Search Results (110)

A long-wavelength geoidal geometry reflects mainly lateral density variations in the Earth’s mantle, with the most pronounced features of the Indian Ocean Geoid Low and the West Pacific and North Atlantic Geoid Highs. Despite this spatial pattern being clearly manifested in the global geoidal geometry determined from gravity-dedicated satellite missions, the gravitational signature of the deep mantle could be refined by modelling and subsequently removing the gravitational contribution of lithospheric geometry and density structure. Nonetheless, the expected large uncertainties in available lithospheric density models (CRUST1.0, LITHO1.0) limit, to some extent, the possibility of realistically reproducing the gravitational signature of the deep mantle. To address this issue, we inspect an alternative approach. Realizing that the gravity geopotential field (i.e., gravity potential) is smoother than its gradient (i.e., gravity), we apply the integral operator to geopotential and then investigate the spatial pattern of this functional (i.e., radially integrated geopotential). Results show that this mathematical operation enhances a long-wavelength signature of the deep mantle by filtering out the gravitational contribution of the lithosphere. This finding is explained by the fact that in the definition of this functional, spherical harmonics of geopotential are scaled by the factor 1/n (where n is the degree of spherical harmonics), thus lessening the contribution of higher-degree spherical harmonics in the radially integrated geopotential. We also demonstrate that further enhancement of the mantle signature in this functional could be achieved based on modelling and subsequent removal of the gravitational contribution of lithospheric geometry and density structure. Full article

This study examines the influence of gravity on different height systems by integrating Global Navigation Satellite Systems (GNSS), leveling, and gravimetric measurements. Although the theoretical influence of gravity on height systems is well known, empirical studies that quantify these effects along steep terrain are rare—particularly within the Croatian reference systems. Geometric leveling, recognized for its precision in geodesy, was employed alongside gravimetric data to analyze the relationship between gravity variations and height differences. The research was conducted along Sljeme Road on Mt. Medvednica, Croatia, where altitude-dependent gravity effects were systematically investigated along an elevation profile with a height difference of about 650 m. GNSS measurements provided positional coordinates referenced to the Croatian Terrestrial Reference System 1996 (HTRS96) (EPSG:4888), while leveling and gravimetric data were analyzed within the Croatian Height Reference System 1971 (HVRS71) (EPSG:5610) and Croatian Gravimetric Reference System 2003 (HGRS03), respectively. The results demonstrate that differences between geometric and normal–orthometric heights become more pronounced at higher elevations but remain at the millimeter level. Notably, the impact of gravity is evident in normal and orthometric heights, with differences from geometric heights reaching up to 3.7 cm at the highest points. Additionally, a comparison between normal and orthometric heights reveals that at the beginning of the leveling line, the difference is around 4 mm. However, as the elevation increases, this difference grows, reaching over 1 cm at the end of the leveling line. The study also confirms the theoretical correlation between the geoid–quasigeoid height difference and terrain elevation, with increasing differences observed at higher altitudes. To examine the consistency of different height determination methods, two approaches were applied: one based on adjustment within the geopotential system, and the other involving direct adjustment in the desired height system, with specific height corrections applied. The results confirmed that the height differences between the two methods were 0, to the tenth of a millimeter, indicating that both methods provided identical results. These findings contribute to a deeper understanding of geodetic height systems and the role of gravity in height determination. Full article

Gravity anomalies play critical roles in geological analysis, geodynamic monitoring, and precise geoid modeling. Obtaining accurate gravity data is challenging, particularly in inaccessible or sparsely covered regions. This study evaluates machine learning (ML) methods—Support Vector Regression (SVR), Gaussian Process Regression (GPR), and Ensemble of Trees—for predicting gravity anomalies in southeastern Kazakhstan and compares their effectiveness with traditional Kriging interpolation. A dataset, consisting of the simple Bouguer anomaly values, latitude, longitude, elevation, normal gravity, and terrain corrections derived from historical maps at a scale of 1:200,000, was utilized. Models were trained and validated using cross-validation techniques, with performance assessed by statistical metrics (RMSE, MAE, R²) and spatial error analysis. Results indicated that the Exponential GPR model demonstrated the highest predictive accuracy, outperforming other ML methods, with 72.9% of predictions having errors below 1 mGal. Kriging showed comparable accuracy and superior robustness against extreme errors. Most prediction errors from all methods were spatially associated with mountainous regions featuring significant elevation changes. While this study demonstrated the effectiveness of machine learning methods for gravity anomaly prediction, their accuracy decreases in complex terrain, indicating the need for further research to improve model performance in such environments. Full article

