Geoid Model Research Articles

A coastal mean sea surface with associated errors in Norway based on new-generation altimetry

Abstract The coastal mean sea surface (MSS) has applications within oceanography as well as geodesy. Together with a geoid model, it forms an important component for geodetic mapping of ocean surface currents that are in geostrophic balance. Furthermore, it forms a bridge between open ocean MSS and in situ measurements of mean sea level at or close to land, it contributes to the mapping of the geoid and the marine gravity field, and it is essential for connecting tidal nautical chart datums to physical height systems or global geodetic reference frames. In this study, we determine a coastal MSS with an associated error field for Norway. The MSS is solely based on new-generation altimetry data, i.e., SAR(In) data from Sentinel-3A and CryoSat-2, as well as Ka-band data from SARAL/AltiKa. The data sets partly overlap in time and cover the time period from 2010 to 2017 inclusive. We have chosen these altimeters because they represent evolutions of conventional altimetry, with reduced footprint sizes as a main benefit. This is especially advantageous in the coastal zone, as a smaller footprint reduces the probability of radar pulses being contaminated by energy backscattered from land areas. The satellite missions were harmonized by applying inter-mission biases determined in a regional crossover analysis. Furthermore, in a zone closer to land than 25 km, we have replaced the global ocean tide model with a regional ocean tide model provided by the Norwegian Mapping Authority (NMA). We use an optimal interpolation technique to determine a coastal MSS grid and discuss it in context of the estimated error field. We assess our coastal MSS by comparison to state-of-the-art MSS products along three sections perpendicular to the coast, as well as ellipsoidal mean sea level as observed by an array of permanent tide gauges within the study area. In addition, we assess a higher-resolution version of our MSS in the NMA testbed for vertical datums, by comparison with temporary tide gauges. We find that the coastal MSS outperforms the global MSS models directly at the coast, with standard deviations of differences of ~ 8 cm to the tide gauges, compared to 14–22 cm, obtained with the global MSS models. All MSS models largely agree along three sections perpendicular to the coast, with standard deviations of differences of 2–4 cm. The higher-resolution version of the coastal MSS performs similarly to the coastal MSS in comparison with the temporary tide gauges (standard deviation of differences of ~ 8 cm), but its formal error field also quantifies large uncertainties at the coast and in the fjords, mainly due to the lack of altimetry observations. A trustworthy error field is decisive for the combination of altimetry with other sea-level observations.

Relationship between Polynomial Geometric Surfaces Terms and Observation Points Numbers and Effect in the Accuracy of Geometric Geoid Models

The application of the geometric method of local geoid model determination which requires the fitting of geometric surfaces to known geoid heights to enable geoid heights of new points to be interpolated involves the use of least squares technique for computation of the models' parameters. The selection of polynomial geometric surfaces depends on the size of the study area, the variation of the geoid heights and the number of measurement points. The accuracy of the geometric geoid model increases as the number of observation points approximates the number of geometric surface terms. But in most cases, the number of observation points is not considered. To this effect, this paper presents the relationship between geometric surfaces terms and observation points numbers and effect in the accuracy of geometric geoid models. A total of 23 points of known local gravimetric geoid heights were used. Two polynomial geometric (third and fifth degrees) surfaces were fitted to the geoid heights at various observation point numbers and compared to determine the relationship between the number of model terms and that of observation points and effect in the accuracy of the models. Least squares adjustment technique was applied to obtain the model parameters. The differences between the models and the known geoid heights of the points were computed and used to obtain the RMSEs as well as the accuracy of the models. The obtained results showed that the accuracy of the polynomial geometric geoid models tends to the highest as the number of measurement points approximates the number of the model terms and in a unique solution where the number of observation points is equal to the number of the polynomial geometric model terms, the model accuracy is highest. The paper recommends that the geometric method of local geoid model determination should be strictly applied in small areas. Where the method will be applied in considerable large areas, higher degrees polynomial geometric surfaces with a larger number of terms approximating the number of observation points should be applied. This will enable a proper fit of the polynomial surface to the known geoid heights, as well as high accuracy to be obtained.

