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Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine's, Poisson's, and Stokes's integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations. [ABSTRACT FROM AUTHOR]

The launch of gravity-dedicated satellite missions at the beginning of the new millennium led to an accuracy improvement of global Earth gravity field models (GGMs). One of these missions was the Gravity field and steady-state Ocean Circulation Explorer (GOCE) launched in 2009. As the first European Space Agency's Earth Explorer Mission, the satellite carried a novel instrument, a 3-D gradiometer, which allowed measurement of the second-order directional derivatives of the gravitational potential (gravitational gradients) with a uniform quality and a near-global coverage. The main mission goal was to determine the static Earth's gravity field with the ambitious precision of 1-2 cm in terms of geoid heights and 1 mGal in terms of gravity anomalies for spatial resolution of 100 km (half wavelength at the equator). More than three years of the outstanding measurements resulted in three levels of data products (Level 0, Level 1b and Level 2), six releases of GGMs, and several global grids of gravitational gradients. The grids, which represent a step between gravitational gradients measured directly along the GOCE orbit and those represented by GGMs, found their usage mainly in geophysical applications. In this contribution, we validate the official Level 2 product GRD_SPW_2 using height anomalies over two test areas located in central and northern Europe (Czechia/Slovakia and Norway). A mathematical model based on the least-squares spectral weighting is employed with corresponding spectral weights estimated for validation of gravitational gradient grids. This model continues gravitational gradients from the mean orbital altitude of GOCE down to the irregular Earth's surface (not to a sphere) and transforms them to height anomalies in one computational step. Analytical downward continuation errors of the model are estimated using a closed-loop test. Prior to the comparison of height anomalies estimated from gravitational gradients with their reference values derived from Global Navigation Satellite Systems (GNSS)/levelling over the two test areas, the gravitational gradients and reference data are corrected for all systematic effects such as the tide system conversion. Moreover, the high-frequency part of the gravitational signal is estimated and subtracted from reference data as it is attenuated in the gravitational gradients measured by GOCE. A relative improvement between the release 6 and release 2 gradient grids reaches 48 % in terms of height anomalies in Czechia/Slovakia. The relative improvement in Norway is even more significant and reaches 55 % . The release 6 of the official Level 2 product GRD_SPW_2 gained the absolute accuracy with the standard deviation of 8.7 cm over Czechia/Slovakia and 9.3 cm over Norway. Article Highlights: A mathematical model based on the least-squares spectral weighting for validation of the official Level 2 product GRD_SPW_2 is presented Possibilities of the spectral combination method for the downward continuation of gravitational gradients and their transformation into height anomalies are investigated A review of quality of the official Level 2 product GRD_SPW_2 with respect to GNSS/levelling points over two test areas (Czechia/Slovakia and Norway) is provided [ABSTRACT FROM AUTHOR]

This study investigates the influence of strain rate on plastic deformation developed in the flange of a steel road barrier. This effect can be investigated by the use of the uniaxial tensile test. It was found that the strain rate increases yield as well as ultimate strength and gently drops down the elongation at break. Moreover, the accelerated strain rate is connected with matrix heating and increasing the Taylor–Quinney coefficient. Despite the valuable matrix heating and the higher Taylor–Quinney coefficient at the higher strain rates, samples necking is initiated earlier and dislocation density is higher. Flange grains become preferentially aligned along the direction of uniaxial stress, especially at the higher plastic strains. Finally, surface Zn protective layer delamination is initiated quite early beyond the yielding. It is considered that the cracks are due to the different response of the Zn allayer and underlying steel matrix on the plastic straining. Increasing strain rate attenuates the degree of Zn layer delamination. [ABSTRACT FROM AUTHOR]

Physical geodesy applies potential theory to study the Earth’s gravitational field in space outside and up to a few km inside the Earth’s mass. Among various tools offered by this theory, boundary-value problems are particularly popular for the transformation or continuation of gravitational field parameters across space. Traditional problems, formulated and solved as early as in the nineteenth century, have been gradually supplemented with new problems, as new observational methods and data are available. In most cases, the emphasis is on formulating a functional relationship involving two functions in 3-D space; the values of one function are searched but unobservable; the values of the other function are observable but with errors. Such mathematical models (observation equations) are referred to as deterministic. Since observed data burdened with observational errors are used for their solutions, the relevant stochastic models must be formulated to provide uncertainties of the estimated parameters against which their quality can be evaluated. This article discusses the boundary-value problems of potential theory formulated for gravitational data currently or in the foreseeable future used by physical geodesy. Their solutions in the form of integral formulas and integral equations are reviewed, practical estimators applicable to numerical solutions of the deterministic models are formulated, and their related stochastic models are introduced. Deterministic and stochastic models represent a complete solution to problems in physical geodesy providing estimates of unknown parameters and their error variances (mean squared errors). On the other hand, analyses of error covariances can reveal problems related to the observed data and/or the design of the mathematical models. Numerical experiments demonstrate the applicability of stochastic models in practice. [ABSTRACT FROM AUTHOR]

