On the equivalence of spherical splines with least-squares collocation and Stokes’s formula for regional geoid computation

This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi: 10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.     

Geocenter variations derived from a combined processing of LEO- and ground-based GPS observations

GNSS observations provided by the global tracking network of the International GNSS Service (IGS, Dow et al. in J Geod 83(3):191–198, 2009) play an important role in the realization of a unique terrestrial reference frame that is accurate enough to allow a detailed monitoring of the Earth’s system. Combining these ground-based data with GPS observations tracked by high-quality dual-frequency receivers on-board low earth orbiters (LEOs) is a promising way to further improve the realization of the terrestrial reference frame and the estimation of geocenter coordinates, GPS satellite orbits and Earth rotation parameters. To assess the scope of the improvement on the geocenter coordinates, we processed a network of 53 globally distributed and stable IGS stations together with four LEOs (GRACE-A, GRACE-B, OSTM/Jason-2 and GOCE) over a time interval of 3 years (2010–2012). To ensure fully consistent solutions, the zero-difference phase observations of the ground stations and LEOs were processed in a common least-squares adjustment, estimating all the relevant parameters such as GPS and LEO orbits, station coordinates, Earth rotation parameters and geocenter motion. We present the significant impact of the individual LEO and a combination of all four LEOs on the geocenter coordinates. The formal errors are reduced by around 20% due to the inclusion of one LEO into the ground-only solution, while in a solution with four LEOs LEO-specific characteristics are significantly reduced. We compare the derived geocenter coordinates w.r.t. LAGEOS results and external solutions based on GPS and SLR data. We found good agreement in the amplitudes of all components; however, the phases in x- and z-direction do not agree well.     

A new evil waveforms evaluating method for new BDS navigation signals

With the advent of new global navigation satellite systems (GNSSs) and new signals, GNSS users will rely more on them to obtain higher-accuracy positioning. Evil waveform monitoring and assessment are of great importance for GNSS to achieve its positioning, velocity, and timing service with high accuracy. However, the advent of new navigation signals introduces the necessity to extend the traditional analyzing techniques already accepted for binary phase-shift keying modulation to new techniques. First, the well-known second-order step thread model adopted by the International Civil Aviation Organization is introduced. Then the extended new general thread models are developed for the new binary offset carrier modulated signals. However, no research has been done on navigation signal waveform symmetry yet. Simulation results showed that, waveform asymmetry may also cause tracking errors, range biases, and position errors in GNSS receivers. It is thus imperative that the asymmetry be quantified to enable the design of appropriate error budgets and mitigation strategies for various application fields. A novel evil waveform analysis method, called waveform rising and falling edge symmetry (WRaFES) method, is proposed. Based on this WRaFES method, the correlation metrics are provided to detect asymmetric correlation peaks distorted by received signal asymmetry. Then the statistical properties of the proposed methods are analyzed, and a proper deformation detection threshold is calculated. Finally, both simulation results and experimentally measured results of Beidou navigation satellite system (BDS) M1-S B1Cd signal are given, which show the effectiveness and robustness of the proposed thread models.     

Position-domain integrity risk-based ambiguity validation for the integer bootstrap estimator

Integrity monitoring for ambiguity resolution is of significance for utilizing the high-precision carrier phase differential positioning for safety–critical navigational applications. The integer bootstrap estimator can provide an analytical probability density function, which enables the precise evaluation of the integrity risk for ambiguity validation. In order to monitor the effect of unknown ambiguity bias on the integer bootstrap estimator, the position-domain integrity risk of the integer bootstrapped baseline is evaluated under the complete failure modes by using the worst-case protection principle. Furthermore, a partial ambiguity resolution method is developed in order to satisfy the predefined integrity risk requirement. Static and kinematic experiments are carried out to test the proposed method by comparing with the traditional ratio test method and the protection level-based method. The static experimental result has shown that the proposed method can achieve a significant global availability improvement by 51% at most. The kinematic result reveals that the proposed method obtains the best balance between the positioning accuracy and the continuity performance.     

An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and <Emphasis Type="Italic">t</Emphasis>-distributed deviations

In this paper, we investigate a linear regression time series model of possibly outlier-afflicted observations and autocorrelated random deviations. This colored noise is represented by a covariance-stationary autoregressive (AR) process, in which the independent error components follow a scaled (Student’s) t-distribution. This error model allows for the stochastic modeling of multiple outliers and for an adaptive robust maximum likelihood (ML) estimation of the unknown regression and AR coefficients, the scale parameter, and the degree of freedom of the t-distribution. This approach is meant to be an extension of known estimators, which tend to focus only on the regression model, or on the AR error model, or on normally distributed errors. For the purpose of ML estimation, we derive an expectation conditional maximization either algorithm, which leads to an easy-to-implement version of iteratively reweighted least squares. The estimation performance of the algorithm is evaluated via Monte Carlo simulations for a Fourier as well as a spline model in connection with AR colored noise models of different orders and with three different sampling distributions generating the white noise components. We apply the algorithm to a vibration dataset recorded by a high-accuracy, single-axis accelerometer, focusing on the evaluation of the estimated AR colored noise model.     

