Keywords: filters, deconvolution, seismic accelerograms, wavelet, denoising, thresholding.

Seismic accelerograms are the convolution of ground motion with the transfer function of the accelerometer and structure on which the accelerometer is mounted. The accelerometer response can be corrected in the time or frequency domain to recover an estimate of the ground motion of the seismic event. This correction of the accelerogram is essential for non-linear, finite element, timehistory, analysis of various civil engineering structures, as the uncorrected record is modified in amplitude, phase and frequency. This paper briefly discusses issues of the de- convolution of seismic data and then introduces threshold de-noising using the stationary wavelet transform in order to reduce the unwanted frequencies. The results are compared with optimal filters as well as the more standard band-pass filtering techniques.

De-convolution of the instrument response from seismic accelerograms is an essential step in the correction of seismic data and is used in the UEL methods A & B [1] as well as those of others [2,3,4,5]. Methods of de-convolution include time domain differential mapping [4] and frequency-domain de-convolution [1,2] using the Fast Fourier Transform (FFT) algorithm. The paper shows that de-convolution using the FFT produces an almost flat response characteristic over a larger frequency range than those methods using differential mapping.

It was demonstrated in [1], by comparing power spectral densities of various earthquakes, that band-pass filtering can remove too much energy. The consequences of this were manifest in the estimates of the total acceleration response spectra that produced errors of the order of up to 20% for certain seismic data. In order to try and mitigate this error one approach was to implement a least squares adaptive algorithm [1]. Another approach, included and compared in this paper, is the application of the stationary wavelet transform [10] to de-noise seismic signals. The stationary wavelet transform (SWT) is shift invariant [9] and has the structure of an un-decimated filter bank [8] and is better suited to de-noising than the discrete wavelet transform (DWT). The latter is not shift-invariant and would degrade any de-noising using a thresholding [6] scheme.

In summary, when applied to strong-motion seismic data, the results produced thus far demonstrate that, over the Nyquist cycle, this approach removes less energy as expected with a wavelet transform. Moreover this approach is less computationally intensive than the previous adaptive methods used. Hence it is suggested that using the wavelet transform is a more efficient than adaptive methods and more effective than band-pass methods at removing unwanted frequencies in seismic accelerograms.

1

N.A. Alexander, A.A. Chanerley, N. Goorvadoo, "A Review of Procedures used for the Correction of Seismic Data", Sept 19th-21st, 2001, Eisenstadt-Vienna, Austria, Proc of the 8th International Conference on Civil & Structural Engineering ISBN 0-948749-75-X doi:10.4203/ccp.73.39

2

A. Kumar, S. Basu, B. Chandra, "Blacs-A new Correction Scheme of Analog Accelerograms. Part-1: details of Scheme" Bull. Indian Society of Earthquake Technology, Paper No 348, Vol. 32, No.2 June 1995, pp 33-50

3

A. Kumar, S. Basu, B. Chandra, "The effects of Band Limited interpolation of non-uniform samples on records of Analog Accelerograms" Bull. Indian Society of Earthquake Technology, Paper No 327, Vol. 29, No.4 Dec 1992, pp 53-67

4

S.S. Sunder, J.J. Connor, "Processing Strong Motion Earthquake Signals", Bulletin of the Seismological Society of America, Vol. 72. No.2, pp. 643-661, April 1982

5

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

6

D. Donoho, "De-noising by soft-thresholding", IEEE Transactions on Information Theory", 41(3):613-627, May 1995. doi:10.1109/18.382009

7

D. Donoho, I. Johnstone, "Wavelet Shrinkage: Asymptopia ?", Journal Royal Statistical Soc. B., 57(2): 301-337, 1995.

8

M. Lang, H. Guo, J. Odegard, C. Burrus, R.Wells. "Noise reduction using an un-decimated discrete wavelet transform", IEEE Signal Processing Letters, 3(1): 10-12, January 1996. doi:10.1109/97.475823

9

R. Coifman, D. Donoho, "Translation Invariant de-noising", Wavelets and Statistics, Lecture Notes in Statistics 103: Springer-Verlag, p125-150, 1995

10

G. Nason, B. Silverman, "The stationary wavelet transform and some statistical applications", Wavelets and Statistics, Lecture Notes in Statistics 103: Springer-Verlag, p281-299, 1995

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