Geoid Undulations Research Articles

Precise Geoid Determination in the Eastern Swiss Alps Using Geodetic Astronomy and GNSS/Leveling Methods.

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.

Planning, emphasis and proposal for implementation of gravimetric control network in Israel

AbstractThis study outlines the planning, emphases, and proposed implementation of a gravimetric control network for Israel. The establishment of a comprehensive and accurate gravimetric control network is crucial for various geodetic applications, including geoid determination, gravity field modelling, vertical datum establishment, and geophysical studies. This study aims to provide a framework for the development and implementation of such a network in Israel. The proposed network is divided into two orders. The first order, also known as the absolute order or zero order, comprises four stations situated in geologically stable and tranquil locations. The second order, referred to as the first order, is established through relative gravimetric measurements. This order will encompass 29 stations positioned in areas that are both stable and secure. The zero and first order stations will be evenly distributed throughout the country to ensure homogeneous coverage. The proposed gravimetric network is in conjunction with the requirements to improve the geoid undulation model of Israel. Through the establishment of a small gravimetric network and the implementation of relative gravimetric measurements, we were able to determine the optimal control point density and establish a well-structured methodology for measurement, analysis, and the adjustment process.

Impact of the UNB topographical density model on precise geoid determination in the high mountainous region

A precise gravimetric geoid model is computed utilising Stokes’s formula, supposing an absence of topography above the geoid. Subsequently, the geoid model undergoes a simple correction for topographic masses, the constant density is taken as 2670 kg/m3. Notably, the true density of topographical mass deviates by approximately ±20% from this constant value. Recently, the University of New Brunswick in Canada released a global topographical density model at a 30 arc-second resolution. This paper investigates the impact of incorporating this model on the precision of the gravimetric geoid within a mountainous region in the Colorado test region. Numerical findings reveal that variations in geoid undulation attributable to this model can extend to a few decimetres, a discrepancy that cannot be neglected in geoid modelling with one-centimetre precision. It is therefore recommended that the considerable impact of topographic density fluctuations on geoid determination be taken into account, particularly in mountainous regions.

Geoid Undulation Model as Vertical Reference in Indonesia

Indonesia released a new regional geoid model in 2020—the Indonesian Geoid 2020 (INAGEOID2020). It covers the Indonesian region with a spatial resolution of 0.01 × 0.01 degree with the unit in meters. The model was generated through a series of data and computations. Three components of gravity data, i.e., the observed free-air anomaly, the long-wave from the global geoid model, and the short-wave from the terrain model, were employed. The computation was performed using the Remove-Compute-Restore technique with the Fast Fourier Transformation approach. The output was then fitted to the geoid at tide stations by adding a fitting plane to the geoid model. The fitting plane was constructed based on the difference between the geoid model and each tide gauge benchmark. The final geoid model was evaluated by comparing the model with the reference data. Based on quality metrics, the accuracy of INAGEOID2020 varied between 6 cm to 29 cm. Any interested user can use this gridded geoid model to convert geodetic to orthometric heights and vice versa.

A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces

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.

ESTABLISHMENT OF ORTHOMETRIC HEIGHT CONTROLS IN THE EASTERN PART OF KOGI STATE USING GNSS AND EGM 2008

The aim of this study was to model the orthometric heights in parts of Kogi East, Kogi State using GNSS and EGM 2008 with the objectives of establishing the best geoid undulation for height determination. The Hardwares used for this study include; Hi target V.30 dual frequency GNSS receiver and its accessories, Data logger, Computer hardware and software and Computer – HP Laptop and internet facilities. Softwares used included; Arc GIS, Google map, microsoft word, Microsoft excel, Hi target Geomatic offices, CSRS – PPP website andGeoid Eval Calculator. GNSS observations were conducted on 6 station monument. The dual frequency v30 Hi target instrument was used in the static mode to make observations on all the established points at an epoch rate of 30 seconds; for a period of one hour to one hour thirty minutes (session) per station. The secondary data were the EGM 2008, obtained from Geoid eval platform in https://www.geographiclib.sourceforge.ion/cgi-bin/GeoidEval. Geoid Eval platform makes provision for the geographical coordinates to be inputed before the geoidal-ellipsoidal separation values can be provided. Geoid Eval computes the height of the geoid above the WGS 84 ellipsoid using interpolation in a grid of values for the earth gravity models. To obtain the orthometric height, the geoid value is subtracted from the ellipsoidal height obtained from the GNSS-PPP observation. The accuracy of the geoid value is 1mm and that of the GNSS-PPP was adjudged to be 3cm respectively. It was recommended that, more orthometric heights be distributed for monitoring of the earth’s dynamisms.

