EIGEN-6C4 Gravity Research Articles

A New Combination Approach for Gibbs Phenomenon Suppression in Regional Validation of Global Gravity Field Model: A Case Study in North China

A global gravity field model (GGM) is essential to be validated with ground-based or airborne observational data for the accurate application of the GGM at a regional scale. Furthermore, accurately understanding the commission errors between the GGM and observational data are crucial for improving regional gravity fields. Taking the North China region as an example, to circumvent the omission errors, it is necessary to unify the spatial resolutions of the EIGEN-6C4 model and terrestrial gravity observational data to 110 km (determined by the distribution of gravity stations) by employing the spherical harmonic function for the EIGEN-6C4 model and the Slepian basis function for the gravity data, respectively. However, the application of spherical harmonic function expansions in the gravity model results in the Gibbs phenomenon, which may be a primary factor contributing to commission errors and impedes the accurate validation of the EIGEN-6C4 model with terrestrial gravity data. To effectively mitigate this issue, this study proposes a combination approach of window function filtering and regional eigenvalue constraint (based on the Slepian basis). Utilizing the EIGEN-6C4 gravity model to derive the gravity disturbance field at a resolution of 110 km (with spherical harmonic expansion up to the 180th degree and order), the combination approach effectively suppresses over 90% of high-degree (above the 120th degree) Gibbs phenomena. This approach also reduces signal leakage outside the region, thus enhancing the spatial accuracy of the regional gravity disturbance field. A subsequent comparison of the regional gravity disturbance field derived from the true model and terrestrial gravity data in North China indicates excellent consistency, with a root mean squared error (RMSE) of 0.80 mGal. This validation confirms that the combined approach of window function filtering and regional eigenvalue constraints effectively mitigates the Gibbs phenomenon and yields precise regional gravity fields. This approach is anticipated to significantly benefit scientific applications such as improving the accuracy of regional elevation benchmarks and accurately inverting the Earth’s internal structure.

Crustal structure of the Tian Shan Orogen and its adjacent areas inferred from EIGEN-6C4 gravity field data

The study of crustal structure plays a crucial role in understanding the tectonic development and seismicity of orogens. In this contribution, we analyzed the gravitational responses caused by the subsurface structure of the Tian Shan orogen using global EIGEN-6C4 gravity field data. The quality of the EIGEN-6C4 gravity data in the eastern Tian Shan area was evaluated, and we employed the wavelet multi-scale analysis method in combination with an existing P-wave velocity model to reveal the structures at different scales. The gravity Moho of the study area that best fits the seismological Moho was obtained from the filtered third-order wavelet approach component after several tests using different reference depths and density contrasts. The results reveal significant lateral discontinuities in the lower crust and upper mantle structures across the eastern and western Tian Shan orogens. The Moho beneath the western Tian Shan is about 9 km deeper compared to the eastern Tian Shan. Towards the Tarim and Junggar basins, the Moho decreases to about 45 km in depth. The analysis of the isostatic anomaly suggests that the eastern and western Tian Shan possess varying capabilities for isostatic adjustment. Furthermore, it indicates a gradual weakening of mantle upwelling in the Tian Shan orogen from west to east, implying that different stages of mantle dynamics may have occurred in the western and eastern Tian Shan. The north Tarim fault, Nikolaev-North Nalati fault, and Kazakh fault serve as channels that control magma upwelling in the Tian Shan orogen. The distribution of earthquakes differs between the eastern and western Tian Shan, with the western Tian Shan exhibiting greater gravitational potential energy. These variations between the eastern and western Tian Shan may be attributed to the clockwise rotation of the Tarim craton.

Study on effective elastic thickness of lithosphere and its tectonic significance in surrounding area of Ordos Block

