Earthquake hypocentre determination by analytical presentation of the Wadati’s graphical method (four seismological stations)

(in Croatian)

Abstract

In the series of papers about the shallow and deep earthquakes Japanese seismologist K. Wadati presented a graphical method for the earthquake hypocentre determination. Due to its simplicity, it soon became well known as one of the best methods for locating near earthquakes. In the first part of this paper (Paragraphs 1., 2., 3.) all the details of Wadati’s graphical method are presented. In Paragraph 4. original Wadati’s procedure is presented in the analytical form. The solution of the equations (Eqs. 1.-4) of the seismic waves propagation is sought in the fixed rectangular coordinates. After the necessary transformations the equations (15) are obtained. This equation is linear in four unknowns: x_o, y_o, R_o² and c². To solve the system it is necessary to form the equations for four seismological stations. The system (16) combined with the equation (17) completely defines the position of the hypocentre and the value of the constant c. In Appendix A a method of coordinate transformations is presented according to C. F. Richter (1958). It is somewhat simplified due to the linearity of coefficients A^C and B in the latitudes of Yugoslavia.

To complete the analysis of the data of an earthquakes, in Paragraph 5. the graphical and the analytical method of the hypocentral time determination are presented. The graphical method is presented by the well known Wadati’s diagram (Wadati, 1933) (Fig. 8 as example) while the analytical method consists in the straight line fitting to the observational data by the least squares method (Eq. 25., 28., -30.).

       The describe method for the hypocentre determination cannot be applied when all four seismological stations lie on the straight line. This restriction is discussed in Appendix C where it is shown that in the case of linearity between the station coordinates the corresponding determination (3.C) is vanishing.

       In Paragraph 6. an example is presented. The hypocentre of the earthquake of 05. 02. 1969. at 04h 25m GMT is determined applying both graphical (Fig. 3.) and the analytical procedure. Some difficulties that may arise in the graphical procedure are also discussed, in particular the case when the ration m_ik (Eq. (5)) approaches unity.

       It is shown that the analytical method supersedes the graphical one in the precision of the results obtained, convenience of the procedure and the speed of its application, especially when an electronic computer is available.