Evaluation of gravitational curvatures of a tesseroid in spherical integral kernels

Proper understanding of how the Earth’s mass distributions and redistributions influence the Earth’s gravity field-related functionals is crucial for numerous applications in geodesy, geophysics and related geosciences. Calculations of the gravitational curvatures (GC) have been proposed in geodesy in recent years. In view of future satellite missions, the sixth-order developments of the gradients are becoming requisite. In this paper, a set of 3D integral GC formulas of a tesseroid mass body have been provided by spherical integral kernels in the spatial domain. Based on the Taylor series expansion approach, the numerical expressions of the 3D GC formulas are provided up to sixth order. Moreover, numerical experiments demonstrate the correctness of the 3D Taylor series approach for the GC formulas with order as high as sixth order. Analogous to other gravitational effects (e.g., gravitational potential, gravity vector, gravity gradient tensor), numerically it is found that there exist the very-near-area problem and polar singularity problem in the GC east–east–radial, north–north–radial and radial–radial–radial components in spatial domain, and compared to the other gravitational effects, the relative approximation errors of the GC components are larger due to not only the influence of the geocentric distance but also the influence of the latitude. This study shows that the magnitude of each term for the nonzero GC functionals by a grid resolution 15\(^{{\prime } }\,\times \) 15\(^{{\prime }}\) at GOCE satellite height can reach of about 10\(^{-16}\) m\(^{-1}\) s\(^{2}\) for zero order, 10\(^{-24 }\) or 10\(^{-23}\) m\(^{-1}\) s\(^{2}\) for second order, 10\(^{-29}\) m\(^{-1}\) s\(^{2}\) for fourth order and 10\(^{-35}\) or 10\(^{-34}\) m\(^{-1}\) s\(^{2}\) for sixth order, respectively.     

Recent advances in accessibility research: Representation,methodology and applications

 In this article we examine recent advances in accessibility research and their implications for future studies. We base our discussion on three intersecting dimensions that are useful for evaluating the contribution of recent studies: representation, methodology and applications. Various examples are selected to show that research concerned with representation and methodological problem solving is often applied to issues of broad concern in policy and planning. It is, however, not clear that the simultaneous treatment of representation, methodological and application issues has ever been fully worked out. The questions raised in this article may serve as a foundation for addressing issues pertinent to accurate representation, improved model building, and more rigorous applications in accessibility research. Received: 9 December 2002 / Accepted: 10 February 2003     

GPS tomography in the polar cap: comparison with ionosondes and in situ spacecraft data

Tomographic 4D reconstructions of ionospheric anomalies appearing in the high-latitude polar cap region are compared with plasma density measurements by digital ionosonde located near the north magnetic pole at Eureka station and with in situ plasma measurements on-board DMSP spacecraft. The moderate magnetic storm of 14–17 October 2002 is taken as an example of a geomagnetic disturbance which generates large-scale ionospheric plasma anomalies at mid-latitudes and in the polar cap region. Comparison of the GPS tomographic reconstructions over Eureka station with the ionosonde measurements of the F layer peak densities indicates that the GPS tomography correctly predicts the time of arrival and passage of the ionospheric tongue of ionization over the magnetic pole area, although the tomographic technique appears to under-estimate the value of F peak plasma density. Comparison with the in situ plasma measurements by the DMSP SSIES instruments shows that the GPS tomography correctly reproduces the large-scale spatial structure of ionospheric anomalies over a wide range of latitudes from mid-latitudes to the high-latitude polar cap region, though the tomographic reconstructions tend to over-estimate the density of the topside ionosphere at 840 km DMSP orbit. This study is essential for understanding the quality and limitations of the tomographic reconstruction techniques, particularly in high-latitude regions where GPS TEC measurements and other ionospheric data sources are limited.     

TAZ-level variation in work trip mode choice between 1990 and 2000 and the presence of rail transit

Rail transit continues to be a popular alternative for cities as a tool for alleviating automobile congestion and for redeveloping areas into transit and pedestrian-friendly environments. Ideally, rail transit will draw trips away from cars, but the quantitative work that tests this notion has often been either case studies of neighborhoods, in which conclusions are tough to generalize, or citywide comparisons where important spatial variation is lost in aggregation. This study seeks to narrow this gap in the research by using multivariate analysis of covariance to isolate the effect of covariates and cities on changes in work trip mode choice at the traffic analysis zone (TAZ) level for nine cities between 1990 and 2000. The results suggest differences by city in the change at the TAZ level of the proportion of people driving alone and taking transit. Increases in transit usage were associated with cities that built rail transit, while increases in automobile commuting and decreases in transit usage were associated with cities that did not.     

A new GPS/BDS tropospheric delay resolution approach for monitoring deformation in super high-rise buildings

A new approach for deformation monitoring of super high-rise building using GPS/BDS technology is proposed for the case when prior coordinates are known and the baseline is short but has a large height difference. The approach is based on the ambiguity function method (AFM). Considering that the double-differenced (DD) troposphere delay residual error cannot be ignored, the relative zenith tropospheric delay (RZTD) parameter is introduced into the original AFM equation. Thus, the RZTD and 3D coordinate parameters are together obtained through the modified AFM (MAFM). Due to the low computational efficiency of conventional AFM, an improved particle swarm optimization (IPSO) algorithm is used to search the four optimal parameters X/Y/Z/RZTD and replaces the grid search method. In this study, GPS/BDS deformation monitoring data for buildings with approximately 290 m height difference were used to verify the feasibility of the proposed MAFM. Numerical results show a single-epoch average computation time of approximately 0.3 s, which meets the requirements of near-real-time dynamic monitoring. The average accuracy of the GPS single-epoch RZTD solution is better than 1 cm, the combined GPS/BDS MAFM performance outperforms the GPS-only system, and using multi-epoch observations can further improve the accuracy of the RZTD solution. After RZTD correction, GPS/BDS monitoring precision can be improved, particularly the height dimension, whose precision is improved by approximately 6 cm.