GeoMed2,the geoid of the Mediterranean: work in progress

Geodesy can provide valuable information on marine current estimation based on the combination of gravity and altimetry. Gravity is standardly used to estimate the geoid undulation, i.e. the height of the geoid over a given reference ellipsoid. As it is well known, the geoid undulation over the oceans is closely related to the Mean Sea Surface (MSS) with discrepancies that can reach 1–2 m at global scale. By satellite altimetry, one can get the MSS and then estimate the Mean Dynamic Topography (MDT) as the difference between the MSS and the geoid undulation. As the MDT is related to the ocean circulation, information on the ocean circulation to be compared with oceanographic estimates can be provided using these geodetic measurements. In this context, the GeoMed2 project aims at estimating a high-accuracy and high-resolution geoid model for the Mediterranean Sea based on land and marine gravity data and on recent Global Geopotential Models. In this paper, the processing methodology based on the well-known remove–compute–restore approach for the determination of the geoid in the Mediterranean area is presented. In a pre-processing step, all available gravity observations for the wider Mediterranean basin have been collected, validated, homogenized, and unified in terms of their horizontal and gravity system. In this way, a reliable gravity database to be used for the determination of the geoid has been prepared. This data set has been used in computing a gravimetric geoid estimate based on which the MDT over the Mediterranean Sea was obtained. The results of this computation were then revised, commented and compared with other existing MDT solutions. By these comparisons, it can be concluded that the geodetic computed MDT is not yet satisfactory since it is too noisy. This is possibly due to some inconsistencies still present in the gravity data used for estimating the geoid undulation and to the adopted MSS which seems to be too smooth over the Mediterranean area.

Determination of the Geoid–Quasigeoid Separation Using GGI Method

The determination of the geoid–quasigeoid separation (GQS) is most often based on the use of Bouguer gravity anomalies or disturbances with additional corrections, which allow for the determination of so-called complete or accurate GQS values. This study presents analyses related to an attempt to determine accurate GQS values using the GGI approach (based on the geophysical gravity inversion technique). This approach allows for the modeling of various parameters of the gravity field, and it also enables the determination of the GQS or geoid undulations. Such capabilities of the method have not yet been tested. In this study, the details of the GGI solution in terms of determining the GQS and the first results from tests performed in the area of the Colorado 1 cm geoid computation experiment are presented. The GQS values determined by the GGI approach were compared with the reference values determined previously using the complete classical approach. The differences between the compared values were small, with a standard deviation of 0.007 m, and the maximum differences reached 0.075 m. The analyses also revealed the significant impact of changes in the density of topographic masses on both the geoid undulations and GQS values determined using the GGI approach.

Development of a hybrid geoid model using a global gravity field model over Sri Lanka

Abstract Sri Lanka is still in need of a well-defined local geoid model. This geoidal void has made present-day Global Navigation Satellite System (GNSS) surveys heavily dependent on Global Geopotential Models (GGMs) for height determination. Further, in many instances, the accuracy of GGMs have shown drawbacks in elevation determination over Sri Lanka. Therefore, the study focused on developing a hybrid geoid model (HGM) for Sri Lanka by integrating the available GGMs. Five high-resolution (2190°) GGMs; EGM2008, EIGEN-6C4, GECO, XGM2019e-2159, and SGG-UGM2 were employed to extract GGM-derived geoid undulation for 21 Fundamental Benchmarks (FBMs). The residuals (geoid height deviation) were calculated relative to the observed geoid undulation using GNSS/leveling on the FBMs. The data set was clustered based on topography, and residuals were adjusted using weighted least squares adjustment (LSA). The uneven distribution of the FBMs promotes topography-based clustering. EIGEN-6C4 is found to be the robust GGM for Sri Lanka to develop a hybrid approach, with a 0.001 m RMS value of estimated residuals in LSA. The resulting HGM was interpolated at 1 arc-second grid resolution (30 m × 30 m) using the Inverse Distance Weighted Interpolation. Regression lines were generated for the interpolated HGM with respect to the interpolated observed geoid undulation for 9 transects along the parallel passing through Mount Pedro and for the 16 transects along the meridian. The coefficient of determination on both lines is 0.999. HGM generated by EIGEN-6C4 has shown reliable RMS gradient and intercept values of 8.860078 × 10−9 and 0.0039239, respectively, in first-order polynomial fitting.

