Showing papers in "Journal of Geodesy in 2018"

VMF3/GPT3: refined discrete and empirical troposphere mapping functions

Abstract: Incorrect modeling of troposphere delays is one of the major error sources for space geodetic techniques such as Global Navigation Satellite Systems (GNSS) or Very Long Baseline Interferometry (VLBI). Over the years, many approaches have been devised which aim at mapping the delay of radio waves from zenith direction down to the observed elevation angle, so-called mapping functions. This paper contains a new approach intended to refine the currently most important discrete mapping function, the Vienna Mapping Functions 1 (VMF1), which is successively referred to as Vienna Mapping Functions 3 (VMF3). It is designed in such a way as to eliminate shortcomings in the empirical coefficients b and c and in the tuning for the specific elevation angle of $$3^{\circ }$$ . Ray-traced delays of the ray-tracer RADIATE serve as the basis for the calculation of new mapping function coefficients. Comparisons of modeled slant delays demonstrate the ability of VMF3 to approximate the underlying ray-traced delays more accurately than VMF1 does, in particular at low elevation angles. In other words, when requiring highest precision, VMF3 is to be preferable to VMF1. Aside from revising the discrete form of mapping functions, we also present a new empirical model named Global Pressure and Temperature 3 (GPT3) on a $$5^{\circ }\times 5^{\circ }$$ as well as a $$1^{\circ }\times 1^{\circ }$$ global grid, which is generally based on the same data. Its main components are hydrostatic and wet empirical mapping function coefficients derived from special averaging techniques of the respective (discrete) VMF3 data. In addition, GPT3 also contains a set of meteorological quantities which are adopted as they stand from their predecessor, Global Pressure and Temperature 2 wet. Thus, GPT3 represents a very comprehensive troposphere model which can be used for a series of geodetic as well as meteorological and climatological purposes and is fully consistent with VMF3.

Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle

Abstract: In the context of the International GNSS Service (IGS), several IGS Ionosphere Associated Analysis Centers have developed different techniques to provide global ionospheric maps (GIMs) of vertical total electron content (VTEC) since 1998. In this paper we present a comparison of the performances of all the GIMs created in the frame of IGS. Indeed we compare the classical ones (for the ionospheric analysis centers CODE, ESA/ESOC, JPL and UPC) with the new ones (NRCAN, CAS, WHU). To assess the quality of them in fair and completely independent ways, two assessment methods are used: a direct comparison to altimeter data (VTEC-altimeter) and to the difference of slant total electron content (STEC) observed in independent ground reference stations (dSTEC-GPS). The main conclusion of this study, performed during one solar cycle, is the consistency of the results between so many different GIM techniques and implementations.

Multi-GNSS phase delay estimation and PPP ambiguity resolution: GPS, BDS, GLONASS, Galileo

Abstract: This paper focuses on the precise point positioning (PPP) ambiguity resolution (AR) using the observations acquired from four systems: GPS, BDS, GLONASS, and Galileo (GCRE). A GCRE four-system uncalibrated phase delay (UPD) estimation model and multi-GNSS undifferenced PPP AR method were developed in order to utilize the observations from all systems. For UPD estimation, the GCRE-combined PPP solutions of the globally distributed MGEX and IGS stations are performed to obtain four-system float ambiguities and then UPDs of GCRE satellites can be precisely estimated from these ambiguities. The quality of UPD products in terms of temporal stability and residual distributions is investigated for GPS, BDS, GLONASS, and Galileo satellites, respectively. The BDS satellite-induced code biases were corrected for GEO, IGSO, and MEO satellites before the UPD estimation. The UPD results of global and regional networks were also evaluated for Galileo and BDS, respectively. As a result of the frequency-division multiple-access strategy of GLONASS, the UPD estimation was performed using a network of homogeneous receivers including three commonly used GNSS receivers (TRIMBLE NETR9, JAVAD TRE_G3TH DELTA, and LEICA). Data recorded from 140 MGEX and IGS stations for a 30-day period in January in 2017 were used to validate the proposed GCRE UPD estimation and multi-GNSS dual-frequency PPP AR. Our results show that GCRE four-system PPP AR enables the fastest time to first fix (TTFF) solutions and the highest accuracy for all three coordinate components compared to the single and dual system. An average TTFF of 9.21 min with $$7{^{\circ }}$$ cutoff elevation angle can be achieved for GCRE PPP AR, which is much shorter than that of GPS (18.07 min), GR (12.10 min), GE (15.36 min) and GC (13.21 min). With observations length of 10 min, the positioning accuracy of the GCRE fixed solution is 1.84, 1.11, and 1.53 cm, while the GPS-only result is 2.25, 1.29, and 9.73 cm for the east, north, and vertical components, respectively. When the cutoff elevation angle is increased to $$30{^{\circ }}$$ , the GPS-only PPP AR results are very unreliable, while 13.44 min of TTFF is still achievable for GCRE four-system solutions.

Distributional theory for the DIA method

Abstract: The DIA method for the detection, identification and adaptation of model misspecifications combines estimation with testing. The aim of the present contribution is to introduce a unifying framework for the rigorous capture of this combination. By using a canonical model formulation and a partitioning of misclosure space, we show that the whole estimation–testing scheme can be captured in one single DIA estimator. We study the characteristics of this estimator and discuss some of its distributional properties. With the distribution of the DIA estimator provided, one can then study all the characteristics of the combined estimation and testing scheme, as well as analyse how they propagate into final outcomes. Examples are given, as well as a discussion on how the distributional properties compare with their usage in practice.

