Showing papers in "Journal of Geodesy in 2001"

Adaptively robust filtering for kinematic geodetic positioning

Abstract: The Kalman filter has been applied extensively in the area of kinematic geodetic positioning. The reliability of the linear filtering results, however, is reduced when the kinematic model noise is not accurately modeled in filtering or the measurement noises at any measurement epoch are not normally distributed. A new adaptively robust filtering is proposed based on the robust M (maximum-likelihood-type) estimation. It consists in weighting the influence of the updated parameters in accordance with the magnitude of discrepancy between the updated parameters and the robust estimates obtained from the kinematic measurements and in weighting individual measurements at each discrete epoch. The new procedure is different from functional model-error compensation; it changes the covariance matrix or equivalently changes the weight matrix of the predicted parameters to cover the model errors. A general estimator for an adaptively robust filter is developed, which includes the estimators of the classical Kalman filter, adaptive Kalman filter, robust filter, sequential least-squares adjustment and robust sequential adjustment. The procedure can not only resist the influence of outlying kinematic model errors, but also controls the effects of measurement outliers. In addition to the robustness, the feasibility of implementing the new filter is achieved by using the equivalent weights of the measurements and the predicted state parameters. A numerical example is given to demonstrate the ideas involved.

Accuracy of GPS-derived relative positions as a function of interstation distance and observing-session duration

Abstract: Ten days of GPS data from 1998 were processed to determine how the accuracy of a derived three-dimensional relative position vector between GPS antennas depends on the chord distance (denoted L) between these antennas and on the duration of the GPS observing session (denoted T). It was found that the dependence of accuracy on L is negligibly small when (a) using the `final' GPS satellite orbits disseminated by the International GPS Service, (b) fixing integer ambiguities, (c) estimating appropriate neutral-atmosphere-delay parameters, (d) 26 km ≤ L ≤ 300 km, and (e) 4 h ≤T ≤ 24 h. Under these same conditions, the standard error for the relative position in the north–south dimension (denoted S n and expressed in mm) is adequately approximated by the equation S n =k n /T 0.5 with k n =9.5 ± 2.1 mm · h0.5 and T expressed in hours. Similarly, the standard errors for the relative position in the east–west and in the up-down dimensions are adequately approximated by the equations S e =k e /T 0.5 and S u =k u /T 0.5, respectively, with k e =9.9 ± 3.1 mm · h0.5 and k u =36.5 ± 9.1 mm · h0.5.

The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data

Abstract: The AUSGeoid98 gravimetric geoid model of Australia has been computed using data from the EGM96 global geopotential model, the 1996 release of the Australian gravity database, a nationwide digital elevation model, and satellite altimeter-derived marine gravity anomalies. The geoid heights are on a 2 by 2 arc-minute grid with respect to the GRS80 ellipsoid, and residual geoid heights were computed using the 1-D fast Fourier transform technique. This has been adapted to include a deterministically modified kernel over a spherical cap of limited spatial extent in the generalised Stokes scheme. Comparisons of AUSGeoid98 with GPS and Australian Height Datum (AHD) heights across the continent give an RMS agreement of ±0.364 m, although this apparently large value is attributed partly to distortions in the AHD.

A review of algebraic constraints in terrestrial reference frame datum definition

Abstract: Station coordinates are combined with velocities estimated by space geodesy techniques to produce the International Terrestrial Reference System. The input is sets of coordinates and velocities calculated by International Earth Rotation Service analysis centers using space geodesy techniques. The working reference system of individual analysis centers is generally conventionally defined. However, the implications of such processing can have an effect on the resulting combined set. The problem of datum definition as a function of coordinate combinations is reviewed. In particular, the problem of minimum constraints is clearly emphasized and the reference system effect is defined. The goal is to build a process that could be used generally to remove uncertainties in the underlying coordinate system without disturbing the underlying information with additional unnecessary information.

