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Advancements in Moment Tensor Determination

Explore a new waveform inversion technique used to determine seismic parameters of earthquakes, including moment tensor and seismic source depth. Learn about methodologies, data selection, and inversion techniques.

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Advancements in Moment Tensor Determination

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  1. 12th International Congress of the Geological Society of Greece MOMENT TENSOR DETERMINATION USING A NEW WAVEFORM INVERSION TECHNIQUE Alexandra Moshou, Panayotis Papadimitriou and Kostas Makropoulos Department of Geophysics and Geothermics National and Kapodistrian University of Athens Patra, 19 – 22 May 2010

  2. Maintaskofthisstudy Determination of Seismic parameters of an earthquake • Seismic Moment Tensor, Mij • Seismic source • Depth Methodology Body wave modeling and regional modeling by calculation synthetics seismograms and fitting with the corresponding observed

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

  4. Methodology Local – Regional Data • Selection of data Hellenic Unified Seismological Network Δ<6° • Deconvolution of Instrument Response and band passed filtered 0.01 Hz – 0.1Hz 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 • Inversion of the selected waveforms

  5. Calculationofsyntheticseismograms(teleseismic distances) ρ :the density at the source c: the velocity of P, S-waves g(Δ,h) : geometric spreading r0 : the radius of the earth Ri : the radiation pattern in case of P, SH, SV-waves (i=1, 2, 3) respectively the moment rate

  6. Calculationofsyntheticseismograms • 6 Green’ s function • elementary focal mechanisms

  7. Moment Tensor Inversion where a1,…,a5 are the components of the model m

  8. MomentTensorInversion • n < 5 : under – determined system • n > 5 : over – determinedsystem G non – square matrix pseudo inverse GTG = square matrix

  9. 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): LINEAR LEAST SQUARES PROBLEM Given Amxn , m>n find x such that:

  10. MatrixDecompositions

  11. SINGULAR VALUE DECOMPOSITION ALL MATRICES HAVE A 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

  12. Singularvaluedecomposition Any real matrix G (mxn) can be decomposed in three parts G = U·L·VT where : • U,V : n x n, m x m orthogonal matrices respectively • The columns of U,V are the eigenvectors GT·G, G·GT 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 :

  13. determination Determination of Seismic Parameters Inverse problem • Forward 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 the parameters of the model by the method of trial and error numerical methods

  14. Applications • Methoni earthquake 2008/02/14 (GMT 10:09 MW = 6.7) Teleseismic distances • Crete, earthquake 2009/07/01 (Mw = 6.2) • Seismic Sequence Efpalio 2010/01/18 and 2010/01/22 (GMT 15:56, Mw = 5.1 and GMT 00:42, Mw=5.1) • Crete earthquake 2010/01/26 (GMT 09:30, Mw = 4.4 • Aegean (Mytilini) earthquake 20100511 (GMT 05:26, Mw = 3.8) Regionaldistances

  15. The February 14, 2008 Methoni (GMT 10:09) earthquake Mw=6.7 (36.540° Ν, 21.770° Ε) Intermediate event d=35 km Double sources • Two hours after the main event (2008/02/14 – GMT 10:09) an other strong earthquake occurred at the same region with magnitude Mw=6.0 • A strong number of aftershocks followed the main event of 14 February 2008 • From these events, earthquakes with magnitude Mw > 3.8 was studied

  16. The February 14, 2008 Methoni (GMT 10:09) earthquake Mw=6.7 DC = 90%, CLVD=10% Red = observed waveforms Blue = synthetics waveforms

  17. The seismic parameters for events Mw > 3.7 was calculated The July 1, 2009 Crete (GMT 09:30) earthquake Mw=6.2 (33.920 Ν, 25.770 Ε) Intermediateevent d=35 km Simple trapezoidal source • A large number of aftershocks was occurred after the main shock

  18. TheJuly 1, 2009 (GMT 09:30) CreteearthquakeMw= 6.2 DC=95%, CLVD=5% Red = observed waveforms Blue = synthetics waveforms

  19. TheApril 26, 2010 Taiwan earthquakeMw= 6.5

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

  21. The 18 January 2010, earthquake Efpalio (Mw=5.1) 38.401°N 21.913°E DC = 91%, CLVD=9% Red = observed waveforms Blue = synthetics waveforms

  22. The 22 January 2010, earthquake Efpalio (Mw=5.1) 38.420°N 21.969° E DC = 87%, CLVD = 13% Red = observed waveforms Blue = synthetics waveforms

  23. The January 26, 2010 Crete earthquake (Mw=4.4) DC=85%, CLVD=15% Red = observed waveforms Blue = synthetics waveforms

  24. The 15 May, 2010 Mytilini earthquake (Mw=3.8) DC = 75%, CLVD=25% Red = observed waveforms Blue = synthetics waveforms

  25. The 19 May, 2010 (Mw=3.8) Methoni earthquake The 18 May, 2010 (Mw=3.8) Kythira earthquake

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

  27. 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 (GT·G) or (G·GT). For this reason this method is more stable than others, while Cholesky is faster.

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

  29. THANK YOU FOR YOUR ATTENTION THE END

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