# Journal of Geodesy

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### Impact of network constraining on the terrestrial reference frame realization based on SLR observations to LAGEOS

Thu, 10/17/2019 - 00:00
Abstract

The Satellite Laser Ranging (SLR) network struggles with some major limitations including an inhomogeneous global station distribution and uneven performance of SLR sites. The International Laser Ranging Service (ILRS) prepares the time-variable list of the most well-performing stations denoted as ‘core sites’ and recommends using them for the terrestrial reference frame (TRF) datum realization in SLR processing. Here, we check how different approaches of the TRF datum realization using minimum constraint conditions (MCs) and the selection of datum-defining stations affect the estimated SLR station coordinates, the terrestrial scale, Earth rotation parameters (ERPs), and geocenter coordinates (GCC). The analyses are based on the processing of the SLR observations to LAGEOS-1/-2 collected between 2010 and 2018. We show that it is essential to reject outlying stations from the reference frame realization to maintain a high quality of SLR-based products. We test station selection criteria based on the Helmert transformation of the network w.r.t. the a priori SLRF2014 coordinates to reject misbehaving stations from the list of datum-defining stations. The 25 mm threshold is optimal to eliminate the epoch-wise temporal deviations and to provide a proper number of datum-defining stations. According to the station selection algorithm, we found that some of the stations that are not included in the list of ILRS core sites could be taken into account as potential core stations in the TRF datum realization. When using a robust station selection for the datum definition, we can improve the station coordinate repeatability by 8%, 4%, and 6%, for the North, East and Up components, respectively. The global distribution of datum-defining stations is also crucial for the estimation of ERPs and GCC. When excluding just two core stations from the SLR network, the amplitude of the annual signal in the GCC estimates is changed by up to 2.2 mm, and the noise of the estimated pole coordinates is substantially increased.

### On detection of observation faults in the observation and position domains for positioning of intelligent transport systems

Tue, 10/15/2019 - 00:00
Abstract

Intelligent transportation systems (ITS) depend on global navigation satellite systems (GNSS) as a major positioning sensor, where the sensor should be able to detect and exclude faulty observations to support its reliability. In this article, two fault detection and exclusion (FDE) approaches are discussed. The first is its application in the observation domain using Chi-square test in Kalman filter processing. The second approach discusses FDE testing in the positioning domain using the solution separation (SS) method, where new FDE forms are presented that are tailored for ITS. In the first form, the test is parameterized along the direction of motion of the vehicle and in the cross-direction, which are relevant to applications that require lane identification and collision alert. A combined test is next established. Another form of the test is presented considering the maximum possible positioning error, and finally a direction-independent test. A new test that can be implemented in the urban environment is presented, which takes into account multipath effects that could disrupt the zero-mean normal distribution assumption of the positioning errors. Additionally, a test is presented to check that the position error resulting from the remaining measurements lies within acceptable limits. The proposed methods are demonstrated through a kinematic test run in various environments that may be experienced in ITS.

### Preface to the second special issue on Laser Ranging

Thu, 10/10/2019 - 00:00

### Time bias service: analysis and monitoring of satellite orbit prediction quality

Wed, 10/09/2019 - 00:00
Abstract

The performance of satellite laser ranging (SLR) station operations relies to a large extent on the quality of the required satellite orbit predictions. Poor predictions with large along-track offsets, so-called time biases, increase the target acquisition time and thus reduce the performance of stations and the International Laser Ranging (ILRS) network as a whole. There is currently no established process to evaluate or monitor the quality of predictions. This paper presents a method for such a process that uses normal point data uploaded to data centers by ILRS stations worldwide. The first analysis results show systematic trends over time for most targets and prediction providers. These trends were used to predict the development of time bias values. We also present a service that provides these predicted values for the latest satellite orbit predictions of selected targets and providers in real time. Using these values during tracking allows for faster target acquisition and thus better tracking performance at ILRS SLR stations. Through monitoring, the service further enables stations to select the best available predictions during tracking and to notify prediction providers if issues are encountered. This tool benefits not only the stations by improving their tracking performance but also allows for prediction improvement and greater support of missions.

