Geoid Undulations Research Articles

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.

Accuracy assessment of the PGM17 global geopotential model: a case study of Egypt and Northeast Africa

Global geopotential models play a fundamental role in height transformation to convert the Global Navigation Satellite Systems (GNSS)-based ellipsoidal heights into orthometric heights in several geosciences applications. The U.S. National Geospatial-Intelligence Agency (NGA) is preparing to release the Earth Geopotential Model 2020 (EGM2020) in the near future. Prior to that official release, two primary geopotential models (PGM17 and PGM18) will be distributed to selective members for assessment in different regions worldwide. Within the activities of that working group, the new PGM17 geopotential model has been obtained and judged over a regional scale in northeast Africa, and over a national basis of Egypt. The assessment has been carried out using terrestrial gravity and GPS/leveling datasets within a GIS environment. The utilized dataset consists of 231 terrestrial gravity points in northeast Africa, 247 measured gravity points in Egypt, and 978 GNSS/leveling stations in Egypt. Three other GGMs have been included in the comparison stage, namely GECO, EIGEN-6C4, and EGM2008. For each model, grids of gravity anomalies and geoidal undulations have been computed and, then, compared against the corresponding values at known terrestrial data. Next, statistical measures of mean, range, and standard deviations of the discrepancies have been analyzed. For northeast Africa area, the accomplished results indicated that the PGM17 attains 100% improvements over the old EGM2008, while the GECO, and EIGEN-6C4 models resulted in improvements of 11%, and 1% respectively. Accordingly, it can be concluded that PGM17 represents the gravitational field over northeast Africa in a more enhanced manner. On a national scale, the GGMs assessment has been performed, over Egypt, in two steps: using gravity data, and using GNSS/leveling data. The findings emphasize that PGM17 noticeably enhance the performance of EGM2008, when compared against terrestrial gravity stations in Egypt, by almost 55%. Furthermore, the assessment over GPS/leveling stations concluded that the overall improvements of PGM17 are approximately 54%. Other GGMs did not produce a comparable improvement level in both case studies. Thus, it can be concluded that PGM17 is significantly precise than the other investigated GGMs in representing the gravitational field over Egypt in particular and northeast Africa in general.

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.

Crustal and Mantle Structure Beneath the Southern Payenia Volcanic Province Using Gravity and Magnetic Data

AbstractThe Auca Mahuida volcanic field lies on the southernmost Payenia Volcanic Province, one of the broadest retroarc volcanic plateaux in the southern Central Andes (~38°S). This voluminous basaltic flooding of Quaternary age was originated from a deep asthenospheric source, interpreted as a mantle plume product of changing slab dynamics. The geometry of this source is deduced from magnetotelluric data, but the limited spatial coverage of this array does not allow a detailed resolution of this anomaly. In order to present a detailed geometry of the conductive anomaly and related crustal magmatic bodies, we used multiple data sources. We combined Magnetic and Bouguer anomalies, Curie isotherm depth (Tc), Elastic Thickness (Te) and Moho depth derived from the Global Earth Magnetic Anomaly Grid (EMAG2) and terrestrial gravity measurements, all together in a holistic geophysical analysis. The magnetic data depict a nearly 200‐km‐in‐diameter circular anomaly that would correspond to a dense body according to the Bouguer anomaly. Geoid data from the Gravity Field Model (EIGEN‐6c4) have been filtered in order to isolate deeper mass influences and visualize the asthenospheric upwelling previously described from magnetotelluric data. Moho inversion yields a crustal attenuation at 36‐ to 32‐km depth coinciding with Te below 20‐km depth and a shallow Tc (≤15‐km depth) at the site where Geoid positive undulation was calculated. Finally, surface analysis allowed defining a topographic swell, compatible with the dimensions of the identified magnetic anomaly, where the main rivers deviated, potentially due to a recent base level change.