The Surface Water and Ocean Topography (SWOT) satellite delivers an unprecedented spatial resolution, offering new opportunities for advanced marine gravity field modeling. This study investigates the application of SWOT observational data by computing deflections of the vertical (DOVs) using the eight-directional geoid gradient method, followed by gravity field inversion through the inverse Vening–Meinesz (IVM) formula. Experimental results in the South China Sea region demonstrate that SWOT DOVs, based on 19 observation cycles, achieved accuracies of 0.86 arcseconds for the east–west component $\eta$ and 0.77 arcseconds for the north–south component $\xi$. The marine gravity field inversion accuracy reached 4.97 mGal, comparable to the multi-source altimetry-derived model SIO_v32.1. Further analysis reveals that the primary contributions of SWOT DOVs are observed within the 3.5–20 km wavelength band, with cross-track systematic errors identified as the key factor influencing both DOV calculations and gravity anomaly inversion. Additionally, extending the SWOT observation period enhances DOV accuracy, particularly for the $\eta$. These findings highlight the potential of SWOT data in advancing high-resolution marine gravity field modeling. Full article

Lakes are a crucial component of inland water bodies, and changes in their water levels serve as key indicators of global climate change. Traditional methods of lake water level monitoring rely heavily on hydrological stations, but there are problems such as regional representativeness, data stability, and high maintenance costs. The satellite altimeter is an essential tool in lake research, with the Synthetic Aperture Radar (SAR) altimeter offering a high spatial resolution. This enables precise and quantitative observations of lake water levels on a large scale. In this study, we used Sentinel-3A SAR Radar Altimeter (SRAL) data to establish a more reasonable lake height inversion algorithm for satellite-derived lake heights. Subsequently, using this technology, a systematic analysis study was conducted with Qinghai Lake as the case study area. By employing regional filtering, threshold filtering, and altimeter range filtering techniques, we obtained effective satellite altimeter height measurements of the lake surface height. To enhance the accuracy of the data, we combined these measurements with GPS buoy-based geoid data from Qinghai Lake, normalizing lake surface height data from different periods and locations to a fixed reference point. A dataset based on SAR altimeter data was then constructed to track lake surface height changes in Qinghai Lake. Using data from the Sentinel-3A altimeter’s 067 pass over Qinghai Lake, which has spanned 96 cycles since its launch in 2016, we analyzed over seven years of lake surface height variations. The results show that the lake surface height exhibits distinct seasonal patterns, peaking in September and October and reaching its lowest levels in April and May. From 2016 to 2023, Qinghai Lake showed a general upward trend, with an increase of 2.41 m in lake surface height, corresponding to a rate of 30.0 cm per year. Specifically, from 2016 to 2020, the lake surface height rose at a rate of 47.2 cm per year, while from 2020 to 2022, the height remained relatively stable. Full article

The determination of physical heights is of key importance for a wide spectrum of geoscientific applications and, in particular, for engineering projects. The main scope of the present work is focused on the determination of a high-accuracy and high-resolution gravimetric and hybrid geoid model, to determine orthometric heights without the need of conventional leveling. Both historical and newly acquired gravity data have been collected during dedicated gravity campaigns, around the location of a dedicated GNSS network as well as areas where the existing land gravity database presented voids. Geoid determination was based on the classical remove–compute–restore (RCR) technique and spectral and stochastic approaches. The low frequencies have been modeled based on the XGM2019e global geopotential model (GGM) and the topographic effects have been evaluated with the residual terrain model (RTM) reduction. The evaluation of the final geoid model was performed over 462 GNSS/leveling benchmarks (BMs), where the newly determined gravimetric geoid has shown an improvement of 3.1 cm, in the std of the differences to the GNSS/leveling BMs, compared to the latest national geoid model. A deterministic and stochastic fit to the GNSS/leveling data has been performed, investigating various choices for the parametric models and analytical covariance functions. The scope was to determine a hybrid geoid model, tailored to the area and GNSS/leveling data, which will be the one used for the direct estimation of high-accuracy orthometric heights from GNSS observations. After the deterministic fit, a std to the GNSS/leveling data of 10.1 cm has been achieved, with 54.8% and 83.1% of the absolute height differences being below the 1 cm and 2 cm per square root km of baseline length. The final hybrid geoid model, i.e., after the stochastic treatment of the adjusted residuals, gave a std of the difference to the GNSS/leveling data of 1.1 cm, with 99.8% and 99.9% of the height difference being smaller than the 1 cm and 2 cm standard errors, thus achieving a 1 cm accuracy regional geoid. Full article

