**D.
Malytskyy**^{1}**,
Dr. Sci. (Phys.-Math.), Prof.****,
****E-mail:
dmytro@cb-igph.lviv.ua;**

**O.
Muyla**^{1}**,
Cand. Sci. (Phys.-Math.), Research Associate****,
****E-mail:
orestaro@gmail.com;**

**O.
Hrytsaj**^{1}**,
Postgraduate Student****,
****E-mail:
grycaj.oksana@gmail.com;**

**À.
Pavlova**^{1}**,
Ph.D.;**

**Î.
Astashkina**^{1}**,
Cand. Sci. (Geol.), Research Associate****,
****E-mail:
sac1@ukr.net;**

**O.
Obidina**^{1}**,
Postgraduate Student****,
****E-mail:
jane.det@yandex.ua;**

**Å.
Kozlovskyy**^{1}**,
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.