Deriving a Geoid Undulation Network over Egypt by Merging Data of Different Global Geo-Potential Models

Deriving a Geoid Undulation Network over Egypt by Merging Data of Different Global Geo-Potential Models

Optimal biased Kriging: Homeogram tapering and applications to geoid undulations in Korea

AbstractThis article studies the Optimal Biased Kriging (OBK) approach which is an alternative geostatistical method that gives up the unbiasedness condition of Ordinary Kriging (OK) to gain an improved Mean Squared Prediction Error (MSPE). The system of equations for the optimal linear biased Kriging predictor is derived and itsMSPE is compared with that of Ordinary Kriging. A major impediment in implementing this system of equations and performing Kriging interpolation with massive datasets is the inversion of the spatial coherency matrix. This problem is investigated and a novel method, called “homeogram tapering”, which exploits spatial sorting techniques to create sparse matrices for efficient matrix inversion, is described. Finally, as an application, results from experiments performed on a geoid undulation dataset from Korea are presented. A precise geoid is usually the indispensable basis for meaningful hydrological studies over wide areas. These experiments use the theory presented here along with a relatively new spatial coherency measure, called the homeogram, also known as the non-centered covariance function.

How to Calculate Bouguer Gravity Data in Planetary Studies

In terrestrial studies, Bouguer gravity data is routinely computed by adopting various numerical schemes, starting from the most basic concept of approximating the actual topography by an infinite Bouguer plate, through adding a planar terrain correction to account for a local/regional terrain geometry, to more advanced schemes that involve the computation of the topographic gravity correction by taking into consideration a gravitational contribution of the whole topography while adopting a spherical (or ellipsoidal) approximation. Moreover, the topographic density information has significantly improved the gravity forward modeling and interpretations, especially in polar regions (by accounting for a density contrast of polar glaciers) and in regions characterized by a complex geological structure. Whereas in geodetic studies (such as a gravimetric geoid modeling) only the gravitational contribution of topographic masses above the geoid is computed and subsequently removed from observed (free-air) gravity data, geophysical studies focusing on interpreting the Earth’s inner structure usually require the application of additional stripping gravity corrections that account for known anomalous density structures in order to reveal an unknown (and sought) density structure or density interface. In planetary studies, numerical schemes applied to compile Bouguer gravity maps might differ from terrestrial studies due to two reasons. While in terrestrial studies the topography is defined by physical heights above the geoid surface (i.e., the geoid-referenced topography), in planetary studies the topography is commonly described by geometric heights above the geometric reference surface (i.e., the geometric-referenced topography). Moreover, large parts of a planetary surface have negative heights. This obviously has implications on the computation of the topographic gravity correction and consequently Bouguer gravity data because in this case the application of this correction not only removes the gravitational contribution of a topographic mass surplus, but also compensates for a topographic mass deficit. In this study, we examine numerically possible options of computing the topographic gravity correction and consequently the Bouguer gravity data in planetary applications. In agreement with a theoretical definition of the Bouguer gravity correction, the Bouguer gravity maps compiled based on adopting the geoid-referenced topography are the most relevant. In this case, the application of the topographic gravity correction removes only the gravitational contribution of the topography. Alternative options based on using geometric heights, on the other hand, subtract an additional gravitational signal, spatially closely correlated with the geoidal undulations, that is often attributed to deep mantle density heterogeneities, mantle plumes or other phenomena that are not directly related to a topographic density distribution.

GLOBAL GEOID ADJUSTMENT ON LOCAL AREA FOR GIS APPLICATIONS USING GNSS PERMANENT STATION COORDINATES

Orthometric heights, useful for many engineering and geoscience applications, can be obtained by GPS (Global Positioning System) surveys only when an accurate geoid undulation model (that supplies the vertical separation between the geoid and WGS84 ellipsoid) is available for the considered topic area. Global geoid height models (i.e., EGM2008), deriving from satellite gravity measurements suitably integrated with other data are free available on web, but their accuracy is often not sufficient for the user’s purposes. More accurate local models can nevertheless be acquired, but often only for a fee. GPS/levelling surveys are suitable for determining a local, accurate geoid model, but may be too expensive. This paper aims to demonstrate that GNSS (Global Navigation Satellite System) Permanent Station documents (monographs), freely available on the web and supplying orthometric and ellipsoidal heights, permit to calculate precise geoidal undulations useful to perform global geoid modelling on a local area. In fact, in this study 25 GNSS Permanent Stations (GNSS PS), located in North-Western Italy are considered: the differences between GNSS PS geoidal heights and the corresponding EGM2008 1′ × 1′ ones are used as a starting dataset for Ordinary Kriging applications. The resulting model is summed to the EGM2008 1′ × 1′, obtaining a better-performed model of the interest area. The accuracy tests demonstrate that the resulting model is better than EGM2008 grids to produce contours from a GPS dataset for large-scale mapping.