A large number of fully normalized associated Legendre function (fnALF) calculations are required to compute Earth’s gravity field elements using ultra high-order gravity field coefficient models. In the surveying and mapping industry, researchers typically rely on CPU-based systems for these calculations, which leads to limitations in execution speed and power efficiency. Although modern CPUs improve instruction execution efficiency through instruction-level parallelism, the constraints of a shared memory architecture impose further limitations on the execution speed and power efficiency. This results in exponential increases in computation time as demand rises alongside high power consumption. In this article, we present a new computational implementation of an fnALF based on the ZYNQ platform. We design a task-parallel “pipeline” architecture which converts the original serial logic into a more efficient hardware implementation, and we utilize a redundant calculation layer to handle repetitive coefficient computations separately. The experimental results demonstrate that our system achieved accurate and rapid calculations. Under the only one geocentric residual latitude condition, we measured the computation times for spherical harmonic coefficient degrees of 360, 720, and 1080 to be 0.155922 s, 0.520950 s, and 1.401609 s, respectively. In the case of the multiple geocentric residual latitudes condition, our design generally yielded efficiency gains of over three times those of MATLAB R2020b implementation. Additionally, our calculated results were used to determine the geoid height in the field with an error of less than $\pm 0.1\mathrm{m}$, confirming the reliability of our computations. Full article

Astrogeodetic deflections of the vertical (DoVs) are close indicators of the slope of the geoid. Thus, DoVs observed along horizontal profiles may be integrated to create geoid undulation profiles. In this study, we collected DoV data in the Eastern Swiss Alps using a Swiss Digital Zenith Camera, the COmpact DIgital Astrometric Camera (CODIAC), and two total station-based QDaedalus systems. In the mountainous terrain of the Eastern Swiss Alps, the geoid profile was established at 15 benchmarks over a two-week period in June 2021. The elevation along the profile ranges from 1185 to 1800 m, with benchmark spacing ranging from 0.55 km to 2.10 km. The DoV, gravity, GNSS, and levelling measurements were conducted on these 15 benchmarks. The collected gravity data were primarily used for corrections of the DoV-based geoid profiles, accounting for variations in station height and the geoid-quasigeoid separation. The GNSS/levelling and DoV data were both used to compute geoid heights. These geoid heights are compared with the Swiss Geoid Model 2004 (CHGeo2004) and two global gravity field models (EGM2008 and XGM2019e). Our study demonstrates that absolute geoid heights derived from GNSS/leveling data achieve centimeter-level accuracy, underscoring the precision of this method. Comparisons with CHGeo2004 predictions reveal a strong correlation, closely aligning with both GNSS/leveling and DoV-derived results. Additionally, the differential geoid height analysis highlights localized variations in the geoid surface, further validating the robustness of CHGeo2004 in capturing fine-scale geoid heights. These findings confirm the reliability of both absolute and differential geoid height calculations for precise geoid modeling in complex mountainous terrains. Full article

Numerical tesseroid and radial-type approaches are presented and compared in terms of their efficiency for deriving the regional geoid height, vertical gravity, and gradiometric anomalies from sea floor topography grids. The vertical gradient function is particularly suitable for representing shorter wavelengths of gravity, typically less than 10 km. These two modeling methods were applied to the Great Meteor guyot in the Atlantic Ocean using its bathymetry. To optimize the computation of high-resolution gravity anomalies, the Armadillo, GSL, and OpenMP libraries were used to provide an environment for fast vector implementation, numerical integration for tesseroid calculation, and parallelization for loop iterations, resulting in a computation speed increase. The tesseroid and radial methods remain equivalent up to a resolution of about 1 min, with the radial method being faster when dealing with a large number of model points for the geoid. Aside from optimization enabling high-resolution gravity simulations, these fast modeling data can be used as the main operators in gravimetric inversion or to reduce the terrain effects in gravity observations, revealing gravity and sedimentary layers. Full article

