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In this contribution, we will discuss a problem of regional gravitational field modelling by the spectral combination of spaceborne first-, second- and third-order radial derivatives of the gravitational potential at the mean altitude of 250 km. Some of the mentioned measurements are not observable yet, so we will synthesize them from a global geopotential model over a test area. The main goal of a numerical experiment is to compute the gravitational field in a regional area from spatially restricted higher-order radial derivatives of disturbing potential, and two problems arise. The first is how to estimate local spectral properties, i.e., degree-order variances of local data, and the second is the effect of the omitted distant zone data, i.e., (spatial) truncation error. The spherical harmonic power spectrum from local data will be recovered by 2D-DFT, and the truncation error will be calculated from a global geopotential model. For this purpose, a mathematical model based on the spectral combination method of heterogeneous gravity field data and a global geopotential model is developed. The correctness of derived spectral weights suitable for spectral downward continuation will be verified by a closed-loop test and by direct comparison with the global solution. Moreover, different sizes of a data area (area covered by simulated satellite measurements) to minimize the truncation error and various truncation degrees of a global geopotential model will be tested.

Spherical harmonic expansions are routinely used to represent the gravitational potential and its higher-order spatial derivatives in global geodetic, geophysical, and planetary science applications. The convergence domain of external spherical harmonic expansions is the space outside the minimum Brillouin sphere (the smallest sphere containing all masses of the planetary body). Nevertheless, these expansions are commonly employed inside this bounding surface without any corrections. Justification of this procedure has been debated for several decades, but conclusions among scholars are indefinite and even contradictory.In this contribution, we examine the use of external spherical harmonic expansions for the gravitational field modelling inside the minimum Brillouin sphere. We employ the most recent lunar topographic LOLA (Lunar Orbiter Laser Altimeter) products and the measurements of the lunar gravitational field by the GRAIL (Gravity Recovery and Interior Laboratory) satellite mission. We analyse selected quantities calculated from the most recent GRAIL-derived gravitational field models and forward-modelled (topography-inferred) quantities synthesised by internal/external spherical harmonic expansions. The comparison is performed in the spectral domain (in terms of degree variances depending on the spherical harmonic degree) and in the spatial domain (in terms of spatial maps). To our knowledge, GRAIL is the first gravitational sensor ever, which helped to resolve the long-lasting convergence/divergence problem for the analytical downward continuation of the external spherical harmonic expansions, see (Šprlák and Han, 2021). ReferencesŠprlák M, Han S-C (2021) On the Use of Spherical Harmonic Series Inside the Minimum Brillouin Sphere: Theoretical Review and Evaluation by GRAIL and LOLA Satellite Data. Earth-Science Reviews, 222, 103739, https://doi.org/10.1016/j.earscirev.2021.103739.

This paper investigates the rate of transformation induced plasticity in TRIP steel (TRansformation-Induced Plasticity) after plastic straining by the use of Barkhausen noise emission. The samples were subjected to a variable degree of plastic straining and analysed by the use of conventional techniques such SEM, XRD, as well as microhardness in order to investigate residual stress and microstructural alterations initiated by the uniaxial tensile test. Barkhausen noise emission is analysed as a function of plastic straining as well as in the direction of the exerted load and interpreted with respect to the aforementioned microstructure and stress alterations. It was found that Barkhausen noise markedly decreases along with increasing plastic straining, up to 20%, followed by a strain region in which the evolution of Barkhausen noise reaches saturation. Samples after the tensile test exhibited marked magnetic anisotropy since the Barkhausen noise emission in the direction perpendicular to the tensile stress remained less affected. Apart from the effective value of Barkhausen noise, the Barkhausen noise envelopes were also analysed.

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.

The increasing load of road infrastructure is a current problem in many industrialized regions of Europe. Road freight transport is widely represented in such areas. In combination with high traffic volume from individual and public passenger transport and at the same time insufficient capacity of the road infrastructure, it causes traffic jams and delays. One of the other negative effects in this case is CO2 emissions. That is why the TRANS TRITIA project was created, which aims to make more efficient utilization of intermodal freight transport in order to make transport more environmentally friendly. This goal is also defined in the Europe 2020 strategies and the EU White Paper on Transport. The intention is to transfer 30% of road freight transport over 300 km by 2030 to other modes of transport, such as for rail or shipping, and by 2050 it should be more than 50%. This paper describes the process of implementing the TRANS TRITIA project.

