Journal of Geodesy

For precise orbit determination (POD) and precise applications with POD products, one of the critical issues is the modeling of non-conservative forces acting on satellites. Since the official publication of Galileo satellite metadata in 2017, analytical models including the box-wing model and thermal thrust models have been established to absorb a substantial amount of solar radiation pressure (SRP) and thermal thrust. These models serve as the foundation for the best overall modeling approach, combining the analytical box-wing model and thermal thrust model with parameterization of the remaining non-conservative perturbing forces using various optimized Empirical CODE Orbit Models (ECOMs) of the Center for Orbit Determination in Europe (CODE). Firstly, we have demonstrated the significance of the second-order signals in the D direction and the first-order signals in the B direction through spectral analyses of the pure box-wing model, which are consistent with the currently recommended 7-parameter Empirical CODE Orbit Model 2 (ECOM2). In spite of this, we still found that degradation in orbit accuracy frequently occurs during deep eclipse seasons when using the ECOM2 model. We confirm a high-frequency signal existing in the fluctuating orbit overlap differences through the spectral analysis. Considering this, the ECOM2 force model should be extended to higher order and adapted to absorb the remaining effects of potential perturbing forces. After extending the ECOM2 force model to the sixth order in the Sun direction, we demonstrated the significance of fourth- and sixth-order sine terms for deep eclipses. Due to the higher-order periodic terms, the averaged RMS values of orbit overlap difference over deep eclipses can be reduced from 5.3, 10.8, and 23.8 cm to 3.2, 3.9, and 9.9 cm for in-orbit validation (IOV) satellites, from 5.0, 8.6, and 17.7 cm to 3.0, 3.0, and 7.1 cm for the first generation of full operational capability (FOC-1) satellites, and from 5.4, 8.6, and 19.0 cm to 3.6, 3.6, and 7.4 cm for the second generation of FOC (FOC-2) satellites, in the radial, cross-track, and along-track directions, respectively. Fluctuations with a peak amplitude of approximately 0.4 nm/s2 in the bias in the solar panel axis (Y) direction (Y-bias) are effectively mitigated by the higher-order terms. Due to the higher-order terms, the vertical positioning errors during kinematic precise point positioning (PPP) convergence can be improved from 42.3 to 37.1 cm at the 95.5% confidence level. Meanwhile, a low correlation level of up to 0.02 is found between the newly introduced higher-order parameters and earth rotation parameters (ERPs).

Carrier phase integer ambiguities must be determined for BDS-3/inertial navigation system (INS) tightly coupled (TC) integration to achieve centimetre-level positioning accuracy. However, cycle slip breaks the consistency of the integer ambiguities. Conventional multi-frequency cycle slip methods use the pseudorange; thus, requiring improvement when applied to kinematic situations. Furthermore, a concise and nonprior information-dependent model is crucial for real-time processing. In this study, an inertial-aided BDS-3 cycle slip detection and repair (I-CDR) method was developed. First, a BDS-3/INS TC model with I-CDR was created. The ionospheric delays were modelled as part of the TC states; therefore, they could be estimated and eliminated. Investigations were conducted on the effects of carrier phase noise, residual ionosphere delay, and INS-predicted position error on combined cycle slip detection (CCD) accuracy. The optimal CCDs under various frequency available configurations were determined. The effectiveness of I-CDR was demonstrated using land vehicle test data. The false alarm ratio was less than 1.0%, and the missed detection ratio was almost zero even in situations with challenging abundant 1-cycle slips in random epochs. Furthermore, the right determination ratio reached 100%. In addition, BDS-3 signal loss-recovery cases were simulated, and all cycle slips for all satellites could be repaired within 40s. I-CDR exhibits outstanding cycle slip detection and repair performance for dense 1-cycle slip and signal loss-recovery cases, demonstrating its suitability for BDS-3/INS TC integration.