Random Simulation and GPS Decorrelation

Abstract: (i) A random simulation approach is proposed, which is at the centre of a numerical comparison of the performances of different GPS decorrelation methods. The most significant advantage of the approach is that it does not depend on nor favour any particular satellite–receiver geometry and weighting system. (ii) An inverse integer Cholesky decorrelation method is proposed, which will be shown to out-perform the integer Gaussian decorrelation and the Lenstra, Lenstra and Lovasz (LLL) algorithm, and thus indicates that the integer Gaussian decorrelation is not the best decorrelation technique and that further improvement is possible. (iii) The performance study of the LLL algorithm is the first of its kind and the results have shown that the algorithm can indeed be used for decorrelation, but that it performs worse than the integer Gaussian decorrelation and the inverse integer Cholesky decorrelation. (iv) Simulations have also shown that no decorrelation techniques available to date can guarantee a smaller condition number, especially in the case of high dimension, although reducing the condition number is the goal of decorrelation.

Integer estimation in the presence of biases

Abstract: Carrier phase ambiguity resolution is the key to fast and high-precision GNSS (Global Navigation Satellite System) kinematic positioning. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. Very high success rates are therefore required for ambiguity resolution to be reliable. Biases which are unaccounted for will lower the success rate and thus increase the chance of unsuccessful ambiguity resolution. The performance of integer ambiguity estimation in the presence of such biases is studied. Particular attention is given to integer rounding, integer bootstrapping and integer least squares. Lower and upper bounds, as well as an exact and easy-to-compute formula for the bias-affected success rate, are presented. These results will enable the evaluation of the bias robustness of ambiguity resolution.

ITRF coordinates and plate velocities from repeated GPS campaigns in Antarctica – an analysis based on different individual solutions

Abstract: Scientific Committee on Antarctic Research (SCAR) The GPS campaigns of the between 1995 and 1998 provide a valuable data set which is used to link Antarctica with the ITRF (International Terrestrial Reference Frame), and to gain the first detailed insights into the tectonic behaviour of the Antarctic plate. The data analysis is based on seven individual solutions with four different software packages (Bernese, GAMIT/GLOBK, GIPSY/OASIS II, GEONAP), and provides the possibility to study the existing software noise. The combined solution yields an accuracy of 1 cm for the horizontal and 2 cm for the vertical coordinate components within ITRF96. Based on repeated observations during a total time span of three years, it is possible to determine detailed regional deformations in the area of the Antarctic Peninsula and to obtain an estimate for the rotation vector of the whole Antarctic plate. The main findings of a very comprehensive project of, in total, seven working groups are summarized and presented for the first time. The main emphasis is on the geodetic aspects of the project, i.e. the analysis of the same data set with different software packages, and on the final results from repeated observations, namely station coordinates and horizontal velocities in ITRF96.

GEOID99 and G99SSS: 1-arc-minute geoid models for the United States

Abstract: Two new geoid models have been computed for the United States and its territories. The first model is the purely gravimetric G99SSS model, approximating the geopotential surface W 0=62636856.88 m2/s2 and referenced to the geocentric GRS-80 ellipsoid whose origin coincides with the ITRF96(1997.0) origin. The other model is the hybrid GEOID99 model, which encompasses all gravimetric information of G99SSS as well as the vertical datum information of 6169 GPS-derived NAD 83 (North American Datum 1983) ellipsoid heights co-located with spirit-leveled NAVD 88 (North American Vertical Datum 1988) Helmert orthometric heights. The coverage of both models in the conterminous United States (CONUS) is from 24°N to 58°N latitude and 230°E to 300°E longitude. Long-wavelength geoid structure was controlled by the EGM96 model coefficients, medium-scale information by 2.6 million gravity measurements, and local features through the use of 30-arc-second and (recently created) 3-arc-second digital elevation models. In addition to new elevation data, there were corrected errors in the old elevation data, better satellite altimetry data, and ellipsoidal corrections in G99SSS and GEOID99 that were not applied in G96SSS and GEOID96. The GEOID99 model replaces the GEOID96 model as the primary conversion surface between NAD 83 ellipsoid heights and NAVD 88 Helmert orthometric heights. While GEOID96 had the ability to convert absolute heights (of the 1996 GPS-on-bench-mark data set, or GPSBMs) at the ±5.3-cm level, GEOID99 works with the latest GPSBMs at the ±4.6-cm level. The improvement from 5.5 to 4.6 cm in RMS further translates into almost 2 cm of additional accuracy in single-tie differential GPS-based leveling over most short baselines. Of this 4.6 cm, some 2.0 to 2.5 cm is attributed to correlated errors in the gravimetric geoid and local GPS errors, decorrelating around 40 km. The GPS-derived ellipsoid heights in CONUS have changed since 1996 due to an increased number of observations and new adjustments of previously observed areas. This new GPS data is reflected in GEOID99, but not GEOID96, which effectively renders GEOID96 out of date (in an absolute sense), so that the absolute agreement between GEOID96 and the 1999 set of GPSBMs is only at the ±6.5-cm level. In addition to the geoid models in CONUS, gravimetric geoid models were produced in Alaska, Hawaii, and Puerto Rico and the American Virgin Islands. Surface deflection of the vertical models were also computed and show agreement with astro-geodetically determined deflections below the 1-arc-second level.