In geophysical and geodetic studies, gravity inversion is typically performed such that observed gravity values are first continued downward onto a regular (planar, spherical or spheroidal) surface by solving an inverse integral transform, which originates from a classical solution to the first boundary-value problem in potential theory. A typical example is continuing gravity observed at the topographic surface down to the mean sea level (geoid). Nowadays, gravity-dedicated satellite missions and aerial gravimetry provide gravity data above the topographic surface in addition to classical terrestrial gravity observations. For specific purposes (such as gravity data combination and validation, or quasigeoid determination), satellite and aerial gravity observations have to be continued to the irregular topographic surface. In this study, we address this issue by formulating a functional model for a spectral downward continuation of selected gravitational field quantities to an irregular topographic surface. Moreover, we generalize this functional model to allow for transformation between different types of gravitational field quantities. In particular, we derive spectral weights for estimation of the disturbing potential or disturbing/anomalous gravity at the Earth's surface by combining the first-, second- and third-order radial gradients of the disturbing potential (disturbing gradients). The correctness of the developed combined spectral estimator is verified in a closed-loop test based on synthetic satellite disturbing gradients. The combined spectral estimator is applied to simulated satellite disturbing gradients polluted by a realistic Gaussian noise. Results of the numerical experiments show that the combined spectral estimator puts the highest importance on the least polluted disturbing gradient, while the contribution of the least accurate disturbing gradient is negligible. An important advantage of this spectral combination method is that no matrix inversion with numerical instabilities requiring regularization is needed. [ABSTRACT FROM AUTHOR]

Elastic thickness (Te) is a parameter representing the lithospheric strength with respect to the loading. Those places, having large values of elastic thickness, flexes less. In this paper, the on-orbit measured gravitational gradients of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission are used for determining the elastic thickness over Africa. A forward computational method is developed based on the Vening Meinesz-Moritz (VMM) and flexural theories of isostasy to find a mathematical relation between the second-order derivative of the Earth's gravity field measured by the GOCE satellite and mechanical properties of the lithosphere. The loading of topography and bathymetry, sediments and crystalline masses are computed from CRUST1.0, in addition to estimates of laterally-variable density of the upper mantle, Young's modulus and Poisson's ratio. The second-order radial derivatives of the gravitational potential are synthesised from the crustal model and different a priori values of elastic thickness to find which one matches the GOCE on-orbit gradient. This method is developed in terms of spherical harmonics and performed at any point along the GOCE orbit without using any planar approximation. Our map of Te over Africa shows that the intra-continental hotspots and volcanoes, such as Ahaggar, Tibesti, Darfur, Cameroon volcanic line and Libya are connected by corridors of low Te. The high values of Te are mainly associated with the cratonic areas of Congo, Chad and the Western African basin. [ABSTRACT FROM AUTHOR]

Four solutions of the spherical gravitational curvature boundary-value problems can be exploited for the determination of the Earth's gravitational potential. In this paper we discuss the combination of simulated satellite gravitational curvatures, that is, components of the third-order gravitational tensor, by merging these solutions using the spectral combination method. For this purpose, integral estimators of biased- and unbiased-types are derived. In numerical studies, we investigate the performance of the developed mathematical models for the gravitational field modelling in the area of Central Europe based on simulated satellite measurements. First, we verify the correctness of the integral estimators for the spectral downward continuation by a closed-loop test. Estimated errors of the combined solution are about eight orders smaller than those from the individual solutions. Second, we perform a numerical experiment by considering the Gaussian noise with the standard deviation of 6.5 × 10−17 m−1 s−2 in the input data at the satellite altitude of 250 km above the mean Earth sphere. This value of standard deviation is equivalent to a signal-to-noise ratio of 10. Superior results with respect to the global geopotential model TIM-r5 (Brockmann et al. 2014) are obtained by the spectral downward continuation of the vertical–vertical–vertical component with the standard deviation of 2.104 m2 s−2, but the root mean square error is the largest and reaches 9.734 m2 s−2. Using the spectral combination of all gravitational curvatures the root mean square error is more than 400 times smaller but the standard deviation reaches 17.234 m2 s−2. The combination of more components decreases the root mean square error of the corresponding solutions while the standard deviations of the combined solutions do not improve as compared to the solution from the vertical–vertical–vertical component. The presented method represents a weight mean in the spectral domain that minimizes the root mean square error of the combined solutions and improves standard deviation of the solution based only on the least accurate components. [ABSTRACT FROM AUTHOR]