Elastická hrúbka litosféry (Te) je parameter predstavujúci litosférickú pevnosť vzhľadom na zaťaženie. Tieto miesta s veľkými hodnotami elastickej hrúbky sa ohýbajú menej. V tomto príspevku sa na určenie hrúbky litosféry v Afrike používajú gravitačné gradienty merané družicou GOCE. Výpočtová metóda bola odvodená na základe teórie podľa Vening Meinesz-Moritz (VMM) a teórie ohybu izostázy na nájdenie matematického vzťahu medzi druhými deriváciami tiažového potenciálu meraného družicou GOCE a mechanickými vlastnosťami litosféry a je založená na priamom modelovaní. Topografia a batymetria bola modelovaná pomocou modelu DTM2006.0. Efekty zaťaženia sedimentov a kryštalických hmôt boli odhadnuté z modelu CRUST1.0. Z rovnakého modelu boli odhadnuté laterálne zmeny hustoty horného plášťa, hodnoty Youngovho modulu a Poissonovho pomeru. Druhé derivácie gravitačného potenciálu boli generované z modelu kôry a rôznej hodnoty elastickej hrúbky litosféry, aby sa zistilo, ktorý z nich zodpovedá gradientu GOCE na obežnej dráhe. Táto metóda bola odvodená v tvare sférických harmonických funkcií a použiteľná v akomkoľvek bode pozdĺž orbity GOCE bez použitia akejkoľvek rovinnej aproximácie. Naše hodnoty Te nad Afrikou ukazujú, že vnútrozemské hotspoty a sopky, ako sú Ahaggar, Tibesti, Darfúr, sopečná linka Kamerun a Líbya, sú spojené koridormi nízkeho Te. Vysoké hodnoty Te sú prevažne spojené s kronickými oblasťami Konga, Čadu a západnej Afriky. The lithospheric strength with respect to the loading is represented by a parameter called elastic thickness (Te) and places with larger value of Te flex less. In this contribution, we use the in-orbit vertical gravitational gradients measured by Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite 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 vertical derivative of the gravitational potential and mechanical properties of the lithosphere. This method is developed in terms of spherical harmonics. Loading effects of topography and bathymetry, sediments and crystalline masses are calculated 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 vertical 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 in-orbit vertical gradient. Our map of Te over Africa shows that the high values of Te are mainly associated with the cratonic areas of Congo, Chad and the Western African basin while the intra-continental hotspots and volcanoes, such as Ahaggar, Tibesti, Darfur, Cameroon volcanic line and Libya are connected by corridors of low Te.

Monitoring of corrosion extent is critical in bridge operation and civil buildings. Reducing the effective cross-sectional area of components can redistribute the stress state and increase the true stress in the critical parts of civil structures. Unexpected rupture of bodies can occur as a result of their over-stressing. However, it is not possible to take relevant samples from these structures for a laboratory analysis during their lifetime. For this reason, this study investigates the potential of magnetic Barkhausen noise analysis for real-time corrosion extent monitoring. This study demonstrates that Barkhausen noise emission drops down along with an increasing degree of corrosion of steel S460MC. Progressive and remarkable decrease of Barkhausen noise and alteration of extracted features result from increasing surface roughness and superimposing influence of the increasing thickness of the near-surface layers containing mainly iron oxides. Furthermore, it was found that corrosion on the surface also alters the relation between tensile stress and Barkhausen noise within the interaction volume.