Based on EIGEN6C4 gravity field model data and ETOPO1 topographic data, we adopt the correlation analysis method based on Fan wavelet to calculate and analyze the distribution characteristics of effective elastic thickness (Te) and load ratio (F) of lithosphere in surrounding of Ordos block. On this basis, the relationship between the lithosphere Te and geological structure, seismic distribution, terrestrial heat flow and crustal deformation is analyzed. The results show that the lithosphere Te in the surrounding area of the Ordos block is obviously different among the tectonic blocks. The lithosphere Te value is low in the south of Qinling orogenic belt and the east of Fen-Wei graben basin. The maximum lithosphere Te value is distributed in the northwest and northeast margin of Ordos block. The lithosphere Te value is negatively correlated with the load ratio F. Earthquakes mainly occur in areas with low lithosphere Te value. Small lithosphere Te value indicates low strength of lithosphere. In such areas, the ability to resist crustal deformation is weak, and it is easy to generate stress accumulation and breed earthquakes. The area with high terrestrial heat flow value corresponds to low lithosphere Te value, showing a negative correlation. However, this negative correlation does not exist in the northwest margin of Ordos Block and Alashan border area, and the eastern margin of Ordos Lvliangshan area. Haiyuan-Liupanshan area has strong tectonic activity, frequent earthquakes and low lithosphere Te value. The internal strain accumulation of Ordos block is small, and the lithosphere Te value is high. The strain rate of the northern Shanxi fault depression zone is large, but the lithosphere Te value is also large, it is speculated that the terrestrial heat flow is the main factor affecting the lithosphere Te value.

New insights on the tectonic framework using EIGEN6C4 gravity data, seismicity, and finite element stress analysis: An attempt to map earthquake vulnerable zones in parts of North-East India and surroundings

Shillong Plateau and its surroundings experience frequent seismic activities characterised by the complex tectonic setup covering different parts of the Eastern Himalaya, Indo-Burma ranges and their conflux at the Assam syntaxis; which need further multivariate investigations. The present study deals with a combined analysis of EIGEN6C4 Bouguer gravity data, 56 years (1964–2020) seismicity data, 11 stations GPS velocity data (1996–2015) and focal mechanism of twelve earthquake events (1996–2017). We have enhanced Bouguer gravity data using the Theta derivative. Upward continuations of the Bouguer gravity were estimated at the heights of 5 km, 10 km, 20 km, 40 km, 60 km, and 100 km, which helped to infer the approximate persistent depth of the structures. Theta derivatives for the upward continued maps have also been estimated to enhance structures and fault zones at different approximated depths. Large numbers of smaller lineaments and shallow crustal smaller blocks are delineated at ∼2.5 km depth, whereas medium grade lineaments are mostly limited within the depth range of 2.5–25 km. However, relatively longer crustal-scale lineaments are highlighted at a greater depth of ∼30 km or more. Euler depth solutions of Bouguer gravity indicate a density heterogeneity interface in the depth range of 12–23 km that corroborates well with upward continuation results and past seismicity data. We have estimated Moho depth using the Parker-Oldenburg algorithm from the Bouguer gravity data. It delineates nine thin crustal structural features (S1–S9), twelve thick crustal structural features (D1-D12) and eleven crustal-scale lineaments (L1-L11) based on the distinct variations of Moho depths and their extensions. The b-value analysis indicates the highest order of b-value in the 15–30 km depth range, mainly in Mikir hill (∼1.7), the eastern part of Shillong Plateau (1.5) and the Indo-Burma ranges (2.3) that perhaps correlate well with the heterogeneity interface delineated by Euler depth solutions. The Eastern Himalaya generally exhibits characteristic low-Bouguer anomaly, high lineament density, low to moderate b-value, and deep-Moho/thick crust. The seismic stress model estimated using the finite element approach exhibits higher-order stress (∼34 MPa) around the central part of Shillong Plateau and at the northern part of Mikir hill with an SW-NE trend. Seismic stress correlates well with the regional lineament trend, GPS velocity/strain direction and low b-value, the joint analysis help to delineate a possible zone of future large earthquakes.

Airborne hyperspectroscopy using AVIRIS- NG data for detecting surface signatures of kimberlite and kimberlite clan of rocks in Bundelkhand Craton, India

Abstract We have utilized spectral bands of Advanced Visible Infrared Imaging Spectroradiometer Next Generation sensor (AVIRIS-NG) to identify surface signatures of new kimberlite. The main focus of the work is to identify the surface signatures of probable kimberlite occurrence and to substantiate these surface signatures as the prospective areas of detailed exploration. Since, all the discovered kimberlites were reported within ferruginous quartzite (Fe-quartzite), we prepared an AVIRIS-NG image composite to demarcate the spatial extension of ferruginous quartziteThe AVIRIS-NG image spectrum of a known kimberlite was also used, as a reference, to identify the surface signatures of new kimberlite within Fe-quartzite. We applied matched filtering and adaptive coherence estimator algorithms to detect surface signatures of kimberlite. The detected surface anomalies of kimberlite in each spectral map are consistent in shape, number and spatial extent. Subsequently, we integrated these maps into a unified surface anomaly map of kimberlite using multiplication operator for identifying the surface anomalies of kimberlite. We observed that the image spectra of these surface anomalies are similar to the laboratory and image spectra of a known pipe. Most of the detected new anomalies are circular and associated with geological lineaments delineated in the residual gravity anomaly map derived from the EIGEN 6C4 gravity model. The surface anomalies of kimberlite, identified in this research, substantiate the importance of detected anomalies for further exploration.