GNSS satellite transmit power and its impact on orbit determination

Abstract: Antenna thrust is a small acceleration acting on Global Navigation Satellite System satellites caused by the transmission of radio navigation signals. Knowledge about the transmit power and the mass of the satellites is required for the computation of this effect. The actual transmit power can be obtained from measurements with a high-gain antenna and knowledge about the properties of the transmit and receive antennas as well as losses along the propagation path. Transmit power measurements for different types of GPS, GLONASS, Galileo, and BeiDou-2 satellites were taken with a 30-m dish antenna of the German Aerospace Center (DLR) located at its ground station in Weilheim. For GPS, total L-band transmit power levels of 50–240 W were obtained, 20–135 W for GLONASS, 95–265 W for Galileo, and 130–185 W for BeiDou-2. The transmit power differs usually only slightly for individual spacecraft within one satellite block. An exception are the GLONASS-M satellites where six subgroups with different transmit power levels could be identified. Considering the antenna thrust in precise orbit determination of GNSS satellites decreases the orbital radius by 1–27 mm depending on the transmit power, the satellite mass, and the orbital period.

Precise orbit determination of the Sentinel-3A altimetry satellite using ambiguity-fixed GPS carrier phase observations

Abstract: The Sentinel-3 mission takes routine measurements of sea surface heights and depends crucially on accurate and precise knowledge of the spacecraft. Orbit determination with a targeted uncertainty of less than 2 cm in radial direction is supported through an onboard Global Positioning System (GPS) receiver, a Doppler Orbitography and Radiopositioning Integrated by Satellite instrument, and a complementary laser retroreflector for satellite laser ranging. Within this study, the potential of ambiguity fixing for GPS-only precise orbit determination (POD) of the Sentinel-3 spacecraft is assessed. A refined strategy for carrier phase generation out of low-level measurements is employed to cope with half-cycle ambiguities in the tracking of the Sentinel-3 GPS receiver that have so far inhibited ambiguity-fixed POD solutions. Rather than explicitly fixing double-difference phase ambiguities with respect to a network of terrestrial reference stations, a single-receiver ambiguity resolution concept is employed that builds on dedicated GPS orbit, clock, and wide-lane bias products provided by the CNES/CLS (Centre National d’Etudes Spatiales/Collecte Localisation Satellites) analysis center of the International GNSS Service. Compared to float ambiguity solutions, a notably improved precision can be inferred from laser ranging residuals. These decrease from roughly 9 mm down to 5 mm standard deviation for high-grade stations on average over low and high elevations. Furthermore, the ambiguity-fixed orbits offer a substantially improved cross-track accuracy and help to identify lateral offsets in the GPS antenna or center-of-mass (CoM) location. With respect to altimetry, the improved orbit precision also benefits the global consistency of sea surface measurements. However, modeling of the absolute height continues to rely on proper dynamical models for the spacecraft motion as well as ground calibrations for the relative position of the altimeter reference point and the CoM.

Joint estimation of vertical total electron content (VTEC) and satellite differential code biases (SDCBs) using low-cost receivers

Abstract: Vertical total electron content (VTEC) parameters estimated using global navigation satellite system (GNSS) data are of great interest for ionosphere sensing. Satellite differential code biases (SDCBs) account for one source of error which, if left uncorrected, can deteriorate performance of positioning, timing and other applications. The customary approach to estimate VTEC along with SDCBs from dual-frequency GNSS data, hereinafter referred to as DF approach, consists of two sequential steps. The first step seeks to retrieve ionospheric observables through the carrier-to-code leveling technique. This observable, related to the slant total electron content (STEC) along the satellite–receiver line-of-sight, is biased also by the SDCBs and the receiver differential code biases (RDCBs). By means of thin-layer ionospheric model, in the second step one is able to isolate the VTEC, the SDCBs and the RDCBs from the ionospheric observables. In this work, we present a single-frequency (SF) approach, enabling the joint estimation of VTEC and SDCBs using low-cost receivers; this approach is also based on two steps and it differs from the DF approach only in the first step, where we turn to the precise point positioning technique to retrieve from the single-frequency GNSS data the ionospheric observables, interpreted as the combination of the STEC, the SDCBs and the biased receiver clocks at the pivot epoch. Our numerical analyses clarify how SF approach performs when being applied to GPS L1 data collected by a single receiver under both calm and disturbed ionospheric conditions. The daily time series of zenith VTEC estimates has an accuracy ranging from a few tenths of a TEC unit (TECU) to approximately 2 TECU. For 73–96% of GPS satellites in view, the daily estimates of SDCBs do not deviate, in absolute value, more than 1 ns from their ground truth values published by the Centre for Orbit Determination in Europe.

