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Until recently, the information about the topographic density distribution has been limited to only certain regions and some countries, while missing in the global context. The UNB_TopoDens is the first model that provides the information about a lateral topographic density globally. The analysis of this model also reveals that the average topographic density for the entire continental landmass (excluding polar glaciers) is 2247 kg m−3. This density differs significantly from the value of 2670 kg m−3 that is typically adopted to represent the continental upper crustal density. In this study, we use the UNB_TopoDens density model to inspect how the topographic density variations affect interpretational properties of Bouguer gravity maps. Since this model provides also the information about density uncertainties of individual lithologies (main rock types), we estimate the corresponding errors in the Bouguer gravity data. Despite a new estimate of the average topographic density corresponds to relative changes of ∼16 per cent in values of the topographic gravity correction, these changes do not affect interpretational properties of Bouguer gravity maps. The anomalous topographic density distribution (taken with respect to the average density of 2247 kg m−3), however, modifies the Bouguer gravity pattern. We demonstrate that the gravitational contribution of anomalous topographic density is globally mostly within ±25 mGal, but much large values are detected in Himalaya, Tibet, central Andes and along the East African Rift System. Our estimates also indicate that errors in the Bouguer gravity data attributed to topographic density uncertainties are mostly less than ±15 mGal, but in mountainous regions could reach large values exceeding even ±50 mGal. Unarguably, the UNB_TopoDens model provides an improved information about the global topographic density variations and their uncertainties. Nevertheless, much more in situ measurements of rock density samples together with detailed 3-D geological models are still necessary to understand better the actual density distribution within the whole topography, particularly to mention a density change with depth. [ABSTRACT FROM AUTHOR]

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Integral transformations of gravitational gradients onto a Gravity Recovery And Climate Experiment (GRACE) type of observable are derived in this article. The gravitational gradients represent components of the gravitational tensor in the local north-oriented frame. The GRACE type of observable corresponds to a difference between two gravitational vectors as projected onto the line of sight between the two GRACE satellites. In total, three integral transformations relating vertical-vertical, vertical-horizontal and horizontal-horizontal gravitational gradients with the GRACE type of observable are provided. Spectral and closed forms of corresponding isotropic kernels are derived for each transformation. Special cases show that the integral transformations are general and relate gravitational gradients to many other quantities of the gravitational field, such as the gravitational vector, and its radial and tangential components. Correctness of the mathematical derivations is validated in a closed-loop simulation using synthetic data. [ABSTRACT FROM AUTHOR]

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme. [ABSTRACT FROM AUTHOR]

We developed and applied a novel numerical scheme for a gravimetric forward modelling of the Earth's crustal density structures based entirely on methods for a spherical analysis and synthesis of the gravitational field. This numerical scheme utilises expressions for the gravitational potentials and their radial derivatives generated by the homogeneous or laterally varying mass density layers with a variable height/depth and thickness given in terms of spherical harmonics. We used these expressions to compute globally the complete crust-corrected Earth's gravity field and its contribution generated by the Earth's crust. The gravimetric forward modelling of large known mass density structures within the Earth's crust is realised by using global models of the Earth's gravity field (EGM2008), topography/bathymetry (DTM2006.0), continental ice-thickness (ICE-5G), and crustal density structures (CRUST2.0). The crust-corrected gravity field is obtained after modelling and subtracting the gravitational contribution of the Earth's crust from the EGM2008 gravity data. These refined gravity data mainly comprise information on the Moho interface and mantle lithosphere. Numerical results also reveal that the gravitational contribution of the Earth's crust varies globally from 1,843 to 12,010 mGal. This gravitational signal is strongly correlated with the crustal thickness with its maxima in mountainous regions (Himalayas, Tibetan Plateau and Andes) with the presence of large isostatic compensation. The corresponding minima over the open oceans are due to the thin and heavier oceanic crust. [ABSTRACT FROM AUTHOR]

spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton's integral. In this study we utilise Molodensky's truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky's truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data. [ABSTRACT FROM AUTHOR]