The spatial distribution of the strike angles derived from EIGEN 6C4 gravity model - a new possibility for oil and gas exploration

The spatial distribution of the strike angles derived from EIGEN 6C4 gravity model - a new possibility for oil and gas exploration

The spatial distribution of the strike angles derived from EIGEN 6C4 gravity model a new possibility for oil and gas exploration

We discovered a correlation between the 'combed' strike angles (one of the gravity functions derived from a gravity field model) and regions with oil, gas, shale gas deposits or ground water, paleo...

A Gravity Search for Oil and Gas and Groundwater in Egypt Using the Strike Angles Derived from EIGEN 6C4

We correlate the gravity aspects (descriptors), namely the strike angles, derived from a recent gravity field model, with the known oil, gas and groundwater deposits/reservoirs and hypothetical paleolakes with the locations of archaeological sites. This allows us to extrapolate the investigation, by analogy, to unknown regions. The gravity aspects, derived from the EIGEN 6C4 gravity field model, are used, together with EMAG 2 magnetic anomalies and ETOPO 1 topography model, for the investigation of oil, gas and water deposits in Egypt. One of the gravity aspects, s/c strike angle, is significantly combed (oriented in one direction locally) in places where the known deposits exist. However, they are combed also in some other places. This may be used as a guide as to where to seek new and promising deposits. Accounting for the combed strike angles and the relationship between gravity anomalies and height differences, we reconstructed potential paleolakes under thick sand layers in the Great Sand Sea, Western Egypt (our previous work), and between Kharga and Toshka, Southern Egypt (this work), consistent with the known archaeological sites.

New evidence for a thin crust and magmatic underplating beneath the Cambay rift basin, Western India through modelling of EIGEN-6C4 gravity data

The Cambay rift basin (CRB) is an intracratonic rift in the western part of India. The basin assumes great importance in petroleum exploration owing to the presence of thick hydrocarbon bearing sedimentary rocks. Previous investigations using deep seismic soundings (DSS), gravity and heat flow data reveal that the CRB is characterised by a thin crust, high heat flow and high density lower crust. In this study, a detailed crustal structure of the basin is presented by performing a 2.5D density modelling of the EIGEN-6C4 gravity data. Present study attempt to find a plausible explanation for the variation in the Bouguer anomaly (BA) values from +20 to −50 mGal within the basin. It refined the crustal model that is constrained using results from radial average power spectrum (RAPS) analysis of gravity data along with previous seismological and geophysical studies, which reveals that the values of average sedimentary and Deccan Traps thickness are in the order of 4–5 and 1.5–3 km, respectively, along the rift. It also presents possible evidences for a high density underplated layer of thickness 7–15 km along the central part of the CRB. To study the deep-seated features, upward continuation of the BA is carried out at heights of 30, 40 and 50 km. The extension of underplating layer is noticed in the present crustal model and in the upward continued BA in the western part, while it merges with the Moho in eastern part of the CRB. The Moho depths, varying from 31 to 37 km, are found to be shallower inside the CRB than the surroundings. It is inferred that the high BA values in the basin are due to the combined effect of the high density underplated layer in the lower crust and a shallow Moho.

Delineation of subsurface geological fractures in the Cambay rift and surrounding regions of NW India: an integrated approach using satellite derived EIGEN-6C4 gravity data

The Cambay rift (CR) is a rich petroleum province of north-western India and extensively studied for petroleum exploration and mapping the sub-surface structure through various geophysical studies. However, the CR is not studied so far towards mapping the geological structures, like faults/lineaments and estimating their source depths. In the present study, the lineaments in the CR have been delineated and their source depths are estimated using satellite-derived EIGEN-6C4 gravity data, though different modern processing techniques (Total Horizontal Derivative (THD), Analytical Signal (AS), Tilt Derivative (TDR) and Euler deconvolution). The majority of the identified lineaments are trending in NW-SE and NE-SW direction and their source depths range between 4.4 and 15 km, indicative of shallow subsurface features in this area. This study provides an updated tectonic map, which offers a better understanding of the structural framework and tectonic setting of the area.