We investigate a numerical performance of four different schemes applied to a regional recovery of the gravity anomalies from the third-order gravitational tensor components (assumed to be observable in the future) synthetized at the satellite altitude of 200 km above the mean sphere. The first approach is based on applying a regional inversion without modelling the far-zone contribution or long-wavelength support. In the second approach we separate integral formulas into two parts, that is, the effects of the third-order disturbing tensor data within near and far zones. Whereas the far-zone contribution is evaluated by using existing global geopotential model (GGM) with spectral weights given by truncation error coefficients, the near-zone contribution is solved by applying a regional inversion. We then extend this approach for a smoothing procedure, in which we remove the gravitational contributions of the topographic-isostatic and atmospheric masses. Finally, we apply the remove-compute-restore (r-c-r) scheme in order to reduce the far-zone contribution by subtracting the reference (long-wavelength) gravity field, which is computed for maximum degree 80. We apply these four numerical schemes to a regional recovery of the gravity anomalies from individual components of the third-order gravitational tensor as well as from their combinations, while applying two different levels of a white noise. We validated our results with respect to gravity anomalies evaluated at the mean sphere from EGM2008 up to the degree 250. Not surprisingly, better fit in terms of standard deviation (STD) was attained using lower level of noise. The worst results were gained applying classical approach, STD values of our solution from Tzzz are 1.705 mGal (noise value with a standard deviation 0.01 x 10 -15m-1s-2) and 2.005 mGal (noise value with a standard deviation 0.05 x 10-15m-1s-2), while the superior from r-c-r up to the degree 80, STD fit of gravity anomalies from Tzzz with respect to the same counterpart from EGM2008 is 0.510 mGal (noise value with a standard deviation 0.01 x 10-15m-1s-2) and 1.190 mGal (noise value with a standard deviation 0.05 x 10-15m-1s-2). [ABSTRACT FROM AUTHOR]

In this article we discuss a regional recovery of gravity disturbances at the mean geocentric sphere approximating the Earth over the area of Central Europe from satellite gravitational gradients. For this purpose, we derive integral formulas which allow converting the gravity disturbances onto the disturbing gravitational gradients in the local north-oriented frame (LNOF). The derived formulas are free of singularities in case of r ≠ R . We then investigate three numerical approaches for solving their inverses. In the initial approach, the integral formulas are firstly modified for solving individually the near- and distant-zone contributions. While the effect of the near-zone gravitational gradients is solved as an inverse problem, the effect of the distant-zone gravitational gradients is computed by numerical integration from the global gravitational model (GGM) TIM-r4. In the second approach, we further elaborate the first scenario by reducing measured gravitational gradients for gravitational effects of topographic masses. In the third approach, we apply additional modification by reducing gravitational gradients for the reference GGM. In all approaches we determine the gravity disturbances from each of the four accurately measured gravitational gradients separately as well as from their combination. Our regional gravitational field solutions are based on the GOCE EGG_TRF_2 gravitational gradients collected within the period from November 1 2009 until January 11 2010. Obtained results are compared with EGM2008, DIR-r1, TIM-r1 and SPW-r1. The best fit, in terms of RMS (2.9 mGal), is achieved for EGM2008 while using the third approach which combine all four well-measured gravitational gradients. This is explained by the fact that a-priori information about the Earth’s gravitational field up to the degree and order 180 was used. [ABSTRACT FROM AUTHOR]

Regional recovery of the disturbing gravitational potential in the area of Central Europe from satellite gravitational gradients data is discussed in this contribution. The disturbing gravitational potential is obtained by inverting surface integral formulas which transform the disturbing gravitational potential onto disturbing gravitational gradients in the spherical local north-oriented frame. Two numerical approaches that solve the inverse problem are considered. In the first approach, the integral formulas are rigorously decomposed into two parts, that is, the effects of the gradient data within near and distant zones. While the effect of the near zone data is sought as an inverse problem, the effect of the distant zone data is synthesized from the global gravitational model GGM05S using spectral weights given by truncation error coefficients up to the degree 150. In the second approach, a reference gravitational field up to the degree 180 is applied to reduce and smooth measured gravitational gradients. In both cases we recovered the disturbing gravitational potential from each of the four well-measured gravitational gradients of the GOCE satellite separately as well as from their combination. Obtained results are compared with the EGM2008, DIR-r2, TIM-r2 and SPW-r2 global gravitational models. The best fit was achieved for EGM2008 and the second approach combining all four well-measured gravitational gradients with rms of 1.231 m2 s-2. [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]

