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We used the geological map and published rock density measurements to compile the digital rock density model for the Hong Kong territories. We then estimated the average density for the whole territory. According to our result, the rock density values in Hong Kong vary from 2101 to 2681 kg·m−3. These density values are typically smaller than the average density of 2670 kg·m−3, often adopted to represent the average density of the upper continental crust in physical geodesy and gravimetric geophysics applications. This finding reflects that the geological configuration in Hong Kong is mainly formed by light volcanic formations and lava flows with overlying sedimentary deposits at many locations, while the percentage of heavier metamorphic rocks is very low (less than 1%). This product will improve the accuracy of a detailed geoid model and orthometric heights.

New gravity and precise levelling measurements have been performed throughout the Hong Kong territories to modernize a vertical geodetic datum that is currently realized by heights of levelling benchmarks defined in the Hong Kong Principal Datum (HKPD). Modernization of the HKPD involved delivering various products, including new detailed geoid and quasigeoid models and newly determined orthometric and normal heights of levelling benchmarks. In this study, we present the result of gravimetric quasigeoid modelling. The method used to compute a detailed gravimetric quasigeoid model is based on the finite-element method to solve the geodetic boundary-value problem with oblique derivative boundary conditions considered directly at computational nodes on the discretized Earth’s topography. The result of a gravimetric quasigeoid modelling shows a good agreement with a geometric quasigeoid model at the Global Navigation Satellite System (GNSS)-levelling benchmarks. The standard deviation of differences between the gravimetric and geometric quasigeoid heights of ±3.3 cm is compatible with the expected accuracy of gravity, levelling, and GNSS measurements.

We investigate the evolution of stress fields during the supercontinent cycle using the 2D Cartesian geometry model of thermochemical convection with the non-Newtonian rheology in the presence of floating deformable continents. In the course of the simulation, the supercontinent cycle is implemented several times. The number of continents considered in our model as a function of time oscillates around 3. The lifetime of a supercontinent depends on its dimension. Our results suggest that immediately before a supercontinent breakup, the over-lithostatic horizontal stresses in it (referring to the mean value by the computational area) are tensile and can reach −250 MPa. At the same time, a vast area beneath a supercontinent with an upward flow exhibits clearly the over-lithostatic compressive horizontal stresses of 50–100 МРа. The reason for the difference in stresses in the supercontinent and the underlying mantle is a sharp difference in their viscosity. In large parts of the mantle, the over-lithostatic horizontal stresses are in the range of ±25 MPa, while the horizontal stresses along subduction zones and continental margins are significantly larger. During the process of continent-to-continent collisions, the compressive stresses can approximately reach 130 MPa, while within the subcontinental mantle, the tensile over-lithostatic stresses are about −50 MPa. The dynamic topography reflects the main features of the supercontinent cycle and correlates with real ones. Before the breakup and immediately after the disintegration of the supercontinent, continents experience maximum uplift. During the supercontinent cycle, topographic heights of continents typically vary within the interval of about ±1.5 km, relatively to a mean value. Topographic maxima of orogenic formations to about 2–4 km are detected along continent-to-continent collisions as well as when adjacent subduction zones interact with continental margins.

The determination of accurate orthometric or normal heights remains one of the main challenges for the geodetic community in Ethiopia. These heights are required for geodetic and geodynamic scientific research as well as for extensive engineering applications. The main objective of this study is to estimate the geoid-to-quasi geoid separation (GQS) in Ethiopia (ETH-GQS). Such separation would be required for the conversion between geoid and quasigeoid models, which is mandatory for the determination of accurate geodetic heights in mountain regions. The airborne free-air gravity anomalies and the topographic information retrieved from the SRTM3 (Shuttle Radar Topography Mission of a spatial resolution 3 arc-second) digital elevation model were used to compute the ETH-GQS model according to the Sjöberg's strict formula for the geoid-to-quasigeoid separation. The ETH-GQS was then validated using GNSS-levelling data as well as geoid heights determined from different Global Geopotential Models (GGMs), namely the EGM2008, EIGEN-6C4 and GECO. The results reveal that the standard deviation of differences between the geoid heights obtained from the EIGEN-6C4 model and the geometric geoid heights obtained from GNSS-levelling data were improved by ∼75% (i.e. from ∼24 to ∼6 cm) when considering GQS values obtained from the ETH-GQS.

