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.