This paper focuses on the retrieval of refractivity fields from GNSS measurements by means of least-squares collocation. Collocation adjustment estimates parameters that relate delays and refractivity without relying on a grid. It contains functional and stochastic models that define the characteristics of the retrieved refractivity fields. This work aims at emphasizing the capabilities and limitations of the collocation method in modeling refractivity and to present it as a valuable alternative to GNSS tomography. Initially, we analyze the stochastic models in collocation and compare the theoretical errors of collocation with those of tomography. We emphasize the low variability of collocation formal variances/covariances compared to tomography and its lower dependence on a-priori fields. Then, based on real and simulated data, we investigate the importance of station resolution and station heights for collocation. Increasing the network resolution, for example, from 10 to 2 km, results in improved a-posteriori statistics, including a 10% reduction in the error statistic for the retrieved refractivity up to 6 km. In addition, using additional stations at higher altitudes has an impact on the retrieved refractivity fields of about 1 ppm in terms of standard deviation up to 6 km, and a bias reduction of more than 3 ppm up to 3 km. Furthermore, we compare refractivity fields retrieved through tomography and collocation, where data of the COSMO weather model are utilized in a closed-loop validation mode to simulate tropospheric delays and validate the retrieved profiles. While tomography estimates are less biased, collocation captures relative changes in refractivity more effectively among the voxels within one height level. Finally, we apply tomography and collocation to test their capabilities to detect an approaching weather front. Both methods can sense the weather front, but their atmospheric structures appear more similar when the GNSS network has a well-distributed height coverage.

Deformation phenomena on Earth are inherently three dimensional. With SAR interferometry (InSAR), in many practical situations the maximum number of observations is two (ascending and descending), resulting in an infinite number of possible displacement estimates. Here we propose a practical solution to this underdeterminancy problem in the form of the strapdown approach. With the strapdown approach, it is possible to obtain “3D-global/2D-local” solutions, by using minimal and largely undisputed contextual information, on the expected driving mechanisms and/or spatial geometry. It is a generic method that defines a local reference system with transversal, longitudinal, and normal (TLN) axes, with displacement occurring in the transversal-normal plane only. Since the orientation of the local frame is based on the physics of the problem at hand, the strapdown approach gives physically more relevant estimates compared to conventional approaches. Moreover, using an a-priori uncertainty approximation on the orientation of the local frame it is possible to assess the precision of the final estimates. As a result, appropriate cartographic visualization using a vector map with confidence ellipses enables an improved interpretation of the results.

Accurate information on satellite antenna phase center offsets (PCOs) and phase variations (PVs) is indispensable for high-precision geodetic applications. In the absence of consistent pre-flight calibrations, satellite antenna PCOs and PVs of global navigation satellite systems are commonly estimated based on observations from a global network, constraining the scale to a given reference frame. As part of this estimation, flatness and zero-mean conditions need to be applied to unambiguously separate PCOs, PVs, and constant phase ambiguities. Within this study, we analytically investigate the impact of different boresight-angle-dependent weighting functions for PV minimization, and we compare antenna models generated with different observation-based weighting schemes with those based on uniform weighting. For the case of the GPS IIR/-M and III satellites, systematic differences of 10 mm in the PVs and 65 cm in the corresponding PCOs are identified. In addition, new antenna models for the different blocks of BeiDou-3 satellites in medium Earth orbit are derived using different processing schemes. As a drawback of traditional approaches estimating PCOs and PVs consecutively in distinct steps, it is shown that different, albeit self-consistent, PCO/PV pairs may result depending on whether PCOs or PVs are estimated first. This apparent discrepancy can be attributed to potentially inconsistent weighting functions in the individual processing steps. Use of a single-step process is therefore proposed, in which a dedicated constraint for PCO-PV separation is applied in the solution of the normal equations. Finally, the impact of neglecting phase patterns in precise point positioning applications is investigated. In addition to an overall increase of the position scatter, the occurrence of systematic height biases is illustrated. While observation-based weighting in the pattern estimation can help to avoid such biases, the possible benefit depends critically on the specific elevation-dependent weighting applied in the user’s positioning model. As such, the practical advantage of such antenna models would remain limited, and uniform weighting is recommended as a lean and transparent approach for the pattern estimation of satellite antenna models from observations.

An attractive feature of PPP-RTK is the possibility of reducing the amount of data that needs to be transferred to users. By leveraging the state-space Representation (SSR) of the corrections, the correction provider (i.e., a GNSS network) can consider distinct transfer rates for each of the individual corrections according to their temporal characteristics. Reducing the transfer rates comes at the cost of delivering time-delayed corrections, urging the user to time predict the corrections to bridge the gap between the corrections’ generation time and the positioning time. Consequently, the user Kalman filter needs to be equipped with a strategy to account for the errors caused by such predictions, minimizing the precision loss of the user parameter solutions. In this contribution, we apply a processing strategy for both the network and user filters to handle the latency of corrections. This enables the network to update corrections over longer time-intervals. To have the strategy applicable to regional networks, an ionosphere-weighted model is adopted for the corresponding observations, delivering minimum-variance spatially predicted ionospheric corrections to users. It is shown that certain components of the network filter’s dynamic model are duplicated and should be excluded from processing. To illustrate the performance of the strategy at work, three globally distributed regional networks are employed, and maximum correction latencies to meet different positioning criteria are evaluated. In terms of both the positioning precision and time-to-first-fix (TTFF), the strategy is numerically shown to outperform the user processing case in which the uncertainty of corrections is discarded.