Combined Global Gravitational Models (GGMs) are being used in numerous geoscience applications, most notably for gravimetric geoid modeling (in geodesy) and for geological mapping and geophysical explorations (in the Earth’s sciences). The aim of this study is to evaluate the suitability of different combined GGMs that could be used for the geological mapping of middle belt region and Southeastern Nigeria. For this purpose, we digitized geological maps of Afikpo and Anambra Basins to evaluate geological signatures implied by gravity field quantities (Bouguer gravity anomalies and vertical gravity gradient) derived from the EGM2008, EIGEN-6C4, GECO, SGG-UGM-1 and XGM2019e_2159 gravitational models. We also stochastically evaluated the performance of these GGMs by computing their Root-Mean-Square (RMS) fit with ground-based gravity measurements. The results show that the EIGEN-6C4 and XGM2019e_2159 models have the best RMS fit with the ground-based gravity data. A spatial pattern in Bouguer gravity maps (compiled using these two models) generally closely agrees with a geological configuration of the basins, while also exhibiting some more detailed geological features. Interestingly, however, despite the XGM2019e has the best fit and better mimics major geological features, the gravity image from this model does not exhibit a sediment signature in a portion of the Afikpo basin. A possible reason is that the topographic information used to recover a higher-frequency gravity spectrum of this model might suppress a gravitational signature of subsurface density structures. A comprehensive interpretation of geological features thus requires a careful analysis of existing GGMs, terrestrial gravity data as well as all other reliable geological and geophysical information.

It is a well-known fact that the long-wavelength terrestrial geoid undulations are mainly attributed to deep mantle density heterogeneities, while more detailed features in the geoid geometry are associated with the topography and the lithospheric density structure. To enhance a gravitational signature of mantle density heterogeneities below the lithosphere, the gravitational contributions of topography and lithospheric density heterogeneities should be modelled and subsequently removed from Earth's gravity field. The refined gravity field obtained after this numerical procedure is more suitable for a recovery of a mantle density structure (below the lithosphere). Following this idea, methods for a spherical harmonic analysis and synthesis of gravity field and lithospheric density structures are presented, and a theoretical relation between gravity field and mass density structure is formulated. Since a gravimetric recovery of inner density structure has a non-unique solution, we propose an alternative method based on a conversion of seismic velocities to mass densities. A forward modelling approach is then employed to find the mantle density configuration that generates the gravitational field that best approximates the corresponding refined gravitational field obtained from observed gravity field after subtracting the gravitational signal of the lithosphere. Keywords: Asthenosphere, Crust, Gravity field, Lithosphere, Mantle

The Atlas region in northwest Africa is characterized by the Quaternary volcanism and elevated topography with past complex tectonic mitigation between the African and European plates. Geodynamics of this atypical region has left indubitably imprints in crustal architectonics, mainly regarding the crustal thickness as well as the crustal density structure. The knowledge of crustal thickness variations is of a significant interest, since it provides a crucial constraint to geodynamic and geophysical modelling of this region. In this study, we use gravity, topographic, bathymetric and sediment data together with results of seismic surveys to image the Moho topography beneath the Atlas region. The Bouguer gravity anomalies used for a gravimetric Moho recovery are obtained from the free-air gravity anomalies after subtracting the gravitational contributions of topography, bathymetry and sediments. The regional gravimetric Moho inversion constrained on seismic data is carried out by applying a regularized inversion technique based on Gauss-Newton's formulation of improved Bott's method, while adopting Earth's spherical approximation. The numerical result reveals relatively significant Moho depth variations in the Moroccan Atlas, with minima of approximately 24 km along continental margins of the Mediterranean Sea and maxima exceeding 51 km beneath the Rif Cordillera. The Moho depth beneath the West African Craton varies from 32 km in its southern margin to 45 km beneath the Middle Atlas. The Tell Atlas is characterized by the shallow Moho depth of approximately 22 km and further deepening to 42 km towards the northern edge of the Aures Mountains. Our findings indicate a limited tectonic shortening of the High Atlas with the crustal thickness mostly within 36–42 km. Topographic discrepancies between the Rif Cordillera and the Atlas Mountains suggest that the hypothesis of isostatic compensation cannot be fully established. Keywords: Atlas, Gravity, Inversion, Moho, Seismic data