Computations of Geoid Undulation from Comparison of GNSS/Levelling with EGM 2008 for Geodetic Applications

Computations of Geoid Undulation from Comparison of GNSS/Levelling with EGM 2008 for Geodetic Applications

Computations of Geoid Undulation from Comparison of GNSS/Levelling with EGM 2008 for Geodetic Applications

Consistency is an important characteristic in height systems which the mean sea level (msl) surface cannot guarantee. Only a geoid surface can provide height consistency. The quality of geoid undulation (N) will obviously affect the resulting orthometric height (H) determined from GNSS. The geoid undulation may be global, regional/national and local. Online software CSRS-PPP was used for post processing rinex data. 𝑁𝐸𝐺𝑀2008 was computed from AllTrans EGM2008 geoid calculator while h was used to compute 𝑁𝐺𝑃𝑆 from the relationship N= h-H. H is the existing orthometric height. Twenty-four controls with FCT 260 P as base reference station were used for this study. The computed standard deviation of differences in 𝑁𝐺𝑃𝑆 − 𝑁𝐸𝐺𝑀2008 (σ) is used as accuracy indicator and σ =0.419m .The root mean square error (RMSE) is 0.934m. This indicates the quality and reliability of the geoid undulation from the EGM2008 model. Comparing the observed 𝑁𝐺𝑁𝑆𝑆 and 𝑁𝐸𝐺𝑀2008, the use of global models may not satisfy the accuracy level of orthometric height desired for local applications in the FCT, Abuja. GNSS (GPS) may be used along with local geoid model as a way to acquire acceptable orthometric height. The smaller the 𝑁𝐺𝑃𝑆-𝑁𝐸𝐺𝑀2008 makes it better model. The range of 1.585m from (𝑁𝐺𝑃𝑆-𝑁𝐸𝐺𝑀2008) in this study is a strong indication that global models should be avoided as much as possible in local applications.

Computations of Geoid Undulation from Comparison of GNSS/Levelling with EGM 2008 for Geodetic Applications

Computations of Geoid Undulation from Comparison of GNSS/Levelling with EGM 2008 for Geodetic Applications

A Gravimetric Approach for the Determination of Orthometric Heights in Akure Environs, Ondo State, Nigeria

An important component in position determination of any point is its height. Orthometric height determination has a significant role in geodesy, and it has wide-ranging application in numerous fields and activities. Orthometric height is the height above or below the geoid along the gravity plumb line. Orthometric height system is preferred to ellipsoidal height by many because of its relationship with the mean sea level (MSL). Determination of orthometric height for Akure environs was carried out using gravimetric approach by determining geoidal undulation via Stokes integral and ellipsoidal heights via GPS observations. A total of 59 stations within the study area were occupied for gravity observation using Lacoste and Romberg (G-512 series) and its complete accessories. South GNSS instrument was used for position and ellipsoidal height determination in static mode. In order to determine orthometric heights of the study area, the difference between the tailored geoid height (N Taylor ) and the ellipsoidal heights (h GPS ) were computed. The result shows that the determined orthometric height had an accuracy of 10.6121m standard deviation with a standard error of 1.38159m. The contour map and the 3D surface map of the computed orthometric heights were produced for the study area using kriging gridding method embedded in Surfer 11 software. The study recommends the use of gravimetric method in orthometric height determination when the tailored geoid height is computed. Keywords: Orthometric height, ellipsoidal height, Gravimetric approach, Gravity values, and geoidal height

Practical Local Geoid Model Determination for Mean Sea Level Heights of Surveys and Stable Building Projects

A local geoid model for Evboriaria, Benin City using the geometric (GPS/Levelling) method was determined for calculation of mean sea level heights. Fifty points were established for the model and ten points were used for interpolation. The geoid heights were determined by finding the difference between the observed orthometric heights and the ellipsoidal heights. The polynomial regression model D was used for the interpolation of the orthometric heights. The computed mean standard deviation between the observed orthometric heights and the interpolated orthometric heights was ± 21cm. A mean geoidal undulation of 28.410m was computed using the gravimetric method. The computed orthometric heights using the gravimetry mean geoidal undulation were compared with the observed orthometric heights and seen to be identical. It is recommended that orthometric heights of project areas should be determined from GPS observations with the local geoid model of the area also determined.