This paper studies the contribution of airborne gravity data to improvement of gravimetric geoid modelling across the mountainous area in Colorado, USA. First, airborne gravity data was processed, filtered, and downward-continued. Then, three gravity anomaly grids were prepared; the first grid only from the terrestrial gravity data, the second grid only from the downward-continued airborne gravity data, and the third grid from combined downward-continued airborne and terrestrial gravity data. Gravimetric geoid models with the three gravity anomaly grids were determined using the least-squares modification of Stokes’ formula with additive corrections (LSMSA) method. The absolute and relative accuracy of the computed gravimetric geoid models was estimated on GNSS/levelling points. Results exhibit the accuracy improved by 1.1 cm or 20% in terms of standard deviation when airborne and terrestrial gravity data was used for geoid computation, compared to the geoid model computed only from terrestrial gravity data. Finally, the spectral analysis of surface gravity anomaly grids and geoid models was performed, which provided insights into specific wavelength bands in which airborne gravity data contributed and improved the power spectrum., Journal of Geodesy, 95 (5), ISSN:0949-7714, ISSN:1432-1394

Cílem tohoto příspěvku je modelování zemského gravitačního pole Země kombinující povrchové gravitační poruchy a údaje z družicové gravitační gradiometrie (SGG) založené na kolokací nejmenších čtverců (MNČ) nad prázdnými oblastmi v Íránu, kde stávající údaje o gravitaci nejsou poskytovány nebo dostupný. Tíhové poruchy jsou získány na geocentrické sféře, která je uvnitř topografických hmot společnou inverzí čtyř vysoce přesných složek (Vxx, Vyy, Vzz, Vxz) dat GOCE SGG za použití druhých deriváci Hotinova integrálu. Globální geopotenciální model GOCE (GGM) GO_CONS_GCF_2_TIM_R5 až do stupně a řádu 60 se použije ke snížení účinku vzdálené zóny. Model zbytkového terénu (RTM) je odstraněn, aby se odhadl vliv chyby vynechání GGM a harmonizoval prostor mimo hraniční plochu. Gravitační poruchy založené na GOCE na geocentrické sféře se pokračují vzhůru na zemský povrch na základě rovinné aproximace Poissonova integrálu a jsou odečítány od pozemských gravitačních měření pro výpočet reziduálních tíhových poruch. Reziduální povrchové tíhové poruchy jsou interpolována v různých velikostech buněk podle rozložení vstupních gravitačních dat a jsou stanoveny polohy prázdných buněk, které představují prázdné oblasti. Zbytkové hodnoty, které nejsou korelovány s výškou, jsou aplikovány na predikci zbytkového gravitačního rušení pro každou prázdnou buňku na základě metody MNČ s využitím analytické lokální kovarianční funkce. Vypočtené výsledky přinesly zlepšení o 18,1% až 24,6% ve srovnání s jedinými řešeními GGM na bázi RTM pro dvě vybrané testovací oblasti v Íránu. Objective of this paper is regional gravity field modelling of the Earth combining surface gravity disturbances and satellite gravity gradiometry (SGG) data based on the least-squares collocation (LSC) over the void areas in Iran, where the existing gravity data are not intensified or available. The gravity disturbances are obtained at minimum geocentric sphere, which is inside the topographic masses by joint inversion of the four high-accuracy components (Vxx, Vyy, Vzz, Vxz) of the GOCE SGG data using second derivatives of the Hotine integral. The GOCE-only global geopotential model (GGM) GO_CONS_GCF_2_TIM_R5 up to degree and order 60 is applied to reduce the effect of the omitted distant-zone, which is restored after calculation. The residual terrain model (RTM) is removed to estimate the effect of omission error of the GGM and provide mass-free space outside the boundary surface. The GOCE-based gravity disturbances at minimum geocentric sphere are continued upward on the Earth's surface based on planar approximation of the Poisson integral and subtracted from terrestrial gravity measurements to compute the residual gravity disturbances. The surface residual gravity disturbances are gridded in different cell-sizes according to the distribution of input gravity data and the locations of the empty cells that represent the void areas are determined. The height-uncorrelated residual values are applied to predict the residual gravity disturbance for every empty cell based on the LSC method using analytical local covariance function. The calculated results yielded an improvement of about 18.1% to 24.6% in gravity field recovery compared to the only RTM-corrected GGM-based solutions for two selected test areas in Iran.

