Geoid Height Research Articles

Gravity recovery from SWOT altimetry using geoid height and geoid gradient

The Surface Water and Ocean Topography (SWOT) altimeter mission can measure high-resolution wide-swath sea surface heights (SSHs) that may greatly improve the current accuracy and spatial resolution of marine gravity from nadir-looking altimeters. To investigate the potential of SWOT in recovering high-quality marine gravity and how SWOT observation errors should be treated to optimize the accuracy of gravity anomaly from SWOT observations, we create high-wavenumber SSH components from multi-beam depths in the northern South China Sea (SCS) and simulate SWOT SSH errors. To cross-validate gravity signals and avoid gravity errors from SWOT, we use two computational methods (inverse Vening-Meinesz formula, IVM and inverse Stokes' integral, ISM) and recommend separate optimal data processing strategies when using geoid gradients (GGs) and geoid heights (GHs) for gravity recovery. The use of GGs (for IVM) effectively eliminates systematic errors in gravity derivation. If GHs (for ISM) are used in gravity recovery, the tilt in wide-swath SSHs should be removed before gravity computation, and the recovered gravity must be filtered (post-processed) to avoid artificial gravity signals due to the SSH errors. Our assessments using mgal-accuracy shipborne gravity anomalies in the northern SCS show that multiple-cycle SWOT observations can deliver high-quality marine gravity anomalies. IVM is more robust than ISM in resisting random and systematic errors in SWOT. Our processing strategies can be used for the gravity validation of SSHs from SWOT's fast-sampling and science phases.

Comparison of different DEMs for hydrological studies in the mountainous areas

The optical and radar satellite data based Global Digital Elevation Models (DEMs) serve as a main data source in hydrological studies today. Open access Remote Sensing (RS) products have increasing accuracy and improved spatial resolution. The main research purposes are to study the use of AW3D30, ASTER GDEM V003 and SRTMGL1 DEMs in modeling mountainous lake basin and comparative analysis of their spatial accuracy. Topographic maps and very high resolution satellite imagery of the study area have been applied for the spatial accuracy verification and surface referencing. The relative accuracies of three different DEMs were analyzed using 1284 elevation points on the terrain surfaces. Comparison of these DEMs with 30 m spatial resolution confirms that an AW3D30 DEM has a higher vertical accuracy compared to other models and can be used for modeling river and lake basins over high mountainous terrain. The detected unreliable sections of the modeled lake basin boundary were analyzed and corrected based on the basin delineation by different DEMs. Analysis of spatial accuracy of the delineated Issyk-Kul Lake basin provide further incentive for DEM modification based on higher resolution remote sensing data, considering the applied geoid height differences as well by using satellite positioning data.

Gravity field of Sarmatia according to satellite data (model EIGEN-6S2) and its interpretation

The paper presents a brief overview of satellite observations of the CHAMP, GRACE and GOCE missions to study the Earth’s global gravity field, and the used mathematical apparatus in the form of an expansion of the geopotential in spherical harmonics. The application of satellite data in various fields of Earth Sciences is considered. As a basic global model of the Earth’s gravity field based on satellite data we used the EIGEN-6S2 model [Rudenko et al, 2014] that combines satellite mission data GRACE and GOCE, and also uses satellite data of LAGEOUS laser ranging. On its basis, the gravity field of Sarmatia was analyzed using the Free Air anomalies, Bouguer anomalies, the second radial derivative of the geopotential and the geoid heights. The geological units of Sarmatia and its surroundings are most clearly manifested in the Free Air anomalies and in the distribution of the second derivative of the geopotential, showing differences in the gravity field pattern of the Ukrainian Shield, the Voronezh Massif, and the Pripyat-Dnieper-Donets basin (PDDB) with characteristic anomalies of the general northwest strike. The continuation of the PDDB in a southeastern direction through the Karpinsky Swell to the northern part of the Caspian Sea confirms the existence of an extended ancient tectonic zone of the Sarmato-Turanian lineament. The geoid within Sarmatia shows in general a regional west-east gradient change from +40 m in the west to -10 m in the east. Such large-scale geoid changes are determined by the Sarmatia position between two global geoid anomalies — the maximum of the North Atlantic and the minimum of the Indian Ocean.