Short note: the experimental geopotential model XGM2016

Abstract: As a precursor study for the upcoming combined Earth Gravitational Model 2020 (EGM2020), the Experimental Gravity Field Model XGM2016, parameterized as a spherical harmonic series up to degree and order 719, is computed. XGM2016 shares the same combination methodology as its predecessor model GOCO05c (Fecher et al. in Surv Geophys 38(3): 571–590, 2017. doi: 10.1007/s10712-016-9406-y). The main difference between these models is that XGM2016 is supported by an improved terrestrial data set of \(15^\prime \times 15^\prime \) gravity anomaly area-means provided by the United States National Geospatial-Intelligence Agency (NGA), resulting in significant upgrades compared to existing combined gravity field models, especially in continental areas such as South America, Africa, parts of Asia, and Antarctica. A combination strategy of relative regional weighting provides for improved performance in near-coastal ocean regions, including regions where the altimetric data are mostly unchanged from previous models. Comparing cumulative height anomalies, from both EGM2008 and XGM2016 at degree/order 719, yields differences of 26 cm in Africa and 40 cm in South America. These differences result from including additional information of satellite data, as well as from the improved ground data in these regions. XGM2016 also yields a smoother Mean Dynamic Topography with significantly reduced artifacts, which indicates an improved modeling of the ocean areas.

Three-frequency BDS precise point positioning ambiguity resolution based on raw observables

Abstract: All BeiDou navigation satellite system (BDS) satellites are transmitting signals on three frequencies, which brings new opportunity and challenges for high-accuracy precise point positioning (PPP) with ambiguity resolution (AR). This paper proposes an effective uncalibrated phase delay (UPD) estimation and AR strategy which is based on a raw PPP model. First, triple-frequency raw PPP models are developed. The observation model and stochastic model are designed and extended to accommodate the third frequency. Then, the UPD is parameterized in raw frequency form while estimating with the high-precision and low-noise integer linear combination of float ambiguity which are derived by ambiguity decorrelation. Third, with UPD corrected, the LAMBDA method is used for resolving full or partial ambiguities which can be fixed. This method can be easily and flexibly extended for dual-, triple- or even more frequency. To verify the effectiveness and performance of triple-frequency PPP AR, tests with real BDS data from 90 stations lasting for 21 days were performed in static mode. Data were processed with three strategies: BDS triple-frequency ambiguity-float PPP, BDS triple-frequency PPP with dual-frequency (B1/B2) and three-frequency AR, respectively. Numerous experiment results showed that compared with the ambiguity-float solution, the performance in terms of convergence time and positioning biases can be significantly improved by AR. Among three groups of solutions, the triple-frequency PPP AR achieved the best performance. Compared with dual-frequency AR, additional the third frequency could apparently improve the position estimations during the initialization phase and under constraint environments when the dual-frequency PPP AR is limited by few satellite numbers.

Initial results of centralized autonomous orbit determination of the new-generation BDS satellites with inter-satellite link measurements

Abstract: Autonomous orbit determination is the ability of navigation satellites to estimate the orbit parameters on-board using inter-satellite link (ISL) measurements This study mainly focuses on data processing of the ISL measurements as a new measurement type and its application on the centralized autonomous orbit determination of the new-generation Beidou navigation satellite system satellites for the first time The ISL measurements are dual one-way measurements that follow a time division multiple access (TDMA) structure The ranging error of the ISL measurements is less than 025 ns This paper proposes a derivation approach to the satellite clock offsets and the geometric distances from TDMA dual one-way measurements without a loss of accuracy The derived clock offsets are used for time synchronization, and the derived geometry distances are used for autonomous orbit determination The clock offsets from the ISL measurements are consistent with the L-band two-way satellite, and time–frequency transfer clock measurements and the detrended residuals vary within 05 ns The centralized autonomous orbit determination is conducted in a batch mode on a ground-capable server for the feasibility study Constant hardware delays are present in the geometric distances and become the largest source of error in the autonomous orbit determination Therefore, the hardware delays are estimated simultaneously with the satellite orbits To avoid uncertainties in the constellation orientation, a ground anchor station that “observes” the satellites with on-board ISL payloads is introduced into the orbit determination The root-mean-square values of orbit determination residuals are within 100 cm, and the standard deviation of the estimated ISL hardware delays is within 02 ns The accuracy of the autonomous orbits is evaluated by analysis of overlap comparison and the satellite laser ranging (SLR) residuals and is compared with the accuracy of the L-band orbits The results indicate that the radial overlap differences between the autonomous orbits are less than 150 cm for the inclined geosynchronous orbit (IGSO) satellites and less than 100 cm for the MEO satellites The SLR residuals are approximately 150 cm for the IGSO satellites and approximately 100 cm for the MEO satellites, representing an improvement over the L-band orbits

Geodetic methods to determine the relativistic redshift at the level of 10^{-18} in the context of international timescales: a review and practical results

Abstract: The frequency stability and uncertainty of the latest generation of optical atomic clocks is now approaching the one part in $$10^{18}$$ level. Comparisons between earthbound clocks at rest must account for the relativistic redshift of the clock frequencies, which is proportional to the corresponding gravity (gravitational plus centrifugal) potential difference. For contributions to international timescales, the relativistic redshift correction must be computed with respect to a conventional zero potential value in order to be consistent with the definition of Terrestrial Time. To benefit fully from the uncertainty of the optical clocks, the gravity potential must be determined with an accuracy of about $$0.1\,\hbox {m}^{2}\,\hbox {s}^{-2}$$ , equivalent to about 0.01 m in height. This contribution focuses on the static part of the gravity field, assuming that temporal variations are accounted for separately by appropriate reductions. Two geodetic approaches are investigated for the derivation of gravity potential values: geometric levelling and the Global Navigation Satellite Systems (GNSS)/geoid approach. Geometric levelling gives potential differences with millimetre uncertainty over shorter distances (several kilometres), but is susceptible to systematic errors at the decimetre level over large distances. The GNSS/geoid approach gives absolute gravity potential values, but with an uncertainty corresponding to about 2 cm in height. For large distances, the GNSS/geoid approach should therefore be better than geometric levelling. This is demonstrated by the results from practical investigations related to three clock sites in Germany and one in France. The estimated uncertainty for the relativistic redshift correction at each site is about $$2 \times 10^{-18}$$ .