In this study, we estimate the ice mass changes, the ice elevation changes and the vertical displacements in Antarctica based on analysis of multi-geodetic datasets that involve the satellite gravimetry (GRACE), the satellite altimetry (ICESat) and the global navigation satellite systems (GNSS). According to our estimates, the total mass change of the Antarctic ice sheet from GRACE data is −162.91 Gt/yr over the investigated period between April 2002 and June 2017. This value was obtained after applying the GIA correction of −98.12 Gt/yr derived from the ICE-5G model of the glacial isostatic adjustment. A more detailed analysis of mass balance changes for three individual drainage regions in Antarctica reveal that the mass loss of the West Antarctic ice sheet was at a rate of −143.11 Gt/yr. The mass loss of the Antarctic Peninsula ice sheet was at a rate of −24.31 Gt/yr. The mass of the East Antarctic ice sheet increased at a rate of 5.29 Gt/yr during the investigated period. When integrated over the entire Antarctic ice sheet, average rates of ice elevation changes over the period from March 2003 to October 2009 derived from ICESat data represent the loss of total ice volume of −155.6 km3. The most prominent features in ice volume changes in Antarctica are characterized by a strong dynamic thinning and ice mass loss in the Amundsen Sea Embayment that is part of the West Antarctic ice sheet. In contrast, coastal regions between Dronning Maud Land and Enderby Land exhibit a minor ice increase, while a minor ice mass loss is observed in Wilkes Land. The vertical load displacement rates estimated from GRACE and GPS data relatively closely agree with the GIA model derived based on the ice-load history and the viscosity profile.

Despite practical limitations of isostatic theories to model the Moho geometry are well-known, gravimetric methods are often used in terrestrial studies of crustal structure in regions with a low seismic data coverage. Moreover, these methods are indispensable in planetary studies. Various gravimetric methods have been proposed. The Airy and Pratt theories are defined based on adopting a local compensation mechanism. The Vening Meinesz theory assumes a regional isostatic flexural model. The Vening Meinesz regional isostatic model generally describes a respond of the lithosphere to a load more realistically than the Airy model over continents. The Pratt method, on the other hand, better describes a compensation mechanism of the oceanic lithosphere. The application of a particular isostatic model also depends on applied numerical procedures, parameters for inversion, input data specifications, and many other aspects. In this study, we address some basic aspects by applying local and regional isostatic models for a Moho recovery. We also conduct a spectral analysis to assess a spectral resolution of gravity data that is optimal for a Moho recovery. Furthermore, we inspect the influence of low-degree spherical harmonics of gravity field on a Moho geometry. Gravimetric results are validated using seismic data at the European plate. Our results confirm a better performance of a regional compensation principle. We also demonstrate that a different thickness of the oceanic and continental crustal thickness should be taken into account as a priori information. Spectral analysis indicates that gravity data used for a Moho inversion should optimally have a spatial resolution between degrees 60 to 180. Results also show that low-degree spherical harmonics do not modify significantly the Moho geometry, particularly over regions with a relatively homogenous structure of deep mantle. Keywords: Crust, Gravity, Moho inversion, Isostasy

The average density of 3300 kg m−3 is often attributed for the asthenosphere. In this study, we inspect this value by estimating the average value of the (upper) asthenosphere based on applying the gravimetric forward modelling of major known lithospheric density structures. The LITHO1.0 global seismic model of the lithospheric density structure is used for this purpose, while considering that the lithosphere-asthenosphere boundary (LAB) is rheological, conventionally taken at the 1300 °C isotherm, above which the mantle behaves in a rigid fashion and below which it behaves in a ductile fashion. According to our result, the average density of the upper asthenosphere is roughly 3400 kg m−3. This density value closely agrees with the corresponding average value 3371 kg m−3 computed based on an empirical density model provided in the Preliminary Reference Earth Model (PREM), while using the LITHO1.0 LAB depth data. We also demonstrate that the sub-lithospheric mantle gravity map exhibits mainly a thermal signature. The most prominent features in this gravity map are mid-oceanic spearing ridges marked by gravity lows, while oceanic subductions in the West Pacific are characterized by the most pronounced gravity highs. Keywords: Asthenosphere, Density, Forward modelling, Gravity, Lithosphere

We compile the GOCE-only satellite model GOSG01S complete to spherical harmonic degree of 220 using Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations along the GOCE orbit based on applying a least-squares analysis. The diagonal components (Vxx, Vyy, Vzz) of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations (ax, ay, az) along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. The analysis of spectral accuracy characteristics of the newly derived gravitational model GOSG01S and the existing models GOTIM04S, GODIR04S, GOSPW04S and JYY_GOCE02S based on their comparison with the ultra-high degree model EIGEN-6C2 reveals a significant consistency at the spectral window approximately between 80 and 190 due to the same period SGG data used to compile these models. The GOCE related satellite gravity models GOSG01S, GOTIM05S, GODIR05S, GOTIM04S, GODIR04S, GOSPW04S, JYY_GOCE02S, EIGEN-6C2 and EGM2008 are also validated by using GPS-leveling data in China and USA. According to the truncation at degree 200, the statistic results show that all GGMs have very similar differences at GPS-leveling points in USA, and all GOCE related gravity models have better performance than EGM2008 in China. This suggests that all these models provide much more information on the gravity field than EGM2008 in areas with low terrestrial gravity coverage. And STDs of height anomaly differences in China for the selected truncation degrees show that GOCE has improved the accuracy of the global models beyond degree 90 and the accuracies of the models improve from 24 cm to 16 cm. STDs of geoid height differences in USA show that GOSG01S model has best consistency comparing with GPS-leveling data for the frequency band of the degree between 20 and 160.