The fractal behaviour of gravity field elements: case study in Egypt

ABSTRACT The fractal behaviour of gravity field elements was investigated. Particularly, the geoidal heights, gravity anomalies and vertical deflection components were considered. Such data were synthesized from the ultra-high degree SGG-UGM-1 geopotential model. The Egyptian Territory was selected as the study area. To supplement the results, parallel fractal analyses were performed on the GTM3A, Earth2012 and SRTM30 terrain data. The regularization, variogram and global Power Spectral Density (PSD) concepts were applied as different tools for fractal analysis. Consistent fractal dimensions resulted from the regularization and variogram algorithms. These two algorithms ensured the fractalness of the gravitational signals, including the geoidal height. The global PSD algorithm failed to give meaningful fractal dimensions for the topography and geoidal heights. However, it yielded realistic dimensions for the other elements. So, fractal dimensions could characterize the roughness and spatial precisions of different gravitational data. Also, multifractal modeling of the geopotential might be a promising task.

A high-resolution gravimetric geoid model for Kingdom of Saudi Arabia

A high-resolution gravimetric geoid model for the Kingdom of Saudi Arabia area was determined. A data set of 459,848 land gravity, 80,632 shipborne marine gravity data, DTU17 altimetry gravity model, and XGM2019e global geopotential model. The computation strategy followed for modelling of the gravimetric geoid is based on the Remove-Compute-Restore with Residual Terrain Model reduction and the 1D- Fast Fourier Transform approach technique. The geoid heights have been determined by using the Stokes integral with Wong–Gore modification. The accuracy of the resulting geoid models was evaluated by comparing them with 5385 GPS/levelling points. The geoid accuracy over all the kingdom is better than 11 cm in STD sense and the comparison in sub-areas obtained accuracy range from 2.8 to 11.9 cm according to the density of gravity observations.

ETH–GM21: A new gravimetric geoid model of Ethiopia developed using the least-squares collocation method

The determination of an accurate national geoid model remains one of the main challenges of geodetic research projects in Ethiopia. The main aim of this study is to develop a new gravimetric geoid model, named the ETH–GM21, over Ethiopia. This new geoid model is based on a combination of terrestrial and airborne gravity data together with the state-of-the art combined Global Geopotential Model (GGM) and the high-resolution SRTM3 (Shuttle Radar Topography Mission with a spatial resolution of 3 arc-second) global digital elevation model. The ETH–GM21 has been determined using the remove-compute-restore (RCR) procedure and the least–squares collocation method. The accuracy of ETH–GM21 gravimetric geoid model has been evaluated using geometric geoid heights obtained from GNSS (Global Navigation Satellite Systems) and levelling data distributed over the country. On the basis of this evaluation the estimated accuracy of geoid heights from the ETH–GM21 geoid model is at the level of 13 cm. The ETH–GM21 can be recommended as a first relatively accurate vertical reference surface for a definition of heights within the whole Ethiopia.

Determination of local gravimetric geoid model over Egypt using LSC and FFT estimation techniques based on different satellite- and ground-based datasets

In this contribution, two gravimetric geoid models for Egypt; EGY-LGM2019 and EGY-FGM2019 are developed using the least-squares collocation and the Fast-Fourier Transformation methods, applying Remove-Restore procedure. Different datasets are used in terms of gravity anomalies obtained from GOCE-based SPW-R5 and EGM2008 models, and from 1015 terrestrial gravity stations. Two gravimetric geoid models, EGY-FLGM2019 derived by the LSC method and EGY-FFGM2019 derived by 1D-FFT method are computed and fitted using the 17 GPS/levelling measurements to adjust the gravimetric geoid model with the national local vertical datum for Egypt. Our findings between the 6 GPS/levelling geoid undulations and the developed fitted geoid models EGY-FFGM2019 and EGY-FLGM2019 are about 7.57 and 18.46 cm, respectively. In addition, the former model improves the geoid heights of about 72-65% over Egypt w.r.t. the EGM2008 and GECO models, respectively. Therefore, the EGY-FFGM2019 model is recommended as a reliable model that improves the geoid heights over Egypt.