A new approach for airborne vector gravimetry using GPS/INS

Abstract: A new method for airborne vector gravimetry using GPS/INS has been developed and the results are presented. The new algorithm uses kinematic accelerations as updates instead of positions or velocities, and all calculations are performed in the inertial frame. Therefore, it is conceptually simpler, easier, more straightforward and computationally less expensive compared to the traditional approach in which the complex navigation equations should be integrated. Moreover, it is a unified approach for determining all three vector components, and no stochastic gravity modeling is required. This approach is based on analyzing the residuals from the Kalman filter of sensor errors, and further processing with wavenumber coefficient filterings is applied in case closely parallel tracks of data are available. An application to actual test-flight data is performed to test the validity of the new algorithm. The results yield an accuracy in the down component of about 3–4 mGal. Also, comparable results are obtained for the horizontal components with accuracies of about 6 mGal. The gravity modeling issue is discussed and alternative methods are presented, none of which improves on the original approach.

Robustness analysis of geodetic horizontal networks

Abstract: Traditional reliability analysis has been augmented with geometrical strength analysis using strain techniques, resulting in the conception of an extension to reliability theory called robustness analysis. To reflect contemporary statistical terminology, robustness is taken to mean insensitivity to gross errors or blunders in the data. Robustness analysis is a natural merger of reliability and strain and is defined as the ability to resist deformations induced by the smallest detectable blunders as determined from internal reliability analysis. The geometrical strength analysis technique is used in order to provide a more complete and detailed description of the potential network deformation in terms of three independent measures representing robustness in scale, orientation, and configuration. These measures are also invariant with respect to “datum” shifts and orientation, and practically invariant to changes in scale. Initial experiences with robustness analysis have shown that it is a very powerful technique capable of providing a detailed point-by-point assessment of the strength of a network.

Precise estimation of residual tropospheric delays using a regional GPS network for real-time kinematic applications

Abstract: A new method called Trop_NetAdjust is described to predict in real time the residual tropospheric delays on the GPS carrier phase observables using the redundant measurements from a network of GPS reference stations. This method can not only enhance the effectiveness and reliability of real-time kinematic users within the network, but also provide a valid approach to tropospheric parameter variation forecasting. Trop_NetAdjust is theoretically based upon LS prediction criteria and enables the prediction of residual tropospheric delays remaining after a standard model has been applied to the raw GPS measurements. Two cases are analyzed, namely a first case when the delay is required for an existing satellite at a new point within the network and a second case when the delay is required for a new satellite. Field tests were conducted using data collected in a network of 11 reference stations covering a 400×600 km region in southern Norway. The results were analyzed in the measurement domain (ionospheric-free double-difference residuals) and showed improvements of 20 to 65% RMS errors using Trop_NetAdjust. The estimates of the Trop_NetAdjust prediction accuracy were also obtained using the covariance analysis method. The agreement was consistently better than 30% when compared with data from a real network.