Geoid Determination and Gravity Works in Nepal

Gravimetric geoid plays the important role in the process of local/regional geoidal undulation determination. This approach uses the residual gravity anomalies determined by the surface gravity measurement using the gravimeter together with best fit geopotential model, with the geoid undulations over the oceans determined from the method of satellite altimetry. Mass distribution, position and elevation are prominent factors affecting the surface gravity. These information in combination with geopotential model helps in satellite orbit determination, oil, mineral and gas exploration supporting in the national economy. The preliminary geoid thus computed using airborne gravity and other surface gravity observation and the accuracy of computed geoid was likely at the 10-20cm in the interior of Nepal but higher near the border due to lack of data in China and India. The geoid thus defined is significantly improved relative to EGM –08 geoid.

Long term variation of sea level anomaly (September 1992-January 2017) in the Indonesian sea from multi-mission satellite altimetry data

Sea level anomaly (SLA) is one of the oceanic parameters that play a crucial role in the ocean dynamics. In the Indonesian region, spatio-temporal characteristic of SLA is significantly affected by local topographic features of the region. In this research, we investigate long-term variation of SLA over the region based on Topex/Poseidon, Jason-1 and Jason-2 data, during the period of Sept 1992 to January 2017. After removing some geophysical effects, time series of SLA values at each point of the collinear tracks are analyzed. The result show the long term MSSH in Indonesia area have increase from western part of Indonesia to eastern part with range -40 m to 80 m. The long term SST in Indonesia have almost same trend with the deviation from geoid undulation full degree with short wavelength type, and this phenomena represent the local effect such as shallow bathymetry and close sea. And for long term SLA, Indonesia shows positive value that the anomaly increasing with variable from western part to eastern part of Indonesia from 2.5 mm/year until 7 mm/yr.

Evaluation of global geopotential models: a case study for India

This paper aims to identify most suitable global geopotential model (GGM) for India, by comparing 15 GGMs developed through 1996 to 2017. The GGM derived geoid undulation values are compared with the geometrical undulation values obtained from GNSS/levelling data on Indian vertical datum. A correction term is added to the computed GGM derived geoid undulation value after fitting three-, four-, five- and seven-parameter models to account for bias and tilt between local geometric Indian vertical datum and global gravimetric vertical datum. The results indicate that EGM2008 model is the best GGM available for India with root-mean-square error (RMSE) of 0.28 m, without model fitting. However, after considering the systematic errors in the two datums with seven-parameter model, GECO GGM has shown significantly better results with RMSE of 0.19 m for India.

Reactivation of the Venezuelan vertical deflection data set from classical astrogeodetic observations

Astrogeodetic vertical deflections (VDs) are gravity field functionals which are independent from any other field observation such as gravity accelerations from gravimetry or geoid undulations from GPS and geometric levelling. They may be useful for the validation of global geopotential models or height transfer via GPS and astronomical levelling. VDs are sensitive to the local mass-distribution, so can be used in geophysical studies, too. Over Southern Hemisphere continents in general and South America in particular, VDs are exceptionally rare. This paper describes the reactivation of a unique VD data set that extends over parts of the Andes Mountains in Venezuela. The VD data was acquired 1983 and 1985 with classical astrogeodetic instrumentation at 24 field stations along a ∼80 km traverse crossing the Cordillera de Mérida with observation site elevations as high as ∼4500 m. To be compatible with modern geocentric gravity field products, the geodetic coordinates of the VD sites were transformed from the historic (non-geocentric) Venezuelan reference system to the geocentric ITRF2014, with residuals smaller than ∼1 m. In the ITRF, the measured VDs have RMS signal strengths of ∼20 arc-seconds (North-South) and ∼14 arc-seconds (East-West), with magnitudes exceeding 60 arc-seconds at one benchmark. The observed VDs were compared against VDs from GRACE, GOCE and EGM2008 data and from the ultra-high resolution GGMplus gravity maps. The GGMplus model was found to capture ∼85 to 90% (in terms of root-mean-square signals) of the measured VD signals. Both VD components are in ∼2 arc-sec agreement with GGMplus. Overall, the agreement between observed VDs and modelled VDs is considered satisfactory, given the VDs were measured in a topographically rugged region, where residual signals may be large and global models are not well supported through regional terrestrial gravity data. The VDs may be useful, e.g., for the assessment of high-frequency constituents of present and future high-degree gravitational models (e.g., EGM2020) and calibration of model commission errors. The Venezuelan VD data is freely available.

