Showing papers in "Journal of Geodesy in 2007"

Short Note: A global model of pressure and temperature for geodetic applications

Abstract: The empirical model GPT (Global Pressure and Temperature), which is based on spherical harmonics up to degree and order nine, provides pressure and temperature at any site in the vicinity of the Earth’s surface. It can be used for geodetic applications such as the determination of a priori hydrostatic zenith delays, reference pressure values for atmospheric loading, or thermal deformation of Very Long Baseline Interferometry (VLBI) radio telescopes. Input parameters of GPT are the station coordinates and the day of the year, thus also allowing one to model the annual variations of the parameters. As an improvement compared with previous models, it reproduces the large pressure anomaly over Antarctica, which can cause station height errors in the analysis of space-geodetic data of up to 1 cm if not considered properly in troposphere modelling. First tests at selected geodetic observing stations show that the pressure biases considerably decrease when using GPT instead of the very simple approaches applied to various Global Navigation Satellite Systems (GNSS) software packages so far. GPT also provides an appropriate model for the annual variability of global temperature.

Calibration errors on experimental slant total electron content (TEC) determined with GPS

Abstract: The Global Positioning System (GPS) has become a powerful tool for ionospheric studies. In addition, ionospheric corrections are necessary for the augmentation systems required for Global Navigation Satellite Systems (GNSS) use. Dual-frequency carrier-phase and code-delay GPS observations are combined to obtain ionospheric observables related to the slant total electron content (sTEC) along the satellite-receiver line-of-sight (LoS). This observable is affected by inter-frequency biases [IFB; often called differential code biases (DCB)] due to the transmitting and the receiving hardware. These biases must be estimated and eliminated from the data in order to calibrate the experimental sTEC obtained from GPS observations. Based on the analysis of single differences of the ionospheric observations obtained from pairs of co-located dual-frequency GPS receivers, this research addresses two major issues: (1) assessing the errors translated from the code-delay to the carrier-phase ionospheric observable by the so-called levelling process, applied to reduce carrier-phase ambiguities from the data; and (2) assessing the short-term stability of receiver IFB. The conclusions achieved are: (1) the levelled carrier-phase ionospheric observable is affected by a systematic error, produced by code-delay multi-path through the levelling procedure; and (2) receiver IFB may experience significant changes during 1 day. The magnitude of both effects depends on the receiver/antenna configuration. Levelling errors found in this research vary from 1.4 total electron content units (TECU) to 5.3 TECU. In addition, intra-day vaiations of code-delay receiver IFB ranging from 1.4 to 8.8 TECU were detected.

Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas

Abstract: The development and numerical values of the new absolute phase-center correction model for GPS receiver and satellite antennas, as adopted by the International GNSS (global navigation satellite systems) Service, are presented. Fixing absolute receiver antenna phase-center corrections to robot-based calibrations, the GeoForschungsZentrum Potsdam (GFZ) and the Technische Universitat Munchen reprocessed more than 10 years of GPS data in order to generate a consistent set of nadir-dependent phase-center variations (PCVs) and offsets in the z-direction pointing toward the Earth for all GPS satellites in orbit during that period. The agreement between the two solutions estimated by independent software packages is better than 1 mm for the PCVs and about 4 cm for the z-offsets. In addition, the long time-series facilitates the study of correlations of the satellite antenna corrections with several other parameters such as the global terrestrial scale or the orientation of the orbital planes with respect to the Sun. Finally, completely reprocessed GPS solutions using different phase-center correction models demonstrate the benefits from switching from relative to absolute antenna phase-center corrections. For example, tropospheric zenith delay biases between GPS and very long baseline interferometry (VLBI), as well as the drift of the terrestrial scale, are reduced and the GPS orbit consistency is improved.

Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models

Abstract: We discuss a new method for approximately decorrelating and non-isotropically filtering the monthly gravity fields provided by the gravity recovery and climate experiment (GRACE) twin-satellite mission. The procedure is more efficient than conventional Gaussian-type isotropic filters in reducing stripes and spurious patterns, while retaining the signal magnitudes. One of the problems that users of GRACE level 2 monthly gravity field solutions fight is the effect of increasing noise in higher frequencies. Simply truncating the spherical harmonic solution at low degrees causes the loss of a significant portion of signal, which is not an option if one is interested in geophysical phenomena on a scale of few hundred to few thousand km. The common approach is to filter the published solutions, that is to convolve them with an isotropic kernel that allows an interpretation as smoothed averaging. The downside of this approach is an amplitude bias and the fact that it neither accounts for the variable data density that increases towards the poles where the orbits converge nor for the anisotropic error correlation structure that the solutions exhibit. Here a relatively simple regularization procedure will be outlined, which allows one to take the latter two effects into account, on the basis of published level 2 products. This leads to a series of approximate decorrelation transformations applied to the monthly solutions, which enable a successive smoothing to reduce the noise in the higher frequencies. This smoothing effect may be used to generate solutions that behave, on average over all possible directions, very close to Gaussian-type filtered ones. The localizing and smoothing properties of our non-isotropic kernels are compared with Gaussian kernels in terms of the kernel variance and the resulting amplitude bias for a standard signal. Examples involving real GRACE level 2 fields as well as geophysical models are used to demonstrate the techniques. With the new method, we find that the characteristic striping pattern in the GRACE solutions are much more reduced than Gaussian-filtered solutions of comparable signal amplitude and root mean square.

A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling

Abstract: The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant (ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms, but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula. Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated.

The International VLBI Service for Geodesy and Astrometry (IVS): current capabilities and future prospects

Abstract: Very Long Baseline Interferometry (VLBI) plays a unique and fundamental role in the maintenance of the global (terrestrial and celestial) reference frames, which are required for precise positioning in many research areas such as the understanding and monitoring of global changes, and for space missions. The International VLBI Service for Geodesy and Astrometry (IVS) coordinates the global VLBI components and resources on an international basis. The service is tasked by the International Association of Geodesy (IAG) and International Astronomical Union (IAU) to provide products for the realization of the Celestial Reference Frame (CRF) through the positions of quasars, to deliver products for the maintenance of the terrestrial reference frame (TRF), such as station positions and their changes with time, and to generate products describing the rotation and orientation of the Earth. In particular, VLBI uniquely provides direct observations of nutation parameters and of the time difference UT1-UTC. This paper summarizes the evolution and current status of the IVS. It points out the activities to improve further on the product quality to meet future service requirements.

An improved and extended GPS-derived 3D velocity field of the glacial isostatic adjustment (GIA) in Fennoscandia

Abstract: We present a new GPS-derived 3D velocity field for the Fennoscandia glacial isostatic adjustment (GIA) area. This new solution is based upon ∼3,000 days of continuous GPS observations obtained from the permanent networks in Fennoscandia. The period encompasses a prolongated phase of stable observation conditions after the northern autumn of 1996. Several significant improvements have led to smaller uncertainties and lower systematic errors in the new solutions compared to our previous results. The GPS satellite elevation cut-off angle was lowered to 10°, we fixed ambiguities to integers where possible, and only a few hardware changes occurred over the entire network. The GAMIT/GLOBK software package was used for the GPS analysis and reference frame realization. Our new results confirmed earlier findings of maximum discrepancies between GIA models and observations in northern Finland. The reason may be related to overestimated ice-sheet thickness and glaciation period in the north. In general, the new solutions are more coherent in the velocity field, as some of the perturbations are now avoided. We compared GPS-derived GIA rates with sea-level rates from tide-gauge observations, repeated precise leveling, and with GIA model computations, which showed consistency.