Effect of topographical density on geoid in the Canadian Rocky Mountains

Abstract: Gravity reduction from the Earth's surface to the geoid requires knowledge of topographical mass density. However, in practice the constant density (2.67 g/cm3) is mostly used to approximate the actual density because of the difficulty and complexity of obtaining the actual density. This approximation introduces errors in the reduced gravity, and consequently, in the geoid. Recently, the geographical information system (GIS) was introduced as an efficient tool to geo-reference actual bedrock densities to digital geological maps.

Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one

Abstract: The topographic and atmospheric effects of gravimetric geoid determination by the modified Stokes formula, which combines terrestrial gravity and a global geopotential model, are presented. Special emphasis is given to the zero- and first-degree effects. The normal potential is defined in the traditional way, such that the disturbing potential in the exterior of the masses contains no zero- and first-degree harmonics. In contrast, it is shown that, as a result of the topographic masses, the gravimetric geoid includes such harmonics of the order of several centimetres. In addition, the atmosphere contributes with a zero-degree harmonic of magnitude within 1 cm.

The impact of magnetic storms on GPS receiver performance

Abstract: During the past decade, geodetic measurements have been greatly improved through the use of space geodesy techniques based on the global positioning system (GPS). The GPS allows for precise surveys, where centimetre-level accuracies may be obtained through differential carrier phase-based positioning algorithms. GPS positioning techniques often rely on the availability of dual-frequency data for formation of ionosphere-free and widelane observables (in the case of ambiguity resolution). While GPS positioning results generally provide adequate accuracies for several precise positioning applications, users may experience degraded positioning accuracies over the next few years. Of particular concern are ionospheric phenomena called magnetic storms, which are characterized by an increased spatial decorrelation of ionosphere range delays and scintillation effects at low and high latitudes. During such events, degradations in differential positioning accuracies are observed, and availability of the L2 observations may be limited through loss of signal lock. The magnitude of such effects is estimated during a global magnetic storm event, and a performance comparison of two survey-grade receivers is conducted. This storm is representative of the activity at solar maximum and several years beyond.

The solution of the general geodetic boundary value problem by least squares

Abstract: A general scheme is given for the solution in a least-squares sense of the geodetic boundary value problem in a spherical, constant-radius approximation, both uniquely and overdetermined, for a large class of observations. The only conditions are that the relation of the observations to the disturbing potential is such that a diagonalization in the spectrum can be found and that the error-covariance function of the observations is isotropic and homogeneous. Most types of observations used in physical geodesy can be adjusted to fit into this approach. Examples are gravity anomalies, deflections of the vertical and the second derivatives of the gravity potential.

Comparison of reliability and geometrical strength criteria in geodetic networks

Abstract: The proper and optimal design and subsequent assessment of geodetic networks is an integral part of most surveying engineering projects Optimization and design are carried out before the measurements are actually made A geodetic network is designed and optimized in terms of high reliability and the results are compared with those obtained by the robustness analysis technique The purpose of an optimal design is to solve for both the network configuration (first-order design) and observations accuracy (second-order design) in order to meet the desired criteria For this purpose, an analytical method is presented for performing the first-order design, second-order design, and/or the combined design In order to evaluate the geometrical strength of a geodetic network, the results of robustness analysis are displayed in terms of robustness in rotation, robustness in shear, and robustness in scale Results showed that the robustness parameters were affected by redundancy numbers The largest robustness parameters were due to the observations with minimum redundancy numbers

Accurate absolute GPS positioning through satellite clock error estimation

Abstract: An algorithm for very accurate absolute positioning through Global Positioning System (GPS) satellite clock estimation has been developed. Using International GPS Service (IGS) precise orbits and measurements, GPS clock errors were estimated at 30-s intervals. Compared to values determined by the Jet Propulsion Laboratory, the agreement was at the level of about 0.1 ns (3 cm). The clock error estimates were then applied to an absolute positioning algorithm in both static and kinematic modes. For the static case, an IGS station was selected and the coordinates were estimated every 30 s. The estimated absolute position coordinates and the known values had a mean difference of up to 18 cm with standard deviation less than 2 cm. For the kinematic case, data obtained every second from a GPS buoy were tested and the result from the absolute positioning was compared to a differential GPS (DGPS) solution. The mean differences between the coordinates estimated by the two methods are less than 40 cm and the standard deviations are less than 25 cm. It was verified that this poorer standard deviation on 1-s position results is due to the clock error interpolation from 30-s estimates with Selective Availability (SA). After SA was turned off, higher-rate clock error estimates (such as 1 s) could be obtained by a simple interpolation with negligible corruption. Therefore, the proposed absolute positioning technique can be used to within a few centimeters' precision at any rate by estimating 30-s satellite clock errors and interpolating them.

