We extend our previous formulation of the gravitational potential of a homogeneous cuboid (right rectangular prism) by deriving uniform analytical expressions for the gravitational acceleration and the full gravitational tensor. The formulation is valid in the interior and exterior of the cuboid and also on its boundary. Classical rectangular–prism formulas express the potential and its derivatives through logarithmic and arctangent kernels whose arguments vanish at faces, edges, and vertices. Although these singularities are removable analytically, their direct numerical evaluation introduces instabilities near the boundaries of the body. We introduce a unified closed-form expressions based on oriented face–distance coordinates combined with analytically regularized sp_log and sp_arctan functions and an alternating vertex-sign summation. This representation produces continuous expressions for the gravitational potential, acceleration, and tensor that remain well defined at interior, boundary, and exterior field points without case distinctions or limiting procedures. Validation using acceleration-field analysis and an analytic Laplacian test confirms exact agreement with Poisson’s equation inside the cuboid and Laplace’s equation outside. The resulting expressions provide stable analytic first and second derivatives suitable for high-accuracy modeling of gravity and gravity-gradient measurements.
This study introduces SUMH25, a regionally refined model that estimates Moho Depth (MD) and Moho Density Contrast (MDC) across the Fennoscandia-Baltic region, incorporating localized geophysical complexity and uncertainty quantification. SUMH25 model covers the entire Fennoscandia-Baltic area, including both continental and adjacent oceanic zones, spanning latitudes from approximately 55°N to 70°N and longitudes from 5°E to 30°E. For the first time this study focuses on this tectonically diverse region by integrating multiple Moho models that provide either MD, MDC or both. These include gravimetric-isostatic models COMHV19, HVMD19 (both with MD, MDC and associated uncertainties), VMM25 and MDC21 model (offering refined MDC estimates), and seismic crustal models CRUST19 (MD only), CRUSRT1.0 (MD and MDC) and the recently introduced ECM1 (primarily MD with support for density refinement). All models are incorporated within a locally adaptive, correlation-aware uncertainty framework. The methodology applies a spatially variable weighted least-squares adjustment model of four MD and three MDC datasets, incorporating pixel-wise variance estimation, region-specific covariance structures, and explicit treatment of inter-model correlations. SUMH25 provides enhanced resolution and reduced uncertainty in both MD and MDC estimates, with standard errors calculated for each 1°×1° grid cell. The MD in continental crust ranges from 36 to 56.7 km, while oceanic crust spans from 11.4 to 35 km. MDC values range from 267 to 550 kg/m3 over the continental crust and 250–445 kg/m3 in oceanic crust. The model also provides detailed uncertainty estimates with MD standard errors typically below 6 km and MDC errors under 100 kg/m3. The new model captures regional variations in Moho geometry and lithospheric heterogeneity, offering improved insight into Fennoscandian tectonic evolution, including the structures of the Baltic Shield and the Caledonian orogen.
The second geodetic boundary value problem plays an important role in physical geodesy, as complex mountainous terrain significantly affects its solution, making accurate computation of topographic effects essential for generating a high-precision geoid. The traditional prism model formulation suffers from singularities at computation points and does not account for the Earth’s curvature. These limitations become more pronounced when high-resolution data are employed and in regions with complex terrain, thereby degrading the accuracy of the computed results. This study adopts the reference ellipsoid as the boundary surface and applies Helmert’s second condensation method to divide the pre-condensation topography into prisms, which are then transformed into rectangular thin plates after condensation. By transforming the computation points to eliminate the influence of Earth’s curvature, an analytical formulation for evaluating the topographic effects, based on a prism-thin plate model that explicitly accounts for the Earth’s curvature is proposed. A typical mountainous region in western China is selected as the study area. Comparisons between the proposed formulation and those of the prism-thin plate model without curvature consideration, the conventional prism model, and the curvature-corrected prism model show numerical consistency. All four models reveal that the direct topographic effects on gravity are predominantly governed by near-field contributions, while the indirect effects on height anomalies and geoid undulations are characterized by cumulative influences. The proposed formulation eliminates singularities while rigorously accounting for Earth’s curvature, establishing unified and numerically stable expressions for the direct and indirect effects under Helmert’s second condensation, thereby providing a robust theoretical and computational framework for high-precision gravity field modeling and quasigeoid refinement.
