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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Paper 217

Using the Method of Total Least Squares for Seismic Correction

A.A. Chanerley+ and N.A. Alexander*

+School of Computing and Technology, University of East London, United Kingdom
*Department of Civil Engineering, University of Bristol, United Kingdom

Full Bibliographic Reference for this paper
A.A. Chanerley, N.A. Alexander, "Using the Method of Total Least Squares for Seismic Correction", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 217, 2005. doi:10.4203/ccp.81.217
Keywords: correction, filter, seismic, wavelet, de-noising, recursive, least squares, band-pass, filtering, filter, inverse filter, convolve, de-convolve.

Summary
Typical techniques used for seismic correction are necessary to: (i) digitise, that is equi-and up-sample the data, (ii) de-trend, (iii) de-noise using the wavelet transform or band-pass filter [1], using digital Butterworth, Chebyshev filters or digital Finite Impulse Response (FIR) filters [8,9,10,11] (iv) correct for instrument characteristics (v) down-sample to an appropriate sampling rate. The sequence of the components (ii) to (v) and the exact algorithms used in these correction techniques may vary significantly, as can the resulting recaptured original ground motion itself.

Recent methods, which describe the de-convolution of an instrument response from seismic data, apply a least-squares based, inverse, system identification method [1,3] with which to de-convolve the instrument response from the ground motion. Previous methods assume a second order, single-degree-of-freedom (SDOF) [8,10] instrument function and apply an inverse filter in the time or frequency domain, with which to de-convolve the instrument. Whereas in other cases, corrected seismic data [4] are not explicitly de-convolved, as a consequence of insufficient instrument parameter data.

The advantage of the least-squares based method is that it does not require any information regarding the instrument; it only requires the data, which the instrument has provided, from which to determine an estimate of the inverse of the instrument response. However, the least squares solution which minimises has been sought with the assumption that the data matrix is 'correct' and that any errors in the problem are in the vector . This paper applies the method of total least squares (TLS), [2,5,6,7] which allows for the fact that both and may be in error.

A range of instruments and their performance are compared using Icelandic seismic data, digitally recorded and with instrument parameters included. These parameters are the viscous damping ratio and the instruments natural frequency. Four instrument types were compared, SMA-1, DCA-333, A-700 and SSA-1, each seismic record has details of the instrument parameters, but only one has some details of the anti-alias filter used in the instrument.

Instrument performance was based on a comparison of instrument frequency responses obtained by de-convolution in the time domain using the QR-RLS, de-convolution in the frequency domain using instrument parameters in the second order SDOF equation and the method of Total Least Squares (TLS). Correlation was good up to the cut-off frequencies, thereafter the roll-off between the methods differed in gradient, with both the TLS and QR-RLS showing steeper gradients than that for the second order, SDOF response. This is consistent with the fact that digital instruments would have an ant-alias filter whose impression is embedded in the data. Details of such filters are not always included in the record.

References
1
A.A. Chanerley and N.A. Alexander, "An Approach to Seismic Correction which includes Wavelet De-noising", in Proceedings of the Third International Conference on Engineering Computational Technology, B.H.V. Topping and Z. Bittnar, (Editors), Civil-Comp Press, Stirling, United Kingdom, paper 44, 2002. doi:10.4203/ccp.75.44
2
G. Golub, C. Reinsch, "Singular Value Decomposition and Least Squares Solutions", Numer. Math. 14, 403-420, 1970. doi:10.1007/BF02163027
3
A.A. Chanerley, N.A. Alexander, "Novel Seismic Correction approaches without instrument data, using adaptive methods and De-Noising", 13th World Conference on Earthquake Engineering, Vancouver, Canada, August 1st -6th, 2004, paper 2664
4
Ambraseys N., et al "Dissemination of European Strong Motion Data, CD collection. European Council, Environment and Climate Research programme", 2000.
5
G. Golub, C.F. Van Loan, "An analysis of the Total Least Squares Problem", SIAM J. Numerical Analysis, 17(6): 883-93, December 1980. doi:10.1137/0717073
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S. van Huffel, J. Vandewalle, "The use of Total Linear Least Squares for Identification and Parameter Estimation", Proc. IFAC/IFORS Symp. on Identification and Parameter Estimation, 1167-72, July 1985.
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S. Van Huffel, J. Vanderwalle, "The Total Least Squares Problem Computational Aspects and Analysis", Siam, vol 9, Frontiers in Applied Mathematics, Philadelphia, 1991.
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O. Khemici, W. Chang, "Frequency domain corrections of Earthquake Accelerograms with experimental verification", San Francisco, USA, Proc of the 8th World Conference on Earthquake Engineering, 1984, Vol 2, pp 103-110.
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J.F.A. Orsmby, "Design of numerical filters with applications in missile data processing", Association for Computing Machinery, Journal 8, 1961.
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A.M. Converse, "BAP: Basic Strong-Motion Accelerograms Processing Software; Version 1.0", USGS, Denver, Colorado. Open-file report 92(296A), 1992
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S. Haykin, "Adaptive Filter Theory", 3rd Ed., Prentice-Hall, pp 562-628, 1996.
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A. Sayed, H. Kaileth, "A state-space approach to adaptive RLS filtering", IEEE signal Process. Mag., vol 11., pp 18-60, 1994. doi:10.1109/79.295229

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