Synthesis, design of a new logistics system, changes the existing system after analysis of the production, marketing, distribution, service systems and their management. The most often analysis applied is systems analysis, SWOT analysis, statistical analysis, multi-criteria analysis. Each analysis has a different objective, the reasons for the application, complexity, and methodology. The modelling of decision-making activities is essential for a model creation - a synthesis of a logistic system, which is mostly applied in a heuristic approach, based on modelling solutions and processes that human experts use for their activities. This article describes the specific methodology of analysis - analysis for heuristics models design. [ABSTRACT FROM AUTHOR]

Results obtained as a solution to the geodetic boundary-value problem, where two different sources of input boundary data are combined together, are presented. On the upper part of the boundary defined by the Earth surface and separated from the Earth masses, the gravity anomalies are derived from GOCE gravity gradiometry data using a transformation and downward continuation procedure. On the bottom part of the boundary, the gravity anomalies derived from the terrestrial or marine gravity measurements are used. The boundary-value problem is then solved numerically by the finite element method. In some cases an independency of our solution from the highdegree global gravity models (e.g. EGM2008) can be considered as the advantage. The standard deviation of our model tested by GNSS/leveling method in Auvergne area is 10 cm. [ABSTRACT FROM AUTHOR]

This study deals with monitoring of Zn layer heterogeneity on the flange of steel road barriers using magnetic measurements. The Barkhausen noise technique is employed for such purpose, and parameters extracted from Barkhausen noise signals are correlated with the true thickness of the Zn layer. The true values of the Zn layer were obtained from the metallographic images, as well as the thickness gauge CM-8825FN (Guangzhou Landtek Instruments Co. Ltd., Guangzhou, China) device. It was observed that the diffusion region lies below the Zn protective layer, which makes the thickness of the Zn layer obtained from the CM-8825FN device thicker than that measured on the metallographic images. For this reason, the chemical gradient of Zn below the Zn layer can be reported, and it affects Barkhausen noise emission. Barkhausen noise decreases along with increasing thickness of the Zn layer, and Barkhausen noise envelopes are shifted to stronger magnetic fields. The number of strong MBN pulses drops down with the increasing thickness of Zn coating at the expense of the increasing number of the weak MBN pulses. The thickness of Zn coating can be polluted by the solidification of Zn melt after galvanizing. The presence of the diffusion layer dims the contrast between ferromagnetic and paramagnetic phases. [ABSTRACT FROM AUTHOR]

Low alloyed steels of low, medium, or high strength are frequently used for many applications in the automotive, civil (bridges), aerospace, and petrochemical industries. A variety of thermomechanical regimes, in which these steels can be produced, enable customization of their matrix with respect to their fatigue resistance, resistance against friction and impact wear, fracture toughness, corrosion resistance, etc. This study analyses the influence of magnetising conditions on Barkhausen noise and other extracted parameters. It was found that the increasing magnetising frequency makes Barkhausen noise weaker, especially in the high strength low alloyed steels, as a result of the decreasing magnetic field in a sample. For this reason, increasing fraction of domain walls is unpinned at the higher frequencies. Barkhausen noise for the high strength low alloyed steels at higher frequencies is remarkably attenuated. Moreover, the different behaviour with respect to direction of the sheet rolling and the transversal direction, can be found due to realignment of the domain walls. This study demonstrates that the position of Barkhausen noise envelopes and the number of Barkhausen noise pulses increase in a systematic manner at the lower magnetising frequencies. Those parameters can be employed for distinction of the low alloyed steels, investigated in this study. However, the increasing magnetising frequency makes attenuation of Barkhausen noise more remarkable for the low alloyed steels of the higher strength. Therefore, the effective value of Barkhausen noise, at the magnetising frequency 750 Hz, in the rolling direction exhibits the systematic descent along with the increasing yield strength. This parameter can be used for distinction of the low alloyed steels after their thermomechanical processing, as well. [ABSTRACT FROM AUTHOR]