Residual least-squares collocation: use of covariance matrices from high-resolution global geopotential models

The paper presents a modified formulation of least-squares collocation. This residual least-squares collocation (RLSC) includes a remove--compute--restore procedure with a high-resolution global geopotential model (GGM) and a topographic gravitational potential model. In contrast to previous approaches, in RLSC, the remaining input residuals are modeled with error covariance matrices instead of signal covariance matrices. Therefore, we include the full variance--covariance information of a high-resolution GGM, namely the XGM2016, to the procedure. The included covariance matrices are anisotropic and location-dependent and enable a realistic error modeling of a target area. This fact represents an advantage over covariance matrices derived from signal degree variances or empirical covariance fitting. Additionally, due to the stochastic modeling of all involved components, RLSC provides realistic accuracy estimates. In a synthetic closed-loop test case with a realistic data distribution in the Andes we demonstrate the advantages of RLSC for regional geoid modeling and quantify the benefit which results mainly from a rigorously handled high-resolution GGM. In terms of root mean square deviations from the true reference solution, RLSC delivers an improvement of about 30% compared to a standard LSC approach, where the benefit is particularly pronounced in areas with a sparse data distribution. This improved performance, together with the fact that the resulting stochastic error estimates better reflect the true errors, might be an important aspect for the application of RLSC to derive gravity potential values and their uncertainties at reference stations of the international height reference system.

Application of the one-step integration method for determination of the regional gravimetric geoid

The regional gravimetric geoid solved using boundary-value problems of the potential theory is usually determined in two computational steps: (1) downward continuing ground gravity data onto the geoid using inverse Poisson’s integral equation in a mass-free space and (2) evaluating geoidal heights by applying Stokes integral to downward continued gravity. In this contribution, the two integration steps are combined in one step and the so-called one-step integration method in spherical approximation is implemented to compute the regional gravimetric geoid model. Advantages of using the one-step integration method instead of the two integration steps include less computational cost, more stable numerical computation and better utilization of input ground gravity data (reduced in each integration step to avoid edge effects). A discrete form of the one-step integral equation is used to convert mean values of ground gravity anomalies into mean values of geoidal heights. To evaluate mean values of the integral kernel in the vicinity of the computation point, a fast and numerically accurate analytical formula is proposed using planar approximation. The proposed formula is tested to determine the regional gravimetric geoid of the Auvergne test area, France. Results show a good agreement of the estimated geoid with geoidal heights estimated at GNSS-levelling reference points, with the standard deviation for the difference of 3.3 cm. Considering the uncertainty of geoidal heights derived at the GNSS/levelling reference points, one can conclude the geoid models computed by the one-step and two-step integration methods have negligible differences. Thus, the one-step method can be recommended for regional geoid modelling with its methodological and numerical advantages.

Alternativa para determinação de altitudes normais-ortométricas na Amazônia Legal Brasileira

El presente trabajo tiene por objetivo desarrollar una metodología alternativa para predicciones de altitudes normales-ortométricas a lo largo de la Amazonia Legal Brasileña, región que impone dificultades logísticas-operacionales para trabajos de campo y en la que hay considerable precariedad en la Red Altimétrica Fundamental Brasileña (RAFB), evidenciando necesidades de mejora de las redes en cuestión para una completa alineación a las actuales recomendaciones y esfuerzos para la consecución de una Red Internacional de Referencia Altimétrica. Una estrategia alternativa se basa en observaciones GNSS y en la utilización de modelos globales y geopotenciales (MGGs); alternativamente se utiliza el modelo geoidal nacional MAPGEO2015. Para ello, se utiliza un enfoque de comparación de tres variantes distintas: a) utilización de métodos absolutos o relativos para determinación altimétrica, b) utilización de extractos de los MGGs obtenidos en diferentes grados de desarrollo y del MAPGEO2015, y c) utilización de las funcionalidades altitud geoidal y anomalía de altitud para modelado de altitud normal-geoidal. Datos referentes a observaciones GNSS son extraídos de las estaciones de la rede SAT-GPS en coincidencia con estaciones RAFB, dotadas de información de altitudes elipsoidal y normal-ortométricas; los extractos de los MGG se obtienen punto a punto para los principales MGG combinados de la actualidad. En vistas de la baja precisión media de datos en el área de estudio y su baja vinculación con la RAFB, se adopta un área de prueba—el estado de Paraná— que se comporta mejor para la elaboración y consolidación de estrategias para aplicar posteriormente en el área de estudio. De los estudios realizados en el área de estudio, se verifica una incuestionable ventaja de utilización del método relativo sobre el absoluto aliado a los MGG, siendo en promedio un 72% mejores para extractos desarrollados hasta grados 2190 y 720; o MAPGEO2015 como una excepción a los resultados, no habiendo

Performance analysis of multi-GNSS static and RTK techniques in estimating height differences