The isle of Gavdos, and its wider area, is one of the few places worldwide where the calibration and validation of altimetric satellites has been carried out during the last, more than, two decades using dedicated techniques at sea and on land. The sea-surface calibration employed for the determination of the bias in the satellite altimeter’s sea-surface height relies on the use of a gravimetric geoid in collocation with data from tide gauges, permanent global navigation satellite system (GNSS) receivers, as well as meteorological and oceanographic sensors. Hence, a high-accuracy and high-resolution gravimetric geoid model in the vicinity of Gavdos and its surrounding area is of vital importance. The existence of such a geoid model resides in the availability of reliable, in terms of accuracy, and dense, in terms of spatial resolution, gravity data. The isle of Gavdos presents varying topographic characteristics with heights larger than 400 m within small spatial distances of ~7 km. The small size of the island and the significant bathymetric variations in its surrounding marine regions make the determination of the gravity field and the geoid a challenging task. Given the above, the objective of the present work was two-fold. First, to collect new land gravity data over the isle of Gavdos in order to complete the existing database and cover parts of the island where voids existed. Relative gravity campaigns have been designed to cover as homogenously as possible the entire island of Gavdos and especially areas where the topographic gradient is large. The second focus was on the determination of a high-resolution, $1\prime \times 1\prime$, and high-accuracy gravimetric geoid for the wider Gavdos area, which will support activities on the determination of the absolute altimetric bias. The relative gravity campaigns have been designed and carried out employing a CG5 relative gravity meter along with geodetic grade GNSS receivers to determine the geodetic position of the acquired observations. Geoid determination has been based on the newly acquired and historical gravity data, GNSS/Leveling observations, and topography and bathymetry databases for the region. The modeling was based on the well-known remove–compute–restore (RCR) method, employing least-squares collocation (LSC) and fast Fourier transform (FFT) methods for the evaluation of the Stokes’ integral. Modeling of the long wavelength contribution has been based on EIGEN6c4 and XGM2019e global geopotential models (GGMs), while for the contribution of the topography, the residual terrain model correction has been employed using both the classical, space domain, and spectral approaches. From the results achieved, the final geoid model accuracy reached the ±1–3 cm level, while in terms of the absolute differences to the GNSS/Leveling data per baseline length, 28.4% of the differences were below the $1\mathrm{c}\mathrm{m}\sqrt{S_{ij} \left[\mathrm{k}\mathrm{m}\right]}$ level and 55.2% below the $2\mathrm{c}\mathrm{m}\sqrt{S_{ij} \left[\mathrm{k}\mathrm{m}\right]}$. The latter improved drastically to 52.8% and 81.1%, respectively, after deterministic fit to GNSS/Leveling data, while in terms of the relative differences, the final geoid reaches relative uncertainties of 11.58 ppm (±1.2 cm) for baselines as short as 0–10 km, which improves to 10.63 ppm (±1.1 cm) after the fit. Full article

This research investigates precision enhancement in regional quasi-geoid refinement through ADS80 tri-linear array scanning stereoscopic imagery for aerial triangulation coupled with GNSS gravity-potential modeling. By acquiring stereoscopic imagery and analyzing triangulation accuracy using an ADS80 camera, we performed this study over the Qinghai–Tibet Plateau’s elevated, desolate terrain, collecting 593 GNSS points following high-precision stereoscopic imagery modeling. By utilizing 12 gravity satellite models, we computed geoid heights and China’s 1985 Yellow Sea elevations for 28 benchmarks and GNSS points, thereby refining the Qinghai Province Quasi-Geoid Model (QPQM) using geometric techniques. The findings reveal that POS-assisted ADS80 stereoscopic imagery yields high-precision triangulation with maximal horizontal and elevation accuracies of 0.083/0.116 cm and 0.053/0.09 cm, respectively, across five control point arrangements. The RMSE of normal heights for 1985, processed via these GNSS points, achieved decimeter precision. By applying error corrections from benchmarks to the 1985 elevation data from gravity satellites and performing weighted averaging, the precision of EGM2008, SGG-UGM-2, and SGG-UGM-1 models improved to 8.61 cm, 9.09 cm, and 9.38 cm, respectively, surpassing the QPQM by 9.22 cm to 9.99 cm. This research demonstrates that the proposed methods can significantly enhance the precision of regional quasi-geoid surfaces. Additionally, these methods offer a novel approach for rapidly establishing regional quasi-geoid models in the uninhabited areas of the Qinghai–Tibet Plateau. Full article