The coefficients of the Global Gravitational Models (GGMs) define the external gravitational field of the Earth. In many geoscience applications and gravity interpretations these coefficients are routinely used to represent the geoid surface and related gravitational field quantities without taking into consideration the internal convergence domain for computing the gravitational field inside the Earth's masses. In this study we discuss this issue and present the numerical approach for computing the gravitational field quantities on the geoid. The proposed numerical approach utilizes spectral expressions for the gravimetric forward modelling of topographic mass density distribution and the indirect gravimetric modelling of mass density heterogeneities inside the geoid. In the numerical realization we demonstrate that for precise applications the differences between the potential values computed at the topographic surface and on the geoid should be taken into consideration especially in mountainous regions with complex geology as well as in polar regions, where these differences in absolute values reach 40 m2 s−2 (or more), which translates into vertical displacements between the geoid and the quasigeoid of about 4 m.

We investigate the relationship between vertical crustal motion and tectonic block configuration. The study is conducted along the active tectonic margin between the Australian and Pacific tectonic plates in New Zealand with a well-defined tectonic block configuration. For this purpose, the rates of vertical crustal motions relative to the ITRF2008 reference frame are estimated based on processing the GPS data (provided by the GeoNET project) collected at 123 continuous and semi-continuous GPS sites. The numerical results confirmed the uplift of the central Southern Alps at the current rate of 4.5 mm/yr. This tectonic uplift is coupled in the South Island by the subsidence on both sides of the Southern Alps. The detected rates of subsidence in the eastern South Island are typically less than 1 mm/yr. The subsidence in the Buller Region (in the northwest South Island) is 1.4–1.5 mm/yr. Except for the Taupo Volcanic Zone and the upper Raukumara Block (in the central and northeast North Island), the subsidence is prevailing in the North Island. The systematic subsidence up to 9 mm/yr is detected along the Dextral Fault Belt (in the lower North Island). The largest localized vertical displacements (between −10 and 17 mm/yr) in the Taupo Volcanic Zone are attributed to active tectonics, volcanisms and geothermal processes in this region. A classification of these vertical tectonic motions with respect to the tectonic block configuration reveals that most of tectonic blocks are systematically uplifted, subsided or tilted, except for regions characterized by a complex pattern of vertical motions attributed to active geothermal and volcanic processes.

The Gravity Recovery and Climate Experiment (GRACE) satellite mission provides a unique opportunity to quantitatively study terrestrial water storage (TWS) variations. In this paper, the terrestrial water storage variations in the Poyang Lake Basin are recovered from the GRACE gravity data from January 2003 to March 2014 and compared with the Global Land Data Assimilation System (GLDAS) hydrological models and satellite altimetry. Furthermore, the impact of soil moisture content from GLDAS and rainfall from the Tropical Rainfall Measuring Mission (TRMM) on TWS variations are investigated. Our results indicate that the TWS variations from GRACE, GLDAS and satellite altimetry have a general consistency. The TWS trends in the Poyang Lake Basin determined from GRACE, GLDAS and satellite altimetry are increasing at 0.0141 km3/a, 0.0328 km3/a and 0.0238 km3/a, respectively during the investigated time period. The TWS is governed mainly by the soil moisture content and dominated primarily by the precipitation but also modulated by the flood season of the Yangtze River as well as the lake and river exchange water.

A generalized mathematical model of the Earth’s density structure is presented in this study. This model is defined based on applying the spectral expressions for a 3-D density distribution within the arbitrary volumetric mass layers. The 3-D density model is then converted into a form which describes the Earth’s density structure by means of the density-contrast interfaces between the volumetric mass layers while additional correction terms are applied to account for radial density changes. The applied numerical schemes utilize methods for a spherical harmonic analysis and synthesis of the global density structure models. The developed the Earth’s density models are then defined in terms of the spherical density and density-contrast functions. We also demonstrate how these Earth’s density models can be applied in the gravimetric forward modeling and discuss some practical aspects of representing mathematically density structures within particular components of the Earth’s interior.