The geoid is an equipotential surface of the static Earth's gravity field which plays a fundamental role in definition of physical heights related to the mean sea level (orthometric heights) in geodesy and which represents a reference surface in many geoscientific studies. Its determination with the cm-level accuracy or better, in particular over dry land, belongs to major tasks of modern geodesy. Traditional data and underlined theory have significantly been affected in recent years by rapid advances in observation techniques. This study reviews gradients of the disturbing gravity potential, both currently available and foreseen, and systematically discusses mathematical models for geoid determination based on gradient data. Fundamentals required for geoid definition and its estimation from measured potential gradients are shortly reviewed at the beginning of the text. Then particular mathematical models based on solutions to boundary-value problems of the potential theory, which include both integral transforms and integral equations, are formulated. Properties of respective integral kernel functions are demonstrated and discussed. With the new mathematical models introduced, new research topics are opened which must be resolved in order to allow for their full-fledged applicability in geoid modelling. Stochastic modelling is also discussed which estimates gradient spatial resolution and accuracy required for geoid modelling with the cm-level accuracy. Results of stochastic modelling suggest that the cm-geoid can be estimated using available gradient data if related problems, namely reduction of gradient data for gravitational effects of all masses outside the geoid and their downward continuation, are solved at the same level of accuracy. [ABSTRACT FROM AUTHOR]

Shipborne gravity can be used to refine altimeter-derived gravity whose accuracy is low in shallow waters and areas with complex submarine topography. As altimeter-derived gravity only within a small radius around the shipborne data can be corrected by traditional methods, a new method based on multi-layer perceptron (MLP) neural network is proposed to refine the altimeter-derived gravity. Input variables of MLP include the positional information at observation points and geophysical information (from our own South China Sea gravity anomaly model (SCSGA) V1.0 and bathymetry model ETOPO1) at grid points around observation points. Output variables of MLP are the refined residual gravity anomalies at observation points. Training shipborne data are classified into four cases to train four MLP models, which are used to predict the refined gravity anomaly model SCSGA V1.1. Then all of the training shipborne data are used for training an MLP model to predict the refined gravity anomaly model SCSGA V1.2. Assessed by testing shipborne data, the accuracy of SCSGA V1.2 is 0.14 mGal higher than that of SCSGA V1.0, and similar to that of SCSGA V1.1. Compared with the original gravity anomaly model (SCSGA V1.0), the accuracy of the refined gravity anomaly model (SCSGA V1.2) by MLP is improved by 4.4% in areas where the training data are concentrated, and also improved by 2.2% in other areas. Therefore, the method of MLP can be used to refine the altimeter-derived gravity model by shipborne gravity, overcoming the problem of limited correction radius for traditional methods. [ABSTRACT FROM AUTHOR]