Geostatistical prediction of a local geometric geoid - kriging and cokriging with the use of EGM2008 geopotential model

We investigate prediction abilities of different variants of kriging and different combinations of data in a local geometric (GNSS/leveling based) geoid modeling. In order to generate local geoid models, we have used GNSS/leveling data and EGM2008 geopotential model. EGM2008 has been used twofold. Firstly, it was used as a basic long wave-length trend to be removed from geoid undulation data to generate a residual field of geoid heights modeled later by kriging (remove-restore technique). Secondly, EGM2008-based undulations were used as a secondary variable in a cokriging prediction procedure (as pseudo-observations). Besides the use of EGM2008, the kriging-based local geometric geoid models were generated only on the basis of raw undulations data. Kriging itself was used in two variants, i.e. ordinary kriging and universal kriging for univariate and bivariate cases (cokriging). The quality of kriging-based prediction for all its variants and all data combinations have been investigated on one fixed validation dataset consisting of 86 points and three training data sets characterized by a different density of sampling. Results of this study indicate that incorporation of EGM08 as a long wave-length trend in kriging prediction procedure outperforms cokriging strategy based on incorporation of EGM08 as a secondary spatially correlated variable.

ERRORS MEASUREMENT OF INTERPOLATION METHODS FOR GEOID MODELS: STUDY CASE IN THE BRAZILIAN REGION

Abstract: The geoid is an equipotential surface regarded as the altimetric reference for geodetic surveys and it therefore, has several practical applications for engineers. In recent decades the geodetic community has concentrated efforts on the development of highly accurate geoid models through modern techniques. These models are supplied through regular grids which users need to make interpolations. Yet, little information can be obtained regarding the most appropriate interpolation method to extract information from the regular grid of geoidal models. The use of an interpolator that does not represent the geoid surface appropriately can impair the quality of geoid undulations and consequently the height transformation. This work aims to quantify the magnitude of error that comes from a regular mesh of geoid models. The analysis consisted of performing a comparison between the interpolation of the MAPGEO2015 program and three interpolation methods: bilinear, cubic spline and neural networks Radial Basis Function. As a result of the experiments, it was concluded that 2.5 cm of the 18 cm error of the MAPGEO2015 validation is caused by the use of interpolations in the 5'x5' grid.

Impact of Nasser Lake on gravity reduction and geoidal heights for Egypt

In the course of the IAG African Geoid Project, it is needed to study the impact of the lakes on the gravity reduction and geoidal heights. The aim of this paper is to study the impact of the water in Nasser Lake on gravity reduction and geoidal heights for Egypt. The determination of the gravimetric geoid is based on the well-known remove-restore technique. The problem of the lakes occurs because the popular programs widely used in practice (e.g., TC-program (Forsberg, 1984)) assume that all positive elevations are filled with rock topography, and all negative elevations are filled with ocean water. This is, however, not true for the case of Nasser Lake, which lies completely above sea level, at about 180 m elevation, with a water depth of about 20 m. The paper presents an approach on estimating the impact of Nasser Lake on gravity reduction and geoidal heights using TC-program with some tricky cases. The results show that the impact of Nasser Lake on both gravity anomalies and geoid undulation is limited to the area of the lake. The impact of Nasser Lake on the gravity anomalies is in the order of sub mgal, while the impact of Nasser lake on the geoid undulation is significant and reaches few centimeters.