### Residual terrain modelling (RTM) in terms of the cap-modified spectral technique: RTM from a new perspective

Tue, 10/01/2019 - 00:00
Abstract

We present two novel approaches to residual terrain modelling (RTM), one of which is practical and the other rather theoretical. The first provides a solution to the harmonic correction issue and the high-frequency error of the spectral filter problem. As its key feature, cap-modified spectral gravity forward modelling is applied to deliver near-zone gravity effects induced by the reference (smooth) topography. Thanks to its spectral nature, the sought gravity can be evaluated at the problematic points inside the smooth topography by regularized downward continuation (solution for the harmonic correction problem), and, at the same time, it is band-limited in spherical harmonics (solution for the high-frequency error). A validation over two mountainous areas, Switzerland and Slovakia, reveals that this technique is at least comparable with two other common RTM variants (RMS agreement up to 0.1 mGal). Finally, we formulate the theoretical RTM concept, showing that the harmonic correction issue and the spectral filter problem are caused by filtering of the mass model in the topography domain. When properly filtering gravity effects in the gravity domain, that is, avoiding the concept of the reference topography, both problems disappear. Limitations of both RTM approaches include the possible divergence effect of spherical harmonic series on the Earth’s surface and a conceptual inconsistency between two involved types of spherical harmonic coefficients. Applications of this study could be found in the development of combined gravity models (the fill-in procedure) or in the spectral enhancement of spherical harmonic gravity models.

### Measuring phase scintillation at different frequencies with conventional GNSS receivers operating at 1 Hz

Tue, 10/01/2019 - 00:00
Abstract

Ionospheric scintillation causes rapid fluctuations of measurements from Global Navigation Satellite Systems (GNSSs), thus threatening space-based communication and geolocation services. The phenomenon is most intense in equatorial regions, around the equinoxes and in maximum solar cycle conditions. Currently, ionospheric scintillation monitoring receivers (ISMRs) measure scintillation with high-pass filter algorithms involving high sampling rates, e.g. 50 Hz, and highly stable clocks, e.g. an ultra-low-noise Oven-Controlled Crystal Oscillator. The present paper evolves phase scintillation indices implemented in conventional geodetic receivers with sampling rates of 1 Hz and rapidly fluctuating clocks. The method is capable to mitigate ISMR artefacts that contaminate the readings of the state-of-the-art phase scintillation index. Our results agree in more than 99.9% within ± 0.05 rad (2 mm) of the ISMRs, with a data set of 8 days which include periods of moderate and strong scintillation. The discrepancies are clearly identified, being associated with data gaps and to cycle-slips in the carrier-phase tracking of ISMR that occur simultaneously with ionospheric scintillation. The technique opens the door to use huge databases available from the International GNSS Service and other centres for scintillation studies. This involves GNSS measurements from hundreds of worldwide-distributed geodetic receivers over more than one Solar Cycle. This overcomes the current limitations of scintillation studies using ISMRs, as only a few tens of ISMRs are available and their data are provided just for short periods of time.

### A VLBI delay model for gravitational deformations of the Onsala 20 m radio telescope and the impact on its global coordinates

Sat, 09/28/2019 - 00:00
Abstract

Deformations of the reflectors of radio telescopes used in geodetic and astrometric VLBI observations belong to the class of systematic error sources which affect the estimated position of the telescope and which necessitates correction at the observation level. The determination of the gravitationally induced deformations requires some effort and needs specific modeling of the impact on the VLBI delay observables. This has been exercised on the Onsala 20 m radio telescope. In this publication, we present an elevation-dependent model for the contributions of the gravitational deformations to the delay observables for application in VLBI data analysis. New is that thermal expansion in some of the contributing components need to be applied also to the gravitational deformation effects. A further novelty is that we can substantiate the validity of and the need for these corrections. Concerning the validity we show that the empirical model used by astronomical colleagues for deliberately shifting the sub-reflector for gain optimization, exactly (within 0.5 mm RMS) matches the measured gravitationally induced displacement of the sub-reflector plus the change in focal length. The other evidence is the impact on the vertical component of the telescope’s coordinates of − 6.1 mm, which reduces the discrepancy determined in the computations of the ITRF2014 to 1.7 mm.