Thermochemical State of the Upper Mantle Beneath South China From Multi‐Observable Probabilistic Inversion

AbstractWe present an upper mantle model of seismic velocities, temperature, density, and bulk composition for the South China Block (SCB) and adjacent areas by jointly inverting Rayleigh wave phase and group dispersion data, absolute elevation, geoid height, and surface heat flow using a probabilistic inversion method. The lithospheric structure of this region is dominated by a thick root (200–240 km) under the western Yangtze and thinner lithosphere (70–120 km) under the eastern SCB, in agreement with independent xenolith data. We reveal for the first time the extent of metasomatism/refertilization in the lithospheric mantle beneath the Yangtze Craton (YC). Our results show pervasive refertilization of the lithospheric mantle associated with the basaltic lavas of the Emeishan large igneous province. We also show that the refertilized mantle identified in the northern boundaries of the YC extends into the interior of the craton further than previously recognized, reaching at least parts of the Sichuan basin. The only region that preserve a depleted mantle signature is the central segment of the Yangtze craton. In contrast, the eastern SCB lithosphere is characterized by an alternating pattern of fast/slow seismic and depleted/fertile anomalies, supporting the hypothesis that the lithospheric mantle has been partly replaced/modified by deep mantle processes during the Phanerozoic. Also, we found no evidence for a linear plume track associated with the Hainan plume that can be used to independently constrain the plate motion history of the SCB.

Jeoid Değişimlerinin ICESat Altimetre Verisi ile Hesaplanan Su Seviyelerine Etkisi

Çalışmada, ICESat uydusundaki GLAS altimetre verileriyle hesaplanan su seviye yüksekliklerine jeoid yüksekliklerinin etkileri Burdur Gölü (BG) üzerinde incelenmiştir. Su yüzeylerinin günlük olarak tespit edilebilmesi için MODIS uydusuna ait karla kaplı alan haritaları kullanılmıştır. ICESat verilerindeki Global EGM2008 jeoid ile hesaplanan ve BG yer gözlem ölçülerinin su seviye farkları -0.96m ile -0.60m aralığında olup hataların ortalama karekökü 0.73m dir. Güncellenen jeoid ile hesaplanan su seviyelerinin hataları -0.20m ile 0.16m aralığında olup hataların ortalama karekökü 0.12m’ye inmiştir. DSİ yer gözlem verileri ile yapılan karşılaştırmada R2 0.98 olarak bulunmuştur. Birbirini takip eden kış ve yaz su seviye farkları 0.74m hesaplanıp maksimum 1.0m lik değişim içinde kalmıştır.

A local lithospheric structure model for Vietnam derived from a high-resolution gravimetric geoid

High-resolution Moho and lithosphere–asthenosphere boundary depth models for Vietnam and its surrounding areas are determined based on a recently released geoid model constructed from surface and satellite gravity data (GEOID_LSC_C model) and on 3ʹʹ resolution topography data (mixed SRTM model). A linear density gradient for the crust and a temperature-dependent density for the lithospheric mantle were used to determine the lithospheric structure under the assumption of local isostasy. In a first step, the impact of correcting elevation data from sedimentary basins to estimate Moho depth has been evaluated using CRUST1.0 model. Results obtained from a test area where seismic data are available, which demonstrated that the sedimentary effect should be considered before the inversion process. The geoid height and elevation-corrected sedimentary layer were filtered to remove signals originating below the lithosphere. The resulting Moho and lithosphere–asthenosphere boundary depth models computed at 1ʹ resolution were evaluated against seismic data as well as global and local lithospheric models available in the study region. These comparisons indicate a consistency of our Moho depth estimation with the seismic data within 1.5 km in standard deviation for the whole Vietnam. This new Moho depth model for the study region represents a significant improvement over the global models CRUST1.0 and GEMMA, which have standard deviations of 3.2 and 3.3 km, respectively, when compared to the seismic data. Even if a detailed geological interpretation of the results is out of scope of this paper, a joint analysis of the obtained models with the high-resolution Bouguer gravity anomaly is finally discussed in terms of the main geological patterns of the study region. The high resolution of our Moho and lithosphere–asthenosphere boundary depth models contribute to better constrain the lithospheric structure as well as tectonic and geodynamic processes of this region. The differences in Moho depth visible in the northeast and southwest sides of the Red River Fault Zone confirmed that the Red River Fault Zone may be considered the boundary between two continental blocks: South China and Indochina blocks. However, no remarkable differences in lithosphere–asthenosphere boundary depth were obtained from our results. This suggests that the Red River Fault Zone developed within the crust and remained a crustal fault.

