Abstract
Three dimensional modeling of the wave field makes possible to obtain a concept on distribution of seismic waves in geological medium with different complexity of its structure. The location of the source of fluctuations may be defined in this case at any depth. Therefore, three dimensional modeling of the wave field may be used while solving the problems of seismology related to the nature of strains in the Earth’s interior, while studying the dangerous phenomena of endogenous origin as well as in the case of studies of the deep structure of the Earth, the boundaries of its division and its tectonic structures.All existing methods of numerical simulation of the wave field are based on the solving the equation of elastodynamics with the help of different methods defined by interfacial and initial conditions and are subdivided into axial-symmetrical and pseudo-spectral. The first ones are based on approximation of structural model as a rotary-symmetrical along the vertical axis of the source of fluctuations and solving of the equation of elastodynamics in sylindric or spheric coordinates and in the second ones while solving the equation of elastodynamics Fourier transformation is used.At the Institute of Geophysics named after S.I. Subbotin NAS of Ukraine a method of finite-differential simulation of the wave field has been elaborated both in two-dimensional and three-dimensional variants. The choice of finite-differential method for proceeding of the wave field is explained by its high stability and accuracy as well as with possibility of its application to the wide class of heterogenous models of the medium. Finite-differential modeling is based on the solving of the scalar equation with application of the spacio-temporal net. While solving the problems of seismology in the case of three-dimensional modeling of the wave field we are to take sphericity of the Earth into account. For this purpose in the differential wave equation a transition from Cartesian to spherical coordinate system has been provided. The theory and algorithm of the elaborated three-dimensional modeling of the wave field with finite differential method taking into account sphericity of the Earth has been considered in the paper with demonstration on the model example.
Highlights
В статье [Пилипенко, Верпаховская, 2019] была исследована возможность применения конечно-разностного метода для кинематического моделирования трехмерного временного поля при решении задач сейсмологии
The theory and algorithm of the elaborated three-dimensional modeling of the wave field with finite differential method taking into account sphericity of the Earth has been considered in the paper with demonstration on the model example
Summary
В статье [Пилипенко, Верпаховская, 2019] была исследована возможность применения конечно-разностного метода для кинематического моделирования трехмерного временного поля при решении задач сейсмологии. Целью моделирования распространения сейсмических волн в сферической Земле является расчет волнового поля на некотором четырехугольном участке поверхности Земли Q (заданном в географических координатах) от эпицентра источника колебаний S, расположенного вне этого участка в точке с координатами ( rqs , θqs , φqs ) , где rs = R − ds , R — радиус Земли, ds — глубина точки возбуждения колебаний, θs — широта и φs — долгота для точки возбуждения. 4. Четырехмерный шаблон пространственно-временной сетки для расчета трехмерного волнового поля, включающий в себя 13 сеточных узлов: j, l, i, k — сеточные координаты, соответствующие осям координат (r, θ, φ, t), кружки — узлы сетки, где волновое поле определено, квадраты — узлы сетки, где волновое поле определяется.
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