Towards Optimal Network for Source Inversion

1. Towards Optimal Network for Source Inversion Lingsen Meng, Jean-Paul Ampuero Seismo Lab,Caltech

2. Source Inversion Validation (SIV): • Finite fault inversion • Different approaches and datasets • Used to study source dynamics, ground motion ,coulomb stress • Huge discrepancies SPICE blind test • Compare and validate current developed models • estimate the actual uncertainty of the model parameters • A blind-test for source inversion approaches • Lead by Martin Mai, Danjiel Schorlemmer, Morgan Page

3. Motivation Best network for next SIV benchmark? Inconsistency due to station distribution Quantification of effectiveness of array geometry (resolution, uncertainty) Tradeoff between information and cost Experiment design example: fault location, magnitude 7, N sensors available: where do we install the sensors to achieve the most reliable source imaging?

4. Outline Review resolution and uncertainty of linear inverse theory Optimal array geometry from prediction theory Statistical approach for Nonlinear survey design Optimal network for earthquake location Best network for next SIV benchmark? Summary

5. Linear source inversion NT*Nm NT Slip on the fault Nt Green’s function seismograms Nt*Ns × = NT*Nm Nt*Ns (+constraints)

6. Linear resolution and Uncertainty resolution • R = higher resolution when the diagnal term of resolution matrix appoach to 1 • Data errors propagate in to model with amplification of 1/eigenvalue uncertainty Eigenvalues(diag matrix) Eigenvectors(diag matrix) Generalized inverse (truncating small eigenvalues)

7. quality measures of eigenspectrum crosswell tomography(curtis et al,1999) regular optimal

8. Optimal array geometry from prediction theory(Iida,1990) • Known strike, dip ,rake, rise time, rupture time. invert for slip on each subfault • Quality metric Similar to related to the std of slip of subfaults • Not an exhaustive search on array geometries: tested few geometries with simple parameterization(Ns, radius , azimuthal coverage)

9. Number of station Array radius Uncertainty and Ns,radius error Two concentric rings, R/r = 2, Ns/2 stations on each ring Free parameters: Ns and R • Inverse root Dependent on number of stations • Array radius good between 0.75-2 fault length

10. Fan array Free parameters: Azimuth coverage (Phi), fixed station density or fixed number Uncertainty and Azimuthal coverage error Azimuthal coverage • Inverse root Dependent on azimuthal coverage which contribute a lot to the inversion

11. Test for optimization of geometry Optimal geometry Testing geometry strike slip Dip slip

12. Summary and Limitations • Green’s function of homogeneous half space • Non-linear effect positivity constraint rupture time , rise time • exhaustive optimization needed • Linear prediction from propagation of errors • Quantification of uncertainties from array parameters • non-exhaustive optimization of network geometry

13. Non-linear experimentdesign theory(Curtis,2004) Measure of information: negative entropy Maximizing (Model dependence on data and design) = Minimizing (data dependence on design) Only forward modeling required to optimize the design , still expensive

14. Optimal network for Earthquake location • D-optimum ( )criteria for optimal network(Kijko,1977) • Optimal network for aftershocks ( Hardt and Scherbaum,1994) Joint optimization for location, focal mechanism, tomography • Maximize for multiple sources (Steinberg et al,1995) W: error correlation matrix for the sites stations surround the epicenter • Online network optimization software(lomax & Curtis, 2004) • Double-difference and bootstrap(Bai et al,2006) need stations close to the epicenter not too large azimuthal gap

15. Possible solutions rule of thumbs good azimuth coverage denser closer to the fault too many stations not realistic realistic network (SIV 1 , the Tottori earthquake) a network minimize the contribution to errors from station geometry (focused design on the true slip distribution) Best network for benchmark

16. Solution?:Focused experiment design Focused metric crosswell tomography(curtis et al,1999) Projection of uncertainty on subspace Design focused on the subfaults actually slipped ! Keeping the errors from stations min on specific source

17. Isochron theory Spudich et al, 1984 Schmedes & Archuleta, 2008 Contour of sum of travel time and rupture time = constant Relate the amplitude of seismogram to slips on each subfault

18. Network quality based on Isochron theory Station distribution(N=6) Stacked map for all stations each station Inverse of area between neighboring isochrons = Sensitivity of each segment of seismograms to the slip on each subfaults Geometrical spreading, attenuation , signal to coda ratio , radiation pattern

19. Discussion • What criterion should guide the selection of a station distribution for next SIV benchmark : • Realistic ? • Minimizes model uncertainties for any source? • Minimizes model uncertainties for a specific source? • Effective optimization scheme to search optimal network ,especially non-linear problem (isochron metric) • Network optimization for tomography (seistivity Kernels)

20. Thank you

21. Isochron backprojection station distribution test Jakka et al, 2010

22. Hypocenter at x=5

23. Station distribution(N=6) Single station inverse of Isocron band area Stacked map

24. Optimal array geometry from prediction theory(Iida,1990)

25. Fan array Free parameters: Azimuth coverage (Phi), fixed station density or fixed number Uncertainty and Azimuthal coverage error Azimuthal coverage Component of seismogram • Inverse root Dependent on azimuthal coverage • Horizontal component of seismogram contribute to strike-slip fault; vertical to dip-slip

26. Not on S for Non-linear experimentdesign theory(Curtis,2004) Measure of information: negative entropy Partition a vector into two parts maximize Not on S minimize Only forward modeling required, still expensive