Collocation and FFT-based geoid estimation within the Colorado 1 cm geoid experiment

In the frame of the International Association of Geodesy Joint Working Group 2.2.2 “The 1 cm geoid experiment”, terrestrial and airborne gravity datasets along with GPS/leveling data were made available for the comparison of different geoid modeling methods and techniques in the wider area of Colorado, USA. We discuss the methods and procedures we followed for computing gravimetric quasi-geoid and geoid models and geopotential values from the available datasets. The procedures followed were based on the remove-compute-restore approach using XGM2016 as a reference geopotential model. The higher frequencies of the gravity field were computed via the residual terrain correction, using (a) the CGIAR-CSI SRTM digital elevation model with the classical technique and (b) a spectral one. Least-Squares Collocation was used for the downward continuation of the airborne data and for gridding. Finally, the geoid models were obtained by applying Least-Squares Collocation and spherical FFT-based methods, while the influence of the orthometric height correction on geoid heights was taken into account by employing simple and complete Bouguer reductions. All results were evaluated with available GPS/leveling benchmarks. Moreover, potential values were determined in support of the International Height Reference System/Frame. From the results acquired, a final accuracy of 5–7 cm for the determined geoid models was achieved depending on the adopted method and data combination, without considering the accuracy of the GPS/leveling data used for their evaluation. The contribution of the airborne gravity data was deemed as limited in combination solutions although the airborne only solution provided equal level of accuracy to the terrestrial and the combined ones. Better consistency was obtained on the points of the GSVS17 line, when compared to the GPS/leveling data, where an accuracy of 2.4 cm and 2.8 cm was reached for the FFT and LSC based methods, respectively.

Potential and scientific requirements of optical clock networks for validating satellite-derived time-variable gravity data

SUMMARY The Gravity Recovery and Climate Experiment (GRACE) and GRACE-FO missions have provided an unprecedented quantification of large-scale changes in the water cycle. However, it is still an open problem of how these missions’ data can be referenced to a ground truth. Meanwhile, stationary optical clocks show fractional instabilities below 10−18 when averaged over an hour, and continue to be improved in terms of stability and accuracy, uptime and transportability. The frequency of a clock is affected by the gravitational redshift, and thus depends on the local geopotential; a relative frequency change of 10−18 corresponds to a geoid height change of about 1 cm. Here we suggest that this effect could be exploited for sensing large-scale temporal geopotential changes via a network of clocks distributed at the Earth’s surface. In fact, several projects have already proposed to create an ensemble of optical clocks connected across Europe via optical fibre links. Our hypothesis is that a clock network with collocated GNSS receivers spread over Europe—for which the physical infrastructure is already partly in place—would enable us to determine temporal variations of the Earth’s gravity field at timescales of days and beyond, and thus provide a new means for validating satellite missions such as GRACE-FO or a future gravity mission. Here, we show through simulations how glacial, hydrological and atmospheric variations over Europe could be observed with clock comparisons in a future network that follows current design concepts in the metrology community. We assume different scenarios for clock and GNSS uncertainties and find that even under conservative assumptions—a clock error of 10−18 and vertical height control error of 1.4 mm for daily measurements—hydrological signals at the annual timescale and atmospheric signals down to the weekly timescale could be observed.