Commentary on Klokočník, J., Kostelecký, and Bezděk, A. 2019. The putative Saginaw impact structure, Michigan, Lake Huron, in the light of gravity aspects derived from recent EIGEN 6C4 gravity field model. Journal of Great Lakes Research 45:12–20

Commentary on Klokočník, J., Kostelecký, and Bezděk, A. 2019. The putative Saginaw impact structure, Michigan, Lake Huron, in the light of gravity aspects derived from recent EIGEN 6C4 gravity field model. Journal of Great Lakes Research 45:12–20

Response to comment by Schaetzl et al. (2019) on “The putative Saginaw impact structure, Michigan, Lake Huron, in the light of gravity aspects derived from recent EIGEN 6C4 gravity field model”

Response to comment by Schaetzl et al. (2019) on “The putative Saginaw impact structure, Michigan, Lake Huron, in the light of gravity aspects derived from recent EIGEN 6C4 gravity field model”

A new crustal thickness model for mainland China derived from EIGEN-6C4 gravity data

In this study, we present a new crustal thickness model of mainland China created from the inversion of EIGEN-6C4 global gravity model. We took advantage of a high-resolution gravimetric data and the regularized inversion, based on the Gauss-Newton formulation of Bott’s equation to resolve the Moho topography. The inversion process uses tesseroids to account for the earth curvature. Moho depth modeling for such a massive landmass cannot ignore the earth’s curvature. The Bouguer effect, terrain effect, and gravity effect due to sediments were also corrected and calculated in a spherical approximation. The resulting crustal thickness model was validated with 2992 seismological data points and compared with continental-scale crustal thickness models for mainland China. The final crustal thickness model shows a minimum value of ∼26 km in Eastern China to a thick crust of over 75 km under the Tibetan Plateau. Nevertheless, the results also show a considerable difference in unconstrained areas of the Tibetan plateau with a crustal thickness of ∼20 km and ∼6 km in sedimentary areas. The tectonic terranes show similar depth estimates for all the continental scale models of mainland China, but the lateral variation differs considerably. The results also expose two East-West fold belt trends on the Tibetan Plateau, a 44 km thick crust between Western and Eastern China, and a 37 km NNE trend line, which separates the stable eastern cratonic blocks from the thinned crustal blocks. Overall, our high-resolution model yields an improved representation of crustal features over previously gravity crustal thickness models of mainland China.

The putative Saginaw impact structure, Michigan, Lake Huron, in the light of gravity aspects derived from recent EIGEN 6C4 gravity field model

Abstract The hypothetical impact structure in the Saginaw Bay (Michigan, USA, Lake Huron) has been tested by the gravity data derived from the recent gravity field model EIGEN 6C4 (expanded to degree and order 2190, with ground resolution of ~9 km). The following gravity field aspects were used: the gravity disturbances/anomalies, second derivatives of the disturbing potential (Marussi tensor), two of three gravity invariants, their specific ratio (known as 2D factor), the strike angles, and the virtual deformations. These gravity aspects are sensitive in various ways to the underground density contrasts. For the Saginaw Bay area, we confirm that we do not see any typical impact crater in terms of gravity disturbance or the radial second order derivative, possibly because of the thick layer of the ice located at the place and time of the impact. But the “combed” strike angles (one type of the gravity aspects we use) disclose a trace of high pressure to the SE/S/SW of the Bay and may be due to an impacting body. Thus, we provide circumstantial evidence of the Younger Dryas impact hypothesis.