We estimate the elastic thickness of a continental lithosphere by using two approaches that combine the Vening Meinesz-Moritz (VMM) regional isostatic principle with isostatic flexure models formulated based on solving flexural differential equations for a thin elastic shell with and without considering a shell curvature. To model the response of the lithosphere on a load more realistically, we also consider lithospheric density heterogeneities. Resulting expressions describe a functional relation between gravity field quantities and mechanical properties of the lithosphere, namely Young’s modulus and Poisson’s ratio that are computed from seismic velocity models in prior of estimating the lithospheric elastic thickness. Our numerical study in central Eurasia reveals that both results have a similar spatial pattern, despite exhibiting also some large localized differences due to disregarding the shell curvature. Results show that cratonic formations of North China and Tarim Cratons, Turan Platform as well as parts of Siberian Craton are characterized by the maximum lithospheric elastic thickness. Indian Craton, on the other hand, is not clearly manifested. Minima of the elastic thickness typically correspond with locations of active continental tectonic margins, major orogens (Tibet, Himalaya and parts of Central Asian Orogenic Belt) and an extended continental crust. These findings generally support the hypothesis that tectonically active zones and orogens have a relatively small lithospheric strength, resulting in a significant respond of the lithosphere on various tectonic loads, compared to a large lithospheric strength of cratonic formations.

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.

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. First Online: 06 November 2018Funders: Czech Science Foundation, GA18-06943S

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) was the first mission which carried a novel instrument, gradiometer, which allowed to measure the second-order directional derivatives of the gravitational potential or gravitational gradients with uniform quality and a near-global coverage. More than three years of the outstanding measurements resulted in two levels of data products (Level 1b and Level 2), six releases of global gravitational models (GGMs), and several grids of gravitational gradients (see, e.g., ESA-funded GOCE+ GeoExplore project or Space-wise GOCE products). The grids of gravitational gradients represent a step between gravitational gradients measured directly along the GOCE orbit and data directly from GGMs. One could use grids of gravitational gradients for geodetic as well as for geophysical applications. In this contribution, we are going to validate the official Level 2 product GRD_SPW_2 by terrestrial gravity disturbances and GNSS/levelling over two test areas located in Europe, namely in Norway and former Czechoslovakia (now Czechia and Slovakia). GRD_SPW_2 product contains all six gravity gradients at satellite altitude from the space-wise approach computed only from GOCE data for the available time span (r-2, r-4, and r-5) and provided on a 0.2 degree grid. A mathematical model based on a least-squares spectral weighting will be developed and the corresponding spectral weights will be presented for the validation of gravitational gradients grids. This model allows us to continue downward gravitational gradients grids to an irregular topographic surface (not to a mean sphere) and transform them into gravity disturbances and/or geoidal heights in one step. Before we compared results obtained by spectral downward continuation, we had to remove the high-frequency part of the gravitational signal from terrestrial data because in gravitational gradients measured at GOCE satellite altitude is attenuated. To do so we employ EGM2008 up to d/o 2160 and the residual terrain model correction (RTC) has been a) interpolated from ERTM2160 gravity model, b) synthesised from dV_ELL_Earth2014_5480_plusGRS80, c) calculated from a residual topographic model by forward modelling in the space domain.

Monitoring of the stress state of bridges by the use of the Barkhausen noise technique has been already introduced and this method can be adapted for monitoring of component’s overstressing. Measurement of Barkhausen noise on real bridges is carried out through the coating applied as a layer to increase the corrosion resistance of bodies. However, it was found that the thickness of the coating could vary, which in turn affects the Barkhausen noise signals and makes it difficult to assess the real stress state. For this reason, this paper deals with attenuation of Barkhausen noise emission due to variable thicknesses of coatings on the steel S460MC. It was found that increasing the thickness progressively decreases the Barkhausen noise emission and shifts the Barkhausen noise envelopes to the higher magnetic fields. Furthermore, the thickness of the coating also affects the relationship between the tensile stress and the Barkhausen noise.

This manuscript aims to create large-scale calculations of agro-climatic factors from global climatic data with high granularity-climatic ERA5-Land dataset from the Copernicus Climate Change Service in particular. First, we analyze existing approaches used for agro-climatic factor calculation and formulate a frame for our calculations. Then we describe the design of our methods for calculation and visualization of certain agro-climatic factors. We then run two case studies. Firstly, the case study of Kojčice validates the uncertainty of input data by in-situ sensors. Then, the case study of the Pilsen region presents certain agro-climatic factors calculated for a representative point of the area and visualizes their time-variability in graphs. Maps represent a spatial distribution of the chosen factors for the Pilsen region. The calculated agro-climatic factors are frost dates, frost-free periods, growing degree units, heat stress units, number of growing days, number of optimal growing days, dates of fall nitrogen application, precipitation, evapotranspiration, and runoff sums together as water balance and solar radiation. The algorithms are usable anywhere in the world, especially in temperate and subtropical zones.