This study addresses the crucial detection of offsets in GNSS time series for accurate velocity estimations, vital in geosciences and engineering. Existing methods are challenged by correlated noise and can suffer from over-segmentation or high false positive rates. A novel methodology is proposed that combines a modified bilateral filter and an adapted implementation of the Pruned Exact Linear Time (PELT) algorithm, optimizing the detection of abrupt changes in the presence of complex noise. The technique was evaluated using a set of 3450 synthetic time series and 45 real time series from Global Navigation Satellite System (GNSS) stations located in Northern Europe. In the synthetic time series, the success threshold of 80 % was exceeded for the horizontal component with offsets of 1.2 mm, and the 90 % threshold for offsets of 1.4 mm. For the vertical component, 80 % success was achieved with offsets of 4 mm and 90 % with offsets of 5 mm. Regarding the real time-series, success percentages of 87.1 % were obtained for the East component, 74.4 % for the North component, and 74.2 % for the vertical component (Up). This hybrid technique enhances the identification of real offsets in GNSS time series, simplifying monitoring and paving the way for fully automated workflows.
Anomalous systematic position offsets have been noted in GNSS time series from a variety of locations around the globe. In many regions, these time series anomalies coincide with the presence of cloud streets in visible satellite imagery, implicating atmospheric lee waves as the source of the position offsets. We report here on seasonal anomalous position offsets from the San Bernardino Basin of southern California that occur during clear skies. Modeling of the troposphere on days with anomalous positions shows the presence of atmospheric lee waves. Ray tracing through these tropospheric models suggests that atmospheric lee waves create low elevation perturbations to microwave refractivity of as much as 10 %. These regions of perturbed refractivity reoccur in consistent locations over the San Bernardino Basin and, from the perspective of individual GNSS antenna, alter the refractivity to mimic spatial offset of the antenna phase center. This refractivity pattern is unlike the pattern estimated using linear and higher-order atmospheric gradients during GNSS position processing and results in apparent position offsets in the GNSS time series correlated with the orientation of the lee waves.
Air traffic control is vital in ensuring safe and efficient airspace usage by aircraft. In the event of a strike by air traffic controllers, flights can be delayed, cancelled, and rerouted. While there is previous research in the geospatial analysis of flight data, the approaches and tools used are complex and specialized, leaving a gap for an accessible and easily adaptable practice for the geospatial analysis of flight data, primarily using geographic information systems. We successfully bridged this gap with the methods of this article, increasing the accessibility of future geospatial flight data analysis. The flight data we used is broadcast by aircraft in regular intervals, including both spatial information and supplemental aircraft information. By detecting differences in the air traffic patterns between the day of an air traffic control strike in France and other reference days (used for normalization), we were able to determine both the spatial change of flight density in and around France and the difference in the number of flights that flew through French airspace on the day of the strike. On the day of the strike, the area around French airspace experienced a higher density of flights. In total, French airspace experienced around 2.7 %–7.1 % more air traffic on normal days than on the day of the strike. Mapping flight densities and systematically comparing them between the observed days highlighted corridors within French airspace that experienced significantly higher air traffic, presumably as a consequence of strike management, simplifying the routes that aircraft took in comparison to normal operations. The most prominent corridor consists of a line from Barcelona to western Italy, with another one spanning the distance between the greater Paris area and the border of French and Spanish airspace near the Atlantic coastline.
This article studies the radio frequency interference (RFI) source localization in a dynamic collaborative navigation scenario using carrier-to-noise density ratio (C/N0) measurements available from most of the low-cost commercial-off-the-shelf global navigation satellite system receivers. The work is a part of a robust SmartNav positioning engine that aims to provide highly accurate positioning using low cost sensors. Cases of jamming source localization using both a single node and multiple cooperating nodes have been considered. For the jammer localization using a single node, a synthetic array-based approach has been applied whose principle suggests that a receiver's output at different positions can be assumed as an output of sensor array elements. In the case of multiple cooperating nodes, the centralized approaches based on combining the measurements from all the nodes are presented. The performance of both approaches has been evaluated though a series of simulations, where dynamic wideband interference signal source has been considered. The article presents the implementation details and results achieved by each approach and discusses the impact of different factors influencing the efficacy of the considered techniques. While results indicate that these techniques are mainly useful for coarse localization of the source, this can still be sufficient for improving the situational awareness of the cooperating nodes enabling them to adopt for safer trajectories.
