inc.     

云南及周边地区GNSS数据处理方法优化

采用麻省理工学院开发的GAMIT／GLOBK软件,将2015年-2016年全球347个IGS站观测数据分七个子网解算,得到一个固定的参考框架来解算云南及周边地区的35个全球卫星导航系统（GNSS）基准站的坐标,测站坐标均方根误差水平方向在0.7 mm以内,垂直方向在0.3 mm以内,水平方向的坐标重复性精度在5 mm以内,垂向坐标的重复性精度大多数在2.5 cm以内;与在ITRF2014下解算的测站坐标、基线长度、水平速度场结果对比表明:测站坐标存在系统误差,水平方向上的差异在8.5 mm以内,垂直方向上在3 cm以内;基线长度差异在2 mm以内,水平速度场在数值上存在毫米级的差异,方向上基本一致．      

＂Digital Region＂ in the Context of the ＂Grid Computing＂

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.     

The adjustment of intersecting chains of triangulation using a medium sized computer

Summary A method of adjusting intersecting chains of triangulation by direct elimination, using variation of coordinates, is given which is suitable for a computer of modest storage capacity (about 4000–8000 words). The network is considered to be made up of “nodes” (where two or more chains intersect) and “links” (which connect two nodes). The normal equations for the points in each link are computed, the coefficients of the unknown coordinates of the associated nodes being treated as extra right hand sides. These equations are then solved to express the coordinates of any link station in terms of the associated nodes. The process is repeated for all links. The normal equations for the nodes are then set up and substitution made for all “link” terms. The method is particularly powerful for the adjustment of a few, long, interconnected chains since the reduced normal equation coefficients are then banded about the diagonal, the semi band width being a little greater than the largest number of unknowns in the nodes along a single tie chain. For a computer of 7000 words capacity it is possible to solve for a band width of over 100 unknowns i.e. more than 50 stations.     

Global plate tectonics and the secular motion of the Pole

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.     

The convergence problem of collocation solutions in the framework of the stochastic interpretation

The problem of the convergence of the collocation solution to the true gravity field was defined long ago (Tscherning in Boll Geod Sci Affini 39:221–252, 1978) and some results were derived, in particular by Krarup (Boll Geod Sci Affini 40:225–240, 1981). The problem is taken up again in the context of the stochastic interpretation of collocation theory and some new results are derived, showing that, when the potential T can be really continued down to a Bjerhammar sphere, we have a quite general convergence property in the noiseless case. When noise is present in data, still reasonable convergence results hold true.

“Democrito che ’l mondo a caso pone” “Democritus who made the world stochastic” Dante Alighieri, La Divina Commedia, Inferno, IV – 136     

Nonlinear least-squares method via an isomorphic geometrical setup

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.     

Polar motion modeling,analysis, and prediction with time dependent harmonic coefficients

A time dependent amplitude model was proposed for the analysis and prediction of polar motion time series. The formulation was implemented to analyze part of the new combined solution, EOP (IERS) C 04, daily polar motion time series of 14 years length using a statistical model with first order autoregressive disturbances. A new solution approach, where the serial correlations of the disturbances are eliminated by sequentially differencing the measurements, was used to estimate the model parameters using weighted least squares. The new model parsimoniously represents the 14-year time series with 0.5 mas rms fit, close to the reported 0.1 mas observed pole position precisions for the x and y components. The model can also predict 6 months into the future with less than 4 mas rms prediction error for both polar motion components, and down to sub mas for one-step ahead prediction as validated using a set of daily time series data that are not used in the estimation. This study is dedicated to the memory of Prof. Urho Uotila (1923–2006) whose teaching of “Adjustment Computations” over the years influenced so much, so many of us who had the privilege of being his students.     

A detailed study of the duality relation for the least squares adjustment in euclidean spaces

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.     

Performance of Singapore Integrated Multiple Reference Station Network (SIMRSN) for RTK Positioning

Conventional RTK positioning is usable, but requires the use of a local base station. It is also restricted by the effects of the de-correlate atmospheric refraction on the GPS signal, which limits the use of the RTK positioning up to distances of 10–15 km from the reference station to the user. With a multiple reference station network approach, precise RTK positioning capability may be extended for a much larger area. The Singapore Integrated Multiple Reference Station Network (SIMRSN) has been established for this purpose. Using an existing method termed linear combination method, the multiple reference station network corrections are generated for the user on an epoch by epoch, satellite-by-satellite basis. A residual-based adaptive Kalman filter is proposed to improve network correction availability. The VRS concept is used to transmit corrections to the user for RTK positioning. Field tests were conducted to demonstrate the general performance of SIMRSN. The tests confirmed that RTK positioning within a multiple reference station network can provide the user with better than 3 cm in horizontal position, the height accuracy is in the range of 1–7 cm, and the average TTF (Time To Fix) are 46 and 76 s during GLC station and LPR station tests, respectively. With this highly efficient survey technique, the user needs only to be equipped with a single GPS receiver. This reduces equipment and manpower costs compared with traditional RTK positioning using a specially set up reference station. The multiple reference station framework also paves the way for various other applications beyond the traditional surveying profession. Electronic Publication     

Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data

At the beginning of the twenty-first century, a technological change took place in geodetic astronomy by the development of Digital Zenith Camera Systems (DZCS). Such instruments provide vertical deflection data at an angular accuracy level of 0.̋1 and better. Recently, DZCS have been employed for the collection of dense sets of astrogeodetic vertical deflection data in several test areas in Germany with high-resolution digital terrain model (DTM) data (10–50 m resolution) available. These considerable advancements motivate a new analysis of the method of astronomical-topographic levelling, which uses DTM data for the interpolation between the astrogeodetic stations. We present and analyse a least-squares collocation technique that uses DTM data for the accurate interpolation of vertical deflection data. The combination of both data sets allows a precise determination of the gravity field along profiles, even in regions with a rugged topography. The accuracy of the method is studied with particular attention on the density of astrogeodetic stations. The error propagation rule of astronomical levelling is empirically derived. It accounts for the signal omission that increases with the station spacing. In a test area located in the German Alps, the method was successfully applied to the determination of a quasigeoid profile of 23 km length. For a station spacing from a few 100 m to about 2 km, the accuracy of the quasigeoid was found to be about 1–2 mm, which corresponds to a relative accuracy of about 0.05−0.1 ppm. Application examples are given, such as the local and regional validation of gravity field models computed from gravimetric data and the economic gravity field determination in geodetically less covered regions.     

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

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     

On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E _Y = (X−E _X ) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E _Y and E _X . Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.     

The corridor correction concept

Atmospheric delays are contributors to the GNSS error budget in precise GNSS positioning that can reduce positioning accuracy considerably if not compensated appropriately. Both ionospheric and tropospheric delay corrections can be determined with help of reference stations in active GNSS networks. One approach to interpolate these error terms to the user’s location that is employed in Germany’s SAPOS network is the determination of area correction parameters (ACP, German: “Fl?chenkorrekturparameter—FKP”). A 2D interpolation scheme using data from at least 3 reference stations surrounding the rover is employed. A modification of this method was developed which only makes use of as few as 2 reference stations and provides 1D linear correction parameters along a “corridor” in which the user’s rover is moving. We present the results of a feasibility study portraying results from use of corridor correction parameters for precise RTK-like positioning. The differences to the reference coordinates (3D) attained in average for 1 h of data employing selected network nodes in Germany are between 0.8 and 2.0 cm, which compares well with the traditional area correction method that yields an error of 0.7 up to 1.1 cm.     

Long-wavelength global gravity field models: GRIM4-S4, GRIM4-C4

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