Geoid lze vypočítat z kvazigeoidu pomocí aplikace separace geoid-kvazigeoid. Separace geoidu na kvazigeoid je také nutná pro sjednocení výškových systémů. Informace o skutečném rozdělení topografické hustoty jsou potřebné k přesnému určení geoidu a ortometrické výšky. V této studii odhadujeme účinek (laterální) anomální topografické hustoty na separaci geoid-kvazigeoid. To bylo možné po vydání prvního globálního modelu laterální topografické hustoty UNB_TopoDens. Tento model poskytuje také informace o nejistotách topografické hustoty. Podle našich odhadů pomocí modelu UNB_TopoDens se účinek anomální topografické hustoty na separaci geoid-kvazigeoid globálně pohybuje většinou v rozmezí +/- 0,02 m. V horských oblastech (zejména v Himalájích a Tibetu) by tento účinek mohl dosáhnout (nebo dokonce překročit) +/- 0,1 m. Analýza také odhaluje, že chyby ve vypočítaných hodnotách separace geoid-kvazigeoid připisované nejistotám hustoty UNB_TopoDens (poskytované ve smyslu standardních odchylek pro reprezentativní litologie) jsou celosvětově většinou v řádu několika centimetrů. V částech horských oblastí s velkou nejistotou topografické hustoty však mohou tyto chyby překročit +/- 0,1 m. Zde se zabýváme dalším zásadním aspektem. Podle modelu UNB_TopoDens je průměrná topografická hustota pro celou pevninu (kromě polárních ledovců) 2247 kg m^(-3). Tato průměrná hustota je podstatně menší než hodnota 2670 kg m^(-3), která se obvykle používá v geodetických a geofyzikálních aplikacích. Pokud bude tento nový odhad potvrzen nezávisle, může mít dopad na Helmertovy ortometrické výšky (přijatý v mnoha zemích pro realizaci vertikálního referenčního systému). Změny ortometrických výšek podle Helmerta v důsledku přijetí tohoto nového odhadu průměrné topografické hustoty jsou systematické a dosahují několika decimetrů. Proto navrhujeme použít gravimetrický model geoidu pro praktickou realizaci vertikálních systémů, která zahrnuje informace o topografické hustotě v zemích, kde jsou použity Helmertovy ortometrické výšky. Toto doporučení je plně kompatibilní s moderními koncepty realizace vertikálních systémů založených na použití gravimetrického geoidu a geodetických (elipsoidních) výšek. Rovněž řešíme nesrovnalosti ve výpočtech Helmertových ortometrických výšek a gravimetrických geoidů a navrhujeme použít pouze přesně vypočítané orthometrické výšky (včetně členu s proměnnou topografickou hustotou) ke kombinování (nebo přizpůsobení) gravimetrického geoidu s geometrickými výškami geoidu na srovnávacích bodech GPS-nivelace. 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.

The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km $$\times $$ 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1′ $$\times $$ 1′ grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sanchez et al. in J Geodesy 95(3):1. https://doi.org/10.1007/s00190-021-01481-0 , 2021). Three comparisons are performed: the area comparison to show the model precision, the comparison with the GSVS17 data to estimate the relative accuracy of the models, and the differential quasigeoid (slope) comparison with GSVS17 to assess the relative accuracy of the height anomalies at different baseline lengths. The results show that the precision of the 1′ × 1′ models over the complete area is about 2 cm, while the accuracy estimates along the GSVS17 profile range from 1.2 cm to 3.4 cm. Considering that the GSVS17 does not pass the roughest terrain, we estimate that the quasigeoid can be computed with an accuracy of ~ 2 cm in Colorado. The slope comparisons show that RMS values of the differences vary from 2 to 8 cm in all baseline lengths. Although the 2-cm precision and 2-cm relative accuracy have been estimated in such a rugged region, the experiment has not reached the 1-cm accuracy goal. At this point, the different accuracy estimates are not a proof of the superiority of one methodology over another because the model precision and accuracy of the GSVS17-derived height anomalies are at a similar level. It appears that the differences are not primarily caused by differences in theory, but that they originate mostly from numerical computations and/or data processing techniques. Consequently, recommendations to improve the model precision toward the 1-cm accuracy are also given in this paper.