### Gravitational deformation of ring-focus antennas for VGOS: first investigations at the Onsala twin telescopes project

Sat, 09/28/2019 - 00:00
Abstract

The receiving properties of radio telescopes used in geodetic and astrometric very long baseline interferometry (VLBI) depend on the surface quality and stability of the main reflector. Deformations of the main reflector as well as changes in the sub-reflector position affect the geometrical ray path length significantly. The deformation pattern and its impact on the VLBI results of conventional radio telescopes have been studied by several research groups using holography, laser tracker, close-range photogrammetry and laser scanner methods. Signal path variations (SPV) of up to 1 cm were reported, which cause, when unaccounted for, systematic biases of the estimated vertical positions of the radio telescopes in the geodetic VLBI analysis and potentially even affect the estimated scale of derived global geodetic reference frames. As a result of the realization of the VLBI 2010 agenda, the geodetic VLBI network is currently extended by several new radio telescopes, which are of a more compact and stiffer design and are able to move faster than conventional radio telescopes. These new telescopes will form the backbone of the next generation geodetic VLBI system, often referred to as VGOS (VLBI Global Observing System). In this investigation, for the first time the deformation pattern of this new generation of radio telescopes for VGOS is studied. ONSA13NE, one of the Onsala twin telescopes at the Onsala Space Observatory, was observed in several elevation angles using close-range photogrammetry. In general, these methods require a crane for preparing the reflector as well as for the data collection. To reduce the observation time and the technical effort during the measurement process, an unmanned aircraft system (UAS) was used for the first time. Using this system, the measurement campaign per elevation angle took less than 30 min. The collected data were used to model the geometrical ray path and its variations. Depending on the distance from the optical axis, the ray path length varies in a range of about $$\pm \,1\,\hbox {mm}$$ . To combine the ray path variations, an illumination function was introduced as weighting function. The resulting total SPV is about $$- \,0.5$$ mm. A simple elevation-dependent SPV model is presented that can easily be used and implemented in VLBI data analysis software packages to correct for gravitational deformation in VGOS radio telescopes. The uncertainty is almost $$200\,\upmu \hbox {m}$$ ( $$2\sigma$$ ) and is derived by Monte Carlo simulations applied to the entire analysis process.

### Correcting surface loading at the observation level: impact on global GNSS and VLBI station networks

Thu, 09/26/2019 - 00:00
Abstract

Time-dependent mass variations of the near-surface geophysical fluids in atmosphere, oceans and the continental hydrosphere lead to systematic and significant load-induced deformations of the Earth’s crust. The Earth System Modeling group of Deutsches GeoForschungsZentrum (ESMGFZ) provides vertical and horizontal surface deformations based on numerical models of the global geophysical fluids in atmosphere, oceans and the continental hydrosphere with a spatial resolution of $$0.5^\circ$$ and a temporal sampling of down to 3 h (Dill and Dobslaw in J Geophys Res 118(9):5008–5017, 2013. https://doi.org/10.1002/jgrb.50353). The assessment of conventionally—i.e. without consideration of non-tidal loading models—processed global GNSS datasets reveals that large parts of the residual station coordinates are indeed related to surface loading effects. Residuals explained by the models often have a pronounced annual component, but variability at other periodicities also contributes to generally high correlations for 7-day averages. More than 10 years of observations from about 400 GNSS and 33 VLBI stations were specifically reprocessed for this study to incorporate non-tidal loading correction models at the observation level. Comparisons with the corresponding conventional processing schemes indicate that the coordinate repeatabilities and residual annual amplitudes decrease by up to 13 mm and 7 mm, respectively, when ESMGFZ’s loading models are applied. In addition, the standard deviation of the daily estimated vertical coordinate is reduced by up to 6.8 mm. The network solutions also allow for an assessment of surface loading effects on GNSS satellite orbits, resulting in radial translations of up to 4 mm and Earth orientation parameters (EOP). In particular, the VLBI-based EOP estimates are critically susceptible to surface loading effects, with root-mean-squared differences reaching of up to 0.2 mas for polar motion, and 10 µs for UT1-UTC.