Beyond Exact Repeat Missions: Embracing Geodetic Altimetry for Inland Water Monitoring and Modeling

Satellite radar altimetry has become an essential tool for hydrological research over the past 2 decades. Unlike imagery that quantifies the water dynamics from the horizontal dimension, altimetry observations deliver vertical information, which is crucial for hydrological research. Geodetic mission (GM) altimetry data, characterized by long repeat periods compared to exact repeat missions, have been underutilized in hydrology. This paper aims to advocate for integrating geodetic altimetry data into inland water research and bring awareness to its potential applications. We demonstrated the values of GM data from CryoSat-2, Saral/AltiKa, Jason-1, and Jason-2 through 3 applications: (a) large-scale lake monitoring for improved understanding of regional and global hydrological cycles, (b) retrieval of local geoid undulations over large lakes to enhance the precision of individual water level observations, and (c) mapping river longitudinal profiles and construction of time series of river levels. We believe that the findings of this study will pave the way for more comprehensive utilization of geodetic altimetry data in diverse hydrological applications, ultimately advancing inland water altimetry.

Utilization of Mean Dynamic Topography for GNSS-Leveling Applications (Case Study: Jakarta Area)

Conventionally, the determination of height that has physical meaning for a location or area can be determined geodetically by measuring the height difference relative to a reference point using the spirit-leveling method. This type of height, orthometric height, is essential to explain any physical phenomena, e.g., determining which direction fluid flows, which is used in many scientific and engineering applications. The coverage area is one limitation of conventional leveling, using waterpass precise leveling. The wider the coverage area, the more time and cost-consuming it will be. Another alternative to determine the height at a particular location is using the Global Navigation Satellite System – Leveling (GNSS-Leveling) method, which can be used to resolve the drawbacks of the previous method. GNSS-Leveling method is able to provide an efficient height determination solution if an accurate geoid undulation model and Mean Dynamic Topography (MDT) model are available over the area of interest. MDT is the height deviation between the geoid surface and the mean sea surface (MSS). Information related to geoid undulation and MDT is important to ensure any GNSS-Leveling measurement referring to the local Mean Sea Level (MSL) surface, which is widely needed in various applications. Applying the EGM2008 geoid model and altimetric-derived MDT solution to GNSS-Leveling measurements in the Jakarta area, the average orthometric height difference relative to the reference value is estimated to be about 5 cm. This shows that GNSS-Leveling is a promising solution when combined with geoid and MDT models.

Modelling local geoid undulations using unmanned aerial vehicles (UAVS): a case study of the Federal University of Technology, Akure, Nigeria

The study was aimed at developing a geoid model using Unmanned Aerial Vehicle (UAV) technology. To accomplish this, a UAV was deployed to capture imagery of the study area from a height of 150m, with a ground resolution of 4.19cm. A total of 3737 images were obtained, covering an area of 725.804 hectares. The existing ellipsoidal and orthometric heights were used to georeferenced the acquired images. For the analysis, 35 points were utilized, with 20 points designated as ground control points (GCPs) and the remaining 15 points as check points (CPs). Using the UAV-derived Digital Terrain Models (DTMs), a dataset comprising 18,492 points was generated for both ellipsoidal (h) and orthometric (H) heights. The differences between these heights, referred to as geoid heights (N), were calculated as N = h - H for all 18,492 points. These geoid heights were subsequently employed to generate a geoid model, including contour maps and 3D maps, of the study area. To assess the accuracy of the UAV-derived geoid heights, a root mean square error (RMSE) analysis was performed by comparing them with the existing geoid heights and was found to be 0.113 m. The scientific novelty and practical significance are in the development of a local geoid model of the study area with centimetre-level precision. Thus, the output of this study can be used for a wide range of applications, including land management, construction, and environmental impact assessments in the study area.