The remove–compute–restore (RCR) approach is widely used in local quasi-geoid modeling. However, the classical RCR approach usually does not take into account the noise of the satellite-only global gravity field model (GGM), which may lead to a suboptimal result. This paper presents an approach for local quasi-geoid modeling based on spherical radial basis functions that combines local noisy datasets and a noisy satellite-only GGM. This approach includes an RCR procedure using a satellite-only GGM. This is a direct approach that takes the spherical harmonic coefficients of satellite-only GGM as a noisy dataset and includes the corresponding full-noise covariance matrix in the least-squares estimation, aiming to obtain a statistically optimal local quasi-geoid model. The direct approach goes beyond the indirect approach, which treats the height anomalies generated from the satellite-only GGM as a noisy dataset. However, the generated GGM height anomaly dataset is not an equivalent representation of the satellite-only GGM, which may result in the loss of information from the satellite-only GGM. Through mathematical deduction, we demonstrate the theoretical consistency between the direct approach and the indirect approach. The direct approach also has an advantage over the indirect approach in terms of computational complexity due to the simpler algorithm. We conducted a synthetic closed-loop test with a real data distribution in Colorado, and numerical results demonstrated the advantage of the direct approach in local quasi-geoid modeling. In terms of the root mean square of the differences between the predicted values and the true reference values, the direct approach provided an improvement of approximately 14% compared to the indirect approach. Full article

The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications. Full article

This article investigates the role of different approaches and interpolation methods in gridding terrestrial gravity anomalies. In this regard, first of all, simple and complete Bouguer anomalies are considered in gravity data gridding. In the comparison results of gridding these two Bouguer anomaly datasets, the effect of the high-frequency contribution of topographic gravitation (by means of the terrain correction) is clarified. After that, the role of the used interpolation algorithm on the resulting grid of mean gravity anomalies and hence on the geoid modeling accuracy is inspected. For this purpose, four different interpolation methods including geostatistical Kriging, nearest neighbor, inverse distance to a power (IDP), and artificial neural networks (ANNs) are applied. Here, the IDP and nearest neighbor methods represent simple-structured algorithms among the interpolation methods tested in this study. The ANN method, on the other hand, is preferred as a complex, optimization-based soft computing method that has been applied in recent years. In addition, the geostatistical Kriging method is one of the conventional methods that is mostly applied for gridding gravity data in geodesy and geophysics. The calculated gravity anomalies in grids are employed in high-resolution geoid model computations using the least squares modifications of Stokes formula with additive corrections (LSMSA) technique. The investigations are carried out using the test datasets of Auvergne, France that are provided by the International Service for the Geoid for scientific research. It is concluded that the interpolation algorithms affect the gravity gridding results and hence the geoid model determination. The ANN method does not provide superior results compared to the conventional algorithms in gravity gridding. The geoid model with 4.1 cm accuracy is computed in the test area. Full article

The technology for determining a point’s coordinates on the earth’s surface using the global navigation satellite system (GNSS) is becoming the norm along with ground-based methods. In this case, determining coordinates does not cause any particular difficulties. However, to identify normal heights using this technology with a given accuracy, special research is required. The fact is that satellite determinations of geodetic heights (h) over an ellipsoid surface differ from ground-based measurements of normal height (H^N) over a quasi-geoid surface by a certain value called quasi-geoid height or height anomaly (ζ). In relation to determining heights of a certain territory, the concept of geoid height (N) is usually operated when dealing with a geoid model. In this work, geodetic and normal heights are determined for five control points in three different regions in Lebanon, where measurements are carried out using GNSS technology and geometric levelling. The obtained quasi-geoid heights are compared with geoid heights derived from the global Earth model EGM2008. The results obtained showed that, in the absence of gravimetric data, the combination of global Earth model data, geometric levelling for selected areas, and satellite determinations allows for the creation of a highly accurate altitude network for mountainous areas. Full article