Satellite-Derived Regional Apparent Thermal Inertia and Gravity for Mapping Different Rock Types in Parts of Banswara, Rajasthan

In this study, utility of satellite derived Apparent Thermal Inertia (ATI) and residual gravity data have been analyzed for delineating the major rock types of Aravalli Supergroup. Density is a common contributing factor responsible for ATI and gravity variations of rock types. In addition to density, thermal inertia is also controlled by specific heat and thermal conductivity. In the present study, MODIS (Moderate resolution imaging spectrometer) day and consecutive night time land surface temperature (LST), albedo data have been processed to derive ATI image for the study area. On the other hand, Bouguer gravity generated from EIGEN6C4 gravity model data is processed to derive the residual gravity over the study area. It has been observed that ATI values of different rock types are influenced by the variations in weathering pattern and forest cover developed over the rock in addition to the thermophysical parameters of rocks. Forest cover and agricultural activity are responsible for shifting the ATI values to higher side for any rocks due to evapotranspiration; which is responsible for reducing day and night land surface temperature difference. ATI pixels( collected from large rock pavements) of different rocks can however, be used to discriminate them from each other. Conjugate analysis of residual gravity and ATI has been proved useful to discriminate different rock types of spatially well distributed rock exposures. It is found that ATI and gravity values are well correlated for the spatially large surface exposures of rocks. In regional scale, low frequency changes of ATI broadly matches with the changes observed on the corresponding spatial profiles of residual gravity with some localized variations. This is due to the fact that shallow surface gravity anomaly is cumulative effect of vertical and horizontal variations of density in contrast to ATI which is more sensitive to near surface density, conductivity and specific heat variations of rocks. Moreover, high frequency variations imprinted on regional ATI spatial profiles are due to the cumulative effect of complex thermophysical properties of surface elements (weathering residuum, surface moisture variations) density variation and intrinsic thermophysical properties of rocks measured at regional scale. High frequency changes of ATI do not correspond well with residual gravity variation.

ASTER multispectral bands, ground magnetic data, ground spectroscopy and space-based EIGEN6C4 gravity data model for identifying potential zones for gold sulphide mineralization in Bhukia, Rajasthan, India

We have used visible near infrared (VNIR) and shortwave infrared (SWIR) spectral bands of Advanced spaceborne thermal emission and reflectance radiometer (ASTER) to identify prospective zones of mineralization associated with the gold deposit in Bhukia area, Rajasthan. Lineament density, the extent of influence of regional structures/lineament, updated lithological map and the reference magnetic anomaly map are used for prospectivity mapping based on Fuzzy-logic based gamma overlay analysis. We have used high pass filtered (5 × 5 windows) image of selected bands of ASTER to delineate geological lineaments; which are believed to have a significant role in localizing hydrothermal fluid movement and also have influenced the ore deposition. In this regard, two evidential layers are prepared for geological structures. One layer is prepared to delineate a zone of influence of geological structures and other is on lineament density. For this, we also have used ground magnetic data and EIGEN6C4 model based Bouguer gravity data to delineate regional structures; which are combined with the ASTER-derived lineaments to derive composite lineament density map. Principle component image of ASTER bands, suitable false color composite (FCC) images of selected bands are derived to update litho-contacts of reference geological map to use as an evidential layer. Weightages for the themes, on favorable structure, lineament density, and lithology are assigned based on the geological understanding of their relative influence on the formation of the deposit. Weightage given to each layer is rescaled to the range of 0–1 using suitable fuzzy function. In addition to the above-mentioned geogenic factors, magnetic anomaly map is also used as an evidential layer. In this regard, empirical relation between a magnetic anomaly and gold sulphides is used as the reference to give gradational higher weightages to high magnetic anomaly zones; which are occurring within the spatial extent of the host rocks. Highly prospective zone in the Fuzzy prospective map prepared using above evidential layers is segmented for “potential or priority areas” and consequently validated. In this regard, high Zr/Hf anomaly map, Fe2O3 oxide anomaly map, “basement-highs” derived from EIGEN6C4 gravity model and ASTER-derived gossan exposure map are used to identify few priority areas within the highly prospective zone of prospect map. Gossan exposure map used in above purpose has been derived based on the implementation of the matched filtering method on ASTER image using gossan broadband spectra as a reference. Highly prospective zone identified using few regional evidence layers are validated using the localized evidences; which have been used to identify few priority areas within the highly prospective zones. Present approach of conjugate use of regional and local evidence layers to identify the priority areas within the highly prospective zone can be regarded as an innovative method to implement for gold exploration in any part of the world.