### A modified phase clock/bias model to improve PPP ambiguity resolution at Wuhan University

Tue, 09/24/2019 - 00:00
Abstract

Precise point positioning ambiguity resolution (PPP-AR) is a valuable tool for high-precision geodetic observations, while phase bias products are critical to implementing GNSS PPP-AR. Based on the conventional integer clock and uncalibrated phase delay (UPD) models, we proposed a modified phase clock/bias model to enable undifferenced ambiguity fixing where it is the phase clocks, rather than the International GNSS Service (IGS) legacy clocks, which are estimated in a network solution by first correcting carrier-phase data for both pre-resolved integer ambiguities and predetermined phase biases. Such phase clock/bias product is compatible with IGS legacy clock and code bias products as opposed to the integer clock model, while ensuring more accurate daily positions in contrast to the UPD model. We carried out precise point positioning (PPP) ambiguity fixing using 1 year of GPS data from about 500 stations and took the IGS weekly solutions as benchmarks: The phase clock/bias model reproduced the positioning achievement of the integer clock model without biasing pseudorange processing, whereas improving markedly the east component of daily positions by 20% compared to the UPD model; interestingly, negligible differences exist between the UPD-based hourly positions and those based on the phase clock/bias model, corroborating that the UPD model is a good approximation to the phase clock/bias model in case of short observation periods. Finally, since phase biases are linearly dependent on clocks, we suggest to compute daily phase bias products, instead of the usual 15-min UPDs, by driving all their temporal variations to the phase clocks, which will greatly facilitate ambiguity-fixed PPP (ftp://igs.gnsswhu.cn/pub/whu/phasebias).

### Multipath extraction and mitigation for high-rate multi-GNSS precise point positioning

Tue, 09/24/2019 - 00:00
Abstract

Multipath effect on carrier-phase observation is one of the bottlenecks for mm-level applications when using precise point positioning (PPP). Hence, we extract the multipath directly from raw carrier-phase residuals of GPS, GLONASS, Galileo, and BDS, by using PPP technique. Although the residuals for one frequency assimilate the errors from other frequencies, which is caused by error adjustment by the least squares estimator, the primary component of residuals is multipath. The results indicate that the residuals between frequencies have a significant linear negative correlation and synchronous time lag for each system. Besides BDS Geostationary Earth Orbit satellites, the residuals for other satellites can establish accurate mathematic relationship between the frequencies. For GLONASS, the residuals of R1 frequency recovered from R2 frequency with the mathematical relationship are better than 0.1 mm accuracy, which means the effect of inter-frequency bias can be neglected. These regularities double-reduce the complexity of data processing. Based on the multipath distribution, we propose a modified Multipath Hemispherical Map model (M-MHM), which constructs grids from residuals and is divided into three equal-elevation angle parts with an optimal resolution 0.2° × 0.2° × 1° from numerous experiments. In addition, the multipath manifests great consistency among satellites for GPS, GLONASS, and Galileo systems when elevation angles are higher than 15°, while is more satellite dependent for BDS. Although GPS L1 frequency is identical to Galileo E1, the model still has some systematic bias between GPS and Galileo. Compared with sidereal filtering and original MHM model, the M-MHM brings the highest improvement in both residual variance reduction and positioning accuracy. The positioning accuracy is on average 12% improvement compared to MHM and 29% improvement compared to SF. For four systems combined solutions with the M-MHM model, can reach an accuracy of 0.75, 0.55, and 2.08 cm in the east, north, and up components.

### Results from a GRACE/GRACE-FO attitude reconstruction Kalman filter

Fri, 09/20/2019 - 00:00
Abstract

This paper outlines JPL’s V03 GRACE attitude processing strategy, characterizes the accelerometer angular measurement error profile, analyzes impact upon GRACE time-varying gravity field products as part of a complete mission reprocessing campaign, and presents implications for linear acceleration measurements. A Kalman filter-based strategy fuses star camera and angular acceleration measurements to reconstruct spacecraft attitude with reduced high-frequency noise and fewer gaps and corrects a pair of processing errors. Running data from tailored accelerometer characterization maneuvers in 2017, K-band calibration maneuvers in 2003, and nominal mission operations through our Kalman filter, estimated aliasing factors from linear to angular acceleration account for multiple forms of observed noise. During most of the mission, V03 produced very limited gains in our gravity field products, but during early and late mission high error regimes the reduction in high-frequency attitude noise substantially damped systematic gravity solution effects (latitudinal bands) and noise (stripes).