Development and testing of a new ray-tracing approach to GNSS carrier-phase multipath modelling

Abstract: Multipath is one of the most important error sources in Global Navigation Satellite System (GNSS) carrier-phase-based precise relative positioning. Its theoretical maximum is a quarter of the carrier wavelength (about 4.8 cm for the Global Positioning System (GPS) L1 carrier) and, although it rarely reaches this size, it must clearly be mitigated if millimetre-accuracy positioning is to be achieved. In most static applications, this may be accomplished by averaging over a sufficiently long period of observation, but in kinematic applications, a modelling approach must be used. This paper is concerned with one such approach: the use of ray-tracing to reconstruct the error and therefore remove it. In order to apply such an approach, it is necessary to have a detailed understanding of the signal transmitted from the satellite, the reflection process, the antenna characteristics and the way that the reflected and direct signal are processed within the receiver. This paper reviews all of these and introduces a formal ray-tracing method for multipath estimation based on precise knowledge of the satellite–reflector–antenna geometry and of the reflector material and antenna characteristics. It is validated experimentally using GPS signals reflected from metal, water and a brick building, and is shown to be able to model most of the main multipath characteristics. The method will have important practical applications for correcting for multipath in well-constrained environments (such as at base stations for local area GPS networks, at International GNSS Service (IGS) reference stations, and on spacecraft), and it can be used to simulate realistic multipath errors for various performance analyses in high-precision positioning.

Higher order ionospheric effects in precise GNSS positioning

Abstract: With the increasing number of precise navigation and positioning applications using Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS), higher order ionospheric effects and their correction become more and more important. Whereas the first-order error can be completely eliminated by a linear combination of dual- frequency measurements, the second- and third-order residual effects remain uncorrected in this approach. To quantify the second-order residual effect, a simple formula has been derived for GNSS users in Germany. Our proposed correction algorithm reduces the second-order effects to a residual error of fractions of 1 mm up to 2 mm at a vertical total electron content level of 1018 electrons/m2 (100 TECU), depending on satellite azimuth and elevation angles. The correction formula can be implemented in real-time applications as it does not require the knowledge of the geomagnetic field or the electron density distribution in the ionosphere along the signal path. It is expected that the correction will enable more accurate positioning using the line-of-sight carrier-phase measurements.

The topographic bias by analytical continuation in physical geodesy

Abstract: This study emphasizes that the harmonic downward continuation of an external representation of the Earth's gravity potential to sea level through the topographic masses implies a topographic bias. ...

GPS sidereal filtering: coordinate- and carrier-phase-level strategies

Abstract: Multipath error is considered one of the major errors affecting GPS observations. One can benefit from the repetition of satellite geometry approximately every sidereal day, and apply filtering to help minimize this error. For GPS data at 1 s interval processed using a double-difference strategy, using the day-to-day coordinate or carrier-phase residual autocorrelation determined with a 10-h window leads to the steadiest estimates of the error-repeat lag, although a window as short as 2 h can produce an acceptable value with > 97% of the optimal lag’s correlation. We conclude that although the lag may vary with time, such variation is marginal and there is little advantage in using a satellite-specific or other time-varying lag in double-difference processing. We filter the GPS data either by stacking a number of days of processed coordinate residuals using the optimum “sidereal” lag (23 h 55 m 54 s), and removing these stacked residuals from the day in question (coordinate space), or by a similar method using double-difference carrier-phase residuals (observational space). Either method results in more consistent and homogeneous set of coordinates throughout the dataset compared with unfiltered processing. Coordinate stacking reduces geometry-related repeating errors (mainly multipath) better than carrier-phase residual stacking, although the latter takes less processing time to achieve final filtered coordinates. Thus, the optimal stacking method will depend on whether coordinate precision or computational time is the over-riding criterion.

Attenuation effect on seasonal basin-scale water storage changes from GRACE time-variable gravity

Abstract: In order to effectively recover surface mass or geoid height changes from the gravity recovery and climate experiment (GRACE) time-variable gravity models, spatial smoothing is required to minimize errors from noise. Spatial smoothing, such as Gaussian smoothing, not only reduces the noise but also attenuates the real signals. Here we investigate possible amplitude attenuations and phase changes of seasonal water storage variations in four drainage basins (Amazon, Mississippi, Ganges and Zambezi) using an advanced global land data assimilation system. It appears that Gaussian smoothing significantly affects GRACE-estimated basin-scale seasonal water storage changes, e.g., in the case of 800 km smoothing, annual amplitudes are reduced by about 25-40%, while annual phases are shifted by up to 10°. With these effects restored, GRACE-estimated water storage changes are consistently larger than model estimates, indicating that the land surface model appears to underestimate terrestrial water storage change. Our analysis based on simulation suggests that normalized attenuation effects (from Gaussian smoothing) on seasonal water storage change are relatively insensitive to the magnitude of the true signal. This study provides a numerical approach that can be used to restore seasonal water storage change in the basins from spatially smoothed GRACE data. © Springer-Verlag 2006.

