Milan Hvoždara

Potential field of a stationary electric current in a stratified medium with a three-dimensional perturbing body

SummaryThe potential of the electric field of a stationary current in a two-layered Earth is calculated by applying Greens formula in the case where a three-dimensional inhomogeneity of different conductivity is located in the basement of the layer. It is proved that the potential outside and inside the perturbing body can be calculated from the potential of an electric double-layer distributed on the surface of this body. An integral equation of the Fredholm type is derived for the surface density of the double-layer, together with some of its integral properties. A similar procedure can be applied to computing the magnetic anomalies of three-dimensional magnetized bodies, geothermal anomalies due to three-dimensional inhomogeneities of different heat conductivity, as well as to potential problems of theoretical electrical engineering.

Electric and magnetic field of a stationary current in a stratified medium with a three-dimensional conductivity inhomogeneity

SummaryNew formulae have been derived for calculating the electric and magnetic field generated by an external source of a stationary electrical current in a stratified medium with a three-dimensional disturbing body in one of the layers. The potential of the anomalous current field is expressed in terms of the potential of a dipole layer, distributed over the surface of the disturbing body. The density of the dipole layer is calculated with the aid of the scalar Fredholm integral equation. The components of the magnetic field intensity are calculated from the superposition of the magnetic effects of elementary electric dipoles distributed over the surface of the disturbing body. The results of the numerical computation of the anomalous electric and magnetic field on the surface of the halfspace inside which a block of different electric conductivity is located, are given, the exciting field being homogeneous.

Algorithm for solving 3-D problems of em induction by means of a vector integral equation

РезюмеПре¶rt;ложен aлzорumм решенuя зa¶rt;aчu элеmромazнumноŭ uн¶rt;укцuu ¶rt;ля zaрмонuческu nеременноzо возбуж¶rt;aющzо nоля в nолуnросmрaнсmве с mрехмерным возмущaющuм mелом. Иссле¶rt;овaны своŭсmвa элеменmов mензорноŭ функцuu nо элеменmaрным обьемным ячеŭкaм, нa коmорые рaз¶rt;елено возмущaющее mело. Полученные nрu вьчuсленuях резульmamы оmносяmся к nроmяженноŭ, высокоnрово¶rt;ящеŭ nрuзме, являющеŭся о¶rt;ноŭ uз мо¶rt;елеŭ nроекma COMMEMI.SummaryAn algorithm is presented for solving the problem of electromagnetic induction in a halspace with a 3-D perturbing body under a harmonically variable exciting field. There are analysed properties of the Greens tensor function elements as well as their integrals in elementary prismatic volumes into which the body is divided. The results, obtained from numerical computations, are given for a long highly conducting prism, which is one of the models in the COMMEMI project.

Electromagnetic induction of a three-dimensional conductivity inhomogeneity in a two-layer earth

SummaryThe results of numerical computations of the electromagnetic field induced in a two-layer model of the Earth with a three-dimensional inhomogeneity (a block) in the subsurface layer are given. Several recommendations are given which have enabled the solution of a complicated system of integral equations and the computation of the field at the Earths surface in an effective way. The analysis of the obtained solution has proved that, in an anomalous electromagnetic field, the field of the horizontal electric dipole, oriented in the direction of the exciting field, is predominant. A number of practically usable diagrams and approximative formulae is given.

Electromagnetic induction of a three-dimensional conductivity inhomogeneity in the two-layered earth

SummaryThe calculation of the elements of Greens tensor function is presented for solving the problem of the electromagnetic induction by means of a vector integral equation. A two-layered Earth is considered as the medium, the surface layer including a three-dimensional conductivity inhomogeneity. Use is made of the boundary condition requiring the vertical component of the electric current to be zero at the Earths surface which partly simplifies the theoretical computation. Long-period asymptotics of the individual complicated functions, occurring in Greens tensor function as well as in the tensor function required to calculate the components of the anomalous magnetic field at the surface of the halfspace, were effected. With the aid of these asymptotics one can obtain estimates of the functions occurring in the theoretical analysis of the problem.

Thermo-elastic deformations due to the annual temperature variation at the tidal station in Vyhne

РезюмеВычuслены mермоуnругuе ¶rt;еформaцuu, вызвaнныего¶rt;овоŭ вaрuaцuеŭ mемnерamуры nоверхносmu скaльного nолуnросmрнaсmвa. Временныŭ хо¶rt; эmuх ¶rt;еформaцuŭ срaвнен с хо¶rt;ом ме¶rt;ленных ¶rt;еформaцuŭ, uзмеренных нa nрuлuвноŭ сmaнцuu Выгне (ЧССР).SummaryThe thermo-elastic deformations due to the annual temperature variation are computed. The time variation of these deformations is compared with the variation of the slow deformations observed at the tidal station in Vyhne.

Electromagnetic induction in a halfspace with an oblique interface

РезюмеДается приблизительное решение проблемы электромагнитной индукции в полупространстве, которое разделено наклонной плоскостью на две клиновидных областей. В одной из них предполагается проводимость идеальная, в другой-конечной величины. Выведены интегральные выражения для поверхностных импедансов и нормальной компоненты магнитного поля на дневной поверхности клина конечной проводимости, приводятся вычисления их модулей в зависимости от угла наклона плоскости раздела, расстояния от ребра клина и периода изменения с временем возбуждающего поля. Даны некоторые выводы относительно магнитотеллурическосо метода.

Refraction Effect in Geothermal Heat Flow Due to a 3-D Prism in a Two-Layered Earth

A mathematical model of the disturbance of the stationary geothermal field due to a three-dimensional perturbing body embedded near the surface of a two layered earth. The theoretical analysis is based on the generalized theory of the double-layer potential, similar to the boundary integral method likely to direct current geoelectricity problems. Special attention is paid to cases in which a pan of the bodys surface is tangential to the planar boundaries of the substratum or of the surface of the earth The numerical calculations are performed for a three-dimensional prism using methods similar to the boundary element method (BEM). Numerous graphs are shown for the disturbance of the temperature field and of the heat flow on the surface of the earth and along vertical profiles (boreholes).

Stationary approximation of the magnetotelluric field in a halfspace with a vertical boundary and three-dimensional perturbing body

SummaryBased on the generalized potential of a double layer, integral formulae have been derived for calculating the stationary approximation of the magnetotelluric field in a halfspace divided by a vertical boundary into two quarterspaces, one of which contains a three-dimensional perturbing body. The appropriate boundary integral equation and other surface integrals have been computed for a perturbing body in the shape of the three-dimensional prism located at the vertical boundary, or in contact with the said boundary. The exciting electrical field is assumed to be homogeneous and perpendicular to the vertical boundary. Isoline graphs of the electrical and magnetic fields on the surface of the halfspace have been plotted and their anomalies are discussed.

Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations

Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations We present a simple derivation of the interior and exterior gravitational potentials due to oblate spheroid and also its gravity field components by using the fundamental solution of the Laplace equation in oblate spheroidal coordinates. Application of the method of separation of variables provides an expression for the potential in terms of oblate spheroidal harmonics of degree n = 0, 2. This solution is more concise and suitable for the numerical calculations in comparison with infinite series in spherical harmonics. Also presented are the computations in the form of potential isolines inside and outside the spheroid, as well as for the gravity field components. These reveal some interesting properties of the gravity field of this fundamental geophysical body useful for the applied gravimetry.