Gravity data used for a recovery of the Moho depths should (optimally) comprise only the gravitational signal of the Moho geometry. This theoretical assumption is typically not required in classical isostatic models, which are applied in gravimetric inverse methods for a recovery of the Moho interface. To overcome this theoretical deficiency, we formulate the gravimetric inverse problem for the consolidated crust-stripped gravity disturbances, which have (theoretically) a maximum correlation with the Moho geometry, while the gravitational contributions of anomalous density structures within the lithosphere and sub-lithosphere mantle (including the core-mantle boundary) should be subtracted from these gravity data. In the absence of a reliable 3-D Earth’s density model, our definitions are limited to the crustal and upper mantle density structures. The gravimetric forward modeling technique is applied to compute these gravity data using available models of major known anomalous crustal and upper mantle density structures. The gravimetric inverse problem is defined by means of the (non-linear) Fredholm integral equation of the first kind. After linearization of the integral equation, the solution to the gravimetric inverse problem is given in a frequency domain. The inverse problem is formulated for a generalized crustal compensation model. It implies that the compensation equilibrium is (theoretically) attained by both, the variable depth and density of compensation. A theoretical definition of this generalized crustal compensation model and a formulation of the gravimetric inverse problem for finding the Moho depths are given in this study.

The gravimetric inverse problem for finding the Moho density contrast is formulated in this study. The solution requires that the crust density structure and the Moho depths are a priori known, for instance, from results of seismic studies. The relation between the isostatic gravity data (i.e., the complete-crust stripped isostatic gravity disturbances) and the Moho density contrast is defined by means of the Fredholm integral equation of the first kind. The closed analytical solution of the integral equation is given. Alternative expressions for solving the inverse problem of isostasy are defined in frequency domain. The isostatic gravity data are computed utilizing methods for a spherical harmonic analysis and synthesis of the gravity field. For this purpose, we define various spherical functions, which define the crust density structures and the Moho interface globally.

We evaluate the new regional quasigeoid model (OTG12) for New Zealand using the method which utilises the discretised integral equation approach for computing the near-zone contribution. The far-zone contribution is computed by modified spherical harmonics of the geopotential. Adopting the remove-compute-restore computation scheme, the near- and far-zone contributions are computed for the residual height anomalies, while the reference height anomalies are evaluated using a global gravitational model. For the numerical realisation, the GOCO-02S coefficients complete to degree 55 of spherical harmonics are used to generate the reference gravity field. The 1×1 arc-min grid data of terrestrial gravity anomalies are used to compute the near-zone contribution. The far-zone contribution is evaluated using the EGM2008 coefficients at the frequency bound between degrees 56 and 2160 of spherical harmonics. The newly-adjusted levelling data are used to validate the available regional geoid/quasigeoid models. The accuracy assessment of gravimetric solutions at GNSS-levelling testing network reveals that the accuracy of OTG12 is compatible with the existing regional geoid/quasigeoid models; the standard deviation (STD) of residuals between the geometric and gravimetric height anomalies is 13 cm (after applying the 3-parameter correction model).

The empirical model of the depth-dependent density change within the upper continental mantle is derived in this study. The density of the upper(most) mantle underlying the continental crust is obtained from the estimated values of the crust–mantle (Moho) density contrast. Since the continental crustal thickness varies significantly, these upper mantle density values to a large extent reflect the density changes with depth. The estimation of the Moho density contrast is done through solving Moritz’s generalization of the Vening-Meinesz inverse problem of isostasy. The solution combines gravity and seismic data in the least-squares estimation model. The estimated upper mantle density (beneath the continental crust) varies between 2770 and 3649 kg/m3. The upper mantle density increases almost proportionally with depth at a rate of 13±2 kg/m3 per 1 km at the investigated depth interval from 6 to 58 km.

We introduce the generic expressions for computing the gravitational field (potential and its radial derivative) generated by an arbitrary density (contrast) layer with a variable depth and thickness having a laterally-distributed radial density variation. The information on the geometry and density distribution of a volumetric mass layer is described by means of spherical harmonics. These generic expressions can uniformly be applied to model all major known density structures within the Earth’s interior using methods for a spherical harmonic analysis and synthesis of the gravitational field. This is demonstrated on specific examples given for various density models commonly adopted for the approximation of crust density structures.

We propose a numerical scheme which efficiently combines various existing methods of solving the Newton’s volume integral. It utilises the analytical solution of Newton’s integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky’s truncation coefficients to a spectral representation of Newton’s integral. The weak singularity of Newton’s integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the M¨obius transformation with an enhanced integration grid resolution at vicinity of the observation point.