Gravity data have been applied for modelling and interpretation studies in geosciences. This contribution reviews currently observable and foreseen gravity data represented by gradients of the gravitational potential. Functional models linking 3-D mass density distribution functions to potential gradients of up to the third order are formulated using volume integrals of the Newtonian type with unitless kernel functions expressed both analytically and using infinite series of associated Legendre functions. Spatial and spectral properties of the kernel functions are analysed and sensitivity of the gradients to particular mass density distributions is studied. Two particular mass density distribution models are used in numerical experiments: a local 3-D mass density model representing shallow mass density variations and a global mass model represented by Earth's upper sediments with lateral mass density variations. Computed values of the gradients demonstrate their different sensitivities to particular mass density distributions which change with an increasing distance of the gradients from gravitating masses. Third-order gradients are particularly useful for studying near subsurface or shallow density structures such as caves, caverns, salt domes, sediment basement morphology, continental margins or buried fault systems that could be identified spatially more closely. Thus, higher-order gradients would offer an interesting tool for mass density mapping once their observability with the sufficient accuracy and resolution is realized. [ABSTRACT FROM AUTHOR]

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]

In context of the vertical datum unification, the geoid-to-quasi-geoid separation has been of significant interest in recent years, because most of existing local vertical datums are realized in the system of either normal or orthometric heights. Nevertheless, the normal-orthometric heights are still used in many other countries where the normal gravity values along leveling lines were adopted instead of the observed gravity. Whereas the conversion between the orthometric and normal heights is defined by means of the mean gravity disturbances (i.e. differences between the mean values of the actual and normal gravity) along the plumbline within the topography, differences between the normal and normal-orthometric heights can be described by means of the surface gravity disturbances. Since the normal gravity field does not reflect the topographic masses and actual mass density distribution inside the Earth, the definition of gravity represents a principal aspect for a realization of particular vertical datum. To address this issue in this study, we investigate effects of the Earth's inner density structure on the surface and mean gravity disturbances, and discuss their impact on the vertical datum realization. These two gravity field quantities are computed globally with a spectral resolution complete to a spherical harmonic degree 2160 using the global gravity, terrain, ice-thickness, inland bathymetry and crustal structure models. Our results reveal that both, the surface and mean gravity disturbances mostly comprise the gravitational signal of topography and masses distributed below the geoid surface. Moreover, in polar areas, a significant contribution comes from large glaciers. In contrast, the contributions of anomalous density distribution within the topography attributed to major lakes, sediments and bedrock density variations are much less pronounced. We also demonstrate that the mean gravity disturbances within the topography are significantly modified compared to the corresponding surface values mainly due to topographic elevation and terrain geometry as well as the presence of large glaciers in polar regions. Changes of the vertical gravity gradient within the topography attributed to the masses distributed below the geoid (dominated mainly by the isostatic signature and the long-wavelength gravitational signature of deep mantle density heterogeneities) are, on the other hand, relatively small. Despite differences between the normal and normal-orthometric heights could directly be assessed from the surface gravity disturbances only when taken along leveling lines with information about the spirit leveling height differences, our results indicate that differences between these two height systems can be significant. [ABSTRACT FROM AUTHOR]

In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth's gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of $${\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}$$ Hz $$^{-1/2}$$ . [ABSTRACT FROM AUTHOR]

Third-order gradients of the gravitational potential (gravitational curvatures) have already found some applications in geosciences. Observability of these parameters, describing the Earth's gravitational field in a more complex way than any other currently available gravitational parameter, such as gravitational acceleration (first-order gradient) or gravitational (second-order) gradient, is currently discussed by physicists. Moreover, first designs of observational devices (sensors) have already been proposed. The spherical harmonic analysis and synthesis are the common tools used by geoscientists to study spectral properties of various functionals of the Earth's gravitational potential. However, the conventional spherical harmonic expansions of the gravitational curvatures in the local north-oriented reference frame have rather complicated forms that depend on the first-, second- and third-order derivatives of the associated Legendre functions. Moreover, some of these expansions also contain singular terms at the poles. In this paper, the conventional series are transformed to new simpler and non-singular forms based on relations between the associated Legendre functions and their derivatives. Numerical experiments demonstrate the applicability and correctness of the new expressions. [ABSTRACT FROM AUTHOR]

The usability of the magnetic Barkhausen noise (MBN) technique for monitoring the critical state of plastic straining in the case of interstitial free (IF) steel is demonstrated. IF steels are frequently employed in the automotive industry (e.g., automotive body parts—doors, trunks, wheel arches, etc.), and rupture of components made of these steels occurs during cold forming. Alteration of microstructure expressed in many terms and the corresponding MBN features after the variable degree of plastic strains developed by the uniaxial tensile loading are investigated. It is reported that the effective value of MBN, as well as the shift of MBN envelopes toward higher magnetic fields, can be proposed as the suitable MBN features alerting the incoming matrix breakage. Moreover, the higher MBN in the direction of exerted load at the expense of the perpendicular direction can be found as a result of domain walls realignment. [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]