ABSTRACTEstablishing reliable elevation differences is imperative for most geoscience and engineering applications. This work has traditionally been accomplished through spirit leveling techniques; however, surveyors have been utilizing satellite positioning systems in measuring height differences for more than a decade. Yet the quality of these heights needs to be evaluated in order to adopt them in different applications. In this article, we present the outcome of an accuracy assessment of height differences obtained with static and RTK surveys. Twenty control points with an average baseline length of 1 km were occupied with dual-frequency GNSS receivers for different time periods. Collected signals were processed using open-source software and verified with an online processing tool. Heights were estimated by processing the GPS and the GLONASS data individually, and combined (i.e. GNSS). Height differences were determined and compared with those measured by spirit levels and corrected through geoid models. Best results were achieved by combining GPS and GLONASS solutions for both static and RTK surveys. Solutions with either GPS or GLONASS satellites were comparable, but in most cases, the GPS solutions performed better. For the static surveys, longer occupation provided much accurate height differences. Inconsistencies among 10 different RTK surveys were minimum for the GPS + GLONASS solutions and worst for the GLONASS solutions. The ANOVA, LSD, F, and χ² statistical tests confirmed our findings at the 95% confidence level.

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

Spatially distributed studies may follow two broad approaches: points that are randomly located and points that are clustered or clumped together. They are discussed within a discipline described as geo-statistics. Geo-statistics is applied in order to interpolate geoid undulation (N) and hence generate through a geoid model orthometric heights from scattered/random or closely/clustered located controls. Control points are coordinated points within a primary/secondary geodetic survey network. Kriging method was adopted to produce topographical maps of the both scattered and closely scenarios. Accuracy computed revealed that standard deviation (σ) of multiquadratic and bicubic models in the study area are respectively 11cm and 14cm in lopsided control study area while over clustered distributed located controls are respectively 12cm and 15cm. Standard deviation with the lowest values among the determined geometric geoid models is at all times preferable scattered/random than closely/clumped scenario. This implies that the multiquadratic models can be applied across the entire study area with high accuracy/reliability irrespective of spatial distribution of the points. Hence, the accuracy of the models are better when the total number of points distributed within the entire study area was used than when a limited number of points within a particular part of the study area was used.

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling

Residual terrain modelling (RTM) plays a key role for short-scale gravity modelling in physical geodesy, e.g. for interpolation of observed gravity and augmentation of global geopotential models (GGMs). However, approximation errors encountered in RTM computation schemes are little investigated. The goal of the present paper is to examine widely used classical RTM techniques in order to provide insights into RTM-specific approximation errors and the resulting RTM accuracy. This is achieved by introducing a new, independent RTM technique as baseline that relies on the combination of (1) a full-scale global numerical integration in the spatial domain and (2) ultra-high-degree spectral forward modelling. The global integration provides the full gravity signal of the complete (detailed) topography, and the spectral modelling that of the RTM reference topography. As a main benefit, the RTM baseline technique inherently solves the “non-harmonicity problem” encountered in classical RTM techniques for points inside the reference topography. The new technique is utilized in a closed-loop type testing regime for in-depth examination of four variants of classical RTM techniques used in the literature which are all affected by one or two types of RTM-specific approximation errors. These are errors due to the (1) harmonic correction (HC) needed for points located inside the reference topography, (2) mass simplification, (3) vertical computation point inconsistency, and (4) neglect of terrain correction (TC) of the reference topography. For the Himalaya Mountains and the European Alps, and a degree-2160 reference topography, RTM approximation errors are quantified. As key finding, approximation errors associated with the standard HC ( $$ 4\pi G\rho H_{\text{P}}^{\text{RTM}} ) $$ may reach amplitudes of ~ 10 mGal for points located deep inside the reference topography. We further show that the popular RTM approximation ( $$ 2\pi G\rho H_{\text{P}}^{\text{RTM}} - {\text{TC}} $$ ) suffers from severe errors that may reach ~ 90 mGal amplitudes in rugged terrain. As a general conclusion, the RTM baseline technique allows inspecting present and future RTM techniques down to the sub-mGal level, thus improving our understanding of technique characteristics and errors. We expect the insights to be useful for future RTM applications, e.g. in geoid modelling using remove–compute–restore techniques, and in the development of new GGMs or high-resolution augmentations thereof.