This article presents a large-scale surface gravity network in Colombia, developed during a 3-year collaboration between Ohio State University (OSU) and Colombia’s national mapping agency, the Instituto Geográfico Agustín Codazzi. The network spans approximately one-third of the country’s surface area and consists of 498 stations, including 22 absolute gravity (AG) constraints. We applied the OSU field protocol and adjustment technique, extending it to address the challenge of traffic-induced noise at benchmarks located near major roads. Our methodology employed measurements with an unusually high degree of redundancy, with 87% of the gravity lines surveyed using a minimum of four relative gravimeters. Additionally, we utilized both automatic (Scintrex CG-6) and manual (LaCoste and Romberg Model G) instruments to leverage their complementary strengths in different environments. The resulting gravity values were determined with a typical uncertainty (1 sigma) of ± 0.03 mGal. The leave-one-out cross-validation tests using the AG stations demonstrated the robustness of our solution, with all residuals statistically indistinguishable from zero. Our findings show that the combination of enhanced observational redundancy and the use of multiple gravimeter types effectively mitigates measurement challenges under suboptimal conditions, with only a modest increase in field time.
Surface displacement caused by natural and anthropogenic activities poses a significant risk to subsurface pipelines, particularly in areas experiencing subsidence. Stress and strain induced by deformation can lead to pipeline buckling and potential vulnerability. This study applies a multidisciplinary approach integrating interferometric synthetic aperture radar (InSAR) deformation analysis, well data, and geological context to assess the risk to oil and gas pipelines in Iran's Qazvin plain. The focus is on evaluating the impact of an unconfined aquifer, which has a lower risk of subsidence compared to a confined aquifer, on pipeline infrastructure. We analyzed multitemporal Sentinel-1 data collected between 2014 and 2021. Our findings reveal that 2,400 km2 of the study area is experiencing subsidence with vertical rates reaching up to 14 cm/year with an average S k {S}_{{\rm{k}}} , or skeletal storage value of 0.04. We explore the reasons for this unexpectedly high subsidence, finding that compressible unconfined aquifers with historically low water levels are also at risk for significant inelastic, or permanent, subsidence. We also evaluate pipeline profiles, which show spatial subsidence, much higher than typically seen in unconfined aquifers, with variations of up to 1 m, indicating long-term risks to the infrastructure. This study demonstrates the potential of InSAR techniques in assessing the risk to critical infrastructure, such as oil and gas pipelines, in regions with land deformation due to groundwater withdrawal. Our approach underscores the importance of continuous monitoring and offers valuable insights for addressing the challenges posed by subsidence on pipeline infrastructure.
We have, using altimetry observations, investigated the nature and magnitude of variations in the local mean dynamic topography (MDT) along the Norwegian coast. Following our findings, we argue that the MDT is well approximated by the nearest observation and demonstrate how the results can be improved using variogram based kriging. Cross validation at tide gauges along the coast confirms the applicability of the approach and, therefore, gives us some confidence in its ability to weigh altimetry and tide gauge observations appropriately. Accordingly, altimetry is a potentially very important source of information on mean sea level outside the Norwegian baseline.
The Nordic and Baltic Sea region is topographically varied, making gravity potential field modelling in the region a challenge. With new gravity data in and around the Baltic Sea, this study investigates improvements in the accuracy of quasigeoid determination in the region. The quasigeoid determination method is the remove-compute-restore using a spherical fast Fourier transform to solve Molodensky's integral, using a Wong-Gore kernel modification. Different kernel modification ranges of the Stokes' kernel are tested. The gravimetric quasigeoids are compared to global navigation satellite systems (GNSS)-levelling datasets in the region to assess the accuracy of the models. The best fitting model has a normalised root mean square error (RMSE) of the residuals to GNSS-levelling height anomalies for a subregion of below 2 cm, while the normalised RMSE to GNSS-levelling height anomalies in the whole region is 2.8 cm, matching the official regional gravimetric Nordic geodetic commission quasigeoid from 2015. A simple planar tilt correction of the residuals improves the fit in Denmark and Latvia. The correction indicates systematic tilts in the national GNSS-levelling networks of said countries. The investigation of different kernel modification ranges shows that the long transition intervals are less sensitive to the choice of bounding degrees, compared to short intervals, and more robustly give a better fit to the GNSS-levelling data in the region.