Steel rope wires represent the main bearing components of bridges whose long-term operation depends on loading conditions, corrosion attack, and/or pre-stressing. Corrosion attack especially can remarkably reduce the effective cross-sectional area, which in turn over-stresses the wires and redistributes stress to the neighboring wires. The premature collapse of many bridges is very often caused by wire rupture as a result of their poor corrosion protection. For these reasons, various processes&mdash, such as galvanizing, phosphating, etc.&mdash, have been applied to steel wires to increase their resistance against corrosion. However, these processes can alter the microstructure, especially in the near-surface regions. The Barkhausen noise technique has been already reported as a suitable technique for investigating corrosion extent and true pre-stress in the steel rope wires. This study reports that non-homogeneity of the surface state of wires undergoing different surface treatment makes it more difficult to assess the true stress state and increase the uncertainty of Barkhausen noise measurement. Barkhausen noise signals are correlated with metallographic and SEM observations as well as microhardness measurements. The non-homogeneity of the surface state of wires is also investigated by the use of chemical mapping and linear chemical analyses.

Geodetic boundary-value problems (BVPs) and their solutions are important tools for describing and modelling the Earth’s gravitational field. Many geodetic BVPs have been formulated based on gravitational observables measured by different sensors on the ground or moving platforms (i.e. aeroplanes, satellites). Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface integrals with Green’s kernel functions. When solving this problem for higher-order derivatives of the gravitational potential as boundary conditions, more than one solution is obtained. Solutions to gravimetric, gradiometric and gravitational curvature BVPs (Martinec 2003; Šprlák and Novák 2016), respectively, lead to two, three and four formulas. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.In this contribution we present combinations of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green’s kernel functions by a spectral combination method. We investigate an optimal combination of different orders and directional derivatives of potential. The spectral combination method is used to combine terrestrial data with global geopotential models in order to calculate geoid/quasigeoid surface. We consider that the first-, second- and third-order directional derivatives are measured at the satellite altitude and we continue them downward to the Earth’s surface and convert them to the disturbing gravitational potential, gravity disturbances and gravity anomalies. The spectral combination method thus serves in our numerical procedures as the downward continuation technique. This requires to derive the corresponding spectral weights for the n-component estimator (n = 1, 2, … 9) and to provide a generalized formula for evaluation of spectral weights for an arbitrary N-component estimator. Properties of the corresponding combinations are investigated in both, spatial and spectral domains.

V geofyzikálních a geodetických studiích je inverze tíhových dat obvykle prováděna tak, že měřené gravitační hodnoty jsou nejprve pokračovány směrem dolů na pravidelný (rovinný, sférický nebo sféroidní) povrch řešením inverzní integrální transformace, která vychází z klasického řešení první úlohy teorie potenciálu. Typickým příkladem je prodlužování gravitace pozorované na topografickém povrchu dolů na střední hladinu moře (geoid). V dnešní době poskytují gravitační družicové mise a letecká gravimetrie kromě klasických povrchových gravitačních dat i gravitační data nad topografickým povrchem. Pro specifické účely (např. kombinace a validace dat nebo řešení kvazigeoidu) se musí data prodloužit na nepravidelný topografický povrch. V této studii se zabýváme problémem vytvořením funkčního modelu pro spektrální prodlužování vybraných parametrů gravitačního pole směrem k nepravidelnému topografickému povrchu směrem dolů. Navíc zobecňujeme tento funkční model, abychom umožnili transformaci mezi různými typy parametrů gravitačního pole. Zejména odvozujeme spektrální váhy pro odhad poruchového potenciálu nebo poruchové / anomální gravitace na zemském povrchu kombinováním radiálních gradientů prvního, druhého a třetího řádu rušivého potenciálu (rušivé gradienty). Správnost vyvinutého kombinovaného spektrálního odhadu se ověřuje v testu založeným na syntetických datech. 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.