We derive the expressions for computing the ice density contrast stripping corrections to the topography corrected gravity field quantities by means of the spherical harmonics. The expressions in the spectral representation utilize two types of the spherical functions, namely the spherical height functions and the newly introduced lower-bound ice functions. The spherical height functions describe the global geometry of the upper topographic bound. The spherical lower-bound ice functions combined with the spherical height functions describe the global thickness of the continental ice sheet. The newly derived formulas are utilized in the forward modelling of the gravitational field quantities generated by the ice density contrast. The 30×30 arc-sec global elevation data from GTOPO30 are used to generate the global elevation model (GEM) coefficients. The spatially averaged global elevation data from GTOPO30 and the 2×2 arc-deg ice-thickness data from the CRUST 2.0 global crustal model are used to generate the global lower-bound ice model (GIM) coefficients. The mean value of the ice density contrast 1753 kg/m3 (i.e., difference of the reference constant density of the continental upper crust 2670 kg/m3and the density of glacial ice 917 kg/m3) is adopted. The numerical examples are given for the gravitational potential and attraction generated by the ice density contrast computed globally with a low-degree spectral resolution complete to degree and order 90 of the GEM and GIM coefficients.

We compile a quasigeoid model at the study area of New Zealand using the boundary element method (BEM). The direct BEM formulation for the Laplace equation is applied to obtain a numerical solution to the linearized fixed gravimetric boundaryvalue problem in points at the Earth’s surface. The numerical scheme uses the collocation method with linear basis functions. It involves a discretisation of the Earth’s surface which is considered as a fixed boundary. The surface gravity disturbances represent the oblique derivative boundary condition. The geocentric positions of the collocation points are determined combining the digital elevation data and the a priori quasigeoid model (onshore) and the mean sea surface topography (offshore). In our numerical realization, we use the global elevation data from SRTM30PLUS V5.0, the detailed DTM of New Zealand, the EGM2008 quasigeoid heights, and the mean sea surface topography from the DNSC08 marine database. The gravity disturbances are computed using two heterogeneous gravity data sets: the altimetry-derived gravity anomalies from the DNSC08 gravity database (offshore) and the observed ground gravity anomalies from the GNS Science gravity database (onshore). The transformation of gravity anomalies to gravity disturbances is realized using the quasigeoid heights calculated from the EGM2008 global geopotential model. The new experimental quasigeoid model NZQM2010 is compiled at the study area of New Zealand bounded by the parallels of 34 and 47.5 arc-deg southern latitude and the meridians of 166 and 179 arc-deg eastern longitude. The least-squares analysis is applied to combine the gravimetric solution with GPS-levelling data using a 7-parameter model. NZQM2010 is validated using GPS-levelling data and compared with the existing regional and global quasigeoid models NZGeoid2009 and EGM2008.

The currently available global geopotential models and the global elevation and bathymetry data allow modelling the topography-corrected and bathymetry stripped reference gravity field to a very high spectral resolution (up to degree 2160 of spherical harmonics) using methods for a spherical harmonic analysis and synthesis of the gravity field. When modelling the topography-corrected and crust-density-contrast stripped reference gravity field, additional stripping corrections are applied due to the ice, sediment and other major known density contrasts within the Earth’s crust. The currently available data of global crustal density structures have, however, a very low resolution and accuracy. The compilation of the global crust density contrast stripped gravity field is thus limited to a low spectral resolution, typically up to degree 180 of spherical harmonics. In this study we derive the expressions used in forward modelling of the bathymetry-generated gravitational field quantities and the corresponding bathymetric stripping corrections to gravity field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The expressions for the potential and its radial derivative are formulated for the adopted constant (average) ocean saltwater density contrast and for the spherical approximation of the geoid surface. These newly derived expressions are utilized in numerical examples to compute the gravitational potential and attraction generated by the ocean density contrast. The computation is realized globally on a 1 × 1 arc-deg geographical grid at the Earth’s surface.

We compile the global maps of atmospheric effects on the gravity field quantities using the spherical harmonic representation of the gravitational field. A simple atmospheric density distribution is assumed within a lower atmosphere (< 6 km). Dis- regarding temporal and lateral atmospheric density variations, the radial atmospheric density model is defined as a function of the nominal atmospheric density at the sea level and the height. For elevations above 6 km, the atmospheric density distribution from the United States Standard Atmosphere 1976 is adopted. The 5 × 5 arc-min global elevation data from the ETOPO5 are used to generate the global elevation model coefficients. These coefficients (which represent the geometry of the lower bound of atmospheric masses) are utilized to compute the atmospheric effects with a spectral resolution complete to degree and order 180. The atmospheric effects on gravity disturbances, gravity anomalies and geoid undulations are evaluated globally on a 1 × 1 arc-deg grid.

