Alexandra Moshou, Panayotis Papadimitriou and Kostas Makropoulos MOMENT TENSOR DETERMINATION USING A NEW WAVEFORM INVERSION TECHNIQUE Department of Geophysics and Geothermics National and Kapodistrian University of Athens Patra, 19 – 22 May th International Congress of the Geological Society of Greece

Main task of this study Determination of Seismic parameters of an earthquake 1.Seismic Moment Tensor, M ij 2.Seismic source 3.Depth Methodology Body wave modeling and regional modeling by calculation synthetics seismograms and fitting with the corresponding observed

Methodology Teleseismic Data  Selection of dataGlobal Seismological Network 30°<Δ<90°  Deconvolution of Instrument Response and band passed filtered 0.01 Hz – 0.2Hz  Inversion of the selected waveforms  Generation of five elementary Green’s Functions Different methodologies MT5 Mc Caffrey (1988) Kikuchi and Kanamori (1982, 1986, 1991)

Methodology Local – Regional Data  Selection of data  Deconvolution of Instrument Response and band passed filtered 0.01 Hz – 0.1Hz  Inversion of the selected waveforms Hellenic Unified Seismological Network Small earthquakes was modeling successfully  Generation of Green’s Functions using the Descrete wavenumber technique (Axitra, Bouchon et al., 1981) Zahradnik et al (2003), Isola Δ<6°

Calculation of synthetic seismograms (teleseismic distances) ρ : the density at the source c: the velocity of P, S-waves g(Δ,h) : geometric spreading r 0 : the radius of the earth R i : the radiation pattern in case of P, SH, SV-waves (i=1, 2, 3) respectively the moment rate

 6 Green’ s function  elementary focal mechanisms Calculation of synthetic seismograms

Moment Tensor Inversion where a 1,…,a 5 are the components of the model m

Moment Tensor Inversion n < 5 : under – determined system n > 5 : over – determined system G non – square matrix pseudo inverse G T G = square matrix

The Linear Least Squares Problem  In general, Ax = b with m > n has no solution  Instead, try to minimize the residual r = b − Ax  With the 2-norm we obtain the linear least squares problem (LSP): Given A mxn, m>n find x such that: LINEAR LEAST SQUARES PROBLEM

Matrix Decompositions DIRECT METHODSITERATIVE METHODS

SINGULAR VALUE DECOMPOSITION We’d like to more formally introduce you to Singular Value Decomposition (SVD) and some of its applications SVD is a type of factorization for a rectangular real or complex matrix ALL MATRICES HAVE A SINGULAR VALUE DECOMPOSITION

► U,V : n x n, m x m orthogonal matrices respectively ► The columns of U,V are the eigenvectors G T ·G, G·G T respectively ► Λ : unique n x m diagonal matrix, with real, non – zero and non – negative elements λi,(singular values of G) i = 1,2, …, min (m, n) > 0, in order : Singular value decomposition Any real matrix G (mxn) can be decomposed in three parts G = U·L · V T where :

Forward problem Inverse problem d : a vector of length m, which corresponds the observed displacements G : a non – square matrix, with dimensions mxn, whose elements are a set of five elementary Green’s functions m : a vector of length n, which corresponds the moment tensor elements determination the parameters of the model by the method of trial and error numerical methods Determination of Seismic Parameters

Applications Teleseismic distances Methoni earthquake 2008/02/14 (GMT 10:09 M W = 6.7) Crete, earthquake 2009/07/01 (M w = 6.2) Regional distances Seismic Sequence Efpalio 2010/01/18 and 2010/01/22 (GMT 15:56, M w = 5.1 and GMT 00:42, M w =5.1) Crete earthquake 2010/01/26 (GMT 09:30, M w = 4.4 Aegean (Mytilini) earthquake (GMT 05:26, M w = 3.8)

 A strong number of aftershocks followed the main event of 14 February 2008  From these events, earthquakes with magnitude M w > 3.8 was studied  Two hours after the main event (2008/02/14 – GMT 10:09) an other strong earthquake occurred at the same region with magnitude M w =6.0 (36.540° Ν, ° Ε) Intermediate event d=35 km Double sources The February 14, 2008 Methoni (GMT 10:09) earthquake M w =6.7

Red = observed waveforms Blue = synthetics waveforms Seismic parameters First solution Final solution Inversion Strike, φ300°305°290° Dip, δ20°25°35° Rake, λ60°65°61° Depth, d32 km35 km DC = 90%, CLVD=10%

( Ν, Ε)  The seismic parameters for events M w > 3.7 was calculated  A large number of aftershocks was occurred after the main shock The July 1, 2009 Crete (GMT 09:30) earthquake M w =6.2 Intermediate event d=35 km Simple trapezoidal source

The July 1, 2009 (GMT 09:30) Crete earthquake M w = 6.2 Red = observed waveforms Blue = synthetics waveforms Seismic parameters First Solution Final Solution Inversion Strike, φ90°92°101° Dip, δ65°67°62° Rake, λ86°80°87° Depth, d20 km22 km DC=95%, CLVD=5%

The April 26, 2010 Taiwan earthquake M w = 6.5

The sequence of Efpalio earthquake 18 – 22 January 2010 The stars denote the two main events

The 18 January 2010, earthquake Efpalio (M w =5.1) °N °E Red = observed waveforms Blue = synthetics waveforms Seismic parameters First Solution Final Solution Inversion Strike, φ90°95° 80 ° Dip, δ40°43° 50 ° Rike, λ-100°-95° -92 ° Depth, d (km)1013 DC = 91%, CLVD=9%

The 22 January 2010, earthquake Efpalio (M w =5.1) °N ° E Red = observed waveforms Blue = synthetics waveforms Seismic parameters First Solution Final Solution Inversion Strike, φ243°250°265° Dip, δ47°50°30° Rake, λ-90°-100°-115° Depth, d (km)1512 DC = 87%, CLVD = 13%

The January 26, 2010 Crete earthquake (M w =4.4) Red = observed waveforms Blue = synthetics waveforms Seismic Parameters First Solution Final Solution Inversion Strike, φ50°40°30° Dip, δ77°75°80° Rake, λ-180°-170°-135° Depth, d (km)2125 DC=85%, CLVD=15%

The 15 May, 2010 Mytilini earthquake (M w =3.8) Red = observed waveforms Blue = synthetics waveforms Seismic parameters First Solution Final Solution Inversion Strike, φ100° (±15)102° (±6)115° Dip, δ77° (±10)80° (±5)65° Rake, λ-135° (±10)-130° (±6)-125° Depth, d (km)1011 DC = 75%, CLVD=25%

The 19 May, 2010 (M w =3.8) Methoni earthquake The 18 May, 2010 (M w =3.8) Kythira earthquake

Earthquakes modeled using regional – local data Earthquakes modeled using teleseismic data Conclusions

Conclusions A new procedure has been developed, to determine the source parameters of earthquakes, located in teleseismic or local distances The inversion is based on numerical methods, QR decomposition, Cholesky decomposition using normal equations and singular value decomposition. The method of Singular Value Decomposition is based on the eigenvalues and eigenvectors of the matrix (G T ·G) or (G·G T ). For this reason this method is more stable than others, while Cholesky is faster.

The numerical methods resulted, in most cases, Double Couple more than 85% The proposed methodology was also successfully applied to strong earthquakes worldwide Conclusions

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