This article investigates the strain and stress tensors of the Afar region using the integration of global navigation satellite system (GNSS) and persistent scatterer interferometric synthetic aperture radar (PSInSAR) data. We first estimated the three-dimensional displacement field of the Afar area using combined GPS and PSInSAR data for the period 2006-2009. Secondly, the variance component estimation is applied to the heterogeneous GNSS and PSInSAR data and the displacement field is computed. Then, the behaviour of crustal movement within the study area is investigated by dividing the region into several small triangles or tetrahedrons using the Delaunay triangulation and applying the finite element method (FEM) to estimate the strain and stress tensors for each tetrahedron. The average stress tensor elements tau x x , tau y y {\tau }_{xx},\hspace{.25em}{\tau }_{yy} , and tau z z {\tau }_{zz} of the study area are 0.067, 0.121, and 0.126 MPa in the North, East, and Up coordinates, respectively. The result also shows that the nearby post-diking deformation has been followed by the Afar Tectono-Magmatic Event that happened in September 2005. Graphical abstract
Ionospheric electron density (IED) and vertical total electron content (VTEC) are two of the most important characteristics that interpret the ionosphere. Today, satellite geodesy plays an important role in providing these data. Apart from monitoring the ionosphere by means of Global Navigation Satellite System observations, predictions of these ionospheric characteristics have also been taken into consideration. The majority of prediction methods in the ionosphere are focused on VTEC prediction. The continuity equation in the ionosphere is a spatiotemporal partial differential equation that can be used to predict both IED and VTEC. In this study, we address this issue during May 8, 2016, which was a day of high geomagnetic activity. For this purpose, first, high-accuracy IED grids with a spatial resolution of 0.5 degrees x 0.5 degrees in longitude and latitude, 50 km in altitude, and a temporal resolution of 10 min are prepared. These IED grids are called analysis grids and are derived by assimilation of GPS-derived VTECs into the background IED grids provided by international reference ionosphere. Second, the analysis grids can be utilized as the initial and boundary values in the continuity equation to predict IED for the next 3 h with a step of 10 min. The analysis grids are constructed over the Iran region, and the performance of the continuity equation in the ionosphere prediction is investigated during the strongest geomagnetic storm of 2016. Finally, the accuracy of the predictions has been evaluated in two ways. First, analysis grids with high accuracy are available for each epoch in which the prediction is presented. On average, the root mean square (RMS) of the differences between the predicted IEDs and the analysis IEDs is 1.66 x 1010 el/m3 for 1-h predictions and 5.33 x 1010 el/m3 for 3-h predictions. Second, by numerical integration of the predicted grids in the vertical direction, VTECs are estimated and compared with VTEC values of test data that were already known. On average, the RMS of their difference was 1.29 total electron content unit (TECU) for 1-h predictions and 2.57 TECU for 3-h predictions. An acceptable accuracy for both IED and VTEC prediction by a continuity equation has been obtained up to 3 h on the most active ionospheric day. Graphical abstract
In America, height systems were established during the twentieth century by connecting the leveling network to a tide gauge. In 1997, the Geodetic Reference System for the Americas (former Geocentric Reference System for South America) created Working Group III, called Vertical Datum, to assist in establishing a unified vertical reference system for the Americas. In this context, this article aims to trace the evolution of height systems in America, especially South America, and present the current situation in the region to contribute to the International Height Reference Frame. A brief chronological description of the systems of South American countries and the efforts developed by Working Group III over the past 25 years is presented. A bibliometric study assesses the scientific community’s involvement in this field. The results revealed that some countries have been standing out in engagement in recent years. Finally, an evaluation is performed between recent global geopotential models and a regional gravity field model at the 17 stations that comprise the International Height Reference Frame in Latin America. The results indicate that the Andes region requires efforts in gravity densification and improvement of global models.