Comparisons of homogeneously reprocessed GPS and VLBI long time-series of troposphere zenith delays and gradients

Abstract: Troposphere parameters estimated from space-geodetic techniques, like the Global Positioning System (GPS) or Very Long Baseline Interferometry (VLBI), can be used to monitor the atmospheric water vapor content. Although the troposphere can only be monitored at discrete locations, the distribution of the instruments, at least the GPS antennas, can be assumed to be quasi-global. Critical in the data analysis are systematic effects within each single technique that significantly degrade the accuracy and especially the long-term stability of the zenith delay determination. In this paper, consistent time-series of troposphere zenith delays and gradients from homogeneously reprocessed GPS and VLBI solutions are compared for a time period of 11 years. The homogeneity of these completely reprocessed time-series is essential to avoid misinterpretations due to individual model changes. Co-located sites are used to investigate systematic effects and the long-term behavior of the two space-geodetic techniques. Both techniques show common signals in the troposphere parameters at a very high level of precision. The biases between the troposphere zenith delays are at the level of a few millimeters. On the other hand, long-term trends significantly differ for the two techniques, preventing climatological interpretations at present. Tests assume these differences to be due to mathematical artifacts such as different sampling rates and unmodeled semi-annual signals with varying amplitudes.

Estimability analysis of variance and covariance components

Abstract: Although variance and covariance components have been extensively investigated and a number of elegant formulae to compute them have been derived, nothing is known, without any ambiguity, about their estimability in the case of a fully unknown variance–covariance matrix. We prove that variance and covariance components in this case are not estimable, thus clarifying the ambiguity of the literature on the topic and correcting some erroneous statements in the literature. We also give a new theorem on the estimability of a linear function of variance and covariance components. Then we propose a new method to estimate the variance–covariance matrix with special structure, which can presumably be represented by, at most, r(r + 1)/2 independent parameters to guarantee its estimability in such a subspace, by directly implementing the positive definiteness of the matrix as constraint to the restricted maximum likelihood method, where r is the number of redundant measurements. Therefore, our estimates of the variance and covariance components always reconstruct a positive definite matrix and are always physically meaningful.

Correcting superconducting gravity time-series using rainfall modelling at the Vienna and Membach stations and application to Earth tide analysis

Abstract: We demonstrate the possibility to improve the signal-to-noise ratio of superconducting gravity time-series by correcting for local hydrological effects. Short-term atmospheric events associated with heavy rain induce step-like gravity signals that deteriorate the frequency spectrum estimates. Based on 4D modeling constrained by high temporal resolution rain gauge data, rainfall admittances for the Vienna and Membach superconducting gravity stations are calculated. This allows routine correction for Newtonian rain water effects, which reduces the standard deviation of residuals after tidal parameter adjustment by 10%. It also improves the correction of steps of instrumental origin when they coincide with step-like water mass signals.

The geopotential value W 0 for specifying the relativistic atomic time scale and a global vertical reference system

Abstract: The TOPEX/Poseidon (T/P) satellite alti- meter mission marked a new era in determining the geopotential constant W0. On the basis of T/P data during 1993–2003 (cycles 11–414), long-term variations in W0 have been investigated. The rounded value W0 = 62636856.0 ± 0.5) m2s−2 has already been adopted by the International Astronomical Union for the definition of the constant LG = W0/c2 = 6.969290134 × 10−10 (where c is the speed of light), which is required for the realization of the relativistic atomic time scale. The constant LG, based on the above value of W0, is also included in the 2003 International Earth Rotation and Reference Frames Service conventions. It has also been suggested that W0 is used to specify a global vertical reference system (GVRS). W0 ensures the consistency with the International Terrestrial Reference System, i.e. after adopting W0, along with the geocentric gravitational constant (GM), the Earth’s rotational velocity (ω) and the second zonal geopotential coefficient (J2) as primary constants (parameters), then the ellipsoidal parameters (a,α) can be computed and adopted as derived parameters. The scale of the International Terrestrial Reference Frame 2000 (ITRF2000) has also been specified with the use of W0 to be consistent with the geocentric coordinate time. As an example of using W0 for a GVRS realization, the geopotential difference between the adopted W0 and the geopotential at the Rimouski tide-gauge point, specifying the North American Vertical Datum 1988 (NAVD88), has been estimated.

