Thermo-elastic strains and stresses play a considerable role in the stress state of the lithosphere and its dynamics, especially at pronounced positive geothermal anomalies. Topography has a significant effect on ground deformation. Two methods for including the topographic effects in the thermo-viscoelastic model are described. First we use an approximate methodology which assumes that the main effect of topography is due to the distance from the source to the free surface and permits to have an analytical solution very attractive for solving the inverse problem. A numerical solution (for 2D plain strain case) is also computed using finite element method (FEM). The numerical method allows to include the local shape of the topography in the modeling. In the numerical model the buried magmatic body is represented by a finite volume thermal source. The temperature distribution is computed by the higher-degree FEM. For analytical as well as numerical model solution only the forces of thermal origin are considered. The results show that for the volcanic areas with prominent topography, its effect on the perturbation of the thermo-viscoelastic solution (deformation and total gravity anomaly) can be quite significant. In consequence, neglecting the topography could give erroneous results in the estimated source parameters.
A generalized mathematical model of the Earth’s density structure is presented in this study. This model is defined based on applying the spectral expressions for a 3-D density distribution within the arbitrary volumetric mass layers. The 3-D density model is then converted into a form which describes the Earth’s density structure by means of the density-contrast interfaces between the volumetric mass layers while additional correction terms are applied to account for radial density changes. The applied numerical schemes utilize methods for a spherical harmonic analysis and synthesis of the global density structure models. The developed the Earth’s density models are then defined in terms of the spherical density and density-contrast functions. We also demonstrate how these Earth’s density models can be applied in the gravimetric forward modeling and discuss some practical aspects of representing mathematically density structures within particular components of the Earth’s interior.
Global atmospheric effects on the gravity field quantities
We compile the global maps of atmospheric effects on the gravity field quantities using the spherical harmonic representation of the gravitational field. A simple atmospheric density distribution is assumed within a lower atmosphere (< 6 km). Disregarding temporal and lateral atmospheric density variations, the radial atmospheric density model is defined as a function of the nominal atmospheric density at the sea level and the height. For elevations above 6 km, the atmospheric density distribution from the United States Standard Atmosphere 1976 is adopted. The 5 × 5 arc-min global elevation data from the ETOPO5 are used to generate the global elevation model coefficients. These coefficients (which represent the geometry of the lower bound of atmospheric masses) are utilized to compute the atmospheric effects with a spectral resolution complete to degree and order 180. The atmospheric effects on gravity disturbances, gravity anomalies and geoid undulations are evaluated globally on a 1 × 1 arc-deg grid.
Vladimír Pohánka, Peter Vajda and Jaroslava Pánisová
Here we investigate the applicability of the harmonic inversion method to time-lapse gravity changes observed in volcanic areas. We carry out our study on gravity changes occured over the period of 2004–2005 during the unrest of the Central Volcanic Complex on Tenerife, Canary Islands. The harmonic inversion method is unique in that it calculates the solution of the form of compact homogeneous source bodies via the mediating 3-harmonic function called quasigravitation. The latter is defined in the whole subsurface domain and it is a linear integral transformation of the surface gravity field. At the beginning the seeds of the future source bodies are introduced: these are quasi-spherical bodies located at the extrema of the quasigravitation (calculated from the input gravity data) and their differential densities are free parameters preselected by the interpreter. In the following automatic iterative process the source bodies change their size and shape according to the local values of quasigravitation (calculated in each iterative step from the residual surface gravity field); the process stops when the residual surface gravity field is sufficiently small. In the case of inverting temporal gravity changes, the source bodies represent the volumetric domains of temporal mass-density changes. The focus of the presented work is to investigate the dependence of the size and shape of the found source bodies on their differential densities. We do not aim here (yet) at interpreting the found solutions in terms of volcanic processes associated with intruding or rejuvenating magma and/or migrating volatiles.
Gerassimos Manoussakis, Romylos Korakitis and Paraskevas Milas
The components of the Eötvös matrix are useful for various geodetic applications, such as interpolation of the elements of the deflection of the vertical, determination of gravity anomalies and determination of geoid heights. A torsion balance instrument is customarily used to determine the Eötvös components. In this work, we show that it is possible to estimate the Eötvös components at a point on the Earth’s physical surface using gravity measurements at three nearby points, comprising a very small network. In the first part, we present the method in detail, while in the second part we demonstrate a numerical example. We conclude that this method is able to estimate the components of the Eötvös matrix with satisfactory accuracy.
Agnieszka Opala-Berdzik, Bogdan Bacik, Joanna Cieślińska-Świder, Michał Plewa and Monika Gajewska
The Influence of Pregnancy on the Location of the Center of Gravity in Standing Position
The purpose of the study was to compare the average location of the center of gravity vertical projection in sagittal plane in women at the beginning of and in advanced pregnancy as well as after delivery. The experiment was performed with the use of a force platform during four test sessions. A group of 44 women (8-16 weeks of pregnancy) participated in the initial test session. In the following sessions the number of the subjects reduced mainly due to medical and childcare problems: 33 women were tested in late pregnancy (2-3 weeks before delivery), and 39 women were tested two and six months after delivery.