### Evidence of daily hydrological loading in GPS time series over Europe

Tue, 09/17/2019 - 00:00
Abstract

Loading deformations from atmospheric, oceanic, and hydrological mass changes mask geophysical processes such as land subsidence and tectonic or volcanic deformation. While it is known that hydrological loading plays a role at seasonal time scales, here we demonstrate evidence that also fast water storage changes contribute to daily Global Positioning System (GPS) height time series. So far, no clear strategy, i.e., no single conventional hydrological model, has been proposed for removing hydrological deformation from daily GPS height time series. Hydrological model predictions of total water storage anomalies tend to diverge and (substantially) deviate from Gravity Recovery and Climate Experiment (GRACE) observations, which however have a limited spatial and temporal resolution. Here, we suggest to overcome these limitations by assimilating GRACE data into a high-resolution (12.5 km) hydrological model. We tested this approach over Europe, and we found that accounting for daily hydrological mass changes reduces the root mean square scatter of GPS height time series almost by a factor of two when compared to monthly hydrological mass changes. We suggest that a GRACE-assimilating hydrological model would provide a promising option for removing hydrology-induced vertical deformation from GPS time series also at the global scale.

### Lunar Laser Ranging: a tool for general relativity, lunar geophysics and Earth science

Tue, 09/17/2019 - 00:00
Abstract

Only a few sites on Earth are technically equipped to carry out Lunar Laser Ranging (LLR) to retroreflector arrays on the surface of the Moon. Despite the weak signal, they have successfully provided LLR range data for about 49 years, generating about 26,000 normal points. Recent system upgrades and new observatories have made millimeter-level range accuracy achievable. Based on appropriate modeling and sophisticated data analysis, LLR is able to determine many parameters associated with Earth–Moon dynamics, involving the lunar ephemeris, lunar physics, the Moon’s interior, reference frames and Earth orientation parameters. LLR has also become one of the strongest tools for testing Einstein’s theory of general relativity in the solar system. By extending the standard solution, it is possible to solve for parameters related to gravitational physics, like the temporal variation of the gravitational constant, metric parameters as well as the strong equivalence principle, preferred-frame effects and standard-model extensions. This paper provides a review about LLR measurement and analysis. After a short historical overview, we describe the key findings of LLR, the apparatus and technologies involved, the requisite modeling techniques, some recent results and future prospects on all fronts. We expect continued improvements in LLR, maintaining its lead in contributing to science.

### A global vertical datum defined by the conventional geoid potential and the Earth ellipsoid parameters

Thu, 09/12/2019 - 00:00
Abstract

The geoid, according to the classical Gauss–Listing definition, is, among infinite equipotential surfaces of the Earth’s gravity field, the equipotential surface that in a least squares sense best fits the undisturbed mean sea level. This equipotential surface, except for its zero-degree harmonic, can be characterized using the Earth’s global gravity models (GGM). Although, nowadays, satellite altimetry technique provides the absolute geoid height over oceans that can be used to calibrate the unknown zero-degree harmonic of the gravimetric geoid models, this technique cannot be utilized to estimate the geometric parameters of the mean Earth ellipsoid (MEE). The main objective of this study is to perform a joint estimation of W0, which defines the zero datum of vertical coordinates, and the MEE parameters relying on a new approach and on the newest gravity field, mean sea surface and mean dynamic topography models. As our approach utilizes both satellite altimetry observations and a GGM model, we consider different aspects of the input data to evaluate the sensitivity of our estimations to the input data. Unlike previous studies, our results show that it is not sufficient to use only the satellite-component of a quasi-stationary GGM to estimate W0. In addition, our results confirm a high sensitivity of the applied approach to the altimetry-based geoid heights, i.e., mean sea surface and mean dynamic topography models. Moreover, as W0 should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth’s gravity field changes as well as the time-dependent sea level changes on the estimation of W0. Our computations resulted in the geoid potential W0 = 62636848.102 ± 0.004 m2 s−2 and the semi-major and minor axes of the MEE, a = 6378137.678 ± 0.0003 m and b = 6356752.964 ± 0.0005 m, which are 0.678 and 0.650 m larger than those axes of GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3 s−2.