Precise geoid computation using Stokes-Helmert's scheme and strict integrals of topographic effects

The high-precision local geoid model was computed based on the improved Stokes-Helmert′s boundary value problem and strict integrals of topographic effects. This proposed method involves three steps. First, the mathematical form of Stokes-Helmert′s boundary value problem was derived, and strict computational formulas regarding topographic effects were provided to overcome the disadvantage of planar approximations. Second, a gravimetric geoid model was constructed using the proposed Stokes-Helmert′s scheme with a heterogeneous data set. Third, a least squares adjustment method combined with a multi-surface function model was employed to remove the bias between the gravimetric geoid model and the GNSS/leveling data and to refine the final local geoid model. The accuracy of the final geoid model was evaluated using independent GNSS/leveling data. Numerical results show that an external precision of 1.45 cm is achievable.

Formulation and validation of a global laterally varying topographical density model

The distribution of masses within topography is extremely important for the evaluation of a rigorous geoid model. Most present day models of topographical density are either determined via gravity field inversions or have too coarse a resolution to be useful on a regional level. The ideal solution would be to have a 3-dimensional (3D) density model but such a dataset does not currently exist and would be quite complicated and expensive to develop. As a substitute, a 2-dimensional (2D) topographical density model is presented here through the application of a high resolution lithological map. By assigning probable surface density values and error estimates to the lithologies based on geological data, the Global Lithology Model (GLiM) is transformed into a laterally varying topographical density model (UNB_TopoDens). This model is then validated against present-day global and regional topographical density models. Apart from some water and ice covered regions, the UNB_TopoDens model agrees with all regional and global models at our disposal with the differences all falling within the 95% confidence region associated with the UNB_Topodens model. UNB_TopoDens is the first laterally varying topographical density model with associated error estimates that spans the globe at a 30″ x 30″ resolution and is therefore extremely useful for investigations of the crust and mantle, isostasy, gravimetric studies and various other disciplines.

IRG2018: A regional geoid model in Iran using Least Squares Collocation

The aim of this study is to determine an accurate geoid model for Iran based on the Least Squares Collocation method in the framework of the Remove — Compute — Restore technique. In areas suffering from a lack of homogeneous and accurate gravity anomaly data, as is the case of Iran, the choice of the most compatible global gravity model has a significant impact on the estimated form of the geoid. Different combined and satellite-only global gravity models were therefore analyzed for Iran, and EIGEN6C4 was selected as the best one. The Shuttle Radar Topography Mission height model was used for the residual terrain correction. The covariance modeling, a crucial step in the Least Squares Collocation method, was based on two strategies. In the first, the study area was divided into four sub-areas, and then an individual empirical covariance was computed and a covariance model fitted to each of them. In the second, an empirical covariance was computed using all terrestrial gravity data, and a unique covariance model was fitted to it. Despite some border effects, the former strategy showed slightly better performance according to the resulting statistics, and therefore it was preferred for the estimation of the geoid model called IRG2018. To remove the offset of IRG2018 with respect to GNSS/Leveling-derived geoid heights, two alternative approaches were tested: subtracting a fitting polynomial surface or directly using the GNSS/Leveling data as an input to the IRG2018 computation process. Evaluation of the results, based on an independent control set of approximately half of available GNSS/Leveling points, showed an advantage of the latter approach, with an estimated accuracy of about 20 cm in terms of RMS.

Construction of regional geoid using a virtual spherical harmonics model

Abstract The spherical cap harmonics (SCH) method can be used in regional geoid modeling. The core of this approach is the computation of its associated Legendre functions (ALF) with non-integer degree. However, it is unlikely to obtain a large number of zero-root values for the non-integer ALF. To overcome this problem, a new approach called virtual spherical harmonics (VSH) is proposed in this paper to transform the cap range into the whole sphere so that unlimited numbers of zero-root values can be obtained. The new approach was tested using four cap ranges with the radii of 30 ∘ {30^{\circ }} , 15 ∘ {15^{\circ }} , 10 ∘ {10^{\circ }} and 5 ∘ {5^{\circ }} , and geoid undulations for each of the regions are calculated from EGM2008. For each of the regions, the geoid undulations were used to construct three models with three different degrees of 20, 30 and 40. Numerical results showed that with the increase in the degree of the VSH model, the value of the maximum error decreases; and the maximum error of the model was less than 1 mm while the maximum degree is 40.