GPS satellite clock determination in case of inter-frequency clock biases for triple-frequency precise point positioning

Abstract: Significant time-varying inter-frequency clock biases (IFCBs) within GPS observations prevent the application of the legacy L1/L2 ionosphere-free clock products on L5 signals. Conventional approaches overcoming this problem are to estimate L1/L5 ionosphere-free clocks in addition to their L1/L2 counterparts or to compute IFCBs between the L1/L2 and L1/L5 clocks which are later modeled through a harmonic analysis. In contrast, we start from the undifferenced uncombined GNSS model and propose an alternative approach where a second satellite clock parameter dedicated to the L5 signals is estimated along with the legacy L1/L2 clock. In this manner, we do not need to rely on the correlated L1/L2 and L1/L5 ionosphere-free observables which complicates triple-frequency GPS stochastic models, or account for the unfavorable time-varying hardware biases in undifferenced GPS functional models since they can be absorbed by the L5 clocks. An extra advantage over the ionosphere-free model is that external ionosphere constraints can potentially be introduced to improve PPP. With 27 days of triple-frequency GPS data from globally distributed stations, we find that the RMS of the positioning differences between our GPS model and all conventional models is below 1 mm for all east, north and up components, demonstrating the effectiveness of our model in addressing triple-frequency observations and time-varying IFCBs. Moreover, we can combine the L1/L2 and L5 clocks derived from our model to calculate precisely the L1/L5 clocks which in practice only depart from their legacy counterparts by less than 0.006 ns in RMS. Our triple-frequency GPS model proves convenient and efficient in combating time-varying IFCBs and can be generalized to more than three frequency signals for satellite clock determination.

Real-Time Precise Point Positioning (RTPPP) with raw observations and its application in real-time regional ionospheric VTEC modeling

Abstract: Precise Point Positioning (PPP) is an absolute positioning technology mainly used in post data processing With the continuously increasing demand for real-time high-precision applications in positioning, timing, retrieval of atmospheric parameters, etc, Real-Time PPP (RTPPP) and its applications have drawn more and more research attention in recent years This study focuses on the models, algorithms and ionospheric applications of RTPPP on the basis of raw observations, in which high-precision slant ionospheric delays are estimated among others in real time For this purpose, a robust processing strategy for multi-station RTPPP with raw observations has been proposed and realized, in which real-time data streams and State-Space-Representative (SSR) satellite orbit and clock corrections are used With the RTPPP-derived slant ionospheric delays from a regional network, a real-time regional ionospheric Vertical Total Electron Content (VTEC) modeling method is proposed based on Adjusted Spherical Harmonic Functions and a Moving-Window Filter SSR satellite orbit and clock corrections from different IGS analysis centers are evaluated Ten globally distributed real-time stations are used to evaluate the positioning performances of the proposed RTPPP algorithms in both static and kinematic modes RMS values of positioning errors in static/kinematic mode are 52/155, 47/174 and 128/466 mm, for north, east and up components, respectively Real-time slant ionospheric delays from RTPPP are compared with those from the traditional Carrier-to-Code Leveling (CCL) method, in terms of function model, formal precision and between-receiver differences of short baseline Results show that slant ionospheric delays from RTPPP are more precise and have a much better convergence performance than those from the CCL method in real-time processing 30 real-time stations from the Asia-Pacific Reference Frame network are used to model the ionospheric VTECs over Australia in real time, with slant ionospheric delays from both RTPPP and CCL methods for comparison RMS of the VTEC differences between RTPPP/CCL method and CODE final products is 091/109 TECU, and RMS of the VTEC differences between RTPPP and CCL methods is 067 TECU Slant Total Electron Contents retrieved from different VTEC models are also validated with epoch-differenced Geometry-Free combinations of dual-frequency phase observations, and mean RMS values are 214, 233 and 207 TECU for RTPPP method, CCL method and CODE final products, respectively This shows the superiority of RTPPP-derived slant ionospheric delays in real-time ionospheric VTEC modeling

Validation of Galileo orbits using SLR with a focus on satellites launched into incorrect orbital planes