We investigate globally the correlation of the step-wise consolidated crust- stripped gravity field quantities with the topography, bathymetry, and the Moho bound- ary. Global correlations are quantified in terms of Pearson’s correlation coefficient. The elevation and bathymetry data from the ETOPO5 are used to estimate the correlation of the gravity field quantities with the topography and bathymetry. The 2×2 arc-deg discrete data of the Moho depth from the global crustal model CRUST 2.0 are used to estimate the correlation of the gravity field quantities with the Moho boundary. The results reveal that the topographically corrected gravity field quantities have the highest absolute correlation with the topography. The negative correlation of the topographically corrected gravity disturbances with the topography over the continents reaches –0.97. The ocean, ice and sediment density contrasts stripped and topographically corrected gravity field quantities have the highest correlation with the bathymetry (ocean bottom relief ). The correlation of the ocean, ice and sediment density contrasts stripped and topographically corrected gravity disturbances over the oceans reaches 0.93. The consolidated crust-stripped grav- ity field quantities have the highest absolute correlation with the Moho boundary. In particular, the global correlation of the consolidated crust-stripped gravity disturbances with the Moho boundary is found to be –0.92. Among all the investigated gravity field quantities, the consolidated crust-stripped gravity disturbances are thus the best suited for a refinement of the Moho density interface by means of the gravimetric modeling or inversion.

We compile global maps of the step-wise topography corrected and crustal components stripped geoids based on the geopotential model EGM’08 complete to spheri- cal harmonic degree 180 and the CRUST 2.0 global crustal model. The spectral resolution complete to degree 180 is used to compute the primary indirect bathymetric stripping and topographic effects on the geoid, while degree 90 for the primary indirect ice stripping effect. The primary indirect stripping effects of the soft and hard sediments, and the upper, middle and lower consolidated crust components are forward modeled in spatial form using the 2 × 2 arc-deg discrete data of the CRUST 2.0 model. The ocean, ice, sediment and consolidated crust density contrasts are defined relative to the adopted ref- erence crustal density of 2670 kg/m3 . Finally we compute and apply the primary indirect stripping effect of the density contrast (relative to the mantle) of the reference crust. The constant value of −520 kg/m3 is adopted for this density contrast relative to the mantle. All data are evaluated on a 1 × 1 arc-deg geographical grid. The complete crust-stripped geoidal undulations, globally having a range of approximately 1.5 km, contain the gravi- tational signal coming from the global mantle lithosphere (upper mantle) morphology and density composition, and from the sub-lithospheric density heterogeneities. Large errors in the complete crust-stripped geoid are expected due to uncertainties of the CRUST 2.0 model, i.e., due to deviations of the CRUST 2.0 model density from the real earth’s crustal density and due to the Moho-boundary uncertainties.

In the theory of the orthometric height, the mean value of gravity along the plumbline between the geoid and the earth's surface is defined as the integral mean. To determine the mean gravity from the gravity observations realized at the physical surface of the earth, the actual topographical density distribution and vertical change of gravity with depth have to be known. In Helmert's (1890) definition of the orthometric height, the assumption of the linear change of normal gravity is used adopting the constant topographical density distribution. The mean value of gravity is then approximately evaluated so that the observed gravity of a point at the earth's surface is reduced to the mid‐point of the plumbline by Poincaré‐Prey's gravity gradient. To avoid the problems related to the determination of mean gravity, Molodensky (1945) formulated the different concept. In his theory of the normal height, the mean value of the normal gravity along the ellipsoidal normal between the ellipsoid surface and telluroid is considered. The mean normal gravity is then evaluated explicitly without any hypothesis about the topographical density distribution and vertical gradient of actual gravity. In this paper, the corrections to Helmert's orthometric height are formulated based on the comparison of the integral mean of gravity and Poincaré‐Prey's gravity reduction. As follows from the results of the numerical investigation, the orthometric heights can also be determined with a reasonable accuracy if the sufficient information about topographical density and gravity are available. Article in English

