Earth and Planetary Science

How can the Choice of Integration Method Optimize the Earthquake Finite Fault Simulation: A Case Study of 2004 Mw 6.1 Parkfield Earthquake

Ameneh Houshmandviki

Geophysics, International Institute of Earthquake Engineering and Seismology, Tehran 19537-14453, Iran

Anooshiravan Ansari

Earthquake Engineering, International Institute of Earthquake Engineering and Seismology, Tehran 19537-14453, Iran

DOI: https://doi.org/10.36956/eps.v4i1.1665

Received: 6 January 2025 | Revised: 21 March 2025 | Accepted: 28 March 2025 | Published Online: 3 April 2025

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Abstract

Earthquake finite fault simulations confront time-consuming and complex calculations. Therefore, finding methods that result in rapid calculations generalized with sufficient accuracy is predominantly necessary. Different methods of finite fault surface integrals for the 2004 Mw 6.1 Parkfield earthquake have become the subject of the current study to achieve a fast and accurate calculation of earthquake finite fault simulation. Calculations are performed considering fault elements carried out with constant and variable source parameters, while interpolation functions can also be considered. The investigations implemented in this research show that changing the conventional trapezoidal integration method into Gaussian integration on optimal element size could reduce the estimated time of calculations. The surface integral can be done only on one Gaussian point, while the required time for calculations can decrease considerably. To simplify the complex structure of Green's function calculations, a constant Green's function can be assumed in a half-space, with a time shift corresponding to the arrival time of the fault rupture representing the Green's function for other sub-faults.

Keywords: Rapid and Accurate Simulation; Parkfield Earthquake; Surface Integral; Source Parameter; Interpolation

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References

[1] Udias, A., 1999. Principle of Seismology. Cambridge University Press: Cambridge, UK. pp. 29–40.

[2] Aki, K., Richards, P.G., 2002. Quantitative seismology, 2nd Ed. University Science Books: Sausalito, CA, USA. pp. 37–41.

[3] Hartzell, S., 1978. Earthquake aftershocks as greens functions. Geophysical Research Letters. 5(1), 1–4.

[4] Irikura, K., 1978. Semi-empirical estimation of strong ground motions during large earthquakes. Bulletin of the Disaster Prevention Research Institute. 33, 63–104.

[5] Boore, D., 1983. Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra. Bulletin of the Seismological Society of America. 73(6A), 1865–1894.

[6] Zeng, Y., Anderson, J.G., Yu, G., 1994. A composite source model for computing realistic synthetic strong ground motions. Geophysical Research Letters. 21(8), 725–728.

[7] Hartzell, S., Harmsen, S., Frankel, S.L., 1999. Calculation of broadband time histories of ground motion: Comparison of methods and validation using strong-ground motion from the 1994 Northridge earthquake. Bulletin of the Seismological Society of America. 89(6), 1484–1504. DOI: https://doi.org/10.1785/BSSA0890061484

[8] Custodio, S., Liu, P., Archuleta, R., 2005. The 2004 Mw 6.0 Parkfield, California, earthquake: Inversion of near-source ground motion using multiple data sets. Geophysical Research Letters. 32(23), L23312. DOI: https://doi.org/10.1029/2005GL024417

[9] Liu, P., Archuleta, R.J., 2004. A new nonlinear finite fault inversion with three-dimensional Green’s functions: application to the 1989 Loma Prieta, California, earthquake. Journal of Geophysical Research. 109(B2), B02318. DOI: https://doi.org/10.1029/2003JB002625

[10] Babuska, I., Suri, M., 1994. The p and h-p version of the finite element method, basic principles, and properties. Society for industrial and applied mathematics. SIAM Review. 36(4), 578–632. DOI: https://doi.org/10.1137/1036141

[11] Olson, A., Apsel, R., 1982. Finite faults and inverse theory with applications to the 1979 Imperial Valley earthquake. Bulletin of the Seismological Society of America. 72(6), 1969–2001.

[12] Custodio, S., 2007. Earthquake rupture and ground-motions: The 2004 Mw 6 Parkfield earthquake [PhD thesis]. Santa Barbara, CA: University of California. pp. 11–13.

[13] Ji, C., 2004. Slip history the 2004 (Mw 5.9) Park-field earthquake (Single-Plane Model). Available from: https://www.tectonics.caltech.edu/slip_history/2004_ca/parkfield2.html (Accessed 10 January 2020).

[14] Dreger, D.S., Gee, L., Lomband, P., et al., 2005. Strong ground motions: Application to the 2003 Mw 6.5 San Simeon and 2004 Mw 6 Parkfield earthquakes. Seismological Research Letters. 76.

[15] Johanson, I.A., Fielding, E.J., Rolandone, F., et al., 2006. Coseismic and Postseismic slip of the 2004 Parkfield earthquake from space-geodetic data. Bulletin of the Seismological Society of America. 96(4B), S269–S282. DOI: https://doi.org/10.1785/0120050818

[16] Mendoza, C., Hartzell, S., 2008. Finite-Fault Analysis of the 2004 Parkfield, California Earthquake Using Pnl Waveforms. Bulletin of the Seismological Society of America. 98(6), 2746–2755. DOI: https://doi.org/10.1785/0120080111

[17] Thurber, C., Roecker, S., Roberts, K., et al., 2003. Earthquake locations and three-dimensional fault zone structure along the creeping station of the San Andreas fault near Parkfield, CA: preparing for SAFOD. Geophysical Research Letters. 30(3), 1112.

[18] Cotton, F., Coutant O., 1997. Dynamic stress variations due to shear faults in a plane-layered medium. Geophysical Journal International. 128(3), 676–688. DOI: https://doi.org/10.1111/j.1365-246X.1997.tb05328.x

[19] Bouchon, M., 1981. A simple method to calculate Green’s functions for elastic layered media. Bulletin of the Seismological Society of America. 71(4), 959–971. DOI: https://doi.org/10.1785/BSSA0710040959

[20] Thurber, C., Zhang, H., Waldhauser, F., et al., 2006. Three-dimensional compressional wavespeed model, earthquake relocations, and focal mechanisms for the Parkfield, California, region. Bulletin of the Seismological Society of America. 96(4B), S38–S49. DOI: https://doi.org/10.1785/0120050825

[21] Kristekova, M., Kristek, J. Moczo, P., 2009. Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of signals. Geophysical Journal International. 178(2), 813–825. DOI: https://doi.org/10.1111/j.1365-246X.2009.04177.x