Abstract: The space segment of the European Global Navigation Satellite System (GNSS) Galileo consists of In-Orbit Validation (IOV) and Full Operational Capability (FOC) spacecraft. The first pair of FOC satellites was launched into an incorrect, highly eccentric orbital plane with a lower than nominal inclination angle. All Galileo satellites are equipped with satellite laser ranging (SLR) retroreflectors which allow, for example, for the assessment of the orbit quality or for the SLR–GNSS co-location in space. The number of SLR observations to Galileo satellites has been continuously increasing thanks to a series of intensive campaigns devoted to SLR tracking of GNSS satellites initiated by the International Laser Ranging Service. This paper assesses systematic effects and quality of Galileo orbits using SLR data with a main focus on Galileo satellites launched into incorrect orbits. We compare the SLR observations with respect to microwave-based Galileo orbits generated by the Center for Orbit Determination in Europe (CODE) in the framework of the International GNSS Service Multi-GNSS Experiment for the period 2014.0–2016.5. We analyze the SLR signature effect, which is characterized by the dependency of SLR residuals with respect to various incidence angles of laser beams for stations equipped with single-photon and multi-photon detectors. Surprisingly, the CODE orbit quality of satellites in the incorrect orbital planes is not worse than that of nominal FOC and IOV orbits. The RMS of SLR residuals is even lower by 5.0 and 1.5 mm for satellites in the incorrect orbital planes than for FOC and IOV satellites, respectively. The mean SLR offsets equal $$-44.9, -35.0$$ , and $$-22.4$$ mm for IOV, FOC, and satellites in the incorrect orbital plane. Finally, we found that the empirical orbit models, which were originally designed for precise orbit determination of GNSS satellites in circular orbits, provide fully appropriate results also for highly eccentric orbits with variable linear and angular velocities.

Long-term prediction of polar motion using a combined SSA and ARMA model

Abstract: To meet the need for real-time and high-accuracy predictions of polar motion (PM), the singular spectrum analysis (SSA) and the autoregressive moving average (ARMA) model are combined for short- and long-term PM prediction. According to the SSA results for PM and the SSA prediction algorithm, the principal components of PM were predicted by SSA, and the remaining components were predicted by the ARMA model. In applying this proposed method, multiple sets of PM predictions were made with lead times of two years, based on an IERS 08 C04 series. The observations and predictions of the principal components correlated well, and the SSA $$+$$ ARMA model effectively predicted the PM. For 360-day lead time predictions, the root-mean-square errors (RMSEs) of PMx and PMy were 20.67 and 20.42 mas, respectively, which were less than the 24.46 and 24.78 mas predicted by IERS Bulletin A. The RMSEs of PMx and PMy in the 720-day lead time predictions were 28.61 and 27.95 mas, respectively.

Modeling tropospheric wet delays with national GNSS reference network in China for BeiDou precise point positioning

Abstract: During past decades, precise point positioning (PPP) has been proven to be a well-known positioning technique for centimeter or decimeter level accuracy. However, it needs long convergence time to get high-accuracy positioning, which limits the prospects of PPP, especially in real-time applications. It is expected that the PPP convergence time can be reduced by introducing high-quality external information, such as ionospheric or tropospheric corrections. In this study, several methods for tropospheric wet delays modeling over wide areas are investigated. A new, improved model is developed, applicable in real-time applications in China. Based on the GPT2w model, a modified parameter of zenith wet delay exponential decay wrt. height is introduced in the modeling of the real-time tropospheric delay. The accuracy of this tropospheric model and GPT2w model in different seasons is evaluated with cross-validation, the root mean square of the zenith troposphere delay (ZTD) is 1.2 and 3.6 cm on average, respectively. On the other hand, this new model proves to be better than the tropospheric modeling based on water-vapor scale height; it can accurately express tropospheric delays up to 10 km altitude, which potentially has benefits in many real-time applications. With the high-accuracy ZTD model, the augmented PPP convergence performance for BeiDou navigation satellite system (BDS) and GPS is evaluated. It shows that the contribution of the high-quality ZTD model on PPP convergence performance has relation with the constellation geometry. As BDS constellation geometry is poorer than GPS, the improvement for BDS PPP is more significant than that for GPS PPP. Compared with standard real-time PPP, the convergence time is reduced by 2–7 and 20–50% for the augmented BDS PPP, while GPS PPP only improves about 6 and 18% (on average), in horizontal and vertical directions, respectively. When GPS and BDS are combined, the geometry is greatly improved, which is good enough to get a reliable PPP solution, the augmentation PPP improves insignificantly comparing with standard PPP.

Statistically optimal estimation of Greenland Ice Sheet mass variations from GRACE monthly solutions using an improved mascon approach.

Abstract: We present an improved mascon approach to transform monthly spherical harmonic solutions based on GRACE satellite data into mass anomaly estimates in Greenland. The GRACE-based spherical harmonic coefficients are used to synthesize gravity anomalies at satellite altitude, which are then inverted into mass anomalies per mascon. The limited spectral content of the gravity anomalies is properly accounted for by applying a low-pass filter as part of the inversion procedure to make the functional model spectrally consistent with the data. The full error covariance matrices of the monthly GRACE solutions are properly propagated using the law of covariance propagation. Using numerical experiments, we demonstrate the importance of a proper data weighting and of the spectral consistency between functional model and data. The developed methodology is applied to process real GRACE level-2 data (CSR RL05). The obtained mass anomaly estimates are integrated over five drainage systems, as well as over entire Greenland. We find that the statistically optimal data weighting reduces random noise by 35–69%, depending on the drainage system. The obtained mass anomaly time-series are de-trended to eliminate the contribution of ice discharge and are compared with de-trended surface mass balance (SMB) time-series computed with the Regional Atmospheric Climate Model (RACMO 2.3). We show that when using a statistically optimal data weighting in GRACE data processing, the discrepancies between GRACE-based estimates of SMB and modelled SMB are reduced by 24–47%.