Since marine seismic studies are relatively sparse and unevenly distributed, detailed tomographic images of the Moho geometry under large parts of the world's oceans and marginal seas are not yet available. Marine gravity data is, therefore, often used to detect the Moho depth in these regions. Alternatively, Airy's isostatic theory can be applied for this purpose. In this study, we compare different isostatic and gravimetric methods for a Moho recovery under the oceanic crust and continental margins, particularly focusing on a numerical performance of Airy, Vening Meinesz–Moritz (VMM), direct gravity inversion, and generalized (for the Earth's spherical approximation) Parker–Oldenburg methods. Numerical experiments are conducted to estimate the Moho depth beneath the Indian Ocean. Results reveal that, among these investigated methods, the VMM model is probably the most suitable for a gravimetric Moho recovery beneath the oceanic crust and continental margins, when taking into consideration the lithospheric mantle density information. This method could to some extent model realistically a Moho geometry beneath mid-oceanic spreading ridges, oceanic subductions, most of oceanic volcanic formations, and marine sediment deposits. Nonetheless, this model still cannot fully reproduce a gradual Moho deepening caused by a conductive cooling and a subsequent isostatic rebalance of the oceanic lithosphere, which can functionally be described by a Moho deepening with the increasing ocean-floor age. Results also indicate that the Airy method typically overestimates the Moho depth under oceanic volcanic formations, while the direct gravity inversion and generalized Parker–Oldenburg methods could not reproduce more detailed features in the Moho geometry. Since Pratt's theory better describes a large-scale isostatic mechanism of the oceanic lithosphere by means of compensation density variations, but does not account for additional changes in compensation depth (i.e., Moho depth) that are caused by these density changes, we tested a possibility of combining Pratt and Airy's isostatic theories in order to estimate the Moho depth under the oceanic crust. Even this combined model cannot fully reproduce a gradual Moho deepening with the increasing ocean-floor age. [ABSTRACT FROM AUTHOR]

The geoid can be computed from the quasigeoid by applying the geoid-to-quasigeoid separation. The geoid-to-quasigeoid separation is also needed for a vertical datum unification. Information about the actual topographic density distribution is required to determine accurately the geoid and orthometric heights. In this study, we estimate the effect of (lateral) anomalous topographic density on the geoid-to-quasigeoid separation. This became possible after releasing the first global lateral topographic density model UNB_TopoDens. This model provides also information about topographic density uncertainties. According to our estimates by using the UNB_TopoDens model, the effect of anomalous topographic density on the geoid-to-quasigeoid separation globally varies mostly within ±0.02 m. In mountainous regions (particularly in the Himalayas and Tibet), this effect could reach (or even exceed) ±0.1 m. The analysis also reveals that the errors in computed values of the geoid-to-quasigeoid separation attributed to the UNB_TopoDens density uncertainties (provided in terms of standard deviations for representative lithologies) are globally mostly within a few centimeters. In parts of mountainous regions with large topographic density uncertainties, however, these errors might exceed ±0.1 m. There is another crucial aspect we address here. According to the UNB_TopoDens model, the average topographic density for the whole landmass (except for polar glaciers) is 2247 kg m−3. This average density is considerably smaller than the value of 2670 kg m−3 that is typically used in geodetic and geophysical applications. This new estimate, if confirmed independently, might have implications on Helmert orthometric heights (adopted in many countries for a vertical datum realization). Changes in Helmert orthometric heights due to adopting this new estimate of the average topographic density are systematic and reach several decimeters. We, therefore, propose to use a gravimetric geoid model for a practical realization of vertical datums that incorporates the topographic density information in countries where Helmert orthometric heights are adopted. This recommendation is fully compatible with modern concepts for a vertical datum realization based on using a gravimetric geoid model and geodetic (ellipsoidal) heights. We also address inconsistencies in computations of Helmert orthometric heights and gravimetric geoid models, and propose to use only accurately computed orthometric heights (including variable topographic density term) to combine (or fit) a gravimetric geoid model with geometric geoid heights at GPS-leveling benchmarks. [ABSTRACT FROM AUTHOR]