Is the instrumental drift of superconducting gravimeters a linear or exponential function of time

Abstract: The instrumental drift of the superconducting gravimeter in Membach, Belgium, is estimated using 9 years of co-located and episodic absolute gravity measurements. We show that the best model of the long-term drift of the SG-C021 is an exponential. The thermal levelers used to compensate tilts are unlikely to induce the observed drift. Rather, the capacitance bridge, magnetic variations, gas adsorption on the levitating sphere, or helium gas pressure variations around it are most likely the possible combined causes of the observed instrumental drift. In practice, either linear or exponential drift models are equivalent as long as the record duration does not exceed 10 years. For longer records, this study demonstrates that an exponential models the drift better than a simple linear trend.

On the computation and approximation of ultra-high-degree spherical harmonic series

Abstract: Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (l,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: \(m = \ell \sin \theta\), where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.

Indirect approach to invariant point determination for SLR and VLBI systems: an assessment

Abstract: We assess the accuracy of some indirect approaches to invariant point (IVP), or system reference point, determination of satellite laser ranging (SLR) and very long baseline interferometry (VLBI) systems using both observed and simulated survey data sets. Indirect IVP determination involves the observation of targets located on these systems during specific rotational sequences and by application of geometrical models that describe the target motion during these sequences. Of concern is that most SLR and VLBI systems have limited rotational freedom thereby placing constraint on the reliability of parameter estimation, including the IVP position. We assess two current approaches to IVP analysis using survey data observed at the Yarragadee (Australia) SLR and the Medicina (Italy) VLBI sites and also simulated data of a large rotationally constrained (azimuth-elevation) VLBI system. To improve reliability we introduce and assess some new geometric conditions, including inter-axis, inter-circle and inter-target conditions, to existing IVP analysis strategies. The error component of a local tie specifically associated with the indirect determination of SLR and VLBI IVP is less than 0.5 mm. For systems with significant rotational limits we find that the inter-axis and inter-circle conditions are critical to the computation of unbiased IVP coordinates at the sub-millimetre level. When the inter-axis and inter-circle geometric conditions are not imposed, we retrieve biased vertical coordinates of the IVP (in our simulated VLBI system) in the range of 1.2–3.4 mm. Using the new geometric conditions we also find that the axis-offset estimates can be recovered at the sub- millimetre accuracy (0.5 mm).

Troposphere gradients from the ECMWF in VLBI analysis

Abstract: Modeling path delays in the neutral atmosphere for the analysis of Very Long Baseline Interferometry (VLBI) observations has been improved significantly in recent years by the use of elevation-dependent mapping functions based on data from numerical weather models. In this paper, we present a fast way of extracting both, hydrostatic and wet, linear horizontal gradients for the troposphere from data of the European Centre for Medium-range Weather Forecasts (ECMWF) model, as it is realized at the Vienna University of Technology on a routine basis for all stations of the International GNSS (Global Navigation Satellite Systems) Service (IGS) and International VLBI Service for Geodesy and Astrometry (IVS) stations. This approach only uses information about the refractivity gradients at the site vertical, but no information from the line-of-sight. VLBI analysis of the CONT02 and CONT05 campaigns, as well as all IVS-R1 and IVS-R4 sessions in the first half of 2006, shows that fixing these a priori gradients improves the repeatability for 74% (40 out of 54) of the VLBI baseline lengths compared to fixing zero or constant a priori gradients, and improves the repeatability for the majority of baselines compared to estimating 24-h offsets for the gradients. Only if 6-h offsets are estimated, the baseline length repeatabilities significantly improve, no matter which a priori gradients are used.