A neural network model for predicting weighted mean temperature

Abstract: Water vapor is an important element of the Earth’s atmosphere, and most of it concentrates at the bottom of the troposphere. Knowledge of the water vapor measured by Global Navigation Satellite Systems (GNSS) is an important direction of GNSS research. In particular, when the zenith wet delay is converted to precipitable water vapor, the weighted mean temperature $$T_\mathrm{m}$$ is a variable parameter to be determined in this conversion. The purpose of the study is getting a more accurate $$T_\mathrm{m}$$ model for global users by a combination of two different characteristics of $$T_\mathrm{m}$$ (i.e., the $$T_\mathrm{m}$$ seasonal variations and the relationships between $$T_\mathrm{m}$$ and surface meteorological elements). The modeling process was carried out by using the neural network technology. A multilayer feedforward neural network model (the NN) was established. The NN model is used with measurements of only surface temperature $$T_\mathrm{S}$$ . The NN was validated and compared with four other published global $$T_\mathrm{m}$$ models. The results show that the NN performed better than any of the four compared models on the global scale.

Refined discrete and empirical horizontal gradients in VLBI analysis

Abstract: Missing or incorrect consideration of azimuthal asymmetry of troposphere delays is a considerable error source in space geodetic techniques such as Global Navigation Satellite Systems (GNSS) or Very Long Baseline Interferometry (VLBI). So-called horizontal troposphere gradients are generally utilized for modeling such azimuthal variations and are particularly required for observations at low elevation angles. Apart from estimating the gradients within the data analysis, which has become common practice in space geodetic techniques, there is also the possibility to determine the gradients beforehand from different data sources than the actual observations. Using ray-tracing through Numerical Weather Models (NWMs), we determined discrete gradient values referred to as GRAD for VLBI observations, based on the standard gradient model by Chen and Herring (J Geophys Res 102(B9):20489-20502, 1997. 10.1029/97JB01739) and also for new, higher-order gradient models. These gradients are produced on the same data basis as the Vienna Mapping Functions 3 (VMF3) (Landskron and Bohm in J Geod, 2017. 10.1007/s00190-017-1066-2), so they can also be regarded as the VMF3 gradients as they are fully consistent with each other. From VLBI analyses of the Vienna VLBI and Satellite Software (VieVS), it becomes evident that baseline length repeatabilities (BLRs) are improved on average by 5% when using a priori gradients GRAD instead of estimating the gradients. The reason for this improvement is that the gradient estimation yields poor results for VLBI sessions with a small number of observations, while the GRAD a priori gradients are unaffected from this. We also developed a new empirical gradient model applicable for any time and location on Earth, which is included in the Global Pressure and Temperature 3 (GPT3) model. Although being able to describe only the systematic component of azimuthal asymmetry and no short-term variations at all, even these empirical a priori gradients slightly reduce (improve) the BLRs with respect to the estimation of gradients. In general, this paper addresses that a priori horizontal gradients are actually more important for VLBI analysis than previously assumed, as particularly the discrete model GRAD as well as the empirical model GPT3 are indeed able to refine and improve the results.

Adaptive Kalman filter based on variance component estimation for the prediction of ionospheric delay in aiding the cycle slip repair of GNSS triple-frequency signals

Abstract: In order to incorporate the time smoothness of ionospheric delay to aid the cycle slip detection, an adaptive Kalman filter is developed based on variance component estimation. The correlations between measurements at neighboring epochs are fully considered in developing a filtering algorithm for colored measurement noise. Within this filtering framework, epoch-differenced ionospheric delays are predicted. Using this prediction, the potential cycle slips are repaired for triple-frequency signals of global navigation satellite systems. Cycle slips are repaired in a stepwise manner; i.e., for two extra wide lane combinations firstly and then for the third frequency. In the estimation for the third frequency, a stochastic model is followed in which the correlations between the ionospheric delay prediction errors and the errors in the epoch-differenced phase measurements are considered. The implementing details of the proposed method are tabulated. A real BeiDou Navigation Satellite System data set is used to check the performance of the proposed method. Most cycle slips, no matter trivial or nontrivial, can be estimated in float values with satisfactorily high accuracy and their integer values can hence be correctly obtained by simple rounding. To be more specific, all manually introduced nontrivial cycle slips are correctly repaired.

Estimation and analysis of the short-term variations of multi-GNSS receiver differential code biases using global ionosphere maps