The results showed the statisticaly significant (p<0,05) posterior displacement of the projection of the center of gravity of the lenth of approximately 4 mm in late pregnancy comparing to the beginning of pregnancy. The displacement may be the result of the body's adaptation to the increased mass in the anterior trunk area in late pregnancy. No discrepancy was found when comparing the average center of gravity location in the early pregnancy and after delivery.
We concluded that the change of the center of gravity location in late pregnancy is temporary and two months after delivery the vertical projection of the center of gravity is located as it was at the beginning of pregnancy.
Alexander P. Karpik, Vadim F. Kanushin, Irina G. Ganagina, Denis N. Goldobin, Nikolay S. Kosarev and Alexandra M. Kosareva
In the context of the rapid development of environmental research technologies and techniques to solve scientific and practical problems in different fields of knowledge including geosciences, the study of Earth’s gravity field models is still important today. The results of gravity anomaly modelling calculated by the current geopotential models data were compared with the independent terrestrial gravity data for the two territories located in West Siberia and Kazakhstan. Statistical characteristics of comparison results for the models under study were obtained. The results of investigations show that about 70% of the differences between the gravity anomaly values calculated by recent global geopotential models and those observed at the points in flat areas are within ±10 mGal, in mountainous areas are within ±20 mGal.
Martin Krajňák, Miroslav Bielik, Irina Makarenko, Olga Legostaeva, Vitaly I. Starostenko and Marián Bošanský
During the last years, many new results related to the thickness of the sedimentary fill and other geophysical and geological constrains on the structure of the Turčianska Kotlina Basin have been obtained. It allowed us to calculate the first, original stripped gravity map in this basin. To obtain this map the 3D gravity effect of the basin sedimentary fill had to be calculated. The gravity effect of the sediments was determined for two different density conditions. Firstly, the average density of the sediments was constant (2.45 g cm−3). Secondly, we supposed that the density of sediments varies exponentially from 2.00 g cm−3 on the surface up to 2.67 g cm−3 on the pre-Tertiary basement. On the maps of the calculated gravity effects it can be observed that the maximum amplitudes reach about -12 mGal. The gravity effect is larger in the case when the densities of the sedimentary fill vary exponentially. After subtraction of the gravity effects from the map of complete Bouguer gravity map, the resultant stripped gravity maps were defined. The detailed analysis of these maps indicates that the northern part of the pre-Tertiary basement of the basin could be built mostly by the Mesozoic rocks, which belong to the Hronic and Fatric units. The structure of the basement in the southern part of the basin seems to be more complicated. This feature probably reflects a presence of Neogene volcanites in the basin basement belonging to the Kremnické vrchy Mts. The picture of the gravity field in the stripped gravity map predicts that the Paleogene sediments probably do not build the sedimentary fill in the southern part of the basin. Finally, taking into account the differences in the size of the gravity gradients along the basin margins it could be suggested that the dipping of the Veľká Fatra Mts. beneath the Turčianska Kotlina basin is gentle in comparison with the Lučanská Malá Fatra.
We have developed a simple and fast quantitative method for depth and shape determination from residual gravity anomalies due to simple geometrical bodies (semi-infinite vertical cylinder, horizontal cylinder, and sphere). The method is based on defining the anomaly value at two characteristic points and their corresponding distances on the anomaly profile. Using all possible combinations of the two characteristic points and their corresponding distances, a statistical procedure is developed for automated determination of the best shape and depth parameters of the buried structure from gravity data. A least-squares procedure is also formulated to estimate the amplitude coefficient which is related to the radius and density contrast of the buried structure. The method is applied to synthetic data with and without random errors and tested on two field examples from the USA and Germany. In all cases examined, the estimated depths and shapes are found to be in good agreement with actual values. The present method has the capability of minimizing the effect of random noise in data points to enhance the interpretation of results.
The gravimetric inverse problem for finding the Moho density contrast is formulated in this study. The solution requires that the crust density structure and the Moho depths are a priori known, for instance, from results of seismic studies. The relation between the isostatic gravity data (i.e., the complete-crust stripped isostatic gravity disturbances) and the Moho density contrast is defined by means of the Fredholm integral equation of the first kind. The closed analytical solution of the integral equation is given. Alternative expressions for solving the inverse problem of isostasy are defined in frequency domain. The isostatic gravity data are computed utilizing methods for a spherical harmonic analysis and synthesis of the gravity field. For this purpose, we define various spherical functions, which define the crust density structures and the Moho interface globally.