### Improved Fourier modeling of gravity fields caused by polyhedral bodies: with applications to asteroid Bennu and comet 67P/Churyumov–Gerasimenko

Tue, 09/03/2019 - 00:00
Abstract

This paper presents an improved algorithm for the 2D and 3D Fourier forward modeling of gravity fields caused by polyhedral bodies with constant and exponential density distributions. Three modifications have been made to the Fourier forward algorithm introduced in a previous paper. First, vertex-based Fourier-domain expressions are used instead of the original face-based Fourier-domain expressions, which simplify the computation of the anomaly spectrum considerably, especially in 3D modeling problems. Second, instead of using a pure Gauss-FFT sampling of the anomaly spectrum, we apply an improved sampling strategy by combining a nonuniform spherical sampling with a low-order Gauss-FFT sampling. In this way, the number of samplings required in the Fourier domain reduces to about $$\frac{1}{3}$$ and $$\frac{1}{7}$$ of those required in a pure Gauss-FFT algorithm for 2D and 3D modeling problems, respectively. A significant acceleration over the original algorithm is achieved. Third, we incorporate all three types of nonuniform fast Fourier transform algorithms to transform directly a uniform or nonuniform anomaly spectrum to gravity fields either on a regular grid, or at a set of arbitrary positions. Extra interpolation operations are no longer needed. Synthetic numerical tests show that for gravity vector components, the new algorithm runs about 3 times faster in 2D modeling and 7 times faster in 3D modeling than the original ones, while maintaining the same level of accuracy. For the gravity potential, the new algorithm is significantly superior to the pure Gauss-FFT solution both in numerical accuracy and in efficiency. We apply this novel approach to compute the gravitational fields of asteroid 101955 Bennu and comet $$67\hbox {P/Churyumov}$$ –Gerasimenko. The 2D algorithm works very efficiently for the computation of gravity fields on horizontal planes. The 3D algorithm is valid both outside, on, and inside the source’s bounding surface, with relative errors less than 0.1% for the gravity potential and less than 2% for the gravity vector. By comparing to modeling results of analytical and spherical harmonic-based solutions, we generally conclude that the Fourier-based algorithm introduced here is an attractive alternative to these conventional solutions, especially for nonspherical, irregularly shaped bodies with complex geometries.

### Assessment of forecast Vienna Mapping Function 1 for real-time tropospheric delay modeling in GNSS

Sun, 09/01/2019 - 00:00
Abstract

The accurate modeling of tropospheric path delay is significant for data processing of radio space-geodetic techniques such as Global Navigation Satellite System (GNSS) and Very-Long-Baseline Interferometry (VLBI). The Vienna Mapping Function 1 (VMF1) model, based on continuous update of Numerical Weather Prediction (NWP) data from the European Centre for Medium-Range Weather Forecasts (ECMWF), is recommended for this purpose in post-processing. The VMF1 coefficients determined from forecast data of the ECMWF are now readily and freely available. However, little or no implementation of this forecast VMF1 (VMF1-FC) model in real-time GNSS or VLBI applications has occurred. This study investigates the performance of the VMF1-FC model in terms of its three components which are critical for the modeling of tropospheric path delay: the Zenith Hydrostatic Delay (ZHD), the Zenith Wet Delay (ZWD) and mapping functions. All three components are assessed in the context of GNSS Precise Point Positioning (PPP) using 28 global stations over a 70-day period. The Zenith Total Delays (ZTD) derived with the VMF1-FC (implemented in real-time PPP) are shown to agree well with the tropospheric delay product from the Center for Orbit Determination Europe (CODE). Root mean square (RMS) errors associated with these ZTD estimates are < 10 mm at all 28 stations. The results also show that the VMF1-FC model performs better than empirical models such as the widely used Global Pressure and Temperature 2 (GPT2) and GPT2 wet (GPT2w), with smaller RMS errors associated with the ZTD estimates. It is recommended that VMF1-FC be applied for future tropospheric delay modeling in real-time GNSS and VLBI applications.