Abstract: Care should be taken to minimize the adverse impact of differential code biases (DCBs) on global navigation satellite systems (GNSS)-derived ionospheric information determinations. For the sake of convenience, satellite and receiver DCB products provided by the International GNSS Service (IGS) are treated as constants over a period of 24 h (Li et al. (2014)). However, if DCB estimates show remarkable intra-day variability, the DCBs estimated as constants over 1-day period will partially account for ionospheric modeling error; in this case DCBs will be required to be estimated over shorter time period. Therefore, it is important to further gain insight into the short-term variation characteristics of receiver DCBs. In this contribution, the IGS combined global ionospheric maps and the German Aerospace Center (DLR)-provided satellite DCBs are used in the improved method to determine the multi-GNSS receiver DCBs with an hourly time resolution. The intra-day stability of the receiver DCBs is thereby analyzed in detail. Based on 1 month of data collected within the multi-GNSS experiment of the IGS, a good agreement within the receiver DCBs is found between the resulting receiver DCB estimates and multi-GNSS DCB products from the DLR at a level of 0.24 ns for GPS, 0.28 ns for GLONASS, 0.28 ns for BDS, and 0.30 ns for Galileo. Although most of the receiver DCBs are relatively stable over a 1-day period, large fluctuations (more than 9 ns between two consecutive hours) within the receiver DCBs can be found. We also demonstrate the impact of the significant short-term variations in receiver DCBs on the extraction of ionospheric total electron content (TEC), at a level of 12.96 TECu (TEC unit). Compared to daily receiver DCB estimates, the hourly DCB estimates obtained from this study can reflect the short-term variations of the DCB estimates more dedicatedly. The main conclusion is that preliminary analysis of characteristics of receiver DCB variations over short-term intervals should be finished prior to estimate daily multi-GNSS receiver DCB products.

An efficient solution of real-time data processing for multi-GNSS network

Abstract: Global navigation satellite systems (GNSS) are acting as an indispensable tool for geodetic research and global monitoring of the Earth, and they have been rapidly developed over the past few years with abundant GNSS networks, modern constellations, and significant improvement in mathematic models of data processing. However, due to the increasing number of satellites and stations, the computational efficiency becomes a key issue and it could hamper the further development of GNSS applications. In this contribution, this problem is overcome from the aspects of both dense linear algebra algorithms and GNSS processing strategy. First, in order to fully explore the power of modern microprocessors, the square root information filter solution based on the blocked QR factorization employing as many matrix–matrix operations as possible is introduced. In addition, the algorithm complexity of GNSS data processing is further decreased by centralizing the carrier-phase observations and ambiguity parameters, as well as performing the real-time ambiguity resolution and elimination. Based on the QR factorization of the simulated matrix, we can conclude that compared to unblocked QR factorization, the blocked QR factorization can greatly improve processing efficiency with a magnitude of nearly two orders on a personal computer with four 3.30 GHz cores. Then, with 82 globally distributed stations, the processing efficiency is further validated in multi-GNSS (GPS/BDS/Galileo) satellite clock estimation. The results suggest that it will take about 31.38 s per epoch for the unblocked method. While, without any loss of accuracy, it only takes 0.50 and 0.31 s for our new algorithm per epoch for float and fixed clock solutions, respectively.

A procedure for the significance testing of unmodeled errors in GNSS observations

Abstract: It is a crucial task to establish a precise mathematical model for global navigation satellite system (GNSS) observations in precise positioning. Due to the spatiotemporal complexity of, and limited knowledge on, systematic errors in GNSS observations, some residual systematic errors would inevitably remain even after corrected with empirical model and parameterization. These residual systematic errors are referred to as unmodeled errors. However, most of the existing studies mainly focus on handling the systematic errors that can be properly modeled and then simply ignore the unmodeled errors that may actually exist. To further improve the accuracy and reliability of GNSS applications, such unmodeled errors must be handled especially when they are significant. Therefore, a very first question is how to statistically validate the significance of unmodeled errors. In this research, we will propose a procedure to examine the significance of these unmodeled errors by the combined use of the hypothesis tests. With this testing procedure, three components of unmodeled errors, i.e., the nonstationary signal, stationary signal and white noise, are identified. The procedure is tested by using simulated data and real BeiDou datasets with varying error sources. The results show that the unmodeled errors can be discriminated by our procedure with approximately 90% confidence. The efficiency of the proposed procedure is further reassured by applying the time-domain Allan variance analysis and frequency-domain fast Fourier transform. In summary, the spatiotemporally correlated unmodeled errors are commonly existent in GNSS observations and mainly governed by the residual atmospheric biases and multipath. Their patterns may also be impacted by the receiver.

The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates

Abstract: We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional $${\sim }$$ 280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at $$1^{\prime \prime }\times 1^{\prime \prime }$$ resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to $${\sim }100$$ mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.

Least-squares cross-wavelet analysis and its applications in geophysical time series

Abstract: The least-squares wavelet analysis, an alternative to the classical wavelet analysis, was introduced in order to analyze unequally spaced and non-stationary time series exhibiting components with variable amplitude and frequency over time. There are a few methods such as cross-wavelet transform and wavelet coherence that can analyze two time series together. However, these methods cannot generally be used to analyze unequally spaced and non-stationary time series with associated covariance matrices that may have trends and/or datum shifts. A new method of analyzing two time series together, namely the least-squares cross-wavelet analysis, is developed and applied to study the disturbances in the gravitational gradients observed by GOCE satellite that arise from plasma flow in the ionosphere represented by Poynting flux. The proposed method also shows its outstanding performance on the Westford–Wettzell very long baseline interferometry baseline length and temperature series.

An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations

Abstract: 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.