We compile a new geoid model at the computation area of New Zealand and its continental shelf using the method developed at the Royal Institute of Technology (KTH) in Stockholm. This method utilizes the least-squares modification of the Stokes integral for the biased, unbiased, and optimum stochastic solutions. The modified Bruns-Stokes integral combines the regional terrestrial gravity data with a global geopotential model (GGM). Four additive corrections are calculated and applied to the approximate geoid heights in order to obtain the gravimetric geoid. These four additive corrections account for the combined direct and indirect effects of topography and atmosphere, the contribution of the downward continuation reduction, and the formulation of the Stokes problem in the spherical approximation. The gravimetric geoid model is computed using two heterogonous gravity data sets: the altimetry-derived gravity anomalies from the DNSC08 marine gravity database (offshore) and the ground gravity measurements from the GNS Science gravity database (onshore). The GGM coefficients are taken from EIGEN-GRACE02S complete to degree 65 of spherical harmonics. The topographic heights are generated from the 1×1 arc-sec detailed digital terrain model (DTM) of New Zealand and from the 30×30 arc-sec global elevation data of SRTM30_PLUS V5.0. The least-squares analysis is applied to combine the gravity and GPS-levelling data using a 7-parameter model. The fit of the KTH geoid model with GPS-levelling data in New Zealand is 7 cm in terms of the standard deviation (STD) of differences. This STD fit is the same as the STD fit of the NZGeoid2009, which is the currently adopted official quasigeoid model for New Zealand. Santrauka Stokholmo Karališkajame technologijos institute (KTH) sukurtu metodu apskaičiuotas naujas Naujosios Zelandijos ir kontinentinio šelfo geoido modelis. Taikoma Stokso integralo mažiausiųjų kvadratų modifikacija, įvertinant paklaidas ir jų nevertinant bei ieškant optimalių stochastinių sprendinių. Modifikuotas Bruno ir Stokso integralas sieja regioninius žemyninius gravimetrinius duomenis su globaliuoju geopotencialo modeliu (GGM). Gravimetriniam geoidui gauti skaičiuojamos keturios papildomos pataisos: topografinės situacijos ir atmosferos tiesioginės ir netiesioginės įtakos, redukcijos įtakos ir Stokso integralo taikymo sferiniam paviršiui. Gravimetrinis geoido modelis apskaičiuotas pagal du duomenų rinkinius: DNSC08 jūrinių gravimetrinių duomenų bazėje (šelfas) esančias altimetriniu metodu nustatytas sunkio pagreičio anomalijas ir žemyninės dalies gravimetrinių matavimų duomenis iš GNS gravimetrinės duomenų bazės (pakrantė). GGM koeficientai imti iš EIGEN-GRACE02S modelio sferinių iki 65 laipsnio harmonikų. Topografiniai aukščiai sugeneruoti iš Naujosios Zelandijos 1×1 sekundės detaliojo skaitmeninio reljefo modelio ir iš 30×30 sekundžių globaliojo aukščių modelio SRTM30_PLUS V5.0. Gravimetriniams ir GPS niveliacijos duomenims sujungti taikytas mažiausiųjų kvadratų 7 parametrų metodas. KTH metodu sudaryto geoido modelio vidutinė kvadratinė paklaida 7 cm. Tai sutampa su NZGeoid 2009 geoido modelio, taikomo Naujoje Zelandijoje, tikslumu. Резюме Модель геоида континентального шельфа Новой Зеландии построена с применением метода, созданного в Королевском технологическом институте Стокгольма. Данный метод основан на модификации решения интеграла Стокса методом наименьших квадратов с оценкой или без оценки погрешностей и поиском оптимальных статистических решений. Модифицированный интеграл БрунаСтокса объединяет региональные надземные гравиметрические данные с глобальной геопотенциальной моделью (GGM). Для определения гравиметрического геоида вычисляются дополнительные поправки прямого и косвенного влияния топографии и атмосферы, редукции и применения проблемы Стокса для сферической поверхности. Гравиметрическая модель геоида вычисляется на основе двух баз данных: альтиметрическим методом определенных аномалий силы тяжести в базе морских гравиметрических данных DNSC08 (шельф) и надземной части гравиметрических измерений из базы данных GNS. Коэффициенты GGM взяты из сферических гармоник до 65 степени модели EIGENGRACEO2S. Топографические высоты сгенерированы из детальной цифровой модели рельефа Новой Зеландии с сеткой 1×1 секунду и из глобальной модели высот SRTM30_PLUSv5.0 с сеткой 30×30 секунд. Для объединения гравиметрических и GPSнивелирных данных применялся метод наименьших квадратов с 7 параметрами. Среднеквадратическая погрешность модели геоида, созданной по методу КТН, равна 7 см. Точность аналогична точности применяемой в Новой Зеландии модели геоида NZGeoid2009. Reikšminiai žodžiai: geoidas, gravimetrija, mažiausiųjų kvadratų metodas, Stokso integralas, UDK 528.21 Ключевые слова: геоид, гравиметрия, способ наименьших квадратов, интеграл Стокса, УДК 528.21