Linear drift and periodic variations observed in long time series of polar motion

Abstract: Two long time series were analysed: the C01 series of the International Earth Rotation Service and the pole series obtained by re-analysis of the classical astronomical observations using the HIPPARCOS reference frame. The linear drift of the pole was determined to be 3.31 ± 0.05 milliarcseconds/year towards 76.1 ±\(\) 0.80° west longitude. For the least-squares fit the a priori correlations between simultaneous pole coordinates x p , y p were taken into account, and the weighting function was calculated by estimating empirical variance components. The decadal variations of the pole path were investigated by Fourier and wavelet analysis. Using sliding windows, the periods and amplitudes of the Chandler wobble and annual wobble were determined. Typical periods in the variable Chandler wobble and annual wobble parameters were obtained from wavelet analyses.

Terrain correction computations for a densely sampled DTM in the Bavarian Alps

Abstract: Various methods are presented for terrain correction computations referred to a digital terrain model (DTM) of a part of the Bavarian Alps to the southwest of Munich. The most commonly used fast Fourier transform (FFT) techniques exhibit some shortcomings for the zone adjacent to the computation point. A combination method is proposed which applies FFT for the whole DTM except for a small zone surrounding the computation point. In the latter an analytical method is employed by decomposing the terrain into elementary bodies. Different modelling techniques are tested, with the arbitrary polyhedron being the most precise one. With the proposed method both the exactness of the analytical expressions and the efficiency and speed of the FFT technique can be exploited. It is shown that this modified approach has two positive effects. First, the convergence of the FFT series is regained, which is not always the case when slopes greater than 45∘ exist in the terrain. Second, it approximates the terrain correction as computed by the classical prism summation method better than its linear approximation as given by FFT.

Modeling and estimation of C1-P1 bias in GPS receivers

Abstract: Modern dual-frequency global positioning system (GPS) receivers are capable of providing direct measurements of both L1 C/A (C1) and P code (P1) without the use of the Y-codes under Anti-Spoofing. A discrepancy or bias between the C1 and P1 measurements from these receivers has however been of concern to operators and users of GPS reference networks. For the purpose of modeling and estimation, the nature and characteristics of the discrepancy must be investigated. The research results presented indicate that the discrepancy between the C1 and P1 measurements contains two different types of components: one is of constant type while another is time variant. A method has been developed for their modeling and estimation. The residual C1–P1 time series after a satellite-dependent bias removal agree at a few-centimeter level, indicating the effectiveness of the proposed model. This allows the C1–P1 discrepancy, both constant and non-constant components, to be removed from GPS reference network solutions. Numerical results are provided to support the analysis.

Geoid, topography, and the Bouguer plate or shell

Abstract: Topography plays an important role in solving many geodetic and geophysical problems. In the evaluation of a topographical effect, a planar model, a spherical model or an even more sophisticated model can be used. In most applications, the planar model is considered appropriate: recall the evaluation of gravity reductions of the free-air, Poincare–Prey or Bouguer kind. For some applications, such as the evaluation of topographical effects in gravimetric geoid computations, it is preferable or even necessary to use at least the spherical model of topography. In modelling the topographical effect, the bulk of the effect comes from the Bouguer plate, in the case of the planar model, or from the Bouguer shell, in the case of the spherical model. The difference between the effects of the Bouguer plate and the Bouguer shell is studied, while the effect of the rest of topography, the terrain, is discussed elsewhere. It is argued that the classical Bouguer plate gravity reduction should be considered as a mathematical construction with unclear physical meaning. It is shown that if the reduction is understood to be reducing observed gravity onto the geoid through the Bouguer plate/shell then both models give practically identical answers, as associated with Poincare's and Prey's work. It is shown why only the spherical model should be used in the evaluation of topographical effects in the Stokes–Helmert solution of Stokes' boundary-value problem. The reason for this is that the Bouguer plate model does not allow for a physically acceptable condensation scheme for the topography.