Deriving time-series three-dimensional displacements of mining areas from a single-geometry InSAR dataset

Abstract: This paper presents a method for deriving time-series three-dimensional (3-D) displacements of mining areas from a single-geometry interferometric synthetic aperture radar (InSAR) dataset (hereafter referred to as the SGI-based method). This is mainly aimed at overcoming the limitation of the traditional multi-temporal InSAR techniques that require SAR data from at least three significantly different imaging geometries to fully retrieve time-series 3-D displacements of mining areas. The SGI-based method first generates the multi-temporal observations of the mining-induced vertical subsidence from the single-geometry InSAR data, using a previously developed method for retrieving 3-D mining-related displacements from a single InSAR pair (SIP). The weighted least-squares solutions of the time series of vertical subsidence are estimated from these generated multi-temporal observations of vertical subsidence. Finally, the time series of horizontal motions in the east and north directions are estimated using the proportional relationship between the horizontal motion and the subsidence gradient of the mining area, on the basis of the SGI-derived time series of vertical subsidence. Seven ascending ALOS PALSAR images from the Datong mining area of China were used to test the proposed SGI-based method. The results suggest that the SGI-based method is effective. The SGI-based method not only extends the SIP-based method to time-series 3-D displacement retrieval from a single-geometry InSAR dataset, but also limits the uncertainty propagation from InSAR-derived deformation to the estimated 3-D displacements.

Combination of GRACE monthly gravity field solutions from different processing strategies

Abstract: We combine the publicly available GRACE monthly gravity field time series to produce gravity fields with reduced systematic errors. We first compare the monthly gravity fields in the spatial domain in terms of signal and noise. Then, we combine the individual gravity fields with comparable signal content, but diverse noise characteristics. We test five different weighting schemes: equal weights, non-iterative coefficient-wise, order-wise, or field-wise weights, and iterative field-wise weights applying variance component estimation (VCE). The combined solutions are evaluated in terms of signal and noise in the spectral and spatial domains. Compared to the individual contributions, they in general show lower noise. In case the noise characteristics of the individual solutions differ significantly, the weighted means are less noisy, compared to the arithmetic mean: The non-seasonal variability over the oceans is reduced by up to 7.7% and the root mean square (RMS) of the residuals of mass change estimates within Antarctic drainage basins is reduced by 18.1% on average. The field-wise weighting schemes in general show better performance, compared to the order- or coefficient-wise weighting schemes. The combination of the full set of considered time series results in lower noise levels, compared to the combination of a subset consisting of the official GRACE Science Data System gravity fields only: The RMS of coefficient-wise anomalies is smaller by up to 22.4% and the non-seasonal variability over the oceans by 25.4%. This study was performed in the frame of the European Gravity Service for Improved Emergency Management (EGSIEM; http://www.egsiem.eu ) project. The gravity fields provided by the EGSIEM scientific combination service ( ftp://ftp.aiub.unibe.ch/EGSIEM/ ) are combined, based on the weights derived by VCE as described in this article.

Stochastic modeling for time series InSAR: with emphasis on atmospheric effects

Abstract: Despite the many applications of time series interferometric synthetic aperture radar (TS-InSAR) techniques in geophysical problems, error analysis and assessment have been largely overlooked. Tropospheric propagation error is still the dominant error source of InSAR observations. However, the spatiotemporal variation of atmospheric effects is seldom considered in the present standard TS-InSAR techniques, such as persistent scatterer interferometry and small baseline subset interferometry. The failure to consider the stochastic properties of atmospheric effects not only affects the accuracy of the estimators, but also makes it difficult to assess the uncertainty of the final geophysical results. To address this issue, this paper proposes a network-based variance–covariance estimation method to model the spatiotemporal variation of tropospheric signals, and to estimate the temporal variance–covariance matrix of TS-InSAR observations. The constructed stochastic model is then incorporated into the TS-InSAR estimators both for parameters (e.g., deformation velocity, topography residual) estimation and uncertainty assessment. It is an incremental and positive improvement to the traditional weighted least squares methods to solve the multitemporal InSAR time series. The performance of the proposed method is validated by using both simulated and real datasets.

Solution of the weighted symmetric similarity transformations based on quaternions

Abstract: A new method through Gauss–Helmert model of adjustment is presented for the solution of the similarity transformations, either 3D or 2D, in the frame of errors-in-variables (EIV) model. EIV model assumes that all the variables in the mathematical model are contaminated by random errors. Total least squares estimation technique may be used to solve the EIV model. Accounting for the heteroscedastic uncertainty both in the target and the source coordinates, that is the more common and general case in practice, leads to a more realistic estimation of the transformation parameters. The presented algorithm can handle the heteroscedastic transformation problems, i.e., positions of the both target and the source points may have full covariance matrices. Therefore, there is no limitation such as the isotropic or the homogenous accuracy for the reference point coordinates. The developed algorithm takes the advantage of the quaternion definition which uniquely represents a 3D rotation matrix. The transformation parameters: scale, translations, and the quaternion (so that the rotation matrix) along with their covariances, are iteratively estimated with rapid convergence. Moreover, prior least squares (LS) estimation of the unknown transformation parameters is not required to start the iterations. We also show that the developed method can also be used to estimate the 2D similarity transformation parameters by simply treating the problem as a 3D transformation problem with zero (0) values assigned for the z-components of both target and source points. The efficiency of the new algorithm is presented with the numerical examples and comparisons with the results of the previous studies which use the same data set. Simulation experiments for the evaluation and comparison of the proposed and the conventional weighted LS (WLS) method is also presented.