We propose a benchmark for comparing gap-filling techniques used on global time-variable gravity field time-series. The Gravity Recovery and Climate Experiment (GRACE) and the GRACE Follow-On missions provide products to study the Earth’s time-variable gravity field. However, the presence of missing months in the measurements poses challenges for understanding specific Earth processes through the gravity field. We reproduce, adapt, and compare satellite-monitoring and interpolation techniques for filling these missing months in GRACE and GRACE Follow-On products on a global scale. Satellite-monitoring techniques utilize solutions from Swarm and satellite laser ranging, while interpolation techniques rely on GRACE and/or Swarm solutions. We assess a wide range of interpolation techniques, including least-squares fitting, principal component analysis, singular spectrum analysis, multichannel singular spectrum analysis, auto-regressive models, and the incorporation of prior data in these techniques. To inter-compare these techniques, we employ a remove-and-restore approach, removing existing GRACE products and predicting missing months using interpolation techniques. We provide detailed comparisons of the techniques and discuss their strengths and limitations. The auto-regressive interpolation technique delivers the best score according to our evaluation metric. The interpolation based on a least-squares fitting of constant, trend, annual, and semi-annual cycles offers a simple and effective prediction with a good score. Through this assessment, we establish a starting benchmark for gap-filling techniques in Earth’s time-variable gravity field analysis.

Inverting fault geometry and slip distribution simultaneously with geodetic observations based on Bayesian theory is becoming increasingly prevalent. A widely used approach, proposed by (Fukuda and Johnson, Geophys J Int 181:1441–1458, 2010) (F-J method), employs the least-squares method to solve the linear parameters of slip distribution after sampling the nonlinear parameters, including fault geometry, data weights and smoothing factor. Here, we present a modified version of the F-J method (MF-J method), which treats data weights and the smoothing factor as hyperparameters not directly linked to surface deformation. Additionally, we introduce the variance component estimation (VCE) method to resolve these hyperparameters. To validate the effectiveness of the MF-J method, we conducted inversion tests using both synthetic data and a real earthquake case. In our comparison of the MF-J and F-J methods using synthetic experiments, we found that the F-J method's inversion results for fault geometry were highly sensitive to the initial values and step sizes of hyperparameters, whereas the MF-J method exhibited greater robustness and stability. The MF-J method also exhibited a higher and more stable acceptance rate, enabling convergence to simulated values and ensuring greater accuracy of the parameter estimation. Furthermore, treating the fault length and width as unknown parameters and solving them simultaneously with other fault geometry parameters and hyperparameters using the MF-J method successfully resolved the issue of non-uniqueness in fault location solutions caused by the excessively large no-slip areas. In the 2017 Mw 7.3 Sarpol-e Zahab earthquake case study, the MF-J method produced a fault slip distribution with low uncertainty that accurately fit surface observation data, aligning with results from other research institutions. This validated the method's applicability and robustness in real-world scenarios. Additionally, we inferred that the second slip asperity was caused by early afterslip.

The 3D ionosphere structure is of interest in many fields such as radio frequency communication and global navigation satellite system (GNSS) applications. However, the limited temporal and spatial coverage of measurements poses a challenge for 3D electron density modeling. To overcome this challenge, we explore the use of kriging interpolation technique. The kriging interpolation is performed to obtain 3D representation of the ionosphere over electron density measurements retrieved by GNSS radio-occultation (RO) data. RO measurements are first reduced to “shape function,” the ratio of electron density to vertical total electron content (VTEC), aiming to create a background model. Then, the empirical residual semivariogram is analyzed for variation characteristics of the shape functions under different solar geomagnetic conditions. Finally, 3D kriging is adopted for shape function interpolation. Compared to the modeling results without kriging, the maximum root mean square error (RMSE) reduction reaches \(3.4\times {10}^{-4}~\text {km}^{-1}\) , which amounts to \(3.4\times {10}^{11}~\text {el/m}^{3}\) of electron density when VTEC is assumed as 100 TECU. This improvement accounts for 17.8% of root mean square (RMS) of shape function.