The 5 mm geoid model for Estonia computed by the least squares modified Stokes’s formula

Computational stages of a new high-resolution 5 mm quasigeoid model for Estonia are explained. Certain requirements for the quality and coverage of gravity data in the context of 5 mm geoid modelling were fulfilled. The present gravity data coverage over the target area is 1 point per 10 km2, less dense in adjacent land and marine areas. The average uncertainty estimate of gravity data within the target area is about 0.5 … 0.75 mGal. The gravity data were merged, cleaned, analysed, reduced and gridded to yield a seamless gravity anomaly field. A least squares modified Stokes formula (Sjöberg, L.E., 1991. Refined Least Squares Modification of Stokes Formula”. Manuscripta Geodaetica, 16, 367–375.) that combines local terrestrial gravity anomalies and the global geopotential model derived long-wavelength component in a truncated Stokes’s integral yielded the best gravimetric geoid modelling results. The combined uncertainty for most of the Estonian GNSS-levelling points is not exceeding 5 mm, allowing an adequate verification of the geoid modelling quality. Inter-comparison of the geoid model, GNSS-derived and spirit-levelled heights at discrete points was conducted for geoid modelling assessment. A two stage stochastic spatial prediction was applied to obtain an optimal fit between precise GNSS-levelling data and the geoid model. The location-specific post-fitting uncertainties of the resulting model EST-GEOID2017 revealed standard deviation of 4.2 mm, i.e. the same level as the accuracy of the used GNSS-levelling control points.

Derivation of Theoretical Gravity Model on the Clarke 1880 Ellipsoid for Practical Local Geoid Model Determination

Derivation of Theoretical Gravity Model on the Clarke 1880 Ellipsoid for Practical Local Geoid Model Determination

Assessment of the latest generation GOCE-based global gravity field models using height and free-air gravity anomalies over South Africa

The latest Gravity field and steady-state Ocean Circulation Explorer (GOCE) global gravity field models (GGMs) are evaluated over South Africa using 118 GPS/levelling, 507 gravity and 88,968 SAGEOID10 data points. It is observed that GOCE data can improve quasi-geoid model in the long-to-medium wavelength components over South Africa. GOCE-based GGMs perform better than EGM2008 in recovering height anomalies up to 240°. Related results are obtained for free-air gravity anomalies, but the improvement over EGM2008 by GOCE data is only up to 180°. TIM5 (up to 240°) is selected as a candidate GOCE model and combined with EGM2008 (from 241 up to 2190°). This model (TIM5/EGM2008) is evaluated by observed data (GPS/levelling and gravity) and the latest hybrid quasi geoid model over South Africa (SAGEOID10). The solutions converge at 1200 and 1800° for height and free-air gravity anomalies, respectively, when TIM5/EGM2008 is used. Results show that TIM5/EGM2008 can effectively recover height and free-air gravity anomalies over South Africa. The standard deviations of the differences between observed (118 GPS/levelling and 507 gravity points) and TIM5/EGM2008 implied height and free-air gravity anomalies are ± 23.7 cm and ± 9.0 mGal, respectively. In addition, the standard deviation of the differences between SAGEOID10 and TIM5/EGM2008 implied height anomalies (at 88,968 grid points) is ± 21.7 cm. The results for height anomalies from TIM5/EGM2008 are comparable with the latest gravimetric quasi-geoid model over South Africa.

Comparing Coincident Elevation and Freeboard From IceBridge and Five Different CryoSat-2 Retrackers

The airborne Operation IceBridge and spaceborne CryoSat-2 missions observe polar sea ice at different spatial and temporal scales as well as with different sensor suites. Comparison of data products from IceBridge and CryoSat-2 is complicated by the fact that they use different geophysical corrections: reference ellipsoid, geoid model, tide model, and atmospheric corrections to derive surface elevation and sea-ice freeboard. In this paper, we compare sea-ice surface elevation and freeboard using eight coincident CryoSat-2, Airborne Topographic Mapper (ATM), and Land, Vegetation, and Ice Sensor (LVIS) observations with direct IceBridge underflights of CryoSat-2 ground tracks. We apply identical geophysical corrections to CryoSat-2 and IceBridge data to eliminate elevation biases due to these differences and focus on differences due to retracker performance. The IceBridge ATM and LVIS elevation and freeboard and Snow Radar snow depth data sets are averaged to each CryoSat-2 footprint for comparison. With snow depth measurements, we are able to compare elevations and freeboards at the snow/ice interface for five different CryoSat-2 retrackers (ESA, GSFCv1, AWI, JPL, and GSFCv2) and IceBridge. The overall mean of freeboard differences between GSFCv2, ESA, AWI, JPL retrackers, and ATM are in agreement within 0.05 m. However, the five different CryoSat-2 retrackers show distinct differences in mean elevation over leads and over floes. This suggests that the physical interpretation of the different retrackers needs to be considered depending on usage, for example, elevations from CryoSat-2 retrackers need to be carefully calibrated before comparing with elevation from other satellites for long-term surface elevation trends.