A Comparison of Gravimetric, Isostatic, and Spectral Decomposition Methods for a Possible Enhancement of the Mantle Signature in the Long-Wavelength Geoidal Geometry

A long-wavelength geoidal geometry characterizes the most pronounced features of the Indian Ocean geoid low and the West Pacific and North Atlantic geoid highs. These large geoid undulations (globally roughly within ±100 m) are mainly attributed to a deep mantle structure and large lithospheric density and geometry variations (such as the African superswell), while maximum geoid modifications by a topographic relief of Himalaya and Tibet are up to ~30 m. To enhance the mantle signature in a long-wavelength geoidal geometry, gravimetric, isostatic, and spectral decomposition methods can be applied. In this study, we demonstrate that isostatic schemes yield isostatic geoid models that closely resemble a long-wavelength geoidal geometry. The gravimetric method, on the other hand, modifies the mantle geoid significantly. Further modifications of the mantle geoid by removing gravitational contributions of lithospheric mantle density and lithospheric thickness variations should (optimally) enhance the signature of the deep mantle in the sub-lithospheric mantle geoid. Our results confirm this assumption by revealing (large-scale) positive anomalies in the Central Pacific and along the Atlantic Ocean that are coupled by two negative anomalies in the East Pacific and South Eurasia. A gravimetric method thus better enhances the mantle signature in the geoidal geometry than isostatic and spatial decomposition methods. Nonetheless, our results also indicate the presence of possibly large errors in geoid modelling results that limit their full implementation in gravimetric studies of the Earth’s mantle density structure without using tomographic images of the mantle and additional geophysical, geothermal, and geochemical constraints.

The Upper Mantle Geoid for Lithospheric Structure and Dynamics

AbstractGeoid anomalies offer crucial information on the internal density structure of the Earth, and thus, on its constitution and dynamic state. In order to interpret geoid undulations in terms of depth, magnitude and lateral extension of density anomalies in the lithosphere and upper mantle, the effects of lower mantle density anomalies need to be removed from the full geoid (thus obtaining the residual “upper mantle geoid”). However, how to achieve this seemingly simple filtering exercise has eluded consensus for decades in the solid Earth community. While there is wide agreement regarding the causative masses of degrees >10 in spherical harmonic expansions of the upper mantle geoid, those contributing to degrees <7–8 remain ambiguous. Here we use spherical harmonic analysis and recent tomography and density models from joint seismic‐geodynamic inversions to derive a representative upper mantle geoid, including the contributions from low harmonic degrees. We show that the upper mantle geoid contains important contributions from degrees 5 and 6 and interpret the causative masses as arising from the coupling between the long‐wavelength lithospheric structure and the sublithospheric upper mantle convection pattern. Importantly, the contributions from degrees 3 < l < 8 do not show a simple power‐law behavior (e.g., Kaula's rule), which precludes the use of standard filtering techniques in the spectral domain. Our upper mantle geoid model will be useful in studies of (a) lithospheric structure, (b) dynamic topography and mantle viscosity, (c) lithosphere‐asthenosphere interactions and (d) the global stress field within the lithosphere and its associated hazards.

Vertical accuracy evaluation free access digital elevation models (DEMs): case Fergana Valley in Uzbekistan

In this study, the vertical accuracy of the Shuttle Radar Topography Mission Digital Elevation Model Version 2.0 (SRTM30), the Advanced Spaceborne Thermal Emission and Reflection Radiometer Global DEM Version 2.0 (ASTER GDEM2), and Advanced Land Observing Satellite World 3D Digital Surface Model Version 2.1 (ALOS AW3D30) was statistically assessed using GPS data. The Fergana Valley area was chosen as a study region, where the land surface can reflect tectonic processes. The values of ellipsoidal heights of 27 points of the regional GPS network were chosen as reference data. The geometric approach using GPS/leveling data and EGM96 global geopotential model-based geoid undulations was applied for geoid surface fitting. The geoid height corrections range ranged from –0.66 m to 0.87 m. Root-Mean-Square errors of ~10.0 m, ~16.4 m, and ~6.6 m was obtained for SRTM30, ASTER GDEM2, and ALOS AW3D30, respectively. It was found that compared with the reference model, all the global DEMs in mountainous areas generally overestimated elevation and the value of vertical accuracy at a 90% confidence level by 3-6 meters exceeded the declared by distributors. But ALOS AW3D30 proved to be the most accurate DEM that best represents the topography of the earth’s surface and could be used for some engineering applications in Fergana Valley.