Thermo-mechanical analysis of the Andean lithosphere over the Chilean-Pampean flat-slab region

Flat subduction segments are generally related to cold lithospheric zones, which suggest low Heat flow values and therefore a deeper Curie isotherm. In this study we perform a geophysical analysis by using high - resolution magnetic data, obtaining shallower Curie point depths through the Chilean-Pampean flat-slab zone. Then using temperatures from borehole data and Curie Point Depths, we performed a Heat Flow map which indicates the existence of mean to high thermal anomalies across the Chilean - Pampean flat-slab. These results were contrasted with Elastic Thickness map (Te) obtained from the Eigen-6C4 gravity model. Low Te values acquired indicate zones with weak mechanical behavior in correspondence to thermal anomalous zones. This spatial correlation suggests that the local high thermal anomalies are conditioning the mechanical behavior of the lithosphere in the study area and the locations of geothermal sources. Overall, our results indicate that the thermal structure of the flat-slab segment is more heterogeneous than previously thought. This has implications for evolution, maturity and fluid circulation in foreland basins of the Main and Frontal cordilleras and the Precordillera thrust belts. Finally, further studies will allow improving our database as well as extending our understanding of the thermal structure of the Chilean-Pampean flat-slab.

High-resolution Moho model for Greenland from EIGEN-6C4 gravity data

The crust–mantle boundary (the Moho) is a first order interface in the Earth and the depth to the Moho is therefore well studied in most regions. However, below regions which are covered by large ice sheets, such as Greenland and Antarctica, the Moho is only partly known and seismic data are difficult to obtain. Here, we take advantage of the global gravity model EIGEN-6C4, together with the Parker-Oldenburg algorithm, to estimate the depth to the Moho beneath Greenland and surroundings. The available free-air gravity data are corrected for the topographic effect and the effect of sedimentary basins. We also correct for the effect on gravity due to the weight of the ice sheet and the accompanying deflection of the Earth's surface, which has not previously been taken into account in gravity studies of currently glaciated regions. Our final Moho depth model for Greenland has an associated uncertainty of ±4.5km for areas with sedimentary basins and ±4km for areas without sedimentary basins. The model shows maximum Moho depths below east Greenland of up to 55km and values less than 20km offshore east Greenland. There is a marked increase in Moho depth of 10–15km from northern to central Greenland, indicating a significant change in geology. A deep Moho at the northern coast of Greenland towards Ellesmere Island might be related to the location of the hot-spot track. Our Moho model is consistent with previous models, but has a higher lateral resolution of 0.1° and covers the entire area of on- and offshore Greenland.

Contribution of mass density heterogeneities to the quasigeoid-to-geoid separation

The geoid-to-quasigeoid separation is often computed only approximately as a function of the simple planar Bouguer gravity anomaly and the height of the computation point while disregarding the contributions of terrain geometry and anomalous topographic density as well as the sub-geoid masses. In this study we demonstrate that these contributions are significant and, therefore, should be taken into consideration when investigating the relation between the normal and orthometric heights particularly in the mountainous, polar and geologically complex regions. These contributions are evaluated by applying the spectral expressions for gravimetric forward modelling and using the EIGEN-6C4 gravity model, the Earth2014 datasets of terrain, ice thickness and inland bathymetry and the CRUST1.0 sediment and (consolidated) crustal density data. Since the global crustal density models currently available (e.g. CRUST1.0) have a limited accuracy and resolution, the comparison of individual density contributions is—for consistency—realized with a limited spectral resolution up to a spherical harmonic degree 360 (or 180). The results reveal that the topographic contribution globally varies between $$-$$ 0.33 and 0.57 m, with maxima in Himalaya and Tibet. The contribution of ice considerably modifies the geoid-to-quasigeoid separation over large parts of Antarctica and Greenland, where it reaches $$\sim $$ 0.2 m. The contributions of sediments and bedrock are less pronounced, with the values typically varying only within a few centimetres. These results, however, have still possibly large uncertainties due to the lack of information on the actual sediment and bedrock density. The contribution of lakes is mostly negligible; its maxima over the Laurentian Great Lakes and the Baikal Lake reach only several millimetres. The contribution of the sub-geoid masses is significant. It is everywhere negative and reaches extreme values of $$-$$ 4.43 m. According to our estimates, the geoid-to-quasigeoid separation globally varies within $$-$$ 4.19 and 0.26 m while the corresponding values computed according to a classical definition are only negative and reach extreme values of $$-$$ 3.5 m. A comparison of these results reveals that inaccuracies caused by disregarding the terrain geometry and mass density heterogeneities distributed within the topography and below the geoid surface can reach $$\pm $$ 2 m or more in the mountainous regions.