Combined Earth orientation parameters based on homogeneous and continuous VLBI and GPS data

Abstract: The CONT02 campaign is of great interest for studies combining very long baseline interferometry (VLBI) with other space-geodetic techniques, because of the continuously available VLBI observations over 2 weeks in October 2002 from a homogeneous network. Especially, the combination with the Global Positioning System (GPS) offers a broad spectrum of common parameters. We combined station coordinates, Earth orientation parameters (EOPs) and troposphere parameters consistently in one solution using technique- specific datum-free normal equation systems. In this paper, we focus on the analyses concerning the EOPs, whereas the comparison and combination of the troposphere parameters and station coordinates are covered in a companion paper in Journal of Geodesy. In order to demonstrate the potential of the VLBI and GPS space-geodetic techniques, we chose a sub-daily resolution for polar motion (PM) and universal time (UT). A consequence of this solution set-up is the presence of a one-to-one correlation between the nutation angles and a retrograde diurnal signal in PM. The Bernese GPS Software used for the combination provides a constraining approach to handle this singularity. Simulation studies involving both nutation offsets and rates helped to get a deeper understanding of this singularity. With a rigorous combination of UT1–UTC and length of day (LOD) from VLBI and GPS, we showed that such a combination works very well and does not suffer from the systematic effects present in the GPS-derived LOD values. By means of wavelet analyses and the formal errors of the estimates, we explain this important result. The same holds for the combination of nutation offsets and rates. The local geodetic ties between GPS and VLBI antennas play an essential role within the inter-technique combination. Several studies already revealed non-negligible discrepancies between the terrestrial measurements and the space-geodetic solutions. We demonstrate to what extent these discrepancies propagate into the combined EOP solution.

Effect of different tropospheric mapping functions on the TRF, CRF and position time-series estimated from VLBI

Abstract: This paper compares estimated terrestrial reference frames (TRF) and celestial reference frames (CRF) as well as position time-series in terms of systematic differences, scale, annual signals and station position repeatabilities using four different tropospheric mapping functions (MF): The NMF (Niell Mapping Function) and the recently developed GMF (Global Mapping Function) consist of easy-to-handle stand-alone formulae, whereas the IMF (Isobaric Mapping Function) and the VMF1 (Vienna Mapping Function 1) are determined from numerical weather models. All computations were performed at the Deutsches Geodatisches Forschungsinstitut (DGFI) using the OCCAM 6.1 and DOGS-CS software packages for Very Long Baseline Interferometry (VLBI) data from 1984 until 2005. While it turned out that CRF estimates only slightly depend on the MF used, showing small systematic effects up to 0.025 mas, some station heights of the computed TRF change by up to 13 mm. The best agreement was achieved for the VMF1 and GMF results concerning the TRFs, and for the VMF1 and IMF results concerning scale variations and position time-series. The amplitudes of the annual periodical signals in the time-series of estimated heights differ by up to 5 mm. The best precision in terms of station height repeatability is found for the VMF1, which is 5–7% better than for the other MFs.

Influence of ambiguity precision on the success rate of GNSS integer ambiguity bootstrapping

Abstract: In this contribution, we study the dependence of the bootstrapped success rate on the precision of the GNSS carrier phase ambiguities. Integer bootstrapping is, because of its ease of computation, a popular method for resolving the integer ambiguities. The method is however known to be suboptimal, because it only takes part of the information from the ambiguity variance matrix into account. This raises the question in what way the bootstrapped success rate is sensitive to changes in precision of the ambiguities. We consider two different cases. (1) The effect of improving the ambiguity precision, and (2) the effect of using an approximate ambiguity variance matrix. As a by-product, we also prove that integer bootstrapping is optimal within the restricted class of sequential integer estimators.