Assessment of latest global gravity field models by GNSS/Levelling Geoid

This paper focuses on making a comparing of GNSS/Levelling data and data obtained from global geopotential models. For comparison, geoid undulations obtained by GNSS/Levelling method and geoid undulations obtained from global geopotential models have been used. As global geopotential models, SGG-UGM-2, XGM2019e_2159, GO_CONS_GCF_2_TIM_R6e, ITSG-Grace2018s, EIGEN-GRGS.RL04.MEAN-FIELD, GOCO06s, GO_CONS_GCF_2_TIM_R6, GO_CONS_GCF_2_DIR_R6 GGMs are used. The data sets used in the improvements of the models are altimetry, satellite, location data and topography. The disparities between the geoid undulations obtained from the GNSS/Levelling method and geoid undulations obtained from global geoid models have been taken. Some statistical criteria for these differences have been calculated. These criteria, such as smallest, biggest, average, standard deviation, Root Mean Square RMS statistical values of deviations between GNSS/Levelling geoid and global geopotential models, are taken into consideration when comparing the models. According to the comparison, the global gravity field model that best fits the GNSS/Levelling is selected.

Can GNSS Derived Height Replace Levelling Height? - A Case of Low-Land of Nepal

Orthometric height is the generally adopted type of height worldwide and in geomatics community. Precise levelling has been the method of obtaining orthometric height in past for most of the country, so as the Nepal. However, due to wide usage of Global Navigation Satellite System (GNSS), the alternative approach of combining GNSS derived ellipsoidal height with geoid undulation to get GNSS derived orthometric height, has been used extensively. In Nepal, this technique was officially adopted in 2020 for Everest height measurement and understood as the efficient way to comply with levelling height. In this study, GNSS surveying was conducted on 15 stations located at the lowland region of Nepal and orthometric heights were obtained from GNSS and geoid method. When compared GNSS derived orthometric height with precise levelling height, the difference remained within threshold of 5cm for majority of observation stations. However, these differences are not sufficient to support the standards set for the third order levelling by Survey Department (SD). The accuracy of GNSS derived orthometric height can be significantly affected by various environment and existing resources such as existing accuracy of geoid, nature of precise levelling height. Considering the revisit upon these conditions, we expect GNSS-levelling as a strong alternative to time consuming, tedious, and costly precise levelling which is most suitable method of obtaining orthometric height in lowland topography at a precision less than 4 cm.

Horizontal gravity disturbance vector in atmospheric dynamics

All meteorological and oceanographic textbooks unintentionally define the Earth gravitation outside the solid Earth through treating the Earth as a point-mass located at the Earth center containing the entire Earth mass or assuming the Earth with uniform mass density. Combination of such simplified gravitational and centrifugal accelerations leads to the effective gravity geff. The geopotential surface corresponding to geff (called the effective geopotential surface) is the ellipsoidal surface. The true Earth gravitation outside the solid Earth is the volume integration of gravitation of all point-masses inside the Earth. Combination of true gravitational and centrifugal accelerations leads to the true gravity g. The geopotential surface corresponding to g (called the true geopotential surface) and respect to the Earth ellipsoidal reference surface is the geoid, which varies approximately from -106 m to 86 m worldwide. The true gravity g minus the effective gravity geff is the gravity disturbance vector, δg = g – geff, which is a major variable in geodesy and solid Earth dynamics. Contrarily, δg is neglected completely in dynamics of atmosphere and ocean. In this study, the true gravity g replaces the effective gravity geff in atmospheric dynamic equations. Seven non-dimensional numbers are proposed to identify the magnitude of horizontal gravity disturbance vector (δg)⊥versus traditional horizontal forcing terms such as pressure gradient force and Coriolis force. These non-dimensional numbers are calculated from two publicly available and independent datasets with the geoid undulation (N) from the static gravity field model EIGEN-6C4 and long-term mean geopotential height (Z), wind velocity (u, v), and temperature (T) at 12 pressure levels in troposphere from the NCEP/NCAR reanalyzed climatology. The results demonstrate (δg)⊥ nonnegligible in hydrostatic equilibrium, geostrophic wind, geostrophic vorticity, Q vector, and Omega equation.