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.

Až donedávna byla informace o distribuci topografické hustoty omezena pouze na určité regiony a některé země, zatímco v globálním kontextu chyběla. UNB_TopoDens je první model, který poskytuje informace o laterální topografické hustotě po celém světě. Analýza tohoto modelu ukazuje, že průměrná topografická hustota pro celou kontinentální pevninu (s výjimkou polárních ledovců) je 2247 kg m-3. Tato hustota se významně liší od hodnoty 2670 kg m-3, která se obvykle používá pro znázornění kontinentální hustoty horních kůr. V této studii používáme model hustoty UNB_TopoDens ke kontrole, jak změny topografické hustoty ovlivňují interpretační vlastnosti Bouguerových tíhových dat. 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.

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.

This paper investigates the potential of a non-destructive magnetic technique based on Barkhausen noise emission for the monitoring of prestressing bars with respect to their undesired over-stressing. Barkhausen noise signals are correlated with tensile stress, residual stresses, and microhardness measurements. It was found that prestressing bars exhibit strong magnetic anisotropy which becomes more pronounced along with the increasing degree of the bar’s over-stressing. Barkhausen noise emission becomes strongly attenuated in the direction of the tensile stress at the expense of the perpendicular direction. However, the Barkhausen noise emission in the direction of the tensile stress exhibits a continuous and remarkable decrease, whereas the Barkhausen noise steep increase for lower degrees of over-stressing is followed by early saturation for higher over-stressing. This study demonstrates that the Barkhausen noise technique is capable of distinguishing between the prestressing bars loaded below yielding, and those which are over-stressed.

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.

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.

In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundary-value problem can formally be formulated for the vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal components of the third-order gravitational tensor. Each integral equation provides an independent set of spherical harmonic coefficients because each component of the third-order gravitational tensor is sensitive to gravitational changes in the different directions. In this contribution, estimations of spherical harmonic coefficients of the gravitational potential are carried out by combining four solutions of the gravitational curvature boundary-value problem using three methods, namely an arithmetic mean, a weighted mean and a conditional adjustment model. Since the third-order gradients of the gravitational potential are not yet observed by satellite sensors, we synthesise them at the satellite altitude of 250 km from a global gravitational model up to the degree 360 while adding a Gaussian noise with the standard deviation of 6.3 × 10−19 m−1 s−2. Results of the numerical analysis reveal that the arithmetic mean model provides the best solution in terms of the RMS fit between predicted and reference values. We explain this result by the facts that the conditions only create additional stochastic bindings between estimated parameters and that more complex numerical schemes for the error propagation are unnecessary in the presence of only a random noise.

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.

Litosférická struktura pod Indickým oceánem je pravděpodobně nejsložitější, ale zároveň nejméně pochopená mezi světovými oceány. Výsledky tomografických, geochemických, magnetických a jiných průzkumů svědčí o jeho komplexní geologické historii. Seismické průzkumy byly primárním zdrojem informací o litosférické struktuře pod Indickým oceánem, ale tyto experimenty jsou soustředěny hlavně na místech s vysokým geofyzikálním zájmem. Údaje o mořské gravitaci získané zpracováním měření družicové výškoměry naopak poskytují podrobný obraz reliéfu celého mořského dna, čímž se dále rozšiřují znalosti o jeho tvorbě, tektonismu a vulkanismu. V této studii používáme údaje o gravitační, bathymetrické, mořské sedimentu a litosférické hustotě pro sestavení gravitačních map Bouguer a pláště. Poté použijeme obě gravitační mapy k interpretaci litosférické struktury pod Indickým oceánem. Bouguerova gravitační mapa odhaluje hlavní tektonické a sopečné rysy, které jsou prostorově korelovány s odchylkami tloušťky kůry. Mapa gravitačního pláště vykazuje hlavně tepelný podpis litosférického pláště. Gravitační minima v této gravitační mapě značí výrazně aktivní oceánské divergentní tektonické okraje podél středových, jihovýchodních a jihozápadních indických hřebenů, včetně také Carlsbergského hřebene. Gravitační minima se rozprostírá podél Rudého moře - Adenského zálivu a východoafrických trhlin, což potvrzuje spojení mezi středo oceánskými hřebeny šíření (v Indickém oceánu) a kontinentálními trhlinami (ve východní Africe). Kombinovaná interpretace Bouguerových a plášťových gravitačních map potvrzuje kolizní původ horských pásem podél kontinentálních trhlin ve východní Africe. Důkaz o jižním rozšíření východoafrického riftového systému a jeho spojení s jihozápadním indickým hřbetem v mapě gravitačního pláště chybí. Podobně se neprojevuje pokračující rozpad složené indo-australské desky. Chybějící tepelný podpis v gravitační mapě pláště na těchto dvou místech je vysvětlen skutečností, že jižní hranice Nubian-Somálska (tj. Lwandleova deska) a Indo-australská hranice (tj. Capricornova deska) jsou rozptýlené zóny konvergence, charakterizované nízkou deformací a seismicitou v důsledku velmi pomalých rychlostí relativních pohybů ubytovaných přes tyto hranice. Chybí také jasný projev tepelného podpisu horkých míst uvnitř destičky v mapě gravitace pláště. Toto zjištění souhlasí s důkazy z přímých měření tepelného toku, které nenaznačují přítomnost významné pozitivní teplotní anomálie ve srovnání s oceánskou litosférou podobného věku. 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.