Shipborne gravimetry in the Baltic Sea: data processing strategies, crucial findings and preliminary geoid determination tests

Shipborne gravimetry is an essential method to measure the Earth’s gravity field in the coastal and offshore areas. It has the special advantages of high-accuracy and high-resolution measurements in coastal areas compared to other techniques (e.g., satellite gravimetry, airborne gravimetry, and altimetry) used to obtain information about the gravity field. In this paper, we present the data processing strategies of shipborne gravimetry in GFZ. One key point is that the most suitable filter parameters to eliminate disturbing accelerations are determined by studying the GNSS-derived kinematic vertical accelerations and the measurement differences at crossover points. Apart from that, two crucial issues impacting on shipborne gravimetry are the seiches in some harbors and the squat effect in the shallow water. We identified that inclusion of GNSS-derived kinematic vertical accelerations can help to improve the shipborne gravimetry results at these special cases in the Baltic Sea. In the absence of the GNSS-derived vertical accelerations, the cutoff wavelength of the low-pass filter should be large enough to filter out these disturbing acceleration signals which causes a coarser spatial resolution of the gravity measurements. Therefore, the GNSS-derived kinematic vertical accelerations are very useful for optimum shipborne gravimetry. Finally, our shipborne gravimetry measurements are successfully used to verify the previous gravimetry data and improve the current geoid models in the Baltic Sea.

Comparison of Three Gravimetric-Geometric Geoid Models for Best Local Geoid Model of Benin City, Nigeria

The conversion of geometric as well as ellipsoidal heights from GNSS observations to practical heights for engineering constructions has necessitated the determination of the local geoid model of areas. Benin City is a developing area which requires a local geoid model for conversion of geometric heights to orthometric heights for physical developments in the area. This paper is on the best local geoid model of Benin City, Nigeria by comparing three gravimetric-geometric geoid models of the study area. GNSS and gravimetric observations were carried out on 49 points to respectively obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid, subtracted from the absolute gravity values and corrected for the air (free air) to obtain the free air gravity anomalies of the points. The computed free air gravity anomalies were applied to compute the geoid heights of the points using the integration of the modified Stokes integral. Three geometric geoid surfaces (plane, second degree and third degree surfaces) were fitted to the computed gravimetric geoid heights using the least squares technique to obtain the gravimetric-geometric geoid models of the study area. The RMSE of the three gravimetric-geometric geoid models were computed to determine their (the models) accuracy. The three gravimetric-geometric geoid models were compared using their accuracy to obtain the most suitable geoid model of the study area. The results of the comparison showed that the third degree gravimetric-geometric geoid model is most suitable for application in the study area. It is recommended that ellipsoidal heights obtained from GNSS observation within Benin City, Nigeria should be converted to orthometric heights using the third degree geoid model.

Geoid determination with Hotine’s integral based on terrestrial gravity data in Semarang city

GNSS surveys can accurately determine the geodetic position, but the elevation values do not have physical meaning for the purposes of engineering work. Physical height can be determined through the gravimetric geoid model representing the relationship between the geometric and physical shapes of the earth. Theoretically, the gravimetric geoid model is a solution of Boundary Value Problems (BVP) to obtain the boundary value in the form of the geoid surface. Before the era of GNSS, a BVP was calculated using Stokes or Molodensky approaches. Another approach that was quite difficult to be practiced in that era was Hotine approach. This approach requires gravity disturbance data on surface of the earth or on the geoid itself. This paper presents an accurate gravimetric geoid modelling based on terrestrial gravity data in Semarang using Hotine’s approach. The main gravity data was measured in March 2016 using Scintrex CG-5 gravimeter which has an accuracy of 5 microgal. It covered an area about 1 arc degree squares. Additional gravity data for improving the accuracy of the geoid models was measured by some government and private agencies using analogue gravimeters. It covered Jawa island. The global geopotential model was GECO with maximum degree 2190. The results showed that the computation of Hotine’s integral for modelling the geoid of Semarang city had an accuracy of ± 0.044 m. In the region of land subsidence, the geoid model is quite prospective to be implemented as the Vertical Reference System (VRS). Measurement of elevation of a distant point using geoid can be completed in a matter of hours, while using spirit levelling completed within a matter of days.