Assessments of recent Global Geopotential Models based on GPS/levelling and gravity data along coastal zones of Egypt

Abstract The orthometric height has an essential role in a variety of civil engineering projects and it is defined as the length of the curved plumbline from a point (on the earth surface) to its intersection with the geoid surface. Leveling process is considered as the most accurate technique for obtaining these heights. However, regardless of its potentials, it is tedious, costly, and time consuming. Recently many organizations and research centers have developed multi Global Geopotential Models (GGMs) depending on several types of available gravity and height datasets to estimate orthometric heights from GNSS measurements. In this study, we present an evaluation and assessment of the accuracy of five of recent and popular GGMS: XGM2016, XGM2019e, EIGEN-6C4, GO_CONS_GCF_2_TIM_R6e, and EGM2008 using actual 145 GNSS/leveling points and 96 terrestrial gravity points. The goal of this research is to find the best fit model along the study area located along the coastal zones of Egypt with distances of about 1,970 km for further determination of geoid modeling at regional scale. The selection of these areas basically was due to their developmental, urban, and economical importance and their continuous need for protection works to fight against the coastal erosion caused by climate change and global warming. The results indicated that for geoid undulation, GO_CONS_GCF_2_TIM_R6e model is the best fit GGM for the estimation of geoid model along Mediterranean Sea coastal line, while XGM2019e_2159 model is the best suitable for coastal line of the Red Sea. And regarding the gravity anomalies, the most reliable GGMs for this study area are XGM2019e_2159 and EIGEN-6C4 for Bouguer and free-air gravity anomaly, respectively.

Performance Comparison of Deterministic and Stochastic Modifications in Stokes’s and Hotine’s Formulas: The Case of Jilin Province, China

The high-precision regional geoid model provides important fundamental geospatial information for developing and applying many disciplines. Deterministic and stochastic modifications are applied to Stokes’s and Hotine’s formulas of geoid modeling to reduce errors. Based on the Experimental Geopotential Model 2019 (XGM2019), this paper used Stokes’s and Hotine’s formulas to analyze the variation of global root mean square error (RMSE) with modification parameters for two deterministic (Wong and Gore; and Vaníček and Kleusberg) and three stochastic modifications (biased, unbiased, and optimum). Taking the quasigeoid refinement of Jilin Province as an example, the global RMSE, approximate geoid undulation, and additive corrections were calculated. The parameter analysis and the global RMSE calculation showed that the variation of the modification limits and the terrestrial gravity data error variance had a centimeter-level effect on the global RMSE. In contrast, the impact of the integration radius was relatively small. The stochastic modifications were better than the deterministic ones in calculating the global RMSE. The global RMSE of Hotine’s formula was smaller than that of Stokes’s, and its unbiased and optimum modifications reached the minimum value of 12.9 mm. The validation of XGM2019 and the refined quasigeoid based on the high accuracy GPS/leveling points showed that the standard deviation (STD) of XGM2019 was 5.8 cm in Jilin Province, and the refined optimal quasigeoid model was 2.9 cm. Stokes’s and Hotine’s formulas provided the same accuracy in the study area. In the western plain area, the accuracy of the deterministic modifications was 2.0 cm, which was about 0.4 cm higher than that of the stochastic modifications. In the eastern mountainous area, the stochastic modifications were better than the deterministic ones, and the accuracy was about 3.2 cm. Stokes’s and Hotine’s formulas based on deterministic and stochastic modifications significantly improve the accuracy of the XGM2019. The deterministic and stochastic modifications show millimeter-level differences in plain and mountainous areas.