Effects of the spherical terrain on gravity and the geoid

Abstract: The determination of the gravimetric geoid is based on the magnitude of gravity observed at the topographical surface and applied in two boundary value problems of potential theory: the Dirichlet problem (for downward continuation of gravity anomalies from the topography to the geoid) and the Stokes problem (for transformation of gravity anomalies into the disturbing gravity potential at the geoid). Since both problems require involved functions to be harmonic everywhere outside the geoid, proper reduction of gravity must be applied. This contribution deals with far-zone effects of the global terrain on gravity and the geoid in the Stokes–Helmert scheme. A spherical harmonic model of the global topography and a Molodenskij-type spectral approach are used for a derivation of suitable computational formulae. Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the significance of these effects in precise (i.e. centimetre) geoid computations. Their omission can be responsible for a long-frequency bias in the geoid, especially over mountainous areas. Due to the rough topography of the testing area, these numerical values can be used as maximum global estimates of the effects (maybe with the exception of the Himalayas). This study is a continuation of efforts to model adequately the topographical effects on gravity and the geoid, especially of a comparing the effects of the planar topographical plate and the spherical topographical shell on gravity and the geoid [Vanicek, Novak, Martinec (2001) J Geod 75: 210–215].

ITRS, PZ-90 and WGS 84: current realizations and the related transformation parameters

Abstract: The first results of the International GLONASS Experiment 1998 (IGEX-98) campaign have provided significant material to illustrate the mutual benefits of the GLONASS system and the realization of the International Terrestrial Reference System (ITRS). A specific aspect, namely the relationship between the World Geodetic System 1984 (WGS 84) and the PZ-90 system using ITRS as a primary standard, is investigated. A review of current works is carried out. A transformation strategy is proposed for the three systems based on recent results from IGEX-98 and an independent set of transformation parameters derived by the Jet Propulsion Laboratory from ITRF97 and PZ-90 coordinates for 16 global stations.

Analytical solar radiation pressure modelling for GLONASS using a pixel array

Abstract: A new method for calculating analytical solar radiation pressure models for GNSS spacecraft has been developed. The method simulates the flux of light from the Sun using a pixel array. The method can cope with a high level of complexity in the spacecraft structure and models effects due to reflected light. Models have been calculated and tested for the Russhar global navigation satellite system GLONASS IIv spacecraft. Results are presented using numerical integration of the force model and long-arc satellite laser ranging (SLR) analysis. The integrated trajectory differs from a precise orbit calculated using a network of global tracking stations by circa 2 m root mean square over a 160 000-km arc. The observed − computed residuals for the 400-day SLR arc are circa 28 mm.

An improvement to ionospheric delay correction for single-frequency GPS users - the APR-I scheme

Abstract: For the commonly used GPS wide-area augmentation systems (WAAS) with a grid ionospheric model, the efficient modelling of ionospheric delays in real time, for single-frequency GPS users, is still a crucial issue which needs further research. This is particularly necessary when differential ionospheric delay corrections cannot be broadcast, when users cannot receive them, or when there are ionospheric anomalies. Ionospheric delays have a severe effect on navigation performance of single-frequency receivers. A new scheme is proposed which can efficiently address the above problems. The robust recurrence technique is based on the efficient combination of single-frequency GPS observations by users and the high-precision differential ionospheric delay corrections from WAAS. Its effectiveness is verified with examples.

Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

Abstract: In many geoscientific applications data are irregularly distributed and not globally available, e.g. caps around the poles which are uncovered due to non-polar satellite orbits, or signals being defined solely on bounded regions on the globe. Starting from a sequence of base functions with global support, which in the present case is composed of spherical harmonics being initially non-orthogonal on a bounded subdomain, a set of functions is generated that constitutes an orthonormal basis. Different approaches to realize this transformation are studied and compared with respect to numerical stability and computational effort, and the corresponding effects on the coefficient recovery are investigated. A number of synthetic tests demonstrate the applicability, the benefit, but also the limitations, of this method.

On the accuracy of modified Stokes's integration in high-frequency gravimetric geoid determination

Abstract: Two numerical techniques are used in recent regional high-frequency geoid computations in Canada: discrete numerical integration and fast Fourier transform. These two techniques have been tested for their numerical accuracy using a synthetic gravity field. The synthetic field was generated by artificially extending the EGM96 spherical harmonic coefficients to degree 2160, which is commensurate with the regular 5′ geographical grid used in Canada. This field was used to generate self-consistent sets of synthetic gravity anomalies and synthetic geoid heights with different degree variance spectra, which were used as control on the numerical geoid computation techniques. Both the discrete integration and the fast Fourier transform were applied within a 6∘ spherical cap centered at each computation point. The effect of the gravity data outside the spherical cap was computed using the spheroidal Molodenskij approach. Comparisons of these geoid solutions with the synthetic geoid heights over western Canada indicate that the high-frequency geoid can be computed with an accuracy of approximately 1 cm using the modified Stokes technique, with discrete numerical integration giving a slightly, though not significantly, better result than fast Fourier transform.

Sea level change in Hong Kong from tide gauge measurements of 1954–1999

Abstract: Tide gauge records of Hong Kong covering the past 45 years (19540–19990) are adopted to analyze the basic features of sea level changes in the region Data sets of atmospheric pressure, southern oscillation index and sea surface temperature during the same time span are also used to determine the possible link between the sea level changes in Hong Kong and local and global geophysical processes Results indicate that the sea level of Hong Kong has a rising trend of 19 ± 04 mm per year, and that there is an upward offset of about 15 cm in the pre-19570 tide gauge records The effect of local atmospheric pressure variations on the amplitude of the annual sea level change is about 30% of the amplitude that is calculated after the effect is corrected It is also found that the interannual variations in the sea level of Hong Kong are related to El Nino and La Nina events that happen frequently in the tropical Pacific

Gravitational attraction of local crustal masses in spherical coordinates

Abstract: The gravitational attractions of terrestrial masses and condensed terrestrial masses were modeled in local regions of gravity stations in different ways. These differences in the models included the type of coordinate frame (Cartesian versus spherical), grid spacing (30 vs 3 arcseconds), and the shape of the terrain (“flat-topped” vs “sloped-topped” prisms). The effect of each of these variables is quantified for its overall impact on Helmert gravity anomalies. The combined effect of removing the masses and restoring the condensed masses is also compared to classical terrain corrections for suitability in computing Helmert anomalies. Some detailed conclusions are drawn from these test computations. The effect of the Earth's curvature has both a near-field effect (due to the differences in volume and shape between rectangular and spherical prisms) and a far-field effect (due to physical location of masses below the horizon). The near-field effect can achieve 0.4 mGal in the Rocky mountains, and affect the geoid by up to 7.5 cm. Additionally, the approximation of the terrain by flat-topped prisms (even at fine spacings such as 3 arcseconds) is inappropriate for terrain near the station, where errors of 20 mGal have been computed using 30-arcsecond data. It is concluded that when 30-arcsecond terrain is allowed to have a more curved (bilinear) prism top, its gravitational attraction is a significantly closer approximation of 3-arcsecond terrain, even for the prism surrounding the station, as compared to the case of 30-arcsecond flat-topped prisms. It is suggested that classical terrain corrections, for many reasons, should not be used to compute Helmert anomalies. Considering only the accuracy, and not the speed, of the computations, the following conclusions are drawn: terrain effects computed inside a local “cap” should be done exclusively in spherical coordinates with a 3-arcsecond Digital Elevation Model (DEM) out to 0.2∘ radius, and then a 30-arcsecond DEM from 0.2 out to 3.5∘. In all cases, bilinearly shaped prism tops should be used.