Local gravity field modelling demands high-quality gravity data as well as an appropriate mathematical model. Particularly in coastal areas, there may be different types of gravity observations available, for instance, terrestrial, aerial, marine gravity, and satellite altimetry data. Thus, it is important to develop a proper tool to merge the different data types for local gravity field modelling and determination of the geoid. In this study, radial basis functions, as a commonly useful tool for gravity data integration, are employed to model the gravity potential field of the southern part of Iran using terrestrial gravity anomalies, gravity anomalies derived from re-tracked satellite altimetry, marine gravity anomalies, and gravity anomalies synthesized from an Earth gravity model. Reference GNSS/levelling (geometric) geoidal heights are used to evaluate the accuracy of the estimated local gravity field model. The gravimetric geoidal heights are in acceptable agreement with the geometric ones in terms of the standard deviation and the mean value which are 4.1 and 12 cm, respectively. Besides, the reference benchmark of the national first-order levelling network of Iran is located in the study area. The derived gravity model was used to compute the gravity potential difference at this point and then transformed into a height difference which results in the value of the shift of this benchmark with respect to the geoid. The estimated shift shows a good agreement with previously published studies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Recent applications in Earth sciences require geoid models to be determined with a sub-centimetre internal error. Regional models of the geoid are usually determined using discrete gravity values measured at and/or outside the Earth, and global models of the Earth gravity field and topographic surface. In this article, we review previous studies that (to some extent) discuss the estimation of the geoid internal error, and provide formulations and methodologies required for a comprehensive formal propagation of errors of gravity data and global models through a mathematical model used for regional geoid determination. The mathematical model is based on combining the inverse Poisson integral equation and the Hotine integral transform in the Helmert harmonic space; also called the one-step integration method. Calculations and tests are performed in one of the most challenging test areas ("the Colorado test area") using ground and airborne gravity observations, a global digital terrain model (DTM) for topographic effects on gravity and the geoid, and a global Earth gravitational model (EGM) for the long-wavelength components of gravity and the geoid. There are three main contributors to the total internal error of the geoid height, namely those associated with the EGM (for estimating the long-wavelength geoid height), DTM heights (for evaluation of the topographic effects on observed gravity and the geoid height), and gravity observations (for determining the short-wavelength components of the geoid height). The geoid errors stemming from the EGM formal variances of its spherical harmonic coefficients amount to 0.3 cm of the total internal error budget of the geoid. The mean value of the standard deviation of the geoid height stemming from the topographic effects (due to uncertainties of DTM heights) is 1.0 cm. The observation errors and spatial distribution (resolution) of regional ground and airborne gravity observations, i.e., the design of the project, are the dominant contributors to the internal error of the geoid height. The internal error estimate of the geoid model in the Colorado test area computed on a 1′ × 1′ grid is 2.7 cm which agrees with the differences between various geoid models that contributed to the Colorado 1-cm geoid experiment. Determining a regional geoid model with a sub-centimetre internal error in areas of rough topography, such as that of Colorado, requires high-accuracy and high-resolution gravity observations. To assess the accuracy of regional gravity measurements required for sub-centimetre geoid models, we simulate airborne gravity measurements in the Colorado test area at a practical lower flight altitude, and improve the spatial distribution of existing ground gravity observations by filling in gaps using synthetic (EGM-based) gravity disturbances. Results show that carrying out airborne gravity surveys with a gentle drape approach at an average flight altitude between 300 and 500 m above the Earth's surface provides airborne gravity measurements with a mean standard deviation of 0.75 mGal at 2.2 km spatial resolution. Thus, a sub-centimetre regional gravimetric geoid model in the Colorado test area would be achievable using the proposed configuration of the ground and airborne gravity observations and more accurate DTMs. [ABSTRACT FROM AUTHOR]

Downward continuation of the potential data (gravity or magnetic) helps to interpret contributing sources by transforming the physical signal to their neighborhood. This paper applies three continuation approaches (Landweber's iteration, the direct method and the inverse-matrix approach with Tikhonov's regularization), based on the Poisson integral equation, and four integration schemes to the second vertical derivative of the anomalous potential $$T_{\rm zz}$$ obtained from GOCE data. In the experiments, $$T_{\rm zz}$$ was downward continued for 250 km in the area of Central Europe with special attention to edge effects. For the integration schemes, which use prior information outside the region of interest to reduce edge effects, the best agreement with TIM-r4 was achieved by the inverse-matrix approach with Tikhonov's regularization (RMS = 1.15 eotvos). On the contrary, without prior information the iterative approach performed with RMS = 1.17 eotvos. [ABSTRACT FROM AUTHOR]

Abstract: In classical isostatic models for a gravimetric recovery of the Moho parameters (i.e., Moho depths and density contrast) the isostatic gravity anomalies are usually defined based on the assumption that the topographic mass surplus and the ocean mass deficiency are compensated within the Earth''s crust. As acquired in this study, this assumption yields large disagreements between isostatic and seismic Moho models. To assess the effects not accounted for in classical isostatic models, we conduct a number of numerical experiments using available global gravity and crustal structure models. First, we compute the gravitational contributions of mass density contrasts due to ice and sediments, and subsequently evaluate respective changes in the Moho geometry. Residual differences between the gravimetric and seismic Moho models are then used to predict a remaining non-isostatic gravity signal, which is mainly attributed to unmodeled density structures and other geophysical phenomena. We utilize three recently developed computational schemes in our numerical studies. The apparatus of spherical harmonic analysis and synthesis is applied in forward modeling of the isostatic gravity disturbances. The Moho depths are estimated globally on a 1 arc-deg equiangular grid by solving the Vening-Meinesz Moritz inverse problem of isostasy. The same estimation model is applied to evaluate the differences between the isostatic and seismic models. We demonstrate that the application of the ice and sediment density contrasts stripping gravity corrections is essential for a more accurate determination of the Moho geometry. We also show that the application of the additional non-isostatic correction further improves the agreement between the Moho models derived based on gravity and seismic data. Our conclusions are based on comparing the gravimetric results with the CRUST2.0 global crustal model compiled using results of seismic surveys. [Copyright &y& Elsevier]