D. Malytskyy1, Dr. Sci. (Phys.-Math.), Prof., E-mail: dmytro@cb-igph.lviv.ua;

O. Muyla1, Cand. Sci. (Phys.-Math.), Research Associate, E-mail: orestaro@gmail.com;

O. Hrytsaj1, Postgraduate Student, E-mail: grycaj.oksana@gmail.com;

. Pavlova1, Ph.D.;

. Astashkina1, Cand. Sci. (Geol.), Research Associate, E-mail: sac1@ukr.net;

O. Obidina1, Postgraduate Student, E-mail: jane.det@yandex.ua;

. Kozlovskyy1, Ph.D


EXTENDED SOURCE: MODELING RESULTS AND PROSPECTS OF APPLICATION FOR SEISMOLOGY PROBLEMS


1 Carpathian Branch of Subbotin Institute of Geophysics NAS of Ukraine, 3-b Naukova St., Lviv, Ukraine, 79060


The solution of the direct problem is presented for the displacement field on the free surface of layered isotropic medium using the matrix method. The results of the direct problem are used to determine the seismic moment tensor. An extended source is considered as a set of point sources, each one is presented by seismic moment tensor. An important aspect is that for the solution of inverse problem an analytical value of the direct problem is used, i.e. inversion for seismic tensor is realized by using solutions for displacement fields. The solution for extended sources is based in the fact that the wave field from such a source is the superposition of displacement fields from each point source. Thus, the statement of the direct problem is to determine the wave field on the free surface of layered half-space when the earthquake's focus is represented as an extended source in space and time. A method is described which determines the displacement field on the free surface in the spectral domain using the values of the shift for elementary sources as well as rise time and rupture time. Matrix method is used in case of seismic waves in horizontal layered half-space where heterogeneous medium is simulated by homogeneous isotropic layers with parallel boundaries. The earthquake's focus as an extended source is placed in a uniform layer. We have shown the transition from a redefined system of equations for determining a slip vector to the solution for the generalized inverse problem. The results of the inverse problem for determining the rupture plane were tested on the example of the events that took place near Malta (24.04.2011: 13h02m12s, 35.92N, 14.95E, Mw4.0)). For this event, the determination of the rise time and rupture time is shown. Correctness of the inverse problem is provided by determining of a functional in which the norm is minimized between the real data and parameters that are obtained using the proposed method. For a singular matrix it is suggested to use a singular decomposition.

Keywords: extended source, seismic field, seismic moment tensor of the earthquake, isotropic medium.


References:

1. Aki K., Richards P., (1983). Quantitative seismology. Theory and methods. Moscow, Mir, 520 p. (In Russian).

2. Verbitsky T.Z., Pochinaiko R.S., Starodub Y.P., Fedorishin O.S., (1985). Mathematical modelling in seismic exploration. Kiev, Science Thought – Naukova Dumka, 275 p. (In Russian).

3. Malytskyy D.V., (2010). Analytical and numerical approaches to the calculation of time depending on the seismic moment tensor components. Geoinformatics, 1, 79-86. (In Ukrainian).

4. Malytskyy D.V., (1998). Basic principles of solving of the seismology dynamic problems based on recurrent approach. Geophysics Journal, 5, 96-98. (In Ukrainian).

5. Malytskyy D.V., (2005). About the seismic waves source. Geophysics Journal, 27 (2), 304-308. (In Ukrainian).

6. Malytskyy D.V., (1994). The recursive method for the inverse dynamic problem solving of seismic exploration in a vertically inhomogeneous medium. Geophysics Journal, 16 (3), 61-66. (In Russian).

7. Malytskyy D.V., Muyla O.O., (2007). About the application of the matrix method and its modifications for the seismic waves study in layered media. Theoretical and applied aspects of geoinformatics, 124-136. (In Ukrainian).

8. Chen Ji, (2005). Computer Simulation of Earth Movement that Spawned the Tsunami. California Institute of Technology.

9. Malytskyy D., D`Amico S., (2015). Moment tensor solutions through waveforms inversion. Messina, Mistral Servise sas, 25 p.

10. Müller G., (1985). The reflectivity method: a tutorial. Geophys. J., 58, 153–174.