Postseismic Deformation Following the 1994 Northridge Earthquake Identified Using the Localized Hartley Transform Filter

  • K. F. Tiampo
  • Dawit Assefa
  • J. Fernández
  • L. Mansinha
  • H. Rasmussen
Part of the Pageoph Topical Volumes book series (PTV)


Here we present a new mathematical tool, the localized (HL); Bracewell, 1990), that allows for the filtering of 1-D time series through the identification of the power at various spatial and temporal wavelengths. Its application to and the associated results are presented from its application to continuous Global Positioning System (GPS) data from southern California for the time period 1994 through 2006. The HL transform filter removes the high-frequency components of the data and effectively isolates the longer period signal. This long-period signal is modeled as time-dependent postseismic deformation using the viscoelastic-gravitational model of (2004) for six stations selected for their proximity to the Northridge earthquake. The x-, y-, and z-components of the postseismic deformation are compared to the filtered data. Results suggest that this long-period deformation is a result of postseismic relaxation and that the HL transform filter provides an important new technique for the filtering of geophysical data consisting of the superposition of the effects of numerous complex sources at a variety of spatial and temporal scales.

Key words

Hartley transform GPS time series analysis spatio-temporal filtering postseismic deformation Northridge earthquake 


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  1. Belbachir, N., Chilton, T., Dunn, M., Nunkeser, M., Sidhom, S., and Szajnowski, G. (2003), Image Compression using the Hartley Transform, Pattern Recognition and Image Processing Group, Inst. Comput. Aided Automation, Vienna, Austria, Tech. Rep. PRIP-TR-087.Google Scholar
  2. Bracewell, R. N. (1983), Discrete Hartley transform, J Opt Soc Am 73, 1832–1835.CrossRefGoogle Scholar
  3. Bracewell, R. N., The Hartley transform (Oxford University Press, New York. (1986)).Google Scholar
  4. Bracewell, R. N. (1990), Assessing the Hartley Transform: Correspondence, IEEE Trans. on Acou. Speech and Sig. Proc. 38, 2174–2176.CrossRefGoogle Scholar
  5. Bock, Y., Wdowinski, S., Fang, P., Zhang, J., Williams, S., Johnson, H., Behr, J., Genrich, J., Dean, J., VAN Domselaar, M., Agnew, D., Wyatt, F., Stark, K., Oral, B., Hudnut, K., King, R., Herring, T., Dinardo, S., Young, W., Jackson, D., AND Gurtner, W. (1997), Southern California Permanent GPS Geodetic Array: Continuous measurements of crustal deformation between the 1992 Landers and 1994 Northridge earthquakes, J Geophys Res 102,18, 013–18,033.Google Scholar
  6. Chilton, E., and Hassanain, E. (2006), Phase estimation of minimum phase systems using the Hartley phase Spectrum, Proc. of 24th IASTED Int. Conf. on Sig. Proc., Pattern Recognition and Applications, pp. 171–176.Google Scholar
  7. Dong, D., Fang, P., Bock, Y., Webb, F., Prawirodirdjo, L., Kedar, S., and Jamason, P. (2006), Spatiotemporal filtering using principal component analysis and Karhunen-Loeve expansion approaches for regional GPS network analysis, J Geophys Res 111, doi: 10.1029/2005JB003806.Google Scholar
  8. Dong, D., Herring, T.A., and King, R. A. (1998), Estimating regional deformation from a combination of space and terrestrial geodetic data, J. Geod. 72, 200–214.CrossRefGoogle Scholar
  9. Dong, D., Fang, P., Bock, Y., Cheng M. K., and Miyazaki, S. (2002), Anatomy of apparent seasonal variations from GPS-derived site position time series, J Geophys Res, 107, doi:10.1029/2001JB000573.Google Scholar
  10. Donnellan, A. and Lyzenga, G. A. (1998), GPS Observations of fault afterslip and upper crustal deformation following the Northridge earthquake, J Geophys Res 103,21, 285–21, 297.Google Scholar
  11. Donnellan, A., Parker, J. W., AND scpeltzer, G. (2002), Combined GPS and InSAR models of postseismic deformation from the northridge Earthquake, Pure Appl. Geophys. 159, 2261–2270.CrossRefGoogle Scholar
  12. Fernández, J. and Rundle, J. B. (2004), Postseismic viscoelastic-gravitational half-space computations: Problems and solutions, Geophys. Res. Lt. 31, doi: 10.1029/2004GL019654.Google Scholar
  13. Fernández, J., Yu, T.-T., and Rundle, J. B. (1996a), Horizontal viscoelastic-gravitational displacement due to a rectangular dipping thrust fault in a layered earth model, J. Geophys. Res. 101(B6), 13581–13594 (Correction, J. Geophys. Res. 103, 30,283-30,286, 1998).CrossRefGoogle Scholar
  14. Fernández, J., Yu, T.-T., and Rundle, J. B. (1996b), Deformation produced by a rectangular dipping fault in a viscoelastic-gravitational layered Earth model. Part I: Thrust fault, Comput Geosci, 22, 735–750 (Correction, Comput. Geosci. 25, 301-307, 1999).CrossRefGoogle Scholar
  15. Hartley, R. V. L. (1942), A More Symmetrical Fourier analysis applied to transmission problems,Proceed. IRE., 30, 144–150.CrossRefGoogle Scholar