ПРО ПРЕДСТАВЛЕННЯ АКФ STHA-ПОЛІНОМАМИ ДЛЯ МОДЕЛЮВАННЯ РЕГІОНАЛЬНОГО ГРАВІТАЦІЙНОГО ПОЛЯ ЗЕМЛІ

The current Baltic height system in Ukraine is outdated and requires modernization. The current resolution of Cabinet of Ministers of Ukraine states that the implementation of topographic, geodetic and cartographic works starting from January 1, 2023 will be carried out using UELN/EVRS2000 height system. For successful integration of our height system into European one, it is necessary to construct a high-precision model of geoid surface on territory of Ukraine, which should fit well with European geoid model EGG2015. This will allow using advanced satellite technologies to determine gravity-dependent heights. The main methods of constructing regional gravitational field of Earth include method of geopotential decomposition into series of spherical functions with fractional indices (in particular, STHA-method) and least-squares collocation method. However, to date, no relationship has been established between them. The construction of ACF using STHApolynomials, which wavelength is much shorter than Legendre polynomials and depends on area of studied region, will significantly improve accuracy of regional models of Earth's gravitational field. The approach to construction of ACF using STHA-polynomials can be very promising precisely because wavelength of these functions coincides with area of studied region, and all other properties coincide with properties of Legendre polynomials. That is why it is necessary to perform ACF calculations using Legendre polynomials and STHApolynomials and analyze results. The solution of this problem is realized by constructing and analyzing covariance and cross-covariance functions of geopotential functionals using Legendre polynomials and STHA-polynomials according to Tscherning-Rapp model. A number of covariance and crossover-covariance functions are calculated and compared using Legendre polynomials and STHA-polynomials according to Tscherning-Rapp model. The values of geoid heights obtained from GNSS-observations at points of SGN of I, II and III classes in territory of Lviv region are taken as input data. A method for constructing a local ACF using STHA-polynomials is proposed and tested. STHA-polynomials have been found to be significantly more efficient than Legendre polynomials because their domain coincides with that of region under study. For example, it is shown that for Lviv region decomposition of covariance functions in a series by Legendre polynomials up to order 600 is equivalent to decomposition in a series by STHA-polynomials up to order 8.

Validating Geoid Models with Marine GNSS Measurements, Sea Surface Models, and Additional Gravity Observations in the Gulf of Finland

Traditionally, geoid models have been validated using GNSS-levelling benchmarks on land only. As such benchmarks cannot be established offshore, marine areas of geoid models must be evaluated in a different way. In this research, we present a marine GNSS/gravity campaign where existing geoid models were validated at sea areas by GNSS measurements in combination with sea surface models. Additionally, a new geoid model, calculated using the newly collected marine gravity data, was validated. The campaign was carried out with the marine geology research catamaran Geomari (operated by the Geological Survey of Finland), which sailed back and forth the eastern part of the Finnish territorial waters of the Gulf of Finland during the early summer of 2018. From the GNSS and sea surface data we were able to obtain geoid heights at sea areas with an accuracy of a few centimetres. When the GNSS derived geoid heights are compared with geoid heights from the geoid models differences between the respective models are seen in the most eastern and southern parts of the campaign area. The new gravity data changed the geoid model heights by up to 15 cm in areas of sparse/non-existing gravity data.

Determination of Orthometric Height using GNSS and EGM Data: a Scenario of the Federal University of Technology Akure

