Open AccessThe paper presents results of the mathematical synthesis of the method of passive location of a seismic wave source. The method employs measurements of regular attenuation of seismic oscillation amplitudes. If it is impossible to determine the location of a seismic event by means of direct measurements, indirect measurements are needed. A priori information for the mathematical synthesis was obtained from functional equations showing inverse proportions of measured amplitudes, arbitrary effective attenuation coefficients and corresponding coordinates. An original method was applied to process the data. The method providing for passive location of seismic waves sources has been developed; it is called the radial basic method. In the one-dimensional case, a distance is determined on the basis of seismic oscillation amplitudes measured by two seismographs that are located at a known base distance coinciding with the direction to the source of seismic waves. The distance is calculated from the receiver that is nearest to the source. If the base distance and the direct line between the seismograph and the seismic wave source do not coincide, a projection of the distance between the receivers to the given straight line is taken into account.
Three seismographs were placed at mutually perpendicular base distances in a plane (i.e. the two-dimensional space). This allowed us to obtain an analytical equation for determining the direction to the seismic wave source using measured amplitudes. The value of the angle is taken into account when calculating the distance.
For the seismic wave source located in the three-dimensional space, transition equations for combined coordinate systems (i.e. the Descartes (Cartesian)), at the axes of which the seismographs were placed, and the spherical coordinate systems were applied, and analytical equations were obtained for determination of coordinates, such as distance/polar radius, elevation angle/latitude, and bearing angle/longitude.
To analyze the application of the radial basic method of passive location, an absolute error resulting from indirect measurement was calculated. This method is a special case of determining the statistical characteristics of the sought function (coordinates) out of random values, which are the measured amplitudes of seismic oscillations. In the obtained analytical expressions for determination of the mean square deviation of the distance and direction to the seismic wave source, noise from seismographs is taken into account, as well as external noise manifested as microseisms and other interfering seismic waves. For the three-dimensional space, equations are derived for calculation of RMS distance, elevation and bearing angles.
In cases where distances to seismic wave sources, directions, coefficients that factor properties of the medium, and base distances are known, the radial basic method allows us to determine effective coefficients of seismic wave attenuation and capacities of seismic wave sources.