The article summarises findings from laboratory testing of asphalt mixtures using reclaimed asphalt containing polymer-modified binder and subsequent technicoeconomical considerations of their use in pavement management system. Tested mixtures had 0%, 15%, and 40% content of reclaimed asphalt (RA) containing polymer-modified bitumen (PMB) obtained by milling from surfacing layers of existing PA, AC11, and SMA11 pavement layers. A complete description of these mixtures is given, and testing methods used are described. The mixtures were tested for air void content, ITSR water sensitivity test, plastic deformation wheel tracking test, stiffness of the tested mixtures, and two-point bending test to ascertain fatigue. Concise and succinct conclusions from laboratory testing are derived and used as an input in the second part of the article. It is found that the addition of RA containing PMB increased stiffness modulus of the final mixture, which increases resistance to plastic deformation and leads to higher brittleness at low temperatures. It is found that virgin PMB also increases fatigue resistance as opposed to PMB content from reclaimed materials. In the second part of the article, findings from laboratory testing are addressed in the context of their practical use in pavement management. The key element identified, having a direct impact on pavement performance models, was the stiffness modulus for mixtures with different PMB-reclaimed asphalt contents. A method is described to evaluate pavement construction properties related to wearing course materials via the pavement performance model. Practical use of this method is described and applied in a case study. In this case study, the proposed method is used to evaluate the issues regarding practical use of asphalt mixtures with different ratios of reclaimed asphalt containing PMB and economic implications of their use. It is found that pavement performance of surfacing mixtures with reclaimed asphalt containing PMB is significantly better for plastic deformation at the cost of earlier initiation and progression of surface distress due to cracking and potholing. This paper suggests that due to cracking and potholing, periodic maintenance costs increase for pavements with reclaimed asphalt material containing PMB; however, they are outweighed by lower procurement cost and longer life expectancy due to slower plastic deformation of the pavement. [ABSTRACT FROM AUTHOR]

The lithospheric structure beneath the Indian Ocean is probably the most complicated, but at the same time, the least understood among world's oceans. Results of tomographic, geochemical, magnetic and other surveys provide the evidence of its complex geological history. Seismic surveys have been a primary source of information about the lithospheric structure beneath the Indian Ocean, but these experiments are mainly concentrated at locations of a high geophysical interest. Marine gravity data obtained from processing the satellite altimetry measurements, on the other hand, deliver a detailed image of the whole seafloor relief, advancing further the knowledge about its formation, tectonism and volcanism. In this study, we use gravitational, bathymetric, marine sediment and lithospheric density structure data to compile the Bouguer and mantle gravity maps. We then use both gravity maps to interpret the lithospheric structure beneath the Indian Ocean. The Bouguer gravity map reveals major tectonic and volcanic features that are spatially correlated with crustal thickness variations. The mantle gravity map exhibits mainly a thermal signature of the lithospheric mantle. Gravity lows in this gravity map mark distinctively active oceanic divergent tectonic margins along the Central, Southeast and Southwest Indian Ridges including also the Carlsberg Ridge. Gravity lows extend along the Red Sea–Gulf of Aden and East African Rift Systems, confirming a connection between mid-oceanic spreading ridges (in the Indian Ocean) and continental rift systems (in East Africa). The combined interpretation of the Bouguer and mantle gravity maps confirms a non-collisional origin of mountain ranges along continental rift systems in East Africa. The evidence of a southern extension of the East African Rift System and its link with the Southwest Indian Ridge in the mantle gravity map is absent. Similarly, the ongoing breakup of the composite Indo-Australian plate is not manifested. A missing thermal signature in the mantle gravity map at these two locations is explained by the fact that the southern Nubian-Somalian plate boundary (i.e., the Lwandle plate) and the Indo-Australian plate boundary (i.e., the Capricorn plate) are diffuse zones of convergence, characterized by low deformation and seismicity due to very slow rates of relative motions accommodated across these boundaries. The clear manifestation of the thermal signature of intraplate hot spots in the mantle gravity map is also absent. This finding agrees with the evidence from direct heat flow measurements that do not indicate the presence of a significant positive temperature anomaly compared to the oceanic lithosphere of a similar age. [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]

Considering that Venus has a size very similar to Earth, thermal evolution of both planets should be comparable. Nonetheless, there is no clear evidence of plate tectonics or plate motions on Venus. Instead, various surface deformations attributed to volcanism, resurfacing, localized subduction and other geologic processes were recognized on the planet. In this study we attempt to classify the origin of lithospheric forces on Venus based on using topographic and gravity information. For this purpose, we also estimate the Venusian crustal thickness. In agreement with findings from previous studies, the signature of past or recent global tectonism in the body-force vector pattern on Venus is absent, while exhibiting only regional anomalies. The maximum intensity inferred in the Atla and Beta Regios is likely attributed to mantle upwelling. This is also confirmed by the gravity-topography spectral correlation and admittance analysis that shows the isostatic relaxation of these volcanic regions. The regional body-force pattern in the Bell Regio suggests that a much less pronounced force intensity there is possibly related to crustal load of lava flows. Elsewhere, the body-force intensity is relatively weak, with slightly more pronounced intensity around the Ishtar Terra and the Arthemis Chasmata. The body-force pattern in the Arthemis Chasmata supports the hypothesis that coronae structures are the result of mantle upwelling and the subsequent (localized) plume-induced subduction with only limited horizontal crustal motions. The prevailing divergent pattern of body-force vectors in the Ishtar Terra region suggests the presence of tensional forces due to the downwelling mantle flow that is responsible for a crustal thickening along the Freyja and Maxwell Montes. Except for the Atla and Beta Regios where the isostasy is relaxed by the (active) mantle plumes, the crustal thickness is spatially highly correlated with the topography, with a thin crust under the plains and a thick crust under the plateaus. The maximum Moho depth under the Maxwell Montes in the Ishtar Terra exceeds 90 km. [ABSTRACT FROM AUTHOR]