The advent of Global Navigation Satellite System (GNSS) and Earth Gravitational Model (EGM)particularly Global Positioning System (GPS) has modernized geodetic surveying in providing horizontal and vertical positions of points with a sub-meter level of accuracy over the reference ellipsoid. The GPS gives ellipsoidal heights which makes the conversion of the heights to orthometric heights possible by incorporating a geoid model. The conventional method of determining orthometric height is tedious, time-consuming, and labour intensive. This study entails the determination of orthometric height using GNSS and EGM data. A total of forty-nine (49) stations selected within the study area were occupied for GPS observation using South DGPS instrument in static mode for the position and ellipsoidal height determination. The geoidal height values of the GPS derived data were computed using GeoidEval utility software with reference to three different EGMs (EGM2008, EGM96 and EGM84). In order to determine the orthometric heights of the selected stations, the difference between the EGM geoidal height values (NEGM) and the ellipsoidal heights were computed. The results show that the orthometric height obtained with respect to EGM2008 gives better results with a standard deviation of 9.530m and a standard error of 1.361m. The study reveals that the use of GNSS and EGM data for orthometric height determination is less expensive, less tedious, accurate and time-saving compared to the conventional approach of geodetic and spirit levelling.

The earth\u2019s gravity field recovery using the third invariant of the gravity gradient tensor from GOCE

Due to the independence of the gradiometer instrument’s orientation in space, the second invariant I_2 of gravity gradients in combination with individual gravity gradients are demonstrated to be valid for gravity field determination. In this contribution, we develop a novel gravity field model named I3GG, which is built mainly based on three novel elements: (1) proposing to utilize the third invariant I_3 of the gravity field and steady-state ocean circulation explorer (GOCE) gravity gradient tensor, instead of using the I_2, similar to the previous studies; (2) applying an alternative two-dimensional fast fourier transform (2D FFT) method; (3) showing the advantages of I_3 over I_2 in the effect of measurement noise from the theoretical and practical computations. For the purpose of implementing the linearization of the third invariant, this study employs the theory of boundary value problems with sphere approximation at an accuracy level of O(J_2^2cdot T_{ij}). In order to efficiently solve the boundary value problems, we proposed an alternative method of 2D FFT, which uses the coherent sampling theory to obtain the relationship between the 2D FFT and the third invariant measurements and uses the pseudo-inverse via QR factorization to transform the 2D Fourier coefficients to spherical harmonic ones. Based on the GOCE gravity gradient data of the nominal mission phase, a novel global gravity field model (I3GG) is derived up to maximum degree/order 240, corresponding to a spatial resolution of 83 km at the equator. Moreover, in order to investigate the differences of gravity field determination between I_3 with I_2, we applied the same processing strategy on the second invariant measurements of the GOCE mission and we obtained another gravity field model (I2GG) with a maximum degree of 220, which is 20 degrees lower than that of I3GG. The root-mean-square (RMS) values of geoid differences indicates that the effects of measurement noise of I3GG is about 20% lower than that on I2GG when compared to the gravity field model EGM2008 (Earth Gravitational Model 2008) or EIGEN-5C (EIGEN: European Improved Gravity model of the Earth by New techniques). Then the accuracy of I3GG is evaluated independently by comparison the RMS differences between Global Navigation Satellite System (GNSS)/leveling data and the model-derived geoid heights. Meanwhile, the re-calibrated GOCE data released in 2018 is also dealt with and the corresponding result also shows the similar characteristics.

RETRACTED ARTICLE: Ocean dynamic equations with the real gravity

Two different treatments in ocean dynamics are found between the gravity and pressure gradient force. Vertical component is 5–6 orders of magnitude larger than horizontal components for the pressure gradient force in large-scale motion, and for the gravity in any scale motion. The horizontal pressure gradient force is considered as a dominant force in oceanic motion from planetary to small scales. However, the horizontal gravity is omitted in oceanography completely. A non-dimensional C number (ratio between the horizontal gravity and the Coriolis force) is used to identify the importance of horizontal gravity in the ocean dynamics. Unexpectedly large C number with the global mean around 24 is obtained using the community datasets of the marine geoid height and ocean surface currents. New large-scale ocean dynamic equations with the real gravity are presented such as hydrostatic balance, geostrophic equilibrium, thermal wind, equipotential coordinate system, and vorticity equation.