### A GPS relative positioning quality control algorithm considering both code and phase observation errors

Sun, 09/01/2019 - 00:00
Abstract

GPS relative positioning performance highly relies on the quality control algorithm. Previous efforts were mainly made with emphasis on the phase cycle slip detection in the preprocessing stage and the posterior residual check to re-weight observations. Very limited work focused on the code observation error and its effect on GPS relative positioning. This paper proposes a GPS relative positioning quality control algorithm that considers both code and phase errors of dual-frequency observations for geodetic and navigation receivers. In addition to the phase cycle slip detection in the preprocessing stage, a posterior code residual check is developed that has priority over the posterior phase residual check. If the posterior code residual check fails, no posterior phase residual check is needed. In this sense, the effect of code observation error on the phase ambiguity estimation and subsequently high-precision positioning can be investigated. Three dedicated static and kinematic experiments were carried out to assess the proposed method in terms of the ambiguity and positioning solutions, respectively. As for Experiment #1 under good observation environment, the proposed method provided a similar performance as the conventional method not taking the code observation error into account. However, Experiment #2 characterized by the tree-surrounded observation environment and Experiment #3 for kinematic positioning above the lake surface indicated that the proposed method could provide significant improvements over the conventional method. The ignorance of code observation error would deteriorate the phase ambiguity estimation and subsequently lead to worse positioning convergence and precision.

### A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling

Sun, 09/01/2019 - 00:00
Abstract

Residual terrain modelling (RTM) plays a key role for short-scale gravity modelling in physical geodesy, e.g. for interpolation of observed gravity and augmentation of global geopotential models (GGMs). However, approximation errors encountered in RTM computation schemes are little investigated. The goal of the present paper is to examine widely used classical RTM techniques in order to provide insights into RTM-specific approximation errors and the resulting RTM accuracy. This is achieved by introducing a new, independent RTM technique as baseline that relies on the combination of (1) a full-scale global numerical integration in the spatial domain and (2) ultra-high-degree spectral forward modelling. The global integration provides the full gravity signal of the complete (detailed) topography, and the spectral modelling that of the RTM reference topography. As a main benefit, the RTM baseline technique inherently solves the “non-harmonicity problem” encountered in classical RTM techniques for points inside the reference topography. The new technique is utilized in a closed-loop type testing regime for in-depth examination of four variants of classical RTM techniques used in the literature which are all affected by one or two types of RTM-specific approximation errors. These are errors due to the (1) harmonic correction (HC) needed for points located inside the reference topography, (2) mass simplification, (3) vertical computation point inconsistency, and (4) neglect of terrain correction (TC) of the reference topography. For the Himalaya Mountains and the European Alps, and a degree-2160 reference topography, RTM approximation errors are quantified. As key finding, approximation errors associated with the standard HC ( $$4\pi G\rho H_{\text{P}}^{\text{RTM}} )$$ may reach amplitudes of ~ 10 mGal for points located deep inside the reference topography. We further show that the popular RTM approximation ( $$2\pi G\rho H_{\text{P}}^{\text{RTM}} - {\text{TC}}$$ ) suffers from severe errors that may reach ~ 90 mGal amplitudes in rugged terrain. As a general conclusion, the RTM baseline technique allows inspecting present and future RTM techniques down to the sub-mGal level, thus improving our understanding of technique characteristics and errors. We expect the insights to be useful for future RTM applications, e.g. in geoid modelling using remove–compute–restore techniques, and in the development of new GGMs or high-resolution augmentations thereof.

### Correction to: Regularization and error characterization of GRACE mascons

Sun, 09/01/2019 - 00:00

In the originally published version of the article, Eq. (15) is shown incorrectly.

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