Gravitační údaje josu používány pro modelovaní a interpretační studie v geovědách. Tento příspěvek analyzuje v současné době pozorovatelná i předpokládaná gravitační data reprezentovaná gradienty gravitačního potenciálu. Funkční modely mapující 3D měrnou hmotnost na gradienty potenciálu až do třetího řádu jsou formulovány za použití objemových integrálů newtonovského typu s jednotkovými jádrovými funkcemi vyjádřenými analyticky i pomocí nekonečné řady přidružených Legendreovy funkcí. Analyzují se prostorové a spektrální vlastnosti integračních jader a studuje se citlivost gradientů na konkrétní rozložení hmot. V numerických experimentech jsou použity dva konkrétní modely distribuce hmot: lokální trojrozměrný model hustoty a globální model představovaný horními sedimenty s laterální změnou měrné hmotnosti. Vypočítané hodnoty gradientů demonstrují jejich různé citlivosti na konkrétní rozložení hmot, které se mění s rostoucí vzdáleností gradientů od gravitujících hmot. Gradienty třetího řádu jsou zvláště užitečné pro studium mělkých hmot blízkých povrchu Země, jako jsou jeskyně, solné dómy, sedimenty nebo kontinentální okraje. Gradienty vyššího řádu mohou tedy být zajímavým nástrojem pro mapování hmot, jakmile bude zajištěna jejich pozorovatelnost s dostatečnou přesností a rozlišením. 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.

This paper deals with monitoring of the rupture of S235 steel used in ship structures under uniaxial loading by the use of a non-destructive magnetic method based on Barkhausen noise emission. Component failure in constructions of marine or civil engineering can be found during their long-term operation or/and over-stressing. Especially the corrosion process can significantly reduce effective cross-section area of structures, which in turn increases true stresses. This study demonstrates that Barkhausen noise emission and extracted parameters can reliably reveal structure over-stressing. A remarkable decrease of Barkhausen noise near the true yield stress alerts the high risk of incoming breakage. It was found that the evolution of Barkhausen noise is driven by the steel hardening (expressed in terms of hardness and the corresponding dislocation density) and the superimposing stress state.

This paper deals with the monitoring of steel wires used for pre-stressed concrete steel strands and their over-stressing. These components, as the main bearing elements, represent high-strength structures that are usually pre-stressed in bridge constructions. This study employs the Barkhausen noise technique to prevent the over-stressing of these wires and demonstrates that Barkhausen noise emission decreases remarkably with increasing degree of over-stressing. It was found that over-stressing beyond the yield strength increases dislocation density, which in turn increases interference between the domain walls in motion. Increasing the pinning strength of the matrix reduces Barkhausen noise emission and an increasing degree of over-stressing can be directly linked with irreversible alteration of Barkhausen noise and extracted parameters such as the number of Barkhausen pulses and their magnitude as well as Barkhausen noise envelopes.

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.

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 $$^{-1/2}$$ .