Airborne mobile mapping systems are crucial in various geodetic applications. A key aspect of these systems is the accurate estimation of exterior orientation parameters (EOPs), which is achieved through the integration of global navigation satellite systems (GNSSs) and inertial measurement unit (IMU) technologies. One critical component in this integration is the lever arm (LA), the vector that connects the GNSS antenna and the IMU center. The uncertainty (standard deviation) in LA measurements can introduce errors in the EOP estimation, thereby affecting the overall system performance. However, how much the EOP estimation is affected by LA measurement uncertainty is examined in this study based on calibration data (test flight) using the TerrainMapper 2 system collected by Lantm & auml;teriet in Sweden. The findings reveal that LA uncertainties have minimal influence on attitude and negligible impacts on position in terms of standard deviation (SD) if the LA is measured with an accuracy of less than 2-3 cm. Additionally, the research explores the combined effects of virtual reference station-rover baseline length and dilution of precision on positioning accuracy and their correlation with LA uncertainty, providing further insights into the complexities of EOP estimation. By advancing GNSS/IMU integration techniques, this study contributes to the enhancement of geodetic technologies customized for airborne mobile mapping applications. Graphical abstract
Topography is a problem in geoid determination by the Stokes formula, a high degree Earth Gravitational Model (EGM), or for a combination thereof. Herein, we consider this problem in analytical/harmonic downward continuation of the external potential at point P to a geocentric spherical sea level approximation in geoid determination as well as to a sphere through the footpoint at the topography of the normal through P. Decomposing the topographic bias into a Bouguer shell component and a terrain component, we derive these components. It is shown that there is no terrain bias outside a spherical dome of base radius equal to the height H P of P above the sphere, and the height of the dome is about 0.4 x H P . In the case of dealing with an EGM, utilizing Molodensky truncation coefficients is one way to cope with the bias.
This research evaluates the use of the iPad Pro 11 with LiDAR sensor (LS) for geospatial surveying in Ecuador's artisanal and small-scale mining (ASM) sector, a key contributor to the national economy; however, it is hindered by outdated equipment and environmental challenges. It was conducted at "La Zamorana," an underground gold mine. The study compares the efficacy of iPad LS against traditional surveying methods across five phases, including control point establishment, traditional and LiDAR surveying, data postprocessing, and an electronic survey assessing ASM's technological readiness. The findings indicate that the iPad LS outperforms traditional methods in detail and accuracy, particularly in elevation measurements, with most discrepancies under 5 cm. The precision of the LiDAR methodology is highlighted by the closer alignment of points to control points than traditional surveying methods. In summary, the iPad Pro 11 with LS shows promise as an affordable ASM tool for geospatial surveying in ASM in Ecuador. The research emphasizes how receptive the industry is to new technology developments. However, further research is recommended to explore the technology's effectiveness in diverse mining environments, ensuring its comprehensive applicability in ASM.
In this article, we address the potential effects of construction activities around a permanent Global Navigation Satellite System (GNSS) station. The study is based on a consistent time series of GNSS observations collected over a span of almost 13 years and comprises an analysis in both observation and coordinate domains. The analysis in the observation domain revealed a significant increase of the code multipath for the C1C signal of GPS and GLONASS. In terms of standard deviations the values increased by 37 and 31%, respectively. No significant increase was observed for other signals. Interestingly, the analysis showed that the Galileo E5 AltBOC signal outperformed the other signals in distinguishing direct from indirect signals. In the coordinate domain, the relative GNSS solution results demonstrated an increase in root mean square error for the eastern component during the same period. Noteworthy, the observed increase in both domains coincided with the period of the highest level of construction activity around the GNSS station. These findings suggest that the construction activity near the station impacted the quality of GNSS observations and that the Galileo E5 AltBOC signal was better at compensating for local changes than other signals.
This study investigates the impact of observation session duration and station latitude on the precision of Precise Point Positioning (PPP). Global Positioning System-only Receiver Independent Exchange files from 516 continuous stations spanning latitudes from 90°N to 90°S across the Americas (30°–130°W) were binned into sessions of 1, 2, 3, 4, 6, and 12 h and processed using the GPSPACE PPP software. These sub-daily solutions, along with the original 24 h ones, were compared against a reference coordinate derived from an extended linear trajectory model for each station. Analysis of the results, considering both accuracy and precision, was conducted across the entire latitude range of the study. We found that the precision of a PPP solution approximately follows a power-law relationship with observation duration. Parameters for these power-law relationships were determined for all latitude ranges that allow users to predict a result’s uncertainty as a function of session length. Findings indicate that longer observation sessions lead to reduced positioning errors, with vertical scatter decreasing with increasing latitude. Since all these stations are characterized by good to excellent sky view, our power-law rule-of-thumb provides a lower bound on the occupation time needed to achieve target positioning precision at locations with poorer sky visibility.
