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PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
A New Approach to Seismic Correction using Recursive Least Squares and Wavelet De-Noising
A.A. Chanerley+ and N.A. Alexander*
+School of Computing & Technology, University of East London, England
A.A. Chanerley, N.A. Alexander, "A New Approach to Seismic Correction using Recursive Least Squares and Wavelet De-Noising", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 107, 2003. doi:10.4203/ccp.77.107
Keywords: correction, filter, seismic, wavelet, de-noising, recursive, least squares,.
This paper begins with a brief introduction to some methods used to correct seismic data [1,2,3]. Most corrected seismic data  assume a 2nd order, single-degree-of-freedom (SDOF) instrument function with which to de-convolve the instrument response from the ground motion. Other corrected seismic data  is not explicitly de-convolved, citing as reason insufficient instrument information with which to de-convolve the data. Whereas this latter approach may facilitate ease of processing, the estimate of the ground motion cannot be entirely reliable. This paper discusses a relatively straightforward implementation of the well-known recursive least squares (RLS) algorithm in the context of a system identification problem. The resulting inverse filter is then applied to the data in order to de-convolve the instrument response.
The RLS algorithm was chosen in preference to the least mean squares (LMS) adaptive algorithm. The RLS algorithm has only one parameter to adjust in a relatively straightforward manner. This parameter is exponentially weighted and is called a forgetting factor, , in the literature. On application it reduces the effect due to previous error values. The RLS algorithm is dependent on the incoming data samples rather than the statistics of the ensemble average as in the case of the LMS algorithm. This means that the coefficients will be optimal for the given data without making any assumptions regarding the statistics of the process, also the algorithm has a higher rate of convergence than the LMS.
The RLS has a variant, which is used to produce the results in this presentation. It is numerically more stable than the direct algorithm. This is the QR decomposition-based RLS algorithm deduced directly from the square-root Kalman filter counterpart. The QR-RLS adapts by first updating the square root of the correlation matrix, , and then updating the filter weights using .
The papers then discuss the implementation of the translation invariant wavelet transform [5,6] in order to de-noise  rather than filter the resulting seismic data. Ideally, noise errors should be removed before any instrument correction is applied, since de-convolution may amplify the noise within a seismic data set. However standard procedures for correcting seismic data [2,3,4] apply a band-pass filter on the resulting data available, after performing a 2nd order instrument de-convolution. This is necessary since pre-filtering first, would render de-convolution after, a redundant exercise. Nevertheless, even post-filtering after de-convolution as is the general case alters the data set and for the same reason, can no longer adequately represents the true ground motion, since the filtering will remove some of the true ground motion data from the seismic set.
Wavelet de-noising however removes only those signals whose amplitudes are below a certain threshold. De-noising is not frequency selective and cannot affect the data in the same way. Therefore it is proposed that wavelet de-noising be implemented prior to de-convolution. It is considered that even if the specification on the type of instrument used to record the seismic event is available, then the QR-RLS algorithm is still a better choice for inverse filtering the resulting data to obtain a better representation of the true ground motion.
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