The spherical radial basis function (SRBF) approach, widely used in gravity modeling, is theoretically surveyed from a viewpoint of random field theory. Let the gravity potential be a random field which is represented as an integral functional of another random field, namely an isotropic Gaussian random field (IGRF) on a sphere inside the Bjerhammar sphere with the SRBF as the integral kernel. When the integration is approximated by a discrete sum within a local region, one gets the widely applicable SRBF model. With this theoretical study, the following two findings are made. First, the IGRF implies a Gaussian prior on the spherical harmonic coefficients (SHCs) of the gravity potential; for this prior the SHCs are independent with each other and their variances are degree-only dependent. This should be reminiscent of two well-known priors, namely the power-law Kaula’s rule and the asymptotic power-law Tscherning-Rapp model. Second, the IGRF-SRBF representation is non-unique. Benefiting from this redundant representation, one can employ a simple IGRF, e.g., the simplest white field, and then design the SRBF accordingly to represent a potential with desired prior statistical properties. This can simplify the corresponding SRBF modeling significantly; to be more specific, the regularization matrix in parameter estimation of the SRBF modeling can be chosen to be a diagonal matrix, or even the naïve identity matrix.

In recent years, coseismic velocity from high-rate global navigation satellite systems (GNSS) carrier phase data has been widely utilized to estimate instrumental seismic intensity, thereby guiding earthquake early warning and emergency response. However, using carrier phase data only yields displacement, displacement increment, and average velocity but not instantaneous velocity at the epoch level. In large earthquakes, using average velocity over a brief time span (e.g., 1 s) to quantify instantaneous coseismic velocity is less reliable for recovering accurate deformation dynamics, especially for the near-field region. In this study, we first introduce GNSS raw Doppler-based instantaneous velocity into seismology, expanding carrier phase-based traditional GNSS seismology. We also propose a new integrated GNSS velocity estimation method that employs a Kalman filter to integrate raw Doppler-based instantaneous velocity and carrier phase-based average velocity. The GNSS data from shake table experiments and two real-world earthquake events (i.e., the 2016 Mw 6.6 Norcia earthquake and the 2011 Mw 9.1 Tohoku-oki earthquake) are used to investigate the impact of high-rate GNSS raw Doppler on capturing coseismic velocity waveforms and predicting instrumental seismic intensity. The simulated sine wave experiment results indicate that the accuracy of instantaneous and average velocity for the 1 Hz sampling rate case is 1.20 cm/s and 12.67 cm/s, respectively. A similar case holds for the simulated quake wave experiment. The retrospective analysis of the ultra-high-rate (20 Hz) GNSS data for the Norcia earthquake shows the average velocities exhibit more aliasing and have a smaller peak ground velocity value than instantaneous velocities in all cases (i.e., 1, 2, 4, 5, 10, and 20 Hz). For the 2011 Mw 9.1 Tohoku-oki earthquake, results show that incorporating raw Doppler data enhances the consistency between the GNSS intensity map and the United States Geological Survey intensity map for near-field regions. Therefore, high-rate GNSS RD data as it becomes more widely available should be incorporated into data processing of high-rate GNSS seismology to capture more accurate instantaneous coseismic velocity waveforms and predict more realistic instrumental seismic intensity in future analyses.

Global Navigation Satellite Systems (GNSS) applications require computation of the geometric range between the satellite vehicle at the time-of-signal transmission and the receiver antenna location at the time-of-signal reception. This computation requires attention to the frames of reference due to the rotation of the Earth-Centered Earth-Fixed (ECEF) frame during the time-of-signal propagation. Three range computation approaches are commonplace and will be discussed herein. The first is the Global Positioning System Interface Control Document recommendation to rotate the ECEF frames to a common reference time. The other two are forms of the Sagnac correction. The Sagnac derivations already in the literature are either limited to stationary receivers or lack the connection between the Earth-centered inertial (ECI) and ECEF frames. Neither form of the Sagnac correction exactly reproduces the geometric range. They are approximations. The literature does not currently contain an analysis of the error involved in using either form of the Sagnac correction. This article makes two contributions: (1) it presents derivations for both forms of the Sagnac correction that are valid for moving receivers and that maintain the connection between the ECI and ECEF frames; and (2) it analyzes the error of the Sagnac correction for orbits of different radius. The analysis shows that Sagnac corrections introduce range errors less than \(7.57\times 10^{-4}\) meters for GNSS satellites at medium Earth orbit.