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,

O. Kutniv1, Engineer, E-mail: okutniv@yahoo.com,

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


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

The authors present a moment tensor inversion of waveforms, which is more robust and yields more stable and more accurate results than standard approaches. The inversion is solved in two steps. First, a point source of seismic waves is considered, with defined location and origin time. Matrix method is used to solve the problem of wave propagation in the medium modeled as a horizontally layered heterogeneous elastic structure (isotropic and/or anisotropic). In order to allow the source mechanism to change with time each moment tensor component has its own time history. The source is described by the full moment tensor Mlm A numerical technique developed based on forward modeling is used for the inversion of the observed waveforms for the components of moment tensor and the earthquake source-time function (STF(t)). The method provides a good estimate for the complete mechanism when records are treated, which corresponds to a velocity model contained inside the interpolation range. The method of waveform inversion using only direct P- and S-waves at stations that we have developed allows us to retrieve the moment tensor of a point source as a function of time.

We computed the moment tensor solutions also using the graphic method. The traditional graphical method is based on the P-waves prior arrival using information about fuzzy first motion and the S/P amplitude ratio. The polarities between P-waves first motion were defined from complete records on seismograms taking into account the possible inversion of the sign on the z-component. A logarithm of the S/P amplitude ratio is calculated using seismic data received at each station from the three components. Input data for the azimuth and take-off angle are calculated by software packages for each event. Finally, the proposed moment tensor inversion is tested on real data for the earthquakes of 24.04.2011 (13h02m12s, 35.92°N, 14.95°E (near Malta), Mw4.0) and 29.12.2013 (17h09m0.04s, 41.37°N, 14.45°E (Southern Italy), Mw4.9).

Keywords: matrix method, moment time function, earthquake mechanism, tensor of seismic moment.


1. Dziewonski A.M, Chou T.A.,Woodhouse J.H., (1981). Determination of earthquake source parameters from waveform data for studies of regional and global seismicity. J.geophys.Res., 86, 2825-2852.

2. Godano M., Bardainne T., Regnier M., Deschamps A., (2011). Moment tensor determination by nonlinear inversion of amplitudes. Bull.seism. Soc.Am., 101, 366-378.

3. Hardebeck J.L., Shearer P.M., (2003). Using S/P amplitude ratios to constrain the focal mechanisms of small earthquakes. Bull.seism. Soc.Am., 93, 2432-2444.

4. Kikuchi M., Kanamori H., (1991). Inversion of complex body waves-III. Bull.seism. Soc.Am., 81, 2335-2350.

5. Malytskyy D., Kozlovskyy E., (2014). Seismic waves in layered media. J. of Earth Science and Engineering, 4, 311-325.

6. Malytskyy D.V., (2010). Analytic-numerical approaches to the calculation of seismic moment tensor as a function of time. Geoinformatika, 1, 79-85. (In Ukrainian).

7. Malytskyy D., Muyla O., Pavlova A., Hrytsaj O., (2013). Determining the focal mechanism of an earthquake in the Transcarpathian region of Ukraine. Visnyk of Taras Shevchenko National University of Kyiv: Geology, 4(63), 38-44.

8. Miller A.D., Julian B.R., Foulger G.R., (1998). Three-dimensional seismic structure and moment tensors of non-double-couple earthquakes at the Hengill-Grensdalur volcanic complex, Iceland. Geophys. J. Int., 133, 309-325.

9. Sileny J., Panza G.F., Campus P., (1992). Waveform inversion for point source moment tensor retrieval with variable hypocentral depth and structural model. Geophys. J. Int., 109, 259-274.

10. Sipkin S.A., (1986). Estimation of earthquake source parameters by the inversion of waveform data: Global seismicity, 1981-1983. Bull.seism. Soc.Am., 76, 1515-1541.

11. Vavrychuk V., Kuhn D., (2012). Moment tensor inversion of waveforms: a two-step time frequency approach. Geophys. J. Int., 190, 1761-1776.