  16. Hudnut, K. W., Shen, S., Murray, M., Mcclusky, S., King, R., Herring, T., Hager, B., Feng, Y., Fang, P., Donnellan, A., and Bock, Y. (1996), Co-seismic displacements of the 1994 Northridge, California, earthquake, Bull Seismol Soc Am 86, S19–S33.Google Scholar
  17. Kuhl, H. and Sacchi, M. D. (1999), Least-squares split-step migration using the Hartley transform, Univ. of Alberta, SEG Expanded Abstracts, 1548–1551.Google Scholar
  18. Legrand, L., Diebold, H., Tosser, A. J., and Dusserre, L. (1990), Computation of functional angiographic images with the Hartley transform, Computers in Cardiology Proceedings, 423–426.Google Scholar
  19. Liu J.-C. and Lin T. P. (1993), Short-time Hartley transform, IEEE Proc.-F (140)171–174.Google Scholar
  20. Mansinha, L., Stockwell, R. G., and Lowe, R.P. (1997a), Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms, Physica A, 239, 286–295.CrossRefGoogle Scholar
  21. Mansinha, L., Stockwell, R. G., Lowe, R. P., Eramanian, M., and Schincariol, R. A. (1997b), Local S-spectrum analysis of 1-D and 2-D data, Phys. of the Earth and Planet. Int., 103, 329–336.CrossRefGoogle Scholar
  22. Nikolaidis, R. (2002), Observation of Geodetic and Seismic Deformation with the Global Positioning System, Ph.D. Thesis, University of California, San Diego.Google Scholar
  23. Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half space, Bull. Seismol. Soc. Am. 75, 1135–1154.Google Scholar
  24. Pinnegar, C. R., and Mansinha, L. (2004), Time-frequency localization with the Hartley S-Transform, Sig. Proc. 84, 2437–2442.CrossRefGoogle Scholar
  25. Pollitz, F. F. (1997), Gravitational viscoelastic postseismic relaxation in a layered spherical earth, J Geophys Res 102, 17921–17941.CrossRefGoogle Scholar
  26. Rundle, J. B. (1981), Vertical displacements from a rectangular fault in a layered elastic-gravitational media, J Phys Earth 29, 173–186.Google Scholar
  27. Rundle, J. B. (1982), Viscoelastic-gravitational deformation by a rectangular thrust fault in a layered Earth, J. Geophys. Res. 87, 7787–7796.CrossRefGoogle Scholar
  28. The SCIGN Project Report to NSF (1998), Southern California Earthquake Center.Google Scholar
  29. Stockwell R. G., Mansinha L., and Lowe R. P. (1996), Localization of the Complex Spectrum: The S transform, IEEE, Trans. on Sig. Proc. 44, 998–1001.CrossRefGoogle Scholar
  30. Stockwell, R. G. (1999), S-Transform Analysis of Gravity Wave Activity, Ph.D. Dissertation, Dept. of Physics and Astronomy, The University of Western Ontario, London, Ont., Canada.Google Scholar
  31. Stockwell, R. G. (2007), A basis for efficient representation of the S-transform, Digital Signal Processing 17, 371–393.CrossRefGoogle Scholar
  32. Theussl, T., Tobler, R. F., and Groller, E. (2000), The multi-dimensional Hartley transform as a basis for volume rendering, Winter School of Computer Graphics, The 8th Internat. Conf. in Central Europe on Computer Graphics, Visualization and Interactive Digital Media, Proceedings abstract.Google Scholar
  33. Unruh, J. R., Twiss, R. J., and Hauksson, E. (1997), Kinematics of postseismic relaxation from aftershock focal mechanisms of the 1994 Northridge, California earthquake, J. Geophys. Res. 102,24,589–24,603.Google Scholar
  34. Wald, D. J., Heaton, T. H., and Hudnut, K. W. (1996), The slip history of the 1994 Northridge, California, earthquake determined from strong-motion, teleseismic, GPS, and leveling data, Bull Seismol Soc Am 86, S49–S70.Google Scholar
  35. Watson, K. M., Bock, Y., and Sandwell, D. T. (2002), Satellite interferometric observations of displacements associated with seasonal groundwater, J. Geophys. Res. 107, doi:10.1029/2001JB000470.Google Scholar
  36. Williams, S. D. P., Bock, Y., Fang, P., Jamason, P., Nikolaidis, R. M., Prawirodirdjo, L., Miller, M. AND Johnson, D. J. (2004), Error analysis of continuous GPS position time series, J. Geophys. Res. 109, B03412, doi:10.1029/2003JB002741.CrossRefGoogle Scholar
  37. Zumberge, J. F., Heflin, M. B., Jefferson, D. C., Watkins, M.M., and Webb, F.H. (1997), Precise point positioning for the efficient and robust analysis of GPS data from large networks, J. Geophys. Res. 102, 5005–5017.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2008

Authors and Affiliations

  • K. F. Tiampo
    • 1
  • Dawit Assefa
    • 2
  • J. Fernández
    • 3
  • L. Mansinha
    • 1
  • H. Rasmussen
    • 4
  1. 1.Department of Earth SciencesUniversity of Western OntarioLondonCanada
  2. 2.Radiation Physics Department, Radiation Medicine ProgramPrincess Margaret HospitalTorontoCanada
  3. 3.Institutio de Astronomía y Geodesia (CSIC-UCM), Facultad de CC. MatemáticasCiudad UniversitariaMadridSpain
  4. 4.Department of Applied MathematicsThe University of Western OntarioLondonCanada

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