In planetary sciences, the geodetic (geometric) heights defined with respect to the reference surface (the sphere or the ellipsoid) or with respect to the center of the planet/moon are typically used for mapping topographic surface, compilation of global topographic models, detailed mapping of potential landing sites, and other space science and engineering purposes. Nevertheless, certain applications, such as studies of gravity-driven mass movements, require the physical heights to be defined with respect to the equipotential surface. Taking the analogy with terrestrial height systems, the realization of height systems for telluric planets and moons could be done by means of defining the orthometric and geoidal heights. In this case, however, the definition of the orthometric heights in principle differs. Whereas the terrestrial geoid is described as an equipotential surface that best approximates the mean sea level, such a definition for planets/moons is irrelevant in the absence of (liquid) global oceans. A more natural choice for planets and moons is to adopt the geoidal equipotential surface that closely approximates the geometric reference surface (the sphere or the ellipsoid). In this study, we address these aspects by proposing a more accurate approach for defining the orthometric heights for telluric planets and moons from available topographic and gravity models, while adopting the average crustal density in the absence of reliable crustal density models. In particular, we discuss a proper treatment of topographic masses in the context of gravimetric geoid determination. In numerical studies, we investigate differences between the geodetic and orthometric heights, represented by the geoidal heights, on Mercury, Venus, Mars, and Moon. Our results reveal that these differences are significant. The geoidal heights on Mercury vary from − 132 to 166 m. On Venus, the geoidal heights are between − 51 and 137 m with maxima on this planet at Atla Regio and Beta Regio. The largest geoid undulations between − 747 and 1685 m were found on Mars, with the extreme positive geoidal heights under Olympus Mons in Tharsis region. Large variations in the geoidal geometry are also confirmed on the Moon, with the geoidal heights ranging from − 298 to 461 m. For comparison, the terrestrial geoid undulations are mostly within ± 100 m. We also demonstrate that a commonly used method for computing the geoidal heights that disregards the differences between the gravity field outside and inside topographic masses yields relatively large errors. According to our estimates, these errors are − 0.3/+ 3.4 m for Mercury, 0.0/+ 13.3 m for Venus, − 1.4/+ 125.6 m for Mars, and − 5.6/+ 45.2 m for the Moon. [ABSTRACT FROM AUTHOR]

In this article boundary-value problems of potential theory, that are required for transforming and continuing selected parameters of a potential field, are discussed. By the potential field parameters we understand the potential and its gradients (potential gradients) up to the third order. In particular, integral equations defined for a spherical boundary transforming potential gradients of different orders and continuing potential gradients of the same order through the 3-D space are reviewed and classified. This mathematical apparatus can be used for any harmonic potential, such as electric, magnetic or gravitational, under the assumption of its conservativeness, i.e., neglecting possible temporal variations. Integral transforms are discussed in context of geoscience applications, namely in terms of the Earth's gravitational field; however, the article can serve as a general reference for integral transforms of potential field parameters in any scientific or engineering area where potential fields are used. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The remarkable drop of MBN envelope width and the shift of MBN envelope above the critical threshold also alert incoming rupture. In article 2100597, Neslusan and co-workers demonstrate the usability of the magnetic Barkhausen noise (MBN) technique for monitoring the critical state of plastic straining in the case of interstitial free (IF) steel used in automotive body parts. [Extracted from the article]

This study investigates the microstructure, residual stress state, and the corresponding magnetic anisotropy of the ship structure samples made of S235 steel after uniaxial tensile deformation. A non-destructive magnetic technique based on Barkhausen noise is employed for fast and reliable monitoring of samples exposed to the variable degrees of plastic straining. It was found that the progressively developed plastic straining of the matrix results in an alteration of the easy axis of magnetization, stress anisotropy (expressed in residual stresses state) as well as the corresponding Barkhausen noise emission. Moreover, remarkable non-homogeneity can be found within the plastically strained region, especially when the localized plastic straining takes place. [ABSTRACT FROM AUTHOR]