A Fast Method for Calculation of Marine Gravity Anomaly

Gravity data have been playing an important role in marine exploration and research. However, obtaining gravity data over an extensive marine area is expensive and inefficient. In reality, marine gravity anomalies are usually calculated from satellite altimetry data. Over the years, numerous methods have been presented for achieving this purpose, most of which are time-consuming due to the integral calculation over a global region and the singularity problem. This paper proposes a fast method for the calculation of marine gravity anomalies. The proposed method introduces a novel scheme to solve the singularity problem and implements the parallel technique based on a graphics processing unit (GPU) for fast calculation. The details for the implementation of the proposed method are described, and it is tested using the geoid height undulation from the Earth Gravitational Model 2008 (EGM2008). The accuracy of the presented method is evaluated by comparing it with marine shipboard gravity data. Its efficiency is demonstrated through comparison with the conventional sequential method. The tests demonstrate that the proposed method can be employed for accurately calculating marine gravity anomalies and provides an advantage on computational efficiency.

Implications of the temporal variations of geoid heights over South American vertical reference frames

The temporal variations in a vertical reference frame depend on the variations in the reference surface (e.g., the geoid), and in the surface where the benchmarks are located. The mass redistribution on and beneath the Earth’s surface generates changes in the geoid, which is the reference surface for orthometric heights. To model the phenomena and processes that affect Earth’s gravity field, the maintenance of a high-precision global physical reference frame that considers its temporal variations is necessary. In this contribution, we estimate and analyze the impact of three geophysical signals, namely mega-earthquakes (Mw > 7.5), hydrological loadings, and glacial isostatic adjustment over the South American vertical reference frames based on the Gravity Recovery and Climate Experiment monthly solutions, glacial isostatic adjustment, and hydrological models. Our results reveal that long-term trends range from −0.5 to 0.3 mm/year, which implies possible height variations up to 10 mm over 20 years. These results point out the possibilities for maintaining modern vertical reference frames if variations in the trends (e.g., due to accelerations) are disregarded. The seasonal variations (from peak to peak), co-seismic deformation, and post-seismic deformation reach 31 mm, 3.8 mm, and 0.8 mm, respectively. According to wavelet analysis based on the independent components (IC) obtained from independent component analysis, it was observed that the predominant signal has one cycle per year. The first four ICs represent 94.4% of the geoid variability. The spatial patterns of the four modes indicated that geoid variations are concentrated in the Amazon, Tocantins, Parana, and Sao Francisco river basins. Temporal variations of the geoid in South America reach magnitudes that should be considered in the evolution of height systems to accomplish the requirements of a modern height system (accuracy on the order of 10 mm or better).

Applications of numerical integration in geodesy and geophysics

In this paper, two applications of numerical integration in geodesy and geophysics are presented. In the first application, the Molodenskij truncation coefficients for the Abel-Poisson kernel are computed using eleven different numerical integration procedures, namely two-, three-, four-, and five-point Gaussian, Gauss–Kronrod, trapezoidal rule, Simpson and its adaptive mode, Romberg, Lobatto, and Sard’s approximating functional numerical integration methods. The coefficients are computed for truncation degree 90, and truncation radius $$6^\circ$$ . The results are then compared with an independent method for calculating these coefficients. It is shown that numerical integration methods represent better accuracy. In the second application, the gravity accelerations at sea surface in Qeshm in southern Iran are calculated using the spherical spline numerical integration method. The formulae for spherical spline numerical integration in two different modes weighted and without weight are derived. The special case when the weight of the integral is the so-called Stokes’ kernel is thoroughly investigated. Then, the results are used to generate gravity accelerations. First, the geoid height from the sum of the mean sea level and sea surface topography is calculated. Then, a spherical spline analytical representation—with unknown coefficients—is considered for the gravity anomaly. In the next step, using the Stokes’ formula for the integral relation between geoid height and gravity anomaly, the unknown coefficients in the previous step are calculated and subsequently the gravity anomalies are derived. Adding the gravity of the reference ellipsoid to the gravity anomalies, the actual gravity accelerations at sea surface in Qeshm are calculated. To analyze the accuracy, the derived values are compared with the values observed by shipborne gravimetry. It is shown that using Bernstein polynomials as basis function for calculating numerical integration has a better accuracy than other numerical integration methods of the same degree.