A Comparison of GPS, VLBI and Model Estimates of Ocean Tide Loading Displacements

Abstract: In recent years, ocean tide loading displacements (OTLD) have been measured using the Global Positioning System (GPS) and Very Long Baseline Interferometry (VLBI) This study assesses the accuracy of GPS measurements of OTLD by comparison with VLBI measurements and estimates derived from numerical ocean tide models A daily precise point positioning (PPP) analysis was carried out on ∼11 years of GPS data for each of 25 sites that have previous OTLD estimates based on data from co-located VLBI sites Ambiguities were fixed to integer values where possible The resulting daily estimates of OTLD, at eight principal diurnal and semi-diurnal tidal frequencies, were combined to give GPS measurements of OTLD at each site The 3D GPS and VLBI measurements of OTLD were compared with estimates computed (by convolution with Green’s functions) from five modern ocean tide models (CSR40, FES2004, GOT002, NAO99b and TPXO62) The GPS/model agreement is shown to be similar to the VLBI/model agreement In the important radial direction, the GPS/model misfit is shown to be smaller than the VLBI/model misfit for seven of the eight tidal constituents; the exception being the K2 constituent Fixing of GPS carrier-phase ambiguities to integer values resulted in a marginal improvement to the GPS/model agreement Statistically, it is shown there is no significance to the difference between the fit of the GPS and VLBI measurements of OTLD to modelled values Equally, differences in fit of either the complete set of GPS or VLBI estimates to the five sets of model-derived values cannot be identified with statistical significance It is thus concluded that, overall, we cannot distinguish between GPS and VLBI measurements of OTLD, and that at the global scale, present ocean tide models are accurate to within the current measurement noise of these techniques

Effect of heteroscedasticity and heterogeneousness on outlier detection for geodetic networks

Abstract: The robustness of an outlier detection method strongly depends on the weights of observations, i.e., the type of the stochastic model applied (homoscedasticity, heteroscedasticity and heterogeneousness). In this paper, we have investigated how the reliability of the robust methods and tests for outliers changes depending on the weights of the observations in geodetic networks. Furthermore, the contribution of the directions and distances to horizontal control network with regard to reliability are investigated separately. The concept of a breakdown point is used as a global measure of robustness against outliers. The mean success rate (MSR) is found to be a practical tool for confirming the breakdown point. Many different “good” data samples are generated for each network and then deliberately contaminated using a Monte-Carlo simulation. Six robust methods and Baarda’s test are applied to the corrupted samples and the degree of corruption is varied. The performance of each method is measured using both local and global MSRs. Our research shows: (1) The MSRs of Baarda’s test change depending on the strength of the heteroscedasticity, but do not change for trilateration and leveling networks, (2) the global MSRs of robust methods do not differ considerably from the local ones

Combination of long time-series of troposphere zenith delays observed by VLBI

Abstract: Within the International Very Long Baseline Interferometry (VLBI) Service for Geodesy and Astrometry (IVS), long time-series of zenith wet and total troposphere delays have been combined at the level of parameter estimates. The data sets were submitted by eight IVS Analysis Centers (ACs) and cover January 1984 to December 2004. In this paper, the combination method is presented and the time-series submitted by the eight IVS ACs are compared with each other. The combined zenith delays are compared with time-series provided by the International Global Navigation Satellite System (GNSS) Service (IGS), and with zenith delays derived from the European Centre for Medium-Range Weather Forecasts (ECMWF). Before the combination, outliers are eliminated from the individual time-series using the robust BIBER (bounded influence by standardized residuals) estimator. For each station and AC, relative weight factors are obtained by variance component estimation. The mean bias of the IVS ACs’ time-series with respect to the IVS combined time-series is 0.89 mm and the mean root mean square is 7.67 mm. Small differences between stations and ACs can be found, which are due to the inhomogeneous analysis options, different parameterizations, and different treatment of missing in-situ pressure records. Compared to the IGS zenith total delays, the combined IVS series show small positive mean biases and different long-term trends. Zenith wet delays from the ECMWF are used to validate the IVS combined series. Inconsistencies, e.g., long-term inhomogeneity of the in-situ pressure data used for the determination of VLBI zenith delays, are identified.

Tropospheric parameters: combination studies based on homogeneous VLBI and GPS data

Abstract: The combination of tropospheric parameters derived from different space-geodetic techniques has not been of large interest in geodesy so far. However, due to the high correlation between station coordinates and tropospheric parameters, the latter should not be neglected in combinations. This paper deals with the comparison and combination of tropospheric parameters derived from global positioning system (GPS) and very long baseline interferometry (VLBI) observations stemming from a 15-day campaign of continuous VLBI observations in 2002 (CONT02). The observation data of both techniques were processed homogeneously to avoid systematic differences between the solutions. We compared the tropospheric estimates of GPS and VLBI at eight co-location sites and found a very good agreement in the temporal behavior of the tropospheric zenith path delays (ZPD), reflected by correlation factors up to 0.98. Following this, a combination of the tropospheric parameters was performed. We demonstrate that the combination of tropospheric parameters leads to a stabilization of combined station networks. This becomes visible in the improvement of the repeatabilities of the station height components. Furthermore, the potential use of independent data from water vapor radiometers (WVRs) to validate space-technique-derived tropospheric parameters was investigated. Correlation coefficients of 0.95 or better were estimated between the tropospheric parameters of WVR and GPS or VLBI. Additionally, the utility of the tropospheric parameters for validation of local tie vectors was investigated. Both tropospheric zenith delays and tropospheric gradients were found to be very suitable to validate the height component and the horizontal components of the local tie, respectively.

Determination of the free core nutation period from tidal gravity observations of the GGP superconducting gravimeter network

Abstract: This study is based on 25 long time-series of tidal gravity observations recorded with superconduct- ing gravimeters at 20 stations belonging to the Global Geodynamic Project (GGP). We investigate the diurnal waves around the liquid core resonance, i.e., K1, ψ1 and ϕ1, to determine the free core nutation (FCN) period, and compare these experimental results with models of the Earth response to the tidal forces. For this purpose, it is necessary to compute corrected amplitude factors andphasedifferencesbysubtractingtheoceantideload- ing (OTL) effect. To determine this loading effect for each wave, it was thus necessary to interpolate the con- tribution of the smaller oceanic constituents from the four well determined diurnal waves, i.e., Q1,O1,P1,K1. It was done for 11 different ocean tide models: SCW80, CSR3.0, CSR4.0, FES95.2, FES99, FES02, TPXO2, ORI96, AG95, NAO99 and GOT00. The numerical re- sults show that no model is decisively better than the others and that a mean tidal loading vector gives the most stable solution for a study of the liquid core res- onance. We compared solutions based on the mean of the 11 ocean models to subsets of six models used in a previous study and five more recent ones. The cali- bration errors put a limit on the accuracy of our global results at the level of ±0.1%, although the tidal factors of O1 and K1 are determined with an internal precision of close to 0.05%. The results for O1 more closely fit

Modeling thermal deformation of VLBI antennas with a new temperature model

Abstract: Temperature variations at very long baseline interferometry (VLBI) sites cause thermal deformations of the VLBI antennas and corresponding displacements of the VLBI reference points The thermal deformation effects typically contain seasonal and daily signatures The amplitudes of the annual vertical motion of the antenna reference point can reach several millimeters, depending on the design of the antenna structure, on the material, and on the environmental effects such as global station position, station height and climatology effects Simple methods to correct this effect use the difference of the environmental temperature with respect to a defined reference temperature, the antenna dimensions, the elevation of the antenna, the material of antenna structure Applying these simple models for thermal deformation in the VLBI data analysis improves the baseline length repeatability by 35% A comparison of these simple models with local thermal deformation measurements at the antennas in Onsala and Wettzell show that the local measurements and the modeled corrections agree well when the temperature of the antenna structure is used, but agree less good when the surrounding air temperatures are used To overcome this problem we present a method to model temperature penetration into the antenna structures, that allows to model thermal deformation effects that agree with the observed vertical deformation of the Onsala and Wettzell radio telescopes with a root mean square deviation of 007 and 013 mm, respectively Possible implementations in the VLBI analysis are presented, and the definition of an adequate reference temperature is discussed

Best prediction in linear models with mixed integer/real unknowns: theory and application

Abstract: In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).